Embed Size (px)
Citation preview
The Pennsylvania State University
The Graduate School
ADVENTURES IN HIGH DIMENSIONS UNDERSTANDING
GLASS FOR THE 21ST CENTURY
A Dissertation in
Material Science and Engineering
by
Collin James Wilkinson
copy 2021 Collin James Wilkinson
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
May 2021
ii
The dissertation of Collin Wilkinson was reviewed and approved by the following
John Mauro
Professor of Materials Science and Engineering
Chair Intercollege Graduate Degree Program
Associate Head for Graduate Education Materials Science and Engineering
Dissertation Advisor
Chair of Committee
Seong Kim
Professor of Chemical Engineering
Professor of Materials Science and Engineering
Ismaila Dabo
Associate Professor of Materials Science and Engineering
Susan Sinnott
Professor of Materials Science and Engineering
Professor of Chemistry
Head of the Department of Materials Science and Engineering
iii
Abstract
Glass is infinitely variable This complexity stands as a promising technology for the 21st
century since the need for environmentally friendly materials has reached a critical point due to
climate change However such a wide range of variability makes new glass compositions difficult
to design The difficulty is only exaggerated when considering that not only is there an infinite
variability in the compositional space but also an infinite variability thermal history of a glass and
in the crystallinity of glass-cearmics This means that even for a simple binary glass there are at
least 3 dimensions that have to be optimized To resolve this difficulty it is shown that energy
landscapes can capture all three sets of complexity (composition thermal history and crystallinity)
The explicit energy landscape optimization however has a large computational cost To
circumvent the cost of the energy landscape mapping we present new research that allows for
physical predictions of key properties These methods are divided into two categories
compositional models and thermal history models Both models for composition and thermal
history are derived from energy landscapes Software for each method is presented As a
conclusion applications of the newly created models are discussed
iv
Table of Contents
List of Figures vi
List of Tables xi
Acknowledgments xii
Chapter 1 The Difficulty of Optimizing Glass 1
11 Energy Landscapes 6 12 Fictive Temperature 8 13 Topological Constraint Theory 15 14 Machine Learning 19 15 Goals of this Dissertation 20
Chapter 2 Software for Enabling the Study of Glass 22
21 ExplorerPy 22 22 RelaxPy 28
Chapter 3 Understanding Nucleation in Liquids 30
31 Crystallization Methods 33 31A Mapping and Classifying the Landscape 35 31B Kinetic Term for CNT 38 31C Degeneracy calculations 41 31D Free Energy Difference 42 31E Interfacial Energy 44
32 Results amp Discussion 46 33 Conclusions 51
Chapter 4 Expanding the Current State of Relaxation 52
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition 52 41A Relativistic Liquid 56 41B Relativistic Observer 60
42 Temperature and Compositional Dependence of the Stretching Exponent 64 42A Deriving a Model 67 42B Experimental Validation 73
43 Conclusion 81
Chapter 5 Glass Kinetics Without Fictive Temperature 82
51 Background of the Adam Gibbs Relationship 82
v
52 Methods 84 53 Results 86
53A Adam-Gibbs Validation 86 53B MYEGA Validation 87 53C Adam-Gibbs and Structural Relaxation 89 53D Landscape Features 91
54 Topography-Property Relations 94 55 Barrier Free Description of Thermodynamics 99 57 Toy Landscapes for the Design of Glasses and Glass Ceramics 102 57 Discussion 107
Chapter 6 Enabling the Prediction of Glass Properties 109
61 Controlling Surface Reactivity 109 62 Elastic Modulus Prediction 125 63 Ionic Conductivity 135 64 Machine Learning Expansion 145
Chapter 7 Designing Green Glasses for the 21st Century 152
71 Glass Electrolytes 152 72 Hydrogen Fuel Cell Glasses 158
Chapter 8 Conclusions 167
References 169
vi
List of Figures
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists
the equilibrium liquid which as it is quenched can either become a super cooled
liquid or crystallize Crystallization causes a discontinuity in the volume The super-
cooled liquid upon further quenching departs from equilibrium and transitions into
the glassy state Reproduced from Fundamentals of Inorganic Glass Science with
permission from the author1 5
Figure 2 The schematic for the flow of the program Beginning in the top right corner
and running until the condition in the pink box is satisfied Yellow diamonds
represent checks and blue operations 25
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above
and then plotted using PyConnect The plot is a disconnectivity graph where each
terminating line represents an inherent structure and tracing where two lines meet
describes the activation barrier The potentials are taken from the BKS potential 94 27
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-
axis being an arbitrary phase space and the y-axis being the potential energy
calculated from the Pedone et al potentials36 The colors are indicative of the
crystallinity where the blue basin is the initial starting configuration The landscape
shows the lowering of energy associated with partially crystallizing the sample (B)
The energy landscape relationship between the cutoff for crystalline and super-
cooled liquid states for 256 atoms is shown This shows a clear drastic energy
change occurring around the cutoff value of 10 Aring 37
Figure 5 An example interfacial structure between the crystalline phase on the left and
the last sequential SCLglass phase on the right for a barium disilicate system The
gray atoms are barium silicon is shown in red and blue represents oxygen 43
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a
function of temperature for each system size The experimental values for kinetics
and thermodynamics come from ref125 and from the heat capacity data taken from 126 respectively It is clear to see that driving force shows good agreement across all
systems however the kinetics terms only converge for the 256 and 512 atoms
systems (Bottom) The fit used to calculate the interfacial energy as a function of
temperature 45
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted
using the model presented in this work The data referenced can be found in refs 113128129 47
Figure 8 The surface energy with respect to temperature for the work presented here 50
Figure 9 An Angell diagram created using the MYEGA expression 55 With a very strong
glass (m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) 17134135
vii
The infinite temperature limit is from the work of Zheng et al54 and the glass
transition temperature is from the Angell definition 54
Figure 10 The relativistic glass transition temperature for B2O3 glass 57
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different
fractions of light speed All of the viscosities approaching the universal temperature
limit for viscosity 59
Figure 12 The modulus needed to satisfy the condition for the glass transition 61
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at
v=044c (Bottom) Viscosity plot showing a dramatic reduction of the viscosity as
the observer approaches the speed of light 63
Figure 14 b predicted and from literature showing good agreement with a total root-
mean-square error of 01 The fit for organic systems is given by 66 ln Kln = minus
and for inorganic systems by 75ln Kln = minus 72
Figure 15 The equilibrium model proposed with the experimental points showing good
agreement between the experimentally measured data points and the equilibrium
derived model RMSE was 002 for Corningcopy JadeTM (A) and less than 001 for
SG80 (B) (C) The model fit for B2O3 experimental data154155 The fragility and glass
transition temperature of the B2O3 are taken from the work of Mauro et al17 75
Figure 16 The stretching exponent calculated as described in the text for a Gaussian
distribution of barriers This plot shows that the distribution of barriers has a large
effect on the stretching exponent A Tg cannot be described since there is no
vibrational frequency included in the model though the glass transition temperature
should be the same for all distributions since the mean relaxation time is the same
for all distributions at all temperatures The deviation is given in ln eV units 78
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent
Each color designates one term in the series (Bottom) The output from RelaxPy
v20 showing the stretching exponent effects on the relaxation prediction of
Corningcopy JadeTM glass Each quadrant shows one property that is of interest for
relaxation experiments In particular it is interesting to see the dynamics of the
stretching exponent during a typical quench 80
Figure 18 The viscosity (left) and landscape (right) predictions for three common
systems The first system is newly calculated in this work while the latter two come
from our previous works7289 It is seen that the viscosity predicted from the AG
model is very accurately able to reproduce the experimental viscosity curves from
the MYEGA model The last system is a potential energy landscape while the others
are enthalpy landscapes 87
Figure 19 The configurational entropy comparisons between the three major viscosity
models which validates the main underlying assumption of the MYEGA model The
viii
VFT and AM are unable to capture the physics of configurational entropy therefore
ruling 89
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the
configurational entropy from the enthalpy landscape with the barrier 00155 eV
(compared to the shear barrier of 00149 eV) and the infinite limit allowed to vary
(10-263 Pamiddots for bulk viscosity) In Figure 18 the configurational entropy is
confirmed for the shear viscosity thus confirming the AG for both shear and bulk
viscosities Sidebottom data are from Ref 154 90
Figure 21 (A) The histogram for the energy minima for each of the test systems showing
a good fit with the log normal distribution This distribution will then be a valid
form to calculate the enthalpy distribution of the model presented in the next
session (B) The configurational entropy from the model showing the accuracy of
the scaling of the entropy predicted by the MYEGA model The S value was fit
for each system 93
Figure 22 (A) Comparison between the randomized method (histogram) and the
deterministic method (vertical lines) showing good agreement between the
maximum in the histogram and the value predicted by the deterministic technique
validating the approach It is worth noting that the 100-basin distribution is a very
wide distribution where the total number of basins is less than the number of points
used in the calculation This is done for a variable number of basins with the number
of basins shown in the legend (B) The dependence of fragility and the glass
transition temperature vs the distribution of states and the number of basins 99
Figure 23 The driving forces for different example glasses calculated using a
combination of the MAP model RelaxPy and the toy landscape model The
parameters for each glass can be found in Table 4 102
Figure 24 The predicted outputs from the toy landscape method showing predictions of
the enthalpy and entropy under a standard quench for barium disilicate This
prediction does not require fictive temperature or any such assumptions about the
evolution of the non-equilibrium behavior 105
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume
in nucleation is assumed to be on the order of one cubic angstrom while the a
parameter in growth is assumed to be around one nm both are in good agreement
for estimates in literature The values for the orange points are taken from these
works128167 107
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the
hydration models Color scheme Si atom (ivory) O atom (red) and Na atom (blue)
The z-axis is elongated to allow space for an insert of water 114
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the
top surface in contact with water is shown here 116
ix
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example
of the electronic-structure DFT calculation using semilocal exchange-correlation
functionals finding the binding energy of a water lsquopixelrsquo to the surface 119
Figure 29 Example contour surface showing the average coordination per atom on the
glass surface for the first run at 300 ps 121
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of
constraints for surface atoms at the local pixel Results show a distinct maximum in
which there is a near hydrophilic-hydrophobic transition of the surface The error
bars represent the standard deviation A second system with 1500 atoms was
performed to show convergence of the ReaxFF MD results 123
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values
of the model predictions are 641 GPa for constraint density 313 GPa for free
energy density and 774 GPa for angular constraint density 128
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from
minimum to maximum error The free energy density model gives the most accurate
results The constraint density has a RMSE of 61 GPa the angular density has a
RMSE of 20 GPa and the energy density has a RMSE of 59 GPa 129
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and
experiment for 10 Na2O 90 B2O3 Using the previously fitted onset temperatures
the only free parameters are then the vibrational frequency and the heating time in
which their product was fitted to be 14000 Where each dip in the modulus
corresponds to a constraint no longer being rigid as heated through each onset The
onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The
data was fit using a least-squares method and the resultant fit is shown as the
calculated method The fit has an R2 of 094 (B) The contribution from each
constraint to the overall modulus 131
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in
Fig 3) and experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of
093 and (B) xLi2Omiddot(100-x)B2O3 glasses with an R2 of 0986 133
Figure 35 The structure for the initial minimum energy configuration showing the boron
(blue) network with interconnecting oxygens (red) and the interstitial sodium ions
(yellow) 139
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and
sodium is ivory The barrier is overestimated compared to experimental data this
could be from several sources of error such as potential fitting thermal history
fluctuations or sampling too few transitions The line is drawn as a guide to the eye 139
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function
of reaction coordinates The middle shows the local deformation around the ion of
x
any atom that moves in between inherent structure mandating a relaxation force
The color shows the degree of deformation 141
Figure 38 Different network formers and the prediction of the activation barrier from our
model compared with activation barriers from literature (A) Sodium silicate
predictions and experimental values241 the error is calculated from the error in the
fragility when fitting the data (B) Lithium phosphate activation energy235 predicted
with topological constraint theory and compared with the experimental values (C)
Predictions over two different systems of alkali borates232 sodium and lithium with
a reported R2 of 097 143
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top
left) shows the distribution of the infinite temperature limit in the database after
limits exerted on the system (Top right) The distribution of the infinite viscosity
limit vs the glass transition (Bottom left) The relationship between fragility and
infinite temperature limit of viscosity (Bottom right) The infinite temperature limit
of viscosity vs the key metric predicted by SR 149
Figure 40 The prediction of the scaling of the activation barriers for a common sodium
borosilicate system 157
Figure 41 The relationship between the glass transition and the proton conductivity This
is justified two ways one through the relationship of the entropy of diffusion and
glass formation (the Adam-Gibbs model) and through the fact that water is known to
depress the glass transition 162
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine
the glass forming region Over top the constraints at the glass transition provided by
each oxide species is listed Since the objective is to decrease Tg while staying in the
glass forming region we will attempt to minimize use of elements that increase the
glass transition (nc gt 17) 166
xi
List of Tables
Table 1 The key properties considered for commercial application The optical properties
have been omitted since it is physical unrealistic to expect quantitative predictions of
quantum-controlled phenomena from a classical description of glass structure 18
Table 2 Variable definitions 65
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy
JadeTM glass and Sylvania Incorporatedrsquos SG80 74
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition
temperature (500 K) were kept the same while the total number of basins were
allowed to vary 101
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K 112
Table 6 System configurations for sodium silicate glass-water reactions 117
Table 7 Fitted values from this analysis compared to those reported in the literature The
disparity between the constraints evaluated with molecular dynamics most likely
come from the speed in which the samples are quenched 132
Table 8 Hyperparameters for different neural networks after hyper-optimizations 151
Table 9 A table with some ionic conductivity models and the parameters needed for
them as well as the disadvantages for each These are not the only models but are
representative of those commonly used in literature 154
Table 10 The predicted compositions based on the optimization scheme proposed 157
Table 11 The compositions synthesized in this work These compositions were predicted
by minimizing the cost function described in Eq (123) OP is the variant that was
melted after OP partially crystallized B-OP appeared to have surface nucleation in
some spots but was cut and removed before APS treatment 165
xii
Acknowledgments
ldquoTo Love Another Person is to See the Face of Godrdquo -Victor Hugo Les Miserables
I have been truly blessed to love and be loved by a plethora of mentors friends colleagues and
family across my academic journey I have to first thank my advisor Dr John Mauro who has
been as kind patient and caring as any mentor I have ever met and in the process has made me a
better scientist and a better person His mentorship was built on top of those who first found and
shaped me into a scientist Dr Ugur Akgun and Dr Steve Feller without whom my scientific career
would not be possible Along with these individuals Irsquove had the pleasure to learn so much about
glass and life from Dr Madoka Ona Dr Firdevs Duru Dr Mario Affatigato Dr Doug Allan Dr
Ozgur Gulbiten and Dr Seong Kim
Rebecca Welch Arron Potter and Anthony DeCeanne are my partners in all things
Rebecca is my partner in research and in life Her opinion support and patience has been
indispensable to me Arron is my oldest colleague and friend he is the one that I share the highs of
research and friendship with Anthony has put up with me non-stop for about 5 years now He is
the most patient and kind friend you could ask for Without these three individuals I could not have
done anything presented in this work I additionally have to thank Karan Doss Aubrey Fry Caio
Bragatto Daniel Cassar Brenna Gorin Mikkel Bodker Greg Palmer Katie Kirchner Kuo-Hao
Lee and Yongjian Yang I have the privilege to call these individuals both my friends and
colleagues
The last people I have to thank are my family I want to thank my brothers (Sam amp Quinn)
my mom and my dad My mom has helped me explore the universe through a love of reading and
her constant unwavering support My dad was the first to show me the wonders of science with
rockets as a kid as a teenager he supported my fledgling interest as he drove hundreds of miles to
every corner of this country and he is the person for whom this dissertation is dedicated
Chapter 1
The Difficulty of Optimizing Glass
Glass is a complex world-changing material Though it has existed for thousands of years
the surface of its true potential is only just being scratched To unlock the potential of glass for new
applications and to enable the maximum benefit to society we must be able to design new glasses
with wide ranging properties faster and cheaper than ever before [1]ndash[5] This is particularly
important now that the need for ecologically responsible materials is increasing due to climate
change In order to facilitate such glass design we must return to first principles and build a picture
of glass from the ground up encompassing both compositional and thermal history dependencies
into our understanding of glass properties[3] [6] The present accepted description of glass is a
non-crystalline non-ergodic non-equilibrium material that appears solid and is continuously
relaxing towards the super-cooled equilibrium liquid state[1] This definition gives insights into the
nature of glass and includes points of particular interest in this dissertation
The first point to consider is that glass is non-crystalline meaning that it contains no long-
range order This gives glass one of its key advantages being infinitely variable Since a glass is
not limited by having to reach stoichiometric crystalline structures there is no limit to the number
of possible glasses or glass structures This gives glass its possibilites Glass has been considered
as a candidate for everything from hydrogen fuel cells[7] to batteries[8] and for applications
ranging from commercial smart phone covers[9] to bioactive applications[10] The infinite
variability also explains the difficulty of designing glasses Consider a system with three
constituents for example SiO2 Na2O and B2O3 This then means there are two independent
compositional dimensions that must be fully explored to find an optimal composition for an
application If this space were discretized to 1 mol spacing that would give over 5000 unique
2
glass candidates that would need to be studied to reach the optimal design A five-component glass
would correspondingly have over 9 million unique glasses More generally for a glass with up to
C components there are 1C minus dimensions over which to optimize If instead one considers a
crystal there is not a continuously variable space but instead only discrete points
The other ramification of glass being non-crystalline is that it must bypass the region of
crystallization at a rate much faster than the rate of nucleation[11] This is inherent in the definition
of glass and is shown in Figure 1 Figure 1 is called the volume-temperature (VT) diagram and is
key to understanding the nature of glass The bypassing of the crystallization region leads to a
super-cooled liquid and during quenching the material departs from equilibrium and enters the
glassy state The region of departure is called the glass transition temperature range or simply the
glass transition To understand why this occurs we must first understand the concept of ergodicity
Ergodicity was defined by Boltzmann to mean that the time-average value of a property is
equal to the ensemble-average value of that property[12] [13] Now that ergodicity is defined we
can start to consider the implications of the non-ergodic nature of glass Since it is non-ergodic the
ensemble average will not equal the time average on human timescales however the ergodic
hypothesis says that in the limit of long time they will be equal[12] In order for both statements to
be true glass must evolve to be ergodic and thus is inherently unstable To explore this concept
further we must understand the glass transition process and define a timescale If there exists a
liquid at some temperature T and is then perturbed to T dT+ there will be some inherent time
associated with its relaxation towards equilibrium which is described by struct the structural
relaxation time There is a separate relaxation time when the system is mechanically perturbed
called the stress relaxation time stress To then understand whether the resulting material is a glass
or a liquid we need to compare struct with the observation time If the observation time is much
longer than the relaxation time then what is observed over the course of a experiment is the
3
equilibrium liquid properties Conversely if the observation time is much shorter than the
relaxation time then the observed properties will not be ergodic and as such we will not sample
the properties of the equilibrium liquid We define this non-ergodic state as the glassy state
However since in the limit of long time the system must become ergodic once again then over time
all glasses will return to an ergodic nature through a process called relaxation [14]
The last point considered in the definition of the glass is that it is non-equilibrium Not only
is glass non-equilibrium it is also unstable and continuously progressing towards the super-cooled
liquid state (eg relaxation) Relaxation is inherent in all glasses and we can see this for a model
system in the VT diagram (Figure 1) In this diagram there are two glasses (of the same
composition) quenched at different rates (both of which bypass crystallization) In the slow-cooled
glass there is a lower volume (higher density) compared to the fast-cooled glass This difference is
due to the fact that the slow-cooled glass has been allowed to relax more fully compared to the
other This adds at least one additional dimension of optimization (such as quench rate from a high
temperature to room temperature) in which glasses must be designed in
This dissertation is dedicated to understanding the infinite variability of glass with respect
to crystallization composition and thermal history with tools being implemented and designed so
that the challenge of optimizing over these vast spaces can be done with higher efficiency lower
cost and with less experimental work load These tools are all inspired by the energy landscape
description of glass To design the optimal glass for any application we must understand the
relationship between the 1C minus compositional dimensions a minimum of one thermal history
dimension and the properties of a material There are many ways to explore these relationships
each with its advantages and disadvantages The methods to optimize over these thermal history
and compositional dimensions include energy landscapes[15] fictive temperature[16] topological
constraint theory[17] and machine learning[3] Of these energy landscapes are the most
fundamental with each other technique being related back to the energy landscape in some form
4
Though they are the most fundamental they are also the most difficult time consuming and have
the largest barrier to entry for use To then build our understanding of each technique we will start
with energy landscapes then explore how through a series of assumptions we can arrive at the other
methods
5
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists the
equilibrium liquid which as it is quenched can either become a super cooled liquid or crystallize
Crystallization causes a discontinuity in the volume The super-cooled liquid upon further
quenching departs from equilibrium and transitions into the glassy state Reproduced from
Fundamentals of Inorganic Glasses with permission from the author[1]
6
11 Energy Landscapes
Energy landscapes were first developed by Goldstein and Stillinger who proposed the
concept as a way to understand the evolution of a material under complex conditions [18]ndash[20]
Since Stillingerrsquos seminal work [18] energy landscapes have become important in the study of
protein folding [15] [21] [22] network glasses [6] [13] and other complex materials The concept
of energy landscapes has become commonplace and is readily evoked when describing the time
evolution of a material [20] [22]ndash[26] Despite energy landscapes being commonplace in some
fields the procedure for mapping a landscape remains difficult due to the associated computational
challenges [15] [27] To map an energy landscape one must find a set of local minima and the first
order saddle point connecting each pairwise combination of those minima Each minimum is called
an inherent structure and represents a structurallychemically stable state of the material A basin
is the set of structures that converge to the same inherent structure upon minimization [18]
In glass science the energy landscape is used to justify complex behaviours which are most
commonly calculated using molecular dynamics simulations (MD) [13] [28] [29] It is easy to
understand why MD is used so frequently used when considering the ease of use of common MD
packages For example LAMMPS [30] is a common user-friendly MD program with a low barrier
to entry allowing individuals with little programming experience to run complex MD simulations
Inspired by this accessibility as well as the explicit Python bridge to LAMMPS we have developed
a software that can be used to explore energy landscapes with a variety of exploration techniques
Similar softwares [15] [31]ndash[35] have been created to map landscapes however each program has
limitations and may not be suitable for all purposes while ours is a general purpose software
The calculation of the energy for landscapes is done using an empirical interatomic
potential that can be fit using either experimental data and a series of MD simulations or using ab
initio methods One common potential form is the Lennard-Jones form (LJ)
7
12 6
4LJVr r
= minus
(1)
in which and are fitting parameters for every pair-wise set of atoms in a system V is the
potential energy contributed by the two-atom interactions and r is the distance between the two
ions in question Though the LJ form is simple and easy to parameterize it is unable to recreate
complicated material behaviours for a real glass As such a more complicated potential is often
used such as a Morse potential A commonly used silicate glass potential is the Pedone potential
which consists of Morse potential with an additional columbic term and repulsive term [36] and is
given by
( )2
2
0 121 exp 1
i j
Pedone
Z Z e CV Y a r r
r r = + minus minus minus minus +
(2)
Z is the charge of a given ion e is the charge of an electron and Y C a and r0 are fitting parameters
It is also worth noting that the Pedone et al potentials are only accurate when considering the
interactions between each individual ion species with oxygen For instance Si-O Na-O and O-O
are all explicitly parameterized while Si-Na only interacts coulombically (the first term)
Interatomic potentials are discussed here only briefly but there are extensive resources available
for those readers who wish to learn more[37]ndash[39]
Once an energy landscape has been mapped the thermodynamics and kinetics for a given
system can be easily calculated [33] [40] The thermodynamic driving force is given by the
difference between the free energy of the current inherent structure and the free energy of the lowest
state while the kinetics is the rate in which the system crosses the barrier between the basins This
can be calculated explicitly for arbitrary timescales and temperatures with another software
KineticPy [41] The energy landscape provides insights into the detailed nature of material
behaviour Though energy landscapes are very powerful they are also difficult to map To
approximate the behavior of the landscape then we use approximations such as fictive temperature
8
topological constraint theory (TCT) or machine learning Fictive temperature is a way to estimate
the effects of the thermal history dimension[42] while TCT is used to estimate the compositional
dependence of properties[43]
12 Fictive Temperature
Fictive temperature is an approximation used to find the change in the occupational
probabilities on an energy landscape[16] [44] Instead of considering every possible combination
of occupational probabilities like the energy landscape it instead approximates the occupational
probability using a single temperature at which the landscape would be in equilibrium It is
essentially a measure of how closefar a glass is from equilibrium This approximation allows for
us to calculate the relaxation dynamics of a glass without needing any computationally heavy
energy landscapes However this single parameter description oversimplifies relaxation[14] [16]
The functional form for glass relaxation dates back to 1854 when Kohlrausch fit the
residual charge of a Leyden jar with[45]
( ) (0)expt
g t g
= minus
(3)
where t is time τ is the relaxation time constant β is the stretching exponent and g(t) is the
relaxation function of some property (eg volume conductivity viscosity etc) Eq (3) is referred
to as the stretched exponential relaxation function (SER) The stretching exponent is bounded from
0 lt β le 1 where the upper limit (β = 1) represents a simple exponential decay and fractional values
of β represent stretched exponential decay
For over a century the physical origin of the stretched exponential relaxation form was one
of the greatest unsolved problems in physics until 1982 when Grassberger and Proccacia[46] were
able to derive the general form based on a model of randomly distributed traps that annihilate
9
excitations and the diffusion of the excitations through a network This model however provided
no physical meaning for β which was still treated as an empirical fitting parameter In 1994
Phillips[47] [48] was able to extend the diffusion trap model by showing that β=1 at high
temperatures and at low temperatures the stretching exponent can be derived based on the effective
dimensionality of available relaxation pathways He in turn expressed the stretching exponent
2
df
df =
+ (4)
where d is the dimensionality of the network and f is the fraction of relaxation pathways available
He then proposed a set of lsquomagicrsquo numbers for common scenarios Assuming a three-dimensional
network with all pathways activated (d = 3f = 1) a value of β= 35 is obtained The second case is
a three dimensional network with only half of the relaxation pathways activated (d =3f = 12)
yielding a value of β= 37 A third magic value β= 12 was found for a two dimensional technique
with all pathways activated (d = 2f = 1) β = 35 occurs for stress relaxation of glasses under a load
because both long- and short-range activation pathways are activated whereas a value of β = 37 is
obtained for structural relaxation of a glass without an applied stress The value was confirmed by
Welch et al[9] when the value was measured over a period of 15 years at more than 600C below
the glass transition temperature of Corning Gorilla Glass Though this model was able to reproduce
the limiting values it was criticized widely in the community Critics argued that the model was
simply created such that it reproduced the stretching exponent values of certain experiments and
ignored many other experimental results that appeared to disagree with the model The model also
fails to interpolate between the low temperature values Phillips predicted and the high temperature
limit
The work from Phillips and the stretched exponent gave a form for relaxation however
there remained no method to instantaneously state the distance from equilibrium of a glass To
account for this problem an additional thermodynamic variable (or order parameter) called fictive
10
temperature (Tf) was proposed Early research on relaxation in glasses dates back to the work of
Tool and Eichlin in 1932[49] and Tool in 1946[50] whose works originally proposed a temperature
at which a glass system could be in equilibrium without any atomic rearrangement Tool suggested
that this fictive temperature was sufficient to understand the thermodynamics of a glassy system
Originally the fictive temperature was treated as a single value that was some function of thermal
history (Tf[T(t)]) The evolution of fictive temperature as proposed by Tool is then given by
)
( ) ( ( ))
(
f f
f
dT T t T T t
dt T T
minus= (5)
In which for stress relaxation is given by
( )(( ) ( ))fT t T
G
T t = (6)
In Eq (6) is the shear viscosity and G is the shear modulus This means that to predict the time
evolution of a glass only three things are required shear modulus the viscosity as a function of
temperature and fictive temperature and the stretching exponent Though fictive temperature
qualitatively reproduced the results they were looking for subsequent experiments have shown that
the concept of a single fictive temperature is inadequate[51] [52]
A key experiment was performed by Ritland in 1956[51] Ritland took several samples
with different thermal histories but the same measured fictive temperature thus based on Toolrsquos
equation both should have had identical relaxation properties Ritland however showed that the
refractive index evolved differently between samples To account for these differences
the use of multiple fictive temperatures was suggested[52] This worked because the stretched
exponential form could be approximated with a Prony series
( ) exp expi i
N
i
Kt w K t minus minus
(7)
11
In the Prony series N is the number of terms while iw and iK are fitting parameters This means
that each term could represent one fictive temperature and this collection of fictive temperatures
would reproduce the overall experimentally observed form Ritlandrsquos experiment showed that
materials held isothermally at their fictive temperatures with varying thermal histories can give
different results for the relaxation of a given property Thus as a glass network relaxes the fictive
temperature will also shift Narayanaswamy[52] was able to apply Ritlandrsquos concept in a highly
successful engineering model with multiple fictive temperatures but the physical meaning of
multiple fictive temperatures remains elusive
In general there is reason to be skeptical of the concept of fictive temperature Fictive
temperature was not derived but instead was an empirical approximation that Tool needed to gain
an early quantitative understanding Recently Mauro et al[16] used energy landscapes to try to
understand the underlying validity of fictive temperature fT implies that the occupational
probability ( ip ) is given by
1
exp ii
Hp
Q kT
= minus
(8)
Q is the partition function k is Boltzmannrsquos constant and iH is the enthalpy of the given state
However when the actual probability evolution predicted by fictive temperature was compared to
the probability calculated through a systematic Monte Carlo approach the results were found to be
drastically different (even when many fictive temperatures were tried) This implied that fictive
temperature is insufficient to describe the glassy state and must be replaced with a new method
Despite the inadequacies of fictive temperature state-of-the-art modeling techniques for
glass relaxation still relies on the concept In these techniques a set of fictive temperatures (usually
12) are used to calculate 12 parallel relaxing fictive temperatures[42] This is because there has
been no alternative method able to reach a prediction of the evolution of glass under different
12
thermal histories The current state-of-the-art model is called the Mauro-Allan-Potuzak (MAP)[44]
model of non-equilibrium viscosity and was derived from a more fundamental expression for
viscosity called the Adam-Gibbs model (AG) which is given by[53]
10 10) log( )
log ( f
c f
BT T
TS T T = + (9)
B is the barrier for a cooperative rearrangement 10log is the infinite temperature limit of
viscosity and according to Zheng et al[54] it is approximately 10-293 Pa s and cS is the
configurational Adam-Gibbs entropy This entropy has long since been a source of debate and will
be addressed later in chapter 5
In order to calculate the equilibrium part of the viscosity the MAP model relies on an earlier
Mauro Yue Ellison Gupta and Allan (MYEGA) equilibrium viscosity model[55] This model
was derived based on earlier work by Naumis[23] [56] Gupta and Mauro[57] and the AG model
These models showed that the configurational entropy can be related to the degrees of freedom
lncS fNk= (10)
f is the degrees of freedom per atom N is the number of atoms and is the number of degenerate
states per floppy mode To then build in temperature dependence they assumed that the degrees of
freedom can be modeled using an Arrhenius form (with activation barrier H )
( ) expH
f T dkT
= minus
(11)
When Eqs (9) (10) and (11) are combined into an equilibrium form the equilibrium viscosity is
then given by
10 10 el pog og xlC H
T kT
+
=
(12)
Where C is equivalent to
13
ln
BC
dkN=
(13)
Through a change in variables this can be written in terms of three common variables that allow
for all viscosity equations to be written with three variables specified by Angel[58] the glass
transition temperature the fragility and 10log The glass transition temperature is defined as
the isokom where the liquid has a viscosity of 1210 Pa s The fragility is defined as[58]
10log
g
g
T T
dm
Td
T
=
(14)
The three most common viscosity models (MYEGA Vogel-Fulcher-Tammann (VFT)[59] and
Avramov-Milchev (AM)[60]) are listed below in terms of the same parameters
10 10 10
10
log (12 log ) exp 1 11 o
log2 l g
g gT Tm
T T
+ minus minus minus
minus =
(15)
( )
2
10
10 10
10
12 logl log
1 1
og
2 logg
Tm
T
minus+
minus + minus
=
(16)
and
( )10(12 log
10 10 1
)
0l gog l 12 logo
m
gT
T
minus
+ minus=
(17)
To incorporate non-equilibrium effects with these equilibrium models the MAP model
proposes
10 10 10log ( ) (1 ) log ( )log MYEGA g ne f gT m T mx x T T T = + minus (18)
14
10log ne is the non-equilibrium viscosity whose form and compositional dependence can be found
elsewhere[42] [44] [61] x is the ergodicity parameter and is of crucial importance to
understanding glass regardless of the model being used[12] In the MAP model it is given by
( )( )
3244min
max
f
f
m
T Tx
T T
=
(19)
Though the MAP model was derived from the best understanding of energy landscapes of glasses
at the time it has a few flaws that prevent it from perfectly predicting the different relaxation
behaviors of glass A few of the short-comings are
1 The method ignores crystallization For a comprehensive model showing the response of
glass to temperature the possibility of crystallization must be included Currently no
relaxation model includes this possibility
2 It is built around the concept of fictive temperature This could be replaced in a future
model but currently at present it carries with it all the failures of the fictive temperature
picture of glass dynamics adding only the relaxation time coming from landscape-derived
values
3 The MAP model does not consider temperature dependence of the stretching exponent
This is typically circumvented by choosing a fixed stretching exponent
4 The parameters needed for the non-equilibrium viscosity in the MAP model are
experimentally challenging to measure To circumvent this Guo et al[61] published
methods to estimate the values based on fragility and the glass transition but the
approximations were only tested on a few samples with similar fragilities and chemistries
5 The MAP model only predicts stress relaxation Since the publication of the MAP model
it has been shown that only stress relaxation time is related to the shear viscosity[62]
Structural relaxation must be incorporated in future models
15
For a complete understanding of the thermal-history dependence of glass all of these questions
must be rigorously answered
13 Topological Constraint Theory
TCT is one way to estimate the compositional dependence of glasses It uses a set of
assumptions that connects the underlying structure to glass properties through understanding of the
energy landscape These assumptions include the temperature dependence of the rigidity the same
assumptions used in the derivation of the MYEGA models and that the effects of thermal history
on structure are minimal TCT was originally proposed by Gupta and Cooper (GCTCT) to try to
understand the role of rigid polytopes on glasses properties[63] Gupta and Cooper fundamentally
understood that the ability for glasses to form was reliant on the transformation of a liquid with
many degrees of freedom to a glass with a fixed degree of freedom Since the structure of the glass
is the same as the structure of the liquid at the glass transition and the degrees of freedom of the
glass ( f ) changes during the transition to glass it was possible to gain insights into the
transformation process This approach is fundamentally an approximation of the highly
multidimensional landscape where only approximate structural information is used to understand
the entropy of the energy landscape Though the core idea was useful it was ldquoclunkyrdquo and difficult
to make calculations with
To improve upon the ideas of GCTCT Phillips and Thorpe (PTTCT)[64] proposed a
mathematically equivalent form where the degrees of freedom could be simply found by
considering the number of constraints ( cn ) around each network-forming atom in the glass They
showed that the degrees of freedom were related to the number of constraints by
cf d n= minus (20)
16
The number of constraints could then be calculated by considering the two-body and three-body
interactions (first and second terms) as a function of the mean coordination of the network forming
species
2 32
c
rn r= + minus (21)
This led to a qualitative language to discuss glass with a few quantitative insights The biggest
quantitative insight comes in the form of the lsquoBoolchand Intermediate Phasersquo[65]ndash[68] If it has
positive degrees of freedom then a network is lsquofloppyrsquo and retains high kinetic energy (as it flips
back and forth between two inherent structures) If the system has negative degrees of freedom
then the network is stressed-rigid and there is additional energy stored in the stress associated with
over-coordination If the number of constraints is equal to the number of dimensions then the
resulting system will have zero degrees of freedom and there is no stored stressed energy or kinetic
energy from flipping between basins Boolchand went searching for this perfectly lsquoisostaticrsquo glass
and found a non-zero window of compositions that were close to isostatic but had properties
drastically different compared to both stress rigid and floppy glasses This is now called the
Boolchand Intermediate Phase
Despite the success in the prediction of the intermediate phase TCT remained largely
qualitative until the work by Naumis[23] [56] and Gupta amp Mauro[57] Naumis pointed out that
the degrees of freedom are a lsquovalleyrsquo of continuous low energy so that the structure can freely move
between basins of the landscape Since these valleys are deformation pathways Naumis pointed
out that they will dominate the configurational entropy of the system Building on this Mauro and
Gupta wrote that since the entropy is proportional (Eq (10)) we could then write the glass transition
temperature as a function of the degrees of freedom and the AG model
( )10o1 ln2 l g
gTB
f Nkminus=
(22)
17
Since everything in Eq (22) is approximately a constant between compositions except for f we
could then write the expressions in terms of reference values (subscript r) and accurately scale the
glass transition of new compositions by knowing the changing structures
( ) g r c r
g
c
T d nT
d n
minus=
minus (23)
The last insight provided by Mauro and Gupta was that the number of constraints change as a
function of temperature They realized that each constraint has a temperature dependence which is
controlled by the interaction energy of the constraints This makes intuitive sense because as the
thermal energy becomes greater than that of the barriers additional valleys in the energy landscape
form This also allowed them to expanded TCT to include quantitative predictions of fragility
The understanding of the relationships between the configurational entropy the number of
constraints the degrees of freedom and the viscous properties lead to a revolution in understanding
glass properties Soon after the work by Mauro amp Gupta multiple works came out predicting
mechanical chemical and viscous properties of glass knowing only the underlying structure
Though this framework is powerful it is not a complete description without further
parameterizations with each model requiring some fitting parameters or reference values In
addition there are still key properties that are missing from the standard topological approach that
are needed to design commercial glasses
Ultimately MGTCT needs be expanded to include the properties that are needed for
commercial applications A list of composition dependent properties needed for understanding the
performance of commercial glasses are shown in Table 1 Properties taken from Ref [3] show what
properties must be controlled to manufacture technological glasses All models take additional
inputs and there are multiple ways to link outputs from some models to inputs of the other Choice
of model vary depending on the use case
18
Table 1 The key properties considered for commercial application The optical properties have
been omitted since it is physical unrealistic to expect quantitative predictions of quantum-
controlled phenomena from a classical description of glass structure
Properties TCT Machine Learning Other methods
Glass Transition [57] [69] [70] Energy Landscapes [71]
Fragility [57] [69] This work This work
Relaxation MAP model [44] [61]
Crystallization This work [72]
MDMC [73]ndash[75]
Melting Temperature This work MDThis work[72]
Chemical Durability [76] [77]
This work [78]
[79] [80] Linear Methods [81]
Density [82] MD Packing [83]
CTE This work MD
Hardness [5] [84] [85] MD
Youngrsquos Modulus [86] This
work [87]
[82] This work MD
Activation Barrier for Conductivity MD This work [88]
Batch Cost Economic Calculation
Ion Exchange Properties Empirical Relationships [2]
19
14 Machine Learning
All the methods discussed so far are physically informed techniques The techniques
presented are inherently powerful because their very nature is informed by reality Unfortunately
this means that some knowledge about the system is required These techniques are useless without
the required parameterizations However a class of algorithms called lsquoMachine Learningrsquo (ML)
require no such parameterization Instead ML uses large quantities of data to determine trends
linking any physically connected input to output allowing for accurate predictions of properties
across wide compositional spaces This is favorable with a large of volume data because the
required inputs are minimal to achieve a prediction The difficulty (and often expense) in this
technique is gathering large quantities of high-quality data to enable this method
The predominant ML method used in this work is neural networks (NN) Neural networks
work by constructing a network of neurons This method was designed in such a way to mimic a
human brain so the network is elastic enough to be modified improving results Each neuron takes
in a series of inputs received from the previous layer (or input data) multiplies by a weighting
factor sums the values and then performs some lsquoactivation functionrsquo on them By changing the
weights it is then possible to recreate any function given sufficient data This is a preferred
technique because it packages into a simple easy-to-use software Neural Networks (once trained)
can be used by anyone stored in a small file can recreate any function is computationally cheap
and does not require as much data as some other ML techniques
This is an interesting method to estimate the effects of the composition on the landscape
and corresponding properties because we cannot understand the underlying mathematics It is
fundamentally a numerical process Thus even though we know that the properties of a glass are
dominated by the topography of the landscape how the NN figures out the relationship between
the chemistry and properties is not understood It is possible to then use the NN to give us the
20
physical parameters we need for other models building multiple methods together to achieve new
insights
15 Goals of this Dissertation
To enable the next generation of glass design we must enable a new era of computationally
methods being used to produce accurate predictions of the behavior of glass This includes both the
thermal history dependence of glass (in terms of relaxation and crystallization) and the
compositional dependence especially those properties listed in Table 1 The goals of this thesis will
be to address some of these outstanding questions in a systematic way that is practical for an
external user to understand and implement Predicting thermal history effects are covered in
Chapters 3-5 while compositional effects are predicted in Chapters 6-7 The breakdown of each
following Chapter as it relates to these questions is listed below
21
bull Chapter 2 Software Unfortunately there is no standardized software for the
calculation of relaxation or energy landscapes I present two new codes to
standardize these calculations so that the process is reproduceable and standardized
bull Chapter 3 Crystallization This section will address new computational ways to
calculate crystallization focusing on predicting nucleation rates and how to improve
our nucleation calculations
bull Chapter 4 Expanding the Current State of Relaxation This chapter will focus on
expanding the MAP model with a temperature dependent form of the stretching
exponent as well as understanding the role of ergodicity in glass relaxation
bull Chapter 5 Relaxation and Crystallization without Fictive Temperature Chapter 5
will focus on discussing relaxation and crystallization without fictive temperature
by proposing a new relaxation model that is reliant on creating lsquoToy Landscapesrsquo
This approach removes assumptions about the evolution of the occupational
probabilities
bull Chapter 6 Enabling Prediction of Properties This chapter will use TCT and ML to
address the compositional dependance of different key properties needed for
enabling the faster design of new glass compositions
bull Chapter 7 Designing lsquoGreenrsquo Glasses Applying information learned throughout
this dissertation to create new glass compositions
bull Chapter 8 Conclusions
22
Chapter 2
Software for Enabling the Study of Glass
To being our systematic investigation it is important to have a suite of tools that
implement current state-of-the-art models Though many tools exist for structure and
topological constraint theory there were no codes that existed for the calculation of
relaxation using the MAP model and there were no codes that were capable of exploring
the landscapes that we are interested in To correct this we present two new novel codes
ExplorerPy for mapping energy landscapes and RelaxPy for calculating MAP
dynamics[42] [89]
21 ExplorerPy
Mapping an energy landscape allows for modeling a range of traditionally inaccessible
processes such as glass relaxation The advantage of this software (over other codes to map energy
landscapes) is that it only requires knowledge of LAMMPS to run Also our software contains a
choice of three methods to map the landscape The first of the three different methods implemented
to explore energy landscapes in this work is eigenvector following [90] Eigenvector following has
been used in the past to generate landscapes that predict the glass transition and protein folding
pathways [44] [91] This implementation follows the work of Mauro et al [90] [92] and uses an
independent Lagrange multiplier for every eigenvector Eigenvector following excels at finding
small barriers quickly however it rarely finds unique structural changes in complex materials
(especially in systems with periodic boundary conditions) The second method that we have
implemented following in the steps of Niblett et al [27] is a combination of molecular dynamics
and nudged elastic band methods (NEB) [34] [35] To find the transition points MD is run at a
23
fixed temperature for a fixed amount of time the structure is minimized and then NEB is used to
find the saddle point between the initial structure and the new inherent structure Details of the
nudged elastic band method can be found elsewhere [34] Using these techniques it is possible to
map a sufficiently large landscape to make kinetic and thermodynamic measurements possible
These two methods benefit from the fact that they work well when the volume is varied and together
enable computationally inexpensive mapping of enthalpy landscapes [71] [92] (where the volume
is no longer fixed and instead the pressure is specified) However it is not recommended to generate
a large enthalpy landscape from MD in this software due to the computational cost involved
Although molecular dynamics is a powerful technique there is a large computational cost to
simulate a small amount of time
The third method is a new technique created in an attempt to circumvent the computational
cost of MD while still finding large structural changes We have named this technique ldquoeigen
treibanrdquo (ET) This technique relies on shoving along an eigenvector with a multiplier rather than
stepping After the system is shoved the system is minimized and if the minimized structure exhibits
a displacement in atomic position is larger than a threshold a NEB calculation is performed to find
the saddle point By shoving in the direction of an eigenvector the goal is to reproduce large changes
due to vibrational modes The fixed multiplier (m0) is calculated by taking a target distance (d) and
the largest displacement magnitude out of the atoms j (vij) in a given eigenvector direction i (ai)
m0
=d
max[nij] (24)
To convert to the displacement magnitude from an eigenvector in three dimensions we use the
expression
vij
= ai[3 j]2 +a
i[3 j +1]2 + a
i[3 j + 2]2 (25)
The dimension j has the length of the number of atoms in the systems
24
This software uses the system and potentials created from a LAMMPS input
scripts and as such has a wide range of access to different potentials and possible systems The
only input that has to be provided is the file which would be used for molecular dynamics in
LAMMPS If the script has been used for molecular dynamics simply by eliminating any
thermostats or barostats as well as any ldquorunrdquo commands it is possible to use it as an input script to
ExplorerPy (it is also recommended to eliminate any calls to ldquoThermordquo because it may interfere
with the extraction of variables from the underlying LAMMPS framework) If the LAMMPS input
script successfully sets up a simulation all other variables and parameters can be controlled from
command line options as described in the ldquoIllustrative Examplerdquo section The initial structure will
be taken from the input script as well
ExplorerPy is a software that has been written in Python with modular functions and ease
of use in mind It can be run from a Linux terminal with all major options being set via the command
line The software then generates a list of inherent structure energies stored in the mindat file and
stores the transition point energy for each pairwise inherent structure in the tsddat file All of the
transition points and minimum are stored in the same directory from which it is run as xyz files
The software architecture is shown in Figure 2 Details of the stepping procedure for eigenvector
following can be found elsewhere[93] The variables ldquoMD frequencyrdquo and ldquoET frequencyrdquo are set
as command line options at the time of the launch of the software
25
Figure 2 The schematic for the flow of the program Beginning in the top right corner and running
until the condition in the pink box is satisfied Yellow diamonds represent checks and blue
operations
The software is capable of mapping an energy landscape for an arbitrary system ie for
any system that can be simulated in LAMMPS This allows for accurate prediction of the evolution
of complex thermodynamics and kinetics The software requires no programming and is
controllable via command line options ExplorerPy prints all of the basins transition states inherent
volumes as well as information about the curvature of the system so that the vibrational frequency
is calculable This is all of the information required to calculate a complex range of phenomena
over a specified long timescale ExplorerPy is only set up to work within the LAMMPS metal units
To run the software there are only a few parameters that have to be set for each mode The
force threshold (-force) curvature threshold (-curve) and step size (-stepsize) when running the
command line option have to be set to allow for a stable exploration of a given landscape
These seem to be potential-specific and for a given potential appear to be fairly stable It is best to
start large for the force threshold and small for the eigen threshold and step size then modify as
26
necessary If eigenvector following is not being used there is no need to adjust these values Beyond
this there are several optional command line options (only the file (-file) command is required)
-file [file input] this is a standard lammps input if the input sets up a lammps run
it will work to set this up There should not be any lsquothermorsquo calls in the lammps
script
-press [specified pressure in atmospheres] if this is specified the system will
always be minimized to this pressure while if not specified the volume remains
fixed
-num [number of basins to explore]
-time [max time to run in hours] if this time is reached before the number of basins
are found the simulation will stop
-thresh [minimum energy for barrier to keep]
-center if this is given as a flag the system will be set to the anchor location each
step
-anchor [atom selection for the position to be held constant]
-md [frequency of molecular dynamics search] If a custom MD LAMMPS file is
not provided it defaults to starting all of the atoms at 3000 K and running for 50
fs
-shove [frequency of eigen treiban]
-custom [file used for MD exploration]
-help [List all commands possible in software with these descriptions]
An example run would then be done the following way
[mpirun -np 5 python37 Explorerpy -file inSiO2 -num 1000 -force 10 -curve 1e-5 -stepsize 012
-press 00 -md 11 -time 48]
This would start a run with five parallel processors searching for 1000 basins with an MD
step being ran every 11 searches with a maximum time of 48 hours The results of this command
are shown in Figure 3 Below we have attached an example of inSiO2 the output and additional
scripts related to the example has been uploaded to the associated GitHub
27
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above and then
plotted using PyConnect The plot is a disconnectivity graph where each terminating line
represents an inherent structure and tracing where two lines meet describes the activation barrier
The potentials are taken from the BKS potential [94]
httpsgithubcomlsmeetonpyconnect
28
22 RelaxPy
RelaxPy script is written using Python syntax and relies on the numpy scipy and
matplotlib libraries It solves the lack of a unified relaxation code in our community and is based
on the MAP model[44] [61] interpretation of multiple fictive temperatures and the shear
modulusviscosity form of the Maxwell equation[62] [95] It is designed to be run within a Linux
terminal with command line options or inside a Python interpreter It can be used by calling the
name of the script followed by the input file the desired output file name and finally an optional
tag for all fictive components to be displayed as well as printed to a specified output file (formatted
as comma separated values)
The RelaxPy package consists of an algorithm that iterates over time in order to determine
the values of viscosity (η) relaxation time (τ ) and current fictive temperature (Tf ) given the user-
supplied thermal history and material property values The purpose of this code is to enable
calculation of the relaxation behaviorrsquos change over time by using the MAP model using only
experimentally-attainable parameters Toolrsquos equation is solved for each term in a Prony series
approximation of relaxation finding the change in each Tfi component for a given time (t) to
accurately reproduce the stretched exponential form The program then iterates over the entire
thermal history of the sample using a user-specified time step dt allowing for calculation of the
overall fictive temperature over time The values for the Prony series fit (wi and Ki) are taken from
the database originally generated by Mauro and Mauro[96] and expnded to include any value of
the stretching exponent The values were fit by Mauro and Mauro with a hybrid fitting method
based on the number of terms in the desired Prony series and the magic value from Phillips The
thermal history is also defined in the original input file using linear interpolation The initial fictive
temperature components are assumed to be at equilibrium (Tfi = T(0))
29
To solve for the non-equilibrium viscosity used the MAP expression is used with the approximate
scaling of values proposed by Guo et al[61] The software is capable of performing the calculations
needed for a variety of thermal histories and processing methods desired by both industry and
academia with a focus on the fictive temperature This software is then fundamentally limited by
the issues associated with the MAP model and the compositional approximations made For
instance this software will need to be further modified to include structural relaxation when such
a model is proposed[62]
An example input code is listed below
Denotes Comments
Tg 794 0 m 368 C 1494 dt 10 Beta 37
Start Temp End Temp Time [s]
1000 10 1300
10 100 1000
100 30 500
Assumes Linear Interpolation
This example input states that over the space of 1300 seconds an initial glass sample is cooled from
1000 C to 10 C Then it is heated to 100C over 1000 seconds and then finally down to room
temperature in the space of 500 seconds All temperatures are given in Celsius
This software aims to provide the first open-source software for modeling of glass
relaxation behavior using the MAP model of viscosity by creating a package that can easily and
quickly be used to approximate the fictive temperature components of a glass as well as the
composite fictive temperature RelaxPy allows for researchers to calculate the evolution of
macroscopic properties of glass during a relaxation process (assuming a fixed stretching exponent)
and will facilitate the calculation based on a user-supplied thermal history RelaxPy can be
seamlessly adopted by most glass research groups and removes the guessing associated with
determining a fictive temperature allowing for a fully quantitative comparable discussion of Tf
simultaneously creating a tool that can easily be expanded for the entire community as new physics
is discovered
30
Chapter 3
Understanding Nucleation in Liquids
Crystallization cannot be ignored in any material system at temperatures below its liquidus
temperature although extreme resistance to crystallization is reported in some organic and
inorganic substances[97] Crystallization (and by extension nucleation) is critical in systems
ranging from ice to thin films[98]ndash[100] Glasses are non-crystalline substances therefore
crystallization kinetics will determine the necessary thermal history to avoid crystallization and
obtain a glassy state Despite the absolute importance of crystallization to the study and design of
glasses there is no current method to predict the nucleation of crystals in this event One of the main
challenges in the production of glass-ceramics is finding the optimal thermal history for the desired
crystallite distributions There are two phenomenological steps in the process of phase
transformation from a supercooled liquid to a crystal a nucleation step and a growth step The
growth step is described well by Wilson-Frenkelacutes theory[1] but the underlying physics of
nucleation remains elusive A stable nucleus is the precursor to a crystal comprising a nano-sized
periodic assemblage of atoms A clear understanding of the relationship between liquid kinetics
and crystal nucleation has not been established despite decades of research
The study of nucleation is particularly critical for glass-ceramics Since their discovery by
S Donald Stookey glass-ceramics have become prevalent in our society through products such as
stovetops and Corningware[73] [101] [102] Glass-ceramics are a composite material comprising
at least one crystalline phase embedded in a parent glassy phase benefitting from the properties of
both the glassy and crystalline phases[1] [103] One of the reasons glass-ceramics are ubiquitous
is their relatively easy production glass-ceramics can be formed through standard glass forming
procedures using additional heat treatments Additionally there is a wide range of accessible
31
properties through tuning both the chemical composition of the parent glass and the thermal history
and hence the microstructure of the heat-treated glass-ceramics[101] [103]ndash[106]
Glass-ceramic production consists of first synthesizing a glass and then generating a
microstructure through the two-stage ceramming process[73] [103] [107] The nucleation step
remains poorly understood with classical nucleation theory (CNT) giving widely varying results
depending on how the parameters are determined[1] [73] [108] The inability to consistently get
accurate predictions for a nucleation curve means that choosing an appropriate nucleation
temperature must be done experimentally which can prove to be one of the most challenging and
time-consuming aspects of properly designing a glass-ceramic The difficulty is due to the
magnitudes of the crystal nucleation rates not being predictable with CNT or any other current
model
This difficulty in designing a material is apparent when considering the high dimensional
phase space of parameters that can adjusted We haver presesnted some tools developed to help
optimize over this lsquoglassyrsquo space such as RelaxPy[42] KineticPy[41] topological constraint
theory[109] or various machine learning models[4] [70] [110] [111] However a glass-ceramic
system has at least four additional phase dimensions (two for the nucleationgrowth temperatures
and two for nucleationgrowth times) that need to be optimized (depending on the parameterization
of the experiment it is also possible to add another four dimensions relating the heating and cooling
between each step) Therefore a glass-ceramic system has a minimum of d+5 dimensions over
which the system must be optimized and yet those four additional dimensions are the least
understood Designing methods to easily optimize over this large space with feasible computational
methods is of utmost importance to reaching a new generation of custom materials
Experimental studies of nucleation have used techniques ranging from differential
scanning calorimetry (DSC) to electron microscopy to try to capture predictive capability in the
lsquoceramicrsquo phase-space (the four additional dimensions) These studies have successfully captured
32
the shape of the nucleation curve but fail to provide the information needed to calculate the
temperature-dependence of the nucleation rate The shape of the curve can be determined through
DSC and with great effort an approximation of the rate can be made however this technique is too
laborusome to be efficient[112]ndash[114] Since the parameters have been inaccessible to traditional
computational approaches and experimental determination all previous studies of nucleation have
required at least one variable to be fit If a nucleation curve could be predicted only using a
computational method or with an efficient simple DSC method the rate at which new glass-
ceramics products are created would increase In addition the cost of producing glass-ceramics
would likely decrease and more complex microstructures and therefore more unique properties
could be explored
Crystal nucleation has been historically difficult to simulate and measure experimentally
because the phenomena happen on spatial scales of nanometers but over many orders of magnitude
of time scales CNT is the most widely applied model despite its supposed failure To test CNTrsquos
assumptions an interest in computational work has grown Recent approaches using grand
canonical Monte Carlo have successfully predicted the short time dynamics of nuclei
formation[73] This approach involves use of a continuous solvation model for the liquid and has
enabled greater computational insights however it remains computationally expensive[74]
Molecular dynamics (MD) fails to capture nucleation in glass-forming systems due to the need for
longer spatial and temporal scales to capture nucleation effects accurately Some attempts have
been made using Monte Carlo MD or empirical approaches to replace classical nucleation theory
(CNT) with some alternate method where the parameters can be directly measured[73] [115]ndash
[117] All nucleation models and Monte Carlo models attempt to describe the rate that a
supercooled liquid system transitions to a crystalline structure as a function of temperature
Explicitly mapping the energy landscape would fundamentally explain the underlying
physics of nucleation An energy landscape of a system is given by
33
1 1 1 2 2 2( )N NNU U x y z x y z x y z= (26)
where U is the potential energy and x y and z are the locations of each of the N atoms in a periodic
box The locations of these atoms give rise to an energy as parameterized by an interatomic
potential An energy landscape can be constructed by mapping the relationship of the local minima
and the first order saddle points between pairs of minima22-27 Energy landscapes have become
crucial to studies of biomolecules glasses and catalysts[21] [26] [44] because landscapes show
atomic configurational changes and calculate the associated kinetics and thermodynamics In this
study we will explore the energy landscape of a barium disilicate system capturing the glassy and
crystalline basins enabling a test of CNT by calculating all needed parameters (interfacial free
energy kinetic barrier and free energy difference of the supercooled liquid and the equilibrium
state)
31 Crystallization Methods
CNT expresses the steady-state nucleation rate (I) as the product of a kinetic and a
thermodynamic factor
( )exp eI Z D TW
kT
minus =
(27)
where Ze is the Zeldovich factor which incorporates the probability of a critical sized nucleus that
will form a new phase independent of other nuclei D(T) is the kinetic factor often given by an
Arrhenius expression over sufficiently small temperature ranges W is the work associated with
generating a thermodynamically stable nucleus that will continue to grow rather than dissipate (ie
the critical work) if temperature and pressure are kept constant k is Boltzmannrsquos constant and T is
absolute temperature The Zeldovich factor is typically treated as a constant with a value ~01[118]
34
Here the kinetic term is expressed as D(T) since in this work the kinetics are explicitly calculated
at every temperature and it is not treated as a simple Arrhenius expression In CNT the work W is
given by the work to generate a spherical isotropic nucleus[1] which consists of a volume term
and a surface energy term It is given by
3 24
( ) 4 ( )3
W r G T r T = + (28)
where G is the Gibbs free energy difference between the crystalline state and the corresponding
liquid r is the radius and is the interfacial energy between the crystalline and liquid phases Eq
(28) neglects the strain energy term that arises from the formation of a solid nucleus in the
supercooled liquid It is assumed that the liquid relaxes fast enough so that strain energy is not
significant for nucleation in liquids[11]
Determining the critical radius (ie taking the derivative of Eq (28) with respect to the
radius and setting the expression equal to 0) allows one to solve for the critical work and the
expression becomes
3
2
16
)3(
( )
( )W
T
G T
=
(29)
Three temperature dependent functions remain D G and Though there are existing methods
in literature to approximate each of these quantities the approximations often include large errors
and assumptions about the behavior of the system[119] Often experimentally the kinetic function
is approximated with the Stokes-Einstein relationship the G term is calculated from heat
capacity curves (assuming that the macroscopic energy is similar to the nanoscopic energy) and
is fitted[119] In the calculation presented here only configurational energies are considered
while vibrational contributions are ignored Ignoring the vibrational component and the reliance on
empirical potentials are the most likely sources of error in this work
35
The energy landscape for this investigation was generated using ExplorerPy[89] starting
with the orthorhombic barium disilicate crystalline structure[89] Barium disilicate was chosen
since it is a commonly studied glass-ceramic with multiple reliable experimental datasets being
found in literature[75] [113] [120] The rest of this section is divided into four parts starting with
a discussion of mapping the landscape and the following three sections are a discussion of
calculating D G and respectively
31A Mapping and Classifying the Landscape
The ExplorerPy energy landscape exploration employed the Eigen-Trieban technique in
the canonical ensemble with systems consisting of 128 256 and 512 atoms[89] The number of
atoms was chosen to achieve convergence of the results while also balancing the large
computational cost associated with energy landscape exploration A smaller system was also
calculated with 64 atoms however this systems was too small to accurately discern energies
enough to make the nucleation calculation The Eigen-Trieban technique is a newly developed
method to explore energy landscapes using a method of shoving along the eigenvectors and
minimizing the potential energy which is followed by using nudged elastic band calculations to
find the associated transition points[89] This technique allows for larger transitions to be found
while retaining the ability to calculate the vibrational frequencies The Pedone et al potentials[36]
were used and the volume of the cell was fixed due to simulation instability when pressure was
fixed The Pedone et al potentials were chosen since they have reproduced key parameters for both
the crystalline and glassy state in previous Monte Carlo studies as well as in the training set of the
potential[73] [74] [121] Though the Eigen-Trieban method may work well for predicting
crystallization and finding large barriers it may be insufficient to describe glass relaxation or
36
viscosity since it rarely finds the small structural transitions that dominate the Adam-Gibbs
relationship Note that these exceptions are inconsequential for predictions of nucleation found in
the work presented here33
The barium disilicate system was explored for 2500 basins for 128 and 256 atoms and 4000
for 512 atoms since more basins were needed to converge the liquid state distribution of energies
This number of basins was shown to be sufficient since multiple paths from the initial crystalline
starting point to SCL states were obtained The distribution of energies of the SCL were found to
converge and the resulting energy landscape is seen in Figure 4
A crystal parameter was then defined from the root mean square displacement (RMSD) for
each atom from the nearest lattice site of the same species ( ) This average value is used to
characterize whether a basin is a crystal or SCL The RMSD approach is not the only method to
assign a crystallinity value[122] however it is one of the simplest and as such was chosen for
this application To then categorize each basin as either a crystalline basin or SCL basin a plot of
the minimum energy of the SCL states versus the cutoff of RMSD was produced This plot for 256
atoms shows a clear transition in the energies around a cutoff value of 1 Aring which is therefore
taken as the cutoff radius of which basins are considered crystals The same analysis was repeated
for each system size This analysis is shown in Figure 4 Any basin with a RMSD less than the
cutoff value is considered a crystalline state This cutoff was found to be generally insensitive to
the results with any values ranging between 09 Aring and 11 Aring changing the resulting prediction of
nucleation by less than one order of magnitude but shifting the peak temperature by up to 40 K
This cutoff also reproduced the energy difference between the SCL and crystal in molecular
dynamics simulations of the materials
37
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-axis being
an arbitrary phase space and the y-axis being the potential energy calculated from the Pedone et al
potentials[36] The colors are indicative of the crystallinity where the blue basin is the initial
starting configuration The landscape shows the lowering of energy associated with partially
crystallizing the sample (B) The energy landscape relationship between the cutoff for crystalline
and super-cooled liquid states for 256 atoms is shown This shows a clear drastic energy change
occurring around the cutoff value of 10 Aring
[
38
31B Kinetic Term for CNT
The transition rate was calculated by the inverse of the average relaxation time from the
SCL to the crystalline states The relaxation time ( ) is defined by the relaxation function ( )
0 exp
t
= minus
(30)
where t is time and 0 is the scaling of the relaxation function The reason we are able to
calculate the kinetics of nucleation but not viscosity is because the nucleation kinetics are
dominated by the transitions between two states that we specifically searched for when exploring
the energy landscapes while the viscosity is defined by the entropy dominating small transitions
described by the AG
The AG relationship shows that the viscosity is dominated by the configurational entropy
of the system Experimentally the kinetic term is often calculated with the Stokes-Einstein
relationship
( )6
kTD
aT
= (31)
where a is the radius of the diffusing species and is the viscosity of the liquid Interestingly this
approximation gives reasonable results when investigating experimental data since the viscosity is
a measurement of the entropy of small cooperative liquid rearrangements Conversely the kinetics
that govern nucleation are large structural rearrangements from a supercooled liquid to a crystal It
is nonetheless possible that these cooperative rearrangements and the large diffusion processes for
nucleation are on the same order of magnitude which is why the Stokes-Einstein relationship gives
39
a reasonable approximation in experimental parameterizations When the Stokes-Einstein
relationship is used there is an implicit assumption that the large nucleating rearrangements are
proportional to the small barriers that govern viscosity[53]
A mean relaxation time can be found by averaging over a system of master equations
( )j
ii ji j j ij i
i
dpg K p g K p
dt
= minus (32)
In this expression g is the degeneracy of a microstate pi is the probability of occupying the
microstate and Kij is a transition rate between inherent structure i and inherent structure j The
equilibrium probability of occupying a microstate can be given by a Boltzmann probability (where
Z is the partition function and U is the potential energy of the inherent structure)
1
exp ii i
Up g
Z kT
= minus
(33)
By substituting the value of jp in Eq (32) for a two-state model ( 1 21 i jp p= == + ) we write
( )( )1ii ji i j ij i
dpg K p g K p
dt= minus minus (34)
Isolating the probabilities and splitting the differential we have
( )1
i
i ji i j ij i
dpdt
g K p g K p=
minus minus (35)
which when solved via integration takes the form
ln ( )
( )
i ji i ji j ij i
i ji j ij
g K g K g K pt
g K g K
minus + =
minus + (36)
Solving for the occupational probability yields
( )expi ji j ij i ji
i
i ji j ij
g K g K g K tp
g K g K
minus minus + =
+ (37)
This shows an analytical solution for the relaxation time in a two-state model
40
1
i ji j ijKg g K
+
= (38)
The two-state model approximation is made to enable the calculation of the mean relaxation time
between the liquid basins to the crystalline basins The transition rates between the basins are given
by the vibrational frequency ( ) and the barrier between the states ( ijU )
exp ij
ij
UK
kT
minus=
(39)
The probability of transitioning from a given liquid basin i to a crystalline basin j is given by
1
expij
ij
Up
Y kT
minus =
(40)
where Y is the partition function with respect to all transitions from the current SCL basin to all
crystalline basins as only these transitions are considered to calculate the relaxation time
expj
j
iY
k
U
T
= minus
(41)
The total expected relaxation time of the system making the transition is then given by the
probability of occupying a basin the probability of transitioning from one basin to another specific
basin and the associated relaxation time
1
i ij
j i ji j ji i i
p pg K g K
=
+ (42)
The first summation is over the liquid states and the second summation is performed just over
crystalline states to capture the mean relaxation time from a supercooled liquid to a crystal We
fixed the vibrational frequency to a value of 17 times 1014 Hz (ie about one jump attempt per 5
femtoseconds) which was calculated using the average of the eigenvalues of each basin ( ) and
the average mass of the atoms (m) using
41
1
2 m
= (43)
31C Degeneracy calculations
To obtain an appropriate value for the degeneracy an understanding the phase-space
volume of the basin is needed since the degeneracy is linked to the volume of the basin[40] By
taking the displacement between the basin coordinates of each atom and the transition point we can
define a radius of the basin R Using R and a spherical approximation we can calculate the phase-
space volume degeneracies
3 23
12
NN
i i
cry
Ng R
g
+
=
(44)
In which ( )x is Eulerrsquos gamma function The degeneracy is then normalized by a constant such
that the liquid-liquid state transitions had a value of 100 s at the glass transition (980 K) to be in
agreement with Angell[58] The normalization step allows for a quantitative prediction of the
liquidus temperature to confirm the accuracy of our model The liquidus value found is
approximately 1200 K 1715 K and 1950 K for 128 256 and 512 atoms respectively The
experimentally determined liquidus temperatures range from 1680 K to 1710 K[113] [115] [120]
It is worth noting that changing the exponent drastically changes the thermodynamics while having
minor effects on the kinetics and the normalization value drastically changes the kinetics while
keeping the thermodynamics nearly constant We chose the denominator to satisfy the liquid
relaxation constraint It is also feasible to choose the exponent to satisfy a constraint on the liquidus
temperature The normalization value is an important to consider because for larger system a
change in 001 can result in a change in the kinetics by ~10 orders of magnitude
42
31D Free Energy Difference
Due to limitations in the exploration of the energy landscape requiring a fixed volume we
will approximate the difference in the Gibbs free energy by calculating the difference of the
Helmholtz free energy This approximation is reasonable due to the similar densities between the
crystalline and glassy phases under ambient pressure[73] [74] The first step is determining the
mean energy difference of all the basins and the second step was repeating the process for only the
liquid basins The mean energy was calculated with the following expression[123]
( )ln ln ii
i
i i iG kTp p U kT g p= + minus (45)
In Eq (45) i is the set of relevant basins for the conditions we are interested in The difference
between the mean free energy of the liquid basins and the entire landscape basin energies is the
G parameter This free energy difference is typically experimentally calculated by integrating
heat capacities however numerous approximations exist These approximations are often linear
approximations normalized to the liquidus temperature In this work we find that the driving force
appears to be predominantly linear which is in good agreement with the approximations that are
often used for the driving force[124]
43
Figure 5 An example interfacial structure between the crystalline phase on the left and the last
sequential SCLglass phase on the right for a barium disilicate system The gray atoms are barium
silicon is shown in red and blue represents oxygen
44
31E Interfacial Energy
To calculate the interfacial energy of the interface between the crystalline phase and the
SCL the initial basin (pure crystalline) and the last basin (glass basin) were annealed at 1700 K for
1 ns and then cooled to a temperature of 800 K over a period of 10 ns The same process was
repeated for a composite where the initial stable crystal and the glass were placed next to each
otherThe structure was multiplied 10 times to create a wall of a large interface to reduce energetic
fluctuations A small section of the composite structure is shown in Figure 5 The difference
between the addition of the energy of the crystalline phase and glassy phase and the composite
gives the energy of the interface which was normalized to the size of the surface area Multiple
arrangements of the composite were tested but anisotropy was found to be minimal with the
fluctuations of the energy being larger than the differences from different tested directions The
final expression used for the interfacial free energy (in eVAngstrom2) is
610 00 35047 040T minus +=
45
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a function of
temperature for each system size The experimental values for kinetics and thermodynamics come
from ref[125] and from the heat capacity data taken from [126] respectively It is clear to see that
driving force shows good agreement across all systems however the kinetics terms only converge
46
for the 256 and 512 atoms systems (Bottom) The fit used to calculate the interfacial energy as a
function of temperature
32 Results amp Discussion
The results for the free energy difference the kinetic parameter and the surface energy are
shown in Figure 6 These parameters as a function of temperature are used to calculate the
nucleation rates in the expression
3
2( )
01 16exp
3( )I
TT G k
= minus
(46)
The 01 being an approximate value of the Zeldovich factor Using the parameters shown in Figure
6 and solving Eq (46) the nucleation rates are shown in Figure 7 along with experimental data for
validation At first glance we notice that there is about 7 order of magnitude differences between
data and prediction Considering that theses potentials were trained on glass and were not intended
for this purpose we consider this as excellent agreement This method also shows agreement below
the peak but the accuracy falls off more quickly below the peak temperature It is likely that the
data below the peak is severely underestimated due to a failure to reach steady state nucleation
rates Reliable steady state nucleation rates are not available for this system and as such a direct
comparison below the peak temperature is not directly available It is worth mentioning that the
failure of CNT of predicting the order of magnitude of nucleation rates is well reported and can
amount to dozens of orders of magnitude when a constant value of is used[75] [113] [115]
[119] [127] thus the proposed method has an advantage when comparing with the current
classical nucleation theory toolset Moreover the prediction of Tmax (~950 K) is also very close to
the experimental value (985 K)
47
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted using the
model presented in this work The data referenced can be found in refs [113] [128] [129]
48
To further compare CNT and the model presented here we decided to compare the results
of the fitted and this workrsquos calculated surface energies as seen in Figure 8 The results show good
agreement all achieving the same order of magnitude implying that experimental methods can
achieve a close approximation to the values required for CNT The prediction of nucleation is
extremely sensitive to the value of the surface energy with small changes leading to orders of
magnitude change It stands to reason then that further work that needs to be approached through
computation and experiments is developing a rigorous model to predict the interfacial free energy
The other assumption that is used experimentally is the Stokes-Einstein relationship To
compare the landscape results of the transition rate from the SCL to crystalline states and the
Stokes-Einstein relationship the liquid state relaxation was calculated The small barriers that
govern the Adam-Gibbs relationship were not found with this exploration method hence
quantitative comparisons are not possible (and relaxation calculations should not be made)
however the relaxation times can be qualitatively compared The liquid state relaxation rate is
proportional to the viscosity and is given by the expression[44]
j
i
i
l
ij
p
K =
(47)
To understand the validity of the Stokes-Einstein relationship we can write a series of theoretically
true proportionalities
( )l
T TD T
(48)
The values of the experimental kinetics fail to align with the kinetics calculated in this work This
is expected due to the fact that the viscosityliquid relaxation is dominated by the set of small
transitions in the landscape while the nucleation transition is defined by the transition of one region
to another with a substantially higher activation barrier This comparison may also account for
49
some part of the experimental overestimation below the peak with the kinetic term calculated
dropping off at a faster rate than that calculated from the experimental viscosity Figure 6 also
shows evidence that it may be inappropriate to use the Stokes-Einstein relationship for nucleation
and as such we may need to re-visit the experimental models that often rely on the assumption of
the Stokes-Einstein relationship This observation casts some doubt in the legitimacy of describing
the kinetic factor as a diffusivity which is standard for CNT For more information on this CNT
failure please see Refs [75] [113] [115] [119] [125] [127]
Nucleation is a complicated concept to investigate and model Unlike the crystalline phase
where the atomic positions are clearly defined the details of the atomic coordinates and bond
configurations of the liquid sites must be described in terms of statistical distributions Due to this
added complication mean-field distributions are typically used to describe non-crystalline
sites[14] [16] [108] [130]ndash[133] The incorporation of fluctuations could more accurately predict
each liquid sitersquos transition to the crystalline phase Fluctuations in the degeneracy could be one
reason that the nucleation rate is over predicted when the temperature is less than the peak
nucleation temperature A given G varies with the environment and depends on the specific
liquid state not the mean value Therefore the rate of nucleation from all supercooled liquid states
to the crystal state is not constant which is why fluctuations need to be considered and why energy
landscapes are such a powerful technique This is especially highlighted when considering that
different degeneracies yield drastically different heat capacity Energy landscapes are extremely
useful because they consider a large distribution in free energies and show all structural fluctuations
possible Fluctuations in cluster size and degeneracy influence Adam-Gibbs configurational
entropy which directly impact the transition rate between the crystal and liquid state
50
Figure 8 The surface energy with respect to temperature for the work presented here
0
0002
0004
0006
0008
001
0012
0014
800 850 900 950 1000 1050 1100
Su
rface
En
ergy (
eVAring
2)
Temperature [K]
This Work
Xia et al (2017) Ref 23
Fokin et al (2016) Ref 41
51
33 Conclusions
Though this is not the first time CNT has been examined computationally in an attempt to
understand if the theory is valid[74] [116] [117] it is the first parameterization of each variable
independently in an attempt to calculate the resulting nucleation rate Not only are all the variables
calculated when used in the expression for CNT the liquidus temperature is reproduced and the
nucleation rate is reasonably predicted This is as far as the authors know also the first time a real
energy landscape for a complicated glass-ceramic system is being reported This method leveraging
energy landscapes alongside CNT enables for a reasonably fast calculation of the nucleation curve
of a glass forming liquid In future work this model could be used for either developing commercial
glass-ceramics or for predicting the critical rate needed to maintain a glass This model can also
provide additional insights into the physics of nucleation and the validity of CNT without the
constraints of fitting parameters This method fundamentally has the power to connect the 3N
dimensions of the energy landscape to the compositional dimensions and thus allowing for a unique
powerful method of predicting nucleation parameters as a result simplifying the crystalline
dimensions in which a glassglass-ceramic must be optimized over
Chapter 4
Expanding the Current State of Relaxation
Before we consider a new approach to relaxation (optimizing our understanding of
the 3N dimensions) there are first missing pieces in the current state specifically the temperature
and compositional dependence of the stretching exponent However even before expanding the
current state of the art relaxation methods we also need to understand ergodicity In this section we
will start with a simple though experiment to understand the temporal effects of relaxation and then
in the second section combine it with fictive temperature to create a new model for the stretched
exponent
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition
The prophet Deborah sang in the book of Judges ldquoThe mountains flowed before the Lordrdquo
[6] [13] However the question the prophet failed to ask was what happens if the mountains are
moving at relativistic speeds The focus of this section is to highlight the temporal effects on
ergodicity in the context of special relativity and the reference frame of the observer In order to
understand relativistic effects on materials one must consider a well-characterized material that
actively relaxes but appears solid on our timescale viz pure B2O3 glass [134] [135] Glass is a
particularly good candidate to study the effects of time dilation on a material since it undergoes a
kinetic transition known as the glass transition (Tg) which has multiple definitions that will be
considered
53
The first definition is in terms of the aptly named ldquoDeborah Numberrdquo which was proposed
by Reiner [136] to offer a view of fluidity and equilibrium mechanics It was later expanded to
explain the origin of the glass transition and the transition from an ergodic liquid system to a non-
ergodic solid-like glassy state[12] which is the fundamental definition of the glass transition [11]
[12] [137] It is important to note here that the breakdown of ergodicity is relative to the
observation timescale and is reflected in the Deborah number (D) which is defined as
Dt
= (49)
where t is the (external) observation time and is the relaxation time of the material which can be
expressed through the Maxwell relation [42] By definition the Deborah Number equals 1 at the
glass transition temperature Tg [13] [137]
Another definition of the glass transition temperature is due to Angell who defined Tg as
the temperature where the viscosity of the supercooled liquid is 1012 Pa s [58] [138] The Angell
diagram is an important plot relevant to viscosity and the glass transition where the abscissa is TgT
and the ordinate axis is log10( ) In the Angell diagram the value of the viscosity at Tg and in the
limit of infinite temperature are fixed and the difference in the temperature scaling of the TgT-
normalized viscosity is the slope of the log10( ) curve at TgT = 1 which Angell defined as the
fragility (m) of the supercooled liquid Using this definition one can establish the consistency of
the Angell plot at relativistic speeds An example of the Angell plot is shown in Figure 9
54
Figure 9 An Angell diagram created using the MYEGA expression [55] With a very strong glass
(m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) [17] [134] [135] The
infinite temperature limit is from the work of Zheng et al[54] and the glass transition temperature
is from the Angell definition
55
Using the two previous definitions it is expressed
1gT
GtD
= = (50)
which can be re-written as
(
(
)
)g
g
tG T
T= (51)
In order to solve this equation one must know the shear modulus at the glass transition temperature
recent improvements in topological constraint theory have led to the ability to predict properties
with varying temperature and composition Using a recent topological constraint model for elastic
moduli proposed by Wilkinson et al [139] one can predict the temperature dependence of the
modulus
( ) ( ) ( ) ( ) ( )( ) ( )Δ Δ Δ Ac c C Ac
c c
F n q T F n q T F n q T x Nn q T x NdG dGG
d F M d F M
+ + = =
(52)
In Eq (5) nc is the number of topological constraints (c) qc is the constraint onset function as
described by Mauro et al [69] Fc is the free energy of the constraint is the density of the
system M is the molar mass and 119889119866
119889120549119865119888 is a scaling factor This expression was evaluated as a
function of temperature with the total modulus coming from Kodama et al [135] and is equal to 8
GPa at room temperature The B2O3 structure consists entirely of three-coordinated boron [140]
[141]
Using the modulus predicted in Eq (52) numerical observation time is expressed
56
12
9
10 Pa s141 s
709 10 Pat = =
(53)
Using this description of the static (vc=0 where v is the speed of the glass and c is the
speed of light) glass transition it is possible now to describe the relativistic glass transition for
which there are considered two separate cases It is worth noting that this is not the first case study
of relativistic viscosity a fairly large basis of literature dealing with ideal fluids near large
gravitational bodies [142] There is also some work also relating the relaxation time in a
gravitational field to the nonequilibrium entropy relaxation time [143] [144] This work
emphasizes the theoretical effects of special relativity on real liquids their glass transitions and
the properties there dependent on
41A Relativistic Liquid
The first case is when a sample of a glass-forming system moves past a stationary observer
at a speed approaching light giving the new observation time as
0tt
= (54)
where t0 = 141 s (as previously calculated) and g is the Lorentz factor
2
2
1
1v
c
=
minus
(55)
The Maxwell relation can be modified to account for relativistic effects by combining with Eq (54)
0
()
)(
g
gG TT
t
= (56)
57
This gives a condition that is temperature dependent to predict the new glass transition as a function
of v however in order to accurately describe the relativistic behavior density must be considered
Density will change by a factor of 2 because both the mass and the volume will be affected and
as such
0) (( )g gG TT t = (57)
The prediction then for the glass transition as a function of the speed of light is shown in Figure
10
Figure 10 The relativistic glass transition temperature for B2O3 glass
58
Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature
which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) expression [55] [145] The MYEGA equation needs 2 other parameters
besides Tg to generate the viscosity curve fragility (m) which is the slope at Tg in the Angell plot
and the infinite temperature viscosity () The fragility will not change because one assumption
in this model is the consistency of the Angell plot at all relativistic velocities even relativistic The
infinite temperature viscosity which also will not change because this is the minimum possible
viscosity for a liquid The new viscosity curves for various speeds are shown in Figure 11
59
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different fractions of
light speed All of the viscosities approaching the universal temperature limit for viscosity
60
By using the Deborah Number it has been shown that at relativistic speeds will change the viscosity
in order to maintain the Deborah number and in this case it is shown that when the Lord would
move past the mountains at relativistic speeds the mountains would not actually flow but instead
hold still (even more so than they currently do) From this it is inferred that any liquid moving past
the earth at relativistic speeds will appear to be more solid and eventually appear as a glass
41B Relativistic Observer
In this case we consider an observer moving past a sample of glass at relativistic speeds
giving
0t t = (58)
which when the relativistic shear modulus 2 factor is included can be written as
( )
( )0
g
g
TG T
t
= (59)
Solving in the same manner as before it is shown which shear modulus is needed to satisfy the
condition in Figure 12
61
Figure 12 The modulus needed to satisfy the condition for the glass transition
62
In Figure 12 the shear modulus at the glass transition temperature is shown however the
constraint theory mechanism being implemented to calculate the shear modulus has a built in upper
limit when calculating the Tg past the point in which all the constraints are intact Due to this
limitation there is an upper limit to our prediction Nonetheless the calculated Tg over the available
range are shown in Figure 13 as well as the related viscosities It is seen clearly that at higher
velocities the glass transition shifts towards zero more dramatically at the higher temperatures
This leads to the interesting result that if we take the limit of an observer moving past the earth at
close to the speed of light all glass would appear to be a liquid
63
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at v=044c
(Bottom) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches
the speed of light
64
Applying the mechanisms proposed by Einstein to the concept of the Deborah number it
has been shown that the viscosities will appear to change dramatically as a glass-forming system
approaches relativistic speeds This allows us to build an intuitive understanding of the differences
between glass and liquids Specifically any material can appear either more or less fluid based on
the velocity with the velocity with which they travel relative to the observer This serves to highlight
the importance of the role of the observer on the glass transition and why even small changes in
DSC scan rates or sampling frequency can lead to drastically different measured glass transitions
42 Temperature and Compositional Dependence of the Stretching Exponent
The mathematical form for glass relaxation ( ) was originally proposed by Kohlrausch in
1854 based on the decay of charge in a Leyden jar[9] [14] [29] [42] [45] [96] [146]
( ) ( )( )exp t t
= minus (60)
where t is time t is the relaxation time of the system and b is the dimensionless stretching exponent
Eq(1) was originally proposed empirically and is known as the stretched exponential relaxation
(SER) function[42] [96] The relaxation time t depends on the composition temperature thermal
history pressure and pressure history of the glass as well as the property being measured[62]
[147] For example the stress relaxation time of a glass can be written as[62]
( )( )
ff
s
f f
T T P P
G T T P P
= (61)
where h is the shear viscosity and G is the shear modulus of the glass-forming system[42]
(Variables are defined in Table 2) In addition to using Eq(61) to describe stress relaxation time
structural relaxation time has recently been hypothesized follow Eq(61) with bulk viscosity and
65
the difference of bulk modulus between infinite and 0 frequency replacing shear viscosity and shear
modulus[62] Both h and G vary with the temperature (T) and thermal history of the glass (as
quantified via the fictive temperature Tf) as well as the pressure (P) and its corresponding history
(fictive pressure Pf)[16]
To connect the stretching exponent to its physical origins we can begin with a result from
Richert and Richert[148] that relates to an underlying structural relaxation time distribution and
then develop equations that determine from known quantities Their expression
2 22
ln 2
1
6
minus=
(62)
relates to the variance of the logarithm ( )2
ln of the structural relaxation time For reference
all variables are defined in Table 2 The physical origins of the relaxation time are related to the
configurational entropy (Sc) as shown by the Adam-Gibbs equation[53] [55] [149]
ln lnc
B
TS = + (63)
where is the infinite temperature relaxation time T is the absolute temperature and B represents
the energy barrier for relaxation
Table 2 Variable definitions
Variable Definition
t ts Structural relaxation time stress relaxation time
b Kohlrausch exponent (ie the stretching exponent)
T Tf Tfi Absolute temperature fictive temp partial fictive temp
P Pf Pressure fictive pressure
66
kB Boltzmannrsquos constant
wi ki i=1 to N Prony series parameters
B Adam-Gibbs relaxation barrier
Sc Adam-Gibbs configurational entropy
B Distribution of activation barriers
2 Variance
Na Number of atoms
f Topological degrees of freedom per atom
Degenerate configurations per floppy mode
H Enthalpy barrier for Relaxation
d Number of dimensions
f Fraction of activated relaxation pathways
S Adam-Gibbs entropy in the infinite-temperature limit
m Kinetic fragility index
m0 The limit of a strong liquid
Grouped unknowns 6
ln
B
a BN k
A Proportionality between fragility and distribution of barriers
x A given composition
125640
S
A
Intercept of the linear model of
67
42A Deriving a Model
Combining Eqs(62) and (63) Gupta and Mauro[150] proposed that the variance of the energy
barriers for relaxation (2
B ) could be rewritten as
2 2
22
2
1
6( )B c TTS
minus=
(64)
Solving for the stretching exponent
( )22 (
6
( )
)
c
B c
S T
T
T
TS
= +
(65)
The work of Naumis et al[23] [56] has shown that the configurational entropy is proportional to
the topological degrees of freedom in the network a result that was used in the derivation of
temperature-dependent constraint theory[57] and the MYEGA (Mauro-Yue-Ellison-Gupta-Allan)
equation for the relaxation of supercooled liquids[55] The MYEGA equation was derived by
expressing the configurational entropy as
( ) ( ) lna BcS T x f T x N k= (66)
In Eq (66) f is the topological degrees of freedom per atom Na is the number of atoms kB is
Boltzmannrsquos constant x is a given composition and is the number of degenerate configurations
per floppy mode The temperature dependence of the topological degrees of freedom was
approximated using a simple two-state model
( )( ) 3exp
B
H xf x T df
k T
minus= =
(67)
Here H(x) is the enthalpy barrier for relaxation d is the dimensionality of the system and f is the
68
fraction of activated relaxation pathways Combining Eqs (65) and (67) as well as condensing the
unknowns into the term defined by
6
ln
B
a BN k
=
(68)
we get
( )
22
df
dfT
=
+
(69)
This can be compared to the prediction made by Phillips[48] for the stretching exponent at
temperatures below the glass transition
2
df
df =
+ (70)
Comparing the two expressions of Eq(69) and Eq(70) they would agree if
4 4dfT
= + (71)
We will show later (Figure 12) an example where extrapolating our model prediction for to room
temperature (with T in Eq(71) replaced by fictive temperature Tf) gives a result close to the Phillips
value of Eq(70) When the temperature T in Eq(69) is high (much larger than ) then Eq(69) gives
us rarr1 Thus the model can interpolate between the low-temperature (Phillips) value and the
high-temperature value of 1 In our current model we do not have access to a value for in Eq(68)
so this model cannot assess whether the Phillips room temperature value for is universal ie
whether Eq(71) is always satisfied at room temperature
Zheng et al[151] showed that the Mauro-Allan-Potuzak (MAP) model for the relaxation
time of the nonequilibrium glassy state[44] implies that the configurational entropy can be written
as a function of thermal history (Tf) the fragility index (m) the fragility limit of a strong liquid
69
(m0) the glass transition temperature (Tg) and the limit of infinite temperature configurational
entropy ( S ) as
0
) exp 1 0
(ln1
c f
f
gT
TS m
TS
m
= minus minus
(72)
The qualitative relationship between the distribution of activation barriers and fragility was
proposed by Stillinger[18] [19] who suggested that the energy landscape of a strong liquid (low
fragility) has a small distribution of activation barriers and that a higher fragility is associated with
a broader distribution of activation barriers ie a higher variance of the activation barriers This
leads to a phenomenological relationship that is here assumed to be valid based on Stillingerrsquos work
on energy landscapes[18] [19]
2
0
1B
mA
m
= minus
(73)
where A is some constant of proportionality This expression was chosen because in the limit of a
strong liquid there is an infinitely sharp distribution of activation barriers Combining Eqs(65)
(72) and (73) we obtain the temperature dependence of the stretching exponent for a liquid
( )
0
2 2 2 2
0 0
exp 1
( ) 2
6 ln10 1 exp 1
g
f
f
f
g
f
f
T mT S
T mT
Tm mA T S
m T m
minus minus
=
minus + minus minus
(74)
Fictive temperature appears in this new expression because we are deliberately expressing this
function as an equilibrium model Rewriting Eq (74) in terms of one unknown () and grouping
constants
70
( ) 0
2 2
0 0
exp 1
2exp 11
g
f
f
f
g
f
f
T m
T m
T m
T
T
Tm
T
m
m
minus minus
minus minus
=
minus +
(75)
Eq (75) is an expression for the stretching exponent as a function of thermal history glass
transition and fragility index with only one unknown In Eq (75)
( ) 12
6 ln10
S
A
= (76)
The only unknown for the compositional dependence of the stretching exponent is the value of D
The composition-dependent part of Eq(76) is Syen
since we approximate A to be independent of
composition The fragility dependence of Syen
was proposed by Guo et al [61]
Syen(x) = S
yen(xref
)expm(x) -m(x
ref)
m0
eacute
eumlecircecirc
ugrave
ucircuacuteuacute
(77)
where x is composition and xref is a reference composition in the same glass family Seeking the
simplest possible expression to approximate the unknown D we take the natural logarithm of
Eq(76) and of Eq(77) and combine them to get
0
ln lnm
m = + (78)
with the additional definition
( )( )
( )12
0
ln l6 ln10
nref refS x m x
mA
minus
=
(79)
The result is that ln D varies linearly with fragility m and the intercept ln D is independent of
composition (depends on one reference composition)
In Figure 9 we plot the data of Boumlhmer et al[152] vs the predicted exponent The work of
71
Boumlhmer et al is the summary of the literature data relating fragility index to the stretching exponent
at the glass transition which we use to fit values to Eq (78) The dataset included in their work
covers chalcogenide oxide and organic glasses The fitting (consisting of least-squares
minimization of the difference between the predicted stretching exponent and that which was
reported) was done twice once for organic and once for inorganic systems During the fit it was
assumed that Tg=Tf Some assumption about thermal history was necessary since the individual
thermal histories or Tf values are not known for this whole collection since we are trying to track
overall trends in values vs composition this reasonable simplifying assumption is consistent
with our program Using Eqs(75) and (78) the temperature and compositional dependence of the
stretching exponent can be described with only one free parameter
72
Figure 14 b predicted and from literature showing good agreement with a total root-mean-square
error of 01 The fit for organic systems is given by 66 ln Kln = minus and for inorganic systems by
75ln Kln = minus
73
42B Experimental Validation
Experimental density measurements were made using Corningcopy JadeTM glass as described
elsewhere[146] Changes in the density are normalized to obtain the relaxation functions which
are fit with Eq (60) The resulting values of are shown in Table 3 A comparison in Figure 15
is shown with data obtained on a soda-lime silicate SG80 where the stretching exponent was
obtained using the measurement of the released enthalpy as a function of isothermal annealing time
during relaxation below the glass transition The measurement of the released enthalpy[153] relies
on the change in the excess heat capacity in the glass transition range as a function of annealing
time Normalized released enthalpy relaxation functions for each temperature were fit with Eq(60)
to obtain the stretched exponent values for SG80 In Figure 15 parts A and B are samples measured
below Tg while in part C the samples measured are at Tg and above The values were then fit with
Eq(75) where the only free parameter was D with an equilibrium assumption (Tf = T)
74
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy JadeTM
glass and Sylvania Incorporatedrsquos SG80
Glass Temperature (K) Tg (K) TgT m D (K-1) Beta t (s)
Corningcopy JadeTM 1031 1074 1042 32 0002 0504 7249
Corningcopy JadeTM 1008 1074 1065 32 0002 0447 32458
Corningcopy JadeTM 973 1074 1103 32 0002 0395 419214
SG80 800 800 1 36 00057 0632 92
SG80 783 800 1021 36 00057 0593 182
75
Figure 15 The equilibrium model proposed with the experimental points showing good agreement
between the experimentally measured data points and the equilibrium derived model RMSE was
002 for Corningcopy JadeTM (A) and less than 001 for SG80 (B) (C) The model fit for B2O3
experimental data[154] [155] The fragility and glass transition temperature of the B2O3 are taken
from the work of Mauro et al[17]
(
A)
(
B)
(
C)
76
A combination of the MAP model for non-equilibrium shear viscosity[44] the model
presented here primarily in Eq(75) and the model presented by Wilkinson et al[139] for the
temperature dependence of elastic modulus allows for fully quantitative modeling of stress
relaxation behavior The missing model required to understand structural relaxation is the bulk
viscosity curve[62] All previous relaxation (structural or stress) models[42] [61] have relied on
approximations that use a constant exponent and on a constant (temperature-independent)
modulus value whereas here every parameter of Eq(60) may be modeled as a function of
temperature Furthermore in combination with the relaxation models described by Guo et al[61]
in which multiple fictive temperatures are described using a Prony series and a temperature-
dependent modulus one can construct a relaxation curve accounting for the temperature
dependence and thermal history dependence of all relevant parameters
( )
( )( )
1
( )( )exp exp
( )) (
fTN
i
i
f
f
f fi
G T k T tG T tw
T TT
T T
=
minus minus
(80)
Here wi and ki are fitting parameters that are completely determined by the value of and each
term in the Prony series is denoted with the subscript i Usually 8 or 12 terms are included in the
Prony series[96] Each term in the Prony series is assigned a partial fictive temperature (Tfi) whose
relaxation is described by the simple exponential in the Prony series Equilibrium conditions are
assumed at the start of the simulation which allows for a known set of starting probabilities within
the energy landscape interpretation of relaxation The same model also includes the fragility index
and glass transition dependence of the non-equilibrium shear viscosity This method is
implemented in RelaxPy[42] as discussed in the next section This serves as an approximation for
the evolution of the of the non-equilibrium state however the temperature dependence of the bulk
viscosity and a replacement for fictive temperature need to be quantified to improve the
understanding of the underlying physics[1] [16] [62]
77
Separately we can explore the relationship between this model given by Eq(75) and the underlying
energy landscape In order to better understand the stretching exponent consider that there exists
a full set of parallel relaxation modes within an energy landscape The relaxation modes are then
weighted by the occupational probability corresponding to a particular mode This gives a series of
transition rates with some probability prefactor and an associated relaxation time (scaled from the
mean relaxation time) which gives rise to a Prony series form of Eq(80) Thus the distribution of
relaxation times (or barriers) determines the evolution of the stretching exponent while the average
barrier determines the mean relaxation time If this Prony series description in turn describes the
stretching exponent we arrive at a physical description and origin of the stretching exponent As
the temperature approaches infinity even though there is a distribution of activation barriers the
distribution of relaxation times approaches a Dirac delta function and the stretching exponent
approaches one (a simple exponential decay) As the temperature decreases the distribution of
relaxation times broadens due to the wide distribution of barriers however due to the broken
ergodic nature of glass the value of the stretching exponent will be controlled by the instantaneous
occupational probability
The activation barriers may be modeled eg using a numerically random set of Gaussian
distribution of barriers[156] The probability of selecting an individual transition is then calculated
using a Boltzmann weighting function the transition also has an associated relaxation time with it
The Prony series is then recreated with the probability multiplied with the simple exponential
relaxation form The sum of all of these terms is then fit with the stretched exponential form Figure
11 shows the stretching exponent calculated for the mean activation energy of 10 eV and three
choices of standard deviation 25 2 15 ande e e minus minus minus= eV as shown in the legend where Eq(60) has
been fit to the Prony series calculated from an equilibrium Boltzmann sampling of a Gaussian
distribution of barriers The temperature dependence for the stretching exponent matches the form
78
derived earlier in this work This stretching exponent shape has been shown previously though not
in a closed-form solution[156] This secondary method not only validates the generalized form of
Eq(75) but also highlights the validity of using the configurational entropy and barrier distributions
as the underlying metric controlling the temperature dependence of the stretching exponent
Figure 16 The stretching exponent calculated as described in the text for a Gaussian distribution
of barriers This plot shows that the distribution of barriers has a large effect on the stretching
exponent A Tg cannot be described since there is no vibrational frequency included in the model
though the glass transition temperature should be the same for all distributions since the mean
relaxation time is the same for all distributions at all temperatures The deviation is given in ln eV
units
79
To create software to model this complex behavior the authorsrsquo existing software RelaxPy[42] has
been modified into a new version RelaxPy v20 Instead of fixed values for the Prony series the
values are chosen dynamically to match the stretching exponent as predicted by Eq(75) The
database of values wi() and ki() was created with a fixed number of terms N in Eq(80) (we chose
N=12) This database is available in the same Github repository as the software The parameters
were fit by starting with the values obtained for =37 reported by Mauro and Mauro[96] to make
the new version of the model smoothly reproduce a prior optimal fit then was stepped by 001
(when less than 095) or by 0001 (when greater than 095) Using the previously optimized values
of wi and ki for starting values for each new the wi and ki values were then varied to minimize
the root-mean-square error between the stretching exponent form and the Prony series form ie to
satisfy Eq(80) with least error This list of compiled values makes up the contents of a database
that the software accesses The values of wi and ki used for a given are calculated by finding the
closest value in the database to the given value of This allows for an efficient implementation
of relaxation modeling Figure 12 shows the fitted parameters to the Prony series and an example
RelaxPy output for Corningcopy JadeTM glass undergoing quenching at a rate of 10 Kmin Note that
the room temperature stretching exponent value is 046 a close match the Phillipsrsquos ldquomagicrdquo value
of 37 (043)[157] This prediction for Corningcopy JadeTM is a non-equilibrium prediction in contrast
to the equilibrium high-temperature prediction shown in
80
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent Each color
designates one term in the series (Bottom) The output from RelaxPy v20 showing the stretching
exponent effects on the relaxation prediction of Corningcopy JadeTM glass Each quadrant shows one
property that is of interest for relaxation experiments In particular it is interesting to see the
dynamics of the stretching exponent during a typical quench
81
43 Conclusion
In this section we have sought to expand the current understanding of relaxation and improve the
models that currently exists We have proposed a model that accounts for the effects of special
relativity as well as a model was derived through an understanding of the distribution of barriers
for relaxation and the temperature dependence of the Adam-Gibbs entropy The model outlined
herein describes the temperature dependence of the stretching exponent in glass relaxation
The work done on the relativistic effects on relaxation illustrates the crucial role of the observer on
all relaxation models and offers an extreme in which to test common relaxation models The model
for the stretching exponent not only considers the extremes at high and low temperature (when
compared to the glass transition) as in the Phillips model but also for any intermediate temperature
as a function of the fictive temperature This model does not have any explicit temperature
dependence since it was assumed that an equilibrium model works well to describe the
instantaneous distribution of relaxation times Given the physical argument and the success of this
model when tested by experiments and by another model it is at least reasonable to formulate the
temperature-dependence of in terms of its fictive-temperature-dependence as we have done here
Including both T and Tf is possible to consider but lies outside our current scope Using previously
derived compositional dependence for the MAP model a fragility index dependence of the
stretching exponent was defined and tested The model was confirmed using multiple experimental
datasets In addition a theoretical comparison to a distribution of landscape activation barriers was
found to reproduce the same trends as the model
Chapter 5
Glass Kinetics Without Fictive Temperature
Fictive temperature as discussed through this text is unable to capture key physical
phenomena[16] [42] [51] To circumvent this limitation we rely on two different frameworks
The Adam-Gibbs relationship and generalized features of the energy landscape These two features
allow us to create a method that is completely generalizable and does not involve fictive
temperature This method creates fake landscapes that generally capture the trends of glass based
on experimental measurements This new method we have called lsquoToy Landscapesrsquo
51 Background of the Adam Gibbs Relationship
Experimental evidence in support of the AG model comes from the work of Richet[158]
in which they compared the predicted configurational entropy from viscosity curves with that
obtained from DSC The configurational entropy was calculated from DSC data using
0
0( )p p vi
T
T
b
c c
C CS S T dT
T
minus= + (81)
In Eq (81) Cp is the isobaric heat capacity Cpvib is the vibrational heat capacity and T0 is the initial
temperature from which the configurational entropy is integrated The difference between the total
heat capacity and the vibrational heat capacity is the heat capacity due to structural rearrangements
Richet found good agreement between the measured viscosity curve and calorimetric
configurational entropy at temperatures above the glass transition confirming validity of the AG
relationship When applying Eq equation reference goes here all temperatures must remain above
the glass transition ie in the liquid or supercooled liquid state The calculation is invalid upon
83
cooling through the glass transition due to the breakdown of ergodicity[12] [13] [159] and the
irreversibility of the glass transition process Application of Eq (6) to temperatures below the glass
transition leads to an incorrect calculation of excess entropy in the glassy state[160]
In this study we expand on these previous works to understand the relationship between
viscosity and the underlying energy landscape and provide new insights into the thermodynamics
of liquids through a simplified ldquotoy landscaperdquo model constructed from experimentally measured
viscosity parameters The energy landscape framework is especially helpful for elucidating the
thermodynamics and kinetics of supercooled liquid and glassy systems Energy landscapes describe
the evolution of all atomic transitions for kinetic processes in a system To perform such
calculations information about the inherent structures and transition points in a landscape must be
known In this work we solve the inverse problem of deducing realistic landscape parameters using
the AG relationship and experimentally measured data This information is used to construct a ldquotoy
landscaperdquo model to describe glass relaxation processes To achieve an accurate relaxation model
the following steps are taken to ensure the validity of the results
A Confirm the AG model for shear viscosity using energyenthalpy landscape
calculations
B Validate the assumptions made by the MYEGA model
C Confirm the AG model for bulk viscosity
D Explore the fundamental relations between viscous properties and topography of the
landscape
E Use the knowledge gained to propose a new thermodynamic model to calculate the
driving force for glass relaxation and the scaling of the liquidrsquos free energy
This chapter is organized into sections devoted to each of the topics above preceded by a
methods section All of the sections are presented to validate a new model of glass relaxation that
is sufficiently complex to achieve a calculation of the thermodynamics and kinetics of the liquid
This new model will be based on understanding the landscape from experimental properties and
we have called this approach the ldquotoy landscaperdquo model
84
52 Methods
Potential energy landscapes (fixed volume) and enthalpy landscapes (fixed pressure) are
powerful modeling techniques where atomic configurations are mapped in a 3N-dimensional phase
space (or 3N+1 dimensions for enthalpy landscapes)[89] The energy landscape approach is based
on mapping the continuous 3N dimensional space to a discrete set of energy minima (called
ldquoinherent structuresrdquo) and first-order saddle points (ldquotransition pointsrdquo) To perform this mapping
a systematic search for inherent structures is performed while also mapping the lowest-energy
transition points between each pairwise combination of adjoining minima The landscape itself is
partitioned into basins which represent the set of all configurations that minimize to a common
inherent structure[161] This combination of the basin and transition point information results in a
topographic map of the 3N phase space (either a partial or whole mapping) that can be used to gain
insights into the physics of any atomistic system such as proteins[91] liquids[18] [19] glasses[1]
[93] and nucleating crystals At equilibrium the probability distribution for occupying the various
basins in the landscape is given by[13]
exp
exp
i
j
i
i
j
j
Eg
kTp
Eg
kT
minus
= minus
(82)
where Ei is the potential energy (or enthalpy) of the inherent structure gi is the degeneracy of the
basin i is the basin index and pi is the probability of occupying a given basin i The denominator
is a normalization factor given by the summation over all basins Mauro et al[44] previously used
this formalism to study the non-equilibrium viscosity calculate the evolution of configurational
heat capacity and elucidate the long-time relaxation kinetics of selenium glass[44] [93] [160] For
equilibrium systems the configurational entropy is given by the Gibbs formula
85
lnc i
i
iS k p p= minus (83)
In order to validate the AG model the landscapes of several common systems were
explored using ExplorerPy[89] ExplorerPy is a software program built specifically to map
landscapes using the LAMMPS molecular dynamics package[30] The three systems we explored
are B2O3[162] (150 atoms and 2000 basins) SiO2[162] (150 atoms and 1000 basins) and BaO-
2SiO2[121] (128 atoms and 1500 basins) All systems used periodic boundary conditions The
initial structures for B2O3 and SiO2 were constructed by placing atoms corresponding to the
chemistries of the systems on a random grid (with 1 Aring spacing) corresponding to a density of 20
gcm3 and equilibrated at 2500 K then quenching at a rate of 1 Kps to room temperature using the
Wang et al[162] potentials in the NPT (constant pressure number of atoms and temperature)
ensemble The energy landscape of the BaO-2SiO2 system was taken from our previous work[72]
The SiO2 (constant pressure exploration) system was explored using ET with a push distance of 03
Aring and a pressure of 10 atm B2O3 was explored using isobaric MD at a temperature of 1500 K for
500-time steps of 1 fs also at a pressure of 1 atm The minimization (after MD or ET) was handled
by the LAMMPS[30] steepest descent algorithm until the certainty in energy was smaller than 10-
4 eV Both used a nudged elastic band spring constant of 10 eVAring2 The energy distribution for all
systems was fully sampled which was confirmed by randomly checking if half the basin
distributions matched the full distribution It was also found that multiple structures in each
exploration were discovered more than once and a slightly smaller B2O3 system (130 atoms) was
also explored and showed identical results to the 150-atom distribution The insights gained from
the interaction between the experimentally accessible viscous properties and the landscapes
presented here will be generalized to create a simplified ldquotoy landscaperdquo model to capture the
thermodynamics and kinetics of the glass-forming system
86
53 Results
53A Adam-Gibbs Validation
To predict viscosity using the Adam-Gibbs expression there are only three required
parameters the configurational entropy (as a function of temperature) the barrier for cooperative
rearrangements and the infinite-temperature limit of viscosity Using a landscape the equilibrium
configurational entropy can be obtained using Eq (83) The infinite-temperature limit of shear
viscosity is given by the work of Zheng et al[54] where a systematic study of MYEGA fits to
viscosity data revealed a common value of 10-293 Pamiddots This infinite temperature value of viscosity
is used for all the experimental and computational viscosity curves shown in this work The final
parameter is the mean barrier for a cooperative rearrangement In Figure 18 the barrier for
cooperative rearrangements was chosen to accurately produce a viscosity of 1012 Pamiddots at the
experimentally measured value of the glass transition temperature From these parameter values
the predicted Adam-Gibbs viscosity curve is plotted The universal infinite limit fragility and glass
transition values from references were inserted into the MYEGA equation to plot the
lsquoExperimentalrsquo curves for B2O3 SiO2 and BaOmiddot2SiO2 respectively[58] [69] [120] When
examining Figure 18 it is clear that the AG model is able to fundamentally recapture the viscosity
87
Figure 18 The viscosity (left) and landscape (right) predictions for three common systems The
first system is newly calculated in this work while the latter two come from our previous works[72]
[89] It is seen that the viscosity predicted from the AG model is very accurately able to reproduce
the experimental viscosity curves from the MYEGA model The last system is a potential energy
landscape while the others are enthalpy landscapes
53B MYEGA Validation
Now that we have confirmed that the configurational entropy of the landscape works well
to describe the viscosity via the AG relationship a definitive temperature-dependent form is needed
in order to make more general predictions The key approximation made in deriving the MYEGA
model is that the configurational entropy follows an Arrhenius form By testing this Arrhenius
88
form we can evaluate this assumption in the derivation of the MYEGA model while simultaneously
gaining insights into the thermodynamics of the liquid and supercooled liquid states A strong liquid
( 15m ) has a configurational entropy that is nearly constant as a function of temperature while a
fragile liquid ( 15m ) is a liquid in which the configurational entropy is highly temperature
dependent Since we are testing the MYEGA equation it is also worthwhile to consider alternate
viscosity expressions Assuming validity of the Adam-Gibbs equation the configurational entropy
deduced from the viscosity experiments is given by
( )10 01log l) og(
c
BS
T T minus= (84)
where the viscosity function is given by a viscosity model such as Vogel-Fulcher-Tammann[59]
(VFT) equation the Avramov-Milchev[60] [163] (AM) equation or the MYEGA expression
The resulting entropy predicted by each viscosity model (using experimental data for
fragility and the glass transition) is compared to the calculated configurational entropy from the
landscape for B2O3 in ( 518gT = K and 33m = [69] were used for the experimental values) The
figure clearly shows that the MYEGA equation which assumes that configurational entropy scales
in an Arrhenius fashion reproduces the calculated configurational entropy from the landscape the
most accurately out of the three considered viscosity models This also means that the
configurational entropy form of the AM and VFT models are not physical and cannot be used to
recreate physical results Since the temperature dependence of the entropy has a fixed mathematical
form and is dominated by the distribution of basins we now understand that the distribution of
basins must also have a fixed form and that form must control experimentally accessible properties
This is the key insight that will allow for the development of the toy landscape model
89
53C Adam-Gibbs and Structural Relaxation
Although the Adam-Gibbs model has been validated for shear viscosity and has provided
the functional basis of the MYEGA model let us now consider whether the temperature
dependence of the configurational entropy can also predict the bulk viscosity (also known as the
volume viscosity) Recent work has highlighted the importance of bulk viscosity viz that the
structural relaxation time is proportional to the bulk viscosity[62] Hence the temperature
dependence of the bulk viscosity provides key information needed to predict structural relaxation
times[62] Following Scherer[164] the relationship between the structural relaxation time ( structural
) and the bulk viscosity ( B ) is given by
( )0B structuralK K = minus (85)
Figure 19 The configurational entropy comparisons between the three major viscosity models
which validates the main underlying assumption of the MYEGA model The VFT and AM are
unable to capture the physics of configurational entropy therefore ruling
90
where K is the infinite frequency bulk modulus and 0K is the zero-frequency bulk modulus In
order to calculate bulk viscosity for B2O3 the zero-frequency and infinite-frequency modulus is
taken from these works respectively [165] [166] The experimental structural relaxation times were
taken from the work of Sidebottom et al[154] In Figure 20 the bulk viscosity is shown as well as
the Adam-Gibbs comparison
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the configurational
entropy from the enthalpy landscape with the barrier 00155 eV (compared to the shear barrier of
00149 eV) and the infinite limit allowed to vary (10-263 Pamiddots for bulk viscosity) In Figure 18 the
configurational entropy is confirmed for the shear viscosity thus confirming the AG for both shear
and bulk viscosities Sidebottom data are from Ref [154]
In Figure 20 the fitted values of B and 10log can be used to gain insights into the
difference between structural and stress relaxations since the configurational entropy is universal
between the two The value of Bk (the barrier for cooperative re-arrangements in natural units) for
shear viscosity was 173 K while for bulk viscosity it was 180 K This difference of 7 K in the
barrier leads to an order of magnitude difference in the fitted viscosity curves This is a small
91
difference and explains the variance in the values at Tg The other difference is in the infinite
temperature value of viscosity which is -293 log(Pamiddots) for shear viscosity (from Zheng et al) and
-263 log(Pamiddots) for bulk viscosity This implies that the MYEGA raw form is accurate to predict
the structural relaxation time as well as the stress relaxation (though the values of the pre-
exponential factor and the composite constant will change between the two forms) However the
form in terms of fragility and the glass transition will need to be modified to accurately describe
the structural relaxation time The form assumes the viscosity of interest has a value of 1012 Pamiddots at
the glass transition temperature for shear viscosity which does not have to be the value for the bulk
viscosity curves at the glass transition Further research is needed to understand the appropriate
bulk viscosity at the glass transition and the infinite temperature of bulk viscosity
53D Landscape Features
As shown here the configurational entropy is governed by a Boltzmann sampling of a
probabilistic distribution of states n(E) This distribution of the inherent structures is empirically
found to follow a log-normal distribution as shown in Figure 21 While we are not claiming a
deeper meaning to the underlying origin of the log-normal distribution we may adopt this form of
the probability density function since it accurately describes the underlying distribution of
microstates This is the key insight needed to develop a new model because this means that the
fragility and glass transition are fundamentally related to the parameters of the distribution of basins
on the landscape (number of basins and standard distribution of the enthalpy)
The configurational entropy in the MYEGA model is given by
10
exp 1ln10 12 log
g
c
TS mS
T
= minus minus
minus
(86)
92
where S is the infinite temperature limit of the configurational entropy In Figure 21 we also
show the computed value of Eq (83) as a function of temperature compared to that of Eq (86)
which shows excellent agreement between the configurational entropy and the MYEGA model
This test also supports Stillingerrsquos[18] [19] view of fragility being correlated to the lsquoroughnessrsquo of
the landscape topography According to Stillingerrsquos view if the distribution of inherent structure
energies is narrow then the configurational entropy cannot have a large temperature dependence
leading to a low fragility Alternatively if the distribution is broad then there must be a temperature
dependence to the configurational entropy which necessarily results in a higher fragility
93
Figure 21 (A) The histogram for the energy minima for each of the test systems showing a good fit
with the log normal distribution This distribution will then be a valid form to calculate the enthalpy
distribution of the model presented in the next session (B) The configurational entropy from the
model showing the accuracy of the scaling of the entropy predicted by the MYEGA model The
S value was fit for each system
(A)
(
B)
(B)
94
54 Topography-Property Relations
Now that we have confirmed the validity of the Adam-Gibbs and MYEGA equations from
the energy landscape approach we can draw some key conclusions regarding the thermodynamic
properties of liquids For example heat capacity has been related to fragility[112] and is of
importance for glass manufacturing since it is related to the cost of heating and forming a glass
Isobaric heat capacity is defined as
p
P
HC
T
=
(87)
where H is enthalpy T is temperature and P is pressure The mean configurational enthalpy
calculated from an enthalpy landscape is given by
conf i conf iH p H= (88)
The heat capacity has contributions from both vibrational and configurational modes in liquids
while in glasses the configurational mode is mostly lost For a reversible process the
configurational heat capacity can be written as a function of the configurational entropy
c
p conf
P
SC T
T
=
(89)
By combining Eq (86) and Eq (89) we can write the expression for configurational heat capacity
of a liquid as
10 10
1 exp 1 exp12 log 12 log
g g
p conf
T Tm m S HC S H
T T T T
= minus minus minus = minus
minus minus
(90)
From fluctuation theory the heat capacity can be expressed in terms of the variance of the enthalpy
(2
H )
95
2
2
1p conf HC
kT= (91)
It is important to note that the variance of the enthalpy is not equivalent to the variance of the log-
normal distribution The variance of the enthalpy includes a Boltzmann distribution over the log-
normal distributed basins This gives a direct relationship between the enthalpy fluctuations and
the viscosity parameters If we further adopt the log-normal distribution as the effective distribution
for all liquid basins of all enthalpy landscapes then the distribution is only a function of the
common viscosity parameters ( S m gT and 10log ) The viscosity parameters will only
change the variance of the distribution since the mean of the distribution is inconsequential (the
zero-point enthalpy can be adjusted to an arbitrary value without changing the calculation of the
configurational entropy) The parameters of the log-normal distribution can then be fitted to
reproduce the configurational entropyrsquos temperature dependence thus creating a lsquotoyrsquo enthalpy
landscape for us to play with for the purpose of thermodynamic and kinetic calculations Generating
the landscape is a numerical calculation as the number of basins and standard deviation must be
chosen such to reproduce key phenomena
To quantify the dependence of these parameters on the degeneracy and temperature it is
helpful to develop a descriptive formal model The cumulative probability distribution of a normal
distribution (with zero mean) is given by
10log )1
1 er(
f2 2
Hp
+
=
(92)
where is the standard deviation of the log-normal distribution erf is the error function H is the
enthalpy and p is the probability To then calculate the equal probability spacing for a log-normal
distribution we solve for H for the set of probabilities p which range from 2 to 1 2minus
96
with a step-size of This then gives enthalpies of basins equally spaced by the cumulative
probability distribution and as a result each basin has equal degeneracy
1 2 r2 1e f10
pH
minus minus = (93)
This set of basins carries with it an Arrhenius activation barrier for the configurational entropy that
is only a function of and the degeneracy of each basin (if the is chosen such that the energy
distributions converge) This creates a model that fully describes the thermodynamics of the liquid
and though it currently lacks any kinetic component it offers the possibility to be a useful tool
when used in conjunction with Eqs (82) (83) and (88) This new tool is denoted the ldquotoy landscape
modelrdquo (TLM) It is called the toy landscape model because we are generating a landscape that is
not the real landscape but is enough to reproduce key experimental properties without needing
expensive computational calculations It is a lsquotoyrsquo for us to play with without needing to worry
about upfront computational cost
To confirm the validity of TLM and the way it is constructed we can calculate the
activation barrier from this deterministic method and for a random set of inherent structures with
the same number of basins and distributions For this purpose we used temperatures ranging from
1000 to 2000 K for both the deterministic and random models with a step size of 0001 for TLM
Excellent agreement between the random and deterministic methods is shown in Figure 22A This
analysis allows us to understand that the thermodynamics of the system are fundamentally linked
to the viscous flow behavior Though this technique was created to target viscosity and relaxation
it is generally applicable to all configurational thermodynamics of liquids and glasses It is worth
noting that all the thermodynamics will be relative to a 0 eV energy state defined at equilibrium at
0 K and as such the enthalpy differences at some fixed temperature must be known to compare
over compositional spaces
97
In Figure 22B we see the same method being used to understand the relationship between
the viscous parameters and the parameters of the landscape This is the key to the numerical
calculation of the lsquotoy landscapersquo model The toy landscape is a numerical calculation where we
use experimentally accessible values to generate a landscape whose thermodynamics should
reproduce key thermodynamic phenomena In Figure 22B we show that the fragility and glass
transition are systematically changed as the number of basins and standard deviation of the enthalpy
is changed When this idea is combined with the basin calculation in Eq (93) (giving equal
degeneracies for all basins) then the complete set of enthalpy basins is easily attainable giving
access to the fundamental thermodynamics of the system
98
(
A)
(B)
(A)
99
Figure 22 (A) Comparison between the randomized method (histogram) and the deterministic
method (vertical lines) showing good agreement between the maximum in the histogram and the
value predicted by the deterministic technique validating the approach It is worth noting that the
100-basin distribution is a very wide distribution where the total number of basins is less than the
number of points used in the calculation This is done for a variable number of basins with the
number of basins shown in the legend (B) The dependence of fragility and the glass transition
temperature vs the distribution of states and the number of basins
55 Barrier Free Description of Thermodynamics
Throughout this work we have systematically explored the Adam-Gibbs model of
viscosity An important insight that we have gained is that the configurational entropy is free of any
barriers This is of particular interest because the Arrhenius slope of the activation barrier is linked
to the glass transition and fragility The glass transition temperature is known to be related to the
set of activation barriers as it corresponds to a breakdown of ergodicity What is found here is that
the fragility is the ldquoconversion factorrdquo to account for the slope of the configurational entropy
(which as previously identified is related to the distribution of basins) This means that the barriers
themselves do not matter until the material is close to the breakdown of ergodicity and thus the
equilibrium liquid thermodynamics and kinetics can be described without explicit information
about the saddle points in the landscape
Understanding the role of the topography in the properties of glasses and liquids lead to
the most important insights derived from the AG model Since the configurational entropy is the
same for both stress and structural relaxation simple viscous parameters allow us to understand
fundamental thermodynamic quantities of the system without needing to map computationally
expensive energyenthalpy landscapes The process of converting experimentally accessible
properties to a usable landscape for thermodynamic calculations is what we are calling the toy
landscape model To show the utility of toy landscapes we can calculate the driving force for glass
relaxation This information can give insights into the physics of relaxation by dynamically
100
calculating the Gibbs free energy of the equilibrium liquid at any temperature which defines a
driving force for glass relaxation To do this calculation of the driving force for glass relaxation
we consider the ergodicity parameter ( x ) defined by the Mauro-Allan-Potuzak (MAP) model of
glass relaxation as the ratio between the glass entropy and the equilibrium liquid entropy
liquiss dglaS x S= (94)
To get information about the enthalpy we consider an instantaneous Boltzmann distribution on the
landscape given by an equilibrium distribution at some elevated temperature fT
1
exp
exp
iglass i i
fii
f
HH H g
kTHg
kT
= minus
minus
(95)
This formalism in conjunction with TLM gives us the ability to explore the temperature
and compositional dependence of the driving force for relaxation The enthalpy of the liquid is
calculated from an equilibrium distribution on the landscape The driving force ( ) is defined as
( ) ( )( ) ( )1iliquglas ds f l quii dH H TS xT T = minus + minus (96)
For our purposes this gives a nice approximation for the driving force improving on previous
approximations given by ( fT Tminus )[16] [42] TLM does not include a kinetic feature to calculate
the occupational probability in each basin however when coupled to a kinetic model such as the
MAP model[44] a complete description of the underlying thermodynamics of glass relaxation can
be considered For this calculation of the driving force the kinetics are calculated through the MAP
model but this is not necessarily a requirement In Figure 23 the compositional and temperature
dependence of the driving force is shown for three glasses with identical glass transitions and
varying distributions of basins (as listed in Table 1) The calculations are performed using the
RelaxPy script and MAP model with the stretching exponent fixed at the Phillips value of 37
101
explanations of which are available elsewhere[42] The systems were quenched from 700 K to 300
K at a rate of 10 Kmin Figure 23 clearly shows how simply changing the total number of basins
creates a larger driving force for glass relaxation This simple change of the number of basins
drastically increases the entropic contribution and due to the loss of ergodicity explains the increase
of the driving force of relaxation The kinetics could be calculated through any relaxation model or
directly from the landscape if some barrier approximations are made
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition temperature (500 K)
were kept the same while the total number of basins were allowed to vary
Sample Total Basins [-] Distribution of States [log eV] 1
1493g
mT
minus
[K]
Fragility [-]
1 100 02 837 40
2 500 02 537 31
3 1000 02 466 28
102
Figure 23 The driving forces for different example glasses calculated using a combination of the
MAP model RelaxPy and the toy landscape model The parameters for each glass can be found in
Table 4
57 Toy Landscapes for the Design of Glasses and Glass Ceramics
The concept of a toy landscape can go even further building a method to predict the
dynamics of a glass system based on simple inputs This is possible because in the previous section
we have related the distribution of basins on an energy landscape to the experimental viscosity
which allows for thermodynamic insights Building on this we can assume that the barriers (H )
for transitioning between states is equal to that of the barrier mean for the viscous relaxation
ln10gH mkT= (97)
This means that the only key factor needed to then predict the dynamics of a landscape using the
metabasin approach is the vibrational frequency However the relaxation time at the glass transition
103
is known to be a constant of 100s and as such we can choose a vibrational frequency that reproduces
this fixed point This gives an entire description of relaxation without any need for fictive
temperature An example of the results predicted through this method are shown in Figure 24 This
prediction only required knowledge of the glass transition temperature and the fragility and from
this a pure prediction of relaxation is made
104
105
Figure 24 The predicted outputs from the toy landscape method showing predictions of the enthalpy
and entropy under a standard quench for barium disilicate This prediction does not require fictive
temperature or any such assumptions about the evolution of the non-equilibrium behavior
This method is also not limited to predicting the relaxation of glasses but can also capture
the crystallization It is known from the earlier work on nucleation presented in Chapter 3 that a
simple two state model was able to capture the underlying physics of the system Thus if we know
the free energy of the crystal and barrier to crystallization we can find predict key crystallization
106
phenomena By fitting the heat flow peak magnitude and location this gives enough information to
infer the barrier and the enthalpy of the crystal The degeneracy (and thus the entropy) of the crystal
can then be calculated through knowledge of the liquidus temperature
ex(
p)cry SCL l
cry
lk
G TH
T
minus=
(98)
This then gives everything for the model presented in Chapter 3 from only 1 calorimetry
experiment the glass transition the fragility and the liquidus temperature To predict nucleation a
series of estimates were done running the toy landscape at different temperatures until the
temperature with the fastest crystallization rate was found This was then assumed to be the
maximum rate of nucleation The value of the interfacial energy was then chosen in such a way so
that the predicted curve reproduces the peak at the same temperature This gives all the parameters
needed for a prediction using CNT It is also possible to make a prediction of the growth rate of
crystals using this expression
2
1 exp6
GU
kT
a kT
minus minus
=
(99)
The results for these estimates are shown in Figure 25 The reason why growth is more accurate
compared to nucleation is that the kinetic factor of growth is governed by viscosity while nucleation
is governed by a kinetic factor that includes an assumed vibrational frequency
These methods have a plethora of assumptions built in such as that the vibrational
frequency of basins undergoing relaxation is equal to the vibrational frequency when undergoing
crystallization however as more information is gained about a particular system (such as more
crystallization peak as a function of temperature) the information could be built into the system to
improve crystallization predictions If the peak size is known as a function of scan ratetemperature
even the enthalpy as a function of temperature could be known This provides a whole system for
the real estimates of crystallization can be improved
107
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume in
nucleation is assumed to be on the order of one cubic angstrom while the a parameter in growth is
assumed to be around one nm both are in good agreement for estimates in literature The values
for the orange points are taken from these works[128] [167]
57 Discussion
The impact of the Adam-Gibbs model on glass science is hard to overstate and it has
become foundational to ideas that will enable the design of next generation glasses The Adam-
Gibbs equation still has more to offer the glass community in terms of fundamental understanding
and practical applications In this work we have shown that the Adam-Gibbs relationship is
consistent with both shear and bulk viscosities Also we have validated the main assumption of the
MYEGA model giving insights into the scaling of configurational entropy from the underlying
enthalpy landscape Using this knowledge of the relationship between the Adam-Gibbs model with
the underlying enthalpy landscapes we have developed a simplified ldquotoy landscaperdquo approach for
modeling glass relaxation The toy landscape is constructed numerically by understanding the
distribution of inherent structures and transition points in the landscape and then relating these
distributions to experimentally measured parameters of the MYEGA viscosity model This
1 00
1600
1400
1200
1000
00
600
400
200
000
0000 110000 130000 1 0000 1 0000
(m
s)
Temperature
Growth PredictionCassar et al
0
2
4
6
10
12
14
00 1100 1300 1 00
log 1
0(I) n
ucl
ei s
m3
Temperature
Predicted RateRodrigues et al
4
3
2
1
0
00 1000 1100 1200 1300
log
10(I m
ax(I))
nuclei s
m3
Temperature elvin
108
simplified landscape model can then be used to calculate the underlying thermodynamics and
kinetics of the system and represents a practical application of energy landscapes to understand
glass relaxation nucleation and other relevant thermodynamic properties
This also closes our discussion on understanding the thermal history effects on glass What
we have shown is that the dynamics of a glass system can be understood through energy landscapes
and fundamentally the same insights can be recreated through lsquoToy Landscapesrsquo This approach
allows fundamentally for the complex 3N dimensional landscape to be summarized and abbreviated
into a smaller landscape that can easily be implemented for any glass However this new
parameterization relies on knowing some key properties that are only due to the location in the
compositional phase space and for a complete picture of the system to be created we must also
create a sufficient approximation for the compositional phase space shown in the next 2 chapters
Chapter 6
Enabling the Prediction of Glass Properties
So far models have been created that describe the very nature of the thermal history effects on
glasses However as an input to these models we have to know about the compositional dependence
of key parameters namely the glass transition and the fragility In this chapter we extend
compositional models to other key properties needed for the design of glasses Here these key
properties that have not been readily predictable are approached using both machine learning and
TCT models The TCT models are focused on surface reactivity Youngrsquos modulus and ionic
conductivity The ML addresses CTE fragility melting point as well as Youngrsquos modulus
61 Controlling Surface Reactivity
Glass surfaces especially their interactions with water[168]ndash[171] are of the utmost importance to
nuclear waste glass cover glass and many other modern applications[106] It is generally accepted
that the outermost surface of oxide glasses readily reacts with water molecules where water
molecules chemisorb on an as-formed glass surface and populate it with surface hydroxyls Recent
models attempt to quantitatively explain the change in properties observed on hydroxylated glass
surfaces[77] [168] [172] [173] and most agree that the surface readily becomes hydrated because
it is more stable to have a bonded hydroxyl group than to have dangling bonds In glasswater
interactions hydrolysis and diffusion lead to two forms of adsorbed water chemisorbed and
physisorbed[171] [174]ndash[176] Both affect the network differently and are controlled by different
processes physisorption by hydrogen bonding[175] and chemisorption by reaction with the surface
to form bonded hydroxyl groups
110
equiv Si minus O minus Si equiv +H2O rarr equiv Si minus OH + OH minus Si equiv
Although binding energy studies have been performed using molecular dynamics simulations[177]
a specific study of the surface reactivity in tandem has not been conducted Recent advancements
in reactive force field modeling allow the direct observation of both diffusion and surface
reactivity[38] [178]ndash[180] and several studies have found surface reactivity constants consistent
with those reported experimentally[173] [178] [181]
Recently TCT has been used to predict various bulk properties of glassy systems such as
Vickers hardness and fragility[17] [182] [183] Based on the TCT description of the glass network
glassy materials have been postulated to have a so-called lsquointermediate phasersquo in which the atomic
structure of a material will self-organize so as to be isostatically constrained (n=3) as explored
extensively in various works[65]ndash[67] [133] [184] [185] In this case it has been shown that such
materials exhibit anomalous behavior in certain properties such as an high hardness and lower free
energy[43] [67] [77]
TCT has also been used to understand the impact of water on bulk glass structure and
properties Potter et al[168] found that by accounting for the impact of chemisorbed (dissociated)
water on glass network connectivity (viz breaking Si-O-Si bonds) and the impact of physisorbed
(molecular) water on strain of the Si-O-Si bond angle the glass transition temperature (Tg) could
be predicted This model could be further expanded to include other properties (such as
modulus[139] or dissolution kinetics[77]) predicted by TCT such as the work done by Liu et
al[186] on calcium-silicate-hydrate gel An extensive study relating topological constraints[6]
[17] [57] and surface energy was performed by Yu et al[172] specifically focusing on the
transition from hydrophilic to hydrophobic behavior on silica surfaces Their work used reactive
molecular dynamics to model the change in surface energy of a silicate glass and then correlate it
with the number of surface constraints present with the work focusing primarily on the global
average of the surface
111
To investigate the effects of glass network topology on the surface reactivity we model
the hydration of a silicate glass was modelled using molecular dynamics simulations according to
the following procedures Initially bulk sodium silicate glasses were simulated (150 atoms with a
molar composition of 70SiO2middot30Na2O) using the Teter potential[187] For a bulk glass this pair
potential is known to accurately simulate sodium silicate glasses and has been extensively studied
to investigate various properties including structural features transport of sodium ions and
vibrational density of states[188] The size of the system is initially set to achieve the
experimentally measured density of 2466 gcm3 [189] A total of 35 Si-atoms 85 O-atoms and 30
Na-atoms were randomly inserted within a periodic cubic box of a = 12686 Å and the initial
configuration was energy-minimized to avoid any overlaps prior to the glass formation The glass
was held for 05 ns with a constant number of atoms volume and energy (a microcanonical or
NVE ensemble) Since a thermal fluctuation of ~400 K (from 2000 K to 2400 K) was observed
during the microcanonical run the glass was allowed to evolve further at 2400 K for 05 ns with a
constant number of atoms volume and temperature (a canonical or NVT ensemble) Within the
same ensemble the melted system was then cooled with a constant cooling rate of 05 Kps and
after the temperature had reached 300 K it was equilibrated for 1 ns Finally a constant pressure
of 1 atm was applied to the system (NPT ensemble) for another 1 ns to ensure no significant
fluctuations of the density during the equilibration process Three different sodium silicate glasses
of the same nominal composition were constructed via the aforementioned procedure using
different initial atomic positions These repeated simulations compensate for the limitation of small
sample size by allowing us to capture the statistical behavior of glass surfaces during the glass-
water reactions The final densities of the sodium silicate are presented in Table 5 These final
structures were used as starting configurations for the glass-water reactions MD simulations with
the Teter potential creating three glass networks were carried out using the Large-scale
AtomicMolecular Massively Parallel Simulator (LAMMPS) package[30]
112
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K
Simulation box dimension a (cubic) [Å] Density [gcm3]
Run1 12979 2303
Run2 12630 2499
Run3 12737 2437
Initial dimension a=12686 Å
Experimental density (70SiO2middot30Na2O mol) 2466 gcm3
113
After the bulk sodium silicate glasses were obtained a reactive potential was employed to model
the glass surface and subsequent glass-water interface as the system of interest for this study
required characterization of the reactive processes during the simulation of surface phenomena In
this study all reactive MD simulations are performed with the NaSiOH parameterization using
the ReaxFF reactive force field framework15 The ReaxFF parameters are trained using a first-
principles data set that describes water interaction at the sodium silicate glass-water interface
Further details of the ReaxFF methodology and its potential forms can be found in earlier
publications by van Duin et al[38] [179] [181] [190]
114
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the hydration
models Color scheme Si atom (ivory) O atom (red) and Na atom (blue) The z-axis is elongated
to allow space for an insert of water
115
As shown in Figure 26 a free surface was first created by expanding the c parameters of each of
the equilibrated bulk glasses This process results in a vacuum region above and below the two
surfaces of the glass slab along the z-direction These exposed surfaces were relaxed at 300 K for
100 ps with the ReaxFF reactive force field Following the relaxation of both top and bottom
surfaces the vacuum region was filled with water molecules The number of water molecules that
are inserted in the vacuum region was controlled to have a density of ~099 gcm3 In addition all
water molecules were kept at least 2 Å from the uppermost and lowermost atoms of the glass
surface to prevent any initial close contact with the surface Glass-water reaction simulations were
carried out at 300 K with the configurations shown in Table 6 for 500 ps in the NVT ensemble
From these runs trajectories at every 100 ps were obtained for the surface reactivity analyses
Figure 27 shows the initial and final positions of glass surface reaction with water at 0 and 500 ps
respectively
116
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the top surface
in contact with water is shown here
a) b)
117
Table 6 System configurations for sodium silicate glass-water reactions
Simulation cell [Å3] Number of water molecules
Run1 12979 times 12979 times 3894 146
Run2 12630 times 12630 times 3789 134
Run3 12737 times 12737 times 3821 138
118
The hydrated surfaces of sodium silicate glasses from each run were used to investigate the
adsorption behavior of a water molecule Since the time evolution of surface reactivity is of key
interest the binding energy of a water molecule to the hydrated surface was calculated at 0 100
200 300 400 ps In order to evaluate the local heterogeneity of reactivity imposed by varied glass
surface structures each surface was divided into a 10 by 10 grid and the binding energy at each site
was mapped across the grid The water molecule position in the z-direction from top and bottom
surface was maintained to be 20 Å from the outermost atom during the energy calculation The
binding energies (Eb) were calculated as below where the energy of a surface with adsorbed water
(Esw) was taken with respect to the surface energy (Es) and the energy of molecular water (Ew)
b s wswE E E E= minus minus (100)
A negative binding energy would indicate that water adsorption to the surface site at the
corresponding grid is thermodynamically favorable The binding energy map may then be used to
locate where water binding would be most stable based on the ReaxFF calculations From three
independent glass-water reaction boxes a total of six hydrated surfaces are obtained (a top and
bottom surface from each run) increasing the statistical reliability of this analysis in the ensemble
despite the small overall system size of any one box The binding energy mapping process is shown
in Figure 28
119
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example of the
electronic-structure DFT calculation using semilocal exchange-correlation functionals finding the
binding energy of a water lsquopixelrsquo to the surface
~20 Å
surface
10times10 grid
120
Similar calculations are carried out at an electronic structure level using the DFT framework as
implemented within the Vienna Ab-initio Simulation Package (VASP) software Total energies are
computed from the static geometries obtained from the reacted ReaxFF structures using the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional with projector augmented wave
(PAW) pseudopotentials to describe the ion cores A kinetic energy cutoff of 500 eV was used for
the expansion of the plane waves Due to the large non-crystalline structure the Brillouin zone is
only sampled at the Γ-point with 01 eV of Methfessel-Paxton smearing to help electronic
convergence Due to the computational cost associated with these DFT simulations the results were
primarily used to elucidate the electronic interactions between the water molecule and the glass
surface
To count the topological constraints in the structure generated from the ReaxFF MD
simulations an algorithm was developed to systematically exclude all non-bridging oxygens
alkali and hydrogens from the structure The number of surface rigid constraints (in the pixel)
around each network-forming atom was then calculated an example contour plot generated using
this procedure is shown in Figure 29 The surfaces were then converted to a 10times10 pixel grid and
the average coordination of network-forming atoms in each pixel calculated The depth of each
pixel taken was to be 4 Aring below the surface
121
Figure 29 Example contour surface showing the average coordination per atom on the glass
surface for the first run at 300 ps
122
Figure 30 shows the results from the MD binding energy studies where a clear transition in the
region around 30 constraintsatom is shown This region is isostatic because the number of
constraints is equal to the number of translational degrees of freedom and the width highlighted in
the figure is evidence for the existence of an intermediate phase In order to confirm that the binding
energies had indeed converged a larger system of 1500 atoms was simulated Good quantitative
agreement was found between the 150- and 1500-atom systems In this work the isostatic behavior
occurs over a wide range in which certain silicon network sites dramatically increase the binding
sites (binding energy approaching 0) The state or structure with nc outside of the intermediate range
will readily interact with water Only those in the isostatic region are likely to remain unaffected
after the interaction with water thus they are the rate limiting species in dissolution of silica glass
123
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of constraints for
surface atoms at the local pixel Results show a distinct maximum in which there is a near
hydrophilic-hydrophobic transition of the surface The error bars represent the standard deviation
A second system with 1500 atoms was performed to show convergence of the ReaxFF MD results
⟨119889119867minus119874119866119897119886119904119904⟩ = 345 Aring
⟨119889119867minus119874119866119897119886119904119904⟩ = 248 Aring
124
The intermediate phase result is intuitive due to the implicit stability that comes with having an
isostatic phase which is both energetically favorable and stress-free The larger negative binding
energies in the over-constrained regions arise from large differences between the free energies of
the chemically bonded state (chemisorbed) and unbound state (water in near proximity to the
surface but not bonded) in these regions in this region the chemical reaction with the water
molecule allows the glass surface to reduce the number of incompatible constraints which in turn
alleviates local stress in the network In other words a local region undergoes an alleviation of
stress when Q4 units convert to Q3 units The larger negative binding energy values in the under-
constrained regions can trace their origin to the sites available for bonding with the water molecule
Their high number of degrees of freedom allows for the facile reaction of water ie the only energy
cost is for the oxygen to dissociate from the network The isostatic network within the intermediate
phase is able to preserve its structural integrity because the energy barrier to deform the network is
high and there is no localized stress to create an additional driving force for reaction with water
Indeed the isostatic network is the most energetically favorable arrangement of a non-crystalline
structure This intermediate phase is governed by the fluctuations of the topology that arise with
the minimization of stresses These stresses change the local topology (within the range of the
fluctuations) so that they become isostatic[68] This leads to a flat region where there is a constant
binding energy at the surface Though it appears to be an intermediate phase it is important to note
that this may instead be simply a manifestation of an isostatic threshold (a peak in binding energies
rather than a plateau)
The average number of surface constraints has been shown in previous work to be largely
controlled by the annealing time[172] The surface constraints may then be used to predict surface
reactivity However it is worth noting that the local number of constraints is not constant in
time[17] Furthermore Potter et al[168] showed that molecular water in the network can radically
shift the free energy of the γ constraint Therefore the number of surface constraints is dynamic
125
during the influx of water Shifting constraint energy will also alter the fragility which in turn alters
the diffusion activation enthalpy In the future this may provide a path for developing more durable
glasses ndash currently it shows that a homogeneous intermediate phase (isostatic) surface should be
targeted to achieve maximum chemical durability
62 Elastic Modulus Prediction
Tailoring the elastic properties of glass is important in the design of new advanced compositions to
control the stiffness and damage resistance of a variety of glass products[2] [191]ndash[193] Elastic
moduli are defined based on the ratio of the stress exerted upon a material to the resulting strain
While various elastic moduli can be defined (eg Youngrsquos modulus shear modulus bulk modulus
and Poissonrsquos ratio) for an isotropic material such as glass only two of these quantities are
mutually independent[1] [191] [194]
Previous attempts to model elastic moduli have focused on either computationally costly
molecular dynamics simulations or empirical fitting methods The work by Makishima and
Mackenzie[195] offered a possible solution for predicting elastic moduli based on the dissociation
energy per unit volume of the glass however topological changes in the glass network are ignored
and the model cannot account for the temperature dependence of modulus With MD simulations
elastic moduli can be obtained by applying a stress and measuring the resulting strain on the
simulation cell assuming that accurate interatomic potentials are available[2] Machine learning
has also been applied to model elastic moduli using experimentally measured composition-property
databases with a high predictive ability being achieved[2] [196] Previous analytical modeling
techniques related to the topology of the glass-forming network have shown good qualitative
agreement with compositional trends in modulus but have a lack of quantitative accuracy[64]
[197]ndash[199]
126
It is common in the glass community to assume that Youngrsquos modulus may scale linearly
with hardness While this is clearly an oversimplification let us begin with previously derived
models for glass hardness as a potential starting point since glass hardness has received
considerable attention in the context of topological constraint theory [85] [192] [200] Several
models have been proposed to explain the origin of glass hardness all of which have a linear form
( ) vv
dHH n x n
dn= minus (101)
where Hv is the Vickers hardness x is chemical composition nrsquo is the critical level of constraints
needed for the substance to provide mechanical resistance to the indenter in three dimensions and
n is a measure of the constraint rigidity Recent work on glass hardness[192] has shown that n can
most accurately be defined in terms of either the density of rigid constraints (or alternatively the
density of rigid angular constraints) given by
)( )
( )( )
(c An xn
x Nx
M x
= (102)
where nc is either the number of constraints per atom or number of angular constraints per atom ρ
is the density of the composition M is the molar mass and NA is Avogadrorsquos number Since
hardness and elastic modulus are often considered to be correlated one might surmise that the
elastic modulus might also be some function of angular constraint density or total constraint
density However such an approach is not able to give quantitatively accurate predictions of
modulus
Here we propose an improved model of Youngrsquos modulus based on the free energy density of the
topological constraints
int
( ) ( ))
(
(
) i constr
c i
a s
Ai in x q T
M x
x NF F
=
= (103)
where qn(T) is the onset function for each associated constraint to give temperature dependence to
127
the predictions and is determined by the free energy ∆Fn The calculated values of Δ119865c are then
used to calculate Youngrsquos modulus (E) by
( )c c
c
FdE
EF
xd
F
= minus (104)
In this new model the Youngrsquos modulus is controlled by the free energy parameters related to the
strength of each constraint Each parameter in this model captures a key physical phenomenon with
each parameter corresponding to a physical property Moreover we demonstrate that a common
set of parameters can simultaneously capture both the temperature and compositional dependence
of modulus as well as predict other properties such as the glass transition temperature and fragility
To demonstrate the validity of this temperature-dependent constraint model Youngrsquos
modulus data are collected from literature for lithium borate[195] sodium borate[195] and
germanium selenide glass systems[201] Additionally the sodium phosphosilicate[202] system
was experimentally determined following the procedure described by Zheng et al[183] using room
temperature resonant ultrasound spectroscopy on prisms of dimensions 10 mm times 8 mm times 6 mm
Analytical topological constraint models have already been published for each of these
systems[17] [57] [202]
The proposed models for glass elasticity (the models previously proposed for hardness
based on constraint density and angular constraint density as well as the new model for modulus
based on the free energy density of constraints) were optimized in each case leaving the intercept
and slope (119889119864
119889119899
119889119864
119889119880119888) as adjustable parameters The constraint onset temperatures were also optimized
for the free energy density model Figure 31 and Figure 32 show the results for each model in the
phosphosilicate and sodium borate systems respectively
128
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values of the model
predictions are 641 GPa for constraint density 313 GPa for free energy density and 774 GPa
for angular constraint density
129
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from minimum
to maximum error The free energy density model gives the most accurate results The constraint
density has a RMSE of 61 GPa the angular density has a RMSE of 20 GPa and the energy density
has a RMSE of 59 GPa
130
We have also evaluated the predictive ability of the model in terms of the temperature dependence
of the Youngrsquos modulus sing the 10 Na2Omiddot90 B2O3 data reported by Jaccani and
Huang[191] the model and experimental predictions of the temperature dependence of the Youngrsquos
modulus are plotted in Figure 33 Here the constraint onset temperatures are determined from the
room temperature modulus data The number of escape attempts was optimized since this controls
the width of each transition and is thermal history dependent
131
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and experiment for
10 Na2O 90 B2O3 Using the previously fitted onset temperatures the only free parameters are
then the vibrational frequency and the heating time in which their product was fitted to be 14000
Where each dip in the modulus corresponds to a constraint no longer being rigid as heated through
each onset The onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The data was fit
using a least-squares method and the resultant fit is shown as the calculated method The fit has an
R2 of 094 (B) The contribution from each constraint to the overall modulus
It is also possible to validate the model by comparing the free parameters fitted to the Youngrsquos
modulus data vs that of previously reported data as shown in Table 7 Results for two additional
systems are plotted in Figure 34 to show the general validity of the free energy density model
132
Table 7 Fitted values from this analysis compared to those reported in the literature The disparity
between the constraints evaluated with molecular dynamics most likely come from the speed in
which the samples are quenched
Value Fitted Value (K) Literature (K) Method Citation
Silicate Onset 2212 1986 MD (Potter et
al[168])
Silicate Onset 818 1600 MD (Potter et
al[168])
Silicate Onset 450-500 810 MD (Potter et
al[168])
Borate Onset 921 Not Reported Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 715 740-760 Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 393 328 Fitting Parameter
(Mauro Gupta and
Loucks[69])
133
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in Fig 3) and
experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of 093 and (B) xLi2Omiddot(100-
x)B2O3 glasses with an R2 of 0986
(
A)
(
B)
134
In the previous work of Zheng et al[192] concerning hardness of glass the authors write
ldquoIt should be noted that each bond constraint corresponds to a certain energy since different kind
of bond has different bonding energyhellip and thus the constraint density also represents an energy
per unit volume In other words hardness is correlated to the energy per unit volume Our findings
for the borosilicate and phosphosilicate systems are further evidence in support of this argument
since both the total constraint density and angular constraint density approaches give better
prediction of glass hardness compared to models based on number of atomic constraintsrdquo
which makes it clear that the prediction of the hardnessmodulus should be closely related to the
onset temperatures (ie free energies) of the associated constraints The model that was previously
proposed could be extended to modulus if all constraints had equal amounts of potential energy
but due to the drastic change in strengths between the constraints the prediction fails
Bauchy et al[85] showed also that hardness in the calcium-silicate-hydrate system is
controlled by the angular constraints which led to the development of angular constraint density
as the governing control for hardness When the analysis of the density of angular constraints is
considered through the free energy view it becomes apparent why this method works effectively
for some systems Models for predicting the hardness and elastic modulus of glass attempt to
explicitly connect the rigid bond energy to the macroscopic properties of the system The
assumption for the hardness models is that the energies for breaking each type of constraint are all
approximately equal and hence only the number of the constraints matters The argument can then
be extended for elasticity since elastic modulus is a bulk material property with the approximation
of all constraints being equal strength the number of constraints per volume should be related To
correct for this approximation when a weighted sum of free energies is used the estimation
becomes significantly more accurate Using the density of the glass a precise free energy per
volume of the rigid constraints can be calculated The knowledge that the energy of the bonds is
tied to the elastic modulus has been widely known but had not been quantified nor placed within
the context of topological constraint theory[203]
135
63 Ionic Conductivity
Ionic conductivity 120590 is related to the number of charge carriers n and the mobility of the
carriers 120583 as they diffuse through a network by[1]
Ze n = (105)
where Ze is the charge of the conducting species Since the charge-carrying ion is constant within
a given glass family the ionic conductivity depends only on the mobility and the concentration of
the charge carriers present However it is not typically feasible to measure 120583 and n independently
of each other and thus it is unclear which variable scales in an Arrhenius fashion thereby
controlling the ionic conductivity of a glass In the weak electrolyte model by Ravaine and
Souquet[204]ndash[207] the mobility of the charge carriers is considered constant while the
concentration scales in an Arrhenius function Conversely the strong electrolyte model assumes a
constant number of charge carriers while the mobility follows an Arrhenius function[208] [209]
Recent MD simulations from Welch et al[210] support the weak electrolyte interpretation for alkali
silicate glasses concluding that ionic conductivity is dictated by the concentration of charge
carriers Previous reports in the literature also supporting this view have proposed several
hypotheses to explain this conclusion[208] [209] [211]ndash[213] however further exploration is
needed
The simplest form of the temperature dependence of free ions is given by
0 exp an nE
kT
= minus
(106)
Here n0 is the total possible number of mobile ions and Ea is the activation barrier for an ion to
diffuse It is often assumed that the activation enthalpy is related to the energy needed to deform
the network temporarily to allow for ion motion[193] [209] [210] [212] however it is our
hypothesis that there is a coupling between the ion hopping event and the relaxation of the local
136
network allows for a permanent relaxation or deformation along ion channels This means that
there is some cooperative relaxation of the network along the diffusion path in conjunction with an
ionic hopping event These cooperative relaxations are discussed explicitly in the Adam-Gibbs
formalism[53] for describing the relationship between viscosity and configurational entropy Sc
expc
B
TS
=
(107)
where B is an activation barrier is the viscosity and is the viscosity of the liquid in the limit
of infinite temperature In the high-temperature (low-viscosity) liquid state there is a well-defined
relationship between the viscosity and diffusion (Stokes-Einstein relation) However this
relationship breaks down in the low-temperature (high-viscosity) glassy phase
Herein we present a new model for predictions for the compositional dependence of ionic
conductivity We can write the activation free energy barrier Eac for an ionic hopping event as
a c c c cE T S= minus (108)
Here is an enthalpic barrier for the ion to hop to a neighboring site CT is the configurational
temperature (a value that describes the distribution of energies of the ions) and cS is the entropy
of the activation barriers These quantities are associated with configurational changes in the glass
indicated by the subscript c In the weak electrolyte model all charge carriers have the same
mobility and hence the entropic effects are dominant This indicates a small value of an
assumption that will be validated by results later in this work leading to a simplified approximation
for the activation energy at temperatures below the glass transition but suffiently high temperatures
such that
a c c cE T S (109)
In the case of a nonequilibrium glass at low temperature the system becomes trapped in a localized
137
region of the energy landscape known as a ldquometabasinrdquo with slow inter-metabasin transitions[156]
[161] The MAP model of the nonequilibrium viscosity of glass relies on this partial confinement
within metabasins where variation of the activation barrier for relaxation (H) is related to the
fragility of the system by[44]
1
ln10g
dm
dH kT= (110)
Here m is the fragility index (as proposed by Angell[58]) and Tg is the glass transition temperature
Splitting the differential in Eq (110) and integrating we can write
ln10gmH kT= (111)
This relationship was also proposed by Moynihan et al[214] for the activation enthalpy of
relaxation of a glass[214] [215] Setting the Arrhenius form of the barrier described in Eq (111)
equal to the Adam-Gibbs equation (Eq (107)) we write
exp exp c
H B
kT TS
=
(112)
Solving Eq (112) for the configurational entropy and inserting the result into Eq (109) we obtain
the simple relationship
2
ln10
ca
g
T BE
k T m (113)
Dividing the above expression with that of a reference state r and assuming that B and is cT are
constant with respect to compositional variation we have
g r ra
a r g
T mE
TE m= (114)
This new equation is the first to relate the activation free energy for ionic hopping in a glass
to the glass transition temperature and fragility of its corresponding supercooled liquid It also
138
predicts a coupling between ionic diffusion and the cooperative rearrangements described by the
Adam-Gibbs relation This possibility was first proposed by Ngai and Martin[216] who showed a
correlation between the product of the Kohlrausch stretching exponent ( ) and the activation
barrier for relaxation with the activation barrier for ionic conductivity This implies that the
diffusion in glasses is not governed by an elastic component but instead controlled by the
surrounding cooperative rearrangements ie the -relaxation of the glass[14] in good qualitative
agreement with previous results shown by Potuzak et al[217] Further evidence comes from the
polymer community where the decoupling of relaxation and conductivity is reported with work
showing that the decoupling of viscosity and diffusion varies with the fragility and the glass
transition temperature[218] This is a general relationship which we demonstrate later to be valid
for binary alkali borate alkali silicates and alkali phosphate glasses as well as validating the
atomistic diffusion method using computational techniques
In order to test this proposed relationship between relaxation and diffusion through
simulation a glass with the composition 10 Na2O-90 B2O3 (mol ) was synthesized in molecular
dynamics with 1150 atoms using the potentials of Wang et al[162] The system was quenched at a
rate of 1 Kps from randomly placed atoms at 2500 K in the NPT ensemble to room temperature
as shown in Figure 35[210] To find the transition point energy of the diffusing ion a nudged elastic
band (NEB) calculation was performed An alkali ion was chosen and its closest alkali neighbor
was moved to another location so that the NEB calculation could be performed between those two
sites All calculations were carried out using the LAMMPS software package Eleven reaction
coordinates were used in the NEB calculation and the results are shown in Figure 36
139
Figure 35 The structure for the initial minimum energy configuration showing the boron (blue)
network with interconnecting oxygens (red) and the interstitial sodium ions (yellow)
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and sodium is
ivory The barrier is overestimated compared to experimental data this could be from several
sources of error such as potential fitting thermal history fluctuations or sampling too few
transitions The line is drawn as a guide to the eye
140
Although the transition point energy is key for determining the activation barrier for diffusion here
we are also interested in the dynamics of the atoms around the mobile ion Since we have
hypothesized here that the ion diffusion is dependent on the ability for the local region to relax and
deform instead of elastically straining we expect a local deformation (relaxation) along the ion
path Figure 37 shows that along the ion diffusion tunnel there is indeed a permanent displacement
of the atoms
141
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function of
reaction coordinates The middle shows the local deformation around the ion of any atom that
moves in between inherent structure mandating a relaxation force The color shows the degree of
deformation
142
In the final configuration of Figure 37 the permanent deformation required to lower the
energy of the system is clearly seen If it were purely an elastic response to dilate a pathway the
displacement between two inherent structures should be confined to changes with only the mobile
ion however this is not observed Instead along the pathway of the ion movement the network
forming atoms also deform and change local positions to minimize the energy implying a net
relaxation around the ion ldquopathwaysrdquo To apply this and show the validity of Eq (114) the sodium
borate lithium borate lithium phosphate and sodium silicate systems were considered with the
model results compared to experimental values in Figure 38 To confirm this model literature values
for activation barriers have been used The experimental activation barriers for the borates are taken
from the work of Martin[213] while the fragilities and the glass transition temperatures are from
Nemilov[219] The viscosity curve of the silicate system was fit with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) equation of viscosity[55] using the data from Poole[220] and Knoche et
al[221] with the activation barriers from Martinsen[222] Lithium phosphate activation data was
taken from Ngai and Martin[216] with the glass transition and fragility taken from Hermansen et
al and predicted values from topological constraint theory[223]
143
Figure 38 Different network formers and the prediction of the activation barrier from our model
compared with activation barriers from literature (A) Sodium silicate predictions and experimental
values[222] the error is calculated from the error in the fragility when fitting the data (B) Lithium
phosphate activation energy[216] predicted with topological constraint theory and compared with
the experimental values (C) Predictions over two different systems of alkali borates[213] sodium
and lithium with a reported R2 of 097
144
It is worth noting that this model is predicting a non-equilibrium kinetic property using
equilibrium viscosity parameters a connection that has been demonstrated in the past for viscosity
and relaxation models[61] In this work we are applying the same principles to the diffusion of
ions in a network The diffusion and viscosity in the case of equilibrium are known to be intimately
linked using the Stokes-Einstein expression (which breaks down for highly cooperative
rearrangements near and below the glass transition temperature[224]) Here we are no longer
connecting the two properties explicitly but instead connecting the activation barrier for diffusive
hopping to the two key parameters governing the viscous flow of the supercooled liquid state
The failure of the Stokes-Einstein expression near the glass transition is a well-known
problem and has been studied extensively for purposes of predicting crystallization rates[167] This
occurs because there is a breakdown of the ergodicity of a system around the glass transition In
the MAP model[44] the viscosity is controlled by the ergodic parameter x and is given by
min( )
max( )
f
p
f
T Tx
T T
=
(115)
Here the exponent p is related to the sharpness of the ergodic to non-ergodic transition Using the
ergodic parameter the overall viscosity ( ) is expressed as a linear combination of equilibrium
viscosity ( eq ) and non-equilibrium viscosity ( ne ) such that
10 10 10log ) log ( ) (1 ) log ( )( f eq f ne fx T T x T TT T = + minus (116)
By combining the Stokes-Einstein expression and the model proposed in this work we can define
the diffusion of an ion through the network as
( )10 10 10 0log ( 1 log exp ) l6 )
og(
b af
f
kx
T ED T T D
a T Tx
kT
minus + minus =
(117)
Here a is an empirical constant related to the size of the diffusing species k is Boltzmannrsquos
145
constant aE is the activation barrier calculated with Eq (114) and D0 is the limit of diffusion as
the temperature approaches infinity D0 should be chosen such that at fT T= the diffusion is
continuous This can be justified because the preexponential factor is related to the entropy of the
system and the fictive temperature is the temperature at which the configurational entropy of the
transition barriers becomes nearly constant Only one parameter is left unknown a since the
viscosity and fictive temperature can be calculated with an arbitrary thermal history using a
relaxation modeling tool such as RelaxPy[42] Implicit in RelaxPy are the assumptions pertaining
to the viscosity made by Guo et al[61] A similar model was first proposed by Cassar et al[167]
for modeling diffusion during crystallization although their model used an empirical hyperbolic
tangent function to approximate the ergodic factor In contrast the current model can be considered
explicitly as a function of thermal history and composition
64 Machine Learning Expansion
Though the TCT models presented here for different properties are immensely useful they
do have limitations the greatest of which is that intimate knowledge of the glassy structure is
required to parameterize these models This is due to the fact that the landscape is fundamentally
informed by the underlying structure of the network and as such to predict the dynamics of the
landscape information on the structure must be incorporated in some form However there is an
alternate method in which the physical origin is not considered in the development of the model
machine learning Machine learning comes in many forms and has become crucial to the study of
materials in the modern era There is a plethora of machine learning techniques however in this
paper we are focused on four key methods linear fitting (LF) methods random forest (RF)
symbolic regression (SR) and neural networks (NN)[3] [80] [225] [226] These methods are
146
explored extensively in literature and the exact underlying mathematics is beyond the scope of this
work In this work these methods have been used to create models for number of constraints (LF)
glass stability (RF) fragility (NN) melting point (NN) the infinite limit of viscosity (SR) Youngrsquos
modulus (NN) and coefficient of thermal expansion (NN)
To enable a linear fitting method for the number of constraints a simple steepest descent
algorithm was used to find the number of constraints due to each component There is no direct
experimental way to measure the number of constraints so instead a technique leveraging some
other property must be used to approximate the number of constraints In this case hardness is
related to the number of constraints through
25v cH A n= minus (118)
In which A is an empirical scaling parameter that is dependent on load indenter geometry and
glass family In this linear approach to determining constraints cn can be given by
0
compon
k
c k k
ents
cn x n=
= (119)
Combining this expression with a hardness database grouped by composition (in this case Sciglass)
allows for an approximate number of constraints of the system but due to A being a variable there
must be a fixed constraint in the system and every set of data is given a unqiue value of A The
fixed constraint in this system is that silicon atoms are four coordinated The value of A is chosen
such that the error between the prediction and the hardness is minimized The total process is as
follows
Choose a set of c kn
1 Calculate number of constraints for all compositions
2 Iterate over every dataset in Sciglass with gt 1 datapoints and scale by a factor of A
3 Calculate total error for steepest descent algorithm
4 Descent error slope
5 Repeat until error has been minimized
147
This then leads to an estimate for the number of constraints for every atomic species which can
then be used to estimate important properties such as chemical durability hardness and
qualitatively can predict glass stability
Two other techniques that are used are RF and SR When implemented a grid search for
the hyperparameters were used with a testing set used to determine the efficiency of each set of
hyperparameters Additionally each model was trained repeatedly and the lowest error was taken
The RF method was used to predict glass stability while undergoing a novel processing technique
called alkali-proton substitution (APS) The SR method was used to understand the compositional
dependence of the infinite temperature viscosity limit used in 3-parameter viscosity models To do
this the Sciglass database was taken (as implemented in GlassPy[227]) and sets of data that
included more than eight datapoints were fit to the MYEGA equation If the RMSE error of the fit
was less than 001 Pa s the fragility was greater than 17 and the value of the log infinite limit was
293 5minus it was accepted into the final database Each of these criteria were added due to the well-
known fact that this database is full of error and imprecise measurements The symbolic regression
was then performed through the grid search and the equation that performed best and reappeared is
given as
10loggT
mA B = + (120)
With A and B being parameters that varied depending on the hyper-parameters the infinite limit
threshold and the accuracy threshold used when creating the database We are currently in the
process of refining values for A and B is ongoing
This method allows for an easy prediction of the infinite limit so if the glass transition and
fragility are predicted through some other means (such as ML or TCT) a better prediction of the
third parameter is known without just accepting -293 as the mean Plots of the infinite limit vs the
148
fragility glass transition and Tgm showing the clear dependance of the infinite limit on the term
Tgm are shown in Figure 39
149
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top left) shows
the distribution of the infinite temperature limit in the database after limits exerted on the system
(Top right) The distribution of the infinite viscosity limit vs the glass transition (Bottom left) The
relationship between fragility and infinite temperature limit of viscosity (Bottom right) The infinite
temperature limit of viscosity vs the key metric predicted by SR
150
NN are perhaps the most widely discussed ML tool applied in literature This makes sense
given they are universal function approximators and are quite easy to train given sufficient data
The most difficult part of creating a NN is adjusting the hyperparameters so that good accuracy can
be found without causing overfitting To avoid overfitting the data is divided into a testing set then
a training set (just as for RF and SR) but instead of doing a grid search for the hyperparameters we
used gradient boosted regression trees to find the optimal set of parameters This consists of
creating an initial tree then evaluating it based on a small set of data and finding the minimum
point then running a new set with perturbations around that point to create another tree This is
repeated until the value converges to a small error In this work the value being determined is the
lowest error of 100 sequentially trained neural nets with a set of hyperparameters This is
particularly useful because there is no concern for the type of parameters (string int floats) that
determine the value at the end so one can do a mixed parameter optimization that includes every
conceivable variable The values being optimized are given by
bull Learning rate [10-5 10-1] (float on logarithmic scale)
bull Decay rate for Adam Optimization [10-6 10-2] (float on logarithmic scale)
bull Hidden layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull Output layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull The number of nodes in each layer [81024] (int)
bull Batch size [1256] (int)
bull Patience [5250] (int)
In Table 8 we have listed the hyper-parameters that were used for the fragility melting point
Youngrsquos modulus and co-efficient of thermal expansion (CTE)
151
Table 8 Hyperparameters for different neural networks after hyper-optimizations
Model Fragility Melting
Temperature
Youngrsquos
Modulus
CTE
Learning Rate 0035 0018 0027 00091
Decay Rate 000293 000095 000161 0002
Hidden Layer RELU RELU RELU RELU
Output Layer Linear Linear SELU RELU
Number of Hidden
Layers
2 1 1 1
Number of Nodes 235 506 514 470
Batch Size 4 118 211 125
Patience 25 250 23 22
RMSE 5 [-] 805 [K] 68 [GPa] 96 x 10-7 [1 K]
152
Chapter 7
Designing Green Glasses for the 21st Century
Thus far the work presented in this dissertation has enabled a new set of design tools for
everything from crystallization glass relaxation and optimizing over the ideal compositional space
we can use them for designing new glasses Building on these models and insights new applications
of glass are enabled specifically three glasses an ion conducting glass for batteries a proton
conducting glass for hydrogen fuel cells and a commodity glass for consumers All three are
currently undergoing testing but experimental results have been reported back as of yet Overall
both the hydrogen and ion conducting glasses have been designed thorough a combination of ML
and underlying physical models and as such hare reported herein
71 Glass Electrolytes
An emerging high-interest application of glass is solid state batteries Glasses offer a
possible solution to the growing energy crisis however to realize this new paradigm new research
methods are needed The main barrier for any new batteries to be commercialized is a high
conductivity at room temperature (gt 10-3 S cm) high stability ease of processing and ultimately
an alternative must be cheaper than current liquid state Li-ion batteries glass is an attractive
candidate due to the innate stability a simple processing technique the infinite variability and the
relatively low cost of production In addition the structure of a glass is a liquid-like structure which
may encourage ion migration like that seen in liquids Despite all of these advantages only a few
compositions have realized the requirements but further research is needed to find optimal
compositions
153
To optimize the composition for activation barriers we must understand the relationship
between conductivity and structure Structural effects can be propagated in two different methods
based on the Arrhrenius expression for conductivity
0 exp aE
kT
minus =
(121)
The two methods are through 0 (the infinite temperature conductivity) and the activation barrier
( aE ) The pre-exponential factor is approximately a constant for simliar compositions while the
activation barrier varies dramatically over small compositional spaces making it the larger of the
two concerns Many models have been presented to predict the compositional dependence of the
activation barrier the Anderson-Stuart (AS) model the Christensen-Martin-Anderson-Stuart
(CMAS) model the weak electrolyte (WE) model Kohlrausch exponent model (KEM) from Ngai
et al[216] and there is the model proposed in this dissertation which we will call the Wilkinson
viscous cooperative conductivity model (WVCC)[8] [88] [204] [207]ndash[209]
To optimize machine learning (ML) is a powerful technique that is implemented The best
method to train the ML is to use the direct relationship between the compositions and the activation
barrier but this is not currently feasible due to a lack of central depository or database for this
information To make the prediction of the activation barrier we need use some model that is reliant
on commonly accesible parameters This rules out all models for this type of analysis except since
WVCC since the other models require some fitting parameters not accessible Each models required
parameters fitted values (please note that the models typically include approximations for these
values) and caveats are shown in Table 9
154
Table 9 A table with some ionic conductivity models and the parameters needed for them as well
as the disadvantages for each These are not the only models but are representative of those
commonly used in literature
Model Required Parameters Fit Parameters Caveats
AS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
lsquoMadelungrsquo Constant
Covalency Parameter
Doorway Radius1
The Madelung
Constant is not smooth
as a function of
composition and the
fitting parameters scale
the activation energy
non-proportionally
CMAS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
Dielectric Permittivity
lsquoMadelungrsquo Constant
Doorway Radius1
Jump Distance of Ion1
The same issues as AS
but in addition the
dielectric permittivity
lacks a database as
well
WE Charge Concentration Equilibrium Coefficient1 The equilibrium
coefficient drastically
changes the prediction
non-linearly
KEM Stretching Exponent Proportionality
Coefficient
There is not enough
data available to know
the compositional
dependence of the
stretching exponent
WVCC Glass Transition
Fragility Index
Proportionality
Coefficient
Due to the
proportionality
constant the activation
barrier can only be
known in a local
composition range
1 There are ways to approximate this value based on additional data
155
The WVCC model predicts that the activation barrier for ionic conductivity is given by
a
g
EA
mT= (122)
in which A is a proportionality constant This means that to predict the glass behavior we donrsquot
need to know things about the glass but instead merely about the liquid state where the viscosity is
readily available Leveraging this technique if the viscosity information as a function of
compositions is available then the local glass with the lowest activation barrier can be found by
minimizing 1 gT m To access this information multiple options are available such as topological
constraint theory but in this work we will leverage ML since it is easily applied to a large
compositional space
To get the viscosity we will use neural networks (NN) A well trained NN for the glass
transition is readily available and as such will be used for our prediction of gT In the previous
section there was presented a trained fragility NN which enables a complete optimization of the
local composition based on WVCC To generate a database of fragility values that the NN was
trained on SciGlass viscosity values were fit with MYEGA model The data had to fit following
criteria
1 The total number of data points for fitting had to be greater than 5 from the same literature
source
2 The root-mean-square-error of the fit had to be less than 001 Pa s
3 The infinite temperature value of viscosity had to be within 3 orders of magnitude of the
accepted logarithmic value (-293 log(Pa s))
An additional neural network recently created by Cassar[228] was also used for an independent
comparison Cassarrsquos NN was based on a hybrid physical ML work and may offer better insights
when extrapolating far from experimental datapoints In this study to prove this concept we will
only focus on P2O5 B2O3 Al2O3 and Li2O
156
This gives an entire method to predict glass battery candidates This technique will find all
of the local minimum but due to the fact that there is a proportionality constant it is unknown which
local minimum is the absolute minimum This means that the composition with the lowest
activation barrier in the space is identified but another technique is needed to investigate which one
of the identified compositions is best The method follows
1 Randomize an initial glass candidate with the content of being normalized
2 Run the glass candidate through the NN to find values of fragility and the glass transition
3 Perturb the composition to find the gradient
4 Step down the gradient
5 If the new gradient is 0 this is a local minimum and a possible glass candidate return to 1
Otherwise return to step 3
This method cannot find global optima because there is no guarantee that the
proportionality constant will stay constant across wide compositional spaces To narrow it down
an additional technique to compare the predicted values is needed (such as another activation
barrier model molecular dynamics simulations a regression algorithm or simply melt the proposed
compositions) To further pair down in this work a k-means algorithm was used along with the
assumption that the proportionality constant was slowly changing over compositional space This
allowed us to take the best glass in each grouping predicted by k-means The glasses we have
predicted (using both the separate predictor NN and the method by Cassar et al[228]) are listed
below in Table 10 while an example prediction for 1000 gmT (which should scale with the
activation barrier) in the Na2O ndash B2O3 ndash SiO2 system is shown in Figure 40 The glasses are currently
undergoing melting at Coe College and Iowa State It is worth noting that composition D is a well-
known composition that is known to exhibit criteria close to the criteria for conductivity If this
method shows that these compositions are better than other compositions in the family found in
literature then it can be expanded to include a wider range of component and hopefully finding a
universal candidate for glass batteries
157
Table 10 The predicted compositions based on the optimization scheme proposed
Compositions Li2O Al2O3 P2O5 B2O3
A- This work NN 55 31 0 14
B- This work NN 44 1 5 50
C- Cassar 60 26 2 13
D- Cassar 50 0 0 50
Figure 40 The prediction of the scaling of the activation barriers for a common sodium borosilicate
system
72 Hydrogen Fuel Cell Glasses
An emerging high-interest application of glass that is proton-conducting intermediate
temperature fuel cells[7] [229] A fuel cell in short is a device that turns chemical fuel into
electricity without the use of combustion It is often described as a chemical battery due to the
similarity of a fuel construction with batteries One of the crucial components to a working
intermediate range fuel cell is an electrolyte with a high proton migration The electrolyte must be
stable over a wide range of temperatures (up to 673 K) and a proton conductivity greater than 10-2
S cm-1 at the operating temperature[7] Previously phosphate glasses near the metaphosphate
compositions were used since they carried the appropriate number of residual protons for
conductivity studies However this restricted the phase space to only n-2 dimensions drastically
reducing the degrees of freedom of the problem and artificially restricting possible solutions In
2013[230] to access the full phase space a new method was developed where a sodium phosphate
glass could have all the sodium ions replaced with protons This allows high concentrations of
protons to be achieved with a variety of starting compositions A previous report[230] has
summarized successful samples these data are used as a training set for this study with additional
data on compositions that have failed taken from internal theses
To choose the candidate material for each model a cost function is defined The cost function
is an analytical function whose inputs are the fraction of each oxide component and the output is a
value that rates the composition The cost function consists as of many properties that are of interest
In the application of oxide electrolytes for an intermediate range fuel cell the two main properties
to worry about are the ionic conductivity at high temperatures (we will choose 473 K due to this
being the low end in which these materials are considered) and the relative stability of each phase
159
This is especially important since the material will undergo the APS (alkali-proton substitution) in
which the sodium will be replaced with protons in the bulk of the material The cost function used
in this work is given by
473 1( ) log 000T K Sx = minus= minus (123)
In which S is either a 0 or 1 (representing a crystal or a glass respectively) and is the
ionic conductivity The function is arbitrary and could be expanded to include as many terms are
as needed in the goal of this glass The stability is heavily weighted because it is more important
than the conductivity and any sample that is not stable should not be considered In order to predict
each of these samples a hybrid physicalempirical approach is used For predicting the ionic
conductivity of a proton-swapped glass we began with an observed relationship between the glass
transition temperature and the conductivity at 473 K This empirical relationship is shown in Figure
41 The relationship although empirical can provide some insights into the physics of proton
transferred glasses (no universality is claimed in this work) The activation barrier for these glasses
has widely remained unchanged as noted in a previous work due to the fact that the activation
barrier is simply to disassociate the proton with the non-bridging oxygen (NBO) and once the
proton is free interacts rarely with the network until it is rebounded to said network Thus the pre-
factor can be understood as being related to the degeneracy of proton conduction pathways When
the network has lower configurational entropy there are fewer pathways for the proton to travel in
this case it is known that the configurational entropy is then the dominating effect for the glass
transition[53]
The relationship between the glass transition temperature and the configurational entropy
can be expanded through the Stokes-Einstein relationship evaluated at the glass transition
22
1273 10
6 10g
g
T T g
kTnZeD nZe
anT
minus
= = = (124)
160
In which a is approximately the size of a proton and n is the number of charge carriers (which is
approximately a constant according to the weak electrolyte theory) This determines the intercept
of the glassy form proton conductivity and then the activation barrier (slope) is found to be
approximately the same for all proton conducting glasses meaning that the intercept is the dominant
effect on the behavior of the glass and the only variable controlling the intercept is the glass
transition
Leveraging the configurational entropyrsquos relationship with the glass transition along with
topological constraint theory we can write an expression for the glass transition and as such gain
predictive power for proton conductivity To predict the glass transition temperature (Tg) we start
with the Adam Gibbs model of viscosity[53] Topological constraint theory then states that the
degrees of freedom of the network is proportional to the configurational entropy[23] [56] [57]
Using the liquid state definition of the glass transition (1210 = Pa s) and the well-known infinite
temperature of viscosity we can then write
1
1493 lng
BT
kf=
(125)
Where f is the degrees of freedom Letting the constants be equal to A and rewriting the expression
in terms of constraints we arrive at a predictive formula for the glass transition[109]
3
g
x c x
x
AT
m n=
minus (126)
Where nc is the number of constraints provided by each component at the glass transition associated
with each component x and their molar fraction mx The value of the constraints was then linearly
parameterized to the glass transition data from literature[7] [230] [231] This is not the most
explicit approach to counting constraints however it is the most convenient when considering
large phase spaces being optimized over This glass transition temperature is then converted to
conductivity using the empirical translation shown in Figure 41
161
The other term in our cost function is that of stability The question of what forms glass is
a notorious question that goes back to the first serious days of research into materials It has been
shown that no one metric is a good predictor and no metric is universal However one must be used
to rule out bad compositions from the start To do this a Random Forest method is implemented
Though this may fail in edge cases it will at least help identify the right area to explore A
comprehensive review of random forest methods and machine learning methods for prediction of
glass properties can be found elsewhere[3] [4] [225] The amount of information previously
obtained about what forms glass in the compositional family we will be working on is not enough
to use random forest methods on just the fraction of each phase In order to circumvent this issue
we chose four physical parameters that would be used as features for the predictions
bull Total mol oxide modifier This was chosen because it is well known that modifiers break
up the network and most glass forming theories are related to the network percentages We
considered the oxide versions of H Na Ba Sr and Ca as the modifiers
bull Total mol network former Similarly it is well known the network influences glass
forming We considered B Ge and P as the network forming cations
bull Mean cation charge Though there has been previous work on this topic it is rarely
considered as a lsquonormalrsquo predictor of glass forming capability It is included here because
these glasses are mainly invert and as such a higher field strength will increase the cohesive
nature of the network
bull Entropy of mixing This is an important parameter to consider with so many components
it may be stabilized by the entropy of mixing The entropy of mixing (Sm) was calculated
using the Gibbs entropy
The Gibbs entropy is given by
lnm x
x
xS k m m= minus (127)
The random forest model consisted of 100 trees with a maximum depth of 3 The results of both
the random forest model the topological predictions the optimization and the resultant glass are
shown in the results section The optimization was limited to the range of each component that has
been explored experimentally as well as an additional constraint of at least 5 mol rare earth oxide
and at least 5 mol of boron oxide or germania this is justified elsewhere[231]
162
Figure 41 The relationship between the glass transition and the proton conductivity This is justified
two ways one through the relationship of the entropy of diffusion and glass formation (the Adam-
Gibbs model) and through the fact that water is known to depress the glass transition
163
To explain the results we will first start with the models listed to confirm their validity
The first model is the prediction of the glass transition temperature Figure 42 shows the accuracy
of the glass transition prediction as well as the relative contribution of each component to the glass
transition In this figure any component that has a number of constraints greater then PO52 will
increase the glass transition and as such decrease the conductivity Thus it is preferable to construct
a glass with only components whose constraints at Tg are less than PO52 as long as the glass
remains stable Interestingly whether the data includes only the glass transitions of protonated glass
or all glasses determines the role of La Barium and sodium both decrease the glass transition
playing the role of the more traditional modifier however they do provide some rigidity to the
network Protons have a net negative effect which we believe to be a result of the APS process
where they are introduced as the sodium is leaving so this negative effect corresponds to a breaking
of the rigidity that exists It is also interesting to note that adding a boron phase to the sample
increases the glass transition though not as much as some of smaller alkaline earth samples
assumingly due to their increased field strength The worst network former appears to be GeO2
To calibrate the accuracy of the random forest model a confusion matrix is shown It shows
that the RF always predict that a glass will be a glass however there is some error when it forms a
crystal with 23 of the time being misidentified Though random forest methods are empirical we
can derive some physical meaning by looking at the relative importance of each feature From the
random forest study of stability we are able to rank the relative importance of each feature used
The mean cation charge mixing entropy and percent network formers are all of considerable
importance while the percent modifiers is less so
Once these models were developed and checked the reliability of the cost function is
confirmed After which the minimization of the cost function was performed and one glass was
chosen as optimal Boron was found to be preferential in the glass but was limited since the
compositional space was bound by previous samples (an additional sample is undergoing
164
characterization where the boron content was slightly increased) The optimized sample when
synthesized had some visible nucleation so the optimization while redone with limiting the
component maxima to just the limits of those where compositions in which APS was successfully
performed This resulted in the OP composition The three samples currently are undergoing
experimental characterization are listed in Table 11
165
Table 11 The compositions synthesized in this work These compositions were predicted by
minimizing the cost function described in Eq (123) OP is the variant that was melted after OP
partially crystallized B-OP appeared to have surface nucleation in some spots but was cut and
removed before APS treatment
Name PO52 NaO12 BaO LaO32 GeO2 BO32
OP 44 36 10 5 3 2
OP 49 36 5 5 3 2
B-VOP 44 36 10 5 1 4
166
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine the glass
forming region Over top the constraints at the glass transition provided by each oxide species is
listed Since the objective is to decrease Tg while staying in the glass forming region we will
attempt to minimize use of elements that increase the glass transition temperature (nc gt 17)
167
Chapter 8
Conclusions
The rapid rate of new information in the current scientific climate and the infinite
variability of glass both stand as both unique challenges and opportunities In this dissertation we
have divided the influence of energy landscapes on the effects of properties into two independent
spaces the compositional and the thermal history dimensions By building models that succinctly
and accurately describe the dynamics of different hyper-coordinate (composition thermal history
crystallinity) changes the feasibility of designing new glasses for the challenges of the 21st century
is obtainable
Before designing new models the current state-of-the-art models need to be implemented
and understood To reach this end two softwares based on previous work were created RelaxPy
and ExplorerPy RelaxPy is an implementation of the MAP model and though the MAP model is
powerful when predicting the dynamics of the glass RelaxPy ultimately showed that fictive
temperature is insufficient at capturing the underlying physics and is intensive to parameterize
ExplorerPy was created to standardize the approach to mapping energy landscapes Energy
landscapes served as the key method of understanding the deeper dynamics of glasses glass-
ceramics and liquids
The thermal history dependence of glass with respect to both relaxation and crystallization
has been incorporated into the existence of new models called ldquotoy landscapesrdquo The toy landscapes
have built upon the physics of previous models such as the MAP model however due to the lower
parametrization cost and the built-in increased physicality toy landscapes pose as a tool to increase
our understanding and speed-up the rate of new glass discoveries This tool can deal with the
complexity of the higher dimensional spaces due to crystallization and relaxation without
assumptions concerning fictive temperature
168
The compositional dimensions are the remaining dimensions that must be optimized when
designing a glass or glass-ceramic Building on previous work models have been developed that
enable predictions of Youngrsquos modulus surface reactivity and ionic conductivity However there
are additional properties that are needed and for those we have proposed novel machine learning
approaches This dissertation has not enabled the design of glass for every application However
it lays the groundwork and approaches to design glasses for societyrsquos growing needs Presented as
well are the methods used to design glasses that could satisfy the requirements for solid state glass
electrolytes and hydrogen fuel cells All of these techniques together promise to be a powerful new
framework to build the glasses of the future
169
References
[1] A K Varshneya and J C Mauro Fundamentals of Inorganic Glasses 3rd ed Elsevier
2019
[2] J C Mauro A Tandia K D Vargheese Y Z Mauro and M M Smedskjaer
ldquoAccelerating the Design of Functional Glasses through Modelingrdquo Chem Mater vol
28 no 12 pp 4267ndash4277 2016 doi 101021acschemmater6b01054
[3] J C Mauro ldquoDecoding the glass genomerdquo Curr Opin Solid State Mater Sci pp 1ndash7
2017 doi 101016jcossms201709001
[4] H Liu Z Fu Yang X Xu and M Bauchy ldquoMachine learning for glass science and
engineering A reviewrdquo J Non Cryst Solids no March p 119419 2019 doi
101016jjnoncrysol201904039
[5] M M Smedskjaer C Hermansen and R E Youngman ldquoTopological engineering of
glasses using temperature-dependent constraintsrdquo MRS Bull vol 42 no 01 pp 29ndash33
2017 doi 101557mrs2016299
[6] J C Mauro and M M Smedskjaer ldquoStatistical mechanics of glassrdquo J Non Cryst Solids
vol 396ndash397 pp 41ndash53 2014 doi 101016jjnoncrysol201404009
[7] T Omata et al ldquoProton transport properties of proton-conducting phosphate glasses at
their glass transition temperaturesrdquo Phys Chem Chem Phys vol 21 no 20 pp 10744ndash
10749 2019 doi 101039c9cp01502g
[8] Z A Grady C J Wilkinson C A Randall and J C Mauro ldquoEmerging Role of Non-
crystalline Electrolytes in Solid-State Battery Researchrdquo Front Energy Res vol 8 p
218 2020 doi 103389fenrg202000218
[9] R C Welch et al ldquoDynamics of glass relaxation at room temperaturerdquo Phys Rev Lett
170
vol 110 no 26 p 265901 2013 doi 101103PhysRevLett110265901
[10] M Montazerian E D Zanotto and J C Mauro ldquoModel-driven design of bioactive
glasses from molecular dynamics through machine learningrdquo Int Mater Rev vol 0 no
0 pp 1ndash25 2019 doi 1010800950660820191694779
[11] E D Zanotto and J C Mauro ldquoThe glassy state of matter Its definition and ultimate
faterdquo J Non Cryst Solids vol 471 pp 490ndash495 2017 doi
101016jjnoncrysol201705019
[12] R G Palmer ldquoBroken ergodicityrdquo Adv Phys vol 31 no 6 pp 669ndash735 1982 doi
10108000018738200101438
[13] J C Mauro P Gupta and R J Loucks ldquoContinuously broken ergodicityrdquo J Chem
Phys vol 126 no 18 p 184511 2007 doi 10106312731774
[14] M Micoulaut ldquoRelaxation and physical aging in network glasses A reviewrdquo Reports
Prog Phys vol 79 no 6 p 066504 2016 doi 1010880034-4885796066504
[15] D J Wales and H A Scheraga ldquoGlobal optimization of clusters crystals and
biomoleculesrdquo Science (80- ) vol 285 no 5432 pp 1368ndash1372 1999 doi
101126science28554321368
[16] J C Mauro R J Loucks and P Gupta ldquoFictive temperature and the glassy staterdquo J
Am Ceram Soc vol 92 no 1 pp 75ndash86 2009 doi 101111j1551-2916200802851x
[17] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 2009 doi 10106313152432
[18] F H Stillinger and T A Weber ldquoHidden structure in liquidsrdquo Phys Rev A vol 25 no
2 pp 978ndash989 1982 doi 101103PhysRevA25978
[19] F H Stillinger ldquoSupercooled liquids glass transitions and the auzmann paradoxrdquo J
Chem Phys vol 88 no 12 pp 7818ndash7825 1988 doi 1010631454295
171
[20] S Neelamraju C Oligschleger and J C Schoumln ldquoThe threshold algorithm Description of
the methodology and new developmentsrdquo J Chem Phys vol 147 no 15 p 152713
2017 doi 10106314985912
[21] D Prada-Gracia J Goacutemez-Gardentildees P Echenique and F Falo ldquoExploring the free
energy landscape From dynamics to networks and backrdquo PLoS Comput Biol vol 5 no
6 p e1000415 2009 doi 101371journalpcbi1000415
[22] E Pollak A Auerbach and P Talknerz ldquoObservations on rate theory for rugged energy
landscapesrdquo Biophys J vol 95 no 9 pp 4258ndash4265 2008 doi
101529biophysj108136358
[23] G G Naumis ldquoEnergy landscape and rigidityrdquo Phys Rev E vol 71 p 026114 2005
doi 101103PhysRevE71026114
[24] S J Benkovic G G Hammes and S Hammes-Schiffer ldquoFree-energy landscape of
enzyme catalysisrdquo Biochemistry vol 47 no 11 pp 3317ndash3321 2008 doi
101021bi800049z
[25] M Roca B Messer D Hilvert and A Warshel ldquoOn the relationship between folding and
chemical landscapes in enzyme catalysisrdquo Proc Natl Acad Sci U S A vol 105 no 37
pp 13877ndash13882 2008 doi 101073pnas0803405105
[26] P Li G Henkelman J A eith and J Johnson ldquoElucidation of aqueous solvent-
mediated hydrogen-transfer reactions by ab initio molecular dynamics and nudged elastic-
band studies of NaBH4 hydrolysisrdquo J Phys Chem C vol 118 no 37 pp 21385ndash21399
2014 doi 101021jp507872d
[27] S P Niblett M Biedermann D J Wales and V De Souza ldquoPathways for diffusion in
the potential energy landscape of the network glass former SiO2rdquo J Chem Phys vol
147 no 15 p 152726 2017 doi 10106315005924
[28] Y Yu M Wang M M Smedskjaer J C Mauro G Sant and M Bauchy
172
ldquoThermometer Effect Origin of the Mixed Alkali Effect in Glass Relaxationrdquo Phys Rev
Lett vol 119 no 9 pp 1ndash5 2017 doi 101103PhysRevLett119095501
[29] Y Yu M Wang D Zhang B Wang G Sant and M Bauchy ldquoStretched Exponential
Relaxation of Glasses at Low Temperaturerdquo Phys Rev Lett vol 115 no 16 p 165901
2015 doi 101103PhysRevLett115165901
[30] Steve Plimpton ldquoFast Parallel Algorithms for Short-Range Molecular Dynamicsrdquo J
Comput Phys vol 117 no 1 pp 1ndash19 1995 doi
httpsdoiorg101006jcph19951039
[31] N Mousseau and G T Barkema ldquoTraveling through potential energy landscapes of
disordered materials The activation-relaxation techniquerdquo Phys Rev E vol 57 no 2
pp 2419ndash2424 1998 doi 101103PhysRevE572419
[32] F El-Mellouhi N Mousseau and L J Lewis ldquo inetic activation-relaxation technique
An off-lattice self-learning kinetic Monte Carlo algorithmrdquo Phys Rev B vol 78 p
153202 2008 doi 101103PhysRevB78153202
[33] D J Wales ldquoDiscrete path samplingrdquo Mol Phys vol 100 no 20 pp 3285ndash3305 2002
doi 10108000268970210162691
[34] G Henkelman and H Joacutensson ldquoImproved tangent estimate in the nudged elastic band
method for finding minimum energy paths and saddle pointsrdquo J Chem Phys vol 113
no 10 pp 9978ndash9985 2000 doi 10106314961868
[35] D Sheppard P Xiao W Chemelewski D D Johnson and G Henkelman ldquoA
generalized solid-state nudged elastic band methodrdquo J Chem Phys vol 136 p 074103
2012 doi 10106313684549
[36] A Pedone G Malavasi M C Menziani A N Cormack A V and N York ldquoA New
Self-Consistent Empirical Interatomic Potential Model for Oxides Silicates and Silica-
Based Glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
173
[37] A Takada C R A Catlow and G D Price ldquoComputer modelling of B2O3 II Molecular
dynamics simulations of vitreous structuresrdquo J Phys Condens Matter vol 7 no 46 pp
8693ndash8722 1995 doi 1010880953-8984746004
[38] A C T Van Duin S Dasgupta F Lorant and W A Goddard ldquoReaxFF A reactive
force field for hydrocarbonsrdquo J Phys Chem A vol 105 no 41 pp 9396ndash9409 2001
doi 101021jp004368u
[39] J C Mauro and A Varshneya ldquoModel interaction potentials for selenium from ab initio
molecular simulationsrdquo Phys Rev B vol 71 p 214105 2005 doi
101103PhysRevB71214105
[40] J C Mauro R J Loucks J Balakrishnan and S Raghavan ldquoMonte Carlo method for
computing density of states and quench probability of potential energy and enthalpy
landscapesrdquo J Chem Phys vol 126 no 19 2007 doi 10106312733674
[41] Y Z Mauro C J Wilkinson and J C Mauro ldquo ineticPy A tool to calculate long-time
kinetics in energy landscapes with broken ergodicityrdquo SoftwareX vol 11 p 100393
2020 doi 101016jsoftx2019100393
[42] C J Wilkinson Y Z Mauro and J C Mauro ldquoRelaxPy Python code for modeling of
glass relaxation behaviorrdquo SoftwareX vol 7 pp 255ndash258 2018 doi
101016jsoftx201807008
[43] J C Mauro ldquoTopological constraint theory of glassrdquo Am Ceram Soc Bull vol 90 no
4 pp 31ndash37 2011 doi 101039c3ee40810h
[44] J C Mauro D C Allan and M Potuzak ldquoNonequilibrium viscosity of glassrdquo Phys Rev
B vol 80 p 094204 2009 doi 101103PhysRevB80094204
[45] M Cardona R V Chamberlin and W Marx ldquoThe history of the stretched exponential
functionrdquo Ann der Phys vol 16 no 12 pp 842ndash845 2007 doi
101002andp200710269
174
[46] P Grassberger and I Procaccia ldquoThe long time properties of diffusion in a medium with
static trapsrdquo J Chem Phys vol 77 no 1982 pp 6281ndash6284 1982 doi
1010631443832
[47] J C Phillips ldquoMicroscopic aspects of Stretched Exponential Relaxation (SER) in
homogeneous molecular and network glasses and polymersrdquo J Non Cryst Solids vol
357 no 22ndash23 pp 3853ndash3865 2011 doi 101016jjnoncrysol201108001
[48] J C Phillips ldquo ohlrausch explained The solution to a problem that is 1 0 years oldrdquo J
Stat Phys vol 77 no 3ndash4 pp 945ndash947 1994 doi 101007BF02179472
[49] A Q Tool and C G Eichlin ldquoVariations caused in the heating curves of glass caused by
heat treatmentrdquo J Am Ceram Soc vol 14 pp 276ndash308 1931
[50] A Q Tool ldquoRelation Between Inelastic Deformability and Thermal Expansion of Glass in
Its Annealing Rangerdquo J Am Ceram Soc vol 29 no 9 pp 240ndash253 1946 doi
101111j1151-29161946tb11592x
[51] H N Ritland ldquoLimitations of the Fictive Temperature Conceptrdquo J Am Ceram Soc vol
39 no 12 pp 403ndash406 Dec 1956 doi 101111j1151-29161956tb15613x
[52] O S Narayanaswamy ldquoA model of structural relaxation in glassrdquo J Am Ceram Soc
vol 54 no 10 pp 491ndash498 1971 doi 101111j1151-29161971tb12186x
[53] G Adam and J Gibbs ldquoOn the Temperature Dependence of Cooperative Relaxation
Properties in Glass‐Forming Liquidsrdquo J Chem Phys vol 43 no 1 pp 139ndash146 1965
doi 10106311696442
[54] Q Zheng J C Mauro A J Ellison M Potuzak and Y Yue ldquo niversality of the high-
temperature viscosity limit of silicate liquidsrdquo Phys Rev B vol 83 no 21 p 212202
2011 doi 101103PhysRevB83212202
[55] J C Mauro Y Yue A J Ellison P Gupta and D C Allan ldquoViscosity of glass-
forming liquidsrdquo Proc Natl Acad Sci vol 106 no 47 pp 19780ndash19784 2009 doi
175
101073pnas0911705106
[56] G G Naumis ldquoGlass transition phenomenology and flexibility An approach using the
energy landscape formalismrdquo J Non Cryst Solids vol 352 no 42-49 SPEC ISS pp
4865ndash4870 2006 doi 101016jjnoncrysol200601160
[57] P Gupta and J C Mauro ldquoComposition dependence of glass transition temperature
and fragility I A topological model incorporating temperature-dependent constraintsrdquo J
Chem Phys vol 130 no 9 p 094503 2009 doi 10106313077168
[58] C A Angell ldquoRelaxation in liquids polymers and plastic crystals - strongfragile patterns
and problemsrdquo J Non Cryst Solids vol 13 no 31 pp 131ndash133 1991 doi
1010160022-3093(91)90266-9
[59] G S Fulcher ldquoAnalysis of Recent Measurements of the Viscosity of Glassesrdquo J Am
Ceram Soc vol 75 no 5 pp 1043ndash1055 1992 doi 101111j1151-
29161992tb05536x
[60] I Avramov ldquoViscosity in disordered mediardquo J Non Cryst Solids vol 351 no 40ndash42
pp 3163ndash3173 Oct 2005 doi 101016JJNONCRYSOL200508021
[61] X Guo J C Mauro D C Allan and M M Smedskjaer ldquoPredictive model for the
composition dependence of glassy dynamicsrdquo J Am Ceram Soc vol 101 pp 1169ndash
1179 2018 doi 101111jace15272
[62] Doss C J Wilkinson Y Yang H Lee L Huang and J C Mauro ldquoMaxwell
relaxation time for nonexponential α-relaxation phenomena in glassy systemsrdquo J Am
Ceram Soc vol 103 no 6 pp 3590ndash3599 2020 doi 101111jace17051
[63] P Gupta and A R Cooper ldquoTopologically disordered networks of rigid polytopesrdquo J
Non Cryst Solids 1990 doi 1010160022-3093(90)90768-H
[64] J C Phillips and M F Thorpe ldquoConstraint theory vector percolation and glass
formationrdquo Solid State Commun vol 53 no 8 pp 699ndash702 1985 doi 1010160038-
176
1098(85)90381-3
[65] X Feng W Bresser and P Boolchand ldquoDirect Evidence for Stiffness Threshold in
Chalcogenide Glassesrdquo Phys Rev Lett vol 78 no 23 pp 4422ndash4425 1997 doi
101103PhysRevLett784422
[66] W Bresser P Boolchand and P Suranyi ldquoRigidity Percolation and Molecular Clustering
in Network Glassesrdquo Phys Rev Lett vol 56 no 23 pp 2493ndash2496 1986 doi
101103PhysRevLett562493
[67] Y Vaills T Qu M Micoulaut F Chaimbault and P Boolchand ldquoDirect evidence of
rigidity loss and self-organization in silicate glassesrdquo J Phys Condens Matter vol 17
no 32 pp 4889ndash4896 2005 doi 1010880953-89841732003
[68] A irchner and J C Mauro ldquoStatistical Mechanical Model of the Self-Organized
Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologiesrdquo
Front Mater vol 6 p 11 2019 doi 103389fmats201900011
[69] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 Jun 2009 doi 10106313152432
[70] D R Cassar A C P L F de Carvalho and E D Zanotto ldquoPredicting glass transition
temperatures using neural networksrdquo Acta Mater vol 159 pp 249ndash256 Oct 2018 doi
101016JACTAMAT201808022
[71] J C Mauro and R J Loucks ldquoSelenium glass transition A model based on the enthalpy
landscape approach and nonequilibrium statistical mechanicsrdquo Phys Rev B vol 76 no
17 p 174202 2007 doi 101103PhysRevB76174202
[72] C J Wilkinson et al ldquoEnergy Landscape Modeling of Crystal Nucleationrdquo Nat Comput
Mater p Submitted 2020
[73] M E Mc enzie and J C Mauro ldquoHybrid Monte Carlo technique for modeling of crystal
177
nucleation and application to lithium disilicate glass-ceramicsrdquo Comput Mater Sci vol
149 no January pp 202ndash207 2018 doi 101016jcommatsci201803034
[74] M E McKenzie et al ldquoImplicit glass model for simulation of crystal nucleation for glass-
ceramicsrdquo npj Comput Mater vol 4 no 1 pp 1ndash7 2018 doi 101038s41524-018-
0116-5
[75] E D Zanotto and P F James ldquoExperimental tests of the classical nucleation theory for
glassesrdquo J Non Cryst Solids vol 74 no 2ndash3 pp 373ndash394 1985 doi 1010160022-
3093(85)90080-8
[76] Y Yu N M A Krishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[77] I Pignatelli A Kumar M Bauchy and G Sant ldquoTopological control on silicatesrsquo
dissolution kineticsrdquo Langmuir vol 32 pp 4434ndash4439 2016 doi
101021acslangmuir6b00359
[78] C J Wilkinson et al ldquoTopological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phaserdquo J Phys Chem Lett vol 10 pp
3955ndash3960 2019 doi 101021acsjpclett9b01275
[79] Y Zhang A Li B Deng and Hughes ldquoData-driven predictive models for chemical
durability of oxide glass under different chemical conditionsrdquo npj Mater Degrad vol 4
no 1 pp 1ndash11 2020 doi 101038s41529-020-0118-x
[80] N M Anoop Krishnan S Mangalathu M M Smedskjaer A Tandia H Burton and M
Bauchy ldquoPredicting the dissolution kinetics of silicate glasses using machine learningrdquo J
Non Cryst Solids vol 487 no February pp 37ndash45 2018 doi
101016jjnoncrysol201802023
[81] R M Potter J Hoffman and J Hadley ldquoAn update of the equation for predicting the
178
dissolution rate of glass fibers from their chemical compositionsrdquo Inhal Toxicol vol 29
no 4 pp 145ndash146 2017 doi 1010800895837820171321702
[82] B Deng ldquoMachine learning on density and elastic property of oxide glasses driven by
large datasetrdquo J Non Cryst Solids vol 529 no August 2019 p 119768 2020 doi
101016jjnoncrysol2019119768
[83] S Feller N Lower and M Affatigato ldquoDensity as a probe of oxide glass structurerdquo
Phys Chem Glas vol 42 no 3 pp 240ndash246 2001 Accessed Jul 19 2018 [Online]
Available
httpswwwingentaconnectcomcontentsgtpcg200100000042000000034203240
[84] Q Zheng and H Zeng ldquoProgress in modeling of glass properties using topological
constraint theoryrdquo Int J Appl Glas Sci vol 11 no 3 pp 432ndash441 2020 doi
101111ijag15105
[85] M Bauchy M Javad Abdolhosseini Qomi C Bichara F-J Ulm and R J-M Pellenq
ldquoRigidity Transition in Materials Hardness is Driven by Weak Atomic Constraintsrdquo
2015 doi 101103PhysRevLett114125502
[86] Yang X Xu B Yang B Cook H Ramos and M Bauchy ldquoPrediction of Silicate
Glassesrsquo Stiffness by High-Throughput Molecular Dynamics Simulations and Machine
Learningrdquo Accessed Mar 13 201 Online Available
httpsarxivorgpdf190109323pdf
[87] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological constraint model for
the elasticity of glass-forming systemsrdquo J Non Cryst Solids X vol 2 2019 doi
101016jnocx2019100019
[88] C J Wilkinson K Doss D R Cassar R S Welch C B Bragatto and J C Mauro
ldquoPredicting Ionic Diffusion in Glass from Its Relaxation Behaviorrdquo J Phys Chem B vol
124 no 6 pp 1099ndash1103 2020 doi 101021acsjpcb9b10645
179
[89] C Wilkinson and J C Mauro ldquoExplorerpy Mapping the energy landscapes of complex
materialsrdquo SoftwareX vol Submitted 2020
[90] J C Mauro R J Loucks and J Balakrishnan ldquoA simplified eigenvector-following
technique for locating transition points in an energy landscaperdquo J Phys Chem A vol
109 no 42 pp 9578ndash9583 2005 doi 101021jp053581t
[91] Roumlder and D J Wales ldquoAnalysis of the b to b-CR Transition in biquitinrdquo
Biochemistry vol 57 no 43 pp 6180ndash6186 2018 doi 101021acsbiochem8b00770
[92] J C Mauro R J Loucks and J Balakrishnan ldquoSplit-step eigenvector-following
technique for exploring enthalpy landscapes at absolute zerordquo J Phys Chem B vol 110
no 10 pp 5005ndash5011 2006 doi 101021jp056803w
[93] J C Mauro R J Loucks J Balakrishnan and A Varshneya ldquoMapping the potential
energy landscapes of selenium clustersrdquo J Non Cryst Solids vol 353 pp 1268ndash1273
2007 doi 101016jjnoncrysol200609062
[94] B W H van Beest G J Kramer and R A van Santen ldquoForce fields for silicas and
aluminophosphates based on ab initio calculationsrdquo Phys Rev Lett vol 64 no 16 pp
1955ndash1958 Apr 1990 doi 101103PhysRevLett641955
[95] Q Zheng and J C Mauro ldquoViscosity of glass-forming systemsrdquo J Am Ceram Soc vol
100 no 1 pp 6ndash25 2017 doi 101111jace14678
[96] J C Mauro and Y Z Mauro ldquoOn the Prony Series Representation of Stretched
Exponential Relaxationrdquo Physica A vol 506 pp 75ndash87 2018 doi
101016jphysa201804047
[97] E D Zanotto and D R Cassar ldquoThe microscopic origin of the extreme glass-forming
ability of Albite and B2O3rdquo Sci Rep vol 7 no February pp 1ndash13 2017 doi
101038srep43022
[98] P Pedevilla S J Cox B Slater and A Michaelides ldquoCan Ice-Like Structures Form on
180
Non-Ice-Like Substrates The Example of the K-feldspar Microclinerdquo J Phys Chem C
vol 120 no 12 pp 6704ndash6713 2016 doi 101021acsjpcc6b01155
[99] O Bjoumlrneholm et al ldquoWater at Interfacesrdquo Chem Rev vol 116 no 13 pp 7698ndash7726
2016 doi 101021acschemrev6b00045
[100] Reichelt ldquoNucleation and growth of thin filmsrdquo Vacuum vol 38 no 12 pp 1083ndash
1099 Jan 1988 doi 1010160042-207X(88)90004-8
[101] G H Beall ldquoDr S Donald (Don) Stookey (1 1 ndash2014) Pioneering researcher and
adventurerrdquo Front Mater vol 3 no August pp 1ndash8 2016 doi
103389fmats201600037
[102] J C Mauro C S Philip D J Vaughn and M S Pambianchi ldquoGlass science in the
nited States Current status and future directionsrdquo Int J Appl Glas Sci vol 5 no 1
pp 2ndash15 2014 doi 101111ijag12058
[103] J Deubener et al ldquo pdated definition of glass-ceramicsrdquo J Non Cryst Solids vol 501
no January pp 3ndash10 2018 doi 101016jjnoncrysol201801033
[104] X Hao ldquoA review on the dielectric materials for high energy-storage applicationrdquo J Adv
Dielectr vol 03 no 01 p 1330001 2013 doi 101142s2010135x13300016
[105] J Deubener G Helsch A Moiseev and H Bornhoumlft ldquoGlasses for solar energy
conversion systemsrdquo J Eur Ceram Soc vol 29 no 7 pp 1203ndash1210 2009 doi
101016jjeurceramsoc200808009
[106] G S Frankel et al ldquoA comparative review of the aqueous corrosion of glasses crystalline
ceramics and metalsrdquo npj Mater Degrad vol 2 no 1 p 15 2018 doi
101038s41529-018-0037-2
[107] G H Beall ldquoDesign and properties of glass-ceramicsrdquo Annu Rev Mater Sci vol 22
no 1 pp 91ndash119 1992 doi 101146annurevms22080192000515
[108] J Deubener ldquoConfigurational entropy and crystal nucleation of silicate glassesrdquo Phys
181
Chem Glas vol 45 pp 61ndash63 2004
[109] J C Mauro A J Ellison D C Allan and M M Smedskjaer ldquoTopological model for the
viscosity of multicomponent glass-forming liquidsrdquo Int J Appl Glas Sci vol 4 no 4
pp 408ndash413 2013 doi 101111ijag12009
[110] M M Smedskjaer J C Mauro S Sen and Y Yue ldquoQuantitative design of glassy
materials using temperature-dependent constraint theoryrdquo Chem Mater vol 22 no 18
pp 5358ndash5365 2010 doi 101021cm1016799
[111] Y T Sun H Y Bai M Z Li and W H Wang ldquoMachine Learning Approach for
Prediction and Understanding of Glass-Forming Abilityrdquo J Phys Chem Lett vol 8 no
14 pp 3434ndash3439 2017 doi 101021acsjpclett7b01046
[112] Q Zheng et al ldquo nderstanding Glass through Differential Scanning Calorimetryrdquo Chem
Rev vol 119 no 13 pp 7848ndash7939 2019 doi 101021acschemrev8b00510
[113] X Xia I Dutta J C Mauro B G Aitken and F elton ldquoTemperature dependence of
crystal nucleation in BaOmiddot2SiO2 and 5BaOmiddot8SiO2 glasses using differential thermal
analysisrdquo J Non Cryst Solids vol 459 pp 45ndash50 2017 doi
101016jjnoncrysol201612032
[114] V M Fokin A A Cabral R M C V Reis M L F Nascimento and E D Zanotto
ldquoCritical assessment of DTA-DSC methods for the study of nucleation kinetics in
glassesrdquo J Non Cryst Solids vol 356 no 6ndash8 pp 358ndash367 2010 doi
101016jjnoncrysol200911038
[115] D C Van Hoesen X Xia M E Mc enzie and F elton ldquoModeling nonisothermal
crystallization in a BaO∙2SiO2 glassrdquo J Am Ceram Soc no December 2019 pp 2471ndash
2482 2019 doi 101111jace16979
[116] S C C Prado J P Rino and E D Zanotto ldquoSuccessful test of the classical nucleation
theory by molecular dynamic simulations of BaSrdquo Comput Mater Sci vol 161 no
182
January pp 99ndash106 2019 doi 101016jcommatsci201901023
[117] A O Tipeev and E D Zanotto ldquoNucleation kinetics in supercooled Ni 0Ti 0 Computer
simulation data corroborate the validity of the Classical Nucleation Theoryrdquo Chem Phys
Lett vol 735 no August p 136749 2019 doi 101016jcplett2019136749
[118] K F Kelton and A L Greer Nucleation in Condensed Matter Applications in Materials
and Biology 1st ed Pergamon 2010
[119] V M Fokin E D Zanotto N S Yuritsyn and J W P Schmelzer ldquoHomogeneous
crystal nucleation in silicate glasses A 40 years perspectiverdquo J Non Cryst Solids vol
352 no 26ndash27 pp 2681ndash2714 2006 doi 101016jjnoncrysol200602074
[120] A M Rodrigues D R Cassar V M Fokin and E D Zanotto ldquoCrystal growth and
viscous flow in barium disilicate glassrdquo J Non Cryst Solids vol 479 pp 55ndash61 2018
doi 101016jjnoncrysol201710007
[121] A Pedone G Malavasi M C Menziani A N Cormack and U Segre ldquoA new selft-
consistent empirical interatomic potential model for oxides silicates and silica-based
glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
[122] J J Maldonis A D Banadaki S Patala and P M Voyles ldquoShort-range order structure
motifs learned from an atomistic model of a Zr 0Cu4 Al metallic glassrdquo Acta Mater
vol 175 pp 35ndash45 2019 doi 101016jactamat201905002
[123] A irchner S Goyal M E Mc enzie J T Harris and J C Mauro ldquoStatistical
Description of the Thermodynamics of Glass-Forming Liquidsrdquo Physica A vol 559 p
125059 2020 doi 101016jphysa2020125059
[124] D R Cassar ldquoCrystallization Driving Force of Supercooled Oxide Liquidsrdquo Int J Appl
Glas Sci vol 7 no 3 pp 262ndash269 2016 doi 101111ijag12218
[125] V M Fokin A S Abyzov E D Zanotto D R Cassar A M Rodrigues and J W P
Schmelzer ldquoCrystal nucleation in glass-forming liquids Variation of the size of the
183
lsquostructural unitsrsquo with temperaturerdquo J Non Cryst Solids vol 447 pp 35ndash44 2016 doi
101016jjnoncrysol201605017
[126] M Lenoir A Grandjean Y Linard B Cochain and D R Neuville ldquoThe influence of
SiB substitution and of the nature of network-modifying cations on the properties and
structure of borosilicate glasses and meltsrdquo Chem Geol vol 256 no 3ndash4 pp 316ndash325
2008 doi 101016jchemgeo200807002
[127] P Gupta D R Cassar and E D Zanotto ldquoRole of dynamic heterogeneities in crystal
nucleation kinetics in an oxide supercooled liquidrdquo J Chem Phys vol 145 no 21 2016
doi 10106314964674
[128] A M Rodrigues ldquoDiffusion Processes Crystallization and Viscous Flow in Barium
Disilicate Glassrdquo Dissertation 2014
[129] E D Zanotto ldquoThe effects of amorphous phase sepatration on crystal nucleation in baria-
silica and lithia-silica glassesrdquo Dissertation 1982
[130] D Vargheese A Tandia and J C Mauro ldquoOrigin of dynamical heterogeneities in
calcium aluminosilicate liquidsrdquo J Chem Phys vol 132 no 19 p 194501 May 2010
doi 10106313429880
[131] K Deenamma Vargheese A Tandia and J C Mauro ldquoStatistics of modifier distributions
in mixed network glassesrdquo J Chem Phys vol 132 p 24507 2010 doi
10106313429880
[132] M Micoulaut ldquoThe Deep Effect of Topology on Glass Relaxationrdquo Physics (College
Park Md) vol 6 p 72 2013 doi 101103Physics672
[133] M Micoulaut ldquoRigidity and intermediate phases in glasses driven by speciationrdquo Phys
Rev B vol 74 p 184208 2006 doi 101103PhysRevB74184208
[134] M Kodama S Kojima S Feller and M Affatigato ldquoThe occurrence of minima in the
Borate anomaly anharmonicity and fragility in lithium borate glassesrdquo Phys Chem
184
Glas vol 46 no 2 pp 190ndash193 2005
[135] Y Fukawa et al ldquoVelocity of Sound and Elastic Properties of Li2O-B2O3 Glassesrdquo Jpn
J Appl Phys vol 34 p 2570 1995 doi httpsdoiorg101143JJAP342570
[136] M Reiner ldquoThe Deborah Numberrdquo Physics Today vol 17 no 1 p 62 1964 doi
10106313051374
[137] P K Gupta and J C Mauro ldquoThe laboratory glass transitionrdquo J Chem Phys vol 126
no 22 p 2240504 2007 doi 10106312738471
[138] R Richert and C A Angell ldquoDynamics of glass-forming liquids V On the link between
molecular dynamics and configurational entropyrdquo J Chem Phys vol 108 no 21 pp
9016ndash9026 1998 doi 1010631476348
[139] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological Constraint Model for
the Elasticity of Glass-Forming Systemsrdquo J Non Cryst Solids X vol 2 p 100019 2019
doi 101016jnocx2019100019
[140] S Feller S Bista A OrsquoDonovan-Zavada T Mullenbach M Franke and M Affatigato
ldquoPacking in alkali and alkaline earth borosilicate glass systemsrdquo Phys Chem Glas Eur
J Glas Sci Technol Part B vol 50 no 3 pp 224ndash228 2009
[141] P J Bray S Feller G E Jellison and Y H Yun ldquoB10 NMR studies of the structure of
borate glassesrdquo J Non Cryst Solids vol 38ndash39 no PART 1 pp 93ndash98 May 1980 doi
1010160022-3093(80)90400-7
[142] N Andersson and G L Comer ldquoRelativistic Fluid Dynamics Physics for Many Different
Scales Living Reviews in Relativityrdquo Living Rev Relativ vol 10 p 1 2007 Accessed
Feb 18 2019 [Online] Available httpwwwlivingreviewsorglrr-2007-
1httprelativitylivingreviewsorghttpwwwmathssotonacukstaffAnderssonhttpw
wwslueducollegesASphysicsprofscomerhtml
[143] P Ilg and H C Oumlttinger ldquoNonequilibrium relativistic thermodynamics in bulk viscous
185
cosmologyrdquo Phys Rev D - Part Fields Gravit Cosmol vol 61 no 2 p 023510 2000
doi 101103PhysRevD61023510
[144] P Romatschke ldquoRelativistic viscous fluid dynamics and non-equilibrium entropyrdquo Class
Quantum Gravity vol 27 no 2 p 025006 Jan 2010 doi 1010880264-
9381272025006
[145] D C Allan ldquoInverting the MYEGA equation for viscosityrdquo J Non Cryst Solids vol
358 no 2 pp 440ndash442 Jan 2012 doi 101016JJNONCRYSOL201109036
[146] M Potuzak R C Welch and J C Mauro ldquoTopological origin of stretched exponential
relaxation in glassrdquo J Chem Phys vol 135 no 21 p 214502 Dec 2011 doi
10106313664744
[147] L Ding M Thieme S Demouchy C unisch and B J P aus ldquoEffect of pressure and
temperature on viscosity of a borosilicate glassrdquo J Am Ceram Soc vol 101 no 9 pp
3936ndash3946 2018 doi 101111jace15588
[148] R Richert and M Richert ldquoDynamic heterogeneity spatially distributed stretched-
exponential patterns and transient dispersions in solvation dynamicsrdquo Phys Rev E vol
58 no 1 pp 779ndash784 1998 doi 101103PhysRevE58779
[149] P Richet ldquoViscosity and configurational entropy of silicate meltsrdquo Geochim Cosmochim
Acta vol 48 no 3 pp 471ndash483 Mar 1984 doi 1010160016-7037(84)90275-8
[150] P Gupta and J C Mauro ldquoTwo factors governing fragility Stretching exponent and
configurational entropyrdquo Phys Rev E vol 78 no 6 p 063501 2008 doi
101103PhysRevE78062501
[151] Z Zheng J C Mauro and D C Allan ldquoModeling of delayed elasticity in glassrdquo J Non
Cryst Solids vol 500 no August pp 432ndash442 2018 doi
101016jjnoncrysol201808033
[152] R Boumlhmer L Ngai C A Angell and D J Plazek ldquoNonexponential relaxations in
186
strong and fragile glass formersrdquo J Chem Phys vol 99 no 5 pp 4201ndash4209 1993
doi 1010631466117
[153] O Gulbiten J C Mauro and P Lucas ldquoRelaxation of enthalpy fluctuations during sub-
Tg annealing of glassy seleniumrdquo J Chem Phys vol 138 no 24 p 244504 2013 doi
10106314811488
[154] D Sidebottom R Bergman L Boumlrjesson and L M Torell ldquoTwo-step relaxation decay
in a strong glass formerrdquo Phys Rev Lett vol 71 no 14 pp 2260ndash2263 1993 doi
101103PhysRevLett712260
[155] G M Bartenev and V A Lomovskoi ldquoRelaxation time spectra and the peculiarities of
the process of boron anhydride glass transitionrdquo J Non Cryst Solids vol 146 no C pp
225ndash232 1992 doi 101016S0022-3093(05)80495-8
[156] A Heuer ldquoExploring the potential energy landscape of glass-forming systems From
inherent structures via metabasins to macroscopic transportrdquo J Phys Condens Matter
vol 20 no 37 p 373101 2008 doi 1010880953-89842037373101
[157] J C Phillips ldquoStretched exponential relaxation in molecular and electronic gasesrdquo
Reports Prog Phys vol 59 pp 1133ndash1207 1996
[158] P Richet ldquoResidual and configurational entropy Quantitative checks through applications
of Adam-Gibbs theory to the viscosity of silicate meltsrdquo J Non Cryst Solids vol 355
no 10ndash12 pp 628ndash635 2009 doi 101016jjnoncrysol200901027
[159] C J Wilkinson Doss G Palmer and J C Mauro ldquoThe relativistic glass transition A
thought experimentrdquo J Non-Crystalline Solids X vol 2 2019 doi
101016jnocx2019100018
[160] J C Mauro R J Loucks and S Sen ldquoHeat capacity enthalpy fluctuations and
configurational entropy in broken ergodic systemsrdquo J Chem Phys vol 133 no 16 pp
1ndash9 2010 doi 10106313499326
187
[161] J C Mauro R J Loucks and P Gupta ldquoMetabasin approach for computing the
master equation dynamics of systems with broken ergodicityrdquo J Phys Chem A vol 111
no 32 pp 7957ndash7965 2007 doi 101021jp0731194
[162] M Wang N M Anoop Krishnan B Wang M M Smedskjaer J C Mauro and M
Bauchy ldquoA new transferable interatomic potential for molecular dynamics simulations of
borosilicate glassesrdquo J Non Cryst Solids vol 498 no December 2017 pp 294ndash304
2018 doi 101016jjnoncrysol201804063
[163] I Avramov and A Milchev ldquoEffect of disorder on diffusion and viscosity in condensed
systemsrdquo J Non Cryst Solids vol 104 no 2ndash3 pp 253ndash260 Sep 1988 doi
1010160022-3093(88)90396-1
[164] G Scherer Relaxation in Glass and Composites Krieger Publishing Company 1992
[165] L Huang J Nicholas J Kieffer and J Bass ldquoPolyamorphic transitions in vitreous B2O3
under pressurerdquo J Phys Condens Matter vol 20 no 7 2008 doi 1010880953-
8984207075107
[166] W Capps P B Macedo B OrsquoMeara and T A Litovitz ldquoTemperature Dependecne of
the High-Frequency Moduli of Vitreous B2O3rdquo J Chem Phys vol 45 p 3431 1966
doi 107868s0002337x1402002x
[167] D R Cassar A M Rodrigues M L F Nascimento and E D Zanotto ldquoThe diffusion
coefficient controlling crystal growth in a silicate glass-formerrdquo Int J Appl Glas Sci
vol 9 no 3 pp 373ndash382 Jul 2018 doi 101111ijag12319
[168] A R Potter C J Wilkinson S H im and J C Mauro ldquoEffect of Water on
Topological Constraints in Silica Glassrdquo Scr Mater vol 160 pp 48ndash52 2019 doi
101016jscriptamat201809041
[169] D B Asay and S H im ldquoEvolution of the adsorbed water layer structure on silicon
oxide at room temperaturerdquo J Phys Chem B vol 109 pp 16760ndash16763 2005 doi
188
101021jp053042o
[170] M Tomozawa ldquoWater diffusion oxygen vacancy annihilation and structural relaxation in
silica glassesrdquo J Non Cryst Solids vol 179 pp 162ndash169 1994 doi 1010160022-
3093(94)90693-9
[171] S Kapoor R E Youngman K Zakharchuk A Yaremchenko N J Smith and A Goel
ldquoStructural and Chemical Approach toward nderstanding the Aqueous Corrosion of
Sodium Aluminoborate Glassesrdquo J Phys Chem B vol 122 pp 10913ndash10927 2018 doi
101021acsjpcb8b06155
[172] Y Yu N M A rishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[173] Y Yu B Wang M Wang G Sant and M Bauchy ldquoReactive Molecular Dynamics
Simulations of Sodium Silicate Glasses mdash Toward an Improved Understanding of the
Structurerdquo Int J Appl Glas Sci vol 8 no 3 pp 276ndash284 2017 doi
101111ijag12248
[174] E Stolper ldquoThe speciation of water in silicate meltsrdquo Geochim Cosmochim Acta vol
46 no 12 pp 2609ndash2620 Dec 1982 doi 1010160016-7037(82)90381-7
[175] M G Mesko P A Schader and J E Shelby ldquoWater solubility and diffusion in sodium
silicate meltsrdquo Phys Chem Glas vol 43 no 6 pp 283ndash290 2002 doi
101016jsemradonc201010001
[176] M Tomozawa M Takata J Acocella E Bruce Watson and T Takamori ldquoThermal
properties of Na2Omiddot3SiO2 glasses with high water contentrdquo J Non Cryst Solids vol 56
no 1ndash3 pp 343ndash348 Jul 1983 doi 1010160022-3093(83)90491-X
[177] E A Leed and C G Pantano ldquoComputer modeling of water adsorption on silica and
silicate glass fracture surfacesrdquo J Non Cryst Solids vol 325 pp 48ndash60 2003 doi
189
101016S0022-3093(03)00361-2
[178] J C Fogarty H M Aktulga A Y Grama A C T van Duin and S A Pandit ldquoA
reactive molecular dynamics simulation of the silica-water interfacerdquo J Chem Phys vol
132 no 17 p 174704 2010 doi 10106313407433
[179] S H Hahn et al ldquoDevelopment of a ReaxFF Reactive Force Field for NaSiOx Water
Systems and Its Application to Sodium and Proton Self-Diffusionrdquo J Phys Chem C vol
122 no 34 pp 19613ndash19624 2018 doi 101021acsjpcc8b05852
[180] T S Mahadevan W Sun and J Du ldquoDevelopment of Water Reactive Potentials for
Sodium Silicate Glassesrdquo J Phys Chem B vol 123 pp 4452ndash4461 2019 doi
101021acsjpcb9b02216
[181] A C T van Duin A Strachan S Stewman Q Zhang X Xu and W A Goddard III
ldquoReaxFF SiO Reactive Force Field for Silicon and Silicon Oxide Systemsrdquo J Phys
Chem A vol 107 no 19 pp 3803ndash3811 2003 doi 101021jp0276303
[182] M M Smedskjaer J C Mauro R E Youngman C L Hogue M Potuzak and Y Yue
ldquoTopological principles of borosilicate glass chemistryrdquo J Phys Chem B vol 115 no
44 pp 12930ndash12946 2011 doi 101021jp208796b
[183] Q Zheng M Potuzak J C Mauro M M Smedskjaer R E Youngman and Y Yue
ldquoComposition-structure-property relationships in boroaluminosilicate glassesrdquo J Non
Cryst Solids vol 358 pp 993ndash1002 2012 doi 101016jjnoncrysol201201030
[184] K Rompicharla D I Novita P Chen P Boolchand M Micoulaut and W Huff
ldquoAbrupt boundaries of intermediate phases and space filling in oxide glassesrdquo J Phys
Condens Matter vol 20 p 202101 2008 doi 1010880953-89842020202101
[185] M Micoulaut and J C Phillips ldquoRings and rigidity transitions in network glassesrdquo Phys
Rev B vol 67 p 104204 2003 doi 101103PhysRevB67104204
[186] H Liu S Dong L Tang N M A rishnan G Sant and M Bauchy ldquoEffects of
190
polydispersity and disorder on the mechanical properties of hydrated silicate gelsrdquo J
Mech Phys Solids vol 122 pp 555ndash565 Jan 2019 doi 101016JJMPS201810003
[187] A Tilocca N H De Leeuw and A N Cormack ldquoShell-model molecular dynamics
calculations of modified silicate glassesrdquo Phys Rev B - Condens Matter Mater Phys
vol 73 p 104209 2006 doi 101103PhysRevB73104209
[188] J Du and A N Cormack ldquoThe medium range structure of sodium silicate glasses A
molecular dynamics simulationrdquo J Non Cryst Solids vol 349 no 1ndash3 pp 66ndash79 2004
doi 101016jjnoncrysol200408264
[189] N P Bansal and R H Doremus Handbook of glass properties Academic Press 1986
[190] S H Hahn and A C T van Duin ldquoSurface Reactivity and Leaching of a Sodium Silicate
Glass under an Aqueous Environment A ReaxFF Molecular Dynamics Studyrdquo J Phys
Chem C p acsjpcc9b02940 2019 doi 101021acsjpcc9b02940
[191] S P Jaccani and L Huang ldquo nderstanding Sodium Borate Glasses and Melts from Their
Elastic Response to Temperaturerdquo Int J Appl Glas Sci vol 7 no 4 pp 452ndash463 Dec
2016 doi 101111ijag12250
[192] Q Zheng Y Yue and J C Mauro ldquoDensity of topological constraints as a metric for
predicting glass hardnessrdquo Appl Phys Lett vol 111 no 1 p 011907 Jul 2017 doi
10106314991971
[193] J C Dyre ldquoElastic models for the non-Arrhenius relaxation time of glass-forming
liquidsrdquo in AIP Conference Proceedings 2006 vol 832 pp 113ndash117 doi
10106312204470
[194] Y T Cheng and C M Cheng ldquoRelationships between hardness elastic modulus and the
work of indentationrdquo Appl Phys Lett vol 73 no 5 pp 614ndash616 Jul 1998 doi
1010631121873
[195] A Makishima and J D Mackenzie ldquoDirect calculation of Youngrsquos modulus of glassrdquo J
191
Non Cryst Solids vol 12 no 1 pp 35ndash45 1973 doi 1010160022-3093(73)90053-7
[196] G Pilania C Wang X Jiang S Rajasekaran and R Ramprasad ldquoAccelerating materials
property predictions using machine learningrdquo Sci Rep vol 3 pp 1ndash6 2013 doi
101038srep02810
[197] H He and M Thorpe ldquoElastic properties of glassesrdquo Phys Rev Lett vol 54 no 19 pp
2107ndash2110 1985 Accessed Aug 29 2018 [Online] Available
httpsjournalsapsorgprlpdf101103PhysRevLett542107
[198] M Thorpe ldquoElastic properties of glassesrdquo Cambridge Arch 1985 doi 101557PROC-
61-49
[199] J C Phillips ldquoChemical Bonding Internal Surfaces and the Topology of Non-Crystalline
Solidsrdquo Phys status solidi vol 101 no 2 pp 473ndash479 Oct 1980 doi
101002pssb2221010204
[200] M M Smedskjaer J C Mauro and Y Yue ldquoPrediction of glass hardness using
temperature-dependent constraint theoryrdquo Phys Rev Lett vol 105 no 11 2010 doi
101103PhysRevLett105115503
[201] J D Musgraves J Hu and L Calvez Springer Handbook of Glass Springer US 2021
[202] C Hermansen X Guo R E Youngman J C Mauro M M Smedskjaer and Y Yue
ldquoStructure-topology-property correlations of sodium phosphosilicate glassesrdquo J Chem
Phys vol 143 no 6 p 064510 Aug 2015 doi 10106314928330
[203] L Wondraczek et al ldquoTowards ltrastrong Glassesrdquo Adv Mater vol 23 no 39 pp
4578ndash4586 Oct 2011 doi 101002adma201102795
[204] D Ravaine ldquoGlasses as solid electrolytesrdquo J Non Cryst Solids vol 38ndash39 no PART 1
pp 353ndash358 1980 doi 1010160022-3093(80)90444-5
[205] D Ravaine and J L Souquet ldquoA thermodynamic approach to ionic conductivity in oxide
glasses Part 1 Correlation of the ionic conductivity with the chemical potential of alkali
192
oxide in oxide glassesrdquo Phys Chem Glas vol 18 no 2 pp 27ndash31 1977
[206] J Swenson and L Boumlrjesson ldquoCorrelation between Free Volume and Ionic Conductivity
in Fast Ion Conducting Glassesrdquo Phys Rev Lett vol 77 no 17 pp 3569ndash3572 1996
doi 101103PhysRevLett773569
[207] C B Bragatto A C M Rodrigues and J L Souquet ldquoDissociation Equilibrium and
Charge Carrier Formation in AgI-AgPO3 Glassesrdquo J Phys Chem C vol 121 no 25 pp
13507ndash13514 2017 doi 101021acsjpcc7b02477
[208] S W Martin R Christensen G Olson J ieffer and W Wang ldquoNew Interpretation of
Na Ion Conduction in and the Structures and Properties of Sodium Borosilicate Mixed
Glass Former Glassesrdquo J Phys Chem C vol 140 pp 6343ndash6352 2019 doi
101021acsjpcc8b11735
[209] O L Anderson and D A Stuart ldquoCalculation of Activation Energy of Ionic Conductivity
in Silica Glasses by Classical Methodsrdquo J Am Ceram Soc vol 37 no 12 pp 573ndash580
1954 doi 101111j1151-29161954tb13991x
[210] R S Welch C J Wilkinson J C Mauro and C B Bragatto ldquoCharge Carrier Mobility
of Alkali Silicate Glasses Calculated by Molecular Dynamicsrdquo Front Mater vol 6 p
121 May 2019 doi 103389fmats201900121
[211] Y J Zhang ldquoEntropy and ionic conductivityrdquo Phys A Stat Mech its Appl vol 391 no
19 pp 4470ndash4475 2012 doi 101016jphysa201204021
[212] J C Dyre ldquoOn the mechanism of glass ionic conductivityrdquo J Non Cryst Solids vol 88
no 2ndash3 pp 271ndash280 1986 doi 101016S0022-3093(86)80030-8
[213] M Steve ldquoIonic Conduction in Phosphate Glassesrdquo JAmCeramSoc vol 74 no 8 pp
1767ndash1784 1991 doi 101111j1151-29161991tb07788x
[214] C T Moynihan A J Easteal J Wilder and J Tucker ldquoDependence of the Glass
Transition Temperature on Heating and Cooling Raterdquo J Phys Chem vol 78 no 26 pp
193
2673ndash2677 1974 Accessed Jul 13 2018 [Online] Available
httpspubsacsorgsharingguidelines
[215] Ito C T Moynihan and C A Angell ldquoThermodynamic determination of fragility in
liquids anda fragile-to-strong liquid transition inwaterrdquo Lett to Nat vol 398 no April
p 492 1999
[216] L Ngai and S W Martin ldquoCorrelation between the activation enthalpy and
Kohlrausch exponent for ionic conductivity in oxide glassesrdquo Phys Rev B vol 40 no
15 pp 10550ndash10556 1989 doi 101103PhysRevB4010550
[217] M Potuzak X Guo M M Smedskjaer and J C Mauro ldquoAre the dynamics of a glass
embedded in its elastic propertiesrdquo J Chem Phys vol 138 no 12 2013 doi
10106314730525
[218] A L Agapov and A P Sokolov ldquoDecoupling ionic conductivity from structural
relaxation A way to solid polymer electrolytesrdquo Macromolecules vol 44 no 11 pp
4410ndash4414 2011 doi 101021ma2001096
[219] S V Nemilov ldquoA Structural Investigation of Glasses in the B2O3ndashNa2O System by the
Viscosimetric Methodrdquo Izv Akad Nauk SSSR Neorg Mater vol 2 no 2 pp 349ndash359
1966
[220] J P Poole ldquoLow‐Temperature Viscosity of Alkali Silicate Glassesrdquo J Am Ceram Soc
vol 32 no 7 pp 230ndash233 1949 doi 101111j1151-29161949tb18952x
[221] R noche D B Dingwell F A Seifert and S L Webb ldquoNon-linear properties of
supercooled liquids in the system Na2O SiO2rdquo Chem Geol vol 116 pp 1ndash16 1994 doi
1010160009-2541(94)90154-6
[222] W Martinsen ldquoSelected properties of sodium silicate glasses and their structural
significancerdquo Dissertation 1969
[223] C Hermansen J C Mauro and Y Yue ldquoA model for phosphate glass topology
194
considering the modifying ion sub-networkrdquo J Chem Phys vol 140 no 15 2014 doi
10106314870764
[224] M L F Nascimento and E Dutra Zanotto ldquoDoes viscosity describe the kinetic barrier for
crystal growth from the liquidus to the glass transitionrdquo J Chem Phys vol 133 no 17
p 174701 2010 doi 10106313490793
[225] D Denisko and M M Hoffman ldquoClassification and interaction in random forestsrdquo Proc
Natl Acad Sci U S A vol 115 no 8 pp 1690ndash1692 2018 doi
101073pnas1800256115
[226] G Varney C Dema B E Gul C J Wilkinson and Akgun ldquo se of machine learning
in CARNA proton imagerrdquo in Progress in Biomedical Optics and Imaging - Proceedings
of SPIE 2019 vol 10948 doi 101117122512565
[227] ldquoMDLregSciGlass-6 MDL Information Systems San Leandro CArdquo 2003
[228] D R Cassar ldquoViscNet Neural Network for predicting the fragility index and the
temperature-dependency of viscosityrdquo pp 1ndash33 2020
[229] D J L Brett A Atkinson N P Brandon and S J Skinner ldquoIntermediate temperature
solid oxide fuel cellsrdquo Chem Soc Rev vol 37 no 8 pp 1568ndash1578 2008 doi
101039b612060c
[230] T Ishiyama S Suzuki J Nishii T Yamashita H awazoe and T Omata ldquoProton
conducting tungsten phosphate glass and its application in intermediate temperature fuel
cellsrdquo Solid State Ionics vol 262 pp 856ndash859 2014 doi 101016jssi201310055
[231] T Yamaguchi et al ldquoProton-conducting phosphate glass and its melt exhibiting high
electrical conductivity at intermediate temperaturesrdquo J Mater Chem A vol 6 no 46 pp
23628ndash23637 2018 doi 101039C8TA08162J
VITA
Collin Wilkinson was born in Mt Carroll IL He attended Coe College as an undergraduate and
studied glass resulting in a physics bachelorrsquos degree in 201 He worked with Dr Ugur Akgun
and Dr Steve Feller on modeling different key properties of glass propertiesresponses to external
stimuli Collin joined the group of Dr John Mauro at Pennsylvania State University in 2018
List of publications written by first author (or co-first author) Collin while at Penn State
1 CJ Wilkinson JC Mauro Explorerpy Mapping the Energy Landscapes of Complex
Materials SoftwareX Submitted
2 CJ Wilkinson DR Cassar AV DeCeanne KA Kirchner ME McKenzie ED Zanotto and
JC Mauro Energy Landscape Modeling of Crystal Nucleation NPJ Computational
Materialis Submitted
3 CJ Wilkinson K Doss O Gulbiten DC Allan JC Mauro Fragility and Temperature
Dependence of Stretched Exponential Relaxation in Glass-Forming Systems JACERS
Submitted
4 CJ Wilkinson JC Mauro Comment on ldquoThe Fragility of Alkali Silicate Glass Melts Part
of a niversal Topological Patterrdquo by DL Sidebottom JNCS (2020) ( 2 ) 11
5 CJ Wilkinson K Doss DR Cassar RS Welch CB Bragatto JC Mauro Predicting Ionic
Diffusion in Glass From its Relaxation Behavior Journal of Physical Chemistry B (2020)
124 (6) 1099-1103
6 Z Ding CJ Wilkinson J Zheng Y Lin H Liu J Shen SH Kim Y Yue J Rent CJ Mauro
Q Zheng Topological Understanding of the Mixed Alkaline Earth Effect in Glass JNCS
(2020) (526) 119696
7 CJ Wilkinson AR Potter RS Welch CB Bragatto Q Zheng M Bauchy M Affatigato SA
Feller JC Mauro Topological Origins of Mixed Alkali Effect in Glass Journal of Physical
Chemistry B (2019) 123 (34) 7482
8 CJ Wilkinson K Doss SH Hahn Nathan Keilbart AR Potter NJ Smith I Dabo ACT Van
Duin SH Kim JC Mauro Topological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phase JPCL (2019) 10 (14) 3955
9 CJ Wilkinson K Doss G Palmer JC Mauro The Relativistic Glass Transition A Through
Experiment JNCSX (2019) 100018
10 CJ Wilkinson Q Zheng L Huang JC Mauro Topological Constraint Model for the
Elasticity of Glass-Forming Systems JNCSX (2019) 100019
11 AR Potter CJ Wilkinson SH Kim JC Mauro Effect of Water on Toplogical Constraints
in Silica Glass Scripta Materialia (2019) (160) 48-52
12 CJ Wilkinson E Pakhomenko MR Jesuit AV DeCeanne B Hauke M Packard SA Feller
JC Mauro Topological Constraint Model of Alkali Tellurite Glasses JNCS (2018) (502)
172
13 CJ Wilkinson YZ Mauro JC Mauro RelaxPy Python code for Modeling of Glass
Relaxation Behavior SoftwareX (2018) (7) 255
ii
The dissertation of Collin Wilkinson was reviewed and approved by the following
John Mauro
Professor of Materials Science and Engineering
Chair Intercollege Graduate Degree Program
Associate Head for Graduate Education Materials Science and Engineering
Dissertation Advisor
Chair of Committee
Seong Kim
Professor of Chemical Engineering
Professor of Materials Science and Engineering
Ismaila Dabo
Associate Professor of Materials Science and Engineering
Susan Sinnott
Professor of Materials Science and Engineering
Professor of Chemistry
Head of the Department of Materials Science and Engineering
iii
Abstract
Glass is infinitely variable This complexity stands as a promising technology for the 21st
century since the need for environmentally friendly materials has reached a critical point due to
climate change However such a wide range of variability makes new glass compositions difficult
to design The difficulty is only exaggerated when considering that not only is there an infinite
variability in the compositional space but also an infinite variability thermal history of a glass and
in the crystallinity of glass-cearmics This means that even for a simple binary glass there are at
least 3 dimensions that have to be optimized To resolve this difficulty it is shown that energy
landscapes can capture all three sets of complexity (composition thermal history and crystallinity)
The explicit energy landscape optimization however has a large computational cost To
circumvent the cost of the energy landscape mapping we present new research that allows for
physical predictions of key properties These methods are divided into two categories
compositional models and thermal history models Both models for composition and thermal
history are derived from energy landscapes Software for each method is presented As a
conclusion applications of the newly created models are discussed
iv
Table of Contents
List of Figures vi
List of Tables xi
Acknowledgments xii
Chapter 1 The Difficulty of Optimizing Glass 1
11 Energy Landscapes 6 12 Fictive Temperature 8 13 Topological Constraint Theory 15 14 Machine Learning 19 15 Goals of this Dissertation 20
Chapter 2 Software for Enabling the Study of Glass 22
21 ExplorerPy 22 22 RelaxPy 28
Chapter 3 Understanding Nucleation in Liquids 30
31 Crystallization Methods 33 31A Mapping and Classifying the Landscape 35 31B Kinetic Term for CNT 38 31C Degeneracy calculations 41 31D Free Energy Difference 42 31E Interfacial Energy 44
32 Results amp Discussion 46 33 Conclusions 51
Chapter 4 Expanding the Current State of Relaxation 52
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition 52 41A Relativistic Liquid 56 41B Relativistic Observer 60
42 Temperature and Compositional Dependence of the Stretching Exponent 64 42A Deriving a Model 67 42B Experimental Validation 73
43 Conclusion 81
Chapter 5 Glass Kinetics Without Fictive Temperature 82
51 Background of the Adam Gibbs Relationship 82
v
52 Methods 84 53 Results 86
53A Adam-Gibbs Validation 86 53B MYEGA Validation 87 53C Adam-Gibbs and Structural Relaxation 89 53D Landscape Features 91
54 Topography-Property Relations 94 55 Barrier Free Description of Thermodynamics 99 57 Toy Landscapes for the Design of Glasses and Glass Ceramics 102 57 Discussion 107
Chapter 6 Enabling the Prediction of Glass Properties 109
61 Controlling Surface Reactivity 109 62 Elastic Modulus Prediction 125 63 Ionic Conductivity 135 64 Machine Learning Expansion 145
Chapter 7 Designing Green Glasses for the 21st Century 152
71 Glass Electrolytes 152 72 Hydrogen Fuel Cell Glasses 158
Chapter 8 Conclusions 167
References 169
vi
List of Figures
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists
the equilibrium liquid which as it is quenched can either become a super cooled
liquid or crystallize Crystallization causes a discontinuity in the volume The super-
cooled liquid upon further quenching departs from equilibrium and transitions into
the glassy state Reproduced from Fundamentals of Inorganic Glass Science with
permission from the author1 5
Figure 2 The schematic for the flow of the program Beginning in the top right corner
and running until the condition in the pink box is satisfied Yellow diamonds
represent checks and blue operations 25
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above
and then plotted using PyConnect The plot is a disconnectivity graph where each
terminating line represents an inherent structure and tracing where two lines meet
describes the activation barrier The potentials are taken from the BKS potential 94 27
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-
axis being an arbitrary phase space and the y-axis being the potential energy
calculated from the Pedone et al potentials36 The colors are indicative of the
crystallinity where the blue basin is the initial starting configuration The landscape
shows the lowering of energy associated with partially crystallizing the sample (B)
The energy landscape relationship between the cutoff for crystalline and super-
cooled liquid states for 256 atoms is shown This shows a clear drastic energy
change occurring around the cutoff value of 10 Aring 37
Figure 5 An example interfacial structure between the crystalline phase on the left and
the last sequential SCLglass phase on the right for a barium disilicate system The
gray atoms are barium silicon is shown in red and blue represents oxygen 43
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a
function of temperature for each system size The experimental values for kinetics
and thermodynamics come from ref125 and from the heat capacity data taken from 126 respectively It is clear to see that driving force shows good agreement across all
systems however the kinetics terms only converge for the 256 and 512 atoms
systems (Bottom) The fit used to calculate the interfacial energy as a function of
temperature 45
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted
using the model presented in this work The data referenced can be found in refs 113128129 47
Figure 8 The surface energy with respect to temperature for the work presented here 50
Figure 9 An Angell diagram created using the MYEGA expression 55 With a very strong
glass (m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) 17134135
vii
The infinite temperature limit is from the work of Zheng et al54 and the glass
transition temperature is from the Angell definition 54
Figure 10 The relativistic glass transition temperature for B2O3 glass 57
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different
fractions of light speed All of the viscosities approaching the universal temperature
limit for viscosity 59
Figure 12 The modulus needed to satisfy the condition for the glass transition 61
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at
v=044c (Bottom) Viscosity plot showing a dramatic reduction of the viscosity as
the observer approaches the speed of light 63
Figure 14 b predicted and from literature showing good agreement with a total root-
mean-square error of 01 The fit for organic systems is given by 66 ln Kln = minus
and for inorganic systems by 75ln Kln = minus 72
Figure 15 The equilibrium model proposed with the experimental points showing good
agreement between the experimentally measured data points and the equilibrium
derived model RMSE was 002 for Corningcopy JadeTM (A) and less than 001 for
SG80 (B) (C) The model fit for B2O3 experimental data154155 The fragility and glass
transition temperature of the B2O3 are taken from the work of Mauro et al17 75
Figure 16 The stretching exponent calculated as described in the text for a Gaussian
distribution of barriers This plot shows that the distribution of barriers has a large
effect on the stretching exponent A Tg cannot be described since there is no
vibrational frequency included in the model though the glass transition temperature
should be the same for all distributions since the mean relaxation time is the same
for all distributions at all temperatures The deviation is given in ln eV units 78
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent
Each color designates one term in the series (Bottom) The output from RelaxPy
v20 showing the stretching exponent effects on the relaxation prediction of
Corningcopy JadeTM glass Each quadrant shows one property that is of interest for
relaxation experiments In particular it is interesting to see the dynamics of the
stretching exponent during a typical quench 80
Figure 18 The viscosity (left) and landscape (right) predictions for three common
systems The first system is newly calculated in this work while the latter two come
from our previous works7289 It is seen that the viscosity predicted from the AG
model is very accurately able to reproduce the experimental viscosity curves from
the MYEGA model The last system is a potential energy landscape while the others
are enthalpy landscapes 87
Figure 19 The configurational entropy comparisons between the three major viscosity
models which validates the main underlying assumption of the MYEGA model The
viii
VFT and AM are unable to capture the physics of configurational entropy therefore
ruling 89
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the
configurational entropy from the enthalpy landscape with the barrier 00155 eV
(compared to the shear barrier of 00149 eV) and the infinite limit allowed to vary
(10-263 Pamiddots for bulk viscosity) In Figure 18 the configurational entropy is
confirmed for the shear viscosity thus confirming the AG for both shear and bulk
viscosities Sidebottom data are from Ref 154 90
Figure 21 (A) The histogram for the energy minima for each of the test systems showing
a good fit with the log normal distribution This distribution will then be a valid
form to calculate the enthalpy distribution of the model presented in the next
session (B) The configurational entropy from the model showing the accuracy of
the scaling of the entropy predicted by the MYEGA model The S value was fit
for each system 93
Figure 22 (A) Comparison between the randomized method (histogram) and the
deterministic method (vertical lines) showing good agreement between the
maximum in the histogram and the value predicted by the deterministic technique
validating the approach It is worth noting that the 100-basin distribution is a very
wide distribution where the total number of basins is less than the number of points
used in the calculation This is done for a variable number of basins with the number
of basins shown in the legend (B) The dependence of fragility and the glass
transition temperature vs the distribution of states and the number of basins 99
Figure 23 The driving forces for different example glasses calculated using a
combination of the MAP model RelaxPy and the toy landscape model The
parameters for each glass can be found in Table 4 102
Figure 24 The predicted outputs from the toy landscape method showing predictions of
the enthalpy and entropy under a standard quench for barium disilicate This
prediction does not require fictive temperature or any such assumptions about the
evolution of the non-equilibrium behavior 105
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume
in nucleation is assumed to be on the order of one cubic angstrom while the a
parameter in growth is assumed to be around one nm both are in good agreement
for estimates in literature The values for the orange points are taken from these
works128167 107
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the
hydration models Color scheme Si atom (ivory) O atom (red) and Na atom (blue)
The z-axis is elongated to allow space for an insert of water 114
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the
top surface in contact with water is shown here 116
ix
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example
of the electronic-structure DFT calculation using semilocal exchange-correlation
functionals finding the binding energy of a water lsquopixelrsquo to the surface 119
Figure 29 Example contour surface showing the average coordination per atom on the
glass surface for the first run at 300 ps 121
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of
constraints for surface atoms at the local pixel Results show a distinct maximum in
which there is a near hydrophilic-hydrophobic transition of the surface The error
bars represent the standard deviation A second system with 1500 atoms was
performed to show convergence of the ReaxFF MD results 123
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values
of the model predictions are 641 GPa for constraint density 313 GPa for free
energy density and 774 GPa for angular constraint density 128
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from
minimum to maximum error The free energy density model gives the most accurate
results The constraint density has a RMSE of 61 GPa the angular density has a
RMSE of 20 GPa and the energy density has a RMSE of 59 GPa 129
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and
experiment for 10 Na2O 90 B2O3 Using the previously fitted onset temperatures
the only free parameters are then the vibrational frequency and the heating time in
which their product was fitted to be 14000 Where each dip in the modulus
corresponds to a constraint no longer being rigid as heated through each onset The
onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The
data was fit using a least-squares method and the resultant fit is shown as the
calculated method The fit has an R2 of 094 (B) The contribution from each
constraint to the overall modulus 131
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in
Fig 3) and experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of
093 and (B) xLi2Omiddot(100-x)B2O3 glasses with an R2 of 0986 133
Figure 35 The structure for the initial minimum energy configuration showing the boron
(blue) network with interconnecting oxygens (red) and the interstitial sodium ions
(yellow) 139
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and
sodium is ivory The barrier is overestimated compared to experimental data this
could be from several sources of error such as potential fitting thermal history
fluctuations or sampling too few transitions The line is drawn as a guide to the eye 139
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function
of reaction coordinates The middle shows the local deformation around the ion of
x
any atom that moves in between inherent structure mandating a relaxation force
The color shows the degree of deformation 141
Figure 38 Different network formers and the prediction of the activation barrier from our
model compared with activation barriers from literature (A) Sodium silicate
predictions and experimental values241 the error is calculated from the error in the
fragility when fitting the data (B) Lithium phosphate activation energy235 predicted
with topological constraint theory and compared with the experimental values (C)
Predictions over two different systems of alkali borates232 sodium and lithium with
a reported R2 of 097 143
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top
left) shows the distribution of the infinite temperature limit in the database after
limits exerted on the system (Top right) The distribution of the infinite viscosity
limit vs the glass transition (Bottom left) The relationship between fragility and
infinite temperature limit of viscosity (Bottom right) The infinite temperature limit
of viscosity vs the key metric predicted by SR 149
Figure 40 The prediction of the scaling of the activation barriers for a common sodium
borosilicate system 157
Figure 41 The relationship between the glass transition and the proton conductivity This
is justified two ways one through the relationship of the entropy of diffusion and
glass formation (the Adam-Gibbs model) and through the fact that water is known to
depress the glass transition 162
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine
the glass forming region Over top the constraints at the glass transition provided by
each oxide species is listed Since the objective is to decrease Tg while staying in the
glass forming region we will attempt to minimize use of elements that increase the
glass transition (nc gt 17) 166
xi
List of Tables
Table 1 The key properties considered for commercial application The optical properties
have been omitted since it is physical unrealistic to expect quantitative predictions of
quantum-controlled phenomena from a classical description of glass structure 18
Table 2 Variable definitions 65
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy
JadeTM glass and Sylvania Incorporatedrsquos SG80 74
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition
temperature (500 K) were kept the same while the total number of basins were
allowed to vary 101
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K 112
Table 6 System configurations for sodium silicate glass-water reactions 117
Table 7 Fitted values from this analysis compared to those reported in the literature The
disparity between the constraints evaluated with molecular dynamics most likely
come from the speed in which the samples are quenched 132
Table 8 Hyperparameters for different neural networks after hyper-optimizations 151
Table 9 A table with some ionic conductivity models and the parameters needed for
them as well as the disadvantages for each These are not the only models but are
representative of those commonly used in literature 154
Table 10 The predicted compositions based on the optimization scheme proposed 157
Table 11 The compositions synthesized in this work These compositions were predicted
by minimizing the cost function described in Eq (123) OP is the variant that was
melted after OP partially crystallized B-OP appeared to have surface nucleation in
some spots but was cut and removed before APS treatment 165
xii
Acknowledgments
ldquoTo Love Another Person is to See the Face of Godrdquo -Victor Hugo Les Miserables
I have been truly blessed to love and be loved by a plethora of mentors friends colleagues and
family across my academic journey I have to first thank my advisor Dr John Mauro who has
been as kind patient and caring as any mentor I have ever met and in the process has made me a
better scientist and a better person His mentorship was built on top of those who first found and
shaped me into a scientist Dr Ugur Akgun and Dr Steve Feller without whom my scientific career
would not be possible Along with these individuals Irsquove had the pleasure to learn so much about
glass and life from Dr Madoka Ona Dr Firdevs Duru Dr Mario Affatigato Dr Doug Allan Dr
Ozgur Gulbiten and Dr Seong Kim
Rebecca Welch Arron Potter and Anthony DeCeanne are my partners in all things
Rebecca is my partner in research and in life Her opinion support and patience has been
indispensable to me Arron is my oldest colleague and friend he is the one that I share the highs of
research and friendship with Anthony has put up with me non-stop for about 5 years now He is
the most patient and kind friend you could ask for Without these three individuals I could not have
done anything presented in this work I additionally have to thank Karan Doss Aubrey Fry Caio
Bragatto Daniel Cassar Brenna Gorin Mikkel Bodker Greg Palmer Katie Kirchner Kuo-Hao
Lee and Yongjian Yang I have the privilege to call these individuals both my friends and
colleagues
The last people I have to thank are my family I want to thank my brothers (Sam amp Quinn)
my mom and my dad My mom has helped me explore the universe through a love of reading and
her constant unwavering support My dad was the first to show me the wonders of science with
rockets as a kid as a teenager he supported my fledgling interest as he drove hundreds of miles to
every corner of this country and he is the person for whom this dissertation is dedicated
Chapter 1
The Difficulty of Optimizing Glass
Glass is a complex world-changing material Though it has existed for thousands of years
the surface of its true potential is only just being scratched To unlock the potential of glass for new
applications and to enable the maximum benefit to society we must be able to design new glasses
with wide ranging properties faster and cheaper than ever before [1]ndash[5] This is particularly
important now that the need for ecologically responsible materials is increasing due to climate
change In order to facilitate such glass design we must return to first principles and build a picture
of glass from the ground up encompassing both compositional and thermal history dependencies
into our understanding of glass properties[3] [6] The present accepted description of glass is a
non-crystalline non-ergodic non-equilibrium material that appears solid and is continuously
relaxing towards the super-cooled equilibrium liquid state[1] This definition gives insights into the
nature of glass and includes points of particular interest in this dissertation
The first point to consider is that glass is non-crystalline meaning that it contains no long-
range order This gives glass one of its key advantages being infinitely variable Since a glass is
not limited by having to reach stoichiometric crystalline structures there is no limit to the number
of possible glasses or glass structures This gives glass its possibilites Glass has been considered
as a candidate for everything from hydrogen fuel cells[7] to batteries[8] and for applications
ranging from commercial smart phone covers[9] to bioactive applications[10] The infinite
variability also explains the difficulty of designing glasses Consider a system with three
constituents for example SiO2 Na2O and B2O3 This then means there are two independent
compositional dimensions that must be fully explored to find an optimal composition for an
application If this space were discretized to 1 mol spacing that would give over 5000 unique
2
glass candidates that would need to be studied to reach the optimal design A five-component glass
would correspondingly have over 9 million unique glasses More generally for a glass with up to
C components there are 1C minus dimensions over which to optimize If instead one considers a
crystal there is not a continuously variable space but instead only discrete points
The other ramification of glass being non-crystalline is that it must bypass the region of
crystallization at a rate much faster than the rate of nucleation[11] This is inherent in the definition
of glass and is shown in Figure 1 Figure 1 is called the volume-temperature (VT) diagram and is
key to understanding the nature of glass The bypassing of the crystallization region leads to a
super-cooled liquid and during quenching the material departs from equilibrium and enters the
glassy state The region of departure is called the glass transition temperature range or simply the
glass transition To understand why this occurs we must first understand the concept of ergodicity
Ergodicity was defined by Boltzmann to mean that the time-average value of a property is
equal to the ensemble-average value of that property[12] [13] Now that ergodicity is defined we
can start to consider the implications of the non-ergodic nature of glass Since it is non-ergodic the
ensemble average will not equal the time average on human timescales however the ergodic
hypothesis says that in the limit of long time they will be equal[12] In order for both statements to
be true glass must evolve to be ergodic and thus is inherently unstable To explore this concept
further we must understand the glass transition process and define a timescale If there exists a
liquid at some temperature T and is then perturbed to T dT+ there will be some inherent time
associated with its relaxation towards equilibrium which is described by struct the structural
relaxation time There is a separate relaxation time when the system is mechanically perturbed
called the stress relaxation time stress To then understand whether the resulting material is a glass
or a liquid we need to compare struct with the observation time If the observation time is much
longer than the relaxation time then what is observed over the course of a experiment is the
3
equilibrium liquid properties Conversely if the observation time is much shorter than the
relaxation time then the observed properties will not be ergodic and as such we will not sample
the properties of the equilibrium liquid We define this non-ergodic state as the glassy state
However since in the limit of long time the system must become ergodic once again then over time
all glasses will return to an ergodic nature through a process called relaxation [14]
The last point considered in the definition of the glass is that it is non-equilibrium Not only
is glass non-equilibrium it is also unstable and continuously progressing towards the super-cooled
liquid state (eg relaxation) Relaxation is inherent in all glasses and we can see this for a model
system in the VT diagram (Figure 1) In this diagram there are two glasses (of the same
composition) quenched at different rates (both of which bypass crystallization) In the slow-cooled
glass there is a lower volume (higher density) compared to the fast-cooled glass This difference is
due to the fact that the slow-cooled glass has been allowed to relax more fully compared to the
other This adds at least one additional dimension of optimization (such as quench rate from a high
temperature to room temperature) in which glasses must be designed in
This dissertation is dedicated to understanding the infinite variability of glass with respect
to crystallization composition and thermal history with tools being implemented and designed so
that the challenge of optimizing over these vast spaces can be done with higher efficiency lower
cost and with less experimental work load These tools are all inspired by the energy landscape
description of glass To design the optimal glass for any application we must understand the
relationship between the 1C minus compositional dimensions a minimum of one thermal history
dimension and the properties of a material There are many ways to explore these relationships
each with its advantages and disadvantages The methods to optimize over these thermal history
and compositional dimensions include energy landscapes[15] fictive temperature[16] topological
constraint theory[17] and machine learning[3] Of these energy landscapes are the most
fundamental with each other technique being related back to the energy landscape in some form
4
Though they are the most fundamental they are also the most difficult time consuming and have
the largest barrier to entry for use To then build our understanding of each technique we will start
with energy landscapes then explore how through a series of assumptions we can arrive at the other
methods
5
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists the
equilibrium liquid which as it is quenched can either become a super cooled liquid or crystallize
Crystallization causes a discontinuity in the volume The super-cooled liquid upon further
quenching departs from equilibrium and transitions into the glassy state Reproduced from
Fundamentals of Inorganic Glasses with permission from the author[1]
6
11 Energy Landscapes
Energy landscapes were first developed by Goldstein and Stillinger who proposed the
concept as a way to understand the evolution of a material under complex conditions [18]ndash[20]
Since Stillingerrsquos seminal work [18] energy landscapes have become important in the study of
protein folding [15] [21] [22] network glasses [6] [13] and other complex materials The concept
of energy landscapes has become commonplace and is readily evoked when describing the time
evolution of a material [20] [22]ndash[26] Despite energy landscapes being commonplace in some
fields the procedure for mapping a landscape remains difficult due to the associated computational
challenges [15] [27] To map an energy landscape one must find a set of local minima and the first
order saddle point connecting each pairwise combination of those minima Each minimum is called
an inherent structure and represents a structurallychemically stable state of the material A basin
is the set of structures that converge to the same inherent structure upon minimization [18]
In glass science the energy landscape is used to justify complex behaviours which are most
commonly calculated using molecular dynamics simulations (MD) [13] [28] [29] It is easy to
understand why MD is used so frequently used when considering the ease of use of common MD
packages For example LAMMPS [30] is a common user-friendly MD program with a low barrier
to entry allowing individuals with little programming experience to run complex MD simulations
Inspired by this accessibility as well as the explicit Python bridge to LAMMPS we have developed
a software that can be used to explore energy landscapes with a variety of exploration techniques
Similar softwares [15] [31]ndash[35] have been created to map landscapes however each program has
limitations and may not be suitable for all purposes while ours is a general purpose software
The calculation of the energy for landscapes is done using an empirical interatomic
potential that can be fit using either experimental data and a series of MD simulations or using ab
initio methods One common potential form is the Lennard-Jones form (LJ)
7
12 6
4LJVr r
= minus
(1)
in which and are fitting parameters for every pair-wise set of atoms in a system V is the
potential energy contributed by the two-atom interactions and r is the distance between the two
ions in question Though the LJ form is simple and easy to parameterize it is unable to recreate
complicated material behaviours for a real glass As such a more complicated potential is often
used such as a Morse potential A commonly used silicate glass potential is the Pedone potential
which consists of Morse potential with an additional columbic term and repulsive term [36] and is
given by
( )2
2
0 121 exp 1
i j
Pedone
Z Z e CV Y a r r
r r = + minus minus minus minus +
(2)
Z is the charge of a given ion e is the charge of an electron and Y C a and r0 are fitting parameters
It is also worth noting that the Pedone et al potentials are only accurate when considering the
interactions between each individual ion species with oxygen For instance Si-O Na-O and O-O
are all explicitly parameterized while Si-Na only interacts coulombically (the first term)
Interatomic potentials are discussed here only briefly but there are extensive resources available
for those readers who wish to learn more[37]ndash[39]
Once an energy landscape has been mapped the thermodynamics and kinetics for a given
system can be easily calculated [33] [40] The thermodynamic driving force is given by the
difference between the free energy of the current inherent structure and the free energy of the lowest
state while the kinetics is the rate in which the system crosses the barrier between the basins This
can be calculated explicitly for arbitrary timescales and temperatures with another software
KineticPy [41] The energy landscape provides insights into the detailed nature of material
behaviour Though energy landscapes are very powerful they are also difficult to map To
approximate the behavior of the landscape then we use approximations such as fictive temperature
8
topological constraint theory (TCT) or machine learning Fictive temperature is a way to estimate
the effects of the thermal history dimension[42] while TCT is used to estimate the compositional
dependence of properties[43]
12 Fictive Temperature
Fictive temperature is an approximation used to find the change in the occupational
probabilities on an energy landscape[16] [44] Instead of considering every possible combination
of occupational probabilities like the energy landscape it instead approximates the occupational
probability using a single temperature at which the landscape would be in equilibrium It is
essentially a measure of how closefar a glass is from equilibrium This approximation allows for
us to calculate the relaxation dynamics of a glass without needing any computationally heavy
energy landscapes However this single parameter description oversimplifies relaxation[14] [16]
The functional form for glass relaxation dates back to 1854 when Kohlrausch fit the
residual charge of a Leyden jar with[45]
( ) (0)expt
g t g
= minus
(3)
where t is time τ is the relaxation time constant β is the stretching exponent and g(t) is the
relaxation function of some property (eg volume conductivity viscosity etc) Eq (3) is referred
to as the stretched exponential relaxation function (SER) The stretching exponent is bounded from
0 lt β le 1 where the upper limit (β = 1) represents a simple exponential decay and fractional values
of β represent stretched exponential decay
For over a century the physical origin of the stretched exponential relaxation form was one
of the greatest unsolved problems in physics until 1982 when Grassberger and Proccacia[46] were
able to derive the general form based on a model of randomly distributed traps that annihilate
9
excitations and the diffusion of the excitations through a network This model however provided
no physical meaning for β which was still treated as an empirical fitting parameter In 1994
Phillips[47] [48] was able to extend the diffusion trap model by showing that β=1 at high
temperatures and at low temperatures the stretching exponent can be derived based on the effective
dimensionality of available relaxation pathways He in turn expressed the stretching exponent
2
df
df =
+ (4)
where d is the dimensionality of the network and f is the fraction of relaxation pathways available
He then proposed a set of lsquomagicrsquo numbers for common scenarios Assuming a three-dimensional
network with all pathways activated (d = 3f = 1) a value of β= 35 is obtained The second case is
a three dimensional network with only half of the relaxation pathways activated (d =3f = 12)
yielding a value of β= 37 A third magic value β= 12 was found for a two dimensional technique
with all pathways activated (d = 2f = 1) β = 35 occurs for stress relaxation of glasses under a load
because both long- and short-range activation pathways are activated whereas a value of β = 37 is
obtained for structural relaxation of a glass without an applied stress The value was confirmed by
Welch et al[9] when the value was measured over a period of 15 years at more than 600C below
the glass transition temperature of Corning Gorilla Glass Though this model was able to reproduce
the limiting values it was criticized widely in the community Critics argued that the model was
simply created such that it reproduced the stretching exponent values of certain experiments and
ignored many other experimental results that appeared to disagree with the model The model also
fails to interpolate between the low temperature values Phillips predicted and the high temperature
limit
The work from Phillips and the stretched exponent gave a form for relaxation however
there remained no method to instantaneously state the distance from equilibrium of a glass To
account for this problem an additional thermodynamic variable (or order parameter) called fictive
10
temperature (Tf) was proposed Early research on relaxation in glasses dates back to the work of
Tool and Eichlin in 1932[49] and Tool in 1946[50] whose works originally proposed a temperature
at which a glass system could be in equilibrium without any atomic rearrangement Tool suggested
that this fictive temperature was sufficient to understand the thermodynamics of a glassy system
Originally the fictive temperature was treated as a single value that was some function of thermal
history (Tf[T(t)]) The evolution of fictive temperature as proposed by Tool is then given by
)
( ) ( ( ))
(
f f
f
dT T t T T t
dt T T
minus= (5)
In which for stress relaxation is given by
( )(( ) ( ))fT t T
G
T t = (6)
In Eq (6) is the shear viscosity and G is the shear modulus This means that to predict the time
evolution of a glass only three things are required shear modulus the viscosity as a function of
temperature and fictive temperature and the stretching exponent Though fictive temperature
qualitatively reproduced the results they were looking for subsequent experiments have shown that
the concept of a single fictive temperature is inadequate[51] [52]
A key experiment was performed by Ritland in 1956[51] Ritland took several samples
with different thermal histories but the same measured fictive temperature thus based on Toolrsquos
equation both should have had identical relaxation properties Ritland however showed that the
refractive index evolved differently between samples To account for these differences
the use of multiple fictive temperatures was suggested[52] This worked because the stretched
exponential form could be approximated with a Prony series
( ) exp expi i
N
i
Kt w K t minus minus
(7)
11
In the Prony series N is the number of terms while iw and iK are fitting parameters This means
that each term could represent one fictive temperature and this collection of fictive temperatures
would reproduce the overall experimentally observed form Ritlandrsquos experiment showed that
materials held isothermally at their fictive temperatures with varying thermal histories can give
different results for the relaxation of a given property Thus as a glass network relaxes the fictive
temperature will also shift Narayanaswamy[52] was able to apply Ritlandrsquos concept in a highly
successful engineering model with multiple fictive temperatures but the physical meaning of
multiple fictive temperatures remains elusive
In general there is reason to be skeptical of the concept of fictive temperature Fictive
temperature was not derived but instead was an empirical approximation that Tool needed to gain
an early quantitative understanding Recently Mauro et al[16] used energy landscapes to try to
understand the underlying validity of fictive temperature fT implies that the occupational
probability ( ip ) is given by
1
exp ii
Hp
Q kT
= minus
(8)
Q is the partition function k is Boltzmannrsquos constant and iH is the enthalpy of the given state
However when the actual probability evolution predicted by fictive temperature was compared to
the probability calculated through a systematic Monte Carlo approach the results were found to be
drastically different (even when many fictive temperatures were tried) This implied that fictive
temperature is insufficient to describe the glassy state and must be replaced with a new method
Despite the inadequacies of fictive temperature state-of-the-art modeling techniques for
glass relaxation still relies on the concept In these techniques a set of fictive temperatures (usually
12) are used to calculate 12 parallel relaxing fictive temperatures[42] This is because there has
been no alternative method able to reach a prediction of the evolution of glass under different
12
thermal histories The current state-of-the-art model is called the Mauro-Allan-Potuzak (MAP)[44]
model of non-equilibrium viscosity and was derived from a more fundamental expression for
viscosity called the Adam-Gibbs model (AG) which is given by[53]
10 10) log( )
log ( f
c f
BT T
TS T T = + (9)
B is the barrier for a cooperative rearrangement 10log is the infinite temperature limit of
viscosity and according to Zheng et al[54] it is approximately 10-293 Pa s and cS is the
configurational Adam-Gibbs entropy This entropy has long since been a source of debate and will
be addressed later in chapter 5
In order to calculate the equilibrium part of the viscosity the MAP model relies on an earlier
Mauro Yue Ellison Gupta and Allan (MYEGA) equilibrium viscosity model[55] This model
was derived based on earlier work by Naumis[23] [56] Gupta and Mauro[57] and the AG model
These models showed that the configurational entropy can be related to the degrees of freedom
lncS fNk= (10)
f is the degrees of freedom per atom N is the number of atoms and is the number of degenerate
states per floppy mode To then build in temperature dependence they assumed that the degrees of
freedom can be modeled using an Arrhenius form (with activation barrier H )
( ) expH
f T dkT
= minus
(11)
When Eqs (9) (10) and (11) are combined into an equilibrium form the equilibrium viscosity is
then given by
10 10 el pog og xlC H
T kT
+
=
(12)
Where C is equivalent to
13
ln
BC
dkN=
(13)
Through a change in variables this can be written in terms of three common variables that allow
for all viscosity equations to be written with three variables specified by Angel[58] the glass
transition temperature the fragility and 10log The glass transition temperature is defined as
the isokom where the liquid has a viscosity of 1210 Pa s The fragility is defined as[58]
10log
g
g
T T
dm
Td
T
=
(14)
The three most common viscosity models (MYEGA Vogel-Fulcher-Tammann (VFT)[59] and
Avramov-Milchev (AM)[60]) are listed below in terms of the same parameters
10 10 10
10
log (12 log ) exp 1 11 o
log2 l g
g gT Tm
T T
+ minus minus minus
minus =
(15)
( )
2
10
10 10
10
12 logl log
1 1
og
2 logg
Tm
T
minus+
minus + minus
=
(16)
and
( )10(12 log
10 10 1
)
0l gog l 12 logo
m
gT
T
minus
+ minus=
(17)
To incorporate non-equilibrium effects with these equilibrium models the MAP model
proposes
10 10 10log ( ) (1 ) log ( )log MYEGA g ne f gT m T mx x T T T = + minus (18)
14
10log ne is the non-equilibrium viscosity whose form and compositional dependence can be found
elsewhere[42] [44] [61] x is the ergodicity parameter and is of crucial importance to
understanding glass regardless of the model being used[12] In the MAP model it is given by
( )( )
3244min
max
f
f
m
T Tx
T T
=
(19)
Though the MAP model was derived from the best understanding of energy landscapes of glasses
at the time it has a few flaws that prevent it from perfectly predicting the different relaxation
behaviors of glass A few of the short-comings are
1 The method ignores crystallization For a comprehensive model showing the response of
glass to temperature the possibility of crystallization must be included Currently no
relaxation model includes this possibility
2 It is built around the concept of fictive temperature This could be replaced in a future
model but currently at present it carries with it all the failures of the fictive temperature
picture of glass dynamics adding only the relaxation time coming from landscape-derived
values
3 The MAP model does not consider temperature dependence of the stretching exponent
This is typically circumvented by choosing a fixed stretching exponent
4 The parameters needed for the non-equilibrium viscosity in the MAP model are
experimentally challenging to measure To circumvent this Guo et al[61] published
methods to estimate the values based on fragility and the glass transition but the
approximations were only tested on a few samples with similar fragilities and chemistries
5 The MAP model only predicts stress relaxation Since the publication of the MAP model
it has been shown that only stress relaxation time is related to the shear viscosity[62]
Structural relaxation must be incorporated in future models
15
For a complete understanding of the thermal-history dependence of glass all of these questions
must be rigorously answered
13 Topological Constraint Theory
TCT is one way to estimate the compositional dependence of glasses It uses a set of
assumptions that connects the underlying structure to glass properties through understanding of the
energy landscape These assumptions include the temperature dependence of the rigidity the same
assumptions used in the derivation of the MYEGA models and that the effects of thermal history
on structure are minimal TCT was originally proposed by Gupta and Cooper (GCTCT) to try to
understand the role of rigid polytopes on glasses properties[63] Gupta and Cooper fundamentally
understood that the ability for glasses to form was reliant on the transformation of a liquid with
many degrees of freedom to a glass with a fixed degree of freedom Since the structure of the glass
is the same as the structure of the liquid at the glass transition and the degrees of freedom of the
glass ( f ) changes during the transition to glass it was possible to gain insights into the
transformation process This approach is fundamentally an approximation of the highly
multidimensional landscape where only approximate structural information is used to understand
the entropy of the energy landscape Though the core idea was useful it was ldquoclunkyrdquo and difficult
to make calculations with
To improve upon the ideas of GCTCT Phillips and Thorpe (PTTCT)[64] proposed a
mathematically equivalent form where the degrees of freedom could be simply found by
considering the number of constraints ( cn ) around each network-forming atom in the glass They
showed that the degrees of freedom were related to the number of constraints by
cf d n= minus (20)
16
The number of constraints could then be calculated by considering the two-body and three-body
interactions (first and second terms) as a function of the mean coordination of the network forming
species
2 32
c
rn r= + minus (21)
This led to a qualitative language to discuss glass with a few quantitative insights The biggest
quantitative insight comes in the form of the lsquoBoolchand Intermediate Phasersquo[65]ndash[68] If it has
positive degrees of freedom then a network is lsquofloppyrsquo and retains high kinetic energy (as it flips
back and forth between two inherent structures) If the system has negative degrees of freedom
then the network is stressed-rigid and there is additional energy stored in the stress associated with
over-coordination If the number of constraints is equal to the number of dimensions then the
resulting system will have zero degrees of freedom and there is no stored stressed energy or kinetic
energy from flipping between basins Boolchand went searching for this perfectly lsquoisostaticrsquo glass
and found a non-zero window of compositions that were close to isostatic but had properties
drastically different compared to both stress rigid and floppy glasses This is now called the
Boolchand Intermediate Phase
Despite the success in the prediction of the intermediate phase TCT remained largely
qualitative until the work by Naumis[23] [56] and Gupta amp Mauro[57] Naumis pointed out that
the degrees of freedom are a lsquovalleyrsquo of continuous low energy so that the structure can freely move
between basins of the landscape Since these valleys are deformation pathways Naumis pointed
out that they will dominate the configurational entropy of the system Building on this Mauro and
Gupta wrote that since the entropy is proportional (Eq (10)) we could then write the glass transition
temperature as a function of the degrees of freedom and the AG model
( )10o1 ln2 l g
gTB
f Nkminus=
(22)
17
Since everything in Eq (22) is approximately a constant between compositions except for f we
could then write the expressions in terms of reference values (subscript r) and accurately scale the
glass transition of new compositions by knowing the changing structures
( ) g r c r
g
c
T d nT
d n
minus=
minus (23)
The last insight provided by Mauro and Gupta was that the number of constraints change as a
function of temperature They realized that each constraint has a temperature dependence which is
controlled by the interaction energy of the constraints This makes intuitive sense because as the
thermal energy becomes greater than that of the barriers additional valleys in the energy landscape
form This also allowed them to expanded TCT to include quantitative predictions of fragility
The understanding of the relationships between the configurational entropy the number of
constraints the degrees of freedom and the viscous properties lead to a revolution in understanding
glass properties Soon after the work by Mauro amp Gupta multiple works came out predicting
mechanical chemical and viscous properties of glass knowing only the underlying structure
Though this framework is powerful it is not a complete description without further
parameterizations with each model requiring some fitting parameters or reference values In
addition there are still key properties that are missing from the standard topological approach that
are needed to design commercial glasses
Ultimately MGTCT needs be expanded to include the properties that are needed for
commercial applications A list of composition dependent properties needed for understanding the
performance of commercial glasses are shown in Table 1 Properties taken from Ref [3] show what
properties must be controlled to manufacture technological glasses All models take additional
inputs and there are multiple ways to link outputs from some models to inputs of the other Choice
of model vary depending on the use case
18
Table 1 The key properties considered for commercial application The optical properties have
been omitted since it is physical unrealistic to expect quantitative predictions of quantum-
controlled phenomena from a classical description of glass structure
Properties TCT Machine Learning Other methods
Glass Transition [57] [69] [70] Energy Landscapes [71]
Fragility [57] [69] This work This work
Relaxation MAP model [44] [61]
Crystallization This work [72]
MDMC [73]ndash[75]
Melting Temperature This work MDThis work[72]
Chemical Durability [76] [77]
This work [78]
[79] [80] Linear Methods [81]
Density [82] MD Packing [83]
CTE This work MD
Hardness [5] [84] [85] MD
Youngrsquos Modulus [86] This
work [87]
[82] This work MD
Activation Barrier for Conductivity MD This work [88]
Batch Cost Economic Calculation
Ion Exchange Properties Empirical Relationships [2]
19
14 Machine Learning
All the methods discussed so far are physically informed techniques The techniques
presented are inherently powerful because their very nature is informed by reality Unfortunately
this means that some knowledge about the system is required These techniques are useless without
the required parameterizations However a class of algorithms called lsquoMachine Learningrsquo (ML)
require no such parameterization Instead ML uses large quantities of data to determine trends
linking any physically connected input to output allowing for accurate predictions of properties
across wide compositional spaces This is favorable with a large of volume data because the
required inputs are minimal to achieve a prediction The difficulty (and often expense) in this
technique is gathering large quantities of high-quality data to enable this method
The predominant ML method used in this work is neural networks (NN) Neural networks
work by constructing a network of neurons This method was designed in such a way to mimic a
human brain so the network is elastic enough to be modified improving results Each neuron takes
in a series of inputs received from the previous layer (or input data) multiplies by a weighting
factor sums the values and then performs some lsquoactivation functionrsquo on them By changing the
weights it is then possible to recreate any function given sufficient data This is a preferred
technique because it packages into a simple easy-to-use software Neural Networks (once trained)
can be used by anyone stored in a small file can recreate any function is computationally cheap
and does not require as much data as some other ML techniques
This is an interesting method to estimate the effects of the composition on the landscape
and corresponding properties because we cannot understand the underlying mathematics It is
fundamentally a numerical process Thus even though we know that the properties of a glass are
dominated by the topography of the landscape how the NN figures out the relationship between
the chemistry and properties is not understood It is possible to then use the NN to give us the
20
physical parameters we need for other models building multiple methods together to achieve new
insights
15 Goals of this Dissertation
To enable the next generation of glass design we must enable a new era of computationally
methods being used to produce accurate predictions of the behavior of glass This includes both the
thermal history dependence of glass (in terms of relaxation and crystallization) and the
compositional dependence especially those properties listed in Table 1 The goals of this thesis will
be to address some of these outstanding questions in a systematic way that is practical for an
external user to understand and implement Predicting thermal history effects are covered in
Chapters 3-5 while compositional effects are predicted in Chapters 6-7 The breakdown of each
following Chapter as it relates to these questions is listed below
21
bull Chapter 2 Software Unfortunately there is no standardized software for the
calculation of relaxation or energy landscapes I present two new codes to
standardize these calculations so that the process is reproduceable and standardized
bull Chapter 3 Crystallization This section will address new computational ways to
calculate crystallization focusing on predicting nucleation rates and how to improve
our nucleation calculations
bull Chapter 4 Expanding the Current State of Relaxation This chapter will focus on
expanding the MAP model with a temperature dependent form of the stretching
exponent as well as understanding the role of ergodicity in glass relaxation
bull Chapter 5 Relaxation and Crystallization without Fictive Temperature Chapter 5
will focus on discussing relaxation and crystallization without fictive temperature
by proposing a new relaxation model that is reliant on creating lsquoToy Landscapesrsquo
This approach removes assumptions about the evolution of the occupational
probabilities
bull Chapter 6 Enabling Prediction of Properties This chapter will use TCT and ML to
address the compositional dependance of different key properties needed for
enabling the faster design of new glass compositions
bull Chapter 7 Designing lsquoGreenrsquo Glasses Applying information learned throughout
this dissertation to create new glass compositions
bull Chapter 8 Conclusions
22
Chapter 2
Software for Enabling the Study of Glass
To being our systematic investigation it is important to have a suite of tools that
implement current state-of-the-art models Though many tools exist for structure and
topological constraint theory there were no codes that existed for the calculation of
relaxation using the MAP model and there were no codes that were capable of exploring
the landscapes that we are interested in To correct this we present two new novel codes
ExplorerPy for mapping energy landscapes and RelaxPy for calculating MAP
dynamics[42] [89]
21 ExplorerPy
Mapping an energy landscape allows for modeling a range of traditionally inaccessible
processes such as glass relaxation The advantage of this software (over other codes to map energy
landscapes) is that it only requires knowledge of LAMMPS to run Also our software contains a
choice of three methods to map the landscape The first of the three different methods implemented
to explore energy landscapes in this work is eigenvector following [90] Eigenvector following has
been used in the past to generate landscapes that predict the glass transition and protein folding
pathways [44] [91] This implementation follows the work of Mauro et al [90] [92] and uses an
independent Lagrange multiplier for every eigenvector Eigenvector following excels at finding
small barriers quickly however it rarely finds unique structural changes in complex materials
(especially in systems with periodic boundary conditions) The second method that we have
implemented following in the steps of Niblett et al [27] is a combination of molecular dynamics
and nudged elastic band methods (NEB) [34] [35] To find the transition points MD is run at a
23
fixed temperature for a fixed amount of time the structure is minimized and then NEB is used to
find the saddle point between the initial structure and the new inherent structure Details of the
nudged elastic band method can be found elsewhere [34] Using these techniques it is possible to
map a sufficiently large landscape to make kinetic and thermodynamic measurements possible
These two methods benefit from the fact that they work well when the volume is varied and together
enable computationally inexpensive mapping of enthalpy landscapes [71] [92] (where the volume
is no longer fixed and instead the pressure is specified) However it is not recommended to generate
a large enthalpy landscape from MD in this software due to the computational cost involved
Although molecular dynamics is a powerful technique there is a large computational cost to
simulate a small amount of time
The third method is a new technique created in an attempt to circumvent the computational
cost of MD while still finding large structural changes We have named this technique ldquoeigen
treibanrdquo (ET) This technique relies on shoving along an eigenvector with a multiplier rather than
stepping After the system is shoved the system is minimized and if the minimized structure exhibits
a displacement in atomic position is larger than a threshold a NEB calculation is performed to find
the saddle point By shoving in the direction of an eigenvector the goal is to reproduce large changes
due to vibrational modes The fixed multiplier (m0) is calculated by taking a target distance (d) and
the largest displacement magnitude out of the atoms j (vij) in a given eigenvector direction i (ai)
m0
=d
max[nij] (24)
To convert to the displacement magnitude from an eigenvector in three dimensions we use the
expression
vij
= ai[3 j]2 +a
i[3 j +1]2 + a
i[3 j + 2]2 (25)
The dimension j has the length of the number of atoms in the systems
24
This software uses the system and potentials created from a LAMMPS input
scripts and as such has a wide range of access to different potentials and possible systems The
only input that has to be provided is the file which would be used for molecular dynamics in
LAMMPS If the script has been used for molecular dynamics simply by eliminating any
thermostats or barostats as well as any ldquorunrdquo commands it is possible to use it as an input script to
ExplorerPy (it is also recommended to eliminate any calls to ldquoThermordquo because it may interfere
with the extraction of variables from the underlying LAMMPS framework) If the LAMMPS input
script successfully sets up a simulation all other variables and parameters can be controlled from
command line options as described in the ldquoIllustrative Examplerdquo section The initial structure will
be taken from the input script as well
ExplorerPy is a software that has been written in Python with modular functions and ease
of use in mind It can be run from a Linux terminal with all major options being set via the command
line The software then generates a list of inherent structure energies stored in the mindat file and
stores the transition point energy for each pairwise inherent structure in the tsddat file All of the
transition points and minimum are stored in the same directory from which it is run as xyz files
The software architecture is shown in Figure 2 Details of the stepping procedure for eigenvector
following can be found elsewhere[93] The variables ldquoMD frequencyrdquo and ldquoET frequencyrdquo are set
as command line options at the time of the launch of the software
25
Figure 2 The schematic for the flow of the program Beginning in the top right corner and running
until the condition in the pink box is satisfied Yellow diamonds represent checks and blue
operations
The software is capable of mapping an energy landscape for an arbitrary system ie for
any system that can be simulated in LAMMPS This allows for accurate prediction of the evolution
of complex thermodynamics and kinetics The software requires no programming and is
controllable via command line options ExplorerPy prints all of the basins transition states inherent
volumes as well as information about the curvature of the system so that the vibrational frequency
is calculable This is all of the information required to calculate a complex range of phenomena
over a specified long timescale ExplorerPy is only set up to work within the LAMMPS metal units
To run the software there are only a few parameters that have to be set for each mode The
force threshold (-force) curvature threshold (-curve) and step size (-stepsize) when running the
command line option have to be set to allow for a stable exploration of a given landscape
These seem to be potential-specific and for a given potential appear to be fairly stable It is best to
start large for the force threshold and small for the eigen threshold and step size then modify as
26
necessary If eigenvector following is not being used there is no need to adjust these values Beyond
this there are several optional command line options (only the file (-file) command is required)
-file [file input] this is a standard lammps input if the input sets up a lammps run
it will work to set this up There should not be any lsquothermorsquo calls in the lammps
script
-press [specified pressure in atmospheres] if this is specified the system will
always be minimized to this pressure while if not specified the volume remains
fixed
-num [number of basins to explore]
-time [max time to run in hours] if this time is reached before the number of basins
are found the simulation will stop
-thresh [minimum energy for barrier to keep]
-center if this is given as a flag the system will be set to the anchor location each
step
-anchor [atom selection for the position to be held constant]
-md [frequency of molecular dynamics search] If a custom MD LAMMPS file is
not provided it defaults to starting all of the atoms at 3000 K and running for 50
fs
-shove [frequency of eigen treiban]
-custom [file used for MD exploration]
-help [List all commands possible in software with these descriptions]
An example run would then be done the following way
[mpirun -np 5 python37 Explorerpy -file inSiO2 -num 1000 -force 10 -curve 1e-5 -stepsize 012
-press 00 -md 11 -time 48]
This would start a run with five parallel processors searching for 1000 basins with an MD
step being ran every 11 searches with a maximum time of 48 hours The results of this command
are shown in Figure 3 Below we have attached an example of inSiO2 the output and additional
scripts related to the example has been uploaded to the associated GitHub
27
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above and then
plotted using PyConnect The plot is a disconnectivity graph where each terminating line
represents an inherent structure and tracing where two lines meet describes the activation barrier
The potentials are taken from the BKS potential [94]
httpsgithubcomlsmeetonpyconnect
28
22 RelaxPy
RelaxPy script is written using Python syntax and relies on the numpy scipy and
matplotlib libraries It solves the lack of a unified relaxation code in our community and is based
on the MAP model[44] [61] interpretation of multiple fictive temperatures and the shear
modulusviscosity form of the Maxwell equation[62] [95] It is designed to be run within a Linux
terminal with command line options or inside a Python interpreter It can be used by calling the
name of the script followed by the input file the desired output file name and finally an optional
tag for all fictive components to be displayed as well as printed to a specified output file (formatted
as comma separated values)
The RelaxPy package consists of an algorithm that iterates over time in order to determine
the values of viscosity (η) relaxation time (τ ) and current fictive temperature (Tf ) given the user-
supplied thermal history and material property values The purpose of this code is to enable
calculation of the relaxation behaviorrsquos change over time by using the MAP model using only
experimentally-attainable parameters Toolrsquos equation is solved for each term in a Prony series
approximation of relaxation finding the change in each Tfi component for a given time (t) to
accurately reproduce the stretched exponential form The program then iterates over the entire
thermal history of the sample using a user-specified time step dt allowing for calculation of the
overall fictive temperature over time The values for the Prony series fit (wi and Ki) are taken from
the database originally generated by Mauro and Mauro[96] and expnded to include any value of
the stretching exponent The values were fit by Mauro and Mauro with a hybrid fitting method
based on the number of terms in the desired Prony series and the magic value from Phillips The
thermal history is also defined in the original input file using linear interpolation The initial fictive
temperature components are assumed to be at equilibrium (Tfi = T(0))
29
To solve for the non-equilibrium viscosity used the MAP expression is used with the approximate
scaling of values proposed by Guo et al[61] The software is capable of performing the calculations
needed for a variety of thermal histories and processing methods desired by both industry and
academia with a focus on the fictive temperature This software is then fundamentally limited by
the issues associated with the MAP model and the compositional approximations made For
instance this software will need to be further modified to include structural relaxation when such
a model is proposed[62]
An example input code is listed below
Denotes Comments
Tg 794 0 m 368 C 1494 dt 10 Beta 37
Start Temp End Temp Time [s]
1000 10 1300
10 100 1000
100 30 500
Assumes Linear Interpolation
This example input states that over the space of 1300 seconds an initial glass sample is cooled from
1000 C to 10 C Then it is heated to 100C over 1000 seconds and then finally down to room
temperature in the space of 500 seconds All temperatures are given in Celsius
This software aims to provide the first open-source software for modeling of glass
relaxation behavior using the MAP model of viscosity by creating a package that can easily and
quickly be used to approximate the fictive temperature components of a glass as well as the
composite fictive temperature RelaxPy allows for researchers to calculate the evolution of
macroscopic properties of glass during a relaxation process (assuming a fixed stretching exponent)
and will facilitate the calculation based on a user-supplied thermal history RelaxPy can be
seamlessly adopted by most glass research groups and removes the guessing associated with
determining a fictive temperature allowing for a fully quantitative comparable discussion of Tf
simultaneously creating a tool that can easily be expanded for the entire community as new physics
is discovered
30
Chapter 3
Understanding Nucleation in Liquids
Crystallization cannot be ignored in any material system at temperatures below its liquidus
temperature although extreme resistance to crystallization is reported in some organic and
inorganic substances[97] Crystallization (and by extension nucleation) is critical in systems
ranging from ice to thin films[98]ndash[100] Glasses are non-crystalline substances therefore
crystallization kinetics will determine the necessary thermal history to avoid crystallization and
obtain a glassy state Despite the absolute importance of crystallization to the study and design of
glasses there is no current method to predict the nucleation of crystals in this event One of the main
challenges in the production of glass-ceramics is finding the optimal thermal history for the desired
crystallite distributions There are two phenomenological steps in the process of phase
transformation from a supercooled liquid to a crystal a nucleation step and a growth step The
growth step is described well by Wilson-Frenkelacutes theory[1] but the underlying physics of
nucleation remains elusive A stable nucleus is the precursor to a crystal comprising a nano-sized
periodic assemblage of atoms A clear understanding of the relationship between liquid kinetics
and crystal nucleation has not been established despite decades of research
The study of nucleation is particularly critical for glass-ceramics Since their discovery by
S Donald Stookey glass-ceramics have become prevalent in our society through products such as
stovetops and Corningware[73] [101] [102] Glass-ceramics are a composite material comprising
at least one crystalline phase embedded in a parent glassy phase benefitting from the properties of
both the glassy and crystalline phases[1] [103] One of the reasons glass-ceramics are ubiquitous
is their relatively easy production glass-ceramics can be formed through standard glass forming
procedures using additional heat treatments Additionally there is a wide range of accessible
31
properties through tuning both the chemical composition of the parent glass and the thermal history
and hence the microstructure of the heat-treated glass-ceramics[101] [103]ndash[106]
Glass-ceramic production consists of first synthesizing a glass and then generating a
microstructure through the two-stage ceramming process[73] [103] [107] The nucleation step
remains poorly understood with classical nucleation theory (CNT) giving widely varying results
depending on how the parameters are determined[1] [73] [108] The inability to consistently get
accurate predictions for a nucleation curve means that choosing an appropriate nucleation
temperature must be done experimentally which can prove to be one of the most challenging and
time-consuming aspects of properly designing a glass-ceramic The difficulty is due to the
magnitudes of the crystal nucleation rates not being predictable with CNT or any other current
model
This difficulty in designing a material is apparent when considering the high dimensional
phase space of parameters that can adjusted We haver presesnted some tools developed to help
optimize over this lsquoglassyrsquo space such as RelaxPy[42] KineticPy[41] topological constraint
theory[109] or various machine learning models[4] [70] [110] [111] However a glass-ceramic
system has at least four additional phase dimensions (two for the nucleationgrowth temperatures
and two for nucleationgrowth times) that need to be optimized (depending on the parameterization
of the experiment it is also possible to add another four dimensions relating the heating and cooling
between each step) Therefore a glass-ceramic system has a minimum of d+5 dimensions over
which the system must be optimized and yet those four additional dimensions are the least
understood Designing methods to easily optimize over this large space with feasible computational
methods is of utmost importance to reaching a new generation of custom materials
Experimental studies of nucleation have used techniques ranging from differential
scanning calorimetry (DSC) to electron microscopy to try to capture predictive capability in the
lsquoceramicrsquo phase-space (the four additional dimensions) These studies have successfully captured
32
the shape of the nucleation curve but fail to provide the information needed to calculate the
temperature-dependence of the nucleation rate The shape of the curve can be determined through
DSC and with great effort an approximation of the rate can be made however this technique is too
laborusome to be efficient[112]ndash[114] Since the parameters have been inaccessible to traditional
computational approaches and experimental determination all previous studies of nucleation have
required at least one variable to be fit If a nucleation curve could be predicted only using a
computational method or with an efficient simple DSC method the rate at which new glass-
ceramics products are created would increase In addition the cost of producing glass-ceramics
would likely decrease and more complex microstructures and therefore more unique properties
could be explored
Crystal nucleation has been historically difficult to simulate and measure experimentally
because the phenomena happen on spatial scales of nanometers but over many orders of magnitude
of time scales CNT is the most widely applied model despite its supposed failure To test CNTrsquos
assumptions an interest in computational work has grown Recent approaches using grand
canonical Monte Carlo have successfully predicted the short time dynamics of nuclei
formation[73] This approach involves use of a continuous solvation model for the liquid and has
enabled greater computational insights however it remains computationally expensive[74]
Molecular dynamics (MD) fails to capture nucleation in glass-forming systems due to the need for
longer spatial and temporal scales to capture nucleation effects accurately Some attempts have
been made using Monte Carlo MD or empirical approaches to replace classical nucleation theory
(CNT) with some alternate method where the parameters can be directly measured[73] [115]ndash
[117] All nucleation models and Monte Carlo models attempt to describe the rate that a
supercooled liquid system transitions to a crystalline structure as a function of temperature
Explicitly mapping the energy landscape would fundamentally explain the underlying
physics of nucleation An energy landscape of a system is given by
33
1 1 1 2 2 2( )N NNU U x y z x y z x y z= (26)
where U is the potential energy and x y and z are the locations of each of the N atoms in a periodic
box The locations of these atoms give rise to an energy as parameterized by an interatomic
potential An energy landscape can be constructed by mapping the relationship of the local minima
and the first order saddle points between pairs of minima22-27 Energy landscapes have become
crucial to studies of biomolecules glasses and catalysts[21] [26] [44] because landscapes show
atomic configurational changes and calculate the associated kinetics and thermodynamics In this
study we will explore the energy landscape of a barium disilicate system capturing the glassy and
crystalline basins enabling a test of CNT by calculating all needed parameters (interfacial free
energy kinetic barrier and free energy difference of the supercooled liquid and the equilibrium
state)
31 Crystallization Methods
CNT expresses the steady-state nucleation rate (I) as the product of a kinetic and a
thermodynamic factor
( )exp eI Z D TW
kT
minus =
(27)
where Ze is the Zeldovich factor which incorporates the probability of a critical sized nucleus that
will form a new phase independent of other nuclei D(T) is the kinetic factor often given by an
Arrhenius expression over sufficiently small temperature ranges W is the work associated with
generating a thermodynamically stable nucleus that will continue to grow rather than dissipate (ie
the critical work) if temperature and pressure are kept constant k is Boltzmannrsquos constant and T is
absolute temperature The Zeldovich factor is typically treated as a constant with a value ~01[118]
34
Here the kinetic term is expressed as D(T) since in this work the kinetics are explicitly calculated
at every temperature and it is not treated as a simple Arrhenius expression In CNT the work W is
given by the work to generate a spherical isotropic nucleus[1] which consists of a volume term
and a surface energy term It is given by
3 24
( ) 4 ( )3
W r G T r T = + (28)
where G is the Gibbs free energy difference between the crystalline state and the corresponding
liquid r is the radius and is the interfacial energy between the crystalline and liquid phases Eq
(28) neglects the strain energy term that arises from the formation of a solid nucleus in the
supercooled liquid It is assumed that the liquid relaxes fast enough so that strain energy is not
significant for nucleation in liquids[11]
Determining the critical radius (ie taking the derivative of Eq (28) with respect to the
radius and setting the expression equal to 0) allows one to solve for the critical work and the
expression becomes
3
2
16
)3(
( )
( )W
T
G T
=
(29)
Three temperature dependent functions remain D G and Though there are existing methods
in literature to approximate each of these quantities the approximations often include large errors
and assumptions about the behavior of the system[119] Often experimentally the kinetic function
is approximated with the Stokes-Einstein relationship the G term is calculated from heat
capacity curves (assuming that the macroscopic energy is similar to the nanoscopic energy) and
is fitted[119] In the calculation presented here only configurational energies are considered
while vibrational contributions are ignored Ignoring the vibrational component and the reliance on
empirical potentials are the most likely sources of error in this work
35
The energy landscape for this investigation was generated using ExplorerPy[89] starting
with the orthorhombic barium disilicate crystalline structure[89] Barium disilicate was chosen
since it is a commonly studied glass-ceramic with multiple reliable experimental datasets being
found in literature[75] [113] [120] The rest of this section is divided into four parts starting with
a discussion of mapping the landscape and the following three sections are a discussion of
calculating D G and respectively
31A Mapping and Classifying the Landscape
The ExplorerPy energy landscape exploration employed the Eigen-Trieban technique in
the canonical ensemble with systems consisting of 128 256 and 512 atoms[89] The number of
atoms was chosen to achieve convergence of the results while also balancing the large
computational cost associated with energy landscape exploration A smaller system was also
calculated with 64 atoms however this systems was too small to accurately discern energies
enough to make the nucleation calculation The Eigen-Trieban technique is a newly developed
method to explore energy landscapes using a method of shoving along the eigenvectors and
minimizing the potential energy which is followed by using nudged elastic band calculations to
find the associated transition points[89] This technique allows for larger transitions to be found
while retaining the ability to calculate the vibrational frequencies The Pedone et al potentials[36]
were used and the volume of the cell was fixed due to simulation instability when pressure was
fixed The Pedone et al potentials were chosen since they have reproduced key parameters for both
the crystalline and glassy state in previous Monte Carlo studies as well as in the training set of the
potential[73] [74] [121] Though the Eigen-Trieban method may work well for predicting
crystallization and finding large barriers it may be insufficient to describe glass relaxation or
36
viscosity since it rarely finds the small structural transitions that dominate the Adam-Gibbs
relationship Note that these exceptions are inconsequential for predictions of nucleation found in
the work presented here33
The barium disilicate system was explored for 2500 basins for 128 and 256 atoms and 4000
for 512 atoms since more basins were needed to converge the liquid state distribution of energies
This number of basins was shown to be sufficient since multiple paths from the initial crystalline
starting point to SCL states were obtained The distribution of energies of the SCL were found to
converge and the resulting energy landscape is seen in Figure 4
A crystal parameter was then defined from the root mean square displacement (RMSD) for
each atom from the nearest lattice site of the same species ( ) This average value is used to
characterize whether a basin is a crystal or SCL The RMSD approach is not the only method to
assign a crystallinity value[122] however it is one of the simplest and as such was chosen for
this application To then categorize each basin as either a crystalline basin or SCL basin a plot of
the minimum energy of the SCL states versus the cutoff of RMSD was produced This plot for 256
atoms shows a clear transition in the energies around a cutoff value of 1 Aring which is therefore
taken as the cutoff radius of which basins are considered crystals The same analysis was repeated
for each system size This analysis is shown in Figure 4 Any basin with a RMSD less than the
cutoff value is considered a crystalline state This cutoff was found to be generally insensitive to
the results with any values ranging between 09 Aring and 11 Aring changing the resulting prediction of
nucleation by less than one order of magnitude but shifting the peak temperature by up to 40 K
This cutoff also reproduced the energy difference between the SCL and crystal in molecular
dynamics simulations of the materials
37
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-axis being
an arbitrary phase space and the y-axis being the potential energy calculated from the Pedone et al
potentials[36] The colors are indicative of the crystallinity where the blue basin is the initial
starting configuration The landscape shows the lowering of energy associated with partially
crystallizing the sample (B) The energy landscape relationship between the cutoff for crystalline
and super-cooled liquid states for 256 atoms is shown This shows a clear drastic energy change
occurring around the cutoff value of 10 Aring
[
38
31B Kinetic Term for CNT
The transition rate was calculated by the inverse of the average relaxation time from the
SCL to the crystalline states The relaxation time ( ) is defined by the relaxation function ( )
0 exp
t
= minus
(30)
where t is time and 0 is the scaling of the relaxation function The reason we are able to
calculate the kinetics of nucleation but not viscosity is because the nucleation kinetics are
dominated by the transitions between two states that we specifically searched for when exploring
the energy landscapes while the viscosity is defined by the entropy dominating small transitions
described by the AG
The AG relationship shows that the viscosity is dominated by the configurational entropy
of the system Experimentally the kinetic term is often calculated with the Stokes-Einstein
relationship
( )6
kTD
aT
= (31)
where a is the radius of the diffusing species and is the viscosity of the liquid Interestingly this
approximation gives reasonable results when investigating experimental data since the viscosity is
a measurement of the entropy of small cooperative liquid rearrangements Conversely the kinetics
that govern nucleation are large structural rearrangements from a supercooled liquid to a crystal It
is nonetheless possible that these cooperative rearrangements and the large diffusion processes for
nucleation are on the same order of magnitude which is why the Stokes-Einstein relationship gives
39
a reasonable approximation in experimental parameterizations When the Stokes-Einstein
relationship is used there is an implicit assumption that the large nucleating rearrangements are
proportional to the small barriers that govern viscosity[53]
A mean relaxation time can be found by averaging over a system of master equations
( )j
ii ji j j ij i
i
dpg K p g K p
dt
= minus (32)
In this expression g is the degeneracy of a microstate pi is the probability of occupying the
microstate and Kij is a transition rate between inherent structure i and inherent structure j The
equilibrium probability of occupying a microstate can be given by a Boltzmann probability (where
Z is the partition function and U is the potential energy of the inherent structure)
1
exp ii i
Up g
Z kT
= minus
(33)
By substituting the value of jp in Eq (32) for a two-state model ( 1 21 i jp p= == + ) we write
( )( )1ii ji i j ij i
dpg K p g K p
dt= minus minus (34)
Isolating the probabilities and splitting the differential we have
( )1
i
i ji i j ij i
dpdt
g K p g K p=
minus minus (35)
which when solved via integration takes the form
ln ( )
( )
i ji i ji j ij i
i ji j ij
g K g K g K pt
g K g K
minus + =
minus + (36)
Solving for the occupational probability yields
( )expi ji j ij i ji
i
i ji j ij
g K g K g K tp
g K g K
minus minus + =
+ (37)
This shows an analytical solution for the relaxation time in a two-state model
40
1
i ji j ijKg g K
+
= (38)
The two-state model approximation is made to enable the calculation of the mean relaxation time
between the liquid basins to the crystalline basins The transition rates between the basins are given
by the vibrational frequency ( ) and the barrier between the states ( ijU )
exp ij
ij
UK
kT
minus=
(39)
The probability of transitioning from a given liquid basin i to a crystalline basin j is given by
1
expij
ij
Up
Y kT
minus =
(40)
where Y is the partition function with respect to all transitions from the current SCL basin to all
crystalline basins as only these transitions are considered to calculate the relaxation time
expj
j
iY
k
U
T
= minus
(41)
The total expected relaxation time of the system making the transition is then given by the
probability of occupying a basin the probability of transitioning from one basin to another specific
basin and the associated relaxation time
1
i ij
j i ji j ji i i
p pg K g K
=
+ (42)
The first summation is over the liquid states and the second summation is performed just over
crystalline states to capture the mean relaxation time from a supercooled liquid to a crystal We
fixed the vibrational frequency to a value of 17 times 1014 Hz (ie about one jump attempt per 5
femtoseconds) which was calculated using the average of the eigenvalues of each basin ( ) and
the average mass of the atoms (m) using
41
1
2 m
= (43)
31C Degeneracy calculations
To obtain an appropriate value for the degeneracy an understanding the phase-space
volume of the basin is needed since the degeneracy is linked to the volume of the basin[40] By
taking the displacement between the basin coordinates of each atom and the transition point we can
define a radius of the basin R Using R and a spherical approximation we can calculate the phase-
space volume degeneracies
3 23
12
NN
i i
cry
Ng R
g
+
=
(44)
In which ( )x is Eulerrsquos gamma function The degeneracy is then normalized by a constant such
that the liquid-liquid state transitions had a value of 100 s at the glass transition (980 K) to be in
agreement with Angell[58] The normalization step allows for a quantitative prediction of the
liquidus temperature to confirm the accuracy of our model The liquidus value found is
approximately 1200 K 1715 K and 1950 K for 128 256 and 512 atoms respectively The
experimentally determined liquidus temperatures range from 1680 K to 1710 K[113] [115] [120]
It is worth noting that changing the exponent drastically changes the thermodynamics while having
minor effects on the kinetics and the normalization value drastically changes the kinetics while
keeping the thermodynamics nearly constant We chose the denominator to satisfy the liquid
relaxation constraint It is also feasible to choose the exponent to satisfy a constraint on the liquidus
temperature The normalization value is an important to consider because for larger system a
change in 001 can result in a change in the kinetics by ~10 orders of magnitude
42
31D Free Energy Difference
Due to limitations in the exploration of the energy landscape requiring a fixed volume we
will approximate the difference in the Gibbs free energy by calculating the difference of the
Helmholtz free energy This approximation is reasonable due to the similar densities between the
crystalline and glassy phases under ambient pressure[73] [74] The first step is determining the
mean energy difference of all the basins and the second step was repeating the process for only the
liquid basins The mean energy was calculated with the following expression[123]
( )ln ln ii
i
i i iG kTp p U kT g p= + minus (45)
In Eq (45) i is the set of relevant basins for the conditions we are interested in The difference
between the mean free energy of the liquid basins and the entire landscape basin energies is the
G parameter This free energy difference is typically experimentally calculated by integrating
heat capacities however numerous approximations exist These approximations are often linear
approximations normalized to the liquidus temperature In this work we find that the driving force
appears to be predominantly linear which is in good agreement with the approximations that are
often used for the driving force[124]
43
Figure 5 An example interfacial structure between the crystalline phase on the left and the last
sequential SCLglass phase on the right for a barium disilicate system The gray atoms are barium
silicon is shown in red and blue represents oxygen
44
31E Interfacial Energy
To calculate the interfacial energy of the interface between the crystalline phase and the
SCL the initial basin (pure crystalline) and the last basin (glass basin) were annealed at 1700 K for
1 ns and then cooled to a temperature of 800 K over a period of 10 ns The same process was
repeated for a composite where the initial stable crystal and the glass were placed next to each
otherThe structure was multiplied 10 times to create a wall of a large interface to reduce energetic
fluctuations A small section of the composite structure is shown in Figure 5 The difference
between the addition of the energy of the crystalline phase and glassy phase and the composite
gives the energy of the interface which was normalized to the size of the surface area Multiple
arrangements of the composite were tested but anisotropy was found to be minimal with the
fluctuations of the energy being larger than the differences from different tested directions The
final expression used for the interfacial free energy (in eVAngstrom2) is
610 00 35047 040T minus +=
45
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a function of
temperature for each system size The experimental values for kinetics and thermodynamics come
from ref[125] and from the heat capacity data taken from [126] respectively It is clear to see that
driving force shows good agreement across all systems however the kinetics terms only converge
46
for the 256 and 512 atoms systems (Bottom) The fit used to calculate the interfacial energy as a
function of temperature
32 Results amp Discussion
The results for the free energy difference the kinetic parameter and the surface energy are
shown in Figure 6 These parameters as a function of temperature are used to calculate the
nucleation rates in the expression
3
2( )
01 16exp
3( )I
TT G k
= minus
(46)
The 01 being an approximate value of the Zeldovich factor Using the parameters shown in Figure
6 and solving Eq (46) the nucleation rates are shown in Figure 7 along with experimental data for
validation At first glance we notice that there is about 7 order of magnitude differences between
data and prediction Considering that theses potentials were trained on glass and were not intended
for this purpose we consider this as excellent agreement This method also shows agreement below
the peak but the accuracy falls off more quickly below the peak temperature It is likely that the
data below the peak is severely underestimated due to a failure to reach steady state nucleation
rates Reliable steady state nucleation rates are not available for this system and as such a direct
comparison below the peak temperature is not directly available It is worth mentioning that the
failure of CNT of predicting the order of magnitude of nucleation rates is well reported and can
amount to dozens of orders of magnitude when a constant value of is used[75] [113] [115]
[119] [127] thus the proposed method has an advantage when comparing with the current
classical nucleation theory toolset Moreover the prediction of Tmax (~950 K) is also very close to
the experimental value (985 K)
47
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted using the
model presented in this work The data referenced can be found in refs [113] [128] [129]
48
To further compare CNT and the model presented here we decided to compare the results
of the fitted and this workrsquos calculated surface energies as seen in Figure 8 The results show good
agreement all achieving the same order of magnitude implying that experimental methods can
achieve a close approximation to the values required for CNT The prediction of nucleation is
extremely sensitive to the value of the surface energy with small changes leading to orders of
magnitude change It stands to reason then that further work that needs to be approached through
computation and experiments is developing a rigorous model to predict the interfacial free energy
The other assumption that is used experimentally is the Stokes-Einstein relationship To
compare the landscape results of the transition rate from the SCL to crystalline states and the
Stokes-Einstein relationship the liquid state relaxation was calculated The small barriers that
govern the Adam-Gibbs relationship were not found with this exploration method hence
quantitative comparisons are not possible (and relaxation calculations should not be made)
however the relaxation times can be qualitatively compared The liquid state relaxation rate is
proportional to the viscosity and is given by the expression[44]
j
i
i
l
ij
p
K =
(47)
To understand the validity of the Stokes-Einstein relationship we can write a series of theoretically
true proportionalities
( )l
T TD T
(48)
The values of the experimental kinetics fail to align with the kinetics calculated in this work This
is expected due to the fact that the viscosityliquid relaxation is dominated by the set of small
transitions in the landscape while the nucleation transition is defined by the transition of one region
to another with a substantially higher activation barrier This comparison may also account for
49
some part of the experimental overestimation below the peak with the kinetic term calculated
dropping off at a faster rate than that calculated from the experimental viscosity Figure 6 also
shows evidence that it may be inappropriate to use the Stokes-Einstein relationship for nucleation
and as such we may need to re-visit the experimental models that often rely on the assumption of
the Stokes-Einstein relationship This observation casts some doubt in the legitimacy of describing
the kinetic factor as a diffusivity which is standard for CNT For more information on this CNT
failure please see Refs [75] [113] [115] [119] [125] [127]
Nucleation is a complicated concept to investigate and model Unlike the crystalline phase
where the atomic positions are clearly defined the details of the atomic coordinates and bond
configurations of the liquid sites must be described in terms of statistical distributions Due to this
added complication mean-field distributions are typically used to describe non-crystalline
sites[14] [16] [108] [130]ndash[133] The incorporation of fluctuations could more accurately predict
each liquid sitersquos transition to the crystalline phase Fluctuations in the degeneracy could be one
reason that the nucleation rate is over predicted when the temperature is less than the peak
nucleation temperature A given G varies with the environment and depends on the specific
liquid state not the mean value Therefore the rate of nucleation from all supercooled liquid states
to the crystal state is not constant which is why fluctuations need to be considered and why energy
landscapes are such a powerful technique This is especially highlighted when considering that
different degeneracies yield drastically different heat capacity Energy landscapes are extremely
useful because they consider a large distribution in free energies and show all structural fluctuations
possible Fluctuations in cluster size and degeneracy influence Adam-Gibbs configurational
entropy which directly impact the transition rate between the crystal and liquid state
50
Figure 8 The surface energy with respect to temperature for the work presented here
0
0002
0004
0006
0008
001
0012
0014
800 850 900 950 1000 1050 1100
Su
rface
En
ergy (
eVAring
2)
Temperature [K]
This Work
Xia et al (2017) Ref 23
Fokin et al (2016) Ref 41
51
33 Conclusions
Though this is not the first time CNT has been examined computationally in an attempt to
understand if the theory is valid[74] [116] [117] it is the first parameterization of each variable
independently in an attempt to calculate the resulting nucleation rate Not only are all the variables
calculated when used in the expression for CNT the liquidus temperature is reproduced and the
nucleation rate is reasonably predicted This is as far as the authors know also the first time a real
energy landscape for a complicated glass-ceramic system is being reported This method leveraging
energy landscapes alongside CNT enables for a reasonably fast calculation of the nucleation curve
of a glass forming liquid In future work this model could be used for either developing commercial
glass-ceramics or for predicting the critical rate needed to maintain a glass This model can also
provide additional insights into the physics of nucleation and the validity of CNT without the
constraints of fitting parameters This method fundamentally has the power to connect the 3N
dimensions of the energy landscape to the compositional dimensions and thus allowing for a unique
powerful method of predicting nucleation parameters as a result simplifying the crystalline
dimensions in which a glassglass-ceramic must be optimized over
Chapter 4
Expanding the Current State of Relaxation
Before we consider a new approach to relaxation (optimizing our understanding of
the 3N dimensions) there are first missing pieces in the current state specifically the temperature
and compositional dependence of the stretching exponent However even before expanding the
current state of the art relaxation methods we also need to understand ergodicity In this section we
will start with a simple though experiment to understand the temporal effects of relaxation and then
in the second section combine it with fictive temperature to create a new model for the stretched
exponent
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition
The prophet Deborah sang in the book of Judges ldquoThe mountains flowed before the Lordrdquo
[6] [13] However the question the prophet failed to ask was what happens if the mountains are
moving at relativistic speeds The focus of this section is to highlight the temporal effects on
ergodicity in the context of special relativity and the reference frame of the observer In order to
understand relativistic effects on materials one must consider a well-characterized material that
actively relaxes but appears solid on our timescale viz pure B2O3 glass [134] [135] Glass is a
particularly good candidate to study the effects of time dilation on a material since it undergoes a
kinetic transition known as the glass transition (Tg) which has multiple definitions that will be
considered
53
The first definition is in terms of the aptly named ldquoDeborah Numberrdquo which was proposed
by Reiner [136] to offer a view of fluidity and equilibrium mechanics It was later expanded to
explain the origin of the glass transition and the transition from an ergodic liquid system to a non-
ergodic solid-like glassy state[12] which is the fundamental definition of the glass transition [11]
[12] [137] It is important to note here that the breakdown of ergodicity is relative to the
observation timescale and is reflected in the Deborah number (D) which is defined as
Dt
= (49)
where t is the (external) observation time and is the relaxation time of the material which can be
expressed through the Maxwell relation [42] By definition the Deborah Number equals 1 at the
glass transition temperature Tg [13] [137]
Another definition of the glass transition temperature is due to Angell who defined Tg as
the temperature where the viscosity of the supercooled liquid is 1012 Pa s [58] [138] The Angell
diagram is an important plot relevant to viscosity and the glass transition where the abscissa is TgT
and the ordinate axis is log10( ) In the Angell diagram the value of the viscosity at Tg and in the
limit of infinite temperature are fixed and the difference in the temperature scaling of the TgT-
normalized viscosity is the slope of the log10( ) curve at TgT = 1 which Angell defined as the
fragility (m) of the supercooled liquid Using this definition one can establish the consistency of
the Angell plot at relativistic speeds An example of the Angell plot is shown in Figure 9
54
Figure 9 An Angell diagram created using the MYEGA expression [55] With a very strong glass
(m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) [17] [134] [135] The
infinite temperature limit is from the work of Zheng et al[54] and the glass transition temperature
is from the Angell definition
55
Using the two previous definitions it is expressed
1gT
GtD
= = (50)
which can be re-written as
(
(
)
)g
g
tG T
T= (51)
In order to solve this equation one must know the shear modulus at the glass transition temperature
recent improvements in topological constraint theory have led to the ability to predict properties
with varying temperature and composition Using a recent topological constraint model for elastic
moduli proposed by Wilkinson et al [139] one can predict the temperature dependence of the
modulus
( ) ( ) ( ) ( ) ( )( ) ( )Δ Δ Δ Ac c C Ac
c c
F n q T F n q T F n q T x Nn q T x NdG dGG
d F M d F M
+ + = =
(52)
In Eq (5) nc is the number of topological constraints (c) qc is the constraint onset function as
described by Mauro et al [69] Fc is the free energy of the constraint is the density of the
system M is the molar mass and 119889119866
119889120549119865119888 is a scaling factor This expression was evaluated as a
function of temperature with the total modulus coming from Kodama et al [135] and is equal to 8
GPa at room temperature The B2O3 structure consists entirely of three-coordinated boron [140]
[141]
Using the modulus predicted in Eq (52) numerical observation time is expressed
56
12
9
10 Pa s141 s
709 10 Pat = =
(53)
Using this description of the static (vc=0 where v is the speed of the glass and c is the
speed of light) glass transition it is possible now to describe the relativistic glass transition for
which there are considered two separate cases It is worth noting that this is not the first case study
of relativistic viscosity a fairly large basis of literature dealing with ideal fluids near large
gravitational bodies [142] There is also some work also relating the relaxation time in a
gravitational field to the nonequilibrium entropy relaxation time [143] [144] This work
emphasizes the theoretical effects of special relativity on real liquids their glass transitions and
the properties there dependent on
41A Relativistic Liquid
The first case is when a sample of a glass-forming system moves past a stationary observer
at a speed approaching light giving the new observation time as
0tt
= (54)
where t0 = 141 s (as previously calculated) and g is the Lorentz factor
2
2
1
1v
c
=
minus
(55)
The Maxwell relation can be modified to account for relativistic effects by combining with Eq (54)
0
()
)(
g
gG TT
t
= (56)
57
This gives a condition that is temperature dependent to predict the new glass transition as a function
of v however in order to accurately describe the relativistic behavior density must be considered
Density will change by a factor of 2 because both the mass and the volume will be affected and
as such
0) (( )g gG TT t = (57)
The prediction then for the glass transition as a function of the speed of light is shown in Figure
10
Figure 10 The relativistic glass transition temperature for B2O3 glass
58
Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature
which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) expression [55] [145] The MYEGA equation needs 2 other parameters
besides Tg to generate the viscosity curve fragility (m) which is the slope at Tg in the Angell plot
and the infinite temperature viscosity () The fragility will not change because one assumption
in this model is the consistency of the Angell plot at all relativistic velocities even relativistic The
infinite temperature viscosity which also will not change because this is the minimum possible
viscosity for a liquid The new viscosity curves for various speeds are shown in Figure 11
59
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different fractions of
light speed All of the viscosities approaching the universal temperature limit for viscosity
60
By using the Deborah Number it has been shown that at relativistic speeds will change the viscosity
in order to maintain the Deborah number and in this case it is shown that when the Lord would
move past the mountains at relativistic speeds the mountains would not actually flow but instead
hold still (even more so than they currently do) From this it is inferred that any liquid moving past
the earth at relativistic speeds will appear to be more solid and eventually appear as a glass
41B Relativistic Observer
In this case we consider an observer moving past a sample of glass at relativistic speeds
giving
0t t = (58)
which when the relativistic shear modulus 2 factor is included can be written as
( )
( )0
g
g
TG T
t
= (59)
Solving in the same manner as before it is shown which shear modulus is needed to satisfy the
condition in Figure 12
61
Figure 12 The modulus needed to satisfy the condition for the glass transition
62
In Figure 12 the shear modulus at the glass transition temperature is shown however the
constraint theory mechanism being implemented to calculate the shear modulus has a built in upper
limit when calculating the Tg past the point in which all the constraints are intact Due to this
limitation there is an upper limit to our prediction Nonetheless the calculated Tg over the available
range are shown in Figure 13 as well as the related viscosities It is seen clearly that at higher
velocities the glass transition shifts towards zero more dramatically at the higher temperatures
This leads to the interesting result that if we take the limit of an observer moving past the earth at
close to the speed of light all glass would appear to be a liquid
63
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at v=044c
(Bottom) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches
the speed of light
64
Applying the mechanisms proposed by Einstein to the concept of the Deborah number it
has been shown that the viscosities will appear to change dramatically as a glass-forming system
approaches relativistic speeds This allows us to build an intuitive understanding of the differences
between glass and liquids Specifically any material can appear either more or less fluid based on
the velocity with the velocity with which they travel relative to the observer This serves to highlight
the importance of the role of the observer on the glass transition and why even small changes in
DSC scan rates or sampling frequency can lead to drastically different measured glass transitions
42 Temperature and Compositional Dependence of the Stretching Exponent
The mathematical form for glass relaxation ( ) was originally proposed by Kohlrausch in
1854 based on the decay of charge in a Leyden jar[9] [14] [29] [42] [45] [96] [146]
( ) ( )( )exp t t
= minus (60)
where t is time t is the relaxation time of the system and b is the dimensionless stretching exponent
Eq(1) was originally proposed empirically and is known as the stretched exponential relaxation
(SER) function[42] [96] The relaxation time t depends on the composition temperature thermal
history pressure and pressure history of the glass as well as the property being measured[62]
[147] For example the stress relaxation time of a glass can be written as[62]
( )( )
ff
s
f f
T T P P
G T T P P
= (61)
where h is the shear viscosity and G is the shear modulus of the glass-forming system[42]
(Variables are defined in Table 2) In addition to using Eq(61) to describe stress relaxation time
structural relaxation time has recently been hypothesized follow Eq(61) with bulk viscosity and
65
the difference of bulk modulus between infinite and 0 frequency replacing shear viscosity and shear
modulus[62] Both h and G vary with the temperature (T) and thermal history of the glass (as
quantified via the fictive temperature Tf) as well as the pressure (P) and its corresponding history
(fictive pressure Pf)[16]
To connect the stretching exponent to its physical origins we can begin with a result from
Richert and Richert[148] that relates to an underlying structural relaxation time distribution and
then develop equations that determine from known quantities Their expression
2 22
ln 2
1
6
minus=
(62)
relates to the variance of the logarithm ( )2
ln of the structural relaxation time For reference
all variables are defined in Table 2 The physical origins of the relaxation time are related to the
configurational entropy (Sc) as shown by the Adam-Gibbs equation[53] [55] [149]
ln lnc
B
TS = + (63)
where is the infinite temperature relaxation time T is the absolute temperature and B represents
the energy barrier for relaxation
Table 2 Variable definitions
Variable Definition
t ts Structural relaxation time stress relaxation time
b Kohlrausch exponent (ie the stretching exponent)
T Tf Tfi Absolute temperature fictive temp partial fictive temp
P Pf Pressure fictive pressure
66
kB Boltzmannrsquos constant
wi ki i=1 to N Prony series parameters
B Adam-Gibbs relaxation barrier
Sc Adam-Gibbs configurational entropy
B Distribution of activation barriers
2 Variance
Na Number of atoms
f Topological degrees of freedom per atom
Degenerate configurations per floppy mode
H Enthalpy barrier for Relaxation
d Number of dimensions
f Fraction of activated relaxation pathways
S Adam-Gibbs entropy in the infinite-temperature limit
m Kinetic fragility index
m0 The limit of a strong liquid
Grouped unknowns 6
ln
B
a BN k
A Proportionality between fragility and distribution of barriers
x A given composition
125640
S
A
Intercept of the linear model of
67
42A Deriving a Model
Combining Eqs(62) and (63) Gupta and Mauro[150] proposed that the variance of the energy
barriers for relaxation (2
B ) could be rewritten as
2 2
22
2
1
6( )B c TTS
minus=
(64)
Solving for the stretching exponent
( )22 (
6
( )
)
c
B c
S T
T
T
TS
= +
(65)
The work of Naumis et al[23] [56] has shown that the configurational entropy is proportional to
the topological degrees of freedom in the network a result that was used in the derivation of
temperature-dependent constraint theory[57] and the MYEGA (Mauro-Yue-Ellison-Gupta-Allan)
equation for the relaxation of supercooled liquids[55] The MYEGA equation was derived by
expressing the configurational entropy as
( ) ( ) lna BcS T x f T x N k= (66)
In Eq (66) f is the topological degrees of freedom per atom Na is the number of atoms kB is
Boltzmannrsquos constant x is a given composition and is the number of degenerate configurations
per floppy mode The temperature dependence of the topological degrees of freedom was
approximated using a simple two-state model
( )( ) 3exp
B
H xf x T df
k T
minus= =
(67)
Here H(x) is the enthalpy barrier for relaxation d is the dimensionality of the system and f is the
68
fraction of activated relaxation pathways Combining Eqs (65) and (67) as well as condensing the
unknowns into the term defined by
6
ln
B
a BN k
=
(68)
we get
( )
22
df
dfT
=
+
(69)
This can be compared to the prediction made by Phillips[48] for the stretching exponent at
temperatures below the glass transition
2
df
df =
+ (70)
Comparing the two expressions of Eq(69) and Eq(70) they would agree if
4 4dfT
= + (71)
We will show later (Figure 12) an example where extrapolating our model prediction for to room
temperature (with T in Eq(71) replaced by fictive temperature Tf) gives a result close to the Phillips
value of Eq(70) When the temperature T in Eq(69) is high (much larger than ) then Eq(69) gives
us rarr1 Thus the model can interpolate between the low-temperature (Phillips) value and the
high-temperature value of 1 In our current model we do not have access to a value for in Eq(68)
so this model cannot assess whether the Phillips room temperature value for is universal ie
whether Eq(71) is always satisfied at room temperature
Zheng et al[151] showed that the Mauro-Allan-Potuzak (MAP) model for the relaxation
time of the nonequilibrium glassy state[44] implies that the configurational entropy can be written
as a function of thermal history (Tf) the fragility index (m) the fragility limit of a strong liquid
69
(m0) the glass transition temperature (Tg) and the limit of infinite temperature configurational
entropy ( S ) as
0
) exp 1 0
(ln1
c f
f
gT
TS m
TS
m
= minus minus
(72)
The qualitative relationship between the distribution of activation barriers and fragility was
proposed by Stillinger[18] [19] who suggested that the energy landscape of a strong liquid (low
fragility) has a small distribution of activation barriers and that a higher fragility is associated with
a broader distribution of activation barriers ie a higher variance of the activation barriers This
leads to a phenomenological relationship that is here assumed to be valid based on Stillingerrsquos work
on energy landscapes[18] [19]
2
0
1B
mA
m
= minus
(73)
where A is some constant of proportionality This expression was chosen because in the limit of a
strong liquid there is an infinitely sharp distribution of activation barriers Combining Eqs(65)
(72) and (73) we obtain the temperature dependence of the stretching exponent for a liquid
( )
0
2 2 2 2
0 0
exp 1
( ) 2
6 ln10 1 exp 1
g
f
f
f
g
f
f
T mT S
T mT
Tm mA T S
m T m
minus minus
=
minus + minus minus
(74)
Fictive temperature appears in this new expression because we are deliberately expressing this
function as an equilibrium model Rewriting Eq (74) in terms of one unknown () and grouping
constants
70
( ) 0
2 2
0 0
exp 1
2exp 11
g
f
f
f
g
f
f
T m
T m
T m
T
T
Tm
T
m
m
minus minus
minus minus
=
minus +
(75)
Eq (75) is an expression for the stretching exponent as a function of thermal history glass
transition and fragility index with only one unknown In Eq (75)
( ) 12
6 ln10
S
A
= (76)
The only unknown for the compositional dependence of the stretching exponent is the value of D
The composition-dependent part of Eq(76) is Syen
since we approximate A to be independent of
composition The fragility dependence of Syen
was proposed by Guo et al [61]
Syen(x) = S
yen(xref
)expm(x) -m(x
ref)
m0
eacute
eumlecircecirc
ugrave
ucircuacuteuacute
(77)
where x is composition and xref is a reference composition in the same glass family Seeking the
simplest possible expression to approximate the unknown D we take the natural logarithm of
Eq(76) and of Eq(77) and combine them to get
0
ln lnm
m = + (78)
with the additional definition
( )( )
( )12
0
ln l6 ln10
nref refS x m x
mA
minus
=
(79)
The result is that ln D varies linearly with fragility m and the intercept ln D is independent of
composition (depends on one reference composition)
In Figure 9 we plot the data of Boumlhmer et al[152] vs the predicted exponent The work of
71
Boumlhmer et al is the summary of the literature data relating fragility index to the stretching exponent
at the glass transition which we use to fit values to Eq (78) The dataset included in their work
covers chalcogenide oxide and organic glasses The fitting (consisting of least-squares
minimization of the difference between the predicted stretching exponent and that which was
reported) was done twice once for organic and once for inorganic systems During the fit it was
assumed that Tg=Tf Some assumption about thermal history was necessary since the individual
thermal histories or Tf values are not known for this whole collection since we are trying to track
overall trends in values vs composition this reasonable simplifying assumption is consistent
with our program Using Eqs(75) and (78) the temperature and compositional dependence of the
stretching exponent can be described with only one free parameter
72
Figure 14 b predicted and from literature showing good agreement with a total root-mean-square
error of 01 The fit for organic systems is given by 66 ln Kln = minus and for inorganic systems by
75ln Kln = minus
73
42B Experimental Validation
Experimental density measurements were made using Corningcopy JadeTM glass as described
elsewhere[146] Changes in the density are normalized to obtain the relaxation functions which
are fit with Eq (60) The resulting values of are shown in Table 3 A comparison in Figure 15
is shown with data obtained on a soda-lime silicate SG80 where the stretching exponent was
obtained using the measurement of the released enthalpy as a function of isothermal annealing time
during relaxation below the glass transition The measurement of the released enthalpy[153] relies
on the change in the excess heat capacity in the glass transition range as a function of annealing
time Normalized released enthalpy relaxation functions for each temperature were fit with Eq(60)
to obtain the stretched exponent values for SG80 In Figure 15 parts A and B are samples measured
below Tg while in part C the samples measured are at Tg and above The values were then fit with
Eq(75) where the only free parameter was D with an equilibrium assumption (Tf = T)
74
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy JadeTM
glass and Sylvania Incorporatedrsquos SG80
Glass Temperature (K) Tg (K) TgT m D (K-1) Beta t (s)
Corningcopy JadeTM 1031 1074 1042 32 0002 0504 7249
Corningcopy JadeTM 1008 1074 1065 32 0002 0447 32458
Corningcopy JadeTM 973 1074 1103 32 0002 0395 419214
SG80 800 800 1 36 00057 0632 92
SG80 783 800 1021 36 00057 0593 182
75
Figure 15 The equilibrium model proposed with the experimental points showing good agreement
between the experimentally measured data points and the equilibrium derived model RMSE was
002 for Corningcopy JadeTM (A) and less than 001 for SG80 (B) (C) The model fit for B2O3
experimental data[154] [155] The fragility and glass transition temperature of the B2O3 are taken
from the work of Mauro et al[17]
(
A)
(
B)
(
C)
76
A combination of the MAP model for non-equilibrium shear viscosity[44] the model
presented here primarily in Eq(75) and the model presented by Wilkinson et al[139] for the
temperature dependence of elastic modulus allows for fully quantitative modeling of stress
relaxation behavior The missing model required to understand structural relaxation is the bulk
viscosity curve[62] All previous relaxation (structural or stress) models[42] [61] have relied on
approximations that use a constant exponent and on a constant (temperature-independent)
modulus value whereas here every parameter of Eq(60) may be modeled as a function of
temperature Furthermore in combination with the relaxation models described by Guo et al[61]
in which multiple fictive temperatures are described using a Prony series and a temperature-
dependent modulus one can construct a relaxation curve accounting for the temperature
dependence and thermal history dependence of all relevant parameters
( )
( )( )
1
( )( )exp exp
( )) (
fTN
i
i
f
f
f fi
G T k T tG T tw
T TT
T T
=
minus minus
(80)
Here wi and ki are fitting parameters that are completely determined by the value of and each
term in the Prony series is denoted with the subscript i Usually 8 or 12 terms are included in the
Prony series[96] Each term in the Prony series is assigned a partial fictive temperature (Tfi) whose
relaxation is described by the simple exponential in the Prony series Equilibrium conditions are
assumed at the start of the simulation which allows for a known set of starting probabilities within
the energy landscape interpretation of relaxation The same model also includes the fragility index
and glass transition dependence of the non-equilibrium shear viscosity This method is
implemented in RelaxPy[42] as discussed in the next section This serves as an approximation for
the evolution of the of the non-equilibrium state however the temperature dependence of the bulk
viscosity and a replacement for fictive temperature need to be quantified to improve the
understanding of the underlying physics[1] [16] [62]
77
Separately we can explore the relationship between this model given by Eq(75) and the underlying
energy landscape In order to better understand the stretching exponent consider that there exists
a full set of parallel relaxation modes within an energy landscape The relaxation modes are then
weighted by the occupational probability corresponding to a particular mode This gives a series of
transition rates with some probability prefactor and an associated relaxation time (scaled from the
mean relaxation time) which gives rise to a Prony series form of Eq(80) Thus the distribution of
relaxation times (or barriers) determines the evolution of the stretching exponent while the average
barrier determines the mean relaxation time If this Prony series description in turn describes the
stretching exponent we arrive at a physical description and origin of the stretching exponent As
the temperature approaches infinity even though there is a distribution of activation barriers the
distribution of relaxation times approaches a Dirac delta function and the stretching exponent
approaches one (a simple exponential decay) As the temperature decreases the distribution of
relaxation times broadens due to the wide distribution of barriers however due to the broken
ergodic nature of glass the value of the stretching exponent will be controlled by the instantaneous
occupational probability
The activation barriers may be modeled eg using a numerically random set of Gaussian
distribution of barriers[156] The probability of selecting an individual transition is then calculated
using a Boltzmann weighting function the transition also has an associated relaxation time with it
The Prony series is then recreated with the probability multiplied with the simple exponential
relaxation form The sum of all of these terms is then fit with the stretched exponential form Figure
11 shows the stretching exponent calculated for the mean activation energy of 10 eV and three
choices of standard deviation 25 2 15 ande e e minus minus minus= eV as shown in the legend where Eq(60) has
been fit to the Prony series calculated from an equilibrium Boltzmann sampling of a Gaussian
distribution of barriers The temperature dependence for the stretching exponent matches the form
78
derived earlier in this work This stretching exponent shape has been shown previously though not
in a closed-form solution[156] This secondary method not only validates the generalized form of
Eq(75) but also highlights the validity of using the configurational entropy and barrier distributions
as the underlying metric controlling the temperature dependence of the stretching exponent
Figure 16 The stretching exponent calculated as described in the text for a Gaussian distribution
of barriers This plot shows that the distribution of barriers has a large effect on the stretching
exponent A Tg cannot be described since there is no vibrational frequency included in the model
though the glass transition temperature should be the same for all distributions since the mean
relaxation time is the same for all distributions at all temperatures The deviation is given in ln eV
units
79
To create software to model this complex behavior the authorsrsquo existing software RelaxPy[42] has
been modified into a new version RelaxPy v20 Instead of fixed values for the Prony series the
values are chosen dynamically to match the stretching exponent as predicted by Eq(75) The
database of values wi() and ki() was created with a fixed number of terms N in Eq(80) (we chose
N=12) This database is available in the same Github repository as the software The parameters
were fit by starting with the values obtained for =37 reported by Mauro and Mauro[96] to make
the new version of the model smoothly reproduce a prior optimal fit then was stepped by 001
(when less than 095) or by 0001 (when greater than 095) Using the previously optimized values
of wi and ki for starting values for each new the wi and ki values were then varied to minimize
the root-mean-square error between the stretching exponent form and the Prony series form ie to
satisfy Eq(80) with least error This list of compiled values makes up the contents of a database
that the software accesses The values of wi and ki used for a given are calculated by finding the
closest value in the database to the given value of This allows for an efficient implementation
of relaxation modeling Figure 12 shows the fitted parameters to the Prony series and an example
RelaxPy output for Corningcopy JadeTM glass undergoing quenching at a rate of 10 Kmin Note that
the room temperature stretching exponent value is 046 a close match the Phillipsrsquos ldquomagicrdquo value
of 37 (043)[157] This prediction for Corningcopy JadeTM is a non-equilibrium prediction in contrast
to the equilibrium high-temperature prediction shown in
80
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent Each color
designates one term in the series (Bottom) The output from RelaxPy v20 showing the stretching
exponent effects on the relaxation prediction of Corningcopy JadeTM glass Each quadrant shows one
property that is of interest for relaxation experiments In particular it is interesting to see the
dynamics of the stretching exponent during a typical quench
81
43 Conclusion
In this section we have sought to expand the current understanding of relaxation and improve the
models that currently exists We have proposed a model that accounts for the effects of special
relativity as well as a model was derived through an understanding of the distribution of barriers
for relaxation and the temperature dependence of the Adam-Gibbs entropy The model outlined
herein describes the temperature dependence of the stretching exponent in glass relaxation
The work done on the relativistic effects on relaxation illustrates the crucial role of the observer on
all relaxation models and offers an extreme in which to test common relaxation models The model
for the stretching exponent not only considers the extremes at high and low temperature (when
compared to the glass transition) as in the Phillips model but also for any intermediate temperature
as a function of the fictive temperature This model does not have any explicit temperature
dependence since it was assumed that an equilibrium model works well to describe the
instantaneous distribution of relaxation times Given the physical argument and the success of this
model when tested by experiments and by another model it is at least reasonable to formulate the
temperature-dependence of in terms of its fictive-temperature-dependence as we have done here
Including both T and Tf is possible to consider but lies outside our current scope Using previously
derived compositional dependence for the MAP model a fragility index dependence of the
stretching exponent was defined and tested The model was confirmed using multiple experimental
datasets In addition a theoretical comparison to a distribution of landscape activation barriers was
found to reproduce the same trends as the model
Chapter 5
Glass Kinetics Without Fictive Temperature
Fictive temperature as discussed through this text is unable to capture key physical
phenomena[16] [42] [51] To circumvent this limitation we rely on two different frameworks
The Adam-Gibbs relationship and generalized features of the energy landscape These two features
allow us to create a method that is completely generalizable and does not involve fictive
temperature This method creates fake landscapes that generally capture the trends of glass based
on experimental measurements This new method we have called lsquoToy Landscapesrsquo
51 Background of the Adam Gibbs Relationship
Experimental evidence in support of the AG model comes from the work of Richet[158]
in which they compared the predicted configurational entropy from viscosity curves with that
obtained from DSC The configurational entropy was calculated from DSC data using
0
0( )p p vi
T
T
b
c c
C CS S T dT
T
minus= + (81)
In Eq (81) Cp is the isobaric heat capacity Cpvib is the vibrational heat capacity and T0 is the initial
temperature from which the configurational entropy is integrated The difference between the total
heat capacity and the vibrational heat capacity is the heat capacity due to structural rearrangements
Richet found good agreement between the measured viscosity curve and calorimetric
configurational entropy at temperatures above the glass transition confirming validity of the AG
relationship When applying Eq equation reference goes here all temperatures must remain above
the glass transition ie in the liquid or supercooled liquid state The calculation is invalid upon
83
cooling through the glass transition due to the breakdown of ergodicity[12] [13] [159] and the
irreversibility of the glass transition process Application of Eq (6) to temperatures below the glass
transition leads to an incorrect calculation of excess entropy in the glassy state[160]
In this study we expand on these previous works to understand the relationship between
viscosity and the underlying energy landscape and provide new insights into the thermodynamics
of liquids through a simplified ldquotoy landscaperdquo model constructed from experimentally measured
viscosity parameters The energy landscape framework is especially helpful for elucidating the
thermodynamics and kinetics of supercooled liquid and glassy systems Energy landscapes describe
the evolution of all atomic transitions for kinetic processes in a system To perform such
calculations information about the inherent structures and transition points in a landscape must be
known In this work we solve the inverse problem of deducing realistic landscape parameters using
the AG relationship and experimentally measured data This information is used to construct a ldquotoy
landscaperdquo model to describe glass relaxation processes To achieve an accurate relaxation model
the following steps are taken to ensure the validity of the results
A Confirm the AG model for shear viscosity using energyenthalpy landscape
calculations
B Validate the assumptions made by the MYEGA model
C Confirm the AG model for bulk viscosity
D Explore the fundamental relations between viscous properties and topography of the
landscape
E Use the knowledge gained to propose a new thermodynamic model to calculate the
driving force for glass relaxation and the scaling of the liquidrsquos free energy
This chapter is organized into sections devoted to each of the topics above preceded by a
methods section All of the sections are presented to validate a new model of glass relaxation that
is sufficiently complex to achieve a calculation of the thermodynamics and kinetics of the liquid
This new model will be based on understanding the landscape from experimental properties and
we have called this approach the ldquotoy landscaperdquo model
84
52 Methods
Potential energy landscapes (fixed volume) and enthalpy landscapes (fixed pressure) are
powerful modeling techniques where atomic configurations are mapped in a 3N-dimensional phase
space (or 3N+1 dimensions for enthalpy landscapes)[89] The energy landscape approach is based
on mapping the continuous 3N dimensional space to a discrete set of energy minima (called
ldquoinherent structuresrdquo) and first-order saddle points (ldquotransition pointsrdquo) To perform this mapping
a systematic search for inherent structures is performed while also mapping the lowest-energy
transition points between each pairwise combination of adjoining minima The landscape itself is
partitioned into basins which represent the set of all configurations that minimize to a common
inherent structure[161] This combination of the basin and transition point information results in a
topographic map of the 3N phase space (either a partial or whole mapping) that can be used to gain
insights into the physics of any atomistic system such as proteins[91] liquids[18] [19] glasses[1]
[93] and nucleating crystals At equilibrium the probability distribution for occupying the various
basins in the landscape is given by[13]
exp
exp
i
j
i
i
j
j
Eg
kTp
Eg
kT
minus
= minus
(82)
where Ei is the potential energy (or enthalpy) of the inherent structure gi is the degeneracy of the
basin i is the basin index and pi is the probability of occupying a given basin i The denominator
is a normalization factor given by the summation over all basins Mauro et al[44] previously used
this formalism to study the non-equilibrium viscosity calculate the evolution of configurational
heat capacity and elucidate the long-time relaxation kinetics of selenium glass[44] [93] [160] For
equilibrium systems the configurational entropy is given by the Gibbs formula
85
lnc i
i
iS k p p= minus (83)
In order to validate the AG model the landscapes of several common systems were
explored using ExplorerPy[89] ExplorerPy is a software program built specifically to map
landscapes using the LAMMPS molecular dynamics package[30] The three systems we explored
are B2O3[162] (150 atoms and 2000 basins) SiO2[162] (150 atoms and 1000 basins) and BaO-
2SiO2[121] (128 atoms and 1500 basins) All systems used periodic boundary conditions The
initial structures for B2O3 and SiO2 were constructed by placing atoms corresponding to the
chemistries of the systems on a random grid (with 1 Aring spacing) corresponding to a density of 20
gcm3 and equilibrated at 2500 K then quenching at a rate of 1 Kps to room temperature using the
Wang et al[162] potentials in the NPT (constant pressure number of atoms and temperature)
ensemble The energy landscape of the BaO-2SiO2 system was taken from our previous work[72]
The SiO2 (constant pressure exploration) system was explored using ET with a push distance of 03
Aring and a pressure of 10 atm B2O3 was explored using isobaric MD at a temperature of 1500 K for
500-time steps of 1 fs also at a pressure of 1 atm The minimization (after MD or ET) was handled
by the LAMMPS[30] steepest descent algorithm until the certainty in energy was smaller than 10-
4 eV Both used a nudged elastic band spring constant of 10 eVAring2 The energy distribution for all
systems was fully sampled which was confirmed by randomly checking if half the basin
distributions matched the full distribution It was also found that multiple structures in each
exploration were discovered more than once and a slightly smaller B2O3 system (130 atoms) was
also explored and showed identical results to the 150-atom distribution The insights gained from
the interaction between the experimentally accessible viscous properties and the landscapes
presented here will be generalized to create a simplified ldquotoy landscaperdquo model to capture the
thermodynamics and kinetics of the glass-forming system
86
53 Results
53A Adam-Gibbs Validation
To predict viscosity using the Adam-Gibbs expression there are only three required
parameters the configurational entropy (as a function of temperature) the barrier for cooperative
rearrangements and the infinite-temperature limit of viscosity Using a landscape the equilibrium
configurational entropy can be obtained using Eq (83) The infinite-temperature limit of shear
viscosity is given by the work of Zheng et al[54] where a systematic study of MYEGA fits to
viscosity data revealed a common value of 10-293 Pamiddots This infinite temperature value of viscosity
is used for all the experimental and computational viscosity curves shown in this work The final
parameter is the mean barrier for a cooperative rearrangement In Figure 18 the barrier for
cooperative rearrangements was chosen to accurately produce a viscosity of 1012 Pamiddots at the
experimentally measured value of the glass transition temperature From these parameter values
the predicted Adam-Gibbs viscosity curve is plotted The universal infinite limit fragility and glass
transition values from references were inserted into the MYEGA equation to plot the
lsquoExperimentalrsquo curves for B2O3 SiO2 and BaOmiddot2SiO2 respectively[58] [69] [120] When
examining Figure 18 it is clear that the AG model is able to fundamentally recapture the viscosity
87
Figure 18 The viscosity (left) and landscape (right) predictions for three common systems The
first system is newly calculated in this work while the latter two come from our previous works[72]
[89] It is seen that the viscosity predicted from the AG model is very accurately able to reproduce
the experimental viscosity curves from the MYEGA model The last system is a potential energy
landscape while the others are enthalpy landscapes
53B MYEGA Validation
Now that we have confirmed that the configurational entropy of the landscape works well
to describe the viscosity via the AG relationship a definitive temperature-dependent form is needed
in order to make more general predictions The key approximation made in deriving the MYEGA
model is that the configurational entropy follows an Arrhenius form By testing this Arrhenius
88
form we can evaluate this assumption in the derivation of the MYEGA model while simultaneously
gaining insights into the thermodynamics of the liquid and supercooled liquid states A strong liquid
( 15m ) has a configurational entropy that is nearly constant as a function of temperature while a
fragile liquid ( 15m ) is a liquid in which the configurational entropy is highly temperature
dependent Since we are testing the MYEGA equation it is also worthwhile to consider alternate
viscosity expressions Assuming validity of the Adam-Gibbs equation the configurational entropy
deduced from the viscosity experiments is given by
( )10 01log l) og(
c
BS
T T minus= (84)
where the viscosity function is given by a viscosity model such as Vogel-Fulcher-Tammann[59]
(VFT) equation the Avramov-Milchev[60] [163] (AM) equation or the MYEGA expression
The resulting entropy predicted by each viscosity model (using experimental data for
fragility and the glass transition) is compared to the calculated configurational entropy from the
landscape for B2O3 in ( 518gT = K and 33m = [69] were used for the experimental values) The
figure clearly shows that the MYEGA equation which assumes that configurational entropy scales
in an Arrhenius fashion reproduces the calculated configurational entropy from the landscape the
most accurately out of the three considered viscosity models This also means that the
configurational entropy form of the AM and VFT models are not physical and cannot be used to
recreate physical results Since the temperature dependence of the entropy has a fixed mathematical
form and is dominated by the distribution of basins we now understand that the distribution of
basins must also have a fixed form and that form must control experimentally accessible properties
This is the key insight that will allow for the development of the toy landscape model
89
53C Adam-Gibbs and Structural Relaxation
Although the Adam-Gibbs model has been validated for shear viscosity and has provided
the functional basis of the MYEGA model let us now consider whether the temperature
dependence of the configurational entropy can also predict the bulk viscosity (also known as the
volume viscosity) Recent work has highlighted the importance of bulk viscosity viz that the
structural relaxation time is proportional to the bulk viscosity[62] Hence the temperature
dependence of the bulk viscosity provides key information needed to predict structural relaxation
times[62] Following Scherer[164] the relationship between the structural relaxation time ( structural
) and the bulk viscosity ( B ) is given by
( )0B structuralK K = minus (85)
Figure 19 The configurational entropy comparisons between the three major viscosity models
which validates the main underlying assumption of the MYEGA model The VFT and AM are
unable to capture the physics of configurational entropy therefore ruling
90
where K is the infinite frequency bulk modulus and 0K is the zero-frequency bulk modulus In
order to calculate bulk viscosity for B2O3 the zero-frequency and infinite-frequency modulus is
taken from these works respectively [165] [166] The experimental structural relaxation times were
taken from the work of Sidebottom et al[154] In Figure 20 the bulk viscosity is shown as well as
the Adam-Gibbs comparison
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the configurational
entropy from the enthalpy landscape with the barrier 00155 eV (compared to the shear barrier of
00149 eV) and the infinite limit allowed to vary (10-263 Pamiddots for bulk viscosity) In Figure 18 the
configurational entropy is confirmed for the shear viscosity thus confirming the AG for both shear
and bulk viscosities Sidebottom data are from Ref [154]
In Figure 20 the fitted values of B and 10log can be used to gain insights into the
difference between structural and stress relaxations since the configurational entropy is universal
between the two The value of Bk (the barrier for cooperative re-arrangements in natural units) for
shear viscosity was 173 K while for bulk viscosity it was 180 K This difference of 7 K in the
barrier leads to an order of magnitude difference in the fitted viscosity curves This is a small
91
difference and explains the variance in the values at Tg The other difference is in the infinite
temperature value of viscosity which is -293 log(Pamiddots) for shear viscosity (from Zheng et al) and
-263 log(Pamiddots) for bulk viscosity This implies that the MYEGA raw form is accurate to predict
the structural relaxation time as well as the stress relaxation (though the values of the pre-
exponential factor and the composite constant will change between the two forms) However the
form in terms of fragility and the glass transition will need to be modified to accurately describe
the structural relaxation time The form assumes the viscosity of interest has a value of 1012 Pamiddots at
the glass transition temperature for shear viscosity which does not have to be the value for the bulk
viscosity curves at the glass transition Further research is needed to understand the appropriate
bulk viscosity at the glass transition and the infinite temperature of bulk viscosity
53D Landscape Features
As shown here the configurational entropy is governed by a Boltzmann sampling of a
probabilistic distribution of states n(E) This distribution of the inherent structures is empirically
found to follow a log-normal distribution as shown in Figure 21 While we are not claiming a
deeper meaning to the underlying origin of the log-normal distribution we may adopt this form of
the probability density function since it accurately describes the underlying distribution of
microstates This is the key insight needed to develop a new model because this means that the
fragility and glass transition are fundamentally related to the parameters of the distribution of basins
on the landscape (number of basins and standard distribution of the enthalpy)
The configurational entropy in the MYEGA model is given by
10
exp 1ln10 12 log
g
c
TS mS
T
= minus minus
minus
(86)
92
where S is the infinite temperature limit of the configurational entropy In Figure 21 we also
show the computed value of Eq (83) as a function of temperature compared to that of Eq (86)
which shows excellent agreement between the configurational entropy and the MYEGA model
This test also supports Stillingerrsquos[18] [19] view of fragility being correlated to the lsquoroughnessrsquo of
the landscape topography According to Stillingerrsquos view if the distribution of inherent structure
energies is narrow then the configurational entropy cannot have a large temperature dependence
leading to a low fragility Alternatively if the distribution is broad then there must be a temperature
dependence to the configurational entropy which necessarily results in a higher fragility
93
Figure 21 (A) The histogram for the energy minima for each of the test systems showing a good fit
with the log normal distribution This distribution will then be a valid form to calculate the enthalpy
distribution of the model presented in the next session (B) The configurational entropy from the
model showing the accuracy of the scaling of the entropy predicted by the MYEGA model The
S value was fit for each system
(A)
(
B)
(B)
94
54 Topography-Property Relations
Now that we have confirmed the validity of the Adam-Gibbs and MYEGA equations from
the energy landscape approach we can draw some key conclusions regarding the thermodynamic
properties of liquids For example heat capacity has been related to fragility[112] and is of
importance for glass manufacturing since it is related to the cost of heating and forming a glass
Isobaric heat capacity is defined as
p
P
HC
T
=
(87)
where H is enthalpy T is temperature and P is pressure The mean configurational enthalpy
calculated from an enthalpy landscape is given by
conf i conf iH p H= (88)
The heat capacity has contributions from both vibrational and configurational modes in liquids
while in glasses the configurational mode is mostly lost For a reversible process the
configurational heat capacity can be written as a function of the configurational entropy
c
p conf
P
SC T
T
=
(89)
By combining Eq (86) and Eq (89) we can write the expression for configurational heat capacity
of a liquid as
10 10
1 exp 1 exp12 log 12 log
g g
p conf
T Tm m S HC S H
T T T T
= minus minus minus = minus
minus minus
(90)
From fluctuation theory the heat capacity can be expressed in terms of the variance of the enthalpy
(2
H )
95
2
2
1p conf HC
kT= (91)
It is important to note that the variance of the enthalpy is not equivalent to the variance of the log-
normal distribution The variance of the enthalpy includes a Boltzmann distribution over the log-
normal distributed basins This gives a direct relationship between the enthalpy fluctuations and
the viscosity parameters If we further adopt the log-normal distribution as the effective distribution
for all liquid basins of all enthalpy landscapes then the distribution is only a function of the
common viscosity parameters ( S m gT and 10log ) The viscosity parameters will only
change the variance of the distribution since the mean of the distribution is inconsequential (the
zero-point enthalpy can be adjusted to an arbitrary value without changing the calculation of the
configurational entropy) The parameters of the log-normal distribution can then be fitted to
reproduce the configurational entropyrsquos temperature dependence thus creating a lsquotoyrsquo enthalpy
landscape for us to play with for the purpose of thermodynamic and kinetic calculations Generating
the landscape is a numerical calculation as the number of basins and standard deviation must be
chosen such to reproduce key phenomena
To quantify the dependence of these parameters on the degeneracy and temperature it is
helpful to develop a descriptive formal model The cumulative probability distribution of a normal
distribution (with zero mean) is given by
10log )1
1 er(
f2 2
Hp
+
=
(92)
where is the standard deviation of the log-normal distribution erf is the error function H is the
enthalpy and p is the probability To then calculate the equal probability spacing for a log-normal
distribution we solve for H for the set of probabilities p which range from 2 to 1 2minus
96
with a step-size of This then gives enthalpies of basins equally spaced by the cumulative
probability distribution and as a result each basin has equal degeneracy
1 2 r2 1e f10
pH
minus minus = (93)
This set of basins carries with it an Arrhenius activation barrier for the configurational entropy that
is only a function of and the degeneracy of each basin (if the is chosen such that the energy
distributions converge) This creates a model that fully describes the thermodynamics of the liquid
and though it currently lacks any kinetic component it offers the possibility to be a useful tool
when used in conjunction with Eqs (82) (83) and (88) This new tool is denoted the ldquotoy landscape
modelrdquo (TLM) It is called the toy landscape model because we are generating a landscape that is
not the real landscape but is enough to reproduce key experimental properties without needing
expensive computational calculations It is a lsquotoyrsquo for us to play with without needing to worry
about upfront computational cost
To confirm the validity of TLM and the way it is constructed we can calculate the
activation barrier from this deterministic method and for a random set of inherent structures with
the same number of basins and distributions For this purpose we used temperatures ranging from
1000 to 2000 K for both the deterministic and random models with a step size of 0001 for TLM
Excellent agreement between the random and deterministic methods is shown in Figure 22A This
analysis allows us to understand that the thermodynamics of the system are fundamentally linked
to the viscous flow behavior Though this technique was created to target viscosity and relaxation
it is generally applicable to all configurational thermodynamics of liquids and glasses It is worth
noting that all the thermodynamics will be relative to a 0 eV energy state defined at equilibrium at
0 K and as such the enthalpy differences at some fixed temperature must be known to compare
over compositional spaces
97
In Figure 22B we see the same method being used to understand the relationship between
the viscous parameters and the parameters of the landscape This is the key to the numerical
calculation of the lsquotoy landscapersquo model The toy landscape is a numerical calculation where we
use experimentally accessible values to generate a landscape whose thermodynamics should
reproduce key thermodynamic phenomena In Figure 22B we show that the fragility and glass
transition are systematically changed as the number of basins and standard deviation of the enthalpy
is changed When this idea is combined with the basin calculation in Eq (93) (giving equal
degeneracies for all basins) then the complete set of enthalpy basins is easily attainable giving
access to the fundamental thermodynamics of the system
98
(
A)
(B)
(A)
99
Figure 22 (A) Comparison between the randomized method (histogram) and the deterministic
method (vertical lines) showing good agreement between the maximum in the histogram and the
value predicted by the deterministic technique validating the approach It is worth noting that the
100-basin distribution is a very wide distribution where the total number of basins is less than the
number of points used in the calculation This is done for a variable number of basins with the
number of basins shown in the legend (B) The dependence of fragility and the glass transition
temperature vs the distribution of states and the number of basins
55 Barrier Free Description of Thermodynamics
Throughout this work we have systematically explored the Adam-Gibbs model of
viscosity An important insight that we have gained is that the configurational entropy is free of any
barriers This is of particular interest because the Arrhenius slope of the activation barrier is linked
to the glass transition and fragility The glass transition temperature is known to be related to the
set of activation barriers as it corresponds to a breakdown of ergodicity What is found here is that
the fragility is the ldquoconversion factorrdquo to account for the slope of the configurational entropy
(which as previously identified is related to the distribution of basins) This means that the barriers
themselves do not matter until the material is close to the breakdown of ergodicity and thus the
equilibrium liquid thermodynamics and kinetics can be described without explicit information
about the saddle points in the landscape
Understanding the role of the topography in the properties of glasses and liquids lead to
the most important insights derived from the AG model Since the configurational entropy is the
same for both stress and structural relaxation simple viscous parameters allow us to understand
fundamental thermodynamic quantities of the system without needing to map computationally
expensive energyenthalpy landscapes The process of converting experimentally accessible
properties to a usable landscape for thermodynamic calculations is what we are calling the toy
landscape model To show the utility of toy landscapes we can calculate the driving force for glass
relaxation This information can give insights into the physics of relaxation by dynamically
100
calculating the Gibbs free energy of the equilibrium liquid at any temperature which defines a
driving force for glass relaxation To do this calculation of the driving force for glass relaxation
we consider the ergodicity parameter ( x ) defined by the Mauro-Allan-Potuzak (MAP) model of
glass relaxation as the ratio between the glass entropy and the equilibrium liquid entropy
liquiss dglaS x S= (94)
To get information about the enthalpy we consider an instantaneous Boltzmann distribution on the
landscape given by an equilibrium distribution at some elevated temperature fT
1
exp
exp
iglass i i
fii
f
HH H g
kTHg
kT
= minus
minus
(95)
This formalism in conjunction with TLM gives us the ability to explore the temperature
and compositional dependence of the driving force for relaxation The enthalpy of the liquid is
calculated from an equilibrium distribution on the landscape The driving force ( ) is defined as
( ) ( )( ) ( )1iliquglas ds f l quii dH H TS xT T = minus + minus (96)
For our purposes this gives a nice approximation for the driving force improving on previous
approximations given by ( fT Tminus )[16] [42] TLM does not include a kinetic feature to calculate
the occupational probability in each basin however when coupled to a kinetic model such as the
MAP model[44] a complete description of the underlying thermodynamics of glass relaxation can
be considered For this calculation of the driving force the kinetics are calculated through the MAP
model but this is not necessarily a requirement In Figure 23 the compositional and temperature
dependence of the driving force is shown for three glasses with identical glass transitions and
varying distributions of basins (as listed in Table 1) The calculations are performed using the
RelaxPy script and MAP model with the stretching exponent fixed at the Phillips value of 37
101
explanations of which are available elsewhere[42] The systems were quenched from 700 K to 300
K at a rate of 10 Kmin Figure 23 clearly shows how simply changing the total number of basins
creates a larger driving force for glass relaxation This simple change of the number of basins
drastically increases the entropic contribution and due to the loss of ergodicity explains the increase
of the driving force of relaxation The kinetics could be calculated through any relaxation model or
directly from the landscape if some barrier approximations are made
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition temperature (500 K)
were kept the same while the total number of basins were allowed to vary
Sample Total Basins [-] Distribution of States [log eV] 1
1493g
mT
minus
[K]
Fragility [-]
1 100 02 837 40
2 500 02 537 31
3 1000 02 466 28
102
Figure 23 The driving forces for different example glasses calculated using a combination of the
MAP model RelaxPy and the toy landscape model The parameters for each glass can be found in
Table 4
57 Toy Landscapes for the Design of Glasses and Glass Ceramics
The concept of a toy landscape can go even further building a method to predict the
dynamics of a glass system based on simple inputs This is possible because in the previous section
we have related the distribution of basins on an energy landscape to the experimental viscosity
which allows for thermodynamic insights Building on this we can assume that the barriers (H )
for transitioning between states is equal to that of the barrier mean for the viscous relaxation
ln10gH mkT= (97)
This means that the only key factor needed to then predict the dynamics of a landscape using the
metabasin approach is the vibrational frequency However the relaxation time at the glass transition
103
is known to be a constant of 100s and as such we can choose a vibrational frequency that reproduces
this fixed point This gives an entire description of relaxation without any need for fictive
temperature An example of the results predicted through this method are shown in Figure 24 This
prediction only required knowledge of the glass transition temperature and the fragility and from
this a pure prediction of relaxation is made
104
105
Figure 24 The predicted outputs from the toy landscape method showing predictions of the enthalpy
and entropy under a standard quench for barium disilicate This prediction does not require fictive
temperature or any such assumptions about the evolution of the non-equilibrium behavior
This method is also not limited to predicting the relaxation of glasses but can also capture
the crystallization It is known from the earlier work on nucleation presented in Chapter 3 that a
simple two state model was able to capture the underlying physics of the system Thus if we know
the free energy of the crystal and barrier to crystallization we can find predict key crystallization
106
phenomena By fitting the heat flow peak magnitude and location this gives enough information to
infer the barrier and the enthalpy of the crystal The degeneracy (and thus the entropy) of the crystal
can then be calculated through knowledge of the liquidus temperature
ex(
p)cry SCL l
cry
lk
G TH
T
minus=
(98)
This then gives everything for the model presented in Chapter 3 from only 1 calorimetry
experiment the glass transition the fragility and the liquidus temperature To predict nucleation a
series of estimates were done running the toy landscape at different temperatures until the
temperature with the fastest crystallization rate was found This was then assumed to be the
maximum rate of nucleation The value of the interfacial energy was then chosen in such a way so
that the predicted curve reproduces the peak at the same temperature This gives all the parameters
needed for a prediction using CNT It is also possible to make a prediction of the growth rate of
crystals using this expression
2
1 exp6
GU
kT
a kT
minus minus
=
(99)
The results for these estimates are shown in Figure 25 The reason why growth is more accurate
compared to nucleation is that the kinetic factor of growth is governed by viscosity while nucleation
is governed by a kinetic factor that includes an assumed vibrational frequency
These methods have a plethora of assumptions built in such as that the vibrational
frequency of basins undergoing relaxation is equal to the vibrational frequency when undergoing
crystallization however as more information is gained about a particular system (such as more
crystallization peak as a function of temperature) the information could be built into the system to
improve crystallization predictions If the peak size is known as a function of scan ratetemperature
even the enthalpy as a function of temperature could be known This provides a whole system for
the real estimates of crystallization can be improved
107
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume in
nucleation is assumed to be on the order of one cubic angstrom while the a parameter in growth is
assumed to be around one nm both are in good agreement for estimates in literature The values
for the orange points are taken from these works[128] [167]
57 Discussion
The impact of the Adam-Gibbs model on glass science is hard to overstate and it has
become foundational to ideas that will enable the design of next generation glasses The Adam-
Gibbs equation still has more to offer the glass community in terms of fundamental understanding
and practical applications In this work we have shown that the Adam-Gibbs relationship is
consistent with both shear and bulk viscosities Also we have validated the main assumption of the
MYEGA model giving insights into the scaling of configurational entropy from the underlying
enthalpy landscape Using this knowledge of the relationship between the Adam-Gibbs model with
the underlying enthalpy landscapes we have developed a simplified ldquotoy landscaperdquo approach for
modeling glass relaxation The toy landscape is constructed numerically by understanding the
distribution of inherent structures and transition points in the landscape and then relating these
distributions to experimentally measured parameters of the MYEGA viscosity model This
1 00
1600
1400
1200
1000
00
600
400
200
000
0000 110000 130000 1 0000 1 0000
(m
s)
Temperature
Growth PredictionCassar et al
0
2
4
6
10
12
14
00 1100 1300 1 00
log 1
0(I) n
ucl
ei s
m3
Temperature
Predicted RateRodrigues et al
4
3
2
1
0
00 1000 1100 1200 1300
log
10(I m
ax(I))
nuclei s
m3
Temperature elvin
108
simplified landscape model can then be used to calculate the underlying thermodynamics and
kinetics of the system and represents a practical application of energy landscapes to understand
glass relaxation nucleation and other relevant thermodynamic properties
This also closes our discussion on understanding the thermal history effects on glass What
we have shown is that the dynamics of a glass system can be understood through energy landscapes
and fundamentally the same insights can be recreated through lsquoToy Landscapesrsquo This approach
allows fundamentally for the complex 3N dimensional landscape to be summarized and abbreviated
into a smaller landscape that can easily be implemented for any glass However this new
parameterization relies on knowing some key properties that are only due to the location in the
compositional phase space and for a complete picture of the system to be created we must also
create a sufficient approximation for the compositional phase space shown in the next 2 chapters
Chapter 6
Enabling the Prediction of Glass Properties
So far models have been created that describe the very nature of the thermal history effects on
glasses However as an input to these models we have to know about the compositional dependence
of key parameters namely the glass transition and the fragility In this chapter we extend
compositional models to other key properties needed for the design of glasses Here these key
properties that have not been readily predictable are approached using both machine learning and
TCT models The TCT models are focused on surface reactivity Youngrsquos modulus and ionic
conductivity The ML addresses CTE fragility melting point as well as Youngrsquos modulus
61 Controlling Surface Reactivity
Glass surfaces especially their interactions with water[168]ndash[171] are of the utmost importance to
nuclear waste glass cover glass and many other modern applications[106] It is generally accepted
that the outermost surface of oxide glasses readily reacts with water molecules where water
molecules chemisorb on an as-formed glass surface and populate it with surface hydroxyls Recent
models attempt to quantitatively explain the change in properties observed on hydroxylated glass
surfaces[77] [168] [172] [173] and most agree that the surface readily becomes hydrated because
it is more stable to have a bonded hydroxyl group than to have dangling bonds In glasswater
interactions hydrolysis and diffusion lead to two forms of adsorbed water chemisorbed and
physisorbed[171] [174]ndash[176] Both affect the network differently and are controlled by different
processes physisorption by hydrogen bonding[175] and chemisorption by reaction with the surface
to form bonded hydroxyl groups
110
equiv Si minus O minus Si equiv +H2O rarr equiv Si minus OH + OH minus Si equiv
Although binding energy studies have been performed using molecular dynamics simulations[177]
a specific study of the surface reactivity in tandem has not been conducted Recent advancements
in reactive force field modeling allow the direct observation of both diffusion and surface
reactivity[38] [178]ndash[180] and several studies have found surface reactivity constants consistent
with those reported experimentally[173] [178] [181]
Recently TCT has been used to predict various bulk properties of glassy systems such as
Vickers hardness and fragility[17] [182] [183] Based on the TCT description of the glass network
glassy materials have been postulated to have a so-called lsquointermediate phasersquo in which the atomic
structure of a material will self-organize so as to be isostatically constrained (n=3) as explored
extensively in various works[65]ndash[67] [133] [184] [185] In this case it has been shown that such
materials exhibit anomalous behavior in certain properties such as an high hardness and lower free
energy[43] [67] [77]
TCT has also been used to understand the impact of water on bulk glass structure and
properties Potter et al[168] found that by accounting for the impact of chemisorbed (dissociated)
water on glass network connectivity (viz breaking Si-O-Si bonds) and the impact of physisorbed
(molecular) water on strain of the Si-O-Si bond angle the glass transition temperature (Tg) could
be predicted This model could be further expanded to include other properties (such as
modulus[139] or dissolution kinetics[77]) predicted by TCT such as the work done by Liu et
al[186] on calcium-silicate-hydrate gel An extensive study relating topological constraints[6]
[17] [57] and surface energy was performed by Yu et al[172] specifically focusing on the
transition from hydrophilic to hydrophobic behavior on silica surfaces Their work used reactive
molecular dynamics to model the change in surface energy of a silicate glass and then correlate it
with the number of surface constraints present with the work focusing primarily on the global
average of the surface
111
To investigate the effects of glass network topology on the surface reactivity we model
the hydration of a silicate glass was modelled using molecular dynamics simulations according to
the following procedures Initially bulk sodium silicate glasses were simulated (150 atoms with a
molar composition of 70SiO2middot30Na2O) using the Teter potential[187] For a bulk glass this pair
potential is known to accurately simulate sodium silicate glasses and has been extensively studied
to investigate various properties including structural features transport of sodium ions and
vibrational density of states[188] The size of the system is initially set to achieve the
experimentally measured density of 2466 gcm3 [189] A total of 35 Si-atoms 85 O-atoms and 30
Na-atoms were randomly inserted within a periodic cubic box of a = 12686 Å and the initial
configuration was energy-minimized to avoid any overlaps prior to the glass formation The glass
was held for 05 ns with a constant number of atoms volume and energy (a microcanonical or
NVE ensemble) Since a thermal fluctuation of ~400 K (from 2000 K to 2400 K) was observed
during the microcanonical run the glass was allowed to evolve further at 2400 K for 05 ns with a
constant number of atoms volume and temperature (a canonical or NVT ensemble) Within the
same ensemble the melted system was then cooled with a constant cooling rate of 05 Kps and
after the temperature had reached 300 K it was equilibrated for 1 ns Finally a constant pressure
of 1 atm was applied to the system (NPT ensemble) for another 1 ns to ensure no significant
fluctuations of the density during the equilibration process Three different sodium silicate glasses
of the same nominal composition were constructed via the aforementioned procedure using
different initial atomic positions These repeated simulations compensate for the limitation of small
sample size by allowing us to capture the statistical behavior of glass surfaces during the glass-
water reactions The final densities of the sodium silicate are presented in Table 5 These final
structures were used as starting configurations for the glass-water reactions MD simulations with
the Teter potential creating three glass networks were carried out using the Large-scale
AtomicMolecular Massively Parallel Simulator (LAMMPS) package[30]
112
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K
Simulation box dimension a (cubic) [Å] Density [gcm3]
Run1 12979 2303
Run2 12630 2499
Run3 12737 2437
Initial dimension a=12686 Å
Experimental density (70SiO2middot30Na2O mol) 2466 gcm3
113
After the bulk sodium silicate glasses were obtained a reactive potential was employed to model
the glass surface and subsequent glass-water interface as the system of interest for this study
required characterization of the reactive processes during the simulation of surface phenomena In
this study all reactive MD simulations are performed with the NaSiOH parameterization using
the ReaxFF reactive force field framework15 The ReaxFF parameters are trained using a first-
principles data set that describes water interaction at the sodium silicate glass-water interface
Further details of the ReaxFF methodology and its potential forms can be found in earlier
publications by van Duin et al[38] [179] [181] [190]
114
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the hydration
models Color scheme Si atom (ivory) O atom (red) and Na atom (blue) The z-axis is elongated
to allow space for an insert of water
115
As shown in Figure 26 a free surface was first created by expanding the c parameters of each of
the equilibrated bulk glasses This process results in a vacuum region above and below the two
surfaces of the glass slab along the z-direction These exposed surfaces were relaxed at 300 K for
100 ps with the ReaxFF reactive force field Following the relaxation of both top and bottom
surfaces the vacuum region was filled with water molecules The number of water molecules that
are inserted in the vacuum region was controlled to have a density of ~099 gcm3 In addition all
water molecules were kept at least 2 Å from the uppermost and lowermost atoms of the glass
surface to prevent any initial close contact with the surface Glass-water reaction simulations were
carried out at 300 K with the configurations shown in Table 6 for 500 ps in the NVT ensemble
From these runs trajectories at every 100 ps were obtained for the surface reactivity analyses
Figure 27 shows the initial and final positions of glass surface reaction with water at 0 and 500 ps
respectively
116
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the top surface
in contact with water is shown here
a) b)
117
Table 6 System configurations for sodium silicate glass-water reactions
Simulation cell [Å3] Number of water molecules
Run1 12979 times 12979 times 3894 146
Run2 12630 times 12630 times 3789 134
Run3 12737 times 12737 times 3821 138
118
The hydrated surfaces of sodium silicate glasses from each run were used to investigate the
adsorption behavior of a water molecule Since the time evolution of surface reactivity is of key
interest the binding energy of a water molecule to the hydrated surface was calculated at 0 100
200 300 400 ps In order to evaluate the local heterogeneity of reactivity imposed by varied glass
surface structures each surface was divided into a 10 by 10 grid and the binding energy at each site
was mapped across the grid The water molecule position in the z-direction from top and bottom
surface was maintained to be 20 Å from the outermost atom during the energy calculation The
binding energies (Eb) were calculated as below where the energy of a surface with adsorbed water
(Esw) was taken with respect to the surface energy (Es) and the energy of molecular water (Ew)
b s wswE E E E= minus minus (100)
A negative binding energy would indicate that water adsorption to the surface site at the
corresponding grid is thermodynamically favorable The binding energy map may then be used to
locate where water binding would be most stable based on the ReaxFF calculations From three
independent glass-water reaction boxes a total of six hydrated surfaces are obtained (a top and
bottom surface from each run) increasing the statistical reliability of this analysis in the ensemble
despite the small overall system size of any one box The binding energy mapping process is shown
in Figure 28
119
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example of the
electronic-structure DFT calculation using semilocal exchange-correlation functionals finding the
binding energy of a water lsquopixelrsquo to the surface
~20 Å
surface
10times10 grid
120
Similar calculations are carried out at an electronic structure level using the DFT framework as
implemented within the Vienna Ab-initio Simulation Package (VASP) software Total energies are
computed from the static geometries obtained from the reacted ReaxFF structures using the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional with projector augmented wave
(PAW) pseudopotentials to describe the ion cores A kinetic energy cutoff of 500 eV was used for
the expansion of the plane waves Due to the large non-crystalline structure the Brillouin zone is
only sampled at the Γ-point with 01 eV of Methfessel-Paxton smearing to help electronic
convergence Due to the computational cost associated with these DFT simulations the results were
primarily used to elucidate the electronic interactions between the water molecule and the glass
surface
To count the topological constraints in the structure generated from the ReaxFF MD
simulations an algorithm was developed to systematically exclude all non-bridging oxygens
alkali and hydrogens from the structure The number of surface rigid constraints (in the pixel)
around each network-forming atom was then calculated an example contour plot generated using
this procedure is shown in Figure 29 The surfaces were then converted to a 10times10 pixel grid and
the average coordination of network-forming atoms in each pixel calculated The depth of each
pixel taken was to be 4 Aring below the surface
121
Figure 29 Example contour surface showing the average coordination per atom on the glass
surface for the first run at 300 ps
122
Figure 30 shows the results from the MD binding energy studies where a clear transition in the
region around 30 constraintsatom is shown This region is isostatic because the number of
constraints is equal to the number of translational degrees of freedom and the width highlighted in
the figure is evidence for the existence of an intermediate phase In order to confirm that the binding
energies had indeed converged a larger system of 1500 atoms was simulated Good quantitative
agreement was found between the 150- and 1500-atom systems In this work the isostatic behavior
occurs over a wide range in which certain silicon network sites dramatically increase the binding
sites (binding energy approaching 0) The state or structure with nc outside of the intermediate range
will readily interact with water Only those in the isostatic region are likely to remain unaffected
after the interaction with water thus they are the rate limiting species in dissolution of silica glass
123
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of constraints for
surface atoms at the local pixel Results show a distinct maximum in which there is a near
hydrophilic-hydrophobic transition of the surface The error bars represent the standard deviation
A second system with 1500 atoms was performed to show convergence of the ReaxFF MD results
⟨119889119867minus119874119866119897119886119904119904⟩ = 345 Aring
⟨119889119867minus119874119866119897119886119904119904⟩ = 248 Aring
124
The intermediate phase result is intuitive due to the implicit stability that comes with having an
isostatic phase which is both energetically favorable and stress-free The larger negative binding
energies in the over-constrained regions arise from large differences between the free energies of
the chemically bonded state (chemisorbed) and unbound state (water in near proximity to the
surface but not bonded) in these regions in this region the chemical reaction with the water
molecule allows the glass surface to reduce the number of incompatible constraints which in turn
alleviates local stress in the network In other words a local region undergoes an alleviation of
stress when Q4 units convert to Q3 units The larger negative binding energy values in the under-
constrained regions can trace their origin to the sites available for bonding with the water molecule
Their high number of degrees of freedom allows for the facile reaction of water ie the only energy
cost is for the oxygen to dissociate from the network The isostatic network within the intermediate
phase is able to preserve its structural integrity because the energy barrier to deform the network is
high and there is no localized stress to create an additional driving force for reaction with water
Indeed the isostatic network is the most energetically favorable arrangement of a non-crystalline
structure This intermediate phase is governed by the fluctuations of the topology that arise with
the minimization of stresses These stresses change the local topology (within the range of the
fluctuations) so that they become isostatic[68] This leads to a flat region where there is a constant
binding energy at the surface Though it appears to be an intermediate phase it is important to note
that this may instead be simply a manifestation of an isostatic threshold (a peak in binding energies
rather than a plateau)
The average number of surface constraints has been shown in previous work to be largely
controlled by the annealing time[172] The surface constraints may then be used to predict surface
reactivity However it is worth noting that the local number of constraints is not constant in
time[17] Furthermore Potter et al[168] showed that molecular water in the network can radically
shift the free energy of the γ constraint Therefore the number of surface constraints is dynamic
125
during the influx of water Shifting constraint energy will also alter the fragility which in turn alters
the diffusion activation enthalpy In the future this may provide a path for developing more durable
glasses ndash currently it shows that a homogeneous intermediate phase (isostatic) surface should be
targeted to achieve maximum chemical durability
62 Elastic Modulus Prediction
Tailoring the elastic properties of glass is important in the design of new advanced compositions to
control the stiffness and damage resistance of a variety of glass products[2] [191]ndash[193] Elastic
moduli are defined based on the ratio of the stress exerted upon a material to the resulting strain
While various elastic moduli can be defined (eg Youngrsquos modulus shear modulus bulk modulus
and Poissonrsquos ratio) for an isotropic material such as glass only two of these quantities are
mutually independent[1] [191] [194]
Previous attempts to model elastic moduli have focused on either computationally costly
molecular dynamics simulations or empirical fitting methods The work by Makishima and
Mackenzie[195] offered a possible solution for predicting elastic moduli based on the dissociation
energy per unit volume of the glass however topological changes in the glass network are ignored
and the model cannot account for the temperature dependence of modulus With MD simulations
elastic moduli can be obtained by applying a stress and measuring the resulting strain on the
simulation cell assuming that accurate interatomic potentials are available[2] Machine learning
has also been applied to model elastic moduli using experimentally measured composition-property
databases with a high predictive ability being achieved[2] [196] Previous analytical modeling
techniques related to the topology of the glass-forming network have shown good qualitative
agreement with compositional trends in modulus but have a lack of quantitative accuracy[64]
[197]ndash[199]
126
It is common in the glass community to assume that Youngrsquos modulus may scale linearly
with hardness While this is clearly an oversimplification let us begin with previously derived
models for glass hardness as a potential starting point since glass hardness has received
considerable attention in the context of topological constraint theory [85] [192] [200] Several
models have been proposed to explain the origin of glass hardness all of which have a linear form
( ) vv
dHH n x n
dn= minus (101)
where Hv is the Vickers hardness x is chemical composition nrsquo is the critical level of constraints
needed for the substance to provide mechanical resistance to the indenter in three dimensions and
n is a measure of the constraint rigidity Recent work on glass hardness[192] has shown that n can
most accurately be defined in terms of either the density of rigid constraints (or alternatively the
density of rigid angular constraints) given by
)( )
( )( )
(c An xn
x Nx
M x
= (102)
where nc is either the number of constraints per atom or number of angular constraints per atom ρ
is the density of the composition M is the molar mass and NA is Avogadrorsquos number Since
hardness and elastic modulus are often considered to be correlated one might surmise that the
elastic modulus might also be some function of angular constraint density or total constraint
density However such an approach is not able to give quantitatively accurate predictions of
modulus
Here we propose an improved model of Youngrsquos modulus based on the free energy density of the
topological constraints
int
( ) ( ))
(
(
) i constr
c i
a s
Ai in x q T
M x
x NF F
=
= (103)
where qn(T) is the onset function for each associated constraint to give temperature dependence to
127
the predictions and is determined by the free energy ∆Fn The calculated values of Δ119865c are then
used to calculate Youngrsquos modulus (E) by
( )c c
c
FdE
EF
xd
F
= minus (104)
In this new model the Youngrsquos modulus is controlled by the free energy parameters related to the
strength of each constraint Each parameter in this model captures a key physical phenomenon with
each parameter corresponding to a physical property Moreover we demonstrate that a common
set of parameters can simultaneously capture both the temperature and compositional dependence
of modulus as well as predict other properties such as the glass transition temperature and fragility
To demonstrate the validity of this temperature-dependent constraint model Youngrsquos
modulus data are collected from literature for lithium borate[195] sodium borate[195] and
germanium selenide glass systems[201] Additionally the sodium phosphosilicate[202] system
was experimentally determined following the procedure described by Zheng et al[183] using room
temperature resonant ultrasound spectroscopy on prisms of dimensions 10 mm times 8 mm times 6 mm
Analytical topological constraint models have already been published for each of these
systems[17] [57] [202]
The proposed models for glass elasticity (the models previously proposed for hardness
based on constraint density and angular constraint density as well as the new model for modulus
based on the free energy density of constraints) were optimized in each case leaving the intercept
and slope (119889119864
119889119899
119889119864
119889119880119888) as adjustable parameters The constraint onset temperatures were also optimized
for the free energy density model Figure 31 and Figure 32 show the results for each model in the
phosphosilicate and sodium borate systems respectively
128
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values of the model
predictions are 641 GPa for constraint density 313 GPa for free energy density and 774 GPa
for angular constraint density
129
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from minimum
to maximum error The free energy density model gives the most accurate results The constraint
density has a RMSE of 61 GPa the angular density has a RMSE of 20 GPa and the energy density
has a RMSE of 59 GPa
130
We have also evaluated the predictive ability of the model in terms of the temperature dependence
of the Youngrsquos modulus sing the 10 Na2Omiddot90 B2O3 data reported by Jaccani and
Huang[191] the model and experimental predictions of the temperature dependence of the Youngrsquos
modulus are plotted in Figure 33 Here the constraint onset temperatures are determined from the
room temperature modulus data The number of escape attempts was optimized since this controls
the width of each transition and is thermal history dependent
131
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and experiment for
10 Na2O 90 B2O3 Using the previously fitted onset temperatures the only free parameters are
then the vibrational frequency and the heating time in which their product was fitted to be 14000
Where each dip in the modulus corresponds to a constraint no longer being rigid as heated through
each onset The onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The data was fit
using a least-squares method and the resultant fit is shown as the calculated method The fit has an
R2 of 094 (B) The contribution from each constraint to the overall modulus
It is also possible to validate the model by comparing the free parameters fitted to the Youngrsquos
modulus data vs that of previously reported data as shown in Table 7 Results for two additional
systems are plotted in Figure 34 to show the general validity of the free energy density model
132
Table 7 Fitted values from this analysis compared to those reported in the literature The disparity
between the constraints evaluated with molecular dynamics most likely come from the speed in
which the samples are quenched
Value Fitted Value (K) Literature (K) Method Citation
Silicate Onset 2212 1986 MD (Potter et
al[168])
Silicate Onset 818 1600 MD (Potter et
al[168])
Silicate Onset 450-500 810 MD (Potter et
al[168])
Borate Onset 921 Not Reported Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 715 740-760 Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 393 328 Fitting Parameter
(Mauro Gupta and
Loucks[69])
133
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in Fig 3) and
experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of 093 and (B) xLi2Omiddot(100-
x)B2O3 glasses with an R2 of 0986
(
A)
(
B)
134
In the previous work of Zheng et al[192] concerning hardness of glass the authors write
ldquoIt should be noted that each bond constraint corresponds to a certain energy since different kind
of bond has different bonding energyhellip and thus the constraint density also represents an energy
per unit volume In other words hardness is correlated to the energy per unit volume Our findings
for the borosilicate and phosphosilicate systems are further evidence in support of this argument
since both the total constraint density and angular constraint density approaches give better
prediction of glass hardness compared to models based on number of atomic constraintsrdquo
which makes it clear that the prediction of the hardnessmodulus should be closely related to the
onset temperatures (ie free energies) of the associated constraints The model that was previously
proposed could be extended to modulus if all constraints had equal amounts of potential energy
but due to the drastic change in strengths between the constraints the prediction fails
Bauchy et al[85] showed also that hardness in the calcium-silicate-hydrate system is
controlled by the angular constraints which led to the development of angular constraint density
as the governing control for hardness When the analysis of the density of angular constraints is
considered through the free energy view it becomes apparent why this method works effectively
for some systems Models for predicting the hardness and elastic modulus of glass attempt to
explicitly connect the rigid bond energy to the macroscopic properties of the system The
assumption for the hardness models is that the energies for breaking each type of constraint are all
approximately equal and hence only the number of the constraints matters The argument can then
be extended for elasticity since elastic modulus is a bulk material property with the approximation
of all constraints being equal strength the number of constraints per volume should be related To
correct for this approximation when a weighted sum of free energies is used the estimation
becomes significantly more accurate Using the density of the glass a precise free energy per
volume of the rigid constraints can be calculated The knowledge that the energy of the bonds is
tied to the elastic modulus has been widely known but had not been quantified nor placed within
the context of topological constraint theory[203]
135
63 Ionic Conductivity
Ionic conductivity 120590 is related to the number of charge carriers n and the mobility of the
carriers 120583 as they diffuse through a network by[1]
Ze n = (105)
where Ze is the charge of the conducting species Since the charge-carrying ion is constant within
a given glass family the ionic conductivity depends only on the mobility and the concentration of
the charge carriers present However it is not typically feasible to measure 120583 and n independently
of each other and thus it is unclear which variable scales in an Arrhenius fashion thereby
controlling the ionic conductivity of a glass In the weak electrolyte model by Ravaine and
Souquet[204]ndash[207] the mobility of the charge carriers is considered constant while the
concentration scales in an Arrhenius function Conversely the strong electrolyte model assumes a
constant number of charge carriers while the mobility follows an Arrhenius function[208] [209]
Recent MD simulations from Welch et al[210] support the weak electrolyte interpretation for alkali
silicate glasses concluding that ionic conductivity is dictated by the concentration of charge
carriers Previous reports in the literature also supporting this view have proposed several
hypotheses to explain this conclusion[208] [209] [211]ndash[213] however further exploration is
needed
The simplest form of the temperature dependence of free ions is given by
0 exp an nE
kT
= minus
(106)
Here n0 is the total possible number of mobile ions and Ea is the activation barrier for an ion to
diffuse It is often assumed that the activation enthalpy is related to the energy needed to deform
the network temporarily to allow for ion motion[193] [209] [210] [212] however it is our
hypothesis that there is a coupling between the ion hopping event and the relaxation of the local
136
network allows for a permanent relaxation or deformation along ion channels This means that
there is some cooperative relaxation of the network along the diffusion path in conjunction with an
ionic hopping event These cooperative relaxations are discussed explicitly in the Adam-Gibbs
formalism[53] for describing the relationship between viscosity and configurational entropy Sc
expc
B
TS
=
(107)
where B is an activation barrier is the viscosity and is the viscosity of the liquid in the limit
of infinite temperature In the high-temperature (low-viscosity) liquid state there is a well-defined
relationship between the viscosity and diffusion (Stokes-Einstein relation) However this
relationship breaks down in the low-temperature (high-viscosity) glassy phase
Herein we present a new model for predictions for the compositional dependence of ionic
conductivity We can write the activation free energy barrier Eac for an ionic hopping event as
a c c c cE T S= minus (108)
Here is an enthalpic barrier for the ion to hop to a neighboring site CT is the configurational
temperature (a value that describes the distribution of energies of the ions) and cS is the entropy
of the activation barriers These quantities are associated with configurational changes in the glass
indicated by the subscript c In the weak electrolyte model all charge carriers have the same
mobility and hence the entropic effects are dominant This indicates a small value of an
assumption that will be validated by results later in this work leading to a simplified approximation
for the activation energy at temperatures below the glass transition but suffiently high temperatures
such that
a c c cE T S (109)
In the case of a nonequilibrium glass at low temperature the system becomes trapped in a localized
137
region of the energy landscape known as a ldquometabasinrdquo with slow inter-metabasin transitions[156]
[161] The MAP model of the nonequilibrium viscosity of glass relies on this partial confinement
within metabasins where variation of the activation barrier for relaxation (H) is related to the
fragility of the system by[44]
1
ln10g
dm
dH kT= (110)
Here m is the fragility index (as proposed by Angell[58]) and Tg is the glass transition temperature
Splitting the differential in Eq (110) and integrating we can write
ln10gmH kT= (111)
This relationship was also proposed by Moynihan et al[214] for the activation enthalpy of
relaxation of a glass[214] [215] Setting the Arrhenius form of the barrier described in Eq (111)
equal to the Adam-Gibbs equation (Eq (107)) we write
exp exp c
H B
kT TS
=
(112)
Solving Eq (112) for the configurational entropy and inserting the result into Eq (109) we obtain
the simple relationship
2
ln10
ca
g
T BE
k T m (113)
Dividing the above expression with that of a reference state r and assuming that B and is cT are
constant with respect to compositional variation we have
g r ra
a r g
T mE
TE m= (114)
This new equation is the first to relate the activation free energy for ionic hopping in a glass
to the glass transition temperature and fragility of its corresponding supercooled liquid It also
138
predicts a coupling between ionic diffusion and the cooperative rearrangements described by the
Adam-Gibbs relation This possibility was first proposed by Ngai and Martin[216] who showed a
correlation between the product of the Kohlrausch stretching exponent ( ) and the activation
barrier for relaxation with the activation barrier for ionic conductivity This implies that the
diffusion in glasses is not governed by an elastic component but instead controlled by the
surrounding cooperative rearrangements ie the -relaxation of the glass[14] in good qualitative
agreement with previous results shown by Potuzak et al[217] Further evidence comes from the
polymer community where the decoupling of relaxation and conductivity is reported with work
showing that the decoupling of viscosity and diffusion varies with the fragility and the glass
transition temperature[218] This is a general relationship which we demonstrate later to be valid
for binary alkali borate alkali silicates and alkali phosphate glasses as well as validating the
atomistic diffusion method using computational techniques
In order to test this proposed relationship between relaxation and diffusion through
simulation a glass with the composition 10 Na2O-90 B2O3 (mol ) was synthesized in molecular
dynamics with 1150 atoms using the potentials of Wang et al[162] The system was quenched at a
rate of 1 Kps from randomly placed atoms at 2500 K in the NPT ensemble to room temperature
as shown in Figure 35[210] To find the transition point energy of the diffusing ion a nudged elastic
band (NEB) calculation was performed An alkali ion was chosen and its closest alkali neighbor
was moved to another location so that the NEB calculation could be performed between those two
sites All calculations were carried out using the LAMMPS software package Eleven reaction
coordinates were used in the NEB calculation and the results are shown in Figure 36
139
Figure 35 The structure for the initial minimum energy configuration showing the boron (blue)
network with interconnecting oxygens (red) and the interstitial sodium ions (yellow)
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and sodium is
ivory The barrier is overestimated compared to experimental data this could be from several
sources of error such as potential fitting thermal history fluctuations or sampling too few
transitions The line is drawn as a guide to the eye
140
Although the transition point energy is key for determining the activation barrier for diffusion here
we are also interested in the dynamics of the atoms around the mobile ion Since we have
hypothesized here that the ion diffusion is dependent on the ability for the local region to relax and
deform instead of elastically straining we expect a local deformation (relaxation) along the ion
path Figure 37 shows that along the ion diffusion tunnel there is indeed a permanent displacement
of the atoms
141
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function of
reaction coordinates The middle shows the local deformation around the ion of any atom that
moves in between inherent structure mandating a relaxation force The color shows the degree of
deformation
142
In the final configuration of Figure 37 the permanent deformation required to lower the
energy of the system is clearly seen If it were purely an elastic response to dilate a pathway the
displacement between two inherent structures should be confined to changes with only the mobile
ion however this is not observed Instead along the pathway of the ion movement the network
forming atoms also deform and change local positions to minimize the energy implying a net
relaxation around the ion ldquopathwaysrdquo To apply this and show the validity of Eq (114) the sodium
borate lithium borate lithium phosphate and sodium silicate systems were considered with the
model results compared to experimental values in Figure 38 To confirm this model literature values
for activation barriers have been used The experimental activation barriers for the borates are taken
from the work of Martin[213] while the fragilities and the glass transition temperatures are from
Nemilov[219] The viscosity curve of the silicate system was fit with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) equation of viscosity[55] using the data from Poole[220] and Knoche et
al[221] with the activation barriers from Martinsen[222] Lithium phosphate activation data was
taken from Ngai and Martin[216] with the glass transition and fragility taken from Hermansen et
al and predicted values from topological constraint theory[223]
143
Figure 38 Different network formers and the prediction of the activation barrier from our model
compared with activation barriers from literature (A) Sodium silicate predictions and experimental
values[222] the error is calculated from the error in the fragility when fitting the data (B) Lithium
phosphate activation energy[216] predicted with topological constraint theory and compared with
the experimental values (C) Predictions over two different systems of alkali borates[213] sodium
and lithium with a reported R2 of 097
144
It is worth noting that this model is predicting a non-equilibrium kinetic property using
equilibrium viscosity parameters a connection that has been demonstrated in the past for viscosity
and relaxation models[61] In this work we are applying the same principles to the diffusion of
ions in a network The diffusion and viscosity in the case of equilibrium are known to be intimately
linked using the Stokes-Einstein expression (which breaks down for highly cooperative
rearrangements near and below the glass transition temperature[224]) Here we are no longer
connecting the two properties explicitly but instead connecting the activation barrier for diffusive
hopping to the two key parameters governing the viscous flow of the supercooled liquid state
The failure of the Stokes-Einstein expression near the glass transition is a well-known
problem and has been studied extensively for purposes of predicting crystallization rates[167] This
occurs because there is a breakdown of the ergodicity of a system around the glass transition In
the MAP model[44] the viscosity is controlled by the ergodic parameter x and is given by
min( )
max( )
f
p
f
T Tx
T T
=
(115)
Here the exponent p is related to the sharpness of the ergodic to non-ergodic transition Using the
ergodic parameter the overall viscosity ( ) is expressed as a linear combination of equilibrium
viscosity ( eq ) and non-equilibrium viscosity ( ne ) such that
10 10 10log ) log ( ) (1 ) log ( )( f eq f ne fx T T x T TT T = + minus (116)
By combining the Stokes-Einstein expression and the model proposed in this work we can define
the diffusion of an ion through the network as
( )10 10 10 0log ( 1 log exp ) l6 )
og(
b af
f
kx
T ED T T D
a T Tx
kT
minus + minus =
(117)
Here a is an empirical constant related to the size of the diffusing species k is Boltzmannrsquos
145
constant aE is the activation barrier calculated with Eq (114) and D0 is the limit of diffusion as
the temperature approaches infinity D0 should be chosen such that at fT T= the diffusion is
continuous This can be justified because the preexponential factor is related to the entropy of the
system and the fictive temperature is the temperature at which the configurational entropy of the
transition barriers becomes nearly constant Only one parameter is left unknown a since the
viscosity and fictive temperature can be calculated with an arbitrary thermal history using a
relaxation modeling tool such as RelaxPy[42] Implicit in RelaxPy are the assumptions pertaining
to the viscosity made by Guo et al[61] A similar model was first proposed by Cassar et al[167]
for modeling diffusion during crystallization although their model used an empirical hyperbolic
tangent function to approximate the ergodic factor In contrast the current model can be considered
explicitly as a function of thermal history and composition
64 Machine Learning Expansion
Though the TCT models presented here for different properties are immensely useful they
do have limitations the greatest of which is that intimate knowledge of the glassy structure is
required to parameterize these models This is due to the fact that the landscape is fundamentally
informed by the underlying structure of the network and as such to predict the dynamics of the
landscape information on the structure must be incorporated in some form However there is an
alternate method in which the physical origin is not considered in the development of the model
machine learning Machine learning comes in many forms and has become crucial to the study of
materials in the modern era There is a plethora of machine learning techniques however in this
paper we are focused on four key methods linear fitting (LF) methods random forest (RF)
symbolic regression (SR) and neural networks (NN)[3] [80] [225] [226] These methods are
146
explored extensively in literature and the exact underlying mathematics is beyond the scope of this
work In this work these methods have been used to create models for number of constraints (LF)
glass stability (RF) fragility (NN) melting point (NN) the infinite limit of viscosity (SR) Youngrsquos
modulus (NN) and coefficient of thermal expansion (NN)
To enable a linear fitting method for the number of constraints a simple steepest descent
algorithm was used to find the number of constraints due to each component There is no direct
experimental way to measure the number of constraints so instead a technique leveraging some
other property must be used to approximate the number of constraints In this case hardness is
related to the number of constraints through
25v cH A n= minus (118)
In which A is an empirical scaling parameter that is dependent on load indenter geometry and
glass family In this linear approach to determining constraints cn can be given by
0
compon
k
c k k
ents
cn x n=
= (119)
Combining this expression with a hardness database grouped by composition (in this case Sciglass)
allows for an approximate number of constraints of the system but due to A being a variable there
must be a fixed constraint in the system and every set of data is given a unqiue value of A The
fixed constraint in this system is that silicon atoms are four coordinated The value of A is chosen
such that the error between the prediction and the hardness is minimized The total process is as
follows
Choose a set of c kn
1 Calculate number of constraints for all compositions
2 Iterate over every dataset in Sciglass with gt 1 datapoints and scale by a factor of A
3 Calculate total error for steepest descent algorithm
4 Descent error slope
5 Repeat until error has been minimized
147
This then leads to an estimate for the number of constraints for every atomic species which can
then be used to estimate important properties such as chemical durability hardness and
qualitatively can predict glass stability
Two other techniques that are used are RF and SR When implemented a grid search for
the hyperparameters were used with a testing set used to determine the efficiency of each set of
hyperparameters Additionally each model was trained repeatedly and the lowest error was taken
The RF method was used to predict glass stability while undergoing a novel processing technique
called alkali-proton substitution (APS) The SR method was used to understand the compositional
dependence of the infinite temperature viscosity limit used in 3-parameter viscosity models To do
this the Sciglass database was taken (as implemented in GlassPy[227]) and sets of data that
included more than eight datapoints were fit to the MYEGA equation If the RMSE error of the fit
was less than 001 Pa s the fragility was greater than 17 and the value of the log infinite limit was
293 5minus it was accepted into the final database Each of these criteria were added due to the well-
known fact that this database is full of error and imprecise measurements The symbolic regression
was then performed through the grid search and the equation that performed best and reappeared is
given as
10loggT
mA B = + (120)
With A and B being parameters that varied depending on the hyper-parameters the infinite limit
threshold and the accuracy threshold used when creating the database We are currently in the
process of refining values for A and B is ongoing
This method allows for an easy prediction of the infinite limit so if the glass transition and
fragility are predicted through some other means (such as ML or TCT) a better prediction of the
third parameter is known without just accepting -293 as the mean Plots of the infinite limit vs the
148
fragility glass transition and Tgm showing the clear dependance of the infinite limit on the term
Tgm are shown in Figure 39
149
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top left) shows
the distribution of the infinite temperature limit in the database after limits exerted on the system
(Top right) The distribution of the infinite viscosity limit vs the glass transition (Bottom left) The
relationship between fragility and infinite temperature limit of viscosity (Bottom right) The infinite
temperature limit of viscosity vs the key metric predicted by SR
150
NN are perhaps the most widely discussed ML tool applied in literature This makes sense
given they are universal function approximators and are quite easy to train given sufficient data
The most difficult part of creating a NN is adjusting the hyperparameters so that good accuracy can
be found without causing overfitting To avoid overfitting the data is divided into a testing set then
a training set (just as for RF and SR) but instead of doing a grid search for the hyperparameters we
used gradient boosted regression trees to find the optimal set of parameters This consists of
creating an initial tree then evaluating it based on a small set of data and finding the minimum
point then running a new set with perturbations around that point to create another tree This is
repeated until the value converges to a small error In this work the value being determined is the
lowest error of 100 sequentially trained neural nets with a set of hyperparameters This is
particularly useful because there is no concern for the type of parameters (string int floats) that
determine the value at the end so one can do a mixed parameter optimization that includes every
conceivable variable The values being optimized are given by
bull Learning rate [10-5 10-1] (float on logarithmic scale)
bull Decay rate for Adam Optimization [10-6 10-2] (float on logarithmic scale)
bull Hidden layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull Output layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull The number of nodes in each layer [81024] (int)
bull Batch size [1256] (int)
bull Patience [5250] (int)
In Table 8 we have listed the hyper-parameters that were used for the fragility melting point
Youngrsquos modulus and co-efficient of thermal expansion (CTE)
151
Table 8 Hyperparameters for different neural networks after hyper-optimizations
Model Fragility Melting
Temperature
Youngrsquos
Modulus
CTE
Learning Rate 0035 0018 0027 00091
Decay Rate 000293 000095 000161 0002
Hidden Layer RELU RELU RELU RELU
Output Layer Linear Linear SELU RELU
Number of Hidden
Layers
2 1 1 1
Number of Nodes 235 506 514 470
Batch Size 4 118 211 125
Patience 25 250 23 22
RMSE 5 [-] 805 [K] 68 [GPa] 96 x 10-7 [1 K]
152
Chapter 7
Designing Green Glasses for the 21st Century
Thus far the work presented in this dissertation has enabled a new set of design tools for
everything from crystallization glass relaxation and optimizing over the ideal compositional space
we can use them for designing new glasses Building on these models and insights new applications
of glass are enabled specifically three glasses an ion conducting glass for batteries a proton
conducting glass for hydrogen fuel cells and a commodity glass for consumers All three are
currently undergoing testing but experimental results have been reported back as of yet Overall
both the hydrogen and ion conducting glasses have been designed thorough a combination of ML
and underlying physical models and as such hare reported herein
71 Glass Electrolytes
An emerging high-interest application of glass is solid state batteries Glasses offer a
possible solution to the growing energy crisis however to realize this new paradigm new research
methods are needed The main barrier for any new batteries to be commercialized is a high
conductivity at room temperature (gt 10-3 S cm) high stability ease of processing and ultimately
an alternative must be cheaper than current liquid state Li-ion batteries glass is an attractive
candidate due to the innate stability a simple processing technique the infinite variability and the
relatively low cost of production In addition the structure of a glass is a liquid-like structure which
may encourage ion migration like that seen in liquids Despite all of these advantages only a few
compositions have realized the requirements but further research is needed to find optimal
compositions
153
To optimize the composition for activation barriers we must understand the relationship
between conductivity and structure Structural effects can be propagated in two different methods
based on the Arrhrenius expression for conductivity
0 exp aE
kT
minus =
(121)
The two methods are through 0 (the infinite temperature conductivity) and the activation barrier
( aE ) The pre-exponential factor is approximately a constant for simliar compositions while the
activation barrier varies dramatically over small compositional spaces making it the larger of the
two concerns Many models have been presented to predict the compositional dependence of the
activation barrier the Anderson-Stuart (AS) model the Christensen-Martin-Anderson-Stuart
(CMAS) model the weak electrolyte (WE) model Kohlrausch exponent model (KEM) from Ngai
et al[216] and there is the model proposed in this dissertation which we will call the Wilkinson
viscous cooperative conductivity model (WVCC)[8] [88] [204] [207]ndash[209]
To optimize machine learning (ML) is a powerful technique that is implemented The best
method to train the ML is to use the direct relationship between the compositions and the activation
barrier but this is not currently feasible due to a lack of central depository or database for this
information To make the prediction of the activation barrier we need use some model that is reliant
on commonly accesible parameters This rules out all models for this type of analysis except since
WVCC since the other models require some fitting parameters not accessible Each models required
parameters fitted values (please note that the models typically include approximations for these
values) and caveats are shown in Table 9
154
Table 9 A table with some ionic conductivity models and the parameters needed for them as well
as the disadvantages for each These are not the only models but are representative of those
commonly used in literature
Model Required Parameters Fit Parameters Caveats
AS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
lsquoMadelungrsquo Constant
Covalency Parameter
Doorway Radius1
The Madelung
Constant is not smooth
as a function of
composition and the
fitting parameters scale
the activation energy
non-proportionally
CMAS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
Dielectric Permittivity
lsquoMadelungrsquo Constant
Doorway Radius1
Jump Distance of Ion1
The same issues as AS
but in addition the
dielectric permittivity
lacks a database as
well
WE Charge Concentration Equilibrium Coefficient1 The equilibrium
coefficient drastically
changes the prediction
non-linearly
KEM Stretching Exponent Proportionality
Coefficient
There is not enough
data available to know
the compositional
dependence of the
stretching exponent
WVCC Glass Transition
Fragility Index
Proportionality
Coefficient
Due to the
proportionality
constant the activation
barrier can only be
known in a local
composition range
1 There are ways to approximate this value based on additional data
155
The WVCC model predicts that the activation barrier for ionic conductivity is given by
a
g
EA
mT= (122)
in which A is a proportionality constant This means that to predict the glass behavior we donrsquot
need to know things about the glass but instead merely about the liquid state where the viscosity is
readily available Leveraging this technique if the viscosity information as a function of
compositions is available then the local glass with the lowest activation barrier can be found by
minimizing 1 gT m To access this information multiple options are available such as topological
constraint theory but in this work we will leverage ML since it is easily applied to a large
compositional space
To get the viscosity we will use neural networks (NN) A well trained NN for the glass
transition is readily available and as such will be used for our prediction of gT In the previous
section there was presented a trained fragility NN which enables a complete optimization of the
local composition based on WVCC To generate a database of fragility values that the NN was
trained on SciGlass viscosity values were fit with MYEGA model The data had to fit following
criteria
1 The total number of data points for fitting had to be greater than 5 from the same literature
source
2 The root-mean-square-error of the fit had to be less than 001 Pa s
3 The infinite temperature value of viscosity had to be within 3 orders of magnitude of the
accepted logarithmic value (-293 log(Pa s))
An additional neural network recently created by Cassar[228] was also used for an independent
comparison Cassarrsquos NN was based on a hybrid physical ML work and may offer better insights
when extrapolating far from experimental datapoints In this study to prove this concept we will
only focus on P2O5 B2O3 Al2O3 and Li2O
156
This gives an entire method to predict glass battery candidates This technique will find all
of the local minimum but due to the fact that there is a proportionality constant it is unknown which
local minimum is the absolute minimum This means that the composition with the lowest
activation barrier in the space is identified but another technique is needed to investigate which one
of the identified compositions is best The method follows
1 Randomize an initial glass candidate with the content of being normalized
2 Run the glass candidate through the NN to find values of fragility and the glass transition
3 Perturb the composition to find the gradient
4 Step down the gradient
5 If the new gradient is 0 this is a local minimum and a possible glass candidate return to 1
Otherwise return to step 3
This method cannot find global optima because there is no guarantee that the
proportionality constant will stay constant across wide compositional spaces To narrow it down
an additional technique to compare the predicted values is needed (such as another activation
barrier model molecular dynamics simulations a regression algorithm or simply melt the proposed
compositions) To further pair down in this work a k-means algorithm was used along with the
assumption that the proportionality constant was slowly changing over compositional space This
allowed us to take the best glass in each grouping predicted by k-means The glasses we have
predicted (using both the separate predictor NN and the method by Cassar et al[228]) are listed
below in Table 10 while an example prediction for 1000 gmT (which should scale with the
activation barrier) in the Na2O ndash B2O3 ndash SiO2 system is shown in Figure 40 The glasses are currently
undergoing melting at Coe College and Iowa State It is worth noting that composition D is a well-
known composition that is known to exhibit criteria close to the criteria for conductivity If this
method shows that these compositions are better than other compositions in the family found in
literature then it can be expanded to include a wider range of component and hopefully finding a
universal candidate for glass batteries
157
Table 10 The predicted compositions based on the optimization scheme proposed
Compositions Li2O Al2O3 P2O5 B2O3
A- This work NN 55 31 0 14
B- This work NN 44 1 5 50
C- Cassar 60 26 2 13
D- Cassar 50 0 0 50
Figure 40 The prediction of the scaling of the activation barriers for a common sodium borosilicate
system
72 Hydrogen Fuel Cell Glasses
An emerging high-interest application of glass that is proton-conducting intermediate
temperature fuel cells[7] [229] A fuel cell in short is a device that turns chemical fuel into
electricity without the use of combustion It is often described as a chemical battery due to the
similarity of a fuel construction with batteries One of the crucial components to a working
intermediate range fuel cell is an electrolyte with a high proton migration The electrolyte must be
stable over a wide range of temperatures (up to 673 K) and a proton conductivity greater than 10-2
S cm-1 at the operating temperature[7] Previously phosphate glasses near the metaphosphate
compositions were used since they carried the appropriate number of residual protons for
conductivity studies However this restricted the phase space to only n-2 dimensions drastically
reducing the degrees of freedom of the problem and artificially restricting possible solutions In
2013[230] to access the full phase space a new method was developed where a sodium phosphate
glass could have all the sodium ions replaced with protons This allows high concentrations of
protons to be achieved with a variety of starting compositions A previous report[230] has
summarized successful samples these data are used as a training set for this study with additional
data on compositions that have failed taken from internal theses
To choose the candidate material for each model a cost function is defined The cost function
is an analytical function whose inputs are the fraction of each oxide component and the output is a
value that rates the composition The cost function consists as of many properties that are of interest
In the application of oxide electrolytes for an intermediate range fuel cell the two main properties
to worry about are the ionic conductivity at high temperatures (we will choose 473 K due to this
being the low end in which these materials are considered) and the relative stability of each phase
159
This is especially important since the material will undergo the APS (alkali-proton substitution) in
which the sodium will be replaced with protons in the bulk of the material The cost function used
in this work is given by
473 1( ) log 000T K Sx = minus= minus (123)
In which S is either a 0 or 1 (representing a crystal or a glass respectively) and is the
ionic conductivity The function is arbitrary and could be expanded to include as many terms are
as needed in the goal of this glass The stability is heavily weighted because it is more important
than the conductivity and any sample that is not stable should not be considered In order to predict
each of these samples a hybrid physicalempirical approach is used For predicting the ionic
conductivity of a proton-swapped glass we began with an observed relationship between the glass
transition temperature and the conductivity at 473 K This empirical relationship is shown in Figure
41 The relationship although empirical can provide some insights into the physics of proton
transferred glasses (no universality is claimed in this work) The activation barrier for these glasses
has widely remained unchanged as noted in a previous work due to the fact that the activation
barrier is simply to disassociate the proton with the non-bridging oxygen (NBO) and once the
proton is free interacts rarely with the network until it is rebounded to said network Thus the pre-
factor can be understood as being related to the degeneracy of proton conduction pathways When
the network has lower configurational entropy there are fewer pathways for the proton to travel in
this case it is known that the configurational entropy is then the dominating effect for the glass
transition[53]
The relationship between the glass transition temperature and the configurational entropy
can be expanded through the Stokes-Einstein relationship evaluated at the glass transition
22
1273 10
6 10g
g
T T g
kTnZeD nZe
anT
minus
= = = (124)
160
In which a is approximately the size of a proton and n is the number of charge carriers (which is
approximately a constant according to the weak electrolyte theory) This determines the intercept
of the glassy form proton conductivity and then the activation barrier (slope) is found to be
approximately the same for all proton conducting glasses meaning that the intercept is the dominant
effect on the behavior of the glass and the only variable controlling the intercept is the glass
transition
Leveraging the configurational entropyrsquos relationship with the glass transition along with
topological constraint theory we can write an expression for the glass transition and as such gain
predictive power for proton conductivity To predict the glass transition temperature (Tg) we start
with the Adam Gibbs model of viscosity[53] Topological constraint theory then states that the
degrees of freedom of the network is proportional to the configurational entropy[23] [56] [57]
Using the liquid state definition of the glass transition (1210 = Pa s) and the well-known infinite
temperature of viscosity we can then write
1
1493 lng
BT
kf=
(125)
Where f is the degrees of freedom Letting the constants be equal to A and rewriting the expression
in terms of constraints we arrive at a predictive formula for the glass transition[109]
3
g
x c x
x
AT
m n=
minus (126)
Where nc is the number of constraints provided by each component at the glass transition associated
with each component x and their molar fraction mx The value of the constraints was then linearly
parameterized to the glass transition data from literature[7] [230] [231] This is not the most
explicit approach to counting constraints however it is the most convenient when considering
large phase spaces being optimized over This glass transition temperature is then converted to
conductivity using the empirical translation shown in Figure 41
161
The other term in our cost function is that of stability The question of what forms glass is
a notorious question that goes back to the first serious days of research into materials It has been
shown that no one metric is a good predictor and no metric is universal However one must be used
to rule out bad compositions from the start To do this a Random Forest method is implemented
Though this may fail in edge cases it will at least help identify the right area to explore A
comprehensive review of random forest methods and machine learning methods for prediction of
glass properties can be found elsewhere[3] [4] [225] The amount of information previously
obtained about what forms glass in the compositional family we will be working on is not enough
to use random forest methods on just the fraction of each phase In order to circumvent this issue
we chose four physical parameters that would be used as features for the predictions
bull Total mol oxide modifier This was chosen because it is well known that modifiers break
up the network and most glass forming theories are related to the network percentages We
considered the oxide versions of H Na Ba Sr and Ca as the modifiers
bull Total mol network former Similarly it is well known the network influences glass
forming We considered B Ge and P as the network forming cations
bull Mean cation charge Though there has been previous work on this topic it is rarely
considered as a lsquonormalrsquo predictor of glass forming capability It is included here because
these glasses are mainly invert and as such a higher field strength will increase the cohesive
nature of the network
bull Entropy of mixing This is an important parameter to consider with so many components
it may be stabilized by the entropy of mixing The entropy of mixing (Sm) was calculated
using the Gibbs entropy
The Gibbs entropy is given by
lnm x
x
xS k m m= minus (127)
The random forest model consisted of 100 trees with a maximum depth of 3 The results of both
the random forest model the topological predictions the optimization and the resultant glass are
shown in the results section The optimization was limited to the range of each component that has
been explored experimentally as well as an additional constraint of at least 5 mol rare earth oxide
and at least 5 mol of boron oxide or germania this is justified elsewhere[231]
162
Figure 41 The relationship between the glass transition and the proton conductivity This is justified
two ways one through the relationship of the entropy of diffusion and glass formation (the Adam-
Gibbs model) and through the fact that water is known to depress the glass transition
163
To explain the results we will first start with the models listed to confirm their validity
The first model is the prediction of the glass transition temperature Figure 42 shows the accuracy
of the glass transition prediction as well as the relative contribution of each component to the glass
transition In this figure any component that has a number of constraints greater then PO52 will
increase the glass transition and as such decrease the conductivity Thus it is preferable to construct
a glass with only components whose constraints at Tg are less than PO52 as long as the glass
remains stable Interestingly whether the data includes only the glass transitions of protonated glass
or all glasses determines the role of La Barium and sodium both decrease the glass transition
playing the role of the more traditional modifier however they do provide some rigidity to the
network Protons have a net negative effect which we believe to be a result of the APS process
where they are introduced as the sodium is leaving so this negative effect corresponds to a breaking
of the rigidity that exists It is also interesting to note that adding a boron phase to the sample
increases the glass transition though not as much as some of smaller alkaline earth samples
assumingly due to their increased field strength The worst network former appears to be GeO2
To calibrate the accuracy of the random forest model a confusion matrix is shown It shows
that the RF always predict that a glass will be a glass however there is some error when it forms a
crystal with 23 of the time being misidentified Though random forest methods are empirical we
can derive some physical meaning by looking at the relative importance of each feature From the
random forest study of stability we are able to rank the relative importance of each feature used
The mean cation charge mixing entropy and percent network formers are all of considerable
importance while the percent modifiers is less so
Once these models were developed and checked the reliability of the cost function is
confirmed After which the minimization of the cost function was performed and one glass was
chosen as optimal Boron was found to be preferential in the glass but was limited since the
compositional space was bound by previous samples (an additional sample is undergoing
164
characterization where the boron content was slightly increased) The optimized sample when
synthesized had some visible nucleation so the optimization while redone with limiting the
component maxima to just the limits of those where compositions in which APS was successfully
performed This resulted in the OP composition The three samples currently are undergoing
experimental characterization are listed in Table 11
165
Table 11 The compositions synthesized in this work These compositions were predicted by
minimizing the cost function described in Eq (123) OP is the variant that was melted after OP
partially crystallized B-OP appeared to have surface nucleation in some spots but was cut and
removed before APS treatment
Name PO52 NaO12 BaO LaO32 GeO2 BO32
OP 44 36 10 5 3 2
OP 49 36 5 5 3 2
B-VOP 44 36 10 5 1 4
166
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine the glass
forming region Over top the constraints at the glass transition provided by each oxide species is
listed Since the objective is to decrease Tg while staying in the glass forming region we will
attempt to minimize use of elements that increase the glass transition temperature (nc gt 17)
167
Chapter 8
Conclusions
The rapid rate of new information in the current scientific climate and the infinite
variability of glass both stand as both unique challenges and opportunities In this dissertation we
have divided the influence of energy landscapes on the effects of properties into two independent
spaces the compositional and the thermal history dimensions By building models that succinctly
and accurately describe the dynamics of different hyper-coordinate (composition thermal history
crystallinity) changes the feasibility of designing new glasses for the challenges of the 21st century
is obtainable
Before designing new models the current state-of-the-art models need to be implemented
and understood To reach this end two softwares based on previous work were created RelaxPy
and ExplorerPy RelaxPy is an implementation of the MAP model and though the MAP model is
powerful when predicting the dynamics of the glass RelaxPy ultimately showed that fictive
temperature is insufficient at capturing the underlying physics and is intensive to parameterize
ExplorerPy was created to standardize the approach to mapping energy landscapes Energy
landscapes served as the key method of understanding the deeper dynamics of glasses glass-
ceramics and liquids
The thermal history dependence of glass with respect to both relaxation and crystallization
has been incorporated into the existence of new models called ldquotoy landscapesrdquo The toy landscapes
have built upon the physics of previous models such as the MAP model however due to the lower
parametrization cost and the built-in increased physicality toy landscapes pose as a tool to increase
our understanding and speed-up the rate of new glass discoveries This tool can deal with the
complexity of the higher dimensional spaces due to crystallization and relaxation without
assumptions concerning fictive temperature
168
The compositional dimensions are the remaining dimensions that must be optimized when
designing a glass or glass-ceramic Building on previous work models have been developed that
enable predictions of Youngrsquos modulus surface reactivity and ionic conductivity However there
are additional properties that are needed and for those we have proposed novel machine learning
approaches This dissertation has not enabled the design of glass for every application However
it lays the groundwork and approaches to design glasses for societyrsquos growing needs Presented as
well are the methods used to design glasses that could satisfy the requirements for solid state glass
electrolytes and hydrogen fuel cells All of these techniques together promise to be a powerful new
framework to build the glasses of the future
169
References
[1] A K Varshneya and J C Mauro Fundamentals of Inorganic Glasses 3rd ed Elsevier
2019
[2] J C Mauro A Tandia K D Vargheese Y Z Mauro and M M Smedskjaer
ldquoAccelerating the Design of Functional Glasses through Modelingrdquo Chem Mater vol
28 no 12 pp 4267ndash4277 2016 doi 101021acschemmater6b01054
[3] J C Mauro ldquoDecoding the glass genomerdquo Curr Opin Solid State Mater Sci pp 1ndash7
2017 doi 101016jcossms201709001
[4] H Liu Z Fu Yang X Xu and M Bauchy ldquoMachine learning for glass science and
engineering A reviewrdquo J Non Cryst Solids no March p 119419 2019 doi
101016jjnoncrysol201904039
[5] M M Smedskjaer C Hermansen and R E Youngman ldquoTopological engineering of
glasses using temperature-dependent constraintsrdquo MRS Bull vol 42 no 01 pp 29ndash33
2017 doi 101557mrs2016299
[6] J C Mauro and M M Smedskjaer ldquoStatistical mechanics of glassrdquo J Non Cryst Solids
vol 396ndash397 pp 41ndash53 2014 doi 101016jjnoncrysol201404009
[7] T Omata et al ldquoProton transport properties of proton-conducting phosphate glasses at
their glass transition temperaturesrdquo Phys Chem Chem Phys vol 21 no 20 pp 10744ndash
10749 2019 doi 101039c9cp01502g
[8] Z A Grady C J Wilkinson C A Randall and J C Mauro ldquoEmerging Role of Non-
crystalline Electrolytes in Solid-State Battery Researchrdquo Front Energy Res vol 8 p
218 2020 doi 103389fenrg202000218
[9] R C Welch et al ldquoDynamics of glass relaxation at room temperaturerdquo Phys Rev Lett
170
vol 110 no 26 p 265901 2013 doi 101103PhysRevLett110265901
[10] M Montazerian E D Zanotto and J C Mauro ldquoModel-driven design of bioactive
glasses from molecular dynamics through machine learningrdquo Int Mater Rev vol 0 no
0 pp 1ndash25 2019 doi 1010800950660820191694779
[11] E D Zanotto and J C Mauro ldquoThe glassy state of matter Its definition and ultimate
faterdquo J Non Cryst Solids vol 471 pp 490ndash495 2017 doi
101016jjnoncrysol201705019
[12] R G Palmer ldquoBroken ergodicityrdquo Adv Phys vol 31 no 6 pp 669ndash735 1982 doi
10108000018738200101438
[13] J C Mauro P Gupta and R J Loucks ldquoContinuously broken ergodicityrdquo J Chem
Phys vol 126 no 18 p 184511 2007 doi 10106312731774
[14] M Micoulaut ldquoRelaxation and physical aging in network glasses A reviewrdquo Reports
Prog Phys vol 79 no 6 p 066504 2016 doi 1010880034-4885796066504
[15] D J Wales and H A Scheraga ldquoGlobal optimization of clusters crystals and
biomoleculesrdquo Science (80- ) vol 285 no 5432 pp 1368ndash1372 1999 doi
101126science28554321368
[16] J C Mauro R J Loucks and P Gupta ldquoFictive temperature and the glassy staterdquo J
Am Ceram Soc vol 92 no 1 pp 75ndash86 2009 doi 101111j1551-2916200802851x
[17] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 2009 doi 10106313152432
[18] F H Stillinger and T A Weber ldquoHidden structure in liquidsrdquo Phys Rev A vol 25 no
2 pp 978ndash989 1982 doi 101103PhysRevA25978
[19] F H Stillinger ldquoSupercooled liquids glass transitions and the auzmann paradoxrdquo J
Chem Phys vol 88 no 12 pp 7818ndash7825 1988 doi 1010631454295
171
[20] S Neelamraju C Oligschleger and J C Schoumln ldquoThe threshold algorithm Description of
the methodology and new developmentsrdquo J Chem Phys vol 147 no 15 p 152713
2017 doi 10106314985912
[21] D Prada-Gracia J Goacutemez-Gardentildees P Echenique and F Falo ldquoExploring the free
energy landscape From dynamics to networks and backrdquo PLoS Comput Biol vol 5 no
6 p e1000415 2009 doi 101371journalpcbi1000415
[22] E Pollak A Auerbach and P Talknerz ldquoObservations on rate theory for rugged energy
landscapesrdquo Biophys J vol 95 no 9 pp 4258ndash4265 2008 doi
101529biophysj108136358
[23] G G Naumis ldquoEnergy landscape and rigidityrdquo Phys Rev E vol 71 p 026114 2005
doi 101103PhysRevE71026114
[24] S J Benkovic G G Hammes and S Hammes-Schiffer ldquoFree-energy landscape of
enzyme catalysisrdquo Biochemistry vol 47 no 11 pp 3317ndash3321 2008 doi
101021bi800049z
[25] M Roca B Messer D Hilvert and A Warshel ldquoOn the relationship between folding and
chemical landscapes in enzyme catalysisrdquo Proc Natl Acad Sci U S A vol 105 no 37
pp 13877ndash13882 2008 doi 101073pnas0803405105
[26] P Li G Henkelman J A eith and J Johnson ldquoElucidation of aqueous solvent-
mediated hydrogen-transfer reactions by ab initio molecular dynamics and nudged elastic-
band studies of NaBH4 hydrolysisrdquo J Phys Chem C vol 118 no 37 pp 21385ndash21399
2014 doi 101021jp507872d
[27] S P Niblett M Biedermann D J Wales and V De Souza ldquoPathways for diffusion in
the potential energy landscape of the network glass former SiO2rdquo J Chem Phys vol
147 no 15 p 152726 2017 doi 10106315005924
[28] Y Yu M Wang M M Smedskjaer J C Mauro G Sant and M Bauchy
172
ldquoThermometer Effect Origin of the Mixed Alkali Effect in Glass Relaxationrdquo Phys Rev
Lett vol 119 no 9 pp 1ndash5 2017 doi 101103PhysRevLett119095501
[29] Y Yu M Wang D Zhang B Wang G Sant and M Bauchy ldquoStretched Exponential
Relaxation of Glasses at Low Temperaturerdquo Phys Rev Lett vol 115 no 16 p 165901
2015 doi 101103PhysRevLett115165901
[30] Steve Plimpton ldquoFast Parallel Algorithms for Short-Range Molecular Dynamicsrdquo J
Comput Phys vol 117 no 1 pp 1ndash19 1995 doi
httpsdoiorg101006jcph19951039
[31] N Mousseau and G T Barkema ldquoTraveling through potential energy landscapes of
disordered materials The activation-relaxation techniquerdquo Phys Rev E vol 57 no 2
pp 2419ndash2424 1998 doi 101103PhysRevE572419
[32] F El-Mellouhi N Mousseau and L J Lewis ldquo inetic activation-relaxation technique
An off-lattice self-learning kinetic Monte Carlo algorithmrdquo Phys Rev B vol 78 p
153202 2008 doi 101103PhysRevB78153202
[33] D J Wales ldquoDiscrete path samplingrdquo Mol Phys vol 100 no 20 pp 3285ndash3305 2002
doi 10108000268970210162691
[34] G Henkelman and H Joacutensson ldquoImproved tangent estimate in the nudged elastic band
method for finding minimum energy paths and saddle pointsrdquo J Chem Phys vol 113
no 10 pp 9978ndash9985 2000 doi 10106314961868
[35] D Sheppard P Xiao W Chemelewski D D Johnson and G Henkelman ldquoA
generalized solid-state nudged elastic band methodrdquo J Chem Phys vol 136 p 074103
2012 doi 10106313684549
[36] A Pedone G Malavasi M C Menziani A N Cormack A V and N York ldquoA New
Self-Consistent Empirical Interatomic Potential Model for Oxides Silicates and Silica-
Based Glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
173
[37] A Takada C R A Catlow and G D Price ldquoComputer modelling of B2O3 II Molecular
dynamics simulations of vitreous structuresrdquo J Phys Condens Matter vol 7 no 46 pp
8693ndash8722 1995 doi 1010880953-8984746004
[38] A C T Van Duin S Dasgupta F Lorant and W A Goddard ldquoReaxFF A reactive
force field for hydrocarbonsrdquo J Phys Chem A vol 105 no 41 pp 9396ndash9409 2001
doi 101021jp004368u
[39] J C Mauro and A Varshneya ldquoModel interaction potentials for selenium from ab initio
molecular simulationsrdquo Phys Rev B vol 71 p 214105 2005 doi
101103PhysRevB71214105
[40] J C Mauro R J Loucks J Balakrishnan and S Raghavan ldquoMonte Carlo method for
computing density of states and quench probability of potential energy and enthalpy
landscapesrdquo J Chem Phys vol 126 no 19 2007 doi 10106312733674
[41] Y Z Mauro C J Wilkinson and J C Mauro ldquo ineticPy A tool to calculate long-time
kinetics in energy landscapes with broken ergodicityrdquo SoftwareX vol 11 p 100393
2020 doi 101016jsoftx2019100393
[42] C J Wilkinson Y Z Mauro and J C Mauro ldquoRelaxPy Python code for modeling of
glass relaxation behaviorrdquo SoftwareX vol 7 pp 255ndash258 2018 doi
101016jsoftx201807008
[43] J C Mauro ldquoTopological constraint theory of glassrdquo Am Ceram Soc Bull vol 90 no
4 pp 31ndash37 2011 doi 101039c3ee40810h
[44] J C Mauro D C Allan and M Potuzak ldquoNonequilibrium viscosity of glassrdquo Phys Rev
B vol 80 p 094204 2009 doi 101103PhysRevB80094204
[45] M Cardona R V Chamberlin and W Marx ldquoThe history of the stretched exponential
functionrdquo Ann der Phys vol 16 no 12 pp 842ndash845 2007 doi
101002andp200710269
174
[46] P Grassberger and I Procaccia ldquoThe long time properties of diffusion in a medium with
static trapsrdquo J Chem Phys vol 77 no 1982 pp 6281ndash6284 1982 doi
1010631443832
[47] J C Phillips ldquoMicroscopic aspects of Stretched Exponential Relaxation (SER) in
homogeneous molecular and network glasses and polymersrdquo J Non Cryst Solids vol
357 no 22ndash23 pp 3853ndash3865 2011 doi 101016jjnoncrysol201108001
[48] J C Phillips ldquo ohlrausch explained The solution to a problem that is 1 0 years oldrdquo J
Stat Phys vol 77 no 3ndash4 pp 945ndash947 1994 doi 101007BF02179472
[49] A Q Tool and C G Eichlin ldquoVariations caused in the heating curves of glass caused by
heat treatmentrdquo J Am Ceram Soc vol 14 pp 276ndash308 1931
[50] A Q Tool ldquoRelation Between Inelastic Deformability and Thermal Expansion of Glass in
Its Annealing Rangerdquo J Am Ceram Soc vol 29 no 9 pp 240ndash253 1946 doi
101111j1151-29161946tb11592x
[51] H N Ritland ldquoLimitations of the Fictive Temperature Conceptrdquo J Am Ceram Soc vol
39 no 12 pp 403ndash406 Dec 1956 doi 101111j1151-29161956tb15613x
[52] O S Narayanaswamy ldquoA model of structural relaxation in glassrdquo J Am Ceram Soc
vol 54 no 10 pp 491ndash498 1971 doi 101111j1151-29161971tb12186x
[53] G Adam and J Gibbs ldquoOn the Temperature Dependence of Cooperative Relaxation
Properties in Glass‐Forming Liquidsrdquo J Chem Phys vol 43 no 1 pp 139ndash146 1965
doi 10106311696442
[54] Q Zheng J C Mauro A J Ellison M Potuzak and Y Yue ldquo niversality of the high-
temperature viscosity limit of silicate liquidsrdquo Phys Rev B vol 83 no 21 p 212202
2011 doi 101103PhysRevB83212202
[55] J C Mauro Y Yue A J Ellison P Gupta and D C Allan ldquoViscosity of glass-
forming liquidsrdquo Proc Natl Acad Sci vol 106 no 47 pp 19780ndash19784 2009 doi
175
101073pnas0911705106
[56] G G Naumis ldquoGlass transition phenomenology and flexibility An approach using the
energy landscape formalismrdquo J Non Cryst Solids vol 352 no 42-49 SPEC ISS pp
4865ndash4870 2006 doi 101016jjnoncrysol200601160
[57] P Gupta and J C Mauro ldquoComposition dependence of glass transition temperature
and fragility I A topological model incorporating temperature-dependent constraintsrdquo J
Chem Phys vol 130 no 9 p 094503 2009 doi 10106313077168
[58] C A Angell ldquoRelaxation in liquids polymers and plastic crystals - strongfragile patterns
and problemsrdquo J Non Cryst Solids vol 13 no 31 pp 131ndash133 1991 doi
1010160022-3093(91)90266-9
[59] G S Fulcher ldquoAnalysis of Recent Measurements of the Viscosity of Glassesrdquo J Am
Ceram Soc vol 75 no 5 pp 1043ndash1055 1992 doi 101111j1151-
29161992tb05536x
[60] I Avramov ldquoViscosity in disordered mediardquo J Non Cryst Solids vol 351 no 40ndash42
pp 3163ndash3173 Oct 2005 doi 101016JJNONCRYSOL200508021
[61] X Guo J C Mauro D C Allan and M M Smedskjaer ldquoPredictive model for the
composition dependence of glassy dynamicsrdquo J Am Ceram Soc vol 101 pp 1169ndash
1179 2018 doi 101111jace15272
[62] Doss C J Wilkinson Y Yang H Lee L Huang and J C Mauro ldquoMaxwell
relaxation time for nonexponential α-relaxation phenomena in glassy systemsrdquo J Am
Ceram Soc vol 103 no 6 pp 3590ndash3599 2020 doi 101111jace17051
[63] P Gupta and A R Cooper ldquoTopologically disordered networks of rigid polytopesrdquo J
Non Cryst Solids 1990 doi 1010160022-3093(90)90768-H
[64] J C Phillips and M F Thorpe ldquoConstraint theory vector percolation and glass
formationrdquo Solid State Commun vol 53 no 8 pp 699ndash702 1985 doi 1010160038-
176
1098(85)90381-3
[65] X Feng W Bresser and P Boolchand ldquoDirect Evidence for Stiffness Threshold in
Chalcogenide Glassesrdquo Phys Rev Lett vol 78 no 23 pp 4422ndash4425 1997 doi
101103PhysRevLett784422
[66] W Bresser P Boolchand and P Suranyi ldquoRigidity Percolation and Molecular Clustering
in Network Glassesrdquo Phys Rev Lett vol 56 no 23 pp 2493ndash2496 1986 doi
101103PhysRevLett562493
[67] Y Vaills T Qu M Micoulaut F Chaimbault and P Boolchand ldquoDirect evidence of
rigidity loss and self-organization in silicate glassesrdquo J Phys Condens Matter vol 17
no 32 pp 4889ndash4896 2005 doi 1010880953-89841732003
[68] A irchner and J C Mauro ldquoStatistical Mechanical Model of the Self-Organized
Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologiesrdquo
Front Mater vol 6 p 11 2019 doi 103389fmats201900011
[69] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 Jun 2009 doi 10106313152432
[70] D R Cassar A C P L F de Carvalho and E D Zanotto ldquoPredicting glass transition
temperatures using neural networksrdquo Acta Mater vol 159 pp 249ndash256 Oct 2018 doi
101016JACTAMAT201808022
[71] J C Mauro and R J Loucks ldquoSelenium glass transition A model based on the enthalpy
landscape approach and nonequilibrium statistical mechanicsrdquo Phys Rev B vol 76 no
17 p 174202 2007 doi 101103PhysRevB76174202
[72] C J Wilkinson et al ldquoEnergy Landscape Modeling of Crystal Nucleationrdquo Nat Comput
Mater p Submitted 2020
[73] M E Mc enzie and J C Mauro ldquoHybrid Monte Carlo technique for modeling of crystal
177
nucleation and application to lithium disilicate glass-ceramicsrdquo Comput Mater Sci vol
149 no January pp 202ndash207 2018 doi 101016jcommatsci201803034
[74] M E McKenzie et al ldquoImplicit glass model for simulation of crystal nucleation for glass-
ceramicsrdquo npj Comput Mater vol 4 no 1 pp 1ndash7 2018 doi 101038s41524-018-
0116-5
[75] E D Zanotto and P F James ldquoExperimental tests of the classical nucleation theory for
glassesrdquo J Non Cryst Solids vol 74 no 2ndash3 pp 373ndash394 1985 doi 1010160022-
3093(85)90080-8
[76] Y Yu N M A Krishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[77] I Pignatelli A Kumar M Bauchy and G Sant ldquoTopological control on silicatesrsquo
dissolution kineticsrdquo Langmuir vol 32 pp 4434ndash4439 2016 doi
101021acslangmuir6b00359
[78] C J Wilkinson et al ldquoTopological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phaserdquo J Phys Chem Lett vol 10 pp
3955ndash3960 2019 doi 101021acsjpclett9b01275
[79] Y Zhang A Li B Deng and Hughes ldquoData-driven predictive models for chemical
durability of oxide glass under different chemical conditionsrdquo npj Mater Degrad vol 4
no 1 pp 1ndash11 2020 doi 101038s41529-020-0118-x
[80] N M Anoop Krishnan S Mangalathu M M Smedskjaer A Tandia H Burton and M
Bauchy ldquoPredicting the dissolution kinetics of silicate glasses using machine learningrdquo J
Non Cryst Solids vol 487 no February pp 37ndash45 2018 doi
101016jjnoncrysol201802023
[81] R M Potter J Hoffman and J Hadley ldquoAn update of the equation for predicting the
178
dissolution rate of glass fibers from their chemical compositionsrdquo Inhal Toxicol vol 29
no 4 pp 145ndash146 2017 doi 1010800895837820171321702
[82] B Deng ldquoMachine learning on density and elastic property of oxide glasses driven by
large datasetrdquo J Non Cryst Solids vol 529 no August 2019 p 119768 2020 doi
101016jjnoncrysol2019119768
[83] S Feller N Lower and M Affatigato ldquoDensity as a probe of oxide glass structurerdquo
Phys Chem Glas vol 42 no 3 pp 240ndash246 2001 Accessed Jul 19 2018 [Online]
Available
httpswwwingentaconnectcomcontentsgtpcg200100000042000000034203240
[84] Q Zheng and H Zeng ldquoProgress in modeling of glass properties using topological
constraint theoryrdquo Int J Appl Glas Sci vol 11 no 3 pp 432ndash441 2020 doi
101111ijag15105
[85] M Bauchy M Javad Abdolhosseini Qomi C Bichara F-J Ulm and R J-M Pellenq
ldquoRigidity Transition in Materials Hardness is Driven by Weak Atomic Constraintsrdquo
2015 doi 101103PhysRevLett114125502
[86] Yang X Xu B Yang B Cook H Ramos and M Bauchy ldquoPrediction of Silicate
Glassesrsquo Stiffness by High-Throughput Molecular Dynamics Simulations and Machine
Learningrdquo Accessed Mar 13 201 Online Available
httpsarxivorgpdf190109323pdf
[87] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological constraint model for
the elasticity of glass-forming systemsrdquo J Non Cryst Solids X vol 2 2019 doi
101016jnocx2019100019
[88] C J Wilkinson K Doss D R Cassar R S Welch C B Bragatto and J C Mauro
ldquoPredicting Ionic Diffusion in Glass from Its Relaxation Behaviorrdquo J Phys Chem B vol
124 no 6 pp 1099ndash1103 2020 doi 101021acsjpcb9b10645
179
[89] C Wilkinson and J C Mauro ldquoExplorerpy Mapping the energy landscapes of complex
materialsrdquo SoftwareX vol Submitted 2020
[90] J C Mauro R J Loucks and J Balakrishnan ldquoA simplified eigenvector-following
technique for locating transition points in an energy landscaperdquo J Phys Chem A vol
109 no 42 pp 9578ndash9583 2005 doi 101021jp053581t
[91] Roumlder and D J Wales ldquoAnalysis of the b to b-CR Transition in biquitinrdquo
Biochemistry vol 57 no 43 pp 6180ndash6186 2018 doi 101021acsbiochem8b00770
[92] J C Mauro R J Loucks and J Balakrishnan ldquoSplit-step eigenvector-following
technique for exploring enthalpy landscapes at absolute zerordquo J Phys Chem B vol 110
no 10 pp 5005ndash5011 2006 doi 101021jp056803w
[93] J C Mauro R J Loucks J Balakrishnan and A Varshneya ldquoMapping the potential
energy landscapes of selenium clustersrdquo J Non Cryst Solids vol 353 pp 1268ndash1273
2007 doi 101016jjnoncrysol200609062
[94] B W H van Beest G J Kramer and R A van Santen ldquoForce fields for silicas and
aluminophosphates based on ab initio calculationsrdquo Phys Rev Lett vol 64 no 16 pp
1955ndash1958 Apr 1990 doi 101103PhysRevLett641955
[95] Q Zheng and J C Mauro ldquoViscosity of glass-forming systemsrdquo J Am Ceram Soc vol
100 no 1 pp 6ndash25 2017 doi 101111jace14678
[96] J C Mauro and Y Z Mauro ldquoOn the Prony Series Representation of Stretched
Exponential Relaxationrdquo Physica A vol 506 pp 75ndash87 2018 doi
101016jphysa201804047
[97] E D Zanotto and D R Cassar ldquoThe microscopic origin of the extreme glass-forming
ability of Albite and B2O3rdquo Sci Rep vol 7 no February pp 1ndash13 2017 doi
101038srep43022
[98] P Pedevilla S J Cox B Slater and A Michaelides ldquoCan Ice-Like Structures Form on
180
Non-Ice-Like Substrates The Example of the K-feldspar Microclinerdquo J Phys Chem C
vol 120 no 12 pp 6704ndash6713 2016 doi 101021acsjpcc6b01155
[99] O Bjoumlrneholm et al ldquoWater at Interfacesrdquo Chem Rev vol 116 no 13 pp 7698ndash7726
2016 doi 101021acschemrev6b00045
[100] Reichelt ldquoNucleation and growth of thin filmsrdquo Vacuum vol 38 no 12 pp 1083ndash
1099 Jan 1988 doi 1010160042-207X(88)90004-8
[101] G H Beall ldquoDr S Donald (Don) Stookey (1 1 ndash2014) Pioneering researcher and
adventurerrdquo Front Mater vol 3 no August pp 1ndash8 2016 doi
103389fmats201600037
[102] J C Mauro C S Philip D J Vaughn and M S Pambianchi ldquoGlass science in the
nited States Current status and future directionsrdquo Int J Appl Glas Sci vol 5 no 1
pp 2ndash15 2014 doi 101111ijag12058
[103] J Deubener et al ldquo pdated definition of glass-ceramicsrdquo J Non Cryst Solids vol 501
no January pp 3ndash10 2018 doi 101016jjnoncrysol201801033
[104] X Hao ldquoA review on the dielectric materials for high energy-storage applicationrdquo J Adv
Dielectr vol 03 no 01 p 1330001 2013 doi 101142s2010135x13300016
[105] J Deubener G Helsch A Moiseev and H Bornhoumlft ldquoGlasses for solar energy
conversion systemsrdquo J Eur Ceram Soc vol 29 no 7 pp 1203ndash1210 2009 doi
101016jjeurceramsoc200808009
[106] G S Frankel et al ldquoA comparative review of the aqueous corrosion of glasses crystalline
ceramics and metalsrdquo npj Mater Degrad vol 2 no 1 p 15 2018 doi
101038s41529-018-0037-2
[107] G H Beall ldquoDesign and properties of glass-ceramicsrdquo Annu Rev Mater Sci vol 22
no 1 pp 91ndash119 1992 doi 101146annurevms22080192000515
[108] J Deubener ldquoConfigurational entropy and crystal nucleation of silicate glassesrdquo Phys
181
Chem Glas vol 45 pp 61ndash63 2004
[109] J C Mauro A J Ellison D C Allan and M M Smedskjaer ldquoTopological model for the
viscosity of multicomponent glass-forming liquidsrdquo Int J Appl Glas Sci vol 4 no 4
pp 408ndash413 2013 doi 101111ijag12009
[110] M M Smedskjaer J C Mauro S Sen and Y Yue ldquoQuantitative design of glassy
materials using temperature-dependent constraint theoryrdquo Chem Mater vol 22 no 18
pp 5358ndash5365 2010 doi 101021cm1016799
[111] Y T Sun H Y Bai M Z Li and W H Wang ldquoMachine Learning Approach for
Prediction and Understanding of Glass-Forming Abilityrdquo J Phys Chem Lett vol 8 no
14 pp 3434ndash3439 2017 doi 101021acsjpclett7b01046
[112] Q Zheng et al ldquo nderstanding Glass through Differential Scanning Calorimetryrdquo Chem
Rev vol 119 no 13 pp 7848ndash7939 2019 doi 101021acschemrev8b00510
[113] X Xia I Dutta J C Mauro B G Aitken and F elton ldquoTemperature dependence of
crystal nucleation in BaOmiddot2SiO2 and 5BaOmiddot8SiO2 glasses using differential thermal
analysisrdquo J Non Cryst Solids vol 459 pp 45ndash50 2017 doi
101016jjnoncrysol201612032
[114] V M Fokin A A Cabral R M C V Reis M L F Nascimento and E D Zanotto
ldquoCritical assessment of DTA-DSC methods for the study of nucleation kinetics in
glassesrdquo J Non Cryst Solids vol 356 no 6ndash8 pp 358ndash367 2010 doi
101016jjnoncrysol200911038
[115] D C Van Hoesen X Xia M E Mc enzie and F elton ldquoModeling nonisothermal
crystallization in a BaO∙2SiO2 glassrdquo J Am Ceram Soc no December 2019 pp 2471ndash
2482 2019 doi 101111jace16979
[116] S C C Prado J P Rino and E D Zanotto ldquoSuccessful test of the classical nucleation
theory by molecular dynamic simulations of BaSrdquo Comput Mater Sci vol 161 no
182
January pp 99ndash106 2019 doi 101016jcommatsci201901023
[117] A O Tipeev and E D Zanotto ldquoNucleation kinetics in supercooled Ni 0Ti 0 Computer
simulation data corroborate the validity of the Classical Nucleation Theoryrdquo Chem Phys
Lett vol 735 no August p 136749 2019 doi 101016jcplett2019136749
[118] K F Kelton and A L Greer Nucleation in Condensed Matter Applications in Materials
and Biology 1st ed Pergamon 2010
[119] V M Fokin E D Zanotto N S Yuritsyn and J W P Schmelzer ldquoHomogeneous
crystal nucleation in silicate glasses A 40 years perspectiverdquo J Non Cryst Solids vol
352 no 26ndash27 pp 2681ndash2714 2006 doi 101016jjnoncrysol200602074
[120] A M Rodrigues D R Cassar V M Fokin and E D Zanotto ldquoCrystal growth and
viscous flow in barium disilicate glassrdquo J Non Cryst Solids vol 479 pp 55ndash61 2018
doi 101016jjnoncrysol201710007
[121] A Pedone G Malavasi M C Menziani A N Cormack and U Segre ldquoA new selft-
consistent empirical interatomic potential model for oxides silicates and silica-based
glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
[122] J J Maldonis A D Banadaki S Patala and P M Voyles ldquoShort-range order structure
motifs learned from an atomistic model of a Zr 0Cu4 Al metallic glassrdquo Acta Mater
vol 175 pp 35ndash45 2019 doi 101016jactamat201905002
[123] A irchner S Goyal M E Mc enzie J T Harris and J C Mauro ldquoStatistical
Description of the Thermodynamics of Glass-Forming Liquidsrdquo Physica A vol 559 p
125059 2020 doi 101016jphysa2020125059
[124] D R Cassar ldquoCrystallization Driving Force of Supercooled Oxide Liquidsrdquo Int J Appl
Glas Sci vol 7 no 3 pp 262ndash269 2016 doi 101111ijag12218
[125] V M Fokin A S Abyzov E D Zanotto D R Cassar A M Rodrigues and J W P
Schmelzer ldquoCrystal nucleation in glass-forming liquids Variation of the size of the
183
lsquostructural unitsrsquo with temperaturerdquo J Non Cryst Solids vol 447 pp 35ndash44 2016 doi
101016jjnoncrysol201605017
[126] M Lenoir A Grandjean Y Linard B Cochain and D R Neuville ldquoThe influence of
SiB substitution and of the nature of network-modifying cations on the properties and
structure of borosilicate glasses and meltsrdquo Chem Geol vol 256 no 3ndash4 pp 316ndash325
2008 doi 101016jchemgeo200807002
[127] P Gupta D R Cassar and E D Zanotto ldquoRole of dynamic heterogeneities in crystal
nucleation kinetics in an oxide supercooled liquidrdquo J Chem Phys vol 145 no 21 2016
doi 10106314964674
[128] A M Rodrigues ldquoDiffusion Processes Crystallization and Viscous Flow in Barium
Disilicate Glassrdquo Dissertation 2014
[129] E D Zanotto ldquoThe effects of amorphous phase sepatration on crystal nucleation in baria-
silica and lithia-silica glassesrdquo Dissertation 1982
[130] D Vargheese A Tandia and J C Mauro ldquoOrigin of dynamical heterogeneities in
calcium aluminosilicate liquidsrdquo J Chem Phys vol 132 no 19 p 194501 May 2010
doi 10106313429880
[131] K Deenamma Vargheese A Tandia and J C Mauro ldquoStatistics of modifier distributions
in mixed network glassesrdquo J Chem Phys vol 132 p 24507 2010 doi
10106313429880
[132] M Micoulaut ldquoThe Deep Effect of Topology on Glass Relaxationrdquo Physics (College
Park Md) vol 6 p 72 2013 doi 101103Physics672
[133] M Micoulaut ldquoRigidity and intermediate phases in glasses driven by speciationrdquo Phys
Rev B vol 74 p 184208 2006 doi 101103PhysRevB74184208
[134] M Kodama S Kojima S Feller and M Affatigato ldquoThe occurrence of minima in the
Borate anomaly anharmonicity and fragility in lithium borate glassesrdquo Phys Chem
184
Glas vol 46 no 2 pp 190ndash193 2005
[135] Y Fukawa et al ldquoVelocity of Sound and Elastic Properties of Li2O-B2O3 Glassesrdquo Jpn
J Appl Phys vol 34 p 2570 1995 doi httpsdoiorg101143JJAP342570
[136] M Reiner ldquoThe Deborah Numberrdquo Physics Today vol 17 no 1 p 62 1964 doi
10106313051374
[137] P K Gupta and J C Mauro ldquoThe laboratory glass transitionrdquo J Chem Phys vol 126
no 22 p 2240504 2007 doi 10106312738471
[138] R Richert and C A Angell ldquoDynamics of glass-forming liquids V On the link between
molecular dynamics and configurational entropyrdquo J Chem Phys vol 108 no 21 pp
9016ndash9026 1998 doi 1010631476348
[139] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological Constraint Model for
the Elasticity of Glass-Forming Systemsrdquo J Non Cryst Solids X vol 2 p 100019 2019
doi 101016jnocx2019100019
[140] S Feller S Bista A OrsquoDonovan-Zavada T Mullenbach M Franke and M Affatigato
ldquoPacking in alkali and alkaline earth borosilicate glass systemsrdquo Phys Chem Glas Eur
J Glas Sci Technol Part B vol 50 no 3 pp 224ndash228 2009
[141] P J Bray S Feller G E Jellison and Y H Yun ldquoB10 NMR studies of the structure of
borate glassesrdquo J Non Cryst Solids vol 38ndash39 no PART 1 pp 93ndash98 May 1980 doi
1010160022-3093(80)90400-7
[142] N Andersson and G L Comer ldquoRelativistic Fluid Dynamics Physics for Many Different
Scales Living Reviews in Relativityrdquo Living Rev Relativ vol 10 p 1 2007 Accessed
Feb 18 2019 [Online] Available httpwwwlivingreviewsorglrr-2007-
1httprelativitylivingreviewsorghttpwwwmathssotonacukstaffAnderssonhttpw
wwslueducollegesASphysicsprofscomerhtml
[143] P Ilg and H C Oumlttinger ldquoNonequilibrium relativistic thermodynamics in bulk viscous
185
cosmologyrdquo Phys Rev D - Part Fields Gravit Cosmol vol 61 no 2 p 023510 2000
doi 101103PhysRevD61023510
[144] P Romatschke ldquoRelativistic viscous fluid dynamics and non-equilibrium entropyrdquo Class
Quantum Gravity vol 27 no 2 p 025006 Jan 2010 doi 1010880264-
9381272025006
[145] D C Allan ldquoInverting the MYEGA equation for viscosityrdquo J Non Cryst Solids vol
358 no 2 pp 440ndash442 Jan 2012 doi 101016JJNONCRYSOL201109036
[146] M Potuzak R C Welch and J C Mauro ldquoTopological origin of stretched exponential
relaxation in glassrdquo J Chem Phys vol 135 no 21 p 214502 Dec 2011 doi
10106313664744
[147] L Ding M Thieme S Demouchy C unisch and B J P aus ldquoEffect of pressure and
temperature on viscosity of a borosilicate glassrdquo J Am Ceram Soc vol 101 no 9 pp
3936ndash3946 2018 doi 101111jace15588
[148] R Richert and M Richert ldquoDynamic heterogeneity spatially distributed stretched-
exponential patterns and transient dispersions in solvation dynamicsrdquo Phys Rev E vol
58 no 1 pp 779ndash784 1998 doi 101103PhysRevE58779
[149] P Richet ldquoViscosity and configurational entropy of silicate meltsrdquo Geochim Cosmochim
Acta vol 48 no 3 pp 471ndash483 Mar 1984 doi 1010160016-7037(84)90275-8
[150] P Gupta and J C Mauro ldquoTwo factors governing fragility Stretching exponent and
configurational entropyrdquo Phys Rev E vol 78 no 6 p 063501 2008 doi
101103PhysRevE78062501
[151] Z Zheng J C Mauro and D C Allan ldquoModeling of delayed elasticity in glassrdquo J Non
Cryst Solids vol 500 no August pp 432ndash442 2018 doi
101016jjnoncrysol201808033
[152] R Boumlhmer L Ngai C A Angell and D J Plazek ldquoNonexponential relaxations in
186
strong and fragile glass formersrdquo J Chem Phys vol 99 no 5 pp 4201ndash4209 1993
doi 1010631466117
[153] O Gulbiten J C Mauro and P Lucas ldquoRelaxation of enthalpy fluctuations during sub-
Tg annealing of glassy seleniumrdquo J Chem Phys vol 138 no 24 p 244504 2013 doi
10106314811488
[154] D Sidebottom R Bergman L Boumlrjesson and L M Torell ldquoTwo-step relaxation decay
in a strong glass formerrdquo Phys Rev Lett vol 71 no 14 pp 2260ndash2263 1993 doi
101103PhysRevLett712260
[155] G M Bartenev and V A Lomovskoi ldquoRelaxation time spectra and the peculiarities of
the process of boron anhydride glass transitionrdquo J Non Cryst Solids vol 146 no C pp
225ndash232 1992 doi 101016S0022-3093(05)80495-8
[156] A Heuer ldquoExploring the potential energy landscape of glass-forming systems From
inherent structures via metabasins to macroscopic transportrdquo J Phys Condens Matter
vol 20 no 37 p 373101 2008 doi 1010880953-89842037373101
[157] J C Phillips ldquoStretched exponential relaxation in molecular and electronic gasesrdquo
Reports Prog Phys vol 59 pp 1133ndash1207 1996
[158] P Richet ldquoResidual and configurational entropy Quantitative checks through applications
of Adam-Gibbs theory to the viscosity of silicate meltsrdquo J Non Cryst Solids vol 355
no 10ndash12 pp 628ndash635 2009 doi 101016jjnoncrysol200901027
[159] C J Wilkinson Doss G Palmer and J C Mauro ldquoThe relativistic glass transition A
thought experimentrdquo J Non-Crystalline Solids X vol 2 2019 doi
101016jnocx2019100018
[160] J C Mauro R J Loucks and S Sen ldquoHeat capacity enthalpy fluctuations and
configurational entropy in broken ergodic systemsrdquo J Chem Phys vol 133 no 16 pp
1ndash9 2010 doi 10106313499326
187
[161] J C Mauro R J Loucks and P Gupta ldquoMetabasin approach for computing the
master equation dynamics of systems with broken ergodicityrdquo J Phys Chem A vol 111
no 32 pp 7957ndash7965 2007 doi 101021jp0731194
[162] M Wang N M Anoop Krishnan B Wang M M Smedskjaer J C Mauro and M
Bauchy ldquoA new transferable interatomic potential for molecular dynamics simulations of
borosilicate glassesrdquo J Non Cryst Solids vol 498 no December 2017 pp 294ndash304
2018 doi 101016jjnoncrysol201804063
[163] I Avramov and A Milchev ldquoEffect of disorder on diffusion and viscosity in condensed
systemsrdquo J Non Cryst Solids vol 104 no 2ndash3 pp 253ndash260 Sep 1988 doi
1010160022-3093(88)90396-1
[164] G Scherer Relaxation in Glass and Composites Krieger Publishing Company 1992
[165] L Huang J Nicholas J Kieffer and J Bass ldquoPolyamorphic transitions in vitreous B2O3
under pressurerdquo J Phys Condens Matter vol 20 no 7 2008 doi 1010880953-
8984207075107
[166] W Capps P B Macedo B OrsquoMeara and T A Litovitz ldquoTemperature Dependecne of
the High-Frequency Moduli of Vitreous B2O3rdquo J Chem Phys vol 45 p 3431 1966
doi 107868s0002337x1402002x
[167] D R Cassar A M Rodrigues M L F Nascimento and E D Zanotto ldquoThe diffusion
coefficient controlling crystal growth in a silicate glass-formerrdquo Int J Appl Glas Sci
vol 9 no 3 pp 373ndash382 Jul 2018 doi 101111ijag12319
[168] A R Potter C J Wilkinson S H im and J C Mauro ldquoEffect of Water on
Topological Constraints in Silica Glassrdquo Scr Mater vol 160 pp 48ndash52 2019 doi
101016jscriptamat201809041
[169] D B Asay and S H im ldquoEvolution of the adsorbed water layer structure on silicon
oxide at room temperaturerdquo J Phys Chem B vol 109 pp 16760ndash16763 2005 doi
188
101021jp053042o
[170] M Tomozawa ldquoWater diffusion oxygen vacancy annihilation and structural relaxation in
silica glassesrdquo J Non Cryst Solids vol 179 pp 162ndash169 1994 doi 1010160022-
3093(94)90693-9
[171] S Kapoor R E Youngman K Zakharchuk A Yaremchenko N J Smith and A Goel
ldquoStructural and Chemical Approach toward nderstanding the Aqueous Corrosion of
Sodium Aluminoborate Glassesrdquo J Phys Chem B vol 122 pp 10913ndash10927 2018 doi
101021acsjpcb8b06155
[172] Y Yu N M A rishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[173] Y Yu B Wang M Wang G Sant and M Bauchy ldquoReactive Molecular Dynamics
Simulations of Sodium Silicate Glasses mdash Toward an Improved Understanding of the
Structurerdquo Int J Appl Glas Sci vol 8 no 3 pp 276ndash284 2017 doi
101111ijag12248
[174] E Stolper ldquoThe speciation of water in silicate meltsrdquo Geochim Cosmochim Acta vol
46 no 12 pp 2609ndash2620 Dec 1982 doi 1010160016-7037(82)90381-7
[175] M G Mesko P A Schader and J E Shelby ldquoWater solubility and diffusion in sodium
silicate meltsrdquo Phys Chem Glas vol 43 no 6 pp 283ndash290 2002 doi
101016jsemradonc201010001
[176] M Tomozawa M Takata J Acocella E Bruce Watson and T Takamori ldquoThermal
properties of Na2Omiddot3SiO2 glasses with high water contentrdquo J Non Cryst Solids vol 56
no 1ndash3 pp 343ndash348 Jul 1983 doi 1010160022-3093(83)90491-X
[177] E A Leed and C G Pantano ldquoComputer modeling of water adsorption on silica and
silicate glass fracture surfacesrdquo J Non Cryst Solids vol 325 pp 48ndash60 2003 doi
189
101016S0022-3093(03)00361-2
[178] J C Fogarty H M Aktulga A Y Grama A C T van Duin and S A Pandit ldquoA
reactive molecular dynamics simulation of the silica-water interfacerdquo J Chem Phys vol
132 no 17 p 174704 2010 doi 10106313407433
[179] S H Hahn et al ldquoDevelopment of a ReaxFF Reactive Force Field for NaSiOx Water
Systems and Its Application to Sodium and Proton Self-Diffusionrdquo J Phys Chem C vol
122 no 34 pp 19613ndash19624 2018 doi 101021acsjpcc8b05852
[180] T S Mahadevan W Sun and J Du ldquoDevelopment of Water Reactive Potentials for
Sodium Silicate Glassesrdquo J Phys Chem B vol 123 pp 4452ndash4461 2019 doi
101021acsjpcb9b02216
[181] A C T van Duin A Strachan S Stewman Q Zhang X Xu and W A Goddard III
ldquoReaxFF SiO Reactive Force Field for Silicon and Silicon Oxide Systemsrdquo J Phys
Chem A vol 107 no 19 pp 3803ndash3811 2003 doi 101021jp0276303
[182] M M Smedskjaer J C Mauro R E Youngman C L Hogue M Potuzak and Y Yue
ldquoTopological principles of borosilicate glass chemistryrdquo J Phys Chem B vol 115 no
44 pp 12930ndash12946 2011 doi 101021jp208796b
[183] Q Zheng M Potuzak J C Mauro M M Smedskjaer R E Youngman and Y Yue
ldquoComposition-structure-property relationships in boroaluminosilicate glassesrdquo J Non
Cryst Solids vol 358 pp 993ndash1002 2012 doi 101016jjnoncrysol201201030
[184] K Rompicharla D I Novita P Chen P Boolchand M Micoulaut and W Huff
ldquoAbrupt boundaries of intermediate phases and space filling in oxide glassesrdquo J Phys
Condens Matter vol 20 p 202101 2008 doi 1010880953-89842020202101
[185] M Micoulaut and J C Phillips ldquoRings and rigidity transitions in network glassesrdquo Phys
Rev B vol 67 p 104204 2003 doi 101103PhysRevB67104204
[186] H Liu S Dong L Tang N M A rishnan G Sant and M Bauchy ldquoEffects of
190
polydispersity and disorder on the mechanical properties of hydrated silicate gelsrdquo J
Mech Phys Solids vol 122 pp 555ndash565 Jan 2019 doi 101016JJMPS201810003
[187] A Tilocca N H De Leeuw and A N Cormack ldquoShell-model molecular dynamics
calculations of modified silicate glassesrdquo Phys Rev B - Condens Matter Mater Phys
vol 73 p 104209 2006 doi 101103PhysRevB73104209
[188] J Du and A N Cormack ldquoThe medium range structure of sodium silicate glasses A
molecular dynamics simulationrdquo J Non Cryst Solids vol 349 no 1ndash3 pp 66ndash79 2004
doi 101016jjnoncrysol200408264
[189] N P Bansal and R H Doremus Handbook of glass properties Academic Press 1986
[190] S H Hahn and A C T van Duin ldquoSurface Reactivity and Leaching of a Sodium Silicate
Glass under an Aqueous Environment A ReaxFF Molecular Dynamics Studyrdquo J Phys
Chem C p acsjpcc9b02940 2019 doi 101021acsjpcc9b02940
[191] S P Jaccani and L Huang ldquo nderstanding Sodium Borate Glasses and Melts from Their
Elastic Response to Temperaturerdquo Int J Appl Glas Sci vol 7 no 4 pp 452ndash463 Dec
2016 doi 101111ijag12250
[192] Q Zheng Y Yue and J C Mauro ldquoDensity of topological constraints as a metric for
predicting glass hardnessrdquo Appl Phys Lett vol 111 no 1 p 011907 Jul 2017 doi
10106314991971
[193] J C Dyre ldquoElastic models for the non-Arrhenius relaxation time of glass-forming
liquidsrdquo in AIP Conference Proceedings 2006 vol 832 pp 113ndash117 doi
10106312204470
[194] Y T Cheng and C M Cheng ldquoRelationships between hardness elastic modulus and the
work of indentationrdquo Appl Phys Lett vol 73 no 5 pp 614ndash616 Jul 1998 doi
1010631121873
[195] A Makishima and J D Mackenzie ldquoDirect calculation of Youngrsquos modulus of glassrdquo J
191
Non Cryst Solids vol 12 no 1 pp 35ndash45 1973 doi 1010160022-3093(73)90053-7
[196] G Pilania C Wang X Jiang S Rajasekaran and R Ramprasad ldquoAccelerating materials
property predictions using machine learningrdquo Sci Rep vol 3 pp 1ndash6 2013 doi
101038srep02810
[197] H He and M Thorpe ldquoElastic properties of glassesrdquo Phys Rev Lett vol 54 no 19 pp
2107ndash2110 1985 Accessed Aug 29 2018 [Online] Available
httpsjournalsapsorgprlpdf101103PhysRevLett542107
[198] M Thorpe ldquoElastic properties of glassesrdquo Cambridge Arch 1985 doi 101557PROC-
61-49
[199] J C Phillips ldquoChemical Bonding Internal Surfaces and the Topology of Non-Crystalline
Solidsrdquo Phys status solidi vol 101 no 2 pp 473ndash479 Oct 1980 doi
101002pssb2221010204
[200] M M Smedskjaer J C Mauro and Y Yue ldquoPrediction of glass hardness using
temperature-dependent constraint theoryrdquo Phys Rev Lett vol 105 no 11 2010 doi
101103PhysRevLett105115503
[201] J D Musgraves J Hu and L Calvez Springer Handbook of Glass Springer US 2021
[202] C Hermansen X Guo R E Youngman J C Mauro M M Smedskjaer and Y Yue
ldquoStructure-topology-property correlations of sodium phosphosilicate glassesrdquo J Chem
Phys vol 143 no 6 p 064510 Aug 2015 doi 10106314928330
[203] L Wondraczek et al ldquoTowards ltrastrong Glassesrdquo Adv Mater vol 23 no 39 pp
4578ndash4586 Oct 2011 doi 101002adma201102795
[204] D Ravaine ldquoGlasses as solid electrolytesrdquo J Non Cryst Solids vol 38ndash39 no PART 1
pp 353ndash358 1980 doi 1010160022-3093(80)90444-5
[205] D Ravaine and J L Souquet ldquoA thermodynamic approach to ionic conductivity in oxide
glasses Part 1 Correlation of the ionic conductivity with the chemical potential of alkali
192
oxide in oxide glassesrdquo Phys Chem Glas vol 18 no 2 pp 27ndash31 1977
[206] J Swenson and L Boumlrjesson ldquoCorrelation between Free Volume and Ionic Conductivity
in Fast Ion Conducting Glassesrdquo Phys Rev Lett vol 77 no 17 pp 3569ndash3572 1996
doi 101103PhysRevLett773569
[207] C B Bragatto A C M Rodrigues and J L Souquet ldquoDissociation Equilibrium and
Charge Carrier Formation in AgI-AgPO3 Glassesrdquo J Phys Chem C vol 121 no 25 pp
13507ndash13514 2017 doi 101021acsjpcc7b02477
[208] S W Martin R Christensen G Olson J ieffer and W Wang ldquoNew Interpretation of
Na Ion Conduction in and the Structures and Properties of Sodium Borosilicate Mixed
Glass Former Glassesrdquo J Phys Chem C vol 140 pp 6343ndash6352 2019 doi
101021acsjpcc8b11735
[209] O L Anderson and D A Stuart ldquoCalculation of Activation Energy of Ionic Conductivity
in Silica Glasses by Classical Methodsrdquo J Am Ceram Soc vol 37 no 12 pp 573ndash580
1954 doi 101111j1151-29161954tb13991x
[210] R S Welch C J Wilkinson J C Mauro and C B Bragatto ldquoCharge Carrier Mobility
of Alkali Silicate Glasses Calculated by Molecular Dynamicsrdquo Front Mater vol 6 p
121 May 2019 doi 103389fmats201900121
[211] Y J Zhang ldquoEntropy and ionic conductivityrdquo Phys A Stat Mech its Appl vol 391 no
19 pp 4470ndash4475 2012 doi 101016jphysa201204021
[212] J C Dyre ldquoOn the mechanism of glass ionic conductivityrdquo J Non Cryst Solids vol 88
no 2ndash3 pp 271ndash280 1986 doi 101016S0022-3093(86)80030-8
[213] M Steve ldquoIonic Conduction in Phosphate Glassesrdquo JAmCeramSoc vol 74 no 8 pp
1767ndash1784 1991 doi 101111j1151-29161991tb07788x
[214] C T Moynihan A J Easteal J Wilder and J Tucker ldquoDependence of the Glass
Transition Temperature on Heating and Cooling Raterdquo J Phys Chem vol 78 no 26 pp
193
2673ndash2677 1974 Accessed Jul 13 2018 [Online] Available
httpspubsacsorgsharingguidelines
[215] Ito C T Moynihan and C A Angell ldquoThermodynamic determination of fragility in
liquids anda fragile-to-strong liquid transition inwaterrdquo Lett to Nat vol 398 no April
p 492 1999
[216] L Ngai and S W Martin ldquoCorrelation between the activation enthalpy and
Kohlrausch exponent for ionic conductivity in oxide glassesrdquo Phys Rev B vol 40 no
15 pp 10550ndash10556 1989 doi 101103PhysRevB4010550
[217] M Potuzak X Guo M M Smedskjaer and J C Mauro ldquoAre the dynamics of a glass
embedded in its elastic propertiesrdquo J Chem Phys vol 138 no 12 2013 doi
10106314730525
[218] A L Agapov and A P Sokolov ldquoDecoupling ionic conductivity from structural
relaxation A way to solid polymer electrolytesrdquo Macromolecules vol 44 no 11 pp
4410ndash4414 2011 doi 101021ma2001096
[219] S V Nemilov ldquoA Structural Investigation of Glasses in the B2O3ndashNa2O System by the
Viscosimetric Methodrdquo Izv Akad Nauk SSSR Neorg Mater vol 2 no 2 pp 349ndash359
1966
[220] J P Poole ldquoLow‐Temperature Viscosity of Alkali Silicate Glassesrdquo J Am Ceram Soc
vol 32 no 7 pp 230ndash233 1949 doi 101111j1151-29161949tb18952x
[221] R noche D B Dingwell F A Seifert and S L Webb ldquoNon-linear properties of
supercooled liquids in the system Na2O SiO2rdquo Chem Geol vol 116 pp 1ndash16 1994 doi
1010160009-2541(94)90154-6
[222] W Martinsen ldquoSelected properties of sodium silicate glasses and their structural
significancerdquo Dissertation 1969
[223] C Hermansen J C Mauro and Y Yue ldquoA model for phosphate glass topology
194
considering the modifying ion sub-networkrdquo J Chem Phys vol 140 no 15 2014 doi
10106314870764
[224] M L F Nascimento and E Dutra Zanotto ldquoDoes viscosity describe the kinetic barrier for
crystal growth from the liquidus to the glass transitionrdquo J Chem Phys vol 133 no 17
p 174701 2010 doi 10106313490793
[225] D Denisko and M M Hoffman ldquoClassification and interaction in random forestsrdquo Proc
Natl Acad Sci U S A vol 115 no 8 pp 1690ndash1692 2018 doi
101073pnas1800256115
[226] G Varney C Dema B E Gul C J Wilkinson and Akgun ldquo se of machine learning
in CARNA proton imagerrdquo in Progress in Biomedical Optics and Imaging - Proceedings
of SPIE 2019 vol 10948 doi 101117122512565
[227] ldquoMDLregSciGlass-6 MDL Information Systems San Leandro CArdquo 2003
[228] D R Cassar ldquoViscNet Neural Network for predicting the fragility index and the
temperature-dependency of viscosityrdquo pp 1ndash33 2020
[229] D J L Brett A Atkinson N P Brandon and S J Skinner ldquoIntermediate temperature
solid oxide fuel cellsrdquo Chem Soc Rev vol 37 no 8 pp 1568ndash1578 2008 doi
101039b612060c
[230] T Ishiyama S Suzuki J Nishii T Yamashita H awazoe and T Omata ldquoProton
conducting tungsten phosphate glass and its application in intermediate temperature fuel
cellsrdquo Solid State Ionics vol 262 pp 856ndash859 2014 doi 101016jssi201310055
[231] T Yamaguchi et al ldquoProton-conducting phosphate glass and its melt exhibiting high
electrical conductivity at intermediate temperaturesrdquo J Mater Chem A vol 6 no 46 pp
23628ndash23637 2018 doi 101039C8TA08162J
VITA
Collin Wilkinson was born in Mt Carroll IL He attended Coe College as an undergraduate and
studied glass resulting in a physics bachelorrsquos degree in 201 He worked with Dr Ugur Akgun
and Dr Steve Feller on modeling different key properties of glass propertiesresponses to external
stimuli Collin joined the group of Dr John Mauro at Pennsylvania State University in 2018
List of publications written by first author (or co-first author) Collin while at Penn State
1 CJ Wilkinson JC Mauro Explorerpy Mapping the Energy Landscapes of Complex
Materials SoftwareX Submitted
2 CJ Wilkinson DR Cassar AV DeCeanne KA Kirchner ME McKenzie ED Zanotto and
JC Mauro Energy Landscape Modeling of Crystal Nucleation NPJ Computational
Materialis Submitted
3 CJ Wilkinson K Doss O Gulbiten DC Allan JC Mauro Fragility and Temperature
Dependence of Stretched Exponential Relaxation in Glass-Forming Systems JACERS
Submitted
4 CJ Wilkinson JC Mauro Comment on ldquoThe Fragility of Alkali Silicate Glass Melts Part
of a niversal Topological Patterrdquo by DL Sidebottom JNCS (2020) ( 2 ) 11
5 CJ Wilkinson K Doss DR Cassar RS Welch CB Bragatto JC Mauro Predicting Ionic
Diffusion in Glass From its Relaxation Behavior Journal of Physical Chemistry B (2020)
124 (6) 1099-1103
6 Z Ding CJ Wilkinson J Zheng Y Lin H Liu J Shen SH Kim Y Yue J Rent CJ Mauro
Q Zheng Topological Understanding of the Mixed Alkaline Earth Effect in Glass JNCS
(2020) (526) 119696
7 CJ Wilkinson AR Potter RS Welch CB Bragatto Q Zheng M Bauchy M Affatigato SA
Feller JC Mauro Topological Origins of Mixed Alkali Effect in Glass Journal of Physical
Chemistry B (2019) 123 (34) 7482
8 CJ Wilkinson K Doss SH Hahn Nathan Keilbart AR Potter NJ Smith I Dabo ACT Van
Duin SH Kim JC Mauro Topological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phase JPCL (2019) 10 (14) 3955
9 CJ Wilkinson K Doss G Palmer JC Mauro The Relativistic Glass Transition A Through
Experiment JNCSX (2019) 100018
10 CJ Wilkinson Q Zheng L Huang JC Mauro Topological Constraint Model for the
Elasticity of Glass-Forming Systems JNCSX (2019) 100019
11 AR Potter CJ Wilkinson SH Kim JC Mauro Effect of Water on Toplogical Constraints
in Silica Glass Scripta Materialia (2019) (160) 48-52
12 CJ Wilkinson E Pakhomenko MR Jesuit AV DeCeanne B Hauke M Packard SA Feller
JC Mauro Topological Constraint Model of Alkali Tellurite Glasses JNCS (2018) (502)
172
13 CJ Wilkinson YZ Mauro JC Mauro RelaxPy Python code for Modeling of Glass
Relaxation Behavior SoftwareX (2018) (7) 255
iii
Abstract
Glass is infinitely variable This complexity stands as a promising technology for the 21st
century since the need for environmentally friendly materials has reached a critical point due to
climate change However such a wide range of variability makes new glass compositions difficult
to design The difficulty is only exaggerated when considering that not only is there an infinite
variability in the compositional space but also an infinite variability thermal history of a glass and
in the crystallinity of glass-cearmics This means that even for a simple binary glass there are at
least 3 dimensions that have to be optimized To resolve this difficulty it is shown that energy
landscapes can capture all three sets of complexity (composition thermal history and crystallinity)
The explicit energy landscape optimization however has a large computational cost To
circumvent the cost of the energy landscape mapping we present new research that allows for
physical predictions of key properties These methods are divided into two categories
compositional models and thermal history models Both models for composition and thermal
history are derived from energy landscapes Software for each method is presented As a
conclusion applications of the newly created models are discussed
iv
Table of Contents
List of Figures vi
List of Tables xi
Acknowledgments xii
Chapter 1 The Difficulty of Optimizing Glass 1
11 Energy Landscapes 6 12 Fictive Temperature 8 13 Topological Constraint Theory 15 14 Machine Learning 19 15 Goals of this Dissertation 20
Chapter 2 Software for Enabling the Study of Glass 22
21 ExplorerPy 22 22 RelaxPy 28
Chapter 3 Understanding Nucleation in Liquids 30
31 Crystallization Methods 33 31A Mapping and Classifying the Landscape 35 31B Kinetic Term for CNT 38 31C Degeneracy calculations 41 31D Free Energy Difference 42 31E Interfacial Energy 44
32 Results amp Discussion 46 33 Conclusions 51
Chapter 4 Expanding the Current State of Relaxation 52
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition 52 41A Relativistic Liquid 56 41B Relativistic Observer 60
42 Temperature and Compositional Dependence of the Stretching Exponent 64 42A Deriving a Model 67 42B Experimental Validation 73
43 Conclusion 81
Chapter 5 Glass Kinetics Without Fictive Temperature 82
51 Background of the Adam Gibbs Relationship 82
v
52 Methods 84 53 Results 86
53A Adam-Gibbs Validation 86 53B MYEGA Validation 87 53C Adam-Gibbs and Structural Relaxation 89 53D Landscape Features 91
54 Topography-Property Relations 94 55 Barrier Free Description of Thermodynamics 99 57 Toy Landscapes for the Design of Glasses and Glass Ceramics 102 57 Discussion 107
Chapter 6 Enabling the Prediction of Glass Properties 109
61 Controlling Surface Reactivity 109 62 Elastic Modulus Prediction 125 63 Ionic Conductivity 135 64 Machine Learning Expansion 145
Chapter 7 Designing Green Glasses for the 21st Century 152
71 Glass Electrolytes 152 72 Hydrogen Fuel Cell Glasses 158
Chapter 8 Conclusions 167
References 169
vi
List of Figures
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists
the equilibrium liquid which as it is quenched can either become a super cooled
liquid or crystallize Crystallization causes a discontinuity in the volume The super-
cooled liquid upon further quenching departs from equilibrium and transitions into
the glassy state Reproduced from Fundamentals of Inorganic Glass Science with
permission from the author1 5
Figure 2 The schematic for the flow of the program Beginning in the top right corner
and running until the condition in the pink box is satisfied Yellow diamonds
represent checks and blue operations 25
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above
and then plotted using PyConnect The plot is a disconnectivity graph where each
terminating line represents an inherent structure and tracing where two lines meet
describes the activation barrier The potentials are taken from the BKS potential 94 27
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-
axis being an arbitrary phase space and the y-axis being the potential energy
calculated from the Pedone et al potentials36 The colors are indicative of the
crystallinity where the blue basin is the initial starting configuration The landscape
shows the lowering of energy associated with partially crystallizing the sample (B)
The energy landscape relationship between the cutoff for crystalline and super-
cooled liquid states for 256 atoms is shown This shows a clear drastic energy
change occurring around the cutoff value of 10 Aring 37
Figure 5 An example interfacial structure between the crystalline phase on the left and
the last sequential SCLglass phase on the right for a barium disilicate system The
gray atoms are barium silicon is shown in red and blue represents oxygen 43
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a
function of temperature for each system size The experimental values for kinetics
and thermodynamics come from ref125 and from the heat capacity data taken from 126 respectively It is clear to see that driving force shows good agreement across all
systems however the kinetics terms only converge for the 256 and 512 atoms
systems (Bottom) The fit used to calculate the interfacial energy as a function of
temperature 45
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted
using the model presented in this work The data referenced can be found in refs 113128129 47
Figure 8 The surface energy with respect to temperature for the work presented here 50
Figure 9 An Angell diagram created using the MYEGA expression 55 With a very strong
glass (m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) 17134135
vii
The infinite temperature limit is from the work of Zheng et al54 and the glass
transition temperature is from the Angell definition 54
Figure 10 The relativistic glass transition temperature for B2O3 glass 57
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different
fractions of light speed All of the viscosities approaching the universal temperature
limit for viscosity 59
Figure 12 The modulus needed to satisfy the condition for the glass transition 61
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at
v=044c (Bottom) Viscosity plot showing a dramatic reduction of the viscosity as
the observer approaches the speed of light 63
Figure 14 b predicted and from literature showing good agreement with a total root-
mean-square error of 01 The fit for organic systems is given by 66 ln Kln = minus
and for inorganic systems by 75ln Kln = minus 72
Figure 15 The equilibrium model proposed with the experimental points showing good
agreement between the experimentally measured data points and the equilibrium
derived model RMSE was 002 for Corningcopy JadeTM (A) and less than 001 for
SG80 (B) (C) The model fit for B2O3 experimental data154155 The fragility and glass
transition temperature of the B2O3 are taken from the work of Mauro et al17 75
Figure 16 The stretching exponent calculated as described in the text for a Gaussian
distribution of barriers This plot shows that the distribution of barriers has a large
effect on the stretching exponent A Tg cannot be described since there is no
vibrational frequency included in the model though the glass transition temperature
should be the same for all distributions since the mean relaxation time is the same
for all distributions at all temperatures The deviation is given in ln eV units 78
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent
Each color designates one term in the series (Bottom) The output from RelaxPy
v20 showing the stretching exponent effects on the relaxation prediction of
Corningcopy JadeTM glass Each quadrant shows one property that is of interest for
relaxation experiments In particular it is interesting to see the dynamics of the
stretching exponent during a typical quench 80
Figure 18 The viscosity (left) and landscape (right) predictions for three common
systems The first system is newly calculated in this work while the latter two come
from our previous works7289 It is seen that the viscosity predicted from the AG
model is very accurately able to reproduce the experimental viscosity curves from
the MYEGA model The last system is a potential energy landscape while the others
are enthalpy landscapes 87
Figure 19 The configurational entropy comparisons between the three major viscosity
models which validates the main underlying assumption of the MYEGA model The
viii
VFT and AM are unable to capture the physics of configurational entropy therefore
ruling 89
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the
configurational entropy from the enthalpy landscape with the barrier 00155 eV
(compared to the shear barrier of 00149 eV) and the infinite limit allowed to vary
(10-263 Pamiddots for bulk viscosity) In Figure 18 the configurational entropy is
confirmed for the shear viscosity thus confirming the AG for both shear and bulk
viscosities Sidebottom data are from Ref 154 90
Figure 21 (A) The histogram for the energy minima for each of the test systems showing
a good fit with the log normal distribution This distribution will then be a valid
form to calculate the enthalpy distribution of the model presented in the next
session (B) The configurational entropy from the model showing the accuracy of
the scaling of the entropy predicted by the MYEGA model The S value was fit
for each system 93
Figure 22 (A) Comparison between the randomized method (histogram) and the
deterministic method (vertical lines) showing good agreement between the
maximum in the histogram and the value predicted by the deterministic technique
validating the approach It is worth noting that the 100-basin distribution is a very
wide distribution where the total number of basins is less than the number of points
used in the calculation This is done for a variable number of basins with the number
of basins shown in the legend (B) The dependence of fragility and the glass
transition temperature vs the distribution of states and the number of basins 99
Figure 23 The driving forces for different example glasses calculated using a
combination of the MAP model RelaxPy and the toy landscape model The
parameters for each glass can be found in Table 4 102
Figure 24 The predicted outputs from the toy landscape method showing predictions of
the enthalpy and entropy under a standard quench for barium disilicate This
prediction does not require fictive temperature or any such assumptions about the
evolution of the non-equilibrium behavior 105
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume
in nucleation is assumed to be on the order of one cubic angstrom while the a
parameter in growth is assumed to be around one nm both are in good agreement
for estimates in literature The values for the orange points are taken from these
works128167 107
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the
hydration models Color scheme Si atom (ivory) O atom (red) and Na atom (blue)
The z-axis is elongated to allow space for an insert of water 114
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the
top surface in contact with water is shown here 116
ix
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example
of the electronic-structure DFT calculation using semilocal exchange-correlation
functionals finding the binding energy of a water lsquopixelrsquo to the surface 119
Figure 29 Example contour surface showing the average coordination per atom on the
glass surface for the first run at 300 ps 121
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of
constraints for surface atoms at the local pixel Results show a distinct maximum in
which there is a near hydrophilic-hydrophobic transition of the surface The error
bars represent the standard deviation A second system with 1500 atoms was
performed to show convergence of the ReaxFF MD results 123
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values
of the model predictions are 641 GPa for constraint density 313 GPa for free
energy density and 774 GPa for angular constraint density 128
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from
minimum to maximum error The free energy density model gives the most accurate
results The constraint density has a RMSE of 61 GPa the angular density has a
RMSE of 20 GPa and the energy density has a RMSE of 59 GPa 129
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and
experiment for 10 Na2O 90 B2O3 Using the previously fitted onset temperatures
the only free parameters are then the vibrational frequency and the heating time in
which their product was fitted to be 14000 Where each dip in the modulus
corresponds to a constraint no longer being rigid as heated through each onset The
onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The
data was fit using a least-squares method and the resultant fit is shown as the
calculated method The fit has an R2 of 094 (B) The contribution from each
constraint to the overall modulus 131
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in
Fig 3) and experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of
093 and (B) xLi2Omiddot(100-x)B2O3 glasses with an R2 of 0986 133
Figure 35 The structure for the initial minimum energy configuration showing the boron
(blue) network with interconnecting oxygens (red) and the interstitial sodium ions
(yellow) 139
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and
sodium is ivory The barrier is overestimated compared to experimental data this
could be from several sources of error such as potential fitting thermal history
fluctuations or sampling too few transitions The line is drawn as a guide to the eye 139
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function
of reaction coordinates The middle shows the local deformation around the ion of
x
any atom that moves in between inherent structure mandating a relaxation force
The color shows the degree of deformation 141
Figure 38 Different network formers and the prediction of the activation barrier from our
model compared with activation barriers from literature (A) Sodium silicate
predictions and experimental values241 the error is calculated from the error in the
fragility when fitting the data (B) Lithium phosphate activation energy235 predicted
with topological constraint theory and compared with the experimental values (C)
Predictions over two different systems of alkali borates232 sodium and lithium with
a reported R2 of 097 143
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top
left) shows the distribution of the infinite temperature limit in the database after
limits exerted on the system (Top right) The distribution of the infinite viscosity
limit vs the glass transition (Bottom left) The relationship between fragility and
infinite temperature limit of viscosity (Bottom right) The infinite temperature limit
of viscosity vs the key metric predicted by SR 149
Figure 40 The prediction of the scaling of the activation barriers for a common sodium
borosilicate system 157
Figure 41 The relationship between the glass transition and the proton conductivity This
is justified two ways one through the relationship of the entropy of diffusion and
glass formation (the Adam-Gibbs model) and through the fact that water is known to
depress the glass transition 162
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine
the glass forming region Over top the constraints at the glass transition provided by
each oxide species is listed Since the objective is to decrease Tg while staying in the
glass forming region we will attempt to minimize use of elements that increase the
glass transition (nc gt 17) 166
xi
List of Tables
Table 1 The key properties considered for commercial application The optical properties
have been omitted since it is physical unrealistic to expect quantitative predictions of
quantum-controlled phenomena from a classical description of glass structure 18
Table 2 Variable definitions 65
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy
JadeTM glass and Sylvania Incorporatedrsquos SG80 74
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition
temperature (500 K) were kept the same while the total number of basins were
allowed to vary 101
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K 112
Table 6 System configurations for sodium silicate glass-water reactions 117
Table 7 Fitted values from this analysis compared to those reported in the literature The
disparity between the constraints evaluated with molecular dynamics most likely
come from the speed in which the samples are quenched 132
Table 8 Hyperparameters for different neural networks after hyper-optimizations 151
Table 9 A table with some ionic conductivity models and the parameters needed for
them as well as the disadvantages for each These are not the only models but are
representative of those commonly used in literature 154
Table 10 The predicted compositions based on the optimization scheme proposed 157
Table 11 The compositions synthesized in this work These compositions were predicted
by minimizing the cost function described in Eq (123) OP is the variant that was
melted after OP partially crystallized B-OP appeared to have surface nucleation in
some spots but was cut and removed before APS treatment 165
xii
Acknowledgments
ldquoTo Love Another Person is to See the Face of Godrdquo -Victor Hugo Les Miserables
I have been truly blessed to love and be loved by a plethora of mentors friends colleagues and
family across my academic journey I have to first thank my advisor Dr John Mauro who has
been as kind patient and caring as any mentor I have ever met and in the process has made me a
better scientist and a better person His mentorship was built on top of those who first found and
shaped me into a scientist Dr Ugur Akgun and Dr Steve Feller without whom my scientific career
would not be possible Along with these individuals Irsquove had the pleasure to learn so much about
glass and life from Dr Madoka Ona Dr Firdevs Duru Dr Mario Affatigato Dr Doug Allan Dr
Ozgur Gulbiten and Dr Seong Kim
Rebecca Welch Arron Potter and Anthony DeCeanne are my partners in all things
Rebecca is my partner in research and in life Her opinion support and patience has been
indispensable to me Arron is my oldest colleague and friend he is the one that I share the highs of
research and friendship with Anthony has put up with me non-stop for about 5 years now He is
the most patient and kind friend you could ask for Without these three individuals I could not have
done anything presented in this work I additionally have to thank Karan Doss Aubrey Fry Caio
Bragatto Daniel Cassar Brenna Gorin Mikkel Bodker Greg Palmer Katie Kirchner Kuo-Hao
Lee and Yongjian Yang I have the privilege to call these individuals both my friends and
colleagues
The last people I have to thank are my family I want to thank my brothers (Sam amp Quinn)
my mom and my dad My mom has helped me explore the universe through a love of reading and
her constant unwavering support My dad was the first to show me the wonders of science with
rockets as a kid as a teenager he supported my fledgling interest as he drove hundreds of miles to
every corner of this country and he is the person for whom this dissertation is dedicated
Chapter 1
The Difficulty of Optimizing Glass
Glass is a complex world-changing material Though it has existed for thousands of years
the surface of its true potential is only just being scratched To unlock the potential of glass for new
applications and to enable the maximum benefit to society we must be able to design new glasses
with wide ranging properties faster and cheaper than ever before [1]ndash[5] This is particularly
important now that the need for ecologically responsible materials is increasing due to climate
change In order to facilitate such glass design we must return to first principles and build a picture
of glass from the ground up encompassing both compositional and thermal history dependencies
into our understanding of glass properties[3] [6] The present accepted description of glass is a
non-crystalline non-ergodic non-equilibrium material that appears solid and is continuously
relaxing towards the super-cooled equilibrium liquid state[1] This definition gives insights into the
nature of glass and includes points of particular interest in this dissertation
The first point to consider is that glass is non-crystalline meaning that it contains no long-
range order This gives glass one of its key advantages being infinitely variable Since a glass is
not limited by having to reach stoichiometric crystalline structures there is no limit to the number
of possible glasses or glass structures This gives glass its possibilites Glass has been considered
as a candidate for everything from hydrogen fuel cells[7] to batteries[8] and for applications
ranging from commercial smart phone covers[9] to bioactive applications[10] The infinite
variability also explains the difficulty of designing glasses Consider a system with three
constituents for example SiO2 Na2O and B2O3 This then means there are two independent
compositional dimensions that must be fully explored to find an optimal composition for an
application If this space were discretized to 1 mol spacing that would give over 5000 unique
2
glass candidates that would need to be studied to reach the optimal design A five-component glass
would correspondingly have over 9 million unique glasses More generally for a glass with up to
C components there are 1C minus dimensions over which to optimize If instead one considers a
crystal there is not a continuously variable space but instead only discrete points
The other ramification of glass being non-crystalline is that it must bypass the region of
crystallization at a rate much faster than the rate of nucleation[11] This is inherent in the definition
of glass and is shown in Figure 1 Figure 1 is called the volume-temperature (VT) diagram and is
key to understanding the nature of glass The bypassing of the crystallization region leads to a
super-cooled liquid and during quenching the material departs from equilibrium and enters the
glassy state The region of departure is called the glass transition temperature range or simply the
glass transition To understand why this occurs we must first understand the concept of ergodicity
Ergodicity was defined by Boltzmann to mean that the time-average value of a property is
equal to the ensemble-average value of that property[12] [13] Now that ergodicity is defined we
can start to consider the implications of the non-ergodic nature of glass Since it is non-ergodic the
ensemble average will not equal the time average on human timescales however the ergodic
hypothesis says that in the limit of long time they will be equal[12] In order for both statements to
be true glass must evolve to be ergodic and thus is inherently unstable To explore this concept
further we must understand the glass transition process and define a timescale If there exists a
liquid at some temperature T and is then perturbed to T dT+ there will be some inherent time
associated with its relaxation towards equilibrium which is described by struct the structural
relaxation time There is a separate relaxation time when the system is mechanically perturbed
called the stress relaxation time stress To then understand whether the resulting material is a glass
or a liquid we need to compare struct with the observation time If the observation time is much
longer than the relaxation time then what is observed over the course of a experiment is the
3
equilibrium liquid properties Conversely if the observation time is much shorter than the
relaxation time then the observed properties will not be ergodic and as such we will not sample
the properties of the equilibrium liquid We define this non-ergodic state as the glassy state
However since in the limit of long time the system must become ergodic once again then over time
all glasses will return to an ergodic nature through a process called relaxation [14]
The last point considered in the definition of the glass is that it is non-equilibrium Not only
is glass non-equilibrium it is also unstable and continuously progressing towards the super-cooled
liquid state (eg relaxation) Relaxation is inherent in all glasses and we can see this for a model
system in the VT diagram (Figure 1) In this diagram there are two glasses (of the same
composition) quenched at different rates (both of which bypass crystallization) In the slow-cooled
glass there is a lower volume (higher density) compared to the fast-cooled glass This difference is
due to the fact that the slow-cooled glass has been allowed to relax more fully compared to the
other This adds at least one additional dimension of optimization (such as quench rate from a high
temperature to room temperature) in which glasses must be designed in
This dissertation is dedicated to understanding the infinite variability of glass with respect
to crystallization composition and thermal history with tools being implemented and designed so
that the challenge of optimizing over these vast spaces can be done with higher efficiency lower
cost and with less experimental work load These tools are all inspired by the energy landscape
description of glass To design the optimal glass for any application we must understand the
relationship between the 1C minus compositional dimensions a minimum of one thermal history
dimension and the properties of a material There are many ways to explore these relationships
each with its advantages and disadvantages The methods to optimize over these thermal history
and compositional dimensions include energy landscapes[15] fictive temperature[16] topological
constraint theory[17] and machine learning[3] Of these energy landscapes are the most
fundamental with each other technique being related back to the energy landscape in some form
4
Though they are the most fundamental they are also the most difficult time consuming and have
the largest barrier to entry for use To then build our understanding of each technique we will start
with energy landscapes then explore how through a series of assumptions we can arrive at the other
methods
5
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists the
equilibrium liquid which as it is quenched can either become a super cooled liquid or crystallize
Crystallization causes a discontinuity in the volume The super-cooled liquid upon further
quenching departs from equilibrium and transitions into the glassy state Reproduced from
Fundamentals of Inorganic Glasses with permission from the author[1]
6
11 Energy Landscapes
Energy landscapes were first developed by Goldstein and Stillinger who proposed the
concept as a way to understand the evolution of a material under complex conditions [18]ndash[20]
Since Stillingerrsquos seminal work [18] energy landscapes have become important in the study of
protein folding [15] [21] [22] network glasses [6] [13] and other complex materials The concept
of energy landscapes has become commonplace and is readily evoked when describing the time
evolution of a material [20] [22]ndash[26] Despite energy landscapes being commonplace in some
fields the procedure for mapping a landscape remains difficult due to the associated computational
challenges [15] [27] To map an energy landscape one must find a set of local minima and the first
order saddle point connecting each pairwise combination of those minima Each minimum is called
an inherent structure and represents a structurallychemically stable state of the material A basin
is the set of structures that converge to the same inherent structure upon minimization [18]
In glass science the energy landscape is used to justify complex behaviours which are most
commonly calculated using molecular dynamics simulations (MD) [13] [28] [29] It is easy to
understand why MD is used so frequently used when considering the ease of use of common MD
packages For example LAMMPS [30] is a common user-friendly MD program with a low barrier
to entry allowing individuals with little programming experience to run complex MD simulations
Inspired by this accessibility as well as the explicit Python bridge to LAMMPS we have developed
a software that can be used to explore energy landscapes with a variety of exploration techniques
Similar softwares [15] [31]ndash[35] have been created to map landscapes however each program has
limitations and may not be suitable for all purposes while ours is a general purpose software
The calculation of the energy for landscapes is done using an empirical interatomic
potential that can be fit using either experimental data and a series of MD simulations or using ab
initio methods One common potential form is the Lennard-Jones form (LJ)
7
12 6
4LJVr r
= minus
(1)
in which and are fitting parameters for every pair-wise set of atoms in a system V is the
potential energy contributed by the two-atom interactions and r is the distance between the two
ions in question Though the LJ form is simple and easy to parameterize it is unable to recreate
complicated material behaviours for a real glass As such a more complicated potential is often
used such as a Morse potential A commonly used silicate glass potential is the Pedone potential
which consists of Morse potential with an additional columbic term and repulsive term [36] and is
given by
( )2
2
0 121 exp 1
i j
Pedone
Z Z e CV Y a r r
r r = + minus minus minus minus +
(2)
Z is the charge of a given ion e is the charge of an electron and Y C a and r0 are fitting parameters
It is also worth noting that the Pedone et al potentials are only accurate when considering the
interactions between each individual ion species with oxygen For instance Si-O Na-O and O-O
are all explicitly parameterized while Si-Na only interacts coulombically (the first term)
Interatomic potentials are discussed here only briefly but there are extensive resources available
for those readers who wish to learn more[37]ndash[39]
Once an energy landscape has been mapped the thermodynamics and kinetics for a given
system can be easily calculated [33] [40] The thermodynamic driving force is given by the
difference between the free energy of the current inherent structure and the free energy of the lowest
state while the kinetics is the rate in which the system crosses the barrier between the basins This
can be calculated explicitly for arbitrary timescales and temperatures with another software
KineticPy [41] The energy landscape provides insights into the detailed nature of material
behaviour Though energy landscapes are very powerful they are also difficult to map To
approximate the behavior of the landscape then we use approximations such as fictive temperature
8
topological constraint theory (TCT) or machine learning Fictive temperature is a way to estimate
the effects of the thermal history dimension[42] while TCT is used to estimate the compositional
dependence of properties[43]
12 Fictive Temperature
Fictive temperature is an approximation used to find the change in the occupational
probabilities on an energy landscape[16] [44] Instead of considering every possible combination
of occupational probabilities like the energy landscape it instead approximates the occupational
probability using a single temperature at which the landscape would be in equilibrium It is
essentially a measure of how closefar a glass is from equilibrium This approximation allows for
us to calculate the relaxation dynamics of a glass without needing any computationally heavy
energy landscapes However this single parameter description oversimplifies relaxation[14] [16]
The functional form for glass relaxation dates back to 1854 when Kohlrausch fit the
residual charge of a Leyden jar with[45]
( ) (0)expt
g t g
= minus
(3)
where t is time τ is the relaxation time constant β is the stretching exponent and g(t) is the
relaxation function of some property (eg volume conductivity viscosity etc) Eq (3) is referred
to as the stretched exponential relaxation function (SER) The stretching exponent is bounded from
0 lt β le 1 where the upper limit (β = 1) represents a simple exponential decay and fractional values
of β represent stretched exponential decay
For over a century the physical origin of the stretched exponential relaxation form was one
of the greatest unsolved problems in physics until 1982 when Grassberger and Proccacia[46] were
able to derive the general form based on a model of randomly distributed traps that annihilate
9
excitations and the diffusion of the excitations through a network This model however provided
no physical meaning for β which was still treated as an empirical fitting parameter In 1994
Phillips[47] [48] was able to extend the diffusion trap model by showing that β=1 at high
temperatures and at low temperatures the stretching exponent can be derived based on the effective
dimensionality of available relaxation pathways He in turn expressed the stretching exponent
2
df
df =
+ (4)
where d is the dimensionality of the network and f is the fraction of relaxation pathways available
He then proposed a set of lsquomagicrsquo numbers for common scenarios Assuming a three-dimensional
network with all pathways activated (d = 3f = 1) a value of β= 35 is obtained The second case is
a three dimensional network with only half of the relaxation pathways activated (d =3f = 12)
yielding a value of β= 37 A third magic value β= 12 was found for a two dimensional technique
with all pathways activated (d = 2f = 1) β = 35 occurs for stress relaxation of glasses under a load
because both long- and short-range activation pathways are activated whereas a value of β = 37 is
obtained for structural relaxation of a glass without an applied stress The value was confirmed by
Welch et al[9] when the value was measured over a period of 15 years at more than 600C below
the glass transition temperature of Corning Gorilla Glass Though this model was able to reproduce
the limiting values it was criticized widely in the community Critics argued that the model was
simply created such that it reproduced the stretching exponent values of certain experiments and
ignored many other experimental results that appeared to disagree with the model The model also
fails to interpolate between the low temperature values Phillips predicted and the high temperature
limit
The work from Phillips and the stretched exponent gave a form for relaxation however
there remained no method to instantaneously state the distance from equilibrium of a glass To
account for this problem an additional thermodynamic variable (or order parameter) called fictive
10
temperature (Tf) was proposed Early research on relaxation in glasses dates back to the work of
Tool and Eichlin in 1932[49] and Tool in 1946[50] whose works originally proposed a temperature
at which a glass system could be in equilibrium without any atomic rearrangement Tool suggested
that this fictive temperature was sufficient to understand the thermodynamics of a glassy system
Originally the fictive temperature was treated as a single value that was some function of thermal
history (Tf[T(t)]) The evolution of fictive temperature as proposed by Tool is then given by
)
( ) ( ( ))
(
f f
f
dT T t T T t
dt T T
minus= (5)
In which for stress relaxation is given by
( )(( ) ( ))fT t T
G
T t = (6)
In Eq (6) is the shear viscosity and G is the shear modulus This means that to predict the time
evolution of a glass only three things are required shear modulus the viscosity as a function of
temperature and fictive temperature and the stretching exponent Though fictive temperature
qualitatively reproduced the results they were looking for subsequent experiments have shown that
the concept of a single fictive temperature is inadequate[51] [52]
A key experiment was performed by Ritland in 1956[51] Ritland took several samples
with different thermal histories but the same measured fictive temperature thus based on Toolrsquos
equation both should have had identical relaxation properties Ritland however showed that the
refractive index evolved differently between samples To account for these differences
the use of multiple fictive temperatures was suggested[52] This worked because the stretched
exponential form could be approximated with a Prony series
( ) exp expi i
N
i
Kt w K t minus minus
(7)
11
In the Prony series N is the number of terms while iw and iK are fitting parameters This means
that each term could represent one fictive temperature and this collection of fictive temperatures
would reproduce the overall experimentally observed form Ritlandrsquos experiment showed that
materials held isothermally at their fictive temperatures with varying thermal histories can give
different results for the relaxation of a given property Thus as a glass network relaxes the fictive
temperature will also shift Narayanaswamy[52] was able to apply Ritlandrsquos concept in a highly
successful engineering model with multiple fictive temperatures but the physical meaning of
multiple fictive temperatures remains elusive
In general there is reason to be skeptical of the concept of fictive temperature Fictive
temperature was not derived but instead was an empirical approximation that Tool needed to gain
an early quantitative understanding Recently Mauro et al[16] used energy landscapes to try to
understand the underlying validity of fictive temperature fT implies that the occupational
probability ( ip ) is given by
1
exp ii
Hp
Q kT
= minus
(8)
Q is the partition function k is Boltzmannrsquos constant and iH is the enthalpy of the given state
However when the actual probability evolution predicted by fictive temperature was compared to
the probability calculated through a systematic Monte Carlo approach the results were found to be
drastically different (even when many fictive temperatures were tried) This implied that fictive
temperature is insufficient to describe the glassy state and must be replaced with a new method
Despite the inadequacies of fictive temperature state-of-the-art modeling techniques for
glass relaxation still relies on the concept In these techniques a set of fictive temperatures (usually
12) are used to calculate 12 parallel relaxing fictive temperatures[42] This is because there has
been no alternative method able to reach a prediction of the evolution of glass under different
12
thermal histories The current state-of-the-art model is called the Mauro-Allan-Potuzak (MAP)[44]
model of non-equilibrium viscosity and was derived from a more fundamental expression for
viscosity called the Adam-Gibbs model (AG) which is given by[53]
10 10) log( )
log ( f
c f
BT T
TS T T = + (9)
B is the barrier for a cooperative rearrangement 10log is the infinite temperature limit of
viscosity and according to Zheng et al[54] it is approximately 10-293 Pa s and cS is the
configurational Adam-Gibbs entropy This entropy has long since been a source of debate and will
be addressed later in chapter 5
In order to calculate the equilibrium part of the viscosity the MAP model relies on an earlier
Mauro Yue Ellison Gupta and Allan (MYEGA) equilibrium viscosity model[55] This model
was derived based on earlier work by Naumis[23] [56] Gupta and Mauro[57] and the AG model
These models showed that the configurational entropy can be related to the degrees of freedom
lncS fNk= (10)
f is the degrees of freedom per atom N is the number of atoms and is the number of degenerate
states per floppy mode To then build in temperature dependence they assumed that the degrees of
freedom can be modeled using an Arrhenius form (with activation barrier H )
( ) expH
f T dkT
= minus
(11)
When Eqs (9) (10) and (11) are combined into an equilibrium form the equilibrium viscosity is
then given by
10 10 el pog og xlC H
T kT
+
=
(12)
Where C is equivalent to
13
ln
BC
dkN=
(13)
Through a change in variables this can be written in terms of three common variables that allow
for all viscosity equations to be written with three variables specified by Angel[58] the glass
transition temperature the fragility and 10log The glass transition temperature is defined as
the isokom where the liquid has a viscosity of 1210 Pa s The fragility is defined as[58]
10log
g
g
T T
dm
Td
T
=
(14)
The three most common viscosity models (MYEGA Vogel-Fulcher-Tammann (VFT)[59] and
Avramov-Milchev (AM)[60]) are listed below in terms of the same parameters
10 10 10
10
log (12 log ) exp 1 11 o
log2 l g
g gT Tm
T T
+ minus minus minus
minus =
(15)
( )
2
10
10 10
10
12 logl log
1 1
og
2 logg
Tm
T
minus+
minus + minus
=
(16)
and
( )10(12 log
10 10 1
)
0l gog l 12 logo
m
gT
T
minus
+ minus=
(17)
To incorporate non-equilibrium effects with these equilibrium models the MAP model
proposes
10 10 10log ( ) (1 ) log ( )log MYEGA g ne f gT m T mx x T T T = + minus (18)
14
10log ne is the non-equilibrium viscosity whose form and compositional dependence can be found
elsewhere[42] [44] [61] x is the ergodicity parameter and is of crucial importance to
understanding glass regardless of the model being used[12] In the MAP model it is given by
( )( )
3244min
max
f
f
m
T Tx
T T
=
(19)
Though the MAP model was derived from the best understanding of energy landscapes of glasses
at the time it has a few flaws that prevent it from perfectly predicting the different relaxation
behaviors of glass A few of the short-comings are
1 The method ignores crystallization For a comprehensive model showing the response of
glass to temperature the possibility of crystallization must be included Currently no
relaxation model includes this possibility
2 It is built around the concept of fictive temperature This could be replaced in a future
model but currently at present it carries with it all the failures of the fictive temperature
picture of glass dynamics adding only the relaxation time coming from landscape-derived
values
3 The MAP model does not consider temperature dependence of the stretching exponent
This is typically circumvented by choosing a fixed stretching exponent
4 The parameters needed for the non-equilibrium viscosity in the MAP model are
experimentally challenging to measure To circumvent this Guo et al[61] published
methods to estimate the values based on fragility and the glass transition but the
approximations were only tested on a few samples with similar fragilities and chemistries
5 The MAP model only predicts stress relaxation Since the publication of the MAP model
it has been shown that only stress relaxation time is related to the shear viscosity[62]
Structural relaxation must be incorporated in future models
15
For a complete understanding of the thermal-history dependence of glass all of these questions
must be rigorously answered
13 Topological Constraint Theory
TCT is one way to estimate the compositional dependence of glasses It uses a set of
assumptions that connects the underlying structure to glass properties through understanding of the
energy landscape These assumptions include the temperature dependence of the rigidity the same
assumptions used in the derivation of the MYEGA models and that the effects of thermal history
on structure are minimal TCT was originally proposed by Gupta and Cooper (GCTCT) to try to
understand the role of rigid polytopes on glasses properties[63] Gupta and Cooper fundamentally
understood that the ability for glasses to form was reliant on the transformation of a liquid with
many degrees of freedom to a glass with a fixed degree of freedom Since the structure of the glass
is the same as the structure of the liquid at the glass transition and the degrees of freedom of the
glass ( f ) changes during the transition to glass it was possible to gain insights into the
transformation process This approach is fundamentally an approximation of the highly
multidimensional landscape where only approximate structural information is used to understand
the entropy of the energy landscape Though the core idea was useful it was ldquoclunkyrdquo and difficult
to make calculations with
To improve upon the ideas of GCTCT Phillips and Thorpe (PTTCT)[64] proposed a
mathematically equivalent form where the degrees of freedom could be simply found by
considering the number of constraints ( cn ) around each network-forming atom in the glass They
showed that the degrees of freedom were related to the number of constraints by
cf d n= minus (20)
16
The number of constraints could then be calculated by considering the two-body and three-body
interactions (first and second terms) as a function of the mean coordination of the network forming
species
2 32
c
rn r= + minus (21)
This led to a qualitative language to discuss glass with a few quantitative insights The biggest
quantitative insight comes in the form of the lsquoBoolchand Intermediate Phasersquo[65]ndash[68] If it has
positive degrees of freedom then a network is lsquofloppyrsquo and retains high kinetic energy (as it flips
back and forth between two inherent structures) If the system has negative degrees of freedom
then the network is stressed-rigid and there is additional energy stored in the stress associated with
over-coordination If the number of constraints is equal to the number of dimensions then the
resulting system will have zero degrees of freedom and there is no stored stressed energy or kinetic
energy from flipping between basins Boolchand went searching for this perfectly lsquoisostaticrsquo glass
and found a non-zero window of compositions that were close to isostatic but had properties
drastically different compared to both stress rigid and floppy glasses This is now called the
Boolchand Intermediate Phase
Despite the success in the prediction of the intermediate phase TCT remained largely
qualitative until the work by Naumis[23] [56] and Gupta amp Mauro[57] Naumis pointed out that
the degrees of freedom are a lsquovalleyrsquo of continuous low energy so that the structure can freely move
between basins of the landscape Since these valleys are deformation pathways Naumis pointed
out that they will dominate the configurational entropy of the system Building on this Mauro and
Gupta wrote that since the entropy is proportional (Eq (10)) we could then write the glass transition
temperature as a function of the degrees of freedom and the AG model
( )10o1 ln2 l g
gTB
f Nkminus=
(22)
17
Since everything in Eq (22) is approximately a constant between compositions except for f we
could then write the expressions in terms of reference values (subscript r) and accurately scale the
glass transition of new compositions by knowing the changing structures
( ) g r c r
g
c
T d nT
d n
minus=
minus (23)
The last insight provided by Mauro and Gupta was that the number of constraints change as a
function of temperature They realized that each constraint has a temperature dependence which is
controlled by the interaction energy of the constraints This makes intuitive sense because as the
thermal energy becomes greater than that of the barriers additional valleys in the energy landscape
form This also allowed them to expanded TCT to include quantitative predictions of fragility
The understanding of the relationships between the configurational entropy the number of
constraints the degrees of freedom and the viscous properties lead to a revolution in understanding
glass properties Soon after the work by Mauro amp Gupta multiple works came out predicting
mechanical chemical and viscous properties of glass knowing only the underlying structure
Though this framework is powerful it is not a complete description without further
parameterizations with each model requiring some fitting parameters or reference values In
addition there are still key properties that are missing from the standard topological approach that
are needed to design commercial glasses
Ultimately MGTCT needs be expanded to include the properties that are needed for
commercial applications A list of composition dependent properties needed for understanding the
performance of commercial glasses are shown in Table 1 Properties taken from Ref [3] show what
properties must be controlled to manufacture technological glasses All models take additional
inputs and there are multiple ways to link outputs from some models to inputs of the other Choice
of model vary depending on the use case
18
Table 1 The key properties considered for commercial application The optical properties have
been omitted since it is physical unrealistic to expect quantitative predictions of quantum-
controlled phenomena from a classical description of glass structure
Properties TCT Machine Learning Other methods
Glass Transition [57] [69] [70] Energy Landscapes [71]
Fragility [57] [69] This work This work
Relaxation MAP model [44] [61]
Crystallization This work [72]
MDMC [73]ndash[75]
Melting Temperature This work MDThis work[72]
Chemical Durability [76] [77]
This work [78]
[79] [80] Linear Methods [81]
Density [82] MD Packing [83]
CTE This work MD
Hardness [5] [84] [85] MD
Youngrsquos Modulus [86] This
work [87]
[82] This work MD
Activation Barrier for Conductivity MD This work [88]
Batch Cost Economic Calculation
Ion Exchange Properties Empirical Relationships [2]
19
14 Machine Learning
All the methods discussed so far are physically informed techniques The techniques
presented are inherently powerful because their very nature is informed by reality Unfortunately
this means that some knowledge about the system is required These techniques are useless without
the required parameterizations However a class of algorithms called lsquoMachine Learningrsquo (ML)
require no such parameterization Instead ML uses large quantities of data to determine trends
linking any physically connected input to output allowing for accurate predictions of properties
across wide compositional spaces This is favorable with a large of volume data because the
required inputs are minimal to achieve a prediction The difficulty (and often expense) in this
technique is gathering large quantities of high-quality data to enable this method
The predominant ML method used in this work is neural networks (NN) Neural networks
work by constructing a network of neurons This method was designed in such a way to mimic a
human brain so the network is elastic enough to be modified improving results Each neuron takes
in a series of inputs received from the previous layer (or input data) multiplies by a weighting
factor sums the values and then performs some lsquoactivation functionrsquo on them By changing the
weights it is then possible to recreate any function given sufficient data This is a preferred
technique because it packages into a simple easy-to-use software Neural Networks (once trained)
can be used by anyone stored in a small file can recreate any function is computationally cheap
and does not require as much data as some other ML techniques
This is an interesting method to estimate the effects of the composition on the landscape
and corresponding properties because we cannot understand the underlying mathematics It is
fundamentally a numerical process Thus even though we know that the properties of a glass are
dominated by the topography of the landscape how the NN figures out the relationship between
the chemistry and properties is not understood It is possible to then use the NN to give us the
20
physical parameters we need for other models building multiple methods together to achieve new
insights
15 Goals of this Dissertation
To enable the next generation of glass design we must enable a new era of computationally
methods being used to produce accurate predictions of the behavior of glass This includes both the
thermal history dependence of glass (in terms of relaxation and crystallization) and the
compositional dependence especially those properties listed in Table 1 The goals of this thesis will
be to address some of these outstanding questions in a systematic way that is practical for an
external user to understand and implement Predicting thermal history effects are covered in
Chapters 3-5 while compositional effects are predicted in Chapters 6-7 The breakdown of each
following Chapter as it relates to these questions is listed below
21
bull Chapter 2 Software Unfortunately there is no standardized software for the
calculation of relaxation or energy landscapes I present two new codes to
standardize these calculations so that the process is reproduceable and standardized
bull Chapter 3 Crystallization This section will address new computational ways to
calculate crystallization focusing on predicting nucleation rates and how to improve
our nucleation calculations
bull Chapter 4 Expanding the Current State of Relaxation This chapter will focus on
expanding the MAP model with a temperature dependent form of the stretching
exponent as well as understanding the role of ergodicity in glass relaxation
bull Chapter 5 Relaxation and Crystallization without Fictive Temperature Chapter 5
will focus on discussing relaxation and crystallization without fictive temperature
by proposing a new relaxation model that is reliant on creating lsquoToy Landscapesrsquo
This approach removes assumptions about the evolution of the occupational
probabilities
bull Chapter 6 Enabling Prediction of Properties This chapter will use TCT and ML to
address the compositional dependance of different key properties needed for
enabling the faster design of new glass compositions
bull Chapter 7 Designing lsquoGreenrsquo Glasses Applying information learned throughout
this dissertation to create new glass compositions
bull Chapter 8 Conclusions
22
Chapter 2
Software for Enabling the Study of Glass
To being our systematic investigation it is important to have a suite of tools that
implement current state-of-the-art models Though many tools exist for structure and
topological constraint theory there were no codes that existed for the calculation of
relaxation using the MAP model and there were no codes that were capable of exploring
the landscapes that we are interested in To correct this we present two new novel codes
ExplorerPy for mapping energy landscapes and RelaxPy for calculating MAP
dynamics[42] [89]
21 ExplorerPy
Mapping an energy landscape allows for modeling a range of traditionally inaccessible
processes such as glass relaxation The advantage of this software (over other codes to map energy
landscapes) is that it only requires knowledge of LAMMPS to run Also our software contains a
choice of three methods to map the landscape The first of the three different methods implemented
to explore energy landscapes in this work is eigenvector following [90] Eigenvector following has
been used in the past to generate landscapes that predict the glass transition and protein folding
pathways [44] [91] This implementation follows the work of Mauro et al [90] [92] and uses an
independent Lagrange multiplier for every eigenvector Eigenvector following excels at finding
small barriers quickly however it rarely finds unique structural changes in complex materials
(especially in systems with periodic boundary conditions) The second method that we have
implemented following in the steps of Niblett et al [27] is a combination of molecular dynamics
and nudged elastic band methods (NEB) [34] [35] To find the transition points MD is run at a
23
fixed temperature for a fixed amount of time the structure is minimized and then NEB is used to
find the saddle point between the initial structure and the new inherent structure Details of the
nudged elastic band method can be found elsewhere [34] Using these techniques it is possible to
map a sufficiently large landscape to make kinetic and thermodynamic measurements possible
These two methods benefit from the fact that they work well when the volume is varied and together
enable computationally inexpensive mapping of enthalpy landscapes [71] [92] (where the volume
is no longer fixed and instead the pressure is specified) However it is not recommended to generate
a large enthalpy landscape from MD in this software due to the computational cost involved
Although molecular dynamics is a powerful technique there is a large computational cost to
simulate a small amount of time
The third method is a new technique created in an attempt to circumvent the computational
cost of MD while still finding large structural changes We have named this technique ldquoeigen
treibanrdquo (ET) This technique relies on shoving along an eigenvector with a multiplier rather than
stepping After the system is shoved the system is minimized and if the minimized structure exhibits
a displacement in atomic position is larger than a threshold a NEB calculation is performed to find
the saddle point By shoving in the direction of an eigenvector the goal is to reproduce large changes
due to vibrational modes The fixed multiplier (m0) is calculated by taking a target distance (d) and
the largest displacement magnitude out of the atoms j (vij) in a given eigenvector direction i (ai)
m0
=d
max[nij] (24)
To convert to the displacement magnitude from an eigenvector in three dimensions we use the
expression
vij
= ai[3 j]2 +a
i[3 j +1]2 + a
i[3 j + 2]2 (25)
The dimension j has the length of the number of atoms in the systems
24
This software uses the system and potentials created from a LAMMPS input
scripts and as such has a wide range of access to different potentials and possible systems The
only input that has to be provided is the file which would be used for molecular dynamics in
LAMMPS If the script has been used for molecular dynamics simply by eliminating any
thermostats or barostats as well as any ldquorunrdquo commands it is possible to use it as an input script to
ExplorerPy (it is also recommended to eliminate any calls to ldquoThermordquo because it may interfere
with the extraction of variables from the underlying LAMMPS framework) If the LAMMPS input
script successfully sets up a simulation all other variables and parameters can be controlled from
command line options as described in the ldquoIllustrative Examplerdquo section The initial structure will
be taken from the input script as well
ExplorerPy is a software that has been written in Python with modular functions and ease
of use in mind It can be run from a Linux terminal with all major options being set via the command
line The software then generates a list of inherent structure energies stored in the mindat file and
stores the transition point energy for each pairwise inherent structure in the tsddat file All of the
transition points and minimum are stored in the same directory from which it is run as xyz files
The software architecture is shown in Figure 2 Details of the stepping procedure for eigenvector
following can be found elsewhere[93] The variables ldquoMD frequencyrdquo and ldquoET frequencyrdquo are set
as command line options at the time of the launch of the software
25
Figure 2 The schematic for the flow of the program Beginning in the top right corner and running
until the condition in the pink box is satisfied Yellow diamonds represent checks and blue
operations
The software is capable of mapping an energy landscape for an arbitrary system ie for
any system that can be simulated in LAMMPS This allows for accurate prediction of the evolution
of complex thermodynamics and kinetics The software requires no programming and is
controllable via command line options ExplorerPy prints all of the basins transition states inherent
volumes as well as information about the curvature of the system so that the vibrational frequency
is calculable This is all of the information required to calculate a complex range of phenomena
over a specified long timescale ExplorerPy is only set up to work within the LAMMPS metal units
To run the software there are only a few parameters that have to be set for each mode The
force threshold (-force) curvature threshold (-curve) and step size (-stepsize) when running the
command line option have to be set to allow for a stable exploration of a given landscape
These seem to be potential-specific and for a given potential appear to be fairly stable It is best to
start large for the force threshold and small for the eigen threshold and step size then modify as
26
necessary If eigenvector following is not being used there is no need to adjust these values Beyond
this there are several optional command line options (only the file (-file) command is required)
-file [file input] this is a standard lammps input if the input sets up a lammps run
it will work to set this up There should not be any lsquothermorsquo calls in the lammps
script
-press [specified pressure in atmospheres] if this is specified the system will
always be minimized to this pressure while if not specified the volume remains
fixed
-num [number of basins to explore]
-time [max time to run in hours] if this time is reached before the number of basins
are found the simulation will stop
-thresh [minimum energy for barrier to keep]
-center if this is given as a flag the system will be set to the anchor location each
step
-anchor [atom selection for the position to be held constant]
-md [frequency of molecular dynamics search] If a custom MD LAMMPS file is
not provided it defaults to starting all of the atoms at 3000 K and running for 50
fs
-shove [frequency of eigen treiban]
-custom [file used for MD exploration]
-help [List all commands possible in software with these descriptions]
An example run would then be done the following way
[mpirun -np 5 python37 Explorerpy -file inSiO2 -num 1000 -force 10 -curve 1e-5 -stepsize 012
-press 00 -md 11 -time 48]
This would start a run with five parallel processors searching for 1000 basins with an MD
step being ran every 11 searches with a maximum time of 48 hours The results of this command
are shown in Figure 3 Below we have attached an example of inSiO2 the output and additional
scripts related to the example has been uploaded to the associated GitHub
27
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above and then
plotted using PyConnect The plot is a disconnectivity graph where each terminating line
represents an inherent structure and tracing where two lines meet describes the activation barrier
The potentials are taken from the BKS potential [94]
httpsgithubcomlsmeetonpyconnect
28
22 RelaxPy
RelaxPy script is written using Python syntax and relies on the numpy scipy and
matplotlib libraries It solves the lack of a unified relaxation code in our community and is based
on the MAP model[44] [61] interpretation of multiple fictive temperatures and the shear
modulusviscosity form of the Maxwell equation[62] [95] It is designed to be run within a Linux
terminal with command line options or inside a Python interpreter It can be used by calling the
name of the script followed by the input file the desired output file name and finally an optional
tag for all fictive components to be displayed as well as printed to a specified output file (formatted
as comma separated values)
The RelaxPy package consists of an algorithm that iterates over time in order to determine
the values of viscosity (η) relaxation time (τ ) and current fictive temperature (Tf ) given the user-
supplied thermal history and material property values The purpose of this code is to enable
calculation of the relaxation behaviorrsquos change over time by using the MAP model using only
experimentally-attainable parameters Toolrsquos equation is solved for each term in a Prony series
approximation of relaxation finding the change in each Tfi component for a given time (t) to
accurately reproduce the stretched exponential form The program then iterates over the entire
thermal history of the sample using a user-specified time step dt allowing for calculation of the
overall fictive temperature over time The values for the Prony series fit (wi and Ki) are taken from
the database originally generated by Mauro and Mauro[96] and expnded to include any value of
the stretching exponent The values were fit by Mauro and Mauro with a hybrid fitting method
based on the number of terms in the desired Prony series and the magic value from Phillips The
thermal history is also defined in the original input file using linear interpolation The initial fictive
temperature components are assumed to be at equilibrium (Tfi = T(0))
29
To solve for the non-equilibrium viscosity used the MAP expression is used with the approximate
scaling of values proposed by Guo et al[61] The software is capable of performing the calculations
needed for a variety of thermal histories and processing methods desired by both industry and
academia with a focus on the fictive temperature This software is then fundamentally limited by
the issues associated with the MAP model and the compositional approximations made For
instance this software will need to be further modified to include structural relaxation when such
a model is proposed[62]
An example input code is listed below
Denotes Comments
Tg 794 0 m 368 C 1494 dt 10 Beta 37
Start Temp End Temp Time [s]
1000 10 1300
10 100 1000
100 30 500
Assumes Linear Interpolation
This example input states that over the space of 1300 seconds an initial glass sample is cooled from
1000 C to 10 C Then it is heated to 100C over 1000 seconds and then finally down to room
temperature in the space of 500 seconds All temperatures are given in Celsius
This software aims to provide the first open-source software for modeling of glass
relaxation behavior using the MAP model of viscosity by creating a package that can easily and
quickly be used to approximate the fictive temperature components of a glass as well as the
composite fictive temperature RelaxPy allows for researchers to calculate the evolution of
macroscopic properties of glass during a relaxation process (assuming a fixed stretching exponent)
and will facilitate the calculation based on a user-supplied thermal history RelaxPy can be
seamlessly adopted by most glass research groups and removes the guessing associated with
determining a fictive temperature allowing for a fully quantitative comparable discussion of Tf
simultaneously creating a tool that can easily be expanded for the entire community as new physics
is discovered
30
Chapter 3
Understanding Nucleation in Liquids
Crystallization cannot be ignored in any material system at temperatures below its liquidus
temperature although extreme resistance to crystallization is reported in some organic and
inorganic substances[97] Crystallization (and by extension nucleation) is critical in systems
ranging from ice to thin films[98]ndash[100] Glasses are non-crystalline substances therefore
crystallization kinetics will determine the necessary thermal history to avoid crystallization and
obtain a glassy state Despite the absolute importance of crystallization to the study and design of
glasses there is no current method to predict the nucleation of crystals in this event One of the main
challenges in the production of glass-ceramics is finding the optimal thermal history for the desired
crystallite distributions There are two phenomenological steps in the process of phase
transformation from a supercooled liquid to a crystal a nucleation step and a growth step The
growth step is described well by Wilson-Frenkelacutes theory[1] but the underlying physics of
nucleation remains elusive A stable nucleus is the precursor to a crystal comprising a nano-sized
periodic assemblage of atoms A clear understanding of the relationship between liquid kinetics
and crystal nucleation has not been established despite decades of research
The study of nucleation is particularly critical for glass-ceramics Since their discovery by
S Donald Stookey glass-ceramics have become prevalent in our society through products such as
stovetops and Corningware[73] [101] [102] Glass-ceramics are a composite material comprising
at least one crystalline phase embedded in a parent glassy phase benefitting from the properties of
both the glassy and crystalline phases[1] [103] One of the reasons glass-ceramics are ubiquitous
is their relatively easy production glass-ceramics can be formed through standard glass forming
procedures using additional heat treatments Additionally there is a wide range of accessible
31
properties through tuning both the chemical composition of the parent glass and the thermal history
and hence the microstructure of the heat-treated glass-ceramics[101] [103]ndash[106]
Glass-ceramic production consists of first synthesizing a glass and then generating a
microstructure through the two-stage ceramming process[73] [103] [107] The nucleation step
remains poorly understood with classical nucleation theory (CNT) giving widely varying results
depending on how the parameters are determined[1] [73] [108] The inability to consistently get
accurate predictions for a nucleation curve means that choosing an appropriate nucleation
temperature must be done experimentally which can prove to be one of the most challenging and
time-consuming aspects of properly designing a glass-ceramic The difficulty is due to the
magnitudes of the crystal nucleation rates not being predictable with CNT or any other current
model
This difficulty in designing a material is apparent when considering the high dimensional
phase space of parameters that can adjusted We haver presesnted some tools developed to help
optimize over this lsquoglassyrsquo space such as RelaxPy[42] KineticPy[41] topological constraint
theory[109] or various machine learning models[4] [70] [110] [111] However a glass-ceramic
system has at least four additional phase dimensions (two for the nucleationgrowth temperatures
and two for nucleationgrowth times) that need to be optimized (depending on the parameterization
of the experiment it is also possible to add another four dimensions relating the heating and cooling
between each step) Therefore a glass-ceramic system has a minimum of d+5 dimensions over
which the system must be optimized and yet those four additional dimensions are the least
understood Designing methods to easily optimize over this large space with feasible computational
methods is of utmost importance to reaching a new generation of custom materials
Experimental studies of nucleation have used techniques ranging from differential
scanning calorimetry (DSC) to electron microscopy to try to capture predictive capability in the
lsquoceramicrsquo phase-space (the four additional dimensions) These studies have successfully captured
32
the shape of the nucleation curve but fail to provide the information needed to calculate the
temperature-dependence of the nucleation rate The shape of the curve can be determined through
DSC and with great effort an approximation of the rate can be made however this technique is too
laborusome to be efficient[112]ndash[114] Since the parameters have been inaccessible to traditional
computational approaches and experimental determination all previous studies of nucleation have
required at least one variable to be fit If a nucleation curve could be predicted only using a
computational method or with an efficient simple DSC method the rate at which new glass-
ceramics products are created would increase In addition the cost of producing glass-ceramics
would likely decrease and more complex microstructures and therefore more unique properties
could be explored
Crystal nucleation has been historically difficult to simulate and measure experimentally
because the phenomena happen on spatial scales of nanometers but over many orders of magnitude
of time scales CNT is the most widely applied model despite its supposed failure To test CNTrsquos
assumptions an interest in computational work has grown Recent approaches using grand
canonical Monte Carlo have successfully predicted the short time dynamics of nuclei
formation[73] This approach involves use of a continuous solvation model for the liquid and has
enabled greater computational insights however it remains computationally expensive[74]
Molecular dynamics (MD) fails to capture nucleation in glass-forming systems due to the need for
longer spatial and temporal scales to capture nucleation effects accurately Some attempts have
been made using Monte Carlo MD or empirical approaches to replace classical nucleation theory
(CNT) with some alternate method where the parameters can be directly measured[73] [115]ndash
[117] All nucleation models and Monte Carlo models attempt to describe the rate that a
supercooled liquid system transitions to a crystalline structure as a function of temperature
Explicitly mapping the energy landscape would fundamentally explain the underlying
physics of nucleation An energy landscape of a system is given by
33
1 1 1 2 2 2( )N NNU U x y z x y z x y z= (26)
where U is the potential energy and x y and z are the locations of each of the N atoms in a periodic
box The locations of these atoms give rise to an energy as parameterized by an interatomic
potential An energy landscape can be constructed by mapping the relationship of the local minima
and the first order saddle points between pairs of minima22-27 Energy landscapes have become
crucial to studies of biomolecules glasses and catalysts[21] [26] [44] because landscapes show
atomic configurational changes and calculate the associated kinetics and thermodynamics In this
study we will explore the energy landscape of a barium disilicate system capturing the glassy and
crystalline basins enabling a test of CNT by calculating all needed parameters (interfacial free
energy kinetic barrier and free energy difference of the supercooled liquid and the equilibrium
state)
31 Crystallization Methods
CNT expresses the steady-state nucleation rate (I) as the product of a kinetic and a
thermodynamic factor
( )exp eI Z D TW
kT
minus =
(27)
where Ze is the Zeldovich factor which incorporates the probability of a critical sized nucleus that
will form a new phase independent of other nuclei D(T) is the kinetic factor often given by an
Arrhenius expression over sufficiently small temperature ranges W is the work associated with
generating a thermodynamically stable nucleus that will continue to grow rather than dissipate (ie
the critical work) if temperature and pressure are kept constant k is Boltzmannrsquos constant and T is
absolute temperature The Zeldovich factor is typically treated as a constant with a value ~01[118]
34
Here the kinetic term is expressed as D(T) since in this work the kinetics are explicitly calculated
at every temperature and it is not treated as a simple Arrhenius expression In CNT the work W is
given by the work to generate a spherical isotropic nucleus[1] which consists of a volume term
and a surface energy term It is given by
3 24
( ) 4 ( )3
W r G T r T = + (28)
where G is the Gibbs free energy difference between the crystalline state and the corresponding
liquid r is the radius and is the interfacial energy between the crystalline and liquid phases Eq
(28) neglects the strain energy term that arises from the formation of a solid nucleus in the
supercooled liquid It is assumed that the liquid relaxes fast enough so that strain energy is not
significant for nucleation in liquids[11]
Determining the critical radius (ie taking the derivative of Eq (28) with respect to the
radius and setting the expression equal to 0) allows one to solve for the critical work and the
expression becomes
3
2
16
)3(
( )
( )W
T
G T
=
(29)
Three temperature dependent functions remain D G and Though there are existing methods
in literature to approximate each of these quantities the approximations often include large errors
and assumptions about the behavior of the system[119] Often experimentally the kinetic function
is approximated with the Stokes-Einstein relationship the G term is calculated from heat
capacity curves (assuming that the macroscopic energy is similar to the nanoscopic energy) and
is fitted[119] In the calculation presented here only configurational energies are considered
while vibrational contributions are ignored Ignoring the vibrational component and the reliance on
empirical potentials are the most likely sources of error in this work
35
The energy landscape for this investigation was generated using ExplorerPy[89] starting
with the orthorhombic barium disilicate crystalline structure[89] Barium disilicate was chosen
since it is a commonly studied glass-ceramic with multiple reliable experimental datasets being
found in literature[75] [113] [120] The rest of this section is divided into four parts starting with
a discussion of mapping the landscape and the following three sections are a discussion of
calculating D G and respectively
31A Mapping and Classifying the Landscape
The ExplorerPy energy landscape exploration employed the Eigen-Trieban technique in
the canonical ensemble with systems consisting of 128 256 and 512 atoms[89] The number of
atoms was chosen to achieve convergence of the results while also balancing the large
computational cost associated with energy landscape exploration A smaller system was also
calculated with 64 atoms however this systems was too small to accurately discern energies
enough to make the nucleation calculation The Eigen-Trieban technique is a newly developed
method to explore energy landscapes using a method of shoving along the eigenvectors and
minimizing the potential energy which is followed by using nudged elastic band calculations to
find the associated transition points[89] This technique allows for larger transitions to be found
while retaining the ability to calculate the vibrational frequencies The Pedone et al potentials[36]
were used and the volume of the cell was fixed due to simulation instability when pressure was
fixed The Pedone et al potentials were chosen since they have reproduced key parameters for both
the crystalline and glassy state in previous Monte Carlo studies as well as in the training set of the
potential[73] [74] [121] Though the Eigen-Trieban method may work well for predicting
crystallization and finding large barriers it may be insufficient to describe glass relaxation or
36
viscosity since it rarely finds the small structural transitions that dominate the Adam-Gibbs
relationship Note that these exceptions are inconsequential for predictions of nucleation found in
the work presented here33
The barium disilicate system was explored for 2500 basins for 128 and 256 atoms and 4000
for 512 atoms since more basins were needed to converge the liquid state distribution of energies
This number of basins was shown to be sufficient since multiple paths from the initial crystalline
starting point to SCL states were obtained The distribution of energies of the SCL were found to
converge and the resulting energy landscape is seen in Figure 4
A crystal parameter was then defined from the root mean square displacement (RMSD) for
each atom from the nearest lattice site of the same species ( ) This average value is used to
characterize whether a basin is a crystal or SCL The RMSD approach is not the only method to
assign a crystallinity value[122] however it is one of the simplest and as such was chosen for
this application To then categorize each basin as either a crystalline basin or SCL basin a plot of
the minimum energy of the SCL states versus the cutoff of RMSD was produced This plot for 256
atoms shows a clear transition in the energies around a cutoff value of 1 Aring which is therefore
taken as the cutoff radius of which basins are considered crystals The same analysis was repeated
for each system size This analysis is shown in Figure 4 Any basin with a RMSD less than the
cutoff value is considered a crystalline state This cutoff was found to be generally insensitive to
the results with any values ranging between 09 Aring and 11 Aring changing the resulting prediction of
nucleation by less than one order of magnitude but shifting the peak temperature by up to 40 K
This cutoff also reproduced the energy difference between the SCL and crystal in molecular
dynamics simulations of the materials
37
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-axis being
an arbitrary phase space and the y-axis being the potential energy calculated from the Pedone et al
potentials[36] The colors are indicative of the crystallinity where the blue basin is the initial
starting configuration The landscape shows the lowering of energy associated with partially
crystallizing the sample (B) The energy landscape relationship between the cutoff for crystalline
and super-cooled liquid states for 256 atoms is shown This shows a clear drastic energy change
occurring around the cutoff value of 10 Aring
[
38
31B Kinetic Term for CNT
The transition rate was calculated by the inverse of the average relaxation time from the
SCL to the crystalline states The relaxation time ( ) is defined by the relaxation function ( )
0 exp
t
= minus
(30)
where t is time and 0 is the scaling of the relaxation function The reason we are able to
calculate the kinetics of nucleation but not viscosity is because the nucleation kinetics are
dominated by the transitions between two states that we specifically searched for when exploring
the energy landscapes while the viscosity is defined by the entropy dominating small transitions
described by the AG
The AG relationship shows that the viscosity is dominated by the configurational entropy
of the system Experimentally the kinetic term is often calculated with the Stokes-Einstein
relationship
( )6
kTD
aT
= (31)
where a is the radius of the diffusing species and is the viscosity of the liquid Interestingly this
approximation gives reasonable results when investigating experimental data since the viscosity is
a measurement of the entropy of small cooperative liquid rearrangements Conversely the kinetics
that govern nucleation are large structural rearrangements from a supercooled liquid to a crystal It
is nonetheless possible that these cooperative rearrangements and the large diffusion processes for
nucleation are on the same order of magnitude which is why the Stokes-Einstein relationship gives
39
a reasonable approximation in experimental parameterizations When the Stokes-Einstein
relationship is used there is an implicit assumption that the large nucleating rearrangements are
proportional to the small barriers that govern viscosity[53]
A mean relaxation time can be found by averaging over a system of master equations
( )j
ii ji j j ij i
i
dpg K p g K p
dt
= minus (32)
In this expression g is the degeneracy of a microstate pi is the probability of occupying the
microstate and Kij is a transition rate between inherent structure i and inherent structure j The
equilibrium probability of occupying a microstate can be given by a Boltzmann probability (where
Z is the partition function and U is the potential energy of the inherent structure)
1
exp ii i
Up g
Z kT
= minus
(33)
By substituting the value of jp in Eq (32) for a two-state model ( 1 21 i jp p= == + ) we write
( )( )1ii ji i j ij i
dpg K p g K p
dt= minus minus (34)
Isolating the probabilities and splitting the differential we have
( )1
i
i ji i j ij i
dpdt
g K p g K p=
minus minus (35)
which when solved via integration takes the form
ln ( )
( )
i ji i ji j ij i
i ji j ij
g K g K g K pt
g K g K
minus + =
minus + (36)
Solving for the occupational probability yields
( )expi ji j ij i ji
i
i ji j ij
g K g K g K tp
g K g K
minus minus + =
+ (37)
This shows an analytical solution for the relaxation time in a two-state model
40
1
i ji j ijKg g K
+
= (38)
The two-state model approximation is made to enable the calculation of the mean relaxation time
between the liquid basins to the crystalline basins The transition rates between the basins are given
by the vibrational frequency ( ) and the barrier between the states ( ijU )
exp ij
ij
UK
kT
minus=
(39)
The probability of transitioning from a given liquid basin i to a crystalline basin j is given by
1
expij
ij
Up
Y kT
minus =
(40)
where Y is the partition function with respect to all transitions from the current SCL basin to all
crystalline basins as only these transitions are considered to calculate the relaxation time
expj
j
iY
k
U
T
= minus
(41)
The total expected relaxation time of the system making the transition is then given by the
probability of occupying a basin the probability of transitioning from one basin to another specific
basin and the associated relaxation time
1
i ij
j i ji j ji i i
p pg K g K
=
+ (42)
The first summation is over the liquid states and the second summation is performed just over
crystalline states to capture the mean relaxation time from a supercooled liquid to a crystal We
fixed the vibrational frequency to a value of 17 times 1014 Hz (ie about one jump attempt per 5
femtoseconds) which was calculated using the average of the eigenvalues of each basin ( ) and
the average mass of the atoms (m) using
41
1
2 m
= (43)
31C Degeneracy calculations
To obtain an appropriate value for the degeneracy an understanding the phase-space
volume of the basin is needed since the degeneracy is linked to the volume of the basin[40] By
taking the displacement between the basin coordinates of each atom and the transition point we can
define a radius of the basin R Using R and a spherical approximation we can calculate the phase-
space volume degeneracies
3 23
12
NN
i i
cry
Ng R
g
+
=
(44)
In which ( )x is Eulerrsquos gamma function The degeneracy is then normalized by a constant such
that the liquid-liquid state transitions had a value of 100 s at the glass transition (980 K) to be in
agreement with Angell[58] The normalization step allows for a quantitative prediction of the
liquidus temperature to confirm the accuracy of our model The liquidus value found is
approximately 1200 K 1715 K and 1950 K for 128 256 and 512 atoms respectively The
experimentally determined liquidus temperatures range from 1680 K to 1710 K[113] [115] [120]
It is worth noting that changing the exponent drastically changes the thermodynamics while having
minor effects on the kinetics and the normalization value drastically changes the kinetics while
keeping the thermodynamics nearly constant We chose the denominator to satisfy the liquid
relaxation constraint It is also feasible to choose the exponent to satisfy a constraint on the liquidus
temperature The normalization value is an important to consider because for larger system a
change in 001 can result in a change in the kinetics by ~10 orders of magnitude
42
31D Free Energy Difference
Due to limitations in the exploration of the energy landscape requiring a fixed volume we
will approximate the difference in the Gibbs free energy by calculating the difference of the
Helmholtz free energy This approximation is reasonable due to the similar densities between the
crystalline and glassy phases under ambient pressure[73] [74] The first step is determining the
mean energy difference of all the basins and the second step was repeating the process for only the
liquid basins The mean energy was calculated with the following expression[123]
( )ln ln ii
i
i i iG kTp p U kT g p= + minus (45)
In Eq (45) i is the set of relevant basins for the conditions we are interested in The difference
between the mean free energy of the liquid basins and the entire landscape basin energies is the
G parameter This free energy difference is typically experimentally calculated by integrating
heat capacities however numerous approximations exist These approximations are often linear
approximations normalized to the liquidus temperature In this work we find that the driving force
appears to be predominantly linear which is in good agreement with the approximations that are
often used for the driving force[124]
43
Figure 5 An example interfacial structure between the crystalline phase on the left and the last
sequential SCLglass phase on the right for a barium disilicate system The gray atoms are barium
silicon is shown in red and blue represents oxygen
44
31E Interfacial Energy
To calculate the interfacial energy of the interface between the crystalline phase and the
SCL the initial basin (pure crystalline) and the last basin (glass basin) were annealed at 1700 K for
1 ns and then cooled to a temperature of 800 K over a period of 10 ns The same process was
repeated for a composite where the initial stable crystal and the glass were placed next to each
otherThe structure was multiplied 10 times to create a wall of a large interface to reduce energetic
fluctuations A small section of the composite structure is shown in Figure 5 The difference
between the addition of the energy of the crystalline phase and glassy phase and the composite
gives the energy of the interface which was normalized to the size of the surface area Multiple
arrangements of the composite were tested but anisotropy was found to be minimal with the
fluctuations of the energy being larger than the differences from different tested directions The
final expression used for the interfacial free energy (in eVAngstrom2) is
610 00 35047 040T minus +=
45
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a function of
temperature for each system size The experimental values for kinetics and thermodynamics come
from ref[125] and from the heat capacity data taken from [126] respectively It is clear to see that
driving force shows good agreement across all systems however the kinetics terms only converge
46
for the 256 and 512 atoms systems (Bottom) The fit used to calculate the interfacial energy as a
function of temperature
32 Results amp Discussion
The results for the free energy difference the kinetic parameter and the surface energy are
shown in Figure 6 These parameters as a function of temperature are used to calculate the
nucleation rates in the expression
3
2( )
01 16exp
3( )I
TT G k
= minus
(46)
The 01 being an approximate value of the Zeldovich factor Using the parameters shown in Figure
6 and solving Eq (46) the nucleation rates are shown in Figure 7 along with experimental data for
validation At first glance we notice that there is about 7 order of magnitude differences between
data and prediction Considering that theses potentials were trained on glass and were not intended
for this purpose we consider this as excellent agreement This method also shows agreement below
the peak but the accuracy falls off more quickly below the peak temperature It is likely that the
data below the peak is severely underestimated due to a failure to reach steady state nucleation
rates Reliable steady state nucleation rates are not available for this system and as such a direct
comparison below the peak temperature is not directly available It is worth mentioning that the
failure of CNT of predicting the order of magnitude of nucleation rates is well reported and can
amount to dozens of orders of magnitude when a constant value of is used[75] [113] [115]
[119] [127] thus the proposed method has an advantage when comparing with the current
classical nucleation theory toolset Moreover the prediction of Tmax (~950 K) is also very close to
the experimental value (985 K)
47
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted using the
model presented in this work The data referenced can be found in refs [113] [128] [129]
48
To further compare CNT and the model presented here we decided to compare the results
of the fitted and this workrsquos calculated surface energies as seen in Figure 8 The results show good
agreement all achieving the same order of magnitude implying that experimental methods can
achieve a close approximation to the values required for CNT The prediction of nucleation is
extremely sensitive to the value of the surface energy with small changes leading to orders of
magnitude change It stands to reason then that further work that needs to be approached through
computation and experiments is developing a rigorous model to predict the interfacial free energy
The other assumption that is used experimentally is the Stokes-Einstein relationship To
compare the landscape results of the transition rate from the SCL to crystalline states and the
Stokes-Einstein relationship the liquid state relaxation was calculated The small barriers that
govern the Adam-Gibbs relationship were not found with this exploration method hence
quantitative comparisons are not possible (and relaxation calculations should not be made)
however the relaxation times can be qualitatively compared The liquid state relaxation rate is
proportional to the viscosity and is given by the expression[44]
j
i
i
l
ij
p
K =
(47)
To understand the validity of the Stokes-Einstein relationship we can write a series of theoretically
true proportionalities
( )l
T TD T
(48)
The values of the experimental kinetics fail to align with the kinetics calculated in this work This
is expected due to the fact that the viscosityliquid relaxation is dominated by the set of small
transitions in the landscape while the nucleation transition is defined by the transition of one region
to another with a substantially higher activation barrier This comparison may also account for
49
some part of the experimental overestimation below the peak with the kinetic term calculated
dropping off at a faster rate than that calculated from the experimental viscosity Figure 6 also
shows evidence that it may be inappropriate to use the Stokes-Einstein relationship for nucleation
and as such we may need to re-visit the experimental models that often rely on the assumption of
the Stokes-Einstein relationship This observation casts some doubt in the legitimacy of describing
the kinetic factor as a diffusivity which is standard for CNT For more information on this CNT
failure please see Refs [75] [113] [115] [119] [125] [127]
Nucleation is a complicated concept to investigate and model Unlike the crystalline phase
where the atomic positions are clearly defined the details of the atomic coordinates and bond
configurations of the liquid sites must be described in terms of statistical distributions Due to this
added complication mean-field distributions are typically used to describe non-crystalline
sites[14] [16] [108] [130]ndash[133] The incorporation of fluctuations could more accurately predict
each liquid sitersquos transition to the crystalline phase Fluctuations in the degeneracy could be one
reason that the nucleation rate is over predicted when the temperature is less than the peak
nucleation temperature A given G varies with the environment and depends on the specific
liquid state not the mean value Therefore the rate of nucleation from all supercooled liquid states
to the crystal state is not constant which is why fluctuations need to be considered and why energy
landscapes are such a powerful technique This is especially highlighted when considering that
different degeneracies yield drastically different heat capacity Energy landscapes are extremely
useful because they consider a large distribution in free energies and show all structural fluctuations
possible Fluctuations in cluster size and degeneracy influence Adam-Gibbs configurational
entropy which directly impact the transition rate between the crystal and liquid state
50
Figure 8 The surface energy with respect to temperature for the work presented here
0
0002
0004
0006
0008
001
0012
0014
800 850 900 950 1000 1050 1100
Su
rface
En
ergy (
eVAring
2)
Temperature [K]
This Work
Xia et al (2017) Ref 23
Fokin et al (2016) Ref 41
51
33 Conclusions
Though this is not the first time CNT has been examined computationally in an attempt to
understand if the theory is valid[74] [116] [117] it is the first parameterization of each variable
independently in an attempt to calculate the resulting nucleation rate Not only are all the variables
calculated when used in the expression for CNT the liquidus temperature is reproduced and the
nucleation rate is reasonably predicted This is as far as the authors know also the first time a real
energy landscape for a complicated glass-ceramic system is being reported This method leveraging
energy landscapes alongside CNT enables for a reasonably fast calculation of the nucleation curve
of a glass forming liquid In future work this model could be used for either developing commercial
glass-ceramics or for predicting the critical rate needed to maintain a glass This model can also
provide additional insights into the physics of nucleation and the validity of CNT without the
constraints of fitting parameters This method fundamentally has the power to connect the 3N
dimensions of the energy landscape to the compositional dimensions and thus allowing for a unique
powerful method of predicting nucleation parameters as a result simplifying the crystalline
dimensions in which a glassglass-ceramic must be optimized over
Chapter 4
Expanding the Current State of Relaxation
Before we consider a new approach to relaxation (optimizing our understanding of
the 3N dimensions) there are first missing pieces in the current state specifically the temperature
and compositional dependence of the stretching exponent However even before expanding the
current state of the art relaxation methods we also need to understand ergodicity In this section we
will start with a simple though experiment to understand the temporal effects of relaxation and then
in the second section combine it with fictive temperature to create a new model for the stretched
exponent
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition
The prophet Deborah sang in the book of Judges ldquoThe mountains flowed before the Lordrdquo
[6] [13] However the question the prophet failed to ask was what happens if the mountains are
moving at relativistic speeds The focus of this section is to highlight the temporal effects on
ergodicity in the context of special relativity and the reference frame of the observer In order to
understand relativistic effects on materials one must consider a well-characterized material that
actively relaxes but appears solid on our timescale viz pure B2O3 glass [134] [135] Glass is a
particularly good candidate to study the effects of time dilation on a material since it undergoes a
kinetic transition known as the glass transition (Tg) which has multiple definitions that will be
considered
53
The first definition is in terms of the aptly named ldquoDeborah Numberrdquo which was proposed
by Reiner [136] to offer a view of fluidity and equilibrium mechanics It was later expanded to
explain the origin of the glass transition and the transition from an ergodic liquid system to a non-
ergodic solid-like glassy state[12] which is the fundamental definition of the glass transition [11]
[12] [137] It is important to note here that the breakdown of ergodicity is relative to the
observation timescale and is reflected in the Deborah number (D) which is defined as
Dt
= (49)
where t is the (external) observation time and is the relaxation time of the material which can be
expressed through the Maxwell relation [42] By definition the Deborah Number equals 1 at the
glass transition temperature Tg [13] [137]
Another definition of the glass transition temperature is due to Angell who defined Tg as
the temperature where the viscosity of the supercooled liquid is 1012 Pa s [58] [138] The Angell
diagram is an important plot relevant to viscosity and the glass transition where the abscissa is TgT
and the ordinate axis is log10( ) In the Angell diagram the value of the viscosity at Tg and in the
limit of infinite temperature are fixed and the difference in the temperature scaling of the TgT-
normalized viscosity is the slope of the log10( ) curve at TgT = 1 which Angell defined as the
fragility (m) of the supercooled liquid Using this definition one can establish the consistency of
the Angell plot at relativistic speeds An example of the Angell plot is shown in Figure 9
54
Figure 9 An Angell diagram created using the MYEGA expression [55] With a very strong glass
(m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) [17] [134] [135] The
infinite temperature limit is from the work of Zheng et al[54] and the glass transition temperature
is from the Angell definition
55
Using the two previous definitions it is expressed
1gT
GtD
= = (50)
which can be re-written as
(
(
)
)g
g
tG T
T= (51)
In order to solve this equation one must know the shear modulus at the glass transition temperature
recent improvements in topological constraint theory have led to the ability to predict properties
with varying temperature and composition Using a recent topological constraint model for elastic
moduli proposed by Wilkinson et al [139] one can predict the temperature dependence of the
modulus
( ) ( ) ( ) ( ) ( )( ) ( )Δ Δ Δ Ac c C Ac
c c
F n q T F n q T F n q T x Nn q T x NdG dGG
d F M d F M
+ + = =
(52)
In Eq (5) nc is the number of topological constraints (c) qc is the constraint onset function as
described by Mauro et al [69] Fc is the free energy of the constraint is the density of the
system M is the molar mass and 119889119866
119889120549119865119888 is a scaling factor This expression was evaluated as a
function of temperature with the total modulus coming from Kodama et al [135] and is equal to 8
GPa at room temperature The B2O3 structure consists entirely of three-coordinated boron [140]
[141]
Using the modulus predicted in Eq (52) numerical observation time is expressed
56
12
9
10 Pa s141 s
709 10 Pat = =
(53)
Using this description of the static (vc=0 where v is the speed of the glass and c is the
speed of light) glass transition it is possible now to describe the relativistic glass transition for
which there are considered two separate cases It is worth noting that this is not the first case study
of relativistic viscosity a fairly large basis of literature dealing with ideal fluids near large
gravitational bodies [142] There is also some work also relating the relaxation time in a
gravitational field to the nonequilibrium entropy relaxation time [143] [144] This work
emphasizes the theoretical effects of special relativity on real liquids their glass transitions and
the properties there dependent on
41A Relativistic Liquid
The first case is when a sample of a glass-forming system moves past a stationary observer
at a speed approaching light giving the new observation time as
0tt
= (54)
where t0 = 141 s (as previously calculated) and g is the Lorentz factor
2
2
1
1v
c
=
minus
(55)
The Maxwell relation can be modified to account for relativistic effects by combining with Eq (54)
0
()
)(
g
gG TT
t
= (56)
57
This gives a condition that is temperature dependent to predict the new glass transition as a function
of v however in order to accurately describe the relativistic behavior density must be considered
Density will change by a factor of 2 because both the mass and the volume will be affected and
as such
0) (( )g gG TT t = (57)
The prediction then for the glass transition as a function of the speed of light is shown in Figure
10
Figure 10 The relativistic glass transition temperature for B2O3 glass
58
Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature
which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) expression [55] [145] The MYEGA equation needs 2 other parameters
besides Tg to generate the viscosity curve fragility (m) which is the slope at Tg in the Angell plot
and the infinite temperature viscosity () The fragility will not change because one assumption
in this model is the consistency of the Angell plot at all relativistic velocities even relativistic The
infinite temperature viscosity which also will not change because this is the minimum possible
viscosity for a liquid The new viscosity curves for various speeds are shown in Figure 11
59
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different fractions of
light speed All of the viscosities approaching the universal temperature limit for viscosity
60
By using the Deborah Number it has been shown that at relativistic speeds will change the viscosity
in order to maintain the Deborah number and in this case it is shown that when the Lord would
move past the mountains at relativistic speeds the mountains would not actually flow but instead
hold still (even more so than they currently do) From this it is inferred that any liquid moving past
the earth at relativistic speeds will appear to be more solid and eventually appear as a glass
41B Relativistic Observer
In this case we consider an observer moving past a sample of glass at relativistic speeds
giving
0t t = (58)
which when the relativistic shear modulus 2 factor is included can be written as
( )
( )0
g
g
TG T
t
= (59)
Solving in the same manner as before it is shown which shear modulus is needed to satisfy the
condition in Figure 12
61
Figure 12 The modulus needed to satisfy the condition for the glass transition
62
In Figure 12 the shear modulus at the glass transition temperature is shown however the
constraint theory mechanism being implemented to calculate the shear modulus has a built in upper
limit when calculating the Tg past the point in which all the constraints are intact Due to this
limitation there is an upper limit to our prediction Nonetheless the calculated Tg over the available
range are shown in Figure 13 as well as the related viscosities It is seen clearly that at higher
velocities the glass transition shifts towards zero more dramatically at the higher temperatures
This leads to the interesting result that if we take the limit of an observer moving past the earth at
close to the speed of light all glass would appear to be a liquid
63
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at v=044c
(Bottom) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches
the speed of light
64
Applying the mechanisms proposed by Einstein to the concept of the Deborah number it
has been shown that the viscosities will appear to change dramatically as a glass-forming system
approaches relativistic speeds This allows us to build an intuitive understanding of the differences
between glass and liquids Specifically any material can appear either more or less fluid based on
the velocity with the velocity with which they travel relative to the observer This serves to highlight
the importance of the role of the observer on the glass transition and why even small changes in
DSC scan rates or sampling frequency can lead to drastically different measured glass transitions
42 Temperature and Compositional Dependence of the Stretching Exponent
The mathematical form for glass relaxation ( ) was originally proposed by Kohlrausch in
1854 based on the decay of charge in a Leyden jar[9] [14] [29] [42] [45] [96] [146]
( ) ( )( )exp t t
= minus (60)
where t is time t is the relaxation time of the system and b is the dimensionless stretching exponent
Eq(1) was originally proposed empirically and is known as the stretched exponential relaxation
(SER) function[42] [96] The relaxation time t depends on the composition temperature thermal
history pressure and pressure history of the glass as well as the property being measured[62]
[147] For example the stress relaxation time of a glass can be written as[62]
( )( )
ff
s
f f
T T P P
G T T P P
= (61)
where h is the shear viscosity and G is the shear modulus of the glass-forming system[42]
(Variables are defined in Table 2) In addition to using Eq(61) to describe stress relaxation time
structural relaxation time has recently been hypothesized follow Eq(61) with bulk viscosity and
65
the difference of bulk modulus between infinite and 0 frequency replacing shear viscosity and shear
modulus[62] Both h and G vary with the temperature (T) and thermal history of the glass (as
quantified via the fictive temperature Tf) as well as the pressure (P) and its corresponding history
(fictive pressure Pf)[16]
To connect the stretching exponent to its physical origins we can begin with a result from
Richert and Richert[148] that relates to an underlying structural relaxation time distribution and
then develop equations that determine from known quantities Their expression
2 22
ln 2
1
6
minus=
(62)
relates to the variance of the logarithm ( )2
ln of the structural relaxation time For reference
all variables are defined in Table 2 The physical origins of the relaxation time are related to the
configurational entropy (Sc) as shown by the Adam-Gibbs equation[53] [55] [149]
ln lnc
B
TS = + (63)
where is the infinite temperature relaxation time T is the absolute temperature and B represents
the energy barrier for relaxation
Table 2 Variable definitions
Variable Definition
t ts Structural relaxation time stress relaxation time
b Kohlrausch exponent (ie the stretching exponent)
T Tf Tfi Absolute temperature fictive temp partial fictive temp
P Pf Pressure fictive pressure
66
kB Boltzmannrsquos constant
wi ki i=1 to N Prony series parameters
B Adam-Gibbs relaxation barrier
Sc Adam-Gibbs configurational entropy
B Distribution of activation barriers
2 Variance
Na Number of atoms
f Topological degrees of freedom per atom
Degenerate configurations per floppy mode
H Enthalpy barrier for Relaxation
d Number of dimensions
f Fraction of activated relaxation pathways
S Adam-Gibbs entropy in the infinite-temperature limit
m Kinetic fragility index
m0 The limit of a strong liquid
Grouped unknowns 6
ln
B
a BN k
A Proportionality between fragility and distribution of barriers
x A given composition
125640
S
A
Intercept of the linear model of
67
42A Deriving a Model
Combining Eqs(62) and (63) Gupta and Mauro[150] proposed that the variance of the energy
barriers for relaxation (2
B ) could be rewritten as
2 2
22
2
1
6( )B c TTS
minus=
(64)
Solving for the stretching exponent
( )22 (
6
( )
)
c
B c
S T
T
T
TS
= +
(65)
The work of Naumis et al[23] [56] has shown that the configurational entropy is proportional to
the topological degrees of freedom in the network a result that was used in the derivation of
temperature-dependent constraint theory[57] and the MYEGA (Mauro-Yue-Ellison-Gupta-Allan)
equation for the relaxation of supercooled liquids[55] The MYEGA equation was derived by
expressing the configurational entropy as
( ) ( ) lna BcS T x f T x N k= (66)
In Eq (66) f is the topological degrees of freedom per atom Na is the number of atoms kB is
Boltzmannrsquos constant x is a given composition and is the number of degenerate configurations
per floppy mode The temperature dependence of the topological degrees of freedom was
approximated using a simple two-state model
( )( ) 3exp
B
H xf x T df
k T
minus= =
(67)
Here H(x) is the enthalpy barrier for relaxation d is the dimensionality of the system and f is the
68
fraction of activated relaxation pathways Combining Eqs (65) and (67) as well as condensing the
unknowns into the term defined by
6
ln
B
a BN k
=
(68)
we get
( )
22
df
dfT
=
+
(69)
This can be compared to the prediction made by Phillips[48] for the stretching exponent at
temperatures below the glass transition
2
df
df =
+ (70)
Comparing the two expressions of Eq(69) and Eq(70) they would agree if
4 4dfT
= + (71)
We will show later (Figure 12) an example where extrapolating our model prediction for to room
temperature (with T in Eq(71) replaced by fictive temperature Tf) gives a result close to the Phillips
value of Eq(70) When the temperature T in Eq(69) is high (much larger than ) then Eq(69) gives
us rarr1 Thus the model can interpolate between the low-temperature (Phillips) value and the
high-temperature value of 1 In our current model we do not have access to a value for in Eq(68)
so this model cannot assess whether the Phillips room temperature value for is universal ie
whether Eq(71) is always satisfied at room temperature
Zheng et al[151] showed that the Mauro-Allan-Potuzak (MAP) model for the relaxation
time of the nonequilibrium glassy state[44] implies that the configurational entropy can be written
as a function of thermal history (Tf) the fragility index (m) the fragility limit of a strong liquid
69
(m0) the glass transition temperature (Tg) and the limit of infinite temperature configurational
entropy ( S ) as
0
) exp 1 0
(ln1
c f
f
gT
TS m
TS
m
= minus minus
(72)
The qualitative relationship between the distribution of activation barriers and fragility was
proposed by Stillinger[18] [19] who suggested that the energy landscape of a strong liquid (low
fragility) has a small distribution of activation barriers and that a higher fragility is associated with
a broader distribution of activation barriers ie a higher variance of the activation barriers This
leads to a phenomenological relationship that is here assumed to be valid based on Stillingerrsquos work
on energy landscapes[18] [19]
2
0
1B
mA
m
= minus
(73)
where A is some constant of proportionality This expression was chosen because in the limit of a
strong liquid there is an infinitely sharp distribution of activation barriers Combining Eqs(65)
(72) and (73) we obtain the temperature dependence of the stretching exponent for a liquid
( )
0
2 2 2 2
0 0
exp 1
( ) 2
6 ln10 1 exp 1
g
f
f
f
g
f
f
T mT S
T mT
Tm mA T S
m T m
minus minus
=
minus + minus minus
(74)
Fictive temperature appears in this new expression because we are deliberately expressing this
function as an equilibrium model Rewriting Eq (74) in terms of one unknown () and grouping
constants
70
( ) 0
2 2
0 0
exp 1
2exp 11
g
f
f
f
g
f
f
T m
T m
T m
T
T
Tm
T
m
m
minus minus
minus minus
=
minus +
(75)
Eq (75) is an expression for the stretching exponent as a function of thermal history glass
transition and fragility index with only one unknown In Eq (75)
( ) 12
6 ln10
S
A
= (76)
The only unknown for the compositional dependence of the stretching exponent is the value of D
The composition-dependent part of Eq(76) is Syen
since we approximate A to be independent of
composition The fragility dependence of Syen
was proposed by Guo et al [61]
Syen(x) = S
yen(xref
)expm(x) -m(x
ref)
m0
eacute
eumlecircecirc
ugrave
ucircuacuteuacute
(77)
where x is composition and xref is a reference composition in the same glass family Seeking the
simplest possible expression to approximate the unknown D we take the natural logarithm of
Eq(76) and of Eq(77) and combine them to get
0
ln lnm
m = + (78)
with the additional definition
( )( )
( )12
0
ln l6 ln10
nref refS x m x
mA
minus
=
(79)
The result is that ln D varies linearly with fragility m and the intercept ln D is independent of
composition (depends on one reference composition)
In Figure 9 we plot the data of Boumlhmer et al[152] vs the predicted exponent The work of
71
Boumlhmer et al is the summary of the literature data relating fragility index to the stretching exponent
at the glass transition which we use to fit values to Eq (78) The dataset included in their work
covers chalcogenide oxide and organic glasses The fitting (consisting of least-squares
minimization of the difference between the predicted stretching exponent and that which was
reported) was done twice once for organic and once for inorganic systems During the fit it was
assumed that Tg=Tf Some assumption about thermal history was necessary since the individual
thermal histories or Tf values are not known for this whole collection since we are trying to track
overall trends in values vs composition this reasonable simplifying assumption is consistent
with our program Using Eqs(75) and (78) the temperature and compositional dependence of the
stretching exponent can be described with only one free parameter
72
Figure 14 b predicted and from literature showing good agreement with a total root-mean-square
error of 01 The fit for organic systems is given by 66 ln Kln = minus and for inorganic systems by
75ln Kln = minus
73
42B Experimental Validation
Experimental density measurements were made using Corningcopy JadeTM glass as described
elsewhere[146] Changes in the density are normalized to obtain the relaxation functions which
are fit with Eq (60) The resulting values of are shown in Table 3 A comparison in Figure 15
is shown with data obtained on a soda-lime silicate SG80 where the stretching exponent was
obtained using the measurement of the released enthalpy as a function of isothermal annealing time
during relaxation below the glass transition The measurement of the released enthalpy[153] relies
on the change in the excess heat capacity in the glass transition range as a function of annealing
time Normalized released enthalpy relaxation functions for each temperature were fit with Eq(60)
to obtain the stretched exponent values for SG80 In Figure 15 parts A and B are samples measured
below Tg while in part C the samples measured are at Tg and above The values were then fit with
Eq(75) where the only free parameter was D with an equilibrium assumption (Tf = T)
74
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy JadeTM
glass and Sylvania Incorporatedrsquos SG80
Glass Temperature (K) Tg (K) TgT m D (K-1) Beta t (s)
Corningcopy JadeTM 1031 1074 1042 32 0002 0504 7249
Corningcopy JadeTM 1008 1074 1065 32 0002 0447 32458
Corningcopy JadeTM 973 1074 1103 32 0002 0395 419214
SG80 800 800 1 36 00057 0632 92
SG80 783 800 1021 36 00057 0593 182
75
Figure 15 The equilibrium model proposed with the experimental points showing good agreement
between the experimentally measured data points and the equilibrium derived model RMSE was
002 for Corningcopy JadeTM (A) and less than 001 for SG80 (B) (C) The model fit for B2O3
experimental data[154] [155] The fragility and glass transition temperature of the B2O3 are taken
from the work of Mauro et al[17]
(
A)
(
B)
(
C)
76
A combination of the MAP model for non-equilibrium shear viscosity[44] the model
presented here primarily in Eq(75) and the model presented by Wilkinson et al[139] for the
temperature dependence of elastic modulus allows for fully quantitative modeling of stress
relaxation behavior The missing model required to understand structural relaxation is the bulk
viscosity curve[62] All previous relaxation (structural or stress) models[42] [61] have relied on
approximations that use a constant exponent and on a constant (temperature-independent)
modulus value whereas here every parameter of Eq(60) may be modeled as a function of
temperature Furthermore in combination with the relaxation models described by Guo et al[61]
in which multiple fictive temperatures are described using a Prony series and a temperature-
dependent modulus one can construct a relaxation curve accounting for the temperature
dependence and thermal history dependence of all relevant parameters
( )
( )( )
1
( )( )exp exp
( )) (
fTN
i
i
f
f
f fi
G T k T tG T tw
T TT
T T
=
minus minus
(80)
Here wi and ki are fitting parameters that are completely determined by the value of and each
term in the Prony series is denoted with the subscript i Usually 8 or 12 terms are included in the
Prony series[96] Each term in the Prony series is assigned a partial fictive temperature (Tfi) whose
relaxation is described by the simple exponential in the Prony series Equilibrium conditions are
assumed at the start of the simulation which allows for a known set of starting probabilities within
the energy landscape interpretation of relaxation The same model also includes the fragility index
and glass transition dependence of the non-equilibrium shear viscosity This method is
implemented in RelaxPy[42] as discussed in the next section This serves as an approximation for
the evolution of the of the non-equilibrium state however the temperature dependence of the bulk
viscosity and a replacement for fictive temperature need to be quantified to improve the
understanding of the underlying physics[1] [16] [62]
77
Separately we can explore the relationship between this model given by Eq(75) and the underlying
energy landscape In order to better understand the stretching exponent consider that there exists
a full set of parallel relaxation modes within an energy landscape The relaxation modes are then
weighted by the occupational probability corresponding to a particular mode This gives a series of
transition rates with some probability prefactor and an associated relaxation time (scaled from the
mean relaxation time) which gives rise to a Prony series form of Eq(80) Thus the distribution of
relaxation times (or barriers) determines the evolution of the stretching exponent while the average
barrier determines the mean relaxation time If this Prony series description in turn describes the
stretching exponent we arrive at a physical description and origin of the stretching exponent As
the temperature approaches infinity even though there is a distribution of activation barriers the
distribution of relaxation times approaches a Dirac delta function and the stretching exponent
approaches one (a simple exponential decay) As the temperature decreases the distribution of
relaxation times broadens due to the wide distribution of barriers however due to the broken
ergodic nature of glass the value of the stretching exponent will be controlled by the instantaneous
occupational probability
The activation barriers may be modeled eg using a numerically random set of Gaussian
distribution of barriers[156] The probability of selecting an individual transition is then calculated
using a Boltzmann weighting function the transition also has an associated relaxation time with it
The Prony series is then recreated with the probability multiplied with the simple exponential
relaxation form The sum of all of these terms is then fit with the stretched exponential form Figure
11 shows the stretching exponent calculated for the mean activation energy of 10 eV and three
choices of standard deviation 25 2 15 ande e e minus minus minus= eV as shown in the legend where Eq(60) has
been fit to the Prony series calculated from an equilibrium Boltzmann sampling of a Gaussian
distribution of barriers The temperature dependence for the stretching exponent matches the form
78
derived earlier in this work This stretching exponent shape has been shown previously though not
in a closed-form solution[156] This secondary method not only validates the generalized form of
Eq(75) but also highlights the validity of using the configurational entropy and barrier distributions
as the underlying metric controlling the temperature dependence of the stretching exponent
Figure 16 The stretching exponent calculated as described in the text for a Gaussian distribution
of barriers This plot shows that the distribution of barriers has a large effect on the stretching
exponent A Tg cannot be described since there is no vibrational frequency included in the model
though the glass transition temperature should be the same for all distributions since the mean
relaxation time is the same for all distributions at all temperatures The deviation is given in ln eV
units
79
To create software to model this complex behavior the authorsrsquo existing software RelaxPy[42] has
been modified into a new version RelaxPy v20 Instead of fixed values for the Prony series the
values are chosen dynamically to match the stretching exponent as predicted by Eq(75) The
database of values wi() and ki() was created with a fixed number of terms N in Eq(80) (we chose
N=12) This database is available in the same Github repository as the software The parameters
were fit by starting with the values obtained for =37 reported by Mauro and Mauro[96] to make
the new version of the model smoothly reproduce a prior optimal fit then was stepped by 001
(when less than 095) or by 0001 (when greater than 095) Using the previously optimized values
of wi and ki for starting values for each new the wi and ki values were then varied to minimize
the root-mean-square error between the stretching exponent form and the Prony series form ie to
satisfy Eq(80) with least error This list of compiled values makes up the contents of a database
that the software accesses The values of wi and ki used for a given are calculated by finding the
closest value in the database to the given value of This allows for an efficient implementation
of relaxation modeling Figure 12 shows the fitted parameters to the Prony series and an example
RelaxPy output for Corningcopy JadeTM glass undergoing quenching at a rate of 10 Kmin Note that
the room temperature stretching exponent value is 046 a close match the Phillipsrsquos ldquomagicrdquo value
of 37 (043)[157] This prediction for Corningcopy JadeTM is a non-equilibrium prediction in contrast
to the equilibrium high-temperature prediction shown in
80
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent Each color
designates one term in the series (Bottom) The output from RelaxPy v20 showing the stretching
exponent effects on the relaxation prediction of Corningcopy JadeTM glass Each quadrant shows one
property that is of interest for relaxation experiments In particular it is interesting to see the
dynamics of the stretching exponent during a typical quench
81
43 Conclusion
In this section we have sought to expand the current understanding of relaxation and improve the
models that currently exists We have proposed a model that accounts for the effects of special
relativity as well as a model was derived through an understanding of the distribution of barriers
for relaxation and the temperature dependence of the Adam-Gibbs entropy The model outlined
herein describes the temperature dependence of the stretching exponent in glass relaxation
The work done on the relativistic effects on relaxation illustrates the crucial role of the observer on
all relaxation models and offers an extreme in which to test common relaxation models The model
for the stretching exponent not only considers the extremes at high and low temperature (when
compared to the glass transition) as in the Phillips model but also for any intermediate temperature
as a function of the fictive temperature This model does not have any explicit temperature
dependence since it was assumed that an equilibrium model works well to describe the
instantaneous distribution of relaxation times Given the physical argument and the success of this
model when tested by experiments and by another model it is at least reasonable to formulate the
temperature-dependence of in terms of its fictive-temperature-dependence as we have done here
Including both T and Tf is possible to consider but lies outside our current scope Using previously
derived compositional dependence for the MAP model a fragility index dependence of the
stretching exponent was defined and tested The model was confirmed using multiple experimental
datasets In addition a theoretical comparison to a distribution of landscape activation barriers was
found to reproduce the same trends as the model
Chapter 5
Glass Kinetics Without Fictive Temperature
Fictive temperature as discussed through this text is unable to capture key physical
phenomena[16] [42] [51] To circumvent this limitation we rely on two different frameworks
The Adam-Gibbs relationship and generalized features of the energy landscape These two features
allow us to create a method that is completely generalizable and does not involve fictive
temperature This method creates fake landscapes that generally capture the trends of glass based
on experimental measurements This new method we have called lsquoToy Landscapesrsquo
51 Background of the Adam Gibbs Relationship
Experimental evidence in support of the AG model comes from the work of Richet[158]
in which they compared the predicted configurational entropy from viscosity curves with that
obtained from DSC The configurational entropy was calculated from DSC data using
0
0( )p p vi
T
T
b
c c
C CS S T dT
T
minus= + (81)
In Eq (81) Cp is the isobaric heat capacity Cpvib is the vibrational heat capacity and T0 is the initial
temperature from which the configurational entropy is integrated The difference between the total
heat capacity and the vibrational heat capacity is the heat capacity due to structural rearrangements
Richet found good agreement between the measured viscosity curve and calorimetric
configurational entropy at temperatures above the glass transition confirming validity of the AG
relationship When applying Eq equation reference goes here all temperatures must remain above
the glass transition ie in the liquid or supercooled liquid state The calculation is invalid upon
83
cooling through the glass transition due to the breakdown of ergodicity[12] [13] [159] and the
irreversibility of the glass transition process Application of Eq (6) to temperatures below the glass
transition leads to an incorrect calculation of excess entropy in the glassy state[160]
In this study we expand on these previous works to understand the relationship between
viscosity and the underlying energy landscape and provide new insights into the thermodynamics
of liquids through a simplified ldquotoy landscaperdquo model constructed from experimentally measured
viscosity parameters The energy landscape framework is especially helpful for elucidating the
thermodynamics and kinetics of supercooled liquid and glassy systems Energy landscapes describe
the evolution of all atomic transitions for kinetic processes in a system To perform such
calculations information about the inherent structures and transition points in a landscape must be
known In this work we solve the inverse problem of deducing realistic landscape parameters using
the AG relationship and experimentally measured data This information is used to construct a ldquotoy
landscaperdquo model to describe glass relaxation processes To achieve an accurate relaxation model
the following steps are taken to ensure the validity of the results
A Confirm the AG model for shear viscosity using energyenthalpy landscape
calculations
B Validate the assumptions made by the MYEGA model
C Confirm the AG model for bulk viscosity
D Explore the fundamental relations between viscous properties and topography of the
landscape
E Use the knowledge gained to propose a new thermodynamic model to calculate the
driving force for glass relaxation and the scaling of the liquidrsquos free energy
This chapter is organized into sections devoted to each of the topics above preceded by a
methods section All of the sections are presented to validate a new model of glass relaxation that
is sufficiently complex to achieve a calculation of the thermodynamics and kinetics of the liquid
This new model will be based on understanding the landscape from experimental properties and
we have called this approach the ldquotoy landscaperdquo model
84
52 Methods
Potential energy landscapes (fixed volume) and enthalpy landscapes (fixed pressure) are
powerful modeling techniques where atomic configurations are mapped in a 3N-dimensional phase
space (or 3N+1 dimensions for enthalpy landscapes)[89] The energy landscape approach is based
on mapping the continuous 3N dimensional space to a discrete set of energy minima (called
ldquoinherent structuresrdquo) and first-order saddle points (ldquotransition pointsrdquo) To perform this mapping
a systematic search for inherent structures is performed while also mapping the lowest-energy
transition points between each pairwise combination of adjoining minima The landscape itself is
partitioned into basins which represent the set of all configurations that minimize to a common
inherent structure[161] This combination of the basin and transition point information results in a
topographic map of the 3N phase space (either a partial or whole mapping) that can be used to gain
insights into the physics of any atomistic system such as proteins[91] liquids[18] [19] glasses[1]
[93] and nucleating crystals At equilibrium the probability distribution for occupying the various
basins in the landscape is given by[13]
exp
exp
i
j
i
i
j
j
Eg
kTp
Eg
kT
minus
= minus
(82)
where Ei is the potential energy (or enthalpy) of the inherent structure gi is the degeneracy of the
basin i is the basin index and pi is the probability of occupying a given basin i The denominator
is a normalization factor given by the summation over all basins Mauro et al[44] previously used
this formalism to study the non-equilibrium viscosity calculate the evolution of configurational
heat capacity and elucidate the long-time relaxation kinetics of selenium glass[44] [93] [160] For
equilibrium systems the configurational entropy is given by the Gibbs formula
85
lnc i
i
iS k p p= minus (83)
In order to validate the AG model the landscapes of several common systems were
explored using ExplorerPy[89] ExplorerPy is a software program built specifically to map
landscapes using the LAMMPS molecular dynamics package[30] The three systems we explored
are B2O3[162] (150 atoms and 2000 basins) SiO2[162] (150 atoms and 1000 basins) and BaO-
2SiO2[121] (128 atoms and 1500 basins) All systems used periodic boundary conditions The
initial structures for B2O3 and SiO2 were constructed by placing atoms corresponding to the
chemistries of the systems on a random grid (with 1 Aring spacing) corresponding to a density of 20
gcm3 and equilibrated at 2500 K then quenching at a rate of 1 Kps to room temperature using the
Wang et al[162] potentials in the NPT (constant pressure number of atoms and temperature)
ensemble The energy landscape of the BaO-2SiO2 system was taken from our previous work[72]
The SiO2 (constant pressure exploration) system was explored using ET with a push distance of 03
Aring and a pressure of 10 atm B2O3 was explored using isobaric MD at a temperature of 1500 K for
500-time steps of 1 fs also at a pressure of 1 atm The minimization (after MD or ET) was handled
by the LAMMPS[30] steepest descent algorithm until the certainty in energy was smaller than 10-
4 eV Both used a nudged elastic band spring constant of 10 eVAring2 The energy distribution for all
systems was fully sampled which was confirmed by randomly checking if half the basin
distributions matched the full distribution It was also found that multiple structures in each
exploration were discovered more than once and a slightly smaller B2O3 system (130 atoms) was
also explored and showed identical results to the 150-atom distribution The insights gained from
the interaction between the experimentally accessible viscous properties and the landscapes
presented here will be generalized to create a simplified ldquotoy landscaperdquo model to capture the
thermodynamics and kinetics of the glass-forming system
86
53 Results
53A Adam-Gibbs Validation
To predict viscosity using the Adam-Gibbs expression there are only three required
parameters the configurational entropy (as a function of temperature) the barrier for cooperative
rearrangements and the infinite-temperature limit of viscosity Using a landscape the equilibrium
configurational entropy can be obtained using Eq (83) The infinite-temperature limit of shear
viscosity is given by the work of Zheng et al[54] where a systematic study of MYEGA fits to
viscosity data revealed a common value of 10-293 Pamiddots This infinite temperature value of viscosity
is used for all the experimental and computational viscosity curves shown in this work The final
parameter is the mean barrier for a cooperative rearrangement In Figure 18 the barrier for
cooperative rearrangements was chosen to accurately produce a viscosity of 1012 Pamiddots at the
experimentally measured value of the glass transition temperature From these parameter values
the predicted Adam-Gibbs viscosity curve is plotted The universal infinite limit fragility and glass
transition values from references were inserted into the MYEGA equation to plot the
lsquoExperimentalrsquo curves for B2O3 SiO2 and BaOmiddot2SiO2 respectively[58] [69] [120] When
examining Figure 18 it is clear that the AG model is able to fundamentally recapture the viscosity
87
Figure 18 The viscosity (left) and landscape (right) predictions for three common systems The
first system is newly calculated in this work while the latter two come from our previous works[72]
[89] It is seen that the viscosity predicted from the AG model is very accurately able to reproduce
the experimental viscosity curves from the MYEGA model The last system is a potential energy
landscape while the others are enthalpy landscapes
53B MYEGA Validation
Now that we have confirmed that the configurational entropy of the landscape works well
to describe the viscosity via the AG relationship a definitive temperature-dependent form is needed
in order to make more general predictions The key approximation made in deriving the MYEGA
model is that the configurational entropy follows an Arrhenius form By testing this Arrhenius
88
form we can evaluate this assumption in the derivation of the MYEGA model while simultaneously
gaining insights into the thermodynamics of the liquid and supercooled liquid states A strong liquid
( 15m ) has a configurational entropy that is nearly constant as a function of temperature while a
fragile liquid ( 15m ) is a liquid in which the configurational entropy is highly temperature
dependent Since we are testing the MYEGA equation it is also worthwhile to consider alternate
viscosity expressions Assuming validity of the Adam-Gibbs equation the configurational entropy
deduced from the viscosity experiments is given by
( )10 01log l) og(
c
BS
T T minus= (84)
where the viscosity function is given by a viscosity model such as Vogel-Fulcher-Tammann[59]
(VFT) equation the Avramov-Milchev[60] [163] (AM) equation or the MYEGA expression
The resulting entropy predicted by each viscosity model (using experimental data for
fragility and the glass transition) is compared to the calculated configurational entropy from the
landscape for B2O3 in ( 518gT = K and 33m = [69] were used for the experimental values) The
figure clearly shows that the MYEGA equation which assumes that configurational entropy scales
in an Arrhenius fashion reproduces the calculated configurational entropy from the landscape the
most accurately out of the three considered viscosity models This also means that the
configurational entropy form of the AM and VFT models are not physical and cannot be used to
recreate physical results Since the temperature dependence of the entropy has a fixed mathematical
form and is dominated by the distribution of basins we now understand that the distribution of
basins must also have a fixed form and that form must control experimentally accessible properties
This is the key insight that will allow for the development of the toy landscape model
89
53C Adam-Gibbs and Structural Relaxation
Although the Adam-Gibbs model has been validated for shear viscosity and has provided
the functional basis of the MYEGA model let us now consider whether the temperature
dependence of the configurational entropy can also predict the bulk viscosity (also known as the
volume viscosity) Recent work has highlighted the importance of bulk viscosity viz that the
structural relaxation time is proportional to the bulk viscosity[62] Hence the temperature
dependence of the bulk viscosity provides key information needed to predict structural relaxation
times[62] Following Scherer[164] the relationship between the structural relaxation time ( structural
) and the bulk viscosity ( B ) is given by
( )0B structuralK K = minus (85)
Figure 19 The configurational entropy comparisons between the three major viscosity models
which validates the main underlying assumption of the MYEGA model The VFT and AM are
unable to capture the physics of configurational entropy therefore ruling
90
where K is the infinite frequency bulk modulus and 0K is the zero-frequency bulk modulus In
order to calculate bulk viscosity for B2O3 the zero-frequency and infinite-frequency modulus is
taken from these works respectively [165] [166] The experimental structural relaxation times were
taken from the work of Sidebottom et al[154] In Figure 20 the bulk viscosity is shown as well as
the Adam-Gibbs comparison
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the configurational
entropy from the enthalpy landscape with the barrier 00155 eV (compared to the shear barrier of
00149 eV) and the infinite limit allowed to vary (10-263 Pamiddots for bulk viscosity) In Figure 18 the
configurational entropy is confirmed for the shear viscosity thus confirming the AG for both shear
and bulk viscosities Sidebottom data are from Ref [154]
In Figure 20 the fitted values of B and 10log can be used to gain insights into the
difference between structural and stress relaxations since the configurational entropy is universal
between the two The value of Bk (the barrier for cooperative re-arrangements in natural units) for
shear viscosity was 173 K while for bulk viscosity it was 180 K This difference of 7 K in the
barrier leads to an order of magnitude difference in the fitted viscosity curves This is a small
91
difference and explains the variance in the values at Tg The other difference is in the infinite
temperature value of viscosity which is -293 log(Pamiddots) for shear viscosity (from Zheng et al) and
-263 log(Pamiddots) for bulk viscosity This implies that the MYEGA raw form is accurate to predict
the structural relaxation time as well as the stress relaxation (though the values of the pre-
exponential factor and the composite constant will change between the two forms) However the
form in terms of fragility and the glass transition will need to be modified to accurately describe
the structural relaxation time The form assumes the viscosity of interest has a value of 1012 Pamiddots at
the glass transition temperature for shear viscosity which does not have to be the value for the bulk
viscosity curves at the glass transition Further research is needed to understand the appropriate
bulk viscosity at the glass transition and the infinite temperature of bulk viscosity
53D Landscape Features
As shown here the configurational entropy is governed by a Boltzmann sampling of a
probabilistic distribution of states n(E) This distribution of the inherent structures is empirically
found to follow a log-normal distribution as shown in Figure 21 While we are not claiming a
deeper meaning to the underlying origin of the log-normal distribution we may adopt this form of
the probability density function since it accurately describes the underlying distribution of
microstates This is the key insight needed to develop a new model because this means that the
fragility and glass transition are fundamentally related to the parameters of the distribution of basins
on the landscape (number of basins and standard distribution of the enthalpy)
The configurational entropy in the MYEGA model is given by
10
exp 1ln10 12 log
g
c
TS mS
T
= minus minus
minus
(86)
92
where S is the infinite temperature limit of the configurational entropy In Figure 21 we also
show the computed value of Eq (83) as a function of temperature compared to that of Eq (86)
which shows excellent agreement between the configurational entropy and the MYEGA model
This test also supports Stillingerrsquos[18] [19] view of fragility being correlated to the lsquoroughnessrsquo of
the landscape topography According to Stillingerrsquos view if the distribution of inherent structure
energies is narrow then the configurational entropy cannot have a large temperature dependence
leading to a low fragility Alternatively if the distribution is broad then there must be a temperature
dependence to the configurational entropy which necessarily results in a higher fragility
93
Figure 21 (A) The histogram for the energy minima for each of the test systems showing a good fit
with the log normal distribution This distribution will then be a valid form to calculate the enthalpy
distribution of the model presented in the next session (B) The configurational entropy from the
model showing the accuracy of the scaling of the entropy predicted by the MYEGA model The
S value was fit for each system
(A)
(
B)
(B)
94
54 Topography-Property Relations
Now that we have confirmed the validity of the Adam-Gibbs and MYEGA equations from
the energy landscape approach we can draw some key conclusions regarding the thermodynamic
properties of liquids For example heat capacity has been related to fragility[112] and is of
importance for glass manufacturing since it is related to the cost of heating and forming a glass
Isobaric heat capacity is defined as
p
P
HC
T
=
(87)
where H is enthalpy T is temperature and P is pressure The mean configurational enthalpy
calculated from an enthalpy landscape is given by
conf i conf iH p H= (88)
The heat capacity has contributions from both vibrational and configurational modes in liquids
while in glasses the configurational mode is mostly lost For a reversible process the
configurational heat capacity can be written as a function of the configurational entropy
c
p conf
P
SC T
T
=
(89)
By combining Eq (86) and Eq (89) we can write the expression for configurational heat capacity
of a liquid as
10 10
1 exp 1 exp12 log 12 log
g g
p conf
T Tm m S HC S H
T T T T
= minus minus minus = minus
minus minus
(90)
From fluctuation theory the heat capacity can be expressed in terms of the variance of the enthalpy
(2
H )
95
2
2
1p conf HC
kT= (91)
It is important to note that the variance of the enthalpy is not equivalent to the variance of the log-
normal distribution The variance of the enthalpy includes a Boltzmann distribution over the log-
normal distributed basins This gives a direct relationship between the enthalpy fluctuations and
the viscosity parameters If we further adopt the log-normal distribution as the effective distribution
for all liquid basins of all enthalpy landscapes then the distribution is only a function of the
common viscosity parameters ( S m gT and 10log ) The viscosity parameters will only
change the variance of the distribution since the mean of the distribution is inconsequential (the
zero-point enthalpy can be adjusted to an arbitrary value without changing the calculation of the
configurational entropy) The parameters of the log-normal distribution can then be fitted to
reproduce the configurational entropyrsquos temperature dependence thus creating a lsquotoyrsquo enthalpy
landscape for us to play with for the purpose of thermodynamic and kinetic calculations Generating
the landscape is a numerical calculation as the number of basins and standard deviation must be
chosen such to reproduce key phenomena
To quantify the dependence of these parameters on the degeneracy and temperature it is
helpful to develop a descriptive formal model The cumulative probability distribution of a normal
distribution (with zero mean) is given by
10log )1
1 er(
f2 2
Hp
+
=
(92)
where is the standard deviation of the log-normal distribution erf is the error function H is the
enthalpy and p is the probability To then calculate the equal probability spacing for a log-normal
distribution we solve for H for the set of probabilities p which range from 2 to 1 2minus
96
with a step-size of This then gives enthalpies of basins equally spaced by the cumulative
probability distribution and as a result each basin has equal degeneracy
1 2 r2 1e f10
pH
minus minus = (93)
This set of basins carries with it an Arrhenius activation barrier for the configurational entropy that
is only a function of and the degeneracy of each basin (if the is chosen such that the energy
distributions converge) This creates a model that fully describes the thermodynamics of the liquid
and though it currently lacks any kinetic component it offers the possibility to be a useful tool
when used in conjunction with Eqs (82) (83) and (88) This new tool is denoted the ldquotoy landscape
modelrdquo (TLM) It is called the toy landscape model because we are generating a landscape that is
not the real landscape but is enough to reproduce key experimental properties without needing
expensive computational calculations It is a lsquotoyrsquo for us to play with without needing to worry
about upfront computational cost
To confirm the validity of TLM and the way it is constructed we can calculate the
activation barrier from this deterministic method and for a random set of inherent structures with
the same number of basins and distributions For this purpose we used temperatures ranging from
1000 to 2000 K for both the deterministic and random models with a step size of 0001 for TLM
Excellent agreement between the random and deterministic methods is shown in Figure 22A This
analysis allows us to understand that the thermodynamics of the system are fundamentally linked
to the viscous flow behavior Though this technique was created to target viscosity and relaxation
it is generally applicable to all configurational thermodynamics of liquids and glasses It is worth
noting that all the thermodynamics will be relative to a 0 eV energy state defined at equilibrium at
0 K and as such the enthalpy differences at some fixed temperature must be known to compare
over compositional spaces
97
In Figure 22B we see the same method being used to understand the relationship between
the viscous parameters and the parameters of the landscape This is the key to the numerical
calculation of the lsquotoy landscapersquo model The toy landscape is a numerical calculation where we
use experimentally accessible values to generate a landscape whose thermodynamics should
reproduce key thermodynamic phenomena In Figure 22B we show that the fragility and glass
transition are systematically changed as the number of basins and standard deviation of the enthalpy
is changed When this idea is combined with the basin calculation in Eq (93) (giving equal
degeneracies for all basins) then the complete set of enthalpy basins is easily attainable giving
access to the fundamental thermodynamics of the system
98
(
A)
(B)
(A)
99
Figure 22 (A) Comparison between the randomized method (histogram) and the deterministic
method (vertical lines) showing good agreement between the maximum in the histogram and the
value predicted by the deterministic technique validating the approach It is worth noting that the
100-basin distribution is a very wide distribution where the total number of basins is less than the
number of points used in the calculation This is done for a variable number of basins with the
number of basins shown in the legend (B) The dependence of fragility and the glass transition
temperature vs the distribution of states and the number of basins
55 Barrier Free Description of Thermodynamics
Throughout this work we have systematically explored the Adam-Gibbs model of
viscosity An important insight that we have gained is that the configurational entropy is free of any
barriers This is of particular interest because the Arrhenius slope of the activation barrier is linked
to the glass transition and fragility The glass transition temperature is known to be related to the
set of activation barriers as it corresponds to a breakdown of ergodicity What is found here is that
the fragility is the ldquoconversion factorrdquo to account for the slope of the configurational entropy
(which as previously identified is related to the distribution of basins) This means that the barriers
themselves do not matter until the material is close to the breakdown of ergodicity and thus the
equilibrium liquid thermodynamics and kinetics can be described without explicit information
about the saddle points in the landscape
Understanding the role of the topography in the properties of glasses and liquids lead to
the most important insights derived from the AG model Since the configurational entropy is the
same for both stress and structural relaxation simple viscous parameters allow us to understand
fundamental thermodynamic quantities of the system without needing to map computationally
expensive energyenthalpy landscapes The process of converting experimentally accessible
properties to a usable landscape for thermodynamic calculations is what we are calling the toy
landscape model To show the utility of toy landscapes we can calculate the driving force for glass
relaxation This information can give insights into the physics of relaxation by dynamically
100
calculating the Gibbs free energy of the equilibrium liquid at any temperature which defines a
driving force for glass relaxation To do this calculation of the driving force for glass relaxation
we consider the ergodicity parameter ( x ) defined by the Mauro-Allan-Potuzak (MAP) model of
glass relaxation as the ratio between the glass entropy and the equilibrium liquid entropy
liquiss dglaS x S= (94)
To get information about the enthalpy we consider an instantaneous Boltzmann distribution on the
landscape given by an equilibrium distribution at some elevated temperature fT
1
exp
exp
iglass i i
fii
f
HH H g
kTHg
kT
= minus
minus
(95)
This formalism in conjunction with TLM gives us the ability to explore the temperature
and compositional dependence of the driving force for relaxation The enthalpy of the liquid is
calculated from an equilibrium distribution on the landscape The driving force ( ) is defined as
( ) ( )( ) ( )1iliquglas ds f l quii dH H TS xT T = minus + minus (96)
For our purposes this gives a nice approximation for the driving force improving on previous
approximations given by ( fT Tminus )[16] [42] TLM does not include a kinetic feature to calculate
the occupational probability in each basin however when coupled to a kinetic model such as the
MAP model[44] a complete description of the underlying thermodynamics of glass relaxation can
be considered For this calculation of the driving force the kinetics are calculated through the MAP
model but this is not necessarily a requirement In Figure 23 the compositional and temperature
dependence of the driving force is shown for three glasses with identical glass transitions and
varying distributions of basins (as listed in Table 1) The calculations are performed using the
RelaxPy script and MAP model with the stretching exponent fixed at the Phillips value of 37
101
explanations of which are available elsewhere[42] The systems were quenched from 700 K to 300
K at a rate of 10 Kmin Figure 23 clearly shows how simply changing the total number of basins
creates a larger driving force for glass relaxation This simple change of the number of basins
drastically increases the entropic contribution and due to the loss of ergodicity explains the increase
of the driving force of relaxation The kinetics could be calculated through any relaxation model or
directly from the landscape if some barrier approximations are made
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition temperature (500 K)
were kept the same while the total number of basins were allowed to vary
Sample Total Basins [-] Distribution of States [log eV] 1
1493g
mT
minus
[K]
Fragility [-]
1 100 02 837 40
2 500 02 537 31
3 1000 02 466 28
102
Figure 23 The driving forces for different example glasses calculated using a combination of the
MAP model RelaxPy and the toy landscape model The parameters for each glass can be found in
Table 4
57 Toy Landscapes for the Design of Glasses and Glass Ceramics
The concept of a toy landscape can go even further building a method to predict the
dynamics of a glass system based on simple inputs This is possible because in the previous section
we have related the distribution of basins on an energy landscape to the experimental viscosity
which allows for thermodynamic insights Building on this we can assume that the barriers (H )
for transitioning between states is equal to that of the barrier mean for the viscous relaxation
ln10gH mkT= (97)
This means that the only key factor needed to then predict the dynamics of a landscape using the
metabasin approach is the vibrational frequency However the relaxation time at the glass transition
103
is known to be a constant of 100s and as such we can choose a vibrational frequency that reproduces
this fixed point This gives an entire description of relaxation without any need for fictive
temperature An example of the results predicted through this method are shown in Figure 24 This
prediction only required knowledge of the glass transition temperature and the fragility and from
this a pure prediction of relaxation is made
104
105
Figure 24 The predicted outputs from the toy landscape method showing predictions of the enthalpy
and entropy under a standard quench for barium disilicate This prediction does not require fictive
temperature or any such assumptions about the evolution of the non-equilibrium behavior
This method is also not limited to predicting the relaxation of glasses but can also capture
the crystallization It is known from the earlier work on nucleation presented in Chapter 3 that a
simple two state model was able to capture the underlying physics of the system Thus if we know
the free energy of the crystal and barrier to crystallization we can find predict key crystallization
106
phenomena By fitting the heat flow peak magnitude and location this gives enough information to
infer the barrier and the enthalpy of the crystal The degeneracy (and thus the entropy) of the crystal
can then be calculated through knowledge of the liquidus temperature
ex(
p)cry SCL l
cry
lk
G TH
T
minus=
(98)
This then gives everything for the model presented in Chapter 3 from only 1 calorimetry
experiment the glass transition the fragility and the liquidus temperature To predict nucleation a
series of estimates were done running the toy landscape at different temperatures until the
temperature with the fastest crystallization rate was found This was then assumed to be the
maximum rate of nucleation The value of the interfacial energy was then chosen in such a way so
that the predicted curve reproduces the peak at the same temperature This gives all the parameters
needed for a prediction using CNT It is also possible to make a prediction of the growth rate of
crystals using this expression
2
1 exp6
GU
kT
a kT
minus minus
=
(99)
The results for these estimates are shown in Figure 25 The reason why growth is more accurate
compared to nucleation is that the kinetic factor of growth is governed by viscosity while nucleation
is governed by a kinetic factor that includes an assumed vibrational frequency
These methods have a plethora of assumptions built in such as that the vibrational
frequency of basins undergoing relaxation is equal to the vibrational frequency when undergoing
crystallization however as more information is gained about a particular system (such as more
crystallization peak as a function of temperature) the information could be built into the system to
improve crystallization predictions If the peak size is known as a function of scan ratetemperature
even the enthalpy as a function of temperature could be known This provides a whole system for
the real estimates of crystallization can be improved
107
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume in
nucleation is assumed to be on the order of one cubic angstrom while the a parameter in growth is
assumed to be around one nm both are in good agreement for estimates in literature The values
for the orange points are taken from these works[128] [167]
57 Discussion
The impact of the Adam-Gibbs model on glass science is hard to overstate and it has
become foundational to ideas that will enable the design of next generation glasses The Adam-
Gibbs equation still has more to offer the glass community in terms of fundamental understanding
and practical applications In this work we have shown that the Adam-Gibbs relationship is
consistent with both shear and bulk viscosities Also we have validated the main assumption of the
MYEGA model giving insights into the scaling of configurational entropy from the underlying
enthalpy landscape Using this knowledge of the relationship between the Adam-Gibbs model with
the underlying enthalpy landscapes we have developed a simplified ldquotoy landscaperdquo approach for
modeling glass relaxation The toy landscape is constructed numerically by understanding the
distribution of inherent structures and transition points in the landscape and then relating these
distributions to experimentally measured parameters of the MYEGA viscosity model This
1 00
1600
1400
1200
1000
00
600
400
200
000
0000 110000 130000 1 0000 1 0000
(m
s)
Temperature
Growth PredictionCassar et al
0
2
4
6
10
12
14
00 1100 1300 1 00
log 1
0(I) n
ucl
ei s
m3
Temperature
Predicted RateRodrigues et al
4
3
2
1
0
00 1000 1100 1200 1300
log
10(I m
ax(I))
nuclei s
m3
Temperature elvin
108
simplified landscape model can then be used to calculate the underlying thermodynamics and
kinetics of the system and represents a practical application of energy landscapes to understand
glass relaxation nucleation and other relevant thermodynamic properties
This also closes our discussion on understanding the thermal history effects on glass What
we have shown is that the dynamics of a glass system can be understood through energy landscapes
and fundamentally the same insights can be recreated through lsquoToy Landscapesrsquo This approach
allows fundamentally for the complex 3N dimensional landscape to be summarized and abbreviated
into a smaller landscape that can easily be implemented for any glass However this new
parameterization relies on knowing some key properties that are only due to the location in the
compositional phase space and for a complete picture of the system to be created we must also
create a sufficient approximation for the compositional phase space shown in the next 2 chapters
Chapter 6
Enabling the Prediction of Glass Properties
So far models have been created that describe the very nature of the thermal history effects on
glasses However as an input to these models we have to know about the compositional dependence
of key parameters namely the glass transition and the fragility In this chapter we extend
compositional models to other key properties needed for the design of glasses Here these key
properties that have not been readily predictable are approached using both machine learning and
TCT models The TCT models are focused on surface reactivity Youngrsquos modulus and ionic
conductivity The ML addresses CTE fragility melting point as well as Youngrsquos modulus
61 Controlling Surface Reactivity
Glass surfaces especially their interactions with water[168]ndash[171] are of the utmost importance to
nuclear waste glass cover glass and many other modern applications[106] It is generally accepted
that the outermost surface of oxide glasses readily reacts with water molecules where water
molecules chemisorb on an as-formed glass surface and populate it with surface hydroxyls Recent
models attempt to quantitatively explain the change in properties observed on hydroxylated glass
surfaces[77] [168] [172] [173] and most agree that the surface readily becomes hydrated because
it is more stable to have a bonded hydroxyl group than to have dangling bonds In glasswater
interactions hydrolysis and diffusion lead to two forms of adsorbed water chemisorbed and
physisorbed[171] [174]ndash[176] Both affect the network differently and are controlled by different
processes physisorption by hydrogen bonding[175] and chemisorption by reaction with the surface
to form bonded hydroxyl groups
110
equiv Si minus O minus Si equiv +H2O rarr equiv Si minus OH + OH minus Si equiv
Although binding energy studies have been performed using molecular dynamics simulations[177]
a specific study of the surface reactivity in tandem has not been conducted Recent advancements
in reactive force field modeling allow the direct observation of both diffusion and surface
reactivity[38] [178]ndash[180] and several studies have found surface reactivity constants consistent
with those reported experimentally[173] [178] [181]
Recently TCT has been used to predict various bulk properties of glassy systems such as
Vickers hardness and fragility[17] [182] [183] Based on the TCT description of the glass network
glassy materials have been postulated to have a so-called lsquointermediate phasersquo in which the atomic
structure of a material will self-organize so as to be isostatically constrained (n=3) as explored
extensively in various works[65]ndash[67] [133] [184] [185] In this case it has been shown that such
materials exhibit anomalous behavior in certain properties such as an high hardness and lower free
energy[43] [67] [77]
TCT has also been used to understand the impact of water on bulk glass structure and
properties Potter et al[168] found that by accounting for the impact of chemisorbed (dissociated)
water on glass network connectivity (viz breaking Si-O-Si bonds) and the impact of physisorbed
(molecular) water on strain of the Si-O-Si bond angle the glass transition temperature (Tg) could
be predicted This model could be further expanded to include other properties (such as
modulus[139] or dissolution kinetics[77]) predicted by TCT such as the work done by Liu et
al[186] on calcium-silicate-hydrate gel An extensive study relating topological constraints[6]
[17] [57] and surface energy was performed by Yu et al[172] specifically focusing on the
transition from hydrophilic to hydrophobic behavior on silica surfaces Their work used reactive
molecular dynamics to model the change in surface energy of a silicate glass and then correlate it
with the number of surface constraints present with the work focusing primarily on the global
average of the surface
111
To investigate the effects of glass network topology on the surface reactivity we model
the hydration of a silicate glass was modelled using molecular dynamics simulations according to
the following procedures Initially bulk sodium silicate glasses were simulated (150 atoms with a
molar composition of 70SiO2middot30Na2O) using the Teter potential[187] For a bulk glass this pair
potential is known to accurately simulate sodium silicate glasses and has been extensively studied
to investigate various properties including structural features transport of sodium ions and
vibrational density of states[188] The size of the system is initially set to achieve the
experimentally measured density of 2466 gcm3 [189] A total of 35 Si-atoms 85 O-atoms and 30
Na-atoms were randomly inserted within a periodic cubic box of a = 12686 Å and the initial
configuration was energy-minimized to avoid any overlaps prior to the glass formation The glass
was held for 05 ns with a constant number of atoms volume and energy (a microcanonical or
NVE ensemble) Since a thermal fluctuation of ~400 K (from 2000 K to 2400 K) was observed
during the microcanonical run the glass was allowed to evolve further at 2400 K for 05 ns with a
constant number of atoms volume and temperature (a canonical or NVT ensemble) Within the
same ensemble the melted system was then cooled with a constant cooling rate of 05 Kps and
after the temperature had reached 300 K it was equilibrated for 1 ns Finally a constant pressure
of 1 atm was applied to the system (NPT ensemble) for another 1 ns to ensure no significant
fluctuations of the density during the equilibration process Three different sodium silicate glasses
of the same nominal composition were constructed via the aforementioned procedure using
different initial atomic positions These repeated simulations compensate for the limitation of small
sample size by allowing us to capture the statistical behavior of glass surfaces during the glass-
water reactions The final densities of the sodium silicate are presented in Table 5 These final
structures were used as starting configurations for the glass-water reactions MD simulations with
the Teter potential creating three glass networks were carried out using the Large-scale
AtomicMolecular Massively Parallel Simulator (LAMMPS) package[30]
112
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K
Simulation box dimension a (cubic) [Å] Density [gcm3]
Run1 12979 2303
Run2 12630 2499
Run3 12737 2437
Initial dimension a=12686 Å
Experimental density (70SiO2middot30Na2O mol) 2466 gcm3
113
After the bulk sodium silicate glasses were obtained a reactive potential was employed to model
the glass surface and subsequent glass-water interface as the system of interest for this study
required characterization of the reactive processes during the simulation of surface phenomena In
this study all reactive MD simulations are performed with the NaSiOH parameterization using
the ReaxFF reactive force field framework15 The ReaxFF parameters are trained using a first-
principles data set that describes water interaction at the sodium silicate glass-water interface
Further details of the ReaxFF methodology and its potential forms can be found in earlier
publications by van Duin et al[38] [179] [181] [190]
114
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the hydration
models Color scheme Si atom (ivory) O atom (red) and Na atom (blue) The z-axis is elongated
to allow space for an insert of water
115
As shown in Figure 26 a free surface was first created by expanding the c parameters of each of
the equilibrated bulk glasses This process results in a vacuum region above and below the two
surfaces of the glass slab along the z-direction These exposed surfaces were relaxed at 300 K for
100 ps with the ReaxFF reactive force field Following the relaxation of both top and bottom
surfaces the vacuum region was filled with water molecules The number of water molecules that
are inserted in the vacuum region was controlled to have a density of ~099 gcm3 In addition all
water molecules were kept at least 2 Å from the uppermost and lowermost atoms of the glass
surface to prevent any initial close contact with the surface Glass-water reaction simulations were
carried out at 300 K with the configurations shown in Table 6 for 500 ps in the NVT ensemble
From these runs trajectories at every 100 ps were obtained for the surface reactivity analyses
Figure 27 shows the initial and final positions of glass surface reaction with water at 0 and 500 ps
respectively
116
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the top surface
in contact with water is shown here
a) b)
117
Table 6 System configurations for sodium silicate glass-water reactions
Simulation cell [Å3] Number of water molecules
Run1 12979 times 12979 times 3894 146
Run2 12630 times 12630 times 3789 134
Run3 12737 times 12737 times 3821 138
118
The hydrated surfaces of sodium silicate glasses from each run were used to investigate the
adsorption behavior of a water molecule Since the time evolution of surface reactivity is of key
interest the binding energy of a water molecule to the hydrated surface was calculated at 0 100
200 300 400 ps In order to evaluate the local heterogeneity of reactivity imposed by varied glass
surface structures each surface was divided into a 10 by 10 grid and the binding energy at each site
was mapped across the grid The water molecule position in the z-direction from top and bottom
surface was maintained to be 20 Å from the outermost atom during the energy calculation The
binding energies (Eb) were calculated as below where the energy of a surface with adsorbed water
(Esw) was taken with respect to the surface energy (Es) and the energy of molecular water (Ew)
b s wswE E E E= minus minus (100)
A negative binding energy would indicate that water adsorption to the surface site at the
corresponding grid is thermodynamically favorable The binding energy map may then be used to
locate where water binding would be most stable based on the ReaxFF calculations From three
independent glass-water reaction boxes a total of six hydrated surfaces are obtained (a top and
bottom surface from each run) increasing the statistical reliability of this analysis in the ensemble
despite the small overall system size of any one box The binding energy mapping process is shown
in Figure 28
119
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example of the
electronic-structure DFT calculation using semilocal exchange-correlation functionals finding the
binding energy of a water lsquopixelrsquo to the surface
~20 Å
surface
10times10 grid
120
Similar calculations are carried out at an electronic structure level using the DFT framework as
implemented within the Vienna Ab-initio Simulation Package (VASP) software Total energies are
computed from the static geometries obtained from the reacted ReaxFF structures using the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional with projector augmented wave
(PAW) pseudopotentials to describe the ion cores A kinetic energy cutoff of 500 eV was used for
the expansion of the plane waves Due to the large non-crystalline structure the Brillouin zone is
only sampled at the Γ-point with 01 eV of Methfessel-Paxton smearing to help electronic
convergence Due to the computational cost associated with these DFT simulations the results were
primarily used to elucidate the electronic interactions between the water molecule and the glass
surface
To count the topological constraints in the structure generated from the ReaxFF MD
simulations an algorithm was developed to systematically exclude all non-bridging oxygens
alkali and hydrogens from the structure The number of surface rigid constraints (in the pixel)
around each network-forming atom was then calculated an example contour plot generated using
this procedure is shown in Figure 29 The surfaces were then converted to a 10times10 pixel grid and
the average coordination of network-forming atoms in each pixel calculated The depth of each
pixel taken was to be 4 Aring below the surface
121
Figure 29 Example contour surface showing the average coordination per atom on the glass
surface for the first run at 300 ps
122
Figure 30 shows the results from the MD binding energy studies where a clear transition in the
region around 30 constraintsatom is shown This region is isostatic because the number of
constraints is equal to the number of translational degrees of freedom and the width highlighted in
the figure is evidence for the existence of an intermediate phase In order to confirm that the binding
energies had indeed converged a larger system of 1500 atoms was simulated Good quantitative
agreement was found between the 150- and 1500-atom systems In this work the isostatic behavior
occurs over a wide range in which certain silicon network sites dramatically increase the binding
sites (binding energy approaching 0) The state or structure with nc outside of the intermediate range
will readily interact with water Only those in the isostatic region are likely to remain unaffected
after the interaction with water thus they are the rate limiting species in dissolution of silica glass
123
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of constraints for
surface atoms at the local pixel Results show a distinct maximum in which there is a near
hydrophilic-hydrophobic transition of the surface The error bars represent the standard deviation
A second system with 1500 atoms was performed to show convergence of the ReaxFF MD results
⟨119889119867minus119874119866119897119886119904119904⟩ = 345 Aring
⟨119889119867minus119874119866119897119886119904119904⟩ = 248 Aring
124
The intermediate phase result is intuitive due to the implicit stability that comes with having an
isostatic phase which is both energetically favorable and stress-free The larger negative binding
energies in the over-constrained regions arise from large differences between the free energies of
the chemically bonded state (chemisorbed) and unbound state (water in near proximity to the
surface but not bonded) in these regions in this region the chemical reaction with the water
molecule allows the glass surface to reduce the number of incompatible constraints which in turn
alleviates local stress in the network In other words a local region undergoes an alleviation of
stress when Q4 units convert to Q3 units The larger negative binding energy values in the under-
constrained regions can trace their origin to the sites available for bonding with the water molecule
Their high number of degrees of freedom allows for the facile reaction of water ie the only energy
cost is for the oxygen to dissociate from the network The isostatic network within the intermediate
phase is able to preserve its structural integrity because the energy barrier to deform the network is
high and there is no localized stress to create an additional driving force for reaction with water
Indeed the isostatic network is the most energetically favorable arrangement of a non-crystalline
structure This intermediate phase is governed by the fluctuations of the topology that arise with
the minimization of stresses These stresses change the local topology (within the range of the
fluctuations) so that they become isostatic[68] This leads to a flat region where there is a constant
binding energy at the surface Though it appears to be an intermediate phase it is important to note
that this may instead be simply a manifestation of an isostatic threshold (a peak in binding energies
rather than a plateau)
The average number of surface constraints has been shown in previous work to be largely
controlled by the annealing time[172] The surface constraints may then be used to predict surface
reactivity However it is worth noting that the local number of constraints is not constant in
time[17] Furthermore Potter et al[168] showed that molecular water in the network can radically
shift the free energy of the γ constraint Therefore the number of surface constraints is dynamic
125
during the influx of water Shifting constraint energy will also alter the fragility which in turn alters
the diffusion activation enthalpy In the future this may provide a path for developing more durable
glasses ndash currently it shows that a homogeneous intermediate phase (isostatic) surface should be
targeted to achieve maximum chemical durability
62 Elastic Modulus Prediction
Tailoring the elastic properties of glass is important in the design of new advanced compositions to
control the stiffness and damage resistance of a variety of glass products[2] [191]ndash[193] Elastic
moduli are defined based on the ratio of the stress exerted upon a material to the resulting strain
While various elastic moduli can be defined (eg Youngrsquos modulus shear modulus bulk modulus
and Poissonrsquos ratio) for an isotropic material such as glass only two of these quantities are
mutually independent[1] [191] [194]
Previous attempts to model elastic moduli have focused on either computationally costly
molecular dynamics simulations or empirical fitting methods The work by Makishima and
Mackenzie[195] offered a possible solution for predicting elastic moduli based on the dissociation
energy per unit volume of the glass however topological changes in the glass network are ignored
and the model cannot account for the temperature dependence of modulus With MD simulations
elastic moduli can be obtained by applying a stress and measuring the resulting strain on the
simulation cell assuming that accurate interatomic potentials are available[2] Machine learning
has also been applied to model elastic moduli using experimentally measured composition-property
databases with a high predictive ability being achieved[2] [196] Previous analytical modeling
techniques related to the topology of the glass-forming network have shown good qualitative
agreement with compositional trends in modulus but have a lack of quantitative accuracy[64]
[197]ndash[199]
126
It is common in the glass community to assume that Youngrsquos modulus may scale linearly
with hardness While this is clearly an oversimplification let us begin with previously derived
models for glass hardness as a potential starting point since glass hardness has received
considerable attention in the context of topological constraint theory [85] [192] [200] Several
models have been proposed to explain the origin of glass hardness all of which have a linear form
( ) vv
dHH n x n
dn= minus (101)
where Hv is the Vickers hardness x is chemical composition nrsquo is the critical level of constraints
needed for the substance to provide mechanical resistance to the indenter in three dimensions and
n is a measure of the constraint rigidity Recent work on glass hardness[192] has shown that n can
most accurately be defined in terms of either the density of rigid constraints (or alternatively the
density of rigid angular constraints) given by
)( )
( )( )
(c An xn
x Nx
M x
= (102)
where nc is either the number of constraints per atom or number of angular constraints per atom ρ
is the density of the composition M is the molar mass and NA is Avogadrorsquos number Since
hardness and elastic modulus are often considered to be correlated one might surmise that the
elastic modulus might also be some function of angular constraint density or total constraint
density However such an approach is not able to give quantitatively accurate predictions of
modulus
Here we propose an improved model of Youngrsquos modulus based on the free energy density of the
topological constraints
int
( ) ( ))
(
(
) i constr
c i
a s
Ai in x q T
M x
x NF F
=
= (103)
where qn(T) is the onset function for each associated constraint to give temperature dependence to
127
the predictions and is determined by the free energy ∆Fn The calculated values of Δ119865c are then
used to calculate Youngrsquos modulus (E) by
( )c c
c
FdE
EF
xd
F
= minus (104)
In this new model the Youngrsquos modulus is controlled by the free energy parameters related to the
strength of each constraint Each parameter in this model captures a key physical phenomenon with
each parameter corresponding to a physical property Moreover we demonstrate that a common
set of parameters can simultaneously capture both the temperature and compositional dependence
of modulus as well as predict other properties such as the glass transition temperature and fragility
To demonstrate the validity of this temperature-dependent constraint model Youngrsquos
modulus data are collected from literature for lithium borate[195] sodium borate[195] and
germanium selenide glass systems[201] Additionally the sodium phosphosilicate[202] system
was experimentally determined following the procedure described by Zheng et al[183] using room
temperature resonant ultrasound spectroscopy on prisms of dimensions 10 mm times 8 mm times 6 mm
Analytical topological constraint models have already been published for each of these
systems[17] [57] [202]
The proposed models for glass elasticity (the models previously proposed for hardness
based on constraint density and angular constraint density as well as the new model for modulus
based on the free energy density of constraints) were optimized in each case leaving the intercept
and slope (119889119864
119889119899
119889119864
119889119880119888) as adjustable parameters The constraint onset temperatures were also optimized
for the free energy density model Figure 31 and Figure 32 show the results for each model in the
phosphosilicate and sodium borate systems respectively
128
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values of the model
predictions are 641 GPa for constraint density 313 GPa for free energy density and 774 GPa
for angular constraint density
129
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from minimum
to maximum error The free energy density model gives the most accurate results The constraint
density has a RMSE of 61 GPa the angular density has a RMSE of 20 GPa and the energy density
has a RMSE of 59 GPa
130
We have also evaluated the predictive ability of the model in terms of the temperature dependence
of the Youngrsquos modulus sing the 10 Na2Omiddot90 B2O3 data reported by Jaccani and
Huang[191] the model and experimental predictions of the temperature dependence of the Youngrsquos
modulus are plotted in Figure 33 Here the constraint onset temperatures are determined from the
room temperature modulus data The number of escape attempts was optimized since this controls
the width of each transition and is thermal history dependent
131
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and experiment for
10 Na2O 90 B2O3 Using the previously fitted onset temperatures the only free parameters are
then the vibrational frequency and the heating time in which their product was fitted to be 14000
Where each dip in the modulus corresponds to a constraint no longer being rigid as heated through
each onset The onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The data was fit
using a least-squares method and the resultant fit is shown as the calculated method The fit has an
R2 of 094 (B) The contribution from each constraint to the overall modulus
It is also possible to validate the model by comparing the free parameters fitted to the Youngrsquos
modulus data vs that of previously reported data as shown in Table 7 Results for two additional
systems are plotted in Figure 34 to show the general validity of the free energy density model
132
Table 7 Fitted values from this analysis compared to those reported in the literature The disparity
between the constraints evaluated with molecular dynamics most likely come from the speed in
which the samples are quenched
Value Fitted Value (K) Literature (K) Method Citation
Silicate Onset 2212 1986 MD (Potter et
al[168])
Silicate Onset 818 1600 MD (Potter et
al[168])
Silicate Onset 450-500 810 MD (Potter et
al[168])
Borate Onset 921 Not Reported Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 715 740-760 Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 393 328 Fitting Parameter
(Mauro Gupta and
Loucks[69])
133
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in Fig 3) and
experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of 093 and (B) xLi2Omiddot(100-
x)B2O3 glasses with an R2 of 0986
(
A)
(
B)
134
In the previous work of Zheng et al[192] concerning hardness of glass the authors write
ldquoIt should be noted that each bond constraint corresponds to a certain energy since different kind
of bond has different bonding energyhellip and thus the constraint density also represents an energy
per unit volume In other words hardness is correlated to the energy per unit volume Our findings
for the borosilicate and phosphosilicate systems are further evidence in support of this argument
since both the total constraint density and angular constraint density approaches give better
prediction of glass hardness compared to models based on number of atomic constraintsrdquo
which makes it clear that the prediction of the hardnessmodulus should be closely related to the
onset temperatures (ie free energies) of the associated constraints The model that was previously
proposed could be extended to modulus if all constraints had equal amounts of potential energy
but due to the drastic change in strengths between the constraints the prediction fails
Bauchy et al[85] showed also that hardness in the calcium-silicate-hydrate system is
controlled by the angular constraints which led to the development of angular constraint density
as the governing control for hardness When the analysis of the density of angular constraints is
considered through the free energy view it becomes apparent why this method works effectively
for some systems Models for predicting the hardness and elastic modulus of glass attempt to
explicitly connect the rigid bond energy to the macroscopic properties of the system The
assumption for the hardness models is that the energies for breaking each type of constraint are all
approximately equal and hence only the number of the constraints matters The argument can then
be extended for elasticity since elastic modulus is a bulk material property with the approximation
of all constraints being equal strength the number of constraints per volume should be related To
correct for this approximation when a weighted sum of free energies is used the estimation
becomes significantly more accurate Using the density of the glass a precise free energy per
volume of the rigid constraints can be calculated The knowledge that the energy of the bonds is
tied to the elastic modulus has been widely known but had not been quantified nor placed within
the context of topological constraint theory[203]
135
63 Ionic Conductivity
Ionic conductivity 120590 is related to the number of charge carriers n and the mobility of the
carriers 120583 as they diffuse through a network by[1]
Ze n = (105)
where Ze is the charge of the conducting species Since the charge-carrying ion is constant within
a given glass family the ionic conductivity depends only on the mobility and the concentration of
the charge carriers present However it is not typically feasible to measure 120583 and n independently
of each other and thus it is unclear which variable scales in an Arrhenius fashion thereby
controlling the ionic conductivity of a glass In the weak electrolyte model by Ravaine and
Souquet[204]ndash[207] the mobility of the charge carriers is considered constant while the
concentration scales in an Arrhenius function Conversely the strong electrolyte model assumes a
constant number of charge carriers while the mobility follows an Arrhenius function[208] [209]
Recent MD simulations from Welch et al[210] support the weak electrolyte interpretation for alkali
silicate glasses concluding that ionic conductivity is dictated by the concentration of charge
carriers Previous reports in the literature also supporting this view have proposed several
hypotheses to explain this conclusion[208] [209] [211]ndash[213] however further exploration is
needed
The simplest form of the temperature dependence of free ions is given by
0 exp an nE
kT
= minus
(106)
Here n0 is the total possible number of mobile ions and Ea is the activation barrier for an ion to
diffuse It is often assumed that the activation enthalpy is related to the energy needed to deform
the network temporarily to allow for ion motion[193] [209] [210] [212] however it is our
hypothesis that there is a coupling between the ion hopping event and the relaxation of the local
136
network allows for a permanent relaxation or deformation along ion channels This means that
there is some cooperative relaxation of the network along the diffusion path in conjunction with an
ionic hopping event These cooperative relaxations are discussed explicitly in the Adam-Gibbs
formalism[53] for describing the relationship between viscosity and configurational entropy Sc
expc
B
TS
=
(107)
where B is an activation barrier is the viscosity and is the viscosity of the liquid in the limit
of infinite temperature In the high-temperature (low-viscosity) liquid state there is a well-defined
relationship between the viscosity and diffusion (Stokes-Einstein relation) However this
relationship breaks down in the low-temperature (high-viscosity) glassy phase
Herein we present a new model for predictions for the compositional dependence of ionic
conductivity We can write the activation free energy barrier Eac for an ionic hopping event as
a c c c cE T S= minus (108)
Here is an enthalpic barrier for the ion to hop to a neighboring site CT is the configurational
temperature (a value that describes the distribution of energies of the ions) and cS is the entropy
of the activation barriers These quantities are associated with configurational changes in the glass
indicated by the subscript c In the weak electrolyte model all charge carriers have the same
mobility and hence the entropic effects are dominant This indicates a small value of an
assumption that will be validated by results later in this work leading to a simplified approximation
for the activation energy at temperatures below the glass transition but suffiently high temperatures
such that
a c c cE T S (109)
In the case of a nonequilibrium glass at low temperature the system becomes trapped in a localized
137
region of the energy landscape known as a ldquometabasinrdquo with slow inter-metabasin transitions[156]
[161] The MAP model of the nonequilibrium viscosity of glass relies on this partial confinement
within metabasins where variation of the activation barrier for relaxation (H) is related to the
fragility of the system by[44]
1
ln10g
dm
dH kT= (110)
Here m is the fragility index (as proposed by Angell[58]) and Tg is the glass transition temperature
Splitting the differential in Eq (110) and integrating we can write
ln10gmH kT= (111)
This relationship was also proposed by Moynihan et al[214] for the activation enthalpy of
relaxation of a glass[214] [215] Setting the Arrhenius form of the barrier described in Eq (111)
equal to the Adam-Gibbs equation (Eq (107)) we write
exp exp c
H B
kT TS
=
(112)
Solving Eq (112) for the configurational entropy and inserting the result into Eq (109) we obtain
the simple relationship
2
ln10
ca
g
T BE
k T m (113)
Dividing the above expression with that of a reference state r and assuming that B and is cT are
constant with respect to compositional variation we have
g r ra
a r g
T mE
TE m= (114)
This new equation is the first to relate the activation free energy for ionic hopping in a glass
to the glass transition temperature and fragility of its corresponding supercooled liquid It also
138
predicts a coupling between ionic diffusion and the cooperative rearrangements described by the
Adam-Gibbs relation This possibility was first proposed by Ngai and Martin[216] who showed a
correlation between the product of the Kohlrausch stretching exponent ( ) and the activation
barrier for relaxation with the activation barrier for ionic conductivity This implies that the
diffusion in glasses is not governed by an elastic component but instead controlled by the
surrounding cooperative rearrangements ie the -relaxation of the glass[14] in good qualitative
agreement with previous results shown by Potuzak et al[217] Further evidence comes from the
polymer community where the decoupling of relaxation and conductivity is reported with work
showing that the decoupling of viscosity and diffusion varies with the fragility and the glass
transition temperature[218] This is a general relationship which we demonstrate later to be valid
for binary alkali borate alkali silicates and alkali phosphate glasses as well as validating the
atomistic diffusion method using computational techniques
In order to test this proposed relationship between relaxation and diffusion through
simulation a glass with the composition 10 Na2O-90 B2O3 (mol ) was synthesized in molecular
dynamics with 1150 atoms using the potentials of Wang et al[162] The system was quenched at a
rate of 1 Kps from randomly placed atoms at 2500 K in the NPT ensemble to room temperature
as shown in Figure 35[210] To find the transition point energy of the diffusing ion a nudged elastic
band (NEB) calculation was performed An alkali ion was chosen and its closest alkali neighbor
was moved to another location so that the NEB calculation could be performed between those two
sites All calculations were carried out using the LAMMPS software package Eleven reaction
coordinates were used in the NEB calculation and the results are shown in Figure 36
139
Figure 35 The structure for the initial minimum energy configuration showing the boron (blue)
network with interconnecting oxygens (red) and the interstitial sodium ions (yellow)
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and sodium is
ivory The barrier is overestimated compared to experimental data this could be from several
sources of error such as potential fitting thermal history fluctuations or sampling too few
transitions The line is drawn as a guide to the eye
140
Although the transition point energy is key for determining the activation barrier for diffusion here
we are also interested in the dynamics of the atoms around the mobile ion Since we have
hypothesized here that the ion diffusion is dependent on the ability for the local region to relax and
deform instead of elastically straining we expect a local deformation (relaxation) along the ion
path Figure 37 shows that along the ion diffusion tunnel there is indeed a permanent displacement
of the atoms
141
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function of
reaction coordinates The middle shows the local deformation around the ion of any atom that
moves in between inherent structure mandating a relaxation force The color shows the degree of
deformation
142
In the final configuration of Figure 37 the permanent deformation required to lower the
energy of the system is clearly seen If it were purely an elastic response to dilate a pathway the
displacement between two inherent structures should be confined to changes with only the mobile
ion however this is not observed Instead along the pathway of the ion movement the network
forming atoms also deform and change local positions to minimize the energy implying a net
relaxation around the ion ldquopathwaysrdquo To apply this and show the validity of Eq (114) the sodium
borate lithium borate lithium phosphate and sodium silicate systems were considered with the
model results compared to experimental values in Figure 38 To confirm this model literature values
for activation barriers have been used The experimental activation barriers for the borates are taken
from the work of Martin[213] while the fragilities and the glass transition temperatures are from
Nemilov[219] The viscosity curve of the silicate system was fit with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) equation of viscosity[55] using the data from Poole[220] and Knoche et
al[221] with the activation barriers from Martinsen[222] Lithium phosphate activation data was
taken from Ngai and Martin[216] with the glass transition and fragility taken from Hermansen et
al and predicted values from topological constraint theory[223]
143
Figure 38 Different network formers and the prediction of the activation barrier from our model
compared with activation barriers from literature (A) Sodium silicate predictions and experimental
values[222] the error is calculated from the error in the fragility when fitting the data (B) Lithium
phosphate activation energy[216] predicted with topological constraint theory and compared with
the experimental values (C) Predictions over two different systems of alkali borates[213] sodium
and lithium with a reported R2 of 097
144
It is worth noting that this model is predicting a non-equilibrium kinetic property using
equilibrium viscosity parameters a connection that has been demonstrated in the past for viscosity
and relaxation models[61] In this work we are applying the same principles to the diffusion of
ions in a network The diffusion and viscosity in the case of equilibrium are known to be intimately
linked using the Stokes-Einstein expression (which breaks down for highly cooperative
rearrangements near and below the glass transition temperature[224]) Here we are no longer
connecting the two properties explicitly but instead connecting the activation barrier for diffusive
hopping to the two key parameters governing the viscous flow of the supercooled liquid state
The failure of the Stokes-Einstein expression near the glass transition is a well-known
problem and has been studied extensively for purposes of predicting crystallization rates[167] This
occurs because there is a breakdown of the ergodicity of a system around the glass transition In
the MAP model[44] the viscosity is controlled by the ergodic parameter x and is given by
min( )
max( )
f
p
f
T Tx
T T
=
(115)
Here the exponent p is related to the sharpness of the ergodic to non-ergodic transition Using the
ergodic parameter the overall viscosity ( ) is expressed as a linear combination of equilibrium
viscosity ( eq ) and non-equilibrium viscosity ( ne ) such that
10 10 10log ) log ( ) (1 ) log ( )( f eq f ne fx T T x T TT T = + minus (116)
By combining the Stokes-Einstein expression and the model proposed in this work we can define
the diffusion of an ion through the network as
( )10 10 10 0log ( 1 log exp ) l6 )
og(
b af
f
kx
T ED T T D
a T Tx
kT
minus + minus =
(117)
Here a is an empirical constant related to the size of the diffusing species k is Boltzmannrsquos
145
constant aE is the activation barrier calculated with Eq (114) and D0 is the limit of diffusion as
the temperature approaches infinity D0 should be chosen such that at fT T= the diffusion is
continuous This can be justified because the preexponential factor is related to the entropy of the
system and the fictive temperature is the temperature at which the configurational entropy of the
transition barriers becomes nearly constant Only one parameter is left unknown a since the
viscosity and fictive temperature can be calculated with an arbitrary thermal history using a
relaxation modeling tool such as RelaxPy[42] Implicit in RelaxPy are the assumptions pertaining
to the viscosity made by Guo et al[61] A similar model was first proposed by Cassar et al[167]
for modeling diffusion during crystallization although their model used an empirical hyperbolic
tangent function to approximate the ergodic factor In contrast the current model can be considered
explicitly as a function of thermal history and composition
64 Machine Learning Expansion
Though the TCT models presented here for different properties are immensely useful they
do have limitations the greatest of which is that intimate knowledge of the glassy structure is
required to parameterize these models This is due to the fact that the landscape is fundamentally
informed by the underlying structure of the network and as such to predict the dynamics of the
landscape information on the structure must be incorporated in some form However there is an
alternate method in which the physical origin is not considered in the development of the model
machine learning Machine learning comes in many forms and has become crucial to the study of
materials in the modern era There is a plethora of machine learning techniques however in this
paper we are focused on four key methods linear fitting (LF) methods random forest (RF)
symbolic regression (SR) and neural networks (NN)[3] [80] [225] [226] These methods are
146
explored extensively in literature and the exact underlying mathematics is beyond the scope of this
work In this work these methods have been used to create models for number of constraints (LF)
glass stability (RF) fragility (NN) melting point (NN) the infinite limit of viscosity (SR) Youngrsquos
modulus (NN) and coefficient of thermal expansion (NN)
To enable a linear fitting method for the number of constraints a simple steepest descent
algorithm was used to find the number of constraints due to each component There is no direct
experimental way to measure the number of constraints so instead a technique leveraging some
other property must be used to approximate the number of constraints In this case hardness is
related to the number of constraints through
25v cH A n= minus (118)
In which A is an empirical scaling parameter that is dependent on load indenter geometry and
glass family In this linear approach to determining constraints cn can be given by
0
compon
k
c k k
ents
cn x n=
= (119)
Combining this expression with a hardness database grouped by composition (in this case Sciglass)
allows for an approximate number of constraints of the system but due to A being a variable there
must be a fixed constraint in the system and every set of data is given a unqiue value of A The
fixed constraint in this system is that silicon atoms are four coordinated The value of A is chosen
such that the error between the prediction and the hardness is minimized The total process is as
follows
Choose a set of c kn
1 Calculate number of constraints for all compositions
2 Iterate over every dataset in Sciglass with gt 1 datapoints and scale by a factor of A
3 Calculate total error for steepest descent algorithm
4 Descent error slope
5 Repeat until error has been minimized
147
This then leads to an estimate for the number of constraints for every atomic species which can
then be used to estimate important properties such as chemical durability hardness and
qualitatively can predict glass stability
Two other techniques that are used are RF and SR When implemented a grid search for
the hyperparameters were used with a testing set used to determine the efficiency of each set of
hyperparameters Additionally each model was trained repeatedly and the lowest error was taken
The RF method was used to predict glass stability while undergoing a novel processing technique
called alkali-proton substitution (APS) The SR method was used to understand the compositional
dependence of the infinite temperature viscosity limit used in 3-parameter viscosity models To do
this the Sciglass database was taken (as implemented in GlassPy[227]) and sets of data that
included more than eight datapoints were fit to the MYEGA equation If the RMSE error of the fit
was less than 001 Pa s the fragility was greater than 17 and the value of the log infinite limit was
293 5minus it was accepted into the final database Each of these criteria were added due to the well-
known fact that this database is full of error and imprecise measurements The symbolic regression
was then performed through the grid search and the equation that performed best and reappeared is
given as
10loggT
mA B = + (120)
With A and B being parameters that varied depending on the hyper-parameters the infinite limit
threshold and the accuracy threshold used when creating the database We are currently in the
process of refining values for A and B is ongoing
This method allows for an easy prediction of the infinite limit so if the glass transition and
fragility are predicted through some other means (such as ML or TCT) a better prediction of the
third parameter is known without just accepting -293 as the mean Plots of the infinite limit vs the
148
fragility glass transition and Tgm showing the clear dependance of the infinite limit on the term
Tgm are shown in Figure 39
149
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top left) shows
the distribution of the infinite temperature limit in the database after limits exerted on the system
(Top right) The distribution of the infinite viscosity limit vs the glass transition (Bottom left) The
relationship between fragility and infinite temperature limit of viscosity (Bottom right) The infinite
temperature limit of viscosity vs the key metric predicted by SR
150
NN are perhaps the most widely discussed ML tool applied in literature This makes sense
given they are universal function approximators and are quite easy to train given sufficient data
The most difficult part of creating a NN is adjusting the hyperparameters so that good accuracy can
be found without causing overfitting To avoid overfitting the data is divided into a testing set then
a training set (just as for RF and SR) but instead of doing a grid search for the hyperparameters we
used gradient boosted regression trees to find the optimal set of parameters This consists of
creating an initial tree then evaluating it based on a small set of data and finding the minimum
point then running a new set with perturbations around that point to create another tree This is
repeated until the value converges to a small error In this work the value being determined is the
lowest error of 100 sequentially trained neural nets with a set of hyperparameters This is
particularly useful because there is no concern for the type of parameters (string int floats) that
determine the value at the end so one can do a mixed parameter optimization that includes every
conceivable variable The values being optimized are given by
bull Learning rate [10-5 10-1] (float on logarithmic scale)
bull Decay rate for Adam Optimization [10-6 10-2] (float on logarithmic scale)
bull Hidden layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull Output layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull The number of nodes in each layer [81024] (int)
bull Batch size [1256] (int)
bull Patience [5250] (int)
In Table 8 we have listed the hyper-parameters that were used for the fragility melting point
Youngrsquos modulus and co-efficient of thermal expansion (CTE)
151
Table 8 Hyperparameters for different neural networks after hyper-optimizations
Model Fragility Melting
Temperature
Youngrsquos
Modulus
CTE
Learning Rate 0035 0018 0027 00091
Decay Rate 000293 000095 000161 0002
Hidden Layer RELU RELU RELU RELU
Output Layer Linear Linear SELU RELU
Number of Hidden
Layers
2 1 1 1
Number of Nodes 235 506 514 470
Batch Size 4 118 211 125
Patience 25 250 23 22
RMSE 5 [-] 805 [K] 68 [GPa] 96 x 10-7 [1 K]
152
Chapter 7
Designing Green Glasses for the 21st Century
Thus far the work presented in this dissertation has enabled a new set of design tools for
everything from crystallization glass relaxation and optimizing over the ideal compositional space
we can use them for designing new glasses Building on these models and insights new applications
of glass are enabled specifically three glasses an ion conducting glass for batteries a proton
conducting glass for hydrogen fuel cells and a commodity glass for consumers All three are
currently undergoing testing but experimental results have been reported back as of yet Overall
both the hydrogen and ion conducting glasses have been designed thorough a combination of ML
and underlying physical models and as such hare reported herein
71 Glass Electrolytes
An emerging high-interest application of glass is solid state batteries Glasses offer a
possible solution to the growing energy crisis however to realize this new paradigm new research
methods are needed The main barrier for any new batteries to be commercialized is a high
conductivity at room temperature (gt 10-3 S cm) high stability ease of processing and ultimately
an alternative must be cheaper than current liquid state Li-ion batteries glass is an attractive
candidate due to the innate stability a simple processing technique the infinite variability and the
relatively low cost of production In addition the structure of a glass is a liquid-like structure which
may encourage ion migration like that seen in liquids Despite all of these advantages only a few
compositions have realized the requirements but further research is needed to find optimal
compositions
153
To optimize the composition for activation barriers we must understand the relationship
between conductivity and structure Structural effects can be propagated in two different methods
based on the Arrhrenius expression for conductivity
0 exp aE
kT
minus =
(121)
The two methods are through 0 (the infinite temperature conductivity) and the activation barrier
( aE ) The pre-exponential factor is approximately a constant for simliar compositions while the
activation barrier varies dramatically over small compositional spaces making it the larger of the
two concerns Many models have been presented to predict the compositional dependence of the
activation barrier the Anderson-Stuart (AS) model the Christensen-Martin-Anderson-Stuart
(CMAS) model the weak electrolyte (WE) model Kohlrausch exponent model (KEM) from Ngai
et al[216] and there is the model proposed in this dissertation which we will call the Wilkinson
viscous cooperative conductivity model (WVCC)[8] [88] [204] [207]ndash[209]
To optimize machine learning (ML) is a powerful technique that is implemented The best
method to train the ML is to use the direct relationship between the compositions and the activation
barrier but this is not currently feasible due to a lack of central depository or database for this
information To make the prediction of the activation barrier we need use some model that is reliant
on commonly accesible parameters This rules out all models for this type of analysis except since
WVCC since the other models require some fitting parameters not accessible Each models required
parameters fitted values (please note that the models typically include approximations for these
values) and caveats are shown in Table 9
154
Table 9 A table with some ionic conductivity models and the parameters needed for them as well
as the disadvantages for each These are not the only models but are representative of those
commonly used in literature
Model Required Parameters Fit Parameters Caveats
AS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
lsquoMadelungrsquo Constant
Covalency Parameter
Doorway Radius1
The Madelung
Constant is not smooth
as a function of
composition and the
fitting parameters scale
the activation energy
non-proportionally
CMAS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
Dielectric Permittivity
lsquoMadelungrsquo Constant
Doorway Radius1
Jump Distance of Ion1
The same issues as AS
but in addition the
dielectric permittivity
lacks a database as
well
WE Charge Concentration Equilibrium Coefficient1 The equilibrium
coefficient drastically
changes the prediction
non-linearly
KEM Stretching Exponent Proportionality
Coefficient
There is not enough
data available to know
the compositional
dependence of the
stretching exponent
WVCC Glass Transition
Fragility Index
Proportionality
Coefficient
Due to the
proportionality
constant the activation
barrier can only be
known in a local
composition range
1 There are ways to approximate this value based on additional data
155
The WVCC model predicts that the activation barrier for ionic conductivity is given by
a
g
EA
mT= (122)
in which A is a proportionality constant This means that to predict the glass behavior we donrsquot
need to know things about the glass but instead merely about the liquid state where the viscosity is
readily available Leveraging this technique if the viscosity information as a function of
compositions is available then the local glass with the lowest activation barrier can be found by
minimizing 1 gT m To access this information multiple options are available such as topological
constraint theory but in this work we will leverage ML since it is easily applied to a large
compositional space
To get the viscosity we will use neural networks (NN) A well trained NN for the glass
transition is readily available and as such will be used for our prediction of gT In the previous
section there was presented a trained fragility NN which enables a complete optimization of the
local composition based on WVCC To generate a database of fragility values that the NN was
trained on SciGlass viscosity values were fit with MYEGA model The data had to fit following
criteria
1 The total number of data points for fitting had to be greater than 5 from the same literature
source
2 The root-mean-square-error of the fit had to be less than 001 Pa s
3 The infinite temperature value of viscosity had to be within 3 orders of magnitude of the
accepted logarithmic value (-293 log(Pa s))
An additional neural network recently created by Cassar[228] was also used for an independent
comparison Cassarrsquos NN was based on a hybrid physical ML work and may offer better insights
when extrapolating far from experimental datapoints In this study to prove this concept we will
only focus on P2O5 B2O3 Al2O3 and Li2O
156
This gives an entire method to predict glass battery candidates This technique will find all
of the local minimum but due to the fact that there is a proportionality constant it is unknown which
local minimum is the absolute minimum This means that the composition with the lowest
activation barrier in the space is identified but another technique is needed to investigate which one
of the identified compositions is best The method follows
1 Randomize an initial glass candidate with the content of being normalized
2 Run the glass candidate through the NN to find values of fragility and the glass transition
3 Perturb the composition to find the gradient
4 Step down the gradient
5 If the new gradient is 0 this is a local minimum and a possible glass candidate return to 1
Otherwise return to step 3
This method cannot find global optima because there is no guarantee that the
proportionality constant will stay constant across wide compositional spaces To narrow it down
an additional technique to compare the predicted values is needed (such as another activation
barrier model molecular dynamics simulations a regression algorithm or simply melt the proposed
compositions) To further pair down in this work a k-means algorithm was used along with the
assumption that the proportionality constant was slowly changing over compositional space This
allowed us to take the best glass in each grouping predicted by k-means The glasses we have
predicted (using both the separate predictor NN and the method by Cassar et al[228]) are listed
below in Table 10 while an example prediction for 1000 gmT (which should scale with the
activation barrier) in the Na2O ndash B2O3 ndash SiO2 system is shown in Figure 40 The glasses are currently
undergoing melting at Coe College and Iowa State It is worth noting that composition D is a well-
known composition that is known to exhibit criteria close to the criteria for conductivity If this
method shows that these compositions are better than other compositions in the family found in
literature then it can be expanded to include a wider range of component and hopefully finding a
universal candidate for glass batteries
157
Table 10 The predicted compositions based on the optimization scheme proposed
Compositions Li2O Al2O3 P2O5 B2O3
A- This work NN 55 31 0 14
B- This work NN 44 1 5 50
C- Cassar 60 26 2 13
D- Cassar 50 0 0 50
Figure 40 The prediction of the scaling of the activation barriers for a common sodium borosilicate
system
72 Hydrogen Fuel Cell Glasses
An emerging high-interest application of glass that is proton-conducting intermediate
temperature fuel cells[7] [229] A fuel cell in short is a device that turns chemical fuel into
electricity without the use of combustion It is often described as a chemical battery due to the
similarity of a fuel construction with batteries One of the crucial components to a working
intermediate range fuel cell is an electrolyte with a high proton migration The electrolyte must be
stable over a wide range of temperatures (up to 673 K) and a proton conductivity greater than 10-2
S cm-1 at the operating temperature[7] Previously phosphate glasses near the metaphosphate
compositions were used since they carried the appropriate number of residual protons for
conductivity studies However this restricted the phase space to only n-2 dimensions drastically
reducing the degrees of freedom of the problem and artificially restricting possible solutions In
2013[230] to access the full phase space a new method was developed where a sodium phosphate
glass could have all the sodium ions replaced with protons This allows high concentrations of
protons to be achieved with a variety of starting compositions A previous report[230] has
summarized successful samples these data are used as a training set for this study with additional
data on compositions that have failed taken from internal theses
To choose the candidate material for each model a cost function is defined The cost function
is an analytical function whose inputs are the fraction of each oxide component and the output is a
value that rates the composition The cost function consists as of many properties that are of interest
In the application of oxide electrolytes for an intermediate range fuel cell the two main properties
to worry about are the ionic conductivity at high temperatures (we will choose 473 K due to this
being the low end in which these materials are considered) and the relative stability of each phase
159
This is especially important since the material will undergo the APS (alkali-proton substitution) in
which the sodium will be replaced with protons in the bulk of the material The cost function used
in this work is given by
473 1( ) log 000T K Sx = minus= minus (123)
In which S is either a 0 or 1 (representing a crystal or a glass respectively) and is the
ionic conductivity The function is arbitrary and could be expanded to include as many terms are
as needed in the goal of this glass The stability is heavily weighted because it is more important
than the conductivity and any sample that is not stable should not be considered In order to predict
each of these samples a hybrid physicalempirical approach is used For predicting the ionic
conductivity of a proton-swapped glass we began with an observed relationship between the glass
transition temperature and the conductivity at 473 K This empirical relationship is shown in Figure
41 The relationship although empirical can provide some insights into the physics of proton
transferred glasses (no universality is claimed in this work) The activation barrier for these glasses
has widely remained unchanged as noted in a previous work due to the fact that the activation
barrier is simply to disassociate the proton with the non-bridging oxygen (NBO) and once the
proton is free interacts rarely with the network until it is rebounded to said network Thus the pre-
factor can be understood as being related to the degeneracy of proton conduction pathways When
the network has lower configurational entropy there are fewer pathways for the proton to travel in
this case it is known that the configurational entropy is then the dominating effect for the glass
transition[53]
The relationship between the glass transition temperature and the configurational entropy
can be expanded through the Stokes-Einstein relationship evaluated at the glass transition
22
1273 10
6 10g
g
T T g
kTnZeD nZe
anT
minus
= = = (124)
160
In which a is approximately the size of a proton and n is the number of charge carriers (which is
approximately a constant according to the weak electrolyte theory) This determines the intercept
of the glassy form proton conductivity and then the activation barrier (slope) is found to be
approximately the same for all proton conducting glasses meaning that the intercept is the dominant
effect on the behavior of the glass and the only variable controlling the intercept is the glass
transition
Leveraging the configurational entropyrsquos relationship with the glass transition along with
topological constraint theory we can write an expression for the glass transition and as such gain
predictive power for proton conductivity To predict the glass transition temperature (Tg) we start
with the Adam Gibbs model of viscosity[53] Topological constraint theory then states that the
degrees of freedom of the network is proportional to the configurational entropy[23] [56] [57]
Using the liquid state definition of the glass transition (1210 = Pa s) and the well-known infinite
temperature of viscosity we can then write
1
1493 lng
BT
kf=
(125)
Where f is the degrees of freedom Letting the constants be equal to A and rewriting the expression
in terms of constraints we arrive at a predictive formula for the glass transition[109]
3
g
x c x
x
AT
m n=
minus (126)
Where nc is the number of constraints provided by each component at the glass transition associated
with each component x and their molar fraction mx The value of the constraints was then linearly
parameterized to the glass transition data from literature[7] [230] [231] This is not the most
explicit approach to counting constraints however it is the most convenient when considering
large phase spaces being optimized over This glass transition temperature is then converted to
conductivity using the empirical translation shown in Figure 41
161
The other term in our cost function is that of stability The question of what forms glass is
a notorious question that goes back to the first serious days of research into materials It has been
shown that no one metric is a good predictor and no metric is universal However one must be used
to rule out bad compositions from the start To do this a Random Forest method is implemented
Though this may fail in edge cases it will at least help identify the right area to explore A
comprehensive review of random forest methods and machine learning methods for prediction of
glass properties can be found elsewhere[3] [4] [225] The amount of information previously
obtained about what forms glass in the compositional family we will be working on is not enough
to use random forest methods on just the fraction of each phase In order to circumvent this issue
we chose four physical parameters that would be used as features for the predictions
bull Total mol oxide modifier This was chosen because it is well known that modifiers break
up the network and most glass forming theories are related to the network percentages We
considered the oxide versions of H Na Ba Sr and Ca as the modifiers
bull Total mol network former Similarly it is well known the network influences glass
forming We considered B Ge and P as the network forming cations
bull Mean cation charge Though there has been previous work on this topic it is rarely
considered as a lsquonormalrsquo predictor of glass forming capability It is included here because
these glasses are mainly invert and as such a higher field strength will increase the cohesive
nature of the network
bull Entropy of mixing This is an important parameter to consider with so many components
it may be stabilized by the entropy of mixing The entropy of mixing (Sm) was calculated
using the Gibbs entropy
The Gibbs entropy is given by
lnm x
x
xS k m m= minus (127)
The random forest model consisted of 100 trees with a maximum depth of 3 The results of both
the random forest model the topological predictions the optimization and the resultant glass are
shown in the results section The optimization was limited to the range of each component that has
been explored experimentally as well as an additional constraint of at least 5 mol rare earth oxide
and at least 5 mol of boron oxide or germania this is justified elsewhere[231]
162
Figure 41 The relationship between the glass transition and the proton conductivity This is justified
two ways one through the relationship of the entropy of diffusion and glass formation (the Adam-
Gibbs model) and through the fact that water is known to depress the glass transition
163
To explain the results we will first start with the models listed to confirm their validity
The first model is the prediction of the glass transition temperature Figure 42 shows the accuracy
of the glass transition prediction as well as the relative contribution of each component to the glass
transition In this figure any component that has a number of constraints greater then PO52 will
increase the glass transition and as such decrease the conductivity Thus it is preferable to construct
a glass with only components whose constraints at Tg are less than PO52 as long as the glass
remains stable Interestingly whether the data includes only the glass transitions of protonated glass
or all glasses determines the role of La Barium and sodium both decrease the glass transition
playing the role of the more traditional modifier however they do provide some rigidity to the
network Protons have a net negative effect which we believe to be a result of the APS process
where they are introduced as the sodium is leaving so this negative effect corresponds to a breaking
of the rigidity that exists It is also interesting to note that adding a boron phase to the sample
increases the glass transition though not as much as some of smaller alkaline earth samples
assumingly due to their increased field strength The worst network former appears to be GeO2
To calibrate the accuracy of the random forest model a confusion matrix is shown It shows
that the RF always predict that a glass will be a glass however there is some error when it forms a
crystal with 23 of the time being misidentified Though random forest methods are empirical we
can derive some physical meaning by looking at the relative importance of each feature From the
random forest study of stability we are able to rank the relative importance of each feature used
The mean cation charge mixing entropy and percent network formers are all of considerable
importance while the percent modifiers is less so
Once these models were developed and checked the reliability of the cost function is
confirmed After which the minimization of the cost function was performed and one glass was
chosen as optimal Boron was found to be preferential in the glass but was limited since the
compositional space was bound by previous samples (an additional sample is undergoing
164
characterization where the boron content was slightly increased) The optimized sample when
synthesized had some visible nucleation so the optimization while redone with limiting the
component maxima to just the limits of those where compositions in which APS was successfully
performed This resulted in the OP composition The three samples currently are undergoing
experimental characterization are listed in Table 11
165
Table 11 The compositions synthesized in this work These compositions were predicted by
minimizing the cost function described in Eq (123) OP is the variant that was melted after OP
partially crystallized B-OP appeared to have surface nucleation in some spots but was cut and
removed before APS treatment
Name PO52 NaO12 BaO LaO32 GeO2 BO32
OP 44 36 10 5 3 2
OP 49 36 5 5 3 2
B-VOP 44 36 10 5 1 4
166
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine the glass
forming region Over top the constraints at the glass transition provided by each oxide species is
listed Since the objective is to decrease Tg while staying in the glass forming region we will
attempt to minimize use of elements that increase the glass transition temperature (nc gt 17)
167
Chapter 8
Conclusions
The rapid rate of new information in the current scientific climate and the infinite
variability of glass both stand as both unique challenges and opportunities In this dissertation we
have divided the influence of energy landscapes on the effects of properties into two independent
spaces the compositional and the thermal history dimensions By building models that succinctly
and accurately describe the dynamics of different hyper-coordinate (composition thermal history
crystallinity) changes the feasibility of designing new glasses for the challenges of the 21st century
is obtainable
Before designing new models the current state-of-the-art models need to be implemented
and understood To reach this end two softwares based on previous work were created RelaxPy
and ExplorerPy RelaxPy is an implementation of the MAP model and though the MAP model is
powerful when predicting the dynamics of the glass RelaxPy ultimately showed that fictive
temperature is insufficient at capturing the underlying physics and is intensive to parameterize
ExplorerPy was created to standardize the approach to mapping energy landscapes Energy
landscapes served as the key method of understanding the deeper dynamics of glasses glass-
ceramics and liquids
The thermal history dependence of glass with respect to both relaxation and crystallization
has been incorporated into the existence of new models called ldquotoy landscapesrdquo The toy landscapes
have built upon the physics of previous models such as the MAP model however due to the lower
parametrization cost and the built-in increased physicality toy landscapes pose as a tool to increase
our understanding and speed-up the rate of new glass discoveries This tool can deal with the
complexity of the higher dimensional spaces due to crystallization and relaxation without
assumptions concerning fictive temperature
168
The compositional dimensions are the remaining dimensions that must be optimized when
designing a glass or glass-ceramic Building on previous work models have been developed that
enable predictions of Youngrsquos modulus surface reactivity and ionic conductivity However there
are additional properties that are needed and for those we have proposed novel machine learning
approaches This dissertation has not enabled the design of glass for every application However
it lays the groundwork and approaches to design glasses for societyrsquos growing needs Presented as
well are the methods used to design glasses that could satisfy the requirements for solid state glass
electrolytes and hydrogen fuel cells All of these techniques together promise to be a powerful new
framework to build the glasses of the future
169
References
[1] A K Varshneya and J C Mauro Fundamentals of Inorganic Glasses 3rd ed Elsevier
2019
[2] J C Mauro A Tandia K D Vargheese Y Z Mauro and M M Smedskjaer
ldquoAccelerating the Design of Functional Glasses through Modelingrdquo Chem Mater vol
28 no 12 pp 4267ndash4277 2016 doi 101021acschemmater6b01054
[3] J C Mauro ldquoDecoding the glass genomerdquo Curr Opin Solid State Mater Sci pp 1ndash7
2017 doi 101016jcossms201709001
[4] H Liu Z Fu Yang X Xu and M Bauchy ldquoMachine learning for glass science and
engineering A reviewrdquo J Non Cryst Solids no March p 119419 2019 doi
101016jjnoncrysol201904039
[5] M M Smedskjaer C Hermansen and R E Youngman ldquoTopological engineering of
glasses using temperature-dependent constraintsrdquo MRS Bull vol 42 no 01 pp 29ndash33
2017 doi 101557mrs2016299
[6] J C Mauro and M M Smedskjaer ldquoStatistical mechanics of glassrdquo J Non Cryst Solids
vol 396ndash397 pp 41ndash53 2014 doi 101016jjnoncrysol201404009
[7] T Omata et al ldquoProton transport properties of proton-conducting phosphate glasses at
their glass transition temperaturesrdquo Phys Chem Chem Phys vol 21 no 20 pp 10744ndash
10749 2019 doi 101039c9cp01502g
[8] Z A Grady C J Wilkinson C A Randall and J C Mauro ldquoEmerging Role of Non-
crystalline Electrolytes in Solid-State Battery Researchrdquo Front Energy Res vol 8 p
218 2020 doi 103389fenrg202000218
[9] R C Welch et al ldquoDynamics of glass relaxation at room temperaturerdquo Phys Rev Lett
170
vol 110 no 26 p 265901 2013 doi 101103PhysRevLett110265901
[10] M Montazerian E D Zanotto and J C Mauro ldquoModel-driven design of bioactive
glasses from molecular dynamics through machine learningrdquo Int Mater Rev vol 0 no
0 pp 1ndash25 2019 doi 1010800950660820191694779
[11] E D Zanotto and J C Mauro ldquoThe glassy state of matter Its definition and ultimate
faterdquo J Non Cryst Solids vol 471 pp 490ndash495 2017 doi
101016jjnoncrysol201705019
[12] R G Palmer ldquoBroken ergodicityrdquo Adv Phys vol 31 no 6 pp 669ndash735 1982 doi
10108000018738200101438
[13] J C Mauro P Gupta and R J Loucks ldquoContinuously broken ergodicityrdquo J Chem
Phys vol 126 no 18 p 184511 2007 doi 10106312731774
[14] M Micoulaut ldquoRelaxation and physical aging in network glasses A reviewrdquo Reports
Prog Phys vol 79 no 6 p 066504 2016 doi 1010880034-4885796066504
[15] D J Wales and H A Scheraga ldquoGlobal optimization of clusters crystals and
biomoleculesrdquo Science (80- ) vol 285 no 5432 pp 1368ndash1372 1999 doi
101126science28554321368
[16] J C Mauro R J Loucks and P Gupta ldquoFictive temperature and the glassy staterdquo J
Am Ceram Soc vol 92 no 1 pp 75ndash86 2009 doi 101111j1551-2916200802851x
[17] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 2009 doi 10106313152432
[18] F H Stillinger and T A Weber ldquoHidden structure in liquidsrdquo Phys Rev A vol 25 no
2 pp 978ndash989 1982 doi 101103PhysRevA25978
[19] F H Stillinger ldquoSupercooled liquids glass transitions and the auzmann paradoxrdquo J
Chem Phys vol 88 no 12 pp 7818ndash7825 1988 doi 1010631454295
171
[20] S Neelamraju C Oligschleger and J C Schoumln ldquoThe threshold algorithm Description of
the methodology and new developmentsrdquo J Chem Phys vol 147 no 15 p 152713
2017 doi 10106314985912
[21] D Prada-Gracia J Goacutemez-Gardentildees P Echenique and F Falo ldquoExploring the free
energy landscape From dynamics to networks and backrdquo PLoS Comput Biol vol 5 no
6 p e1000415 2009 doi 101371journalpcbi1000415
[22] E Pollak A Auerbach and P Talknerz ldquoObservations on rate theory for rugged energy
landscapesrdquo Biophys J vol 95 no 9 pp 4258ndash4265 2008 doi
101529biophysj108136358
[23] G G Naumis ldquoEnergy landscape and rigidityrdquo Phys Rev E vol 71 p 026114 2005
doi 101103PhysRevE71026114
[24] S J Benkovic G G Hammes and S Hammes-Schiffer ldquoFree-energy landscape of
enzyme catalysisrdquo Biochemistry vol 47 no 11 pp 3317ndash3321 2008 doi
101021bi800049z
[25] M Roca B Messer D Hilvert and A Warshel ldquoOn the relationship between folding and
chemical landscapes in enzyme catalysisrdquo Proc Natl Acad Sci U S A vol 105 no 37
pp 13877ndash13882 2008 doi 101073pnas0803405105
[26] P Li G Henkelman J A eith and J Johnson ldquoElucidation of aqueous solvent-
mediated hydrogen-transfer reactions by ab initio molecular dynamics and nudged elastic-
band studies of NaBH4 hydrolysisrdquo J Phys Chem C vol 118 no 37 pp 21385ndash21399
2014 doi 101021jp507872d
[27] S P Niblett M Biedermann D J Wales and V De Souza ldquoPathways for diffusion in
the potential energy landscape of the network glass former SiO2rdquo J Chem Phys vol
147 no 15 p 152726 2017 doi 10106315005924
[28] Y Yu M Wang M M Smedskjaer J C Mauro G Sant and M Bauchy
172
ldquoThermometer Effect Origin of the Mixed Alkali Effect in Glass Relaxationrdquo Phys Rev
Lett vol 119 no 9 pp 1ndash5 2017 doi 101103PhysRevLett119095501
[29] Y Yu M Wang D Zhang B Wang G Sant and M Bauchy ldquoStretched Exponential
Relaxation of Glasses at Low Temperaturerdquo Phys Rev Lett vol 115 no 16 p 165901
2015 doi 101103PhysRevLett115165901
[30] Steve Plimpton ldquoFast Parallel Algorithms for Short-Range Molecular Dynamicsrdquo J
Comput Phys vol 117 no 1 pp 1ndash19 1995 doi
httpsdoiorg101006jcph19951039
[31] N Mousseau and G T Barkema ldquoTraveling through potential energy landscapes of
disordered materials The activation-relaxation techniquerdquo Phys Rev E vol 57 no 2
pp 2419ndash2424 1998 doi 101103PhysRevE572419
[32] F El-Mellouhi N Mousseau and L J Lewis ldquo inetic activation-relaxation technique
An off-lattice self-learning kinetic Monte Carlo algorithmrdquo Phys Rev B vol 78 p
153202 2008 doi 101103PhysRevB78153202
[33] D J Wales ldquoDiscrete path samplingrdquo Mol Phys vol 100 no 20 pp 3285ndash3305 2002
doi 10108000268970210162691
[34] G Henkelman and H Joacutensson ldquoImproved tangent estimate in the nudged elastic band
method for finding minimum energy paths and saddle pointsrdquo J Chem Phys vol 113
no 10 pp 9978ndash9985 2000 doi 10106314961868
[35] D Sheppard P Xiao W Chemelewski D D Johnson and G Henkelman ldquoA
generalized solid-state nudged elastic band methodrdquo J Chem Phys vol 136 p 074103
2012 doi 10106313684549
[36] A Pedone G Malavasi M C Menziani A N Cormack A V and N York ldquoA New
Self-Consistent Empirical Interatomic Potential Model for Oxides Silicates and Silica-
Based Glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
173
[37] A Takada C R A Catlow and G D Price ldquoComputer modelling of B2O3 II Molecular
dynamics simulations of vitreous structuresrdquo J Phys Condens Matter vol 7 no 46 pp
8693ndash8722 1995 doi 1010880953-8984746004
[38] A C T Van Duin S Dasgupta F Lorant and W A Goddard ldquoReaxFF A reactive
force field for hydrocarbonsrdquo J Phys Chem A vol 105 no 41 pp 9396ndash9409 2001
doi 101021jp004368u
[39] J C Mauro and A Varshneya ldquoModel interaction potentials for selenium from ab initio
molecular simulationsrdquo Phys Rev B vol 71 p 214105 2005 doi
101103PhysRevB71214105
[40] J C Mauro R J Loucks J Balakrishnan and S Raghavan ldquoMonte Carlo method for
computing density of states and quench probability of potential energy and enthalpy
landscapesrdquo J Chem Phys vol 126 no 19 2007 doi 10106312733674
[41] Y Z Mauro C J Wilkinson and J C Mauro ldquo ineticPy A tool to calculate long-time
kinetics in energy landscapes with broken ergodicityrdquo SoftwareX vol 11 p 100393
2020 doi 101016jsoftx2019100393
[42] C J Wilkinson Y Z Mauro and J C Mauro ldquoRelaxPy Python code for modeling of
glass relaxation behaviorrdquo SoftwareX vol 7 pp 255ndash258 2018 doi
101016jsoftx201807008
[43] J C Mauro ldquoTopological constraint theory of glassrdquo Am Ceram Soc Bull vol 90 no
4 pp 31ndash37 2011 doi 101039c3ee40810h
[44] J C Mauro D C Allan and M Potuzak ldquoNonequilibrium viscosity of glassrdquo Phys Rev
B vol 80 p 094204 2009 doi 101103PhysRevB80094204
[45] M Cardona R V Chamberlin and W Marx ldquoThe history of the stretched exponential
functionrdquo Ann der Phys vol 16 no 12 pp 842ndash845 2007 doi
101002andp200710269
174
[46] P Grassberger and I Procaccia ldquoThe long time properties of diffusion in a medium with
static trapsrdquo J Chem Phys vol 77 no 1982 pp 6281ndash6284 1982 doi
1010631443832
[47] J C Phillips ldquoMicroscopic aspects of Stretched Exponential Relaxation (SER) in
homogeneous molecular and network glasses and polymersrdquo J Non Cryst Solids vol
357 no 22ndash23 pp 3853ndash3865 2011 doi 101016jjnoncrysol201108001
[48] J C Phillips ldquo ohlrausch explained The solution to a problem that is 1 0 years oldrdquo J
Stat Phys vol 77 no 3ndash4 pp 945ndash947 1994 doi 101007BF02179472
[49] A Q Tool and C G Eichlin ldquoVariations caused in the heating curves of glass caused by
heat treatmentrdquo J Am Ceram Soc vol 14 pp 276ndash308 1931
[50] A Q Tool ldquoRelation Between Inelastic Deformability and Thermal Expansion of Glass in
Its Annealing Rangerdquo J Am Ceram Soc vol 29 no 9 pp 240ndash253 1946 doi
101111j1151-29161946tb11592x
[51] H N Ritland ldquoLimitations of the Fictive Temperature Conceptrdquo J Am Ceram Soc vol
39 no 12 pp 403ndash406 Dec 1956 doi 101111j1151-29161956tb15613x
[52] O S Narayanaswamy ldquoA model of structural relaxation in glassrdquo J Am Ceram Soc
vol 54 no 10 pp 491ndash498 1971 doi 101111j1151-29161971tb12186x
[53] G Adam and J Gibbs ldquoOn the Temperature Dependence of Cooperative Relaxation
Properties in Glass‐Forming Liquidsrdquo J Chem Phys vol 43 no 1 pp 139ndash146 1965
doi 10106311696442
[54] Q Zheng J C Mauro A J Ellison M Potuzak and Y Yue ldquo niversality of the high-
temperature viscosity limit of silicate liquidsrdquo Phys Rev B vol 83 no 21 p 212202
2011 doi 101103PhysRevB83212202
[55] J C Mauro Y Yue A J Ellison P Gupta and D C Allan ldquoViscosity of glass-
forming liquidsrdquo Proc Natl Acad Sci vol 106 no 47 pp 19780ndash19784 2009 doi
175
101073pnas0911705106
[56] G G Naumis ldquoGlass transition phenomenology and flexibility An approach using the
energy landscape formalismrdquo J Non Cryst Solids vol 352 no 42-49 SPEC ISS pp
4865ndash4870 2006 doi 101016jjnoncrysol200601160
[57] P Gupta and J C Mauro ldquoComposition dependence of glass transition temperature
and fragility I A topological model incorporating temperature-dependent constraintsrdquo J
Chem Phys vol 130 no 9 p 094503 2009 doi 10106313077168
[58] C A Angell ldquoRelaxation in liquids polymers and plastic crystals - strongfragile patterns
and problemsrdquo J Non Cryst Solids vol 13 no 31 pp 131ndash133 1991 doi
1010160022-3093(91)90266-9
[59] G S Fulcher ldquoAnalysis of Recent Measurements of the Viscosity of Glassesrdquo J Am
Ceram Soc vol 75 no 5 pp 1043ndash1055 1992 doi 101111j1151-
29161992tb05536x
[60] I Avramov ldquoViscosity in disordered mediardquo J Non Cryst Solids vol 351 no 40ndash42
pp 3163ndash3173 Oct 2005 doi 101016JJNONCRYSOL200508021
[61] X Guo J C Mauro D C Allan and M M Smedskjaer ldquoPredictive model for the
composition dependence of glassy dynamicsrdquo J Am Ceram Soc vol 101 pp 1169ndash
1179 2018 doi 101111jace15272
[62] Doss C J Wilkinson Y Yang H Lee L Huang and J C Mauro ldquoMaxwell
relaxation time for nonexponential α-relaxation phenomena in glassy systemsrdquo J Am
Ceram Soc vol 103 no 6 pp 3590ndash3599 2020 doi 101111jace17051
[63] P Gupta and A R Cooper ldquoTopologically disordered networks of rigid polytopesrdquo J
Non Cryst Solids 1990 doi 1010160022-3093(90)90768-H
[64] J C Phillips and M F Thorpe ldquoConstraint theory vector percolation and glass
formationrdquo Solid State Commun vol 53 no 8 pp 699ndash702 1985 doi 1010160038-
176
1098(85)90381-3
[65] X Feng W Bresser and P Boolchand ldquoDirect Evidence for Stiffness Threshold in
Chalcogenide Glassesrdquo Phys Rev Lett vol 78 no 23 pp 4422ndash4425 1997 doi
101103PhysRevLett784422
[66] W Bresser P Boolchand and P Suranyi ldquoRigidity Percolation and Molecular Clustering
in Network Glassesrdquo Phys Rev Lett vol 56 no 23 pp 2493ndash2496 1986 doi
101103PhysRevLett562493
[67] Y Vaills T Qu M Micoulaut F Chaimbault and P Boolchand ldquoDirect evidence of
rigidity loss and self-organization in silicate glassesrdquo J Phys Condens Matter vol 17
no 32 pp 4889ndash4896 2005 doi 1010880953-89841732003
[68] A irchner and J C Mauro ldquoStatistical Mechanical Model of the Self-Organized
Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologiesrdquo
Front Mater vol 6 p 11 2019 doi 103389fmats201900011
[69] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 Jun 2009 doi 10106313152432
[70] D R Cassar A C P L F de Carvalho and E D Zanotto ldquoPredicting glass transition
temperatures using neural networksrdquo Acta Mater vol 159 pp 249ndash256 Oct 2018 doi
101016JACTAMAT201808022
[71] J C Mauro and R J Loucks ldquoSelenium glass transition A model based on the enthalpy
landscape approach and nonequilibrium statistical mechanicsrdquo Phys Rev B vol 76 no
17 p 174202 2007 doi 101103PhysRevB76174202
[72] C J Wilkinson et al ldquoEnergy Landscape Modeling of Crystal Nucleationrdquo Nat Comput
Mater p Submitted 2020
[73] M E Mc enzie and J C Mauro ldquoHybrid Monte Carlo technique for modeling of crystal
177
nucleation and application to lithium disilicate glass-ceramicsrdquo Comput Mater Sci vol
149 no January pp 202ndash207 2018 doi 101016jcommatsci201803034
[74] M E McKenzie et al ldquoImplicit glass model for simulation of crystal nucleation for glass-
ceramicsrdquo npj Comput Mater vol 4 no 1 pp 1ndash7 2018 doi 101038s41524-018-
0116-5
[75] E D Zanotto and P F James ldquoExperimental tests of the classical nucleation theory for
glassesrdquo J Non Cryst Solids vol 74 no 2ndash3 pp 373ndash394 1985 doi 1010160022-
3093(85)90080-8
[76] Y Yu N M A Krishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[77] I Pignatelli A Kumar M Bauchy and G Sant ldquoTopological control on silicatesrsquo
dissolution kineticsrdquo Langmuir vol 32 pp 4434ndash4439 2016 doi
101021acslangmuir6b00359
[78] C J Wilkinson et al ldquoTopological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phaserdquo J Phys Chem Lett vol 10 pp
3955ndash3960 2019 doi 101021acsjpclett9b01275
[79] Y Zhang A Li B Deng and Hughes ldquoData-driven predictive models for chemical
durability of oxide glass under different chemical conditionsrdquo npj Mater Degrad vol 4
no 1 pp 1ndash11 2020 doi 101038s41529-020-0118-x
[80] N M Anoop Krishnan S Mangalathu M M Smedskjaer A Tandia H Burton and M
Bauchy ldquoPredicting the dissolution kinetics of silicate glasses using machine learningrdquo J
Non Cryst Solids vol 487 no February pp 37ndash45 2018 doi
101016jjnoncrysol201802023
[81] R M Potter J Hoffman and J Hadley ldquoAn update of the equation for predicting the
178
dissolution rate of glass fibers from their chemical compositionsrdquo Inhal Toxicol vol 29
no 4 pp 145ndash146 2017 doi 1010800895837820171321702
[82] B Deng ldquoMachine learning on density and elastic property of oxide glasses driven by
large datasetrdquo J Non Cryst Solids vol 529 no August 2019 p 119768 2020 doi
101016jjnoncrysol2019119768
[83] S Feller N Lower and M Affatigato ldquoDensity as a probe of oxide glass structurerdquo
Phys Chem Glas vol 42 no 3 pp 240ndash246 2001 Accessed Jul 19 2018 [Online]
Available
httpswwwingentaconnectcomcontentsgtpcg200100000042000000034203240
[84] Q Zheng and H Zeng ldquoProgress in modeling of glass properties using topological
constraint theoryrdquo Int J Appl Glas Sci vol 11 no 3 pp 432ndash441 2020 doi
101111ijag15105
[85] M Bauchy M Javad Abdolhosseini Qomi C Bichara F-J Ulm and R J-M Pellenq
ldquoRigidity Transition in Materials Hardness is Driven by Weak Atomic Constraintsrdquo
2015 doi 101103PhysRevLett114125502
[86] Yang X Xu B Yang B Cook H Ramos and M Bauchy ldquoPrediction of Silicate
Glassesrsquo Stiffness by High-Throughput Molecular Dynamics Simulations and Machine
Learningrdquo Accessed Mar 13 201 Online Available
httpsarxivorgpdf190109323pdf
[87] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological constraint model for
the elasticity of glass-forming systemsrdquo J Non Cryst Solids X vol 2 2019 doi
101016jnocx2019100019
[88] C J Wilkinson K Doss D R Cassar R S Welch C B Bragatto and J C Mauro
ldquoPredicting Ionic Diffusion in Glass from Its Relaxation Behaviorrdquo J Phys Chem B vol
124 no 6 pp 1099ndash1103 2020 doi 101021acsjpcb9b10645
179
[89] C Wilkinson and J C Mauro ldquoExplorerpy Mapping the energy landscapes of complex
materialsrdquo SoftwareX vol Submitted 2020
[90] J C Mauro R J Loucks and J Balakrishnan ldquoA simplified eigenvector-following
technique for locating transition points in an energy landscaperdquo J Phys Chem A vol
109 no 42 pp 9578ndash9583 2005 doi 101021jp053581t
[91] Roumlder and D J Wales ldquoAnalysis of the b to b-CR Transition in biquitinrdquo
Biochemistry vol 57 no 43 pp 6180ndash6186 2018 doi 101021acsbiochem8b00770
[92] J C Mauro R J Loucks and J Balakrishnan ldquoSplit-step eigenvector-following
technique for exploring enthalpy landscapes at absolute zerordquo J Phys Chem B vol 110
no 10 pp 5005ndash5011 2006 doi 101021jp056803w
[93] J C Mauro R J Loucks J Balakrishnan and A Varshneya ldquoMapping the potential
energy landscapes of selenium clustersrdquo J Non Cryst Solids vol 353 pp 1268ndash1273
2007 doi 101016jjnoncrysol200609062
[94] B W H van Beest G J Kramer and R A van Santen ldquoForce fields for silicas and
aluminophosphates based on ab initio calculationsrdquo Phys Rev Lett vol 64 no 16 pp
1955ndash1958 Apr 1990 doi 101103PhysRevLett641955
[95] Q Zheng and J C Mauro ldquoViscosity of glass-forming systemsrdquo J Am Ceram Soc vol
100 no 1 pp 6ndash25 2017 doi 101111jace14678
[96] J C Mauro and Y Z Mauro ldquoOn the Prony Series Representation of Stretched
Exponential Relaxationrdquo Physica A vol 506 pp 75ndash87 2018 doi
101016jphysa201804047
[97] E D Zanotto and D R Cassar ldquoThe microscopic origin of the extreme glass-forming
ability of Albite and B2O3rdquo Sci Rep vol 7 no February pp 1ndash13 2017 doi
101038srep43022
[98] P Pedevilla S J Cox B Slater and A Michaelides ldquoCan Ice-Like Structures Form on
180
Non-Ice-Like Substrates The Example of the K-feldspar Microclinerdquo J Phys Chem C
vol 120 no 12 pp 6704ndash6713 2016 doi 101021acsjpcc6b01155
[99] O Bjoumlrneholm et al ldquoWater at Interfacesrdquo Chem Rev vol 116 no 13 pp 7698ndash7726
2016 doi 101021acschemrev6b00045
[100] Reichelt ldquoNucleation and growth of thin filmsrdquo Vacuum vol 38 no 12 pp 1083ndash
1099 Jan 1988 doi 1010160042-207X(88)90004-8
[101] G H Beall ldquoDr S Donald (Don) Stookey (1 1 ndash2014) Pioneering researcher and
adventurerrdquo Front Mater vol 3 no August pp 1ndash8 2016 doi
103389fmats201600037
[102] J C Mauro C S Philip D J Vaughn and M S Pambianchi ldquoGlass science in the
nited States Current status and future directionsrdquo Int J Appl Glas Sci vol 5 no 1
pp 2ndash15 2014 doi 101111ijag12058
[103] J Deubener et al ldquo pdated definition of glass-ceramicsrdquo J Non Cryst Solids vol 501
no January pp 3ndash10 2018 doi 101016jjnoncrysol201801033
[104] X Hao ldquoA review on the dielectric materials for high energy-storage applicationrdquo J Adv
Dielectr vol 03 no 01 p 1330001 2013 doi 101142s2010135x13300016
[105] J Deubener G Helsch A Moiseev and H Bornhoumlft ldquoGlasses for solar energy
conversion systemsrdquo J Eur Ceram Soc vol 29 no 7 pp 1203ndash1210 2009 doi
101016jjeurceramsoc200808009
[106] G S Frankel et al ldquoA comparative review of the aqueous corrosion of glasses crystalline
ceramics and metalsrdquo npj Mater Degrad vol 2 no 1 p 15 2018 doi
101038s41529-018-0037-2
[107] G H Beall ldquoDesign and properties of glass-ceramicsrdquo Annu Rev Mater Sci vol 22
no 1 pp 91ndash119 1992 doi 101146annurevms22080192000515
[108] J Deubener ldquoConfigurational entropy and crystal nucleation of silicate glassesrdquo Phys
181
Chem Glas vol 45 pp 61ndash63 2004
[109] J C Mauro A J Ellison D C Allan and M M Smedskjaer ldquoTopological model for the
viscosity of multicomponent glass-forming liquidsrdquo Int J Appl Glas Sci vol 4 no 4
pp 408ndash413 2013 doi 101111ijag12009
[110] M M Smedskjaer J C Mauro S Sen and Y Yue ldquoQuantitative design of glassy
materials using temperature-dependent constraint theoryrdquo Chem Mater vol 22 no 18
pp 5358ndash5365 2010 doi 101021cm1016799
[111] Y T Sun H Y Bai M Z Li and W H Wang ldquoMachine Learning Approach for
Prediction and Understanding of Glass-Forming Abilityrdquo J Phys Chem Lett vol 8 no
14 pp 3434ndash3439 2017 doi 101021acsjpclett7b01046
[112] Q Zheng et al ldquo nderstanding Glass through Differential Scanning Calorimetryrdquo Chem
Rev vol 119 no 13 pp 7848ndash7939 2019 doi 101021acschemrev8b00510
[113] X Xia I Dutta J C Mauro B G Aitken and F elton ldquoTemperature dependence of
crystal nucleation in BaOmiddot2SiO2 and 5BaOmiddot8SiO2 glasses using differential thermal
analysisrdquo J Non Cryst Solids vol 459 pp 45ndash50 2017 doi
101016jjnoncrysol201612032
[114] V M Fokin A A Cabral R M C V Reis M L F Nascimento and E D Zanotto
ldquoCritical assessment of DTA-DSC methods for the study of nucleation kinetics in
glassesrdquo J Non Cryst Solids vol 356 no 6ndash8 pp 358ndash367 2010 doi
101016jjnoncrysol200911038
[115] D C Van Hoesen X Xia M E Mc enzie and F elton ldquoModeling nonisothermal
crystallization in a BaO∙2SiO2 glassrdquo J Am Ceram Soc no December 2019 pp 2471ndash
2482 2019 doi 101111jace16979
[116] S C C Prado J P Rino and E D Zanotto ldquoSuccessful test of the classical nucleation
theory by molecular dynamic simulations of BaSrdquo Comput Mater Sci vol 161 no
182
January pp 99ndash106 2019 doi 101016jcommatsci201901023
[117] A O Tipeev and E D Zanotto ldquoNucleation kinetics in supercooled Ni 0Ti 0 Computer
simulation data corroborate the validity of the Classical Nucleation Theoryrdquo Chem Phys
Lett vol 735 no August p 136749 2019 doi 101016jcplett2019136749
[118] K F Kelton and A L Greer Nucleation in Condensed Matter Applications in Materials
and Biology 1st ed Pergamon 2010
[119] V M Fokin E D Zanotto N S Yuritsyn and J W P Schmelzer ldquoHomogeneous
crystal nucleation in silicate glasses A 40 years perspectiverdquo J Non Cryst Solids vol
352 no 26ndash27 pp 2681ndash2714 2006 doi 101016jjnoncrysol200602074
[120] A M Rodrigues D R Cassar V M Fokin and E D Zanotto ldquoCrystal growth and
viscous flow in barium disilicate glassrdquo J Non Cryst Solids vol 479 pp 55ndash61 2018
doi 101016jjnoncrysol201710007
[121] A Pedone G Malavasi M C Menziani A N Cormack and U Segre ldquoA new selft-
consistent empirical interatomic potential model for oxides silicates and silica-based
glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
[122] J J Maldonis A D Banadaki S Patala and P M Voyles ldquoShort-range order structure
motifs learned from an atomistic model of a Zr 0Cu4 Al metallic glassrdquo Acta Mater
vol 175 pp 35ndash45 2019 doi 101016jactamat201905002
[123] A irchner S Goyal M E Mc enzie J T Harris and J C Mauro ldquoStatistical
Description of the Thermodynamics of Glass-Forming Liquidsrdquo Physica A vol 559 p
125059 2020 doi 101016jphysa2020125059
[124] D R Cassar ldquoCrystallization Driving Force of Supercooled Oxide Liquidsrdquo Int J Appl
Glas Sci vol 7 no 3 pp 262ndash269 2016 doi 101111ijag12218
[125] V M Fokin A S Abyzov E D Zanotto D R Cassar A M Rodrigues and J W P
Schmelzer ldquoCrystal nucleation in glass-forming liquids Variation of the size of the
183
lsquostructural unitsrsquo with temperaturerdquo J Non Cryst Solids vol 447 pp 35ndash44 2016 doi
101016jjnoncrysol201605017
[126] M Lenoir A Grandjean Y Linard B Cochain and D R Neuville ldquoThe influence of
SiB substitution and of the nature of network-modifying cations on the properties and
structure of borosilicate glasses and meltsrdquo Chem Geol vol 256 no 3ndash4 pp 316ndash325
2008 doi 101016jchemgeo200807002
[127] P Gupta D R Cassar and E D Zanotto ldquoRole of dynamic heterogeneities in crystal
nucleation kinetics in an oxide supercooled liquidrdquo J Chem Phys vol 145 no 21 2016
doi 10106314964674
[128] A M Rodrigues ldquoDiffusion Processes Crystallization and Viscous Flow in Barium
Disilicate Glassrdquo Dissertation 2014
[129] E D Zanotto ldquoThe effects of amorphous phase sepatration on crystal nucleation in baria-
silica and lithia-silica glassesrdquo Dissertation 1982
[130] D Vargheese A Tandia and J C Mauro ldquoOrigin of dynamical heterogeneities in
calcium aluminosilicate liquidsrdquo J Chem Phys vol 132 no 19 p 194501 May 2010
doi 10106313429880
[131] K Deenamma Vargheese A Tandia and J C Mauro ldquoStatistics of modifier distributions
in mixed network glassesrdquo J Chem Phys vol 132 p 24507 2010 doi
10106313429880
[132] M Micoulaut ldquoThe Deep Effect of Topology on Glass Relaxationrdquo Physics (College
Park Md) vol 6 p 72 2013 doi 101103Physics672
[133] M Micoulaut ldquoRigidity and intermediate phases in glasses driven by speciationrdquo Phys
Rev B vol 74 p 184208 2006 doi 101103PhysRevB74184208
[134] M Kodama S Kojima S Feller and M Affatigato ldquoThe occurrence of minima in the
Borate anomaly anharmonicity and fragility in lithium borate glassesrdquo Phys Chem
184
Glas vol 46 no 2 pp 190ndash193 2005
[135] Y Fukawa et al ldquoVelocity of Sound and Elastic Properties of Li2O-B2O3 Glassesrdquo Jpn
J Appl Phys vol 34 p 2570 1995 doi httpsdoiorg101143JJAP342570
[136] M Reiner ldquoThe Deborah Numberrdquo Physics Today vol 17 no 1 p 62 1964 doi
10106313051374
[137] P K Gupta and J C Mauro ldquoThe laboratory glass transitionrdquo J Chem Phys vol 126
no 22 p 2240504 2007 doi 10106312738471
[138] R Richert and C A Angell ldquoDynamics of glass-forming liquids V On the link between
molecular dynamics and configurational entropyrdquo J Chem Phys vol 108 no 21 pp
9016ndash9026 1998 doi 1010631476348
[139] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological Constraint Model for
the Elasticity of Glass-Forming Systemsrdquo J Non Cryst Solids X vol 2 p 100019 2019
doi 101016jnocx2019100019
[140] S Feller S Bista A OrsquoDonovan-Zavada T Mullenbach M Franke and M Affatigato
ldquoPacking in alkali and alkaline earth borosilicate glass systemsrdquo Phys Chem Glas Eur
J Glas Sci Technol Part B vol 50 no 3 pp 224ndash228 2009
[141] P J Bray S Feller G E Jellison and Y H Yun ldquoB10 NMR studies of the structure of
borate glassesrdquo J Non Cryst Solids vol 38ndash39 no PART 1 pp 93ndash98 May 1980 doi
1010160022-3093(80)90400-7
[142] N Andersson and G L Comer ldquoRelativistic Fluid Dynamics Physics for Many Different
Scales Living Reviews in Relativityrdquo Living Rev Relativ vol 10 p 1 2007 Accessed
Feb 18 2019 [Online] Available httpwwwlivingreviewsorglrr-2007-
1httprelativitylivingreviewsorghttpwwwmathssotonacukstaffAnderssonhttpw
wwslueducollegesASphysicsprofscomerhtml
[143] P Ilg and H C Oumlttinger ldquoNonequilibrium relativistic thermodynamics in bulk viscous
185
cosmologyrdquo Phys Rev D - Part Fields Gravit Cosmol vol 61 no 2 p 023510 2000
doi 101103PhysRevD61023510
[144] P Romatschke ldquoRelativistic viscous fluid dynamics and non-equilibrium entropyrdquo Class
Quantum Gravity vol 27 no 2 p 025006 Jan 2010 doi 1010880264-
9381272025006
[145] D C Allan ldquoInverting the MYEGA equation for viscosityrdquo J Non Cryst Solids vol
358 no 2 pp 440ndash442 Jan 2012 doi 101016JJNONCRYSOL201109036
[146] M Potuzak R C Welch and J C Mauro ldquoTopological origin of stretched exponential
relaxation in glassrdquo J Chem Phys vol 135 no 21 p 214502 Dec 2011 doi
10106313664744
[147] L Ding M Thieme S Demouchy C unisch and B J P aus ldquoEffect of pressure and
temperature on viscosity of a borosilicate glassrdquo J Am Ceram Soc vol 101 no 9 pp
3936ndash3946 2018 doi 101111jace15588
[148] R Richert and M Richert ldquoDynamic heterogeneity spatially distributed stretched-
exponential patterns and transient dispersions in solvation dynamicsrdquo Phys Rev E vol
58 no 1 pp 779ndash784 1998 doi 101103PhysRevE58779
[149] P Richet ldquoViscosity and configurational entropy of silicate meltsrdquo Geochim Cosmochim
Acta vol 48 no 3 pp 471ndash483 Mar 1984 doi 1010160016-7037(84)90275-8
[150] P Gupta and J C Mauro ldquoTwo factors governing fragility Stretching exponent and
configurational entropyrdquo Phys Rev E vol 78 no 6 p 063501 2008 doi
101103PhysRevE78062501
[151] Z Zheng J C Mauro and D C Allan ldquoModeling of delayed elasticity in glassrdquo J Non
Cryst Solids vol 500 no August pp 432ndash442 2018 doi
101016jjnoncrysol201808033
[152] R Boumlhmer L Ngai C A Angell and D J Plazek ldquoNonexponential relaxations in
186
strong and fragile glass formersrdquo J Chem Phys vol 99 no 5 pp 4201ndash4209 1993
doi 1010631466117
[153] O Gulbiten J C Mauro and P Lucas ldquoRelaxation of enthalpy fluctuations during sub-
Tg annealing of glassy seleniumrdquo J Chem Phys vol 138 no 24 p 244504 2013 doi
10106314811488
[154] D Sidebottom R Bergman L Boumlrjesson and L M Torell ldquoTwo-step relaxation decay
in a strong glass formerrdquo Phys Rev Lett vol 71 no 14 pp 2260ndash2263 1993 doi
101103PhysRevLett712260
[155] G M Bartenev and V A Lomovskoi ldquoRelaxation time spectra and the peculiarities of
the process of boron anhydride glass transitionrdquo J Non Cryst Solids vol 146 no C pp
225ndash232 1992 doi 101016S0022-3093(05)80495-8
[156] A Heuer ldquoExploring the potential energy landscape of glass-forming systems From
inherent structures via metabasins to macroscopic transportrdquo J Phys Condens Matter
vol 20 no 37 p 373101 2008 doi 1010880953-89842037373101
[157] J C Phillips ldquoStretched exponential relaxation in molecular and electronic gasesrdquo
Reports Prog Phys vol 59 pp 1133ndash1207 1996
[158] P Richet ldquoResidual and configurational entropy Quantitative checks through applications
of Adam-Gibbs theory to the viscosity of silicate meltsrdquo J Non Cryst Solids vol 355
no 10ndash12 pp 628ndash635 2009 doi 101016jjnoncrysol200901027
[159] C J Wilkinson Doss G Palmer and J C Mauro ldquoThe relativistic glass transition A
thought experimentrdquo J Non-Crystalline Solids X vol 2 2019 doi
101016jnocx2019100018
[160] J C Mauro R J Loucks and S Sen ldquoHeat capacity enthalpy fluctuations and
configurational entropy in broken ergodic systemsrdquo J Chem Phys vol 133 no 16 pp
1ndash9 2010 doi 10106313499326
187
[161] J C Mauro R J Loucks and P Gupta ldquoMetabasin approach for computing the
master equation dynamics of systems with broken ergodicityrdquo J Phys Chem A vol 111
no 32 pp 7957ndash7965 2007 doi 101021jp0731194
[162] M Wang N M Anoop Krishnan B Wang M M Smedskjaer J C Mauro and M
Bauchy ldquoA new transferable interatomic potential for molecular dynamics simulations of
borosilicate glassesrdquo J Non Cryst Solids vol 498 no December 2017 pp 294ndash304
2018 doi 101016jjnoncrysol201804063
[163] I Avramov and A Milchev ldquoEffect of disorder on diffusion and viscosity in condensed
systemsrdquo J Non Cryst Solids vol 104 no 2ndash3 pp 253ndash260 Sep 1988 doi
1010160022-3093(88)90396-1
[164] G Scherer Relaxation in Glass and Composites Krieger Publishing Company 1992
[165] L Huang J Nicholas J Kieffer and J Bass ldquoPolyamorphic transitions in vitreous B2O3
under pressurerdquo J Phys Condens Matter vol 20 no 7 2008 doi 1010880953-
8984207075107
[166] W Capps P B Macedo B OrsquoMeara and T A Litovitz ldquoTemperature Dependecne of
the High-Frequency Moduli of Vitreous B2O3rdquo J Chem Phys vol 45 p 3431 1966
doi 107868s0002337x1402002x
[167] D R Cassar A M Rodrigues M L F Nascimento and E D Zanotto ldquoThe diffusion
coefficient controlling crystal growth in a silicate glass-formerrdquo Int J Appl Glas Sci
vol 9 no 3 pp 373ndash382 Jul 2018 doi 101111ijag12319
[168] A R Potter C J Wilkinson S H im and J C Mauro ldquoEffect of Water on
Topological Constraints in Silica Glassrdquo Scr Mater vol 160 pp 48ndash52 2019 doi
101016jscriptamat201809041
[169] D B Asay and S H im ldquoEvolution of the adsorbed water layer structure on silicon
oxide at room temperaturerdquo J Phys Chem B vol 109 pp 16760ndash16763 2005 doi
188
101021jp053042o
[170] M Tomozawa ldquoWater diffusion oxygen vacancy annihilation and structural relaxation in
silica glassesrdquo J Non Cryst Solids vol 179 pp 162ndash169 1994 doi 1010160022-
3093(94)90693-9
[171] S Kapoor R E Youngman K Zakharchuk A Yaremchenko N J Smith and A Goel
ldquoStructural and Chemical Approach toward nderstanding the Aqueous Corrosion of
Sodium Aluminoborate Glassesrdquo J Phys Chem B vol 122 pp 10913ndash10927 2018 doi
101021acsjpcb8b06155
[172] Y Yu N M A rishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[173] Y Yu B Wang M Wang G Sant and M Bauchy ldquoReactive Molecular Dynamics
Simulations of Sodium Silicate Glasses mdash Toward an Improved Understanding of the
Structurerdquo Int J Appl Glas Sci vol 8 no 3 pp 276ndash284 2017 doi
101111ijag12248
[174] E Stolper ldquoThe speciation of water in silicate meltsrdquo Geochim Cosmochim Acta vol
46 no 12 pp 2609ndash2620 Dec 1982 doi 1010160016-7037(82)90381-7
[175] M G Mesko P A Schader and J E Shelby ldquoWater solubility and diffusion in sodium
silicate meltsrdquo Phys Chem Glas vol 43 no 6 pp 283ndash290 2002 doi
101016jsemradonc201010001
[176] M Tomozawa M Takata J Acocella E Bruce Watson and T Takamori ldquoThermal
properties of Na2Omiddot3SiO2 glasses with high water contentrdquo J Non Cryst Solids vol 56
no 1ndash3 pp 343ndash348 Jul 1983 doi 1010160022-3093(83)90491-X
[177] E A Leed and C G Pantano ldquoComputer modeling of water adsorption on silica and
silicate glass fracture surfacesrdquo J Non Cryst Solids vol 325 pp 48ndash60 2003 doi
189
101016S0022-3093(03)00361-2
[178] J C Fogarty H M Aktulga A Y Grama A C T van Duin and S A Pandit ldquoA
reactive molecular dynamics simulation of the silica-water interfacerdquo J Chem Phys vol
132 no 17 p 174704 2010 doi 10106313407433
[179] S H Hahn et al ldquoDevelopment of a ReaxFF Reactive Force Field for NaSiOx Water
Systems and Its Application to Sodium and Proton Self-Diffusionrdquo J Phys Chem C vol
122 no 34 pp 19613ndash19624 2018 doi 101021acsjpcc8b05852
[180] T S Mahadevan W Sun and J Du ldquoDevelopment of Water Reactive Potentials for
Sodium Silicate Glassesrdquo J Phys Chem B vol 123 pp 4452ndash4461 2019 doi
101021acsjpcb9b02216
[181] A C T van Duin A Strachan S Stewman Q Zhang X Xu and W A Goddard III
ldquoReaxFF SiO Reactive Force Field for Silicon and Silicon Oxide Systemsrdquo J Phys
Chem A vol 107 no 19 pp 3803ndash3811 2003 doi 101021jp0276303
[182] M M Smedskjaer J C Mauro R E Youngman C L Hogue M Potuzak and Y Yue
ldquoTopological principles of borosilicate glass chemistryrdquo J Phys Chem B vol 115 no
44 pp 12930ndash12946 2011 doi 101021jp208796b
[183] Q Zheng M Potuzak J C Mauro M M Smedskjaer R E Youngman and Y Yue
ldquoComposition-structure-property relationships in boroaluminosilicate glassesrdquo J Non
Cryst Solids vol 358 pp 993ndash1002 2012 doi 101016jjnoncrysol201201030
[184] K Rompicharla D I Novita P Chen P Boolchand M Micoulaut and W Huff
ldquoAbrupt boundaries of intermediate phases and space filling in oxide glassesrdquo J Phys
Condens Matter vol 20 p 202101 2008 doi 1010880953-89842020202101
[185] M Micoulaut and J C Phillips ldquoRings and rigidity transitions in network glassesrdquo Phys
Rev B vol 67 p 104204 2003 doi 101103PhysRevB67104204
[186] H Liu S Dong L Tang N M A rishnan G Sant and M Bauchy ldquoEffects of
190
polydispersity and disorder on the mechanical properties of hydrated silicate gelsrdquo J
Mech Phys Solids vol 122 pp 555ndash565 Jan 2019 doi 101016JJMPS201810003
[187] A Tilocca N H De Leeuw and A N Cormack ldquoShell-model molecular dynamics
calculations of modified silicate glassesrdquo Phys Rev B - Condens Matter Mater Phys
vol 73 p 104209 2006 doi 101103PhysRevB73104209
[188] J Du and A N Cormack ldquoThe medium range structure of sodium silicate glasses A
molecular dynamics simulationrdquo J Non Cryst Solids vol 349 no 1ndash3 pp 66ndash79 2004
doi 101016jjnoncrysol200408264
[189] N P Bansal and R H Doremus Handbook of glass properties Academic Press 1986
[190] S H Hahn and A C T van Duin ldquoSurface Reactivity and Leaching of a Sodium Silicate
Glass under an Aqueous Environment A ReaxFF Molecular Dynamics Studyrdquo J Phys
Chem C p acsjpcc9b02940 2019 doi 101021acsjpcc9b02940
[191] S P Jaccani and L Huang ldquo nderstanding Sodium Borate Glasses and Melts from Their
Elastic Response to Temperaturerdquo Int J Appl Glas Sci vol 7 no 4 pp 452ndash463 Dec
2016 doi 101111ijag12250
[192] Q Zheng Y Yue and J C Mauro ldquoDensity of topological constraints as a metric for
predicting glass hardnessrdquo Appl Phys Lett vol 111 no 1 p 011907 Jul 2017 doi
10106314991971
[193] J C Dyre ldquoElastic models for the non-Arrhenius relaxation time of glass-forming
liquidsrdquo in AIP Conference Proceedings 2006 vol 832 pp 113ndash117 doi
10106312204470
[194] Y T Cheng and C M Cheng ldquoRelationships between hardness elastic modulus and the
work of indentationrdquo Appl Phys Lett vol 73 no 5 pp 614ndash616 Jul 1998 doi
1010631121873
[195] A Makishima and J D Mackenzie ldquoDirect calculation of Youngrsquos modulus of glassrdquo J
191
Non Cryst Solids vol 12 no 1 pp 35ndash45 1973 doi 1010160022-3093(73)90053-7
[196] G Pilania C Wang X Jiang S Rajasekaran and R Ramprasad ldquoAccelerating materials
property predictions using machine learningrdquo Sci Rep vol 3 pp 1ndash6 2013 doi
101038srep02810
[197] H He and M Thorpe ldquoElastic properties of glassesrdquo Phys Rev Lett vol 54 no 19 pp
2107ndash2110 1985 Accessed Aug 29 2018 [Online] Available
httpsjournalsapsorgprlpdf101103PhysRevLett542107
[198] M Thorpe ldquoElastic properties of glassesrdquo Cambridge Arch 1985 doi 101557PROC-
61-49
[199] J C Phillips ldquoChemical Bonding Internal Surfaces and the Topology of Non-Crystalline
Solidsrdquo Phys status solidi vol 101 no 2 pp 473ndash479 Oct 1980 doi
101002pssb2221010204
[200] M M Smedskjaer J C Mauro and Y Yue ldquoPrediction of glass hardness using
temperature-dependent constraint theoryrdquo Phys Rev Lett vol 105 no 11 2010 doi
101103PhysRevLett105115503
[201] J D Musgraves J Hu and L Calvez Springer Handbook of Glass Springer US 2021
[202] C Hermansen X Guo R E Youngman J C Mauro M M Smedskjaer and Y Yue
ldquoStructure-topology-property correlations of sodium phosphosilicate glassesrdquo J Chem
Phys vol 143 no 6 p 064510 Aug 2015 doi 10106314928330
[203] L Wondraczek et al ldquoTowards ltrastrong Glassesrdquo Adv Mater vol 23 no 39 pp
4578ndash4586 Oct 2011 doi 101002adma201102795
[204] D Ravaine ldquoGlasses as solid electrolytesrdquo J Non Cryst Solids vol 38ndash39 no PART 1
pp 353ndash358 1980 doi 1010160022-3093(80)90444-5
[205] D Ravaine and J L Souquet ldquoA thermodynamic approach to ionic conductivity in oxide
glasses Part 1 Correlation of the ionic conductivity with the chemical potential of alkali
192
oxide in oxide glassesrdquo Phys Chem Glas vol 18 no 2 pp 27ndash31 1977
[206] J Swenson and L Boumlrjesson ldquoCorrelation between Free Volume and Ionic Conductivity
in Fast Ion Conducting Glassesrdquo Phys Rev Lett vol 77 no 17 pp 3569ndash3572 1996
doi 101103PhysRevLett773569
[207] C B Bragatto A C M Rodrigues and J L Souquet ldquoDissociation Equilibrium and
Charge Carrier Formation in AgI-AgPO3 Glassesrdquo J Phys Chem C vol 121 no 25 pp
13507ndash13514 2017 doi 101021acsjpcc7b02477
[208] S W Martin R Christensen G Olson J ieffer and W Wang ldquoNew Interpretation of
Na Ion Conduction in and the Structures and Properties of Sodium Borosilicate Mixed
Glass Former Glassesrdquo J Phys Chem C vol 140 pp 6343ndash6352 2019 doi
101021acsjpcc8b11735
[209] O L Anderson and D A Stuart ldquoCalculation of Activation Energy of Ionic Conductivity
in Silica Glasses by Classical Methodsrdquo J Am Ceram Soc vol 37 no 12 pp 573ndash580
1954 doi 101111j1151-29161954tb13991x
[210] R S Welch C J Wilkinson J C Mauro and C B Bragatto ldquoCharge Carrier Mobility
of Alkali Silicate Glasses Calculated by Molecular Dynamicsrdquo Front Mater vol 6 p
121 May 2019 doi 103389fmats201900121
[211] Y J Zhang ldquoEntropy and ionic conductivityrdquo Phys A Stat Mech its Appl vol 391 no
19 pp 4470ndash4475 2012 doi 101016jphysa201204021
[212] J C Dyre ldquoOn the mechanism of glass ionic conductivityrdquo J Non Cryst Solids vol 88
no 2ndash3 pp 271ndash280 1986 doi 101016S0022-3093(86)80030-8
[213] M Steve ldquoIonic Conduction in Phosphate Glassesrdquo JAmCeramSoc vol 74 no 8 pp
1767ndash1784 1991 doi 101111j1151-29161991tb07788x
[214] C T Moynihan A J Easteal J Wilder and J Tucker ldquoDependence of the Glass
Transition Temperature on Heating and Cooling Raterdquo J Phys Chem vol 78 no 26 pp
193
2673ndash2677 1974 Accessed Jul 13 2018 [Online] Available
httpspubsacsorgsharingguidelines
[215] Ito C T Moynihan and C A Angell ldquoThermodynamic determination of fragility in
liquids anda fragile-to-strong liquid transition inwaterrdquo Lett to Nat vol 398 no April
p 492 1999
[216] L Ngai and S W Martin ldquoCorrelation between the activation enthalpy and
Kohlrausch exponent for ionic conductivity in oxide glassesrdquo Phys Rev B vol 40 no
15 pp 10550ndash10556 1989 doi 101103PhysRevB4010550
[217] M Potuzak X Guo M M Smedskjaer and J C Mauro ldquoAre the dynamics of a glass
embedded in its elastic propertiesrdquo J Chem Phys vol 138 no 12 2013 doi
10106314730525
[218] A L Agapov and A P Sokolov ldquoDecoupling ionic conductivity from structural
relaxation A way to solid polymer electrolytesrdquo Macromolecules vol 44 no 11 pp
4410ndash4414 2011 doi 101021ma2001096
[219] S V Nemilov ldquoA Structural Investigation of Glasses in the B2O3ndashNa2O System by the
Viscosimetric Methodrdquo Izv Akad Nauk SSSR Neorg Mater vol 2 no 2 pp 349ndash359
1966
[220] J P Poole ldquoLow‐Temperature Viscosity of Alkali Silicate Glassesrdquo J Am Ceram Soc
vol 32 no 7 pp 230ndash233 1949 doi 101111j1151-29161949tb18952x
[221] R noche D B Dingwell F A Seifert and S L Webb ldquoNon-linear properties of
supercooled liquids in the system Na2O SiO2rdquo Chem Geol vol 116 pp 1ndash16 1994 doi
1010160009-2541(94)90154-6
[222] W Martinsen ldquoSelected properties of sodium silicate glasses and their structural
significancerdquo Dissertation 1969
[223] C Hermansen J C Mauro and Y Yue ldquoA model for phosphate glass topology
194
considering the modifying ion sub-networkrdquo J Chem Phys vol 140 no 15 2014 doi
10106314870764
[224] M L F Nascimento and E Dutra Zanotto ldquoDoes viscosity describe the kinetic barrier for
crystal growth from the liquidus to the glass transitionrdquo J Chem Phys vol 133 no 17
p 174701 2010 doi 10106313490793
[225] D Denisko and M M Hoffman ldquoClassification and interaction in random forestsrdquo Proc
Natl Acad Sci U S A vol 115 no 8 pp 1690ndash1692 2018 doi
101073pnas1800256115
[226] G Varney C Dema B E Gul C J Wilkinson and Akgun ldquo se of machine learning
in CARNA proton imagerrdquo in Progress in Biomedical Optics and Imaging - Proceedings
of SPIE 2019 vol 10948 doi 101117122512565
[227] ldquoMDLregSciGlass-6 MDL Information Systems San Leandro CArdquo 2003
[228] D R Cassar ldquoViscNet Neural Network for predicting the fragility index and the
temperature-dependency of viscosityrdquo pp 1ndash33 2020
[229] D J L Brett A Atkinson N P Brandon and S J Skinner ldquoIntermediate temperature
solid oxide fuel cellsrdquo Chem Soc Rev vol 37 no 8 pp 1568ndash1578 2008 doi
101039b612060c
[230] T Ishiyama S Suzuki J Nishii T Yamashita H awazoe and T Omata ldquoProton
conducting tungsten phosphate glass and its application in intermediate temperature fuel
cellsrdquo Solid State Ionics vol 262 pp 856ndash859 2014 doi 101016jssi201310055
[231] T Yamaguchi et al ldquoProton-conducting phosphate glass and its melt exhibiting high
electrical conductivity at intermediate temperaturesrdquo J Mater Chem A vol 6 no 46 pp
23628ndash23637 2018 doi 101039C8TA08162J
VITA
Collin Wilkinson was born in Mt Carroll IL He attended Coe College as an undergraduate and
studied glass resulting in a physics bachelorrsquos degree in 201 He worked with Dr Ugur Akgun
and Dr Steve Feller on modeling different key properties of glass propertiesresponses to external
stimuli Collin joined the group of Dr John Mauro at Pennsylvania State University in 2018
List of publications written by first author (or co-first author) Collin while at Penn State
1 CJ Wilkinson JC Mauro Explorerpy Mapping the Energy Landscapes of Complex
Materials SoftwareX Submitted
2 CJ Wilkinson DR Cassar AV DeCeanne KA Kirchner ME McKenzie ED Zanotto and
JC Mauro Energy Landscape Modeling of Crystal Nucleation NPJ Computational
Materialis Submitted
3 CJ Wilkinson K Doss O Gulbiten DC Allan JC Mauro Fragility and Temperature
Dependence of Stretched Exponential Relaxation in Glass-Forming Systems JACERS
Submitted
4 CJ Wilkinson JC Mauro Comment on ldquoThe Fragility of Alkali Silicate Glass Melts Part
of a niversal Topological Patterrdquo by DL Sidebottom JNCS (2020) ( 2 ) 11
5 CJ Wilkinson K Doss DR Cassar RS Welch CB Bragatto JC Mauro Predicting Ionic
Diffusion in Glass From its Relaxation Behavior Journal of Physical Chemistry B (2020)
124 (6) 1099-1103
6 Z Ding CJ Wilkinson J Zheng Y Lin H Liu J Shen SH Kim Y Yue J Rent CJ Mauro
Q Zheng Topological Understanding of the Mixed Alkaline Earth Effect in Glass JNCS
(2020) (526) 119696
7 CJ Wilkinson AR Potter RS Welch CB Bragatto Q Zheng M Bauchy M Affatigato SA
Feller JC Mauro Topological Origins of Mixed Alkali Effect in Glass Journal of Physical
Chemistry B (2019) 123 (34) 7482
8 CJ Wilkinson K Doss SH Hahn Nathan Keilbart AR Potter NJ Smith I Dabo ACT Van
Duin SH Kim JC Mauro Topological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phase JPCL (2019) 10 (14) 3955
9 CJ Wilkinson K Doss G Palmer JC Mauro The Relativistic Glass Transition A Through
Experiment JNCSX (2019) 100018
10 CJ Wilkinson Q Zheng L Huang JC Mauro Topological Constraint Model for the
Elasticity of Glass-Forming Systems JNCSX (2019) 100019
11 AR Potter CJ Wilkinson SH Kim JC Mauro Effect of Water on Toplogical Constraints
in Silica Glass Scripta Materialia (2019) (160) 48-52
12 CJ Wilkinson E Pakhomenko MR Jesuit AV DeCeanne B Hauke M Packard SA Feller
JC Mauro Topological Constraint Model of Alkali Tellurite Glasses JNCS (2018) (502)
172
13 CJ Wilkinson YZ Mauro JC Mauro RelaxPy Python code for Modeling of Glass
Relaxation Behavior SoftwareX (2018) (7) 255
iv
Table of Contents
List of Figures vi
List of Tables xi
Acknowledgments xii
Chapter 1 The Difficulty of Optimizing Glass 1
11 Energy Landscapes 6 12 Fictive Temperature 8 13 Topological Constraint Theory 15 14 Machine Learning 19 15 Goals of this Dissertation 20
Chapter 2 Software for Enabling the Study of Glass 22
21 ExplorerPy 22 22 RelaxPy 28
Chapter 3 Understanding Nucleation in Liquids 30
31 Crystallization Methods 33 31A Mapping and Classifying the Landscape 35 31B Kinetic Term for CNT 38 31C Degeneracy calculations 41 31D Free Energy Difference 42 31E Interfacial Energy 44
32 Results amp Discussion 46 33 Conclusions 51
Chapter 4 Expanding the Current State of Relaxation 52
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition 52 41A Relativistic Liquid 56 41B Relativistic Observer 60
42 Temperature and Compositional Dependence of the Stretching Exponent 64 42A Deriving a Model 67 42B Experimental Validation 73
43 Conclusion 81
Chapter 5 Glass Kinetics Without Fictive Temperature 82
51 Background of the Adam Gibbs Relationship 82
v
52 Methods 84 53 Results 86
53A Adam-Gibbs Validation 86 53B MYEGA Validation 87 53C Adam-Gibbs and Structural Relaxation 89 53D Landscape Features 91
54 Topography-Property Relations 94 55 Barrier Free Description of Thermodynamics 99 57 Toy Landscapes for the Design of Glasses and Glass Ceramics 102 57 Discussion 107
Chapter 6 Enabling the Prediction of Glass Properties 109
61 Controlling Surface Reactivity 109 62 Elastic Modulus Prediction 125 63 Ionic Conductivity 135 64 Machine Learning Expansion 145
Chapter 7 Designing Green Glasses for the 21st Century 152
71 Glass Electrolytes 152 72 Hydrogen Fuel Cell Glasses 158
Chapter 8 Conclusions 167
References 169
vi
List of Figures
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists
the equilibrium liquid which as it is quenched can either become a super cooled
liquid or crystallize Crystallization causes a discontinuity in the volume The super-
cooled liquid upon further quenching departs from equilibrium and transitions into
the glassy state Reproduced from Fundamentals of Inorganic Glass Science with
permission from the author1 5
Figure 2 The schematic for the flow of the program Beginning in the top right corner
and running until the condition in the pink box is satisfied Yellow diamonds
represent checks and blue operations 25
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above
and then plotted using PyConnect The plot is a disconnectivity graph where each
terminating line represents an inherent structure and tracing where two lines meet
describes the activation barrier The potentials are taken from the BKS potential 94 27
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-
axis being an arbitrary phase space and the y-axis being the potential energy
calculated from the Pedone et al potentials36 The colors are indicative of the
crystallinity where the blue basin is the initial starting configuration The landscape
shows the lowering of energy associated with partially crystallizing the sample (B)
The energy landscape relationship between the cutoff for crystalline and super-
cooled liquid states for 256 atoms is shown This shows a clear drastic energy
change occurring around the cutoff value of 10 Aring 37
Figure 5 An example interfacial structure between the crystalline phase on the left and
the last sequential SCLglass phase on the right for a barium disilicate system The
gray atoms are barium silicon is shown in red and blue represents oxygen 43
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a
function of temperature for each system size The experimental values for kinetics
and thermodynamics come from ref125 and from the heat capacity data taken from 126 respectively It is clear to see that driving force shows good agreement across all
systems however the kinetics terms only converge for the 256 and 512 atoms
systems (Bottom) The fit used to calculate the interfacial energy as a function of
temperature 45
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted
using the model presented in this work The data referenced can be found in refs 113128129 47
Figure 8 The surface energy with respect to temperature for the work presented here 50
Figure 9 An Angell diagram created using the MYEGA expression 55 With a very strong
glass (m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) 17134135
vii
The infinite temperature limit is from the work of Zheng et al54 and the glass
transition temperature is from the Angell definition 54
Figure 10 The relativistic glass transition temperature for B2O3 glass 57
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different
fractions of light speed All of the viscosities approaching the universal temperature
limit for viscosity 59
Figure 12 The modulus needed to satisfy the condition for the glass transition 61
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at
v=044c (Bottom) Viscosity plot showing a dramatic reduction of the viscosity as
the observer approaches the speed of light 63
Figure 14 b predicted and from literature showing good agreement with a total root-
mean-square error of 01 The fit for organic systems is given by 66 ln Kln = minus
and for inorganic systems by 75ln Kln = minus 72
Figure 15 The equilibrium model proposed with the experimental points showing good
agreement between the experimentally measured data points and the equilibrium
derived model RMSE was 002 for Corningcopy JadeTM (A) and less than 001 for
SG80 (B) (C) The model fit for B2O3 experimental data154155 The fragility and glass
transition temperature of the B2O3 are taken from the work of Mauro et al17 75
Figure 16 The stretching exponent calculated as described in the text for a Gaussian
distribution of barriers This plot shows that the distribution of barriers has a large
effect on the stretching exponent A Tg cannot be described since there is no
vibrational frequency included in the model though the glass transition temperature
should be the same for all distributions since the mean relaxation time is the same
for all distributions at all temperatures The deviation is given in ln eV units 78
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent
Each color designates one term in the series (Bottom) The output from RelaxPy
v20 showing the stretching exponent effects on the relaxation prediction of
Corningcopy JadeTM glass Each quadrant shows one property that is of interest for
relaxation experiments In particular it is interesting to see the dynamics of the
stretching exponent during a typical quench 80
Figure 18 The viscosity (left) and landscape (right) predictions for three common
systems The first system is newly calculated in this work while the latter two come
from our previous works7289 It is seen that the viscosity predicted from the AG
model is very accurately able to reproduce the experimental viscosity curves from
the MYEGA model The last system is a potential energy landscape while the others
are enthalpy landscapes 87
Figure 19 The configurational entropy comparisons between the three major viscosity
models which validates the main underlying assumption of the MYEGA model The
viii
VFT and AM are unable to capture the physics of configurational entropy therefore
ruling 89
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the
configurational entropy from the enthalpy landscape with the barrier 00155 eV
(compared to the shear barrier of 00149 eV) and the infinite limit allowed to vary
(10-263 Pamiddots for bulk viscosity) In Figure 18 the configurational entropy is
confirmed for the shear viscosity thus confirming the AG for both shear and bulk
viscosities Sidebottom data are from Ref 154 90
Figure 21 (A) The histogram for the energy minima for each of the test systems showing
a good fit with the log normal distribution This distribution will then be a valid
form to calculate the enthalpy distribution of the model presented in the next
session (B) The configurational entropy from the model showing the accuracy of
the scaling of the entropy predicted by the MYEGA model The S value was fit
for each system 93
Figure 22 (A) Comparison between the randomized method (histogram) and the
deterministic method (vertical lines) showing good agreement between the
maximum in the histogram and the value predicted by the deterministic technique
validating the approach It is worth noting that the 100-basin distribution is a very
wide distribution where the total number of basins is less than the number of points
used in the calculation This is done for a variable number of basins with the number
of basins shown in the legend (B) The dependence of fragility and the glass
transition temperature vs the distribution of states and the number of basins 99
Figure 23 The driving forces for different example glasses calculated using a
combination of the MAP model RelaxPy and the toy landscape model The
parameters for each glass can be found in Table 4 102
Figure 24 The predicted outputs from the toy landscape method showing predictions of
the enthalpy and entropy under a standard quench for barium disilicate This
prediction does not require fictive temperature or any such assumptions about the
evolution of the non-equilibrium behavior 105
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume
in nucleation is assumed to be on the order of one cubic angstrom while the a
parameter in growth is assumed to be around one nm both are in good agreement
for estimates in literature The values for the orange points are taken from these
works128167 107
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the
hydration models Color scheme Si atom (ivory) O atom (red) and Na atom (blue)
The z-axis is elongated to allow space for an insert of water 114
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the
top surface in contact with water is shown here 116
ix
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example
of the electronic-structure DFT calculation using semilocal exchange-correlation
functionals finding the binding energy of a water lsquopixelrsquo to the surface 119
Figure 29 Example contour surface showing the average coordination per atom on the
glass surface for the first run at 300 ps 121
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of
constraints for surface atoms at the local pixel Results show a distinct maximum in
which there is a near hydrophilic-hydrophobic transition of the surface The error
bars represent the standard deviation A second system with 1500 atoms was
performed to show convergence of the ReaxFF MD results 123
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values
of the model predictions are 641 GPa for constraint density 313 GPa for free
energy density and 774 GPa for angular constraint density 128
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from
minimum to maximum error The free energy density model gives the most accurate
results The constraint density has a RMSE of 61 GPa the angular density has a
RMSE of 20 GPa and the energy density has a RMSE of 59 GPa 129
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and
experiment for 10 Na2O 90 B2O3 Using the previously fitted onset temperatures
the only free parameters are then the vibrational frequency and the heating time in
which their product was fitted to be 14000 Where each dip in the modulus
corresponds to a constraint no longer being rigid as heated through each onset The
onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The
data was fit using a least-squares method and the resultant fit is shown as the
calculated method The fit has an R2 of 094 (B) The contribution from each
constraint to the overall modulus 131
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in
Fig 3) and experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of
093 and (B) xLi2Omiddot(100-x)B2O3 glasses with an R2 of 0986 133
Figure 35 The structure for the initial minimum energy configuration showing the boron
(blue) network with interconnecting oxygens (red) and the interstitial sodium ions
(yellow) 139
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and
sodium is ivory The barrier is overestimated compared to experimental data this
could be from several sources of error such as potential fitting thermal history
fluctuations or sampling too few transitions The line is drawn as a guide to the eye 139
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function
of reaction coordinates The middle shows the local deformation around the ion of
x
any atom that moves in between inherent structure mandating a relaxation force
The color shows the degree of deformation 141
Figure 38 Different network formers and the prediction of the activation barrier from our
model compared with activation barriers from literature (A) Sodium silicate
predictions and experimental values241 the error is calculated from the error in the
fragility when fitting the data (B) Lithium phosphate activation energy235 predicted
with topological constraint theory and compared with the experimental values (C)
Predictions over two different systems of alkali borates232 sodium and lithium with
a reported R2 of 097 143
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top
left) shows the distribution of the infinite temperature limit in the database after
limits exerted on the system (Top right) The distribution of the infinite viscosity
limit vs the glass transition (Bottom left) The relationship between fragility and
infinite temperature limit of viscosity (Bottom right) The infinite temperature limit
of viscosity vs the key metric predicted by SR 149
Figure 40 The prediction of the scaling of the activation barriers for a common sodium
borosilicate system 157
Figure 41 The relationship between the glass transition and the proton conductivity This
is justified two ways one through the relationship of the entropy of diffusion and
glass formation (the Adam-Gibbs model) and through the fact that water is known to
depress the glass transition 162
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine
the glass forming region Over top the constraints at the glass transition provided by
each oxide species is listed Since the objective is to decrease Tg while staying in the
glass forming region we will attempt to minimize use of elements that increase the
glass transition (nc gt 17) 166
xi
List of Tables
Table 1 The key properties considered for commercial application The optical properties
have been omitted since it is physical unrealistic to expect quantitative predictions of
quantum-controlled phenomena from a classical description of glass structure 18
Table 2 Variable definitions 65
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy
JadeTM glass and Sylvania Incorporatedrsquos SG80 74
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition
temperature (500 K) were kept the same while the total number of basins were
allowed to vary 101
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K 112
Table 6 System configurations for sodium silicate glass-water reactions 117
Table 7 Fitted values from this analysis compared to those reported in the literature The
disparity between the constraints evaluated with molecular dynamics most likely
come from the speed in which the samples are quenched 132
Table 8 Hyperparameters for different neural networks after hyper-optimizations 151
Table 9 A table with some ionic conductivity models and the parameters needed for
them as well as the disadvantages for each These are not the only models but are
representative of those commonly used in literature 154
Table 10 The predicted compositions based on the optimization scheme proposed 157
Table 11 The compositions synthesized in this work These compositions were predicted
by minimizing the cost function described in Eq (123) OP is the variant that was
melted after OP partially crystallized B-OP appeared to have surface nucleation in
some spots but was cut and removed before APS treatment 165
xii
Acknowledgments
ldquoTo Love Another Person is to See the Face of Godrdquo -Victor Hugo Les Miserables
I have been truly blessed to love and be loved by a plethora of mentors friends colleagues and
family across my academic journey I have to first thank my advisor Dr John Mauro who has
been as kind patient and caring as any mentor I have ever met and in the process has made me a
better scientist and a better person His mentorship was built on top of those who first found and
shaped me into a scientist Dr Ugur Akgun and Dr Steve Feller without whom my scientific career
would not be possible Along with these individuals Irsquove had the pleasure to learn so much about
glass and life from Dr Madoka Ona Dr Firdevs Duru Dr Mario Affatigato Dr Doug Allan Dr
Ozgur Gulbiten and Dr Seong Kim
Rebecca Welch Arron Potter and Anthony DeCeanne are my partners in all things
Rebecca is my partner in research and in life Her opinion support and patience has been
indispensable to me Arron is my oldest colleague and friend he is the one that I share the highs of
research and friendship with Anthony has put up with me non-stop for about 5 years now He is
the most patient and kind friend you could ask for Without these three individuals I could not have
done anything presented in this work I additionally have to thank Karan Doss Aubrey Fry Caio
Bragatto Daniel Cassar Brenna Gorin Mikkel Bodker Greg Palmer Katie Kirchner Kuo-Hao
Lee and Yongjian Yang I have the privilege to call these individuals both my friends and
colleagues
The last people I have to thank are my family I want to thank my brothers (Sam amp Quinn)
my mom and my dad My mom has helped me explore the universe through a love of reading and
her constant unwavering support My dad was the first to show me the wonders of science with
rockets as a kid as a teenager he supported my fledgling interest as he drove hundreds of miles to
every corner of this country and he is the person for whom this dissertation is dedicated
Chapter 1
The Difficulty of Optimizing Glass
Glass is a complex world-changing material Though it has existed for thousands of years
the surface of its true potential is only just being scratched To unlock the potential of glass for new
applications and to enable the maximum benefit to society we must be able to design new glasses
with wide ranging properties faster and cheaper than ever before [1]ndash[5] This is particularly
important now that the need for ecologically responsible materials is increasing due to climate
change In order to facilitate such glass design we must return to first principles and build a picture
of glass from the ground up encompassing both compositional and thermal history dependencies
into our understanding of glass properties[3] [6] The present accepted description of glass is a
non-crystalline non-ergodic non-equilibrium material that appears solid and is continuously
relaxing towards the super-cooled equilibrium liquid state[1] This definition gives insights into the
nature of glass and includes points of particular interest in this dissertation
The first point to consider is that glass is non-crystalline meaning that it contains no long-
range order This gives glass one of its key advantages being infinitely variable Since a glass is
not limited by having to reach stoichiometric crystalline structures there is no limit to the number
of possible glasses or glass structures This gives glass its possibilites Glass has been considered
as a candidate for everything from hydrogen fuel cells[7] to batteries[8] and for applications
ranging from commercial smart phone covers[9] to bioactive applications[10] The infinite
variability also explains the difficulty of designing glasses Consider a system with three
constituents for example SiO2 Na2O and B2O3 This then means there are two independent
compositional dimensions that must be fully explored to find an optimal composition for an
application If this space were discretized to 1 mol spacing that would give over 5000 unique
2
glass candidates that would need to be studied to reach the optimal design A five-component glass
would correspondingly have over 9 million unique glasses More generally for a glass with up to
C components there are 1C minus dimensions over which to optimize If instead one considers a
crystal there is not a continuously variable space but instead only discrete points
The other ramification of glass being non-crystalline is that it must bypass the region of
crystallization at a rate much faster than the rate of nucleation[11] This is inherent in the definition
of glass and is shown in Figure 1 Figure 1 is called the volume-temperature (VT) diagram and is
key to understanding the nature of glass The bypassing of the crystallization region leads to a
super-cooled liquid and during quenching the material departs from equilibrium and enters the
glassy state The region of departure is called the glass transition temperature range or simply the
glass transition To understand why this occurs we must first understand the concept of ergodicity
Ergodicity was defined by Boltzmann to mean that the time-average value of a property is
equal to the ensemble-average value of that property[12] [13] Now that ergodicity is defined we
can start to consider the implications of the non-ergodic nature of glass Since it is non-ergodic the
ensemble average will not equal the time average on human timescales however the ergodic
hypothesis says that in the limit of long time they will be equal[12] In order for both statements to
be true glass must evolve to be ergodic and thus is inherently unstable To explore this concept
further we must understand the glass transition process and define a timescale If there exists a
liquid at some temperature T and is then perturbed to T dT+ there will be some inherent time
associated with its relaxation towards equilibrium which is described by struct the structural
relaxation time There is a separate relaxation time when the system is mechanically perturbed
called the stress relaxation time stress To then understand whether the resulting material is a glass
or a liquid we need to compare struct with the observation time If the observation time is much
longer than the relaxation time then what is observed over the course of a experiment is the
3
equilibrium liquid properties Conversely if the observation time is much shorter than the
relaxation time then the observed properties will not be ergodic and as such we will not sample
the properties of the equilibrium liquid We define this non-ergodic state as the glassy state
However since in the limit of long time the system must become ergodic once again then over time
all glasses will return to an ergodic nature through a process called relaxation [14]
The last point considered in the definition of the glass is that it is non-equilibrium Not only
is glass non-equilibrium it is also unstable and continuously progressing towards the super-cooled
liquid state (eg relaxation) Relaxation is inherent in all glasses and we can see this for a model
system in the VT diagram (Figure 1) In this diagram there are two glasses (of the same
composition) quenched at different rates (both of which bypass crystallization) In the slow-cooled
glass there is a lower volume (higher density) compared to the fast-cooled glass This difference is
due to the fact that the slow-cooled glass has been allowed to relax more fully compared to the
other This adds at least one additional dimension of optimization (such as quench rate from a high
temperature to room temperature) in which glasses must be designed in
This dissertation is dedicated to understanding the infinite variability of glass with respect
to crystallization composition and thermal history with tools being implemented and designed so
that the challenge of optimizing over these vast spaces can be done with higher efficiency lower
cost and with less experimental work load These tools are all inspired by the energy landscape
description of glass To design the optimal glass for any application we must understand the
relationship between the 1C minus compositional dimensions a minimum of one thermal history
dimension and the properties of a material There are many ways to explore these relationships
each with its advantages and disadvantages The methods to optimize over these thermal history
and compositional dimensions include energy landscapes[15] fictive temperature[16] topological
constraint theory[17] and machine learning[3] Of these energy landscapes are the most
fundamental with each other technique being related back to the energy landscape in some form
4
Though they are the most fundamental they are also the most difficult time consuming and have
the largest barrier to entry for use To then build our understanding of each technique we will start
with energy landscapes then explore how through a series of assumptions we can arrive at the other
methods
5
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists the
equilibrium liquid which as it is quenched can either become a super cooled liquid or crystallize
Crystallization causes a discontinuity in the volume The super-cooled liquid upon further
quenching departs from equilibrium and transitions into the glassy state Reproduced from
Fundamentals of Inorganic Glasses with permission from the author[1]
6
11 Energy Landscapes
Energy landscapes were first developed by Goldstein and Stillinger who proposed the
concept as a way to understand the evolution of a material under complex conditions [18]ndash[20]
Since Stillingerrsquos seminal work [18] energy landscapes have become important in the study of
protein folding [15] [21] [22] network glasses [6] [13] and other complex materials The concept
of energy landscapes has become commonplace and is readily evoked when describing the time
evolution of a material [20] [22]ndash[26] Despite energy landscapes being commonplace in some
fields the procedure for mapping a landscape remains difficult due to the associated computational
challenges [15] [27] To map an energy landscape one must find a set of local minima and the first
order saddle point connecting each pairwise combination of those minima Each minimum is called
an inherent structure and represents a structurallychemically stable state of the material A basin
is the set of structures that converge to the same inherent structure upon minimization [18]
In glass science the energy landscape is used to justify complex behaviours which are most
commonly calculated using molecular dynamics simulations (MD) [13] [28] [29] It is easy to
understand why MD is used so frequently used when considering the ease of use of common MD
packages For example LAMMPS [30] is a common user-friendly MD program with a low barrier
to entry allowing individuals with little programming experience to run complex MD simulations
Inspired by this accessibility as well as the explicit Python bridge to LAMMPS we have developed
a software that can be used to explore energy landscapes with a variety of exploration techniques
Similar softwares [15] [31]ndash[35] have been created to map landscapes however each program has
limitations and may not be suitable for all purposes while ours is a general purpose software
The calculation of the energy for landscapes is done using an empirical interatomic
potential that can be fit using either experimental data and a series of MD simulations or using ab
initio methods One common potential form is the Lennard-Jones form (LJ)
7
12 6
4LJVr r
= minus
(1)
in which and are fitting parameters for every pair-wise set of atoms in a system V is the
potential energy contributed by the two-atom interactions and r is the distance between the two
ions in question Though the LJ form is simple and easy to parameterize it is unable to recreate
complicated material behaviours for a real glass As such a more complicated potential is often
used such as a Morse potential A commonly used silicate glass potential is the Pedone potential
which consists of Morse potential with an additional columbic term and repulsive term [36] and is
given by
( )2
2
0 121 exp 1
i j
Pedone
Z Z e CV Y a r r
r r = + minus minus minus minus +
(2)
Z is the charge of a given ion e is the charge of an electron and Y C a and r0 are fitting parameters
It is also worth noting that the Pedone et al potentials are only accurate when considering the
interactions between each individual ion species with oxygen For instance Si-O Na-O and O-O
are all explicitly parameterized while Si-Na only interacts coulombically (the first term)
Interatomic potentials are discussed here only briefly but there are extensive resources available
for those readers who wish to learn more[37]ndash[39]
Once an energy landscape has been mapped the thermodynamics and kinetics for a given
system can be easily calculated [33] [40] The thermodynamic driving force is given by the
difference between the free energy of the current inherent structure and the free energy of the lowest
state while the kinetics is the rate in which the system crosses the barrier between the basins This
can be calculated explicitly for arbitrary timescales and temperatures with another software
KineticPy [41] The energy landscape provides insights into the detailed nature of material
behaviour Though energy landscapes are very powerful they are also difficult to map To
approximate the behavior of the landscape then we use approximations such as fictive temperature
8
topological constraint theory (TCT) or machine learning Fictive temperature is a way to estimate
the effects of the thermal history dimension[42] while TCT is used to estimate the compositional
dependence of properties[43]
12 Fictive Temperature
Fictive temperature is an approximation used to find the change in the occupational
probabilities on an energy landscape[16] [44] Instead of considering every possible combination
of occupational probabilities like the energy landscape it instead approximates the occupational
probability using a single temperature at which the landscape would be in equilibrium It is
essentially a measure of how closefar a glass is from equilibrium This approximation allows for
us to calculate the relaxation dynamics of a glass without needing any computationally heavy
energy landscapes However this single parameter description oversimplifies relaxation[14] [16]
The functional form for glass relaxation dates back to 1854 when Kohlrausch fit the
residual charge of a Leyden jar with[45]
( ) (0)expt
g t g
= minus
(3)
where t is time τ is the relaxation time constant β is the stretching exponent and g(t) is the
relaxation function of some property (eg volume conductivity viscosity etc) Eq (3) is referred
to as the stretched exponential relaxation function (SER) The stretching exponent is bounded from
0 lt β le 1 where the upper limit (β = 1) represents a simple exponential decay and fractional values
of β represent stretched exponential decay
For over a century the physical origin of the stretched exponential relaxation form was one
of the greatest unsolved problems in physics until 1982 when Grassberger and Proccacia[46] were
able to derive the general form based on a model of randomly distributed traps that annihilate
9
excitations and the diffusion of the excitations through a network This model however provided
no physical meaning for β which was still treated as an empirical fitting parameter In 1994
Phillips[47] [48] was able to extend the diffusion trap model by showing that β=1 at high
temperatures and at low temperatures the stretching exponent can be derived based on the effective
dimensionality of available relaxation pathways He in turn expressed the stretching exponent
2
df
df =
+ (4)
where d is the dimensionality of the network and f is the fraction of relaxation pathways available
He then proposed a set of lsquomagicrsquo numbers for common scenarios Assuming a three-dimensional
network with all pathways activated (d = 3f = 1) a value of β= 35 is obtained The second case is
a three dimensional network with only half of the relaxation pathways activated (d =3f = 12)
yielding a value of β= 37 A third magic value β= 12 was found for a two dimensional technique
with all pathways activated (d = 2f = 1) β = 35 occurs for stress relaxation of glasses under a load
because both long- and short-range activation pathways are activated whereas a value of β = 37 is
obtained for structural relaxation of a glass without an applied stress The value was confirmed by
Welch et al[9] when the value was measured over a period of 15 years at more than 600C below
the glass transition temperature of Corning Gorilla Glass Though this model was able to reproduce
the limiting values it was criticized widely in the community Critics argued that the model was
simply created such that it reproduced the stretching exponent values of certain experiments and
ignored many other experimental results that appeared to disagree with the model The model also
fails to interpolate between the low temperature values Phillips predicted and the high temperature
limit
The work from Phillips and the stretched exponent gave a form for relaxation however
there remained no method to instantaneously state the distance from equilibrium of a glass To
account for this problem an additional thermodynamic variable (or order parameter) called fictive
10
temperature (Tf) was proposed Early research on relaxation in glasses dates back to the work of
Tool and Eichlin in 1932[49] and Tool in 1946[50] whose works originally proposed a temperature
at which a glass system could be in equilibrium without any atomic rearrangement Tool suggested
that this fictive temperature was sufficient to understand the thermodynamics of a glassy system
Originally the fictive temperature was treated as a single value that was some function of thermal
history (Tf[T(t)]) The evolution of fictive temperature as proposed by Tool is then given by
)
( ) ( ( ))
(
f f
f
dT T t T T t
dt T T
minus= (5)
In which for stress relaxation is given by
( )(( ) ( ))fT t T
G
T t = (6)
In Eq (6) is the shear viscosity and G is the shear modulus This means that to predict the time
evolution of a glass only three things are required shear modulus the viscosity as a function of
temperature and fictive temperature and the stretching exponent Though fictive temperature
qualitatively reproduced the results they were looking for subsequent experiments have shown that
the concept of a single fictive temperature is inadequate[51] [52]
A key experiment was performed by Ritland in 1956[51] Ritland took several samples
with different thermal histories but the same measured fictive temperature thus based on Toolrsquos
equation both should have had identical relaxation properties Ritland however showed that the
refractive index evolved differently between samples To account for these differences
the use of multiple fictive temperatures was suggested[52] This worked because the stretched
exponential form could be approximated with a Prony series
( ) exp expi i
N
i
Kt w K t minus minus
(7)
11
In the Prony series N is the number of terms while iw and iK are fitting parameters This means
that each term could represent one fictive temperature and this collection of fictive temperatures
would reproduce the overall experimentally observed form Ritlandrsquos experiment showed that
materials held isothermally at their fictive temperatures with varying thermal histories can give
different results for the relaxation of a given property Thus as a glass network relaxes the fictive
temperature will also shift Narayanaswamy[52] was able to apply Ritlandrsquos concept in a highly
successful engineering model with multiple fictive temperatures but the physical meaning of
multiple fictive temperatures remains elusive
In general there is reason to be skeptical of the concept of fictive temperature Fictive
temperature was not derived but instead was an empirical approximation that Tool needed to gain
an early quantitative understanding Recently Mauro et al[16] used energy landscapes to try to
understand the underlying validity of fictive temperature fT implies that the occupational
probability ( ip ) is given by
1
exp ii
Hp
Q kT
= minus
(8)
Q is the partition function k is Boltzmannrsquos constant and iH is the enthalpy of the given state
However when the actual probability evolution predicted by fictive temperature was compared to
the probability calculated through a systematic Monte Carlo approach the results were found to be
drastically different (even when many fictive temperatures were tried) This implied that fictive
temperature is insufficient to describe the glassy state and must be replaced with a new method
Despite the inadequacies of fictive temperature state-of-the-art modeling techniques for
glass relaxation still relies on the concept In these techniques a set of fictive temperatures (usually
12) are used to calculate 12 parallel relaxing fictive temperatures[42] This is because there has
been no alternative method able to reach a prediction of the evolution of glass under different
12
thermal histories The current state-of-the-art model is called the Mauro-Allan-Potuzak (MAP)[44]
model of non-equilibrium viscosity and was derived from a more fundamental expression for
viscosity called the Adam-Gibbs model (AG) which is given by[53]
10 10) log( )
log ( f
c f
BT T
TS T T = + (9)
B is the barrier for a cooperative rearrangement 10log is the infinite temperature limit of
viscosity and according to Zheng et al[54] it is approximately 10-293 Pa s and cS is the
configurational Adam-Gibbs entropy This entropy has long since been a source of debate and will
be addressed later in chapter 5
In order to calculate the equilibrium part of the viscosity the MAP model relies on an earlier
Mauro Yue Ellison Gupta and Allan (MYEGA) equilibrium viscosity model[55] This model
was derived based on earlier work by Naumis[23] [56] Gupta and Mauro[57] and the AG model
These models showed that the configurational entropy can be related to the degrees of freedom
lncS fNk= (10)
f is the degrees of freedom per atom N is the number of atoms and is the number of degenerate
states per floppy mode To then build in temperature dependence they assumed that the degrees of
freedom can be modeled using an Arrhenius form (with activation barrier H )
( ) expH
f T dkT
= minus
(11)
When Eqs (9) (10) and (11) are combined into an equilibrium form the equilibrium viscosity is
then given by
10 10 el pog og xlC H
T kT
+
=
(12)
Where C is equivalent to
13
ln
BC
dkN=
(13)
Through a change in variables this can be written in terms of three common variables that allow
for all viscosity equations to be written with three variables specified by Angel[58] the glass
transition temperature the fragility and 10log The glass transition temperature is defined as
the isokom where the liquid has a viscosity of 1210 Pa s The fragility is defined as[58]
10log
g
g
T T
dm
Td
T
=
(14)
The three most common viscosity models (MYEGA Vogel-Fulcher-Tammann (VFT)[59] and
Avramov-Milchev (AM)[60]) are listed below in terms of the same parameters
10 10 10
10
log (12 log ) exp 1 11 o
log2 l g
g gT Tm
T T
+ minus minus minus
minus =
(15)
( )
2
10
10 10
10
12 logl log
1 1
og
2 logg
Tm
T
minus+
minus + minus
=
(16)
and
( )10(12 log
10 10 1
)
0l gog l 12 logo
m
gT
T
minus
+ minus=
(17)
To incorporate non-equilibrium effects with these equilibrium models the MAP model
proposes
10 10 10log ( ) (1 ) log ( )log MYEGA g ne f gT m T mx x T T T = + minus (18)
14
10log ne is the non-equilibrium viscosity whose form and compositional dependence can be found
elsewhere[42] [44] [61] x is the ergodicity parameter and is of crucial importance to
understanding glass regardless of the model being used[12] In the MAP model it is given by
( )( )
3244min
max
f
f
m
T Tx
T T
=
(19)
Though the MAP model was derived from the best understanding of energy landscapes of glasses
at the time it has a few flaws that prevent it from perfectly predicting the different relaxation
behaviors of glass A few of the short-comings are
1 The method ignores crystallization For a comprehensive model showing the response of
glass to temperature the possibility of crystallization must be included Currently no
relaxation model includes this possibility
2 It is built around the concept of fictive temperature This could be replaced in a future
model but currently at present it carries with it all the failures of the fictive temperature
picture of glass dynamics adding only the relaxation time coming from landscape-derived
values
3 The MAP model does not consider temperature dependence of the stretching exponent
This is typically circumvented by choosing a fixed stretching exponent
4 The parameters needed for the non-equilibrium viscosity in the MAP model are
experimentally challenging to measure To circumvent this Guo et al[61] published
methods to estimate the values based on fragility and the glass transition but the
approximations were only tested on a few samples with similar fragilities and chemistries
5 The MAP model only predicts stress relaxation Since the publication of the MAP model
it has been shown that only stress relaxation time is related to the shear viscosity[62]
Structural relaxation must be incorporated in future models
15
For a complete understanding of the thermal-history dependence of glass all of these questions
must be rigorously answered
13 Topological Constraint Theory
TCT is one way to estimate the compositional dependence of glasses It uses a set of
assumptions that connects the underlying structure to glass properties through understanding of the
energy landscape These assumptions include the temperature dependence of the rigidity the same
assumptions used in the derivation of the MYEGA models and that the effects of thermal history
on structure are minimal TCT was originally proposed by Gupta and Cooper (GCTCT) to try to
understand the role of rigid polytopes on glasses properties[63] Gupta and Cooper fundamentally
understood that the ability for glasses to form was reliant on the transformation of a liquid with
many degrees of freedom to a glass with a fixed degree of freedom Since the structure of the glass
is the same as the structure of the liquid at the glass transition and the degrees of freedom of the
glass ( f ) changes during the transition to glass it was possible to gain insights into the
transformation process This approach is fundamentally an approximation of the highly
multidimensional landscape where only approximate structural information is used to understand
the entropy of the energy landscape Though the core idea was useful it was ldquoclunkyrdquo and difficult
to make calculations with
To improve upon the ideas of GCTCT Phillips and Thorpe (PTTCT)[64] proposed a
mathematically equivalent form where the degrees of freedom could be simply found by
considering the number of constraints ( cn ) around each network-forming atom in the glass They
showed that the degrees of freedom were related to the number of constraints by
cf d n= minus (20)
16
The number of constraints could then be calculated by considering the two-body and three-body
interactions (first and second terms) as a function of the mean coordination of the network forming
species
2 32
c
rn r= + minus (21)
This led to a qualitative language to discuss glass with a few quantitative insights The biggest
quantitative insight comes in the form of the lsquoBoolchand Intermediate Phasersquo[65]ndash[68] If it has
positive degrees of freedom then a network is lsquofloppyrsquo and retains high kinetic energy (as it flips
back and forth between two inherent structures) If the system has negative degrees of freedom
then the network is stressed-rigid and there is additional energy stored in the stress associated with
over-coordination If the number of constraints is equal to the number of dimensions then the
resulting system will have zero degrees of freedom and there is no stored stressed energy or kinetic
energy from flipping between basins Boolchand went searching for this perfectly lsquoisostaticrsquo glass
and found a non-zero window of compositions that were close to isostatic but had properties
drastically different compared to both stress rigid and floppy glasses This is now called the
Boolchand Intermediate Phase
Despite the success in the prediction of the intermediate phase TCT remained largely
qualitative until the work by Naumis[23] [56] and Gupta amp Mauro[57] Naumis pointed out that
the degrees of freedom are a lsquovalleyrsquo of continuous low energy so that the structure can freely move
between basins of the landscape Since these valleys are deformation pathways Naumis pointed
out that they will dominate the configurational entropy of the system Building on this Mauro and
Gupta wrote that since the entropy is proportional (Eq (10)) we could then write the glass transition
temperature as a function of the degrees of freedom and the AG model
( )10o1 ln2 l g
gTB
f Nkminus=
(22)
17
Since everything in Eq (22) is approximately a constant between compositions except for f we
could then write the expressions in terms of reference values (subscript r) and accurately scale the
glass transition of new compositions by knowing the changing structures
( ) g r c r
g
c
T d nT
d n
minus=
minus (23)
The last insight provided by Mauro and Gupta was that the number of constraints change as a
function of temperature They realized that each constraint has a temperature dependence which is
controlled by the interaction energy of the constraints This makes intuitive sense because as the
thermal energy becomes greater than that of the barriers additional valleys in the energy landscape
form This also allowed them to expanded TCT to include quantitative predictions of fragility
The understanding of the relationships between the configurational entropy the number of
constraints the degrees of freedom and the viscous properties lead to a revolution in understanding
glass properties Soon after the work by Mauro amp Gupta multiple works came out predicting
mechanical chemical and viscous properties of glass knowing only the underlying structure
Though this framework is powerful it is not a complete description without further
parameterizations with each model requiring some fitting parameters or reference values In
addition there are still key properties that are missing from the standard topological approach that
are needed to design commercial glasses
Ultimately MGTCT needs be expanded to include the properties that are needed for
commercial applications A list of composition dependent properties needed for understanding the
performance of commercial glasses are shown in Table 1 Properties taken from Ref [3] show what
properties must be controlled to manufacture technological glasses All models take additional
inputs and there are multiple ways to link outputs from some models to inputs of the other Choice
of model vary depending on the use case
18
Table 1 The key properties considered for commercial application The optical properties have
been omitted since it is physical unrealistic to expect quantitative predictions of quantum-
controlled phenomena from a classical description of glass structure
Properties TCT Machine Learning Other methods
Glass Transition [57] [69] [70] Energy Landscapes [71]
Fragility [57] [69] This work This work
Relaxation MAP model [44] [61]
Crystallization This work [72]
MDMC [73]ndash[75]
Melting Temperature This work MDThis work[72]
Chemical Durability [76] [77]
This work [78]
[79] [80] Linear Methods [81]
Density [82] MD Packing [83]
CTE This work MD
Hardness [5] [84] [85] MD
Youngrsquos Modulus [86] This
work [87]
[82] This work MD
Activation Barrier for Conductivity MD This work [88]
Batch Cost Economic Calculation
Ion Exchange Properties Empirical Relationships [2]
19
14 Machine Learning
All the methods discussed so far are physically informed techniques The techniques
presented are inherently powerful because their very nature is informed by reality Unfortunately
this means that some knowledge about the system is required These techniques are useless without
the required parameterizations However a class of algorithms called lsquoMachine Learningrsquo (ML)
require no such parameterization Instead ML uses large quantities of data to determine trends
linking any physically connected input to output allowing for accurate predictions of properties
across wide compositional spaces This is favorable with a large of volume data because the
required inputs are minimal to achieve a prediction The difficulty (and often expense) in this
technique is gathering large quantities of high-quality data to enable this method
The predominant ML method used in this work is neural networks (NN) Neural networks
work by constructing a network of neurons This method was designed in such a way to mimic a
human brain so the network is elastic enough to be modified improving results Each neuron takes
in a series of inputs received from the previous layer (or input data) multiplies by a weighting
factor sums the values and then performs some lsquoactivation functionrsquo on them By changing the
weights it is then possible to recreate any function given sufficient data This is a preferred
technique because it packages into a simple easy-to-use software Neural Networks (once trained)
can be used by anyone stored in a small file can recreate any function is computationally cheap
and does not require as much data as some other ML techniques
This is an interesting method to estimate the effects of the composition on the landscape
and corresponding properties because we cannot understand the underlying mathematics It is
fundamentally a numerical process Thus even though we know that the properties of a glass are
dominated by the topography of the landscape how the NN figures out the relationship between
the chemistry and properties is not understood It is possible to then use the NN to give us the
20
physical parameters we need for other models building multiple methods together to achieve new
insights
15 Goals of this Dissertation
To enable the next generation of glass design we must enable a new era of computationally
methods being used to produce accurate predictions of the behavior of glass This includes both the
thermal history dependence of glass (in terms of relaxation and crystallization) and the
compositional dependence especially those properties listed in Table 1 The goals of this thesis will
be to address some of these outstanding questions in a systematic way that is practical for an
external user to understand and implement Predicting thermal history effects are covered in
Chapters 3-5 while compositional effects are predicted in Chapters 6-7 The breakdown of each
following Chapter as it relates to these questions is listed below
21
bull Chapter 2 Software Unfortunately there is no standardized software for the
calculation of relaxation or energy landscapes I present two new codes to
standardize these calculations so that the process is reproduceable and standardized
bull Chapter 3 Crystallization This section will address new computational ways to
calculate crystallization focusing on predicting nucleation rates and how to improve
our nucleation calculations
bull Chapter 4 Expanding the Current State of Relaxation This chapter will focus on
expanding the MAP model with a temperature dependent form of the stretching
exponent as well as understanding the role of ergodicity in glass relaxation
bull Chapter 5 Relaxation and Crystallization without Fictive Temperature Chapter 5
will focus on discussing relaxation and crystallization without fictive temperature
by proposing a new relaxation model that is reliant on creating lsquoToy Landscapesrsquo
This approach removes assumptions about the evolution of the occupational
probabilities
bull Chapter 6 Enabling Prediction of Properties This chapter will use TCT and ML to
address the compositional dependance of different key properties needed for
enabling the faster design of new glass compositions
bull Chapter 7 Designing lsquoGreenrsquo Glasses Applying information learned throughout
this dissertation to create new glass compositions
bull Chapter 8 Conclusions
22
Chapter 2
Software for Enabling the Study of Glass
To being our systematic investigation it is important to have a suite of tools that
implement current state-of-the-art models Though many tools exist for structure and
topological constraint theory there were no codes that existed for the calculation of
relaxation using the MAP model and there were no codes that were capable of exploring
the landscapes that we are interested in To correct this we present two new novel codes
ExplorerPy for mapping energy landscapes and RelaxPy for calculating MAP
dynamics[42] [89]
21 ExplorerPy
Mapping an energy landscape allows for modeling a range of traditionally inaccessible
processes such as glass relaxation The advantage of this software (over other codes to map energy
landscapes) is that it only requires knowledge of LAMMPS to run Also our software contains a
choice of three methods to map the landscape The first of the three different methods implemented
to explore energy landscapes in this work is eigenvector following [90] Eigenvector following has
been used in the past to generate landscapes that predict the glass transition and protein folding
pathways [44] [91] This implementation follows the work of Mauro et al [90] [92] and uses an
independent Lagrange multiplier for every eigenvector Eigenvector following excels at finding
small barriers quickly however it rarely finds unique structural changes in complex materials
(especially in systems with periodic boundary conditions) The second method that we have
implemented following in the steps of Niblett et al [27] is a combination of molecular dynamics
and nudged elastic band methods (NEB) [34] [35] To find the transition points MD is run at a
23
fixed temperature for a fixed amount of time the structure is minimized and then NEB is used to
find the saddle point between the initial structure and the new inherent structure Details of the
nudged elastic band method can be found elsewhere [34] Using these techniques it is possible to
map a sufficiently large landscape to make kinetic and thermodynamic measurements possible
These two methods benefit from the fact that they work well when the volume is varied and together
enable computationally inexpensive mapping of enthalpy landscapes [71] [92] (where the volume
is no longer fixed and instead the pressure is specified) However it is not recommended to generate
a large enthalpy landscape from MD in this software due to the computational cost involved
Although molecular dynamics is a powerful technique there is a large computational cost to
simulate a small amount of time
The third method is a new technique created in an attempt to circumvent the computational
cost of MD while still finding large structural changes We have named this technique ldquoeigen
treibanrdquo (ET) This technique relies on shoving along an eigenvector with a multiplier rather than
stepping After the system is shoved the system is minimized and if the minimized structure exhibits
a displacement in atomic position is larger than a threshold a NEB calculation is performed to find
the saddle point By shoving in the direction of an eigenvector the goal is to reproduce large changes
due to vibrational modes The fixed multiplier (m0) is calculated by taking a target distance (d) and
the largest displacement magnitude out of the atoms j (vij) in a given eigenvector direction i (ai)
m0
=d
max[nij] (24)
To convert to the displacement magnitude from an eigenvector in three dimensions we use the
expression
vij
= ai[3 j]2 +a
i[3 j +1]2 + a
i[3 j + 2]2 (25)
The dimension j has the length of the number of atoms in the systems
24
This software uses the system and potentials created from a LAMMPS input
scripts and as such has a wide range of access to different potentials and possible systems The
only input that has to be provided is the file which would be used for molecular dynamics in
LAMMPS If the script has been used for molecular dynamics simply by eliminating any
thermostats or barostats as well as any ldquorunrdquo commands it is possible to use it as an input script to
ExplorerPy (it is also recommended to eliminate any calls to ldquoThermordquo because it may interfere
with the extraction of variables from the underlying LAMMPS framework) If the LAMMPS input
script successfully sets up a simulation all other variables and parameters can be controlled from
command line options as described in the ldquoIllustrative Examplerdquo section The initial structure will
be taken from the input script as well
ExplorerPy is a software that has been written in Python with modular functions and ease
of use in mind It can be run from a Linux terminal with all major options being set via the command
line The software then generates a list of inherent structure energies stored in the mindat file and
stores the transition point energy for each pairwise inherent structure in the tsddat file All of the
transition points and minimum are stored in the same directory from which it is run as xyz files
The software architecture is shown in Figure 2 Details of the stepping procedure for eigenvector
following can be found elsewhere[93] The variables ldquoMD frequencyrdquo and ldquoET frequencyrdquo are set
as command line options at the time of the launch of the software
25
Figure 2 The schematic for the flow of the program Beginning in the top right corner and running
until the condition in the pink box is satisfied Yellow diamonds represent checks and blue
operations
The software is capable of mapping an energy landscape for an arbitrary system ie for
any system that can be simulated in LAMMPS This allows for accurate prediction of the evolution
of complex thermodynamics and kinetics The software requires no programming and is
controllable via command line options ExplorerPy prints all of the basins transition states inherent
volumes as well as information about the curvature of the system so that the vibrational frequency
is calculable This is all of the information required to calculate a complex range of phenomena
over a specified long timescale ExplorerPy is only set up to work within the LAMMPS metal units
To run the software there are only a few parameters that have to be set for each mode The
force threshold (-force) curvature threshold (-curve) and step size (-stepsize) when running the
command line option have to be set to allow for a stable exploration of a given landscape
These seem to be potential-specific and for a given potential appear to be fairly stable It is best to
start large for the force threshold and small for the eigen threshold and step size then modify as
26
necessary If eigenvector following is not being used there is no need to adjust these values Beyond
this there are several optional command line options (only the file (-file) command is required)
-file [file input] this is a standard lammps input if the input sets up a lammps run
it will work to set this up There should not be any lsquothermorsquo calls in the lammps
script
-press [specified pressure in atmospheres] if this is specified the system will
always be minimized to this pressure while if not specified the volume remains
fixed
-num [number of basins to explore]
-time [max time to run in hours] if this time is reached before the number of basins
are found the simulation will stop
-thresh [minimum energy for barrier to keep]
-center if this is given as a flag the system will be set to the anchor location each
step
-anchor [atom selection for the position to be held constant]
-md [frequency of molecular dynamics search] If a custom MD LAMMPS file is
not provided it defaults to starting all of the atoms at 3000 K and running for 50
fs
-shove [frequency of eigen treiban]
-custom [file used for MD exploration]
-help [List all commands possible in software with these descriptions]
An example run would then be done the following way
[mpirun -np 5 python37 Explorerpy -file inSiO2 -num 1000 -force 10 -curve 1e-5 -stepsize 012
-press 00 -md 11 -time 48]
This would start a run with five parallel processors searching for 1000 basins with an MD
step being ran every 11 searches with a maximum time of 48 hours The results of this command
are shown in Figure 3 Below we have attached an example of inSiO2 the output and additional
scripts related to the example has been uploaded to the associated GitHub
27
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above and then
plotted using PyConnect The plot is a disconnectivity graph where each terminating line
represents an inherent structure and tracing where two lines meet describes the activation barrier
The potentials are taken from the BKS potential [94]
httpsgithubcomlsmeetonpyconnect
28
22 RelaxPy
RelaxPy script is written using Python syntax and relies on the numpy scipy and
matplotlib libraries It solves the lack of a unified relaxation code in our community and is based
on the MAP model[44] [61] interpretation of multiple fictive temperatures and the shear
modulusviscosity form of the Maxwell equation[62] [95] It is designed to be run within a Linux
terminal with command line options or inside a Python interpreter It can be used by calling the
name of the script followed by the input file the desired output file name and finally an optional
tag for all fictive components to be displayed as well as printed to a specified output file (formatted
as comma separated values)
The RelaxPy package consists of an algorithm that iterates over time in order to determine
the values of viscosity (η) relaxation time (τ ) and current fictive temperature (Tf ) given the user-
supplied thermal history and material property values The purpose of this code is to enable
calculation of the relaxation behaviorrsquos change over time by using the MAP model using only
experimentally-attainable parameters Toolrsquos equation is solved for each term in a Prony series
approximation of relaxation finding the change in each Tfi component for a given time (t) to
accurately reproduce the stretched exponential form The program then iterates over the entire
thermal history of the sample using a user-specified time step dt allowing for calculation of the
overall fictive temperature over time The values for the Prony series fit (wi and Ki) are taken from
the database originally generated by Mauro and Mauro[96] and expnded to include any value of
the stretching exponent The values were fit by Mauro and Mauro with a hybrid fitting method
based on the number of terms in the desired Prony series and the magic value from Phillips The
thermal history is also defined in the original input file using linear interpolation The initial fictive
temperature components are assumed to be at equilibrium (Tfi = T(0))
29
To solve for the non-equilibrium viscosity used the MAP expression is used with the approximate
scaling of values proposed by Guo et al[61] The software is capable of performing the calculations
needed for a variety of thermal histories and processing methods desired by both industry and
academia with a focus on the fictive temperature This software is then fundamentally limited by
the issues associated with the MAP model and the compositional approximations made For
instance this software will need to be further modified to include structural relaxation when such
a model is proposed[62]
An example input code is listed below
Denotes Comments
Tg 794 0 m 368 C 1494 dt 10 Beta 37
Start Temp End Temp Time [s]
1000 10 1300
10 100 1000
100 30 500
Assumes Linear Interpolation
This example input states that over the space of 1300 seconds an initial glass sample is cooled from
1000 C to 10 C Then it is heated to 100C over 1000 seconds and then finally down to room
temperature in the space of 500 seconds All temperatures are given in Celsius
This software aims to provide the first open-source software for modeling of glass
relaxation behavior using the MAP model of viscosity by creating a package that can easily and
quickly be used to approximate the fictive temperature components of a glass as well as the
composite fictive temperature RelaxPy allows for researchers to calculate the evolution of
macroscopic properties of glass during a relaxation process (assuming a fixed stretching exponent)
and will facilitate the calculation based on a user-supplied thermal history RelaxPy can be
seamlessly adopted by most glass research groups and removes the guessing associated with
determining a fictive temperature allowing for a fully quantitative comparable discussion of Tf
simultaneously creating a tool that can easily be expanded for the entire community as new physics
is discovered
30
Chapter 3
Understanding Nucleation in Liquids
Crystallization cannot be ignored in any material system at temperatures below its liquidus
temperature although extreme resistance to crystallization is reported in some organic and
inorganic substances[97] Crystallization (and by extension nucleation) is critical in systems
ranging from ice to thin films[98]ndash[100] Glasses are non-crystalline substances therefore
crystallization kinetics will determine the necessary thermal history to avoid crystallization and
obtain a glassy state Despite the absolute importance of crystallization to the study and design of
glasses there is no current method to predict the nucleation of crystals in this event One of the main
challenges in the production of glass-ceramics is finding the optimal thermal history for the desired
crystallite distributions There are two phenomenological steps in the process of phase
transformation from a supercooled liquid to a crystal a nucleation step and a growth step The
growth step is described well by Wilson-Frenkelacutes theory[1] but the underlying physics of
nucleation remains elusive A stable nucleus is the precursor to a crystal comprising a nano-sized
periodic assemblage of atoms A clear understanding of the relationship between liquid kinetics
and crystal nucleation has not been established despite decades of research
The study of nucleation is particularly critical for glass-ceramics Since their discovery by
S Donald Stookey glass-ceramics have become prevalent in our society through products such as
stovetops and Corningware[73] [101] [102] Glass-ceramics are a composite material comprising
at least one crystalline phase embedded in a parent glassy phase benefitting from the properties of
both the glassy and crystalline phases[1] [103] One of the reasons glass-ceramics are ubiquitous
is their relatively easy production glass-ceramics can be formed through standard glass forming
procedures using additional heat treatments Additionally there is a wide range of accessible
31
properties through tuning both the chemical composition of the parent glass and the thermal history
and hence the microstructure of the heat-treated glass-ceramics[101] [103]ndash[106]
Glass-ceramic production consists of first synthesizing a glass and then generating a
microstructure through the two-stage ceramming process[73] [103] [107] The nucleation step
remains poorly understood with classical nucleation theory (CNT) giving widely varying results
depending on how the parameters are determined[1] [73] [108] The inability to consistently get
accurate predictions for a nucleation curve means that choosing an appropriate nucleation
temperature must be done experimentally which can prove to be one of the most challenging and
time-consuming aspects of properly designing a glass-ceramic The difficulty is due to the
magnitudes of the crystal nucleation rates not being predictable with CNT or any other current
model
This difficulty in designing a material is apparent when considering the high dimensional
phase space of parameters that can adjusted We haver presesnted some tools developed to help
optimize over this lsquoglassyrsquo space such as RelaxPy[42] KineticPy[41] topological constraint
theory[109] or various machine learning models[4] [70] [110] [111] However a glass-ceramic
system has at least four additional phase dimensions (two for the nucleationgrowth temperatures
and two for nucleationgrowth times) that need to be optimized (depending on the parameterization
of the experiment it is also possible to add another four dimensions relating the heating and cooling
between each step) Therefore a glass-ceramic system has a minimum of d+5 dimensions over
which the system must be optimized and yet those four additional dimensions are the least
understood Designing methods to easily optimize over this large space with feasible computational
methods is of utmost importance to reaching a new generation of custom materials
Experimental studies of nucleation have used techniques ranging from differential
scanning calorimetry (DSC) to electron microscopy to try to capture predictive capability in the
lsquoceramicrsquo phase-space (the four additional dimensions) These studies have successfully captured
32
the shape of the nucleation curve but fail to provide the information needed to calculate the
temperature-dependence of the nucleation rate The shape of the curve can be determined through
DSC and with great effort an approximation of the rate can be made however this technique is too
laborusome to be efficient[112]ndash[114] Since the parameters have been inaccessible to traditional
computational approaches and experimental determination all previous studies of nucleation have
required at least one variable to be fit If a nucleation curve could be predicted only using a
computational method or with an efficient simple DSC method the rate at which new glass-
ceramics products are created would increase In addition the cost of producing glass-ceramics
would likely decrease and more complex microstructures and therefore more unique properties
could be explored
Crystal nucleation has been historically difficult to simulate and measure experimentally
because the phenomena happen on spatial scales of nanometers but over many orders of magnitude
of time scales CNT is the most widely applied model despite its supposed failure To test CNTrsquos
assumptions an interest in computational work has grown Recent approaches using grand
canonical Monte Carlo have successfully predicted the short time dynamics of nuclei
formation[73] This approach involves use of a continuous solvation model for the liquid and has
enabled greater computational insights however it remains computationally expensive[74]
Molecular dynamics (MD) fails to capture nucleation in glass-forming systems due to the need for
longer spatial and temporal scales to capture nucleation effects accurately Some attempts have
been made using Monte Carlo MD or empirical approaches to replace classical nucleation theory
(CNT) with some alternate method where the parameters can be directly measured[73] [115]ndash
[117] All nucleation models and Monte Carlo models attempt to describe the rate that a
supercooled liquid system transitions to a crystalline structure as a function of temperature
Explicitly mapping the energy landscape would fundamentally explain the underlying
physics of nucleation An energy landscape of a system is given by
33
1 1 1 2 2 2( )N NNU U x y z x y z x y z= (26)
where U is the potential energy and x y and z are the locations of each of the N atoms in a periodic
box The locations of these atoms give rise to an energy as parameterized by an interatomic
potential An energy landscape can be constructed by mapping the relationship of the local minima
and the first order saddle points between pairs of minima22-27 Energy landscapes have become
crucial to studies of biomolecules glasses and catalysts[21] [26] [44] because landscapes show
atomic configurational changes and calculate the associated kinetics and thermodynamics In this
study we will explore the energy landscape of a barium disilicate system capturing the glassy and
crystalline basins enabling a test of CNT by calculating all needed parameters (interfacial free
energy kinetic barrier and free energy difference of the supercooled liquid and the equilibrium
state)
31 Crystallization Methods
CNT expresses the steady-state nucleation rate (I) as the product of a kinetic and a
thermodynamic factor
( )exp eI Z D TW
kT
minus =
(27)
where Ze is the Zeldovich factor which incorporates the probability of a critical sized nucleus that
will form a new phase independent of other nuclei D(T) is the kinetic factor often given by an
Arrhenius expression over sufficiently small temperature ranges W is the work associated with
generating a thermodynamically stable nucleus that will continue to grow rather than dissipate (ie
the critical work) if temperature and pressure are kept constant k is Boltzmannrsquos constant and T is
absolute temperature The Zeldovich factor is typically treated as a constant with a value ~01[118]
34
Here the kinetic term is expressed as D(T) since in this work the kinetics are explicitly calculated
at every temperature and it is not treated as a simple Arrhenius expression In CNT the work W is
given by the work to generate a spherical isotropic nucleus[1] which consists of a volume term
and a surface energy term It is given by
3 24
( ) 4 ( )3
W r G T r T = + (28)
where G is the Gibbs free energy difference between the crystalline state and the corresponding
liquid r is the radius and is the interfacial energy between the crystalline and liquid phases Eq
(28) neglects the strain energy term that arises from the formation of a solid nucleus in the
supercooled liquid It is assumed that the liquid relaxes fast enough so that strain energy is not
significant for nucleation in liquids[11]
Determining the critical radius (ie taking the derivative of Eq (28) with respect to the
radius and setting the expression equal to 0) allows one to solve for the critical work and the
expression becomes
3
2
16
)3(
( )
( )W
T
G T
=
(29)
Three temperature dependent functions remain D G and Though there are existing methods
in literature to approximate each of these quantities the approximations often include large errors
and assumptions about the behavior of the system[119] Often experimentally the kinetic function
is approximated with the Stokes-Einstein relationship the G term is calculated from heat
capacity curves (assuming that the macroscopic energy is similar to the nanoscopic energy) and
is fitted[119] In the calculation presented here only configurational energies are considered
while vibrational contributions are ignored Ignoring the vibrational component and the reliance on
empirical potentials are the most likely sources of error in this work
35
The energy landscape for this investigation was generated using ExplorerPy[89] starting
with the orthorhombic barium disilicate crystalline structure[89] Barium disilicate was chosen
since it is a commonly studied glass-ceramic with multiple reliable experimental datasets being
found in literature[75] [113] [120] The rest of this section is divided into four parts starting with
a discussion of mapping the landscape and the following three sections are a discussion of
calculating D G and respectively
31A Mapping and Classifying the Landscape
The ExplorerPy energy landscape exploration employed the Eigen-Trieban technique in
the canonical ensemble with systems consisting of 128 256 and 512 atoms[89] The number of
atoms was chosen to achieve convergence of the results while also balancing the large
computational cost associated with energy landscape exploration A smaller system was also
calculated with 64 atoms however this systems was too small to accurately discern energies
enough to make the nucleation calculation The Eigen-Trieban technique is a newly developed
method to explore energy landscapes using a method of shoving along the eigenvectors and
minimizing the potential energy which is followed by using nudged elastic band calculations to
find the associated transition points[89] This technique allows for larger transitions to be found
while retaining the ability to calculate the vibrational frequencies The Pedone et al potentials[36]
were used and the volume of the cell was fixed due to simulation instability when pressure was
fixed The Pedone et al potentials were chosen since they have reproduced key parameters for both
the crystalline and glassy state in previous Monte Carlo studies as well as in the training set of the
potential[73] [74] [121] Though the Eigen-Trieban method may work well for predicting
crystallization and finding large barriers it may be insufficient to describe glass relaxation or
36
viscosity since it rarely finds the small structural transitions that dominate the Adam-Gibbs
relationship Note that these exceptions are inconsequential for predictions of nucleation found in
the work presented here33
The barium disilicate system was explored for 2500 basins for 128 and 256 atoms and 4000
for 512 atoms since more basins were needed to converge the liquid state distribution of energies
This number of basins was shown to be sufficient since multiple paths from the initial crystalline
starting point to SCL states were obtained The distribution of energies of the SCL were found to
converge and the resulting energy landscape is seen in Figure 4
A crystal parameter was then defined from the root mean square displacement (RMSD) for
each atom from the nearest lattice site of the same species ( ) This average value is used to
characterize whether a basin is a crystal or SCL The RMSD approach is not the only method to
assign a crystallinity value[122] however it is one of the simplest and as such was chosen for
this application To then categorize each basin as either a crystalline basin or SCL basin a plot of
the minimum energy of the SCL states versus the cutoff of RMSD was produced This plot for 256
atoms shows a clear transition in the energies around a cutoff value of 1 Aring which is therefore
taken as the cutoff radius of which basins are considered crystals The same analysis was repeated
for each system size This analysis is shown in Figure 4 Any basin with a RMSD less than the
cutoff value is considered a crystalline state This cutoff was found to be generally insensitive to
the results with any values ranging between 09 Aring and 11 Aring changing the resulting prediction of
nucleation by less than one order of magnitude but shifting the peak temperature by up to 40 K
This cutoff also reproduced the energy difference between the SCL and crystal in molecular
dynamics simulations of the materials
37
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-axis being
an arbitrary phase space and the y-axis being the potential energy calculated from the Pedone et al
potentials[36] The colors are indicative of the crystallinity where the blue basin is the initial
starting configuration The landscape shows the lowering of energy associated with partially
crystallizing the sample (B) The energy landscape relationship between the cutoff for crystalline
and super-cooled liquid states for 256 atoms is shown This shows a clear drastic energy change
occurring around the cutoff value of 10 Aring
[
38
31B Kinetic Term for CNT
The transition rate was calculated by the inverse of the average relaxation time from the
SCL to the crystalline states The relaxation time ( ) is defined by the relaxation function ( )
0 exp
t
= minus
(30)
where t is time and 0 is the scaling of the relaxation function The reason we are able to
calculate the kinetics of nucleation but not viscosity is because the nucleation kinetics are
dominated by the transitions between two states that we specifically searched for when exploring
the energy landscapes while the viscosity is defined by the entropy dominating small transitions
described by the AG
The AG relationship shows that the viscosity is dominated by the configurational entropy
of the system Experimentally the kinetic term is often calculated with the Stokes-Einstein
relationship
( )6
kTD
aT
= (31)
where a is the radius of the diffusing species and is the viscosity of the liquid Interestingly this
approximation gives reasonable results when investigating experimental data since the viscosity is
a measurement of the entropy of small cooperative liquid rearrangements Conversely the kinetics
that govern nucleation are large structural rearrangements from a supercooled liquid to a crystal It
is nonetheless possible that these cooperative rearrangements and the large diffusion processes for
nucleation are on the same order of magnitude which is why the Stokes-Einstein relationship gives
39
a reasonable approximation in experimental parameterizations When the Stokes-Einstein
relationship is used there is an implicit assumption that the large nucleating rearrangements are
proportional to the small barriers that govern viscosity[53]
A mean relaxation time can be found by averaging over a system of master equations
( )j
ii ji j j ij i
i
dpg K p g K p
dt
= minus (32)
In this expression g is the degeneracy of a microstate pi is the probability of occupying the
microstate and Kij is a transition rate between inherent structure i and inherent structure j The
equilibrium probability of occupying a microstate can be given by a Boltzmann probability (where
Z is the partition function and U is the potential energy of the inherent structure)
1
exp ii i
Up g
Z kT
= minus
(33)
By substituting the value of jp in Eq (32) for a two-state model ( 1 21 i jp p= == + ) we write
( )( )1ii ji i j ij i
dpg K p g K p
dt= minus minus (34)
Isolating the probabilities and splitting the differential we have
( )1
i
i ji i j ij i
dpdt
g K p g K p=
minus minus (35)
which when solved via integration takes the form
ln ( )
( )
i ji i ji j ij i
i ji j ij
g K g K g K pt
g K g K
minus + =
minus + (36)
Solving for the occupational probability yields
( )expi ji j ij i ji
i
i ji j ij
g K g K g K tp
g K g K
minus minus + =
+ (37)
This shows an analytical solution for the relaxation time in a two-state model
40
1
i ji j ijKg g K
+
= (38)
The two-state model approximation is made to enable the calculation of the mean relaxation time
between the liquid basins to the crystalline basins The transition rates between the basins are given
by the vibrational frequency ( ) and the barrier between the states ( ijU )
exp ij
ij
UK
kT
minus=
(39)
The probability of transitioning from a given liquid basin i to a crystalline basin j is given by
1
expij
ij
Up
Y kT
minus =
(40)
where Y is the partition function with respect to all transitions from the current SCL basin to all
crystalline basins as only these transitions are considered to calculate the relaxation time
expj
j
iY
k
U
T
= minus
(41)
The total expected relaxation time of the system making the transition is then given by the
probability of occupying a basin the probability of transitioning from one basin to another specific
basin and the associated relaxation time
1
i ij
j i ji j ji i i
p pg K g K
=
+ (42)
The first summation is over the liquid states and the second summation is performed just over
crystalline states to capture the mean relaxation time from a supercooled liquid to a crystal We
fixed the vibrational frequency to a value of 17 times 1014 Hz (ie about one jump attempt per 5
femtoseconds) which was calculated using the average of the eigenvalues of each basin ( ) and
the average mass of the atoms (m) using
41
1
2 m
= (43)
31C Degeneracy calculations
To obtain an appropriate value for the degeneracy an understanding the phase-space
volume of the basin is needed since the degeneracy is linked to the volume of the basin[40] By
taking the displacement between the basin coordinates of each atom and the transition point we can
define a radius of the basin R Using R and a spherical approximation we can calculate the phase-
space volume degeneracies
3 23
12
NN
i i
cry
Ng R
g
+
=
(44)
In which ( )x is Eulerrsquos gamma function The degeneracy is then normalized by a constant such
that the liquid-liquid state transitions had a value of 100 s at the glass transition (980 K) to be in
agreement with Angell[58] The normalization step allows for a quantitative prediction of the
liquidus temperature to confirm the accuracy of our model The liquidus value found is
approximately 1200 K 1715 K and 1950 K for 128 256 and 512 atoms respectively The
experimentally determined liquidus temperatures range from 1680 K to 1710 K[113] [115] [120]
It is worth noting that changing the exponent drastically changes the thermodynamics while having
minor effects on the kinetics and the normalization value drastically changes the kinetics while
keeping the thermodynamics nearly constant We chose the denominator to satisfy the liquid
relaxation constraint It is also feasible to choose the exponent to satisfy a constraint on the liquidus
temperature The normalization value is an important to consider because for larger system a
change in 001 can result in a change in the kinetics by ~10 orders of magnitude
42
31D Free Energy Difference
Due to limitations in the exploration of the energy landscape requiring a fixed volume we
will approximate the difference in the Gibbs free energy by calculating the difference of the
Helmholtz free energy This approximation is reasonable due to the similar densities between the
crystalline and glassy phases under ambient pressure[73] [74] The first step is determining the
mean energy difference of all the basins and the second step was repeating the process for only the
liquid basins The mean energy was calculated with the following expression[123]
( )ln ln ii
i
i i iG kTp p U kT g p= + minus (45)
In Eq (45) i is the set of relevant basins for the conditions we are interested in The difference
between the mean free energy of the liquid basins and the entire landscape basin energies is the
G parameter This free energy difference is typically experimentally calculated by integrating
heat capacities however numerous approximations exist These approximations are often linear
approximations normalized to the liquidus temperature In this work we find that the driving force
appears to be predominantly linear which is in good agreement with the approximations that are
often used for the driving force[124]
43
Figure 5 An example interfacial structure between the crystalline phase on the left and the last
sequential SCLglass phase on the right for a barium disilicate system The gray atoms are barium
silicon is shown in red and blue represents oxygen
44
31E Interfacial Energy
To calculate the interfacial energy of the interface between the crystalline phase and the
SCL the initial basin (pure crystalline) and the last basin (glass basin) were annealed at 1700 K for
1 ns and then cooled to a temperature of 800 K over a period of 10 ns The same process was
repeated for a composite where the initial stable crystal and the glass were placed next to each
otherThe structure was multiplied 10 times to create a wall of a large interface to reduce energetic
fluctuations A small section of the composite structure is shown in Figure 5 The difference
between the addition of the energy of the crystalline phase and glassy phase and the composite
gives the energy of the interface which was normalized to the size of the surface area Multiple
arrangements of the composite were tested but anisotropy was found to be minimal with the
fluctuations of the energy being larger than the differences from different tested directions The
final expression used for the interfacial free energy (in eVAngstrom2) is
610 00 35047 040T minus +=
45
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a function of
temperature for each system size The experimental values for kinetics and thermodynamics come
from ref[125] and from the heat capacity data taken from [126] respectively It is clear to see that
driving force shows good agreement across all systems however the kinetics terms only converge
46
for the 256 and 512 atoms systems (Bottom) The fit used to calculate the interfacial energy as a
function of temperature
32 Results amp Discussion
The results for the free energy difference the kinetic parameter and the surface energy are
shown in Figure 6 These parameters as a function of temperature are used to calculate the
nucleation rates in the expression
3
2( )
01 16exp
3( )I
TT G k
= minus
(46)
The 01 being an approximate value of the Zeldovich factor Using the parameters shown in Figure
6 and solving Eq (46) the nucleation rates are shown in Figure 7 along with experimental data for
validation At first glance we notice that there is about 7 order of magnitude differences between
data and prediction Considering that theses potentials were trained on glass and were not intended
for this purpose we consider this as excellent agreement This method also shows agreement below
the peak but the accuracy falls off more quickly below the peak temperature It is likely that the
data below the peak is severely underestimated due to a failure to reach steady state nucleation
rates Reliable steady state nucleation rates are not available for this system and as such a direct
comparison below the peak temperature is not directly available It is worth mentioning that the
failure of CNT of predicting the order of magnitude of nucleation rates is well reported and can
amount to dozens of orders of magnitude when a constant value of is used[75] [113] [115]
[119] [127] thus the proposed method has an advantage when comparing with the current
classical nucleation theory toolset Moreover the prediction of Tmax (~950 K) is also very close to
the experimental value (985 K)
47
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted using the
model presented in this work The data referenced can be found in refs [113] [128] [129]
48
To further compare CNT and the model presented here we decided to compare the results
of the fitted and this workrsquos calculated surface energies as seen in Figure 8 The results show good
agreement all achieving the same order of magnitude implying that experimental methods can
achieve a close approximation to the values required for CNT The prediction of nucleation is
extremely sensitive to the value of the surface energy with small changes leading to orders of
magnitude change It stands to reason then that further work that needs to be approached through
computation and experiments is developing a rigorous model to predict the interfacial free energy
The other assumption that is used experimentally is the Stokes-Einstein relationship To
compare the landscape results of the transition rate from the SCL to crystalline states and the
Stokes-Einstein relationship the liquid state relaxation was calculated The small barriers that
govern the Adam-Gibbs relationship were not found with this exploration method hence
quantitative comparisons are not possible (and relaxation calculations should not be made)
however the relaxation times can be qualitatively compared The liquid state relaxation rate is
proportional to the viscosity and is given by the expression[44]
j
i
i
l
ij
p
K =
(47)
To understand the validity of the Stokes-Einstein relationship we can write a series of theoretically
true proportionalities
( )l
T TD T
(48)
The values of the experimental kinetics fail to align with the kinetics calculated in this work This
is expected due to the fact that the viscosityliquid relaxation is dominated by the set of small
transitions in the landscape while the nucleation transition is defined by the transition of one region
to another with a substantially higher activation barrier This comparison may also account for
49
some part of the experimental overestimation below the peak with the kinetic term calculated
dropping off at a faster rate than that calculated from the experimental viscosity Figure 6 also
shows evidence that it may be inappropriate to use the Stokes-Einstein relationship for nucleation
and as such we may need to re-visit the experimental models that often rely on the assumption of
the Stokes-Einstein relationship This observation casts some doubt in the legitimacy of describing
the kinetic factor as a diffusivity which is standard for CNT For more information on this CNT
failure please see Refs [75] [113] [115] [119] [125] [127]
Nucleation is a complicated concept to investigate and model Unlike the crystalline phase
where the atomic positions are clearly defined the details of the atomic coordinates and bond
configurations of the liquid sites must be described in terms of statistical distributions Due to this
added complication mean-field distributions are typically used to describe non-crystalline
sites[14] [16] [108] [130]ndash[133] The incorporation of fluctuations could more accurately predict
each liquid sitersquos transition to the crystalline phase Fluctuations in the degeneracy could be one
reason that the nucleation rate is over predicted when the temperature is less than the peak
nucleation temperature A given G varies with the environment and depends on the specific
liquid state not the mean value Therefore the rate of nucleation from all supercooled liquid states
to the crystal state is not constant which is why fluctuations need to be considered and why energy
landscapes are such a powerful technique This is especially highlighted when considering that
different degeneracies yield drastically different heat capacity Energy landscapes are extremely
useful because they consider a large distribution in free energies and show all structural fluctuations
possible Fluctuations in cluster size and degeneracy influence Adam-Gibbs configurational
entropy which directly impact the transition rate between the crystal and liquid state
50
Figure 8 The surface energy with respect to temperature for the work presented here
0
0002
0004
0006
0008
001
0012
0014
800 850 900 950 1000 1050 1100
Su
rface
En
ergy (
eVAring
2)
Temperature [K]
This Work
Xia et al (2017) Ref 23
Fokin et al (2016) Ref 41
51
33 Conclusions
Though this is not the first time CNT has been examined computationally in an attempt to
understand if the theory is valid[74] [116] [117] it is the first parameterization of each variable
independently in an attempt to calculate the resulting nucleation rate Not only are all the variables
calculated when used in the expression for CNT the liquidus temperature is reproduced and the
nucleation rate is reasonably predicted This is as far as the authors know also the first time a real
energy landscape for a complicated glass-ceramic system is being reported This method leveraging
energy landscapes alongside CNT enables for a reasonably fast calculation of the nucleation curve
of a glass forming liquid In future work this model could be used for either developing commercial
glass-ceramics or for predicting the critical rate needed to maintain a glass This model can also
provide additional insights into the physics of nucleation and the validity of CNT without the
constraints of fitting parameters This method fundamentally has the power to connect the 3N
dimensions of the energy landscape to the compositional dimensions and thus allowing for a unique
powerful method of predicting nucleation parameters as a result simplifying the crystalline
dimensions in which a glassglass-ceramic must be optimized over
Chapter 4
Expanding the Current State of Relaxation
Before we consider a new approach to relaxation (optimizing our understanding of
the 3N dimensions) there are first missing pieces in the current state specifically the temperature
and compositional dependence of the stretching exponent However even before expanding the
current state of the art relaxation methods we also need to understand ergodicity In this section we
will start with a simple though experiment to understand the temporal effects of relaxation and then
in the second section combine it with fictive temperature to create a new model for the stretched
exponent
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition
The prophet Deborah sang in the book of Judges ldquoThe mountains flowed before the Lordrdquo
[6] [13] However the question the prophet failed to ask was what happens if the mountains are
moving at relativistic speeds The focus of this section is to highlight the temporal effects on
ergodicity in the context of special relativity and the reference frame of the observer In order to
understand relativistic effects on materials one must consider a well-characterized material that
actively relaxes but appears solid on our timescale viz pure B2O3 glass [134] [135] Glass is a
particularly good candidate to study the effects of time dilation on a material since it undergoes a
kinetic transition known as the glass transition (Tg) which has multiple definitions that will be
considered
53
The first definition is in terms of the aptly named ldquoDeborah Numberrdquo which was proposed
by Reiner [136] to offer a view of fluidity and equilibrium mechanics It was later expanded to
explain the origin of the glass transition and the transition from an ergodic liquid system to a non-
ergodic solid-like glassy state[12] which is the fundamental definition of the glass transition [11]
[12] [137] It is important to note here that the breakdown of ergodicity is relative to the
observation timescale and is reflected in the Deborah number (D) which is defined as
Dt
= (49)
where t is the (external) observation time and is the relaxation time of the material which can be
expressed through the Maxwell relation [42] By definition the Deborah Number equals 1 at the
glass transition temperature Tg [13] [137]
Another definition of the glass transition temperature is due to Angell who defined Tg as
the temperature where the viscosity of the supercooled liquid is 1012 Pa s [58] [138] The Angell
diagram is an important plot relevant to viscosity and the glass transition where the abscissa is TgT
and the ordinate axis is log10( ) In the Angell diagram the value of the viscosity at Tg and in the
limit of infinite temperature are fixed and the difference in the temperature scaling of the TgT-
normalized viscosity is the slope of the log10( ) curve at TgT = 1 which Angell defined as the
fragility (m) of the supercooled liquid Using this definition one can establish the consistency of
the Angell plot at relativistic speeds An example of the Angell plot is shown in Figure 9
54
Figure 9 An Angell diagram created using the MYEGA expression [55] With a very strong glass
(m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) [17] [134] [135] The
infinite temperature limit is from the work of Zheng et al[54] and the glass transition temperature
is from the Angell definition
55
Using the two previous definitions it is expressed
1gT
GtD
= = (50)
which can be re-written as
(
(
)
)g
g
tG T
T= (51)
In order to solve this equation one must know the shear modulus at the glass transition temperature
recent improvements in topological constraint theory have led to the ability to predict properties
with varying temperature and composition Using a recent topological constraint model for elastic
moduli proposed by Wilkinson et al [139] one can predict the temperature dependence of the
modulus
( ) ( ) ( ) ( ) ( )( ) ( )Δ Δ Δ Ac c C Ac
c c
F n q T F n q T F n q T x Nn q T x NdG dGG
d F M d F M
+ + = =
(52)
In Eq (5) nc is the number of topological constraints (c) qc is the constraint onset function as
described by Mauro et al [69] Fc is the free energy of the constraint is the density of the
system M is the molar mass and 119889119866
119889120549119865119888 is a scaling factor This expression was evaluated as a
function of temperature with the total modulus coming from Kodama et al [135] and is equal to 8
GPa at room temperature The B2O3 structure consists entirely of three-coordinated boron [140]
[141]
Using the modulus predicted in Eq (52) numerical observation time is expressed
56
12
9
10 Pa s141 s
709 10 Pat = =
(53)
Using this description of the static (vc=0 where v is the speed of the glass and c is the
speed of light) glass transition it is possible now to describe the relativistic glass transition for
which there are considered two separate cases It is worth noting that this is not the first case study
of relativistic viscosity a fairly large basis of literature dealing with ideal fluids near large
gravitational bodies [142] There is also some work also relating the relaxation time in a
gravitational field to the nonequilibrium entropy relaxation time [143] [144] This work
emphasizes the theoretical effects of special relativity on real liquids their glass transitions and
the properties there dependent on
41A Relativistic Liquid
The first case is when a sample of a glass-forming system moves past a stationary observer
at a speed approaching light giving the new observation time as
0tt
= (54)
where t0 = 141 s (as previously calculated) and g is the Lorentz factor
2
2
1
1v
c
=
minus
(55)
The Maxwell relation can be modified to account for relativistic effects by combining with Eq (54)
0
()
)(
g
gG TT
t
= (56)
57
This gives a condition that is temperature dependent to predict the new glass transition as a function
of v however in order to accurately describe the relativistic behavior density must be considered
Density will change by a factor of 2 because both the mass and the volume will be affected and
as such
0) (( )g gG TT t = (57)
The prediction then for the glass transition as a function of the speed of light is shown in Figure
10
Figure 10 The relativistic glass transition temperature for B2O3 glass
58
Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature
which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) expression [55] [145] The MYEGA equation needs 2 other parameters
besides Tg to generate the viscosity curve fragility (m) which is the slope at Tg in the Angell plot
and the infinite temperature viscosity () The fragility will not change because one assumption
in this model is the consistency of the Angell plot at all relativistic velocities even relativistic The
infinite temperature viscosity which also will not change because this is the minimum possible
viscosity for a liquid The new viscosity curves for various speeds are shown in Figure 11
59
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different fractions of
light speed All of the viscosities approaching the universal temperature limit for viscosity
60
By using the Deborah Number it has been shown that at relativistic speeds will change the viscosity
in order to maintain the Deborah number and in this case it is shown that when the Lord would
move past the mountains at relativistic speeds the mountains would not actually flow but instead
hold still (even more so than they currently do) From this it is inferred that any liquid moving past
the earth at relativistic speeds will appear to be more solid and eventually appear as a glass
41B Relativistic Observer
In this case we consider an observer moving past a sample of glass at relativistic speeds
giving
0t t = (58)
which when the relativistic shear modulus 2 factor is included can be written as
( )
( )0
g
g
TG T
t
= (59)
Solving in the same manner as before it is shown which shear modulus is needed to satisfy the
condition in Figure 12
61
Figure 12 The modulus needed to satisfy the condition for the glass transition
62
In Figure 12 the shear modulus at the glass transition temperature is shown however the
constraint theory mechanism being implemented to calculate the shear modulus has a built in upper
limit when calculating the Tg past the point in which all the constraints are intact Due to this
limitation there is an upper limit to our prediction Nonetheless the calculated Tg over the available
range are shown in Figure 13 as well as the related viscosities It is seen clearly that at higher
velocities the glass transition shifts towards zero more dramatically at the higher temperatures
This leads to the interesting result that if we take the limit of an observer moving past the earth at
close to the speed of light all glass would appear to be a liquid
63
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at v=044c
(Bottom) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches
the speed of light
64
Applying the mechanisms proposed by Einstein to the concept of the Deborah number it
has been shown that the viscosities will appear to change dramatically as a glass-forming system
approaches relativistic speeds This allows us to build an intuitive understanding of the differences
between glass and liquids Specifically any material can appear either more or less fluid based on
the velocity with the velocity with which they travel relative to the observer This serves to highlight
the importance of the role of the observer on the glass transition and why even small changes in
DSC scan rates or sampling frequency can lead to drastically different measured glass transitions
42 Temperature and Compositional Dependence of the Stretching Exponent
The mathematical form for glass relaxation ( ) was originally proposed by Kohlrausch in
1854 based on the decay of charge in a Leyden jar[9] [14] [29] [42] [45] [96] [146]
( ) ( )( )exp t t
= minus (60)
where t is time t is the relaxation time of the system and b is the dimensionless stretching exponent
Eq(1) was originally proposed empirically and is known as the stretched exponential relaxation
(SER) function[42] [96] The relaxation time t depends on the composition temperature thermal
history pressure and pressure history of the glass as well as the property being measured[62]
[147] For example the stress relaxation time of a glass can be written as[62]
( )( )
ff
s
f f
T T P P
G T T P P
= (61)
where h is the shear viscosity and G is the shear modulus of the glass-forming system[42]
(Variables are defined in Table 2) In addition to using Eq(61) to describe stress relaxation time
structural relaxation time has recently been hypothesized follow Eq(61) with bulk viscosity and
65
the difference of bulk modulus between infinite and 0 frequency replacing shear viscosity and shear
modulus[62] Both h and G vary with the temperature (T) and thermal history of the glass (as
quantified via the fictive temperature Tf) as well as the pressure (P) and its corresponding history
(fictive pressure Pf)[16]
To connect the stretching exponent to its physical origins we can begin with a result from
Richert and Richert[148] that relates to an underlying structural relaxation time distribution and
then develop equations that determine from known quantities Their expression
2 22
ln 2
1
6
minus=
(62)
relates to the variance of the logarithm ( )2
ln of the structural relaxation time For reference
all variables are defined in Table 2 The physical origins of the relaxation time are related to the
configurational entropy (Sc) as shown by the Adam-Gibbs equation[53] [55] [149]
ln lnc
B
TS = + (63)
where is the infinite temperature relaxation time T is the absolute temperature and B represents
the energy barrier for relaxation
Table 2 Variable definitions
Variable Definition
t ts Structural relaxation time stress relaxation time
b Kohlrausch exponent (ie the stretching exponent)
T Tf Tfi Absolute temperature fictive temp partial fictive temp
P Pf Pressure fictive pressure
66
kB Boltzmannrsquos constant
wi ki i=1 to N Prony series parameters
B Adam-Gibbs relaxation barrier
Sc Adam-Gibbs configurational entropy
B Distribution of activation barriers
2 Variance
Na Number of atoms
f Topological degrees of freedom per atom
Degenerate configurations per floppy mode
H Enthalpy barrier for Relaxation
d Number of dimensions
f Fraction of activated relaxation pathways
S Adam-Gibbs entropy in the infinite-temperature limit
m Kinetic fragility index
m0 The limit of a strong liquid
Grouped unknowns 6
ln
B
a BN k
A Proportionality between fragility and distribution of barriers
x A given composition
125640
S
A
Intercept of the linear model of
67
42A Deriving a Model
Combining Eqs(62) and (63) Gupta and Mauro[150] proposed that the variance of the energy
barriers for relaxation (2
B ) could be rewritten as
2 2
22
2
1
6( )B c TTS
minus=
(64)
Solving for the stretching exponent
( )22 (
6
( )
)
c
B c
S T
T
T
TS
= +
(65)
The work of Naumis et al[23] [56] has shown that the configurational entropy is proportional to
the topological degrees of freedom in the network a result that was used in the derivation of
temperature-dependent constraint theory[57] and the MYEGA (Mauro-Yue-Ellison-Gupta-Allan)
equation for the relaxation of supercooled liquids[55] The MYEGA equation was derived by
expressing the configurational entropy as
( ) ( ) lna BcS T x f T x N k= (66)
In Eq (66) f is the topological degrees of freedom per atom Na is the number of atoms kB is
Boltzmannrsquos constant x is a given composition and is the number of degenerate configurations
per floppy mode The temperature dependence of the topological degrees of freedom was
approximated using a simple two-state model
( )( ) 3exp
B
H xf x T df
k T
minus= =
(67)
Here H(x) is the enthalpy barrier for relaxation d is the dimensionality of the system and f is the
68
fraction of activated relaxation pathways Combining Eqs (65) and (67) as well as condensing the
unknowns into the term defined by
6
ln
B
a BN k
=
(68)
we get
( )
22
df
dfT
=
+
(69)
This can be compared to the prediction made by Phillips[48] for the stretching exponent at
temperatures below the glass transition
2
df
df =
+ (70)
Comparing the two expressions of Eq(69) and Eq(70) they would agree if
4 4dfT
= + (71)
We will show later (Figure 12) an example where extrapolating our model prediction for to room
temperature (with T in Eq(71) replaced by fictive temperature Tf) gives a result close to the Phillips
value of Eq(70) When the temperature T in Eq(69) is high (much larger than ) then Eq(69) gives
us rarr1 Thus the model can interpolate between the low-temperature (Phillips) value and the
high-temperature value of 1 In our current model we do not have access to a value for in Eq(68)
so this model cannot assess whether the Phillips room temperature value for is universal ie
whether Eq(71) is always satisfied at room temperature
Zheng et al[151] showed that the Mauro-Allan-Potuzak (MAP) model for the relaxation
time of the nonequilibrium glassy state[44] implies that the configurational entropy can be written
as a function of thermal history (Tf) the fragility index (m) the fragility limit of a strong liquid
69
(m0) the glass transition temperature (Tg) and the limit of infinite temperature configurational
entropy ( S ) as
0
) exp 1 0
(ln1
c f
f
gT
TS m
TS
m
= minus minus
(72)
The qualitative relationship between the distribution of activation barriers and fragility was
proposed by Stillinger[18] [19] who suggested that the energy landscape of a strong liquid (low
fragility) has a small distribution of activation barriers and that a higher fragility is associated with
a broader distribution of activation barriers ie a higher variance of the activation barriers This
leads to a phenomenological relationship that is here assumed to be valid based on Stillingerrsquos work
on energy landscapes[18] [19]
2
0
1B
mA
m
= minus
(73)
where A is some constant of proportionality This expression was chosen because in the limit of a
strong liquid there is an infinitely sharp distribution of activation barriers Combining Eqs(65)
(72) and (73) we obtain the temperature dependence of the stretching exponent for a liquid
( )
0
2 2 2 2
0 0
exp 1
( ) 2
6 ln10 1 exp 1
g
f
f
f
g
f
f
T mT S
T mT
Tm mA T S
m T m
minus minus
=
minus + minus minus
(74)
Fictive temperature appears in this new expression because we are deliberately expressing this
function as an equilibrium model Rewriting Eq (74) in terms of one unknown () and grouping
constants
70
( ) 0
2 2
0 0
exp 1
2exp 11
g
f
f
f
g
f
f
T m
T m
T m
T
T
Tm
T
m
m
minus minus
minus minus
=
minus +
(75)
Eq (75) is an expression for the stretching exponent as a function of thermal history glass
transition and fragility index with only one unknown In Eq (75)
( ) 12
6 ln10
S
A
= (76)
The only unknown for the compositional dependence of the stretching exponent is the value of D
The composition-dependent part of Eq(76) is Syen
since we approximate A to be independent of
composition The fragility dependence of Syen
was proposed by Guo et al [61]
Syen(x) = S
yen(xref
)expm(x) -m(x
ref)
m0
eacute
eumlecircecirc
ugrave
ucircuacuteuacute
(77)
where x is composition and xref is a reference composition in the same glass family Seeking the
simplest possible expression to approximate the unknown D we take the natural logarithm of
Eq(76) and of Eq(77) and combine them to get
0
ln lnm
m = + (78)
with the additional definition
( )( )
( )12
0
ln l6 ln10
nref refS x m x
mA
minus
=
(79)
The result is that ln D varies linearly with fragility m and the intercept ln D is independent of
composition (depends on one reference composition)
In Figure 9 we plot the data of Boumlhmer et al[152] vs the predicted exponent The work of
71
Boumlhmer et al is the summary of the literature data relating fragility index to the stretching exponent
at the glass transition which we use to fit values to Eq (78) The dataset included in their work
covers chalcogenide oxide and organic glasses The fitting (consisting of least-squares
minimization of the difference between the predicted stretching exponent and that which was
reported) was done twice once for organic and once for inorganic systems During the fit it was
assumed that Tg=Tf Some assumption about thermal history was necessary since the individual
thermal histories or Tf values are not known for this whole collection since we are trying to track
overall trends in values vs composition this reasonable simplifying assumption is consistent
with our program Using Eqs(75) and (78) the temperature and compositional dependence of the
stretching exponent can be described with only one free parameter
72
Figure 14 b predicted and from literature showing good agreement with a total root-mean-square
error of 01 The fit for organic systems is given by 66 ln Kln = minus and for inorganic systems by
75ln Kln = minus
73
42B Experimental Validation
Experimental density measurements were made using Corningcopy JadeTM glass as described
elsewhere[146] Changes in the density are normalized to obtain the relaxation functions which
are fit with Eq (60) The resulting values of are shown in Table 3 A comparison in Figure 15
is shown with data obtained on a soda-lime silicate SG80 where the stretching exponent was
obtained using the measurement of the released enthalpy as a function of isothermal annealing time
during relaxation below the glass transition The measurement of the released enthalpy[153] relies
on the change in the excess heat capacity in the glass transition range as a function of annealing
time Normalized released enthalpy relaxation functions for each temperature were fit with Eq(60)
to obtain the stretched exponent values for SG80 In Figure 15 parts A and B are samples measured
below Tg while in part C the samples measured are at Tg and above The values were then fit with
Eq(75) where the only free parameter was D with an equilibrium assumption (Tf = T)
74
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy JadeTM
glass and Sylvania Incorporatedrsquos SG80
Glass Temperature (K) Tg (K) TgT m D (K-1) Beta t (s)
Corningcopy JadeTM 1031 1074 1042 32 0002 0504 7249
Corningcopy JadeTM 1008 1074 1065 32 0002 0447 32458
Corningcopy JadeTM 973 1074 1103 32 0002 0395 419214
SG80 800 800 1 36 00057 0632 92
SG80 783 800 1021 36 00057 0593 182
75
Figure 15 The equilibrium model proposed with the experimental points showing good agreement
between the experimentally measured data points and the equilibrium derived model RMSE was
002 for Corningcopy JadeTM (A) and less than 001 for SG80 (B) (C) The model fit for B2O3
experimental data[154] [155] The fragility and glass transition temperature of the B2O3 are taken
from the work of Mauro et al[17]
(
A)
(
B)
(
C)
76
A combination of the MAP model for non-equilibrium shear viscosity[44] the model
presented here primarily in Eq(75) and the model presented by Wilkinson et al[139] for the
temperature dependence of elastic modulus allows for fully quantitative modeling of stress
relaxation behavior The missing model required to understand structural relaxation is the bulk
viscosity curve[62] All previous relaxation (structural or stress) models[42] [61] have relied on
approximations that use a constant exponent and on a constant (temperature-independent)
modulus value whereas here every parameter of Eq(60) may be modeled as a function of
temperature Furthermore in combination with the relaxation models described by Guo et al[61]
in which multiple fictive temperatures are described using a Prony series and a temperature-
dependent modulus one can construct a relaxation curve accounting for the temperature
dependence and thermal history dependence of all relevant parameters
( )
( )( )
1
( )( )exp exp
( )) (
fTN
i
i
f
f
f fi
G T k T tG T tw
T TT
T T
=
minus minus
(80)
Here wi and ki are fitting parameters that are completely determined by the value of and each
term in the Prony series is denoted with the subscript i Usually 8 or 12 terms are included in the
Prony series[96] Each term in the Prony series is assigned a partial fictive temperature (Tfi) whose
relaxation is described by the simple exponential in the Prony series Equilibrium conditions are
assumed at the start of the simulation which allows for a known set of starting probabilities within
the energy landscape interpretation of relaxation The same model also includes the fragility index
and glass transition dependence of the non-equilibrium shear viscosity This method is
implemented in RelaxPy[42] as discussed in the next section This serves as an approximation for
the evolution of the of the non-equilibrium state however the temperature dependence of the bulk
viscosity and a replacement for fictive temperature need to be quantified to improve the
understanding of the underlying physics[1] [16] [62]
77
Separately we can explore the relationship between this model given by Eq(75) and the underlying
energy landscape In order to better understand the stretching exponent consider that there exists
a full set of parallel relaxation modes within an energy landscape The relaxation modes are then
weighted by the occupational probability corresponding to a particular mode This gives a series of
transition rates with some probability prefactor and an associated relaxation time (scaled from the
mean relaxation time) which gives rise to a Prony series form of Eq(80) Thus the distribution of
relaxation times (or barriers) determines the evolution of the stretching exponent while the average
barrier determines the mean relaxation time If this Prony series description in turn describes the
stretching exponent we arrive at a physical description and origin of the stretching exponent As
the temperature approaches infinity even though there is a distribution of activation barriers the
distribution of relaxation times approaches a Dirac delta function and the stretching exponent
approaches one (a simple exponential decay) As the temperature decreases the distribution of
relaxation times broadens due to the wide distribution of barriers however due to the broken
ergodic nature of glass the value of the stretching exponent will be controlled by the instantaneous
occupational probability
The activation barriers may be modeled eg using a numerically random set of Gaussian
distribution of barriers[156] The probability of selecting an individual transition is then calculated
using a Boltzmann weighting function the transition also has an associated relaxation time with it
The Prony series is then recreated with the probability multiplied with the simple exponential
relaxation form The sum of all of these terms is then fit with the stretched exponential form Figure
11 shows the stretching exponent calculated for the mean activation energy of 10 eV and three
choices of standard deviation 25 2 15 ande e e minus minus minus= eV as shown in the legend where Eq(60) has
been fit to the Prony series calculated from an equilibrium Boltzmann sampling of a Gaussian
distribution of barriers The temperature dependence for the stretching exponent matches the form
78
derived earlier in this work This stretching exponent shape has been shown previously though not
in a closed-form solution[156] This secondary method not only validates the generalized form of
Eq(75) but also highlights the validity of using the configurational entropy and barrier distributions
as the underlying metric controlling the temperature dependence of the stretching exponent
Figure 16 The stretching exponent calculated as described in the text for a Gaussian distribution
of barriers This plot shows that the distribution of barriers has a large effect on the stretching
exponent A Tg cannot be described since there is no vibrational frequency included in the model
though the glass transition temperature should be the same for all distributions since the mean
relaxation time is the same for all distributions at all temperatures The deviation is given in ln eV
units
79
To create software to model this complex behavior the authorsrsquo existing software RelaxPy[42] has
been modified into a new version RelaxPy v20 Instead of fixed values for the Prony series the
values are chosen dynamically to match the stretching exponent as predicted by Eq(75) The
database of values wi() and ki() was created with a fixed number of terms N in Eq(80) (we chose
N=12) This database is available in the same Github repository as the software The parameters
were fit by starting with the values obtained for =37 reported by Mauro and Mauro[96] to make
the new version of the model smoothly reproduce a prior optimal fit then was stepped by 001
(when less than 095) or by 0001 (when greater than 095) Using the previously optimized values
of wi and ki for starting values for each new the wi and ki values were then varied to minimize
the root-mean-square error between the stretching exponent form and the Prony series form ie to
satisfy Eq(80) with least error This list of compiled values makes up the contents of a database
that the software accesses The values of wi and ki used for a given are calculated by finding the
closest value in the database to the given value of This allows for an efficient implementation
of relaxation modeling Figure 12 shows the fitted parameters to the Prony series and an example
RelaxPy output for Corningcopy JadeTM glass undergoing quenching at a rate of 10 Kmin Note that
the room temperature stretching exponent value is 046 a close match the Phillipsrsquos ldquomagicrdquo value
of 37 (043)[157] This prediction for Corningcopy JadeTM is a non-equilibrium prediction in contrast
to the equilibrium high-temperature prediction shown in
80
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent Each color
designates one term in the series (Bottom) The output from RelaxPy v20 showing the stretching
exponent effects on the relaxation prediction of Corningcopy JadeTM glass Each quadrant shows one
property that is of interest for relaxation experiments In particular it is interesting to see the
dynamics of the stretching exponent during a typical quench
81
43 Conclusion
In this section we have sought to expand the current understanding of relaxation and improve the
models that currently exists We have proposed a model that accounts for the effects of special
relativity as well as a model was derived through an understanding of the distribution of barriers
for relaxation and the temperature dependence of the Adam-Gibbs entropy The model outlined
herein describes the temperature dependence of the stretching exponent in glass relaxation
The work done on the relativistic effects on relaxation illustrates the crucial role of the observer on
all relaxation models and offers an extreme in which to test common relaxation models The model
for the stretching exponent not only considers the extremes at high and low temperature (when
compared to the glass transition) as in the Phillips model but also for any intermediate temperature
as a function of the fictive temperature This model does not have any explicit temperature
dependence since it was assumed that an equilibrium model works well to describe the
instantaneous distribution of relaxation times Given the physical argument and the success of this
model when tested by experiments and by another model it is at least reasonable to formulate the
temperature-dependence of in terms of its fictive-temperature-dependence as we have done here
Including both T and Tf is possible to consider but lies outside our current scope Using previously
derived compositional dependence for the MAP model a fragility index dependence of the
stretching exponent was defined and tested The model was confirmed using multiple experimental
datasets In addition a theoretical comparison to a distribution of landscape activation barriers was
found to reproduce the same trends as the model
Chapter 5
Glass Kinetics Without Fictive Temperature
Fictive temperature as discussed through this text is unable to capture key physical
phenomena[16] [42] [51] To circumvent this limitation we rely on two different frameworks
The Adam-Gibbs relationship and generalized features of the energy landscape These two features
allow us to create a method that is completely generalizable and does not involve fictive
temperature This method creates fake landscapes that generally capture the trends of glass based
on experimental measurements This new method we have called lsquoToy Landscapesrsquo
51 Background of the Adam Gibbs Relationship
Experimental evidence in support of the AG model comes from the work of Richet[158]
in which they compared the predicted configurational entropy from viscosity curves with that
obtained from DSC The configurational entropy was calculated from DSC data using
0
0( )p p vi
T
T
b
c c
C CS S T dT
T
minus= + (81)
In Eq (81) Cp is the isobaric heat capacity Cpvib is the vibrational heat capacity and T0 is the initial
temperature from which the configurational entropy is integrated The difference between the total
heat capacity and the vibrational heat capacity is the heat capacity due to structural rearrangements
Richet found good agreement between the measured viscosity curve and calorimetric
configurational entropy at temperatures above the glass transition confirming validity of the AG
relationship When applying Eq equation reference goes here all temperatures must remain above
the glass transition ie in the liquid or supercooled liquid state The calculation is invalid upon
83
cooling through the glass transition due to the breakdown of ergodicity[12] [13] [159] and the
irreversibility of the glass transition process Application of Eq (6) to temperatures below the glass
transition leads to an incorrect calculation of excess entropy in the glassy state[160]
In this study we expand on these previous works to understand the relationship between
viscosity and the underlying energy landscape and provide new insights into the thermodynamics
of liquids through a simplified ldquotoy landscaperdquo model constructed from experimentally measured
viscosity parameters The energy landscape framework is especially helpful for elucidating the
thermodynamics and kinetics of supercooled liquid and glassy systems Energy landscapes describe
the evolution of all atomic transitions for kinetic processes in a system To perform such
calculations information about the inherent structures and transition points in a landscape must be
known In this work we solve the inverse problem of deducing realistic landscape parameters using
the AG relationship and experimentally measured data This information is used to construct a ldquotoy
landscaperdquo model to describe glass relaxation processes To achieve an accurate relaxation model
the following steps are taken to ensure the validity of the results
A Confirm the AG model for shear viscosity using energyenthalpy landscape
calculations
B Validate the assumptions made by the MYEGA model
C Confirm the AG model for bulk viscosity
D Explore the fundamental relations between viscous properties and topography of the
landscape
E Use the knowledge gained to propose a new thermodynamic model to calculate the
driving force for glass relaxation and the scaling of the liquidrsquos free energy
This chapter is organized into sections devoted to each of the topics above preceded by a
methods section All of the sections are presented to validate a new model of glass relaxation that
is sufficiently complex to achieve a calculation of the thermodynamics and kinetics of the liquid
This new model will be based on understanding the landscape from experimental properties and
we have called this approach the ldquotoy landscaperdquo model
84
52 Methods
Potential energy landscapes (fixed volume) and enthalpy landscapes (fixed pressure) are
powerful modeling techniques where atomic configurations are mapped in a 3N-dimensional phase
space (or 3N+1 dimensions for enthalpy landscapes)[89] The energy landscape approach is based
on mapping the continuous 3N dimensional space to a discrete set of energy minima (called
ldquoinherent structuresrdquo) and first-order saddle points (ldquotransition pointsrdquo) To perform this mapping
a systematic search for inherent structures is performed while also mapping the lowest-energy
transition points between each pairwise combination of adjoining minima The landscape itself is
partitioned into basins which represent the set of all configurations that minimize to a common
inherent structure[161] This combination of the basin and transition point information results in a
topographic map of the 3N phase space (either a partial or whole mapping) that can be used to gain
insights into the physics of any atomistic system such as proteins[91] liquids[18] [19] glasses[1]
[93] and nucleating crystals At equilibrium the probability distribution for occupying the various
basins in the landscape is given by[13]
exp
exp
i
j
i
i
j
j
Eg
kTp
Eg
kT
minus
= minus
(82)
where Ei is the potential energy (or enthalpy) of the inherent structure gi is the degeneracy of the
basin i is the basin index and pi is the probability of occupying a given basin i The denominator
is a normalization factor given by the summation over all basins Mauro et al[44] previously used
this formalism to study the non-equilibrium viscosity calculate the evolution of configurational
heat capacity and elucidate the long-time relaxation kinetics of selenium glass[44] [93] [160] For
equilibrium systems the configurational entropy is given by the Gibbs formula
85
lnc i
i
iS k p p= minus (83)
In order to validate the AG model the landscapes of several common systems were
explored using ExplorerPy[89] ExplorerPy is a software program built specifically to map
landscapes using the LAMMPS molecular dynamics package[30] The three systems we explored
are B2O3[162] (150 atoms and 2000 basins) SiO2[162] (150 atoms and 1000 basins) and BaO-
2SiO2[121] (128 atoms and 1500 basins) All systems used periodic boundary conditions The
initial structures for B2O3 and SiO2 were constructed by placing atoms corresponding to the
chemistries of the systems on a random grid (with 1 Aring spacing) corresponding to a density of 20
gcm3 and equilibrated at 2500 K then quenching at a rate of 1 Kps to room temperature using the
Wang et al[162] potentials in the NPT (constant pressure number of atoms and temperature)
ensemble The energy landscape of the BaO-2SiO2 system was taken from our previous work[72]
The SiO2 (constant pressure exploration) system was explored using ET with a push distance of 03
Aring and a pressure of 10 atm B2O3 was explored using isobaric MD at a temperature of 1500 K for
500-time steps of 1 fs also at a pressure of 1 atm The minimization (after MD or ET) was handled
by the LAMMPS[30] steepest descent algorithm until the certainty in energy was smaller than 10-
4 eV Both used a nudged elastic band spring constant of 10 eVAring2 The energy distribution for all
systems was fully sampled which was confirmed by randomly checking if half the basin
distributions matched the full distribution It was also found that multiple structures in each
exploration were discovered more than once and a slightly smaller B2O3 system (130 atoms) was
also explored and showed identical results to the 150-atom distribution The insights gained from
the interaction between the experimentally accessible viscous properties and the landscapes
presented here will be generalized to create a simplified ldquotoy landscaperdquo model to capture the
thermodynamics and kinetics of the glass-forming system
86
53 Results
53A Adam-Gibbs Validation
To predict viscosity using the Adam-Gibbs expression there are only three required
parameters the configurational entropy (as a function of temperature) the barrier for cooperative
rearrangements and the infinite-temperature limit of viscosity Using a landscape the equilibrium
configurational entropy can be obtained using Eq (83) The infinite-temperature limit of shear
viscosity is given by the work of Zheng et al[54] where a systematic study of MYEGA fits to
viscosity data revealed a common value of 10-293 Pamiddots This infinite temperature value of viscosity
is used for all the experimental and computational viscosity curves shown in this work The final
parameter is the mean barrier for a cooperative rearrangement In Figure 18 the barrier for
cooperative rearrangements was chosen to accurately produce a viscosity of 1012 Pamiddots at the
experimentally measured value of the glass transition temperature From these parameter values
the predicted Adam-Gibbs viscosity curve is plotted The universal infinite limit fragility and glass
transition values from references were inserted into the MYEGA equation to plot the
lsquoExperimentalrsquo curves for B2O3 SiO2 and BaOmiddot2SiO2 respectively[58] [69] [120] When
examining Figure 18 it is clear that the AG model is able to fundamentally recapture the viscosity
87
Figure 18 The viscosity (left) and landscape (right) predictions for three common systems The
first system is newly calculated in this work while the latter two come from our previous works[72]
[89] It is seen that the viscosity predicted from the AG model is very accurately able to reproduce
the experimental viscosity curves from the MYEGA model The last system is a potential energy
landscape while the others are enthalpy landscapes
53B MYEGA Validation
Now that we have confirmed that the configurational entropy of the landscape works well
to describe the viscosity via the AG relationship a definitive temperature-dependent form is needed
in order to make more general predictions The key approximation made in deriving the MYEGA
model is that the configurational entropy follows an Arrhenius form By testing this Arrhenius
88
form we can evaluate this assumption in the derivation of the MYEGA model while simultaneously
gaining insights into the thermodynamics of the liquid and supercooled liquid states A strong liquid
( 15m ) has a configurational entropy that is nearly constant as a function of temperature while a
fragile liquid ( 15m ) is a liquid in which the configurational entropy is highly temperature
dependent Since we are testing the MYEGA equation it is also worthwhile to consider alternate
viscosity expressions Assuming validity of the Adam-Gibbs equation the configurational entropy
deduced from the viscosity experiments is given by
( )10 01log l) og(
c
BS
T T minus= (84)
where the viscosity function is given by a viscosity model such as Vogel-Fulcher-Tammann[59]
(VFT) equation the Avramov-Milchev[60] [163] (AM) equation or the MYEGA expression
The resulting entropy predicted by each viscosity model (using experimental data for
fragility and the glass transition) is compared to the calculated configurational entropy from the
landscape for B2O3 in ( 518gT = K and 33m = [69] were used for the experimental values) The
figure clearly shows that the MYEGA equation which assumes that configurational entropy scales
in an Arrhenius fashion reproduces the calculated configurational entropy from the landscape the
most accurately out of the three considered viscosity models This also means that the
configurational entropy form of the AM and VFT models are not physical and cannot be used to
recreate physical results Since the temperature dependence of the entropy has a fixed mathematical
form and is dominated by the distribution of basins we now understand that the distribution of
basins must also have a fixed form and that form must control experimentally accessible properties
This is the key insight that will allow for the development of the toy landscape model
89
53C Adam-Gibbs and Structural Relaxation
Although the Adam-Gibbs model has been validated for shear viscosity and has provided
the functional basis of the MYEGA model let us now consider whether the temperature
dependence of the configurational entropy can also predict the bulk viscosity (also known as the
volume viscosity) Recent work has highlighted the importance of bulk viscosity viz that the
structural relaxation time is proportional to the bulk viscosity[62] Hence the temperature
dependence of the bulk viscosity provides key information needed to predict structural relaxation
times[62] Following Scherer[164] the relationship between the structural relaxation time ( structural
) and the bulk viscosity ( B ) is given by
( )0B structuralK K = minus (85)
Figure 19 The configurational entropy comparisons between the three major viscosity models
which validates the main underlying assumption of the MYEGA model The VFT and AM are
unable to capture the physics of configurational entropy therefore ruling
90
where K is the infinite frequency bulk modulus and 0K is the zero-frequency bulk modulus In
order to calculate bulk viscosity for B2O3 the zero-frequency and infinite-frequency modulus is
taken from these works respectively [165] [166] The experimental structural relaxation times were
taken from the work of Sidebottom et al[154] In Figure 20 the bulk viscosity is shown as well as
the Adam-Gibbs comparison
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the configurational
entropy from the enthalpy landscape with the barrier 00155 eV (compared to the shear barrier of
00149 eV) and the infinite limit allowed to vary (10-263 Pamiddots for bulk viscosity) In Figure 18 the
configurational entropy is confirmed for the shear viscosity thus confirming the AG for both shear
and bulk viscosities Sidebottom data are from Ref [154]
In Figure 20 the fitted values of B and 10log can be used to gain insights into the
difference between structural and stress relaxations since the configurational entropy is universal
between the two The value of Bk (the barrier for cooperative re-arrangements in natural units) for
shear viscosity was 173 K while for bulk viscosity it was 180 K This difference of 7 K in the
barrier leads to an order of magnitude difference in the fitted viscosity curves This is a small
91
difference and explains the variance in the values at Tg The other difference is in the infinite
temperature value of viscosity which is -293 log(Pamiddots) for shear viscosity (from Zheng et al) and
-263 log(Pamiddots) for bulk viscosity This implies that the MYEGA raw form is accurate to predict
the structural relaxation time as well as the stress relaxation (though the values of the pre-
exponential factor and the composite constant will change between the two forms) However the
form in terms of fragility and the glass transition will need to be modified to accurately describe
the structural relaxation time The form assumes the viscosity of interest has a value of 1012 Pamiddots at
the glass transition temperature for shear viscosity which does not have to be the value for the bulk
viscosity curves at the glass transition Further research is needed to understand the appropriate
bulk viscosity at the glass transition and the infinite temperature of bulk viscosity
53D Landscape Features
As shown here the configurational entropy is governed by a Boltzmann sampling of a
probabilistic distribution of states n(E) This distribution of the inherent structures is empirically
found to follow a log-normal distribution as shown in Figure 21 While we are not claiming a
deeper meaning to the underlying origin of the log-normal distribution we may adopt this form of
the probability density function since it accurately describes the underlying distribution of
microstates This is the key insight needed to develop a new model because this means that the
fragility and glass transition are fundamentally related to the parameters of the distribution of basins
on the landscape (number of basins and standard distribution of the enthalpy)
The configurational entropy in the MYEGA model is given by
10
exp 1ln10 12 log
g
c
TS mS
T
= minus minus
minus
(86)
92
where S is the infinite temperature limit of the configurational entropy In Figure 21 we also
show the computed value of Eq (83) as a function of temperature compared to that of Eq (86)
which shows excellent agreement between the configurational entropy and the MYEGA model
This test also supports Stillingerrsquos[18] [19] view of fragility being correlated to the lsquoroughnessrsquo of
the landscape topography According to Stillingerrsquos view if the distribution of inherent structure
energies is narrow then the configurational entropy cannot have a large temperature dependence
leading to a low fragility Alternatively if the distribution is broad then there must be a temperature
dependence to the configurational entropy which necessarily results in a higher fragility
93
Figure 21 (A) The histogram for the energy minima for each of the test systems showing a good fit
with the log normal distribution This distribution will then be a valid form to calculate the enthalpy
distribution of the model presented in the next session (B) The configurational entropy from the
model showing the accuracy of the scaling of the entropy predicted by the MYEGA model The
S value was fit for each system
(A)
(
B)
(B)
94
54 Topography-Property Relations
Now that we have confirmed the validity of the Adam-Gibbs and MYEGA equations from
the energy landscape approach we can draw some key conclusions regarding the thermodynamic
properties of liquids For example heat capacity has been related to fragility[112] and is of
importance for glass manufacturing since it is related to the cost of heating and forming a glass
Isobaric heat capacity is defined as
p
P
HC
T
=
(87)
where H is enthalpy T is temperature and P is pressure The mean configurational enthalpy
calculated from an enthalpy landscape is given by
conf i conf iH p H= (88)
The heat capacity has contributions from both vibrational and configurational modes in liquids
while in glasses the configurational mode is mostly lost For a reversible process the
configurational heat capacity can be written as a function of the configurational entropy
c
p conf
P
SC T
T
=
(89)
By combining Eq (86) and Eq (89) we can write the expression for configurational heat capacity
of a liquid as
10 10
1 exp 1 exp12 log 12 log
g g
p conf
T Tm m S HC S H
T T T T
= minus minus minus = minus
minus minus
(90)
From fluctuation theory the heat capacity can be expressed in terms of the variance of the enthalpy
(2
H )
95
2
2
1p conf HC
kT= (91)
It is important to note that the variance of the enthalpy is not equivalent to the variance of the log-
normal distribution The variance of the enthalpy includes a Boltzmann distribution over the log-
normal distributed basins This gives a direct relationship between the enthalpy fluctuations and
the viscosity parameters If we further adopt the log-normal distribution as the effective distribution
for all liquid basins of all enthalpy landscapes then the distribution is only a function of the
common viscosity parameters ( S m gT and 10log ) The viscosity parameters will only
change the variance of the distribution since the mean of the distribution is inconsequential (the
zero-point enthalpy can be adjusted to an arbitrary value without changing the calculation of the
configurational entropy) The parameters of the log-normal distribution can then be fitted to
reproduce the configurational entropyrsquos temperature dependence thus creating a lsquotoyrsquo enthalpy
landscape for us to play with for the purpose of thermodynamic and kinetic calculations Generating
the landscape is a numerical calculation as the number of basins and standard deviation must be
chosen such to reproduce key phenomena
To quantify the dependence of these parameters on the degeneracy and temperature it is
helpful to develop a descriptive formal model The cumulative probability distribution of a normal
distribution (with zero mean) is given by
10log )1
1 er(
f2 2
Hp
+
=
(92)
where is the standard deviation of the log-normal distribution erf is the error function H is the
enthalpy and p is the probability To then calculate the equal probability spacing for a log-normal
distribution we solve for H for the set of probabilities p which range from 2 to 1 2minus
96
with a step-size of This then gives enthalpies of basins equally spaced by the cumulative
probability distribution and as a result each basin has equal degeneracy
1 2 r2 1e f10
pH
minus minus = (93)
This set of basins carries with it an Arrhenius activation barrier for the configurational entropy that
is only a function of and the degeneracy of each basin (if the is chosen such that the energy
distributions converge) This creates a model that fully describes the thermodynamics of the liquid
and though it currently lacks any kinetic component it offers the possibility to be a useful tool
when used in conjunction with Eqs (82) (83) and (88) This new tool is denoted the ldquotoy landscape
modelrdquo (TLM) It is called the toy landscape model because we are generating a landscape that is
not the real landscape but is enough to reproduce key experimental properties without needing
expensive computational calculations It is a lsquotoyrsquo for us to play with without needing to worry
about upfront computational cost
To confirm the validity of TLM and the way it is constructed we can calculate the
activation barrier from this deterministic method and for a random set of inherent structures with
the same number of basins and distributions For this purpose we used temperatures ranging from
1000 to 2000 K for both the deterministic and random models with a step size of 0001 for TLM
Excellent agreement between the random and deterministic methods is shown in Figure 22A This
analysis allows us to understand that the thermodynamics of the system are fundamentally linked
to the viscous flow behavior Though this technique was created to target viscosity and relaxation
it is generally applicable to all configurational thermodynamics of liquids and glasses It is worth
noting that all the thermodynamics will be relative to a 0 eV energy state defined at equilibrium at
0 K and as such the enthalpy differences at some fixed temperature must be known to compare
over compositional spaces
97
In Figure 22B we see the same method being used to understand the relationship between
the viscous parameters and the parameters of the landscape This is the key to the numerical
calculation of the lsquotoy landscapersquo model The toy landscape is a numerical calculation where we
use experimentally accessible values to generate a landscape whose thermodynamics should
reproduce key thermodynamic phenomena In Figure 22B we show that the fragility and glass
transition are systematically changed as the number of basins and standard deviation of the enthalpy
is changed When this idea is combined with the basin calculation in Eq (93) (giving equal
degeneracies for all basins) then the complete set of enthalpy basins is easily attainable giving
access to the fundamental thermodynamics of the system
98
(
A)
(B)
(A)
99
Figure 22 (A) Comparison between the randomized method (histogram) and the deterministic
method (vertical lines) showing good agreement between the maximum in the histogram and the
value predicted by the deterministic technique validating the approach It is worth noting that the
100-basin distribution is a very wide distribution where the total number of basins is less than the
number of points used in the calculation This is done for a variable number of basins with the
number of basins shown in the legend (B) The dependence of fragility and the glass transition
temperature vs the distribution of states and the number of basins
55 Barrier Free Description of Thermodynamics
Throughout this work we have systematically explored the Adam-Gibbs model of
viscosity An important insight that we have gained is that the configurational entropy is free of any
barriers This is of particular interest because the Arrhenius slope of the activation barrier is linked
to the glass transition and fragility The glass transition temperature is known to be related to the
set of activation barriers as it corresponds to a breakdown of ergodicity What is found here is that
the fragility is the ldquoconversion factorrdquo to account for the slope of the configurational entropy
(which as previously identified is related to the distribution of basins) This means that the barriers
themselves do not matter until the material is close to the breakdown of ergodicity and thus the
equilibrium liquid thermodynamics and kinetics can be described without explicit information
about the saddle points in the landscape
Understanding the role of the topography in the properties of glasses and liquids lead to
the most important insights derived from the AG model Since the configurational entropy is the
same for both stress and structural relaxation simple viscous parameters allow us to understand
fundamental thermodynamic quantities of the system without needing to map computationally
expensive energyenthalpy landscapes The process of converting experimentally accessible
properties to a usable landscape for thermodynamic calculations is what we are calling the toy
landscape model To show the utility of toy landscapes we can calculate the driving force for glass
relaxation This information can give insights into the physics of relaxation by dynamically
100
calculating the Gibbs free energy of the equilibrium liquid at any temperature which defines a
driving force for glass relaxation To do this calculation of the driving force for glass relaxation
we consider the ergodicity parameter ( x ) defined by the Mauro-Allan-Potuzak (MAP) model of
glass relaxation as the ratio between the glass entropy and the equilibrium liquid entropy
liquiss dglaS x S= (94)
To get information about the enthalpy we consider an instantaneous Boltzmann distribution on the
landscape given by an equilibrium distribution at some elevated temperature fT
1
exp
exp
iglass i i
fii
f
HH H g
kTHg
kT
= minus
minus
(95)
This formalism in conjunction with TLM gives us the ability to explore the temperature
and compositional dependence of the driving force for relaxation The enthalpy of the liquid is
calculated from an equilibrium distribution on the landscape The driving force ( ) is defined as
( ) ( )( ) ( )1iliquglas ds f l quii dH H TS xT T = minus + minus (96)
For our purposes this gives a nice approximation for the driving force improving on previous
approximations given by ( fT Tminus )[16] [42] TLM does not include a kinetic feature to calculate
the occupational probability in each basin however when coupled to a kinetic model such as the
MAP model[44] a complete description of the underlying thermodynamics of glass relaxation can
be considered For this calculation of the driving force the kinetics are calculated through the MAP
model but this is not necessarily a requirement In Figure 23 the compositional and temperature
dependence of the driving force is shown for three glasses with identical glass transitions and
varying distributions of basins (as listed in Table 1) The calculations are performed using the
RelaxPy script and MAP model with the stretching exponent fixed at the Phillips value of 37
101
explanations of which are available elsewhere[42] The systems were quenched from 700 K to 300
K at a rate of 10 Kmin Figure 23 clearly shows how simply changing the total number of basins
creates a larger driving force for glass relaxation This simple change of the number of basins
drastically increases the entropic contribution and due to the loss of ergodicity explains the increase
of the driving force of relaxation The kinetics could be calculated through any relaxation model or
directly from the landscape if some barrier approximations are made
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition temperature (500 K)
were kept the same while the total number of basins were allowed to vary
Sample Total Basins [-] Distribution of States [log eV] 1
1493g
mT
minus
[K]
Fragility [-]
1 100 02 837 40
2 500 02 537 31
3 1000 02 466 28
102
Figure 23 The driving forces for different example glasses calculated using a combination of the
MAP model RelaxPy and the toy landscape model The parameters for each glass can be found in
Table 4
57 Toy Landscapes for the Design of Glasses and Glass Ceramics
The concept of a toy landscape can go even further building a method to predict the
dynamics of a glass system based on simple inputs This is possible because in the previous section
we have related the distribution of basins on an energy landscape to the experimental viscosity
which allows for thermodynamic insights Building on this we can assume that the barriers (H )
for transitioning between states is equal to that of the barrier mean for the viscous relaxation
ln10gH mkT= (97)
This means that the only key factor needed to then predict the dynamics of a landscape using the
metabasin approach is the vibrational frequency However the relaxation time at the glass transition
103
is known to be a constant of 100s and as such we can choose a vibrational frequency that reproduces
this fixed point This gives an entire description of relaxation without any need for fictive
temperature An example of the results predicted through this method are shown in Figure 24 This
prediction only required knowledge of the glass transition temperature and the fragility and from
this a pure prediction of relaxation is made
104
105
Figure 24 The predicted outputs from the toy landscape method showing predictions of the enthalpy
and entropy under a standard quench for barium disilicate This prediction does not require fictive
temperature or any such assumptions about the evolution of the non-equilibrium behavior
This method is also not limited to predicting the relaxation of glasses but can also capture
the crystallization It is known from the earlier work on nucleation presented in Chapter 3 that a
simple two state model was able to capture the underlying physics of the system Thus if we know
the free energy of the crystal and barrier to crystallization we can find predict key crystallization
106
phenomena By fitting the heat flow peak magnitude and location this gives enough information to
infer the barrier and the enthalpy of the crystal The degeneracy (and thus the entropy) of the crystal
can then be calculated through knowledge of the liquidus temperature
ex(
p)cry SCL l
cry
lk
G TH
T
minus=
(98)
This then gives everything for the model presented in Chapter 3 from only 1 calorimetry
experiment the glass transition the fragility and the liquidus temperature To predict nucleation a
series of estimates were done running the toy landscape at different temperatures until the
temperature with the fastest crystallization rate was found This was then assumed to be the
maximum rate of nucleation The value of the interfacial energy was then chosen in such a way so
that the predicted curve reproduces the peak at the same temperature This gives all the parameters
needed for a prediction using CNT It is also possible to make a prediction of the growth rate of
crystals using this expression
2
1 exp6
GU
kT
a kT
minus minus
=
(99)
The results for these estimates are shown in Figure 25 The reason why growth is more accurate
compared to nucleation is that the kinetic factor of growth is governed by viscosity while nucleation
is governed by a kinetic factor that includes an assumed vibrational frequency
These methods have a plethora of assumptions built in such as that the vibrational
frequency of basins undergoing relaxation is equal to the vibrational frequency when undergoing
crystallization however as more information is gained about a particular system (such as more
crystallization peak as a function of temperature) the information could be built into the system to
improve crystallization predictions If the peak size is known as a function of scan ratetemperature
even the enthalpy as a function of temperature could be known This provides a whole system for
the real estimates of crystallization can be improved
107
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume in
nucleation is assumed to be on the order of one cubic angstrom while the a parameter in growth is
assumed to be around one nm both are in good agreement for estimates in literature The values
for the orange points are taken from these works[128] [167]
57 Discussion
The impact of the Adam-Gibbs model on glass science is hard to overstate and it has
become foundational to ideas that will enable the design of next generation glasses The Adam-
Gibbs equation still has more to offer the glass community in terms of fundamental understanding
and practical applications In this work we have shown that the Adam-Gibbs relationship is
consistent with both shear and bulk viscosities Also we have validated the main assumption of the
MYEGA model giving insights into the scaling of configurational entropy from the underlying
enthalpy landscape Using this knowledge of the relationship between the Adam-Gibbs model with
the underlying enthalpy landscapes we have developed a simplified ldquotoy landscaperdquo approach for
modeling glass relaxation The toy landscape is constructed numerically by understanding the
distribution of inherent structures and transition points in the landscape and then relating these
distributions to experimentally measured parameters of the MYEGA viscosity model This
1 00
1600
1400
1200
1000
00
600
400
200
000
0000 110000 130000 1 0000 1 0000
(m
s)
Temperature
Growth PredictionCassar et al
0
2
4
6
10
12
14
00 1100 1300 1 00
log 1
0(I) n
ucl
ei s
m3
Temperature
Predicted RateRodrigues et al
4
3
2
1
0
00 1000 1100 1200 1300
log
10(I m
ax(I))
nuclei s
m3
Temperature elvin
108
simplified landscape model can then be used to calculate the underlying thermodynamics and
kinetics of the system and represents a practical application of energy landscapes to understand
glass relaxation nucleation and other relevant thermodynamic properties
This also closes our discussion on understanding the thermal history effects on glass What
we have shown is that the dynamics of a glass system can be understood through energy landscapes
and fundamentally the same insights can be recreated through lsquoToy Landscapesrsquo This approach
allows fundamentally for the complex 3N dimensional landscape to be summarized and abbreviated
into a smaller landscape that can easily be implemented for any glass However this new
parameterization relies on knowing some key properties that are only due to the location in the
compositional phase space and for a complete picture of the system to be created we must also
create a sufficient approximation for the compositional phase space shown in the next 2 chapters
Chapter 6
Enabling the Prediction of Glass Properties
So far models have been created that describe the very nature of the thermal history effects on
glasses However as an input to these models we have to know about the compositional dependence
of key parameters namely the glass transition and the fragility In this chapter we extend
compositional models to other key properties needed for the design of glasses Here these key
properties that have not been readily predictable are approached using both machine learning and
TCT models The TCT models are focused on surface reactivity Youngrsquos modulus and ionic
conductivity The ML addresses CTE fragility melting point as well as Youngrsquos modulus
61 Controlling Surface Reactivity
Glass surfaces especially their interactions with water[168]ndash[171] are of the utmost importance to
nuclear waste glass cover glass and many other modern applications[106] It is generally accepted
that the outermost surface of oxide glasses readily reacts with water molecules where water
molecules chemisorb on an as-formed glass surface and populate it with surface hydroxyls Recent
models attempt to quantitatively explain the change in properties observed on hydroxylated glass
surfaces[77] [168] [172] [173] and most agree that the surface readily becomes hydrated because
it is more stable to have a bonded hydroxyl group than to have dangling bonds In glasswater
interactions hydrolysis and diffusion lead to two forms of adsorbed water chemisorbed and
physisorbed[171] [174]ndash[176] Both affect the network differently and are controlled by different
processes physisorption by hydrogen bonding[175] and chemisorption by reaction with the surface
to form bonded hydroxyl groups
110
equiv Si minus O minus Si equiv +H2O rarr equiv Si minus OH + OH minus Si equiv
Although binding energy studies have been performed using molecular dynamics simulations[177]
a specific study of the surface reactivity in tandem has not been conducted Recent advancements
in reactive force field modeling allow the direct observation of both diffusion and surface
reactivity[38] [178]ndash[180] and several studies have found surface reactivity constants consistent
with those reported experimentally[173] [178] [181]
Recently TCT has been used to predict various bulk properties of glassy systems such as
Vickers hardness and fragility[17] [182] [183] Based on the TCT description of the glass network
glassy materials have been postulated to have a so-called lsquointermediate phasersquo in which the atomic
structure of a material will self-organize so as to be isostatically constrained (n=3) as explored
extensively in various works[65]ndash[67] [133] [184] [185] In this case it has been shown that such
materials exhibit anomalous behavior in certain properties such as an high hardness and lower free
energy[43] [67] [77]
TCT has also been used to understand the impact of water on bulk glass structure and
properties Potter et al[168] found that by accounting for the impact of chemisorbed (dissociated)
water on glass network connectivity (viz breaking Si-O-Si bonds) and the impact of physisorbed
(molecular) water on strain of the Si-O-Si bond angle the glass transition temperature (Tg) could
be predicted This model could be further expanded to include other properties (such as
modulus[139] or dissolution kinetics[77]) predicted by TCT such as the work done by Liu et
al[186] on calcium-silicate-hydrate gel An extensive study relating topological constraints[6]
[17] [57] and surface energy was performed by Yu et al[172] specifically focusing on the
transition from hydrophilic to hydrophobic behavior on silica surfaces Their work used reactive
molecular dynamics to model the change in surface energy of a silicate glass and then correlate it
with the number of surface constraints present with the work focusing primarily on the global
average of the surface
111
To investigate the effects of glass network topology on the surface reactivity we model
the hydration of a silicate glass was modelled using molecular dynamics simulations according to
the following procedures Initially bulk sodium silicate glasses were simulated (150 atoms with a
molar composition of 70SiO2middot30Na2O) using the Teter potential[187] For a bulk glass this pair
potential is known to accurately simulate sodium silicate glasses and has been extensively studied
to investigate various properties including structural features transport of sodium ions and
vibrational density of states[188] The size of the system is initially set to achieve the
experimentally measured density of 2466 gcm3 [189] A total of 35 Si-atoms 85 O-atoms and 30
Na-atoms were randomly inserted within a periodic cubic box of a = 12686 Å and the initial
configuration was energy-minimized to avoid any overlaps prior to the glass formation The glass
was held for 05 ns with a constant number of atoms volume and energy (a microcanonical or
NVE ensemble) Since a thermal fluctuation of ~400 K (from 2000 K to 2400 K) was observed
during the microcanonical run the glass was allowed to evolve further at 2400 K for 05 ns with a
constant number of atoms volume and temperature (a canonical or NVT ensemble) Within the
same ensemble the melted system was then cooled with a constant cooling rate of 05 Kps and
after the temperature had reached 300 K it was equilibrated for 1 ns Finally a constant pressure
of 1 atm was applied to the system (NPT ensemble) for another 1 ns to ensure no significant
fluctuations of the density during the equilibration process Three different sodium silicate glasses
of the same nominal composition were constructed via the aforementioned procedure using
different initial atomic positions These repeated simulations compensate for the limitation of small
sample size by allowing us to capture the statistical behavior of glass surfaces during the glass-
water reactions The final densities of the sodium silicate are presented in Table 5 These final
structures were used as starting configurations for the glass-water reactions MD simulations with
the Teter potential creating three glass networks were carried out using the Large-scale
AtomicMolecular Massively Parallel Simulator (LAMMPS) package[30]
112
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K
Simulation box dimension a (cubic) [Å] Density [gcm3]
Run1 12979 2303
Run2 12630 2499
Run3 12737 2437
Initial dimension a=12686 Å
Experimental density (70SiO2middot30Na2O mol) 2466 gcm3
113
After the bulk sodium silicate glasses were obtained a reactive potential was employed to model
the glass surface and subsequent glass-water interface as the system of interest for this study
required characterization of the reactive processes during the simulation of surface phenomena In
this study all reactive MD simulations are performed with the NaSiOH parameterization using
the ReaxFF reactive force field framework15 The ReaxFF parameters are trained using a first-
principles data set that describes water interaction at the sodium silicate glass-water interface
Further details of the ReaxFF methodology and its potential forms can be found in earlier
publications by van Duin et al[38] [179] [181] [190]
114
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the hydration
models Color scheme Si atom (ivory) O atom (red) and Na atom (blue) The z-axis is elongated
to allow space for an insert of water
115
As shown in Figure 26 a free surface was first created by expanding the c parameters of each of
the equilibrated bulk glasses This process results in a vacuum region above and below the two
surfaces of the glass slab along the z-direction These exposed surfaces were relaxed at 300 K for
100 ps with the ReaxFF reactive force field Following the relaxation of both top and bottom
surfaces the vacuum region was filled with water molecules The number of water molecules that
are inserted in the vacuum region was controlled to have a density of ~099 gcm3 In addition all
water molecules were kept at least 2 Å from the uppermost and lowermost atoms of the glass
surface to prevent any initial close contact with the surface Glass-water reaction simulations were
carried out at 300 K with the configurations shown in Table 6 for 500 ps in the NVT ensemble
From these runs trajectories at every 100 ps were obtained for the surface reactivity analyses
Figure 27 shows the initial and final positions of glass surface reaction with water at 0 and 500 ps
respectively
116
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the top surface
in contact with water is shown here
a) b)
117
Table 6 System configurations for sodium silicate glass-water reactions
Simulation cell [Å3] Number of water molecules
Run1 12979 times 12979 times 3894 146
Run2 12630 times 12630 times 3789 134
Run3 12737 times 12737 times 3821 138
118
The hydrated surfaces of sodium silicate glasses from each run were used to investigate the
adsorption behavior of a water molecule Since the time evolution of surface reactivity is of key
interest the binding energy of a water molecule to the hydrated surface was calculated at 0 100
200 300 400 ps In order to evaluate the local heterogeneity of reactivity imposed by varied glass
surface structures each surface was divided into a 10 by 10 grid and the binding energy at each site
was mapped across the grid The water molecule position in the z-direction from top and bottom
surface was maintained to be 20 Å from the outermost atom during the energy calculation The
binding energies (Eb) were calculated as below where the energy of a surface with adsorbed water
(Esw) was taken with respect to the surface energy (Es) and the energy of molecular water (Ew)
b s wswE E E E= minus minus (100)
A negative binding energy would indicate that water adsorption to the surface site at the
corresponding grid is thermodynamically favorable The binding energy map may then be used to
locate where water binding would be most stable based on the ReaxFF calculations From three
independent glass-water reaction boxes a total of six hydrated surfaces are obtained (a top and
bottom surface from each run) increasing the statistical reliability of this analysis in the ensemble
despite the small overall system size of any one box The binding energy mapping process is shown
in Figure 28
119
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example of the
electronic-structure DFT calculation using semilocal exchange-correlation functionals finding the
binding energy of a water lsquopixelrsquo to the surface
~20 Å
surface
10times10 grid
120
Similar calculations are carried out at an electronic structure level using the DFT framework as
implemented within the Vienna Ab-initio Simulation Package (VASP) software Total energies are
computed from the static geometries obtained from the reacted ReaxFF structures using the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional with projector augmented wave
(PAW) pseudopotentials to describe the ion cores A kinetic energy cutoff of 500 eV was used for
the expansion of the plane waves Due to the large non-crystalline structure the Brillouin zone is
only sampled at the Γ-point with 01 eV of Methfessel-Paxton smearing to help electronic
convergence Due to the computational cost associated with these DFT simulations the results were
primarily used to elucidate the electronic interactions between the water molecule and the glass
surface
To count the topological constraints in the structure generated from the ReaxFF MD
simulations an algorithm was developed to systematically exclude all non-bridging oxygens
alkali and hydrogens from the structure The number of surface rigid constraints (in the pixel)
around each network-forming atom was then calculated an example contour plot generated using
this procedure is shown in Figure 29 The surfaces were then converted to a 10times10 pixel grid and
the average coordination of network-forming atoms in each pixel calculated The depth of each
pixel taken was to be 4 Aring below the surface
121
Figure 29 Example contour surface showing the average coordination per atom on the glass
surface for the first run at 300 ps
122
Figure 30 shows the results from the MD binding energy studies where a clear transition in the
region around 30 constraintsatom is shown This region is isostatic because the number of
constraints is equal to the number of translational degrees of freedom and the width highlighted in
the figure is evidence for the existence of an intermediate phase In order to confirm that the binding
energies had indeed converged a larger system of 1500 atoms was simulated Good quantitative
agreement was found between the 150- and 1500-atom systems In this work the isostatic behavior
occurs over a wide range in which certain silicon network sites dramatically increase the binding
sites (binding energy approaching 0) The state or structure with nc outside of the intermediate range
will readily interact with water Only those in the isostatic region are likely to remain unaffected
after the interaction with water thus they are the rate limiting species in dissolution of silica glass
123
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of constraints for
surface atoms at the local pixel Results show a distinct maximum in which there is a near
hydrophilic-hydrophobic transition of the surface The error bars represent the standard deviation
A second system with 1500 atoms was performed to show convergence of the ReaxFF MD results
⟨119889119867minus119874119866119897119886119904119904⟩ = 345 Aring
⟨119889119867minus119874119866119897119886119904119904⟩ = 248 Aring
124
The intermediate phase result is intuitive due to the implicit stability that comes with having an
isostatic phase which is both energetically favorable and stress-free The larger negative binding
energies in the over-constrained regions arise from large differences between the free energies of
the chemically bonded state (chemisorbed) and unbound state (water in near proximity to the
surface but not bonded) in these regions in this region the chemical reaction with the water
molecule allows the glass surface to reduce the number of incompatible constraints which in turn
alleviates local stress in the network In other words a local region undergoes an alleviation of
stress when Q4 units convert to Q3 units The larger negative binding energy values in the under-
constrained regions can trace their origin to the sites available for bonding with the water molecule
Their high number of degrees of freedom allows for the facile reaction of water ie the only energy
cost is for the oxygen to dissociate from the network The isostatic network within the intermediate
phase is able to preserve its structural integrity because the energy barrier to deform the network is
high and there is no localized stress to create an additional driving force for reaction with water
Indeed the isostatic network is the most energetically favorable arrangement of a non-crystalline
structure This intermediate phase is governed by the fluctuations of the topology that arise with
the minimization of stresses These stresses change the local topology (within the range of the
fluctuations) so that they become isostatic[68] This leads to a flat region where there is a constant
binding energy at the surface Though it appears to be an intermediate phase it is important to note
that this may instead be simply a manifestation of an isostatic threshold (a peak in binding energies
rather than a plateau)
The average number of surface constraints has been shown in previous work to be largely
controlled by the annealing time[172] The surface constraints may then be used to predict surface
reactivity However it is worth noting that the local number of constraints is not constant in
time[17] Furthermore Potter et al[168] showed that molecular water in the network can radically
shift the free energy of the γ constraint Therefore the number of surface constraints is dynamic
125
during the influx of water Shifting constraint energy will also alter the fragility which in turn alters
the diffusion activation enthalpy In the future this may provide a path for developing more durable
glasses ndash currently it shows that a homogeneous intermediate phase (isostatic) surface should be
targeted to achieve maximum chemical durability
62 Elastic Modulus Prediction
Tailoring the elastic properties of glass is important in the design of new advanced compositions to
control the stiffness and damage resistance of a variety of glass products[2] [191]ndash[193] Elastic
moduli are defined based on the ratio of the stress exerted upon a material to the resulting strain
While various elastic moduli can be defined (eg Youngrsquos modulus shear modulus bulk modulus
and Poissonrsquos ratio) for an isotropic material such as glass only two of these quantities are
mutually independent[1] [191] [194]
Previous attempts to model elastic moduli have focused on either computationally costly
molecular dynamics simulations or empirical fitting methods The work by Makishima and
Mackenzie[195] offered a possible solution for predicting elastic moduli based on the dissociation
energy per unit volume of the glass however topological changes in the glass network are ignored
and the model cannot account for the temperature dependence of modulus With MD simulations
elastic moduli can be obtained by applying a stress and measuring the resulting strain on the
simulation cell assuming that accurate interatomic potentials are available[2] Machine learning
has also been applied to model elastic moduli using experimentally measured composition-property
databases with a high predictive ability being achieved[2] [196] Previous analytical modeling
techniques related to the topology of the glass-forming network have shown good qualitative
agreement with compositional trends in modulus but have a lack of quantitative accuracy[64]
[197]ndash[199]
126
It is common in the glass community to assume that Youngrsquos modulus may scale linearly
with hardness While this is clearly an oversimplification let us begin with previously derived
models for glass hardness as a potential starting point since glass hardness has received
considerable attention in the context of topological constraint theory [85] [192] [200] Several
models have been proposed to explain the origin of glass hardness all of which have a linear form
( ) vv
dHH n x n
dn= minus (101)
where Hv is the Vickers hardness x is chemical composition nrsquo is the critical level of constraints
needed for the substance to provide mechanical resistance to the indenter in three dimensions and
n is a measure of the constraint rigidity Recent work on glass hardness[192] has shown that n can
most accurately be defined in terms of either the density of rigid constraints (or alternatively the
density of rigid angular constraints) given by
)( )
( )( )
(c An xn
x Nx
M x
= (102)
where nc is either the number of constraints per atom or number of angular constraints per atom ρ
is the density of the composition M is the molar mass and NA is Avogadrorsquos number Since
hardness and elastic modulus are often considered to be correlated one might surmise that the
elastic modulus might also be some function of angular constraint density or total constraint
density However such an approach is not able to give quantitatively accurate predictions of
modulus
Here we propose an improved model of Youngrsquos modulus based on the free energy density of the
topological constraints
int
( ) ( ))
(
(
) i constr
c i
a s
Ai in x q T
M x
x NF F
=
= (103)
where qn(T) is the onset function for each associated constraint to give temperature dependence to
127
the predictions and is determined by the free energy ∆Fn The calculated values of Δ119865c are then
used to calculate Youngrsquos modulus (E) by
( )c c
c
FdE
EF
xd
F
= minus (104)
In this new model the Youngrsquos modulus is controlled by the free energy parameters related to the
strength of each constraint Each parameter in this model captures a key physical phenomenon with
each parameter corresponding to a physical property Moreover we demonstrate that a common
set of parameters can simultaneously capture both the temperature and compositional dependence
of modulus as well as predict other properties such as the glass transition temperature and fragility
To demonstrate the validity of this temperature-dependent constraint model Youngrsquos
modulus data are collected from literature for lithium borate[195] sodium borate[195] and
germanium selenide glass systems[201] Additionally the sodium phosphosilicate[202] system
was experimentally determined following the procedure described by Zheng et al[183] using room
temperature resonant ultrasound spectroscopy on prisms of dimensions 10 mm times 8 mm times 6 mm
Analytical topological constraint models have already been published for each of these
systems[17] [57] [202]
The proposed models for glass elasticity (the models previously proposed for hardness
based on constraint density and angular constraint density as well as the new model for modulus
based on the free energy density of constraints) were optimized in each case leaving the intercept
and slope (119889119864
119889119899
119889119864
119889119880119888) as adjustable parameters The constraint onset temperatures were also optimized
for the free energy density model Figure 31 and Figure 32 show the results for each model in the
phosphosilicate and sodium borate systems respectively
128
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values of the model
predictions are 641 GPa for constraint density 313 GPa for free energy density and 774 GPa
for angular constraint density
129
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from minimum
to maximum error The free energy density model gives the most accurate results The constraint
density has a RMSE of 61 GPa the angular density has a RMSE of 20 GPa and the energy density
has a RMSE of 59 GPa
130
We have also evaluated the predictive ability of the model in terms of the temperature dependence
of the Youngrsquos modulus sing the 10 Na2Omiddot90 B2O3 data reported by Jaccani and
Huang[191] the model and experimental predictions of the temperature dependence of the Youngrsquos
modulus are plotted in Figure 33 Here the constraint onset temperatures are determined from the
room temperature modulus data The number of escape attempts was optimized since this controls
the width of each transition and is thermal history dependent
131
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and experiment for
10 Na2O 90 B2O3 Using the previously fitted onset temperatures the only free parameters are
then the vibrational frequency and the heating time in which their product was fitted to be 14000
Where each dip in the modulus corresponds to a constraint no longer being rigid as heated through
each onset The onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The data was fit
using a least-squares method and the resultant fit is shown as the calculated method The fit has an
R2 of 094 (B) The contribution from each constraint to the overall modulus
It is also possible to validate the model by comparing the free parameters fitted to the Youngrsquos
modulus data vs that of previously reported data as shown in Table 7 Results for two additional
systems are plotted in Figure 34 to show the general validity of the free energy density model
132
Table 7 Fitted values from this analysis compared to those reported in the literature The disparity
between the constraints evaluated with molecular dynamics most likely come from the speed in
which the samples are quenched
Value Fitted Value (K) Literature (K) Method Citation
Silicate Onset 2212 1986 MD (Potter et
al[168])
Silicate Onset 818 1600 MD (Potter et
al[168])
Silicate Onset 450-500 810 MD (Potter et
al[168])
Borate Onset 921 Not Reported Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 715 740-760 Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 393 328 Fitting Parameter
(Mauro Gupta and
Loucks[69])
133
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in Fig 3) and
experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of 093 and (B) xLi2Omiddot(100-
x)B2O3 glasses with an R2 of 0986
(
A)
(
B)
134
In the previous work of Zheng et al[192] concerning hardness of glass the authors write
ldquoIt should be noted that each bond constraint corresponds to a certain energy since different kind
of bond has different bonding energyhellip and thus the constraint density also represents an energy
per unit volume In other words hardness is correlated to the energy per unit volume Our findings
for the borosilicate and phosphosilicate systems are further evidence in support of this argument
since both the total constraint density and angular constraint density approaches give better
prediction of glass hardness compared to models based on number of atomic constraintsrdquo
which makes it clear that the prediction of the hardnessmodulus should be closely related to the
onset temperatures (ie free energies) of the associated constraints The model that was previously
proposed could be extended to modulus if all constraints had equal amounts of potential energy
but due to the drastic change in strengths between the constraints the prediction fails
Bauchy et al[85] showed also that hardness in the calcium-silicate-hydrate system is
controlled by the angular constraints which led to the development of angular constraint density
as the governing control for hardness When the analysis of the density of angular constraints is
considered through the free energy view it becomes apparent why this method works effectively
for some systems Models for predicting the hardness and elastic modulus of glass attempt to
explicitly connect the rigid bond energy to the macroscopic properties of the system The
assumption for the hardness models is that the energies for breaking each type of constraint are all
approximately equal and hence only the number of the constraints matters The argument can then
be extended for elasticity since elastic modulus is a bulk material property with the approximation
of all constraints being equal strength the number of constraints per volume should be related To
correct for this approximation when a weighted sum of free energies is used the estimation
becomes significantly more accurate Using the density of the glass a precise free energy per
volume of the rigid constraints can be calculated The knowledge that the energy of the bonds is
tied to the elastic modulus has been widely known but had not been quantified nor placed within
the context of topological constraint theory[203]
135
63 Ionic Conductivity
Ionic conductivity 120590 is related to the number of charge carriers n and the mobility of the
carriers 120583 as they diffuse through a network by[1]
Ze n = (105)
where Ze is the charge of the conducting species Since the charge-carrying ion is constant within
a given glass family the ionic conductivity depends only on the mobility and the concentration of
the charge carriers present However it is not typically feasible to measure 120583 and n independently
of each other and thus it is unclear which variable scales in an Arrhenius fashion thereby
controlling the ionic conductivity of a glass In the weak electrolyte model by Ravaine and
Souquet[204]ndash[207] the mobility of the charge carriers is considered constant while the
concentration scales in an Arrhenius function Conversely the strong electrolyte model assumes a
constant number of charge carriers while the mobility follows an Arrhenius function[208] [209]
Recent MD simulations from Welch et al[210] support the weak electrolyte interpretation for alkali
silicate glasses concluding that ionic conductivity is dictated by the concentration of charge
carriers Previous reports in the literature also supporting this view have proposed several
hypotheses to explain this conclusion[208] [209] [211]ndash[213] however further exploration is
needed
The simplest form of the temperature dependence of free ions is given by
0 exp an nE
kT
= minus
(106)
Here n0 is the total possible number of mobile ions and Ea is the activation barrier for an ion to
diffuse It is often assumed that the activation enthalpy is related to the energy needed to deform
the network temporarily to allow for ion motion[193] [209] [210] [212] however it is our
hypothesis that there is a coupling between the ion hopping event and the relaxation of the local
136
network allows for a permanent relaxation or deformation along ion channels This means that
there is some cooperative relaxation of the network along the diffusion path in conjunction with an
ionic hopping event These cooperative relaxations are discussed explicitly in the Adam-Gibbs
formalism[53] for describing the relationship between viscosity and configurational entropy Sc
expc
B
TS
=
(107)
where B is an activation barrier is the viscosity and is the viscosity of the liquid in the limit
of infinite temperature In the high-temperature (low-viscosity) liquid state there is a well-defined
relationship between the viscosity and diffusion (Stokes-Einstein relation) However this
relationship breaks down in the low-temperature (high-viscosity) glassy phase
Herein we present a new model for predictions for the compositional dependence of ionic
conductivity We can write the activation free energy barrier Eac for an ionic hopping event as
a c c c cE T S= minus (108)
Here is an enthalpic barrier for the ion to hop to a neighboring site CT is the configurational
temperature (a value that describes the distribution of energies of the ions) and cS is the entropy
of the activation barriers These quantities are associated with configurational changes in the glass
indicated by the subscript c In the weak electrolyte model all charge carriers have the same
mobility and hence the entropic effects are dominant This indicates a small value of an
assumption that will be validated by results later in this work leading to a simplified approximation
for the activation energy at temperatures below the glass transition but suffiently high temperatures
such that
a c c cE T S (109)
In the case of a nonequilibrium glass at low temperature the system becomes trapped in a localized
137
region of the energy landscape known as a ldquometabasinrdquo with slow inter-metabasin transitions[156]
[161] The MAP model of the nonequilibrium viscosity of glass relies on this partial confinement
within metabasins where variation of the activation barrier for relaxation (H) is related to the
fragility of the system by[44]
1
ln10g
dm
dH kT= (110)
Here m is the fragility index (as proposed by Angell[58]) and Tg is the glass transition temperature
Splitting the differential in Eq (110) and integrating we can write
ln10gmH kT= (111)
This relationship was also proposed by Moynihan et al[214] for the activation enthalpy of
relaxation of a glass[214] [215] Setting the Arrhenius form of the barrier described in Eq (111)
equal to the Adam-Gibbs equation (Eq (107)) we write
exp exp c
H B
kT TS
=
(112)
Solving Eq (112) for the configurational entropy and inserting the result into Eq (109) we obtain
the simple relationship
2
ln10
ca
g
T BE
k T m (113)
Dividing the above expression with that of a reference state r and assuming that B and is cT are
constant with respect to compositional variation we have
g r ra
a r g
T mE
TE m= (114)
This new equation is the first to relate the activation free energy for ionic hopping in a glass
to the glass transition temperature and fragility of its corresponding supercooled liquid It also
138
predicts a coupling between ionic diffusion and the cooperative rearrangements described by the
Adam-Gibbs relation This possibility was first proposed by Ngai and Martin[216] who showed a
correlation between the product of the Kohlrausch stretching exponent ( ) and the activation
barrier for relaxation with the activation barrier for ionic conductivity This implies that the
diffusion in glasses is not governed by an elastic component but instead controlled by the
surrounding cooperative rearrangements ie the -relaxation of the glass[14] in good qualitative
agreement with previous results shown by Potuzak et al[217] Further evidence comes from the
polymer community where the decoupling of relaxation and conductivity is reported with work
showing that the decoupling of viscosity and diffusion varies with the fragility and the glass
transition temperature[218] This is a general relationship which we demonstrate later to be valid
for binary alkali borate alkali silicates and alkali phosphate glasses as well as validating the
atomistic diffusion method using computational techniques
In order to test this proposed relationship between relaxation and diffusion through
simulation a glass with the composition 10 Na2O-90 B2O3 (mol ) was synthesized in molecular
dynamics with 1150 atoms using the potentials of Wang et al[162] The system was quenched at a
rate of 1 Kps from randomly placed atoms at 2500 K in the NPT ensemble to room temperature
as shown in Figure 35[210] To find the transition point energy of the diffusing ion a nudged elastic
band (NEB) calculation was performed An alkali ion was chosen and its closest alkali neighbor
was moved to another location so that the NEB calculation could be performed between those two
sites All calculations were carried out using the LAMMPS software package Eleven reaction
coordinates were used in the NEB calculation and the results are shown in Figure 36
139
Figure 35 The structure for the initial minimum energy configuration showing the boron (blue)
network with interconnecting oxygens (red) and the interstitial sodium ions (yellow)
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and sodium is
ivory The barrier is overestimated compared to experimental data this could be from several
sources of error such as potential fitting thermal history fluctuations or sampling too few
transitions The line is drawn as a guide to the eye
140
Although the transition point energy is key for determining the activation barrier for diffusion here
we are also interested in the dynamics of the atoms around the mobile ion Since we have
hypothesized here that the ion diffusion is dependent on the ability for the local region to relax and
deform instead of elastically straining we expect a local deformation (relaxation) along the ion
path Figure 37 shows that along the ion diffusion tunnel there is indeed a permanent displacement
of the atoms
141
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function of
reaction coordinates The middle shows the local deformation around the ion of any atom that
moves in between inherent structure mandating a relaxation force The color shows the degree of
deformation
142
In the final configuration of Figure 37 the permanent deformation required to lower the
energy of the system is clearly seen If it were purely an elastic response to dilate a pathway the
displacement between two inherent structures should be confined to changes with only the mobile
ion however this is not observed Instead along the pathway of the ion movement the network
forming atoms also deform and change local positions to minimize the energy implying a net
relaxation around the ion ldquopathwaysrdquo To apply this and show the validity of Eq (114) the sodium
borate lithium borate lithium phosphate and sodium silicate systems were considered with the
model results compared to experimental values in Figure 38 To confirm this model literature values
for activation barriers have been used The experimental activation barriers for the borates are taken
from the work of Martin[213] while the fragilities and the glass transition temperatures are from
Nemilov[219] The viscosity curve of the silicate system was fit with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) equation of viscosity[55] using the data from Poole[220] and Knoche et
al[221] with the activation barriers from Martinsen[222] Lithium phosphate activation data was
taken from Ngai and Martin[216] with the glass transition and fragility taken from Hermansen et
al and predicted values from topological constraint theory[223]
143
Figure 38 Different network formers and the prediction of the activation barrier from our model
compared with activation barriers from literature (A) Sodium silicate predictions and experimental
values[222] the error is calculated from the error in the fragility when fitting the data (B) Lithium
phosphate activation energy[216] predicted with topological constraint theory and compared with
the experimental values (C) Predictions over two different systems of alkali borates[213] sodium
and lithium with a reported R2 of 097
144
It is worth noting that this model is predicting a non-equilibrium kinetic property using
equilibrium viscosity parameters a connection that has been demonstrated in the past for viscosity
and relaxation models[61] In this work we are applying the same principles to the diffusion of
ions in a network The diffusion and viscosity in the case of equilibrium are known to be intimately
linked using the Stokes-Einstein expression (which breaks down for highly cooperative
rearrangements near and below the glass transition temperature[224]) Here we are no longer
connecting the two properties explicitly but instead connecting the activation barrier for diffusive
hopping to the two key parameters governing the viscous flow of the supercooled liquid state
The failure of the Stokes-Einstein expression near the glass transition is a well-known
problem and has been studied extensively for purposes of predicting crystallization rates[167] This
occurs because there is a breakdown of the ergodicity of a system around the glass transition In
the MAP model[44] the viscosity is controlled by the ergodic parameter x and is given by
min( )
max( )
f
p
f
T Tx
T T
=
(115)
Here the exponent p is related to the sharpness of the ergodic to non-ergodic transition Using the
ergodic parameter the overall viscosity ( ) is expressed as a linear combination of equilibrium
viscosity ( eq ) and non-equilibrium viscosity ( ne ) such that
10 10 10log ) log ( ) (1 ) log ( )( f eq f ne fx T T x T TT T = + minus (116)
By combining the Stokes-Einstein expression and the model proposed in this work we can define
the diffusion of an ion through the network as
( )10 10 10 0log ( 1 log exp ) l6 )
og(
b af
f
kx
T ED T T D
a T Tx
kT
minus + minus =
(117)
Here a is an empirical constant related to the size of the diffusing species k is Boltzmannrsquos
145
constant aE is the activation barrier calculated with Eq (114) and D0 is the limit of diffusion as
the temperature approaches infinity D0 should be chosen such that at fT T= the diffusion is
continuous This can be justified because the preexponential factor is related to the entropy of the
system and the fictive temperature is the temperature at which the configurational entropy of the
transition barriers becomes nearly constant Only one parameter is left unknown a since the
viscosity and fictive temperature can be calculated with an arbitrary thermal history using a
relaxation modeling tool such as RelaxPy[42] Implicit in RelaxPy are the assumptions pertaining
to the viscosity made by Guo et al[61] A similar model was first proposed by Cassar et al[167]
for modeling diffusion during crystallization although their model used an empirical hyperbolic
tangent function to approximate the ergodic factor In contrast the current model can be considered
explicitly as a function of thermal history and composition
64 Machine Learning Expansion
Though the TCT models presented here for different properties are immensely useful they
do have limitations the greatest of which is that intimate knowledge of the glassy structure is
required to parameterize these models This is due to the fact that the landscape is fundamentally
informed by the underlying structure of the network and as such to predict the dynamics of the
landscape information on the structure must be incorporated in some form However there is an
alternate method in which the physical origin is not considered in the development of the model
machine learning Machine learning comes in many forms and has become crucial to the study of
materials in the modern era There is a plethora of machine learning techniques however in this
paper we are focused on four key methods linear fitting (LF) methods random forest (RF)
symbolic regression (SR) and neural networks (NN)[3] [80] [225] [226] These methods are
146
explored extensively in literature and the exact underlying mathematics is beyond the scope of this
work In this work these methods have been used to create models for number of constraints (LF)
glass stability (RF) fragility (NN) melting point (NN) the infinite limit of viscosity (SR) Youngrsquos
modulus (NN) and coefficient of thermal expansion (NN)
To enable a linear fitting method for the number of constraints a simple steepest descent
algorithm was used to find the number of constraints due to each component There is no direct
experimental way to measure the number of constraints so instead a technique leveraging some
other property must be used to approximate the number of constraints In this case hardness is
related to the number of constraints through
25v cH A n= minus (118)
In which A is an empirical scaling parameter that is dependent on load indenter geometry and
glass family In this linear approach to determining constraints cn can be given by
0
compon
k
c k k
ents
cn x n=
= (119)
Combining this expression with a hardness database grouped by composition (in this case Sciglass)
allows for an approximate number of constraints of the system but due to A being a variable there
must be a fixed constraint in the system and every set of data is given a unqiue value of A The
fixed constraint in this system is that silicon atoms are four coordinated The value of A is chosen
such that the error between the prediction and the hardness is minimized The total process is as
follows
Choose a set of c kn
1 Calculate number of constraints for all compositions
2 Iterate over every dataset in Sciglass with gt 1 datapoints and scale by a factor of A
3 Calculate total error for steepest descent algorithm
4 Descent error slope
5 Repeat until error has been minimized
147
This then leads to an estimate for the number of constraints for every atomic species which can
then be used to estimate important properties such as chemical durability hardness and
qualitatively can predict glass stability
Two other techniques that are used are RF and SR When implemented a grid search for
the hyperparameters were used with a testing set used to determine the efficiency of each set of
hyperparameters Additionally each model was trained repeatedly and the lowest error was taken
The RF method was used to predict glass stability while undergoing a novel processing technique
called alkali-proton substitution (APS) The SR method was used to understand the compositional
dependence of the infinite temperature viscosity limit used in 3-parameter viscosity models To do
this the Sciglass database was taken (as implemented in GlassPy[227]) and sets of data that
included more than eight datapoints were fit to the MYEGA equation If the RMSE error of the fit
was less than 001 Pa s the fragility was greater than 17 and the value of the log infinite limit was
293 5minus it was accepted into the final database Each of these criteria were added due to the well-
known fact that this database is full of error and imprecise measurements The symbolic regression
was then performed through the grid search and the equation that performed best and reappeared is
given as
10loggT
mA B = + (120)
With A and B being parameters that varied depending on the hyper-parameters the infinite limit
threshold and the accuracy threshold used when creating the database We are currently in the
process of refining values for A and B is ongoing
This method allows for an easy prediction of the infinite limit so if the glass transition and
fragility are predicted through some other means (such as ML or TCT) a better prediction of the
third parameter is known without just accepting -293 as the mean Plots of the infinite limit vs the
148
fragility glass transition and Tgm showing the clear dependance of the infinite limit on the term
Tgm are shown in Figure 39
149
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top left) shows
the distribution of the infinite temperature limit in the database after limits exerted on the system
(Top right) The distribution of the infinite viscosity limit vs the glass transition (Bottom left) The
relationship between fragility and infinite temperature limit of viscosity (Bottom right) The infinite
temperature limit of viscosity vs the key metric predicted by SR
150
NN are perhaps the most widely discussed ML tool applied in literature This makes sense
given they are universal function approximators and are quite easy to train given sufficient data
The most difficult part of creating a NN is adjusting the hyperparameters so that good accuracy can
be found without causing overfitting To avoid overfitting the data is divided into a testing set then
a training set (just as for RF and SR) but instead of doing a grid search for the hyperparameters we
used gradient boosted regression trees to find the optimal set of parameters This consists of
creating an initial tree then evaluating it based on a small set of data and finding the minimum
point then running a new set with perturbations around that point to create another tree This is
repeated until the value converges to a small error In this work the value being determined is the
lowest error of 100 sequentially trained neural nets with a set of hyperparameters This is
particularly useful because there is no concern for the type of parameters (string int floats) that
determine the value at the end so one can do a mixed parameter optimization that includes every
conceivable variable The values being optimized are given by
bull Learning rate [10-5 10-1] (float on logarithmic scale)
bull Decay rate for Adam Optimization [10-6 10-2] (float on logarithmic scale)
bull Hidden layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull Output layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull The number of nodes in each layer [81024] (int)
bull Batch size [1256] (int)
bull Patience [5250] (int)
In Table 8 we have listed the hyper-parameters that were used for the fragility melting point
Youngrsquos modulus and co-efficient of thermal expansion (CTE)
151
Table 8 Hyperparameters for different neural networks after hyper-optimizations
Model Fragility Melting
Temperature
Youngrsquos
Modulus
CTE
Learning Rate 0035 0018 0027 00091
Decay Rate 000293 000095 000161 0002
Hidden Layer RELU RELU RELU RELU
Output Layer Linear Linear SELU RELU
Number of Hidden
Layers
2 1 1 1
Number of Nodes 235 506 514 470
Batch Size 4 118 211 125
Patience 25 250 23 22
RMSE 5 [-] 805 [K] 68 [GPa] 96 x 10-7 [1 K]
152
Chapter 7
Designing Green Glasses for the 21st Century
Thus far the work presented in this dissertation has enabled a new set of design tools for
everything from crystallization glass relaxation and optimizing over the ideal compositional space
we can use them for designing new glasses Building on these models and insights new applications
of glass are enabled specifically three glasses an ion conducting glass for batteries a proton
conducting glass for hydrogen fuel cells and a commodity glass for consumers All three are
currently undergoing testing but experimental results have been reported back as of yet Overall
both the hydrogen and ion conducting glasses have been designed thorough a combination of ML
and underlying physical models and as such hare reported herein
71 Glass Electrolytes
An emerging high-interest application of glass is solid state batteries Glasses offer a
possible solution to the growing energy crisis however to realize this new paradigm new research
methods are needed The main barrier for any new batteries to be commercialized is a high
conductivity at room temperature (gt 10-3 S cm) high stability ease of processing and ultimately
an alternative must be cheaper than current liquid state Li-ion batteries glass is an attractive
candidate due to the innate stability a simple processing technique the infinite variability and the
relatively low cost of production In addition the structure of a glass is a liquid-like structure which
may encourage ion migration like that seen in liquids Despite all of these advantages only a few
compositions have realized the requirements but further research is needed to find optimal
compositions
153
To optimize the composition for activation barriers we must understand the relationship
between conductivity and structure Structural effects can be propagated in two different methods
based on the Arrhrenius expression for conductivity
0 exp aE
kT
minus =
(121)
The two methods are through 0 (the infinite temperature conductivity) and the activation barrier
( aE ) The pre-exponential factor is approximately a constant for simliar compositions while the
activation barrier varies dramatically over small compositional spaces making it the larger of the
two concerns Many models have been presented to predict the compositional dependence of the
activation barrier the Anderson-Stuart (AS) model the Christensen-Martin-Anderson-Stuart
(CMAS) model the weak electrolyte (WE) model Kohlrausch exponent model (KEM) from Ngai
et al[216] and there is the model proposed in this dissertation which we will call the Wilkinson
viscous cooperative conductivity model (WVCC)[8] [88] [204] [207]ndash[209]
To optimize machine learning (ML) is a powerful technique that is implemented The best
method to train the ML is to use the direct relationship between the compositions and the activation
barrier but this is not currently feasible due to a lack of central depository or database for this
information To make the prediction of the activation barrier we need use some model that is reliant
on commonly accesible parameters This rules out all models for this type of analysis except since
WVCC since the other models require some fitting parameters not accessible Each models required
parameters fitted values (please note that the models typically include approximations for these
values) and caveats are shown in Table 9
154
Table 9 A table with some ionic conductivity models and the parameters needed for them as well
as the disadvantages for each These are not the only models but are representative of those
commonly used in literature
Model Required Parameters Fit Parameters Caveats
AS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
lsquoMadelungrsquo Constant
Covalency Parameter
Doorway Radius1
The Madelung
Constant is not smooth
as a function of
composition and the
fitting parameters scale
the activation energy
non-proportionally
CMAS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
Dielectric Permittivity
lsquoMadelungrsquo Constant
Doorway Radius1
Jump Distance of Ion1
The same issues as AS
but in addition the
dielectric permittivity
lacks a database as
well
WE Charge Concentration Equilibrium Coefficient1 The equilibrium
coefficient drastically
changes the prediction
non-linearly
KEM Stretching Exponent Proportionality
Coefficient
There is not enough
data available to know
the compositional
dependence of the
stretching exponent
WVCC Glass Transition
Fragility Index
Proportionality
Coefficient
Due to the
proportionality
constant the activation
barrier can only be
known in a local
composition range
1 There are ways to approximate this value based on additional data
155
The WVCC model predicts that the activation barrier for ionic conductivity is given by
a
g
EA
mT= (122)
in which A is a proportionality constant This means that to predict the glass behavior we donrsquot
need to know things about the glass but instead merely about the liquid state where the viscosity is
readily available Leveraging this technique if the viscosity information as a function of
compositions is available then the local glass with the lowest activation barrier can be found by
minimizing 1 gT m To access this information multiple options are available such as topological
constraint theory but in this work we will leverage ML since it is easily applied to a large
compositional space
To get the viscosity we will use neural networks (NN) A well trained NN for the glass
transition is readily available and as such will be used for our prediction of gT In the previous
section there was presented a trained fragility NN which enables a complete optimization of the
local composition based on WVCC To generate a database of fragility values that the NN was
trained on SciGlass viscosity values were fit with MYEGA model The data had to fit following
criteria
1 The total number of data points for fitting had to be greater than 5 from the same literature
source
2 The root-mean-square-error of the fit had to be less than 001 Pa s
3 The infinite temperature value of viscosity had to be within 3 orders of magnitude of the
accepted logarithmic value (-293 log(Pa s))
An additional neural network recently created by Cassar[228] was also used for an independent
comparison Cassarrsquos NN was based on a hybrid physical ML work and may offer better insights
when extrapolating far from experimental datapoints In this study to prove this concept we will
only focus on P2O5 B2O3 Al2O3 and Li2O
156
This gives an entire method to predict glass battery candidates This technique will find all
of the local minimum but due to the fact that there is a proportionality constant it is unknown which
local minimum is the absolute minimum This means that the composition with the lowest
activation barrier in the space is identified but another technique is needed to investigate which one
of the identified compositions is best The method follows
1 Randomize an initial glass candidate with the content of being normalized
2 Run the glass candidate through the NN to find values of fragility and the glass transition
3 Perturb the composition to find the gradient
4 Step down the gradient
5 If the new gradient is 0 this is a local minimum and a possible glass candidate return to 1
Otherwise return to step 3
This method cannot find global optima because there is no guarantee that the
proportionality constant will stay constant across wide compositional spaces To narrow it down
an additional technique to compare the predicted values is needed (such as another activation
barrier model molecular dynamics simulations a regression algorithm or simply melt the proposed
compositions) To further pair down in this work a k-means algorithm was used along with the
assumption that the proportionality constant was slowly changing over compositional space This
allowed us to take the best glass in each grouping predicted by k-means The glasses we have
predicted (using both the separate predictor NN and the method by Cassar et al[228]) are listed
below in Table 10 while an example prediction for 1000 gmT (which should scale with the
activation barrier) in the Na2O ndash B2O3 ndash SiO2 system is shown in Figure 40 The glasses are currently
undergoing melting at Coe College and Iowa State It is worth noting that composition D is a well-
known composition that is known to exhibit criteria close to the criteria for conductivity If this
method shows that these compositions are better than other compositions in the family found in
literature then it can be expanded to include a wider range of component and hopefully finding a
universal candidate for glass batteries
157
Table 10 The predicted compositions based on the optimization scheme proposed
Compositions Li2O Al2O3 P2O5 B2O3
A- This work NN 55 31 0 14
B- This work NN 44 1 5 50
C- Cassar 60 26 2 13
D- Cassar 50 0 0 50
Figure 40 The prediction of the scaling of the activation barriers for a common sodium borosilicate
system
72 Hydrogen Fuel Cell Glasses
An emerging high-interest application of glass that is proton-conducting intermediate
temperature fuel cells[7] [229] A fuel cell in short is a device that turns chemical fuel into
electricity without the use of combustion It is often described as a chemical battery due to the
similarity of a fuel construction with batteries One of the crucial components to a working
intermediate range fuel cell is an electrolyte with a high proton migration The electrolyte must be
stable over a wide range of temperatures (up to 673 K) and a proton conductivity greater than 10-2
S cm-1 at the operating temperature[7] Previously phosphate glasses near the metaphosphate
compositions were used since they carried the appropriate number of residual protons for
conductivity studies However this restricted the phase space to only n-2 dimensions drastically
reducing the degrees of freedom of the problem and artificially restricting possible solutions In
2013[230] to access the full phase space a new method was developed where a sodium phosphate
glass could have all the sodium ions replaced with protons This allows high concentrations of
protons to be achieved with a variety of starting compositions A previous report[230] has
summarized successful samples these data are used as a training set for this study with additional
data on compositions that have failed taken from internal theses
To choose the candidate material for each model a cost function is defined The cost function
is an analytical function whose inputs are the fraction of each oxide component and the output is a
value that rates the composition The cost function consists as of many properties that are of interest
In the application of oxide electrolytes for an intermediate range fuel cell the two main properties
to worry about are the ionic conductivity at high temperatures (we will choose 473 K due to this
being the low end in which these materials are considered) and the relative stability of each phase
159
This is especially important since the material will undergo the APS (alkali-proton substitution) in
which the sodium will be replaced with protons in the bulk of the material The cost function used
in this work is given by
473 1( ) log 000T K Sx = minus= minus (123)
In which S is either a 0 or 1 (representing a crystal or a glass respectively) and is the
ionic conductivity The function is arbitrary and could be expanded to include as many terms are
as needed in the goal of this glass The stability is heavily weighted because it is more important
than the conductivity and any sample that is not stable should not be considered In order to predict
each of these samples a hybrid physicalempirical approach is used For predicting the ionic
conductivity of a proton-swapped glass we began with an observed relationship between the glass
transition temperature and the conductivity at 473 K This empirical relationship is shown in Figure
41 The relationship although empirical can provide some insights into the physics of proton
transferred glasses (no universality is claimed in this work) The activation barrier for these glasses
has widely remained unchanged as noted in a previous work due to the fact that the activation
barrier is simply to disassociate the proton with the non-bridging oxygen (NBO) and once the
proton is free interacts rarely with the network until it is rebounded to said network Thus the pre-
factor can be understood as being related to the degeneracy of proton conduction pathways When
the network has lower configurational entropy there are fewer pathways for the proton to travel in
this case it is known that the configurational entropy is then the dominating effect for the glass
transition[53]
The relationship between the glass transition temperature and the configurational entropy
can be expanded through the Stokes-Einstein relationship evaluated at the glass transition
22
1273 10
6 10g
g
T T g
kTnZeD nZe
anT
minus
= = = (124)
160
In which a is approximately the size of a proton and n is the number of charge carriers (which is
approximately a constant according to the weak electrolyte theory) This determines the intercept
of the glassy form proton conductivity and then the activation barrier (slope) is found to be
approximately the same for all proton conducting glasses meaning that the intercept is the dominant
effect on the behavior of the glass and the only variable controlling the intercept is the glass
transition
Leveraging the configurational entropyrsquos relationship with the glass transition along with
topological constraint theory we can write an expression for the glass transition and as such gain
predictive power for proton conductivity To predict the glass transition temperature (Tg) we start
with the Adam Gibbs model of viscosity[53] Topological constraint theory then states that the
degrees of freedom of the network is proportional to the configurational entropy[23] [56] [57]
Using the liquid state definition of the glass transition (1210 = Pa s) and the well-known infinite
temperature of viscosity we can then write
1
1493 lng
BT
kf=
(125)
Where f is the degrees of freedom Letting the constants be equal to A and rewriting the expression
in terms of constraints we arrive at a predictive formula for the glass transition[109]
3
g
x c x
x
AT
m n=
minus (126)
Where nc is the number of constraints provided by each component at the glass transition associated
with each component x and their molar fraction mx The value of the constraints was then linearly
parameterized to the glass transition data from literature[7] [230] [231] This is not the most
explicit approach to counting constraints however it is the most convenient when considering
large phase spaces being optimized over This glass transition temperature is then converted to
conductivity using the empirical translation shown in Figure 41
161
The other term in our cost function is that of stability The question of what forms glass is
a notorious question that goes back to the first serious days of research into materials It has been
shown that no one metric is a good predictor and no metric is universal However one must be used
to rule out bad compositions from the start To do this a Random Forest method is implemented
Though this may fail in edge cases it will at least help identify the right area to explore A
comprehensive review of random forest methods and machine learning methods for prediction of
glass properties can be found elsewhere[3] [4] [225] The amount of information previously
obtained about what forms glass in the compositional family we will be working on is not enough
to use random forest methods on just the fraction of each phase In order to circumvent this issue
we chose four physical parameters that would be used as features for the predictions
bull Total mol oxide modifier This was chosen because it is well known that modifiers break
up the network and most glass forming theories are related to the network percentages We
considered the oxide versions of H Na Ba Sr and Ca as the modifiers
bull Total mol network former Similarly it is well known the network influences glass
forming We considered B Ge and P as the network forming cations
bull Mean cation charge Though there has been previous work on this topic it is rarely
considered as a lsquonormalrsquo predictor of glass forming capability It is included here because
these glasses are mainly invert and as such a higher field strength will increase the cohesive
nature of the network
bull Entropy of mixing This is an important parameter to consider with so many components
it may be stabilized by the entropy of mixing The entropy of mixing (Sm) was calculated
using the Gibbs entropy
The Gibbs entropy is given by
lnm x
x
xS k m m= minus (127)
The random forest model consisted of 100 trees with a maximum depth of 3 The results of both
the random forest model the topological predictions the optimization and the resultant glass are
shown in the results section The optimization was limited to the range of each component that has
been explored experimentally as well as an additional constraint of at least 5 mol rare earth oxide
and at least 5 mol of boron oxide or germania this is justified elsewhere[231]
162
Figure 41 The relationship between the glass transition and the proton conductivity This is justified
two ways one through the relationship of the entropy of diffusion and glass formation (the Adam-
Gibbs model) and through the fact that water is known to depress the glass transition
163
To explain the results we will first start with the models listed to confirm their validity
The first model is the prediction of the glass transition temperature Figure 42 shows the accuracy
of the glass transition prediction as well as the relative contribution of each component to the glass
transition In this figure any component that has a number of constraints greater then PO52 will
increase the glass transition and as such decrease the conductivity Thus it is preferable to construct
a glass with only components whose constraints at Tg are less than PO52 as long as the glass
remains stable Interestingly whether the data includes only the glass transitions of protonated glass
or all glasses determines the role of La Barium and sodium both decrease the glass transition
playing the role of the more traditional modifier however they do provide some rigidity to the
network Protons have a net negative effect which we believe to be a result of the APS process
where they are introduced as the sodium is leaving so this negative effect corresponds to a breaking
of the rigidity that exists It is also interesting to note that adding a boron phase to the sample
increases the glass transition though not as much as some of smaller alkaline earth samples
assumingly due to their increased field strength The worst network former appears to be GeO2
To calibrate the accuracy of the random forest model a confusion matrix is shown It shows
that the RF always predict that a glass will be a glass however there is some error when it forms a
crystal with 23 of the time being misidentified Though random forest methods are empirical we
can derive some physical meaning by looking at the relative importance of each feature From the
random forest study of stability we are able to rank the relative importance of each feature used
The mean cation charge mixing entropy and percent network formers are all of considerable
importance while the percent modifiers is less so
Once these models were developed and checked the reliability of the cost function is
confirmed After which the minimization of the cost function was performed and one glass was
chosen as optimal Boron was found to be preferential in the glass but was limited since the
compositional space was bound by previous samples (an additional sample is undergoing
164
characterization where the boron content was slightly increased) The optimized sample when
synthesized had some visible nucleation so the optimization while redone with limiting the
component maxima to just the limits of those where compositions in which APS was successfully
performed This resulted in the OP composition The three samples currently are undergoing
experimental characterization are listed in Table 11
165
Table 11 The compositions synthesized in this work These compositions were predicted by
minimizing the cost function described in Eq (123) OP is the variant that was melted after OP
partially crystallized B-OP appeared to have surface nucleation in some spots but was cut and
removed before APS treatment
Name PO52 NaO12 BaO LaO32 GeO2 BO32
OP 44 36 10 5 3 2
OP 49 36 5 5 3 2
B-VOP 44 36 10 5 1 4
166
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine the glass
forming region Over top the constraints at the glass transition provided by each oxide species is
listed Since the objective is to decrease Tg while staying in the glass forming region we will
attempt to minimize use of elements that increase the glass transition temperature (nc gt 17)
167
Chapter 8
Conclusions
The rapid rate of new information in the current scientific climate and the infinite
variability of glass both stand as both unique challenges and opportunities In this dissertation we
have divided the influence of energy landscapes on the effects of properties into two independent
spaces the compositional and the thermal history dimensions By building models that succinctly
and accurately describe the dynamics of different hyper-coordinate (composition thermal history
crystallinity) changes the feasibility of designing new glasses for the challenges of the 21st century
is obtainable
Before designing new models the current state-of-the-art models need to be implemented
and understood To reach this end two softwares based on previous work were created RelaxPy
and ExplorerPy RelaxPy is an implementation of the MAP model and though the MAP model is
powerful when predicting the dynamics of the glass RelaxPy ultimately showed that fictive
temperature is insufficient at capturing the underlying physics and is intensive to parameterize
ExplorerPy was created to standardize the approach to mapping energy landscapes Energy
landscapes served as the key method of understanding the deeper dynamics of glasses glass-
ceramics and liquids
The thermal history dependence of glass with respect to both relaxation and crystallization
has been incorporated into the existence of new models called ldquotoy landscapesrdquo The toy landscapes
have built upon the physics of previous models such as the MAP model however due to the lower
parametrization cost and the built-in increased physicality toy landscapes pose as a tool to increase
our understanding and speed-up the rate of new glass discoveries This tool can deal with the
complexity of the higher dimensional spaces due to crystallization and relaxation without
assumptions concerning fictive temperature
168
The compositional dimensions are the remaining dimensions that must be optimized when
designing a glass or glass-ceramic Building on previous work models have been developed that
enable predictions of Youngrsquos modulus surface reactivity and ionic conductivity However there
are additional properties that are needed and for those we have proposed novel machine learning
approaches This dissertation has not enabled the design of glass for every application However
it lays the groundwork and approaches to design glasses for societyrsquos growing needs Presented as
well are the methods used to design glasses that could satisfy the requirements for solid state glass
electrolytes and hydrogen fuel cells All of these techniques together promise to be a powerful new
framework to build the glasses of the future
169
References
[1] A K Varshneya and J C Mauro Fundamentals of Inorganic Glasses 3rd ed Elsevier
2019
[2] J C Mauro A Tandia K D Vargheese Y Z Mauro and M M Smedskjaer
ldquoAccelerating the Design of Functional Glasses through Modelingrdquo Chem Mater vol
28 no 12 pp 4267ndash4277 2016 doi 101021acschemmater6b01054
[3] J C Mauro ldquoDecoding the glass genomerdquo Curr Opin Solid State Mater Sci pp 1ndash7
2017 doi 101016jcossms201709001
[4] H Liu Z Fu Yang X Xu and M Bauchy ldquoMachine learning for glass science and
engineering A reviewrdquo J Non Cryst Solids no March p 119419 2019 doi
101016jjnoncrysol201904039
[5] M M Smedskjaer C Hermansen and R E Youngman ldquoTopological engineering of
glasses using temperature-dependent constraintsrdquo MRS Bull vol 42 no 01 pp 29ndash33
2017 doi 101557mrs2016299
[6] J C Mauro and M M Smedskjaer ldquoStatistical mechanics of glassrdquo J Non Cryst Solids
vol 396ndash397 pp 41ndash53 2014 doi 101016jjnoncrysol201404009
[7] T Omata et al ldquoProton transport properties of proton-conducting phosphate glasses at
their glass transition temperaturesrdquo Phys Chem Chem Phys vol 21 no 20 pp 10744ndash
10749 2019 doi 101039c9cp01502g
[8] Z A Grady C J Wilkinson C A Randall and J C Mauro ldquoEmerging Role of Non-
crystalline Electrolytes in Solid-State Battery Researchrdquo Front Energy Res vol 8 p
218 2020 doi 103389fenrg202000218
[9] R C Welch et al ldquoDynamics of glass relaxation at room temperaturerdquo Phys Rev Lett
170
vol 110 no 26 p 265901 2013 doi 101103PhysRevLett110265901
[10] M Montazerian E D Zanotto and J C Mauro ldquoModel-driven design of bioactive
glasses from molecular dynamics through machine learningrdquo Int Mater Rev vol 0 no
0 pp 1ndash25 2019 doi 1010800950660820191694779
[11] E D Zanotto and J C Mauro ldquoThe glassy state of matter Its definition and ultimate
faterdquo J Non Cryst Solids vol 471 pp 490ndash495 2017 doi
101016jjnoncrysol201705019
[12] R G Palmer ldquoBroken ergodicityrdquo Adv Phys vol 31 no 6 pp 669ndash735 1982 doi
10108000018738200101438
[13] J C Mauro P Gupta and R J Loucks ldquoContinuously broken ergodicityrdquo J Chem
Phys vol 126 no 18 p 184511 2007 doi 10106312731774
[14] M Micoulaut ldquoRelaxation and physical aging in network glasses A reviewrdquo Reports
Prog Phys vol 79 no 6 p 066504 2016 doi 1010880034-4885796066504
[15] D J Wales and H A Scheraga ldquoGlobal optimization of clusters crystals and
biomoleculesrdquo Science (80- ) vol 285 no 5432 pp 1368ndash1372 1999 doi
101126science28554321368
[16] J C Mauro R J Loucks and P Gupta ldquoFictive temperature and the glassy staterdquo J
Am Ceram Soc vol 92 no 1 pp 75ndash86 2009 doi 101111j1551-2916200802851x
[17] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 2009 doi 10106313152432
[18] F H Stillinger and T A Weber ldquoHidden structure in liquidsrdquo Phys Rev A vol 25 no
2 pp 978ndash989 1982 doi 101103PhysRevA25978
[19] F H Stillinger ldquoSupercooled liquids glass transitions and the auzmann paradoxrdquo J
Chem Phys vol 88 no 12 pp 7818ndash7825 1988 doi 1010631454295
171
[20] S Neelamraju C Oligschleger and J C Schoumln ldquoThe threshold algorithm Description of
the methodology and new developmentsrdquo J Chem Phys vol 147 no 15 p 152713
2017 doi 10106314985912
[21] D Prada-Gracia J Goacutemez-Gardentildees P Echenique and F Falo ldquoExploring the free
energy landscape From dynamics to networks and backrdquo PLoS Comput Biol vol 5 no
6 p e1000415 2009 doi 101371journalpcbi1000415
[22] E Pollak A Auerbach and P Talknerz ldquoObservations on rate theory for rugged energy
landscapesrdquo Biophys J vol 95 no 9 pp 4258ndash4265 2008 doi
101529biophysj108136358
[23] G G Naumis ldquoEnergy landscape and rigidityrdquo Phys Rev E vol 71 p 026114 2005
doi 101103PhysRevE71026114
[24] S J Benkovic G G Hammes and S Hammes-Schiffer ldquoFree-energy landscape of
enzyme catalysisrdquo Biochemistry vol 47 no 11 pp 3317ndash3321 2008 doi
101021bi800049z
[25] M Roca B Messer D Hilvert and A Warshel ldquoOn the relationship between folding and
chemical landscapes in enzyme catalysisrdquo Proc Natl Acad Sci U S A vol 105 no 37
pp 13877ndash13882 2008 doi 101073pnas0803405105
[26] P Li G Henkelman J A eith and J Johnson ldquoElucidation of aqueous solvent-
mediated hydrogen-transfer reactions by ab initio molecular dynamics and nudged elastic-
band studies of NaBH4 hydrolysisrdquo J Phys Chem C vol 118 no 37 pp 21385ndash21399
2014 doi 101021jp507872d
[27] S P Niblett M Biedermann D J Wales and V De Souza ldquoPathways for diffusion in
the potential energy landscape of the network glass former SiO2rdquo J Chem Phys vol
147 no 15 p 152726 2017 doi 10106315005924
[28] Y Yu M Wang M M Smedskjaer J C Mauro G Sant and M Bauchy
172
ldquoThermometer Effect Origin of the Mixed Alkali Effect in Glass Relaxationrdquo Phys Rev
Lett vol 119 no 9 pp 1ndash5 2017 doi 101103PhysRevLett119095501
[29] Y Yu M Wang D Zhang B Wang G Sant and M Bauchy ldquoStretched Exponential
Relaxation of Glasses at Low Temperaturerdquo Phys Rev Lett vol 115 no 16 p 165901
2015 doi 101103PhysRevLett115165901
[30] Steve Plimpton ldquoFast Parallel Algorithms for Short-Range Molecular Dynamicsrdquo J
Comput Phys vol 117 no 1 pp 1ndash19 1995 doi
httpsdoiorg101006jcph19951039
[31] N Mousseau and G T Barkema ldquoTraveling through potential energy landscapes of
disordered materials The activation-relaxation techniquerdquo Phys Rev E vol 57 no 2
pp 2419ndash2424 1998 doi 101103PhysRevE572419
[32] F El-Mellouhi N Mousseau and L J Lewis ldquo inetic activation-relaxation technique
An off-lattice self-learning kinetic Monte Carlo algorithmrdquo Phys Rev B vol 78 p
153202 2008 doi 101103PhysRevB78153202
[33] D J Wales ldquoDiscrete path samplingrdquo Mol Phys vol 100 no 20 pp 3285ndash3305 2002
doi 10108000268970210162691
[34] G Henkelman and H Joacutensson ldquoImproved tangent estimate in the nudged elastic band
method for finding minimum energy paths and saddle pointsrdquo J Chem Phys vol 113
no 10 pp 9978ndash9985 2000 doi 10106314961868
[35] D Sheppard P Xiao W Chemelewski D D Johnson and G Henkelman ldquoA
generalized solid-state nudged elastic band methodrdquo J Chem Phys vol 136 p 074103
2012 doi 10106313684549
[36] A Pedone G Malavasi M C Menziani A N Cormack A V and N York ldquoA New
Self-Consistent Empirical Interatomic Potential Model for Oxides Silicates and Silica-
Based Glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
173
[37] A Takada C R A Catlow and G D Price ldquoComputer modelling of B2O3 II Molecular
dynamics simulations of vitreous structuresrdquo J Phys Condens Matter vol 7 no 46 pp
8693ndash8722 1995 doi 1010880953-8984746004
[38] A C T Van Duin S Dasgupta F Lorant and W A Goddard ldquoReaxFF A reactive
force field for hydrocarbonsrdquo J Phys Chem A vol 105 no 41 pp 9396ndash9409 2001
doi 101021jp004368u
[39] J C Mauro and A Varshneya ldquoModel interaction potentials for selenium from ab initio
molecular simulationsrdquo Phys Rev B vol 71 p 214105 2005 doi
101103PhysRevB71214105
[40] J C Mauro R J Loucks J Balakrishnan and S Raghavan ldquoMonte Carlo method for
computing density of states and quench probability of potential energy and enthalpy
landscapesrdquo J Chem Phys vol 126 no 19 2007 doi 10106312733674
[41] Y Z Mauro C J Wilkinson and J C Mauro ldquo ineticPy A tool to calculate long-time
kinetics in energy landscapes with broken ergodicityrdquo SoftwareX vol 11 p 100393
2020 doi 101016jsoftx2019100393
[42] C J Wilkinson Y Z Mauro and J C Mauro ldquoRelaxPy Python code for modeling of
glass relaxation behaviorrdquo SoftwareX vol 7 pp 255ndash258 2018 doi
101016jsoftx201807008
[43] J C Mauro ldquoTopological constraint theory of glassrdquo Am Ceram Soc Bull vol 90 no
4 pp 31ndash37 2011 doi 101039c3ee40810h
[44] J C Mauro D C Allan and M Potuzak ldquoNonequilibrium viscosity of glassrdquo Phys Rev
B vol 80 p 094204 2009 doi 101103PhysRevB80094204
[45] M Cardona R V Chamberlin and W Marx ldquoThe history of the stretched exponential
functionrdquo Ann der Phys vol 16 no 12 pp 842ndash845 2007 doi
101002andp200710269
174
[46] P Grassberger and I Procaccia ldquoThe long time properties of diffusion in a medium with
static trapsrdquo J Chem Phys vol 77 no 1982 pp 6281ndash6284 1982 doi
1010631443832
[47] J C Phillips ldquoMicroscopic aspects of Stretched Exponential Relaxation (SER) in
homogeneous molecular and network glasses and polymersrdquo J Non Cryst Solids vol
357 no 22ndash23 pp 3853ndash3865 2011 doi 101016jjnoncrysol201108001
[48] J C Phillips ldquo ohlrausch explained The solution to a problem that is 1 0 years oldrdquo J
Stat Phys vol 77 no 3ndash4 pp 945ndash947 1994 doi 101007BF02179472
[49] A Q Tool and C G Eichlin ldquoVariations caused in the heating curves of glass caused by
heat treatmentrdquo J Am Ceram Soc vol 14 pp 276ndash308 1931
[50] A Q Tool ldquoRelation Between Inelastic Deformability and Thermal Expansion of Glass in
Its Annealing Rangerdquo J Am Ceram Soc vol 29 no 9 pp 240ndash253 1946 doi
101111j1151-29161946tb11592x
[51] H N Ritland ldquoLimitations of the Fictive Temperature Conceptrdquo J Am Ceram Soc vol
39 no 12 pp 403ndash406 Dec 1956 doi 101111j1151-29161956tb15613x
[52] O S Narayanaswamy ldquoA model of structural relaxation in glassrdquo J Am Ceram Soc
vol 54 no 10 pp 491ndash498 1971 doi 101111j1151-29161971tb12186x
[53] G Adam and J Gibbs ldquoOn the Temperature Dependence of Cooperative Relaxation
Properties in Glass‐Forming Liquidsrdquo J Chem Phys vol 43 no 1 pp 139ndash146 1965
doi 10106311696442
[54] Q Zheng J C Mauro A J Ellison M Potuzak and Y Yue ldquo niversality of the high-
temperature viscosity limit of silicate liquidsrdquo Phys Rev B vol 83 no 21 p 212202
2011 doi 101103PhysRevB83212202
[55] J C Mauro Y Yue A J Ellison P Gupta and D C Allan ldquoViscosity of glass-
forming liquidsrdquo Proc Natl Acad Sci vol 106 no 47 pp 19780ndash19784 2009 doi
175
101073pnas0911705106
[56] G G Naumis ldquoGlass transition phenomenology and flexibility An approach using the
energy landscape formalismrdquo J Non Cryst Solids vol 352 no 42-49 SPEC ISS pp
4865ndash4870 2006 doi 101016jjnoncrysol200601160
[57] P Gupta and J C Mauro ldquoComposition dependence of glass transition temperature
and fragility I A topological model incorporating temperature-dependent constraintsrdquo J
Chem Phys vol 130 no 9 p 094503 2009 doi 10106313077168
[58] C A Angell ldquoRelaxation in liquids polymers and plastic crystals - strongfragile patterns
and problemsrdquo J Non Cryst Solids vol 13 no 31 pp 131ndash133 1991 doi
1010160022-3093(91)90266-9
[59] G S Fulcher ldquoAnalysis of Recent Measurements of the Viscosity of Glassesrdquo J Am
Ceram Soc vol 75 no 5 pp 1043ndash1055 1992 doi 101111j1151-
29161992tb05536x
[60] I Avramov ldquoViscosity in disordered mediardquo J Non Cryst Solids vol 351 no 40ndash42
pp 3163ndash3173 Oct 2005 doi 101016JJNONCRYSOL200508021
[61] X Guo J C Mauro D C Allan and M M Smedskjaer ldquoPredictive model for the
composition dependence of glassy dynamicsrdquo J Am Ceram Soc vol 101 pp 1169ndash
1179 2018 doi 101111jace15272
[62] Doss C J Wilkinson Y Yang H Lee L Huang and J C Mauro ldquoMaxwell
relaxation time for nonexponential α-relaxation phenomena in glassy systemsrdquo J Am
Ceram Soc vol 103 no 6 pp 3590ndash3599 2020 doi 101111jace17051
[63] P Gupta and A R Cooper ldquoTopologically disordered networks of rigid polytopesrdquo J
Non Cryst Solids 1990 doi 1010160022-3093(90)90768-H
[64] J C Phillips and M F Thorpe ldquoConstraint theory vector percolation and glass
formationrdquo Solid State Commun vol 53 no 8 pp 699ndash702 1985 doi 1010160038-
176
1098(85)90381-3
[65] X Feng W Bresser and P Boolchand ldquoDirect Evidence for Stiffness Threshold in
Chalcogenide Glassesrdquo Phys Rev Lett vol 78 no 23 pp 4422ndash4425 1997 doi
101103PhysRevLett784422
[66] W Bresser P Boolchand and P Suranyi ldquoRigidity Percolation and Molecular Clustering
in Network Glassesrdquo Phys Rev Lett vol 56 no 23 pp 2493ndash2496 1986 doi
101103PhysRevLett562493
[67] Y Vaills T Qu M Micoulaut F Chaimbault and P Boolchand ldquoDirect evidence of
rigidity loss and self-organization in silicate glassesrdquo J Phys Condens Matter vol 17
no 32 pp 4889ndash4896 2005 doi 1010880953-89841732003
[68] A irchner and J C Mauro ldquoStatistical Mechanical Model of the Self-Organized
Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologiesrdquo
Front Mater vol 6 p 11 2019 doi 103389fmats201900011
[69] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 Jun 2009 doi 10106313152432
[70] D R Cassar A C P L F de Carvalho and E D Zanotto ldquoPredicting glass transition
temperatures using neural networksrdquo Acta Mater vol 159 pp 249ndash256 Oct 2018 doi
101016JACTAMAT201808022
[71] J C Mauro and R J Loucks ldquoSelenium glass transition A model based on the enthalpy
landscape approach and nonequilibrium statistical mechanicsrdquo Phys Rev B vol 76 no
17 p 174202 2007 doi 101103PhysRevB76174202
[72] C J Wilkinson et al ldquoEnergy Landscape Modeling of Crystal Nucleationrdquo Nat Comput
Mater p Submitted 2020
[73] M E Mc enzie and J C Mauro ldquoHybrid Monte Carlo technique for modeling of crystal
177
nucleation and application to lithium disilicate glass-ceramicsrdquo Comput Mater Sci vol
149 no January pp 202ndash207 2018 doi 101016jcommatsci201803034
[74] M E McKenzie et al ldquoImplicit glass model for simulation of crystal nucleation for glass-
ceramicsrdquo npj Comput Mater vol 4 no 1 pp 1ndash7 2018 doi 101038s41524-018-
0116-5
[75] E D Zanotto and P F James ldquoExperimental tests of the classical nucleation theory for
glassesrdquo J Non Cryst Solids vol 74 no 2ndash3 pp 373ndash394 1985 doi 1010160022-
3093(85)90080-8
[76] Y Yu N M A Krishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[77] I Pignatelli A Kumar M Bauchy and G Sant ldquoTopological control on silicatesrsquo
dissolution kineticsrdquo Langmuir vol 32 pp 4434ndash4439 2016 doi
101021acslangmuir6b00359
[78] C J Wilkinson et al ldquoTopological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phaserdquo J Phys Chem Lett vol 10 pp
3955ndash3960 2019 doi 101021acsjpclett9b01275
[79] Y Zhang A Li B Deng and Hughes ldquoData-driven predictive models for chemical
durability of oxide glass under different chemical conditionsrdquo npj Mater Degrad vol 4
no 1 pp 1ndash11 2020 doi 101038s41529-020-0118-x
[80] N M Anoop Krishnan S Mangalathu M M Smedskjaer A Tandia H Burton and M
Bauchy ldquoPredicting the dissolution kinetics of silicate glasses using machine learningrdquo J
Non Cryst Solids vol 487 no February pp 37ndash45 2018 doi
101016jjnoncrysol201802023
[81] R M Potter J Hoffman and J Hadley ldquoAn update of the equation for predicting the
178
dissolution rate of glass fibers from their chemical compositionsrdquo Inhal Toxicol vol 29
no 4 pp 145ndash146 2017 doi 1010800895837820171321702
[82] B Deng ldquoMachine learning on density and elastic property of oxide glasses driven by
large datasetrdquo J Non Cryst Solids vol 529 no August 2019 p 119768 2020 doi
101016jjnoncrysol2019119768
[83] S Feller N Lower and M Affatigato ldquoDensity as a probe of oxide glass structurerdquo
Phys Chem Glas vol 42 no 3 pp 240ndash246 2001 Accessed Jul 19 2018 [Online]
Available
httpswwwingentaconnectcomcontentsgtpcg200100000042000000034203240
[84] Q Zheng and H Zeng ldquoProgress in modeling of glass properties using topological
constraint theoryrdquo Int J Appl Glas Sci vol 11 no 3 pp 432ndash441 2020 doi
101111ijag15105
[85] M Bauchy M Javad Abdolhosseini Qomi C Bichara F-J Ulm and R J-M Pellenq
ldquoRigidity Transition in Materials Hardness is Driven by Weak Atomic Constraintsrdquo
2015 doi 101103PhysRevLett114125502
[86] Yang X Xu B Yang B Cook H Ramos and M Bauchy ldquoPrediction of Silicate
Glassesrsquo Stiffness by High-Throughput Molecular Dynamics Simulations and Machine
Learningrdquo Accessed Mar 13 201 Online Available
httpsarxivorgpdf190109323pdf
[87] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological constraint model for
the elasticity of glass-forming systemsrdquo J Non Cryst Solids X vol 2 2019 doi
101016jnocx2019100019
[88] C J Wilkinson K Doss D R Cassar R S Welch C B Bragatto and J C Mauro
ldquoPredicting Ionic Diffusion in Glass from Its Relaxation Behaviorrdquo J Phys Chem B vol
124 no 6 pp 1099ndash1103 2020 doi 101021acsjpcb9b10645
179
[89] C Wilkinson and J C Mauro ldquoExplorerpy Mapping the energy landscapes of complex
materialsrdquo SoftwareX vol Submitted 2020
[90] J C Mauro R J Loucks and J Balakrishnan ldquoA simplified eigenvector-following
technique for locating transition points in an energy landscaperdquo J Phys Chem A vol
109 no 42 pp 9578ndash9583 2005 doi 101021jp053581t
[91] Roumlder and D J Wales ldquoAnalysis of the b to b-CR Transition in biquitinrdquo
Biochemistry vol 57 no 43 pp 6180ndash6186 2018 doi 101021acsbiochem8b00770
[92] J C Mauro R J Loucks and J Balakrishnan ldquoSplit-step eigenvector-following
technique for exploring enthalpy landscapes at absolute zerordquo J Phys Chem B vol 110
no 10 pp 5005ndash5011 2006 doi 101021jp056803w
[93] J C Mauro R J Loucks J Balakrishnan and A Varshneya ldquoMapping the potential
energy landscapes of selenium clustersrdquo J Non Cryst Solids vol 353 pp 1268ndash1273
2007 doi 101016jjnoncrysol200609062
[94] B W H van Beest G J Kramer and R A van Santen ldquoForce fields for silicas and
aluminophosphates based on ab initio calculationsrdquo Phys Rev Lett vol 64 no 16 pp
1955ndash1958 Apr 1990 doi 101103PhysRevLett641955
[95] Q Zheng and J C Mauro ldquoViscosity of glass-forming systemsrdquo J Am Ceram Soc vol
100 no 1 pp 6ndash25 2017 doi 101111jace14678
[96] J C Mauro and Y Z Mauro ldquoOn the Prony Series Representation of Stretched
Exponential Relaxationrdquo Physica A vol 506 pp 75ndash87 2018 doi
101016jphysa201804047
[97] E D Zanotto and D R Cassar ldquoThe microscopic origin of the extreme glass-forming
ability of Albite and B2O3rdquo Sci Rep vol 7 no February pp 1ndash13 2017 doi
101038srep43022
[98] P Pedevilla S J Cox B Slater and A Michaelides ldquoCan Ice-Like Structures Form on
180
Non-Ice-Like Substrates The Example of the K-feldspar Microclinerdquo J Phys Chem C
vol 120 no 12 pp 6704ndash6713 2016 doi 101021acsjpcc6b01155
[99] O Bjoumlrneholm et al ldquoWater at Interfacesrdquo Chem Rev vol 116 no 13 pp 7698ndash7726
2016 doi 101021acschemrev6b00045
[100] Reichelt ldquoNucleation and growth of thin filmsrdquo Vacuum vol 38 no 12 pp 1083ndash
1099 Jan 1988 doi 1010160042-207X(88)90004-8
[101] G H Beall ldquoDr S Donald (Don) Stookey (1 1 ndash2014) Pioneering researcher and
adventurerrdquo Front Mater vol 3 no August pp 1ndash8 2016 doi
103389fmats201600037
[102] J C Mauro C S Philip D J Vaughn and M S Pambianchi ldquoGlass science in the
nited States Current status and future directionsrdquo Int J Appl Glas Sci vol 5 no 1
pp 2ndash15 2014 doi 101111ijag12058
[103] J Deubener et al ldquo pdated definition of glass-ceramicsrdquo J Non Cryst Solids vol 501
no January pp 3ndash10 2018 doi 101016jjnoncrysol201801033
[104] X Hao ldquoA review on the dielectric materials for high energy-storage applicationrdquo J Adv
Dielectr vol 03 no 01 p 1330001 2013 doi 101142s2010135x13300016
[105] J Deubener G Helsch A Moiseev and H Bornhoumlft ldquoGlasses for solar energy
conversion systemsrdquo J Eur Ceram Soc vol 29 no 7 pp 1203ndash1210 2009 doi
101016jjeurceramsoc200808009
[106] G S Frankel et al ldquoA comparative review of the aqueous corrosion of glasses crystalline
ceramics and metalsrdquo npj Mater Degrad vol 2 no 1 p 15 2018 doi
101038s41529-018-0037-2
[107] G H Beall ldquoDesign and properties of glass-ceramicsrdquo Annu Rev Mater Sci vol 22
no 1 pp 91ndash119 1992 doi 101146annurevms22080192000515
[108] J Deubener ldquoConfigurational entropy and crystal nucleation of silicate glassesrdquo Phys
181
Chem Glas vol 45 pp 61ndash63 2004
[109] J C Mauro A J Ellison D C Allan and M M Smedskjaer ldquoTopological model for the
viscosity of multicomponent glass-forming liquidsrdquo Int J Appl Glas Sci vol 4 no 4
pp 408ndash413 2013 doi 101111ijag12009
[110] M M Smedskjaer J C Mauro S Sen and Y Yue ldquoQuantitative design of glassy
materials using temperature-dependent constraint theoryrdquo Chem Mater vol 22 no 18
pp 5358ndash5365 2010 doi 101021cm1016799
[111] Y T Sun H Y Bai M Z Li and W H Wang ldquoMachine Learning Approach for
Prediction and Understanding of Glass-Forming Abilityrdquo J Phys Chem Lett vol 8 no
14 pp 3434ndash3439 2017 doi 101021acsjpclett7b01046
[112] Q Zheng et al ldquo nderstanding Glass through Differential Scanning Calorimetryrdquo Chem
Rev vol 119 no 13 pp 7848ndash7939 2019 doi 101021acschemrev8b00510
[113] X Xia I Dutta J C Mauro B G Aitken and F elton ldquoTemperature dependence of
crystal nucleation in BaOmiddot2SiO2 and 5BaOmiddot8SiO2 glasses using differential thermal
analysisrdquo J Non Cryst Solids vol 459 pp 45ndash50 2017 doi
101016jjnoncrysol201612032
[114] V M Fokin A A Cabral R M C V Reis M L F Nascimento and E D Zanotto
ldquoCritical assessment of DTA-DSC methods for the study of nucleation kinetics in
glassesrdquo J Non Cryst Solids vol 356 no 6ndash8 pp 358ndash367 2010 doi
101016jjnoncrysol200911038
[115] D C Van Hoesen X Xia M E Mc enzie and F elton ldquoModeling nonisothermal
crystallization in a BaO∙2SiO2 glassrdquo J Am Ceram Soc no December 2019 pp 2471ndash
2482 2019 doi 101111jace16979
[116] S C C Prado J P Rino and E D Zanotto ldquoSuccessful test of the classical nucleation
theory by molecular dynamic simulations of BaSrdquo Comput Mater Sci vol 161 no
182
January pp 99ndash106 2019 doi 101016jcommatsci201901023
[117] A O Tipeev and E D Zanotto ldquoNucleation kinetics in supercooled Ni 0Ti 0 Computer
simulation data corroborate the validity of the Classical Nucleation Theoryrdquo Chem Phys
Lett vol 735 no August p 136749 2019 doi 101016jcplett2019136749
[118] K F Kelton and A L Greer Nucleation in Condensed Matter Applications in Materials
and Biology 1st ed Pergamon 2010
[119] V M Fokin E D Zanotto N S Yuritsyn and J W P Schmelzer ldquoHomogeneous
crystal nucleation in silicate glasses A 40 years perspectiverdquo J Non Cryst Solids vol
352 no 26ndash27 pp 2681ndash2714 2006 doi 101016jjnoncrysol200602074
[120] A M Rodrigues D R Cassar V M Fokin and E D Zanotto ldquoCrystal growth and
viscous flow in barium disilicate glassrdquo J Non Cryst Solids vol 479 pp 55ndash61 2018
doi 101016jjnoncrysol201710007
[121] A Pedone G Malavasi M C Menziani A N Cormack and U Segre ldquoA new selft-
consistent empirical interatomic potential model for oxides silicates and silica-based
glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
[122] J J Maldonis A D Banadaki S Patala and P M Voyles ldquoShort-range order structure
motifs learned from an atomistic model of a Zr 0Cu4 Al metallic glassrdquo Acta Mater
vol 175 pp 35ndash45 2019 doi 101016jactamat201905002
[123] A irchner S Goyal M E Mc enzie J T Harris and J C Mauro ldquoStatistical
Description of the Thermodynamics of Glass-Forming Liquidsrdquo Physica A vol 559 p
125059 2020 doi 101016jphysa2020125059
[124] D R Cassar ldquoCrystallization Driving Force of Supercooled Oxide Liquidsrdquo Int J Appl
Glas Sci vol 7 no 3 pp 262ndash269 2016 doi 101111ijag12218
[125] V M Fokin A S Abyzov E D Zanotto D R Cassar A M Rodrigues and J W P
Schmelzer ldquoCrystal nucleation in glass-forming liquids Variation of the size of the
183
lsquostructural unitsrsquo with temperaturerdquo J Non Cryst Solids vol 447 pp 35ndash44 2016 doi
101016jjnoncrysol201605017
[126] M Lenoir A Grandjean Y Linard B Cochain and D R Neuville ldquoThe influence of
SiB substitution and of the nature of network-modifying cations on the properties and
structure of borosilicate glasses and meltsrdquo Chem Geol vol 256 no 3ndash4 pp 316ndash325
2008 doi 101016jchemgeo200807002
[127] P Gupta D R Cassar and E D Zanotto ldquoRole of dynamic heterogeneities in crystal
nucleation kinetics in an oxide supercooled liquidrdquo J Chem Phys vol 145 no 21 2016
doi 10106314964674
[128] A M Rodrigues ldquoDiffusion Processes Crystallization and Viscous Flow in Barium
Disilicate Glassrdquo Dissertation 2014
[129] E D Zanotto ldquoThe effects of amorphous phase sepatration on crystal nucleation in baria-
silica and lithia-silica glassesrdquo Dissertation 1982
[130] D Vargheese A Tandia and J C Mauro ldquoOrigin of dynamical heterogeneities in
calcium aluminosilicate liquidsrdquo J Chem Phys vol 132 no 19 p 194501 May 2010
doi 10106313429880
[131] K Deenamma Vargheese A Tandia and J C Mauro ldquoStatistics of modifier distributions
in mixed network glassesrdquo J Chem Phys vol 132 p 24507 2010 doi
10106313429880
[132] M Micoulaut ldquoThe Deep Effect of Topology on Glass Relaxationrdquo Physics (College
Park Md) vol 6 p 72 2013 doi 101103Physics672
[133] M Micoulaut ldquoRigidity and intermediate phases in glasses driven by speciationrdquo Phys
Rev B vol 74 p 184208 2006 doi 101103PhysRevB74184208
[134] M Kodama S Kojima S Feller and M Affatigato ldquoThe occurrence of minima in the
Borate anomaly anharmonicity and fragility in lithium borate glassesrdquo Phys Chem
184
Glas vol 46 no 2 pp 190ndash193 2005
[135] Y Fukawa et al ldquoVelocity of Sound and Elastic Properties of Li2O-B2O3 Glassesrdquo Jpn
J Appl Phys vol 34 p 2570 1995 doi httpsdoiorg101143JJAP342570
[136] M Reiner ldquoThe Deborah Numberrdquo Physics Today vol 17 no 1 p 62 1964 doi
10106313051374
[137] P K Gupta and J C Mauro ldquoThe laboratory glass transitionrdquo J Chem Phys vol 126
no 22 p 2240504 2007 doi 10106312738471
[138] R Richert and C A Angell ldquoDynamics of glass-forming liquids V On the link between
molecular dynamics and configurational entropyrdquo J Chem Phys vol 108 no 21 pp
9016ndash9026 1998 doi 1010631476348
[139] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological Constraint Model for
the Elasticity of Glass-Forming Systemsrdquo J Non Cryst Solids X vol 2 p 100019 2019
doi 101016jnocx2019100019
[140] S Feller S Bista A OrsquoDonovan-Zavada T Mullenbach M Franke and M Affatigato
ldquoPacking in alkali and alkaline earth borosilicate glass systemsrdquo Phys Chem Glas Eur
J Glas Sci Technol Part B vol 50 no 3 pp 224ndash228 2009
[141] P J Bray S Feller G E Jellison and Y H Yun ldquoB10 NMR studies of the structure of
borate glassesrdquo J Non Cryst Solids vol 38ndash39 no PART 1 pp 93ndash98 May 1980 doi
1010160022-3093(80)90400-7
[142] N Andersson and G L Comer ldquoRelativistic Fluid Dynamics Physics for Many Different
Scales Living Reviews in Relativityrdquo Living Rev Relativ vol 10 p 1 2007 Accessed
Feb 18 2019 [Online] Available httpwwwlivingreviewsorglrr-2007-
1httprelativitylivingreviewsorghttpwwwmathssotonacukstaffAnderssonhttpw
wwslueducollegesASphysicsprofscomerhtml
[143] P Ilg and H C Oumlttinger ldquoNonequilibrium relativistic thermodynamics in bulk viscous
185
cosmologyrdquo Phys Rev D - Part Fields Gravit Cosmol vol 61 no 2 p 023510 2000
doi 101103PhysRevD61023510
[144] P Romatschke ldquoRelativistic viscous fluid dynamics and non-equilibrium entropyrdquo Class
Quantum Gravity vol 27 no 2 p 025006 Jan 2010 doi 1010880264-
9381272025006
[145] D C Allan ldquoInverting the MYEGA equation for viscosityrdquo J Non Cryst Solids vol
358 no 2 pp 440ndash442 Jan 2012 doi 101016JJNONCRYSOL201109036
[146] M Potuzak R C Welch and J C Mauro ldquoTopological origin of stretched exponential
relaxation in glassrdquo J Chem Phys vol 135 no 21 p 214502 Dec 2011 doi
10106313664744
[147] L Ding M Thieme S Demouchy C unisch and B J P aus ldquoEffect of pressure and
temperature on viscosity of a borosilicate glassrdquo J Am Ceram Soc vol 101 no 9 pp
3936ndash3946 2018 doi 101111jace15588
[148] R Richert and M Richert ldquoDynamic heterogeneity spatially distributed stretched-
exponential patterns and transient dispersions in solvation dynamicsrdquo Phys Rev E vol
58 no 1 pp 779ndash784 1998 doi 101103PhysRevE58779
[149] P Richet ldquoViscosity and configurational entropy of silicate meltsrdquo Geochim Cosmochim
Acta vol 48 no 3 pp 471ndash483 Mar 1984 doi 1010160016-7037(84)90275-8
[150] P Gupta and J C Mauro ldquoTwo factors governing fragility Stretching exponent and
configurational entropyrdquo Phys Rev E vol 78 no 6 p 063501 2008 doi
101103PhysRevE78062501
[151] Z Zheng J C Mauro and D C Allan ldquoModeling of delayed elasticity in glassrdquo J Non
Cryst Solids vol 500 no August pp 432ndash442 2018 doi
101016jjnoncrysol201808033
[152] R Boumlhmer L Ngai C A Angell and D J Plazek ldquoNonexponential relaxations in
186
strong and fragile glass formersrdquo J Chem Phys vol 99 no 5 pp 4201ndash4209 1993
doi 1010631466117
[153] O Gulbiten J C Mauro and P Lucas ldquoRelaxation of enthalpy fluctuations during sub-
Tg annealing of glassy seleniumrdquo J Chem Phys vol 138 no 24 p 244504 2013 doi
10106314811488
[154] D Sidebottom R Bergman L Boumlrjesson and L M Torell ldquoTwo-step relaxation decay
in a strong glass formerrdquo Phys Rev Lett vol 71 no 14 pp 2260ndash2263 1993 doi
101103PhysRevLett712260
[155] G M Bartenev and V A Lomovskoi ldquoRelaxation time spectra and the peculiarities of
the process of boron anhydride glass transitionrdquo J Non Cryst Solids vol 146 no C pp
225ndash232 1992 doi 101016S0022-3093(05)80495-8
[156] A Heuer ldquoExploring the potential energy landscape of glass-forming systems From
inherent structures via metabasins to macroscopic transportrdquo J Phys Condens Matter
vol 20 no 37 p 373101 2008 doi 1010880953-89842037373101
[157] J C Phillips ldquoStretched exponential relaxation in molecular and electronic gasesrdquo
Reports Prog Phys vol 59 pp 1133ndash1207 1996
[158] P Richet ldquoResidual and configurational entropy Quantitative checks through applications
of Adam-Gibbs theory to the viscosity of silicate meltsrdquo J Non Cryst Solids vol 355
no 10ndash12 pp 628ndash635 2009 doi 101016jjnoncrysol200901027
[159] C J Wilkinson Doss G Palmer and J C Mauro ldquoThe relativistic glass transition A
thought experimentrdquo J Non-Crystalline Solids X vol 2 2019 doi
101016jnocx2019100018
[160] J C Mauro R J Loucks and S Sen ldquoHeat capacity enthalpy fluctuations and
configurational entropy in broken ergodic systemsrdquo J Chem Phys vol 133 no 16 pp
1ndash9 2010 doi 10106313499326
187
[161] J C Mauro R J Loucks and P Gupta ldquoMetabasin approach for computing the
master equation dynamics of systems with broken ergodicityrdquo J Phys Chem A vol 111
no 32 pp 7957ndash7965 2007 doi 101021jp0731194
[162] M Wang N M Anoop Krishnan B Wang M M Smedskjaer J C Mauro and M
Bauchy ldquoA new transferable interatomic potential for molecular dynamics simulations of
borosilicate glassesrdquo J Non Cryst Solids vol 498 no December 2017 pp 294ndash304
2018 doi 101016jjnoncrysol201804063
[163] I Avramov and A Milchev ldquoEffect of disorder on diffusion and viscosity in condensed
systemsrdquo J Non Cryst Solids vol 104 no 2ndash3 pp 253ndash260 Sep 1988 doi
1010160022-3093(88)90396-1
[164] G Scherer Relaxation in Glass and Composites Krieger Publishing Company 1992
[165] L Huang J Nicholas J Kieffer and J Bass ldquoPolyamorphic transitions in vitreous B2O3
under pressurerdquo J Phys Condens Matter vol 20 no 7 2008 doi 1010880953-
8984207075107
[166] W Capps P B Macedo B OrsquoMeara and T A Litovitz ldquoTemperature Dependecne of
the High-Frequency Moduli of Vitreous B2O3rdquo J Chem Phys vol 45 p 3431 1966
doi 107868s0002337x1402002x
[167] D R Cassar A M Rodrigues M L F Nascimento and E D Zanotto ldquoThe diffusion
coefficient controlling crystal growth in a silicate glass-formerrdquo Int J Appl Glas Sci
vol 9 no 3 pp 373ndash382 Jul 2018 doi 101111ijag12319
[168] A R Potter C J Wilkinson S H im and J C Mauro ldquoEffect of Water on
Topological Constraints in Silica Glassrdquo Scr Mater vol 160 pp 48ndash52 2019 doi
101016jscriptamat201809041
[169] D B Asay and S H im ldquoEvolution of the adsorbed water layer structure on silicon
oxide at room temperaturerdquo J Phys Chem B vol 109 pp 16760ndash16763 2005 doi
188
101021jp053042o
[170] M Tomozawa ldquoWater diffusion oxygen vacancy annihilation and structural relaxation in
silica glassesrdquo J Non Cryst Solids vol 179 pp 162ndash169 1994 doi 1010160022-
3093(94)90693-9
[171] S Kapoor R E Youngman K Zakharchuk A Yaremchenko N J Smith and A Goel
ldquoStructural and Chemical Approach toward nderstanding the Aqueous Corrosion of
Sodium Aluminoborate Glassesrdquo J Phys Chem B vol 122 pp 10913ndash10927 2018 doi
101021acsjpcb8b06155
[172] Y Yu N M A rishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[173] Y Yu B Wang M Wang G Sant and M Bauchy ldquoReactive Molecular Dynamics
Simulations of Sodium Silicate Glasses mdash Toward an Improved Understanding of the
Structurerdquo Int J Appl Glas Sci vol 8 no 3 pp 276ndash284 2017 doi
101111ijag12248
[174] E Stolper ldquoThe speciation of water in silicate meltsrdquo Geochim Cosmochim Acta vol
46 no 12 pp 2609ndash2620 Dec 1982 doi 1010160016-7037(82)90381-7
[175] M G Mesko P A Schader and J E Shelby ldquoWater solubility and diffusion in sodium
silicate meltsrdquo Phys Chem Glas vol 43 no 6 pp 283ndash290 2002 doi
101016jsemradonc201010001
[176] M Tomozawa M Takata J Acocella E Bruce Watson and T Takamori ldquoThermal
properties of Na2Omiddot3SiO2 glasses with high water contentrdquo J Non Cryst Solids vol 56
no 1ndash3 pp 343ndash348 Jul 1983 doi 1010160022-3093(83)90491-X
[177] E A Leed and C G Pantano ldquoComputer modeling of water adsorption on silica and
silicate glass fracture surfacesrdquo J Non Cryst Solids vol 325 pp 48ndash60 2003 doi
189
101016S0022-3093(03)00361-2
[178] J C Fogarty H M Aktulga A Y Grama A C T van Duin and S A Pandit ldquoA
reactive molecular dynamics simulation of the silica-water interfacerdquo J Chem Phys vol
132 no 17 p 174704 2010 doi 10106313407433
[179] S H Hahn et al ldquoDevelopment of a ReaxFF Reactive Force Field for NaSiOx Water
Systems and Its Application to Sodium and Proton Self-Diffusionrdquo J Phys Chem C vol
122 no 34 pp 19613ndash19624 2018 doi 101021acsjpcc8b05852
[180] T S Mahadevan W Sun and J Du ldquoDevelopment of Water Reactive Potentials for
Sodium Silicate Glassesrdquo J Phys Chem B vol 123 pp 4452ndash4461 2019 doi
101021acsjpcb9b02216
[181] A C T van Duin A Strachan S Stewman Q Zhang X Xu and W A Goddard III
ldquoReaxFF SiO Reactive Force Field for Silicon and Silicon Oxide Systemsrdquo J Phys
Chem A vol 107 no 19 pp 3803ndash3811 2003 doi 101021jp0276303
[182] M M Smedskjaer J C Mauro R E Youngman C L Hogue M Potuzak and Y Yue
ldquoTopological principles of borosilicate glass chemistryrdquo J Phys Chem B vol 115 no
44 pp 12930ndash12946 2011 doi 101021jp208796b
[183] Q Zheng M Potuzak J C Mauro M M Smedskjaer R E Youngman and Y Yue
ldquoComposition-structure-property relationships in boroaluminosilicate glassesrdquo J Non
Cryst Solids vol 358 pp 993ndash1002 2012 doi 101016jjnoncrysol201201030
[184] K Rompicharla D I Novita P Chen P Boolchand M Micoulaut and W Huff
ldquoAbrupt boundaries of intermediate phases and space filling in oxide glassesrdquo J Phys
Condens Matter vol 20 p 202101 2008 doi 1010880953-89842020202101
[185] M Micoulaut and J C Phillips ldquoRings and rigidity transitions in network glassesrdquo Phys
Rev B vol 67 p 104204 2003 doi 101103PhysRevB67104204
[186] H Liu S Dong L Tang N M A rishnan G Sant and M Bauchy ldquoEffects of
190
polydispersity and disorder on the mechanical properties of hydrated silicate gelsrdquo J
Mech Phys Solids vol 122 pp 555ndash565 Jan 2019 doi 101016JJMPS201810003
[187] A Tilocca N H De Leeuw and A N Cormack ldquoShell-model molecular dynamics
calculations of modified silicate glassesrdquo Phys Rev B - Condens Matter Mater Phys
vol 73 p 104209 2006 doi 101103PhysRevB73104209
[188] J Du and A N Cormack ldquoThe medium range structure of sodium silicate glasses A
molecular dynamics simulationrdquo J Non Cryst Solids vol 349 no 1ndash3 pp 66ndash79 2004
doi 101016jjnoncrysol200408264
[189] N P Bansal and R H Doremus Handbook of glass properties Academic Press 1986
[190] S H Hahn and A C T van Duin ldquoSurface Reactivity and Leaching of a Sodium Silicate
Glass under an Aqueous Environment A ReaxFF Molecular Dynamics Studyrdquo J Phys
Chem C p acsjpcc9b02940 2019 doi 101021acsjpcc9b02940
[191] S P Jaccani and L Huang ldquo nderstanding Sodium Borate Glasses and Melts from Their
Elastic Response to Temperaturerdquo Int J Appl Glas Sci vol 7 no 4 pp 452ndash463 Dec
2016 doi 101111ijag12250
[192] Q Zheng Y Yue and J C Mauro ldquoDensity of topological constraints as a metric for
predicting glass hardnessrdquo Appl Phys Lett vol 111 no 1 p 011907 Jul 2017 doi
10106314991971
[193] J C Dyre ldquoElastic models for the non-Arrhenius relaxation time of glass-forming
liquidsrdquo in AIP Conference Proceedings 2006 vol 832 pp 113ndash117 doi
10106312204470
[194] Y T Cheng and C M Cheng ldquoRelationships between hardness elastic modulus and the
work of indentationrdquo Appl Phys Lett vol 73 no 5 pp 614ndash616 Jul 1998 doi
1010631121873
[195] A Makishima and J D Mackenzie ldquoDirect calculation of Youngrsquos modulus of glassrdquo J
191
Non Cryst Solids vol 12 no 1 pp 35ndash45 1973 doi 1010160022-3093(73)90053-7
[196] G Pilania C Wang X Jiang S Rajasekaran and R Ramprasad ldquoAccelerating materials
property predictions using machine learningrdquo Sci Rep vol 3 pp 1ndash6 2013 doi
101038srep02810
[197] H He and M Thorpe ldquoElastic properties of glassesrdquo Phys Rev Lett vol 54 no 19 pp
2107ndash2110 1985 Accessed Aug 29 2018 [Online] Available
httpsjournalsapsorgprlpdf101103PhysRevLett542107
[198] M Thorpe ldquoElastic properties of glassesrdquo Cambridge Arch 1985 doi 101557PROC-
61-49
[199] J C Phillips ldquoChemical Bonding Internal Surfaces and the Topology of Non-Crystalline
Solidsrdquo Phys status solidi vol 101 no 2 pp 473ndash479 Oct 1980 doi
101002pssb2221010204
[200] M M Smedskjaer J C Mauro and Y Yue ldquoPrediction of glass hardness using
temperature-dependent constraint theoryrdquo Phys Rev Lett vol 105 no 11 2010 doi
101103PhysRevLett105115503
[201] J D Musgraves J Hu and L Calvez Springer Handbook of Glass Springer US 2021
[202] C Hermansen X Guo R E Youngman J C Mauro M M Smedskjaer and Y Yue
ldquoStructure-topology-property correlations of sodium phosphosilicate glassesrdquo J Chem
Phys vol 143 no 6 p 064510 Aug 2015 doi 10106314928330
[203] L Wondraczek et al ldquoTowards ltrastrong Glassesrdquo Adv Mater vol 23 no 39 pp
4578ndash4586 Oct 2011 doi 101002adma201102795
[204] D Ravaine ldquoGlasses as solid electrolytesrdquo J Non Cryst Solids vol 38ndash39 no PART 1
pp 353ndash358 1980 doi 1010160022-3093(80)90444-5
[205] D Ravaine and J L Souquet ldquoA thermodynamic approach to ionic conductivity in oxide
glasses Part 1 Correlation of the ionic conductivity with the chemical potential of alkali
192
oxide in oxide glassesrdquo Phys Chem Glas vol 18 no 2 pp 27ndash31 1977
[206] J Swenson and L Boumlrjesson ldquoCorrelation between Free Volume and Ionic Conductivity
in Fast Ion Conducting Glassesrdquo Phys Rev Lett vol 77 no 17 pp 3569ndash3572 1996
doi 101103PhysRevLett773569
[207] C B Bragatto A C M Rodrigues and J L Souquet ldquoDissociation Equilibrium and
Charge Carrier Formation in AgI-AgPO3 Glassesrdquo J Phys Chem C vol 121 no 25 pp
13507ndash13514 2017 doi 101021acsjpcc7b02477
[208] S W Martin R Christensen G Olson J ieffer and W Wang ldquoNew Interpretation of
Na Ion Conduction in and the Structures and Properties of Sodium Borosilicate Mixed
Glass Former Glassesrdquo J Phys Chem C vol 140 pp 6343ndash6352 2019 doi
101021acsjpcc8b11735
[209] O L Anderson and D A Stuart ldquoCalculation of Activation Energy of Ionic Conductivity
in Silica Glasses by Classical Methodsrdquo J Am Ceram Soc vol 37 no 12 pp 573ndash580
1954 doi 101111j1151-29161954tb13991x
[210] R S Welch C J Wilkinson J C Mauro and C B Bragatto ldquoCharge Carrier Mobility
of Alkali Silicate Glasses Calculated by Molecular Dynamicsrdquo Front Mater vol 6 p
121 May 2019 doi 103389fmats201900121
[211] Y J Zhang ldquoEntropy and ionic conductivityrdquo Phys A Stat Mech its Appl vol 391 no
19 pp 4470ndash4475 2012 doi 101016jphysa201204021
[212] J C Dyre ldquoOn the mechanism of glass ionic conductivityrdquo J Non Cryst Solids vol 88
no 2ndash3 pp 271ndash280 1986 doi 101016S0022-3093(86)80030-8
[213] M Steve ldquoIonic Conduction in Phosphate Glassesrdquo JAmCeramSoc vol 74 no 8 pp
1767ndash1784 1991 doi 101111j1151-29161991tb07788x
[214] C T Moynihan A J Easteal J Wilder and J Tucker ldquoDependence of the Glass
Transition Temperature on Heating and Cooling Raterdquo J Phys Chem vol 78 no 26 pp
193
2673ndash2677 1974 Accessed Jul 13 2018 [Online] Available
httpspubsacsorgsharingguidelines
[215] Ito C T Moynihan and C A Angell ldquoThermodynamic determination of fragility in
liquids anda fragile-to-strong liquid transition inwaterrdquo Lett to Nat vol 398 no April
p 492 1999
[216] L Ngai and S W Martin ldquoCorrelation between the activation enthalpy and
Kohlrausch exponent for ionic conductivity in oxide glassesrdquo Phys Rev B vol 40 no
15 pp 10550ndash10556 1989 doi 101103PhysRevB4010550
[217] M Potuzak X Guo M M Smedskjaer and J C Mauro ldquoAre the dynamics of a glass
embedded in its elastic propertiesrdquo J Chem Phys vol 138 no 12 2013 doi
10106314730525
[218] A L Agapov and A P Sokolov ldquoDecoupling ionic conductivity from structural
relaxation A way to solid polymer electrolytesrdquo Macromolecules vol 44 no 11 pp
4410ndash4414 2011 doi 101021ma2001096
[219] S V Nemilov ldquoA Structural Investigation of Glasses in the B2O3ndashNa2O System by the
Viscosimetric Methodrdquo Izv Akad Nauk SSSR Neorg Mater vol 2 no 2 pp 349ndash359
1966
[220] J P Poole ldquoLow‐Temperature Viscosity of Alkali Silicate Glassesrdquo J Am Ceram Soc
vol 32 no 7 pp 230ndash233 1949 doi 101111j1151-29161949tb18952x
[221] R noche D B Dingwell F A Seifert and S L Webb ldquoNon-linear properties of
supercooled liquids in the system Na2O SiO2rdquo Chem Geol vol 116 pp 1ndash16 1994 doi
1010160009-2541(94)90154-6
[222] W Martinsen ldquoSelected properties of sodium silicate glasses and their structural
significancerdquo Dissertation 1969
[223] C Hermansen J C Mauro and Y Yue ldquoA model for phosphate glass topology
194
considering the modifying ion sub-networkrdquo J Chem Phys vol 140 no 15 2014 doi
10106314870764
[224] M L F Nascimento and E Dutra Zanotto ldquoDoes viscosity describe the kinetic barrier for
crystal growth from the liquidus to the glass transitionrdquo J Chem Phys vol 133 no 17
p 174701 2010 doi 10106313490793
[225] D Denisko and M M Hoffman ldquoClassification and interaction in random forestsrdquo Proc
Natl Acad Sci U S A vol 115 no 8 pp 1690ndash1692 2018 doi
101073pnas1800256115
[226] G Varney C Dema B E Gul C J Wilkinson and Akgun ldquo se of machine learning
in CARNA proton imagerrdquo in Progress in Biomedical Optics and Imaging - Proceedings
of SPIE 2019 vol 10948 doi 101117122512565
[227] ldquoMDLregSciGlass-6 MDL Information Systems San Leandro CArdquo 2003
[228] D R Cassar ldquoViscNet Neural Network for predicting the fragility index and the
temperature-dependency of viscosityrdquo pp 1ndash33 2020
[229] D J L Brett A Atkinson N P Brandon and S J Skinner ldquoIntermediate temperature
solid oxide fuel cellsrdquo Chem Soc Rev vol 37 no 8 pp 1568ndash1578 2008 doi
101039b612060c
[230] T Ishiyama S Suzuki J Nishii T Yamashita H awazoe and T Omata ldquoProton
conducting tungsten phosphate glass and its application in intermediate temperature fuel
cellsrdquo Solid State Ionics vol 262 pp 856ndash859 2014 doi 101016jssi201310055
[231] T Yamaguchi et al ldquoProton-conducting phosphate glass and its melt exhibiting high
electrical conductivity at intermediate temperaturesrdquo J Mater Chem A vol 6 no 46 pp
23628ndash23637 2018 doi 101039C8TA08162J
VITA
Collin Wilkinson was born in Mt Carroll IL He attended Coe College as an undergraduate and
studied glass resulting in a physics bachelorrsquos degree in 201 He worked with Dr Ugur Akgun
and Dr Steve Feller on modeling different key properties of glass propertiesresponses to external
stimuli Collin joined the group of Dr John Mauro at Pennsylvania State University in 2018
List of publications written by first author (or co-first author) Collin while at Penn State
1 CJ Wilkinson JC Mauro Explorerpy Mapping the Energy Landscapes of Complex
Materials SoftwareX Submitted
2 CJ Wilkinson DR Cassar AV DeCeanne KA Kirchner ME McKenzie ED Zanotto and
JC Mauro Energy Landscape Modeling of Crystal Nucleation NPJ Computational
Materialis Submitted
3 CJ Wilkinson K Doss O Gulbiten DC Allan JC Mauro Fragility and Temperature
Dependence of Stretched Exponential Relaxation in Glass-Forming Systems JACERS
Submitted
4 CJ Wilkinson JC Mauro Comment on ldquoThe Fragility of Alkali Silicate Glass Melts Part
of a niversal Topological Patterrdquo by DL Sidebottom JNCS (2020) ( 2 ) 11
5 CJ Wilkinson K Doss DR Cassar RS Welch CB Bragatto JC Mauro Predicting Ionic
Diffusion in Glass From its Relaxation Behavior Journal of Physical Chemistry B (2020)
124 (6) 1099-1103
6 Z Ding CJ Wilkinson J Zheng Y Lin H Liu J Shen SH Kim Y Yue J Rent CJ Mauro
Q Zheng Topological Understanding of the Mixed Alkaline Earth Effect in Glass JNCS
(2020) (526) 119696
7 CJ Wilkinson AR Potter RS Welch CB Bragatto Q Zheng M Bauchy M Affatigato SA
Feller JC Mauro Topological Origins of Mixed Alkali Effect in Glass Journal of Physical
Chemistry B (2019) 123 (34) 7482
8 CJ Wilkinson K Doss SH Hahn Nathan Keilbart AR Potter NJ Smith I Dabo ACT Van
Duin SH Kim JC Mauro Topological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phase JPCL (2019) 10 (14) 3955
9 CJ Wilkinson K Doss G Palmer JC Mauro The Relativistic Glass Transition A Through
Experiment JNCSX (2019) 100018
10 CJ Wilkinson Q Zheng L Huang JC Mauro Topological Constraint Model for the
Elasticity of Glass-Forming Systems JNCSX (2019) 100019
11 AR Potter CJ Wilkinson SH Kim JC Mauro Effect of Water on Toplogical Constraints
in Silica Glass Scripta Materialia (2019) (160) 48-52
12 CJ Wilkinson E Pakhomenko MR Jesuit AV DeCeanne B Hauke M Packard SA Feller
JC Mauro Topological Constraint Model of Alkali Tellurite Glasses JNCS (2018) (502)
172
13 CJ Wilkinson YZ Mauro JC Mauro RelaxPy Python code for Modeling of Glass
Relaxation Behavior SoftwareX (2018) (7) 255
v
52 Methods 84 53 Results 86
53A Adam-Gibbs Validation 86 53B MYEGA Validation 87 53C Adam-Gibbs and Structural Relaxation 89 53D Landscape Features 91
54 Topography-Property Relations 94 55 Barrier Free Description of Thermodynamics 99 57 Toy Landscapes for the Design of Glasses and Glass Ceramics 102 57 Discussion 107
Chapter 6 Enabling the Prediction of Glass Properties 109
61 Controlling Surface Reactivity 109 62 Elastic Modulus Prediction 125 63 Ionic Conductivity 135 64 Machine Learning Expansion 145
Chapter 7 Designing Green Glasses for the 21st Century 152
71 Glass Electrolytes 152 72 Hydrogen Fuel Cell Glasses 158
Chapter 8 Conclusions 167
References 169
vi
List of Figures
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists
the equilibrium liquid which as it is quenched can either become a super cooled
liquid or crystallize Crystallization causes a discontinuity in the volume The super-
cooled liquid upon further quenching departs from equilibrium and transitions into
the glassy state Reproduced from Fundamentals of Inorganic Glass Science with
permission from the author1 5
Figure 2 The schematic for the flow of the program Beginning in the top right corner
and running until the condition in the pink box is satisfied Yellow diamonds
represent checks and blue operations 25
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above
and then plotted using PyConnect The plot is a disconnectivity graph where each
terminating line represents an inherent structure and tracing where two lines meet
describes the activation barrier The potentials are taken from the BKS potential 94 27
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-
axis being an arbitrary phase space and the y-axis being the potential energy
calculated from the Pedone et al potentials36 The colors are indicative of the
crystallinity where the blue basin is the initial starting configuration The landscape
shows the lowering of energy associated with partially crystallizing the sample (B)
The energy landscape relationship between the cutoff for crystalline and super-
cooled liquid states for 256 atoms is shown This shows a clear drastic energy
change occurring around the cutoff value of 10 Aring 37
Figure 5 An example interfacial structure between the crystalline phase on the left and
the last sequential SCLglass phase on the right for a barium disilicate system The
gray atoms are barium silicon is shown in red and blue represents oxygen 43
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a
function of temperature for each system size The experimental values for kinetics
and thermodynamics come from ref125 and from the heat capacity data taken from 126 respectively It is clear to see that driving force shows good agreement across all
systems however the kinetics terms only converge for the 256 and 512 atoms
systems (Bottom) The fit used to calculate the interfacial energy as a function of
temperature 45
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted
using the model presented in this work The data referenced can be found in refs 113128129 47
Figure 8 The surface energy with respect to temperature for the work presented here 50
Figure 9 An Angell diagram created using the MYEGA expression 55 With a very strong
glass (m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) 17134135
vii
The infinite temperature limit is from the work of Zheng et al54 and the glass
transition temperature is from the Angell definition 54
Figure 10 The relativistic glass transition temperature for B2O3 glass 57
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different
fractions of light speed All of the viscosities approaching the universal temperature
limit for viscosity 59
Figure 12 The modulus needed to satisfy the condition for the glass transition 61
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at
v=044c (Bottom) Viscosity plot showing a dramatic reduction of the viscosity as
the observer approaches the speed of light 63
Figure 14 b predicted and from literature showing good agreement with a total root-
mean-square error of 01 The fit for organic systems is given by 66 ln Kln = minus
and for inorganic systems by 75ln Kln = minus 72
Figure 15 The equilibrium model proposed with the experimental points showing good
agreement between the experimentally measured data points and the equilibrium
derived model RMSE was 002 for Corningcopy JadeTM (A) and less than 001 for
SG80 (B) (C) The model fit for B2O3 experimental data154155 The fragility and glass
transition temperature of the B2O3 are taken from the work of Mauro et al17 75
Figure 16 The stretching exponent calculated as described in the text for a Gaussian
distribution of barriers This plot shows that the distribution of barriers has a large
effect on the stretching exponent A Tg cannot be described since there is no
vibrational frequency included in the model though the glass transition temperature
should be the same for all distributions since the mean relaxation time is the same
for all distributions at all temperatures The deviation is given in ln eV units 78
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent
Each color designates one term in the series (Bottom) The output from RelaxPy
v20 showing the stretching exponent effects on the relaxation prediction of
Corningcopy JadeTM glass Each quadrant shows one property that is of interest for
relaxation experiments In particular it is interesting to see the dynamics of the
stretching exponent during a typical quench 80
Figure 18 The viscosity (left) and landscape (right) predictions for three common
systems The first system is newly calculated in this work while the latter two come
from our previous works7289 It is seen that the viscosity predicted from the AG
model is very accurately able to reproduce the experimental viscosity curves from
the MYEGA model The last system is a potential energy landscape while the others
are enthalpy landscapes 87
Figure 19 The configurational entropy comparisons between the three major viscosity
models which validates the main underlying assumption of the MYEGA model The
viii
VFT and AM are unable to capture the physics of configurational entropy therefore
ruling 89
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the
configurational entropy from the enthalpy landscape with the barrier 00155 eV
(compared to the shear barrier of 00149 eV) and the infinite limit allowed to vary
(10-263 Pamiddots for bulk viscosity) In Figure 18 the configurational entropy is
confirmed for the shear viscosity thus confirming the AG for both shear and bulk
viscosities Sidebottom data are from Ref 154 90
Figure 21 (A) The histogram for the energy minima for each of the test systems showing
a good fit with the log normal distribution This distribution will then be a valid
form to calculate the enthalpy distribution of the model presented in the next
session (B) The configurational entropy from the model showing the accuracy of
the scaling of the entropy predicted by the MYEGA model The S value was fit
for each system 93
Figure 22 (A) Comparison between the randomized method (histogram) and the
deterministic method (vertical lines) showing good agreement between the
maximum in the histogram and the value predicted by the deterministic technique
validating the approach It is worth noting that the 100-basin distribution is a very
wide distribution where the total number of basins is less than the number of points
used in the calculation This is done for a variable number of basins with the number
of basins shown in the legend (B) The dependence of fragility and the glass
transition temperature vs the distribution of states and the number of basins 99
Figure 23 The driving forces for different example glasses calculated using a
combination of the MAP model RelaxPy and the toy landscape model The
parameters for each glass can be found in Table 4 102
Figure 24 The predicted outputs from the toy landscape method showing predictions of
the enthalpy and entropy under a standard quench for barium disilicate This
prediction does not require fictive temperature or any such assumptions about the
evolution of the non-equilibrium behavior 105
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume
in nucleation is assumed to be on the order of one cubic angstrom while the a
parameter in growth is assumed to be around one nm both are in good agreement
for estimates in literature The values for the orange points are taken from these
works128167 107
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the
hydration models Color scheme Si atom (ivory) O atom (red) and Na atom (blue)
The z-axis is elongated to allow space for an insert of water 114
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the
top surface in contact with water is shown here 116
ix
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example
of the electronic-structure DFT calculation using semilocal exchange-correlation
functionals finding the binding energy of a water lsquopixelrsquo to the surface 119
Figure 29 Example contour surface showing the average coordination per atom on the
glass surface for the first run at 300 ps 121
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of
constraints for surface atoms at the local pixel Results show a distinct maximum in
which there is a near hydrophilic-hydrophobic transition of the surface The error
bars represent the standard deviation A second system with 1500 atoms was
performed to show convergence of the ReaxFF MD results 123
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values
of the model predictions are 641 GPa for constraint density 313 GPa for free
energy density and 774 GPa for angular constraint density 128
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from
minimum to maximum error The free energy density model gives the most accurate
results The constraint density has a RMSE of 61 GPa the angular density has a
RMSE of 20 GPa and the energy density has a RMSE of 59 GPa 129
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and
experiment for 10 Na2O 90 B2O3 Using the previously fitted onset temperatures
the only free parameters are then the vibrational frequency and the heating time in
which their product was fitted to be 14000 Where each dip in the modulus
corresponds to a constraint no longer being rigid as heated through each onset The
onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The
data was fit using a least-squares method and the resultant fit is shown as the
calculated method The fit has an R2 of 094 (B) The contribution from each
constraint to the overall modulus 131
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in
Fig 3) and experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of
093 and (B) xLi2Omiddot(100-x)B2O3 glasses with an R2 of 0986 133
Figure 35 The structure for the initial minimum energy configuration showing the boron
(blue) network with interconnecting oxygens (red) and the interstitial sodium ions
(yellow) 139
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and
sodium is ivory The barrier is overestimated compared to experimental data this
could be from several sources of error such as potential fitting thermal history
fluctuations or sampling too few transitions The line is drawn as a guide to the eye 139
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function
of reaction coordinates The middle shows the local deformation around the ion of
x
any atom that moves in between inherent structure mandating a relaxation force
The color shows the degree of deformation 141
Figure 38 Different network formers and the prediction of the activation barrier from our
model compared with activation barriers from literature (A) Sodium silicate
predictions and experimental values241 the error is calculated from the error in the
fragility when fitting the data (B) Lithium phosphate activation energy235 predicted
with topological constraint theory and compared with the experimental values (C)
Predictions over two different systems of alkali borates232 sodium and lithium with
a reported R2 of 097 143
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top
left) shows the distribution of the infinite temperature limit in the database after
limits exerted on the system (Top right) The distribution of the infinite viscosity
limit vs the glass transition (Bottom left) The relationship between fragility and
infinite temperature limit of viscosity (Bottom right) The infinite temperature limit
of viscosity vs the key metric predicted by SR 149
Figure 40 The prediction of the scaling of the activation barriers for a common sodium
borosilicate system 157
Figure 41 The relationship between the glass transition and the proton conductivity This
is justified two ways one through the relationship of the entropy of diffusion and
glass formation (the Adam-Gibbs model) and through the fact that water is known to
depress the glass transition 162
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine
the glass forming region Over top the constraints at the glass transition provided by
each oxide species is listed Since the objective is to decrease Tg while staying in the
glass forming region we will attempt to minimize use of elements that increase the
glass transition (nc gt 17) 166
xi
List of Tables
Table 1 The key properties considered for commercial application The optical properties
have been omitted since it is physical unrealistic to expect quantitative predictions of
quantum-controlled phenomena from a classical description of glass structure 18
Table 2 Variable definitions 65
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy
JadeTM glass and Sylvania Incorporatedrsquos SG80 74
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition
temperature (500 K) were kept the same while the total number of basins were
allowed to vary 101
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K 112
Table 6 System configurations for sodium silicate glass-water reactions 117
Table 7 Fitted values from this analysis compared to those reported in the literature The
disparity between the constraints evaluated with molecular dynamics most likely
come from the speed in which the samples are quenched 132
Table 8 Hyperparameters for different neural networks after hyper-optimizations 151
Table 9 A table with some ionic conductivity models and the parameters needed for
them as well as the disadvantages for each These are not the only models but are
representative of those commonly used in literature 154
Table 10 The predicted compositions based on the optimization scheme proposed 157
Table 11 The compositions synthesized in this work These compositions were predicted
by minimizing the cost function described in Eq (123) OP is the variant that was
melted after OP partially crystallized B-OP appeared to have surface nucleation in
some spots but was cut and removed before APS treatment 165
xii
Acknowledgments
ldquoTo Love Another Person is to See the Face of Godrdquo -Victor Hugo Les Miserables
I have been truly blessed to love and be loved by a plethora of mentors friends colleagues and
family across my academic journey I have to first thank my advisor Dr John Mauro who has
been as kind patient and caring as any mentor I have ever met and in the process has made me a
better scientist and a better person His mentorship was built on top of those who first found and
shaped me into a scientist Dr Ugur Akgun and Dr Steve Feller without whom my scientific career
would not be possible Along with these individuals Irsquove had the pleasure to learn so much about
glass and life from Dr Madoka Ona Dr Firdevs Duru Dr Mario Affatigato Dr Doug Allan Dr
Ozgur Gulbiten and Dr Seong Kim
Rebecca Welch Arron Potter and Anthony DeCeanne are my partners in all things
Rebecca is my partner in research and in life Her opinion support and patience has been
indispensable to me Arron is my oldest colleague and friend he is the one that I share the highs of
research and friendship with Anthony has put up with me non-stop for about 5 years now He is
the most patient and kind friend you could ask for Without these three individuals I could not have
done anything presented in this work I additionally have to thank Karan Doss Aubrey Fry Caio
Bragatto Daniel Cassar Brenna Gorin Mikkel Bodker Greg Palmer Katie Kirchner Kuo-Hao
Lee and Yongjian Yang I have the privilege to call these individuals both my friends and
colleagues
The last people I have to thank are my family I want to thank my brothers (Sam amp Quinn)
my mom and my dad My mom has helped me explore the universe through a love of reading and
her constant unwavering support My dad was the first to show me the wonders of science with
rockets as a kid as a teenager he supported my fledgling interest as he drove hundreds of miles to
every corner of this country and he is the person for whom this dissertation is dedicated
Chapter 1
The Difficulty of Optimizing Glass
Glass is a complex world-changing material Though it has existed for thousands of years
the surface of its true potential is only just being scratched To unlock the potential of glass for new
applications and to enable the maximum benefit to society we must be able to design new glasses
with wide ranging properties faster and cheaper than ever before [1]ndash[5] This is particularly
important now that the need for ecologically responsible materials is increasing due to climate
change In order to facilitate such glass design we must return to first principles and build a picture
of glass from the ground up encompassing both compositional and thermal history dependencies
into our understanding of glass properties[3] [6] The present accepted description of glass is a
non-crystalline non-ergodic non-equilibrium material that appears solid and is continuously
relaxing towards the super-cooled equilibrium liquid state[1] This definition gives insights into the
nature of glass and includes points of particular interest in this dissertation
The first point to consider is that glass is non-crystalline meaning that it contains no long-
range order This gives glass one of its key advantages being infinitely variable Since a glass is
not limited by having to reach stoichiometric crystalline structures there is no limit to the number
of possible glasses or glass structures This gives glass its possibilites Glass has been considered
as a candidate for everything from hydrogen fuel cells[7] to batteries[8] and for applications
ranging from commercial smart phone covers[9] to bioactive applications[10] The infinite
variability also explains the difficulty of designing glasses Consider a system with three
constituents for example SiO2 Na2O and B2O3 This then means there are two independent
compositional dimensions that must be fully explored to find an optimal composition for an
application If this space were discretized to 1 mol spacing that would give over 5000 unique
2
glass candidates that would need to be studied to reach the optimal design A five-component glass
would correspondingly have over 9 million unique glasses More generally for a glass with up to
C components there are 1C minus dimensions over which to optimize If instead one considers a
crystal there is not a continuously variable space but instead only discrete points
The other ramification of glass being non-crystalline is that it must bypass the region of
crystallization at a rate much faster than the rate of nucleation[11] This is inherent in the definition
of glass and is shown in Figure 1 Figure 1 is called the volume-temperature (VT) diagram and is
key to understanding the nature of glass The bypassing of the crystallization region leads to a
super-cooled liquid and during quenching the material departs from equilibrium and enters the
glassy state The region of departure is called the glass transition temperature range or simply the
glass transition To understand why this occurs we must first understand the concept of ergodicity
Ergodicity was defined by Boltzmann to mean that the time-average value of a property is
equal to the ensemble-average value of that property[12] [13] Now that ergodicity is defined we
can start to consider the implications of the non-ergodic nature of glass Since it is non-ergodic the
ensemble average will not equal the time average on human timescales however the ergodic
hypothesis says that in the limit of long time they will be equal[12] In order for both statements to
be true glass must evolve to be ergodic and thus is inherently unstable To explore this concept
further we must understand the glass transition process and define a timescale If there exists a
liquid at some temperature T and is then perturbed to T dT+ there will be some inherent time
associated with its relaxation towards equilibrium which is described by struct the structural
relaxation time There is a separate relaxation time when the system is mechanically perturbed
called the stress relaxation time stress To then understand whether the resulting material is a glass
or a liquid we need to compare struct with the observation time If the observation time is much
longer than the relaxation time then what is observed over the course of a experiment is the
3
equilibrium liquid properties Conversely if the observation time is much shorter than the
relaxation time then the observed properties will not be ergodic and as such we will not sample
the properties of the equilibrium liquid We define this non-ergodic state as the glassy state
However since in the limit of long time the system must become ergodic once again then over time
all glasses will return to an ergodic nature through a process called relaxation [14]
The last point considered in the definition of the glass is that it is non-equilibrium Not only
is glass non-equilibrium it is also unstable and continuously progressing towards the super-cooled
liquid state (eg relaxation) Relaxation is inherent in all glasses and we can see this for a model
system in the VT diagram (Figure 1) In this diagram there are two glasses (of the same
composition) quenched at different rates (both of which bypass crystallization) In the slow-cooled
glass there is a lower volume (higher density) compared to the fast-cooled glass This difference is
due to the fact that the slow-cooled glass has been allowed to relax more fully compared to the
other This adds at least one additional dimension of optimization (such as quench rate from a high
temperature to room temperature) in which glasses must be designed in
This dissertation is dedicated to understanding the infinite variability of glass with respect
to crystallization composition and thermal history with tools being implemented and designed so
that the challenge of optimizing over these vast spaces can be done with higher efficiency lower
cost and with less experimental work load These tools are all inspired by the energy landscape
description of glass To design the optimal glass for any application we must understand the
relationship between the 1C minus compositional dimensions a minimum of one thermal history
dimension and the properties of a material There are many ways to explore these relationships
each with its advantages and disadvantages The methods to optimize over these thermal history
and compositional dimensions include energy landscapes[15] fictive temperature[16] topological
constraint theory[17] and machine learning[3] Of these energy landscapes are the most
fundamental with each other technique being related back to the energy landscape in some form
4
Though they are the most fundamental they are also the most difficult time consuming and have
the largest barrier to entry for use To then build our understanding of each technique we will start
with energy landscapes then explore how through a series of assumptions we can arrive at the other
methods
5
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists the
equilibrium liquid which as it is quenched can either become a super cooled liquid or crystallize
Crystallization causes a discontinuity in the volume The super-cooled liquid upon further
quenching departs from equilibrium and transitions into the glassy state Reproduced from
Fundamentals of Inorganic Glasses with permission from the author[1]
6
11 Energy Landscapes
Energy landscapes were first developed by Goldstein and Stillinger who proposed the
concept as a way to understand the evolution of a material under complex conditions [18]ndash[20]
Since Stillingerrsquos seminal work [18] energy landscapes have become important in the study of
protein folding [15] [21] [22] network glasses [6] [13] and other complex materials The concept
of energy landscapes has become commonplace and is readily evoked when describing the time
evolution of a material [20] [22]ndash[26] Despite energy landscapes being commonplace in some
fields the procedure for mapping a landscape remains difficult due to the associated computational
challenges [15] [27] To map an energy landscape one must find a set of local minima and the first
order saddle point connecting each pairwise combination of those minima Each minimum is called
an inherent structure and represents a structurallychemically stable state of the material A basin
is the set of structures that converge to the same inherent structure upon minimization [18]
In glass science the energy landscape is used to justify complex behaviours which are most
commonly calculated using molecular dynamics simulations (MD) [13] [28] [29] It is easy to
understand why MD is used so frequently used when considering the ease of use of common MD
packages For example LAMMPS [30] is a common user-friendly MD program with a low barrier
to entry allowing individuals with little programming experience to run complex MD simulations
Inspired by this accessibility as well as the explicit Python bridge to LAMMPS we have developed
a software that can be used to explore energy landscapes with a variety of exploration techniques
Similar softwares [15] [31]ndash[35] have been created to map landscapes however each program has
limitations and may not be suitable for all purposes while ours is a general purpose software
The calculation of the energy for landscapes is done using an empirical interatomic
potential that can be fit using either experimental data and a series of MD simulations or using ab
initio methods One common potential form is the Lennard-Jones form (LJ)
7
12 6
4LJVr r
= minus
(1)
in which and are fitting parameters for every pair-wise set of atoms in a system V is the
potential energy contributed by the two-atom interactions and r is the distance between the two
ions in question Though the LJ form is simple and easy to parameterize it is unable to recreate
complicated material behaviours for a real glass As such a more complicated potential is often
used such as a Morse potential A commonly used silicate glass potential is the Pedone potential
which consists of Morse potential with an additional columbic term and repulsive term [36] and is
given by
( )2
2
0 121 exp 1
i j
Pedone
Z Z e CV Y a r r
r r = + minus minus minus minus +
(2)
Z is the charge of a given ion e is the charge of an electron and Y C a and r0 are fitting parameters
It is also worth noting that the Pedone et al potentials are only accurate when considering the
interactions between each individual ion species with oxygen For instance Si-O Na-O and O-O
are all explicitly parameterized while Si-Na only interacts coulombically (the first term)
Interatomic potentials are discussed here only briefly but there are extensive resources available
for those readers who wish to learn more[37]ndash[39]
Once an energy landscape has been mapped the thermodynamics and kinetics for a given
system can be easily calculated [33] [40] The thermodynamic driving force is given by the
difference between the free energy of the current inherent structure and the free energy of the lowest
state while the kinetics is the rate in which the system crosses the barrier between the basins This
can be calculated explicitly for arbitrary timescales and temperatures with another software
KineticPy [41] The energy landscape provides insights into the detailed nature of material
behaviour Though energy landscapes are very powerful they are also difficult to map To
approximate the behavior of the landscape then we use approximations such as fictive temperature
8
topological constraint theory (TCT) or machine learning Fictive temperature is a way to estimate
the effects of the thermal history dimension[42] while TCT is used to estimate the compositional
dependence of properties[43]
12 Fictive Temperature
Fictive temperature is an approximation used to find the change in the occupational
probabilities on an energy landscape[16] [44] Instead of considering every possible combination
of occupational probabilities like the energy landscape it instead approximates the occupational
probability using a single temperature at which the landscape would be in equilibrium It is
essentially a measure of how closefar a glass is from equilibrium This approximation allows for
us to calculate the relaxation dynamics of a glass without needing any computationally heavy
energy landscapes However this single parameter description oversimplifies relaxation[14] [16]
The functional form for glass relaxation dates back to 1854 when Kohlrausch fit the
residual charge of a Leyden jar with[45]
( ) (0)expt
g t g
= minus
(3)
where t is time τ is the relaxation time constant β is the stretching exponent and g(t) is the
relaxation function of some property (eg volume conductivity viscosity etc) Eq (3) is referred
to as the stretched exponential relaxation function (SER) The stretching exponent is bounded from
0 lt β le 1 where the upper limit (β = 1) represents a simple exponential decay and fractional values
of β represent stretched exponential decay
For over a century the physical origin of the stretched exponential relaxation form was one
of the greatest unsolved problems in physics until 1982 when Grassberger and Proccacia[46] were
able to derive the general form based on a model of randomly distributed traps that annihilate
9
excitations and the diffusion of the excitations through a network This model however provided
no physical meaning for β which was still treated as an empirical fitting parameter In 1994
Phillips[47] [48] was able to extend the diffusion trap model by showing that β=1 at high
temperatures and at low temperatures the stretching exponent can be derived based on the effective
dimensionality of available relaxation pathways He in turn expressed the stretching exponent
2
df
df =
+ (4)
where d is the dimensionality of the network and f is the fraction of relaxation pathways available
He then proposed a set of lsquomagicrsquo numbers for common scenarios Assuming a three-dimensional
network with all pathways activated (d = 3f = 1) a value of β= 35 is obtained The second case is
a three dimensional network with only half of the relaxation pathways activated (d =3f = 12)
yielding a value of β= 37 A third magic value β= 12 was found for a two dimensional technique
with all pathways activated (d = 2f = 1) β = 35 occurs for stress relaxation of glasses under a load
because both long- and short-range activation pathways are activated whereas a value of β = 37 is
obtained for structural relaxation of a glass without an applied stress The value was confirmed by
Welch et al[9] when the value was measured over a period of 15 years at more than 600C below
the glass transition temperature of Corning Gorilla Glass Though this model was able to reproduce
the limiting values it was criticized widely in the community Critics argued that the model was
simply created such that it reproduced the stretching exponent values of certain experiments and
ignored many other experimental results that appeared to disagree with the model The model also
fails to interpolate between the low temperature values Phillips predicted and the high temperature
limit
The work from Phillips and the stretched exponent gave a form for relaxation however
there remained no method to instantaneously state the distance from equilibrium of a glass To
account for this problem an additional thermodynamic variable (or order parameter) called fictive
10
temperature (Tf) was proposed Early research on relaxation in glasses dates back to the work of
Tool and Eichlin in 1932[49] and Tool in 1946[50] whose works originally proposed a temperature
at which a glass system could be in equilibrium without any atomic rearrangement Tool suggested
that this fictive temperature was sufficient to understand the thermodynamics of a glassy system
Originally the fictive temperature was treated as a single value that was some function of thermal
history (Tf[T(t)]) The evolution of fictive temperature as proposed by Tool is then given by
)
( ) ( ( ))
(
f f
f
dT T t T T t
dt T T
minus= (5)
In which for stress relaxation is given by
( )(( ) ( ))fT t T
G
T t = (6)
In Eq (6) is the shear viscosity and G is the shear modulus This means that to predict the time
evolution of a glass only three things are required shear modulus the viscosity as a function of
temperature and fictive temperature and the stretching exponent Though fictive temperature
qualitatively reproduced the results they were looking for subsequent experiments have shown that
the concept of a single fictive temperature is inadequate[51] [52]
A key experiment was performed by Ritland in 1956[51] Ritland took several samples
with different thermal histories but the same measured fictive temperature thus based on Toolrsquos
equation both should have had identical relaxation properties Ritland however showed that the
refractive index evolved differently between samples To account for these differences
the use of multiple fictive temperatures was suggested[52] This worked because the stretched
exponential form could be approximated with a Prony series
( ) exp expi i
N
i
Kt w K t minus minus
(7)
11
In the Prony series N is the number of terms while iw and iK are fitting parameters This means
that each term could represent one fictive temperature and this collection of fictive temperatures
would reproduce the overall experimentally observed form Ritlandrsquos experiment showed that
materials held isothermally at their fictive temperatures with varying thermal histories can give
different results for the relaxation of a given property Thus as a glass network relaxes the fictive
temperature will also shift Narayanaswamy[52] was able to apply Ritlandrsquos concept in a highly
successful engineering model with multiple fictive temperatures but the physical meaning of
multiple fictive temperatures remains elusive
In general there is reason to be skeptical of the concept of fictive temperature Fictive
temperature was not derived but instead was an empirical approximation that Tool needed to gain
an early quantitative understanding Recently Mauro et al[16] used energy landscapes to try to
understand the underlying validity of fictive temperature fT implies that the occupational
probability ( ip ) is given by
1
exp ii
Hp
Q kT
= minus
(8)
Q is the partition function k is Boltzmannrsquos constant and iH is the enthalpy of the given state
However when the actual probability evolution predicted by fictive temperature was compared to
the probability calculated through a systematic Monte Carlo approach the results were found to be
drastically different (even when many fictive temperatures were tried) This implied that fictive
temperature is insufficient to describe the glassy state and must be replaced with a new method
Despite the inadequacies of fictive temperature state-of-the-art modeling techniques for
glass relaxation still relies on the concept In these techniques a set of fictive temperatures (usually
12) are used to calculate 12 parallel relaxing fictive temperatures[42] This is because there has
been no alternative method able to reach a prediction of the evolution of glass under different
12
thermal histories The current state-of-the-art model is called the Mauro-Allan-Potuzak (MAP)[44]
model of non-equilibrium viscosity and was derived from a more fundamental expression for
viscosity called the Adam-Gibbs model (AG) which is given by[53]
10 10) log( )
log ( f
c f
BT T
TS T T = + (9)
B is the barrier for a cooperative rearrangement 10log is the infinite temperature limit of
viscosity and according to Zheng et al[54] it is approximately 10-293 Pa s and cS is the
configurational Adam-Gibbs entropy This entropy has long since been a source of debate and will
be addressed later in chapter 5
In order to calculate the equilibrium part of the viscosity the MAP model relies on an earlier
Mauro Yue Ellison Gupta and Allan (MYEGA) equilibrium viscosity model[55] This model
was derived based on earlier work by Naumis[23] [56] Gupta and Mauro[57] and the AG model
These models showed that the configurational entropy can be related to the degrees of freedom
lncS fNk= (10)
f is the degrees of freedom per atom N is the number of atoms and is the number of degenerate
states per floppy mode To then build in temperature dependence they assumed that the degrees of
freedom can be modeled using an Arrhenius form (with activation barrier H )
( ) expH
f T dkT
= minus
(11)
When Eqs (9) (10) and (11) are combined into an equilibrium form the equilibrium viscosity is
then given by
10 10 el pog og xlC H
T kT
+
=
(12)
Where C is equivalent to
13
ln
BC
dkN=
(13)
Through a change in variables this can be written in terms of three common variables that allow
for all viscosity equations to be written with three variables specified by Angel[58] the glass
transition temperature the fragility and 10log The glass transition temperature is defined as
the isokom where the liquid has a viscosity of 1210 Pa s The fragility is defined as[58]
10log
g
g
T T
dm
Td
T
=
(14)
The three most common viscosity models (MYEGA Vogel-Fulcher-Tammann (VFT)[59] and
Avramov-Milchev (AM)[60]) are listed below in terms of the same parameters
10 10 10
10
log (12 log ) exp 1 11 o
log2 l g
g gT Tm
T T
+ minus minus minus
minus =
(15)
( )
2
10
10 10
10
12 logl log
1 1
og
2 logg
Tm
T
minus+
minus + minus
=
(16)
and
( )10(12 log
10 10 1
)
0l gog l 12 logo
m
gT
T
minus
+ minus=
(17)
To incorporate non-equilibrium effects with these equilibrium models the MAP model
proposes
10 10 10log ( ) (1 ) log ( )log MYEGA g ne f gT m T mx x T T T = + minus (18)
14
10log ne is the non-equilibrium viscosity whose form and compositional dependence can be found
elsewhere[42] [44] [61] x is the ergodicity parameter and is of crucial importance to
understanding glass regardless of the model being used[12] In the MAP model it is given by
( )( )
3244min
max
f
f
m
T Tx
T T
=
(19)
Though the MAP model was derived from the best understanding of energy landscapes of glasses
at the time it has a few flaws that prevent it from perfectly predicting the different relaxation
behaviors of glass A few of the short-comings are
1 The method ignores crystallization For a comprehensive model showing the response of
glass to temperature the possibility of crystallization must be included Currently no
relaxation model includes this possibility
2 It is built around the concept of fictive temperature This could be replaced in a future
model but currently at present it carries with it all the failures of the fictive temperature
picture of glass dynamics adding only the relaxation time coming from landscape-derived
values
3 The MAP model does not consider temperature dependence of the stretching exponent
This is typically circumvented by choosing a fixed stretching exponent
4 The parameters needed for the non-equilibrium viscosity in the MAP model are
experimentally challenging to measure To circumvent this Guo et al[61] published
methods to estimate the values based on fragility and the glass transition but the
approximations were only tested on a few samples with similar fragilities and chemistries
5 The MAP model only predicts stress relaxation Since the publication of the MAP model
it has been shown that only stress relaxation time is related to the shear viscosity[62]
Structural relaxation must be incorporated in future models
15
For a complete understanding of the thermal-history dependence of glass all of these questions
must be rigorously answered
13 Topological Constraint Theory
TCT is one way to estimate the compositional dependence of glasses It uses a set of
assumptions that connects the underlying structure to glass properties through understanding of the
energy landscape These assumptions include the temperature dependence of the rigidity the same
assumptions used in the derivation of the MYEGA models and that the effects of thermal history
on structure are minimal TCT was originally proposed by Gupta and Cooper (GCTCT) to try to
understand the role of rigid polytopes on glasses properties[63] Gupta and Cooper fundamentally
understood that the ability for glasses to form was reliant on the transformation of a liquid with
many degrees of freedom to a glass with a fixed degree of freedom Since the structure of the glass
is the same as the structure of the liquid at the glass transition and the degrees of freedom of the
glass ( f ) changes during the transition to glass it was possible to gain insights into the
transformation process This approach is fundamentally an approximation of the highly
multidimensional landscape where only approximate structural information is used to understand
the entropy of the energy landscape Though the core idea was useful it was ldquoclunkyrdquo and difficult
to make calculations with
To improve upon the ideas of GCTCT Phillips and Thorpe (PTTCT)[64] proposed a
mathematically equivalent form where the degrees of freedom could be simply found by
considering the number of constraints ( cn ) around each network-forming atom in the glass They
showed that the degrees of freedom were related to the number of constraints by
cf d n= minus (20)
16
The number of constraints could then be calculated by considering the two-body and three-body
interactions (first and second terms) as a function of the mean coordination of the network forming
species
2 32
c
rn r= + minus (21)
This led to a qualitative language to discuss glass with a few quantitative insights The biggest
quantitative insight comes in the form of the lsquoBoolchand Intermediate Phasersquo[65]ndash[68] If it has
positive degrees of freedom then a network is lsquofloppyrsquo and retains high kinetic energy (as it flips
back and forth between two inherent structures) If the system has negative degrees of freedom
then the network is stressed-rigid and there is additional energy stored in the stress associated with
over-coordination If the number of constraints is equal to the number of dimensions then the
resulting system will have zero degrees of freedom and there is no stored stressed energy or kinetic
energy from flipping between basins Boolchand went searching for this perfectly lsquoisostaticrsquo glass
and found a non-zero window of compositions that were close to isostatic but had properties
drastically different compared to both stress rigid and floppy glasses This is now called the
Boolchand Intermediate Phase
Despite the success in the prediction of the intermediate phase TCT remained largely
qualitative until the work by Naumis[23] [56] and Gupta amp Mauro[57] Naumis pointed out that
the degrees of freedom are a lsquovalleyrsquo of continuous low energy so that the structure can freely move
between basins of the landscape Since these valleys are deformation pathways Naumis pointed
out that they will dominate the configurational entropy of the system Building on this Mauro and
Gupta wrote that since the entropy is proportional (Eq (10)) we could then write the glass transition
temperature as a function of the degrees of freedom and the AG model
( )10o1 ln2 l g
gTB
f Nkminus=
(22)
17
Since everything in Eq (22) is approximately a constant between compositions except for f we
could then write the expressions in terms of reference values (subscript r) and accurately scale the
glass transition of new compositions by knowing the changing structures
( ) g r c r
g
c
T d nT
d n
minus=
minus (23)
The last insight provided by Mauro and Gupta was that the number of constraints change as a
function of temperature They realized that each constraint has a temperature dependence which is
controlled by the interaction energy of the constraints This makes intuitive sense because as the
thermal energy becomes greater than that of the barriers additional valleys in the energy landscape
form This also allowed them to expanded TCT to include quantitative predictions of fragility
The understanding of the relationships between the configurational entropy the number of
constraints the degrees of freedom and the viscous properties lead to a revolution in understanding
glass properties Soon after the work by Mauro amp Gupta multiple works came out predicting
mechanical chemical and viscous properties of glass knowing only the underlying structure
Though this framework is powerful it is not a complete description without further
parameterizations with each model requiring some fitting parameters or reference values In
addition there are still key properties that are missing from the standard topological approach that
are needed to design commercial glasses
Ultimately MGTCT needs be expanded to include the properties that are needed for
commercial applications A list of composition dependent properties needed for understanding the
performance of commercial glasses are shown in Table 1 Properties taken from Ref [3] show what
properties must be controlled to manufacture technological glasses All models take additional
inputs and there are multiple ways to link outputs from some models to inputs of the other Choice
of model vary depending on the use case
18
Table 1 The key properties considered for commercial application The optical properties have
been omitted since it is physical unrealistic to expect quantitative predictions of quantum-
controlled phenomena from a classical description of glass structure
Properties TCT Machine Learning Other methods
Glass Transition [57] [69] [70] Energy Landscapes [71]
Fragility [57] [69] This work This work
Relaxation MAP model [44] [61]
Crystallization This work [72]
MDMC [73]ndash[75]
Melting Temperature This work MDThis work[72]
Chemical Durability [76] [77]
This work [78]
[79] [80] Linear Methods [81]
Density [82] MD Packing [83]
CTE This work MD
Hardness [5] [84] [85] MD
Youngrsquos Modulus [86] This
work [87]
[82] This work MD
Activation Barrier for Conductivity MD This work [88]
Batch Cost Economic Calculation
Ion Exchange Properties Empirical Relationships [2]
19
14 Machine Learning
All the methods discussed so far are physically informed techniques The techniques
presented are inherently powerful because their very nature is informed by reality Unfortunately
this means that some knowledge about the system is required These techniques are useless without
the required parameterizations However a class of algorithms called lsquoMachine Learningrsquo (ML)
require no such parameterization Instead ML uses large quantities of data to determine trends
linking any physically connected input to output allowing for accurate predictions of properties
across wide compositional spaces This is favorable with a large of volume data because the
required inputs are minimal to achieve a prediction The difficulty (and often expense) in this
technique is gathering large quantities of high-quality data to enable this method
The predominant ML method used in this work is neural networks (NN) Neural networks
work by constructing a network of neurons This method was designed in such a way to mimic a
human brain so the network is elastic enough to be modified improving results Each neuron takes
in a series of inputs received from the previous layer (or input data) multiplies by a weighting
factor sums the values and then performs some lsquoactivation functionrsquo on them By changing the
weights it is then possible to recreate any function given sufficient data This is a preferred
technique because it packages into a simple easy-to-use software Neural Networks (once trained)
can be used by anyone stored in a small file can recreate any function is computationally cheap
and does not require as much data as some other ML techniques
This is an interesting method to estimate the effects of the composition on the landscape
and corresponding properties because we cannot understand the underlying mathematics It is
fundamentally a numerical process Thus even though we know that the properties of a glass are
dominated by the topography of the landscape how the NN figures out the relationship between
the chemistry and properties is not understood It is possible to then use the NN to give us the
20
physical parameters we need for other models building multiple methods together to achieve new
insights
15 Goals of this Dissertation
To enable the next generation of glass design we must enable a new era of computationally
methods being used to produce accurate predictions of the behavior of glass This includes both the
thermal history dependence of glass (in terms of relaxation and crystallization) and the
compositional dependence especially those properties listed in Table 1 The goals of this thesis will
be to address some of these outstanding questions in a systematic way that is practical for an
external user to understand and implement Predicting thermal history effects are covered in
Chapters 3-5 while compositional effects are predicted in Chapters 6-7 The breakdown of each
following Chapter as it relates to these questions is listed below
21
bull Chapter 2 Software Unfortunately there is no standardized software for the
calculation of relaxation or energy landscapes I present two new codes to
standardize these calculations so that the process is reproduceable and standardized
bull Chapter 3 Crystallization This section will address new computational ways to
calculate crystallization focusing on predicting nucleation rates and how to improve
our nucleation calculations
bull Chapter 4 Expanding the Current State of Relaxation This chapter will focus on
expanding the MAP model with a temperature dependent form of the stretching
exponent as well as understanding the role of ergodicity in glass relaxation
bull Chapter 5 Relaxation and Crystallization without Fictive Temperature Chapter 5
will focus on discussing relaxation and crystallization without fictive temperature
by proposing a new relaxation model that is reliant on creating lsquoToy Landscapesrsquo
This approach removes assumptions about the evolution of the occupational
probabilities
bull Chapter 6 Enabling Prediction of Properties This chapter will use TCT and ML to
address the compositional dependance of different key properties needed for
enabling the faster design of new glass compositions
bull Chapter 7 Designing lsquoGreenrsquo Glasses Applying information learned throughout
this dissertation to create new glass compositions
bull Chapter 8 Conclusions
22
Chapter 2
Software for Enabling the Study of Glass
To being our systematic investigation it is important to have a suite of tools that
implement current state-of-the-art models Though many tools exist for structure and
topological constraint theory there were no codes that existed for the calculation of
relaxation using the MAP model and there were no codes that were capable of exploring
the landscapes that we are interested in To correct this we present two new novel codes
ExplorerPy for mapping energy landscapes and RelaxPy for calculating MAP
dynamics[42] [89]
21 ExplorerPy
Mapping an energy landscape allows for modeling a range of traditionally inaccessible
processes such as glass relaxation The advantage of this software (over other codes to map energy
landscapes) is that it only requires knowledge of LAMMPS to run Also our software contains a
choice of three methods to map the landscape The first of the three different methods implemented
to explore energy landscapes in this work is eigenvector following [90] Eigenvector following has
been used in the past to generate landscapes that predict the glass transition and protein folding
pathways [44] [91] This implementation follows the work of Mauro et al [90] [92] and uses an
independent Lagrange multiplier for every eigenvector Eigenvector following excels at finding
small barriers quickly however it rarely finds unique structural changes in complex materials
(especially in systems with periodic boundary conditions) The second method that we have
implemented following in the steps of Niblett et al [27] is a combination of molecular dynamics
and nudged elastic band methods (NEB) [34] [35] To find the transition points MD is run at a
23
fixed temperature for a fixed amount of time the structure is minimized and then NEB is used to
find the saddle point between the initial structure and the new inherent structure Details of the
nudged elastic band method can be found elsewhere [34] Using these techniques it is possible to
map a sufficiently large landscape to make kinetic and thermodynamic measurements possible
These two methods benefit from the fact that they work well when the volume is varied and together
enable computationally inexpensive mapping of enthalpy landscapes [71] [92] (where the volume
is no longer fixed and instead the pressure is specified) However it is not recommended to generate
a large enthalpy landscape from MD in this software due to the computational cost involved
Although molecular dynamics is a powerful technique there is a large computational cost to
simulate a small amount of time
The third method is a new technique created in an attempt to circumvent the computational
cost of MD while still finding large structural changes We have named this technique ldquoeigen
treibanrdquo (ET) This technique relies on shoving along an eigenvector with a multiplier rather than
stepping After the system is shoved the system is minimized and if the minimized structure exhibits
a displacement in atomic position is larger than a threshold a NEB calculation is performed to find
the saddle point By shoving in the direction of an eigenvector the goal is to reproduce large changes
due to vibrational modes The fixed multiplier (m0) is calculated by taking a target distance (d) and
the largest displacement magnitude out of the atoms j (vij) in a given eigenvector direction i (ai)
m0
=d
max[nij] (24)
To convert to the displacement magnitude from an eigenvector in three dimensions we use the
expression
vij
= ai[3 j]2 +a
i[3 j +1]2 + a
i[3 j + 2]2 (25)
The dimension j has the length of the number of atoms in the systems
24
This software uses the system and potentials created from a LAMMPS input
scripts and as such has a wide range of access to different potentials and possible systems The
only input that has to be provided is the file which would be used for molecular dynamics in
LAMMPS If the script has been used for molecular dynamics simply by eliminating any
thermostats or barostats as well as any ldquorunrdquo commands it is possible to use it as an input script to
ExplorerPy (it is also recommended to eliminate any calls to ldquoThermordquo because it may interfere
with the extraction of variables from the underlying LAMMPS framework) If the LAMMPS input
script successfully sets up a simulation all other variables and parameters can be controlled from
command line options as described in the ldquoIllustrative Examplerdquo section The initial structure will
be taken from the input script as well
ExplorerPy is a software that has been written in Python with modular functions and ease
of use in mind It can be run from a Linux terminal with all major options being set via the command
line The software then generates a list of inherent structure energies stored in the mindat file and
stores the transition point energy for each pairwise inherent structure in the tsddat file All of the
transition points and minimum are stored in the same directory from which it is run as xyz files
The software architecture is shown in Figure 2 Details of the stepping procedure for eigenvector
following can be found elsewhere[93] The variables ldquoMD frequencyrdquo and ldquoET frequencyrdquo are set
as command line options at the time of the launch of the software
25
Figure 2 The schematic for the flow of the program Beginning in the top right corner and running
until the condition in the pink box is satisfied Yellow diamonds represent checks and blue
operations
The software is capable of mapping an energy landscape for an arbitrary system ie for
any system that can be simulated in LAMMPS This allows for accurate prediction of the evolution
of complex thermodynamics and kinetics The software requires no programming and is
controllable via command line options ExplorerPy prints all of the basins transition states inherent
volumes as well as information about the curvature of the system so that the vibrational frequency
is calculable This is all of the information required to calculate a complex range of phenomena
over a specified long timescale ExplorerPy is only set up to work within the LAMMPS metal units
To run the software there are only a few parameters that have to be set for each mode The
force threshold (-force) curvature threshold (-curve) and step size (-stepsize) when running the
command line option have to be set to allow for a stable exploration of a given landscape
These seem to be potential-specific and for a given potential appear to be fairly stable It is best to
start large for the force threshold and small for the eigen threshold and step size then modify as
26
necessary If eigenvector following is not being used there is no need to adjust these values Beyond
this there are several optional command line options (only the file (-file) command is required)
-file [file input] this is a standard lammps input if the input sets up a lammps run
it will work to set this up There should not be any lsquothermorsquo calls in the lammps
script
-press [specified pressure in atmospheres] if this is specified the system will
always be minimized to this pressure while if not specified the volume remains
fixed
-num [number of basins to explore]
-time [max time to run in hours] if this time is reached before the number of basins
are found the simulation will stop
-thresh [minimum energy for barrier to keep]
-center if this is given as a flag the system will be set to the anchor location each
step
-anchor [atom selection for the position to be held constant]
-md [frequency of molecular dynamics search] If a custom MD LAMMPS file is
not provided it defaults to starting all of the atoms at 3000 K and running for 50
fs
-shove [frequency of eigen treiban]
-custom [file used for MD exploration]
-help [List all commands possible in software with these descriptions]
An example run would then be done the following way
[mpirun -np 5 python37 Explorerpy -file inSiO2 -num 1000 -force 10 -curve 1e-5 -stepsize 012
-press 00 -md 11 -time 48]
This would start a run with five parallel processors searching for 1000 basins with an MD
step being ran every 11 searches with a maximum time of 48 hours The results of this command
are shown in Figure 3 Below we have attached an example of inSiO2 the output and additional
scripts related to the example has been uploaded to the associated GitHub
27
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above and then
plotted using PyConnect The plot is a disconnectivity graph where each terminating line
represents an inherent structure and tracing where two lines meet describes the activation barrier
The potentials are taken from the BKS potential [94]
httpsgithubcomlsmeetonpyconnect
28
22 RelaxPy
RelaxPy script is written using Python syntax and relies on the numpy scipy and
matplotlib libraries It solves the lack of a unified relaxation code in our community and is based
on the MAP model[44] [61] interpretation of multiple fictive temperatures and the shear
modulusviscosity form of the Maxwell equation[62] [95] It is designed to be run within a Linux
terminal with command line options or inside a Python interpreter It can be used by calling the
name of the script followed by the input file the desired output file name and finally an optional
tag for all fictive components to be displayed as well as printed to a specified output file (formatted
as comma separated values)
The RelaxPy package consists of an algorithm that iterates over time in order to determine
the values of viscosity (η) relaxation time (τ ) and current fictive temperature (Tf ) given the user-
supplied thermal history and material property values The purpose of this code is to enable
calculation of the relaxation behaviorrsquos change over time by using the MAP model using only
experimentally-attainable parameters Toolrsquos equation is solved for each term in a Prony series
approximation of relaxation finding the change in each Tfi component for a given time (t) to
accurately reproduce the stretched exponential form The program then iterates over the entire
thermal history of the sample using a user-specified time step dt allowing for calculation of the
overall fictive temperature over time The values for the Prony series fit (wi and Ki) are taken from
the database originally generated by Mauro and Mauro[96] and expnded to include any value of
the stretching exponent The values were fit by Mauro and Mauro with a hybrid fitting method
based on the number of terms in the desired Prony series and the magic value from Phillips The
thermal history is also defined in the original input file using linear interpolation The initial fictive
temperature components are assumed to be at equilibrium (Tfi = T(0))
29
To solve for the non-equilibrium viscosity used the MAP expression is used with the approximate
scaling of values proposed by Guo et al[61] The software is capable of performing the calculations
needed for a variety of thermal histories and processing methods desired by both industry and
academia with a focus on the fictive temperature This software is then fundamentally limited by
the issues associated with the MAP model and the compositional approximations made For
instance this software will need to be further modified to include structural relaxation when such
a model is proposed[62]
An example input code is listed below
Denotes Comments
Tg 794 0 m 368 C 1494 dt 10 Beta 37
Start Temp End Temp Time [s]
1000 10 1300
10 100 1000
100 30 500
Assumes Linear Interpolation
This example input states that over the space of 1300 seconds an initial glass sample is cooled from
1000 C to 10 C Then it is heated to 100C over 1000 seconds and then finally down to room
temperature in the space of 500 seconds All temperatures are given in Celsius
This software aims to provide the first open-source software for modeling of glass
relaxation behavior using the MAP model of viscosity by creating a package that can easily and
quickly be used to approximate the fictive temperature components of a glass as well as the
composite fictive temperature RelaxPy allows for researchers to calculate the evolution of
macroscopic properties of glass during a relaxation process (assuming a fixed stretching exponent)
and will facilitate the calculation based on a user-supplied thermal history RelaxPy can be
seamlessly adopted by most glass research groups and removes the guessing associated with
determining a fictive temperature allowing for a fully quantitative comparable discussion of Tf
simultaneously creating a tool that can easily be expanded for the entire community as new physics
is discovered
30
Chapter 3
Understanding Nucleation in Liquids
Crystallization cannot be ignored in any material system at temperatures below its liquidus
temperature although extreme resistance to crystallization is reported in some organic and
inorganic substances[97] Crystallization (and by extension nucleation) is critical in systems
ranging from ice to thin films[98]ndash[100] Glasses are non-crystalline substances therefore
crystallization kinetics will determine the necessary thermal history to avoid crystallization and
obtain a glassy state Despite the absolute importance of crystallization to the study and design of
glasses there is no current method to predict the nucleation of crystals in this event One of the main
challenges in the production of glass-ceramics is finding the optimal thermal history for the desired
crystallite distributions There are two phenomenological steps in the process of phase
transformation from a supercooled liquid to a crystal a nucleation step and a growth step The
growth step is described well by Wilson-Frenkelacutes theory[1] but the underlying physics of
nucleation remains elusive A stable nucleus is the precursor to a crystal comprising a nano-sized
periodic assemblage of atoms A clear understanding of the relationship between liquid kinetics
and crystal nucleation has not been established despite decades of research
The study of nucleation is particularly critical for glass-ceramics Since their discovery by
S Donald Stookey glass-ceramics have become prevalent in our society through products such as
stovetops and Corningware[73] [101] [102] Glass-ceramics are a composite material comprising
at least one crystalline phase embedded in a parent glassy phase benefitting from the properties of
both the glassy and crystalline phases[1] [103] One of the reasons glass-ceramics are ubiquitous
is their relatively easy production glass-ceramics can be formed through standard glass forming
procedures using additional heat treatments Additionally there is a wide range of accessible
31
properties through tuning both the chemical composition of the parent glass and the thermal history
and hence the microstructure of the heat-treated glass-ceramics[101] [103]ndash[106]
Glass-ceramic production consists of first synthesizing a glass and then generating a
microstructure through the two-stage ceramming process[73] [103] [107] The nucleation step
remains poorly understood with classical nucleation theory (CNT) giving widely varying results
depending on how the parameters are determined[1] [73] [108] The inability to consistently get
accurate predictions for a nucleation curve means that choosing an appropriate nucleation
temperature must be done experimentally which can prove to be one of the most challenging and
time-consuming aspects of properly designing a glass-ceramic The difficulty is due to the
magnitudes of the crystal nucleation rates not being predictable with CNT or any other current
model
This difficulty in designing a material is apparent when considering the high dimensional
phase space of parameters that can adjusted We haver presesnted some tools developed to help
optimize over this lsquoglassyrsquo space such as RelaxPy[42] KineticPy[41] topological constraint
theory[109] or various machine learning models[4] [70] [110] [111] However a glass-ceramic
system has at least four additional phase dimensions (two for the nucleationgrowth temperatures
and two for nucleationgrowth times) that need to be optimized (depending on the parameterization
of the experiment it is also possible to add another four dimensions relating the heating and cooling
between each step) Therefore a glass-ceramic system has a minimum of d+5 dimensions over
which the system must be optimized and yet those four additional dimensions are the least
understood Designing methods to easily optimize over this large space with feasible computational
methods is of utmost importance to reaching a new generation of custom materials
Experimental studies of nucleation have used techniques ranging from differential
scanning calorimetry (DSC) to electron microscopy to try to capture predictive capability in the
lsquoceramicrsquo phase-space (the four additional dimensions) These studies have successfully captured
32
the shape of the nucleation curve but fail to provide the information needed to calculate the
temperature-dependence of the nucleation rate The shape of the curve can be determined through
DSC and with great effort an approximation of the rate can be made however this technique is too
laborusome to be efficient[112]ndash[114] Since the parameters have been inaccessible to traditional
computational approaches and experimental determination all previous studies of nucleation have
required at least one variable to be fit If a nucleation curve could be predicted only using a
computational method or with an efficient simple DSC method the rate at which new glass-
ceramics products are created would increase In addition the cost of producing glass-ceramics
would likely decrease and more complex microstructures and therefore more unique properties
could be explored
Crystal nucleation has been historically difficult to simulate and measure experimentally
because the phenomena happen on spatial scales of nanometers but over many orders of magnitude
of time scales CNT is the most widely applied model despite its supposed failure To test CNTrsquos
assumptions an interest in computational work has grown Recent approaches using grand
canonical Monte Carlo have successfully predicted the short time dynamics of nuclei
formation[73] This approach involves use of a continuous solvation model for the liquid and has
enabled greater computational insights however it remains computationally expensive[74]
Molecular dynamics (MD) fails to capture nucleation in glass-forming systems due to the need for
longer spatial and temporal scales to capture nucleation effects accurately Some attempts have
been made using Monte Carlo MD or empirical approaches to replace classical nucleation theory
(CNT) with some alternate method where the parameters can be directly measured[73] [115]ndash
[117] All nucleation models and Monte Carlo models attempt to describe the rate that a
supercooled liquid system transitions to a crystalline structure as a function of temperature
Explicitly mapping the energy landscape would fundamentally explain the underlying
physics of nucleation An energy landscape of a system is given by
33
1 1 1 2 2 2( )N NNU U x y z x y z x y z= (26)
where U is the potential energy and x y and z are the locations of each of the N atoms in a periodic
box The locations of these atoms give rise to an energy as parameterized by an interatomic
potential An energy landscape can be constructed by mapping the relationship of the local minima
and the first order saddle points between pairs of minima22-27 Energy landscapes have become
crucial to studies of biomolecules glasses and catalysts[21] [26] [44] because landscapes show
atomic configurational changes and calculate the associated kinetics and thermodynamics In this
study we will explore the energy landscape of a barium disilicate system capturing the glassy and
crystalline basins enabling a test of CNT by calculating all needed parameters (interfacial free
energy kinetic barrier and free energy difference of the supercooled liquid and the equilibrium
state)
31 Crystallization Methods
CNT expresses the steady-state nucleation rate (I) as the product of a kinetic and a
thermodynamic factor
( )exp eI Z D TW
kT
minus =
(27)
where Ze is the Zeldovich factor which incorporates the probability of a critical sized nucleus that
will form a new phase independent of other nuclei D(T) is the kinetic factor often given by an
Arrhenius expression over sufficiently small temperature ranges W is the work associated with
generating a thermodynamically stable nucleus that will continue to grow rather than dissipate (ie
the critical work) if temperature and pressure are kept constant k is Boltzmannrsquos constant and T is
absolute temperature The Zeldovich factor is typically treated as a constant with a value ~01[118]
34
Here the kinetic term is expressed as D(T) since in this work the kinetics are explicitly calculated
at every temperature and it is not treated as a simple Arrhenius expression In CNT the work W is
given by the work to generate a spherical isotropic nucleus[1] which consists of a volume term
and a surface energy term It is given by
3 24
( ) 4 ( )3
W r G T r T = + (28)
where G is the Gibbs free energy difference between the crystalline state and the corresponding
liquid r is the radius and is the interfacial energy between the crystalline and liquid phases Eq
(28) neglects the strain energy term that arises from the formation of a solid nucleus in the
supercooled liquid It is assumed that the liquid relaxes fast enough so that strain energy is not
significant for nucleation in liquids[11]
Determining the critical radius (ie taking the derivative of Eq (28) with respect to the
radius and setting the expression equal to 0) allows one to solve for the critical work and the
expression becomes
3
2
16
)3(
( )
( )W
T
G T
=
(29)
Three temperature dependent functions remain D G and Though there are existing methods
in literature to approximate each of these quantities the approximations often include large errors
and assumptions about the behavior of the system[119] Often experimentally the kinetic function
is approximated with the Stokes-Einstein relationship the G term is calculated from heat
capacity curves (assuming that the macroscopic energy is similar to the nanoscopic energy) and
is fitted[119] In the calculation presented here only configurational energies are considered
while vibrational contributions are ignored Ignoring the vibrational component and the reliance on
empirical potentials are the most likely sources of error in this work
35
The energy landscape for this investigation was generated using ExplorerPy[89] starting
with the orthorhombic barium disilicate crystalline structure[89] Barium disilicate was chosen
since it is a commonly studied glass-ceramic with multiple reliable experimental datasets being
found in literature[75] [113] [120] The rest of this section is divided into four parts starting with
a discussion of mapping the landscape and the following three sections are a discussion of
calculating D G and respectively
31A Mapping and Classifying the Landscape
The ExplorerPy energy landscape exploration employed the Eigen-Trieban technique in
the canonical ensemble with systems consisting of 128 256 and 512 atoms[89] The number of
atoms was chosen to achieve convergence of the results while also balancing the large
computational cost associated with energy landscape exploration A smaller system was also
calculated with 64 atoms however this systems was too small to accurately discern energies
enough to make the nucleation calculation The Eigen-Trieban technique is a newly developed
method to explore energy landscapes using a method of shoving along the eigenvectors and
minimizing the potential energy which is followed by using nudged elastic band calculations to
find the associated transition points[89] This technique allows for larger transitions to be found
while retaining the ability to calculate the vibrational frequencies The Pedone et al potentials[36]
were used and the volume of the cell was fixed due to simulation instability when pressure was
fixed The Pedone et al potentials were chosen since they have reproduced key parameters for both
the crystalline and glassy state in previous Monte Carlo studies as well as in the training set of the
potential[73] [74] [121] Though the Eigen-Trieban method may work well for predicting
crystallization and finding large barriers it may be insufficient to describe glass relaxation or
36
viscosity since it rarely finds the small structural transitions that dominate the Adam-Gibbs
relationship Note that these exceptions are inconsequential for predictions of nucleation found in
the work presented here33
The barium disilicate system was explored for 2500 basins for 128 and 256 atoms and 4000
for 512 atoms since more basins were needed to converge the liquid state distribution of energies
This number of basins was shown to be sufficient since multiple paths from the initial crystalline
starting point to SCL states were obtained The distribution of energies of the SCL were found to
converge and the resulting energy landscape is seen in Figure 4
A crystal parameter was then defined from the root mean square displacement (RMSD) for
each atom from the nearest lattice site of the same species ( ) This average value is used to
characterize whether a basin is a crystal or SCL The RMSD approach is not the only method to
assign a crystallinity value[122] however it is one of the simplest and as such was chosen for
this application To then categorize each basin as either a crystalline basin or SCL basin a plot of
the minimum energy of the SCL states versus the cutoff of RMSD was produced This plot for 256
atoms shows a clear transition in the energies around a cutoff value of 1 Aring which is therefore
taken as the cutoff radius of which basins are considered crystals The same analysis was repeated
for each system size This analysis is shown in Figure 4 Any basin with a RMSD less than the
cutoff value is considered a crystalline state This cutoff was found to be generally insensitive to
the results with any values ranging between 09 Aring and 11 Aring changing the resulting prediction of
nucleation by less than one order of magnitude but shifting the peak temperature by up to 40 K
This cutoff also reproduced the energy difference between the SCL and crystal in molecular
dynamics simulations of the materials
37
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-axis being
an arbitrary phase space and the y-axis being the potential energy calculated from the Pedone et al
potentials[36] The colors are indicative of the crystallinity where the blue basin is the initial
starting configuration The landscape shows the lowering of energy associated with partially
crystallizing the sample (B) The energy landscape relationship between the cutoff for crystalline
and super-cooled liquid states for 256 atoms is shown This shows a clear drastic energy change
occurring around the cutoff value of 10 Aring
[
38
31B Kinetic Term for CNT
The transition rate was calculated by the inverse of the average relaxation time from the
SCL to the crystalline states The relaxation time ( ) is defined by the relaxation function ( )
0 exp
t
= minus
(30)
where t is time and 0 is the scaling of the relaxation function The reason we are able to
calculate the kinetics of nucleation but not viscosity is because the nucleation kinetics are
dominated by the transitions between two states that we specifically searched for when exploring
the energy landscapes while the viscosity is defined by the entropy dominating small transitions
described by the AG
The AG relationship shows that the viscosity is dominated by the configurational entropy
of the system Experimentally the kinetic term is often calculated with the Stokes-Einstein
relationship
( )6
kTD
aT
= (31)
where a is the radius of the diffusing species and is the viscosity of the liquid Interestingly this
approximation gives reasonable results when investigating experimental data since the viscosity is
a measurement of the entropy of small cooperative liquid rearrangements Conversely the kinetics
that govern nucleation are large structural rearrangements from a supercooled liquid to a crystal It
is nonetheless possible that these cooperative rearrangements and the large diffusion processes for
nucleation are on the same order of magnitude which is why the Stokes-Einstein relationship gives
39
a reasonable approximation in experimental parameterizations When the Stokes-Einstein
relationship is used there is an implicit assumption that the large nucleating rearrangements are
proportional to the small barriers that govern viscosity[53]
A mean relaxation time can be found by averaging over a system of master equations
( )j
ii ji j j ij i
i
dpg K p g K p
dt
= minus (32)
In this expression g is the degeneracy of a microstate pi is the probability of occupying the
microstate and Kij is a transition rate between inherent structure i and inherent structure j The
equilibrium probability of occupying a microstate can be given by a Boltzmann probability (where
Z is the partition function and U is the potential energy of the inherent structure)
1
exp ii i
Up g
Z kT
= minus
(33)
By substituting the value of jp in Eq (32) for a two-state model ( 1 21 i jp p= == + ) we write
( )( )1ii ji i j ij i
dpg K p g K p
dt= minus minus (34)
Isolating the probabilities and splitting the differential we have
( )1
i
i ji i j ij i
dpdt
g K p g K p=
minus minus (35)
which when solved via integration takes the form
ln ( )
( )
i ji i ji j ij i
i ji j ij
g K g K g K pt
g K g K
minus + =
minus + (36)
Solving for the occupational probability yields
( )expi ji j ij i ji
i
i ji j ij
g K g K g K tp
g K g K
minus minus + =
+ (37)
This shows an analytical solution for the relaxation time in a two-state model
40
1
i ji j ijKg g K
+
= (38)
The two-state model approximation is made to enable the calculation of the mean relaxation time
between the liquid basins to the crystalline basins The transition rates between the basins are given
by the vibrational frequency ( ) and the barrier between the states ( ijU )
exp ij
ij
UK
kT
minus=
(39)
The probability of transitioning from a given liquid basin i to a crystalline basin j is given by
1
expij
ij
Up
Y kT
minus =
(40)
where Y is the partition function with respect to all transitions from the current SCL basin to all
crystalline basins as only these transitions are considered to calculate the relaxation time
expj
j
iY
k
U
T
= minus
(41)
The total expected relaxation time of the system making the transition is then given by the
probability of occupying a basin the probability of transitioning from one basin to another specific
basin and the associated relaxation time
1
i ij
j i ji j ji i i
p pg K g K
=
+ (42)
The first summation is over the liquid states and the second summation is performed just over
crystalline states to capture the mean relaxation time from a supercooled liquid to a crystal We
fixed the vibrational frequency to a value of 17 times 1014 Hz (ie about one jump attempt per 5
femtoseconds) which was calculated using the average of the eigenvalues of each basin ( ) and
the average mass of the atoms (m) using
41
1
2 m
= (43)
31C Degeneracy calculations
To obtain an appropriate value for the degeneracy an understanding the phase-space
volume of the basin is needed since the degeneracy is linked to the volume of the basin[40] By
taking the displacement between the basin coordinates of each atom and the transition point we can
define a radius of the basin R Using R and a spherical approximation we can calculate the phase-
space volume degeneracies
3 23
12
NN
i i
cry
Ng R
g
+
=
(44)
In which ( )x is Eulerrsquos gamma function The degeneracy is then normalized by a constant such
that the liquid-liquid state transitions had a value of 100 s at the glass transition (980 K) to be in
agreement with Angell[58] The normalization step allows for a quantitative prediction of the
liquidus temperature to confirm the accuracy of our model The liquidus value found is
approximately 1200 K 1715 K and 1950 K for 128 256 and 512 atoms respectively The
experimentally determined liquidus temperatures range from 1680 K to 1710 K[113] [115] [120]
It is worth noting that changing the exponent drastically changes the thermodynamics while having
minor effects on the kinetics and the normalization value drastically changes the kinetics while
keeping the thermodynamics nearly constant We chose the denominator to satisfy the liquid
relaxation constraint It is also feasible to choose the exponent to satisfy a constraint on the liquidus
temperature The normalization value is an important to consider because for larger system a
change in 001 can result in a change in the kinetics by ~10 orders of magnitude
42
31D Free Energy Difference
Due to limitations in the exploration of the energy landscape requiring a fixed volume we
will approximate the difference in the Gibbs free energy by calculating the difference of the
Helmholtz free energy This approximation is reasonable due to the similar densities between the
crystalline and glassy phases under ambient pressure[73] [74] The first step is determining the
mean energy difference of all the basins and the second step was repeating the process for only the
liquid basins The mean energy was calculated with the following expression[123]
( )ln ln ii
i
i i iG kTp p U kT g p= + minus (45)
In Eq (45) i is the set of relevant basins for the conditions we are interested in The difference
between the mean free energy of the liquid basins and the entire landscape basin energies is the
G parameter This free energy difference is typically experimentally calculated by integrating
heat capacities however numerous approximations exist These approximations are often linear
approximations normalized to the liquidus temperature In this work we find that the driving force
appears to be predominantly linear which is in good agreement with the approximations that are
often used for the driving force[124]
43
Figure 5 An example interfacial structure between the crystalline phase on the left and the last
sequential SCLglass phase on the right for a barium disilicate system The gray atoms are barium
silicon is shown in red and blue represents oxygen
44
31E Interfacial Energy
To calculate the interfacial energy of the interface between the crystalline phase and the
SCL the initial basin (pure crystalline) and the last basin (glass basin) were annealed at 1700 K for
1 ns and then cooled to a temperature of 800 K over a period of 10 ns The same process was
repeated for a composite where the initial stable crystal and the glass were placed next to each
otherThe structure was multiplied 10 times to create a wall of a large interface to reduce energetic
fluctuations A small section of the composite structure is shown in Figure 5 The difference
between the addition of the energy of the crystalline phase and glassy phase and the composite
gives the energy of the interface which was normalized to the size of the surface area Multiple
arrangements of the composite were tested but anisotropy was found to be minimal with the
fluctuations of the energy being larger than the differences from different tested directions The
final expression used for the interfacial free energy (in eVAngstrom2) is
610 00 35047 040T minus +=
45
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a function of
temperature for each system size The experimental values for kinetics and thermodynamics come
from ref[125] and from the heat capacity data taken from [126] respectively It is clear to see that
driving force shows good agreement across all systems however the kinetics terms only converge
46
for the 256 and 512 atoms systems (Bottom) The fit used to calculate the interfacial energy as a
function of temperature
32 Results amp Discussion
The results for the free energy difference the kinetic parameter and the surface energy are
shown in Figure 6 These parameters as a function of temperature are used to calculate the
nucleation rates in the expression
3
2( )
01 16exp
3( )I
TT G k
= minus
(46)
The 01 being an approximate value of the Zeldovich factor Using the parameters shown in Figure
6 and solving Eq (46) the nucleation rates are shown in Figure 7 along with experimental data for
validation At first glance we notice that there is about 7 order of magnitude differences between
data and prediction Considering that theses potentials were trained on glass and were not intended
for this purpose we consider this as excellent agreement This method also shows agreement below
the peak but the accuracy falls off more quickly below the peak temperature It is likely that the
data below the peak is severely underestimated due to a failure to reach steady state nucleation
rates Reliable steady state nucleation rates are not available for this system and as such a direct
comparison below the peak temperature is not directly available It is worth mentioning that the
failure of CNT of predicting the order of magnitude of nucleation rates is well reported and can
amount to dozens of orders of magnitude when a constant value of is used[75] [113] [115]
[119] [127] thus the proposed method has an advantage when comparing with the current
classical nucleation theory toolset Moreover the prediction of Tmax (~950 K) is also very close to
the experimental value (985 K)
47
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted using the
model presented in this work The data referenced can be found in refs [113] [128] [129]
48
To further compare CNT and the model presented here we decided to compare the results
of the fitted and this workrsquos calculated surface energies as seen in Figure 8 The results show good
agreement all achieving the same order of magnitude implying that experimental methods can
achieve a close approximation to the values required for CNT The prediction of nucleation is
extremely sensitive to the value of the surface energy with small changes leading to orders of
magnitude change It stands to reason then that further work that needs to be approached through
computation and experiments is developing a rigorous model to predict the interfacial free energy
The other assumption that is used experimentally is the Stokes-Einstein relationship To
compare the landscape results of the transition rate from the SCL to crystalline states and the
Stokes-Einstein relationship the liquid state relaxation was calculated The small barriers that
govern the Adam-Gibbs relationship were not found with this exploration method hence
quantitative comparisons are not possible (and relaxation calculations should not be made)
however the relaxation times can be qualitatively compared The liquid state relaxation rate is
proportional to the viscosity and is given by the expression[44]
j
i
i
l
ij
p
K =
(47)
To understand the validity of the Stokes-Einstein relationship we can write a series of theoretically
true proportionalities
( )l
T TD T
(48)
The values of the experimental kinetics fail to align with the kinetics calculated in this work This
is expected due to the fact that the viscosityliquid relaxation is dominated by the set of small
transitions in the landscape while the nucleation transition is defined by the transition of one region
to another with a substantially higher activation barrier This comparison may also account for
49
some part of the experimental overestimation below the peak with the kinetic term calculated
dropping off at a faster rate than that calculated from the experimental viscosity Figure 6 also
shows evidence that it may be inappropriate to use the Stokes-Einstein relationship for nucleation
and as such we may need to re-visit the experimental models that often rely on the assumption of
the Stokes-Einstein relationship This observation casts some doubt in the legitimacy of describing
the kinetic factor as a diffusivity which is standard for CNT For more information on this CNT
failure please see Refs [75] [113] [115] [119] [125] [127]
Nucleation is a complicated concept to investigate and model Unlike the crystalline phase
where the atomic positions are clearly defined the details of the atomic coordinates and bond
configurations of the liquid sites must be described in terms of statistical distributions Due to this
added complication mean-field distributions are typically used to describe non-crystalline
sites[14] [16] [108] [130]ndash[133] The incorporation of fluctuations could more accurately predict
each liquid sitersquos transition to the crystalline phase Fluctuations in the degeneracy could be one
reason that the nucleation rate is over predicted when the temperature is less than the peak
nucleation temperature A given G varies with the environment and depends on the specific
liquid state not the mean value Therefore the rate of nucleation from all supercooled liquid states
to the crystal state is not constant which is why fluctuations need to be considered and why energy
landscapes are such a powerful technique This is especially highlighted when considering that
different degeneracies yield drastically different heat capacity Energy landscapes are extremely
useful because they consider a large distribution in free energies and show all structural fluctuations
possible Fluctuations in cluster size and degeneracy influence Adam-Gibbs configurational
entropy which directly impact the transition rate between the crystal and liquid state
50
Figure 8 The surface energy with respect to temperature for the work presented here
0
0002
0004
0006
0008
001
0012
0014
800 850 900 950 1000 1050 1100
Su
rface
En
ergy (
eVAring
2)
Temperature [K]
This Work
Xia et al (2017) Ref 23
Fokin et al (2016) Ref 41
51
33 Conclusions
Though this is not the first time CNT has been examined computationally in an attempt to
understand if the theory is valid[74] [116] [117] it is the first parameterization of each variable
independently in an attempt to calculate the resulting nucleation rate Not only are all the variables
calculated when used in the expression for CNT the liquidus temperature is reproduced and the
nucleation rate is reasonably predicted This is as far as the authors know also the first time a real
energy landscape for a complicated glass-ceramic system is being reported This method leveraging
energy landscapes alongside CNT enables for a reasonably fast calculation of the nucleation curve
of a glass forming liquid In future work this model could be used for either developing commercial
glass-ceramics or for predicting the critical rate needed to maintain a glass This model can also
provide additional insights into the physics of nucleation and the validity of CNT without the
constraints of fitting parameters This method fundamentally has the power to connect the 3N
dimensions of the energy landscape to the compositional dimensions and thus allowing for a unique
powerful method of predicting nucleation parameters as a result simplifying the crystalline
dimensions in which a glassglass-ceramic must be optimized over
Chapter 4
Expanding the Current State of Relaxation
Before we consider a new approach to relaxation (optimizing our understanding of
the 3N dimensions) there are first missing pieces in the current state specifically the temperature
and compositional dependence of the stretching exponent However even before expanding the
current state of the art relaxation methods we also need to understand ergodicity In this section we
will start with a simple though experiment to understand the temporal effects of relaxation and then
in the second section combine it with fictive temperature to create a new model for the stretched
exponent
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition
The prophet Deborah sang in the book of Judges ldquoThe mountains flowed before the Lordrdquo
[6] [13] However the question the prophet failed to ask was what happens if the mountains are
moving at relativistic speeds The focus of this section is to highlight the temporal effects on
ergodicity in the context of special relativity and the reference frame of the observer In order to
understand relativistic effects on materials one must consider a well-characterized material that
actively relaxes but appears solid on our timescale viz pure B2O3 glass [134] [135] Glass is a
particularly good candidate to study the effects of time dilation on a material since it undergoes a
kinetic transition known as the glass transition (Tg) which has multiple definitions that will be
considered
53
The first definition is in terms of the aptly named ldquoDeborah Numberrdquo which was proposed
by Reiner [136] to offer a view of fluidity and equilibrium mechanics It was later expanded to
explain the origin of the glass transition and the transition from an ergodic liquid system to a non-
ergodic solid-like glassy state[12] which is the fundamental definition of the glass transition [11]
[12] [137] It is important to note here that the breakdown of ergodicity is relative to the
observation timescale and is reflected in the Deborah number (D) which is defined as
Dt
= (49)
where t is the (external) observation time and is the relaxation time of the material which can be
expressed through the Maxwell relation [42] By definition the Deborah Number equals 1 at the
glass transition temperature Tg [13] [137]
Another definition of the glass transition temperature is due to Angell who defined Tg as
the temperature where the viscosity of the supercooled liquid is 1012 Pa s [58] [138] The Angell
diagram is an important plot relevant to viscosity and the glass transition where the abscissa is TgT
and the ordinate axis is log10( ) In the Angell diagram the value of the viscosity at Tg and in the
limit of infinite temperature are fixed and the difference in the temperature scaling of the TgT-
normalized viscosity is the slope of the log10( ) curve at TgT = 1 which Angell defined as the
fragility (m) of the supercooled liquid Using this definition one can establish the consistency of
the Angell plot at relativistic speeds An example of the Angell plot is shown in Figure 9
54
Figure 9 An Angell diagram created using the MYEGA expression [55] With a very strong glass
(m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) [17] [134] [135] The
infinite temperature limit is from the work of Zheng et al[54] and the glass transition temperature
is from the Angell definition
55
Using the two previous definitions it is expressed
1gT
GtD
= = (50)
which can be re-written as
(
(
)
)g
g
tG T
T= (51)
In order to solve this equation one must know the shear modulus at the glass transition temperature
recent improvements in topological constraint theory have led to the ability to predict properties
with varying temperature and composition Using a recent topological constraint model for elastic
moduli proposed by Wilkinson et al [139] one can predict the temperature dependence of the
modulus
( ) ( ) ( ) ( ) ( )( ) ( )Δ Δ Δ Ac c C Ac
c c
F n q T F n q T F n q T x Nn q T x NdG dGG
d F M d F M
+ + = =
(52)
In Eq (5) nc is the number of topological constraints (c) qc is the constraint onset function as
described by Mauro et al [69] Fc is the free energy of the constraint is the density of the
system M is the molar mass and 119889119866
119889120549119865119888 is a scaling factor This expression was evaluated as a
function of temperature with the total modulus coming from Kodama et al [135] and is equal to 8
GPa at room temperature The B2O3 structure consists entirely of three-coordinated boron [140]
[141]
Using the modulus predicted in Eq (52) numerical observation time is expressed
56
12
9
10 Pa s141 s
709 10 Pat = =
(53)
Using this description of the static (vc=0 where v is the speed of the glass and c is the
speed of light) glass transition it is possible now to describe the relativistic glass transition for
which there are considered two separate cases It is worth noting that this is not the first case study
of relativistic viscosity a fairly large basis of literature dealing with ideal fluids near large
gravitational bodies [142] There is also some work also relating the relaxation time in a
gravitational field to the nonequilibrium entropy relaxation time [143] [144] This work
emphasizes the theoretical effects of special relativity on real liquids their glass transitions and
the properties there dependent on
41A Relativistic Liquid
The first case is when a sample of a glass-forming system moves past a stationary observer
at a speed approaching light giving the new observation time as
0tt
= (54)
where t0 = 141 s (as previously calculated) and g is the Lorentz factor
2
2
1
1v
c
=
minus
(55)
The Maxwell relation can be modified to account for relativistic effects by combining with Eq (54)
0
()
)(
g
gG TT
t
= (56)
57
This gives a condition that is temperature dependent to predict the new glass transition as a function
of v however in order to accurately describe the relativistic behavior density must be considered
Density will change by a factor of 2 because both the mass and the volume will be affected and
as such
0) (( )g gG TT t = (57)
The prediction then for the glass transition as a function of the speed of light is shown in Figure
10
Figure 10 The relativistic glass transition temperature for B2O3 glass
58
Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature
which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) expression [55] [145] The MYEGA equation needs 2 other parameters
besides Tg to generate the viscosity curve fragility (m) which is the slope at Tg in the Angell plot
and the infinite temperature viscosity () The fragility will not change because one assumption
in this model is the consistency of the Angell plot at all relativistic velocities even relativistic The
infinite temperature viscosity which also will not change because this is the minimum possible
viscosity for a liquid The new viscosity curves for various speeds are shown in Figure 11
59
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different fractions of
light speed All of the viscosities approaching the universal temperature limit for viscosity
60
By using the Deborah Number it has been shown that at relativistic speeds will change the viscosity
in order to maintain the Deborah number and in this case it is shown that when the Lord would
move past the mountains at relativistic speeds the mountains would not actually flow but instead
hold still (even more so than they currently do) From this it is inferred that any liquid moving past
the earth at relativistic speeds will appear to be more solid and eventually appear as a glass
41B Relativistic Observer
In this case we consider an observer moving past a sample of glass at relativistic speeds
giving
0t t = (58)
which when the relativistic shear modulus 2 factor is included can be written as
( )
( )0
g
g
TG T
t
= (59)
Solving in the same manner as before it is shown which shear modulus is needed to satisfy the
condition in Figure 12
61
Figure 12 The modulus needed to satisfy the condition for the glass transition
62
In Figure 12 the shear modulus at the glass transition temperature is shown however the
constraint theory mechanism being implemented to calculate the shear modulus has a built in upper
limit when calculating the Tg past the point in which all the constraints are intact Due to this
limitation there is an upper limit to our prediction Nonetheless the calculated Tg over the available
range are shown in Figure 13 as well as the related viscosities It is seen clearly that at higher
velocities the glass transition shifts towards zero more dramatically at the higher temperatures
This leads to the interesting result that if we take the limit of an observer moving past the earth at
close to the speed of light all glass would appear to be a liquid
63
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at v=044c
(Bottom) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches
the speed of light
64
Applying the mechanisms proposed by Einstein to the concept of the Deborah number it
has been shown that the viscosities will appear to change dramatically as a glass-forming system
approaches relativistic speeds This allows us to build an intuitive understanding of the differences
between glass and liquids Specifically any material can appear either more or less fluid based on
the velocity with the velocity with which they travel relative to the observer This serves to highlight
the importance of the role of the observer on the glass transition and why even small changes in
DSC scan rates or sampling frequency can lead to drastically different measured glass transitions
42 Temperature and Compositional Dependence of the Stretching Exponent
The mathematical form for glass relaxation ( ) was originally proposed by Kohlrausch in
1854 based on the decay of charge in a Leyden jar[9] [14] [29] [42] [45] [96] [146]
( ) ( )( )exp t t
= minus (60)
where t is time t is the relaxation time of the system and b is the dimensionless stretching exponent
Eq(1) was originally proposed empirically and is known as the stretched exponential relaxation
(SER) function[42] [96] The relaxation time t depends on the composition temperature thermal
history pressure and pressure history of the glass as well as the property being measured[62]
[147] For example the stress relaxation time of a glass can be written as[62]
( )( )
ff
s
f f
T T P P
G T T P P
= (61)
where h is the shear viscosity and G is the shear modulus of the glass-forming system[42]
(Variables are defined in Table 2) In addition to using Eq(61) to describe stress relaxation time
structural relaxation time has recently been hypothesized follow Eq(61) with bulk viscosity and
65
the difference of bulk modulus between infinite and 0 frequency replacing shear viscosity and shear
modulus[62] Both h and G vary with the temperature (T) and thermal history of the glass (as
quantified via the fictive temperature Tf) as well as the pressure (P) and its corresponding history
(fictive pressure Pf)[16]
To connect the stretching exponent to its physical origins we can begin with a result from
Richert and Richert[148] that relates to an underlying structural relaxation time distribution and
then develop equations that determine from known quantities Their expression
2 22
ln 2
1
6
minus=
(62)
relates to the variance of the logarithm ( )2
ln of the structural relaxation time For reference
all variables are defined in Table 2 The physical origins of the relaxation time are related to the
configurational entropy (Sc) as shown by the Adam-Gibbs equation[53] [55] [149]
ln lnc
B
TS = + (63)
where is the infinite temperature relaxation time T is the absolute temperature and B represents
the energy barrier for relaxation
Table 2 Variable definitions
Variable Definition
t ts Structural relaxation time stress relaxation time
b Kohlrausch exponent (ie the stretching exponent)
T Tf Tfi Absolute temperature fictive temp partial fictive temp
P Pf Pressure fictive pressure
66
kB Boltzmannrsquos constant
wi ki i=1 to N Prony series parameters
B Adam-Gibbs relaxation barrier
Sc Adam-Gibbs configurational entropy
B Distribution of activation barriers
2 Variance
Na Number of atoms
f Topological degrees of freedom per atom
Degenerate configurations per floppy mode
H Enthalpy barrier for Relaxation
d Number of dimensions
f Fraction of activated relaxation pathways
S Adam-Gibbs entropy in the infinite-temperature limit
m Kinetic fragility index
m0 The limit of a strong liquid
Grouped unknowns 6
ln
B
a BN k
A Proportionality between fragility and distribution of barriers
x A given composition
125640
S
A
Intercept of the linear model of
67
42A Deriving a Model
Combining Eqs(62) and (63) Gupta and Mauro[150] proposed that the variance of the energy
barriers for relaxation (2
B ) could be rewritten as
2 2
22
2
1
6( )B c TTS
minus=
(64)
Solving for the stretching exponent
( )22 (
6
( )
)
c
B c
S T
T
T
TS
= +
(65)
The work of Naumis et al[23] [56] has shown that the configurational entropy is proportional to
the topological degrees of freedom in the network a result that was used in the derivation of
temperature-dependent constraint theory[57] and the MYEGA (Mauro-Yue-Ellison-Gupta-Allan)
equation for the relaxation of supercooled liquids[55] The MYEGA equation was derived by
expressing the configurational entropy as
( ) ( ) lna BcS T x f T x N k= (66)
In Eq (66) f is the topological degrees of freedom per atom Na is the number of atoms kB is
Boltzmannrsquos constant x is a given composition and is the number of degenerate configurations
per floppy mode The temperature dependence of the topological degrees of freedom was
approximated using a simple two-state model
( )( ) 3exp
B
H xf x T df
k T
minus= =
(67)
Here H(x) is the enthalpy barrier for relaxation d is the dimensionality of the system and f is the
68
fraction of activated relaxation pathways Combining Eqs (65) and (67) as well as condensing the
unknowns into the term defined by
6
ln
B
a BN k
=
(68)
we get
( )
22
df
dfT
=
+
(69)
This can be compared to the prediction made by Phillips[48] for the stretching exponent at
temperatures below the glass transition
2
df
df =
+ (70)
Comparing the two expressions of Eq(69) and Eq(70) they would agree if
4 4dfT
= + (71)
We will show later (Figure 12) an example where extrapolating our model prediction for to room
temperature (with T in Eq(71) replaced by fictive temperature Tf) gives a result close to the Phillips
value of Eq(70) When the temperature T in Eq(69) is high (much larger than ) then Eq(69) gives
us rarr1 Thus the model can interpolate between the low-temperature (Phillips) value and the
high-temperature value of 1 In our current model we do not have access to a value for in Eq(68)
so this model cannot assess whether the Phillips room temperature value for is universal ie
whether Eq(71) is always satisfied at room temperature
Zheng et al[151] showed that the Mauro-Allan-Potuzak (MAP) model for the relaxation
time of the nonequilibrium glassy state[44] implies that the configurational entropy can be written
as a function of thermal history (Tf) the fragility index (m) the fragility limit of a strong liquid
69
(m0) the glass transition temperature (Tg) and the limit of infinite temperature configurational
entropy ( S ) as
0
) exp 1 0
(ln1
c f
f
gT
TS m
TS
m
= minus minus
(72)
The qualitative relationship between the distribution of activation barriers and fragility was
proposed by Stillinger[18] [19] who suggested that the energy landscape of a strong liquid (low
fragility) has a small distribution of activation barriers and that a higher fragility is associated with
a broader distribution of activation barriers ie a higher variance of the activation barriers This
leads to a phenomenological relationship that is here assumed to be valid based on Stillingerrsquos work
on energy landscapes[18] [19]
2
0
1B
mA
m
= minus
(73)
where A is some constant of proportionality This expression was chosen because in the limit of a
strong liquid there is an infinitely sharp distribution of activation barriers Combining Eqs(65)
(72) and (73) we obtain the temperature dependence of the stretching exponent for a liquid
( )
0
2 2 2 2
0 0
exp 1
( ) 2
6 ln10 1 exp 1
g
f
f
f
g
f
f
T mT S
T mT
Tm mA T S
m T m
minus minus
=
minus + minus minus
(74)
Fictive temperature appears in this new expression because we are deliberately expressing this
function as an equilibrium model Rewriting Eq (74) in terms of one unknown () and grouping
constants
70
( ) 0
2 2
0 0
exp 1
2exp 11
g
f
f
f
g
f
f
T m
T m
T m
T
T
Tm
T
m
m
minus minus
minus minus
=
minus +
(75)
Eq (75) is an expression for the stretching exponent as a function of thermal history glass
transition and fragility index with only one unknown In Eq (75)
( ) 12
6 ln10
S
A
= (76)
The only unknown for the compositional dependence of the stretching exponent is the value of D
The composition-dependent part of Eq(76) is Syen
since we approximate A to be independent of
composition The fragility dependence of Syen
was proposed by Guo et al [61]
Syen(x) = S
yen(xref
)expm(x) -m(x
ref)
m0
eacute
eumlecircecirc
ugrave
ucircuacuteuacute
(77)
where x is composition and xref is a reference composition in the same glass family Seeking the
simplest possible expression to approximate the unknown D we take the natural logarithm of
Eq(76) and of Eq(77) and combine them to get
0
ln lnm
m = + (78)
with the additional definition
( )( )
( )12
0
ln l6 ln10
nref refS x m x
mA
minus
=
(79)
The result is that ln D varies linearly with fragility m and the intercept ln D is independent of
composition (depends on one reference composition)
In Figure 9 we plot the data of Boumlhmer et al[152] vs the predicted exponent The work of
71
Boumlhmer et al is the summary of the literature data relating fragility index to the stretching exponent
at the glass transition which we use to fit values to Eq (78) The dataset included in their work
covers chalcogenide oxide and organic glasses The fitting (consisting of least-squares
minimization of the difference between the predicted stretching exponent and that which was
reported) was done twice once for organic and once for inorganic systems During the fit it was
assumed that Tg=Tf Some assumption about thermal history was necessary since the individual
thermal histories or Tf values are not known for this whole collection since we are trying to track
overall trends in values vs composition this reasonable simplifying assumption is consistent
with our program Using Eqs(75) and (78) the temperature and compositional dependence of the
stretching exponent can be described with only one free parameter
72
Figure 14 b predicted and from literature showing good agreement with a total root-mean-square
error of 01 The fit for organic systems is given by 66 ln Kln = minus and for inorganic systems by
75ln Kln = minus
73
42B Experimental Validation
Experimental density measurements were made using Corningcopy JadeTM glass as described
elsewhere[146] Changes in the density are normalized to obtain the relaxation functions which
are fit with Eq (60) The resulting values of are shown in Table 3 A comparison in Figure 15
is shown with data obtained on a soda-lime silicate SG80 where the stretching exponent was
obtained using the measurement of the released enthalpy as a function of isothermal annealing time
during relaxation below the glass transition The measurement of the released enthalpy[153] relies
on the change in the excess heat capacity in the glass transition range as a function of annealing
time Normalized released enthalpy relaxation functions for each temperature were fit with Eq(60)
to obtain the stretched exponent values for SG80 In Figure 15 parts A and B are samples measured
below Tg while in part C the samples measured are at Tg and above The values were then fit with
Eq(75) where the only free parameter was D with an equilibrium assumption (Tf = T)
74
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy JadeTM
glass and Sylvania Incorporatedrsquos SG80
Glass Temperature (K) Tg (K) TgT m D (K-1) Beta t (s)
Corningcopy JadeTM 1031 1074 1042 32 0002 0504 7249
Corningcopy JadeTM 1008 1074 1065 32 0002 0447 32458
Corningcopy JadeTM 973 1074 1103 32 0002 0395 419214
SG80 800 800 1 36 00057 0632 92
SG80 783 800 1021 36 00057 0593 182
75
Figure 15 The equilibrium model proposed with the experimental points showing good agreement
between the experimentally measured data points and the equilibrium derived model RMSE was
002 for Corningcopy JadeTM (A) and less than 001 for SG80 (B) (C) The model fit for B2O3
experimental data[154] [155] The fragility and glass transition temperature of the B2O3 are taken
from the work of Mauro et al[17]
(
A)
(
B)
(
C)
76
A combination of the MAP model for non-equilibrium shear viscosity[44] the model
presented here primarily in Eq(75) and the model presented by Wilkinson et al[139] for the
temperature dependence of elastic modulus allows for fully quantitative modeling of stress
relaxation behavior The missing model required to understand structural relaxation is the bulk
viscosity curve[62] All previous relaxation (structural or stress) models[42] [61] have relied on
approximations that use a constant exponent and on a constant (temperature-independent)
modulus value whereas here every parameter of Eq(60) may be modeled as a function of
temperature Furthermore in combination with the relaxation models described by Guo et al[61]
in which multiple fictive temperatures are described using a Prony series and a temperature-
dependent modulus one can construct a relaxation curve accounting for the temperature
dependence and thermal history dependence of all relevant parameters
( )
( )( )
1
( )( )exp exp
( )) (
fTN
i
i
f
f
f fi
G T k T tG T tw
T TT
T T
=
minus minus
(80)
Here wi and ki are fitting parameters that are completely determined by the value of and each
term in the Prony series is denoted with the subscript i Usually 8 or 12 terms are included in the
Prony series[96] Each term in the Prony series is assigned a partial fictive temperature (Tfi) whose
relaxation is described by the simple exponential in the Prony series Equilibrium conditions are
assumed at the start of the simulation which allows for a known set of starting probabilities within
the energy landscape interpretation of relaxation The same model also includes the fragility index
and glass transition dependence of the non-equilibrium shear viscosity This method is
implemented in RelaxPy[42] as discussed in the next section This serves as an approximation for
the evolution of the of the non-equilibrium state however the temperature dependence of the bulk
viscosity and a replacement for fictive temperature need to be quantified to improve the
understanding of the underlying physics[1] [16] [62]
77
Separately we can explore the relationship between this model given by Eq(75) and the underlying
energy landscape In order to better understand the stretching exponent consider that there exists
a full set of parallel relaxation modes within an energy landscape The relaxation modes are then
weighted by the occupational probability corresponding to a particular mode This gives a series of
transition rates with some probability prefactor and an associated relaxation time (scaled from the
mean relaxation time) which gives rise to a Prony series form of Eq(80) Thus the distribution of
relaxation times (or barriers) determines the evolution of the stretching exponent while the average
barrier determines the mean relaxation time If this Prony series description in turn describes the
stretching exponent we arrive at a physical description and origin of the stretching exponent As
the temperature approaches infinity even though there is a distribution of activation barriers the
distribution of relaxation times approaches a Dirac delta function and the stretching exponent
approaches one (a simple exponential decay) As the temperature decreases the distribution of
relaxation times broadens due to the wide distribution of barriers however due to the broken
ergodic nature of glass the value of the stretching exponent will be controlled by the instantaneous
occupational probability
The activation barriers may be modeled eg using a numerically random set of Gaussian
distribution of barriers[156] The probability of selecting an individual transition is then calculated
using a Boltzmann weighting function the transition also has an associated relaxation time with it
The Prony series is then recreated with the probability multiplied with the simple exponential
relaxation form The sum of all of these terms is then fit with the stretched exponential form Figure
11 shows the stretching exponent calculated for the mean activation energy of 10 eV and three
choices of standard deviation 25 2 15 ande e e minus minus minus= eV as shown in the legend where Eq(60) has
been fit to the Prony series calculated from an equilibrium Boltzmann sampling of a Gaussian
distribution of barriers The temperature dependence for the stretching exponent matches the form
78
derived earlier in this work This stretching exponent shape has been shown previously though not
in a closed-form solution[156] This secondary method not only validates the generalized form of
Eq(75) but also highlights the validity of using the configurational entropy and barrier distributions
as the underlying metric controlling the temperature dependence of the stretching exponent
Figure 16 The stretching exponent calculated as described in the text for a Gaussian distribution
of barriers This plot shows that the distribution of barriers has a large effect on the stretching
exponent A Tg cannot be described since there is no vibrational frequency included in the model
though the glass transition temperature should be the same for all distributions since the mean
relaxation time is the same for all distributions at all temperatures The deviation is given in ln eV
units
79
To create software to model this complex behavior the authorsrsquo existing software RelaxPy[42] has
been modified into a new version RelaxPy v20 Instead of fixed values for the Prony series the
values are chosen dynamically to match the stretching exponent as predicted by Eq(75) The
database of values wi() and ki() was created with a fixed number of terms N in Eq(80) (we chose
N=12) This database is available in the same Github repository as the software The parameters
were fit by starting with the values obtained for =37 reported by Mauro and Mauro[96] to make
the new version of the model smoothly reproduce a prior optimal fit then was stepped by 001
(when less than 095) or by 0001 (when greater than 095) Using the previously optimized values
of wi and ki for starting values for each new the wi and ki values were then varied to minimize
the root-mean-square error between the stretching exponent form and the Prony series form ie to
satisfy Eq(80) with least error This list of compiled values makes up the contents of a database
that the software accesses The values of wi and ki used for a given are calculated by finding the
closest value in the database to the given value of This allows for an efficient implementation
of relaxation modeling Figure 12 shows the fitted parameters to the Prony series and an example
RelaxPy output for Corningcopy JadeTM glass undergoing quenching at a rate of 10 Kmin Note that
the room temperature stretching exponent value is 046 a close match the Phillipsrsquos ldquomagicrdquo value
of 37 (043)[157] This prediction for Corningcopy JadeTM is a non-equilibrium prediction in contrast
to the equilibrium high-temperature prediction shown in
80
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent Each color
designates one term in the series (Bottom) The output from RelaxPy v20 showing the stretching
exponent effects on the relaxation prediction of Corningcopy JadeTM glass Each quadrant shows one
property that is of interest for relaxation experiments In particular it is interesting to see the
dynamics of the stretching exponent during a typical quench
81
43 Conclusion
In this section we have sought to expand the current understanding of relaxation and improve the
models that currently exists We have proposed a model that accounts for the effects of special
relativity as well as a model was derived through an understanding of the distribution of barriers
for relaxation and the temperature dependence of the Adam-Gibbs entropy The model outlined
herein describes the temperature dependence of the stretching exponent in glass relaxation
The work done on the relativistic effects on relaxation illustrates the crucial role of the observer on
all relaxation models and offers an extreme in which to test common relaxation models The model
for the stretching exponent not only considers the extremes at high and low temperature (when
compared to the glass transition) as in the Phillips model but also for any intermediate temperature
as a function of the fictive temperature This model does not have any explicit temperature
dependence since it was assumed that an equilibrium model works well to describe the
instantaneous distribution of relaxation times Given the physical argument and the success of this
model when tested by experiments and by another model it is at least reasonable to formulate the
temperature-dependence of in terms of its fictive-temperature-dependence as we have done here
Including both T and Tf is possible to consider but lies outside our current scope Using previously
derived compositional dependence for the MAP model a fragility index dependence of the
stretching exponent was defined and tested The model was confirmed using multiple experimental
datasets In addition a theoretical comparison to a distribution of landscape activation barriers was
found to reproduce the same trends as the model
Chapter 5
Glass Kinetics Without Fictive Temperature
Fictive temperature as discussed through this text is unable to capture key physical
phenomena[16] [42] [51] To circumvent this limitation we rely on two different frameworks
The Adam-Gibbs relationship and generalized features of the energy landscape These two features
allow us to create a method that is completely generalizable and does not involve fictive
temperature This method creates fake landscapes that generally capture the trends of glass based
on experimental measurements This new method we have called lsquoToy Landscapesrsquo
51 Background of the Adam Gibbs Relationship
Experimental evidence in support of the AG model comes from the work of Richet[158]
in which they compared the predicted configurational entropy from viscosity curves with that
obtained from DSC The configurational entropy was calculated from DSC data using
0
0( )p p vi
T
T
b
c c
C CS S T dT
T
minus= + (81)
In Eq (81) Cp is the isobaric heat capacity Cpvib is the vibrational heat capacity and T0 is the initial
temperature from which the configurational entropy is integrated The difference between the total
heat capacity and the vibrational heat capacity is the heat capacity due to structural rearrangements
Richet found good agreement between the measured viscosity curve and calorimetric
configurational entropy at temperatures above the glass transition confirming validity of the AG
relationship When applying Eq equation reference goes here all temperatures must remain above
the glass transition ie in the liquid or supercooled liquid state The calculation is invalid upon
83
cooling through the glass transition due to the breakdown of ergodicity[12] [13] [159] and the
irreversibility of the glass transition process Application of Eq (6) to temperatures below the glass
transition leads to an incorrect calculation of excess entropy in the glassy state[160]
In this study we expand on these previous works to understand the relationship between
viscosity and the underlying energy landscape and provide new insights into the thermodynamics
of liquids through a simplified ldquotoy landscaperdquo model constructed from experimentally measured
viscosity parameters The energy landscape framework is especially helpful for elucidating the
thermodynamics and kinetics of supercooled liquid and glassy systems Energy landscapes describe
the evolution of all atomic transitions for kinetic processes in a system To perform such
calculations information about the inherent structures and transition points in a landscape must be
known In this work we solve the inverse problem of deducing realistic landscape parameters using
the AG relationship and experimentally measured data This information is used to construct a ldquotoy
landscaperdquo model to describe glass relaxation processes To achieve an accurate relaxation model
the following steps are taken to ensure the validity of the results
A Confirm the AG model for shear viscosity using energyenthalpy landscape
calculations
B Validate the assumptions made by the MYEGA model
C Confirm the AG model for bulk viscosity
D Explore the fundamental relations between viscous properties and topography of the
landscape
E Use the knowledge gained to propose a new thermodynamic model to calculate the
driving force for glass relaxation and the scaling of the liquidrsquos free energy
This chapter is organized into sections devoted to each of the topics above preceded by a
methods section All of the sections are presented to validate a new model of glass relaxation that
is sufficiently complex to achieve a calculation of the thermodynamics and kinetics of the liquid
This new model will be based on understanding the landscape from experimental properties and
we have called this approach the ldquotoy landscaperdquo model
84
52 Methods
Potential energy landscapes (fixed volume) and enthalpy landscapes (fixed pressure) are
powerful modeling techniques where atomic configurations are mapped in a 3N-dimensional phase
space (or 3N+1 dimensions for enthalpy landscapes)[89] The energy landscape approach is based
on mapping the continuous 3N dimensional space to a discrete set of energy minima (called
ldquoinherent structuresrdquo) and first-order saddle points (ldquotransition pointsrdquo) To perform this mapping
a systematic search for inherent structures is performed while also mapping the lowest-energy
transition points between each pairwise combination of adjoining minima The landscape itself is
partitioned into basins which represent the set of all configurations that minimize to a common
inherent structure[161] This combination of the basin and transition point information results in a
topographic map of the 3N phase space (either a partial or whole mapping) that can be used to gain
insights into the physics of any atomistic system such as proteins[91] liquids[18] [19] glasses[1]
[93] and nucleating crystals At equilibrium the probability distribution for occupying the various
basins in the landscape is given by[13]
exp
exp
i
j
i
i
j
j
Eg
kTp
Eg
kT
minus
= minus
(82)
where Ei is the potential energy (or enthalpy) of the inherent structure gi is the degeneracy of the
basin i is the basin index and pi is the probability of occupying a given basin i The denominator
is a normalization factor given by the summation over all basins Mauro et al[44] previously used
this formalism to study the non-equilibrium viscosity calculate the evolution of configurational
heat capacity and elucidate the long-time relaxation kinetics of selenium glass[44] [93] [160] For
equilibrium systems the configurational entropy is given by the Gibbs formula
85
lnc i
i
iS k p p= minus (83)
In order to validate the AG model the landscapes of several common systems were
explored using ExplorerPy[89] ExplorerPy is a software program built specifically to map
landscapes using the LAMMPS molecular dynamics package[30] The three systems we explored
are B2O3[162] (150 atoms and 2000 basins) SiO2[162] (150 atoms and 1000 basins) and BaO-
2SiO2[121] (128 atoms and 1500 basins) All systems used periodic boundary conditions The
initial structures for B2O3 and SiO2 were constructed by placing atoms corresponding to the
chemistries of the systems on a random grid (with 1 Aring spacing) corresponding to a density of 20
gcm3 and equilibrated at 2500 K then quenching at a rate of 1 Kps to room temperature using the
Wang et al[162] potentials in the NPT (constant pressure number of atoms and temperature)
ensemble The energy landscape of the BaO-2SiO2 system was taken from our previous work[72]
The SiO2 (constant pressure exploration) system was explored using ET with a push distance of 03
Aring and a pressure of 10 atm B2O3 was explored using isobaric MD at a temperature of 1500 K for
500-time steps of 1 fs also at a pressure of 1 atm The minimization (after MD or ET) was handled
by the LAMMPS[30] steepest descent algorithm until the certainty in energy was smaller than 10-
4 eV Both used a nudged elastic band spring constant of 10 eVAring2 The energy distribution for all
systems was fully sampled which was confirmed by randomly checking if half the basin
distributions matched the full distribution It was also found that multiple structures in each
exploration were discovered more than once and a slightly smaller B2O3 system (130 atoms) was
also explored and showed identical results to the 150-atom distribution The insights gained from
the interaction between the experimentally accessible viscous properties and the landscapes
presented here will be generalized to create a simplified ldquotoy landscaperdquo model to capture the
thermodynamics and kinetics of the glass-forming system
86
53 Results
53A Adam-Gibbs Validation
To predict viscosity using the Adam-Gibbs expression there are only three required
parameters the configurational entropy (as a function of temperature) the barrier for cooperative
rearrangements and the infinite-temperature limit of viscosity Using a landscape the equilibrium
configurational entropy can be obtained using Eq (83) The infinite-temperature limit of shear
viscosity is given by the work of Zheng et al[54] where a systematic study of MYEGA fits to
viscosity data revealed a common value of 10-293 Pamiddots This infinite temperature value of viscosity
is used for all the experimental and computational viscosity curves shown in this work The final
parameter is the mean barrier for a cooperative rearrangement In Figure 18 the barrier for
cooperative rearrangements was chosen to accurately produce a viscosity of 1012 Pamiddots at the
experimentally measured value of the glass transition temperature From these parameter values
the predicted Adam-Gibbs viscosity curve is plotted The universal infinite limit fragility and glass
transition values from references were inserted into the MYEGA equation to plot the
lsquoExperimentalrsquo curves for B2O3 SiO2 and BaOmiddot2SiO2 respectively[58] [69] [120] When
examining Figure 18 it is clear that the AG model is able to fundamentally recapture the viscosity
87
Figure 18 The viscosity (left) and landscape (right) predictions for three common systems The
first system is newly calculated in this work while the latter two come from our previous works[72]
[89] It is seen that the viscosity predicted from the AG model is very accurately able to reproduce
the experimental viscosity curves from the MYEGA model The last system is a potential energy
landscape while the others are enthalpy landscapes
53B MYEGA Validation
Now that we have confirmed that the configurational entropy of the landscape works well
to describe the viscosity via the AG relationship a definitive temperature-dependent form is needed
in order to make more general predictions The key approximation made in deriving the MYEGA
model is that the configurational entropy follows an Arrhenius form By testing this Arrhenius
88
form we can evaluate this assumption in the derivation of the MYEGA model while simultaneously
gaining insights into the thermodynamics of the liquid and supercooled liquid states A strong liquid
( 15m ) has a configurational entropy that is nearly constant as a function of temperature while a
fragile liquid ( 15m ) is a liquid in which the configurational entropy is highly temperature
dependent Since we are testing the MYEGA equation it is also worthwhile to consider alternate
viscosity expressions Assuming validity of the Adam-Gibbs equation the configurational entropy
deduced from the viscosity experiments is given by
( )10 01log l) og(
c
BS
T T minus= (84)
where the viscosity function is given by a viscosity model such as Vogel-Fulcher-Tammann[59]
(VFT) equation the Avramov-Milchev[60] [163] (AM) equation or the MYEGA expression
The resulting entropy predicted by each viscosity model (using experimental data for
fragility and the glass transition) is compared to the calculated configurational entropy from the
landscape for B2O3 in ( 518gT = K and 33m = [69] were used for the experimental values) The
figure clearly shows that the MYEGA equation which assumes that configurational entropy scales
in an Arrhenius fashion reproduces the calculated configurational entropy from the landscape the
most accurately out of the three considered viscosity models This also means that the
configurational entropy form of the AM and VFT models are not physical and cannot be used to
recreate physical results Since the temperature dependence of the entropy has a fixed mathematical
form and is dominated by the distribution of basins we now understand that the distribution of
basins must also have a fixed form and that form must control experimentally accessible properties
This is the key insight that will allow for the development of the toy landscape model
89
53C Adam-Gibbs and Structural Relaxation
Although the Adam-Gibbs model has been validated for shear viscosity and has provided
the functional basis of the MYEGA model let us now consider whether the temperature
dependence of the configurational entropy can also predict the bulk viscosity (also known as the
volume viscosity) Recent work has highlighted the importance of bulk viscosity viz that the
structural relaxation time is proportional to the bulk viscosity[62] Hence the temperature
dependence of the bulk viscosity provides key information needed to predict structural relaxation
times[62] Following Scherer[164] the relationship between the structural relaxation time ( structural
) and the bulk viscosity ( B ) is given by
( )0B structuralK K = minus (85)
Figure 19 The configurational entropy comparisons between the three major viscosity models
which validates the main underlying assumption of the MYEGA model The VFT and AM are
unable to capture the physics of configurational entropy therefore ruling
90
where K is the infinite frequency bulk modulus and 0K is the zero-frequency bulk modulus In
order to calculate bulk viscosity for B2O3 the zero-frequency and infinite-frequency modulus is
taken from these works respectively [165] [166] The experimental structural relaxation times were
taken from the work of Sidebottom et al[154] In Figure 20 the bulk viscosity is shown as well as
the Adam-Gibbs comparison
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the configurational
entropy from the enthalpy landscape with the barrier 00155 eV (compared to the shear barrier of
00149 eV) and the infinite limit allowed to vary (10-263 Pamiddots for bulk viscosity) In Figure 18 the
configurational entropy is confirmed for the shear viscosity thus confirming the AG for both shear
and bulk viscosities Sidebottom data are from Ref [154]
In Figure 20 the fitted values of B and 10log can be used to gain insights into the
difference between structural and stress relaxations since the configurational entropy is universal
between the two The value of Bk (the barrier for cooperative re-arrangements in natural units) for
shear viscosity was 173 K while for bulk viscosity it was 180 K This difference of 7 K in the
barrier leads to an order of magnitude difference in the fitted viscosity curves This is a small
91
difference and explains the variance in the values at Tg The other difference is in the infinite
temperature value of viscosity which is -293 log(Pamiddots) for shear viscosity (from Zheng et al) and
-263 log(Pamiddots) for bulk viscosity This implies that the MYEGA raw form is accurate to predict
the structural relaxation time as well as the stress relaxation (though the values of the pre-
exponential factor and the composite constant will change between the two forms) However the
form in terms of fragility and the glass transition will need to be modified to accurately describe
the structural relaxation time The form assumes the viscosity of interest has a value of 1012 Pamiddots at
the glass transition temperature for shear viscosity which does not have to be the value for the bulk
viscosity curves at the glass transition Further research is needed to understand the appropriate
bulk viscosity at the glass transition and the infinite temperature of bulk viscosity
53D Landscape Features
As shown here the configurational entropy is governed by a Boltzmann sampling of a
probabilistic distribution of states n(E) This distribution of the inherent structures is empirically
found to follow a log-normal distribution as shown in Figure 21 While we are not claiming a
deeper meaning to the underlying origin of the log-normal distribution we may adopt this form of
the probability density function since it accurately describes the underlying distribution of
microstates This is the key insight needed to develop a new model because this means that the
fragility and glass transition are fundamentally related to the parameters of the distribution of basins
on the landscape (number of basins and standard distribution of the enthalpy)
The configurational entropy in the MYEGA model is given by
10
exp 1ln10 12 log
g
c
TS mS
T
= minus minus
minus
(86)
92
where S is the infinite temperature limit of the configurational entropy In Figure 21 we also
show the computed value of Eq (83) as a function of temperature compared to that of Eq (86)
which shows excellent agreement between the configurational entropy and the MYEGA model
This test also supports Stillingerrsquos[18] [19] view of fragility being correlated to the lsquoroughnessrsquo of
the landscape topography According to Stillingerrsquos view if the distribution of inherent structure
energies is narrow then the configurational entropy cannot have a large temperature dependence
leading to a low fragility Alternatively if the distribution is broad then there must be a temperature
dependence to the configurational entropy which necessarily results in a higher fragility
93
Figure 21 (A) The histogram for the energy minima for each of the test systems showing a good fit
with the log normal distribution This distribution will then be a valid form to calculate the enthalpy
distribution of the model presented in the next session (B) The configurational entropy from the
model showing the accuracy of the scaling of the entropy predicted by the MYEGA model The
S value was fit for each system
(A)
(
B)
(B)
94
54 Topography-Property Relations
Now that we have confirmed the validity of the Adam-Gibbs and MYEGA equations from
the energy landscape approach we can draw some key conclusions regarding the thermodynamic
properties of liquids For example heat capacity has been related to fragility[112] and is of
importance for glass manufacturing since it is related to the cost of heating and forming a glass
Isobaric heat capacity is defined as
p
P
HC
T
=
(87)
where H is enthalpy T is temperature and P is pressure The mean configurational enthalpy
calculated from an enthalpy landscape is given by
conf i conf iH p H= (88)
The heat capacity has contributions from both vibrational and configurational modes in liquids
while in glasses the configurational mode is mostly lost For a reversible process the
configurational heat capacity can be written as a function of the configurational entropy
c
p conf
P
SC T
T
=
(89)
By combining Eq (86) and Eq (89) we can write the expression for configurational heat capacity
of a liquid as
10 10
1 exp 1 exp12 log 12 log
g g
p conf
T Tm m S HC S H
T T T T
= minus minus minus = minus
minus minus
(90)
From fluctuation theory the heat capacity can be expressed in terms of the variance of the enthalpy
(2
H )
95
2
2
1p conf HC
kT= (91)
It is important to note that the variance of the enthalpy is not equivalent to the variance of the log-
normal distribution The variance of the enthalpy includes a Boltzmann distribution over the log-
normal distributed basins This gives a direct relationship between the enthalpy fluctuations and
the viscosity parameters If we further adopt the log-normal distribution as the effective distribution
for all liquid basins of all enthalpy landscapes then the distribution is only a function of the
common viscosity parameters ( S m gT and 10log ) The viscosity parameters will only
change the variance of the distribution since the mean of the distribution is inconsequential (the
zero-point enthalpy can be adjusted to an arbitrary value without changing the calculation of the
configurational entropy) The parameters of the log-normal distribution can then be fitted to
reproduce the configurational entropyrsquos temperature dependence thus creating a lsquotoyrsquo enthalpy
landscape for us to play with for the purpose of thermodynamic and kinetic calculations Generating
the landscape is a numerical calculation as the number of basins and standard deviation must be
chosen such to reproduce key phenomena
To quantify the dependence of these parameters on the degeneracy and temperature it is
helpful to develop a descriptive formal model The cumulative probability distribution of a normal
distribution (with zero mean) is given by
10log )1
1 er(
f2 2
Hp
+
=
(92)
where is the standard deviation of the log-normal distribution erf is the error function H is the
enthalpy and p is the probability To then calculate the equal probability spacing for a log-normal
distribution we solve for H for the set of probabilities p which range from 2 to 1 2minus
96
with a step-size of This then gives enthalpies of basins equally spaced by the cumulative
probability distribution and as a result each basin has equal degeneracy
1 2 r2 1e f10
pH
minus minus = (93)
This set of basins carries with it an Arrhenius activation barrier for the configurational entropy that
is only a function of and the degeneracy of each basin (if the is chosen such that the energy
distributions converge) This creates a model that fully describes the thermodynamics of the liquid
and though it currently lacks any kinetic component it offers the possibility to be a useful tool
when used in conjunction with Eqs (82) (83) and (88) This new tool is denoted the ldquotoy landscape
modelrdquo (TLM) It is called the toy landscape model because we are generating a landscape that is
not the real landscape but is enough to reproduce key experimental properties without needing
expensive computational calculations It is a lsquotoyrsquo for us to play with without needing to worry
about upfront computational cost
To confirm the validity of TLM and the way it is constructed we can calculate the
activation barrier from this deterministic method and for a random set of inherent structures with
the same number of basins and distributions For this purpose we used temperatures ranging from
1000 to 2000 K for both the deterministic and random models with a step size of 0001 for TLM
Excellent agreement between the random and deterministic methods is shown in Figure 22A This
analysis allows us to understand that the thermodynamics of the system are fundamentally linked
to the viscous flow behavior Though this technique was created to target viscosity and relaxation
it is generally applicable to all configurational thermodynamics of liquids and glasses It is worth
noting that all the thermodynamics will be relative to a 0 eV energy state defined at equilibrium at
0 K and as such the enthalpy differences at some fixed temperature must be known to compare
over compositional spaces
97
In Figure 22B we see the same method being used to understand the relationship between
the viscous parameters and the parameters of the landscape This is the key to the numerical
calculation of the lsquotoy landscapersquo model The toy landscape is a numerical calculation where we
use experimentally accessible values to generate a landscape whose thermodynamics should
reproduce key thermodynamic phenomena In Figure 22B we show that the fragility and glass
transition are systematically changed as the number of basins and standard deviation of the enthalpy
is changed When this idea is combined with the basin calculation in Eq (93) (giving equal
degeneracies for all basins) then the complete set of enthalpy basins is easily attainable giving
access to the fundamental thermodynamics of the system
98
(
A)
(B)
(A)
99
Figure 22 (A) Comparison between the randomized method (histogram) and the deterministic
method (vertical lines) showing good agreement between the maximum in the histogram and the
value predicted by the deterministic technique validating the approach It is worth noting that the
100-basin distribution is a very wide distribution where the total number of basins is less than the
number of points used in the calculation This is done for a variable number of basins with the
number of basins shown in the legend (B) The dependence of fragility and the glass transition
temperature vs the distribution of states and the number of basins
55 Barrier Free Description of Thermodynamics
Throughout this work we have systematically explored the Adam-Gibbs model of
viscosity An important insight that we have gained is that the configurational entropy is free of any
barriers This is of particular interest because the Arrhenius slope of the activation barrier is linked
to the glass transition and fragility The glass transition temperature is known to be related to the
set of activation barriers as it corresponds to a breakdown of ergodicity What is found here is that
the fragility is the ldquoconversion factorrdquo to account for the slope of the configurational entropy
(which as previously identified is related to the distribution of basins) This means that the barriers
themselves do not matter until the material is close to the breakdown of ergodicity and thus the
equilibrium liquid thermodynamics and kinetics can be described without explicit information
about the saddle points in the landscape
Understanding the role of the topography in the properties of glasses and liquids lead to
the most important insights derived from the AG model Since the configurational entropy is the
same for both stress and structural relaxation simple viscous parameters allow us to understand
fundamental thermodynamic quantities of the system without needing to map computationally
expensive energyenthalpy landscapes The process of converting experimentally accessible
properties to a usable landscape for thermodynamic calculations is what we are calling the toy
landscape model To show the utility of toy landscapes we can calculate the driving force for glass
relaxation This information can give insights into the physics of relaxation by dynamically
100
calculating the Gibbs free energy of the equilibrium liquid at any temperature which defines a
driving force for glass relaxation To do this calculation of the driving force for glass relaxation
we consider the ergodicity parameter ( x ) defined by the Mauro-Allan-Potuzak (MAP) model of
glass relaxation as the ratio between the glass entropy and the equilibrium liquid entropy
liquiss dglaS x S= (94)
To get information about the enthalpy we consider an instantaneous Boltzmann distribution on the
landscape given by an equilibrium distribution at some elevated temperature fT
1
exp
exp
iglass i i
fii
f
HH H g
kTHg
kT
= minus
minus
(95)
This formalism in conjunction with TLM gives us the ability to explore the temperature
and compositional dependence of the driving force for relaxation The enthalpy of the liquid is
calculated from an equilibrium distribution on the landscape The driving force ( ) is defined as
( ) ( )( ) ( )1iliquglas ds f l quii dH H TS xT T = minus + minus (96)
For our purposes this gives a nice approximation for the driving force improving on previous
approximations given by ( fT Tminus )[16] [42] TLM does not include a kinetic feature to calculate
the occupational probability in each basin however when coupled to a kinetic model such as the
MAP model[44] a complete description of the underlying thermodynamics of glass relaxation can
be considered For this calculation of the driving force the kinetics are calculated through the MAP
model but this is not necessarily a requirement In Figure 23 the compositional and temperature
dependence of the driving force is shown for three glasses with identical glass transitions and
varying distributions of basins (as listed in Table 1) The calculations are performed using the
RelaxPy script and MAP model with the stretching exponent fixed at the Phillips value of 37
101
explanations of which are available elsewhere[42] The systems were quenched from 700 K to 300
K at a rate of 10 Kmin Figure 23 clearly shows how simply changing the total number of basins
creates a larger driving force for glass relaxation This simple change of the number of basins
drastically increases the entropic contribution and due to the loss of ergodicity explains the increase
of the driving force of relaxation The kinetics could be calculated through any relaxation model or
directly from the landscape if some barrier approximations are made
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition temperature (500 K)
were kept the same while the total number of basins were allowed to vary
Sample Total Basins [-] Distribution of States [log eV] 1
1493g
mT
minus
[K]
Fragility [-]
1 100 02 837 40
2 500 02 537 31
3 1000 02 466 28
102
Figure 23 The driving forces for different example glasses calculated using a combination of the
MAP model RelaxPy and the toy landscape model The parameters for each glass can be found in
Table 4
57 Toy Landscapes for the Design of Glasses and Glass Ceramics
The concept of a toy landscape can go even further building a method to predict the
dynamics of a glass system based on simple inputs This is possible because in the previous section
we have related the distribution of basins on an energy landscape to the experimental viscosity
which allows for thermodynamic insights Building on this we can assume that the barriers (H )
for transitioning between states is equal to that of the barrier mean for the viscous relaxation
ln10gH mkT= (97)
This means that the only key factor needed to then predict the dynamics of a landscape using the
metabasin approach is the vibrational frequency However the relaxation time at the glass transition
103
is known to be a constant of 100s and as such we can choose a vibrational frequency that reproduces
this fixed point This gives an entire description of relaxation without any need for fictive
temperature An example of the results predicted through this method are shown in Figure 24 This
prediction only required knowledge of the glass transition temperature and the fragility and from
this a pure prediction of relaxation is made
104
105
Figure 24 The predicted outputs from the toy landscape method showing predictions of the enthalpy
and entropy under a standard quench for barium disilicate This prediction does not require fictive
temperature or any such assumptions about the evolution of the non-equilibrium behavior
This method is also not limited to predicting the relaxation of glasses but can also capture
the crystallization It is known from the earlier work on nucleation presented in Chapter 3 that a
simple two state model was able to capture the underlying physics of the system Thus if we know
the free energy of the crystal and barrier to crystallization we can find predict key crystallization
106
phenomena By fitting the heat flow peak magnitude and location this gives enough information to
infer the barrier and the enthalpy of the crystal The degeneracy (and thus the entropy) of the crystal
can then be calculated through knowledge of the liquidus temperature
ex(
p)cry SCL l
cry
lk
G TH
T
minus=
(98)
This then gives everything for the model presented in Chapter 3 from only 1 calorimetry
experiment the glass transition the fragility and the liquidus temperature To predict nucleation a
series of estimates were done running the toy landscape at different temperatures until the
temperature with the fastest crystallization rate was found This was then assumed to be the
maximum rate of nucleation The value of the interfacial energy was then chosen in such a way so
that the predicted curve reproduces the peak at the same temperature This gives all the parameters
needed for a prediction using CNT It is also possible to make a prediction of the growth rate of
crystals using this expression
2
1 exp6
GU
kT
a kT
minus minus
=
(99)
The results for these estimates are shown in Figure 25 The reason why growth is more accurate
compared to nucleation is that the kinetic factor of growth is governed by viscosity while nucleation
is governed by a kinetic factor that includes an assumed vibrational frequency
These methods have a plethora of assumptions built in such as that the vibrational
frequency of basins undergoing relaxation is equal to the vibrational frequency when undergoing
crystallization however as more information is gained about a particular system (such as more
crystallization peak as a function of temperature) the information could be built into the system to
improve crystallization predictions If the peak size is known as a function of scan ratetemperature
even the enthalpy as a function of temperature could be known This provides a whole system for
the real estimates of crystallization can be improved
107
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume in
nucleation is assumed to be on the order of one cubic angstrom while the a parameter in growth is
assumed to be around one nm both are in good agreement for estimates in literature The values
for the orange points are taken from these works[128] [167]
57 Discussion
The impact of the Adam-Gibbs model on glass science is hard to overstate and it has
become foundational to ideas that will enable the design of next generation glasses The Adam-
Gibbs equation still has more to offer the glass community in terms of fundamental understanding
and practical applications In this work we have shown that the Adam-Gibbs relationship is
consistent with both shear and bulk viscosities Also we have validated the main assumption of the
MYEGA model giving insights into the scaling of configurational entropy from the underlying
enthalpy landscape Using this knowledge of the relationship between the Adam-Gibbs model with
the underlying enthalpy landscapes we have developed a simplified ldquotoy landscaperdquo approach for
modeling glass relaxation The toy landscape is constructed numerically by understanding the
distribution of inherent structures and transition points in the landscape and then relating these
distributions to experimentally measured parameters of the MYEGA viscosity model This
1 00
1600
1400
1200
1000
00
600
400
200
000
0000 110000 130000 1 0000 1 0000
(m
s)
Temperature
Growth PredictionCassar et al
0
2
4
6
10
12
14
00 1100 1300 1 00
log 1
0(I) n
ucl
ei s
m3
Temperature
Predicted RateRodrigues et al
4
3
2
1
0
00 1000 1100 1200 1300
log
10(I m
ax(I))
nuclei s
m3
Temperature elvin
108
simplified landscape model can then be used to calculate the underlying thermodynamics and
kinetics of the system and represents a practical application of energy landscapes to understand
glass relaxation nucleation and other relevant thermodynamic properties
This also closes our discussion on understanding the thermal history effects on glass What
we have shown is that the dynamics of a glass system can be understood through energy landscapes
and fundamentally the same insights can be recreated through lsquoToy Landscapesrsquo This approach
allows fundamentally for the complex 3N dimensional landscape to be summarized and abbreviated
into a smaller landscape that can easily be implemented for any glass However this new
parameterization relies on knowing some key properties that are only due to the location in the
compositional phase space and for a complete picture of the system to be created we must also
create a sufficient approximation for the compositional phase space shown in the next 2 chapters
Chapter 6
Enabling the Prediction of Glass Properties
So far models have been created that describe the very nature of the thermal history effects on
glasses However as an input to these models we have to know about the compositional dependence
of key parameters namely the glass transition and the fragility In this chapter we extend
compositional models to other key properties needed for the design of glasses Here these key
properties that have not been readily predictable are approached using both machine learning and
TCT models The TCT models are focused on surface reactivity Youngrsquos modulus and ionic
conductivity The ML addresses CTE fragility melting point as well as Youngrsquos modulus
61 Controlling Surface Reactivity
Glass surfaces especially their interactions with water[168]ndash[171] are of the utmost importance to
nuclear waste glass cover glass and many other modern applications[106] It is generally accepted
that the outermost surface of oxide glasses readily reacts with water molecules where water
molecules chemisorb on an as-formed glass surface and populate it with surface hydroxyls Recent
models attempt to quantitatively explain the change in properties observed on hydroxylated glass
surfaces[77] [168] [172] [173] and most agree that the surface readily becomes hydrated because
it is more stable to have a bonded hydroxyl group than to have dangling bonds In glasswater
interactions hydrolysis and diffusion lead to two forms of adsorbed water chemisorbed and
physisorbed[171] [174]ndash[176] Both affect the network differently and are controlled by different
processes physisorption by hydrogen bonding[175] and chemisorption by reaction with the surface
to form bonded hydroxyl groups
110
equiv Si minus O minus Si equiv +H2O rarr equiv Si minus OH + OH minus Si equiv
Although binding energy studies have been performed using molecular dynamics simulations[177]
a specific study of the surface reactivity in tandem has not been conducted Recent advancements
in reactive force field modeling allow the direct observation of both diffusion and surface
reactivity[38] [178]ndash[180] and several studies have found surface reactivity constants consistent
with those reported experimentally[173] [178] [181]
Recently TCT has been used to predict various bulk properties of glassy systems such as
Vickers hardness and fragility[17] [182] [183] Based on the TCT description of the glass network
glassy materials have been postulated to have a so-called lsquointermediate phasersquo in which the atomic
structure of a material will self-organize so as to be isostatically constrained (n=3) as explored
extensively in various works[65]ndash[67] [133] [184] [185] In this case it has been shown that such
materials exhibit anomalous behavior in certain properties such as an high hardness and lower free
energy[43] [67] [77]
TCT has also been used to understand the impact of water on bulk glass structure and
properties Potter et al[168] found that by accounting for the impact of chemisorbed (dissociated)
water on glass network connectivity (viz breaking Si-O-Si bonds) and the impact of physisorbed
(molecular) water on strain of the Si-O-Si bond angle the glass transition temperature (Tg) could
be predicted This model could be further expanded to include other properties (such as
modulus[139] or dissolution kinetics[77]) predicted by TCT such as the work done by Liu et
al[186] on calcium-silicate-hydrate gel An extensive study relating topological constraints[6]
[17] [57] and surface energy was performed by Yu et al[172] specifically focusing on the
transition from hydrophilic to hydrophobic behavior on silica surfaces Their work used reactive
molecular dynamics to model the change in surface energy of a silicate glass and then correlate it
with the number of surface constraints present with the work focusing primarily on the global
average of the surface
111
To investigate the effects of glass network topology on the surface reactivity we model
the hydration of a silicate glass was modelled using molecular dynamics simulations according to
the following procedures Initially bulk sodium silicate glasses were simulated (150 atoms with a
molar composition of 70SiO2middot30Na2O) using the Teter potential[187] For a bulk glass this pair
potential is known to accurately simulate sodium silicate glasses and has been extensively studied
to investigate various properties including structural features transport of sodium ions and
vibrational density of states[188] The size of the system is initially set to achieve the
experimentally measured density of 2466 gcm3 [189] A total of 35 Si-atoms 85 O-atoms and 30
Na-atoms were randomly inserted within a periodic cubic box of a = 12686 Å and the initial
configuration was energy-minimized to avoid any overlaps prior to the glass formation The glass
was held for 05 ns with a constant number of atoms volume and energy (a microcanonical or
NVE ensemble) Since a thermal fluctuation of ~400 K (from 2000 K to 2400 K) was observed
during the microcanonical run the glass was allowed to evolve further at 2400 K for 05 ns with a
constant number of atoms volume and temperature (a canonical or NVT ensemble) Within the
same ensemble the melted system was then cooled with a constant cooling rate of 05 Kps and
after the temperature had reached 300 K it was equilibrated for 1 ns Finally a constant pressure
of 1 atm was applied to the system (NPT ensemble) for another 1 ns to ensure no significant
fluctuations of the density during the equilibration process Three different sodium silicate glasses
of the same nominal composition were constructed via the aforementioned procedure using
different initial atomic positions These repeated simulations compensate for the limitation of small
sample size by allowing us to capture the statistical behavior of glass surfaces during the glass-
water reactions The final densities of the sodium silicate are presented in Table 5 These final
structures were used as starting configurations for the glass-water reactions MD simulations with
the Teter potential creating three glass networks were carried out using the Large-scale
AtomicMolecular Massively Parallel Simulator (LAMMPS) package[30]
112
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K
Simulation box dimension a (cubic) [Å] Density [gcm3]
Run1 12979 2303
Run2 12630 2499
Run3 12737 2437
Initial dimension a=12686 Å
Experimental density (70SiO2middot30Na2O mol) 2466 gcm3
113
After the bulk sodium silicate glasses were obtained a reactive potential was employed to model
the glass surface and subsequent glass-water interface as the system of interest for this study
required characterization of the reactive processes during the simulation of surface phenomena In
this study all reactive MD simulations are performed with the NaSiOH parameterization using
the ReaxFF reactive force field framework15 The ReaxFF parameters are trained using a first-
principles data set that describes water interaction at the sodium silicate glass-water interface
Further details of the ReaxFF methodology and its potential forms can be found in earlier
publications by van Duin et al[38] [179] [181] [190]
114
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the hydration
models Color scheme Si atom (ivory) O atom (red) and Na atom (blue) The z-axis is elongated
to allow space for an insert of water
115
As shown in Figure 26 a free surface was first created by expanding the c parameters of each of
the equilibrated bulk glasses This process results in a vacuum region above and below the two
surfaces of the glass slab along the z-direction These exposed surfaces were relaxed at 300 K for
100 ps with the ReaxFF reactive force field Following the relaxation of both top and bottom
surfaces the vacuum region was filled with water molecules The number of water molecules that
are inserted in the vacuum region was controlled to have a density of ~099 gcm3 In addition all
water molecules were kept at least 2 Å from the uppermost and lowermost atoms of the glass
surface to prevent any initial close contact with the surface Glass-water reaction simulations were
carried out at 300 K with the configurations shown in Table 6 for 500 ps in the NVT ensemble
From these runs trajectories at every 100 ps were obtained for the surface reactivity analyses
Figure 27 shows the initial and final positions of glass surface reaction with water at 0 and 500 ps
respectively
116
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the top surface
in contact with water is shown here
a) b)
117
Table 6 System configurations for sodium silicate glass-water reactions
Simulation cell [Å3] Number of water molecules
Run1 12979 times 12979 times 3894 146
Run2 12630 times 12630 times 3789 134
Run3 12737 times 12737 times 3821 138
118
The hydrated surfaces of sodium silicate glasses from each run were used to investigate the
adsorption behavior of a water molecule Since the time evolution of surface reactivity is of key
interest the binding energy of a water molecule to the hydrated surface was calculated at 0 100
200 300 400 ps In order to evaluate the local heterogeneity of reactivity imposed by varied glass
surface structures each surface was divided into a 10 by 10 grid and the binding energy at each site
was mapped across the grid The water molecule position in the z-direction from top and bottom
surface was maintained to be 20 Å from the outermost atom during the energy calculation The
binding energies (Eb) were calculated as below where the energy of a surface with adsorbed water
(Esw) was taken with respect to the surface energy (Es) and the energy of molecular water (Ew)
b s wswE E E E= minus minus (100)
A negative binding energy would indicate that water adsorption to the surface site at the
corresponding grid is thermodynamically favorable The binding energy map may then be used to
locate where water binding would be most stable based on the ReaxFF calculations From three
independent glass-water reaction boxes a total of six hydrated surfaces are obtained (a top and
bottom surface from each run) increasing the statistical reliability of this analysis in the ensemble
despite the small overall system size of any one box The binding energy mapping process is shown
in Figure 28
119
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example of the
electronic-structure DFT calculation using semilocal exchange-correlation functionals finding the
binding energy of a water lsquopixelrsquo to the surface
~20 Å
surface
10times10 grid
120
Similar calculations are carried out at an electronic structure level using the DFT framework as
implemented within the Vienna Ab-initio Simulation Package (VASP) software Total energies are
computed from the static geometries obtained from the reacted ReaxFF structures using the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional with projector augmented wave
(PAW) pseudopotentials to describe the ion cores A kinetic energy cutoff of 500 eV was used for
the expansion of the plane waves Due to the large non-crystalline structure the Brillouin zone is
only sampled at the Γ-point with 01 eV of Methfessel-Paxton smearing to help electronic
convergence Due to the computational cost associated with these DFT simulations the results were
primarily used to elucidate the electronic interactions between the water molecule and the glass
surface
To count the topological constraints in the structure generated from the ReaxFF MD
simulations an algorithm was developed to systematically exclude all non-bridging oxygens
alkali and hydrogens from the structure The number of surface rigid constraints (in the pixel)
around each network-forming atom was then calculated an example contour plot generated using
this procedure is shown in Figure 29 The surfaces were then converted to a 10times10 pixel grid and
the average coordination of network-forming atoms in each pixel calculated The depth of each
pixel taken was to be 4 Aring below the surface
121
Figure 29 Example contour surface showing the average coordination per atom on the glass
surface for the first run at 300 ps
122
Figure 30 shows the results from the MD binding energy studies where a clear transition in the
region around 30 constraintsatom is shown This region is isostatic because the number of
constraints is equal to the number of translational degrees of freedom and the width highlighted in
the figure is evidence for the existence of an intermediate phase In order to confirm that the binding
energies had indeed converged a larger system of 1500 atoms was simulated Good quantitative
agreement was found between the 150- and 1500-atom systems In this work the isostatic behavior
occurs over a wide range in which certain silicon network sites dramatically increase the binding
sites (binding energy approaching 0) The state or structure with nc outside of the intermediate range
will readily interact with water Only those in the isostatic region are likely to remain unaffected
after the interaction with water thus they are the rate limiting species in dissolution of silica glass
123
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of constraints for
surface atoms at the local pixel Results show a distinct maximum in which there is a near
hydrophilic-hydrophobic transition of the surface The error bars represent the standard deviation
A second system with 1500 atoms was performed to show convergence of the ReaxFF MD results
⟨119889119867minus119874119866119897119886119904119904⟩ = 345 Aring
⟨119889119867minus119874119866119897119886119904119904⟩ = 248 Aring
124
The intermediate phase result is intuitive due to the implicit stability that comes with having an
isostatic phase which is both energetically favorable and stress-free The larger negative binding
energies in the over-constrained regions arise from large differences between the free energies of
the chemically bonded state (chemisorbed) and unbound state (water in near proximity to the
surface but not bonded) in these regions in this region the chemical reaction with the water
molecule allows the glass surface to reduce the number of incompatible constraints which in turn
alleviates local stress in the network In other words a local region undergoes an alleviation of
stress when Q4 units convert to Q3 units The larger negative binding energy values in the under-
constrained regions can trace their origin to the sites available for bonding with the water molecule
Their high number of degrees of freedom allows for the facile reaction of water ie the only energy
cost is for the oxygen to dissociate from the network The isostatic network within the intermediate
phase is able to preserve its structural integrity because the energy barrier to deform the network is
high and there is no localized stress to create an additional driving force for reaction with water
Indeed the isostatic network is the most energetically favorable arrangement of a non-crystalline
structure This intermediate phase is governed by the fluctuations of the topology that arise with
the minimization of stresses These stresses change the local topology (within the range of the
fluctuations) so that they become isostatic[68] This leads to a flat region where there is a constant
binding energy at the surface Though it appears to be an intermediate phase it is important to note
that this may instead be simply a manifestation of an isostatic threshold (a peak in binding energies
rather than a plateau)
The average number of surface constraints has been shown in previous work to be largely
controlled by the annealing time[172] The surface constraints may then be used to predict surface
reactivity However it is worth noting that the local number of constraints is not constant in
time[17] Furthermore Potter et al[168] showed that molecular water in the network can radically
shift the free energy of the γ constraint Therefore the number of surface constraints is dynamic
125
during the influx of water Shifting constraint energy will also alter the fragility which in turn alters
the diffusion activation enthalpy In the future this may provide a path for developing more durable
glasses ndash currently it shows that a homogeneous intermediate phase (isostatic) surface should be
targeted to achieve maximum chemical durability
62 Elastic Modulus Prediction
Tailoring the elastic properties of glass is important in the design of new advanced compositions to
control the stiffness and damage resistance of a variety of glass products[2] [191]ndash[193] Elastic
moduli are defined based on the ratio of the stress exerted upon a material to the resulting strain
While various elastic moduli can be defined (eg Youngrsquos modulus shear modulus bulk modulus
and Poissonrsquos ratio) for an isotropic material such as glass only two of these quantities are
mutually independent[1] [191] [194]
Previous attempts to model elastic moduli have focused on either computationally costly
molecular dynamics simulations or empirical fitting methods The work by Makishima and
Mackenzie[195] offered a possible solution for predicting elastic moduli based on the dissociation
energy per unit volume of the glass however topological changes in the glass network are ignored
and the model cannot account for the temperature dependence of modulus With MD simulations
elastic moduli can be obtained by applying a stress and measuring the resulting strain on the
simulation cell assuming that accurate interatomic potentials are available[2] Machine learning
has also been applied to model elastic moduli using experimentally measured composition-property
databases with a high predictive ability being achieved[2] [196] Previous analytical modeling
techniques related to the topology of the glass-forming network have shown good qualitative
agreement with compositional trends in modulus but have a lack of quantitative accuracy[64]
[197]ndash[199]
126
It is common in the glass community to assume that Youngrsquos modulus may scale linearly
with hardness While this is clearly an oversimplification let us begin with previously derived
models for glass hardness as a potential starting point since glass hardness has received
considerable attention in the context of topological constraint theory [85] [192] [200] Several
models have been proposed to explain the origin of glass hardness all of which have a linear form
( ) vv
dHH n x n
dn= minus (101)
where Hv is the Vickers hardness x is chemical composition nrsquo is the critical level of constraints
needed for the substance to provide mechanical resistance to the indenter in three dimensions and
n is a measure of the constraint rigidity Recent work on glass hardness[192] has shown that n can
most accurately be defined in terms of either the density of rigid constraints (or alternatively the
density of rigid angular constraints) given by
)( )
( )( )
(c An xn
x Nx
M x
= (102)
where nc is either the number of constraints per atom or number of angular constraints per atom ρ
is the density of the composition M is the molar mass and NA is Avogadrorsquos number Since
hardness and elastic modulus are often considered to be correlated one might surmise that the
elastic modulus might also be some function of angular constraint density or total constraint
density However such an approach is not able to give quantitatively accurate predictions of
modulus
Here we propose an improved model of Youngrsquos modulus based on the free energy density of the
topological constraints
int
( ) ( ))
(
(
) i constr
c i
a s
Ai in x q T
M x
x NF F
=
= (103)
where qn(T) is the onset function for each associated constraint to give temperature dependence to
127
the predictions and is determined by the free energy ∆Fn The calculated values of Δ119865c are then
used to calculate Youngrsquos modulus (E) by
( )c c
c
FdE
EF
xd
F
= minus (104)
In this new model the Youngrsquos modulus is controlled by the free energy parameters related to the
strength of each constraint Each parameter in this model captures a key physical phenomenon with
each parameter corresponding to a physical property Moreover we demonstrate that a common
set of parameters can simultaneously capture both the temperature and compositional dependence
of modulus as well as predict other properties such as the glass transition temperature and fragility
To demonstrate the validity of this temperature-dependent constraint model Youngrsquos
modulus data are collected from literature for lithium borate[195] sodium borate[195] and
germanium selenide glass systems[201] Additionally the sodium phosphosilicate[202] system
was experimentally determined following the procedure described by Zheng et al[183] using room
temperature resonant ultrasound spectroscopy on prisms of dimensions 10 mm times 8 mm times 6 mm
Analytical topological constraint models have already been published for each of these
systems[17] [57] [202]
The proposed models for glass elasticity (the models previously proposed for hardness
based on constraint density and angular constraint density as well as the new model for modulus
based on the free energy density of constraints) were optimized in each case leaving the intercept
and slope (119889119864
119889119899
119889119864
119889119880119888) as adjustable parameters The constraint onset temperatures were also optimized
for the free energy density model Figure 31 and Figure 32 show the results for each model in the
phosphosilicate and sodium borate systems respectively
128
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values of the model
predictions are 641 GPa for constraint density 313 GPa for free energy density and 774 GPa
for angular constraint density
129
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from minimum
to maximum error The free energy density model gives the most accurate results The constraint
density has a RMSE of 61 GPa the angular density has a RMSE of 20 GPa and the energy density
has a RMSE of 59 GPa
130
We have also evaluated the predictive ability of the model in terms of the temperature dependence
of the Youngrsquos modulus sing the 10 Na2Omiddot90 B2O3 data reported by Jaccani and
Huang[191] the model and experimental predictions of the temperature dependence of the Youngrsquos
modulus are plotted in Figure 33 Here the constraint onset temperatures are determined from the
room temperature modulus data The number of escape attempts was optimized since this controls
the width of each transition and is thermal history dependent
131
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and experiment for
10 Na2O 90 B2O3 Using the previously fitted onset temperatures the only free parameters are
then the vibrational frequency and the heating time in which their product was fitted to be 14000
Where each dip in the modulus corresponds to a constraint no longer being rigid as heated through
each onset The onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The data was fit
using a least-squares method and the resultant fit is shown as the calculated method The fit has an
R2 of 094 (B) The contribution from each constraint to the overall modulus
It is also possible to validate the model by comparing the free parameters fitted to the Youngrsquos
modulus data vs that of previously reported data as shown in Table 7 Results for two additional
systems are plotted in Figure 34 to show the general validity of the free energy density model
132
Table 7 Fitted values from this analysis compared to those reported in the literature The disparity
between the constraints evaluated with molecular dynamics most likely come from the speed in
which the samples are quenched
Value Fitted Value (K) Literature (K) Method Citation
Silicate Onset 2212 1986 MD (Potter et
al[168])
Silicate Onset 818 1600 MD (Potter et
al[168])
Silicate Onset 450-500 810 MD (Potter et
al[168])
Borate Onset 921 Not Reported Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 715 740-760 Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 393 328 Fitting Parameter
(Mauro Gupta and
Loucks[69])
133
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in Fig 3) and
experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of 093 and (B) xLi2Omiddot(100-
x)B2O3 glasses with an R2 of 0986
(
A)
(
B)
134
In the previous work of Zheng et al[192] concerning hardness of glass the authors write
ldquoIt should be noted that each bond constraint corresponds to a certain energy since different kind
of bond has different bonding energyhellip and thus the constraint density also represents an energy
per unit volume In other words hardness is correlated to the energy per unit volume Our findings
for the borosilicate and phosphosilicate systems are further evidence in support of this argument
since both the total constraint density and angular constraint density approaches give better
prediction of glass hardness compared to models based on number of atomic constraintsrdquo
which makes it clear that the prediction of the hardnessmodulus should be closely related to the
onset temperatures (ie free energies) of the associated constraints The model that was previously
proposed could be extended to modulus if all constraints had equal amounts of potential energy
but due to the drastic change in strengths between the constraints the prediction fails
Bauchy et al[85] showed also that hardness in the calcium-silicate-hydrate system is
controlled by the angular constraints which led to the development of angular constraint density
as the governing control for hardness When the analysis of the density of angular constraints is
considered through the free energy view it becomes apparent why this method works effectively
for some systems Models for predicting the hardness and elastic modulus of glass attempt to
explicitly connect the rigid bond energy to the macroscopic properties of the system The
assumption for the hardness models is that the energies for breaking each type of constraint are all
approximately equal and hence only the number of the constraints matters The argument can then
be extended for elasticity since elastic modulus is a bulk material property with the approximation
of all constraints being equal strength the number of constraints per volume should be related To
correct for this approximation when a weighted sum of free energies is used the estimation
becomes significantly more accurate Using the density of the glass a precise free energy per
volume of the rigid constraints can be calculated The knowledge that the energy of the bonds is
tied to the elastic modulus has been widely known but had not been quantified nor placed within
the context of topological constraint theory[203]
135
63 Ionic Conductivity
Ionic conductivity 120590 is related to the number of charge carriers n and the mobility of the
carriers 120583 as they diffuse through a network by[1]
Ze n = (105)
where Ze is the charge of the conducting species Since the charge-carrying ion is constant within
a given glass family the ionic conductivity depends only on the mobility and the concentration of
the charge carriers present However it is not typically feasible to measure 120583 and n independently
of each other and thus it is unclear which variable scales in an Arrhenius fashion thereby
controlling the ionic conductivity of a glass In the weak electrolyte model by Ravaine and
Souquet[204]ndash[207] the mobility of the charge carriers is considered constant while the
concentration scales in an Arrhenius function Conversely the strong electrolyte model assumes a
constant number of charge carriers while the mobility follows an Arrhenius function[208] [209]
Recent MD simulations from Welch et al[210] support the weak electrolyte interpretation for alkali
silicate glasses concluding that ionic conductivity is dictated by the concentration of charge
carriers Previous reports in the literature also supporting this view have proposed several
hypotheses to explain this conclusion[208] [209] [211]ndash[213] however further exploration is
needed
The simplest form of the temperature dependence of free ions is given by
0 exp an nE
kT
= minus
(106)
Here n0 is the total possible number of mobile ions and Ea is the activation barrier for an ion to
diffuse It is often assumed that the activation enthalpy is related to the energy needed to deform
the network temporarily to allow for ion motion[193] [209] [210] [212] however it is our
hypothesis that there is a coupling between the ion hopping event and the relaxation of the local
136
network allows for a permanent relaxation or deformation along ion channels This means that
there is some cooperative relaxation of the network along the diffusion path in conjunction with an
ionic hopping event These cooperative relaxations are discussed explicitly in the Adam-Gibbs
formalism[53] for describing the relationship between viscosity and configurational entropy Sc
expc
B
TS
=
(107)
where B is an activation barrier is the viscosity and is the viscosity of the liquid in the limit
of infinite temperature In the high-temperature (low-viscosity) liquid state there is a well-defined
relationship between the viscosity and diffusion (Stokes-Einstein relation) However this
relationship breaks down in the low-temperature (high-viscosity) glassy phase
Herein we present a new model for predictions for the compositional dependence of ionic
conductivity We can write the activation free energy barrier Eac for an ionic hopping event as
a c c c cE T S= minus (108)
Here is an enthalpic barrier for the ion to hop to a neighboring site CT is the configurational
temperature (a value that describes the distribution of energies of the ions) and cS is the entropy
of the activation barriers These quantities are associated with configurational changes in the glass
indicated by the subscript c In the weak electrolyte model all charge carriers have the same
mobility and hence the entropic effects are dominant This indicates a small value of an
assumption that will be validated by results later in this work leading to a simplified approximation
for the activation energy at temperatures below the glass transition but suffiently high temperatures
such that
a c c cE T S (109)
In the case of a nonequilibrium glass at low temperature the system becomes trapped in a localized
137
region of the energy landscape known as a ldquometabasinrdquo with slow inter-metabasin transitions[156]
[161] The MAP model of the nonequilibrium viscosity of glass relies on this partial confinement
within metabasins where variation of the activation barrier for relaxation (H) is related to the
fragility of the system by[44]
1
ln10g
dm
dH kT= (110)
Here m is the fragility index (as proposed by Angell[58]) and Tg is the glass transition temperature
Splitting the differential in Eq (110) and integrating we can write
ln10gmH kT= (111)
This relationship was also proposed by Moynihan et al[214] for the activation enthalpy of
relaxation of a glass[214] [215] Setting the Arrhenius form of the barrier described in Eq (111)
equal to the Adam-Gibbs equation (Eq (107)) we write
exp exp c
H B
kT TS
=
(112)
Solving Eq (112) for the configurational entropy and inserting the result into Eq (109) we obtain
the simple relationship
2
ln10
ca
g
T BE
k T m (113)
Dividing the above expression with that of a reference state r and assuming that B and is cT are
constant with respect to compositional variation we have
g r ra
a r g
T mE
TE m= (114)
This new equation is the first to relate the activation free energy for ionic hopping in a glass
to the glass transition temperature and fragility of its corresponding supercooled liquid It also
138
predicts a coupling between ionic diffusion and the cooperative rearrangements described by the
Adam-Gibbs relation This possibility was first proposed by Ngai and Martin[216] who showed a
correlation between the product of the Kohlrausch stretching exponent ( ) and the activation
barrier for relaxation with the activation barrier for ionic conductivity This implies that the
diffusion in glasses is not governed by an elastic component but instead controlled by the
surrounding cooperative rearrangements ie the -relaxation of the glass[14] in good qualitative
agreement with previous results shown by Potuzak et al[217] Further evidence comes from the
polymer community where the decoupling of relaxation and conductivity is reported with work
showing that the decoupling of viscosity and diffusion varies with the fragility and the glass
transition temperature[218] This is a general relationship which we demonstrate later to be valid
for binary alkali borate alkali silicates and alkali phosphate glasses as well as validating the
atomistic diffusion method using computational techniques
In order to test this proposed relationship between relaxation and diffusion through
simulation a glass with the composition 10 Na2O-90 B2O3 (mol ) was synthesized in molecular
dynamics with 1150 atoms using the potentials of Wang et al[162] The system was quenched at a
rate of 1 Kps from randomly placed atoms at 2500 K in the NPT ensemble to room temperature
as shown in Figure 35[210] To find the transition point energy of the diffusing ion a nudged elastic
band (NEB) calculation was performed An alkali ion was chosen and its closest alkali neighbor
was moved to another location so that the NEB calculation could be performed between those two
sites All calculations were carried out using the LAMMPS software package Eleven reaction
coordinates were used in the NEB calculation and the results are shown in Figure 36
139
Figure 35 The structure for the initial minimum energy configuration showing the boron (blue)
network with interconnecting oxygens (red) and the interstitial sodium ions (yellow)
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and sodium is
ivory The barrier is overestimated compared to experimental data this could be from several
sources of error such as potential fitting thermal history fluctuations or sampling too few
transitions The line is drawn as a guide to the eye
140
Although the transition point energy is key for determining the activation barrier for diffusion here
we are also interested in the dynamics of the atoms around the mobile ion Since we have
hypothesized here that the ion diffusion is dependent on the ability for the local region to relax and
deform instead of elastically straining we expect a local deformation (relaxation) along the ion
path Figure 37 shows that along the ion diffusion tunnel there is indeed a permanent displacement
of the atoms
141
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function of
reaction coordinates The middle shows the local deformation around the ion of any atom that
moves in between inherent structure mandating a relaxation force The color shows the degree of
deformation
142
In the final configuration of Figure 37 the permanent deformation required to lower the
energy of the system is clearly seen If it were purely an elastic response to dilate a pathway the
displacement between two inherent structures should be confined to changes with only the mobile
ion however this is not observed Instead along the pathway of the ion movement the network
forming atoms also deform and change local positions to minimize the energy implying a net
relaxation around the ion ldquopathwaysrdquo To apply this and show the validity of Eq (114) the sodium
borate lithium borate lithium phosphate and sodium silicate systems were considered with the
model results compared to experimental values in Figure 38 To confirm this model literature values
for activation barriers have been used The experimental activation barriers for the borates are taken
from the work of Martin[213] while the fragilities and the glass transition temperatures are from
Nemilov[219] The viscosity curve of the silicate system was fit with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) equation of viscosity[55] using the data from Poole[220] and Knoche et
al[221] with the activation barriers from Martinsen[222] Lithium phosphate activation data was
taken from Ngai and Martin[216] with the glass transition and fragility taken from Hermansen et
al and predicted values from topological constraint theory[223]
143
Figure 38 Different network formers and the prediction of the activation barrier from our model
compared with activation barriers from literature (A) Sodium silicate predictions and experimental
values[222] the error is calculated from the error in the fragility when fitting the data (B) Lithium
phosphate activation energy[216] predicted with topological constraint theory and compared with
the experimental values (C) Predictions over two different systems of alkali borates[213] sodium
and lithium with a reported R2 of 097
144
It is worth noting that this model is predicting a non-equilibrium kinetic property using
equilibrium viscosity parameters a connection that has been demonstrated in the past for viscosity
and relaxation models[61] In this work we are applying the same principles to the diffusion of
ions in a network The diffusion and viscosity in the case of equilibrium are known to be intimately
linked using the Stokes-Einstein expression (which breaks down for highly cooperative
rearrangements near and below the glass transition temperature[224]) Here we are no longer
connecting the two properties explicitly but instead connecting the activation barrier for diffusive
hopping to the two key parameters governing the viscous flow of the supercooled liquid state
The failure of the Stokes-Einstein expression near the glass transition is a well-known
problem and has been studied extensively for purposes of predicting crystallization rates[167] This
occurs because there is a breakdown of the ergodicity of a system around the glass transition In
the MAP model[44] the viscosity is controlled by the ergodic parameter x and is given by
min( )
max( )
f
p
f
T Tx
T T
=
(115)
Here the exponent p is related to the sharpness of the ergodic to non-ergodic transition Using the
ergodic parameter the overall viscosity ( ) is expressed as a linear combination of equilibrium
viscosity ( eq ) and non-equilibrium viscosity ( ne ) such that
10 10 10log ) log ( ) (1 ) log ( )( f eq f ne fx T T x T TT T = + minus (116)
By combining the Stokes-Einstein expression and the model proposed in this work we can define
the diffusion of an ion through the network as
( )10 10 10 0log ( 1 log exp ) l6 )
og(
b af
f
kx
T ED T T D
a T Tx
kT
minus + minus =
(117)
Here a is an empirical constant related to the size of the diffusing species k is Boltzmannrsquos
145
constant aE is the activation barrier calculated with Eq (114) and D0 is the limit of diffusion as
the temperature approaches infinity D0 should be chosen such that at fT T= the diffusion is
continuous This can be justified because the preexponential factor is related to the entropy of the
system and the fictive temperature is the temperature at which the configurational entropy of the
transition barriers becomes nearly constant Only one parameter is left unknown a since the
viscosity and fictive temperature can be calculated with an arbitrary thermal history using a
relaxation modeling tool such as RelaxPy[42] Implicit in RelaxPy are the assumptions pertaining
to the viscosity made by Guo et al[61] A similar model was first proposed by Cassar et al[167]
for modeling diffusion during crystallization although their model used an empirical hyperbolic
tangent function to approximate the ergodic factor In contrast the current model can be considered
explicitly as a function of thermal history and composition
64 Machine Learning Expansion
Though the TCT models presented here for different properties are immensely useful they
do have limitations the greatest of which is that intimate knowledge of the glassy structure is
required to parameterize these models This is due to the fact that the landscape is fundamentally
informed by the underlying structure of the network and as such to predict the dynamics of the
landscape information on the structure must be incorporated in some form However there is an
alternate method in which the physical origin is not considered in the development of the model
machine learning Machine learning comes in many forms and has become crucial to the study of
materials in the modern era There is a plethora of machine learning techniques however in this
paper we are focused on four key methods linear fitting (LF) methods random forest (RF)
symbolic regression (SR) and neural networks (NN)[3] [80] [225] [226] These methods are
146
explored extensively in literature and the exact underlying mathematics is beyond the scope of this
work In this work these methods have been used to create models for number of constraints (LF)
glass stability (RF) fragility (NN) melting point (NN) the infinite limit of viscosity (SR) Youngrsquos
modulus (NN) and coefficient of thermal expansion (NN)
To enable a linear fitting method for the number of constraints a simple steepest descent
algorithm was used to find the number of constraints due to each component There is no direct
experimental way to measure the number of constraints so instead a technique leveraging some
other property must be used to approximate the number of constraints In this case hardness is
related to the number of constraints through
25v cH A n= minus (118)
In which A is an empirical scaling parameter that is dependent on load indenter geometry and
glass family In this linear approach to determining constraints cn can be given by
0
compon
k
c k k
ents
cn x n=
= (119)
Combining this expression with a hardness database grouped by composition (in this case Sciglass)
allows for an approximate number of constraints of the system but due to A being a variable there
must be a fixed constraint in the system and every set of data is given a unqiue value of A The
fixed constraint in this system is that silicon atoms are four coordinated The value of A is chosen
such that the error between the prediction and the hardness is minimized The total process is as
follows
Choose a set of c kn
1 Calculate number of constraints for all compositions
2 Iterate over every dataset in Sciglass with gt 1 datapoints and scale by a factor of A
3 Calculate total error for steepest descent algorithm
4 Descent error slope
5 Repeat until error has been minimized
147
This then leads to an estimate for the number of constraints for every atomic species which can
then be used to estimate important properties such as chemical durability hardness and
qualitatively can predict glass stability
Two other techniques that are used are RF and SR When implemented a grid search for
the hyperparameters were used with a testing set used to determine the efficiency of each set of
hyperparameters Additionally each model was trained repeatedly and the lowest error was taken
The RF method was used to predict glass stability while undergoing a novel processing technique
called alkali-proton substitution (APS) The SR method was used to understand the compositional
dependence of the infinite temperature viscosity limit used in 3-parameter viscosity models To do
this the Sciglass database was taken (as implemented in GlassPy[227]) and sets of data that
included more than eight datapoints were fit to the MYEGA equation If the RMSE error of the fit
was less than 001 Pa s the fragility was greater than 17 and the value of the log infinite limit was
293 5minus it was accepted into the final database Each of these criteria were added due to the well-
known fact that this database is full of error and imprecise measurements The symbolic regression
was then performed through the grid search and the equation that performed best and reappeared is
given as
10loggT
mA B = + (120)
With A and B being parameters that varied depending on the hyper-parameters the infinite limit
threshold and the accuracy threshold used when creating the database We are currently in the
process of refining values for A and B is ongoing
This method allows for an easy prediction of the infinite limit so if the glass transition and
fragility are predicted through some other means (such as ML or TCT) a better prediction of the
third parameter is known without just accepting -293 as the mean Plots of the infinite limit vs the
148
fragility glass transition and Tgm showing the clear dependance of the infinite limit on the term
Tgm are shown in Figure 39
149
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top left) shows
the distribution of the infinite temperature limit in the database after limits exerted on the system
(Top right) The distribution of the infinite viscosity limit vs the glass transition (Bottom left) The
relationship between fragility and infinite temperature limit of viscosity (Bottom right) The infinite
temperature limit of viscosity vs the key metric predicted by SR
150
NN are perhaps the most widely discussed ML tool applied in literature This makes sense
given they are universal function approximators and are quite easy to train given sufficient data
The most difficult part of creating a NN is adjusting the hyperparameters so that good accuracy can
be found without causing overfitting To avoid overfitting the data is divided into a testing set then
a training set (just as for RF and SR) but instead of doing a grid search for the hyperparameters we
used gradient boosted regression trees to find the optimal set of parameters This consists of
creating an initial tree then evaluating it based on a small set of data and finding the minimum
point then running a new set with perturbations around that point to create another tree This is
repeated until the value converges to a small error In this work the value being determined is the
lowest error of 100 sequentially trained neural nets with a set of hyperparameters This is
particularly useful because there is no concern for the type of parameters (string int floats) that
determine the value at the end so one can do a mixed parameter optimization that includes every
conceivable variable The values being optimized are given by
bull Learning rate [10-5 10-1] (float on logarithmic scale)
bull Decay rate for Adam Optimization [10-6 10-2] (float on logarithmic scale)
bull Hidden layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull Output layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull The number of nodes in each layer [81024] (int)
bull Batch size [1256] (int)
bull Patience [5250] (int)
In Table 8 we have listed the hyper-parameters that were used for the fragility melting point
Youngrsquos modulus and co-efficient of thermal expansion (CTE)
151
Table 8 Hyperparameters for different neural networks after hyper-optimizations
Model Fragility Melting
Temperature
Youngrsquos
Modulus
CTE
Learning Rate 0035 0018 0027 00091
Decay Rate 000293 000095 000161 0002
Hidden Layer RELU RELU RELU RELU
Output Layer Linear Linear SELU RELU
Number of Hidden
Layers
2 1 1 1
Number of Nodes 235 506 514 470
Batch Size 4 118 211 125
Patience 25 250 23 22
RMSE 5 [-] 805 [K] 68 [GPa] 96 x 10-7 [1 K]
152
Chapter 7
Designing Green Glasses for the 21st Century
Thus far the work presented in this dissertation has enabled a new set of design tools for
everything from crystallization glass relaxation and optimizing over the ideal compositional space
we can use them for designing new glasses Building on these models and insights new applications
of glass are enabled specifically three glasses an ion conducting glass for batteries a proton
conducting glass for hydrogen fuel cells and a commodity glass for consumers All three are
currently undergoing testing but experimental results have been reported back as of yet Overall
both the hydrogen and ion conducting glasses have been designed thorough a combination of ML
and underlying physical models and as such hare reported herein
71 Glass Electrolytes
An emerging high-interest application of glass is solid state batteries Glasses offer a
possible solution to the growing energy crisis however to realize this new paradigm new research
methods are needed The main barrier for any new batteries to be commercialized is a high
conductivity at room temperature (gt 10-3 S cm) high stability ease of processing and ultimately
an alternative must be cheaper than current liquid state Li-ion batteries glass is an attractive
candidate due to the innate stability a simple processing technique the infinite variability and the
relatively low cost of production In addition the structure of a glass is a liquid-like structure which
may encourage ion migration like that seen in liquids Despite all of these advantages only a few
compositions have realized the requirements but further research is needed to find optimal
compositions
153
To optimize the composition for activation barriers we must understand the relationship
between conductivity and structure Structural effects can be propagated in two different methods
based on the Arrhrenius expression for conductivity
0 exp aE
kT
minus =
(121)
The two methods are through 0 (the infinite temperature conductivity) and the activation barrier
( aE ) The pre-exponential factor is approximately a constant for simliar compositions while the
activation barrier varies dramatically over small compositional spaces making it the larger of the
two concerns Many models have been presented to predict the compositional dependence of the
activation barrier the Anderson-Stuart (AS) model the Christensen-Martin-Anderson-Stuart
(CMAS) model the weak electrolyte (WE) model Kohlrausch exponent model (KEM) from Ngai
et al[216] and there is the model proposed in this dissertation which we will call the Wilkinson
viscous cooperative conductivity model (WVCC)[8] [88] [204] [207]ndash[209]
To optimize machine learning (ML) is a powerful technique that is implemented The best
method to train the ML is to use the direct relationship between the compositions and the activation
barrier but this is not currently feasible due to a lack of central depository or database for this
information To make the prediction of the activation barrier we need use some model that is reliant
on commonly accesible parameters This rules out all models for this type of analysis except since
WVCC since the other models require some fitting parameters not accessible Each models required
parameters fitted values (please note that the models typically include approximations for these
values) and caveats are shown in Table 9
154
Table 9 A table with some ionic conductivity models and the parameters needed for them as well
as the disadvantages for each These are not the only models but are representative of those
commonly used in literature
Model Required Parameters Fit Parameters Caveats
AS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
lsquoMadelungrsquo Constant
Covalency Parameter
Doorway Radius1
The Madelung
Constant is not smooth
as a function of
composition and the
fitting parameters scale
the activation energy
non-proportionally
CMAS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
Dielectric Permittivity
lsquoMadelungrsquo Constant
Doorway Radius1
Jump Distance of Ion1
The same issues as AS
but in addition the
dielectric permittivity
lacks a database as
well
WE Charge Concentration Equilibrium Coefficient1 The equilibrium
coefficient drastically
changes the prediction
non-linearly
KEM Stretching Exponent Proportionality
Coefficient
There is not enough
data available to know
the compositional
dependence of the
stretching exponent
WVCC Glass Transition
Fragility Index
Proportionality
Coefficient
Due to the
proportionality
constant the activation
barrier can only be
known in a local
composition range
1 There are ways to approximate this value based on additional data
155
The WVCC model predicts that the activation barrier for ionic conductivity is given by
a
g
EA
mT= (122)
in which A is a proportionality constant This means that to predict the glass behavior we donrsquot
need to know things about the glass but instead merely about the liquid state where the viscosity is
readily available Leveraging this technique if the viscosity information as a function of
compositions is available then the local glass with the lowest activation barrier can be found by
minimizing 1 gT m To access this information multiple options are available such as topological
constraint theory but in this work we will leverage ML since it is easily applied to a large
compositional space
To get the viscosity we will use neural networks (NN) A well trained NN for the glass
transition is readily available and as such will be used for our prediction of gT In the previous
section there was presented a trained fragility NN which enables a complete optimization of the
local composition based on WVCC To generate a database of fragility values that the NN was
trained on SciGlass viscosity values were fit with MYEGA model The data had to fit following
criteria
1 The total number of data points for fitting had to be greater than 5 from the same literature
source
2 The root-mean-square-error of the fit had to be less than 001 Pa s
3 The infinite temperature value of viscosity had to be within 3 orders of magnitude of the
accepted logarithmic value (-293 log(Pa s))
An additional neural network recently created by Cassar[228] was also used for an independent
comparison Cassarrsquos NN was based on a hybrid physical ML work and may offer better insights
when extrapolating far from experimental datapoints In this study to prove this concept we will
only focus on P2O5 B2O3 Al2O3 and Li2O
156
This gives an entire method to predict glass battery candidates This technique will find all
of the local minimum but due to the fact that there is a proportionality constant it is unknown which
local minimum is the absolute minimum This means that the composition with the lowest
activation barrier in the space is identified but another technique is needed to investigate which one
of the identified compositions is best The method follows
1 Randomize an initial glass candidate with the content of being normalized
2 Run the glass candidate through the NN to find values of fragility and the glass transition
3 Perturb the composition to find the gradient
4 Step down the gradient
5 If the new gradient is 0 this is a local minimum and a possible glass candidate return to 1
Otherwise return to step 3
This method cannot find global optima because there is no guarantee that the
proportionality constant will stay constant across wide compositional spaces To narrow it down
an additional technique to compare the predicted values is needed (such as another activation
barrier model molecular dynamics simulations a regression algorithm or simply melt the proposed
compositions) To further pair down in this work a k-means algorithm was used along with the
assumption that the proportionality constant was slowly changing over compositional space This
allowed us to take the best glass in each grouping predicted by k-means The glasses we have
predicted (using both the separate predictor NN and the method by Cassar et al[228]) are listed
below in Table 10 while an example prediction for 1000 gmT (which should scale with the
activation barrier) in the Na2O ndash B2O3 ndash SiO2 system is shown in Figure 40 The glasses are currently
undergoing melting at Coe College and Iowa State It is worth noting that composition D is a well-
known composition that is known to exhibit criteria close to the criteria for conductivity If this
method shows that these compositions are better than other compositions in the family found in
literature then it can be expanded to include a wider range of component and hopefully finding a
universal candidate for glass batteries
157
Table 10 The predicted compositions based on the optimization scheme proposed
Compositions Li2O Al2O3 P2O5 B2O3
A- This work NN 55 31 0 14
B- This work NN 44 1 5 50
C- Cassar 60 26 2 13
D- Cassar 50 0 0 50
Figure 40 The prediction of the scaling of the activation barriers for a common sodium borosilicate
system
72 Hydrogen Fuel Cell Glasses
An emerging high-interest application of glass that is proton-conducting intermediate
temperature fuel cells[7] [229] A fuel cell in short is a device that turns chemical fuel into
electricity without the use of combustion It is often described as a chemical battery due to the
similarity of a fuel construction with batteries One of the crucial components to a working
intermediate range fuel cell is an electrolyte with a high proton migration The electrolyte must be
stable over a wide range of temperatures (up to 673 K) and a proton conductivity greater than 10-2
S cm-1 at the operating temperature[7] Previously phosphate glasses near the metaphosphate
compositions were used since they carried the appropriate number of residual protons for
conductivity studies However this restricted the phase space to only n-2 dimensions drastically
reducing the degrees of freedom of the problem and artificially restricting possible solutions In
2013[230] to access the full phase space a new method was developed where a sodium phosphate
glass could have all the sodium ions replaced with protons This allows high concentrations of
protons to be achieved with a variety of starting compositions A previous report[230] has
summarized successful samples these data are used as a training set for this study with additional
data on compositions that have failed taken from internal theses
To choose the candidate material for each model a cost function is defined The cost function
is an analytical function whose inputs are the fraction of each oxide component and the output is a
value that rates the composition The cost function consists as of many properties that are of interest
In the application of oxide electrolytes for an intermediate range fuel cell the two main properties
to worry about are the ionic conductivity at high temperatures (we will choose 473 K due to this
being the low end in which these materials are considered) and the relative stability of each phase
159
This is especially important since the material will undergo the APS (alkali-proton substitution) in
which the sodium will be replaced with protons in the bulk of the material The cost function used
in this work is given by
473 1( ) log 000T K Sx = minus= minus (123)
In which S is either a 0 or 1 (representing a crystal or a glass respectively) and is the
ionic conductivity The function is arbitrary and could be expanded to include as many terms are
as needed in the goal of this glass The stability is heavily weighted because it is more important
than the conductivity and any sample that is not stable should not be considered In order to predict
each of these samples a hybrid physicalempirical approach is used For predicting the ionic
conductivity of a proton-swapped glass we began with an observed relationship between the glass
transition temperature and the conductivity at 473 K This empirical relationship is shown in Figure
41 The relationship although empirical can provide some insights into the physics of proton
transferred glasses (no universality is claimed in this work) The activation barrier for these glasses
has widely remained unchanged as noted in a previous work due to the fact that the activation
barrier is simply to disassociate the proton with the non-bridging oxygen (NBO) and once the
proton is free interacts rarely with the network until it is rebounded to said network Thus the pre-
factor can be understood as being related to the degeneracy of proton conduction pathways When
the network has lower configurational entropy there are fewer pathways for the proton to travel in
this case it is known that the configurational entropy is then the dominating effect for the glass
transition[53]
The relationship between the glass transition temperature and the configurational entropy
can be expanded through the Stokes-Einstein relationship evaluated at the glass transition
22
1273 10
6 10g
g
T T g
kTnZeD nZe
anT
minus
= = = (124)
160
In which a is approximately the size of a proton and n is the number of charge carriers (which is
approximately a constant according to the weak electrolyte theory) This determines the intercept
of the glassy form proton conductivity and then the activation barrier (slope) is found to be
approximately the same for all proton conducting glasses meaning that the intercept is the dominant
effect on the behavior of the glass and the only variable controlling the intercept is the glass
transition
Leveraging the configurational entropyrsquos relationship with the glass transition along with
topological constraint theory we can write an expression for the glass transition and as such gain
predictive power for proton conductivity To predict the glass transition temperature (Tg) we start
with the Adam Gibbs model of viscosity[53] Topological constraint theory then states that the
degrees of freedom of the network is proportional to the configurational entropy[23] [56] [57]
Using the liquid state definition of the glass transition (1210 = Pa s) and the well-known infinite
temperature of viscosity we can then write
1
1493 lng
BT
kf=
(125)
Where f is the degrees of freedom Letting the constants be equal to A and rewriting the expression
in terms of constraints we arrive at a predictive formula for the glass transition[109]
3
g
x c x
x
AT
m n=
minus (126)
Where nc is the number of constraints provided by each component at the glass transition associated
with each component x and their molar fraction mx The value of the constraints was then linearly
parameterized to the glass transition data from literature[7] [230] [231] This is not the most
explicit approach to counting constraints however it is the most convenient when considering
large phase spaces being optimized over This glass transition temperature is then converted to
conductivity using the empirical translation shown in Figure 41
161
The other term in our cost function is that of stability The question of what forms glass is
a notorious question that goes back to the first serious days of research into materials It has been
shown that no one metric is a good predictor and no metric is universal However one must be used
to rule out bad compositions from the start To do this a Random Forest method is implemented
Though this may fail in edge cases it will at least help identify the right area to explore A
comprehensive review of random forest methods and machine learning methods for prediction of
glass properties can be found elsewhere[3] [4] [225] The amount of information previously
obtained about what forms glass in the compositional family we will be working on is not enough
to use random forest methods on just the fraction of each phase In order to circumvent this issue
we chose four physical parameters that would be used as features for the predictions
bull Total mol oxide modifier This was chosen because it is well known that modifiers break
up the network and most glass forming theories are related to the network percentages We
considered the oxide versions of H Na Ba Sr and Ca as the modifiers
bull Total mol network former Similarly it is well known the network influences glass
forming We considered B Ge and P as the network forming cations
bull Mean cation charge Though there has been previous work on this topic it is rarely
considered as a lsquonormalrsquo predictor of glass forming capability It is included here because
these glasses are mainly invert and as such a higher field strength will increase the cohesive
nature of the network
bull Entropy of mixing This is an important parameter to consider with so many components
it may be stabilized by the entropy of mixing The entropy of mixing (Sm) was calculated
using the Gibbs entropy
The Gibbs entropy is given by
lnm x
x
xS k m m= minus (127)
The random forest model consisted of 100 trees with a maximum depth of 3 The results of both
the random forest model the topological predictions the optimization and the resultant glass are
shown in the results section The optimization was limited to the range of each component that has
been explored experimentally as well as an additional constraint of at least 5 mol rare earth oxide
and at least 5 mol of boron oxide or germania this is justified elsewhere[231]
162
Figure 41 The relationship between the glass transition and the proton conductivity This is justified
two ways one through the relationship of the entropy of diffusion and glass formation (the Adam-
Gibbs model) and through the fact that water is known to depress the glass transition
163
To explain the results we will first start with the models listed to confirm their validity
The first model is the prediction of the glass transition temperature Figure 42 shows the accuracy
of the glass transition prediction as well as the relative contribution of each component to the glass
transition In this figure any component that has a number of constraints greater then PO52 will
increase the glass transition and as such decrease the conductivity Thus it is preferable to construct
a glass with only components whose constraints at Tg are less than PO52 as long as the glass
remains stable Interestingly whether the data includes only the glass transitions of protonated glass
or all glasses determines the role of La Barium and sodium both decrease the glass transition
playing the role of the more traditional modifier however they do provide some rigidity to the
network Protons have a net negative effect which we believe to be a result of the APS process
where they are introduced as the sodium is leaving so this negative effect corresponds to a breaking
of the rigidity that exists It is also interesting to note that adding a boron phase to the sample
increases the glass transition though not as much as some of smaller alkaline earth samples
assumingly due to their increased field strength The worst network former appears to be GeO2
To calibrate the accuracy of the random forest model a confusion matrix is shown It shows
that the RF always predict that a glass will be a glass however there is some error when it forms a
crystal with 23 of the time being misidentified Though random forest methods are empirical we
can derive some physical meaning by looking at the relative importance of each feature From the
random forest study of stability we are able to rank the relative importance of each feature used
The mean cation charge mixing entropy and percent network formers are all of considerable
importance while the percent modifiers is less so
Once these models were developed and checked the reliability of the cost function is
confirmed After which the minimization of the cost function was performed and one glass was
chosen as optimal Boron was found to be preferential in the glass but was limited since the
compositional space was bound by previous samples (an additional sample is undergoing
164
characterization where the boron content was slightly increased) The optimized sample when
synthesized had some visible nucleation so the optimization while redone with limiting the
component maxima to just the limits of those where compositions in which APS was successfully
performed This resulted in the OP composition The three samples currently are undergoing
experimental characterization are listed in Table 11
165
Table 11 The compositions synthesized in this work These compositions were predicted by
minimizing the cost function described in Eq (123) OP is the variant that was melted after OP
partially crystallized B-OP appeared to have surface nucleation in some spots but was cut and
removed before APS treatment
Name PO52 NaO12 BaO LaO32 GeO2 BO32
OP 44 36 10 5 3 2
OP 49 36 5 5 3 2
B-VOP 44 36 10 5 1 4
166
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine the glass
forming region Over top the constraints at the glass transition provided by each oxide species is
listed Since the objective is to decrease Tg while staying in the glass forming region we will
attempt to minimize use of elements that increase the glass transition temperature (nc gt 17)
167
Chapter 8
Conclusions
The rapid rate of new information in the current scientific climate and the infinite
variability of glass both stand as both unique challenges and opportunities In this dissertation we
have divided the influence of energy landscapes on the effects of properties into two independent
spaces the compositional and the thermal history dimensions By building models that succinctly
and accurately describe the dynamics of different hyper-coordinate (composition thermal history
crystallinity) changes the feasibility of designing new glasses for the challenges of the 21st century
is obtainable
Before designing new models the current state-of-the-art models need to be implemented
and understood To reach this end two softwares based on previous work were created RelaxPy
and ExplorerPy RelaxPy is an implementation of the MAP model and though the MAP model is
powerful when predicting the dynamics of the glass RelaxPy ultimately showed that fictive
temperature is insufficient at capturing the underlying physics and is intensive to parameterize
ExplorerPy was created to standardize the approach to mapping energy landscapes Energy
landscapes served as the key method of understanding the deeper dynamics of glasses glass-
ceramics and liquids
The thermal history dependence of glass with respect to both relaxation and crystallization
has been incorporated into the existence of new models called ldquotoy landscapesrdquo The toy landscapes
have built upon the physics of previous models such as the MAP model however due to the lower
parametrization cost and the built-in increased physicality toy landscapes pose as a tool to increase
our understanding and speed-up the rate of new glass discoveries This tool can deal with the
complexity of the higher dimensional spaces due to crystallization and relaxation without
assumptions concerning fictive temperature
168
The compositional dimensions are the remaining dimensions that must be optimized when
designing a glass or glass-ceramic Building on previous work models have been developed that
enable predictions of Youngrsquos modulus surface reactivity and ionic conductivity However there
are additional properties that are needed and for those we have proposed novel machine learning
approaches This dissertation has not enabled the design of glass for every application However
it lays the groundwork and approaches to design glasses for societyrsquos growing needs Presented as
well are the methods used to design glasses that could satisfy the requirements for solid state glass
electrolytes and hydrogen fuel cells All of these techniques together promise to be a powerful new
framework to build the glasses of the future
169
References
[1] A K Varshneya and J C Mauro Fundamentals of Inorganic Glasses 3rd ed Elsevier
2019
[2] J C Mauro A Tandia K D Vargheese Y Z Mauro and M M Smedskjaer
ldquoAccelerating the Design of Functional Glasses through Modelingrdquo Chem Mater vol
28 no 12 pp 4267ndash4277 2016 doi 101021acschemmater6b01054
[3] J C Mauro ldquoDecoding the glass genomerdquo Curr Opin Solid State Mater Sci pp 1ndash7
2017 doi 101016jcossms201709001
[4] H Liu Z Fu Yang X Xu and M Bauchy ldquoMachine learning for glass science and
engineering A reviewrdquo J Non Cryst Solids no March p 119419 2019 doi
101016jjnoncrysol201904039
[5] M M Smedskjaer C Hermansen and R E Youngman ldquoTopological engineering of
glasses using temperature-dependent constraintsrdquo MRS Bull vol 42 no 01 pp 29ndash33
2017 doi 101557mrs2016299
[6] J C Mauro and M M Smedskjaer ldquoStatistical mechanics of glassrdquo J Non Cryst Solids
vol 396ndash397 pp 41ndash53 2014 doi 101016jjnoncrysol201404009
[7] T Omata et al ldquoProton transport properties of proton-conducting phosphate glasses at
their glass transition temperaturesrdquo Phys Chem Chem Phys vol 21 no 20 pp 10744ndash
10749 2019 doi 101039c9cp01502g
[8] Z A Grady C J Wilkinson C A Randall and J C Mauro ldquoEmerging Role of Non-
crystalline Electrolytes in Solid-State Battery Researchrdquo Front Energy Res vol 8 p
218 2020 doi 103389fenrg202000218
[9] R C Welch et al ldquoDynamics of glass relaxation at room temperaturerdquo Phys Rev Lett
170
vol 110 no 26 p 265901 2013 doi 101103PhysRevLett110265901
[10] M Montazerian E D Zanotto and J C Mauro ldquoModel-driven design of bioactive
glasses from molecular dynamics through machine learningrdquo Int Mater Rev vol 0 no
0 pp 1ndash25 2019 doi 1010800950660820191694779
[11] E D Zanotto and J C Mauro ldquoThe glassy state of matter Its definition and ultimate
faterdquo J Non Cryst Solids vol 471 pp 490ndash495 2017 doi
101016jjnoncrysol201705019
[12] R G Palmer ldquoBroken ergodicityrdquo Adv Phys vol 31 no 6 pp 669ndash735 1982 doi
10108000018738200101438
[13] J C Mauro P Gupta and R J Loucks ldquoContinuously broken ergodicityrdquo J Chem
Phys vol 126 no 18 p 184511 2007 doi 10106312731774
[14] M Micoulaut ldquoRelaxation and physical aging in network glasses A reviewrdquo Reports
Prog Phys vol 79 no 6 p 066504 2016 doi 1010880034-4885796066504
[15] D J Wales and H A Scheraga ldquoGlobal optimization of clusters crystals and
biomoleculesrdquo Science (80- ) vol 285 no 5432 pp 1368ndash1372 1999 doi
101126science28554321368
[16] J C Mauro R J Loucks and P Gupta ldquoFictive temperature and the glassy staterdquo J
Am Ceram Soc vol 92 no 1 pp 75ndash86 2009 doi 101111j1551-2916200802851x
[17] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 2009 doi 10106313152432
[18] F H Stillinger and T A Weber ldquoHidden structure in liquidsrdquo Phys Rev A vol 25 no
2 pp 978ndash989 1982 doi 101103PhysRevA25978
[19] F H Stillinger ldquoSupercooled liquids glass transitions and the auzmann paradoxrdquo J
Chem Phys vol 88 no 12 pp 7818ndash7825 1988 doi 1010631454295
171
[20] S Neelamraju C Oligschleger and J C Schoumln ldquoThe threshold algorithm Description of
the methodology and new developmentsrdquo J Chem Phys vol 147 no 15 p 152713
2017 doi 10106314985912
[21] D Prada-Gracia J Goacutemez-Gardentildees P Echenique and F Falo ldquoExploring the free
energy landscape From dynamics to networks and backrdquo PLoS Comput Biol vol 5 no
6 p e1000415 2009 doi 101371journalpcbi1000415
[22] E Pollak A Auerbach and P Talknerz ldquoObservations on rate theory for rugged energy
landscapesrdquo Biophys J vol 95 no 9 pp 4258ndash4265 2008 doi
101529biophysj108136358
[23] G G Naumis ldquoEnergy landscape and rigidityrdquo Phys Rev E vol 71 p 026114 2005
doi 101103PhysRevE71026114
[24] S J Benkovic G G Hammes and S Hammes-Schiffer ldquoFree-energy landscape of
enzyme catalysisrdquo Biochemistry vol 47 no 11 pp 3317ndash3321 2008 doi
101021bi800049z
[25] M Roca B Messer D Hilvert and A Warshel ldquoOn the relationship between folding and
chemical landscapes in enzyme catalysisrdquo Proc Natl Acad Sci U S A vol 105 no 37
pp 13877ndash13882 2008 doi 101073pnas0803405105
[26] P Li G Henkelman J A eith and J Johnson ldquoElucidation of aqueous solvent-
mediated hydrogen-transfer reactions by ab initio molecular dynamics and nudged elastic-
band studies of NaBH4 hydrolysisrdquo J Phys Chem C vol 118 no 37 pp 21385ndash21399
2014 doi 101021jp507872d
[27] S P Niblett M Biedermann D J Wales and V De Souza ldquoPathways for diffusion in
the potential energy landscape of the network glass former SiO2rdquo J Chem Phys vol
147 no 15 p 152726 2017 doi 10106315005924
[28] Y Yu M Wang M M Smedskjaer J C Mauro G Sant and M Bauchy
172
ldquoThermometer Effect Origin of the Mixed Alkali Effect in Glass Relaxationrdquo Phys Rev
Lett vol 119 no 9 pp 1ndash5 2017 doi 101103PhysRevLett119095501
[29] Y Yu M Wang D Zhang B Wang G Sant and M Bauchy ldquoStretched Exponential
Relaxation of Glasses at Low Temperaturerdquo Phys Rev Lett vol 115 no 16 p 165901
2015 doi 101103PhysRevLett115165901
[30] Steve Plimpton ldquoFast Parallel Algorithms for Short-Range Molecular Dynamicsrdquo J
Comput Phys vol 117 no 1 pp 1ndash19 1995 doi
httpsdoiorg101006jcph19951039
[31] N Mousseau and G T Barkema ldquoTraveling through potential energy landscapes of
disordered materials The activation-relaxation techniquerdquo Phys Rev E vol 57 no 2
pp 2419ndash2424 1998 doi 101103PhysRevE572419
[32] F El-Mellouhi N Mousseau and L J Lewis ldquo inetic activation-relaxation technique
An off-lattice self-learning kinetic Monte Carlo algorithmrdquo Phys Rev B vol 78 p
153202 2008 doi 101103PhysRevB78153202
[33] D J Wales ldquoDiscrete path samplingrdquo Mol Phys vol 100 no 20 pp 3285ndash3305 2002
doi 10108000268970210162691
[34] G Henkelman and H Joacutensson ldquoImproved tangent estimate in the nudged elastic band
method for finding minimum energy paths and saddle pointsrdquo J Chem Phys vol 113
no 10 pp 9978ndash9985 2000 doi 10106314961868
[35] D Sheppard P Xiao W Chemelewski D D Johnson and G Henkelman ldquoA
generalized solid-state nudged elastic band methodrdquo J Chem Phys vol 136 p 074103
2012 doi 10106313684549
[36] A Pedone G Malavasi M C Menziani A N Cormack A V and N York ldquoA New
Self-Consistent Empirical Interatomic Potential Model for Oxides Silicates and Silica-
Based Glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
173
[37] A Takada C R A Catlow and G D Price ldquoComputer modelling of B2O3 II Molecular
dynamics simulations of vitreous structuresrdquo J Phys Condens Matter vol 7 no 46 pp
8693ndash8722 1995 doi 1010880953-8984746004
[38] A C T Van Duin S Dasgupta F Lorant and W A Goddard ldquoReaxFF A reactive
force field for hydrocarbonsrdquo J Phys Chem A vol 105 no 41 pp 9396ndash9409 2001
doi 101021jp004368u
[39] J C Mauro and A Varshneya ldquoModel interaction potentials for selenium from ab initio
molecular simulationsrdquo Phys Rev B vol 71 p 214105 2005 doi
101103PhysRevB71214105
[40] J C Mauro R J Loucks J Balakrishnan and S Raghavan ldquoMonte Carlo method for
computing density of states and quench probability of potential energy and enthalpy
landscapesrdquo J Chem Phys vol 126 no 19 2007 doi 10106312733674
[41] Y Z Mauro C J Wilkinson and J C Mauro ldquo ineticPy A tool to calculate long-time
kinetics in energy landscapes with broken ergodicityrdquo SoftwareX vol 11 p 100393
2020 doi 101016jsoftx2019100393
[42] C J Wilkinson Y Z Mauro and J C Mauro ldquoRelaxPy Python code for modeling of
glass relaxation behaviorrdquo SoftwareX vol 7 pp 255ndash258 2018 doi
101016jsoftx201807008
[43] J C Mauro ldquoTopological constraint theory of glassrdquo Am Ceram Soc Bull vol 90 no
4 pp 31ndash37 2011 doi 101039c3ee40810h
[44] J C Mauro D C Allan and M Potuzak ldquoNonequilibrium viscosity of glassrdquo Phys Rev
B vol 80 p 094204 2009 doi 101103PhysRevB80094204
[45] M Cardona R V Chamberlin and W Marx ldquoThe history of the stretched exponential
functionrdquo Ann der Phys vol 16 no 12 pp 842ndash845 2007 doi
101002andp200710269
174
[46] P Grassberger and I Procaccia ldquoThe long time properties of diffusion in a medium with
static trapsrdquo J Chem Phys vol 77 no 1982 pp 6281ndash6284 1982 doi
1010631443832
[47] J C Phillips ldquoMicroscopic aspects of Stretched Exponential Relaxation (SER) in
homogeneous molecular and network glasses and polymersrdquo J Non Cryst Solids vol
357 no 22ndash23 pp 3853ndash3865 2011 doi 101016jjnoncrysol201108001
[48] J C Phillips ldquo ohlrausch explained The solution to a problem that is 1 0 years oldrdquo J
Stat Phys vol 77 no 3ndash4 pp 945ndash947 1994 doi 101007BF02179472
[49] A Q Tool and C G Eichlin ldquoVariations caused in the heating curves of glass caused by
heat treatmentrdquo J Am Ceram Soc vol 14 pp 276ndash308 1931
[50] A Q Tool ldquoRelation Between Inelastic Deformability and Thermal Expansion of Glass in
Its Annealing Rangerdquo J Am Ceram Soc vol 29 no 9 pp 240ndash253 1946 doi
101111j1151-29161946tb11592x
[51] H N Ritland ldquoLimitations of the Fictive Temperature Conceptrdquo J Am Ceram Soc vol
39 no 12 pp 403ndash406 Dec 1956 doi 101111j1151-29161956tb15613x
[52] O S Narayanaswamy ldquoA model of structural relaxation in glassrdquo J Am Ceram Soc
vol 54 no 10 pp 491ndash498 1971 doi 101111j1151-29161971tb12186x
[53] G Adam and J Gibbs ldquoOn the Temperature Dependence of Cooperative Relaxation
Properties in Glass‐Forming Liquidsrdquo J Chem Phys vol 43 no 1 pp 139ndash146 1965
doi 10106311696442
[54] Q Zheng J C Mauro A J Ellison M Potuzak and Y Yue ldquo niversality of the high-
temperature viscosity limit of silicate liquidsrdquo Phys Rev B vol 83 no 21 p 212202
2011 doi 101103PhysRevB83212202
[55] J C Mauro Y Yue A J Ellison P Gupta and D C Allan ldquoViscosity of glass-
forming liquidsrdquo Proc Natl Acad Sci vol 106 no 47 pp 19780ndash19784 2009 doi
175
101073pnas0911705106
[56] G G Naumis ldquoGlass transition phenomenology and flexibility An approach using the
energy landscape formalismrdquo J Non Cryst Solids vol 352 no 42-49 SPEC ISS pp
4865ndash4870 2006 doi 101016jjnoncrysol200601160
[57] P Gupta and J C Mauro ldquoComposition dependence of glass transition temperature
and fragility I A topological model incorporating temperature-dependent constraintsrdquo J
Chem Phys vol 130 no 9 p 094503 2009 doi 10106313077168
[58] C A Angell ldquoRelaxation in liquids polymers and plastic crystals - strongfragile patterns
and problemsrdquo J Non Cryst Solids vol 13 no 31 pp 131ndash133 1991 doi
1010160022-3093(91)90266-9
[59] G S Fulcher ldquoAnalysis of Recent Measurements of the Viscosity of Glassesrdquo J Am
Ceram Soc vol 75 no 5 pp 1043ndash1055 1992 doi 101111j1151-
29161992tb05536x
[60] I Avramov ldquoViscosity in disordered mediardquo J Non Cryst Solids vol 351 no 40ndash42
pp 3163ndash3173 Oct 2005 doi 101016JJNONCRYSOL200508021
[61] X Guo J C Mauro D C Allan and M M Smedskjaer ldquoPredictive model for the
composition dependence of glassy dynamicsrdquo J Am Ceram Soc vol 101 pp 1169ndash
1179 2018 doi 101111jace15272
[62] Doss C J Wilkinson Y Yang H Lee L Huang and J C Mauro ldquoMaxwell
relaxation time for nonexponential α-relaxation phenomena in glassy systemsrdquo J Am
Ceram Soc vol 103 no 6 pp 3590ndash3599 2020 doi 101111jace17051
[63] P Gupta and A R Cooper ldquoTopologically disordered networks of rigid polytopesrdquo J
Non Cryst Solids 1990 doi 1010160022-3093(90)90768-H
[64] J C Phillips and M F Thorpe ldquoConstraint theory vector percolation and glass
formationrdquo Solid State Commun vol 53 no 8 pp 699ndash702 1985 doi 1010160038-
176
1098(85)90381-3
[65] X Feng W Bresser and P Boolchand ldquoDirect Evidence for Stiffness Threshold in
Chalcogenide Glassesrdquo Phys Rev Lett vol 78 no 23 pp 4422ndash4425 1997 doi
101103PhysRevLett784422
[66] W Bresser P Boolchand and P Suranyi ldquoRigidity Percolation and Molecular Clustering
in Network Glassesrdquo Phys Rev Lett vol 56 no 23 pp 2493ndash2496 1986 doi
101103PhysRevLett562493
[67] Y Vaills T Qu M Micoulaut F Chaimbault and P Boolchand ldquoDirect evidence of
rigidity loss and self-organization in silicate glassesrdquo J Phys Condens Matter vol 17
no 32 pp 4889ndash4896 2005 doi 1010880953-89841732003
[68] A irchner and J C Mauro ldquoStatistical Mechanical Model of the Self-Organized
Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologiesrdquo
Front Mater vol 6 p 11 2019 doi 103389fmats201900011
[69] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 Jun 2009 doi 10106313152432
[70] D R Cassar A C P L F de Carvalho and E D Zanotto ldquoPredicting glass transition
temperatures using neural networksrdquo Acta Mater vol 159 pp 249ndash256 Oct 2018 doi
101016JACTAMAT201808022
[71] J C Mauro and R J Loucks ldquoSelenium glass transition A model based on the enthalpy
landscape approach and nonequilibrium statistical mechanicsrdquo Phys Rev B vol 76 no
17 p 174202 2007 doi 101103PhysRevB76174202
[72] C J Wilkinson et al ldquoEnergy Landscape Modeling of Crystal Nucleationrdquo Nat Comput
Mater p Submitted 2020
[73] M E Mc enzie and J C Mauro ldquoHybrid Monte Carlo technique for modeling of crystal
177
nucleation and application to lithium disilicate glass-ceramicsrdquo Comput Mater Sci vol
149 no January pp 202ndash207 2018 doi 101016jcommatsci201803034
[74] M E McKenzie et al ldquoImplicit glass model for simulation of crystal nucleation for glass-
ceramicsrdquo npj Comput Mater vol 4 no 1 pp 1ndash7 2018 doi 101038s41524-018-
0116-5
[75] E D Zanotto and P F James ldquoExperimental tests of the classical nucleation theory for
glassesrdquo J Non Cryst Solids vol 74 no 2ndash3 pp 373ndash394 1985 doi 1010160022-
3093(85)90080-8
[76] Y Yu N M A Krishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[77] I Pignatelli A Kumar M Bauchy and G Sant ldquoTopological control on silicatesrsquo
dissolution kineticsrdquo Langmuir vol 32 pp 4434ndash4439 2016 doi
101021acslangmuir6b00359
[78] C J Wilkinson et al ldquoTopological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phaserdquo J Phys Chem Lett vol 10 pp
3955ndash3960 2019 doi 101021acsjpclett9b01275
[79] Y Zhang A Li B Deng and Hughes ldquoData-driven predictive models for chemical
durability of oxide glass under different chemical conditionsrdquo npj Mater Degrad vol 4
no 1 pp 1ndash11 2020 doi 101038s41529-020-0118-x
[80] N M Anoop Krishnan S Mangalathu M M Smedskjaer A Tandia H Burton and M
Bauchy ldquoPredicting the dissolution kinetics of silicate glasses using machine learningrdquo J
Non Cryst Solids vol 487 no February pp 37ndash45 2018 doi
101016jjnoncrysol201802023
[81] R M Potter J Hoffman and J Hadley ldquoAn update of the equation for predicting the
178
dissolution rate of glass fibers from their chemical compositionsrdquo Inhal Toxicol vol 29
no 4 pp 145ndash146 2017 doi 1010800895837820171321702
[82] B Deng ldquoMachine learning on density and elastic property of oxide glasses driven by
large datasetrdquo J Non Cryst Solids vol 529 no August 2019 p 119768 2020 doi
101016jjnoncrysol2019119768
[83] S Feller N Lower and M Affatigato ldquoDensity as a probe of oxide glass structurerdquo
Phys Chem Glas vol 42 no 3 pp 240ndash246 2001 Accessed Jul 19 2018 [Online]
Available
httpswwwingentaconnectcomcontentsgtpcg200100000042000000034203240
[84] Q Zheng and H Zeng ldquoProgress in modeling of glass properties using topological
constraint theoryrdquo Int J Appl Glas Sci vol 11 no 3 pp 432ndash441 2020 doi
101111ijag15105
[85] M Bauchy M Javad Abdolhosseini Qomi C Bichara F-J Ulm and R J-M Pellenq
ldquoRigidity Transition in Materials Hardness is Driven by Weak Atomic Constraintsrdquo
2015 doi 101103PhysRevLett114125502
[86] Yang X Xu B Yang B Cook H Ramos and M Bauchy ldquoPrediction of Silicate
Glassesrsquo Stiffness by High-Throughput Molecular Dynamics Simulations and Machine
Learningrdquo Accessed Mar 13 201 Online Available
httpsarxivorgpdf190109323pdf
[87] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological constraint model for
the elasticity of glass-forming systemsrdquo J Non Cryst Solids X vol 2 2019 doi
101016jnocx2019100019
[88] C J Wilkinson K Doss D R Cassar R S Welch C B Bragatto and J C Mauro
ldquoPredicting Ionic Diffusion in Glass from Its Relaxation Behaviorrdquo J Phys Chem B vol
124 no 6 pp 1099ndash1103 2020 doi 101021acsjpcb9b10645
179
[89] C Wilkinson and J C Mauro ldquoExplorerpy Mapping the energy landscapes of complex
materialsrdquo SoftwareX vol Submitted 2020
[90] J C Mauro R J Loucks and J Balakrishnan ldquoA simplified eigenvector-following
technique for locating transition points in an energy landscaperdquo J Phys Chem A vol
109 no 42 pp 9578ndash9583 2005 doi 101021jp053581t
[91] Roumlder and D J Wales ldquoAnalysis of the b to b-CR Transition in biquitinrdquo
Biochemistry vol 57 no 43 pp 6180ndash6186 2018 doi 101021acsbiochem8b00770
[92] J C Mauro R J Loucks and J Balakrishnan ldquoSplit-step eigenvector-following
technique for exploring enthalpy landscapes at absolute zerordquo J Phys Chem B vol 110
no 10 pp 5005ndash5011 2006 doi 101021jp056803w
[93] J C Mauro R J Loucks J Balakrishnan and A Varshneya ldquoMapping the potential
energy landscapes of selenium clustersrdquo J Non Cryst Solids vol 353 pp 1268ndash1273
2007 doi 101016jjnoncrysol200609062
[94] B W H van Beest G J Kramer and R A van Santen ldquoForce fields for silicas and
aluminophosphates based on ab initio calculationsrdquo Phys Rev Lett vol 64 no 16 pp
1955ndash1958 Apr 1990 doi 101103PhysRevLett641955
[95] Q Zheng and J C Mauro ldquoViscosity of glass-forming systemsrdquo J Am Ceram Soc vol
100 no 1 pp 6ndash25 2017 doi 101111jace14678
[96] J C Mauro and Y Z Mauro ldquoOn the Prony Series Representation of Stretched
Exponential Relaxationrdquo Physica A vol 506 pp 75ndash87 2018 doi
101016jphysa201804047
[97] E D Zanotto and D R Cassar ldquoThe microscopic origin of the extreme glass-forming
ability of Albite and B2O3rdquo Sci Rep vol 7 no February pp 1ndash13 2017 doi
101038srep43022
[98] P Pedevilla S J Cox B Slater and A Michaelides ldquoCan Ice-Like Structures Form on
180
Non-Ice-Like Substrates The Example of the K-feldspar Microclinerdquo J Phys Chem C
vol 120 no 12 pp 6704ndash6713 2016 doi 101021acsjpcc6b01155
[99] O Bjoumlrneholm et al ldquoWater at Interfacesrdquo Chem Rev vol 116 no 13 pp 7698ndash7726
2016 doi 101021acschemrev6b00045
[100] Reichelt ldquoNucleation and growth of thin filmsrdquo Vacuum vol 38 no 12 pp 1083ndash
1099 Jan 1988 doi 1010160042-207X(88)90004-8
[101] G H Beall ldquoDr S Donald (Don) Stookey (1 1 ndash2014) Pioneering researcher and
adventurerrdquo Front Mater vol 3 no August pp 1ndash8 2016 doi
103389fmats201600037
[102] J C Mauro C S Philip D J Vaughn and M S Pambianchi ldquoGlass science in the
nited States Current status and future directionsrdquo Int J Appl Glas Sci vol 5 no 1
pp 2ndash15 2014 doi 101111ijag12058
[103] J Deubener et al ldquo pdated definition of glass-ceramicsrdquo J Non Cryst Solids vol 501
no January pp 3ndash10 2018 doi 101016jjnoncrysol201801033
[104] X Hao ldquoA review on the dielectric materials for high energy-storage applicationrdquo J Adv
Dielectr vol 03 no 01 p 1330001 2013 doi 101142s2010135x13300016
[105] J Deubener G Helsch A Moiseev and H Bornhoumlft ldquoGlasses for solar energy
conversion systemsrdquo J Eur Ceram Soc vol 29 no 7 pp 1203ndash1210 2009 doi
101016jjeurceramsoc200808009
[106] G S Frankel et al ldquoA comparative review of the aqueous corrosion of glasses crystalline
ceramics and metalsrdquo npj Mater Degrad vol 2 no 1 p 15 2018 doi
101038s41529-018-0037-2
[107] G H Beall ldquoDesign and properties of glass-ceramicsrdquo Annu Rev Mater Sci vol 22
no 1 pp 91ndash119 1992 doi 101146annurevms22080192000515
[108] J Deubener ldquoConfigurational entropy and crystal nucleation of silicate glassesrdquo Phys
181
Chem Glas vol 45 pp 61ndash63 2004
[109] J C Mauro A J Ellison D C Allan and M M Smedskjaer ldquoTopological model for the
viscosity of multicomponent glass-forming liquidsrdquo Int J Appl Glas Sci vol 4 no 4
pp 408ndash413 2013 doi 101111ijag12009
[110] M M Smedskjaer J C Mauro S Sen and Y Yue ldquoQuantitative design of glassy
materials using temperature-dependent constraint theoryrdquo Chem Mater vol 22 no 18
pp 5358ndash5365 2010 doi 101021cm1016799
[111] Y T Sun H Y Bai M Z Li and W H Wang ldquoMachine Learning Approach for
Prediction and Understanding of Glass-Forming Abilityrdquo J Phys Chem Lett vol 8 no
14 pp 3434ndash3439 2017 doi 101021acsjpclett7b01046
[112] Q Zheng et al ldquo nderstanding Glass through Differential Scanning Calorimetryrdquo Chem
Rev vol 119 no 13 pp 7848ndash7939 2019 doi 101021acschemrev8b00510
[113] X Xia I Dutta J C Mauro B G Aitken and F elton ldquoTemperature dependence of
crystal nucleation in BaOmiddot2SiO2 and 5BaOmiddot8SiO2 glasses using differential thermal
analysisrdquo J Non Cryst Solids vol 459 pp 45ndash50 2017 doi
101016jjnoncrysol201612032
[114] V M Fokin A A Cabral R M C V Reis M L F Nascimento and E D Zanotto
ldquoCritical assessment of DTA-DSC methods for the study of nucleation kinetics in
glassesrdquo J Non Cryst Solids vol 356 no 6ndash8 pp 358ndash367 2010 doi
101016jjnoncrysol200911038
[115] D C Van Hoesen X Xia M E Mc enzie and F elton ldquoModeling nonisothermal
crystallization in a BaO∙2SiO2 glassrdquo J Am Ceram Soc no December 2019 pp 2471ndash
2482 2019 doi 101111jace16979
[116] S C C Prado J P Rino and E D Zanotto ldquoSuccessful test of the classical nucleation
theory by molecular dynamic simulations of BaSrdquo Comput Mater Sci vol 161 no
182
January pp 99ndash106 2019 doi 101016jcommatsci201901023
[117] A O Tipeev and E D Zanotto ldquoNucleation kinetics in supercooled Ni 0Ti 0 Computer
simulation data corroborate the validity of the Classical Nucleation Theoryrdquo Chem Phys
Lett vol 735 no August p 136749 2019 doi 101016jcplett2019136749
[118] K F Kelton and A L Greer Nucleation in Condensed Matter Applications in Materials
and Biology 1st ed Pergamon 2010
[119] V M Fokin E D Zanotto N S Yuritsyn and J W P Schmelzer ldquoHomogeneous
crystal nucleation in silicate glasses A 40 years perspectiverdquo J Non Cryst Solids vol
352 no 26ndash27 pp 2681ndash2714 2006 doi 101016jjnoncrysol200602074
[120] A M Rodrigues D R Cassar V M Fokin and E D Zanotto ldquoCrystal growth and
viscous flow in barium disilicate glassrdquo J Non Cryst Solids vol 479 pp 55ndash61 2018
doi 101016jjnoncrysol201710007
[121] A Pedone G Malavasi M C Menziani A N Cormack and U Segre ldquoA new selft-
consistent empirical interatomic potential model for oxides silicates and silica-based
glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
[122] J J Maldonis A D Banadaki S Patala and P M Voyles ldquoShort-range order structure
motifs learned from an atomistic model of a Zr 0Cu4 Al metallic glassrdquo Acta Mater
vol 175 pp 35ndash45 2019 doi 101016jactamat201905002
[123] A irchner S Goyal M E Mc enzie J T Harris and J C Mauro ldquoStatistical
Description of the Thermodynamics of Glass-Forming Liquidsrdquo Physica A vol 559 p
125059 2020 doi 101016jphysa2020125059
[124] D R Cassar ldquoCrystallization Driving Force of Supercooled Oxide Liquidsrdquo Int J Appl
Glas Sci vol 7 no 3 pp 262ndash269 2016 doi 101111ijag12218
[125] V M Fokin A S Abyzov E D Zanotto D R Cassar A M Rodrigues and J W P
Schmelzer ldquoCrystal nucleation in glass-forming liquids Variation of the size of the
183
lsquostructural unitsrsquo with temperaturerdquo J Non Cryst Solids vol 447 pp 35ndash44 2016 doi
101016jjnoncrysol201605017
[126] M Lenoir A Grandjean Y Linard B Cochain and D R Neuville ldquoThe influence of
SiB substitution and of the nature of network-modifying cations on the properties and
structure of borosilicate glasses and meltsrdquo Chem Geol vol 256 no 3ndash4 pp 316ndash325
2008 doi 101016jchemgeo200807002
[127] P Gupta D R Cassar and E D Zanotto ldquoRole of dynamic heterogeneities in crystal
nucleation kinetics in an oxide supercooled liquidrdquo J Chem Phys vol 145 no 21 2016
doi 10106314964674
[128] A M Rodrigues ldquoDiffusion Processes Crystallization and Viscous Flow in Barium
Disilicate Glassrdquo Dissertation 2014
[129] E D Zanotto ldquoThe effects of amorphous phase sepatration on crystal nucleation in baria-
silica and lithia-silica glassesrdquo Dissertation 1982
[130] D Vargheese A Tandia and J C Mauro ldquoOrigin of dynamical heterogeneities in
calcium aluminosilicate liquidsrdquo J Chem Phys vol 132 no 19 p 194501 May 2010
doi 10106313429880
[131] K Deenamma Vargheese A Tandia and J C Mauro ldquoStatistics of modifier distributions
in mixed network glassesrdquo J Chem Phys vol 132 p 24507 2010 doi
10106313429880
[132] M Micoulaut ldquoThe Deep Effect of Topology on Glass Relaxationrdquo Physics (College
Park Md) vol 6 p 72 2013 doi 101103Physics672
[133] M Micoulaut ldquoRigidity and intermediate phases in glasses driven by speciationrdquo Phys
Rev B vol 74 p 184208 2006 doi 101103PhysRevB74184208
[134] M Kodama S Kojima S Feller and M Affatigato ldquoThe occurrence of minima in the
Borate anomaly anharmonicity and fragility in lithium borate glassesrdquo Phys Chem
184
Glas vol 46 no 2 pp 190ndash193 2005
[135] Y Fukawa et al ldquoVelocity of Sound and Elastic Properties of Li2O-B2O3 Glassesrdquo Jpn
J Appl Phys vol 34 p 2570 1995 doi httpsdoiorg101143JJAP342570
[136] M Reiner ldquoThe Deborah Numberrdquo Physics Today vol 17 no 1 p 62 1964 doi
10106313051374
[137] P K Gupta and J C Mauro ldquoThe laboratory glass transitionrdquo J Chem Phys vol 126
no 22 p 2240504 2007 doi 10106312738471
[138] R Richert and C A Angell ldquoDynamics of glass-forming liquids V On the link between
molecular dynamics and configurational entropyrdquo J Chem Phys vol 108 no 21 pp
9016ndash9026 1998 doi 1010631476348
[139] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological Constraint Model for
the Elasticity of Glass-Forming Systemsrdquo J Non Cryst Solids X vol 2 p 100019 2019
doi 101016jnocx2019100019
[140] S Feller S Bista A OrsquoDonovan-Zavada T Mullenbach M Franke and M Affatigato
ldquoPacking in alkali and alkaline earth borosilicate glass systemsrdquo Phys Chem Glas Eur
J Glas Sci Technol Part B vol 50 no 3 pp 224ndash228 2009
[141] P J Bray S Feller G E Jellison and Y H Yun ldquoB10 NMR studies of the structure of
borate glassesrdquo J Non Cryst Solids vol 38ndash39 no PART 1 pp 93ndash98 May 1980 doi
1010160022-3093(80)90400-7
[142] N Andersson and G L Comer ldquoRelativistic Fluid Dynamics Physics for Many Different
Scales Living Reviews in Relativityrdquo Living Rev Relativ vol 10 p 1 2007 Accessed
Feb 18 2019 [Online] Available httpwwwlivingreviewsorglrr-2007-
1httprelativitylivingreviewsorghttpwwwmathssotonacukstaffAnderssonhttpw
wwslueducollegesASphysicsprofscomerhtml
[143] P Ilg and H C Oumlttinger ldquoNonequilibrium relativistic thermodynamics in bulk viscous
185
cosmologyrdquo Phys Rev D - Part Fields Gravit Cosmol vol 61 no 2 p 023510 2000
doi 101103PhysRevD61023510
[144] P Romatschke ldquoRelativistic viscous fluid dynamics and non-equilibrium entropyrdquo Class
Quantum Gravity vol 27 no 2 p 025006 Jan 2010 doi 1010880264-
9381272025006
[145] D C Allan ldquoInverting the MYEGA equation for viscosityrdquo J Non Cryst Solids vol
358 no 2 pp 440ndash442 Jan 2012 doi 101016JJNONCRYSOL201109036
[146] M Potuzak R C Welch and J C Mauro ldquoTopological origin of stretched exponential
relaxation in glassrdquo J Chem Phys vol 135 no 21 p 214502 Dec 2011 doi
10106313664744
[147] L Ding M Thieme S Demouchy C unisch and B J P aus ldquoEffect of pressure and
temperature on viscosity of a borosilicate glassrdquo J Am Ceram Soc vol 101 no 9 pp
3936ndash3946 2018 doi 101111jace15588
[148] R Richert and M Richert ldquoDynamic heterogeneity spatially distributed stretched-
exponential patterns and transient dispersions in solvation dynamicsrdquo Phys Rev E vol
58 no 1 pp 779ndash784 1998 doi 101103PhysRevE58779
[149] P Richet ldquoViscosity and configurational entropy of silicate meltsrdquo Geochim Cosmochim
Acta vol 48 no 3 pp 471ndash483 Mar 1984 doi 1010160016-7037(84)90275-8
[150] P Gupta and J C Mauro ldquoTwo factors governing fragility Stretching exponent and
configurational entropyrdquo Phys Rev E vol 78 no 6 p 063501 2008 doi
101103PhysRevE78062501
[151] Z Zheng J C Mauro and D C Allan ldquoModeling of delayed elasticity in glassrdquo J Non
Cryst Solids vol 500 no August pp 432ndash442 2018 doi
101016jjnoncrysol201808033
[152] R Boumlhmer L Ngai C A Angell and D J Plazek ldquoNonexponential relaxations in
186
strong and fragile glass formersrdquo J Chem Phys vol 99 no 5 pp 4201ndash4209 1993
doi 1010631466117
[153] O Gulbiten J C Mauro and P Lucas ldquoRelaxation of enthalpy fluctuations during sub-
Tg annealing of glassy seleniumrdquo J Chem Phys vol 138 no 24 p 244504 2013 doi
10106314811488
[154] D Sidebottom R Bergman L Boumlrjesson and L M Torell ldquoTwo-step relaxation decay
in a strong glass formerrdquo Phys Rev Lett vol 71 no 14 pp 2260ndash2263 1993 doi
101103PhysRevLett712260
[155] G M Bartenev and V A Lomovskoi ldquoRelaxation time spectra and the peculiarities of
the process of boron anhydride glass transitionrdquo J Non Cryst Solids vol 146 no C pp
225ndash232 1992 doi 101016S0022-3093(05)80495-8
[156] A Heuer ldquoExploring the potential energy landscape of glass-forming systems From
inherent structures via metabasins to macroscopic transportrdquo J Phys Condens Matter
vol 20 no 37 p 373101 2008 doi 1010880953-89842037373101
[157] J C Phillips ldquoStretched exponential relaxation in molecular and electronic gasesrdquo
Reports Prog Phys vol 59 pp 1133ndash1207 1996
[158] P Richet ldquoResidual and configurational entropy Quantitative checks through applications
of Adam-Gibbs theory to the viscosity of silicate meltsrdquo J Non Cryst Solids vol 355
no 10ndash12 pp 628ndash635 2009 doi 101016jjnoncrysol200901027
[159] C J Wilkinson Doss G Palmer and J C Mauro ldquoThe relativistic glass transition A
thought experimentrdquo J Non-Crystalline Solids X vol 2 2019 doi
101016jnocx2019100018
[160] J C Mauro R J Loucks and S Sen ldquoHeat capacity enthalpy fluctuations and
configurational entropy in broken ergodic systemsrdquo J Chem Phys vol 133 no 16 pp
1ndash9 2010 doi 10106313499326
187
[161] J C Mauro R J Loucks and P Gupta ldquoMetabasin approach for computing the
master equation dynamics of systems with broken ergodicityrdquo J Phys Chem A vol 111
no 32 pp 7957ndash7965 2007 doi 101021jp0731194
[162] M Wang N M Anoop Krishnan B Wang M M Smedskjaer J C Mauro and M
Bauchy ldquoA new transferable interatomic potential for molecular dynamics simulations of
borosilicate glassesrdquo J Non Cryst Solids vol 498 no December 2017 pp 294ndash304
2018 doi 101016jjnoncrysol201804063
[163] I Avramov and A Milchev ldquoEffect of disorder on diffusion and viscosity in condensed
systemsrdquo J Non Cryst Solids vol 104 no 2ndash3 pp 253ndash260 Sep 1988 doi
1010160022-3093(88)90396-1
[164] G Scherer Relaxation in Glass and Composites Krieger Publishing Company 1992
[165] L Huang J Nicholas J Kieffer and J Bass ldquoPolyamorphic transitions in vitreous B2O3
under pressurerdquo J Phys Condens Matter vol 20 no 7 2008 doi 1010880953-
8984207075107
[166] W Capps P B Macedo B OrsquoMeara and T A Litovitz ldquoTemperature Dependecne of
the High-Frequency Moduli of Vitreous B2O3rdquo J Chem Phys vol 45 p 3431 1966
doi 107868s0002337x1402002x
[167] D R Cassar A M Rodrigues M L F Nascimento and E D Zanotto ldquoThe diffusion
coefficient controlling crystal growth in a silicate glass-formerrdquo Int J Appl Glas Sci
vol 9 no 3 pp 373ndash382 Jul 2018 doi 101111ijag12319
[168] A R Potter C J Wilkinson S H im and J C Mauro ldquoEffect of Water on
Topological Constraints in Silica Glassrdquo Scr Mater vol 160 pp 48ndash52 2019 doi
101016jscriptamat201809041
[169] D B Asay and S H im ldquoEvolution of the adsorbed water layer structure on silicon
oxide at room temperaturerdquo J Phys Chem B vol 109 pp 16760ndash16763 2005 doi
188
101021jp053042o
[170] M Tomozawa ldquoWater diffusion oxygen vacancy annihilation and structural relaxation in
silica glassesrdquo J Non Cryst Solids vol 179 pp 162ndash169 1994 doi 1010160022-
3093(94)90693-9
[171] S Kapoor R E Youngman K Zakharchuk A Yaremchenko N J Smith and A Goel
ldquoStructural and Chemical Approach toward nderstanding the Aqueous Corrosion of
Sodium Aluminoborate Glassesrdquo J Phys Chem B vol 122 pp 10913ndash10927 2018 doi
101021acsjpcb8b06155
[172] Y Yu N M A rishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[173] Y Yu B Wang M Wang G Sant and M Bauchy ldquoReactive Molecular Dynamics
Simulations of Sodium Silicate Glasses mdash Toward an Improved Understanding of the
Structurerdquo Int J Appl Glas Sci vol 8 no 3 pp 276ndash284 2017 doi
101111ijag12248
[174] E Stolper ldquoThe speciation of water in silicate meltsrdquo Geochim Cosmochim Acta vol
46 no 12 pp 2609ndash2620 Dec 1982 doi 1010160016-7037(82)90381-7
[175] M G Mesko P A Schader and J E Shelby ldquoWater solubility and diffusion in sodium
silicate meltsrdquo Phys Chem Glas vol 43 no 6 pp 283ndash290 2002 doi
101016jsemradonc201010001
[176] M Tomozawa M Takata J Acocella E Bruce Watson and T Takamori ldquoThermal
properties of Na2Omiddot3SiO2 glasses with high water contentrdquo J Non Cryst Solids vol 56
no 1ndash3 pp 343ndash348 Jul 1983 doi 1010160022-3093(83)90491-X
[177] E A Leed and C G Pantano ldquoComputer modeling of water adsorption on silica and
silicate glass fracture surfacesrdquo J Non Cryst Solids vol 325 pp 48ndash60 2003 doi
189
101016S0022-3093(03)00361-2
[178] J C Fogarty H M Aktulga A Y Grama A C T van Duin and S A Pandit ldquoA
reactive molecular dynamics simulation of the silica-water interfacerdquo J Chem Phys vol
132 no 17 p 174704 2010 doi 10106313407433
[179] S H Hahn et al ldquoDevelopment of a ReaxFF Reactive Force Field for NaSiOx Water
Systems and Its Application to Sodium and Proton Self-Diffusionrdquo J Phys Chem C vol
122 no 34 pp 19613ndash19624 2018 doi 101021acsjpcc8b05852
[180] T S Mahadevan W Sun and J Du ldquoDevelopment of Water Reactive Potentials for
Sodium Silicate Glassesrdquo J Phys Chem B vol 123 pp 4452ndash4461 2019 doi
101021acsjpcb9b02216
[181] A C T van Duin A Strachan S Stewman Q Zhang X Xu and W A Goddard III
ldquoReaxFF SiO Reactive Force Field for Silicon and Silicon Oxide Systemsrdquo J Phys
Chem A vol 107 no 19 pp 3803ndash3811 2003 doi 101021jp0276303
[182] M M Smedskjaer J C Mauro R E Youngman C L Hogue M Potuzak and Y Yue
ldquoTopological principles of borosilicate glass chemistryrdquo J Phys Chem B vol 115 no
44 pp 12930ndash12946 2011 doi 101021jp208796b
[183] Q Zheng M Potuzak J C Mauro M M Smedskjaer R E Youngman and Y Yue
ldquoComposition-structure-property relationships in boroaluminosilicate glassesrdquo J Non
Cryst Solids vol 358 pp 993ndash1002 2012 doi 101016jjnoncrysol201201030
[184] K Rompicharla D I Novita P Chen P Boolchand M Micoulaut and W Huff
ldquoAbrupt boundaries of intermediate phases and space filling in oxide glassesrdquo J Phys
Condens Matter vol 20 p 202101 2008 doi 1010880953-89842020202101
[185] M Micoulaut and J C Phillips ldquoRings and rigidity transitions in network glassesrdquo Phys
Rev B vol 67 p 104204 2003 doi 101103PhysRevB67104204
[186] H Liu S Dong L Tang N M A rishnan G Sant and M Bauchy ldquoEffects of
190
polydispersity and disorder on the mechanical properties of hydrated silicate gelsrdquo J
Mech Phys Solids vol 122 pp 555ndash565 Jan 2019 doi 101016JJMPS201810003
[187] A Tilocca N H De Leeuw and A N Cormack ldquoShell-model molecular dynamics
calculations of modified silicate glassesrdquo Phys Rev B - Condens Matter Mater Phys
vol 73 p 104209 2006 doi 101103PhysRevB73104209
[188] J Du and A N Cormack ldquoThe medium range structure of sodium silicate glasses A
molecular dynamics simulationrdquo J Non Cryst Solids vol 349 no 1ndash3 pp 66ndash79 2004
doi 101016jjnoncrysol200408264
[189] N P Bansal and R H Doremus Handbook of glass properties Academic Press 1986
[190] S H Hahn and A C T van Duin ldquoSurface Reactivity and Leaching of a Sodium Silicate
Glass under an Aqueous Environment A ReaxFF Molecular Dynamics Studyrdquo J Phys
Chem C p acsjpcc9b02940 2019 doi 101021acsjpcc9b02940
[191] S P Jaccani and L Huang ldquo nderstanding Sodium Borate Glasses and Melts from Their
Elastic Response to Temperaturerdquo Int J Appl Glas Sci vol 7 no 4 pp 452ndash463 Dec
2016 doi 101111ijag12250
[192] Q Zheng Y Yue and J C Mauro ldquoDensity of topological constraints as a metric for
predicting glass hardnessrdquo Appl Phys Lett vol 111 no 1 p 011907 Jul 2017 doi
10106314991971
[193] J C Dyre ldquoElastic models for the non-Arrhenius relaxation time of glass-forming
liquidsrdquo in AIP Conference Proceedings 2006 vol 832 pp 113ndash117 doi
10106312204470
[194] Y T Cheng and C M Cheng ldquoRelationships between hardness elastic modulus and the
work of indentationrdquo Appl Phys Lett vol 73 no 5 pp 614ndash616 Jul 1998 doi
1010631121873
[195] A Makishima and J D Mackenzie ldquoDirect calculation of Youngrsquos modulus of glassrdquo J
191
Non Cryst Solids vol 12 no 1 pp 35ndash45 1973 doi 1010160022-3093(73)90053-7
[196] G Pilania C Wang X Jiang S Rajasekaran and R Ramprasad ldquoAccelerating materials
property predictions using machine learningrdquo Sci Rep vol 3 pp 1ndash6 2013 doi
101038srep02810
[197] H He and M Thorpe ldquoElastic properties of glassesrdquo Phys Rev Lett vol 54 no 19 pp
2107ndash2110 1985 Accessed Aug 29 2018 [Online] Available
httpsjournalsapsorgprlpdf101103PhysRevLett542107
[198] M Thorpe ldquoElastic properties of glassesrdquo Cambridge Arch 1985 doi 101557PROC-
61-49
[199] J C Phillips ldquoChemical Bonding Internal Surfaces and the Topology of Non-Crystalline
Solidsrdquo Phys status solidi vol 101 no 2 pp 473ndash479 Oct 1980 doi
101002pssb2221010204
[200] M M Smedskjaer J C Mauro and Y Yue ldquoPrediction of glass hardness using
temperature-dependent constraint theoryrdquo Phys Rev Lett vol 105 no 11 2010 doi
101103PhysRevLett105115503
[201] J D Musgraves J Hu and L Calvez Springer Handbook of Glass Springer US 2021
[202] C Hermansen X Guo R E Youngman J C Mauro M M Smedskjaer and Y Yue
ldquoStructure-topology-property correlations of sodium phosphosilicate glassesrdquo J Chem
Phys vol 143 no 6 p 064510 Aug 2015 doi 10106314928330
[203] L Wondraczek et al ldquoTowards ltrastrong Glassesrdquo Adv Mater vol 23 no 39 pp
4578ndash4586 Oct 2011 doi 101002adma201102795
[204] D Ravaine ldquoGlasses as solid electrolytesrdquo J Non Cryst Solids vol 38ndash39 no PART 1
pp 353ndash358 1980 doi 1010160022-3093(80)90444-5
[205] D Ravaine and J L Souquet ldquoA thermodynamic approach to ionic conductivity in oxide
glasses Part 1 Correlation of the ionic conductivity with the chemical potential of alkali
192
oxide in oxide glassesrdquo Phys Chem Glas vol 18 no 2 pp 27ndash31 1977
[206] J Swenson and L Boumlrjesson ldquoCorrelation between Free Volume and Ionic Conductivity
in Fast Ion Conducting Glassesrdquo Phys Rev Lett vol 77 no 17 pp 3569ndash3572 1996
doi 101103PhysRevLett773569
[207] C B Bragatto A C M Rodrigues and J L Souquet ldquoDissociation Equilibrium and
Charge Carrier Formation in AgI-AgPO3 Glassesrdquo J Phys Chem C vol 121 no 25 pp
13507ndash13514 2017 doi 101021acsjpcc7b02477
[208] S W Martin R Christensen G Olson J ieffer and W Wang ldquoNew Interpretation of
Na Ion Conduction in and the Structures and Properties of Sodium Borosilicate Mixed
Glass Former Glassesrdquo J Phys Chem C vol 140 pp 6343ndash6352 2019 doi
101021acsjpcc8b11735
[209] O L Anderson and D A Stuart ldquoCalculation of Activation Energy of Ionic Conductivity
in Silica Glasses by Classical Methodsrdquo J Am Ceram Soc vol 37 no 12 pp 573ndash580
1954 doi 101111j1151-29161954tb13991x
[210] R S Welch C J Wilkinson J C Mauro and C B Bragatto ldquoCharge Carrier Mobility
of Alkali Silicate Glasses Calculated by Molecular Dynamicsrdquo Front Mater vol 6 p
121 May 2019 doi 103389fmats201900121
[211] Y J Zhang ldquoEntropy and ionic conductivityrdquo Phys A Stat Mech its Appl vol 391 no
19 pp 4470ndash4475 2012 doi 101016jphysa201204021
[212] J C Dyre ldquoOn the mechanism of glass ionic conductivityrdquo J Non Cryst Solids vol 88
no 2ndash3 pp 271ndash280 1986 doi 101016S0022-3093(86)80030-8
[213] M Steve ldquoIonic Conduction in Phosphate Glassesrdquo JAmCeramSoc vol 74 no 8 pp
1767ndash1784 1991 doi 101111j1151-29161991tb07788x
[214] C T Moynihan A J Easteal J Wilder and J Tucker ldquoDependence of the Glass
Transition Temperature on Heating and Cooling Raterdquo J Phys Chem vol 78 no 26 pp
193
2673ndash2677 1974 Accessed Jul 13 2018 [Online] Available
httpspubsacsorgsharingguidelines
[215] Ito C T Moynihan and C A Angell ldquoThermodynamic determination of fragility in
liquids anda fragile-to-strong liquid transition inwaterrdquo Lett to Nat vol 398 no April
p 492 1999
[216] L Ngai and S W Martin ldquoCorrelation between the activation enthalpy and
Kohlrausch exponent for ionic conductivity in oxide glassesrdquo Phys Rev B vol 40 no
15 pp 10550ndash10556 1989 doi 101103PhysRevB4010550
[217] M Potuzak X Guo M M Smedskjaer and J C Mauro ldquoAre the dynamics of a glass
embedded in its elastic propertiesrdquo J Chem Phys vol 138 no 12 2013 doi
10106314730525
[218] A L Agapov and A P Sokolov ldquoDecoupling ionic conductivity from structural
relaxation A way to solid polymer electrolytesrdquo Macromolecules vol 44 no 11 pp
4410ndash4414 2011 doi 101021ma2001096
[219] S V Nemilov ldquoA Structural Investigation of Glasses in the B2O3ndashNa2O System by the
Viscosimetric Methodrdquo Izv Akad Nauk SSSR Neorg Mater vol 2 no 2 pp 349ndash359
1966
[220] J P Poole ldquoLow‐Temperature Viscosity of Alkali Silicate Glassesrdquo J Am Ceram Soc
vol 32 no 7 pp 230ndash233 1949 doi 101111j1151-29161949tb18952x
[221] R noche D B Dingwell F A Seifert and S L Webb ldquoNon-linear properties of
supercooled liquids in the system Na2O SiO2rdquo Chem Geol vol 116 pp 1ndash16 1994 doi
1010160009-2541(94)90154-6
[222] W Martinsen ldquoSelected properties of sodium silicate glasses and their structural
significancerdquo Dissertation 1969
[223] C Hermansen J C Mauro and Y Yue ldquoA model for phosphate glass topology
194
considering the modifying ion sub-networkrdquo J Chem Phys vol 140 no 15 2014 doi
10106314870764
[224] M L F Nascimento and E Dutra Zanotto ldquoDoes viscosity describe the kinetic barrier for
crystal growth from the liquidus to the glass transitionrdquo J Chem Phys vol 133 no 17
p 174701 2010 doi 10106313490793
[225] D Denisko and M M Hoffman ldquoClassification and interaction in random forestsrdquo Proc
Natl Acad Sci U S A vol 115 no 8 pp 1690ndash1692 2018 doi
101073pnas1800256115
[226] G Varney C Dema B E Gul C J Wilkinson and Akgun ldquo se of machine learning
in CARNA proton imagerrdquo in Progress in Biomedical Optics and Imaging - Proceedings
of SPIE 2019 vol 10948 doi 101117122512565
[227] ldquoMDLregSciGlass-6 MDL Information Systems San Leandro CArdquo 2003
[228] D R Cassar ldquoViscNet Neural Network for predicting the fragility index and the
temperature-dependency of viscosityrdquo pp 1ndash33 2020
[229] D J L Brett A Atkinson N P Brandon and S J Skinner ldquoIntermediate temperature
solid oxide fuel cellsrdquo Chem Soc Rev vol 37 no 8 pp 1568ndash1578 2008 doi
101039b612060c
[230] T Ishiyama S Suzuki J Nishii T Yamashita H awazoe and T Omata ldquoProton
conducting tungsten phosphate glass and its application in intermediate temperature fuel
cellsrdquo Solid State Ionics vol 262 pp 856ndash859 2014 doi 101016jssi201310055
[231] T Yamaguchi et al ldquoProton-conducting phosphate glass and its melt exhibiting high
electrical conductivity at intermediate temperaturesrdquo J Mater Chem A vol 6 no 46 pp
23628ndash23637 2018 doi 101039C8TA08162J
VITA
Collin Wilkinson was born in Mt Carroll IL He attended Coe College as an undergraduate and
studied glass resulting in a physics bachelorrsquos degree in 201 He worked with Dr Ugur Akgun
and Dr Steve Feller on modeling different key properties of glass propertiesresponses to external
stimuli Collin joined the group of Dr John Mauro at Pennsylvania State University in 2018
List of publications written by first author (or co-first author) Collin while at Penn State
1 CJ Wilkinson JC Mauro Explorerpy Mapping the Energy Landscapes of Complex
Materials SoftwareX Submitted
2 CJ Wilkinson DR Cassar AV DeCeanne KA Kirchner ME McKenzie ED Zanotto and
JC Mauro Energy Landscape Modeling of Crystal Nucleation NPJ Computational
Materialis Submitted
3 CJ Wilkinson K Doss O Gulbiten DC Allan JC Mauro Fragility and Temperature
Dependence of Stretched Exponential Relaxation in Glass-Forming Systems JACERS
Submitted
4 CJ Wilkinson JC Mauro Comment on ldquoThe Fragility of Alkali Silicate Glass Melts Part
of a niversal Topological Patterrdquo by DL Sidebottom JNCS (2020) ( 2 ) 11
5 CJ Wilkinson K Doss DR Cassar RS Welch CB Bragatto JC Mauro Predicting Ionic
Diffusion in Glass From its Relaxation Behavior Journal of Physical Chemistry B (2020)
124 (6) 1099-1103
6 Z Ding CJ Wilkinson J Zheng Y Lin H Liu J Shen SH Kim Y Yue J Rent CJ Mauro
Q Zheng Topological Understanding of the Mixed Alkaline Earth Effect in Glass JNCS
(2020) (526) 119696
7 CJ Wilkinson AR Potter RS Welch CB Bragatto Q Zheng M Bauchy M Affatigato SA
Feller JC Mauro Topological Origins of Mixed Alkali Effect in Glass Journal of Physical
Chemistry B (2019) 123 (34) 7482
8 CJ Wilkinson K Doss SH Hahn Nathan Keilbart AR Potter NJ Smith I Dabo ACT Van
Duin SH Kim JC Mauro Topological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phase JPCL (2019) 10 (14) 3955
9 CJ Wilkinson K Doss G Palmer JC Mauro The Relativistic Glass Transition A Through
Experiment JNCSX (2019) 100018
10 CJ Wilkinson Q Zheng L Huang JC Mauro Topological Constraint Model for the
Elasticity of Glass-Forming Systems JNCSX (2019) 100019
11 AR Potter CJ Wilkinson SH Kim JC Mauro Effect of Water on Toplogical Constraints
in Silica Glass Scripta Materialia (2019) (160) 48-52
12 CJ Wilkinson E Pakhomenko MR Jesuit AV DeCeanne B Hauke M Packard SA Feller
JC Mauro Topological Constraint Model of Alkali Tellurite Glasses JNCS (2018) (502)
172
13 CJ Wilkinson YZ Mauro JC Mauro RelaxPy Python code for Modeling of Glass
Relaxation Behavior SoftwareX (2018) (7) 255
vi
List of Figures
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists
the equilibrium liquid which as it is quenched can either become a super cooled
liquid or crystallize Crystallization causes a discontinuity in the volume The super-
cooled liquid upon further quenching departs from equilibrium and transitions into
the glassy state Reproduced from Fundamentals of Inorganic Glass Science with
permission from the author1 5
Figure 2 The schematic for the flow of the program Beginning in the top right corner
and running until the condition in the pink box is satisfied Yellow diamonds
represent checks and blue operations 25
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above
and then plotted using PyConnect The plot is a disconnectivity graph where each
terminating line represents an inherent structure and tracing where two lines meet
describes the activation barrier The potentials are taken from the BKS potential 94 27
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-
axis being an arbitrary phase space and the y-axis being the potential energy
calculated from the Pedone et al potentials36 The colors are indicative of the
crystallinity where the blue basin is the initial starting configuration The landscape
shows the lowering of energy associated with partially crystallizing the sample (B)
The energy landscape relationship between the cutoff for crystalline and super-
cooled liquid states for 256 atoms is shown This shows a clear drastic energy
change occurring around the cutoff value of 10 Aring 37
Figure 5 An example interfacial structure between the crystalline phase on the left and
the last sequential SCLglass phase on the right for a barium disilicate system The
gray atoms are barium silicon is shown in red and blue represents oxygen 43
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a
function of temperature for each system size The experimental values for kinetics
and thermodynamics come from ref125 and from the heat capacity data taken from 126 respectively It is clear to see that driving force shows good agreement across all
systems however the kinetics terms only converge for the 256 and 512 atoms
systems (Bottom) The fit used to calculate the interfacial energy as a function of
temperature 45
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted
using the model presented in this work The data referenced can be found in refs 113128129 47
Figure 8 The surface energy with respect to temperature for the work presented here 50
Figure 9 An Angell diagram created using the MYEGA expression 55 With a very strong
glass (m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) 17134135
vii
The infinite temperature limit is from the work of Zheng et al54 and the glass
transition temperature is from the Angell definition 54
Figure 10 The relativistic glass transition temperature for B2O3 glass 57
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different
fractions of light speed All of the viscosities approaching the universal temperature
limit for viscosity 59
Figure 12 The modulus needed to satisfy the condition for the glass transition 61
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at
v=044c (Bottom) Viscosity plot showing a dramatic reduction of the viscosity as
the observer approaches the speed of light 63
Figure 14 b predicted and from literature showing good agreement with a total root-
mean-square error of 01 The fit for organic systems is given by 66 ln Kln = minus
and for inorganic systems by 75ln Kln = minus 72
Figure 15 The equilibrium model proposed with the experimental points showing good
agreement between the experimentally measured data points and the equilibrium
derived model RMSE was 002 for Corningcopy JadeTM (A) and less than 001 for
SG80 (B) (C) The model fit for B2O3 experimental data154155 The fragility and glass
transition temperature of the B2O3 are taken from the work of Mauro et al17 75
Figure 16 The stretching exponent calculated as described in the text for a Gaussian
distribution of barriers This plot shows that the distribution of barriers has a large
effect on the stretching exponent A Tg cannot be described since there is no
vibrational frequency included in the model though the glass transition temperature
should be the same for all distributions since the mean relaxation time is the same
for all distributions at all temperatures The deviation is given in ln eV units 78
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent
Each color designates one term in the series (Bottom) The output from RelaxPy
v20 showing the stretching exponent effects on the relaxation prediction of
Corningcopy JadeTM glass Each quadrant shows one property that is of interest for
relaxation experiments In particular it is interesting to see the dynamics of the
stretching exponent during a typical quench 80
Figure 18 The viscosity (left) and landscape (right) predictions for three common
systems The first system is newly calculated in this work while the latter two come
from our previous works7289 It is seen that the viscosity predicted from the AG
model is very accurately able to reproduce the experimental viscosity curves from
the MYEGA model The last system is a potential energy landscape while the others
are enthalpy landscapes 87
Figure 19 The configurational entropy comparisons between the three major viscosity
models which validates the main underlying assumption of the MYEGA model The
viii
VFT and AM are unable to capture the physics of configurational entropy therefore
ruling 89
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the
configurational entropy from the enthalpy landscape with the barrier 00155 eV
(compared to the shear barrier of 00149 eV) and the infinite limit allowed to vary
(10-263 Pamiddots for bulk viscosity) In Figure 18 the configurational entropy is
confirmed for the shear viscosity thus confirming the AG for both shear and bulk
viscosities Sidebottom data are from Ref 154 90
Figure 21 (A) The histogram for the energy minima for each of the test systems showing
a good fit with the log normal distribution This distribution will then be a valid
form to calculate the enthalpy distribution of the model presented in the next
session (B) The configurational entropy from the model showing the accuracy of
the scaling of the entropy predicted by the MYEGA model The S value was fit
for each system 93
Figure 22 (A) Comparison between the randomized method (histogram) and the
deterministic method (vertical lines) showing good agreement between the
maximum in the histogram and the value predicted by the deterministic technique
validating the approach It is worth noting that the 100-basin distribution is a very
wide distribution where the total number of basins is less than the number of points
used in the calculation This is done for a variable number of basins with the number
of basins shown in the legend (B) The dependence of fragility and the glass
transition temperature vs the distribution of states and the number of basins 99
Figure 23 The driving forces for different example glasses calculated using a
combination of the MAP model RelaxPy and the toy landscape model The
parameters for each glass can be found in Table 4 102
Figure 24 The predicted outputs from the toy landscape method showing predictions of
the enthalpy and entropy under a standard quench for barium disilicate This
prediction does not require fictive temperature or any such assumptions about the
evolution of the non-equilibrium behavior 105
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume
in nucleation is assumed to be on the order of one cubic angstrom while the a
parameter in growth is assumed to be around one nm both are in good agreement
for estimates in literature The values for the orange points are taken from these
works128167 107
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the
hydration models Color scheme Si atom (ivory) O atom (red) and Na atom (blue)
The z-axis is elongated to allow space for an insert of water 114
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the
top surface in contact with water is shown here 116
ix
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example
of the electronic-structure DFT calculation using semilocal exchange-correlation
functionals finding the binding energy of a water lsquopixelrsquo to the surface 119
Figure 29 Example contour surface showing the average coordination per atom on the
glass surface for the first run at 300 ps 121
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of
constraints for surface atoms at the local pixel Results show a distinct maximum in
which there is a near hydrophilic-hydrophobic transition of the surface The error
bars represent the standard deviation A second system with 1500 atoms was
performed to show convergence of the ReaxFF MD results 123
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values
of the model predictions are 641 GPa for constraint density 313 GPa for free
energy density and 774 GPa for angular constraint density 128
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from
minimum to maximum error The free energy density model gives the most accurate
results The constraint density has a RMSE of 61 GPa the angular density has a
RMSE of 20 GPa and the energy density has a RMSE of 59 GPa 129
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and
experiment for 10 Na2O 90 B2O3 Using the previously fitted onset temperatures
the only free parameters are then the vibrational frequency and the heating time in
which their product was fitted to be 14000 Where each dip in the modulus
corresponds to a constraint no longer being rigid as heated through each onset The
onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The
data was fit using a least-squares method and the resultant fit is shown as the
calculated method The fit has an R2 of 094 (B) The contribution from each
constraint to the overall modulus 131
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in
Fig 3) and experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of
093 and (B) xLi2Omiddot(100-x)B2O3 glasses with an R2 of 0986 133
Figure 35 The structure for the initial minimum energy configuration showing the boron
(blue) network with interconnecting oxygens (red) and the interstitial sodium ions
(yellow) 139
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and
sodium is ivory The barrier is overestimated compared to experimental data this
could be from several sources of error such as potential fitting thermal history
fluctuations or sampling too few transitions The line is drawn as a guide to the eye 139
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function
of reaction coordinates The middle shows the local deformation around the ion of
x
any atom that moves in between inherent structure mandating a relaxation force
The color shows the degree of deformation 141
Figure 38 Different network formers and the prediction of the activation barrier from our
model compared with activation barriers from literature (A) Sodium silicate
predictions and experimental values241 the error is calculated from the error in the
fragility when fitting the data (B) Lithium phosphate activation energy235 predicted
with topological constraint theory and compared with the experimental values (C)
Predictions over two different systems of alkali borates232 sodium and lithium with
a reported R2 of 097 143
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top
left) shows the distribution of the infinite temperature limit in the database after
limits exerted on the system (Top right) The distribution of the infinite viscosity
limit vs the glass transition (Bottom left) The relationship between fragility and
infinite temperature limit of viscosity (Bottom right) The infinite temperature limit
of viscosity vs the key metric predicted by SR 149
Figure 40 The prediction of the scaling of the activation barriers for a common sodium
borosilicate system 157
Figure 41 The relationship between the glass transition and the proton conductivity This
is justified two ways one through the relationship of the entropy of diffusion and
glass formation (the Adam-Gibbs model) and through the fact that water is known to
depress the glass transition 162
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine
the glass forming region Over top the constraints at the glass transition provided by
each oxide species is listed Since the objective is to decrease Tg while staying in the
glass forming region we will attempt to minimize use of elements that increase the
glass transition (nc gt 17) 166
xi
List of Tables
Table 1 The key properties considered for commercial application The optical properties
have been omitted since it is physical unrealistic to expect quantitative predictions of
quantum-controlled phenomena from a classical description of glass structure 18
Table 2 Variable definitions 65
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy
JadeTM glass and Sylvania Incorporatedrsquos SG80 74
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition
temperature (500 K) were kept the same while the total number of basins were
allowed to vary 101
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K 112
Table 6 System configurations for sodium silicate glass-water reactions 117
Table 7 Fitted values from this analysis compared to those reported in the literature The
disparity between the constraints evaluated with molecular dynamics most likely
come from the speed in which the samples are quenched 132
Table 8 Hyperparameters for different neural networks after hyper-optimizations 151
Table 9 A table with some ionic conductivity models and the parameters needed for
them as well as the disadvantages for each These are not the only models but are
representative of those commonly used in literature 154
Table 10 The predicted compositions based on the optimization scheme proposed 157
Table 11 The compositions synthesized in this work These compositions were predicted
by minimizing the cost function described in Eq (123) OP is the variant that was
melted after OP partially crystallized B-OP appeared to have surface nucleation in
some spots but was cut and removed before APS treatment 165
xii
Acknowledgments
ldquoTo Love Another Person is to See the Face of Godrdquo -Victor Hugo Les Miserables
I have been truly blessed to love and be loved by a plethora of mentors friends colleagues and
family across my academic journey I have to first thank my advisor Dr John Mauro who has
been as kind patient and caring as any mentor I have ever met and in the process has made me a
better scientist and a better person His mentorship was built on top of those who first found and
shaped me into a scientist Dr Ugur Akgun and Dr Steve Feller without whom my scientific career
would not be possible Along with these individuals Irsquove had the pleasure to learn so much about
glass and life from Dr Madoka Ona Dr Firdevs Duru Dr Mario Affatigato Dr Doug Allan Dr
Ozgur Gulbiten and Dr Seong Kim
Rebecca Welch Arron Potter and Anthony DeCeanne are my partners in all things
Rebecca is my partner in research and in life Her opinion support and patience has been
indispensable to me Arron is my oldest colleague and friend he is the one that I share the highs of
research and friendship with Anthony has put up with me non-stop for about 5 years now He is
the most patient and kind friend you could ask for Without these three individuals I could not have
done anything presented in this work I additionally have to thank Karan Doss Aubrey Fry Caio
Bragatto Daniel Cassar Brenna Gorin Mikkel Bodker Greg Palmer Katie Kirchner Kuo-Hao
Lee and Yongjian Yang I have the privilege to call these individuals both my friends and
colleagues
The last people I have to thank are my family I want to thank my brothers (Sam amp Quinn)
my mom and my dad My mom has helped me explore the universe through a love of reading and
her constant unwavering support My dad was the first to show me the wonders of science with
rockets as a kid as a teenager he supported my fledgling interest as he drove hundreds of miles to
every corner of this country and he is the person for whom this dissertation is dedicated
Chapter 1
The Difficulty of Optimizing Glass
Glass is a complex world-changing material Though it has existed for thousands of years
the surface of its true potential is only just being scratched To unlock the potential of glass for new
applications and to enable the maximum benefit to society we must be able to design new glasses
with wide ranging properties faster and cheaper than ever before [1]ndash[5] This is particularly
important now that the need for ecologically responsible materials is increasing due to climate
change In order to facilitate such glass design we must return to first principles and build a picture
of glass from the ground up encompassing both compositional and thermal history dependencies
into our understanding of glass properties[3] [6] The present accepted description of glass is a
non-crystalline non-ergodic non-equilibrium material that appears solid and is continuously
relaxing towards the super-cooled equilibrium liquid state[1] This definition gives insights into the
nature of glass and includes points of particular interest in this dissertation
The first point to consider is that glass is non-crystalline meaning that it contains no long-
range order This gives glass one of its key advantages being infinitely variable Since a glass is
not limited by having to reach stoichiometric crystalline structures there is no limit to the number
of possible glasses or glass structures This gives glass its possibilites Glass has been considered
as a candidate for everything from hydrogen fuel cells[7] to batteries[8] and for applications
ranging from commercial smart phone covers[9] to bioactive applications[10] The infinite
variability also explains the difficulty of designing glasses Consider a system with three
constituents for example SiO2 Na2O and B2O3 This then means there are two independent
compositional dimensions that must be fully explored to find an optimal composition for an
application If this space were discretized to 1 mol spacing that would give over 5000 unique
2
glass candidates that would need to be studied to reach the optimal design A five-component glass
would correspondingly have over 9 million unique glasses More generally for a glass with up to
C components there are 1C minus dimensions over which to optimize If instead one considers a
crystal there is not a continuously variable space but instead only discrete points
The other ramification of glass being non-crystalline is that it must bypass the region of
crystallization at a rate much faster than the rate of nucleation[11] This is inherent in the definition
of glass and is shown in Figure 1 Figure 1 is called the volume-temperature (VT) diagram and is
key to understanding the nature of glass The bypassing of the crystallization region leads to a
super-cooled liquid and during quenching the material departs from equilibrium and enters the
glassy state The region of departure is called the glass transition temperature range or simply the
glass transition To understand why this occurs we must first understand the concept of ergodicity
Ergodicity was defined by Boltzmann to mean that the time-average value of a property is
equal to the ensemble-average value of that property[12] [13] Now that ergodicity is defined we
can start to consider the implications of the non-ergodic nature of glass Since it is non-ergodic the
ensemble average will not equal the time average on human timescales however the ergodic
hypothesis says that in the limit of long time they will be equal[12] In order for both statements to
be true glass must evolve to be ergodic and thus is inherently unstable To explore this concept
further we must understand the glass transition process and define a timescale If there exists a
liquid at some temperature T and is then perturbed to T dT+ there will be some inherent time
associated with its relaxation towards equilibrium which is described by struct the structural
relaxation time There is a separate relaxation time when the system is mechanically perturbed
called the stress relaxation time stress To then understand whether the resulting material is a glass
or a liquid we need to compare struct with the observation time If the observation time is much
longer than the relaxation time then what is observed over the course of a experiment is the
3
equilibrium liquid properties Conversely if the observation time is much shorter than the
relaxation time then the observed properties will not be ergodic and as such we will not sample
the properties of the equilibrium liquid We define this non-ergodic state as the glassy state
However since in the limit of long time the system must become ergodic once again then over time
all glasses will return to an ergodic nature through a process called relaxation [14]
The last point considered in the definition of the glass is that it is non-equilibrium Not only
is glass non-equilibrium it is also unstable and continuously progressing towards the super-cooled
liquid state (eg relaxation) Relaxation is inherent in all glasses and we can see this for a model
system in the VT diagram (Figure 1) In this diagram there are two glasses (of the same
composition) quenched at different rates (both of which bypass crystallization) In the slow-cooled
glass there is a lower volume (higher density) compared to the fast-cooled glass This difference is
due to the fact that the slow-cooled glass has been allowed to relax more fully compared to the
other This adds at least one additional dimension of optimization (such as quench rate from a high
temperature to room temperature) in which glasses must be designed in
This dissertation is dedicated to understanding the infinite variability of glass with respect
to crystallization composition and thermal history with tools being implemented and designed so
that the challenge of optimizing over these vast spaces can be done with higher efficiency lower
cost and with less experimental work load These tools are all inspired by the energy landscape
description of glass To design the optimal glass for any application we must understand the
relationship between the 1C minus compositional dimensions a minimum of one thermal history
dimension and the properties of a material There are many ways to explore these relationships
each with its advantages and disadvantages The methods to optimize over these thermal history
and compositional dimensions include energy landscapes[15] fictive temperature[16] topological
constraint theory[17] and machine learning[3] Of these energy landscapes are the most
fundamental with each other technique being related back to the energy landscape in some form
4
Though they are the most fundamental they are also the most difficult time consuming and have
the largest barrier to entry for use To then build our understanding of each technique we will start
with energy landscapes then explore how through a series of assumptions we can arrive at the other
methods
5
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists the
equilibrium liquid which as it is quenched can either become a super cooled liquid or crystallize
Crystallization causes a discontinuity in the volume The super-cooled liquid upon further
quenching departs from equilibrium and transitions into the glassy state Reproduced from
Fundamentals of Inorganic Glasses with permission from the author[1]
6
11 Energy Landscapes
Energy landscapes were first developed by Goldstein and Stillinger who proposed the
concept as a way to understand the evolution of a material under complex conditions [18]ndash[20]
Since Stillingerrsquos seminal work [18] energy landscapes have become important in the study of
protein folding [15] [21] [22] network glasses [6] [13] and other complex materials The concept
of energy landscapes has become commonplace and is readily evoked when describing the time
evolution of a material [20] [22]ndash[26] Despite energy landscapes being commonplace in some
fields the procedure for mapping a landscape remains difficult due to the associated computational
challenges [15] [27] To map an energy landscape one must find a set of local minima and the first
order saddle point connecting each pairwise combination of those minima Each minimum is called
an inherent structure and represents a structurallychemically stable state of the material A basin
is the set of structures that converge to the same inherent structure upon minimization [18]
In glass science the energy landscape is used to justify complex behaviours which are most
commonly calculated using molecular dynamics simulations (MD) [13] [28] [29] It is easy to
understand why MD is used so frequently used when considering the ease of use of common MD
packages For example LAMMPS [30] is a common user-friendly MD program with a low barrier
to entry allowing individuals with little programming experience to run complex MD simulations
Inspired by this accessibility as well as the explicit Python bridge to LAMMPS we have developed
a software that can be used to explore energy landscapes with a variety of exploration techniques
Similar softwares [15] [31]ndash[35] have been created to map landscapes however each program has
limitations and may not be suitable for all purposes while ours is a general purpose software
The calculation of the energy for landscapes is done using an empirical interatomic
potential that can be fit using either experimental data and a series of MD simulations or using ab
initio methods One common potential form is the Lennard-Jones form (LJ)
7
12 6
4LJVr r
= minus
(1)
in which and are fitting parameters for every pair-wise set of atoms in a system V is the
potential energy contributed by the two-atom interactions and r is the distance between the two
ions in question Though the LJ form is simple and easy to parameterize it is unable to recreate
complicated material behaviours for a real glass As such a more complicated potential is often
used such as a Morse potential A commonly used silicate glass potential is the Pedone potential
which consists of Morse potential with an additional columbic term and repulsive term [36] and is
given by
( )2
2
0 121 exp 1
i j
Pedone
Z Z e CV Y a r r
r r = + minus minus minus minus +
(2)
Z is the charge of a given ion e is the charge of an electron and Y C a and r0 are fitting parameters
It is also worth noting that the Pedone et al potentials are only accurate when considering the
interactions between each individual ion species with oxygen For instance Si-O Na-O and O-O
are all explicitly parameterized while Si-Na only interacts coulombically (the first term)
Interatomic potentials are discussed here only briefly but there are extensive resources available
for those readers who wish to learn more[37]ndash[39]
Once an energy landscape has been mapped the thermodynamics and kinetics for a given
system can be easily calculated [33] [40] The thermodynamic driving force is given by the
difference between the free energy of the current inherent structure and the free energy of the lowest
state while the kinetics is the rate in which the system crosses the barrier between the basins This
can be calculated explicitly for arbitrary timescales and temperatures with another software
KineticPy [41] The energy landscape provides insights into the detailed nature of material
behaviour Though energy landscapes are very powerful they are also difficult to map To
approximate the behavior of the landscape then we use approximations such as fictive temperature
8
topological constraint theory (TCT) or machine learning Fictive temperature is a way to estimate
the effects of the thermal history dimension[42] while TCT is used to estimate the compositional
dependence of properties[43]
12 Fictive Temperature
Fictive temperature is an approximation used to find the change in the occupational
probabilities on an energy landscape[16] [44] Instead of considering every possible combination
of occupational probabilities like the energy landscape it instead approximates the occupational
probability using a single temperature at which the landscape would be in equilibrium It is
essentially a measure of how closefar a glass is from equilibrium This approximation allows for
us to calculate the relaxation dynamics of a glass without needing any computationally heavy
energy landscapes However this single parameter description oversimplifies relaxation[14] [16]
The functional form for glass relaxation dates back to 1854 when Kohlrausch fit the
residual charge of a Leyden jar with[45]
( ) (0)expt
g t g
= minus
(3)
where t is time τ is the relaxation time constant β is the stretching exponent and g(t) is the
relaxation function of some property (eg volume conductivity viscosity etc) Eq (3) is referred
to as the stretched exponential relaxation function (SER) The stretching exponent is bounded from
0 lt β le 1 where the upper limit (β = 1) represents a simple exponential decay and fractional values
of β represent stretched exponential decay
For over a century the physical origin of the stretched exponential relaxation form was one
of the greatest unsolved problems in physics until 1982 when Grassberger and Proccacia[46] were
able to derive the general form based on a model of randomly distributed traps that annihilate
9
excitations and the diffusion of the excitations through a network This model however provided
no physical meaning for β which was still treated as an empirical fitting parameter In 1994
Phillips[47] [48] was able to extend the diffusion trap model by showing that β=1 at high
temperatures and at low temperatures the stretching exponent can be derived based on the effective
dimensionality of available relaxation pathways He in turn expressed the stretching exponent
2
df
df =
+ (4)
where d is the dimensionality of the network and f is the fraction of relaxation pathways available
He then proposed a set of lsquomagicrsquo numbers for common scenarios Assuming a three-dimensional
network with all pathways activated (d = 3f = 1) a value of β= 35 is obtained The second case is
a three dimensional network with only half of the relaxation pathways activated (d =3f = 12)
yielding a value of β= 37 A third magic value β= 12 was found for a two dimensional technique
with all pathways activated (d = 2f = 1) β = 35 occurs for stress relaxation of glasses under a load
because both long- and short-range activation pathways are activated whereas a value of β = 37 is
obtained for structural relaxation of a glass without an applied stress The value was confirmed by
Welch et al[9] when the value was measured over a period of 15 years at more than 600C below
the glass transition temperature of Corning Gorilla Glass Though this model was able to reproduce
the limiting values it was criticized widely in the community Critics argued that the model was
simply created such that it reproduced the stretching exponent values of certain experiments and
ignored many other experimental results that appeared to disagree with the model The model also
fails to interpolate between the low temperature values Phillips predicted and the high temperature
limit
The work from Phillips and the stretched exponent gave a form for relaxation however
there remained no method to instantaneously state the distance from equilibrium of a glass To
account for this problem an additional thermodynamic variable (or order parameter) called fictive
10
temperature (Tf) was proposed Early research on relaxation in glasses dates back to the work of
Tool and Eichlin in 1932[49] and Tool in 1946[50] whose works originally proposed a temperature
at which a glass system could be in equilibrium without any atomic rearrangement Tool suggested
that this fictive temperature was sufficient to understand the thermodynamics of a glassy system
Originally the fictive temperature was treated as a single value that was some function of thermal
history (Tf[T(t)]) The evolution of fictive temperature as proposed by Tool is then given by
)
( ) ( ( ))
(
f f
f
dT T t T T t
dt T T
minus= (5)
In which for stress relaxation is given by
( )(( ) ( ))fT t T
G
T t = (6)
In Eq (6) is the shear viscosity and G is the shear modulus This means that to predict the time
evolution of a glass only three things are required shear modulus the viscosity as a function of
temperature and fictive temperature and the stretching exponent Though fictive temperature
qualitatively reproduced the results they were looking for subsequent experiments have shown that
the concept of a single fictive temperature is inadequate[51] [52]
A key experiment was performed by Ritland in 1956[51] Ritland took several samples
with different thermal histories but the same measured fictive temperature thus based on Toolrsquos
equation both should have had identical relaxation properties Ritland however showed that the
refractive index evolved differently between samples To account for these differences
the use of multiple fictive temperatures was suggested[52] This worked because the stretched
exponential form could be approximated with a Prony series
( ) exp expi i
N
i
Kt w K t minus minus
(7)
11
In the Prony series N is the number of terms while iw and iK are fitting parameters This means
that each term could represent one fictive temperature and this collection of fictive temperatures
would reproduce the overall experimentally observed form Ritlandrsquos experiment showed that
materials held isothermally at their fictive temperatures with varying thermal histories can give
different results for the relaxation of a given property Thus as a glass network relaxes the fictive
temperature will also shift Narayanaswamy[52] was able to apply Ritlandrsquos concept in a highly
successful engineering model with multiple fictive temperatures but the physical meaning of
multiple fictive temperatures remains elusive
In general there is reason to be skeptical of the concept of fictive temperature Fictive
temperature was not derived but instead was an empirical approximation that Tool needed to gain
an early quantitative understanding Recently Mauro et al[16] used energy landscapes to try to
understand the underlying validity of fictive temperature fT implies that the occupational
probability ( ip ) is given by
1
exp ii
Hp
Q kT
= minus
(8)
Q is the partition function k is Boltzmannrsquos constant and iH is the enthalpy of the given state
However when the actual probability evolution predicted by fictive temperature was compared to
the probability calculated through a systematic Monte Carlo approach the results were found to be
drastically different (even when many fictive temperatures were tried) This implied that fictive
temperature is insufficient to describe the glassy state and must be replaced with a new method
Despite the inadequacies of fictive temperature state-of-the-art modeling techniques for
glass relaxation still relies on the concept In these techniques a set of fictive temperatures (usually
12) are used to calculate 12 parallel relaxing fictive temperatures[42] This is because there has
been no alternative method able to reach a prediction of the evolution of glass under different
12
thermal histories The current state-of-the-art model is called the Mauro-Allan-Potuzak (MAP)[44]
model of non-equilibrium viscosity and was derived from a more fundamental expression for
viscosity called the Adam-Gibbs model (AG) which is given by[53]
10 10) log( )
log ( f
c f
BT T
TS T T = + (9)
B is the barrier for a cooperative rearrangement 10log is the infinite temperature limit of
viscosity and according to Zheng et al[54] it is approximately 10-293 Pa s and cS is the
configurational Adam-Gibbs entropy This entropy has long since been a source of debate and will
be addressed later in chapter 5
In order to calculate the equilibrium part of the viscosity the MAP model relies on an earlier
Mauro Yue Ellison Gupta and Allan (MYEGA) equilibrium viscosity model[55] This model
was derived based on earlier work by Naumis[23] [56] Gupta and Mauro[57] and the AG model
These models showed that the configurational entropy can be related to the degrees of freedom
lncS fNk= (10)
f is the degrees of freedom per atom N is the number of atoms and is the number of degenerate
states per floppy mode To then build in temperature dependence they assumed that the degrees of
freedom can be modeled using an Arrhenius form (with activation barrier H )
( ) expH
f T dkT
= minus
(11)
When Eqs (9) (10) and (11) are combined into an equilibrium form the equilibrium viscosity is
then given by
10 10 el pog og xlC H
T kT
+
=
(12)
Where C is equivalent to
13
ln
BC
dkN=
(13)
Through a change in variables this can be written in terms of three common variables that allow
for all viscosity equations to be written with three variables specified by Angel[58] the glass
transition temperature the fragility and 10log The glass transition temperature is defined as
the isokom where the liquid has a viscosity of 1210 Pa s The fragility is defined as[58]
10log
g
g
T T
dm
Td
T
=
(14)
The three most common viscosity models (MYEGA Vogel-Fulcher-Tammann (VFT)[59] and
Avramov-Milchev (AM)[60]) are listed below in terms of the same parameters
10 10 10
10
log (12 log ) exp 1 11 o
log2 l g
g gT Tm
T T
+ minus minus minus
minus =
(15)
( )
2
10
10 10
10
12 logl log
1 1
og
2 logg
Tm
T
minus+
minus + minus
=
(16)
and
( )10(12 log
10 10 1
)
0l gog l 12 logo
m
gT
T
minus
+ minus=
(17)
To incorporate non-equilibrium effects with these equilibrium models the MAP model
proposes
10 10 10log ( ) (1 ) log ( )log MYEGA g ne f gT m T mx x T T T = + minus (18)
14
10log ne is the non-equilibrium viscosity whose form and compositional dependence can be found
elsewhere[42] [44] [61] x is the ergodicity parameter and is of crucial importance to
understanding glass regardless of the model being used[12] In the MAP model it is given by
( )( )
3244min
max
f
f
m
T Tx
T T
=
(19)
Though the MAP model was derived from the best understanding of energy landscapes of glasses
at the time it has a few flaws that prevent it from perfectly predicting the different relaxation
behaviors of glass A few of the short-comings are
1 The method ignores crystallization For a comprehensive model showing the response of
glass to temperature the possibility of crystallization must be included Currently no
relaxation model includes this possibility
2 It is built around the concept of fictive temperature This could be replaced in a future
model but currently at present it carries with it all the failures of the fictive temperature
picture of glass dynamics adding only the relaxation time coming from landscape-derived
values
3 The MAP model does not consider temperature dependence of the stretching exponent
This is typically circumvented by choosing a fixed stretching exponent
4 The parameters needed for the non-equilibrium viscosity in the MAP model are
experimentally challenging to measure To circumvent this Guo et al[61] published
methods to estimate the values based on fragility and the glass transition but the
approximations were only tested on a few samples with similar fragilities and chemistries
5 The MAP model only predicts stress relaxation Since the publication of the MAP model
it has been shown that only stress relaxation time is related to the shear viscosity[62]
Structural relaxation must be incorporated in future models
15
For a complete understanding of the thermal-history dependence of glass all of these questions
must be rigorously answered
13 Topological Constraint Theory
TCT is one way to estimate the compositional dependence of glasses It uses a set of
assumptions that connects the underlying structure to glass properties through understanding of the
energy landscape These assumptions include the temperature dependence of the rigidity the same
assumptions used in the derivation of the MYEGA models and that the effects of thermal history
on structure are minimal TCT was originally proposed by Gupta and Cooper (GCTCT) to try to
understand the role of rigid polytopes on glasses properties[63] Gupta and Cooper fundamentally
understood that the ability for glasses to form was reliant on the transformation of a liquid with
many degrees of freedom to a glass with a fixed degree of freedom Since the structure of the glass
is the same as the structure of the liquid at the glass transition and the degrees of freedom of the
glass ( f ) changes during the transition to glass it was possible to gain insights into the
transformation process This approach is fundamentally an approximation of the highly
multidimensional landscape where only approximate structural information is used to understand
the entropy of the energy landscape Though the core idea was useful it was ldquoclunkyrdquo and difficult
to make calculations with
To improve upon the ideas of GCTCT Phillips and Thorpe (PTTCT)[64] proposed a
mathematically equivalent form where the degrees of freedom could be simply found by
considering the number of constraints ( cn ) around each network-forming atom in the glass They
showed that the degrees of freedom were related to the number of constraints by
cf d n= minus (20)
16
The number of constraints could then be calculated by considering the two-body and three-body
interactions (first and second terms) as a function of the mean coordination of the network forming
species
2 32
c
rn r= + minus (21)
This led to a qualitative language to discuss glass with a few quantitative insights The biggest
quantitative insight comes in the form of the lsquoBoolchand Intermediate Phasersquo[65]ndash[68] If it has
positive degrees of freedom then a network is lsquofloppyrsquo and retains high kinetic energy (as it flips
back and forth between two inherent structures) If the system has negative degrees of freedom
then the network is stressed-rigid and there is additional energy stored in the stress associated with
over-coordination If the number of constraints is equal to the number of dimensions then the
resulting system will have zero degrees of freedom and there is no stored stressed energy or kinetic
energy from flipping between basins Boolchand went searching for this perfectly lsquoisostaticrsquo glass
and found a non-zero window of compositions that were close to isostatic but had properties
drastically different compared to both stress rigid and floppy glasses This is now called the
Boolchand Intermediate Phase
Despite the success in the prediction of the intermediate phase TCT remained largely
qualitative until the work by Naumis[23] [56] and Gupta amp Mauro[57] Naumis pointed out that
the degrees of freedom are a lsquovalleyrsquo of continuous low energy so that the structure can freely move
between basins of the landscape Since these valleys are deformation pathways Naumis pointed
out that they will dominate the configurational entropy of the system Building on this Mauro and
Gupta wrote that since the entropy is proportional (Eq (10)) we could then write the glass transition
temperature as a function of the degrees of freedom and the AG model
( )10o1 ln2 l g
gTB
f Nkminus=
(22)
17
Since everything in Eq (22) is approximately a constant between compositions except for f we
could then write the expressions in terms of reference values (subscript r) and accurately scale the
glass transition of new compositions by knowing the changing structures
( ) g r c r
g
c
T d nT
d n
minus=
minus (23)
The last insight provided by Mauro and Gupta was that the number of constraints change as a
function of temperature They realized that each constraint has a temperature dependence which is
controlled by the interaction energy of the constraints This makes intuitive sense because as the
thermal energy becomes greater than that of the barriers additional valleys in the energy landscape
form This also allowed them to expanded TCT to include quantitative predictions of fragility
The understanding of the relationships between the configurational entropy the number of
constraints the degrees of freedom and the viscous properties lead to a revolution in understanding
glass properties Soon after the work by Mauro amp Gupta multiple works came out predicting
mechanical chemical and viscous properties of glass knowing only the underlying structure
Though this framework is powerful it is not a complete description without further
parameterizations with each model requiring some fitting parameters or reference values In
addition there are still key properties that are missing from the standard topological approach that
are needed to design commercial glasses
Ultimately MGTCT needs be expanded to include the properties that are needed for
commercial applications A list of composition dependent properties needed for understanding the
performance of commercial glasses are shown in Table 1 Properties taken from Ref [3] show what
properties must be controlled to manufacture technological glasses All models take additional
inputs and there are multiple ways to link outputs from some models to inputs of the other Choice
of model vary depending on the use case
18
Table 1 The key properties considered for commercial application The optical properties have
been omitted since it is physical unrealistic to expect quantitative predictions of quantum-
controlled phenomena from a classical description of glass structure
Properties TCT Machine Learning Other methods
Glass Transition [57] [69] [70] Energy Landscapes [71]
Fragility [57] [69] This work This work
Relaxation MAP model [44] [61]
Crystallization This work [72]
MDMC [73]ndash[75]
Melting Temperature This work MDThis work[72]
Chemical Durability [76] [77]
This work [78]
[79] [80] Linear Methods [81]
Density [82] MD Packing [83]
CTE This work MD
Hardness [5] [84] [85] MD
Youngrsquos Modulus [86] This
work [87]
[82] This work MD
Activation Barrier for Conductivity MD This work [88]
Batch Cost Economic Calculation
Ion Exchange Properties Empirical Relationships [2]
19
14 Machine Learning
All the methods discussed so far are physically informed techniques The techniques
presented are inherently powerful because their very nature is informed by reality Unfortunately
this means that some knowledge about the system is required These techniques are useless without
the required parameterizations However a class of algorithms called lsquoMachine Learningrsquo (ML)
require no such parameterization Instead ML uses large quantities of data to determine trends
linking any physically connected input to output allowing for accurate predictions of properties
across wide compositional spaces This is favorable with a large of volume data because the
required inputs are minimal to achieve a prediction The difficulty (and often expense) in this
technique is gathering large quantities of high-quality data to enable this method
The predominant ML method used in this work is neural networks (NN) Neural networks
work by constructing a network of neurons This method was designed in such a way to mimic a
human brain so the network is elastic enough to be modified improving results Each neuron takes
in a series of inputs received from the previous layer (or input data) multiplies by a weighting
factor sums the values and then performs some lsquoactivation functionrsquo on them By changing the
weights it is then possible to recreate any function given sufficient data This is a preferred
technique because it packages into a simple easy-to-use software Neural Networks (once trained)
can be used by anyone stored in a small file can recreate any function is computationally cheap
and does not require as much data as some other ML techniques
This is an interesting method to estimate the effects of the composition on the landscape
and corresponding properties because we cannot understand the underlying mathematics It is
fundamentally a numerical process Thus even though we know that the properties of a glass are
dominated by the topography of the landscape how the NN figures out the relationship between
the chemistry and properties is not understood It is possible to then use the NN to give us the
20
physical parameters we need for other models building multiple methods together to achieve new
insights
15 Goals of this Dissertation
To enable the next generation of glass design we must enable a new era of computationally
methods being used to produce accurate predictions of the behavior of glass This includes both the
thermal history dependence of glass (in terms of relaxation and crystallization) and the
compositional dependence especially those properties listed in Table 1 The goals of this thesis will
be to address some of these outstanding questions in a systematic way that is practical for an
external user to understand and implement Predicting thermal history effects are covered in
Chapters 3-5 while compositional effects are predicted in Chapters 6-7 The breakdown of each
following Chapter as it relates to these questions is listed below
21
bull Chapter 2 Software Unfortunately there is no standardized software for the
calculation of relaxation or energy landscapes I present two new codes to
standardize these calculations so that the process is reproduceable and standardized
bull Chapter 3 Crystallization This section will address new computational ways to
calculate crystallization focusing on predicting nucleation rates and how to improve
our nucleation calculations
bull Chapter 4 Expanding the Current State of Relaxation This chapter will focus on
expanding the MAP model with a temperature dependent form of the stretching
exponent as well as understanding the role of ergodicity in glass relaxation
bull Chapter 5 Relaxation and Crystallization without Fictive Temperature Chapter 5
will focus on discussing relaxation and crystallization without fictive temperature
by proposing a new relaxation model that is reliant on creating lsquoToy Landscapesrsquo
This approach removes assumptions about the evolution of the occupational
probabilities
bull Chapter 6 Enabling Prediction of Properties This chapter will use TCT and ML to
address the compositional dependance of different key properties needed for
enabling the faster design of new glass compositions
bull Chapter 7 Designing lsquoGreenrsquo Glasses Applying information learned throughout
this dissertation to create new glass compositions
bull Chapter 8 Conclusions
22
Chapter 2
Software for Enabling the Study of Glass
To being our systematic investigation it is important to have a suite of tools that
implement current state-of-the-art models Though many tools exist for structure and
topological constraint theory there were no codes that existed for the calculation of
relaxation using the MAP model and there were no codes that were capable of exploring
the landscapes that we are interested in To correct this we present two new novel codes
ExplorerPy for mapping energy landscapes and RelaxPy for calculating MAP
dynamics[42] [89]
21 ExplorerPy
Mapping an energy landscape allows for modeling a range of traditionally inaccessible
processes such as glass relaxation The advantage of this software (over other codes to map energy
landscapes) is that it only requires knowledge of LAMMPS to run Also our software contains a
choice of three methods to map the landscape The first of the three different methods implemented
to explore energy landscapes in this work is eigenvector following [90] Eigenvector following has
been used in the past to generate landscapes that predict the glass transition and protein folding
pathways [44] [91] This implementation follows the work of Mauro et al [90] [92] and uses an
independent Lagrange multiplier for every eigenvector Eigenvector following excels at finding
small barriers quickly however it rarely finds unique structural changes in complex materials
(especially in systems with periodic boundary conditions) The second method that we have
implemented following in the steps of Niblett et al [27] is a combination of molecular dynamics
and nudged elastic band methods (NEB) [34] [35] To find the transition points MD is run at a
23
fixed temperature for a fixed amount of time the structure is minimized and then NEB is used to
find the saddle point between the initial structure and the new inherent structure Details of the
nudged elastic band method can be found elsewhere [34] Using these techniques it is possible to
map a sufficiently large landscape to make kinetic and thermodynamic measurements possible
These two methods benefit from the fact that they work well when the volume is varied and together
enable computationally inexpensive mapping of enthalpy landscapes [71] [92] (where the volume
is no longer fixed and instead the pressure is specified) However it is not recommended to generate
a large enthalpy landscape from MD in this software due to the computational cost involved
Although molecular dynamics is a powerful technique there is a large computational cost to
simulate a small amount of time
The third method is a new technique created in an attempt to circumvent the computational
cost of MD while still finding large structural changes We have named this technique ldquoeigen
treibanrdquo (ET) This technique relies on shoving along an eigenvector with a multiplier rather than
stepping After the system is shoved the system is minimized and if the minimized structure exhibits
a displacement in atomic position is larger than a threshold a NEB calculation is performed to find
the saddle point By shoving in the direction of an eigenvector the goal is to reproduce large changes
due to vibrational modes The fixed multiplier (m0) is calculated by taking a target distance (d) and
the largest displacement magnitude out of the atoms j (vij) in a given eigenvector direction i (ai)
m0
=d
max[nij] (24)
To convert to the displacement magnitude from an eigenvector in three dimensions we use the
expression
vij
= ai[3 j]2 +a
i[3 j +1]2 + a
i[3 j + 2]2 (25)
The dimension j has the length of the number of atoms in the systems
24
This software uses the system and potentials created from a LAMMPS input
scripts and as such has a wide range of access to different potentials and possible systems The
only input that has to be provided is the file which would be used for molecular dynamics in
LAMMPS If the script has been used for molecular dynamics simply by eliminating any
thermostats or barostats as well as any ldquorunrdquo commands it is possible to use it as an input script to
ExplorerPy (it is also recommended to eliminate any calls to ldquoThermordquo because it may interfere
with the extraction of variables from the underlying LAMMPS framework) If the LAMMPS input
script successfully sets up a simulation all other variables and parameters can be controlled from
command line options as described in the ldquoIllustrative Examplerdquo section The initial structure will
be taken from the input script as well
ExplorerPy is a software that has been written in Python with modular functions and ease
of use in mind It can be run from a Linux terminal with all major options being set via the command
line The software then generates a list of inherent structure energies stored in the mindat file and
stores the transition point energy for each pairwise inherent structure in the tsddat file All of the
transition points and minimum are stored in the same directory from which it is run as xyz files
The software architecture is shown in Figure 2 Details of the stepping procedure for eigenvector
following can be found elsewhere[93] The variables ldquoMD frequencyrdquo and ldquoET frequencyrdquo are set
as command line options at the time of the launch of the software
25
Figure 2 The schematic for the flow of the program Beginning in the top right corner and running
until the condition in the pink box is satisfied Yellow diamonds represent checks and blue
operations
The software is capable of mapping an energy landscape for an arbitrary system ie for
any system that can be simulated in LAMMPS This allows for accurate prediction of the evolution
of complex thermodynamics and kinetics The software requires no programming and is
controllable via command line options ExplorerPy prints all of the basins transition states inherent
volumes as well as information about the curvature of the system so that the vibrational frequency
is calculable This is all of the information required to calculate a complex range of phenomena
over a specified long timescale ExplorerPy is only set up to work within the LAMMPS metal units
To run the software there are only a few parameters that have to be set for each mode The
force threshold (-force) curvature threshold (-curve) and step size (-stepsize) when running the
command line option have to be set to allow for a stable exploration of a given landscape
These seem to be potential-specific and for a given potential appear to be fairly stable It is best to
start large for the force threshold and small for the eigen threshold and step size then modify as
26
necessary If eigenvector following is not being used there is no need to adjust these values Beyond
this there are several optional command line options (only the file (-file) command is required)
-file [file input] this is a standard lammps input if the input sets up a lammps run
it will work to set this up There should not be any lsquothermorsquo calls in the lammps
script
-press [specified pressure in atmospheres] if this is specified the system will
always be minimized to this pressure while if not specified the volume remains
fixed
-num [number of basins to explore]
-time [max time to run in hours] if this time is reached before the number of basins
are found the simulation will stop
-thresh [minimum energy for barrier to keep]
-center if this is given as a flag the system will be set to the anchor location each
step
-anchor [atom selection for the position to be held constant]
-md [frequency of molecular dynamics search] If a custom MD LAMMPS file is
not provided it defaults to starting all of the atoms at 3000 K and running for 50
fs
-shove [frequency of eigen treiban]
-custom [file used for MD exploration]
-help [List all commands possible in software with these descriptions]
An example run would then be done the following way
[mpirun -np 5 python37 Explorerpy -file inSiO2 -num 1000 -force 10 -curve 1e-5 -stepsize 012
-press 00 -md 11 -time 48]
This would start a run with five parallel processors searching for 1000 basins with an MD
step being ran every 11 searches with a maximum time of 48 hours The results of this command
are shown in Figure 3 Below we have attached an example of inSiO2 the output and additional
scripts related to the example has been uploaded to the associated GitHub
27
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above and then
plotted using PyConnect The plot is a disconnectivity graph where each terminating line
represents an inherent structure and tracing where two lines meet describes the activation barrier
The potentials are taken from the BKS potential [94]
httpsgithubcomlsmeetonpyconnect
28
22 RelaxPy
RelaxPy script is written using Python syntax and relies on the numpy scipy and
matplotlib libraries It solves the lack of a unified relaxation code in our community and is based
on the MAP model[44] [61] interpretation of multiple fictive temperatures and the shear
modulusviscosity form of the Maxwell equation[62] [95] It is designed to be run within a Linux
terminal with command line options or inside a Python interpreter It can be used by calling the
name of the script followed by the input file the desired output file name and finally an optional
tag for all fictive components to be displayed as well as printed to a specified output file (formatted
as comma separated values)
The RelaxPy package consists of an algorithm that iterates over time in order to determine
the values of viscosity (η) relaxation time (τ ) and current fictive temperature (Tf ) given the user-
supplied thermal history and material property values The purpose of this code is to enable
calculation of the relaxation behaviorrsquos change over time by using the MAP model using only
experimentally-attainable parameters Toolrsquos equation is solved for each term in a Prony series
approximation of relaxation finding the change in each Tfi component for a given time (t) to
accurately reproduce the stretched exponential form The program then iterates over the entire
thermal history of the sample using a user-specified time step dt allowing for calculation of the
overall fictive temperature over time The values for the Prony series fit (wi and Ki) are taken from
the database originally generated by Mauro and Mauro[96] and expnded to include any value of
the stretching exponent The values were fit by Mauro and Mauro with a hybrid fitting method
based on the number of terms in the desired Prony series and the magic value from Phillips The
thermal history is also defined in the original input file using linear interpolation The initial fictive
temperature components are assumed to be at equilibrium (Tfi = T(0))
29
To solve for the non-equilibrium viscosity used the MAP expression is used with the approximate
scaling of values proposed by Guo et al[61] The software is capable of performing the calculations
needed for a variety of thermal histories and processing methods desired by both industry and
academia with a focus on the fictive temperature This software is then fundamentally limited by
the issues associated with the MAP model and the compositional approximations made For
instance this software will need to be further modified to include structural relaxation when such
a model is proposed[62]
An example input code is listed below
Denotes Comments
Tg 794 0 m 368 C 1494 dt 10 Beta 37
Start Temp End Temp Time [s]
1000 10 1300
10 100 1000
100 30 500
Assumes Linear Interpolation
This example input states that over the space of 1300 seconds an initial glass sample is cooled from
1000 C to 10 C Then it is heated to 100C over 1000 seconds and then finally down to room
temperature in the space of 500 seconds All temperatures are given in Celsius
This software aims to provide the first open-source software for modeling of glass
relaxation behavior using the MAP model of viscosity by creating a package that can easily and
quickly be used to approximate the fictive temperature components of a glass as well as the
composite fictive temperature RelaxPy allows for researchers to calculate the evolution of
macroscopic properties of glass during a relaxation process (assuming a fixed stretching exponent)
and will facilitate the calculation based on a user-supplied thermal history RelaxPy can be
seamlessly adopted by most glass research groups and removes the guessing associated with
determining a fictive temperature allowing for a fully quantitative comparable discussion of Tf
simultaneously creating a tool that can easily be expanded for the entire community as new physics
is discovered
30
Chapter 3
Understanding Nucleation in Liquids
Crystallization cannot be ignored in any material system at temperatures below its liquidus
temperature although extreme resistance to crystallization is reported in some organic and
inorganic substances[97] Crystallization (and by extension nucleation) is critical in systems
ranging from ice to thin films[98]ndash[100] Glasses are non-crystalline substances therefore
crystallization kinetics will determine the necessary thermal history to avoid crystallization and
obtain a glassy state Despite the absolute importance of crystallization to the study and design of
glasses there is no current method to predict the nucleation of crystals in this event One of the main
challenges in the production of glass-ceramics is finding the optimal thermal history for the desired
crystallite distributions There are two phenomenological steps in the process of phase
transformation from a supercooled liquid to a crystal a nucleation step and a growth step The
growth step is described well by Wilson-Frenkelacutes theory[1] but the underlying physics of
nucleation remains elusive A stable nucleus is the precursor to a crystal comprising a nano-sized
periodic assemblage of atoms A clear understanding of the relationship between liquid kinetics
and crystal nucleation has not been established despite decades of research
The study of nucleation is particularly critical for glass-ceramics Since their discovery by
S Donald Stookey glass-ceramics have become prevalent in our society through products such as
stovetops and Corningware[73] [101] [102] Glass-ceramics are a composite material comprising
at least one crystalline phase embedded in a parent glassy phase benefitting from the properties of
both the glassy and crystalline phases[1] [103] One of the reasons glass-ceramics are ubiquitous
is their relatively easy production glass-ceramics can be formed through standard glass forming
procedures using additional heat treatments Additionally there is a wide range of accessible
31
properties through tuning both the chemical composition of the parent glass and the thermal history
and hence the microstructure of the heat-treated glass-ceramics[101] [103]ndash[106]
Glass-ceramic production consists of first synthesizing a glass and then generating a
microstructure through the two-stage ceramming process[73] [103] [107] The nucleation step
remains poorly understood with classical nucleation theory (CNT) giving widely varying results
depending on how the parameters are determined[1] [73] [108] The inability to consistently get
accurate predictions for a nucleation curve means that choosing an appropriate nucleation
temperature must be done experimentally which can prove to be one of the most challenging and
time-consuming aspects of properly designing a glass-ceramic The difficulty is due to the
magnitudes of the crystal nucleation rates not being predictable with CNT or any other current
model
This difficulty in designing a material is apparent when considering the high dimensional
phase space of parameters that can adjusted We haver presesnted some tools developed to help
optimize over this lsquoglassyrsquo space such as RelaxPy[42] KineticPy[41] topological constraint
theory[109] or various machine learning models[4] [70] [110] [111] However a glass-ceramic
system has at least four additional phase dimensions (two for the nucleationgrowth temperatures
and two for nucleationgrowth times) that need to be optimized (depending on the parameterization
of the experiment it is also possible to add another four dimensions relating the heating and cooling
between each step) Therefore a glass-ceramic system has a minimum of d+5 dimensions over
which the system must be optimized and yet those four additional dimensions are the least
understood Designing methods to easily optimize over this large space with feasible computational
methods is of utmost importance to reaching a new generation of custom materials
Experimental studies of nucleation have used techniques ranging from differential
scanning calorimetry (DSC) to electron microscopy to try to capture predictive capability in the
lsquoceramicrsquo phase-space (the four additional dimensions) These studies have successfully captured
32
the shape of the nucleation curve but fail to provide the information needed to calculate the
temperature-dependence of the nucleation rate The shape of the curve can be determined through
DSC and with great effort an approximation of the rate can be made however this technique is too
laborusome to be efficient[112]ndash[114] Since the parameters have been inaccessible to traditional
computational approaches and experimental determination all previous studies of nucleation have
required at least one variable to be fit If a nucleation curve could be predicted only using a
computational method or with an efficient simple DSC method the rate at which new glass-
ceramics products are created would increase In addition the cost of producing glass-ceramics
would likely decrease and more complex microstructures and therefore more unique properties
could be explored
Crystal nucleation has been historically difficult to simulate and measure experimentally
because the phenomena happen on spatial scales of nanometers but over many orders of magnitude
of time scales CNT is the most widely applied model despite its supposed failure To test CNTrsquos
assumptions an interest in computational work has grown Recent approaches using grand
canonical Monte Carlo have successfully predicted the short time dynamics of nuclei
formation[73] This approach involves use of a continuous solvation model for the liquid and has
enabled greater computational insights however it remains computationally expensive[74]
Molecular dynamics (MD) fails to capture nucleation in glass-forming systems due to the need for
longer spatial and temporal scales to capture nucleation effects accurately Some attempts have
been made using Monte Carlo MD or empirical approaches to replace classical nucleation theory
(CNT) with some alternate method where the parameters can be directly measured[73] [115]ndash
[117] All nucleation models and Monte Carlo models attempt to describe the rate that a
supercooled liquid system transitions to a crystalline structure as a function of temperature
Explicitly mapping the energy landscape would fundamentally explain the underlying
physics of nucleation An energy landscape of a system is given by
33
1 1 1 2 2 2( )N NNU U x y z x y z x y z= (26)
where U is the potential energy and x y and z are the locations of each of the N atoms in a periodic
box The locations of these atoms give rise to an energy as parameterized by an interatomic
potential An energy landscape can be constructed by mapping the relationship of the local minima
and the first order saddle points between pairs of minima22-27 Energy landscapes have become
crucial to studies of biomolecules glasses and catalysts[21] [26] [44] because landscapes show
atomic configurational changes and calculate the associated kinetics and thermodynamics In this
study we will explore the energy landscape of a barium disilicate system capturing the glassy and
crystalline basins enabling a test of CNT by calculating all needed parameters (interfacial free
energy kinetic barrier and free energy difference of the supercooled liquid and the equilibrium
state)
31 Crystallization Methods
CNT expresses the steady-state nucleation rate (I) as the product of a kinetic and a
thermodynamic factor
( )exp eI Z D TW
kT
minus =
(27)
where Ze is the Zeldovich factor which incorporates the probability of a critical sized nucleus that
will form a new phase independent of other nuclei D(T) is the kinetic factor often given by an
Arrhenius expression over sufficiently small temperature ranges W is the work associated with
generating a thermodynamically stable nucleus that will continue to grow rather than dissipate (ie
the critical work) if temperature and pressure are kept constant k is Boltzmannrsquos constant and T is
absolute temperature The Zeldovich factor is typically treated as a constant with a value ~01[118]
34
Here the kinetic term is expressed as D(T) since in this work the kinetics are explicitly calculated
at every temperature and it is not treated as a simple Arrhenius expression In CNT the work W is
given by the work to generate a spherical isotropic nucleus[1] which consists of a volume term
and a surface energy term It is given by
3 24
( ) 4 ( )3
W r G T r T = + (28)
where G is the Gibbs free energy difference between the crystalline state and the corresponding
liquid r is the radius and is the interfacial energy between the crystalline and liquid phases Eq
(28) neglects the strain energy term that arises from the formation of a solid nucleus in the
supercooled liquid It is assumed that the liquid relaxes fast enough so that strain energy is not
significant for nucleation in liquids[11]
Determining the critical radius (ie taking the derivative of Eq (28) with respect to the
radius and setting the expression equal to 0) allows one to solve for the critical work and the
expression becomes
3
2
16
)3(
( )
( )W
T
G T
=
(29)
Three temperature dependent functions remain D G and Though there are existing methods
in literature to approximate each of these quantities the approximations often include large errors
and assumptions about the behavior of the system[119] Often experimentally the kinetic function
is approximated with the Stokes-Einstein relationship the G term is calculated from heat
capacity curves (assuming that the macroscopic energy is similar to the nanoscopic energy) and
is fitted[119] In the calculation presented here only configurational energies are considered
while vibrational contributions are ignored Ignoring the vibrational component and the reliance on
empirical potentials are the most likely sources of error in this work
35
The energy landscape for this investigation was generated using ExplorerPy[89] starting
with the orthorhombic barium disilicate crystalline structure[89] Barium disilicate was chosen
since it is a commonly studied glass-ceramic with multiple reliable experimental datasets being
found in literature[75] [113] [120] The rest of this section is divided into four parts starting with
a discussion of mapping the landscape and the following three sections are a discussion of
calculating D G and respectively
31A Mapping and Classifying the Landscape
The ExplorerPy energy landscape exploration employed the Eigen-Trieban technique in
the canonical ensemble with systems consisting of 128 256 and 512 atoms[89] The number of
atoms was chosen to achieve convergence of the results while also balancing the large
computational cost associated with energy landscape exploration A smaller system was also
calculated with 64 atoms however this systems was too small to accurately discern energies
enough to make the nucleation calculation The Eigen-Trieban technique is a newly developed
method to explore energy landscapes using a method of shoving along the eigenvectors and
minimizing the potential energy which is followed by using nudged elastic band calculations to
find the associated transition points[89] This technique allows for larger transitions to be found
while retaining the ability to calculate the vibrational frequencies The Pedone et al potentials[36]
were used and the volume of the cell was fixed due to simulation instability when pressure was
fixed The Pedone et al potentials were chosen since they have reproduced key parameters for both
the crystalline and glassy state in previous Monte Carlo studies as well as in the training set of the
potential[73] [74] [121] Though the Eigen-Trieban method may work well for predicting
crystallization and finding large barriers it may be insufficient to describe glass relaxation or
36
viscosity since it rarely finds the small structural transitions that dominate the Adam-Gibbs
relationship Note that these exceptions are inconsequential for predictions of nucleation found in
the work presented here33
The barium disilicate system was explored for 2500 basins for 128 and 256 atoms and 4000
for 512 atoms since more basins were needed to converge the liquid state distribution of energies
This number of basins was shown to be sufficient since multiple paths from the initial crystalline
starting point to SCL states were obtained The distribution of energies of the SCL were found to
converge and the resulting energy landscape is seen in Figure 4
A crystal parameter was then defined from the root mean square displacement (RMSD) for
each atom from the nearest lattice site of the same species ( ) This average value is used to
characterize whether a basin is a crystal or SCL The RMSD approach is not the only method to
assign a crystallinity value[122] however it is one of the simplest and as such was chosen for
this application To then categorize each basin as either a crystalline basin or SCL basin a plot of
the minimum energy of the SCL states versus the cutoff of RMSD was produced This plot for 256
atoms shows a clear transition in the energies around a cutoff value of 1 Aring which is therefore
taken as the cutoff radius of which basins are considered crystals The same analysis was repeated
for each system size This analysis is shown in Figure 4 Any basin with a RMSD less than the
cutoff value is considered a crystalline state This cutoff was found to be generally insensitive to
the results with any values ranging between 09 Aring and 11 Aring changing the resulting prediction of
nucleation by less than one order of magnitude but shifting the peak temperature by up to 40 K
This cutoff also reproduced the energy difference between the SCL and crystal in molecular
dynamics simulations of the materials
37
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-axis being
an arbitrary phase space and the y-axis being the potential energy calculated from the Pedone et al
potentials[36] The colors are indicative of the crystallinity where the blue basin is the initial
starting configuration The landscape shows the lowering of energy associated with partially
crystallizing the sample (B) The energy landscape relationship between the cutoff for crystalline
and super-cooled liquid states for 256 atoms is shown This shows a clear drastic energy change
occurring around the cutoff value of 10 Aring
[
38
31B Kinetic Term for CNT
The transition rate was calculated by the inverse of the average relaxation time from the
SCL to the crystalline states The relaxation time ( ) is defined by the relaxation function ( )
0 exp
t
= minus
(30)
where t is time and 0 is the scaling of the relaxation function The reason we are able to
calculate the kinetics of nucleation but not viscosity is because the nucleation kinetics are
dominated by the transitions between two states that we specifically searched for when exploring
the energy landscapes while the viscosity is defined by the entropy dominating small transitions
described by the AG
The AG relationship shows that the viscosity is dominated by the configurational entropy
of the system Experimentally the kinetic term is often calculated with the Stokes-Einstein
relationship
( )6
kTD
aT
= (31)
where a is the radius of the diffusing species and is the viscosity of the liquid Interestingly this
approximation gives reasonable results when investigating experimental data since the viscosity is
a measurement of the entropy of small cooperative liquid rearrangements Conversely the kinetics
that govern nucleation are large structural rearrangements from a supercooled liquid to a crystal It
is nonetheless possible that these cooperative rearrangements and the large diffusion processes for
nucleation are on the same order of magnitude which is why the Stokes-Einstein relationship gives
39
a reasonable approximation in experimental parameterizations When the Stokes-Einstein
relationship is used there is an implicit assumption that the large nucleating rearrangements are
proportional to the small barriers that govern viscosity[53]
A mean relaxation time can be found by averaging over a system of master equations
( )j
ii ji j j ij i
i
dpg K p g K p
dt
= minus (32)
In this expression g is the degeneracy of a microstate pi is the probability of occupying the
microstate and Kij is a transition rate between inherent structure i and inherent structure j The
equilibrium probability of occupying a microstate can be given by a Boltzmann probability (where
Z is the partition function and U is the potential energy of the inherent structure)
1
exp ii i
Up g
Z kT
= minus
(33)
By substituting the value of jp in Eq (32) for a two-state model ( 1 21 i jp p= == + ) we write
( )( )1ii ji i j ij i
dpg K p g K p
dt= minus minus (34)
Isolating the probabilities and splitting the differential we have
( )1
i
i ji i j ij i
dpdt
g K p g K p=
minus minus (35)
which when solved via integration takes the form
ln ( )
( )
i ji i ji j ij i
i ji j ij
g K g K g K pt
g K g K
minus + =
minus + (36)
Solving for the occupational probability yields
( )expi ji j ij i ji
i
i ji j ij
g K g K g K tp
g K g K
minus minus + =
+ (37)
This shows an analytical solution for the relaxation time in a two-state model
40
1
i ji j ijKg g K
+
= (38)
The two-state model approximation is made to enable the calculation of the mean relaxation time
between the liquid basins to the crystalline basins The transition rates between the basins are given
by the vibrational frequency ( ) and the barrier between the states ( ijU )
exp ij
ij
UK
kT
minus=
(39)
The probability of transitioning from a given liquid basin i to a crystalline basin j is given by
1
expij
ij
Up
Y kT
minus =
(40)
where Y is the partition function with respect to all transitions from the current SCL basin to all
crystalline basins as only these transitions are considered to calculate the relaxation time
expj
j
iY
k
U
T
= minus
(41)
The total expected relaxation time of the system making the transition is then given by the
probability of occupying a basin the probability of transitioning from one basin to another specific
basin and the associated relaxation time
1
i ij
j i ji j ji i i
p pg K g K
=
+ (42)
The first summation is over the liquid states and the second summation is performed just over
crystalline states to capture the mean relaxation time from a supercooled liquid to a crystal We
fixed the vibrational frequency to a value of 17 times 1014 Hz (ie about one jump attempt per 5
femtoseconds) which was calculated using the average of the eigenvalues of each basin ( ) and
the average mass of the atoms (m) using
41
1
2 m
= (43)
31C Degeneracy calculations
To obtain an appropriate value for the degeneracy an understanding the phase-space
volume of the basin is needed since the degeneracy is linked to the volume of the basin[40] By
taking the displacement between the basin coordinates of each atom and the transition point we can
define a radius of the basin R Using R and a spherical approximation we can calculate the phase-
space volume degeneracies
3 23
12
NN
i i
cry
Ng R
g
+
=
(44)
In which ( )x is Eulerrsquos gamma function The degeneracy is then normalized by a constant such
that the liquid-liquid state transitions had a value of 100 s at the glass transition (980 K) to be in
agreement with Angell[58] The normalization step allows for a quantitative prediction of the
liquidus temperature to confirm the accuracy of our model The liquidus value found is
approximately 1200 K 1715 K and 1950 K for 128 256 and 512 atoms respectively The
experimentally determined liquidus temperatures range from 1680 K to 1710 K[113] [115] [120]
It is worth noting that changing the exponent drastically changes the thermodynamics while having
minor effects on the kinetics and the normalization value drastically changes the kinetics while
keeping the thermodynamics nearly constant We chose the denominator to satisfy the liquid
relaxation constraint It is also feasible to choose the exponent to satisfy a constraint on the liquidus
temperature The normalization value is an important to consider because for larger system a
change in 001 can result in a change in the kinetics by ~10 orders of magnitude
42
31D Free Energy Difference
Due to limitations in the exploration of the energy landscape requiring a fixed volume we
will approximate the difference in the Gibbs free energy by calculating the difference of the
Helmholtz free energy This approximation is reasonable due to the similar densities between the
crystalline and glassy phases under ambient pressure[73] [74] The first step is determining the
mean energy difference of all the basins and the second step was repeating the process for only the
liquid basins The mean energy was calculated with the following expression[123]
( )ln ln ii
i
i i iG kTp p U kT g p= + minus (45)
In Eq (45) i is the set of relevant basins for the conditions we are interested in The difference
between the mean free energy of the liquid basins and the entire landscape basin energies is the
G parameter This free energy difference is typically experimentally calculated by integrating
heat capacities however numerous approximations exist These approximations are often linear
approximations normalized to the liquidus temperature In this work we find that the driving force
appears to be predominantly linear which is in good agreement with the approximations that are
often used for the driving force[124]
43
Figure 5 An example interfacial structure between the crystalline phase on the left and the last
sequential SCLglass phase on the right for a barium disilicate system The gray atoms are barium
silicon is shown in red and blue represents oxygen
44
31E Interfacial Energy
To calculate the interfacial energy of the interface between the crystalline phase and the
SCL the initial basin (pure crystalline) and the last basin (glass basin) were annealed at 1700 K for
1 ns and then cooled to a temperature of 800 K over a period of 10 ns The same process was
repeated for a composite where the initial stable crystal and the glass were placed next to each
otherThe structure was multiplied 10 times to create a wall of a large interface to reduce energetic
fluctuations A small section of the composite structure is shown in Figure 5 The difference
between the addition of the energy of the crystalline phase and glassy phase and the composite
gives the energy of the interface which was normalized to the size of the surface area Multiple
arrangements of the composite were tested but anisotropy was found to be minimal with the
fluctuations of the energy being larger than the differences from different tested directions The
final expression used for the interfacial free energy (in eVAngstrom2) is
610 00 35047 040T minus +=
45
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a function of
temperature for each system size The experimental values for kinetics and thermodynamics come
from ref[125] and from the heat capacity data taken from [126] respectively It is clear to see that
driving force shows good agreement across all systems however the kinetics terms only converge
46
for the 256 and 512 atoms systems (Bottom) The fit used to calculate the interfacial energy as a
function of temperature
32 Results amp Discussion
The results for the free energy difference the kinetic parameter and the surface energy are
shown in Figure 6 These parameters as a function of temperature are used to calculate the
nucleation rates in the expression
3
2( )
01 16exp
3( )I
TT G k
= minus
(46)
The 01 being an approximate value of the Zeldovich factor Using the parameters shown in Figure
6 and solving Eq (46) the nucleation rates are shown in Figure 7 along with experimental data for
validation At first glance we notice that there is about 7 order of magnitude differences between
data and prediction Considering that theses potentials were trained on glass and were not intended
for this purpose we consider this as excellent agreement This method also shows agreement below
the peak but the accuracy falls off more quickly below the peak temperature It is likely that the
data below the peak is severely underestimated due to a failure to reach steady state nucleation
rates Reliable steady state nucleation rates are not available for this system and as such a direct
comparison below the peak temperature is not directly available It is worth mentioning that the
failure of CNT of predicting the order of magnitude of nucleation rates is well reported and can
amount to dozens of orders of magnitude when a constant value of is used[75] [113] [115]
[119] [127] thus the proposed method has an advantage when comparing with the current
classical nucleation theory toolset Moreover the prediction of Tmax (~950 K) is also very close to
the experimental value (985 K)
47
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted using the
model presented in this work The data referenced can be found in refs [113] [128] [129]
48
To further compare CNT and the model presented here we decided to compare the results
of the fitted and this workrsquos calculated surface energies as seen in Figure 8 The results show good
agreement all achieving the same order of magnitude implying that experimental methods can
achieve a close approximation to the values required for CNT The prediction of nucleation is
extremely sensitive to the value of the surface energy with small changes leading to orders of
magnitude change It stands to reason then that further work that needs to be approached through
computation and experiments is developing a rigorous model to predict the interfacial free energy
The other assumption that is used experimentally is the Stokes-Einstein relationship To
compare the landscape results of the transition rate from the SCL to crystalline states and the
Stokes-Einstein relationship the liquid state relaxation was calculated The small barriers that
govern the Adam-Gibbs relationship were not found with this exploration method hence
quantitative comparisons are not possible (and relaxation calculations should not be made)
however the relaxation times can be qualitatively compared The liquid state relaxation rate is
proportional to the viscosity and is given by the expression[44]
j
i
i
l
ij
p
K =
(47)
To understand the validity of the Stokes-Einstein relationship we can write a series of theoretically
true proportionalities
( )l
T TD T
(48)
The values of the experimental kinetics fail to align with the kinetics calculated in this work This
is expected due to the fact that the viscosityliquid relaxation is dominated by the set of small
transitions in the landscape while the nucleation transition is defined by the transition of one region
to another with a substantially higher activation barrier This comparison may also account for
49
some part of the experimental overestimation below the peak with the kinetic term calculated
dropping off at a faster rate than that calculated from the experimental viscosity Figure 6 also
shows evidence that it may be inappropriate to use the Stokes-Einstein relationship for nucleation
and as such we may need to re-visit the experimental models that often rely on the assumption of
the Stokes-Einstein relationship This observation casts some doubt in the legitimacy of describing
the kinetic factor as a diffusivity which is standard for CNT For more information on this CNT
failure please see Refs [75] [113] [115] [119] [125] [127]
Nucleation is a complicated concept to investigate and model Unlike the crystalline phase
where the atomic positions are clearly defined the details of the atomic coordinates and bond
configurations of the liquid sites must be described in terms of statistical distributions Due to this
added complication mean-field distributions are typically used to describe non-crystalline
sites[14] [16] [108] [130]ndash[133] The incorporation of fluctuations could more accurately predict
each liquid sitersquos transition to the crystalline phase Fluctuations in the degeneracy could be one
reason that the nucleation rate is over predicted when the temperature is less than the peak
nucleation temperature A given G varies with the environment and depends on the specific
liquid state not the mean value Therefore the rate of nucleation from all supercooled liquid states
to the crystal state is not constant which is why fluctuations need to be considered and why energy
landscapes are such a powerful technique This is especially highlighted when considering that
different degeneracies yield drastically different heat capacity Energy landscapes are extremely
useful because they consider a large distribution in free energies and show all structural fluctuations
possible Fluctuations in cluster size and degeneracy influence Adam-Gibbs configurational
entropy which directly impact the transition rate between the crystal and liquid state
50
Figure 8 The surface energy with respect to temperature for the work presented here
0
0002
0004
0006
0008
001
0012
0014
800 850 900 950 1000 1050 1100
Su
rface
En
ergy (
eVAring
2)
Temperature [K]
This Work
Xia et al (2017) Ref 23
Fokin et al (2016) Ref 41
51
33 Conclusions
Though this is not the first time CNT has been examined computationally in an attempt to
understand if the theory is valid[74] [116] [117] it is the first parameterization of each variable
independently in an attempt to calculate the resulting nucleation rate Not only are all the variables
calculated when used in the expression for CNT the liquidus temperature is reproduced and the
nucleation rate is reasonably predicted This is as far as the authors know also the first time a real
energy landscape for a complicated glass-ceramic system is being reported This method leveraging
energy landscapes alongside CNT enables for a reasonably fast calculation of the nucleation curve
of a glass forming liquid In future work this model could be used for either developing commercial
glass-ceramics or for predicting the critical rate needed to maintain a glass This model can also
provide additional insights into the physics of nucleation and the validity of CNT without the
constraints of fitting parameters This method fundamentally has the power to connect the 3N
dimensions of the energy landscape to the compositional dimensions and thus allowing for a unique
powerful method of predicting nucleation parameters as a result simplifying the crystalline
dimensions in which a glassglass-ceramic must be optimized over
Chapter 4
Expanding the Current State of Relaxation
Before we consider a new approach to relaxation (optimizing our understanding of
the 3N dimensions) there are first missing pieces in the current state specifically the temperature
and compositional dependence of the stretching exponent However even before expanding the
current state of the art relaxation methods we also need to understand ergodicity In this section we
will start with a simple though experiment to understand the temporal effects of relaxation and then
in the second section combine it with fictive temperature to create a new model for the stretched
exponent
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition
The prophet Deborah sang in the book of Judges ldquoThe mountains flowed before the Lordrdquo
[6] [13] However the question the prophet failed to ask was what happens if the mountains are
moving at relativistic speeds The focus of this section is to highlight the temporal effects on
ergodicity in the context of special relativity and the reference frame of the observer In order to
understand relativistic effects on materials one must consider a well-characterized material that
actively relaxes but appears solid on our timescale viz pure B2O3 glass [134] [135] Glass is a
particularly good candidate to study the effects of time dilation on a material since it undergoes a
kinetic transition known as the glass transition (Tg) which has multiple definitions that will be
considered
53
The first definition is in terms of the aptly named ldquoDeborah Numberrdquo which was proposed
by Reiner [136] to offer a view of fluidity and equilibrium mechanics It was later expanded to
explain the origin of the glass transition and the transition from an ergodic liquid system to a non-
ergodic solid-like glassy state[12] which is the fundamental definition of the glass transition [11]
[12] [137] It is important to note here that the breakdown of ergodicity is relative to the
observation timescale and is reflected in the Deborah number (D) which is defined as
Dt
= (49)
where t is the (external) observation time and is the relaxation time of the material which can be
expressed through the Maxwell relation [42] By definition the Deborah Number equals 1 at the
glass transition temperature Tg [13] [137]
Another definition of the glass transition temperature is due to Angell who defined Tg as
the temperature where the viscosity of the supercooled liquid is 1012 Pa s [58] [138] The Angell
diagram is an important plot relevant to viscosity and the glass transition where the abscissa is TgT
and the ordinate axis is log10( ) In the Angell diagram the value of the viscosity at Tg and in the
limit of infinite temperature are fixed and the difference in the temperature scaling of the TgT-
normalized viscosity is the slope of the log10( ) curve at TgT = 1 which Angell defined as the
fragility (m) of the supercooled liquid Using this definition one can establish the consistency of
the Angell plot at relativistic speeds An example of the Angell plot is shown in Figure 9
54
Figure 9 An Angell diagram created using the MYEGA expression [55] With a very strong glass
(m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) [17] [134] [135] The
infinite temperature limit is from the work of Zheng et al[54] and the glass transition temperature
is from the Angell definition
55
Using the two previous definitions it is expressed
1gT
GtD
= = (50)
which can be re-written as
(
(
)
)g
g
tG T
T= (51)
In order to solve this equation one must know the shear modulus at the glass transition temperature
recent improvements in topological constraint theory have led to the ability to predict properties
with varying temperature and composition Using a recent topological constraint model for elastic
moduli proposed by Wilkinson et al [139] one can predict the temperature dependence of the
modulus
( ) ( ) ( ) ( ) ( )( ) ( )Δ Δ Δ Ac c C Ac
c c
F n q T F n q T F n q T x Nn q T x NdG dGG
d F M d F M
+ + = =
(52)
In Eq (5) nc is the number of topological constraints (c) qc is the constraint onset function as
described by Mauro et al [69] Fc is the free energy of the constraint is the density of the
system M is the molar mass and 119889119866
119889120549119865119888 is a scaling factor This expression was evaluated as a
function of temperature with the total modulus coming from Kodama et al [135] and is equal to 8
GPa at room temperature The B2O3 structure consists entirely of three-coordinated boron [140]
[141]
Using the modulus predicted in Eq (52) numerical observation time is expressed
56
12
9
10 Pa s141 s
709 10 Pat = =
(53)
Using this description of the static (vc=0 where v is the speed of the glass and c is the
speed of light) glass transition it is possible now to describe the relativistic glass transition for
which there are considered two separate cases It is worth noting that this is not the first case study
of relativistic viscosity a fairly large basis of literature dealing with ideal fluids near large
gravitational bodies [142] There is also some work also relating the relaxation time in a
gravitational field to the nonequilibrium entropy relaxation time [143] [144] This work
emphasizes the theoretical effects of special relativity on real liquids their glass transitions and
the properties there dependent on
41A Relativistic Liquid
The first case is when a sample of a glass-forming system moves past a stationary observer
at a speed approaching light giving the new observation time as
0tt
= (54)
where t0 = 141 s (as previously calculated) and g is the Lorentz factor
2
2
1
1v
c
=
minus
(55)
The Maxwell relation can be modified to account for relativistic effects by combining with Eq (54)
0
()
)(
g
gG TT
t
= (56)
57
This gives a condition that is temperature dependent to predict the new glass transition as a function
of v however in order to accurately describe the relativistic behavior density must be considered
Density will change by a factor of 2 because both the mass and the volume will be affected and
as such
0) (( )g gG TT t = (57)
The prediction then for the glass transition as a function of the speed of light is shown in Figure
10
Figure 10 The relativistic glass transition temperature for B2O3 glass
58
Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature
which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) expression [55] [145] The MYEGA equation needs 2 other parameters
besides Tg to generate the viscosity curve fragility (m) which is the slope at Tg in the Angell plot
and the infinite temperature viscosity () The fragility will not change because one assumption
in this model is the consistency of the Angell plot at all relativistic velocities even relativistic The
infinite temperature viscosity which also will not change because this is the minimum possible
viscosity for a liquid The new viscosity curves for various speeds are shown in Figure 11
59
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different fractions of
light speed All of the viscosities approaching the universal temperature limit for viscosity
60
By using the Deborah Number it has been shown that at relativistic speeds will change the viscosity
in order to maintain the Deborah number and in this case it is shown that when the Lord would
move past the mountains at relativistic speeds the mountains would not actually flow but instead
hold still (even more so than they currently do) From this it is inferred that any liquid moving past
the earth at relativistic speeds will appear to be more solid and eventually appear as a glass
41B Relativistic Observer
In this case we consider an observer moving past a sample of glass at relativistic speeds
giving
0t t = (58)
which when the relativistic shear modulus 2 factor is included can be written as
( )
( )0
g
g
TG T
t
= (59)
Solving in the same manner as before it is shown which shear modulus is needed to satisfy the
condition in Figure 12
61
Figure 12 The modulus needed to satisfy the condition for the glass transition
62
In Figure 12 the shear modulus at the glass transition temperature is shown however the
constraint theory mechanism being implemented to calculate the shear modulus has a built in upper
limit when calculating the Tg past the point in which all the constraints are intact Due to this
limitation there is an upper limit to our prediction Nonetheless the calculated Tg over the available
range are shown in Figure 13 as well as the related viscosities It is seen clearly that at higher
velocities the glass transition shifts towards zero more dramatically at the higher temperatures
This leads to the interesting result that if we take the limit of an observer moving past the earth at
close to the speed of light all glass would appear to be a liquid
63
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at v=044c
(Bottom) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches
the speed of light
64
Applying the mechanisms proposed by Einstein to the concept of the Deborah number it
has been shown that the viscosities will appear to change dramatically as a glass-forming system
approaches relativistic speeds This allows us to build an intuitive understanding of the differences
between glass and liquids Specifically any material can appear either more or less fluid based on
the velocity with the velocity with which they travel relative to the observer This serves to highlight
the importance of the role of the observer on the glass transition and why even small changes in
DSC scan rates or sampling frequency can lead to drastically different measured glass transitions
42 Temperature and Compositional Dependence of the Stretching Exponent
The mathematical form for glass relaxation ( ) was originally proposed by Kohlrausch in
1854 based on the decay of charge in a Leyden jar[9] [14] [29] [42] [45] [96] [146]
( ) ( )( )exp t t
= minus (60)
where t is time t is the relaxation time of the system and b is the dimensionless stretching exponent
Eq(1) was originally proposed empirically and is known as the stretched exponential relaxation
(SER) function[42] [96] The relaxation time t depends on the composition temperature thermal
history pressure and pressure history of the glass as well as the property being measured[62]
[147] For example the stress relaxation time of a glass can be written as[62]
( )( )
ff
s
f f
T T P P
G T T P P
= (61)
where h is the shear viscosity and G is the shear modulus of the glass-forming system[42]
(Variables are defined in Table 2) In addition to using Eq(61) to describe stress relaxation time
structural relaxation time has recently been hypothesized follow Eq(61) with bulk viscosity and
65
the difference of bulk modulus between infinite and 0 frequency replacing shear viscosity and shear
modulus[62] Both h and G vary with the temperature (T) and thermal history of the glass (as
quantified via the fictive temperature Tf) as well as the pressure (P) and its corresponding history
(fictive pressure Pf)[16]
To connect the stretching exponent to its physical origins we can begin with a result from
Richert and Richert[148] that relates to an underlying structural relaxation time distribution and
then develop equations that determine from known quantities Their expression
2 22
ln 2
1
6
minus=
(62)
relates to the variance of the logarithm ( )2
ln of the structural relaxation time For reference
all variables are defined in Table 2 The physical origins of the relaxation time are related to the
configurational entropy (Sc) as shown by the Adam-Gibbs equation[53] [55] [149]
ln lnc
B
TS = + (63)
where is the infinite temperature relaxation time T is the absolute temperature and B represents
the energy barrier for relaxation
Table 2 Variable definitions
Variable Definition
t ts Structural relaxation time stress relaxation time
b Kohlrausch exponent (ie the stretching exponent)
T Tf Tfi Absolute temperature fictive temp partial fictive temp
P Pf Pressure fictive pressure
66
kB Boltzmannrsquos constant
wi ki i=1 to N Prony series parameters
B Adam-Gibbs relaxation barrier
Sc Adam-Gibbs configurational entropy
B Distribution of activation barriers
2 Variance
Na Number of atoms
f Topological degrees of freedom per atom
Degenerate configurations per floppy mode
H Enthalpy barrier for Relaxation
d Number of dimensions
f Fraction of activated relaxation pathways
S Adam-Gibbs entropy in the infinite-temperature limit
m Kinetic fragility index
m0 The limit of a strong liquid
Grouped unknowns 6
ln
B
a BN k
A Proportionality between fragility and distribution of barriers
x A given composition
125640
S
A
Intercept of the linear model of
67
42A Deriving a Model
Combining Eqs(62) and (63) Gupta and Mauro[150] proposed that the variance of the energy
barriers for relaxation (2
B ) could be rewritten as
2 2
22
2
1
6( )B c TTS
minus=
(64)
Solving for the stretching exponent
( )22 (
6
( )
)
c
B c
S T
T
T
TS
= +
(65)
The work of Naumis et al[23] [56] has shown that the configurational entropy is proportional to
the topological degrees of freedom in the network a result that was used in the derivation of
temperature-dependent constraint theory[57] and the MYEGA (Mauro-Yue-Ellison-Gupta-Allan)
equation for the relaxation of supercooled liquids[55] The MYEGA equation was derived by
expressing the configurational entropy as
( ) ( ) lna BcS T x f T x N k= (66)
In Eq (66) f is the topological degrees of freedom per atom Na is the number of atoms kB is
Boltzmannrsquos constant x is a given composition and is the number of degenerate configurations
per floppy mode The temperature dependence of the topological degrees of freedom was
approximated using a simple two-state model
( )( ) 3exp
B
H xf x T df
k T
minus= =
(67)
Here H(x) is the enthalpy barrier for relaxation d is the dimensionality of the system and f is the
68
fraction of activated relaxation pathways Combining Eqs (65) and (67) as well as condensing the
unknowns into the term defined by
6
ln
B
a BN k
=
(68)
we get
( )
22
df
dfT
=
+
(69)
This can be compared to the prediction made by Phillips[48] for the stretching exponent at
temperatures below the glass transition
2
df
df =
+ (70)
Comparing the two expressions of Eq(69) and Eq(70) they would agree if
4 4dfT
= + (71)
We will show later (Figure 12) an example where extrapolating our model prediction for to room
temperature (with T in Eq(71) replaced by fictive temperature Tf) gives a result close to the Phillips
value of Eq(70) When the temperature T in Eq(69) is high (much larger than ) then Eq(69) gives
us rarr1 Thus the model can interpolate between the low-temperature (Phillips) value and the
high-temperature value of 1 In our current model we do not have access to a value for in Eq(68)
so this model cannot assess whether the Phillips room temperature value for is universal ie
whether Eq(71) is always satisfied at room temperature
Zheng et al[151] showed that the Mauro-Allan-Potuzak (MAP) model for the relaxation
time of the nonequilibrium glassy state[44] implies that the configurational entropy can be written
as a function of thermal history (Tf) the fragility index (m) the fragility limit of a strong liquid
69
(m0) the glass transition temperature (Tg) and the limit of infinite temperature configurational
entropy ( S ) as
0
) exp 1 0
(ln1
c f
f
gT
TS m
TS
m
= minus minus
(72)
The qualitative relationship between the distribution of activation barriers and fragility was
proposed by Stillinger[18] [19] who suggested that the energy landscape of a strong liquid (low
fragility) has a small distribution of activation barriers and that a higher fragility is associated with
a broader distribution of activation barriers ie a higher variance of the activation barriers This
leads to a phenomenological relationship that is here assumed to be valid based on Stillingerrsquos work
on energy landscapes[18] [19]
2
0
1B
mA
m
= minus
(73)
where A is some constant of proportionality This expression was chosen because in the limit of a
strong liquid there is an infinitely sharp distribution of activation barriers Combining Eqs(65)
(72) and (73) we obtain the temperature dependence of the stretching exponent for a liquid
( )
0
2 2 2 2
0 0
exp 1
( ) 2
6 ln10 1 exp 1
g
f
f
f
g
f
f
T mT S
T mT
Tm mA T S
m T m
minus minus
=
minus + minus minus
(74)
Fictive temperature appears in this new expression because we are deliberately expressing this
function as an equilibrium model Rewriting Eq (74) in terms of one unknown () and grouping
constants
70
( ) 0
2 2
0 0
exp 1
2exp 11
g
f
f
f
g
f
f
T m
T m
T m
T
T
Tm
T
m
m
minus minus
minus minus
=
minus +
(75)
Eq (75) is an expression for the stretching exponent as a function of thermal history glass
transition and fragility index with only one unknown In Eq (75)
( ) 12
6 ln10
S
A
= (76)
The only unknown for the compositional dependence of the stretching exponent is the value of D
The composition-dependent part of Eq(76) is Syen
since we approximate A to be independent of
composition The fragility dependence of Syen
was proposed by Guo et al [61]
Syen(x) = S
yen(xref
)expm(x) -m(x
ref)
m0
eacute
eumlecircecirc
ugrave
ucircuacuteuacute
(77)
where x is composition and xref is a reference composition in the same glass family Seeking the
simplest possible expression to approximate the unknown D we take the natural logarithm of
Eq(76) and of Eq(77) and combine them to get
0
ln lnm
m = + (78)
with the additional definition
( )( )
( )12
0
ln l6 ln10
nref refS x m x
mA
minus
=
(79)
The result is that ln D varies linearly with fragility m and the intercept ln D is independent of
composition (depends on one reference composition)
In Figure 9 we plot the data of Boumlhmer et al[152] vs the predicted exponent The work of
71
Boumlhmer et al is the summary of the literature data relating fragility index to the stretching exponent
at the glass transition which we use to fit values to Eq (78) The dataset included in their work
covers chalcogenide oxide and organic glasses The fitting (consisting of least-squares
minimization of the difference between the predicted stretching exponent and that which was
reported) was done twice once for organic and once for inorganic systems During the fit it was
assumed that Tg=Tf Some assumption about thermal history was necessary since the individual
thermal histories or Tf values are not known for this whole collection since we are trying to track
overall trends in values vs composition this reasonable simplifying assumption is consistent
with our program Using Eqs(75) and (78) the temperature and compositional dependence of the
stretching exponent can be described with only one free parameter
72
Figure 14 b predicted and from literature showing good agreement with a total root-mean-square
error of 01 The fit for organic systems is given by 66 ln Kln = minus and for inorganic systems by
75ln Kln = minus
73
42B Experimental Validation
Experimental density measurements were made using Corningcopy JadeTM glass as described
elsewhere[146] Changes in the density are normalized to obtain the relaxation functions which
are fit with Eq (60) The resulting values of are shown in Table 3 A comparison in Figure 15
is shown with data obtained on a soda-lime silicate SG80 where the stretching exponent was
obtained using the measurement of the released enthalpy as a function of isothermal annealing time
during relaxation below the glass transition The measurement of the released enthalpy[153] relies
on the change in the excess heat capacity in the glass transition range as a function of annealing
time Normalized released enthalpy relaxation functions for each temperature were fit with Eq(60)
to obtain the stretched exponent values for SG80 In Figure 15 parts A and B are samples measured
below Tg while in part C the samples measured are at Tg and above The values were then fit with
Eq(75) where the only free parameter was D with an equilibrium assumption (Tf = T)
74
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy JadeTM
glass and Sylvania Incorporatedrsquos SG80
Glass Temperature (K) Tg (K) TgT m D (K-1) Beta t (s)
Corningcopy JadeTM 1031 1074 1042 32 0002 0504 7249
Corningcopy JadeTM 1008 1074 1065 32 0002 0447 32458
Corningcopy JadeTM 973 1074 1103 32 0002 0395 419214
SG80 800 800 1 36 00057 0632 92
SG80 783 800 1021 36 00057 0593 182
75
Figure 15 The equilibrium model proposed with the experimental points showing good agreement
between the experimentally measured data points and the equilibrium derived model RMSE was
002 for Corningcopy JadeTM (A) and less than 001 for SG80 (B) (C) The model fit for B2O3
experimental data[154] [155] The fragility and glass transition temperature of the B2O3 are taken
from the work of Mauro et al[17]
(
A)
(
B)
(
C)
76
A combination of the MAP model for non-equilibrium shear viscosity[44] the model
presented here primarily in Eq(75) and the model presented by Wilkinson et al[139] for the
temperature dependence of elastic modulus allows for fully quantitative modeling of stress
relaxation behavior The missing model required to understand structural relaxation is the bulk
viscosity curve[62] All previous relaxation (structural or stress) models[42] [61] have relied on
approximations that use a constant exponent and on a constant (temperature-independent)
modulus value whereas here every parameter of Eq(60) may be modeled as a function of
temperature Furthermore in combination with the relaxation models described by Guo et al[61]
in which multiple fictive temperatures are described using a Prony series and a temperature-
dependent modulus one can construct a relaxation curve accounting for the temperature
dependence and thermal history dependence of all relevant parameters
( )
( )( )
1
( )( )exp exp
( )) (
fTN
i
i
f
f
f fi
G T k T tG T tw
T TT
T T
=
minus minus
(80)
Here wi and ki are fitting parameters that are completely determined by the value of and each
term in the Prony series is denoted with the subscript i Usually 8 or 12 terms are included in the
Prony series[96] Each term in the Prony series is assigned a partial fictive temperature (Tfi) whose
relaxation is described by the simple exponential in the Prony series Equilibrium conditions are
assumed at the start of the simulation which allows for a known set of starting probabilities within
the energy landscape interpretation of relaxation The same model also includes the fragility index
and glass transition dependence of the non-equilibrium shear viscosity This method is
implemented in RelaxPy[42] as discussed in the next section This serves as an approximation for
the evolution of the of the non-equilibrium state however the temperature dependence of the bulk
viscosity and a replacement for fictive temperature need to be quantified to improve the
understanding of the underlying physics[1] [16] [62]
77
Separately we can explore the relationship between this model given by Eq(75) and the underlying
energy landscape In order to better understand the stretching exponent consider that there exists
a full set of parallel relaxation modes within an energy landscape The relaxation modes are then
weighted by the occupational probability corresponding to a particular mode This gives a series of
transition rates with some probability prefactor and an associated relaxation time (scaled from the
mean relaxation time) which gives rise to a Prony series form of Eq(80) Thus the distribution of
relaxation times (or barriers) determines the evolution of the stretching exponent while the average
barrier determines the mean relaxation time If this Prony series description in turn describes the
stretching exponent we arrive at a physical description and origin of the stretching exponent As
the temperature approaches infinity even though there is a distribution of activation barriers the
distribution of relaxation times approaches a Dirac delta function and the stretching exponent
approaches one (a simple exponential decay) As the temperature decreases the distribution of
relaxation times broadens due to the wide distribution of barriers however due to the broken
ergodic nature of glass the value of the stretching exponent will be controlled by the instantaneous
occupational probability
The activation barriers may be modeled eg using a numerically random set of Gaussian
distribution of barriers[156] The probability of selecting an individual transition is then calculated
using a Boltzmann weighting function the transition also has an associated relaxation time with it
The Prony series is then recreated with the probability multiplied with the simple exponential
relaxation form The sum of all of these terms is then fit with the stretched exponential form Figure
11 shows the stretching exponent calculated for the mean activation energy of 10 eV and three
choices of standard deviation 25 2 15 ande e e minus minus minus= eV as shown in the legend where Eq(60) has
been fit to the Prony series calculated from an equilibrium Boltzmann sampling of a Gaussian
distribution of barriers The temperature dependence for the stretching exponent matches the form
78
derived earlier in this work This stretching exponent shape has been shown previously though not
in a closed-form solution[156] This secondary method not only validates the generalized form of
Eq(75) but also highlights the validity of using the configurational entropy and barrier distributions
as the underlying metric controlling the temperature dependence of the stretching exponent
Figure 16 The stretching exponent calculated as described in the text for a Gaussian distribution
of barriers This plot shows that the distribution of barriers has a large effect on the stretching
exponent A Tg cannot be described since there is no vibrational frequency included in the model
though the glass transition temperature should be the same for all distributions since the mean
relaxation time is the same for all distributions at all temperatures The deviation is given in ln eV
units
79
To create software to model this complex behavior the authorsrsquo existing software RelaxPy[42] has
been modified into a new version RelaxPy v20 Instead of fixed values for the Prony series the
values are chosen dynamically to match the stretching exponent as predicted by Eq(75) The
database of values wi() and ki() was created with a fixed number of terms N in Eq(80) (we chose
N=12) This database is available in the same Github repository as the software The parameters
were fit by starting with the values obtained for =37 reported by Mauro and Mauro[96] to make
the new version of the model smoothly reproduce a prior optimal fit then was stepped by 001
(when less than 095) or by 0001 (when greater than 095) Using the previously optimized values
of wi and ki for starting values for each new the wi and ki values were then varied to minimize
the root-mean-square error between the stretching exponent form and the Prony series form ie to
satisfy Eq(80) with least error This list of compiled values makes up the contents of a database
that the software accesses The values of wi and ki used for a given are calculated by finding the
closest value in the database to the given value of This allows for an efficient implementation
of relaxation modeling Figure 12 shows the fitted parameters to the Prony series and an example
RelaxPy output for Corningcopy JadeTM glass undergoing quenching at a rate of 10 Kmin Note that
the room temperature stretching exponent value is 046 a close match the Phillipsrsquos ldquomagicrdquo value
of 37 (043)[157] This prediction for Corningcopy JadeTM is a non-equilibrium prediction in contrast
to the equilibrium high-temperature prediction shown in
80
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent Each color
designates one term in the series (Bottom) The output from RelaxPy v20 showing the stretching
exponent effects on the relaxation prediction of Corningcopy JadeTM glass Each quadrant shows one
property that is of interest for relaxation experiments In particular it is interesting to see the
dynamics of the stretching exponent during a typical quench
81
43 Conclusion
In this section we have sought to expand the current understanding of relaxation and improve the
models that currently exists We have proposed a model that accounts for the effects of special
relativity as well as a model was derived through an understanding of the distribution of barriers
for relaxation and the temperature dependence of the Adam-Gibbs entropy The model outlined
herein describes the temperature dependence of the stretching exponent in glass relaxation
The work done on the relativistic effects on relaxation illustrates the crucial role of the observer on
all relaxation models and offers an extreme in which to test common relaxation models The model
for the stretching exponent not only considers the extremes at high and low temperature (when
compared to the glass transition) as in the Phillips model but also for any intermediate temperature
as a function of the fictive temperature This model does not have any explicit temperature
dependence since it was assumed that an equilibrium model works well to describe the
instantaneous distribution of relaxation times Given the physical argument and the success of this
model when tested by experiments and by another model it is at least reasonable to formulate the
temperature-dependence of in terms of its fictive-temperature-dependence as we have done here
Including both T and Tf is possible to consider but lies outside our current scope Using previously
derived compositional dependence for the MAP model a fragility index dependence of the
stretching exponent was defined and tested The model was confirmed using multiple experimental
datasets In addition a theoretical comparison to a distribution of landscape activation barriers was
found to reproduce the same trends as the model
Chapter 5
Glass Kinetics Without Fictive Temperature
Fictive temperature as discussed through this text is unable to capture key physical
phenomena[16] [42] [51] To circumvent this limitation we rely on two different frameworks
The Adam-Gibbs relationship and generalized features of the energy landscape These two features
allow us to create a method that is completely generalizable and does not involve fictive
temperature This method creates fake landscapes that generally capture the trends of glass based
on experimental measurements This new method we have called lsquoToy Landscapesrsquo
51 Background of the Adam Gibbs Relationship
Experimental evidence in support of the AG model comes from the work of Richet[158]
in which they compared the predicted configurational entropy from viscosity curves with that
obtained from DSC The configurational entropy was calculated from DSC data using
0
0( )p p vi
T
T
b
c c
C CS S T dT
T
minus= + (81)
In Eq (81) Cp is the isobaric heat capacity Cpvib is the vibrational heat capacity and T0 is the initial
temperature from which the configurational entropy is integrated The difference between the total
heat capacity and the vibrational heat capacity is the heat capacity due to structural rearrangements
Richet found good agreement between the measured viscosity curve and calorimetric
configurational entropy at temperatures above the glass transition confirming validity of the AG
relationship When applying Eq equation reference goes here all temperatures must remain above
the glass transition ie in the liquid or supercooled liquid state The calculation is invalid upon
83
cooling through the glass transition due to the breakdown of ergodicity[12] [13] [159] and the
irreversibility of the glass transition process Application of Eq (6) to temperatures below the glass
transition leads to an incorrect calculation of excess entropy in the glassy state[160]
In this study we expand on these previous works to understand the relationship between
viscosity and the underlying energy landscape and provide new insights into the thermodynamics
of liquids through a simplified ldquotoy landscaperdquo model constructed from experimentally measured
viscosity parameters The energy landscape framework is especially helpful for elucidating the
thermodynamics and kinetics of supercooled liquid and glassy systems Energy landscapes describe
the evolution of all atomic transitions for kinetic processes in a system To perform such
calculations information about the inherent structures and transition points in a landscape must be
known In this work we solve the inverse problem of deducing realistic landscape parameters using
the AG relationship and experimentally measured data This information is used to construct a ldquotoy
landscaperdquo model to describe glass relaxation processes To achieve an accurate relaxation model
the following steps are taken to ensure the validity of the results
A Confirm the AG model for shear viscosity using energyenthalpy landscape
calculations
B Validate the assumptions made by the MYEGA model
C Confirm the AG model for bulk viscosity
D Explore the fundamental relations between viscous properties and topography of the
landscape
E Use the knowledge gained to propose a new thermodynamic model to calculate the
driving force for glass relaxation and the scaling of the liquidrsquos free energy
This chapter is organized into sections devoted to each of the topics above preceded by a
methods section All of the sections are presented to validate a new model of glass relaxation that
is sufficiently complex to achieve a calculation of the thermodynamics and kinetics of the liquid
This new model will be based on understanding the landscape from experimental properties and
we have called this approach the ldquotoy landscaperdquo model
84
52 Methods
Potential energy landscapes (fixed volume) and enthalpy landscapes (fixed pressure) are
powerful modeling techniques where atomic configurations are mapped in a 3N-dimensional phase
space (or 3N+1 dimensions for enthalpy landscapes)[89] The energy landscape approach is based
on mapping the continuous 3N dimensional space to a discrete set of energy minima (called
ldquoinherent structuresrdquo) and first-order saddle points (ldquotransition pointsrdquo) To perform this mapping
a systematic search for inherent structures is performed while also mapping the lowest-energy
transition points between each pairwise combination of adjoining minima The landscape itself is
partitioned into basins which represent the set of all configurations that minimize to a common
inherent structure[161] This combination of the basin and transition point information results in a
topographic map of the 3N phase space (either a partial or whole mapping) that can be used to gain
insights into the physics of any atomistic system such as proteins[91] liquids[18] [19] glasses[1]
[93] and nucleating crystals At equilibrium the probability distribution for occupying the various
basins in the landscape is given by[13]
exp
exp
i
j
i
i
j
j
Eg
kTp
Eg
kT
minus
= minus
(82)
where Ei is the potential energy (or enthalpy) of the inherent structure gi is the degeneracy of the
basin i is the basin index and pi is the probability of occupying a given basin i The denominator
is a normalization factor given by the summation over all basins Mauro et al[44] previously used
this formalism to study the non-equilibrium viscosity calculate the evolution of configurational
heat capacity and elucidate the long-time relaxation kinetics of selenium glass[44] [93] [160] For
equilibrium systems the configurational entropy is given by the Gibbs formula
85
lnc i
i
iS k p p= minus (83)
In order to validate the AG model the landscapes of several common systems were
explored using ExplorerPy[89] ExplorerPy is a software program built specifically to map
landscapes using the LAMMPS molecular dynamics package[30] The three systems we explored
are B2O3[162] (150 atoms and 2000 basins) SiO2[162] (150 atoms and 1000 basins) and BaO-
2SiO2[121] (128 atoms and 1500 basins) All systems used periodic boundary conditions The
initial structures for B2O3 and SiO2 were constructed by placing atoms corresponding to the
chemistries of the systems on a random grid (with 1 Aring spacing) corresponding to a density of 20
gcm3 and equilibrated at 2500 K then quenching at a rate of 1 Kps to room temperature using the
Wang et al[162] potentials in the NPT (constant pressure number of atoms and temperature)
ensemble The energy landscape of the BaO-2SiO2 system was taken from our previous work[72]
The SiO2 (constant pressure exploration) system was explored using ET with a push distance of 03
Aring and a pressure of 10 atm B2O3 was explored using isobaric MD at a temperature of 1500 K for
500-time steps of 1 fs also at a pressure of 1 atm The minimization (after MD or ET) was handled
by the LAMMPS[30] steepest descent algorithm until the certainty in energy was smaller than 10-
4 eV Both used a nudged elastic band spring constant of 10 eVAring2 The energy distribution for all
systems was fully sampled which was confirmed by randomly checking if half the basin
distributions matched the full distribution It was also found that multiple structures in each
exploration were discovered more than once and a slightly smaller B2O3 system (130 atoms) was
also explored and showed identical results to the 150-atom distribution The insights gained from
the interaction between the experimentally accessible viscous properties and the landscapes
presented here will be generalized to create a simplified ldquotoy landscaperdquo model to capture the
thermodynamics and kinetics of the glass-forming system
86
53 Results
53A Adam-Gibbs Validation
To predict viscosity using the Adam-Gibbs expression there are only three required
parameters the configurational entropy (as a function of temperature) the barrier for cooperative
rearrangements and the infinite-temperature limit of viscosity Using a landscape the equilibrium
configurational entropy can be obtained using Eq (83) The infinite-temperature limit of shear
viscosity is given by the work of Zheng et al[54] where a systematic study of MYEGA fits to
viscosity data revealed a common value of 10-293 Pamiddots This infinite temperature value of viscosity
is used for all the experimental and computational viscosity curves shown in this work The final
parameter is the mean barrier for a cooperative rearrangement In Figure 18 the barrier for
cooperative rearrangements was chosen to accurately produce a viscosity of 1012 Pamiddots at the
experimentally measured value of the glass transition temperature From these parameter values
the predicted Adam-Gibbs viscosity curve is plotted The universal infinite limit fragility and glass
transition values from references were inserted into the MYEGA equation to plot the
lsquoExperimentalrsquo curves for B2O3 SiO2 and BaOmiddot2SiO2 respectively[58] [69] [120] When
examining Figure 18 it is clear that the AG model is able to fundamentally recapture the viscosity
87
Figure 18 The viscosity (left) and landscape (right) predictions for three common systems The
first system is newly calculated in this work while the latter two come from our previous works[72]
[89] It is seen that the viscosity predicted from the AG model is very accurately able to reproduce
the experimental viscosity curves from the MYEGA model The last system is a potential energy
landscape while the others are enthalpy landscapes
53B MYEGA Validation
Now that we have confirmed that the configurational entropy of the landscape works well
to describe the viscosity via the AG relationship a definitive temperature-dependent form is needed
in order to make more general predictions The key approximation made in deriving the MYEGA
model is that the configurational entropy follows an Arrhenius form By testing this Arrhenius
88
form we can evaluate this assumption in the derivation of the MYEGA model while simultaneously
gaining insights into the thermodynamics of the liquid and supercooled liquid states A strong liquid
( 15m ) has a configurational entropy that is nearly constant as a function of temperature while a
fragile liquid ( 15m ) is a liquid in which the configurational entropy is highly temperature
dependent Since we are testing the MYEGA equation it is also worthwhile to consider alternate
viscosity expressions Assuming validity of the Adam-Gibbs equation the configurational entropy
deduced from the viscosity experiments is given by
( )10 01log l) og(
c
BS
T T minus= (84)
where the viscosity function is given by a viscosity model such as Vogel-Fulcher-Tammann[59]
(VFT) equation the Avramov-Milchev[60] [163] (AM) equation or the MYEGA expression
The resulting entropy predicted by each viscosity model (using experimental data for
fragility and the glass transition) is compared to the calculated configurational entropy from the
landscape for B2O3 in ( 518gT = K and 33m = [69] were used for the experimental values) The
figure clearly shows that the MYEGA equation which assumes that configurational entropy scales
in an Arrhenius fashion reproduces the calculated configurational entropy from the landscape the
most accurately out of the three considered viscosity models This also means that the
configurational entropy form of the AM and VFT models are not physical and cannot be used to
recreate physical results Since the temperature dependence of the entropy has a fixed mathematical
form and is dominated by the distribution of basins we now understand that the distribution of
basins must also have a fixed form and that form must control experimentally accessible properties
This is the key insight that will allow for the development of the toy landscape model
89
53C Adam-Gibbs and Structural Relaxation
Although the Adam-Gibbs model has been validated for shear viscosity and has provided
the functional basis of the MYEGA model let us now consider whether the temperature
dependence of the configurational entropy can also predict the bulk viscosity (also known as the
volume viscosity) Recent work has highlighted the importance of bulk viscosity viz that the
structural relaxation time is proportional to the bulk viscosity[62] Hence the temperature
dependence of the bulk viscosity provides key information needed to predict structural relaxation
times[62] Following Scherer[164] the relationship between the structural relaxation time ( structural
) and the bulk viscosity ( B ) is given by
( )0B structuralK K = minus (85)
Figure 19 The configurational entropy comparisons between the three major viscosity models
which validates the main underlying assumption of the MYEGA model The VFT and AM are
unable to capture the physics of configurational entropy therefore ruling
90
where K is the infinite frequency bulk modulus and 0K is the zero-frequency bulk modulus In
order to calculate bulk viscosity for B2O3 the zero-frequency and infinite-frequency modulus is
taken from these works respectively [165] [166] The experimental structural relaxation times were
taken from the work of Sidebottom et al[154] In Figure 20 the bulk viscosity is shown as well as
the Adam-Gibbs comparison
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the configurational
entropy from the enthalpy landscape with the barrier 00155 eV (compared to the shear barrier of
00149 eV) and the infinite limit allowed to vary (10-263 Pamiddots for bulk viscosity) In Figure 18 the
configurational entropy is confirmed for the shear viscosity thus confirming the AG for both shear
and bulk viscosities Sidebottom data are from Ref [154]
In Figure 20 the fitted values of B and 10log can be used to gain insights into the
difference between structural and stress relaxations since the configurational entropy is universal
between the two The value of Bk (the barrier for cooperative re-arrangements in natural units) for
shear viscosity was 173 K while for bulk viscosity it was 180 K This difference of 7 K in the
barrier leads to an order of magnitude difference in the fitted viscosity curves This is a small
91
difference and explains the variance in the values at Tg The other difference is in the infinite
temperature value of viscosity which is -293 log(Pamiddots) for shear viscosity (from Zheng et al) and
-263 log(Pamiddots) for bulk viscosity This implies that the MYEGA raw form is accurate to predict
the structural relaxation time as well as the stress relaxation (though the values of the pre-
exponential factor and the composite constant will change between the two forms) However the
form in terms of fragility and the glass transition will need to be modified to accurately describe
the structural relaxation time The form assumes the viscosity of interest has a value of 1012 Pamiddots at
the glass transition temperature for shear viscosity which does not have to be the value for the bulk
viscosity curves at the glass transition Further research is needed to understand the appropriate
bulk viscosity at the glass transition and the infinite temperature of bulk viscosity
53D Landscape Features
As shown here the configurational entropy is governed by a Boltzmann sampling of a
probabilistic distribution of states n(E) This distribution of the inherent structures is empirically
found to follow a log-normal distribution as shown in Figure 21 While we are not claiming a
deeper meaning to the underlying origin of the log-normal distribution we may adopt this form of
the probability density function since it accurately describes the underlying distribution of
microstates This is the key insight needed to develop a new model because this means that the
fragility and glass transition are fundamentally related to the parameters of the distribution of basins
on the landscape (number of basins and standard distribution of the enthalpy)
The configurational entropy in the MYEGA model is given by
10
exp 1ln10 12 log
g
c
TS mS
T
= minus minus
minus
(86)
92
where S is the infinite temperature limit of the configurational entropy In Figure 21 we also
show the computed value of Eq (83) as a function of temperature compared to that of Eq (86)
which shows excellent agreement between the configurational entropy and the MYEGA model
This test also supports Stillingerrsquos[18] [19] view of fragility being correlated to the lsquoroughnessrsquo of
the landscape topography According to Stillingerrsquos view if the distribution of inherent structure
energies is narrow then the configurational entropy cannot have a large temperature dependence
leading to a low fragility Alternatively if the distribution is broad then there must be a temperature
dependence to the configurational entropy which necessarily results in a higher fragility
93
Figure 21 (A) The histogram for the energy minima for each of the test systems showing a good fit
with the log normal distribution This distribution will then be a valid form to calculate the enthalpy
distribution of the model presented in the next session (B) The configurational entropy from the
model showing the accuracy of the scaling of the entropy predicted by the MYEGA model The
S value was fit for each system
(A)
(
B)
(B)
94
54 Topography-Property Relations
Now that we have confirmed the validity of the Adam-Gibbs and MYEGA equations from
the energy landscape approach we can draw some key conclusions regarding the thermodynamic
properties of liquids For example heat capacity has been related to fragility[112] and is of
importance for glass manufacturing since it is related to the cost of heating and forming a glass
Isobaric heat capacity is defined as
p
P
HC
T
=
(87)
where H is enthalpy T is temperature and P is pressure The mean configurational enthalpy
calculated from an enthalpy landscape is given by
conf i conf iH p H= (88)
The heat capacity has contributions from both vibrational and configurational modes in liquids
while in glasses the configurational mode is mostly lost For a reversible process the
configurational heat capacity can be written as a function of the configurational entropy
c
p conf
P
SC T
T
=
(89)
By combining Eq (86) and Eq (89) we can write the expression for configurational heat capacity
of a liquid as
10 10
1 exp 1 exp12 log 12 log
g g
p conf
T Tm m S HC S H
T T T T
= minus minus minus = minus
minus minus
(90)
From fluctuation theory the heat capacity can be expressed in terms of the variance of the enthalpy
(2
H )
95
2
2
1p conf HC
kT= (91)
It is important to note that the variance of the enthalpy is not equivalent to the variance of the log-
normal distribution The variance of the enthalpy includes a Boltzmann distribution over the log-
normal distributed basins This gives a direct relationship between the enthalpy fluctuations and
the viscosity parameters If we further adopt the log-normal distribution as the effective distribution
for all liquid basins of all enthalpy landscapes then the distribution is only a function of the
common viscosity parameters ( S m gT and 10log ) The viscosity parameters will only
change the variance of the distribution since the mean of the distribution is inconsequential (the
zero-point enthalpy can be adjusted to an arbitrary value without changing the calculation of the
configurational entropy) The parameters of the log-normal distribution can then be fitted to
reproduce the configurational entropyrsquos temperature dependence thus creating a lsquotoyrsquo enthalpy
landscape for us to play with for the purpose of thermodynamic and kinetic calculations Generating
the landscape is a numerical calculation as the number of basins and standard deviation must be
chosen such to reproduce key phenomena
To quantify the dependence of these parameters on the degeneracy and temperature it is
helpful to develop a descriptive formal model The cumulative probability distribution of a normal
distribution (with zero mean) is given by
10log )1
1 er(
f2 2
Hp
+
=
(92)
where is the standard deviation of the log-normal distribution erf is the error function H is the
enthalpy and p is the probability To then calculate the equal probability spacing for a log-normal
distribution we solve for H for the set of probabilities p which range from 2 to 1 2minus
96
with a step-size of This then gives enthalpies of basins equally spaced by the cumulative
probability distribution and as a result each basin has equal degeneracy
1 2 r2 1e f10
pH
minus minus = (93)
This set of basins carries with it an Arrhenius activation barrier for the configurational entropy that
is only a function of and the degeneracy of each basin (if the is chosen such that the energy
distributions converge) This creates a model that fully describes the thermodynamics of the liquid
and though it currently lacks any kinetic component it offers the possibility to be a useful tool
when used in conjunction with Eqs (82) (83) and (88) This new tool is denoted the ldquotoy landscape
modelrdquo (TLM) It is called the toy landscape model because we are generating a landscape that is
not the real landscape but is enough to reproduce key experimental properties without needing
expensive computational calculations It is a lsquotoyrsquo for us to play with without needing to worry
about upfront computational cost
To confirm the validity of TLM and the way it is constructed we can calculate the
activation barrier from this deterministic method and for a random set of inherent structures with
the same number of basins and distributions For this purpose we used temperatures ranging from
1000 to 2000 K for both the deterministic and random models with a step size of 0001 for TLM
Excellent agreement between the random and deterministic methods is shown in Figure 22A This
analysis allows us to understand that the thermodynamics of the system are fundamentally linked
to the viscous flow behavior Though this technique was created to target viscosity and relaxation
it is generally applicable to all configurational thermodynamics of liquids and glasses It is worth
noting that all the thermodynamics will be relative to a 0 eV energy state defined at equilibrium at
0 K and as such the enthalpy differences at some fixed temperature must be known to compare
over compositional spaces
97
In Figure 22B we see the same method being used to understand the relationship between
the viscous parameters and the parameters of the landscape This is the key to the numerical
calculation of the lsquotoy landscapersquo model The toy landscape is a numerical calculation where we
use experimentally accessible values to generate a landscape whose thermodynamics should
reproduce key thermodynamic phenomena In Figure 22B we show that the fragility and glass
transition are systematically changed as the number of basins and standard deviation of the enthalpy
is changed When this idea is combined with the basin calculation in Eq (93) (giving equal
degeneracies for all basins) then the complete set of enthalpy basins is easily attainable giving
access to the fundamental thermodynamics of the system
98
(
A)
(B)
(A)
99
Figure 22 (A) Comparison between the randomized method (histogram) and the deterministic
method (vertical lines) showing good agreement between the maximum in the histogram and the
value predicted by the deterministic technique validating the approach It is worth noting that the
100-basin distribution is a very wide distribution where the total number of basins is less than the
number of points used in the calculation This is done for a variable number of basins with the
number of basins shown in the legend (B) The dependence of fragility and the glass transition
temperature vs the distribution of states and the number of basins
55 Barrier Free Description of Thermodynamics
Throughout this work we have systematically explored the Adam-Gibbs model of
viscosity An important insight that we have gained is that the configurational entropy is free of any
barriers This is of particular interest because the Arrhenius slope of the activation barrier is linked
to the glass transition and fragility The glass transition temperature is known to be related to the
set of activation barriers as it corresponds to a breakdown of ergodicity What is found here is that
the fragility is the ldquoconversion factorrdquo to account for the slope of the configurational entropy
(which as previously identified is related to the distribution of basins) This means that the barriers
themselves do not matter until the material is close to the breakdown of ergodicity and thus the
equilibrium liquid thermodynamics and kinetics can be described without explicit information
about the saddle points in the landscape
Understanding the role of the topography in the properties of glasses and liquids lead to
the most important insights derived from the AG model Since the configurational entropy is the
same for both stress and structural relaxation simple viscous parameters allow us to understand
fundamental thermodynamic quantities of the system without needing to map computationally
expensive energyenthalpy landscapes The process of converting experimentally accessible
properties to a usable landscape for thermodynamic calculations is what we are calling the toy
landscape model To show the utility of toy landscapes we can calculate the driving force for glass
relaxation This information can give insights into the physics of relaxation by dynamically
100
calculating the Gibbs free energy of the equilibrium liquid at any temperature which defines a
driving force for glass relaxation To do this calculation of the driving force for glass relaxation
we consider the ergodicity parameter ( x ) defined by the Mauro-Allan-Potuzak (MAP) model of
glass relaxation as the ratio between the glass entropy and the equilibrium liquid entropy
liquiss dglaS x S= (94)
To get information about the enthalpy we consider an instantaneous Boltzmann distribution on the
landscape given by an equilibrium distribution at some elevated temperature fT
1
exp
exp
iglass i i
fii
f
HH H g
kTHg
kT
= minus
minus
(95)
This formalism in conjunction with TLM gives us the ability to explore the temperature
and compositional dependence of the driving force for relaxation The enthalpy of the liquid is
calculated from an equilibrium distribution on the landscape The driving force ( ) is defined as
( ) ( )( ) ( )1iliquglas ds f l quii dH H TS xT T = minus + minus (96)
For our purposes this gives a nice approximation for the driving force improving on previous
approximations given by ( fT Tminus )[16] [42] TLM does not include a kinetic feature to calculate
the occupational probability in each basin however when coupled to a kinetic model such as the
MAP model[44] a complete description of the underlying thermodynamics of glass relaxation can
be considered For this calculation of the driving force the kinetics are calculated through the MAP
model but this is not necessarily a requirement In Figure 23 the compositional and temperature
dependence of the driving force is shown for three glasses with identical glass transitions and
varying distributions of basins (as listed in Table 1) The calculations are performed using the
RelaxPy script and MAP model with the stretching exponent fixed at the Phillips value of 37
101
explanations of which are available elsewhere[42] The systems were quenched from 700 K to 300
K at a rate of 10 Kmin Figure 23 clearly shows how simply changing the total number of basins
creates a larger driving force for glass relaxation This simple change of the number of basins
drastically increases the entropic contribution and due to the loss of ergodicity explains the increase
of the driving force of relaxation The kinetics could be calculated through any relaxation model or
directly from the landscape if some barrier approximations are made
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition temperature (500 K)
were kept the same while the total number of basins were allowed to vary
Sample Total Basins [-] Distribution of States [log eV] 1
1493g
mT
minus
[K]
Fragility [-]
1 100 02 837 40
2 500 02 537 31
3 1000 02 466 28
102
Figure 23 The driving forces for different example glasses calculated using a combination of the
MAP model RelaxPy and the toy landscape model The parameters for each glass can be found in
Table 4
57 Toy Landscapes for the Design of Glasses and Glass Ceramics
The concept of a toy landscape can go even further building a method to predict the
dynamics of a glass system based on simple inputs This is possible because in the previous section
we have related the distribution of basins on an energy landscape to the experimental viscosity
which allows for thermodynamic insights Building on this we can assume that the barriers (H )
for transitioning between states is equal to that of the barrier mean for the viscous relaxation
ln10gH mkT= (97)
This means that the only key factor needed to then predict the dynamics of a landscape using the
metabasin approach is the vibrational frequency However the relaxation time at the glass transition
103
is known to be a constant of 100s and as such we can choose a vibrational frequency that reproduces
this fixed point This gives an entire description of relaxation without any need for fictive
temperature An example of the results predicted through this method are shown in Figure 24 This
prediction only required knowledge of the glass transition temperature and the fragility and from
this a pure prediction of relaxation is made
104
105
Figure 24 The predicted outputs from the toy landscape method showing predictions of the enthalpy
and entropy under a standard quench for barium disilicate This prediction does not require fictive
temperature or any such assumptions about the evolution of the non-equilibrium behavior
This method is also not limited to predicting the relaxation of glasses but can also capture
the crystallization It is known from the earlier work on nucleation presented in Chapter 3 that a
simple two state model was able to capture the underlying physics of the system Thus if we know
the free energy of the crystal and barrier to crystallization we can find predict key crystallization
106
phenomena By fitting the heat flow peak magnitude and location this gives enough information to
infer the barrier and the enthalpy of the crystal The degeneracy (and thus the entropy) of the crystal
can then be calculated through knowledge of the liquidus temperature
ex(
p)cry SCL l
cry
lk
G TH
T
minus=
(98)
This then gives everything for the model presented in Chapter 3 from only 1 calorimetry
experiment the glass transition the fragility and the liquidus temperature To predict nucleation a
series of estimates were done running the toy landscape at different temperatures until the
temperature with the fastest crystallization rate was found This was then assumed to be the
maximum rate of nucleation The value of the interfacial energy was then chosen in such a way so
that the predicted curve reproduces the peak at the same temperature This gives all the parameters
needed for a prediction using CNT It is also possible to make a prediction of the growth rate of
crystals using this expression
2
1 exp6
GU
kT
a kT
minus minus
=
(99)
The results for these estimates are shown in Figure 25 The reason why growth is more accurate
compared to nucleation is that the kinetic factor of growth is governed by viscosity while nucleation
is governed by a kinetic factor that includes an assumed vibrational frequency
These methods have a plethora of assumptions built in such as that the vibrational
frequency of basins undergoing relaxation is equal to the vibrational frequency when undergoing
crystallization however as more information is gained about a particular system (such as more
crystallization peak as a function of temperature) the information could be built into the system to
improve crystallization predictions If the peak size is known as a function of scan ratetemperature
even the enthalpy as a function of temperature could be known This provides a whole system for
the real estimates of crystallization can be improved
107
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume in
nucleation is assumed to be on the order of one cubic angstrom while the a parameter in growth is
assumed to be around one nm both are in good agreement for estimates in literature The values
for the orange points are taken from these works[128] [167]
57 Discussion
The impact of the Adam-Gibbs model on glass science is hard to overstate and it has
become foundational to ideas that will enable the design of next generation glasses The Adam-
Gibbs equation still has more to offer the glass community in terms of fundamental understanding
and practical applications In this work we have shown that the Adam-Gibbs relationship is
consistent with both shear and bulk viscosities Also we have validated the main assumption of the
MYEGA model giving insights into the scaling of configurational entropy from the underlying
enthalpy landscape Using this knowledge of the relationship between the Adam-Gibbs model with
the underlying enthalpy landscapes we have developed a simplified ldquotoy landscaperdquo approach for
modeling glass relaxation The toy landscape is constructed numerically by understanding the
distribution of inherent structures and transition points in the landscape and then relating these
distributions to experimentally measured parameters of the MYEGA viscosity model This
1 00
1600
1400
1200
1000
00
600
400
200
000
0000 110000 130000 1 0000 1 0000
(m
s)
Temperature
Growth PredictionCassar et al
0
2
4
6
10
12
14
00 1100 1300 1 00
log 1
0(I) n
ucl
ei s
m3
Temperature
Predicted RateRodrigues et al
4
3
2
1
0
00 1000 1100 1200 1300
log
10(I m
ax(I))
nuclei s
m3
Temperature elvin
108
simplified landscape model can then be used to calculate the underlying thermodynamics and
kinetics of the system and represents a practical application of energy landscapes to understand
glass relaxation nucleation and other relevant thermodynamic properties
This also closes our discussion on understanding the thermal history effects on glass What
we have shown is that the dynamics of a glass system can be understood through energy landscapes
and fundamentally the same insights can be recreated through lsquoToy Landscapesrsquo This approach
allows fundamentally for the complex 3N dimensional landscape to be summarized and abbreviated
into a smaller landscape that can easily be implemented for any glass However this new
parameterization relies on knowing some key properties that are only due to the location in the
compositional phase space and for a complete picture of the system to be created we must also
create a sufficient approximation for the compositional phase space shown in the next 2 chapters
Chapter 6
Enabling the Prediction of Glass Properties
So far models have been created that describe the very nature of the thermal history effects on
glasses However as an input to these models we have to know about the compositional dependence
of key parameters namely the glass transition and the fragility In this chapter we extend
compositional models to other key properties needed for the design of glasses Here these key
properties that have not been readily predictable are approached using both machine learning and
TCT models The TCT models are focused on surface reactivity Youngrsquos modulus and ionic
conductivity The ML addresses CTE fragility melting point as well as Youngrsquos modulus
61 Controlling Surface Reactivity
Glass surfaces especially their interactions with water[168]ndash[171] are of the utmost importance to
nuclear waste glass cover glass and many other modern applications[106] It is generally accepted
that the outermost surface of oxide glasses readily reacts with water molecules where water
molecules chemisorb on an as-formed glass surface and populate it with surface hydroxyls Recent
models attempt to quantitatively explain the change in properties observed on hydroxylated glass
surfaces[77] [168] [172] [173] and most agree that the surface readily becomes hydrated because
it is more stable to have a bonded hydroxyl group than to have dangling bonds In glasswater
interactions hydrolysis and diffusion lead to two forms of adsorbed water chemisorbed and
physisorbed[171] [174]ndash[176] Both affect the network differently and are controlled by different
processes physisorption by hydrogen bonding[175] and chemisorption by reaction with the surface
to form bonded hydroxyl groups
110
equiv Si minus O minus Si equiv +H2O rarr equiv Si minus OH + OH minus Si equiv
Although binding energy studies have been performed using molecular dynamics simulations[177]
a specific study of the surface reactivity in tandem has not been conducted Recent advancements
in reactive force field modeling allow the direct observation of both diffusion and surface
reactivity[38] [178]ndash[180] and several studies have found surface reactivity constants consistent
with those reported experimentally[173] [178] [181]
Recently TCT has been used to predict various bulk properties of glassy systems such as
Vickers hardness and fragility[17] [182] [183] Based on the TCT description of the glass network
glassy materials have been postulated to have a so-called lsquointermediate phasersquo in which the atomic
structure of a material will self-organize so as to be isostatically constrained (n=3) as explored
extensively in various works[65]ndash[67] [133] [184] [185] In this case it has been shown that such
materials exhibit anomalous behavior in certain properties such as an high hardness and lower free
energy[43] [67] [77]
TCT has also been used to understand the impact of water on bulk glass structure and
properties Potter et al[168] found that by accounting for the impact of chemisorbed (dissociated)
water on glass network connectivity (viz breaking Si-O-Si bonds) and the impact of physisorbed
(molecular) water on strain of the Si-O-Si bond angle the glass transition temperature (Tg) could
be predicted This model could be further expanded to include other properties (such as
modulus[139] or dissolution kinetics[77]) predicted by TCT such as the work done by Liu et
al[186] on calcium-silicate-hydrate gel An extensive study relating topological constraints[6]
[17] [57] and surface energy was performed by Yu et al[172] specifically focusing on the
transition from hydrophilic to hydrophobic behavior on silica surfaces Their work used reactive
molecular dynamics to model the change in surface energy of a silicate glass and then correlate it
with the number of surface constraints present with the work focusing primarily on the global
average of the surface
111
To investigate the effects of glass network topology on the surface reactivity we model
the hydration of a silicate glass was modelled using molecular dynamics simulations according to
the following procedures Initially bulk sodium silicate glasses were simulated (150 atoms with a
molar composition of 70SiO2middot30Na2O) using the Teter potential[187] For a bulk glass this pair
potential is known to accurately simulate sodium silicate glasses and has been extensively studied
to investigate various properties including structural features transport of sodium ions and
vibrational density of states[188] The size of the system is initially set to achieve the
experimentally measured density of 2466 gcm3 [189] A total of 35 Si-atoms 85 O-atoms and 30
Na-atoms were randomly inserted within a periodic cubic box of a = 12686 Å and the initial
configuration was energy-minimized to avoid any overlaps prior to the glass formation The glass
was held for 05 ns with a constant number of atoms volume and energy (a microcanonical or
NVE ensemble) Since a thermal fluctuation of ~400 K (from 2000 K to 2400 K) was observed
during the microcanonical run the glass was allowed to evolve further at 2400 K for 05 ns with a
constant number of atoms volume and temperature (a canonical or NVT ensemble) Within the
same ensemble the melted system was then cooled with a constant cooling rate of 05 Kps and
after the temperature had reached 300 K it was equilibrated for 1 ns Finally a constant pressure
of 1 atm was applied to the system (NPT ensemble) for another 1 ns to ensure no significant
fluctuations of the density during the equilibration process Three different sodium silicate glasses
of the same nominal composition were constructed via the aforementioned procedure using
different initial atomic positions These repeated simulations compensate for the limitation of small
sample size by allowing us to capture the statistical behavior of glass surfaces during the glass-
water reactions The final densities of the sodium silicate are presented in Table 5 These final
structures were used as starting configurations for the glass-water reactions MD simulations with
the Teter potential creating three glass networks were carried out using the Large-scale
AtomicMolecular Massively Parallel Simulator (LAMMPS) package[30]
112
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K
Simulation box dimension a (cubic) [Å] Density [gcm3]
Run1 12979 2303
Run2 12630 2499
Run3 12737 2437
Initial dimension a=12686 Å
Experimental density (70SiO2middot30Na2O mol) 2466 gcm3
113
After the bulk sodium silicate glasses were obtained a reactive potential was employed to model
the glass surface and subsequent glass-water interface as the system of interest for this study
required characterization of the reactive processes during the simulation of surface phenomena In
this study all reactive MD simulations are performed with the NaSiOH parameterization using
the ReaxFF reactive force field framework15 The ReaxFF parameters are trained using a first-
principles data set that describes water interaction at the sodium silicate glass-water interface
Further details of the ReaxFF methodology and its potential forms can be found in earlier
publications by van Duin et al[38] [179] [181] [190]
114
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the hydration
models Color scheme Si atom (ivory) O atom (red) and Na atom (blue) The z-axis is elongated
to allow space for an insert of water
115
As shown in Figure 26 a free surface was first created by expanding the c parameters of each of
the equilibrated bulk glasses This process results in a vacuum region above and below the two
surfaces of the glass slab along the z-direction These exposed surfaces were relaxed at 300 K for
100 ps with the ReaxFF reactive force field Following the relaxation of both top and bottom
surfaces the vacuum region was filled with water molecules The number of water molecules that
are inserted in the vacuum region was controlled to have a density of ~099 gcm3 In addition all
water molecules were kept at least 2 Å from the uppermost and lowermost atoms of the glass
surface to prevent any initial close contact with the surface Glass-water reaction simulations were
carried out at 300 K with the configurations shown in Table 6 for 500 ps in the NVT ensemble
From these runs trajectories at every 100 ps were obtained for the surface reactivity analyses
Figure 27 shows the initial and final positions of glass surface reaction with water at 0 and 500 ps
respectively
116
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the top surface
in contact with water is shown here
a) b)
117
Table 6 System configurations for sodium silicate glass-water reactions
Simulation cell [Å3] Number of water molecules
Run1 12979 times 12979 times 3894 146
Run2 12630 times 12630 times 3789 134
Run3 12737 times 12737 times 3821 138
118
The hydrated surfaces of sodium silicate glasses from each run were used to investigate the
adsorption behavior of a water molecule Since the time evolution of surface reactivity is of key
interest the binding energy of a water molecule to the hydrated surface was calculated at 0 100
200 300 400 ps In order to evaluate the local heterogeneity of reactivity imposed by varied glass
surface structures each surface was divided into a 10 by 10 grid and the binding energy at each site
was mapped across the grid The water molecule position in the z-direction from top and bottom
surface was maintained to be 20 Å from the outermost atom during the energy calculation The
binding energies (Eb) were calculated as below where the energy of a surface with adsorbed water
(Esw) was taken with respect to the surface energy (Es) and the energy of molecular water (Ew)
b s wswE E E E= minus minus (100)
A negative binding energy would indicate that water adsorption to the surface site at the
corresponding grid is thermodynamically favorable The binding energy map may then be used to
locate where water binding would be most stable based on the ReaxFF calculations From three
independent glass-water reaction boxes a total of six hydrated surfaces are obtained (a top and
bottom surface from each run) increasing the statistical reliability of this analysis in the ensemble
despite the small overall system size of any one box The binding energy mapping process is shown
in Figure 28
119
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example of the
electronic-structure DFT calculation using semilocal exchange-correlation functionals finding the
binding energy of a water lsquopixelrsquo to the surface
~20 Å
surface
10times10 grid
120
Similar calculations are carried out at an electronic structure level using the DFT framework as
implemented within the Vienna Ab-initio Simulation Package (VASP) software Total energies are
computed from the static geometries obtained from the reacted ReaxFF structures using the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional with projector augmented wave
(PAW) pseudopotentials to describe the ion cores A kinetic energy cutoff of 500 eV was used for
the expansion of the plane waves Due to the large non-crystalline structure the Brillouin zone is
only sampled at the Γ-point with 01 eV of Methfessel-Paxton smearing to help electronic
convergence Due to the computational cost associated with these DFT simulations the results were
primarily used to elucidate the electronic interactions between the water molecule and the glass
surface
To count the topological constraints in the structure generated from the ReaxFF MD
simulations an algorithm was developed to systematically exclude all non-bridging oxygens
alkali and hydrogens from the structure The number of surface rigid constraints (in the pixel)
around each network-forming atom was then calculated an example contour plot generated using
this procedure is shown in Figure 29 The surfaces were then converted to a 10times10 pixel grid and
the average coordination of network-forming atoms in each pixel calculated The depth of each
pixel taken was to be 4 Aring below the surface
121
Figure 29 Example contour surface showing the average coordination per atom on the glass
surface for the first run at 300 ps
122
Figure 30 shows the results from the MD binding energy studies where a clear transition in the
region around 30 constraintsatom is shown This region is isostatic because the number of
constraints is equal to the number of translational degrees of freedom and the width highlighted in
the figure is evidence for the existence of an intermediate phase In order to confirm that the binding
energies had indeed converged a larger system of 1500 atoms was simulated Good quantitative
agreement was found between the 150- and 1500-atom systems In this work the isostatic behavior
occurs over a wide range in which certain silicon network sites dramatically increase the binding
sites (binding energy approaching 0) The state or structure with nc outside of the intermediate range
will readily interact with water Only those in the isostatic region are likely to remain unaffected
after the interaction with water thus they are the rate limiting species in dissolution of silica glass
123
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of constraints for
surface atoms at the local pixel Results show a distinct maximum in which there is a near
hydrophilic-hydrophobic transition of the surface The error bars represent the standard deviation
A second system with 1500 atoms was performed to show convergence of the ReaxFF MD results
⟨119889119867minus119874119866119897119886119904119904⟩ = 345 Aring
⟨119889119867minus119874119866119897119886119904119904⟩ = 248 Aring
124
The intermediate phase result is intuitive due to the implicit stability that comes with having an
isostatic phase which is both energetically favorable and stress-free The larger negative binding
energies in the over-constrained regions arise from large differences between the free energies of
the chemically bonded state (chemisorbed) and unbound state (water in near proximity to the
surface but not bonded) in these regions in this region the chemical reaction with the water
molecule allows the glass surface to reduce the number of incompatible constraints which in turn
alleviates local stress in the network In other words a local region undergoes an alleviation of
stress when Q4 units convert to Q3 units The larger negative binding energy values in the under-
constrained regions can trace their origin to the sites available for bonding with the water molecule
Their high number of degrees of freedom allows for the facile reaction of water ie the only energy
cost is for the oxygen to dissociate from the network The isostatic network within the intermediate
phase is able to preserve its structural integrity because the energy barrier to deform the network is
high and there is no localized stress to create an additional driving force for reaction with water
Indeed the isostatic network is the most energetically favorable arrangement of a non-crystalline
structure This intermediate phase is governed by the fluctuations of the topology that arise with
the minimization of stresses These stresses change the local topology (within the range of the
fluctuations) so that they become isostatic[68] This leads to a flat region where there is a constant
binding energy at the surface Though it appears to be an intermediate phase it is important to note
that this may instead be simply a manifestation of an isostatic threshold (a peak in binding energies
rather than a plateau)
The average number of surface constraints has been shown in previous work to be largely
controlled by the annealing time[172] The surface constraints may then be used to predict surface
reactivity However it is worth noting that the local number of constraints is not constant in
time[17] Furthermore Potter et al[168] showed that molecular water in the network can radically
shift the free energy of the γ constraint Therefore the number of surface constraints is dynamic
125
during the influx of water Shifting constraint energy will also alter the fragility which in turn alters
the diffusion activation enthalpy In the future this may provide a path for developing more durable
glasses ndash currently it shows that a homogeneous intermediate phase (isostatic) surface should be
targeted to achieve maximum chemical durability
62 Elastic Modulus Prediction
Tailoring the elastic properties of glass is important in the design of new advanced compositions to
control the stiffness and damage resistance of a variety of glass products[2] [191]ndash[193] Elastic
moduli are defined based on the ratio of the stress exerted upon a material to the resulting strain
While various elastic moduli can be defined (eg Youngrsquos modulus shear modulus bulk modulus
and Poissonrsquos ratio) for an isotropic material such as glass only two of these quantities are
mutually independent[1] [191] [194]
Previous attempts to model elastic moduli have focused on either computationally costly
molecular dynamics simulations or empirical fitting methods The work by Makishima and
Mackenzie[195] offered a possible solution for predicting elastic moduli based on the dissociation
energy per unit volume of the glass however topological changes in the glass network are ignored
and the model cannot account for the temperature dependence of modulus With MD simulations
elastic moduli can be obtained by applying a stress and measuring the resulting strain on the
simulation cell assuming that accurate interatomic potentials are available[2] Machine learning
has also been applied to model elastic moduli using experimentally measured composition-property
databases with a high predictive ability being achieved[2] [196] Previous analytical modeling
techniques related to the topology of the glass-forming network have shown good qualitative
agreement with compositional trends in modulus but have a lack of quantitative accuracy[64]
[197]ndash[199]
126
It is common in the glass community to assume that Youngrsquos modulus may scale linearly
with hardness While this is clearly an oversimplification let us begin with previously derived
models for glass hardness as a potential starting point since glass hardness has received
considerable attention in the context of topological constraint theory [85] [192] [200] Several
models have been proposed to explain the origin of glass hardness all of which have a linear form
( ) vv
dHH n x n
dn= minus (101)
where Hv is the Vickers hardness x is chemical composition nrsquo is the critical level of constraints
needed for the substance to provide mechanical resistance to the indenter in three dimensions and
n is a measure of the constraint rigidity Recent work on glass hardness[192] has shown that n can
most accurately be defined in terms of either the density of rigid constraints (or alternatively the
density of rigid angular constraints) given by
)( )
( )( )
(c An xn
x Nx
M x
= (102)
where nc is either the number of constraints per atom or number of angular constraints per atom ρ
is the density of the composition M is the molar mass and NA is Avogadrorsquos number Since
hardness and elastic modulus are often considered to be correlated one might surmise that the
elastic modulus might also be some function of angular constraint density or total constraint
density However such an approach is not able to give quantitatively accurate predictions of
modulus
Here we propose an improved model of Youngrsquos modulus based on the free energy density of the
topological constraints
int
( ) ( ))
(
(
) i constr
c i
a s
Ai in x q T
M x
x NF F
=
= (103)
where qn(T) is the onset function for each associated constraint to give temperature dependence to
127
the predictions and is determined by the free energy ∆Fn The calculated values of Δ119865c are then
used to calculate Youngrsquos modulus (E) by
( )c c
c
FdE
EF
xd
F
= minus (104)
In this new model the Youngrsquos modulus is controlled by the free energy parameters related to the
strength of each constraint Each parameter in this model captures a key physical phenomenon with
each parameter corresponding to a physical property Moreover we demonstrate that a common
set of parameters can simultaneously capture both the temperature and compositional dependence
of modulus as well as predict other properties such as the glass transition temperature and fragility
To demonstrate the validity of this temperature-dependent constraint model Youngrsquos
modulus data are collected from literature for lithium borate[195] sodium borate[195] and
germanium selenide glass systems[201] Additionally the sodium phosphosilicate[202] system
was experimentally determined following the procedure described by Zheng et al[183] using room
temperature resonant ultrasound spectroscopy on prisms of dimensions 10 mm times 8 mm times 6 mm
Analytical topological constraint models have already been published for each of these
systems[17] [57] [202]
The proposed models for glass elasticity (the models previously proposed for hardness
based on constraint density and angular constraint density as well as the new model for modulus
based on the free energy density of constraints) were optimized in each case leaving the intercept
and slope (119889119864
119889119899
119889119864
119889119880119888) as adjustable parameters The constraint onset temperatures were also optimized
for the free energy density model Figure 31 and Figure 32 show the results for each model in the
phosphosilicate and sodium borate systems respectively
128
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values of the model
predictions are 641 GPa for constraint density 313 GPa for free energy density and 774 GPa
for angular constraint density
129
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from minimum
to maximum error The free energy density model gives the most accurate results The constraint
density has a RMSE of 61 GPa the angular density has a RMSE of 20 GPa and the energy density
has a RMSE of 59 GPa
130
We have also evaluated the predictive ability of the model in terms of the temperature dependence
of the Youngrsquos modulus sing the 10 Na2Omiddot90 B2O3 data reported by Jaccani and
Huang[191] the model and experimental predictions of the temperature dependence of the Youngrsquos
modulus are plotted in Figure 33 Here the constraint onset temperatures are determined from the
room temperature modulus data The number of escape attempts was optimized since this controls
the width of each transition and is thermal history dependent
131
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and experiment for
10 Na2O 90 B2O3 Using the previously fitted onset temperatures the only free parameters are
then the vibrational frequency and the heating time in which their product was fitted to be 14000
Where each dip in the modulus corresponds to a constraint no longer being rigid as heated through
each onset The onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The data was fit
using a least-squares method and the resultant fit is shown as the calculated method The fit has an
R2 of 094 (B) The contribution from each constraint to the overall modulus
It is also possible to validate the model by comparing the free parameters fitted to the Youngrsquos
modulus data vs that of previously reported data as shown in Table 7 Results for two additional
systems are plotted in Figure 34 to show the general validity of the free energy density model
132
Table 7 Fitted values from this analysis compared to those reported in the literature The disparity
between the constraints evaluated with molecular dynamics most likely come from the speed in
which the samples are quenched
Value Fitted Value (K) Literature (K) Method Citation
Silicate Onset 2212 1986 MD (Potter et
al[168])
Silicate Onset 818 1600 MD (Potter et
al[168])
Silicate Onset 450-500 810 MD (Potter et
al[168])
Borate Onset 921 Not Reported Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 715 740-760 Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 393 328 Fitting Parameter
(Mauro Gupta and
Loucks[69])
133
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in Fig 3) and
experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of 093 and (B) xLi2Omiddot(100-
x)B2O3 glasses with an R2 of 0986
(
A)
(
B)
134
In the previous work of Zheng et al[192] concerning hardness of glass the authors write
ldquoIt should be noted that each bond constraint corresponds to a certain energy since different kind
of bond has different bonding energyhellip and thus the constraint density also represents an energy
per unit volume In other words hardness is correlated to the energy per unit volume Our findings
for the borosilicate and phosphosilicate systems are further evidence in support of this argument
since both the total constraint density and angular constraint density approaches give better
prediction of glass hardness compared to models based on number of atomic constraintsrdquo
which makes it clear that the prediction of the hardnessmodulus should be closely related to the
onset temperatures (ie free energies) of the associated constraints The model that was previously
proposed could be extended to modulus if all constraints had equal amounts of potential energy
but due to the drastic change in strengths between the constraints the prediction fails
Bauchy et al[85] showed also that hardness in the calcium-silicate-hydrate system is
controlled by the angular constraints which led to the development of angular constraint density
as the governing control for hardness When the analysis of the density of angular constraints is
considered through the free energy view it becomes apparent why this method works effectively
for some systems Models for predicting the hardness and elastic modulus of glass attempt to
explicitly connect the rigid bond energy to the macroscopic properties of the system The
assumption for the hardness models is that the energies for breaking each type of constraint are all
approximately equal and hence only the number of the constraints matters The argument can then
be extended for elasticity since elastic modulus is a bulk material property with the approximation
of all constraints being equal strength the number of constraints per volume should be related To
correct for this approximation when a weighted sum of free energies is used the estimation
becomes significantly more accurate Using the density of the glass a precise free energy per
volume of the rigid constraints can be calculated The knowledge that the energy of the bonds is
tied to the elastic modulus has been widely known but had not been quantified nor placed within
the context of topological constraint theory[203]
135
63 Ionic Conductivity
Ionic conductivity 120590 is related to the number of charge carriers n and the mobility of the
carriers 120583 as they diffuse through a network by[1]
Ze n = (105)
where Ze is the charge of the conducting species Since the charge-carrying ion is constant within
a given glass family the ionic conductivity depends only on the mobility and the concentration of
the charge carriers present However it is not typically feasible to measure 120583 and n independently
of each other and thus it is unclear which variable scales in an Arrhenius fashion thereby
controlling the ionic conductivity of a glass In the weak electrolyte model by Ravaine and
Souquet[204]ndash[207] the mobility of the charge carriers is considered constant while the
concentration scales in an Arrhenius function Conversely the strong electrolyte model assumes a
constant number of charge carriers while the mobility follows an Arrhenius function[208] [209]
Recent MD simulations from Welch et al[210] support the weak electrolyte interpretation for alkali
silicate glasses concluding that ionic conductivity is dictated by the concentration of charge
carriers Previous reports in the literature also supporting this view have proposed several
hypotheses to explain this conclusion[208] [209] [211]ndash[213] however further exploration is
needed
The simplest form of the temperature dependence of free ions is given by
0 exp an nE
kT
= minus
(106)
Here n0 is the total possible number of mobile ions and Ea is the activation barrier for an ion to
diffuse It is often assumed that the activation enthalpy is related to the energy needed to deform
the network temporarily to allow for ion motion[193] [209] [210] [212] however it is our
hypothesis that there is a coupling between the ion hopping event and the relaxation of the local
136
network allows for a permanent relaxation or deformation along ion channels This means that
there is some cooperative relaxation of the network along the diffusion path in conjunction with an
ionic hopping event These cooperative relaxations are discussed explicitly in the Adam-Gibbs
formalism[53] for describing the relationship between viscosity and configurational entropy Sc
expc
B
TS
=
(107)
where B is an activation barrier is the viscosity and is the viscosity of the liquid in the limit
of infinite temperature In the high-temperature (low-viscosity) liquid state there is a well-defined
relationship between the viscosity and diffusion (Stokes-Einstein relation) However this
relationship breaks down in the low-temperature (high-viscosity) glassy phase
Herein we present a new model for predictions for the compositional dependence of ionic
conductivity We can write the activation free energy barrier Eac for an ionic hopping event as
a c c c cE T S= minus (108)
Here is an enthalpic barrier for the ion to hop to a neighboring site CT is the configurational
temperature (a value that describes the distribution of energies of the ions) and cS is the entropy
of the activation barriers These quantities are associated with configurational changes in the glass
indicated by the subscript c In the weak electrolyte model all charge carriers have the same
mobility and hence the entropic effects are dominant This indicates a small value of an
assumption that will be validated by results later in this work leading to a simplified approximation
for the activation energy at temperatures below the glass transition but suffiently high temperatures
such that
a c c cE T S (109)
In the case of a nonequilibrium glass at low temperature the system becomes trapped in a localized
137
region of the energy landscape known as a ldquometabasinrdquo with slow inter-metabasin transitions[156]
[161] The MAP model of the nonequilibrium viscosity of glass relies on this partial confinement
within metabasins where variation of the activation barrier for relaxation (H) is related to the
fragility of the system by[44]
1
ln10g
dm
dH kT= (110)
Here m is the fragility index (as proposed by Angell[58]) and Tg is the glass transition temperature
Splitting the differential in Eq (110) and integrating we can write
ln10gmH kT= (111)
This relationship was also proposed by Moynihan et al[214] for the activation enthalpy of
relaxation of a glass[214] [215] Setting the Arrhenius form of the barrier described in Eq (111)
equal to the Adam-Gibbs equation (Eq (107)) we write
exp exp c
H B
kT TS
=
(112)
Solving Eq (112) for the configurational entropy and inserting the result into Eq (109) we obtain
the simple relationship
2
ln10
ca
g
T BE
k T m (113)
Dividing the above expression with that of a reference state r and assuming that B and is cT are
constant with respect to compositional variation we have
g r ra
a r g
T mE
TE m= (114)
This new equation is the first to relate the activation free energy for ionic hopping in a glass
to the glass transition temperature and fragility of its corresponding supercooled liquid It also
138
predicts a coupling between ionic diffusion and the cooperative rearrangements described by the
Adam-Gibbs relation This possibility was first proposed by Ngai and Martin[216] who showed a
correlation between the product of the Kohlrausch stretching exponent ( ) and the activation
barrier for relaxation with the activation barrier for ionic conductivity This implies that the
diffusion in glasses is not governed by an elastic component but instead controlled by the
surrounding cooperative rearrangements ie the -relaxation of the glass[14] in good qualitative
agreement with previous results shown by Potuzak et al[217] Further evidence comes from the
polymer community where the decoupling of relaxation and conductivity is reported with work
showing that the decoupling of viscosity and diffusion varies with the fragility and the glass
transition temperature[218] This is a general relationship which we demonstrate later to be valid
for binary alkali borate alkali silicates and alkali phosphate glasses as well as validating the
atomistic diffusion method using computational techniques
In order to test this proposed relationship between relaxation and diffusion through
simulation a glass with the composition 10 Na2O-90 B2O3 (mol ) was synthesized in molecular
dynamics with 1150 atoms using the potentials of Wang et al[162] The system was quenched at a
rate of 1 Kps from randomly placed atoms at 2500 K in the NPT ensemble to room temperature
as shown in Figure 35[210] To find the transition point energy of the diffusing ion a nudged elastic
band (NEB) calculation was performed An alkali ion was chosen and its closest alkali neighbor
was moved to another location so that the NEB calculation could be performed between those two
sites All calculations were carried out using the LAMMPS software package Eleven reaction
coordinates were used in the NEB calculation and the results are shown in Figure 36
139
Figure 35 The structure for the initial minimum energy configuration showing the boron (blue)
network with interconnecting oxygens (red) and the interstitial sodium ions (yellow)
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and sodium is
ivory The barrier is overestimated compared to experimental data this could be from several
sources of error such as potential fitting thermal history fluctuations or sampling too few
transitions The line is drawn as a guide to the eye
140
Although the transition point energy is key for determining the activation barrier for diffusion here
we are also interested in the dynamics of the atoms around the mobile ion Since we have
hypothesized here that the ion diffusion is dependent on the ability for the local region to relax and
deform instead of elastically straining we expect a local deformation (relaxation) along the ion
path Figure 37 shows that along the ion diffusion tunnel there is indeed a permanent displacement
of the atoms
141
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function of
reaction coordinates The middle shows the local deformation around the ion of any atom that
moves in between inherent structure mandating a relaxation force The color shows the degree of
deformation
142
In the final configuration of Figure 37 the permanent deformation required to lower the
energy of the system is clearly seen If it were purely an elastic response to dilate a pathway the
displacement between two inherent structures should be confined to changes with only the mobile
ion however this is not observed Instead along the pathway of the ion movement the network
forming atoms also deform and change local positions to minimize the energy implying a net
relaxation around the ion ldquopathwaysrdquo To apply this and show the validity of Eq (114) the sodium
borate lithium borate lithium phosphate and sodium silicate systems were considered with the
model results compared to experimental values in Figure 38 To confirm this model literature values
for activation barriers have been used The experimental activation barriers for the borates are taken
from the work of Martin[213] while the fragilities and the glass transition temperatures are from
Nemilov[219] The viscosity curve of the silicate system was fit with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) equation of viscosity[55] using the data from Poole[220] and Knoche et
al[221] with the activation barriers from Martinsen[222] Lithium phosphate activation data was
taken from Ngai and Martin[216] with the glass transition and fragility taken from Hermansen et
al and predicted values from topological constraint theory[223]
143
Figure 38 Different network formers and the prediction of the activation barrier from our model
compared with activation barriers from literature (A) Sodium silicate predictions and experimental
values[222] the error is calculated from the error in the fragility when fitting the data (B) Lithium
phosphate activation energy[216] predicted with topological constraint theory and compared with
the experimental values (C) Predictions over two different systems of alkali borates[213] sodium
and lithium with a reported R2 of 097
144
It is worth noting that this model is predicting a non-equilibrium kinetic property using
equilibrium viscosity parameters a connection that has been demonstrated in the past for viscosity
and relaxation models[61] In this work we are applying the same principles to the diffusion of
ions in a network The diffusion and viscosity in the case of equilibrium are known to be intimately
linked using the Stokes-Einstein expression (which breaks down for highly cooperative
rearrangements near and below the glass transition temperature[224]) Here we are no longer
connecting the two properties explicitly but instead connecting the activation barrier for diffusive
hopping to the two key parameters governing the viscous flow of the supercooled liquid state
The failure of the Stokes-Einstein expression near the glass transition is a well-known
problem and has been studied extensively for purposes of predicting crystallization rates[167] This
occurs because there is a breakdown of the ergodicity of a system around the glass transition In
the MAP model[44] the viscosity is controlled by the ergodic parameter x and is given by
min( )
max( )
f
p
f
T Tx
T T
=
(115)
Here the exponent p is related to the sharpness of the ergodic to non-ergodic transition Using the
ergodic parameter the overall viscosity ( ) is expressed as a linear combination of equilibrium
viscosity ( eq ) and non-equilibrium viscosity ( ne ) such that
10 10 10log ) log ( ) (1 ) log ( )( f eq f ne fx T T x T TT T = + minus (116)
By combining the Stokes-Einstein expression and the model proposed in this work we can define
the diffusion of an ion through the network as
( )10 10 10 0log ( 1 log exp ) l6 )
og(
b af
f
kx
T ED T T D
a T Tx
kT
minus + minus =
(117)
Here a is an empirical constant related to the size of the diffusing species k is Boltzmannrsquos
145
constant aE is the activation barrier calculated with Eq (114) and D0 is the limit of diffusion as
the temperature approaches infinity D0 should be chosen such that at fT T= the diffusion is
continuous This can be justified because the preexponential factor is related to the entropy of the
system and the fictive temperature is the temperature at which the configurational entropy of the
transition barriers becomes nearly constant Only one parameter is left unknown a since the
viscosity and fictive temperature can be calculated with an arbitrary thermal history using a
relaxation modeling tool such as RelaxPy[42] Implicit in RelaxPy are the assumptions pertaining
to the viscosity made by Guo et al[61] A similar model was first proposed by Cassar et al[167]
for modeling diffusion during crystallization although their model used an empirical hyperbolic
tangent function to approximate the ergodic factor In contrast the current model can be considered
explicitly as a function of thermal history and composition
64 Machine Learning Expansion
Though the TCT models presented here for different properties are immensely useful they
do have limitations the greatest of which is that intimate knowledge of the glassy structure is
required to parameterize these models This is due to the fact that the landscape is fundamentally
informed by the underlying structure of the network and as such to predict the dynamics of the
landscape information on the structure must be incorporated in some form However there is an
alternate method in which the physical origin is not considered in the development of the model
machine learning Machine learning comes in many forms and has become crucial to the study of
materials in the modern era There is a plethora of machine learning techniques however in this
paper we are focused on four key methods linear fitting (LF) methods random forest (RF)
symbolic regression (SR) and neural networks (NN)[3] [80] [225] [226] These methods are
146
explored extensively in literature and the exact underlying mathematics is beyond the scope of this
work In this work these methods have been used to create models for number of constraints (LF)
glass stability (RF) fragility (NN) melting point (NN) the infinite limit of viscosity (SR) Youngrsquos
modulus (NN) and coefficient of thermal expansion (NN)
To enable a linear fitting method for the number of constraints a simple steepest descent
algorithm was used to find the number of constraints due to each component There is no direct
experimental way to measure the number of constraints so instead a technique leveraging some
other property must be used to approximate the number of constraints In this case hardness is
related to the number of constraints through
25v cH A n= minus (118)
In which A is an empirical scaling parameter that is dependent on load indenter geometry and
glass family In this linear approach to determining constraints cn can be given by
0
compon
k
c k k
ents
cn x n=
= (119)
Combining this expression with a hardness database grouped by composition (in this case Sciglass)
allows for an approximate number of constraints of the system but due to A being a variable there
must be a fixed constraint in the system and every set of data is given a unqiue value of A The
fixed constraint in this system is that silicon atoms are four coordinated The value of A is chosen
such that the error between the prediction and the hardness is minimized The total process is as
follows
Choose a set of c kn
1 Calculate number of constraints for all compositions
2 Iterate over every dataset in Sciglass with gt 1 datapoints and scale by a factor of A
3 Calculate total error for steepest descent algorithm
4 Descent error slope
5 Repeat until error has been minimized
147
This then leads to an estimate for the number of constraints for every atomic species which can
then be used to estimate important properties such as chemical durability hardness and
qualitatively can predict glass stability
Two other techniques that are used are RF and SR When implemented a grid search for
the hyperparameters were used with a testing set used to determine the efficiency of each set of
hyperparameters Additionally each model was trained repeatedly and the lowest error was taken
The RF method was used to predict glass stability while undergoing a novel processing technique
called alkali-proton substitution (APS) The SR method was used to understand the compositional
dependence of the infinite temperature viscosity limit used in 3-parameter viscosity models To do
this the Sciglass database was taken (as implemented in GlassPy[227]) and sets of data that
included more than eight datapoints were fit to the MYEGA equation If the RMSE error of the fit
was less than 001 Pa s the fragility was greater than 17 and the value of the log infinite limit was
293 5minus it was accepted into the final database Each of these criteria were added due to the well-
known fact that this database is full of error and imprecise measurements The symbolic regression
was then performed through the grid search and the equation that performed best and reappeared is
given as
10loggT
mA B = + (120)
With A and B being parameters that varied depending on the hyper-parameters the infinite limit
threshold and the accuracy threshold used when creating the database We are currently in the
process of refining values for A and B is ongoing
This method allows for an easy prediction of the infinite limit so if the glass transition and
fragility are predicted through some other means (such as ML or TCT) a better prediction of the
third parameter is known without just accepting -293 as the mean Plots of the infinite limit vs the
148
fragility glass transition and Tgm showing the clear dependance of the infinite limit on the term
Tgm are shown in Figure 39
149
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top left) shows
the distribution of the infinite temperature limit in the database after limits exerted on the system
(Top right) The distribution of the infinite viscosity limit vs the glass transition (Bottom left) The
relationship between fragility and infinite temperature limit of viscosity (Bottom right) The infinite
temperature limit of viscosity vs the key metric predicted by SR
150
NN are perhaps the most widely discussed ML tool applied in literature This makes sense
given they are universal function approximators and are quite easy to train given sufficient data
The most difficult part of creating a NN is adjusting the hyperparameters so that good accuracy can
be found without causing overfitting To avoid overfitting the data is divided into a testing set then
a training set (just as for RF and SR) but instead of doing a grid search for the hyperparameters we
used gradient boosted regression trees to find the optimal set of parameters This consists of
creating an initial tree then evaluating it based on a small set of data and finding the minimum
point then running a new set with perturbations around that point to create another tree This is
repeated until the value converges to a small error In this work the value being determined is the
lowest error of 100 sequentially trained neural nets with a set of hyperparameters This is
particularly useful because there is no concern for the type of parameters (string int floats) that
determine the value at the end so one can do a mixed parameter optimization that includes every
conceivable variable The values being optimized are given by
bull Learning rate [10-5 10-1] (float on logarithmic scale)
bull Decay rate for Adam Optimization [10-6 10-2] (float on logarithmic scale)
bull Hidden layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull Output layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull The number of nodes in each layer [81024] (int)
bull Batch size [1256] (int)
bull Patience [5250] (int)
In Table 8 we have listed the hyper-parameters that were used for the fragility melting point
Youngrsquos modulus and co-efficient of thermal expansion (CTE)
151
Table 8 Hyperparameters for different neural networks after hyper-optimizations
Model Fragility Melting
Temperature
Youngrsquos
Modulus
CTE
Learning Rate 0035 0018 0027 00091
Decay Rate 000293 000095 000161 0002
Hidden Layer RELU RELU RELU RELU
Output Layer Linear Linear SELU RELU
Number of Hidden
Layers
2 1 1 1
Number of Nodes 235 506 514 470
Batch Size 4 118 211 125
Patience 25 250 23 22
RMSE 5 [-] 805 [K] 68 [GPa] 96 x 10-7 [1 K]
152
Chapter 7
Designing Green Glasses for the 21st Century
Thus far the work presented in this dissertation has enabled a new set of design tools for
everything from crystallization glass relaxation and optimizing over the ideal compositional space
we can use them for designing new glasses Building on these models and insights new applications
of glass are enabled specifically three glasses an ion conducting glass for batteries a proton
conducting glass for hydrogen fuel cells and a commodity glass for consumers All three are
currently undergoing testing but experimental results have been reported back as of yet Overall
both the hydrogen and ion conducting glasses have been designed thorough a combination of ML
and underlying physical models and as such hare reported herein
71 Glass Electrolytes
An emerging high-interest application of glass is solid state batteries Glasses offer a
possible solution to the growing energy crisis however to realize this new paradigm new research
methods are needed The main barrier for any new batteries to be commercialized is a high
conductivity at room temperature (gt 10-3 S cm) high stability ease of processing and ultimately
an alternative must be cheaper than current liquid state Li-ion batteries glass is an attractive
candidate due to the innate stability a simple processing technique the infinite variability and the
relatively low cost of production In addition the structure of a glass is a liquid-like structure which
may encourage ion migration like that seen in liquids Despite all of these advantages only a few
compositions have realized the requirements but further research is needed to find optimal
compositions
153
To optimize the composition for activation barriers we must understand the relationship
between conductivity and structure Structural effects can be propagated in two different methods
based on the Arrhrenius expression for conductivity
0 exp aE
kT
minus =
(121)
The two methods are through 0 (the infinite temperature conductivity) and the activation barrier
( aE ) The pre-exponential factor is approximately a constant for simliar compositions while the
activation barrier varies dramatically over small compositional spaces making it the larger of the
two concerns Many models have been presented to predict the compositional dependence of the
activation barrier the Anderson-Stuart (AS) model the Christensen-Martin-Anderson-Stuart
(CMAS) model the weak electrolyte (WE) model Kohlrausch exponent model (KEM) from Ngai
et al[216] and there is the model proposed in this dissertation which we will call the Wilkinson
viscous cooperative conductivity model (WVCC)[8] [88] [204] [207]ndash[209]
To optimize machine learning (ML) is a powerful technique that is implemented The best
method to train the ML is to use the direct relationship between the compositions and the activation
barrier but this is not currently feasible due to a lack of central depository or database for this
information To make the prediction of the activation barrier we need use some model that is reliant
on commonly accesible parameters This rules out all models for this type of analysis except since
WVCC since the other models require some fitting parameters not accessible Each models required
parameters fitted values (please note that the models typically include approximations for these
values) and caveats are shown in Table 9
154
Table 9 A table with some ionic conductivity models and the parameters needed for them as well
as the disadvantages for each These are not the only models but are representative of those
commonly used in literature
Model Required Parameters Fit Parameters Caveats
AS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
lsquoMadelungrsquo Constant
Covalency Parameter
Doorway Radius1
The Madelung
Constant is not smooth
as a function of
composition and the
fitting parameters scale
the activation energy
non-proportionally
CMAS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
Dielectric Permittivity
lsquoMadelungrsquo Constant
Doorway Radius1
Jump Distance of Ion1
The same issues as AS
but in addition the
dielectric permittivity
lacks a database as
well
WE Charge Concentration Equilibrium Coefficient1 The equilibrium
coefficient drastically
changes the prediction
non-linearly
KEM Stretching Exponent Proportionality
Coefficient
There is not enough
data available to know
the compositional
dependence of the
stretching exponent
WVCC Glass Transition
Fragility Index
Proportionality
Coefficient
Due to the
proportionality
constant the activation
barrier can only be
known in a local
composition range
1 There are ways to approximate this value based on additional data
155
The WVCC model predicts that the activation barrier for ionic conductivity is given by
a
g
EA
mT= (122)
in which A is a proportionality constant This means that to predict the glass behavior we donrsquot
need to know things about the glass but instead merely about the liquid state where the viscosity is
readily available Leveraging this technique if the viscosity information as a function of
compositions is available then the local glass with the lowest activation barrier can be found by
minimizing 1 gT m To access this information multiple options are available such as topological
constraint theory but in this work we will leverage ML since it is easily applied to a large
compositional space
To get the viscosity we will use neural networks (NN) A well trained NN for the glass
transition is readily available and as such will be used for our prediction of gT In the previous
section there was presented a trained fragility NN which enables a complete optimization of the
local composition based on WVCC To generate a database of fragility values that the NN was
trained on SciGlass viscosity values were fit with MYEGA model The data had to fit following
criteria
1 The total number of data points for fitting had to be greater than 5 from the same literature
source
2 The root-mean-square-error of the fit had to be less than 001 Pa s
3 The infinite temperature value of viscosity had to be within 3 orders of magnitude of the
accepted logarithmic value (-293 log(Pa s))
An additional neural network recently created by Cassar[228] was also used for an independent
comparison Cassarrsquos NN was based on a hybrid physical ML work and may offer better insights
when extrapolating far from experimental datapoints In this study to prove this concept we will
only focus on P2O5 B2O3 Al2O3 and Li2O
156
This gives an entire method to predict glass battery candidates This technique will find all
of the local minimum but due to the fact that there is a proportionality constant it is unknown which
local minimum is the absolute minimum This means that the composition with the lowest
activation barrier in the space is identified but another technique is needed to investigate which one
of the identified compositions is best The method follows
1 Randomize an initial glass candidate with the content of being normalized
2 Run the glass candidate through the NN to find values of fragility and the glass transition
3 Perturb the composition to find the gradient
4 Step down the gradient
5 If the new gradient is 0 this is a local minimum and a possible glass candidate return to 1
Otherwise return to step 3
This method cannot find global optima because there is no guarantee that the
proportionality constant will stay constant across wide compositional spaces To narrow it down
an additional technique to compare the predicted values is needed (such as another activation
barrier model molecular dynamics simulations a regression algorithm or simply melt the proposed
compositions) To further pair down in this work a k-means algorithm was used along with the
assumption that the proportionality constant was slowly changing over compositional space This
allowed us to take the best glass in each grouping predicted by k-means The glasses we have
predicted (using both the separate predictor NN and the method by Cassar et al[228]) are listed
below in Table 10 while an example prediction for 1000 gmT (which should scale with the
activation barrier) in the Na2O ndash B2O3 ndash SiO2 system is shown in Figure 40 The glasses are currently
undergoing melting at Coe College and Iowa State It is worth noting that composition D is a well-
known composition that is known to exhibit criteria close to the criteria for conductivity If this
method shows that these compositions are better than other compositions in the family found in
literature then it can be expanded to include a wider range of component and hopefully finding a
universal candidate for glass batteries
157
Table 10 The predicted compositions based on the optimization scheme proposed
Compositions Li2O Al2O3 P2O5 B2O3
A- This work NN 55 31 0 14
B- This work NN 44 1 5 50
C- Cassar 60 26 2 13
D- Cassar 50 0 0 50
Figure 40 The prediction of the scaling of the activation barriers for a common sodium borosilicate
system
72 Hydrogen Fuel Cell Glasses
An emerging high-interest application of glass that is proton-conducting intermediate
temperature fuel cells[7] [229] A fuel cell in short is a device that turns chemical fuel into
electricity without the use of combustion It is often described as a chemical battery due to the
similarity of a fuel construction with batteries One of the crucial components to a working
intermediate range fuel cell is an electrolyte with a high proton migration The electrolyte must be
stable over a wide range of temperatures (up to 673 K) and a proton conductivity greater than 10-2
S cm-1 at the operating temperature[7] Previously phosphate glasses near the metaphosphate
compositions were used since they carried the appropriate number of residual protons for
conductivity studies However this restricted the phase space to only n-2 dimensions drastically
reducing the degrees of freedom of the problem and artificially restricting possible solutions In
2013[230] to access the full phase space a new method was developed where a sodium phosphate
glass could have all the sodium ions replaced with protons This allows high concentrations of
protons to be achieved with a variety of starting compositions A previous report[230] has
summarized successful samples these data are used as a training set for this study with additional
data on compositions that have failed taken from internal theses
To choose the candidate material for each model a cost function is defined The cost function
is an analytical function whose inputs are the fraction of each oxide component and the output is a
value that rates the composition The cost function consists as of many properties that are of interest
In the application of oxide electrolytes for an intermediate range fuel cell the two main properties
to worry about are the ionic conductivity at high temperatures (we will choose 473 K due to this
being the low end in which these materials are considered) and the relative stability of each phase
159
This is especially important since the material will undergo the APS (alkali-proton substitution) in
which the sodium will be replaced with protons in the bulk of the material The cost function used
in this work is given by
473 1( ) log 000T K Sx = minus= minus (123)
In which S is either a 0 or 1 (representing a crystal or a glass respectively) and is the
ionic conductivity The function is arbitrary and could be expanded to include as many terms are
as needed in the goal of this glass The stability is heavily weighted because it is more important
than the conductivity and any sample that is not stable should not be considered In order to predict
each of these samples a hybrid physicalempirical approach is used For predicting the ionic
conductivity of a proton-swapped glass we began with an observed relationship between the glass
transition temperature and the conductivity at 473 K This empirical relationship is shown in Figure
41 The relationship although empirical can provide some insights into the physics of proton
transferred glasses (no universality is claimed in this work) The activation barrier for these glasses
has widely remained unchanged as noted in a previous work due to the fact that the activation
barrier is simply to disassociate the proton with the non-bridging oxygen (NBO) and once the
proton is free interacts rarely with the network until it is rebounded to said network Thus the pre-
factor can be understood as being related to the degeneracy of proton conduction pathways When
the network has lower configurational entropy there are fewer pathways for the proton to travel in
this case it is known that the configurational entropy is then the dominating effect for the glass
transition[53]
The relationship between the glass transition temperature and the configurational entropy
can be expanded through the Stokes-Einstein relationship evaluated at the glass transition
22
1273 10
6 10g
g
T T g
kTnZeD nZe
anT
minus
= = = (124)
160
In which a is approximately the size of a proton and n is the number of charge carriers (which is
approximately a constant according to the weak electrolyte theory) This determines the intercept
of the glassy form proton conductivity and then the activation barrier (slope) is found to be
approximately the same for all proton conducting glasses meaning that the intercept is the dominant
effect on the behavior of the glass and the only variable controlling the intercept is the glass
transition
Leveraging the configurational entropyrsquos relationship with the glass transition along with
topological constraint theory we can write an expression for the glass transition and as such gain
predictive power for proton conductivity To predict the glass transition temperature (Tg) we start
with the Adam Gibbs model of viscosity[53] Topological constraint theory then states that the
degrees of freedom of the network is proportional to the configurational entropy[23] [56] [57]
Using the liquid state definition of the glass transition (1210 = Pa s) and the well-known infinite
temperature of viscosity we can then write
1
1493 lng
BT
kf=
(125)
Where f is the degrees of freedom Letting the constants be equal to A and rewriting the expression
in terms of constraints we arrive at a predictive formula for the glass transition[109]
3
g
x c x
x
AT
m n=
minus (126)
Where nc is the number of constraints provided by each component at the glass transition associated
with each component x and their molar fraction mx The value of the constraints was then linearly
parameterized to the glass transition data from literature[7] [230] [231] This is not the most
explicit approach to counting constraints however it is the most convenient when considering
large phase spaces being optimized over This glass transition temperature is then converted to
conductivity using the empirical translation shown in Figure 41
161
The other term in our cost function is that of stability The question of what forms glass is
a notorious question that goes back to the first serious days of research into materials It has been
shown that no one metric is a good predictor and no metric is universal However one must be used
to rule out bad compositions from the start To do this a Random Forest method is implemented
Though this may fail in edge cases it will at least help identify the right area to explore A
comprehensive review of random forest methods and machine learning methods for prediction of
glass properties can be found elsewhere[3] [4] [225] The amount of information previously
obtained about what forms glass in the compositional family we will be working on is not enough
to use random forest methods on just the fraction of each phase In order to circumvent this issue
we chose four physical parameters that would be used as features for the predictions
bull Total mol oxide modifier This was chosen because it is well known that modifiers break
up the network and most glass forming theories are related to the network percentages We
considered the oxide versions of H Na Ba Sr and Ca as the modifiers
bull Total mol network former Similarly it is well known the network influences glass
forming We considered B Ge and P as the network forming cations
bull Mean cation charge Though there has been previous work on this topic it is rarely
considered as a lsquonormalrsquo predictor of glass forming capability It is included here because
these glasses are mainly invert and as such a higher field strength will increase the cohesive
nature of the network
bull Entropy of mixing This is an important parameter to consider with so many components
it may be stabilized by the entropy of mixing The entropy of mixing (Sm) was calculated
using the Gibbs entropy
The Gibbs entropy is given by
lnm x
x
xS k m m= minus (127)
The random forest model consisted of 100 trees with a maximum depth of 3 The results of both
the random forest model the topological predictions the optimization and the resultant glass are
shown in the results section The optimization was limited to the range of each component that has
been explored experimentally as well as an additional constraint of at least 5 mol rare earth oxide
and at least 5 mol of boron oxide or germania this is justified elsewhere[231]
162
Figure 41 The relationship between the glass transition and the proton conductivity This is justified
two ways one through the relationship of the entropy of diffusion and glass formation (the Adam-
Gibbs model) and through the fact that water is known to depress the glass transition
163
To explain the results we will first start with the models listed to confirm their validity
The first model is the prediction of the glass transition temperature Figure 42 shows the accuracy
of the glass transition prediction as well as the relative contribution of each component to the glass
transition In this figure any component that has a number of constraints greater then PO52 will
increase the glass transition and as such decrease the conductivity Thus it is preferable to construct
a glass with only components whose constraints at Tg are less than PO52 as long as the glass
remains stable Interestingly whether the data includes only the glass transitions of protonated glass
or all glasses determines the role of La Barium and sodium both decrease the glass transition
playing the role of the more traditional modifier however they do provide some rigidity to the
network Protons have a net negative effect which we believe to be a result of the APS process
where they are introduced as the sodium is leaving so this negative effect corresponds to a breaking
of the rigidity that exists It is also interesting to note that adding a boron phase to the sample
increases the glass transition though not as much as some of smaller alkaline earth samples
assumingly due to their increased field strength The worst network former appears to be GeO2
To calibrate the accuracy of the random forest model a confusion matrix is shown It shows
that the RF always predict that a glass will be a glass however there is some error when it forms a
crystal with 23 of the time being misidentified Though random forest methods are empirical we
can derive some physical meaning by looking at the relative importance of each feature From the
random forest study of stability we are able to rank the relative importance of each feature used
The mean cation charge mixing entropy and percent network formers are all of considerable
importance while the percent modifiers is less so
Once these models were developed and checked the reliability of the cost function is
confirmed After which the minimization of the cost function was performed and one glass was
chosen as optimal Boron was found to be preferential in the glass but was limited since the
compositional space was bound by previous samples (an additional sample is undergoing
164
characterization where the boron content was slightly increased) The optimized sample when
synthesized had some visible nucleation so the optimization while redone with limiting the
component maxima to just the limits of those where compositions in which APS was successfully
performed This resulted in the OP composition The three samples currently are undergoing
experimental characterization are listed in Table 11
165
Table 11 The compositions synthesized in this work These compositions were predicted by
minimizing the cost function described in Eq (123) OP is the variant that was melted after OP
partially crystallized B-OP appeared to have surface nucleation in some spots but was cut and
removed before APS treatment
Name PO52 NaO12 BaO LaO32 GeO2 BO32
OP 44 36 10 5 3 2
OP 49 36 5 5 3 2
B-VOP 44 36 10 5 1 4
166
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine the glass
forming region Over top the constraints at the glass transition provided by each oxide species is
listed Since the objective is to decrease Tg while staying in the glass forming region we will
attempt to minimize use of elements that increase the glass transition temperature (nc gt 17)
167
Chapter 8
Conclusions
The rapid rate of new information in the current scientific climate and the infinite
variability of glass both stand as both unique challenges and opportunities In this dissertation we
have divided the influence of energy landscapes on the effects of properties into two independent
spaces the compositional and the thermal history dimensions By building models that succinctly
and accurately describe the dynamics of different hyper-coordinate (composition thermal history
crystallinity) changes the feasibility of designing new glasses for the challenges of the 21st century
is obtainable
Before designing new models the current state-of-the-art models need to be implemented
and understood To reach this end two softwares based on previous work were created RelaxPy
and ExplorerPy RelaxPy is an implementation of the MAP model and though the MAP model is
powerful when predicting the dynamics of the glass RelaxPy ultimately showed that fictive
temperature is insufficient at capturing the underlying physics and is intensive to parameterize
ExplorerPy was created to standardize the approach to mapping energy landscapes Energy
landscapes served as the key method of understanding the deeper dynamics of glasses glass-
ceramics and liquids
The thermal history dependence of glass with respect to both relaxation and crystallization
has been incorporated into the existence of new models called ldquotoy landscapesrdquo The toy landscapes
have built upon the physics of previous models such as the MAP model however due to the lower
parametrization cost and the built-in increased physicality toy landscapes pose as a tool to increase
our understanding and speed-up the rate of new glass discoveries This tool can deal with the
complexity of the higher dimensional spaces due to crystallization and relaxation without
assumptions concerning fictive temperature
168
The compositional dimensions are the remaining dimensions that must be optimized when
designing a glass or glass-ceramic Building on previous work models have been developed that
enable predictions of Youngrsquos modulus surface reactivity and ionic conductivity However there
are additional properties that are needed and for those we have proposed novel machine learning
approaches This dissertation has not enabled the design of glass for every application However
it lays the groundwork and approaches to design glasses for societyrsquos growing needs Presented as
well are the methods used to design glasses that could satisfy the requirements for solid state glass
electrolytes and hydrogen fuel cells All of these techniques together promise to be a powerful new
framework to build the glasses of the future
169
References
[1] A K Varshneya and J C Mauro Fundamentals of Inorganic Glasses 3rd ed Elsevier
2019
[2] J C Mauro A Tandia K D Vargheese Y Z Mauro and M M Smedskjaer
ldquoAccelerating the Design of Functional Glasses through Modelingrdquo Chem Mater vol
28 no 12 pp 4267ndash4277 2016 doi 101021acschemmater6b01054
[3] J C Mauro ldquoDecoding the glass genomerdquo Curr Opin Solid State Mater Sci pp 1ndash7
2017 doi 101016jcossms201709001
[4] H Liu Z Fu Yang X Xu and M Bauchy ldquoMachine learning for glass science and
engineering A reviewrdquo J Non Cryst Solids no March p 119419 2019 doi
101016jjnoncrysol201904039
[5] M M Smedskjaer C Hermansen and R E Youngman ldquoTopological engineering of
glasses using temperature-dependent constraintsrdquo MRS Bull vol 42 no 01 pp 29ndash33
2017 doi 101557mrs2016299
[6] J C Mauro and M M Smedskjaer ldquoStatistical mechanics of glassrdquo J Non Cryst Solids
vol 396ndash397 pp 41ndash53 2014 doi 101016jjnoncrysol201404009
[7] T Omata et al ldquoProton transport properties of proton-conducting phosphate glasses at
their glass transition temperaturesrdquo Phys Chem Chem Phys vol 21 no 20 pp 10744ndash
10749 2019 doi 101039c9cp01502g
[8] Z A Grady C J Wilkinson C A Randall and J C Mauro ldquoEmerging Role of Non-
crystalline Electrolytes in Solid-State Battery Researchrdquo Front Energy Res vol 8 p
218 2020 doi 103389fenrg202000218
[9] R C Welch et al ldquoDynamics of glass relaxation at room temperaturerdquo Phys Rev Lett
170
vol 110 no 26 p 265901 2013 doi 101103PhysRevLett110265901
[10] M Montazerian E D Zanotto and J C Mauro ldquoModel-driven design of bioactive
glasses from molecular dynamics through machine learningrdquo Int Mater Rev vol 0 no
0 pp 1ndash25 2019 doi 1010800950660820191694779
[11] E D Zanotto and J C Mauro ldquoThe glassy state of matter Its definition and ultimate
faterdquo J Non Cryst Solids vol 471 pp 490ndash495 2017 doi
101016jjnoncrysol201705019
[12] R G Palmer ldquoBroken ergodicityrdquo Adv Phys vol 31 no 6 pp 669ndash735 1982 doi
10108000018738200101438
[13] J C Mauro P Gupta and R J Loucks ldquoContinuously broken ergodicityrdquo J Chem
Phys vol 126 no 18 p 184511 2007 doi 10106312731774
[14] M Micoulaut ldquoRelaxation and physical aging in network glasses A reviewrdquo Reports
Prog Phys vol 79 no 6 p 066504 2016 doi 1010880034-4885796066504
[15] D J Wales and H A Scheraga ldquoGlobal optimization of clusters crystals and
biomoleculesrdquo Science (80- ) vol 285 no 5432 pp 1368ndash1372 1999 doi
101126science28554321368
[16] J C Mauro R J Loucks and P Gupta ldquoFictive temperature and the glassy staterdquo J
Am Ceram Soc vol 92 no 1 pp 75ndash86 2009 doi 101111j1551-2916200802851x
[17] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 2009 doi 10106313152432
[18] F H Stillinger and T A Weber ldquoHidden structure in liquidsrdquo Phys Rev A vol 25 no
2 pp 978ndash989 1982 doi 101103PhysRevA25978
[19] F H Stillinger ldquoSupercooled liquids glass transitions and the auzmann paradoxrdquo J
Chem Phys vol 88 no 12 pp 7818ndash7825 1988 doi 1010631454295
171
[20] S Neelamraju C Oligschleger and J C Schoumln ldquoThe threshold algorithm Description of
the methodology and new developmentsrdquo J Chem Phys vol 147 no 15 p 152713
2017 doi 10106314985912
[21] D Prada-Gracia J Goacutemez-Gardentildees P Echenique and F Falo ldquoExploring the free
energy landscape From dynamics to networks and backrdquo PLoS Comput Biol vol 5 no
6 p e1000415 2009 doi 101371journalpcbi1000415
[22] E Pollak A Auerbach and P Talknerz ldquoObservations on rate theory for rugged energy
landscapesrdquo Biophys J vol 95 no 9 pp 4258ndash4265 2008 doi
101529biophysj108136358
[23] G G Naumis ldquoEnergy landscape and rigidityrdquo Phys Rev E vol 71 p 026114 2005
doi 101103PhysRevE71026114
[24] S J Benkovic G G Hammes and S Hammes-Schiffer ldquoFree-energy landscape of
enzyme catalysisrdquo Biochemistry vol 47 no 11 pp 3317ndash3321 2008 doi
101021bi800049z
[25] M Roca B Messer D Hilvert and A Warshel ldquoOn the relationship between folding and
chemical landscapes in enzyme catalysisrdquo Proc Natl Acad Sci U S A vol 105 no 37
pp 13877ndash13882 2008 doi 101073pnas0803405105
[26] P Li G Henkelman J A eith and J Johnson ldquoElucidation of aqueous solvent-
mediated hydrogen-transfer reactions by ab initio molecular dynamics and nudged elastic-
band studies of NaBH4 hydrolysisrdquo J Phys Chem C vol 118 no 37 pp 21385ndash21399
2014 doi 101021jp507872d
[27] S P Niblett M Biedermann D J Wales and V De Souza ldquoPathways for diffusion in
the potential energy landscape of the network glass former SiO2rdquo J Chem Phys vol
147 no 15 p 152726 2017 doi 10106315005924
[28] Y Yu M Wang M M Smedskjaer J C Mauro G Sant and M Bauchy
172
ldquoThermometer Effect Origin of the Mixed Alkali Effect in Glass Relaxationrdquo Phys Rev
Lett vol 119 no 9 pp 1ndash5 2017 doi 101103PhysRevLett119095501
[29] Y Yu M Wang D Zhang B Wang G Sant and M Bauchy ldquoStretched Exponential
Relaxation of Glasses at Low Temperaturerdquo Phys Rev Lett vol 115 no 16 p 165901
2015 doi 101103PhysRevLett115165901
[30] Steve Plimpton ldquoFast Parallel Algorithms for Short-Range Molecular Dynamicsrdquo J
Comput Phys vol 117 no 1 pp 1ndash19 1995 doi
httpsdoiorg101006jcph19951039
[31] N Mousseau and G T Barkema ldquoTraveling through potential energy landscapes of
disordered materials The activation-relaxation techniquerdquo Phys Rev E vol 57 no 2
pp 2419ndash2424 1998 doi 101103PhysRevE572419
[32] F El-Mellouhi N Mousseau and L J Lewis ldquo inetic activation-relaxation technique
An off-lattice self-learning kinetic Monte Carlo algorithmrdquo Phys Rev B vol 78 p
153202 2008 doi 101103PhysRevB78153202
[33] D J Wales ldquoDiscrete path samplingrdquo Mol Phys vol 100 no 20 pp 3285ndash3305 2002
doi 10108000268970210162691
[34] G Henkelman and H Joacutensson ldquoImproved tangent estimate in the nudged elastic band
method for finding minimum energy paths and saddle pointsrdquo J Chem Phys vol 113
no 10 pp 9978ndash9985 2000 doi 10106314961868
[35] D Sheppard P Xiao W Chemelewski D D Johnson and G Henkelman ldquoA
generalized solid-state nudged elastic band methodrdquo J Chem Phys vol 136 p 074103
2012 doi 10106313684549
[36] A Pedone G Malavasi M C Menziani A N Cormack A V and N York ldquoA New
Self-Consistent Empirical Interatomic Potential Model for Oxides Silicates and Silica-
Based Glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
173
[37] A Takada C R A Catlow and G D Price ldquoComputer modelling of B2O3 II Molecular
dynamics simulations of vitreous structuresrdquo J Phys Condens Matter vol 7 no 46 pp
8693ndash8722 1995 doi 1010880953-8984746004
[38] A C T Van Duin S Dasgupta F Lorant and W A Goddard ldquoReaxFF A reactive
force field for hydrocarbonsrdquo J Phys Chem A vol 105 no 41 pp 9396ndash9409 2001
doi 101021jp004368u
[39] J C Mauro and A Varshneya ldquoModel interaction potentials for selenium from ab initio
molecular simulationsrdquo Phys Rev B vol 71 p 214105 2005 doi
101103PhysRevB71214105
[40] J C Mauro R J Loucks J Balakrishnan and S Raghavan ldquoMonte Carlo method for
computing density of states and quench probability of potential energy and enthalpy
landscapesrdquo J Chem Phys vol 126 no 19 2007 doi 10106312733674
[41] Y Z Mauro C J Wilkinson and J C Mauro ldquo ineticPy A tool to calculate long-time
kinetics in energy landscapes with broken ergodicityrdquo SoftwareX vol 11 p 100393
2020 doi 101016jsoftx2019100393
[42] C J Wilkinson Y Z Mauro and J C Mauro ldquoRelaxPy Python code for modeling of
glass relaxation behaviorrdquo SoftwareX vol 7 pp 255ndash258 2018 doi
101016jsoftx201807008
[43] J C Mauro ldquoTopological constraint theory of glassrdquo Am Ceram Soc Bull vol 90 no
4 pp 31ndash37 2011 doi 101039c3ee40810h
[44] J C Mauro D C Allan and M Potuzak ldquoNonequilibrium viscosity of glassrdquo Phys Rev
B vol 80 p 094204 2009 doi 101103PhysRevB80094204
[45] M Cardona R V Chamberlin and W Marx ldquoThe history of the stretched exponential
functionrdquo Ann der Phys vol 16 no 12 pp 842ndash845 2007 doi
101002andp200710269
174
[46] P Grassberger and I Procaccia ldquoThe long time properties of diffusion in a medium with
static trapsrdquo J Chem Phys vol 77 no 1982 pp 6281ndash6284 1982 doi
1010631443832
[47] J C Phillips ldquoMicroscopic aspects of Stretched Exponential Relaxation (SER) in
homogeneous molecular and network glasses and polymersrdquo J Non Cryst Solids vol
357 no 22ndash23 pp 3853ndash3865 2011 doi 101016jjnoncrysol201108001
[48] J C Phillips ldquo ohlrausch explained The solution to a problem that is 1 0 years oldrdquo J
Stat Phys vol 77 no 3ndash4 pp 945ndash947 1994 doi 101007BF02179472
[49] A Q Tool and C G Eichlin ldquoVariations caused in the heating curves of glass caused by
heat treatmentrdquo J Am Ceram Soc vol 14 pp 276ndash308 1931
[50] A Q Tool ldquoRelation Between Inelastic Deformability and Thermal Expansion of Glass in
Its Annealing Rangerdquo J Am Ceram Soc vol 29 no 9 pp 240ndash253 1946 doi
101111j1151-29161946tb11592x
[51] H N Ritland ldquoLimitations of the Fictive Temperature Conceptrdquo J Am Ceram Soc vol
39 no 12 pp 403ndash406 Dec 1956 doi 101111j1151-29161956tb15613x
[52] O S Narayanaswamy ldquoA model of structural relaxation in glassrdquo J Am Ceram Soc
vol 54 no 10 pp 491ndash498 1971 doi 101111j1151-29161971tb12186x
[53] G Adam and J Gibbs ldquoOn the Temperature Dependence of Cooperative Relaxation
Properties in Glass‐Forming Liquidsrdquo J Chem Phys vol 43 no 1 pp 139ndash146 1965
doi 10106311696442
[54] Q Zheng J C Mauro A J Ellison M Potuzak and Y Yue ldquo niversality of the high-
temperature viscosity limit of silicate liquidsrdquo Phys Rev B vol 83 no 21 p 212202
2011 doi 101103PhysRevB83212202
[55] J C Mauro Y Yue A J Ellison P Gupta and D C Allan ldquoViscosity of glass-
forming liquidsrdquo Proc Natl Acad Sci vol 106 no 47 pp 19780ndash19784 2009 doi
175
101073pnas0911705106
[56] G G Naumis ldquoGlass transition phenomenology and flexibility An approach using the
energy landscape formalismrdquo J Non Cryst Solids vol 352 no 42-49 SPEC ISS pp
4865ndash4870 2006 doi 101016jjnoncrysol200601160
[57] P Gupta and J C Mauro ldquoComposition dependence of glass transition temperature
and fragility I A topological model incorporating temperature-dependent constraintsrdquo J
Chem Phys vol 130 no 9 p 094503 2009 doi 10106313077168
[58] C A Angell ldquoRelaxation in liquids polymers and plastic crystals - strongfragile patterns
and problemsrdquo J Non Cryst Solids vol 13 no 31 pp 131ndash133 1991 doi
1010160022-3093(91)90266-9
[59] G S Fulcher ldquoAnalysis of Recent Measurements of the Viscosity of Glassesrdquo J Am
Ceram Soc vol 75 no 5 pp 1043ndash1055 1992 doi 101111j1151-
29161992tb05536x
[60] I Avramov ldquoViscosity in disordered mediardquo J Non Cryst Solids vol 351 no 40ndash42
pp 3163ndash3173 Oct 2005 doi 101016JJNONCRYSOL200508021
[61] X Guo J C Mauro D C Allan and M M Smedskjaer ldquoPredictive model for the
composition dependence of glassy dynamicsrdquo J Am Ceram Soc vol 101 pp 1169ndash
1179 2018 doi 101111jace15272
[62] Doss C J Wilkinson Y Yang H Lee L Huang and J C Mauro ldquoMaxwell
relaxation time for nonexponential α-relaxation phenomena in glassy systemsrdquo J Am
Ceram Soc vol 103 no 6 pp 3590ndash3599 2020 doi 101111jace17051
[63] P Gupta and A R Cooper ldquoTopologically disordered networks of rigid polytopesrdquo J
Non Cryst Solids 1990 doi 1010160022-3093(90)90768-H
[64] J C Phillips and M F Thorpe ldquoConstraint theory vector percolation and glass
formationrdquo Solid State Commun vol 53 no 8 pp 699ndash702 1985 doi 1010160038-
176
1098(85)90381-3
[65] X Feng W Bresser and P Boolchand ldquoDirect Evidence for Stiffness Threshold in
Chalcogenide Glassesrdquo Phys Rev Lett vol 78 no 23 pp 4422ndash4425 1997 doi
101103PhysRevLett784422
[66] W Bresser P Boolchand and P Suranyi ldquoRigidity Percolation and Molecular Clustering
in Network Glassesrdquo Phys Rev Lett vol 56 no 23 pp 2493ndash2496 1986 doi
101103PhysRevLett562493
[67] Y Vaills T Qu M Micoulaut F Chaimbault and P Boolchand ldquoDirect evidence of
rigidity loss and self-organization in silicate glassesrdquo J Phys Condens Matter vol 17
no 32 pp 4889ndash4896 2005 doi 1010880953-89841732003
[68] A irchner and J C Mauro ldquoStatistical Mechanical Model of the Self-Organized
Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologiesrdquo
Front Mater vol 6 p 11 2019 doi 103389fmats201900011
[69] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 Jun 2009 doi 10106313152432
[70] D R Cassar A C P L F de Carvalho and E D Zanotto ldquoPredicting glass transition
temperatures using neural networksrdquo Acta Mater vol 159 pp 249ndash256 Oct 2018 doi
101016JACTAMAT201808022
[71] J C Mauro and R J Loucks ldquoSelenium glass transition A model based on the enthalpy
landscape approach and nonequilibrium statistical mechanicsrdquo Phys Rev B vol 76 no
17 p 174202 2007 doi 101103PhysRevB76174202
[72] C J Wilkinson et al ldquoEnergy Landscape Modeling of Crystal Nucleationrdquo Nat Comput
Mater p Submitted 2020
[73] M E Mc enzie and J C Mauro ldquoHybrid Monte Carlo technique for modeling of crystal
177
nucleation and application to lithium disilicate glass-ceramicsrdquo Comput Mater Sci vol
149 no January pp 202ndash207 2018 doi 101016jcommatsci201803034
[74] M E McKenzie et al ldquoImplicit glass model for simulation of crystal nucleation for glass-
ceramicsrdquo npj Comput Mater vol 4 no 1 pp 1ndash7 2018 doi 101038s41524-018-
0116-5
[75] E D Zanotto and P F James ldquoExperimental tests of the classical nucleation theory for
glassesrdquo J Non Cryst Solids vol 74 no 2ndash3 pp 373ndash394 1985 doi 1010160022-
3093(85)90080-8
[76] Y Yu N M A Krishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[77] I Pignatelli A Kumar M Bauchy and G Sant ldquoTopological control on silicatesrsquo
dissolution kineticsrdquo Langmuir vol 32 pp 4434ndash4439 2016 doi
101021acslangmuir6b00359
[78] C J Wilkinson et al ldquoTopological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phaserdquo J Phys Chem Lett vol 10 pp
3955ndash3960 2019 doi 101021acsjpclett9b01275
[79] Y Zhang A Li B Deng and Hughes ldquoData-driven predictive models for chemical
durability of oxide glass under different chemical conditionsrdquo npj Mater Degrad vol 4
no 1 pp 1ndash11 2020 doi 101038s41529-020-0118-x
[80] N M Anoop Krishnan S Mangalathu M M Smedskjaer A Tandia H Burton and M
Bauchy ldquoPredicting the dissolution kinetics of silicate glasses using machine learningrdquo J
Non Cryst Solids vol 487 no February pp 37ndash45 2018 doi
101016jjnoncrysol201802023
[81] R M Potter J Hoffman and J Hadley ldquoAn update of the equation for predicting the
178
dissolution rate of glass fibers from their chemical compositionsrdquo Inhal Toxicol vol 29
no 4 pp 145ndash146 2017 doi 1010800895837820171321702
[82] B Deng ldquoMachine learning on density and elastic property of oxide glasses driven by
large datasetrdquo J Non Cryst Solids vol 529 no August 2019 p 119768 2020 doi
101016jjnoncrysol2019119768
[83] S Feller N Lower and M Affatigato ldquoDensity as a probe of oxide glass structurerdquo
Phys Chem Glas vol 42 no 3 pp 240ndash246 2001 Accessed Jul 19 2018 [Online]
Available
httpswwwingentaconnectcomcontentsgtpcg200100000042000000034203240
[84] Q Zheng and H Zeng ldquoProgress in modeling of glass properties using topological
constraint theoryrdquo Int J Appl Glas Sci vol 11 no 3 pp 432ndash441 2020 doi
101111ijag15105
[85] M Bauchy M Javad Abdolhosseini Qomi C Bichara F-J Ulm and R J-M Pellenq
ldquoRigidity Transition in Materials Hardness is Driven by Weak Atomic Constraintsrdquo
2015 doi 101103PhysRevLett114125502
[86] Yang X Xu B Yang B Cook H Ramos and M Bauchy ldquoPrediction of Silicate
Glassesrsquo Stiffness by High-Throughput Molecular Dynamics Simulations and Machine
Learningrdquo Accessed Mar 13 201 Online Available
httpsarxivorgpdf190109323pdf
[87] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological constraint model for
the elasticity of glass-forming systemsrdquo J Non Cryst Solids X vol 2 2019 doi
101016jnocx2019100019
[88] C J Wilkinson K Doss D R Cassar R S Welch C B Bragatto and J C Mauro
ldquoPredicting Ionic Diffusion in Glass from Its Relaxation Behaviorrdquo J Phys Chem B vol
124 no 6 pp 1099ndash1103 2020 doi 101021acsjpcb9b10645
179
[89] C Wilkinson and J C Mauro ldquoExplorerpy Mapping the energy landscapes of complex
materialsrdquo SoftwareX vol Submitted 2020
[90] J C Mauro R J Loucks and J Balakrishnan ldquoA simplified eigenvector-following
technique for locating transition points in an energy landscaperdquo J Phys Chem A vol
109 no 42 pp 9578ndash9583 2005 doi 101021jp053581t
[91] Roumlder and D J Wales ldquoAnalysis of the b to b-CR Transition in biquitinrdquo
Biochemistry vol 57 no 43 pp 6180ndash6186 2018 doi 101021acsbiochem8b00770
[92] J C Mauro R J Loucks and J Balakrishnan ldquoSplit-step eigenvector-following
technique for exploring enthalpy landscapes at absolute zerordquo J Phys Chem B vol 110
no 10 pp 5005ndash5011 2006 doi 101021jp056803w
[93] J C Mauro R J Loucks J Balakrishnan and A Varshneya ldquoMapping the potential
energy landscapes of selenium clustersrdquo J Non Cryst Solids vol 353 pp 1268ndash1273
2007 doi 101016jjnoncrysol200609062
[94] B W H van Beest G J Kramer and R A van Santen ldquoForce fields for silicas and
aluminophosphates based on ab initio calculationsrdquo Phys Rev Lett vol 64 no 16 pp
1955ndash1958 Apr 1990 doi 101103PhysRevLett641955
[95] Q Zheng and J C Mauro ldquoViscosity of glass-forming systemsrdquo J Am Ceram Soc vol
100 no 1 pp 6ndash25 2017 doi 101111jace14678
[96] J C Mauro and Y Z Mauro ldquoOn the Prony Series Representation of Stretched
Exponential Relaxationrdquo Physica A vol 506 pp 75ndash87 2018 doi
101016jphysa201804047
[97] E D Zanotto and D R Cassar ldquoThe microscopic origin of the extreme glass-forming
ability of Albite and B2O3rdquo Sci Rep vol 7 no February pp 1ndash13 2017 doi
101038srep43022
[98] P Pedevilla S J Cox B Slater and A Michaelides ldquoCan Ice-Like Structures Form on
180
Non-Ice-Like Substrates The Example of the K-feldspar Microclinerdquo J Phys Chem C
vol 120 no 12 pp 6704ndash6713 2016 doi 101021acsjpcc6b01155
[99] O Bjoumlrneholm et al ldquoWater at Interfacesrdquo Chem Rev vol 116 no 13 pp 7698ndash7726
2016 doi 101021acschemrev6b00045
[100] Reichelt ldquoNucleation and growth of thin filmsrdquo Vacuum vol 38 no 12 pp 1083ndash
1099 Jan 1988 doi 1010160042-207X(88)90004-8
[101] G H Beall ldquoDr S Donald (Don) Stookey (1 1 ndash2014) Pioneering researcher and
adventurerrdquo Front Mater vol 3 no August pp 1ndash8 2016 doi
103389fmats201600037
[102] J C Mauro C S Philip D J Vaughn and M S Pambianchi ldquoGlass science in the
nited States Current status and future directionsrdquo Int J Appl Glas Sci vol 5 no 1
pp 2ndash15 2014 doi 101111ijag12058
[103] J Deubener et al ldquo pdated definition of glass-ceramicsrdquo J Non Cryst Solids vol 501
no January pp 3ndash10 2018 doi 101016jjnoncrysol201801033
[104] X Hao ldquoA review on the dielectric materials for high energy-storage applicationrdquo J Adv
Dielectr vol 03 no 01 p 1330001 2013 doi 101142s2010135x13300016
[105] J Deubener G Helsch A Moiseev and H Bornhoumlft ldquoGlasses for solar energy
conversion systemsrdquo J Eur Ceram Soc vol 29 no 7 pp 1203ndash1210 2009 doi
101016jjeurceramsoc200808009
[106] G S Frankel et al ldquoA comparative review of the aqueous corrosion of glasses crystalline
ceramics and metalsrdquo npj Mater Degrad vol 2 no 1 p 15 2018 doi
101038s41529-018-0037-2
[107] G H Beall ldquoDesign and properties of glass-ceramicsrdquo Annu Rev Mater Sci vol 22
no 1 pp 91ndash119 1992 doi 101146annurevms22080192000515
[108] J Deubener ldquoConfigurational entropy and crystal nucleation of silicate glassesrdquo Phys
181
Chem Glas vol 45 pp 61ndash63 2004
[109] J C Mauro A J Ellison D C Allan and M M Smedskjaer ldquoTopological model for the
viscosity of multicomponent glass-forming liquidsrdquo Int J Appl Glas Sci vol 4 no 4
pp 408ndash413 2013 doi 101111ijag12009
[110] M M Smedskjaer J C Mauro S Sen and Y Yue ldquoQuantitative design of glassy
materials using temperature-dependent constraint theoryrdquo Chem Mater vol 22 no 18
pp 5358ndash5365 2010 doi 101021cm1016799
[111] Y T Sun H Y Bai M Z Li and W H Wang ldquoMachine Learning Approach for
Prediction and Understanding of Glass-Forming Abilityrdquo J Phys Chem Lett vol 8 no
14 pp 3434ndash3439 2017 doi 101021acsjpclett7b01046
[112] Q Zheng et al ldquo nderstanding Glass through Differential Scanning Calorimetryrdquo Chem
Rev vol 119 no 13 pp 7848ndash7939 2019 doi 101021acschemrev8b00510
[113] X Xia I Dutta J C Mauro B G Aitken and F elton ldquoTemperature dependence of
crystal nucleation in BaOmiddot2SiO2 and 5BaOmiddot8SiO2 glasses using differential thermal
analysisrdquo J Non Cryst Solids vol 459 pp 45ndash50 2017 doi
101016jjnoncrysol201612032
[114] V M Fokin A A Cabral R M C V Reis M L F Nascimento and E D Zanotto
ldquoCritical assessment of DTA-DSC methods for the study of nucleation kinetics in
glassesrdquo J Non Cryst Solids vol 356 no 6ndash8 pp 358ndash367 2010 doi
101016jjnoncrysol200911038
[115] D C Van Hoesen X Xia M E Mc enzie and F elton ldquoModeling nonisothermal
crystallization in a BaO∙2SiO2 glassrdquo J Am Ceram Soc no December 2019 pp 2471ndash
2482 2019 doi 101111jace16979
[116] S C C Prado J P Rino and E D Zanotto ldquoSuccessful test of the classical nucleation
theory by molecular dynamic simulations of BaSrdquo Comput Mater Sci vol 161 no
182
January pp 99ndash106 2019 doi 101016jcommatsci201901023
[117] A O Tipeev and E D Zanotto ldquoNucleation kinetics in supercooled Ni 0Ti 0 Computer
simulation data corroborate the validity of the Classical Nucleation Theoryrdquo Chem Phys
Lett vol 735 no August p 136749 2019 doi 101016jcplett2019136749
[118] K F Kelton and A L Greer Nucleation in Condensed Matter Applications in Materials
and Biology 1st ed Pergamon 2010
[119] V M Fokin E D Zanotto N S Yuritsyn and J W P Schmelzer ldquoHomogeneous
crystal nucleation in silicate glasses A 40 years perspectiverdquo J Non Cryst Solids vol
352 no 26ndash27 pp 2681ndash2714 2006 doi 101016jjnoncrysol200602074
[120] A M Rodrigues D R Cassar V M Fokin and E D Zanotto ldquoCrystal growth and
viscous flow in barium disilicate glassrdquo J Non Cryst Solids vol 479 pp 55ndash61 2018
doi 101016jjnoncrysol201710007
[121] A Pedone G Malavasi M C Menziani A N Cormack and U Segre ldquoA new selft-
consistent empirical interatomic potential model for oxides silicates and silica-based
glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
[122] J J Maldonis A D Banadaki S Patala and P M Voyles ldquoShort-range order structure
motifs learned from an atomistic model of a Zr 0Cu4 Al metallic glassrdquo Acta Mater
vol 175 pp 35ndash45 2019 doi 101016jactamat201905002
[123] A irchner S Goyal M E Mc enzie J T Harris and J C Mauro ldquoStatistical
Description of the Thermodynamics of Glass-Forming Liquidsrdquo Physica A vol 559 p
125059 2020 doi 101016jphysa2020125059
[124] D R Cassar ldquoCrystallization Driving Force of Supercooled Oxide Liquidsrdquo Int J Appl
Glas Sci vol 7 no 3 pp 262ndash269 2016 doi 101111ijag12218
[125] V M Fokin A S Abyzov E D Zanotto D R Cassar A M Rodrigues and J W P
Schmelzer ldquoCrystal nucleation in glass-forming liquids Variation of the size of the
183
lsquostructural unitsrsquo with temperaturerdquo J Non Cryst Solids vol 447 pp 35ndash44 2016 doi
101016jjnoncrysol201605017
[126] M Lenoir A Grandjean Y Linard B Cochain and D R Neuville ldquoThe influence of
SiB substitution and of the nature of network-modifying cations on the properties and
structure of borosilicate glasses and meltsrdquo Chem Geol vol 256 no 3ndash4 pp 316ndash325
2008 doi 101016jchemgeo200807002
[127] P Gupta D R Cassar and E D Zanotto ldquoRole of dynamic heterogeneities in crystal
nucleation kinetics in an oxide supercooled liquidrdquo J Chem Phys vol 145 no 21 2016
doi 10106314964674
[128] A M Rodrigues ldquoDiffusion Processes Crystallization and Viscous Flow in Barium
Disilicate Glassrdquo Dissertation 2014
[129] E D Zanotto ldquoThe effects of amorphous phase sepatration on crystal nucleation in baria-
silica and lithia-silica glassesrdquo Dissertation 1982
[130] D Vargheese A Tandia and J C Mauro ldquoOrigin of dynamical heterogeneities in
calcium aluminosilicate liquidsrdquo J Chem Phys vol 132 no 19 p 194501 May 2010
doi 10106313429880
[131] K Deenamma Vargheese A Tandia and J C Mauro ldquoStatistics of modifier distributions
in mixed network glassesrdquo J Chem Phys vol 132 p 24507 2010 doi
10106313429880
[132] M Micoulaut ldquoThe Deep Effect of Topology on Glass Relaxationrdquo Physics (College
Park Md) vol 6 p 72 2013 doi 101103Physics672
[133] M Micoulaut ldquoRigidity and intermediate phases in glasses driven by speciationrdquo Phys
Rev B vol 74 p 184208 2006 doi 101103PhysRevB74184208
[134] M Kodama S Kojima S Feller and M Affatigato ldquoThe occurrence of minima in the
Borate anomaly anharmonicity and fragility in lithium borate glassesrdquo Phys Chem
184
Glas vol 46 no 2 pp 190ndash193 2005
[135] Y Fukawa et al ldquoVelocity of Sound and Elastic Properties of Li2O-B2O3 Glassesrdquo Jpn
J Appl Phys vol 34 p 2570 1995 doi httpsdoiorg101143JJAP342570
[136] M Reiner ldquoThe Deborah Numberrdquo Physics Today vol 17 no 1 p 62 1964 doi
10106313051374
[137] P K Gupta and J C Mauro ldquoThe laboratory glass transitionrdquo J Chem Phys vol 126
no 22 p 2240504 2007 doi 10106312738471
[138] R Richert and C A Angell ldquoDynamics of glass-forming liquids V On the link between
molecular dynamics and configurational entropyrdquo J Chem Phys vol 108 no 21 pp
9016ndash9026 1998 doi 1010631476348
[139] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological Constraint Model for
the Elasticity of Glass-Forming Systemsrdquo J Non Cryst Solids X vol 2 p 100019 2019
doi 101016jnocx2019100019
[140] S Feller S Bista A OrsquoDonovan-Zavada T Mullenbach M Franke and M Affatigato
ldquoPacking in alkali and alkaline earth borosilicate glass systemsrdquo Phys Chem Glas Eur
J Glas Sci Technol Part B vol 50 no 3 pp 224ndash228 2009
[141] P J Bray S Feller G E Jellison and Y H Yun ldquoB10 NMR studies of the structure of
borate glassesrdquo J Non Cryst Solids vol 38ndash39 no PART 1 pp 93ndash98 May 1980 doi
1010160022-3093(80)90400-7
[142] N Andersson and G L Comer ldquoRelativistic Fluid Dynamics Physics for Many Different
Scales Living Reviews in Relativityrdquo Living Rev Relativ vol 10 p 1 2007 Accessed
Feb 18 2019 [Online] Available httpwwwlivingreviewsorglrr-2007-
1httprelativitylivingreviewsorghttpwwwmathssotonacukstaffAnderssonhttpw
wwslueducollegesASphysicsprofscomerhtml
[143] P Ilg and H C Oumlttinger ldquoNonequilibrium relativistic thermodynamics in bulk viscous
185
cosmologyrdquo Phys Rev D - Part Fields Gravit Cosmol vol 61 no 2 p 023510 2000
doi 101103PhysRevD61023510
[144] P Romatschke ldquoRelativistic viscous fluid dynamics and non-equilibrium entropyrdquo Class
Quantum Gravity vol 27 no 2 p 025006 Jan 2010 doi 1010880264-
9381272025006
[145] D C Allan ldquoInverting the MYEGA equation for viscosityrdquo J Non Cryst Solids vol
358 no 2 pp 440ndash442 Jan 2012 doi 101016JJNONCRYSOL201109036
[146] M Potuzak R C Welch and J C Mauro ldquoTopological origin of stretched exponential
relaxation in glassrdquo J Chem Phys vol 135 no 21 p 214502 Dec 2011 doi
10106313664744
[147] L Ding M Thieme S Demouchy C unisch and B J P aus ldquoEffect of pressure and
temperature on viscosity of a borosilicate glassrdquo J Am Ceram Soc vol 101 no 9 pp
3936ndash3946 2018 doi 101111jace15588
[148] R Richert and M Richert ldquoDynamic heterogeneity spatially distributed stretched-
exponential patterns and transient dispersions in solvation dynamicsrdquo Phys Rev E vol
58 no 1 pp 779ndash784 1998 doi 101103PhysRevE58779
[149] P Richet ldquoViscosity and configurational entropy of silicate meltsrdquo Geochim Cosmochim
Acta vol 48 no 3 pp 471ndash483 Mar 1984 doi 1010160016-7037(84)90275-8
[150] P Gupta and J C Mauro ldquoTwo factors governing fragility Stretching exponent and
configurational entropyrdquo Phys Rev E vol 78 no 6 p 063501 2008 doi
101103PhysRevE78062501
[151] Z Zheng J C Mauro and D C Allan ldquoModeling of delayed elasticity in glassrdquo J Non
Cryst Solids vol 500 no August pp 432ndash442 2018 doi
101016jjnoncrysol201808033
[152] R Boumlhmer L Ngai C A Angell and D J Plazek ldquoNonexponential relaxations in
186
strong and fragile glass formersrdquo J Chem Phys vol 99 no 5 pp 4201ndash4209 1993
doi 1010631466117
[153] O Gulbiten J C Mauro and P Lucas ldquoRelaxation of enthalpy fluctuations during sub-
Tg annealing of glassy seleniumrdquo J Chem Phys vol 138 no 24 p 244504 2013 doi
10106314811488
[154] D Sidebottom R Bergman L Boumlrjesson and L M Torell ldquoTwo-step relaxation decay
in a strong glass formerrdquo Phys Rev Lett vol 71 no 14 pp 2260ndash2263 1993 doi
101103PhysRevLett712260
[155] G M Bartenev and V A Lomovskoi ldquoRelaxation time spectra and the peculiarities of
the process of boron anhydride glass transitionrdquo J Non Cryst Solids vol 146 no C pp
225ndash232 1992 doi 101016S0022-3093(05)80495-8
[156] A Heuer ldquoExploring the potential energy landscape of glass-forming systems From
inherent structures via metabasins to macroscopic transportrdquo J Phys Condens Matter
vol 20 no 37 p 373101 2008 doi 1010880953-89842037373101
[157] J C Phillips ldquoStretched exponential relaxation in molecular and electronic gasesrdquo
Reports Prog Phys vol 59 pp 1133ndash1207 1996
[158] P Richet ldquoResidual and configurational entropy Quantitative checks through applications
of Adam-Gibbs theory to the viscosity of silicate meltsrdquo J Non Cryst Solids vol 355
no 10ndash12 pp 628ndash635 2009 doi 101016jjnoncrysol200901027
[159] C J Wilkinson Doss G Palmer and J C Mauro ldquoThe relativistic glass transition A
thought experimentrdquo J Non-Crystalline Solids X vol 2 2019 doi
101016jnocx2019100018
[160] J C Mauro R J Loucks and S Sen ldquoHeat capacity enthalpy fluctuations and
configurational entropy in broken ergodic systemsrdquo J Chem Phys vol 133 no 16 pp
1ndash9 2010 doi 10106313499326
187
[161] J C Mauro R J Loucks and P Gupta ldquoMetabasin approach for computing the
master equation dynamics of systems with broken ergodicityrdquo J Phys Chem A vol 111
no 32 pp 7957ndash7965 2007 doi 101021jp0731194
[162] M Wang N M Anoop Krishnan B Wang M M Smedskjaer J C Mauro and M
Bauchy ldquoA new transferable interatomic potential for molecular dynamics simulations of
borosilicate glassesrdquo J Non Cryst Solids vol 498 no December 2017 pp 294ndash304
2018 doi 101016jjnoncrysol201804063
[163] I Avramov and A Milchev ldquoEffect of disorder on diffusion and viscosity in condensed
systemsrdquo J Non Cryst Solids vol 104 no 2ndash3 pp 253ndash260 Sep 1988 doi
1010160022-3093(88)90396-1
[164] G Scherer Relaxation in Glass and Composites Krieger Publishing Company 1992
[165] L Huang J Nicholas J Kieffer and J Bass ldquoPolyamorphic transitions in vitreous B2O3
under pressurerdquo J Phys Condens Matter vol 20 no 7 2008 doi 1010880953-
8984207075107
[166] W Capps P B Macedo B OrsquoMeara and T A Litovitz ldquoTemperature Dependecne of
the High-Frequency Moduli of Vitreous B2O3rdquo J Chem Phys vol 45 p 3431 1966
doi 107868s0002337x1402002x
[167] D R Cassar A M Rodrigues M L F Nascimento and E D Zanotto ldquoThe diffusion
coefficient controlling crystal growth in a silicate glass-formerrdquo Int J Appl Glas Sci
vol 9 no 3 pp 373ndash382 Jul 2018 doi 101111ijag12319
[168] A R Potter C J Wilkinson S H im and J C Mauro ldquoEffect of Water on
Topological Constraints in Silica Glassrdquo Scr Mater vol 160 pp 48ndash52 2019 doi
101016jscriptamat201809041
[169] D B Asay and S H im ldquoEvolution of the adsorbed water layer structure on silicon
oxide at room temperaturerdquo J Phys Chem B vol 109 pp 16760ndash16763 2005 doi
188
101021jp053042o
[170] M Tomozawa ldquoWater diffusion oxygen vacancy annihilation and structural relaxation in
silica glassesrdquo J Non Cryst Solids vol 179 pp 162ndash169 1994 doi 1010160022-
3093(94)90693-9
[171] S Kapoor R E Youngman K Zakharchuk A Yaremchenko N J Smith and A Goel
ldquoStructural and Chemical Approach toward nderstanding the Aqueous Corrosion of
Sodium Aluminoborate Glassesrdquo J Phys Chem B vol 122 pp 10913ndash10927 2018 doi
101021acsjpcb8b06155
[172] Y Yu N M A rishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[173] Y Yu B Wang M Wang G Sant and M Bauchy ldquoReactive Molecular Dynamics
Simulations of Sodium Silicate Glasses mdash Toward an Improved Understanding of the
Structurerdquo Int J Appl Glas Sci vol 8 no 3 pp 276ndash284 2017 doi
101111ijag12248
[174] E Stolper ldquoThe speciation of water in silicate meltsrdquo Geochim Cosmochim Acta vol
46 no 12 pp 2609ndash2620 Dec 1982 doi 1010160016-7037(82)90381-7
[175] M G Mesko P A Schader and J E Shelby ldquoWater solubility and diffusion in sodium
silicate meltsrdquo Phys Chem Glas vol 43 no 6 pp 283ndash290 2002 doi
101016jsemradonc201010001
[176] M Tomozawa M Takata J Acocella E Bruce Watson and T Takamori ldquoThermal
properties of Na2Omiddot3SiO2 glasses with high water contentrdquo J Non Cryst Solids vol 56
no 1ndash3 pp 343ndash348 Jul 1983 doi 1010160022-3093(83)90491-X
[177] E A Leed and C G Pantano ldquoComputer modeling of water adsorption on silica and
silicate glass fracture surfacesrdquo J Non Cryst Solids vol 325 pp 48ndash60 2003 doi
189
101016S0022-3093(03)00361-2
[178] J C Fogarty H M Aktulga A Y Grama A C T van Duin and S A Pandit ldquoA
reactive molecular dynamics simulation of the silica-water interfacerdquo J Chem Phys vol
132 no 17 p 174704 2010 doi 10106313407433
[179] S H Hahn et al ldquoDevelopment of a ReaxFF Reactive Force Field for NaSiOx Water
Systems and Its Application to Sodium and Proton Self-Diffusionrdquo J Phys Chem C vol
122 no 34 pp 19613ndash19624 2018 doi 101021acsjpcc8b05852
[180] T S Mahadevan W Sun and J Du ldquoDevelopment of Water Reactive Potentials for
Sodium Silicate Glassesrdquo J Phys Chem B vol 123 pp 4452ndash4461 2019 doi
101021acsjpcb9b02216
[181] A C T van Duin A Strachan S Stewman Q Zhang X Xu and W A Goddard III
ldquoReaxFF SiO Reactive Force Field for Silicon and Silicon Oxide Systemsrdquo J Phys
Chem A vol 107 no 19 pp 3803ndash3811 2003 doi 101021jp0276303
[182] M M Smedskjaer J C Mauro R E Youngman C L Hogue M Potuzak and Y Yue
ldquoTopological principles of borosilicate glass chemistryrdquo J Phys Chem B vol 115 no
44 pp 12930ndash12946 2011 doi 101021jp208796b
[183] Q Zheng M Potuzak J C Mauro M M Smedskjaer R E Youngman and Y Yue
ldquoComposition-structure-property relationships in boroaluminosilicate glassesrdquo J Non
Cryst Solids vol 358 pp 993ndash1002 2012 doi 101016jjnoncrysol201201030
[184] K Rompicharla D I Novita P Chen P Boolchand M Micoulaut and W Huff
ldquoAbrupt boundaries of intermediate phases and space filling in oxide glassesrdquo J Phys
Condens Matter vol 20 p 202101 2008 doi 1010880953-89842020202101
[185] M Micoulaut and J C Phillips ldquoRings and rigidity transitions in network glassesrdquo Phys
Rev B vol 67 p 104204 2003 doi 101103PhysRevB67104204
[186] H Liu S Dong L Tang N M A rishnan G Sant and M Bauchy ldquoEffects of
190
polydispersity and disorder on the mechanical properties of hydrated silicate gelsrdquo J
Mech Phys Solids vol 122 pp 555ndash565 Jan 2019 doi 101016JJMPS201810003
[187] A Tilocca N H De Leeuw and A N Cormack ldquoShell-model molecular dynamics
calculations of modified silicate glassesrdquo Phys Rev B - Condens Matter Mater Phys
vol 73 p 104209 2006 doi 101103PhysRevB73104209
[188] J Du and A N Cormack ldquoThe medium range structure of sodium silicate glasses A
molecular dynamics simulationrdquo J Non Cryst Solids vol 349 no 1ndash3 pp 66ndash79 2004
doi 101016jjnoncrysol200408264
[189] N P Bansal and R H Doremus Handbook of glass properties Academic Press 1986
[190] S H Hahn and A C T van Duin ldquoSurface Reactivity and Leaching of a Sodium Silicate
Glass under an Aqueous Environment A ReaxFF Molecular Dynamics Studyrdquo J Phys
Chem C p acsjpcc9b02940 2019 doi 101021acsjpcc9b02940
[191] S P Jaccani and L Huang ldquo nderstanding Sodium Borate Glasses and Melts from Their
Elastic Response to Temperaturerdquo Int J Appl Glas Sci vol 7 no 4 pp 452ndash463 Dec
2016 doi 101111ijag12250
[192] Q Zheng Y Yue and J C Mauro ldquoDensity of topological constraints as a metric for
predicting glass hardnessrdquo Appl Phys Lett vol 111 no 1 p 011907 Jul 2017 doi
10106314991971
[193] J C Dyre ldquoElastic models for the non-Arrhenius relaxation time of glass-forming
liquidsrdquo in AIP Conference Proceedings 2006 vol 832 pp 113ndash117 doi
10106312204470
[194] Y T Cheng and C M Cheng ldquoRelationships between hardness elastic modulus and the
work of indentationrdquo Appl Phys Lett vol 73 no 5 pp 614ndash616 Jul 1998 doi
1010631121873
[195] A Makishima and J D Mackenzie ldquoDirect calculation of Youngrsquos modulus of glassrdquo J
191
Non Cryst Solids vol 12 no 1 pp 35ndash45 1973 doi 1010160022-3093(73)90053-7
[196] G Pilania C Wang X Jiang S Rajasekaran and R Ramprasad ldquoAccelerating materials
property predictions using machine learningrdquo Sci Rep vol 3 pp 1ndash6 2013 doi
101038srep02810
[197] H He and M Thorpe ldquoElastic properties of glassesrdquo Phys Rev Lett vol 54 no 19 pp
2107ndash2110 1985 Accessed Aug 29 2018 [Online] Available
httpsjournalsapsorgprlpdf101103PhysRevLett542107
[198] M Thorpe ldquoElastic properties of glassesrdquo Cambridge Arch 1985 doi 101557PROC-
61-49
[199] J C Phillips ldquoChemical Bonding Internal Surfaces and the Topology of Non-Crystalline
Solidsrdquo Phys status solidi vol 101 no 2 pp 473ndash479 Oct 1980 doi
101002pssb2221010204
[200] M M Smedskjaer J C Mauro and Y Yue ldquoPrediction of glass hardness using
temperature-dependent constraint theoryrdquo Phys Rev Lett vol 105 no 11 2010 doi
101103PhysRevLett105115503
[201] J D Musgraves J Hu and L Calvez Springer Handbook of Glass Springer US 2021
[202] C Hermansen X Guo R E Youngman J C Mauro M M Smedskjaer and Y Yue
ldquoStructure-topology-property correlations of sodium phosphosilicate glassesrdquo J Chem
Phys vol 143 no 6 p 064510 Aug 2015 doi 10106314928330
[203] L Wondraczek et al ldquoTowards ltrastrong Glassesrdquo Adv Mater vol 23 no 39 pp
4578ndash4586 Oct 2011 doi 101002adma201102795
[204] D Ravaine ldquoGlasses as solid electrolytesrdquo J Non Cryst Solids vol 38ndash39 no PART 1
pp 353ndash358 1980 doi 1010160022-3093(80)90444-5
[205] D Ravaine and J L Souquet ldquoA thermodynamic approach to ionic conductivity in oxide
glasses Part 1 Correlation of the ionic conductivity with the chemical potential of alkali
192
oxide in oxide glassesrdquo Phys Chem Glas vol 18 no 2 pp 27ndash31 1977
[206] J Swenson and L Boumlrjesson ldquoCorrelation between Free Volume and Ionic Conductivity
in Fast Ion Conducting Glassesrdquo Phys Rev Lett vol 77 no 17 pp 3569ndash3572 1996
doi 101103PhysRevLett773569
[207] C B Bragatto A C M Rodrigues and J L Souquet ldquoDissociation Equilibrium and
Charge Carrier Formation in AgI-AgPO3 Glassesrdquo J Phys Chem C vol 121 no 25 pp
13507ndash13514 2017 doi 101021acsjpcc7b02477
[208] S W Martin R Christensen G Olson J ieffer and W Wang ldquoNew Interpretation of
Na Ion Conduction in and the Structures and Properties of Sodium Borosilicate Mixed
Glass Former Glassesrdquo J Phys Chem C vol 140 pp 6343ndash6352 2019 doi
101021acsjpcc8b11735
[209] O L Anderson and D A Stuart ldquoCalculation of Activation Energy of Ionic Conductivity
in Silica Glasses by Classical Methodsrdquo J Am Ceram Soc vol 37 no 12 pp 573ndash580
1954 doi 101111j1151-29161954tb13991x
[210] R S Welch C J Wilkinson J C Mauro and C B Bragatto ldquoCharge Carrier Mobility
of Alkali Silicate Glasses Calculated by Molecular Dynamicsrdquo Front Mater vol 6 p
121 May 2019 doi 103389fmats201900121
[211] Y J Zhang ldquoEntropy and ionic conductivityrdquo Phys A Stat Mech its Appl vol 391 no
19 pp 4470ndash4475 2012 doi 101016jphysa201204021
[212] J C Dyre ldquoOn the mechanism of glass ionic conductivityrdquo J Non Cryst Solids vol 88
no 2ndash3 pp 271ndash280 1986 doi 101016S0022-3093(86)80030-8
[213] M Steve ldquoIonic Conduction in Phosphate Glassesrdquo JAmCeramSoc vol 74 no 8 pp
1767ndash1784 1991 doi 101111j1151-29161991tb07788x
[214] C T Moynihan A J Easteal J Wilder and J Tucker ldquoDependence of the Glass
Transition Temperature on Heating and Cooling Raterdquo J Phys Chem vol 78 no 26 pp
193
2673ndash2677 1974 Accessed Jul 13 2018 [Online] Available
httpspubsacsorgsharingguidelines
[215] Ito C T Moynihan and C A Angell ldquoThermodynamic determination of fragility in
liquids anda fragile-to-strong liquid transition inwaterrdquo Lett to Nat vol 398 no April
p 492 1999
[216] L Ngai and S W Martin ldquoCorrelation between the activation enthalpy and
Kohlrausch exponent for ionic conductivity in oxide glassesrdquo Phys Rev B vol 40 no
15 pp 10550ndash10556 1989 doi 101103PhysRevB4010550
[217] M Potuzak X Guo M M Smedskjaer and J C Mauro ldquoAre the dynamics of a glass
embedded in its elastic propertiesrdquo J Chem Phys vol 138 no 12 2013 doi
10106314730525
[218] A L Agapov and A P Sokolov ldquoDecoupling ionic conductivity from structural
relaxation A way to solid polymer electrolytesrdquo Macromolecules vol 44 no 11 pp
4410ndash4414 2011 doi 101021ma2001096
[219] S V Nemilov ldquoA Structural Investigation of Glasses in the B2O3ndashNa2O System by the
Viscosimetric Methodrdquo Izv Akad Nauk SSSR Neorg Mater vol 2 no 2 pp 349ndash359
1966
[220] J P Poole ldquoLow‐Temperature Viscosity of Alkali Silicate Glassesrdquo J Am Ceram Soc
vol 32 no 7 pp 230ndash233 1949 doi 101111j1151-29161949tb18952x
[221] R noche D B Dingwell F A Seifert and S L Webb ldquoNon-linear properties of
supercooled liquids in the system Na2O SiO2rdquo Chem Geol vol 116 pp 1ndash16 1994 doi
1010160009-2541(94)90154-6
[222] W Martinsen ldquoSelected properties of sodium silicate glasses and their structural
significancerdquo Dissertation 1969
[223] C Hermansen J C Mauro and Y Yue ldquoA model for phosphate glass topology
194
considering the modifying ion sub-networkrdquo J Chem Phys vol 140 no 15 2014 doi
10106314870764
[224] M L F Nascimento and E Dutra Zanotto ldquoDoes viscosity describe the kinetic barrier for
crystal growth from the liquidus to the glass transitionrdquo J Chem Phys vol 133 no 17
p 174701 2010 doi 10106313490793
[225] D Denisko and M M Hoffman ldquoClassification and interaction in random forestsrdquo Proc
Natl Acad Sci U S A vol 115 no 8 pp 1690ndash1692 2018 doi
101073pnas1800256115
[226] G Varney C Dema B E Gul C J Wilkinson and Akgun ldquo se of machine learning
in CARNA proton imagerrdquo in Progress in Biomedical Optics and Imaging - Proceedings
of SPIE 2019 vol 10948 doi 101117122512565
[227] ldquoMDLregSciGlass-6 MDL Information Systems San Leandro CArdquo 2003
[228] D R Cassar ldquoViscNet Neural Network for predicting the fragility index and the
temperature-dependency of viscosityrdquo pp 1ndash33 2020
[229] D J L Brett A Atkinson N P Brandon and S J Skinner ldquoIntermediate temperature
solid oxide fuel cellsrdquo Chem Soc Rev vol 37 no 8 pp 1568ndash1578 2008 doi
101039b612060c
[230] T Ishiyama S Suzuki J Nishii T Yamashita H awazoe and T Omata ldquoProton
conducting tungsten phosphate glass and its application in intermediate temperature fuel
cellsrdquo Solid State Ionics vol 262 pp 856ndash859 2014 doi 101016jssi201310055
[231] T Yamaguchi et al ldquoProton-conducting phosphate glass and its melt exhibiting high
electrical conductivity at intermediate temperaturesrdquo J Mater Chem A vol 6 no 46 pp
23628ndash23637 2018 doi 101039C8TA08162J
VITA
Collin Wilkinson was born in Mt Carroll IL He attended Coe College as an undergraduate and
studied glass resulting in a physics bachelorrsquos degree in 201 He worked with Dr Ugur Akgun
and Dr Steve Feller on modeling different key properties of glass propertiesresponses to external
stimuli Collin joined the group of Dr John Mauro at Pennsylvania State University in 2018
List of publications written by first author (or co-first author) Collin while at Penn State
1 CJ Wilkinson JC Mauro Explorerpy Mapping the Energy Landscapes of Complex
Materials SoftwareX Submitted
2 CJ Wilkinson DR Cassar AV DeCeanne KA Kirchner ME McKenzie ED Zanotto and
JC Mauro Energy Landscape Modeling of Crystal Nucleation NPJ Computational
Materialis Submitted
3 CJ Wilkinson K Doss O Gulbiten DC Allan JC Mauro Fragility and Temperature
Dependence of Stretched Exponential Relaxation in Glass-Forming Systems JACERS
Submitted
4 CJ Wilkinson JC Mauro Comment on ldquoThe Fragility of Alkali Silicate Glass Melts Part
of a niversal Topological Patterrdquo by DL Sidebottom JNCS (2020) ( 2 ) 11
5 CJ Wilkinson K Doss DR Cassar RS Welch CB Bragatto JC Mauro Predicting Ionic
Diffusion in Glass From its Relaxation Behavior Journal of Physical Chemistry B (2020)
124 (6) 1099-1103
6 Z Ding CJ Wilkinson J Zheng Y Lin H Liu J Shen SH Kim Y Yue J Rent CJ Mauro
Q Zheng Topological Understanding of the Mixed Alkaline Earth Effect in Glass JNCS
(2020) (526) 119696
7 CJ Wilkinson AR Potter RS Welch CB Bragatto Q Zheng M Bauchy M Affatigato SA
Feller JC Mauro Topological Origins of Mixed Alkali Effect in Glass Journal of Physical
Chemistry B (2019) 123 (34) 7482
8 CJ Wilkinson K Doss SH Hahn Nathan Keilbart AR Potter NJ Smith I Dabo ACT Van
Duin SH Kim JC Mauro Topological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phase JPCL (2019) 10 (14) 3955
9 CJ Wilkinson K Doss G Palmer JC Mauro The Relativistic Glass Transition A Through
Experiment JNCSX (2019) 100018
10 CJ Wilkinson Q Zheng L Huang JC Mauro Topological Constraint Model for the
Elasticity of Glass-Forming Systems JNCSX (2019) 100019
11 AR Potter CJ Wilkinson SH Kim JC Mauro Effect of Water on Toplogical Constraints
in Silica Glass Scripta Materialia (2019) (160) 48-52
12 CJ Wilkinson E Pakhomenko MR Jesuit AV DeCeanne B Hauke M Packard SA Feller
JC Mauro Topological Constraint Model of Alkali Tellurite Glasses JNCS (2018) (502)
172
13 CJ Wilkinson YZ Mauro JC Mauro RelaxPy Python code for Modeling of Glass
Relaxation Behavior SoftwareX (2018) (7) 255
vii
The infinite temperature limit is from the work of Zheng et al54 and the glass
transition temperature is from the Angell definition 54
Figure 10 The relativistic glass transition temperature for B2O3 glass 57
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different
fractions of light speed All of the viscosities approaching the universal temperature
limit for viscosity 59
Figure 12 The modulus needed to satisfy the condition for the glass transition 61
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at
v=044c (Bottom) Viscosity plot showing a dramatic reduction of the viscosity as
the observer approaches the speed of light 63
Figure 14 b predicted and from literature showing good agreement with a total root-
mean-square error of 01 The fit for organic systems is given by 66 ln Kln = minus
and for inorganic systems by 75ln Kln = minus 72
Figure 15 The equilibrium model proposed with the experimental points showing good
agreement between the experimentally measured data points and the equilibrium
derived model RMSE was 002 for Corningcopy JadeTM (A) and less than 001 for
SG80 (B) (C) The model fit for B2O3 experimental data154155 The fragility and glass
transition temperature of the B2O3 are taken from the work of Mauro et al17 75
Figure 16 The stretching exponent calculated as described in the text for a Gaussian
distribution of barriers This plot shows that the distribution of barriers has a large
effect on the stretching exponent A Tg cannot be described since there is no
vibrational frequency included in the model though the glass transition temperature
should be the same for all distributions since the mean relaxation time is the same
for all distributions at all temperatures The deviation is given in ln eV units 78
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent
Each color designates one term in the series (Bottom) The output from RelaxPy
v20 showing the stretching exponent effects on the relaxation prediction of
Corningcopy JadeTM glass Each quadrant shows one property that is of interest for
relaxation experiments In particular it is interesting to see the dynamics of the
stretching exponent during a typical quench 80
Figure 18 The viscosity (left) and landscape (right) predictions for three common
systems The first system is newly calculated in this work while the latter two come
from our previous works7289 It is seen that the viscosity predicted from the AG
model is very accurately able to reproduce the experimental viscosity curves from
the MYEGA model The last system is a potential energy landscape while the others
are enthalpy landscapes 87
Figure 19 The configurational entropy comparisons between the three major viscosity
models which validates the main underlying assumption of the MYEGA model The
viii
VFT and AM are unable to capture the physics of configurational entropy therefore
ruling 89
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the
configurational entropy from the enthalpy landscape with the barrier 00155 eV
(compared to the shear barrier of 00149 eV) and the infinite limit allowed to vary
(10-263 Pamiddots for bulk viscosity) In Figure 18 the configurational entropy is
confirmed for the shear viscosity thus confirming the AG for both shear and bulk
viscosities Sidebottom data are from Ref 154 90
Figure 21 (A) The histogram for the energy minima for each of the test systems showing
a good fit with the log normal distribution This distribution will then be a valid
form to calculate the enthalpy distribution of the model presented in the next
session (B) The configurational entropy from the model showing the accuracy of
the scaling of the entropy predicted by the MYEGA model The S value was fit
for each system 93
Figure 22 (A) Comparison between the randomized method (histogram) and the
deterministic method (vertical lines) showing good agreement between the
maximum in the histogram and the value predicted by the deterministic technique
validating the approach It is worth noting that the 100-basin distribution is a very
wide distribution where the total number of basins is less than the number of points
used in the calculation This is done for a variable number of basins with the number
of basins shown in the legend (B) The dependence of fragility and the glass
transition temperature vs the distribution of states and the number of basins 99
Figure 23 The driving forces for different example glasses calculated using a
combination of the MAP model RelaxPy and the toy landscape model The
parameters for each glass can be found in Table 4 102
Figure 24 The predicted outputs from the toy landscape method showing predictions of
the enthalpy and entropy under a standard quench for barium disilicate This
prediction does not require fictive temperature or any such assumptions about the
evolution of the non-equilibrium behavior 105
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume
in nucleation is assumed to be on the order of one cubic angstrom while the a
parameter in growth is assumed to be around one nm both are in good agreement
for estimates in literature The values for the orange points are taken from these
works128167 107
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the
hydration models Color scheme Si atom (ivory) O atom (red) and Na atom (blue)
The z-axis is elongated to allow space for an insert of water 114
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the
top surface in contact with water is shown here 116
ix
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example
of the electronic-structure DFT calculation using semilocal exchange-correlation
functionals finding the binding energy of a water lsquopixelrsquo to the surface 119
Figure 29 Example contour surface showing the average coordination per atom on the
glass surface for the first run at 300 ps 121
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of
constraints for surface atoms at the local pixel Results show a distinct maximum in
which there is a near hydrophilic-hydrophobic transition of the surface The error
bars represent the standard deviation A second system with 1500 atoms was
performed to show convergence of the ReaxFF MD results 123
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values
of the model predictions are 641 GPa for constraint density 313 GPa for free
energy density and 774 GPa for angular constraint density 128
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from
minimum to maximum error The free energy density model gives the most accurate
results The constraint density has a RMSE of 61 GPa the angular density has a
RMSE of 20 GPa and the energy density has a RMSE of 59 GPa 129
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and
experiment for 10 Na2O 90 B2O3 Using the previously fitted onset temperatures
the only free parameters are then the vibrational frequency and the heating time in
which their product was fitted to be 14000 Where each dip in the modulus
corresponds to a constraint no longer being rigid as heated through each onset The
onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The
data was fit using a least-squares method and the resultant fit is shown as the
calculated method The fit has an R2 of 094 (B) The contribution from each
constraint to the overall modulus 131
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in
Fig 3) and experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of
093 and (B) xLi2Omiddot(100-x)B2O3 glasses with an R2 of 0986 133
Figure 35 The structure for the initial minimum energy configuration showing the boron
(blue) network with interconnecting oxygens (red) and the interstitial sodium ions
(yellow) 139
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and
sodium is ivory The barrier is overestimated compared to experimental data this
could be from several sources of error such as potential fitting thermal history
fluctuations or sampling too few transitions The line is drawn as a guide to the eye 139
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function
of reaction coordinates The middle shows the local deformation around the ion of
x
any atom that moves in between inherent structure mandating a relaxation force
The color shows the degree of deformation 141
Figure 38 Different network formers and the prediction of the activation barrier from our
model compared with activation barriers from literature (A) Sodium silicate
predictions and experimental values241 the error is calculated from the error in the
fragility when fitting the data (B) Lithium phosphate activation energy235 predicted
with topological constraint theory and compared with the experimental values (C)
Predictions over two different systems of alkali borates232 sodium and lithium with
a reported R2 of 097 143
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top
left) shows the distribution of the infinite temperature limit in the database after
limits exerted on the system (Top right) The distribution of the infinite viscosity
limit vs the glass transition (Bottom left) The relationship between fragility and
infinite temperature limit of viscosity (Bottom right) The infinite temperature limit
of viscosity vs the key metric predicted by SR 149
Figure 40 The prediction of the scaling of the activation barriers for a common sodium
borosilicate system 157
Figure 41 The relationship between the glass transition and the proton conductivity This
is justified two ways one through the relationship of the entropy of diffusion and
glass formation (the Adam-Gibbs model) and through the fact that water is known to
depress the glass transition 162
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine
the glass forming region Over top the constraints at the glass transition provided by
each oxide species is listed Since the objective is to decrease Tg while staying in the
glass forming region we will attempt to minimize use of elements that increase the
glass transition (nc gt 17) 166
xi
List of Tables
Table 1 The key properties considered for commercial application The optical properties
have been omitted since it is physical unrealistic to expect quantitative predictions of
quantum-controlled phenomena from a classical description of glass structure 18
Table 2 Variable definitions 65
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy
JadeTM glass and Sylvania Incorporatedrsquos SG80 74
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition
temperature (500 K) were kept the same while the total number of basins were
allowed to vary 101
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K 112
Table 6 System configurations for sodium silicate glass-water reactions 117
Table 7 Fitted values from this analysis compared to those reported in the literature The
disparity between the constraints evaluated with molecular dynamics most likely
come from the speed in which the samples are quenched 132
Table 8 Hyperparameters for different neural networks after hyper-optimizations 151
Table 9 A table with some ionic conductivity models and the parameters needed for
them as well as the disadvantages for each These are not the only models but are
representative of those commonly used in literature 154
Table 10 The predicted compositions based on the optimization scheme proposed 157
Table 11 The compositions synthesized in this work These compositions were predicted
by minimizing the cost function described in Eq (123) OP is the variant that was
melted after OP partially crystallized B-OP appeared to have surface nucleation in
some spots but was cut and removed before APS treatment 165
xii
Acknowledgments
ldquoTo Love Another Person is to See the Face of Godrdquo -Victor Hugo Les Miserables
I have been truly blessed to love and be loved by a plethora of mentors friends colleagues and
family across my academic journey I have to first thank my advisor Dr John Mauro who has
been as kind patient and caring as any mentor I have ever met and in the process has made me a
better scientist and a better person His mentorship was built on top of those who first found and
shaped me into a scientist Dr Ugur Akgun and Dr Steve Feller without whom my scientific career
would not be possible Along with these individuals Irsquove had the pleasure to learn so much about
glass and life from Dr Madoka Ona Dr Firdevs Duru Dr Mario Affatigato Dr Doug Allan Dr
Ozgur Gulbiten and Dr Seong Kim
Rebecca Welch Arron Potter and Anthony DeCeanne are my partners in all things
Rebecca is my partner in research and in life Her opinion support and patience has been
indispensable to me Arron is my oldest colleague and friend he is the one that I share the highs of
research and friendship with Anthony has put up with me non-stop for about 5 years now He is
the most patient and kind friend you could ask for Without these three individuals I could not have
done anything presented in this work I additionally have to thank Karan Doss Aubrey Fry Caio
Bragatto Daniel Cassar Brenna Gorin Mikkel Bodker Greg Palmer Katie Kirchner Kuo-Hao
Lee and Yongjian Yang I have the privilege to call these individuals both my friends and
colleagues
The last people I have to thank are my family I want to thank my brothers (Sam amp Quinn)
my mom and my dad My mom has helped me explore the universe through a love of reading and
her constant unwavering support My dad was the first to show me the wonders of science with
rockets as a kid as a teenager he supported my fledgling interest as he drove hundreds of miles to
every corner of this country and he is the person for whom this dissertation is dedicated
Chapter 1
The Difficulty of Optimizing Glass
Glass is a complex world-changing material Though it has existed for thousands of years
the surface of its true potential is only just being scratched To unlock the potential of glass for new
applications and to enable the maximum benefit to society we must be able to design new glasses
with wide ranging properties faster and cheaper than ever before [1]ndash[5] This is particularly
important now that the need for ecologically responsible materials is increasing due to climate
change In order to facilitate such glass design we must return to first principles and build a picture
of glass from the ground up encompassing both compositional and thermal history dependencies
into our understanding of glass properties[3] [6] The present accepted description of glass is a
non-crystalline non-ergodic non-equilibrium material that appears solid and is continuously
relaxing towards the super-cooled equilibrium liquid state[1] This definition gives insights into the
nature of glass and includes points of particular interest in this dissertation
The first point to consider is that glass is non-crystalline meaning that it contains no long-
range order This gives glass one of its key advantages being infinitely variable Since a glass is
not limited by having to reach stoichiometric crystalline structures there is no limit to the number
of possible glasses or glass structures This gives glass its possibilites Glass has been considered
as a candidate for everything from hydrogen fuel cells[7] to batteries[8] and for applications
ranging from commercial smart phone covers[9] to bioactive applications[10] The infinite
variability also explains the difficulty of designing glasses Consider a system with three
constituents for example SiO2 Na2O and B2O3 This then means there are two independent
compositional dimensions that must be fully explored to find an optimal composition for an
application If this space were discretized to 1 mol spacing that would give over 5000 unique
2
glass candidates that would need to be studied to reach the optimal design A five-component glass
would correspondingly have over 9 million unique glasses More generally for a glass with up to
C components there are 1C minus dimensions over which to optimize If instead one considers a
crystal there is not a continuously variable space but instead only discrete points
The other ramification of glass being non-crystalline is that it must bypass the region of
crystallization at a rate much faster than the rate of nucleation[11] This is inherent in the definition
of glass and is shown in Figure 1 Figure 1 is called the volume-temperature (VT) diagram and is
key to understanding the nature of glass The bypassing of the crystallization region leads to a
super-cooled liquid and during quenching the material departs from equilibrium and enters the
glassy state The region of departure is called the glass transition temperature range or simply the
glass transition To understand why this occurs we must first understand the concept of ergodicity
Ergodicity was defined by Boltzmann to mean that the time-average value of a property is
equal to the ensemble-average value of that property[12] [13] Now that ergodicity is defined we
can start to consider the implications of the non-ergodic nature of glass Since it is non-ergodic the
ensemble average will not equal the time average on human timescales however the ergodic
hypothesis says that in the limit of long time they will be equal[12] In order for both statements to
be true glass must evolve to be ergodic and thus is inherently unstable To explore this concept
further we must understand the glass transition process and define a timescale If there exists a
liquid at some temperature T and is then perturbed to T dT+ there will be some inherent time
associated with its relaxation towards equilibrium which is described by struct the structural
relaxation time There is a separate relaxation time when the system is mechanically perturbed
called the stress relaxation time stress To then understand whether the resulting material is a glass
or a liquid we need to compare struct with the observation time If the observation time is much
longer than the relaxation time then what is observed over the course of a experiment is the
3
equilibrium liquid properties Conversely if the observation time is much shorter than the
relaxation time then the observed properties will not be ergodic and as such we will not sample
the properties of the equilibrium liquid We define this non-ergodic state as the glassy state
However since in the limit of long time the system must become ergodic once again then over time
all glasses will return to an ergodic nature through a process called relaxation [14]
The last point considered in the definition of the glass is that it is non-equilibrium Not only
is glass non-equilibrium it is also unstable and continuously progressing towards the super-cooled
liquid state (eg relaxation) Relaxation is inherent in all glasses and we can see this for a model
system in the VT diagram (Figure 1) In this diagram there are two glasses (of the same
composition) quenched at different rates (both of which bypass crystallization) In the slow-cooled
glass there is a lower volume (higher density) compared to the fast-cooled glass This difference is
due to the fact that the slow-cooled glass has been allowed to relax more fully compared to the
other This adds at least one additional dimension of optimization (such as quench rate from a high
temperature to room temperature) in which glasses must be designed in
This dissertation is dedicated to understanding the infinite variability of glass with respect
to crystallization composition and thermal history with tools being implemented and designed so
that the challenge of optimizing over these vast spaces can be done with higher efficiency lower
cost and with less experimental work load These tools are all inspired by the energy landscape
description of glass To design the optimal glass for any application we must understand the
relationship between the 1C minus compositional dimensions a minimum of one thermal history
dimension and the properties of a material There are many ways to explore these relationships
each with its advantages and disadvantages The methods to optimize over these thermal history
and compositional dimensions include energy landscapes[15] fictive temperature[16] topological
constraint theory[17] and machine learning[3] Of these energy landscapes are the most
fundamental with each other technique being related back to the energy landscape in some form
4
Though they are the most fundamental they are also the most difficult time consuming and have
the largest barrier to entry for use To then build our understanding of each technique we will start
with energy landscapes then explore how through a series of assumptions we can arrive at the other
methods
5
Figure 1 The Volume-Temperature (VT) diagram at the highest temperature there exists the
equilibrium liquid which as it is quenched can either become a super cooled liquid or crystallize
Crystallization causes a discontinuity in the volume The super-cooled liquid upon further
quenching departs from equilibrium and transitions into the glassy state Reproduced from
Fundamentals of Inorganic Glasses with permission from the author[1]
6
11 Energy Landscapes
Energy landscapes were first developed by Goldstein and Stillinger who proposed the
concept as a way to understand the evolution of a material under complex conditions [18]ndash[20]
Since Stillingerrsquos seminal work [18] energy landscapes have become important in the study of
protein folding [15] [21] [22] network glasses [6] [13] and other complex materials The concept
of energy landscapes has become commonplace and is readily evoked when describing the time
evolution of a material [20] [22]ndash[26] Despite energy landscapes being commonplace in some
fields the procedure for mapping a landscape remains difficult due to the associated computational
challenges [15] [27] To map an energy landscape one must find a set of local minima and the first
order saddle point connecting each pairwise combination of those minima Each minimum is called
an inherent structure and represents a structurallychemically stable state of the material A basin
is the set of structures that converge to the same inherent structure upon minimization [18]
In glass science the energy landscape is used to justify complex behaviours which are most
commonly calculated using molecular dynamics simulations (MD) [13] [28] [29] It is easy to
understand why MD is used so frequently used when considering the ease of use of common MD
packages For example LAMMPS [30] is a common user-friendly MD program with a low barrier
to entry allowing individuals with little programming experience to run complex MD simulations
Inspired by this accessibility as well as the explicit Python bridge to LAMMPS we have developed
a software that can be used to explore energy landscapes with a variety of exploration techniques
Similar softwares [15] [31]ndash[35] have been created to map landscapes however each program has
limitations and may not be suitable for all purposes while ours is a general purpose software
The calculation of the energy for landscapes is done using an empirical interatomic
potential that can be fit using either experimental data and a series of MD simulations or using ab
initio methods One common potential form is the Lennard-Jones form (LJ)
7
12 6
4LJVr r
= minus
(1)
in which and are fitting parameters for every pair-wise set of atoms in a system V is the
potential energy contributed by the two-atom interactions and r is the distance between the two
ions in question Though the LJ form is simple and easy to parameterize it is unable to recreate
complicated material behaviours for a real glass As such a more complicated potential is often
used such as a Morse potential A commonly used silicate glass potential is the Pedone potential
which consists of Morse potential with an additional columbic term and repulsive term [36] and is
given by
( )2
2
0 121 exp 1
i j
Pedone
Z Z e CV Y a r r
r r = + minus minus minus minus +
(2)
Z is the charge of a given ion e is the charge of an electron and Y C a and r0 are fitting parameters
It is also worth noting that the Pedone et al potentials are only accurate when considering the
interactions between each individual ion species with oxygen For instance Si-O Na-O and O-O
are all explicitly parameterized while Si-Na only interacts coulombically (the first term)
Interatomic potentials are discussed here only briefly but there are extensive resources available
for those readers who wish to learn more[37]ndash[39]
Once an energy landscape has been mapped the thermodynamics and kinetics for a given
system can be easily calculated [33] [40] The thermodynamic driving force is given by the
difference between the free energy of the current inherent structure and the free energy of the lowest
state while the kinetics is the rate in which the system crosses the barrier between the basins This
can be calculated explicitly for arbitrary timescales and temperatures with another software
KineticPy [41] The energy landscape provides insights into the detailed nature of material
behaviour Though energy landscapes are very powerful they are also difficult to map To
approximate the behavior of the landscape then we use approximations such as fictive temperature
8
topological constraint theory (TCT) or machine learning Fictive temperature is a way to estimate
the effects of the thermal history dimension[42] while TCT is used to estimate the compositional
dependence of properties[43]
12 Fictive Temperature
Fictive temperature is an approximation used to find the change in the occupational
probabilities on an energy landscape[16] [44] Instead of considering every possible combination
of occupational probabilities like the energy landscape it instead approximates the occupational
probability using a single temperature at which the landscape would be in equilibrium It is
essentially a measure of how closefar a glass is from equilibrium This approximation allows for
us to calculate the relaxation dynamics of a glass without needing any computationally heavy
energy landscapes However this single parameter description oversimplifies relaxation[14] [16]
The functional form for glass relaxation dates back to 1854 when Kohlrausch fit the
residual charge of a Leyden jar with[45]
( ) (0)expt
g t g
= minus
(3)
where t is time τ is the relaxation time constant β is the stretching exponent and g(t) is the
relaxation function of some property (eg volume conductivity viscosity etc) Eq (3) is referred
to as the stretched exponential relaxation function (SER) The stretching exponent is bounded from
0 lt β le 1 where the upper limit (β = 1) represents a simple exponential decay and fractional values
of β represent stretched exponential decay
For over a century the physical origin of the stretched exponential relaxation form was one
of the greatest unsolved problems in physics until 1982 when Grassberger and Proccacia[46] were
able to derive the general form based on a model of randomly distributed traps that annihilate
9
excitations and the diffusion of the excitations through a network This model however provided
no physical meaning for β which was still treated as an empirical fitting parameter In 1994
Phillips[47] [48] was able to extend the diffusion trap model by showing that β=1 at high
temperatures and at low temperatures the stretching exponent can be derived based on the effective
dimensionality of available relaxation pathways He in turn expressed the stretching exponent
2
df
df =
+ (4)
where d is the dimensionality of the network and f is the fraction of relaxation pathways available
He then proposed a set of lsquomagicrsquo numbers for common scenarios Assuming a three-dimensional
network with all pathways activated (d = 3f = 1) a value of β= 35 is obtained The second case is
a three dimensional network with only half of the relaxation pathways activated (d =3f = 12)
yielding a value of β= 37 A third magic value β= 12 was found for a two dimensional technique
with all pathways activated (d = 2f = 1) β = 35 occurs for stress relaxation of glasses under a load
because both long- and short-range activation pathways are activated whereas a value of β = 37 is
obtained for structural relaxation of a glass without an applied stress The value was confirmed by
Welch et al[9] when the value was measured over a period of 15 years at more than 600C below
the glass transition temperature of Corning Gorilla Glass Though this model was able to reproduce
the limiting values it was criticized widely in the community Critics argued that the model was
simply created such that it reproduced the stretching exponent values of certain experiments and
ignored many other experimental results that appeared to disagree with the model The model also
fails to interpolate between the low temperature values Phillips predicted and the high temperature
limit
The work from Phillips and the stretched exponent gave a form for relaxation however
there remained no method to instantaneously state the distance from equilibrium of a glass To
account for this problem an additional thermodynamic variable (or order parameter) called fictive
10
temperature (Tf) was proposed Early research on relaxation in glasses dates back to the work of
Tool and Eichlin in 1932[49] and Tool in 1946[50] whose works originally proposed a temperature
at which a glass system could be in equilibrium without any atomic rearrangement Tool suggested
that this fictive temperature was sufficient to understand the thermodynamics of a glassy system
Originally the fictive temperature was treated as a single value that was some function of thermal
history (Tf[T(t)]) The evolution of fictive temperature as proposed by Tool is then given by
)
( ) ( ( ))
(
f f
f
dT T t T T t
dt T T
minus= (5)
In which for stress relaxation is given by
( )(( ) ( ))fT t T
G
T t = (6)
In Eq (6) is the shear viscosity and G is the shear modulus This means that to predict the time
evolution of a glass only three things are required shear modulus the viscosity as a function of
temperature and fictive temperature and the stretching exponent Though fictive temperature
qualitatively reproduced the results they were looking for subsequent experiments have shown that
the concept of a single fictive temperature is inadequate[51] [52]
A key experiment was performed by Ritland in 1956[51] Ritland took several samples
with different thermal histories but the same measured fictive temperature thus based on Toolrsquos
equation both should have had identical relaxation properties Ritland however showed that the
refractive index evolved differently between samples To account for these differences
the use of multiple fictive temperatures was suggested[52] This worked because the stretched
exponential form could be approximated with a Prony series
( ) exp expi i
N
i
Kt w K t minus minus
(7)
11
In the Prony series N is the number of terms while iw and iK are fitting parameters This means
that each term could represent one fictive temperature and this collection of fictive temperatures
would reproduce the overall experimentally observed form Ritlandrsquos experiment showed that
materials held isothermally at their fictive temperatures with varying thermal histories can give
different results for the relaxation of a given property Thus as a glass network relaxes the fictive
temperature will also shift Narayanaswamy[52] was able to apply Ritlandrsquos concept in a highly
successful engineering model with multiple fictive temperatures but the physical meaning of
multiple fictive temperatures remains elusive
In general there is reason to be skeptical of the concept of fictive temperature Fictive
temperature was not derived but instead was an empirical approximation that Tool needed to gain
an early quantitative understanding Recently Mauro et al[16] used energy landscapes to try to
understand the underlying validity of fictive temperature fT implies that the occupational
probability ( ip ) is given by
1
exp ii
Hp
Q kT
= minus
(8)
Q is the partition function k is Boltzmannrsquos constant and iH is the enthalpy of the given state
However when the actual probability evolution predicted by fictive temperature was compared to
the probability calculated through a systematic Monte Carlo approach the results were found to be
drastically different (even when many fictive temperatures were tried) This implied that fictive
temperature is insufficient to describe the glassy state and must be replaced with a new method
Despite the inadequacies of fictive temperature state-of-the-art modeling techniques for
glass relaxation still relies on the concept In these techniques a set of fictive temperatures (usually
12) are used to calculate 12 parallel relaxing fictive temperatures[42] This is because there has
been no alternative method able to reach a prediction of the evolution of glass under different
12
thermal histories The current state-of-the-art model is called the Mauro-Allan-Potuzak (MAP)[44]
model of non-equilibrium viscosity and was derived from a more fundamental expression for
viscosity called the Adam-Gibbs model (AG) which is given by[53]
10 10) log( )
log ( f
c f
BT T
TS T T = + (9)
B is the barrier for a cooperative rearrangement 10log is the infinite temperature limit of
viscosity and according to Zheng et al[54] it is approximately 10-293 Pa s and cS is the
configurational Adam-Gibbs entropy This entropy has long since been a source of debate and will
be addressed later in chapter 5
In order to calculate the equilibrium part of the viscosity the MAP model relies on an earlier
Mauro Yue Ellison Gupta and Allan (MYEGA) equilibrium viscosity model[55] This model
was derived based on earlier work by Naumis[23] [56] Gupta and Mauro[57] and the AG model
These models showed that the configurational entropy can be related to the degrees of freedom
lncS fNk= (10)
f is the degrees of freedom per atom N is the number of atoms and is the number of degenerate
states per floppy mode To then build in temperature dependence they assumed that the degrees of
freedom can be modeled using an Arrhenius form (with activation barrier H )
( ) expH
f T dkT
= minus
(11)
When Eqs (9) (10) and (11) are combined into an equilibrium form the equilibrium viscosity is
then given by
10 10 el pog og xlC H
T kT
+
=
(12)
Where C is equivalent to
13
ln
BC
dkN=
(13)
Through a change in variables this can be written in terms of three common variables that allow
for all viscosity equations to be written with three variables specified by Angel[58] the glass
transition temperature the fragility and 10log The glass transition temperature is defined as
the isokom where the liquid has a viscosity of 1210 Pa s The fragility is defined as[58]
10log
g
g
T T
dm
Td
T
=
(14)
The three most common viscosity models (MYEGA Vogel-Fulcher-Tammann (VFT)[59] and
Avramov-Milchev (AM)[60]) are listed below in terms of the same parameters
10 10 10
10
log (12 log ) exp 1 11 o
log2 l g
g gT Tm
T T
+ minus minus minus
minus =
(15)
( )
2
10
10 10
10
12 logl log
1 1
og
2 logg
Tm
T
minus+
minus + minus
=
(16)
and
( )10(12 log
10 10 1
)
0l gog l 12 logo
m
gT
T
minus
+ minus=
(17)
To incorporate non-equilibrium effects with these equilibrium models the MAP model
proposes
10 10 10log ( ) (1 ) log ( )log MYEGA g ne f gT m T mx x T T T = + minus (18)
14
10log ne is the non-equilibrium viscosity whose form and compositional dependence can be found
elsewhere[42] [44] [61] x is the ergodicity parameter and is of crucial importance to
understanding glass regardless of the model being used[12] In the MAP model it is given by
( )( )
3244min
max
f
f
m
T Tx
T T
=
(19)
Though the MAP model was derived from the best understanding of energy landscapes of glasses
at the time it has a few flaws that prevent it from perfectly predicting the different relaxation
behaviors of glass A few of the short-comings are
1 The method ignores crystallization For a comprehensive model showing the response of
glass to temperature the possibility of crystallization must be included Currently no
relaxation model includes this possibility
2 It is built around the concept of fictive temperature This could be replaced in a future
model but currently at present it carries with it all the failures of the fictive temperature
picture of glass dynamics adding only the relaxation time coming from landscape-derived
values
3 The MAP model does not consider temperature dependence of the stretching exponent
This is typically circumvented by choosing a fixed stretching exponent
4 The parameters needed for the non-equilibrium viscosity in the MAP model are
experimentally challenging to measure To circumvent this Guo et al[61] published
methods to estimate the values based on fragility and the glass transition but the
approximations were only tested on a few samples with similar fragilities and chemistries
5 The MAP model only predicts stress relaxation Since the publication of the MAP model
it has been shown that only stress relaxation time is related to the shear viscosity[62]
Structural relaxation must be incorporated in future models
15
For a complete understanding of the thermal-history dependence of glass all of these questions
must be rigorously answered
13 Topological Constraint Theory
TCT is one way to estimate the compositional dependence of glasses It uses a set of
assumptions that connects the underlying structure to glass properties through understanding of the
energy landscape These assumptions include the temperature dependence of the rigidity the same
assumptions used in the derivation of the MYEGA models and that the effects of thermal history
on structure are minimal TCT was originally proposed by Gupta and Cooper (GCTCT) to try to
understand the role of rigid polytopes on glasses properties[63] Gupta and Cooper fundamentally
understood that the ability for glasses to form was reliant on the transformation of a liquid with
many degrees of freedom to a glass with a fixed degree of freedom Since the structure of the glass
is the same as the structure of the liquid at the glass transition and the degrees of freedom of the
glass ( f ) changes during the transition to glass it was possible to gain insights into the
transformation process This approach is fundamentally an approximation of the highly
multidimensional landscape where only approximate structural information is used to understand
the entropy of the energy landscape Though the core idea was useful it was ldquoclunkyrdquo and difficult
to make calculations with
To improve upon the ideas of GCTCT Phillips and Thorpe (PTTCT)[64] proposed a
mathematically equivalent form where the degrees of freedom could be simply found by
considering the number of constraints ( cn ) around each network-forming atom in the glass They
showed that the degrees of freedom were related to the number of constraints by
cf d n= minus (20)
16
The number of constraints could then be calculated by considering the two-body and three-body
interactions (first and second terms) as a function of the mean coordination of the network forming
species
2 32
c
rn r= + minus (21)
This led to a qualitative language to discuss glass with a few quantitative insights The biggest
quantitative insight comes in the form of the lsquoBoolchand Intermediate Phasersquo[65]ndash[68] If it has
positive degrees of freedom then a network is lsquofloppyrsquo and retains high kinetic energy (as it flips
back and forth between two inherent structures) If the system has negative degrees of freedom
then the network is stressed-rigid and there is additional energy stored in the stress associated with
over-coordination If the number of constraints is equal to the number of dimensions then the
resulting system will have zero degrees of freedom and there is no stored stressed energy or kinetic
energy from flipping between basins Boolchand went searching for this perfectly lsquoisostaticrsquo glass
and found a non-zero window of compositions that were close to isostatic but had properties
drastically different compared to both stress rigid and floppy glasses This is now called the
Boolchand Intermediate Phase
Despite the success in the prediction of the intermediate phase TCT remained largely
qualitative until the work by Naumis[23] [56] and Gupta amp Mauro[57] Naumis pointed out that
the degrees of freedom are a lsquovalleyrsquo of continuous low energy so that the structure can freely move
between basins of the landscape Since these valleys are deformation pathways Naumis pointed
out that they will dominate the configurational entropy of the system Building on this Mauro and
Gupta wrote that since the entropy is proportional (Eq (10)) we could then write the glass transition
temperature as a function of the degrees of freedom and the AG model
( )10o1 ln2 l g
gTB
f Nkminus=
(22)
17
Since everything in Eq (22) is approximately a constant between compositions except for f we
could then write the expressions in terms of reference values (subscript r) and accurately scale the
glass transition of new compositions by knowing the changing structures
( ) g r c r
g
c
T d nT
d n
minus=
minus (23)
The last insight provided by Mauro and Gupta was that the number of constraints change as a
function of temperature They realized that each constraint has a temperature dependence which is
controlled by the interaction energy of the constraints This makes intuitive sense because as the
thermal energy becomes greater than that of the barriers additional valleys in the energy landscape
form This also allowed them to expanded TCT to include quantitative predictions of fragility
The understanding of the relationships between the configurational entropy the number of
constraints the degrees of freedom and the viscous properties lead to a revolution in understanding
glass properties Soon after the work by Mauro amp Gupta multiple works came out predicting
mechanical chemical and viscous properties of glass knowing only the underlying structure
Though this framework is powerful it is not a complete description without further
parameterizations with each model requiring some fitting parameters or reference values In
addition there are still key properties that are missing from the standard topological approach that
are needed to design commercial glasses
Ultimately MGTCT needs be expanded to include the properties that are needed for
commercial applications A list of composition dependent properties needed for understanding the
performance of commercial glasses are shown in Table 1 Properties taken from Ref [3] show what
properties must be controlled to manufacture technological glasses All models take additional
inputs and there are multiple ways to link outputs from some models to inputs of the other Choice
of model vary depending on the use case
18
Table 1 The key properties considered for commercial application The optical properties have
been omitted since it is physical unrealistic to expect quantitative predictions of quantum-
controlled phenomena from a classical description of glass structure
Properties TCT Machine Learning Other methods
Glass Transition [57] [69] [70] Energy Landscapes [71]
Fragility [57] [69] This work This work
Relaxation MAP model [44] [61]
Crystallization This work [72]
MDMC [73]ndash[75]
Melting Temperature This work MDThis work[72]
Chemical Durability [76] [77]
This work [78]
[79] [80] Linear Methods [81]
Density [82] MD Packing [83]
CTE This work MD
Hardness [5] [84] [85] MD
Youngrsquos Modulus [86] This
work [87]
[82] This work MD
Activation Barrier for Conductivity MD This work [88]
Batch Cost Economic Calculation
Ion Exchange Properties Empirical Relationships [2]
19
14 Machine Learning
All the methods discussed so far are physically informed techniques The techniques
presented are inherently powerful because their very nature is informed by reality Unfortunately
this means that some knowledge about the system is required These techniques are useless without
the required parameterizations However a class of algorithms called lsquoMachine Learningrsquo (ML)
require no such parameterization Instead ML uses large quantities of data to determine trends
linking any physically connected input to output allowing for accurate predictions of properties
across wide compositional spaces This is favorable with a large of volume data because the
required inputs are minimal to achieve a prediction The difficulty (and often expense) in this
technique is gathering large quantities of high-quality data to enable this method
The predominant ML method used in this work is neural networks (NN) Neural networks
work by constructing a network of neurons This method was designed in such a way to mimic a
human brain so the network is elastic enough to be modified improving results Each neuron takes
in a series of inputs received from the previous layer (or input data) multiplies by a weighting
factor sums the values and then performs some lsquoactivation functionrsquo on them By changing the
weights it is then possible to recreate any function given sufficient data This is a preferred
technique because it packages into a simple easy-to-use software Neural Networks (once trained)
can be used by anyone stored in a small file can recreate any function is computationally cheap
and does not require as much data as some other ML techniques
This is an interesting method to estimate the effects of the composition on the landscape
and corresponding properties because we cannot understand the underlying mathematics It is
fundamentally a numerical process Thus even though we know that the properties of a glass are
dominated by the topography of the landscape how the NN figures out the relationship between
the chemistry and properties is not understood It is possible to then use the NN to give us the
20
physical parameters we need for other models building multiple methods together to achieve new
insights
15 Goals of this Dissertation
To enable the next generation of glass design we must enable a new era of computationally
methods being used to produce accurate predictions of the behavior of glass This includes both the
thermal history dependence of glass (in terms of relaxation and crystallization) and the
compositional dependence especially those properties listed in Table 1 The goals of this thesis will
be to address some of these outstanding questions in a systematic way that is practical for an
external user to understand and implement Predicting thermal history effects are covered in
Chapters 3-5 while compositional effects are predicted in Chapters 6-7 The breakdown of each
following Chapter as it relates to these questions is listed below
21
bull Chapter 2 Software Unfortunately there is no standardized software for the
calculation of relaxation or energy landscapes I present two new codes to
standardize these calculations so that the process is reproduceable and standardized
bull Chapter 3 Crystallization This section will address new computational ways to
calculate crystallization focusing on predicting nucleation rates and how to improve
our nucleation calculations
bull Chapter 4 Expanding the Current State of Relaxation This chapter will focus on
expanding the MAP model with a temperature dependent form of the stretching
exponent as well as understanding the role of ergodicity in glass relaxation
bull Chapter 5 Relaxation and Crystallization without Fictive Temperature Chapter 5
will focus on discussing relaxation and crystallization without fictive temperature
by proposing a new relaxation model that is reliant on creating lsquoToy Landscapesrsquo
This approach removes assumptions about the evolution of the occupational
probabilities
bull Chapter 6 Enabling Prediction of Properties This chapter will use TCT and ML to
address the compositional dependance of different key properties needed for
enabling the faster design of new glass compositions
bull Chapter 7 Designing lsquoGreenrsquo Glasses Applying information learned throughout
this dissertation to create new glass compositions
bull Chapter 8 Conclusions
22
Chapter 2
Software for Enabling the Study of Glass
To being our systematic investigation it is important to have a suite of tools that
implement current state-of-the-art models Though many tools exist for structure and
topological constraint theory there were no codes that existed for the calculation of
relaxation using the MAP model and there were no codes that were capable of exploring
the landscapes that we are interested in To correct this we present two new novel codes
ExplorerPy for mapping energy landscapes and RelaxPy for calculating MAP
dynamics[42] [89]
21 ExplorerPy
Mapping an energy landscape allows for modeling a range of traditionally inaccessible
processes such as glass relaxation The advantage of this software (over other codes to map energy
landscapes) is that it only requires knowledge of LAMMPS to run Also our software contains a
choice of three methods to map the landscape The first of the three different methods implemented
to explore energy landscapes in this work is eigenvector following [90] Eigenvector following has
been used in the past to generate landscapes that predict the glass transition and protein folding
pathways [44] [91] This implementation follows the work of Mauro et al [90] [92] and uses an
independent Lagrange multiplier for every eigenvector Eigenvector following excels at finding
small barriers quickly however it rarely finds unique structural changes in complex materials
(especially in systems with periodic boundary conditions) The second method that we have
implemented following in the steps of Niblett et al [27] is a combination of molecular dynamics
and nudged elastic band methods (NEB) [34] [35] To find the transition points MD is run at a
23
fixed temperature for a fixed amount of time the structure is minimized and then NEB is used to
find the saddle point between the initial structure and the new inherent structure Details of the
nudged elastic band method can be found elsewhere [34] Using these techniques it is possible to
map a sufficiently large landscape to make kinetic and thermodynamic measurements possible
These two methods benefit from the fact that they work well when the volume is varied and together
enable computationally inexpensive mapping of enthalpy landscapes [71] [92] (where the volume
is no longer fixed and instead the pressure is specified) However it is not recommended to generate
a large enthalpy landscape from MD in this software due to the computational cost involved
Although molecular dynamics is a powerful technique there is a large computational cost to
simulate a small amount of time
The third method is a new technique created in an attempt to circumvent the computational
cost of MD while still finding large structural changes We have named this technique ldquoeigen
treibanrdquo (ET) This technique relies on shoving along an eigenvector with a multiplier rather than
stepping After the system is shoved the system is minimized and if the minimized structure exhibits
a displacement in atomic position is larger than a threshold a NEB calculation is performed to find
the saddle point By shoving in the direction of an eigenvector the goal is to reproduce large changes
due to vibrational modes The fixed multiplier (m0) is calculated by taking a target distance (d) and
the largest displacement magnitude out of the atoms j (vij) in a given eigenvector direction i (ai)
m0
=d
max[nij] (24)
To convert to the displacement magnitude from an eigenvector in three dimensions we use the
expression
vij
= ai[3 j]2 +a
i[3 j +1]2 + a
i[3 j + 2]2 (25)
The dimension j has the length of the number of atoms in the systems
24
This software uses the system and potentials created from a LAMMPS input
scripts and as such has a wide range of access to different potentials and possible systems The
only input that has to be provided is the file which would be used for molecular dynamics in
LAMMPS If the script has been used for molecular dynamics simply by eliminating any
thermostats or barostats as well as any ldquorunrdquo commands it is possible to use it as an input script to
ExplorerPy (it is also recommended to eliminate any calls to ldquoThermordquo because it may interfere
with the extraction of variables from the underlying LAMMPS framework) If the LAMMPS input
script successfully sets up a simulation all other variables and parameters can be controlled from
command line options as described in the ldquoIllustrative Examplerdquo section The initial structure will
be taken from the input script as well
ExplorerPy is a software that has been written in Python with modular functions and ease
of use in mind It can be run from a Linux terminal with all major options being set via the command
line The software then generates a list of inherent structure energies stored in the mindat file and
stores the transition point energy for each pairwise inherent structure in the tsddat file All of the
transition points and minimum are stored in the same directory from which it is run as xyz files
The software architecture is shown in Figure 2 Details of the stepping procedure for eigenvector
following can be found elsewhere[93] The variables ldquoMD frequencyrdquo and ldquoET frequencyrdquo are set
as command line options at the time of the launch of the software
25
Figure 2 The schematic for the flow of the program Beginning in the top right corner and running
until the condition in the pink box is satisfied Yellow diamonds represent checks and blue
operations
The software is capable of mapping an energy landscape for an arbitrary system ie for
any system that can be simulated in LAMMPS This allows for accurate prediction of the evolution
of complex thermodynamics and kinetics The software requires no programming and is
controllable via command line options ExplorerPy prints all of the basins transition states inherent
volumes as well as information about the curvature of the system so that the vibrational frequency
is calculable This is all of the information required to calculate a complex range of phenomena
over a specified long timescale ExplorerPy is only set up to work within the LAMMPS metal units
To run the software there are only a few parameters that have to be set for each mode The
force threshold (-force) curvature threshold (-curve) and step size (-stepsize) when running the
command line option have to be set to allow for a stable exploration of a given landscape
These seem to be potential-specific and for a given potential appear to be fairly stable It is best to
start large for the force threshold and small for the eigen threshold and step size then modify as
26
necessary If eigenvector following is not being used there is no need to adjust these values Beyond
this there are several optional command line options (only the file (-file) command is required)
-file [file input] this is a standard lammps input if the input sets up a lammps run
it will work to set this up There should not be any lsquothermorsquo calls in the lammps
script
-press [specified pressure in atmospheres] if this is specified the system will
always be minimized to this pressure while if not specified the volume remains
fixed
-num [number of basins to explore]
-time [max time to run in hours] if this time is reached before the number of basins
are found the simulation will stop
-thresh [minimum energy for barrier to keep]
-center if this is given as a flag the system will be set to the anchor location each
step
-anchor [atom selection for the position to be held constant]
-md [frequency of molecular dynamics search] If a custom MD LAMMPS file is
not provided it defaults to starting all of the atoms at 3000 K and running for 50
fs
-shove [frequency of eigen treiban]
-custom [file used for MD exploration]
-help [List all commands possible in software with these descriptions]
An example run would then be done the following way
[mpirun -np 5 python37 Explorerpy -file inSiO2 -num 1000 -force 10 -curve 1e-5 -stepsize 012
-press 00 -md 11 -time 48]
This would start a run with five parallel processors searching for 1000 basins with an MD
step being ran every 11 searches with a maximum time of 48 hours The results of this command
are shown in Figure 3 Below we have attached an example of inSiO2 the output and additional
scripts related to the example has been uploaded to the associated GitHub
27
Figure 3 The enthalpy landscape for SiO2 This was explored using the command above and then
plotted using PyConnect The plot is a disconnectivity graph where each terminating line
represents an inherent structure and tracing where two lines meet describes the activation barrier
The potentials are taken from the BKS potential [94]
httpsgithubcomlsmeetonpyconnect
28
22 RelaxPy
RelaxPy script is written using Python syntax and relies on the numpy scipy and
matplotlib libraries It solves the lack of a unified relaxation code in our community and is based
on the MAP model[44] [61] interpretation of multiple fictive temperatures and the shear
modulusviscosity form of the Maxwell equation[62] [95] It is designed to be run within a Linux
terminal with command line options or inside a Python interpreter It can be used by calling the
name of the script followed by the input file the desired output file name and finally an optional
tag for all fictive components to be displayed as well as printed to a specified output file (formatted
as comma separated values)
The RelaxPy package consists of an algorithm that iterates over time in order to determine
the values of viscosity (η) relaxation time (τ ) and current fictive temperature (Tf ) given the user-
supplied thermal history and material property values The purpose of this code is to enable
calculation of the relaxation behaviorrsquos change over time by using the MAP model using only
experimentally-attainable parameters Toolrsquos equation is solved for each term in a Prony series
approximation of relaxation finding the change in each Tfi component for a given time (t) to
accurately reproduce the stretched exponential form The program then iterates over the entire
thermal history of the sample using a user-specified time step dt allowing for calculation of the
overall fictive temperature over time The values for the Prony series fit (wi and Ki) are taken from
the database originally generated by Mauro and Mauro[96] and expnded to include any value of
the stretching exponent The values were fit by Mauro and Mauro with a hybrid fitting method
based on the number of terms in the desired Prony series and the magic value from Phillips The
thermal history is also defined in the original input file using linear interpolation The initial fictive
temperature components are assumed to be at equilibrium (Tfi = T(0))
29
To solve for the non-equilibrium viscosity used the MAP expression is used with the approximate
scaling of values proposed by Guo et al[61] The software is capable of performing the calculations
needed for a variety of thermal histories and processing methods desired by both industry and
academia with a focus on the fictive temperature This software is then fundamentally limited by
the issues associated with the MAP model and the compositional approximations made For
instance this software will need to be further modified to include structural relaxation when such
a model is proposed[62]
An example input code is listed below
Denotes Comments
Tg 794 0 m 368 C 1494 dt 10 Beta 37
Start Temp End Temp Time [s]
1000 10 1300
10 100 1000
100 30 500
Assumes Linear Interpolation
This example input states that over the space of 1300 seconds an initial glass sample is cooled from
1000 C to 10 C Then it is heated to 100C over 1000 seconds and then finally down to room
temperature in the space of 500 seconds All temperatures are given in Celsius
This software aims to provide the first open-source software for modeling of glass
relaxation behavior using the MAP model of viscosity by creating a package that can easily and
quickly be used to approximate the fictive temperature components of a glass as well as the
composite fictive temperature RelaxPy allows for researchers to calculate the evolution of
macroscopic properties of glass during a relaxation process (assuming a fixed stretching exponent)
and will facilitate the calculation based on a user-supplied thermal history RelaxPy can be
seamlessly adopted by most glass research groups and removes the guessing associated with
determining a fictive temperature allowing for a fully quantitative comparable discussion of Tf
simultaneously creating a tool that can easily be expanded for the entire community as new physics
is discovered
30
Chapter 3
Understanding Nucleation in Liquids
Crystallization cannot be ignored in any material system at temperatures below its liquidus
temperature although extreme resistance to crystallization is reported in some organic and
inorganic substances[97] Crystallization (and by extension nucleation) is critical in systems
ranging from ice to thin films[98]ndash[100] Glasses are non-crystalline substances therefore
crystallization kinetics will determine the necessary thermal history to avoid crystallization and
obtain a glassy state Despite the absolute importance of crystallization to the study and design of
glasses there is no current method to predict the nucleation of crystals in this event One of the main
challenges in the production of glass-ceramics is finding the optimal thermal history for the desired
crystallite distributions There are two phenomenological steps in the process of phase
transformation from a supercooled liquid to a crystal a nucleation step and a growth step The
growth step is described well by Wilson-Frenkelacutes theory[1] but the underlying physics of
nucleation remains elusive A stable nucleus is the precursor to a crystal comprising a nano-sized
periodic assemblage of atoms A clear understanding of the relationship between liquid kinetics
and crystal nucleation has not been established despite decades of research
The study of nucleation is particularly critical for glass-ceramics Since their discovery by
S Donald Stookey glass-ceramics have become prevalent in our society through products such as
stovetops and Corningware[73] [101] [102] Glass-ceramics are a composite material comprising
at least one crystalline phase embedded in a parent glassy phase benefitting from the properties of
both the glassy and crystalline phases[1] [103] One of the reasons glass-ceramics are ubiquitous
is their relatively easy production glass-ceramics can be formed through standard glass forming
procedures using additional heat treatments Additionally there is a wide range of accessible
31
properties through tuning both the chemical composition of the parent glass and the thermal history
and hence the microstructure of the heat-treated glass-ceramics[101] [103]ndash[106]
Glass-ceramic production consists of first synthesizing a glass and then generating a
microstructure through the two-stage ceramming process[73] [103] [107] The nucleation step
remains poorly understood with classical nucleation theory (CNT) giving widely varying results
depending on how the parameters are determined[1] [73] [108] The inability to consistently get
accurate predictions for a nucleation curve means that choosing an appropriate nucleation
temperature must be done experimentally which can prove to be one of the most challenging and
time-consuming aspects of properly designing a glass-ceramic The difficulty is due to the
magnitudes of the crystal nucleation rates not being predictable with CNT or any other current
model
This difficulty in designing a material is apparent when considering the high dimensional
phase space of parameters that can adjusted We haver presesnted some tools developed to help
optimize over this lsquoglassyrsquo space such as RelaxPy[42] KineticPy[41] topological constraint
theory[109] or various machine learning models[4] [70] [110] [111] However a glass-ceramic
system has at least four additional phase dimensions (two for the nucleationgrowth temperatures
and two for nucleationgrowth times) that need to be optimized (depending on the parameterization
of the experiment it is also possible to add another four dimensions relating the heating and cooling
between each step) Therefore a glass-ceramic system has a minimum of d+5 dimensions over
which the system must be optimized and yet those four additional dimensions are the least
understood Designing methods to easily optimize over this large space with feasible computational
methods is of utmost importance to reaching a new generation of custom materials
Experimental studies of nucleation have used techniques ranging from differential
scanning calorimetry (DSC) to electron microscopy to try to capture predictive capability in the
lsquoceramicrsquo phase-space (the four additional dimensions) These studies have successfully captured
32
the shape of the nucleation curve but fail to provide the information needed to calculate the
temperature-dependence of the nucleation rate The shape of the curve can be determined through
DSC and with great effort an approximation of the rate can be made however this technique is too
laborusome to be efficient[112]ndash[114] Since the parameters have been inaccessible to traditional
computational approaches and experimental determination all previous studies of nucleation have
required at least one variable to be fit If a nucleation curve could be predicted only using a
computational method or with an efficient simple DSC method the rate at which new glass-
ceramics products are created would increase In addition the cost of producing glass-ceramics
would likely decrease and more complex microstructures and therefore more unique properties
could be explored
Crystal nucleation has been historically difficult to simulate and measure experimentally
because the phenomena happen on spatial scales of nanometers but over many orders of magnitude
of time scales CNT is the most widely applied model despite its supposed failure To test CNTrsquos
assumptions an interest in computational work has grown Recent approaches using grand
canonical Monte Carlo have successfully predicted the short time dynamics of nuclei
formation[73] This approach involves use of a continuous solvation model for the liquid and has
enabled greater computational insights however it remains computationally expensive[74]
Molecular dynamics (MD) fails to capture nucleation in glass-forming systems due to the need for
longer spatial and temporal scales to capture nucleation effects accurately Some attempts have
been made using Monte Carlo MD or empirical approaches to replace classical nucleation theory
(CNT) with some alternate method where the parameters can be directly measured[73] [115]ndash
[117] All nucleation models and Monte Carlo models attempt to describe the rate that a
supercooled liquid system transitions to a crystalline structure as a function of temperature
Explicitly mapping the energy landscape would fundamentally explain the underlying
physics of nucleation An energy landscape of a system is given by
33
1 1 1 2 2 2( )N NNU U x y z x y z x y z= (26)
where U is the potential energy and x y and z are the locations of each of the N atoms in a periodic
box The locations of these atoms give rise to an energy as parameterized by an interatomic
potential An energy landscape can be constructed by mapping the relationship of the local minima
and the first order saddle points between pairs of minima22-27 Energy landscapes have become
crucial to studies of biomolecules glasses and catalysts[21] [26] [44] because landscapes show
atomic configurational changes and calculate the associated kinetics and thermodynamics In this
study we will explore the energy landscape of a barium disilicate system capturing the glassy and
crystalline basins enabling a test of CNT by calculating all needed parameters (interfacial free
energy kinetic barrier and free energy difference of the supercooled liquid and the equilibrium
state)
31 Crystallization Methods
CNT expresses the steady-state nucleation rate (I) as the product of a kinetic and a
thermodynamic factor
( )exp eI Z D TW
kT
minus =
(27)
where Ze is the Zeldovich factor which incorporates the probability of a critical sized nucleus that
will form a new phase independent of other nuclei D(T) is the kinetic factor often given by an
Arrhenius expression over sufficiently small temperature ranges W is the work associated with
generating a thermodynamically stable nucleus that will continue to grow rather than dissipate (ie
the critical work) if temperature and pressure are kept constant k is Boltzmannrsquos constant and T is
absolute temperature The Zeldovich factor is typically treated as a constant with a value ~01[118]
34
Here the kinetic term is expressed as D(T) since in this work the kinetics are explicitly calculated
at every temperature and it is not treated as a simple Arrhenius expression In CNT the work W is
given by the work to generate a spherical isotropic nucleus[1] which consists of a volume term
and a surface energy term It is given by
3 24
( ) 4 ( )3
W r G T r T = + (28)
where G is the Gibbs free energy difference between the crystalline state and the corresponding
liquid r is the radius and is the interfacial energy between the crystalline and liquid phases Eq
(28) neglects the strain energy term that arises from the formation of a solid nucleus in the
supercooled liquid It is assumed that the liquid relaxes fast enough so that strain energy is not
significant for nucleation in liquids[11]
Determining the critical radius (ie taking the derivative of Eq (28) with respect to the
radius and setting the expression equal to 0) allows one to solve for the critical work and the
expression becomes
3
2
16
)3(
( )
( )W
T
G T
=
(29)
Three temperature dependent functions remain D G and Though there are existing methods
in literature to approximate each of these quantities the approximations often include large errors
and assumptions about the behavior of the system[119] Often experimentally the kinetic function
is approximated with the Stokes-Einstein relationship the G term is calculated from heat
capacity curves (assuming that the macroscopic energy is similar to the nanoscopic energy) and
is fitted[119] In the calculation presented here only configurational energies are considered
while vibrational contributions are ignored Ignoring the vibrational component and the reliance on
empirical potentials are the most likely sources of error in this work
35
The energy landscape for this investigation was generated using ExplorerPy[89] starting
with the orthorhombic barium disilicate crystalline structure[89] Barium disilicate was chosen
since it is a commonly studied glass-ceramic with multiple reliable experimental datasets being
found in literature[75] [113] [120] The rest of this section is divided into four parts starting with
a discussion of mapping the landscape and the following three sections are a discussion of
calculating D G and respectively
31A Mapping and Classifying the Landscape
The ExplorerPy energy landscape exploration employed the Eigen-Trieban technique in
the canonical ensemble with systems consisting of 128 256 and 512 atoms[89] The number of
atoms was chosen to achieve convergence of the results while also balancing the large
computational cost associated with energy landscape exploration A smaller system was also
calculated with 64 atoms however this systems was too small to accurately discern energies
enough to make the nucleation calculation The Eigen-Trieban technique is a newly developed
method to explore energy landscapes using a method of shoving along the eigenvectors and
minimizing the potential energy which is followed by using nudged elastic band calculations to
find the associated transition points[89] This technique allows for larger transitions to be found
while retaining the ability to calculate the vibrational frequencies The Pedone et al potentials[36]
were used and the volume of the cell was fixed due to simulation instability when pressure was
fixed The Pedone et al potentials were chosen since they have reproduced key parameters for both
the crystalline and glassy state in previous Monte Carlo studies as well as in the training set of the
potential[73] [74] [121] Though the Eigen-Trieban method may work well for predicting
crystallization and finding large barriers it may be insufficient to describe glass relaxation or
36
viscosity since it rarely finds the small structural transitions that dominate the Adam-Gibbs
relationship Note that these exceptions are inconsequential for predictions of nucleation found in
the work presented here33
The barium disilicate system was explored for 2500 basins for 128 and 256 atoms and 4000
for 512 atoms since more basins were needed to converge the liquid state distribution of energies
This number of basins was shown to be sufficient since multiple paths from the initial crystalline
starting point to SCL states were obtained The distribution of energies of the SCL were found to
converge and the resulting energy landscape is seen in Figure 4
A crystal parameter was then defined from the root mean square displacement (RMSD) for
each atom from the nearest lattice site of the same species ( ) This average value is used to
characterize whether a basin is a crystal or SCL The RMSD approach is not the only method to
assign a crystallinity value[122] however it is one of the simplest and as such was chosen for
this application To then categorize each basin as either a crystalline basin or SCL basin a plot of
the minimum energy of the SCL states versus the cutoff of RMSD was produced This plot for 256
atoms shows a clear transition in the energies around a cutoff value of 1 Aring which is therefore
taken as the cutoff radius of which basins are considered crystals The same analysis was repeated
for each system size This analysis is shown in Figure 4 Any basin with a RMSD less than the
cutoff value is considered a crystalline state This cutoff was found to be generally insensitive to
the results with any values ranging between 09 Aring and 11 Aring changing the resulting prediction of
nucleation by less than one order of magnitude but shifting the peak temperature by up to 40 K
This cutoff also reproduced the energy difference between the SCL and crystal in molecular
dynamics simulations of the materials
37
Figure 4 (A) The energy landscape for a 256-atom barium disilicate glass-ceramic the x-axis being
an arbitrary phase space and the y-axis being the potential energy calculated from the Pedone et al
potentials[36] The colors are indicative of the crystallinity where the blue basin is the initial
starting configuration The landscape shows the lowering of energy associated with partially
crystallizing the sample (B) The energy landscape relationship between the cutoff for crystalline
and super-cooled liquid states for 256 atoms is shown This shows a clear drastic energy change
occurring around the cutoff value of 10 Aring
[
38
31B Kinetic Term for CNT
The transition rate was calculated by the inverse of the average relaxation time from the
SCL to the crystalline states The relaxation time ( ) is defined by the relaxation function ( )
0 exp
t
= minus
(30)
where t is time and 0 is the scaling of the relaxation function The reason we are able to
calculate the kinetics of nucleation but not viscosity is because the nucleation kinetics are
dominated by the transitions between two states that we specifically searched for when exploring
the energy landscapes while the viscosity is defined by the entropy dominating small transitions
described by the AG
The AG relationship shows that the viscosity is dominated by the configurational entropy
of the system Experimentally the kinetic term is often calculated with the Stokes-Einstein
relationship
( )6
kTD
aT
= (31)
where a is the radius of the diffusing species and is the viscosity of the liquid Interestingly this
approximation gives reasonable results when investigating experimental data since the viscosity is
a measurement of the entropy of small cooperative liquid rearrangements Conversely the kinetics
that govern nucleation are large structural rearrangements from a supercooled liquid to a crystal It
is nonetheless possible that these cooperative rearrangements and the large diffusion processes for
nucleation are on the same order of magnitude which is why the Stokes-Einstein relationship gives
39
a reasonable approximation in experimental parameterizations When the Stokes-Einstein
relationship is used there is an implicit assumption that the large nucleating rearrangements are
proportional to the small barriers that govern viscosity[53]
A mean relaxation time can be found by averaging over a system of master equations
( )j
ii ji j j ij i
i
dpg K p g K p
dt
= minus (32)
In this expression g is the degeneracy of a microstate pi is the probability of occupying the
microstate and Kij is a transition rate between inherent structure i and inherent structure j The
equilibrium probability of occupying a microstate can be given by a Boltzmann probability (where
Z is the partition function and U is the potential energy of the inherent structure)
1
exp ii i
Up g
Z kT
= minus
(33)
By substituting the value of jp in Eq (32) for a two-state model ( 1 21 i jp p= == + ) we write
( )( )1ii ji i j ij i
dpg K p g K p
dt= minus minus (34)
Isolating the probabilities and splitting the differential we have
( )1
i
i ji i j ij i
dpdt
g K p g K p=
minus minus (35)
which when solved via integration takes the form
ln ( )
( )
i ji i ji j ij i
i ji j ij
g K g K g K pt
g K g K
minus + =
minus + (36)
Solving for the occupational probability yields
( )expi ji j ij i ji
i
i ji j ij
g K g K g K tp
g K g K
minus minus + =
+ (37)
This shows an analytical solution for the relaxation time in a two-state model
40
1
i ji j ijKg g K
+
= (38)
The two-state model approximation is made to enable the calculation of the mean relaxation time
between the liquid basins to the crystalline basins The transition rates between the basins are given
by the vibrational frequency ( ) and the barrier between the states ( ijU )
exp ij
ij
UK
kT
minus=
(39)
The probability of transitioning from a given liquid basin i to a crystalline basin j is given by
1
expij
ij
Up
Y kT
minus =
(40)
where Y is the partition function with respect to all transitions from the current SCL basin to all
crystalline basins as only these transitions are considered to calculate the relaxation time
expj
j
iY
k
U
T
= minus
(41)
The total expected relaxation time of the system making the transition is then given by the
probability of occupying a basin the probability of transitioning from one basin to another specific
basin and the associated relaxation time
1
i ij
j i ji j ji i i
p pg K g K
=
+ (42)
The first summation is over the liquid states and the second summation is performed just over
crystalline states to capture the mean relaxation time from a supercooled liquid to a crystal We
fixed the vibrational frequency to a value of 17 times 1014 Hz (ie about one jump attempt per 5
femtoseconds) which was calculated using the average of the eigenvalues of each basin ( ) and
the average mass of the atoms (m) using
41
1
2 m
= (43)
31C Degeneracy calculations
To obtain an appropriate value for the degeneracy an understanding the phase-space
volume of the basin is needed since the degeneracy is linked to the volume of the basin[40] By
taking the displacement between the basin coordinates of each atom and the transition point we can
define a radius of the basin R Using R and a spherical approximation we can calculate the phase-
space volume degeneracies
3 23
12
NN
i i
cry
Ng R
g
+
=
(44)
In which ( )x is Eulerrsquos gamma function The degeneracy is then normalized by a constant such
that the liquid-liquid state transitions had a value of 100 s at the glass transition (980 K) to be in
agreement with Angell[58] The normalization step allows for a quantitative prediction of the
liquidus temperature to confirm the accuracy of our model The liquidus value found is
approximately 1200 K 1715 K and 1950 K for 128 256 and 512 atoms respectively The
experimentally determined liquidus temperatures range from 1680 K to 1710 K[113] [115] [120]
It is worth noting that changing the exponent drastically changes the thermodynamics while having
minor effects on the kinetics and the normalization value drastically changes the kinetics while
keeping the thermodynamics nearly constant We chose the denominator to satisfy the liquid
relaxation constraint It is also feasible to choose the exponent to satisfy a constraint on the liquidus
temperature The normalization value is an important to consider because for larger system a
change in 001 can result in a change in the kinetics by ~10 orders of magnitude
42
31D Free Energy Difference
Due to limitations in the exploration of the energy landscape requiring a fixed volume we
will approximate the difference in the Gibbs free energy by calculating the difference of the
Helmholtz free energy This approximation is reasonable due to the similar densities between the
crystalline and glassy phases under ambient pressure[73] [74] The first step is determining the
mean energy difference of all the basins and the second step was repeating the process for only the
liquid basins The mean energy was calculated with the following expression[123]
( )ln ln ii
i
i i iG kTp p U kT g p= + minus (45)
In Eq (45) i is the set of relevant basins for the conditions we are interested in The difference
between the mean free energy of the liquid basins and the entire landscape basin energies is the
G parameter This free energy difference is typically experimentally calculated by integrating
heat capacities however numerous approximations exist These approximations are often linear
approximations normalized to the liquidus temperature In this work we find that the driving force
appears to be predominantly linear which is in good agreement with the approximations that are
often used for the driving force[124]
43
Figure 5 An example interfacial structure between the crystalline phase on the left and the last
sequential SCLglass phase on the right for a barium disilicate system The gray atoms are barium
silicon is shown in red and blue represents oxygen
44
31E Interfacial Energy
To calculate the interfacial energy of the interface between the crystalline phase and the
SCL the initial basin (pure crystalline) and the last basin (glass basin) were annealed at 1700 K for
1 ns and then cooled to a temperature of 800 K over a period of 10 ns The same process was
repeated for a composite where the initial stable crystal and the glass were placed next to each
otherThe structure was multiplied 10 times to create a wall of a large interface to reduce energetic
fluctuations A small section of the composite structure is shown in Figure 5 The difference
between the addition of the energy of the crystalline phase and glassy phase and the composite
gives the energy of the interface which was normalized to the size of the surface area Multiple
arrangements of the composite were tested but anisotropy was found to be minimal with the
fluctuations of the energy being larger than the differences from different tested directions The
final expression used for the interfacial free energy (in eVAngstrom2) is
610 00 35047 040T minus +=
45
Figure 6 (Top) The relaxation time and free energy difference (Middle) shown as a function of
temperature for each system size The experimental values for kinetics and thermodynamics come
from ref[125] and from the heat capacity data taken from [126] respectively It is clear to see that
driving force shows good agreement across all systems however the kinetics terms only converge
46
for the 256 and 512 atoms systems (Bottom) The fit used to calculate the interfacial energy as a
function of temperature
32 Results amp Discussion
The results for the free energy difference the kinetic parameter and the surface energy are
shown in Figure 6 These parameters as a function of temperature are used to calculate the
nucleation rates in the expression
3
2( )
01 16exp
3( )I
TT G k
= minus
(46)
The 01 being an approximate value of the Zeldovich factor Using the parameters shown in Figure
6 and solving Eq (46) the nucleation rates are shown in Figure 7 along with experimental data for
validation At first glance we notice that there is about 7 order of magnitude differences between
data and prediction Considering that theses potentials were trained on glass and were not intended
for this purpose we consider this as excellent agreement This method also shows agreement below
the peak but the accuracy falls off more quickly below the peak temperature It is likely that the
data below the peak is severely underestimated due to a failure to reach steady state nucleation
rates Reliable steady state nucleation rates are not available for this system and as such a direct
comparison below the peak temperature is not directly available It is worth mentioning that the
failure of CNT of predicting the order of magnitude of nucleation rates is well reported and can
amount to dozens of orders of magnitude when a constant value of is used[75] [113] [115]
[119] [127] thus the proposed method has an advantage when comparing with the current
classical nucleation theory toolset Moreover the prediction of Tmax (~950 K) is also very close to
the experimental value (985 K)
47
Figure 7 The nucleation curve for barium disilicate for converged systems as predicted using the
model presented in this work The data referenced can be found in refs [113] [128] [129]
48
To further compare CNT and the model presented here we decided to compare the results
of the fitted and this workrsquos calculated surface energies as seen in Figure 8 The results show good
agreement all achieving the same order of magnitude implying that experimental methods can
achieve a close approximation to the values required for CNT The prediction of nucleation is
extremely sensitive to the value of the surface energy with small changes leading to orders of
magnitude change It stands to reason then that further work that needs to be approached through
computation and experiments is developing a rigorous model to predict the interfacial free energy
The other assumption that is used experimentally is the Stokes-Einstein relationship To
compare the landscape results of the transition rate from the SCL to crystalline states and the
Stokes-Einstein relationship the liquid state relaxation was calculated The small barriers that
govern the Adam-Gibbs relationship were not found with this exploration method hence
quantitative comparisons are not possible (and relaxation calculations should not be made)
however the relaxation times can be qualitatively compared The liquid state relaxation rate is
proportional to the viscosity and is given by the expression[44]
j
i
i
l
ij
p
K =
(47)
To understand the validity of the Stokes-Einstein relationship we can write a series of theoretically
true proportionalities
( )l
T TD T
(48)
The values of the experimental kinetics fail to align with the kinetics calculated in this work This
is expected due to the fact that the viscosityliquid relaxation is dominated by the set of small
transitions in the landscape while the nucleation transition is defined by the transition of one region
to another with a substantially higher activation barrier This comparison may also account for
49
some part of the experimental overestimation below the peak with the kinetic term calculated
dropping off at a faster rate than that calculated from the experimental viscosity Figure 6 also
shows evidence that it may be inappropriate to use the Stokes-Einstein relationship for nucleation
and as such we may need to re-visit the experimental models that often rely on the assumption of
the Stokes-Einstein relationship This observation casts some doubt in the legitimacy of describing
the kinetic factor as a diffusivity which is standard for CNT For more information on this CNT
failure please see Refs [75] [113] [115] [119] [125] [127]
Nucleation is a complicated concept to investigate and model Unlike the crystalline phase
where the atomic positions are clearly defined the details of the atomic coordinates and bond
configurations of the liquid sites must be described in terms of statistical distributions Due to this
added complication mean-field distributions are typically used to describe non-crystalline
sites[14] [16] [108] [130]ndash[133] The incorporation of fluctuations could more accurately predict
each liquid sitersquos transition to the crystalline phase Fluctuations in the degeneracy could be one
reason that the nucleation rate is over predicted when the temperature is less than the peak
nucleation temperature A given G varies with the environment and depends on the specific
liquid state not the mean value Therefore the rate of nucleation from all supercooled liquid states
to the crystal state is not constant which is why fluctuations need to be considered and why energy
landscapes are such a powerful technique This is especially highlighted when considering that
different degeneracies yield drastically different heat capacity Energy landscapes are extremely
useful because they consider a large distribution in free energies and show all structural fluctuations
possible Fluctuations in cluster size and degeneracy influence Adam-Gibbs configurational
entropy which directly impact the transition rate between the crystal and liquid state
50
Figure 8 The surface energy with respect to temperature for the work presented here
0
0002
0004
0006
0008
001
0012
0014
800 850 900 950 1000 1050 1100
Su
rface
En
ergy (
eVAring
2)
Temperature [K]
This Work
Xia et al (2017) Ref 23
Fokin et al (2016) Ref 41
51
33 Conclusions
Though this is not the first time CNT has been examined computationally in an attempt to
understand if the theory is valid[74] [116] [117] it is the first parameterization of each variable
independently in an attempt to calculate the resulting nucleation rate Not only are all the variables
calculated when used in the expression for CNT the liquidus temperature is reproduced and the
nucleation rate is reasonably predicted This is as far as the authors know also the first time a real
energy landscape for a complicated glass-ceramic system is being reported This method leveraging
energy landscapes alongside CNT enables for a reasonably fast calculation of the nucleation curve
of a glass forming liquid In future work this model could be used for either developing commercial
glass-ceramics or for predicting the critical rate needed to maintain a glass This model can also
provide additional insights into the physics of nucleation and the validity of CNT without the
constraints of fitting parameters This method fundamentally has the power to connect the 3N
dimensions of the energy landscape to the compositional dimensions and thus allowing for a unique
powerful method of predicting nucleation parameters as a result simplifying the crystalline
dimensions in which a glassglass-ceramic must be optimized over
Chapter 4
Expanding the Current State of Relaxation
Before we consider a new approach to relaxation (optimizing our understanding of
the 3N dimensions) there are first missing pieces in the current state specifically the temperature
and compositional dependence of the stretching exponent However even before expanding the
current state of the art relaxation methods we also need to understand ergodicity In this section we
will start with a simple though experiment to understand the temporal effects of relaxation and then
in the second section combine it with fictive temperature to create a new model for the stretched
exponent
41 A thought experiment to expand our understanding of ergodic phenomenon The
Relativistic Glass Transition
The prophet Deborah sang in the book of Judges ldquoThe mountains flowed before the Lordrdquo
[6] [13] However the question the prophet failed to ask was what happens if the mountains are
moving at relativistic speeds The focus of this section is to highlight the temporal effects on
ergodicity in the context of special relativity and the reference frame of the observer In order to
understand relativistic effects on materials one must consider a well-characterized material that
actively relaxes but appears solid on our timescale viz pure B2O3 glass [134] [135] Glass is a
particularly good candidate to study the effects of time dilation on a material since it undergoes a
kinetic transition known as the glass transition (Tg) which has multiple definitions that will be
considered
53
The first definition is in terms of the aptly named ldquoDeborah Numberrdquo which was proposed
by Reiner [136] to offer a view of fluidity and equilibrium mechanics It was later expanded to
explain the origin of the glass transition and the transition from an ergodic liquid system to a non-
ergodic solid-like glassy state[12] which is the fundamental definition of the glass transition [11]
[12] [137] It is important to note here that the breakdown of ergodicity is relative to the
observation timescale and is reflected in the Deborah number (D) which is defined as
Dt
= (49)
where t is the (external) observation time and is the relaxation time of the material which can be
expressed through the Maxwell relation [42] By definition the Deborah Number equals 1 at the
glass transition temperature Tg [13] [137]
Another definition of the glass transition temperature is due to Angell who defined Tg as
the temperature where the viscosity of the supercooled liquid is 1012 Pa s [58] [138] The Angell
diagram is an important plot relevant to viscosity and the glass transition where the abscissa is TgT
and the ordinate axis is log10( ) In the Angell diagram the value of the viscosity at Tg and in the
limit of infinite temperature are fixed and the difference in the temperature scaling of the TgT-
normalized viscosity is the slope of the log10( ) curve at TgT = 1 which Angell defined as the
fragility (m) of the supercooled liquid Using this definition one can establish the consistency of
the Angell plot at relativistic speeds An example of the Angell plot is shown in Figure 9
54
Figure 9 An Angell diagram created using the MYEGA expression [55] With a very strong glass
(m~17) a highly fragile glass (m~100) and a pure borate glass (m~33) [17] [134] [135] The
infinite temperature limit is from the work of Zheng et al[54] and the glass transition temperature
is from the Angell definition
55
Using the two previous definitions it is expressed
1gT
GtD
= = (50)
which can be re-written as
(
(
)
)g
g
tG T
T= (51)
In order to solve this equation one must know the shear modulus at the glass transition temperature
recent improvements in topological constraint theory have led to the ability to predict properties
with varying temperature and composition Using a recent topological constraint model for elastic
moduli proposed by Wilkinson et al [139] one can predict the temperature dependence of the
modulus
( ) ( ) ( ) ( ) ( )( ) ( )Δ Δ Δ Ac c C Ac
c c
F n q T F n q T F n q T x Nn q T x NdG dGG
d F M d F M
+ + = =
(52)
In Eq (5) nc is the number of topological constraints (c) qc is the constraint onset function as
described by Mauro et al [69] Fc is the free energy of the constraint is the density of the
system M is the molar mass and 119889119866
119889120549119865119888 is a scaling factor This expression was evaluated as a
function of temperature with the total modulus coming from Kodama et al [135] and is equal to 8
GPa at room temperature The B2O3 structure consists entirely of three-coordinated boron [140]
[141]
Using the modulus predicted in Eq (52) numerical observation time is expressed
56
12
9
10 Pa s141 s
709 10 Pat = =
(53)
Using this description of the static (vc=0 where v is the speed of the glass and c is the
speed of light) glass transition it is possible now to describe the relativistic glass transition for
which there are considered two separate cases It is worth noting that this is not the first case study
of relativistic viscosity a fairly large basis of literature dealing with ideal fluids near large
gravitational bodies [142] There is also some work also relating the relaxation time in a
gravitational field to the nonequilibrium entropy relaxation time [143] [144] This work
emphasizes the theoretical effects of special relativity on real liquids their glass transitions and
the properties there dependent on
41A Relativistic Liquid
The first case is when a sample of a glass-forming system moves past a stationary observer
at a speed approaching light giving the new observation time as
0tt
= (54)
where t0 = 141 s (as previously calculated) and g is the Lorentz factor
2
2
1
1v
c
=
minus
(55)
The Maxwell relation can be modified to account for relativistic effects by combining with Eq (54)
0
()
)(
g
gG TT
t
= (56)
57
This gives a condition that is temperature dependent to predict the new glass transition as a function
of v however in order to accurately describe the relativistic behavior density must be considered
Density will change by a factor of 2 because both the mass and the volume will be affected and
as such
0) (( )g gG TT t = (57)
The prediction then for the glass transition as a function of the speed of light is shown in Figure
10
Figure 10 The relativistic glass transition temperature for B2O3 glass
58
Using the Angell plot one can calculate the viscosity using the adjusted glass transition temperature
which is used to understand the dynamics of the equilibrium liquid with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) expression [55] [145] The MYEGA equation needs 2 other parameters
besides Tg to generate the viscosity curve fragility (m) which is the slope at Tg in the Angell plot
and the infinite temperature viscosity () The fragility will not change because one assumption
in this model is the consistency of the Angell plot at all relativistic velocities even relativistic The
infinite temperature viscosity which also will not change because this is the minimum possible
viscosity for a liquid The new viscosity curves for various speeds are shown in Figure 11
59
Figure 11 The equilibrium viscosity curves for a borate glass travelling at different fractions of
light speed All of the viscosities approaching the universal temperature limit for viscosity
60
By using the Deborah Number it has been shown that at relativistic speeds will change the viscosity
in order to maintain the Deborah number and in this case it is shown that when the Lord would
move past the mountains at relativistic speeds the mountains would not actually flow but instead
hold still (even more so than they currently do) From this it is inferred that any liquid moving past
the earth at relativistic speeds will appear to be more solid and eventually appear as a glass
41B Relativistic Observer
In this case we consider an observer moving past a sample of glass at relativistic speeds
giving
0t t = (58)
which when the relativistic shear modulus 2 factor is included can be written as
( )
( )0
g
g
TG T
t
= (59)
Solving in the same manner as before it is shown which shear modulus is needed to satisfy the
condition in Figure 12
61
Figure 12 The modulus needed to satisfy the condition for the glass transition
62
In Figure 12 the shear modulus at the glass transition temperature is shown however the
constraint theory mechanism being implemented to calculate the shear modulus has a built in upper
limit when calculating the Tg past the point in which all the constraints are intact Due to this
limitation there is an upper limit to our prediction Nonetheless the calculated Tg over the available
range are shown in Figure 13 as well as the related viscosities It is seen clearly that at higher
velocities the glass transition shifts towards zero more dramatically at the higher temperatures
This leads to the interesting result that if we take the limit of an observer moving past the earth at
close to the speed of light all glass would appear to be a liquid
63
Figure 13 (Top) The predicted glass transition with the anomalous behavior occurring at v=044c
(Bottom) Viscosity plot showing a dramatic reduction of the viscosity as the observer approaches
the speed of light
64
Applying the mechanisms proposed by Einstein to the concept of the Deborah number it
has been shown that the viscosities will appear to change dramatically as a glass-forming system
approaches relativistic speeds This allows us to build an intuitive understanding of the differences
between glass and liquids Specifically any material can appear either more or less fluid based on
the velocity with the velocity with which they travel relative to the observer This serves to highlight
the importance of the role of the observer on the glass transition and why even small changes in
DSC scan rates or sampling frequency can lead to drastically different measured glass transitions
42 Temperature and Compositional Dependence of the Stretching Exponent
The mathematical form for glass relaxation ( ) was originally proposed by Kohlrausch in
1854 based on the decay of charge in a Leyden jar[9] [14] [29] [42] [45] [96] [146]
( ) ( )( )exp t t
= minus (60)
where t is time t is the relaxation time of the system and b is the dimensionless stretching exponent
Eq(1) was originally proposed empirically and is known as the stretched exponential relaxation
(SER) function[42] [96] The relaxation time t depends on the composition temperature thermal
history pressure and pressure history of the glass as well as the property being measured[62]
[147] For example the stress relaxation time of a glass can be written as[62]
( )( )
ff
s
f f
T T P P
G T T P P
= (61)
where h is the shear viscosity and G is the shear modulus of the glass-forming system[42]
(Variables are defined in Table 2) In addition to using Eq(61) to describe stress relaxation time
structural relaxation time has recently been hypothesized follow Eq(61) with bulk viscosity and
65
the difference of bulk modulus between infinite and 0 frequency replacing shear viscosity and shear
modulus[62] Both h and G vary with the temperature (T) and thermal history of the glass (as
quantified via the fictive temperature Tf) as well as the pressure (P) and its corresponding history
(fictive pressure Pf)[16]
To connect the stretching exponent to its physical origins we can begin with a result from
Richert and Richert[148] that relates to an underlying structural relaxation time distribution and
then develop equations that determine from known quantities Their expression
2 22
ln 2
1
6
minus=
(62)
relates to the variance of the logarithm ( )2
ln of the structural relaxation time For reference
all variables are defined in Table 2 The physical origins of the relaxation time are related to the
configurational entropy (Sc) as shown by the Adam-Gibbs equation[53] [55] [149]
ln lnc
B
TS = + (63)
where is the infinite temperature relaxation time T is the absolute temperature and B represents
the energy barrier for relaxation
Table 2 Variable definitions
Variable Definition
t ts Structural relaxation time stress relaxation time
b Kohlrausch exponent (ie the stretching exponent)
T Tf Tfi Absolute temperature fictive temp partial fictive temp
P Pf Pressure fictive pressure
66
kB Boltzmannrsquos constant
wi ki i=1 to N Prony series parameters
B Adam-Gibbs relaxation barrier
Sc Adam-Gibbs configurational entropy
B Distribution of activation barriers
2 Variance
Na Number of atoms
f Topological degrees of freedom per atom
Degenerate configurations per floppy mode
H Enthalpy barrier for Relaxation
d Number of dimensions
f Fraction of activated relaxation pathways
S Adam-Gibbs entropy in the infinite-temperature limit
m Kinetic fragility index
m0 The limit of a strong liquid
Grouped unknowns 6
ln
B
a BN k
A Proportionality between fragility and distribution of barriers
x A given composition
125640
S
A
Intercept of the linear model of
67
42A Deriving a Model
Combining Eqs(62) and (63) Gupta and Mauro[150] proposed that the variance of the energy
barriers for relaxation (2
B ) could be rewritten as
2 2
22
2
1
6( )B c TTS
minus=
(64)
Solving for the stretching exponent
( )22 (
6
( )
)
c
B c
S T
T
T
TS
= +
(65)
The work of Naumis et al[23] [56] has shown that the configurational entropy is proportional to
the topological degrees of freedom in the network a result that was used in the derivation of
temperature-dependent constraint theory[57] and the MYEGA (Mauro-Yue-Ellison-Gupta-Allan)
equation for the relaxation of supercooled liquids[55] The MYEGA equation was derived by
expressing the configurational entropy as
( ) ( ) lna BcS T x f T x N k= (66)
In Eq (66) f is the topological degrees of freedom per atom Na is the number of atoms kB is
Boltzmannrsquos constant x is a given composition and is the number of degenerate configurations
per floppy mode The temperature dependence of the topological degrees of freedom was
approximated using a simple two-state model
( )( ) 3exp
B
H xf x T df
k T
minus= =
(67)
Here H(x) is the enthalpy barrier for relaxation d is the dimensionality of the system and f is the
68
fraction of activated relaxation pathways Combining Eqs (65) and (67) as well as condensing the
unknowns into the term defined by
6
ln
B
a BN k
=
(68)
we get
( )
22
df
dfT
=
+
(69)
This can be compared to the prediction made by Phillips[48] for the stretching exponent at
temperatures below the glass transition
2
df
df =
+ (70)
Comparing the two expressions of Eq(69) and Eq(70) they would agree if
4 4dfT
= + (71)
We will show later (Figure 12) an example where extrapolating our model prediction for to room
temperature (with T in Eq(71) replaced by fictive temperature Tf) gives a result close to the Phillips
value of Eq(70) When the temperature T in Eq(69) is high (much larger than ) then Eq(69) gives
us rarr1 Thus the model can interpolate between the low-temperature (Phillips) value and the
high-temperature value of 1 In our current model we do not have access to a value for in Eq(68)
so this model cannot assess whether the Phillips room temperature value for is universal ie
whether Eq(71) is always satisfied at room temperature
Zheng et al[151] showed that the Mauro-Allan-Potuzak (MAP) model for the relaxation
time of the nonequilibrium glassy state[44] implies that the configurational entropy can be written
as a function of thermal history (Tf) the fragility index (m) the fragility limit of a strong liquid
69
(m0) the glass transition temperature (Tg) and the limit of infinite temperature configurational
entropy ( S ) as
0
) exp 1 0
(ln1
c f
f
gT
TS m
TS
m
= minus minus
(72)
The qualitative relationship between the distribution of activation barriers and fragility was
proposed by Stillinger[18] [19] who suggested that the energy landscape of a strong liquid (low
fragility) has a small distribution of activation barriers and that a higher fragility is associated with
a broader distribution of activation barriers ie a higher variance of the activation barriers This
leads to a phenomenological relationship that is here assumed to be valid based on Stillingerrsquos work
on energy landscapes[18] [19]
2
0
1B
mA
m
= minus
(73)
where A is some constant of proportionality This expression was chosen because in the limit of a
strong liquid there is an infinitely sharp distribution of activation barriers Combining Eqs(65)
(72) and (73) we obtain the temperature dependence of the stretching exponent for a liquid
( )
0
2 2 2 2
0 0
exp 1
( ) 2
6 ln10 1 exp 1
g
f
f
f
g
f
f
T mT S
T mT
Tm mA T S
m T m
minus minus
=
minus + minus minus
(74)
Fictive temperature appears in this new expression because we are deliberately expressing this
function as an equilibrium model Rewriting Eq (74) in terms of one unknown () and grouping
constants
70
( ) 0
2 2
0 0
exp 1
2exp 11
g
f
f
f
g
f
f
T m
T m
T m
T
T
Tm
T
m
m
minus minus
minus minus
=
minus +
(75)
Eq (75) is an expression for the stretching exponent as a function of thermal history glass
transition and fragility index with only one unknown In Eq (75)
( ) 12
6 ln10
S
A
= (76)
The only unknown for the compositional dependence of the stretching exponent is the value of D
The composition-dependent part of Eq(76) is Syen
since we approximate A to be independent of
composition The fragility dependence of Syen
was proposed by Guo et al [61]
Syen(x) = S
yen(xref
)expm(x) -m(x
ref)
m0
eacute
eumlecircecirc
ugrave
ucircuacuteuacute
(77)
where x is composition and xref is a reference composition in the same glass family Seeking the
simplest possible expression to approximate the unknown D we take the natural logarithm of
Eq(76) and of Eq(77) and combine them to get
0
ln lnm
m = + (78)
with the additional definition
( )( )
( )12
0
ln l6 ln10
nref refS x m x
mA
minus
=
(79)
The result is that ln D varies linearly with fragility m and the intercept ln D is independent of
composition (depends on one reference composition)
In Figure 9 we plot the data of Boumlhmer et al[152] vs the predicted exponent The work of
71
Boumlhmer et al is the summary of the literature data relating fragility index to the stretching exponent
at the glass transition which we use to fit values to Eq (78) The dataset included in their work
covers chalcogenide oxide and organic glasses The fitting (consisting of least-squares
minimization of the difference between the predicted stretching exponent and that which was
reported) was done twice once for organic and once for inorganic systems During the fit it was
assumed that Tg=Tf Some assumption about thermal history was necessary since the individual
thermal histories or Tf values are not known for this whole collection since we are trying to track
overall trends in values vs composition this reasonable simplifying assumption is consistent
with our program Using Eqs(75) and (78) the temperature and compositional dependence of the
stretching exponent can be described with only one free parameter
72
Figure 14 b predicted and from literature showing good agreement with a total root-mean-square
error of 01 The fit for organic systems is given by 66 ln Kln = minus and for inorganic systems by
75ln Kln = minus
73
42B Experimental Validation
Experimental density measurements were made using Corningcopy JadeTM glass as described
elsewhere[146] Changes in the density are normalized to obtain the relaxation functions which
are fit with Eq (60) The resulting values of are shown in Table 3 A comparison in Figure 15
is shown with data obtained on a soda-lime silicate SG80 where the stretching exponent was
obtained using the measurement of the released enthalpy as a function of isothermal annealing time
during relaxation below the glass transition The measurement of the released enthalpy[153] relies
on the change in the excess heat capacity in the glass transition range as a function of annealing
time Normalized released enthalpy relaxation functions for each temperature were fit with Eq(60)
to obtain the stretched exponent values for SG80 In Figure 15 parts A and B are samples measured
below Tg while in part C the samples measured are at Tg and above The values were then fit with
Eq(75) where the only free parameter was D with an equilibrium assumption (Tf = T)
74
Table 3 Measured temperatures and their corresponding relaxation values for Corningcopy JadeTM
glass and Sylvania Incorporatedrsquos SG80
Glass Temperature (K) Tg (K) TgT m D (K-1) Beta t (s)
Corningcopy JadeTM 1031 1074 1042 32 0002 0504 7249
Corningcopy JadeTM 1008 1074 1065 32 0002 0447 32458
Corningcopy JadeTM 973 1074 1103 32 0002 0395 419214
SG80 800 800 1 36 00057 0632 92
SG80 783 800 1021 36 00057 0593 182
75
Figure 15 The equilibrium model proposed with the experimental points showing good agreement
between the experimentally measured data points and the equilibrium derived model RMSE was
002 for Corningcopy JadeTM (A) and less than 001 for SG80 (B) (C) The model fit for B2O3
experimental data[154] [155] The fragility and glass transition temperature of the B2O3 are taken
from the work of Mauro et al[17]
(
A)
(
B)
(
C)
76
A combination of the MAP model for non-equilibrium shear viscosity[44] the model
presented here primarily in Eq(75) and the model presented by Wilkinson et al[139] for the
temperature dependence of elastic modulus allows for fully quantitative modeling of stress
relaxation behavior The missing model required to understand structural relaxation is the bulk
viscosity curve[62] All previous relaxation (structural or stress) models[42] [61] have relied on
approximations that use a constant exponent and on a constant (temperature-independent)
modulus value whereas here every parameter of Eq(60) may be modeled as a function of
temperature Furthermore in combination with the relaxation models described by Guo et al[61]
in which multiple fictive temperatures are described using a Prony series and a temperature-
dependent modulus one can construct a relaxation curve accounting for the temperature
dependence and thermal history dependence of all relevant parameters
( )
( )( )
1
( )( )exp exp
( )) (
fTN
i
i
f
f
f fi
G T k T tG T tw
T TT
T T
=
minus minus
(80)
Here wi and ki are fitting parameters that are completely determined by the value of and each
term in the Prony series is denoted with the subscript i Usually 8 or 12 terms are included in the
Prony series[96] Each term in the Prony series is assigned a partial fictive temperature (Tfi) whose
relaxation is described by the simple exponential in the Prony series Equilibrium conditions are
assumed at the start of the simulation which allows for a known set of starting probabilities within
the energy landscape interpretation of relaxation The same model also includes the fragility index
and glass transition dependence of the non-equilibrium shear viscosity This method is
implemented in RelaxPy[42] as discussed in the next section This serves as an approximation for
the evolution of the of the non-equilibrium state however the temperature dependence of the bulk
viscosity and a replacement for fictive temperature need to be quantified to improve the
understanding of the underlying physics[1] [16] [62]
77
Separately we can explore the relationship between this model given by Eq(75) and the underlying
energy landscape In order to better understand the stretching exponent consider that there exists
a full set of parallel relaxation modes within an energy landscape The relaxation modes are then
weighted by the occupational probability corresponding to a particular mode This gives a series of
transition rates with some probability prefactor and an associated relaxation time (scaled from the
mean relaxation time) which gives rise to a Prony series form of Eq(80) Thus the distribution of
relaxation times (or barriers) determines the evolution of the stretching exponent while the average
barrier determines the mean relaxation time If this Prony series description in turn describes the
stretching exponent we arrive at a physical description and origin of the stretching exponent As
the temperature approaches infinity even though there is a distribution of activation barriers the
distribution of relaxation times approaches a Dirac delta function and the stretching exponent
approaches one (a simple exponential decay) As the temperature decreases the distribution of
relaxation times broadens due to the wide distribution of barriers however due to the broken
ergodic nature of glass the value of the stretching exponent will be controlled by the instantaneous
occupational probability
The activation barriers may be modeled eg using a numerically random set of Gaussian
distribution of barriers[156] The probability of selecting an individual transition is then calculated
using a Boltzmann weighting function the transition also has an associated relaxation time with it
The Prony series is then recreated with the probability multiplied with the simple exponential
relaxation form The sum of all of these terms is then fit with the stretched exponential form Figure
11 shows the stretching exponent calculated for the mean activation energy of 10 eV and three
choices of standard deviation 25 2 15 ande e e minus minus minus= eV as shown in the legend where Eq(60) has
been fit to the Prony series calculated from an equilibrium Boltzmann sampling of a Gaussian
distribution of barriers The temperature dependence for the stretching exponent matches the form
78
derived earlier in this work This stretching exponent shape has been shown previously though not
in a closed-form solution[156] This secondary method not only validates the generalized form of
Eq(75) but also highlights the validity of using the configurational entropy and barrier distributions
as the underlying metric controlling the temperature dependence of the stretching exponent
Figure 16 The stretching exponent calculated as described in the text for a Gaussian distribution
of barriers This plot shows that the distribution of barriers has a large effect on the stretching
exponent A Tg cannot be described since there is no vibrational frequency included in the model
though the glass transition temperature should be the same for all distributions since the mean
relaxation time is the same for all distributions at all temperatures The deviation is given in ln eV
units
79
To create software to model this complex behavior the authorsrsquo existing software RelaxPy[42] has
been modified into a new version RelaxPy v20 Instead of fixed values for the Prony series the
values are chosen dynamically to match the stretching exponent as predicted by Eq(75) The
database of values wi() and ki() was created with a fixed number of terms N in Eq(80) (we chose
N=12) This database is available in the same Github repository as the software The parameters
were fit by starting with the values obtained for =37 reported by Mauro and Mauro[96] to make
the new version of the model smoothly reproduce a prior optimal fit then was stepped by 001
(when less than 095) or by 0001 (when greater than 095) Using the previously optimized values
of wi and ki for starting values for each new the wi and ki values were then varied to minimize
the root-mean-square error between the stretching exponent form and the Prony series form ie to
satisfy Eq(80) with least error This list of compiled values makes up the contents of a database
that the software accesses The values of wi and ki used for a given are calculated by finding the
closest value in the database to the given value of This allows for an efficient implementation
of relaxation modeling Figure 12 shows the fitted parameters to the Prony series and an example
RelaxPy output for Corningcopy JadeTM glass undergoing quenching at a rate of 10 Kmin Note that
the room temperature stretching exponent value is 046 a close match the Phillipsrsquos ldquomagicrdquo value
of 37 (043)[157] This prediction for Corningcopy JadeTM is a non-equilibrium prediction in contrast
to the equilibrium high-temperature prediction shown in
80
Figure 17 (Top) The Prony series parameters as a function of the stretching exponent Each color
designates one term in the series (Bottom) The output from RelaxPy v20 showing the stretching
exponent effects on the relaxation prediction of Corningcopy JadeTM glass Each quadrant shows one
property that is of interest for relaxation experiments In particular it is interesting to see the
dynamics of the stretching exponent during a typical quench
81
43 Conclusion
In this section we have sought to expand the current understanding of relaxation and improve the
models that currently exists We have proposed a model that accounts for the effects of special
relativity as well as a model was derived through an understanding of the distribution of barriers
for relaxation and the temperature dependence of the Adam-Gibbs entropy The model outlined
herein describes the temperature dependence of the stretching exponent in glass relaxation
The work done on the relativistic effects on relaxation illustrates the crucial role of the observer on
all relaxation models and offers an extreme in which to test common relaxation models The model
for the stretching exponent not only considers the extremes at high and low temperature (when
compared to the glass transition) as in the Phillips model but also for any intermediate temperature
as a function of the fictive temperature This model does not have any explicit temperature
dependence since it was assumed that an equilibrium model works well to describe the
instantaneous distribution of relaxation times Given the physical argument and the success of this
model when tested by experiments and by another model it is at least reasonable to formulate the
temperature-dependence of in terms of its fictive-temperature-dependence as we have done here
Including both T and Tf is possible to consider but lies outside our current scope Using previously
derived compositional dependence for the MAP model a fragility index dependence of the
stretching exponent was defined and tested The model was confirmed using multiple experimental
datasets In addition a theoretical comparison to a distribution of landscape activation barriers was
found to reproduce the same trends as the model
Chapter 5
Glass Kinetics Without Fictive Temperature
Fictive temperature as discussed through this text is unable to capture key physical
phenomena[16] [42] [51] To circumvent this limitation we rely on two different frameworks
The Adam-Gibbs relationship and generalized features of the energy landscape These two features
allow us to create a method that is completely generalizable and does not involve fictive
temperature This method creates fake landscapes that generally capture the trends of glass based
on experimental measurements This new method we have called lsquoToy Landscapesrsquo
51 Background of the Adam Gibbs Relationship
Experimental evidence in support of the AG model comes from the work of Richet[158]
in which they compared the predicted configurational entropy from viscosity curves with that
obtained from DSC The configurational entropy was calculated from DSC data using
0
0( )p p vi
T
T
b
c c
C CS S T dT
T
minus= + (81)
In Eq (81) Cp is the isobaric heat capacity Cpvib is the vibrational heat capacity and T0 is the initial
temperature from which the configurational entropy is integrated The difference between the total
heat capacity and the vibrational heat capacity is the heat capacity due to structural rearrangements
Richet found good agreement between the measured viscosity curve and calorimetric
configurational entropy at temperatures above the glass transition confirming validity of the AG
relationship When applying Eq equation reference goes here all temperatures must remain above
the glass transition ie in the liquid or supercooled liquid state The calculation is invalid upon
83
cooling through the glass transition due to the breakdown of ergodicity[12] [13] [159] and the
irreversibility of the glass transition process Application of Eq (6) to temperatures below the glass
transition leads to an incorrect calculation of excess entropy in the glassy state[160]
In this study we expand on these previous works to understand the relationship between
viscosity and the underlying energy landscape and provide new insights into the thermodynamics
of liquids through a simplified ldquotoy landscaperdquo model constructed from experimentally measured
viscosity parameters The energy landscape framework is especially helpful for elucidating the
thermodynamics and kinetics of supercooled liquid and glassy systems Energy landscapes describe
the evolution of all atomic transitions for kinetic processes in a system To perform such
calculations information about the inherent structures and transition points in a landscape must be
known In this work we solve the inverse problem of deducing realistic landscape parameters using
the AG relationship and experimentally measured data This information is used to construct a ldquotoy
landscaperdquo model to describe glass relaxation processes To achieve an accurate relaxation model
the following steps are taken to ensure the validity of the results
A Confirm the AG model for shear viscosity using energyenthalpy landscape
calculations
B Validate the assumptions made by the MYEGA model
C Confirm the AG model for bulk viscosity
D Explore the fundamental relations between viscous properties and topography of the
landscape
E Use the knowledge gained to propose a new thermodynamic model to calculate the
driving force for glass relaxation and the scaling of the liquidrsquos free energy
This chapter is organized into sections devoted to each of the topics above preceded by a
methods section All of the sections are presented to validate a new model of glass relaxation that
is sufficiently complex to achieve a calculation of the thermodynamics and kinetics of the liquid
This new model will be based on understanding the landscape from experimental properties and
we have called this approach the ldquotoy landscaperdquo model
84
52 Methods
Potential energy landscapes (fixed volume) and enthalpy landscapes (fixed pressure) are
powerful modeling techniques where atomic configurations are mapped in a 3N-dimensional phase
space (or 3N+1 dimensions for enthalpy landscapes)[89] The energy landscape approach is based
on mapping the continuous 3N dimensional space to a discrete set of energy minima (called
ldquoinherent structuresrdquo) and first-order saddle points (ldquotransition pointsrdquo) To perform this mapping
a systematic search for inherent structures is performed while also mapping the lowest-energy
transition points between each pairwise combination of adjoining minima The landscape itself is
partitioned into basins which represent the set of all configurations that minimize to a common
inherent structure[161] This combination of the basin and transition point information results in a
topographic map of the 3N phase space (either a partial or whole mapping) that can be used to gain
insights into the physics of any atomistic system such as proteins[91] liquids[18] [19] glasses[1]
[93] and nucleating crystals At equilibrium the probability distribution for occupying the various
basins in the landscape is given by[13]
exp
exp
i
j
i
i
j
j
Eg
kTp
Eg
kT
minus
= minus
(82)
where Ei is the potential energy (or enthalpy) of the inherent structure gi is the degeneracy of the
basin i is the basin index and pi is the probability of occupying a given basin i The denominator
is a normalization factor given by the summation over all basins Mauro et al[44] previously used
this formalism to study the non-equilibrium viscosity calculate the evolution of configurational
heat capacity and elucidate the long-time relaxation kinetics of selenium glass[44] [93] [160] For
equilibrium systems the configurational entropy is given by the Gibbs formula
85
lnc i
i
iS k p p= minus (83)
In order to validate the AG model the landscapes of several common systems were
explored using ExplorerPy[89] ExplorerPy is a software program built specifically to map
landscapes using the LAMMPS molecular dynamics package[30] The three systems we explored
are B2O3[162] (150 atoms and 2000 basins) SiO2[162] (150 atoms and 1000 basins) and BaO-
2SiO2[121] (128 atoms and 1500 basins) All systems used periodic boundary conditions The
initial structures for B2O3 and SiO2 were constructed by placing atoms corresponding to the
chemistries of the systems on a random grid (with 1 Aring spacing) corresponding to a density of 20
gcm3 and equilibrated at 2500 K then quenching at a rate of 1 Kps to room temperature using the
Wang et al[162] potentials in the NPT (constant pressure number of atoms and temperature)
ensemble The energy landscape of the BaO-2SiO2 system was taken from our previous work[72]
The SiO2 (constant pressure exploration) system was explored using ET with a push distance of 03
Aring and a pressure of 10 atm B2O3 was explored using isobaric MD at a temperature of 1500 K for
500-time steps of 1 fs also at a pressure of 1 atm The minimization (after MD or ET) was handled
by the LAMMPS[30] steepest descent algorithm until the certainty in energy was smaller than 10-
4 eV Both used a nudged elastic band spring constant of 10 eVAring2 The energy distribution for all
systems was fully sampled which was confirmed by randomly checking if half the basin
distributions matched the full distribution It was also found that multiple structures in each
exploration were discovered more than once and a slightly smaller B2O3 system (130 atoms) was
also explored and showed identical results to the 150-atom distribution The insights gained from
the interaction between the experimentally accessible viscous properties and the landscapes
presented here will be generalized to create a simplified ldquotoy landscaperdquo model to capture the
thermodynamics and kinetics of the glass-forming system
86
53 Results
53A Adam-Gibbs Validation
To predict viscosity using the Adam-Gibbs expression there are only three required
parameters the configurational entropy (as a function of temperature) the barrier for cooperative
rearrangements and the infinite-temperature limit of viscosity Using a landscape the equilibrium
configurational entropy can be obtained using Eq (83) The infinite-temperature limit of shear
viscosity is given by the work of Zheng et al[54] where a systematic study of MYEGA fits to
viscosity data revealed a common value of 10-293 Pamiddots This infinite temperature value of viscosity
is used for all the experimental and computational viscosity curves shown in this work The final
parameter is the mean barrier for a cooperative rearrangement In Figure 18 the barrier for
cooperative rearrangements was chosen to accurately produce a viscosity of 1012 Pamiddots at the
experimentally measured value of the glass transition temperature From these parameter values
the predicted Adam-Gibbs viscosity curve is plotted The universal infinite limit fragility and glass
transition values from references were inserted into the MYEGA equation to plot the
lsquoExperimentalrsquo curves for B2O3 SiO2 and BaOmiddot2SiO2 respectively[58] [69] [120] When
examining Figure 18 it is clear that the AG model is able to fundamentally recapture the viscosity
87
Figure 18 The viscosity (left) and landscape (right) predictions for three common systems The
first system is newly calculated in this work while the latter two come from our previous works[72]
[89] It is seen that the viscosity predicted from the AG model is very accurately able to reproduce
the experimental viscosity curves from the MYEGA model The last system is a potential energy
landscape while the others are enthalpy landscapes
53B MYEGA Validation
Now that we have confirmed that the configurational entropy of the landscape works well
to describe the viscosity via the AG relationship a definitive temperature-dependent form is needed
in order to make more general predictions The key approximation made in deriving the MYEGA
model is that the configurational entropy follows an Arrhenius form By testing this Arrhenius
88
form we can evaluate this assumption in the derivation of the MYEGA model while simultaneously
gaining insights into the thermodynamics of the liquid and supercooled liquid states A strong liquid
( 15m ) has a configurational entropy that is nearly constant as a function of temperature while a
fragile liquid ( 15m ) is a liquid in which the configurational entropy is highly temperature
dependent Since we are testing the MYEGA equation it is also worthwhile to consider alternate
viscosity expressions Assuming validity of the Adam-Gibbs equation the configurational entropy
deduced from the viscosity experiments is given by
( )10 01log l) og(
c
BS
T T minus= (84)
where the viscosity function is given by a viscosity model such as Vogel-Fulcher-Tammann[59]
(VFT) equation the Avramov-Milchev[60] [163] (AM) equation or the MYEGA expression
The resulting entropy predicted by each viscosity model (using experimental data for
fragility and the glass transition) is compared to the calculated configurational entropy from the
landscape for B2O3 in ( 518gT = K and 33m = [69] were used for the experimental values) The
figure clearly shows that the MYEGA equation which assumes that configurational entropy scales
in an Arrhenius fashion reproduces the calculated configurational entropy from the landscape the
most accurately out of the three considered viscosity models This also means that the
configurational entropy form of the AM and VFT models are not physical and cannot be used to
recreate physical results Since the temperature dependence of the entropy has a fixed mathematical
form and is dominated by the distribution of basins we now understand that the distribution of
basins must also have a fixed form and that form must control experimentally accessible properties
This is the key insight that will allow for the development of the toy landscape model
89
53C Adam-Gibbs and Structural Relaxation
Although the Adam-Gibbs model has been validated for shear viscosity and has provided
the functional basis of the MYEGA model let us now consider whether the temperature
dependence of the configurational entropy can also predict the bulk viscosity (also known as the
volume viscosity) Recent work has highlighted the importance of bulk viscosity viz that the
structural relaxation time is proportional to the bulk viscosity[62] Hence the temperature
dependence of the bulk viscosity provides key information needed to predict structural relaxation
times[62] Following Scherer[164] the relationship between the structural relaxation time ( structural
) and the bulk viscosity ( B ) is given by
( )0B structuralK K = minus (85)
Figure 19 The configurational entropy comparisons between the three major viscosity models
which validates the main underlying assumption of the MYEGA model The VFT and AM are
unable to capture the physics of configurational entropy therefore ruling
90
where K is the infinite frequency bulk modulus and 0K is the zero-frequency bulk modulus In
order to calculate bulk viscosity for B2O3 the zero-frequency and infinite-frequency modulus is
taken from these works respectively [165] [166] The experimental structural relaxation times were
taken from the work of Sidebottom et al[154] In Figure 20 the bulk viscosity is shown as well as
the Adam-Gibbs comparison
Figure 20 Estimated bulk viscosity of B2O3 The fit to the bulk viscosity used the configurational
entropy from the enthalpy landscape with the barrier 00155 eV (compared to the shear barrier of
00149 eV) and the infinite limit allowed to vary (10-263 Pamiddots for bulk viscosity) In Figure 18 the
configurational entropy is confirmed for the shear viscosity thus confirming the AG for both shear
and bulk viscosities Sidebottom data are from Ref [154]
In Figure 20 the fitted values of B and 10log can be used to gain insights into the
difference between structural and stress relaxations since the configurational entropy is universal
between the two The value of Bk (the barrier for cooperative re-arrangements in natural units) for
shear viscosity was 173 K while for bulk viscosity it was 180 K This difference of 7 K in the
barrier leads to an order of magnitude difference in the fitted viscosity curves This is a small
91
difference and explains the variance in the values at Tg The other difference is in the infinite
temperature value of viscosity which is -293 log(Pamiddots) for shear viscosity (from Zheng et al) and
-263 log(Pamiddots) for bulk viscosity This implies that the MYEGA raw form is accurate to predict
the structural relaxation time as well as the stress relaxation (though the values of the pre-
exponential factor and the composite constant will change between the two forms) However the
form in terms of fragility and the glass transition will need to be modified to accurately describe
the structural relaxation time The form assumes the viscosity of interest has a value of 1012 Pamiddots at
the glass transition temperature for shear viscosity which does not have to be the value for the bulk
viscosity curves at the glass transition Further research is needed to understand the appropriate
bulk viscosity at the glass transition and the infinite temperature of bulk viscosity
53D Landscape Features
As shown here the configurational entropy is governed by a Boltzmann sampling of a
probabilistic distribution of states n(E) This distribution of the inherent structures is empirically
found to follow a log-normal distribution as shown in Figure 21 While we are not claiming a
deeper meaning to the underlying origin of the log-normal distribution we may adopt this form of
the probability density function since it accurately describes the underlying distribution of
microstates This is the key insight needed to develop a new model because this means that the
fragility and glass transition are fundamentally related to the parameters of the distribution of basins
on the landscape (number of basins and standard distribution of the enthalpy)
The configurational entropy in the MYEGA model is given by
10
exp 1ln10 12 log
g
c
TS mS
T
= minus minus
minus
(86)
92
where S is the infinite temperature limit of the configurational entropy In Figure 21 we also
show the computed value of Eq (83) as a function of temperature compared to that of Eq (86)
which shows excellent agreement between the configurational entropy and the MYEGA model
This test also supports Stillingerrsquos[18] [19] view of fragility being correlated to the lsquoroughnessrsquo of
the landscape topography According to Stillingerrsquos view if the distribution of inherent structure
energies is narrow then the configurational entropy cannot have a large temperature dependence
leading to a low fragility Alternatively if the distribution is broad then there must be a temperature
dependence to the configurational entropy which necessarily results in a higher fragility
93
Figure 21 (A) The histogram for the energy minima for each of the test systems showing a good fit
with the log normal distribution This distribution will then be a valid form to calculate the enthalpy
distribution of the model presented in the next session (B) The configurational entropy from the
model showing the accuracy of the scaling of the entropy predicted by the MYEGA model The
S value was fit for each system
(A)
(
B)
(B)
94
54 Topography-Property Relations
Now that we have confirmed the validity of the Adam-Gibbs and MYEGA equations from
the energy landscape approach we can draw some key conclusions regarding the thermodynamic
properties of liquids For example heat capacity has been related to fragility[112] and is of
importance for glass manufacturing since it is related to the cost of heating and forming a glass
Isobaric heat capacity is defined as
p
P
HC
T
=
(87)
where H is enthalpy T is temperature and P is pressure The mean configurational enthalpy
calculated from an enthalpy landscape is given by
conf i conf iH p H= (88)
The heat capacity has contributions from both vibrational and configurational modes in liquids
while in glasses the configurational mode is mostly lost For a reversible process the
configurational heat capacity can be written as a function of the configurational entropy
c
p conf
P
SC T
T
=
(89)
By combining Eq (86) and Eq (89) we can write the expression for configurational heat capacity
of a liquid as
10 10
1 exp 1 exp12 log 12 log
g g
p conf
T Tm m S HC S H
T T T T
= minus minus minus = minus
minus minus
(90)
From fluctuation theory the heat capacity can be expressed in terms of the variance of the enthalpy
(2
H )
95
2
2
1p conf HC
kT= (91)
It is important to note that the variance of the enthalpy is not equivalent to the variance of the log-
normal distribution The variance of the enthalpy includes a Boltzmann distribution over the log-
normal distributed basins This gives a direct relationship between the enthalpy fluctuations and
the viscosity parameters If we further adopt the log-normal distribution as the effective distribution
for all liquid basins of all enthalpy landscapes then the distribution is only a function of the
common viscosity parameters ( S m gT and 10log ) The viscosity parameters will only
change the variance of the distribution since the mean of the distribution is inconsequential (the
zero-point enthalpy can be adjusted to an arbitrary value without changing the calculation of the
configurational entropy) The parameters of the log-normal distribution can then be fitted to
reproduce the configurational entropyrsquos temperature dependence thus creating a lsquotoyrsquo enthalpy
landscape for us to play with for the purpose of thermodynamic and kinetic calculations Generating
the landscape is a numerical calculation as the number of basins and standard deviation must be
chosen such to reproduce key phenomena
To quantify the dependence of these parameters on the degeneracy and temperature it is
helpful to develop a descriptive formal model The cumulative probability distribution of a normal
distribution (with zero mean) is given by
10log )1
1 er(
f2 2
Hp
+
=
(92)
where is the standard deviation of the log-normal distribution erf is the error function H is the
enthalpy and p is the probability To then calculate the equal probability spacing for a log-normal
distribution we solve for H for the set of probabilities p which range from 2 to 1 2minus
96
with a step-size of This then gives enthalpies of basins equally spaced by the cumulative
probability distribution and as a result each basin has equal degeneracy
1 2 r2 1e f10
pH
minus minus = (93)
This set of basins carries with it an Arrhenius activation barrier for the configurational entropy that
is only a function of and the degeneracy of each basin (if the is chosen such that the energy
distributions converge) This creates a model that fully describes the thermodynamics of the liquid
and though it currently lacks any kinetic component it offers the possibility to be a useful tool
when used in conjunction with Eqs (82) (83) and (88) This new tool is denoted the ldquotoy landscape
modelrdquo (TLM) It is called the toy landscape model because we are generating a landscape that is
not the real landscape but is enough to reproduce key experimental properties without needing
expensive computational calculations It is a lsquotoyrsquo for us to play with without needing to worry
about upfront computational cost
To confirm the validity of TLM and the way it is constructed we can calculate the
activation barrier from this deterministic method and for a random set of inherent structures with
the same number of basins and distributions For this purpose we used temperatures ranging from
1000 to 2000 K for both the deterministic and random models with a step size of 0001 for TLM
Excellent agreement between the random and deterministic methods is shown in Figure 22A This
analysis allows us to understand that the thermodynamics of the system are fundamentally linked
to the viscous flow behavior Though this technique was created to target viscosity and relaxation
it is generally applicable to all configurational thermodynamics of liquids and glasses It is worth
noting that all the thermodynamics will be relative to a 0 eV energy state defined at equilibrium at
0 K and as such the enthalpy differences at some fixed temperature must be known to compare
over compositional spaces
97
In Figure 22B we see the same method being used to understand the relationship between
the viscous parameters and the parameters of the landscape This is the key to the numerical
calculation of the lsquotoy landscapersquo model The toy landscape is a numerical calculation where we
use experimentally accessible values to generate a landscape whose thermodynamics should
reproduce key thermodynamic phenomena In Figure 22B we show that the fragility and glass
transition are systematically changed as the number of basins and standard deviation of the enthalpy
is changed When this idea is combined with the basin calculation in Eq (93) (giving equal
degeneracies for all basins) then the complete set of enthalpy basins is easily attainable giving
access to the fundamental thermodynamics of the system
98
(
A)
(B)
(A)
99
Figure 22 (A) Comparison between the randomized method (histogram) and the deterministic
method (vertical lines) showing good agreement between the maximum in the histogram and the
value predicted by the deterministic technique validating the approach It is worth noting that the
100-basin distribution is a very wide distribution where the total number of basins is less than the
number of points used in the calculation This is done for a variable number of basins with the
number of basins shown in the legend (B) The dependence of fragility and the glass transition
temperature vs the distribution of states and the number of basins
55 Barrier Free Description of Thermodynamics
Throughout this work we have systematically explored the Adam-Gibbs model of
viscosity An important insight that we have gained is that the configurational entropy is free of any
barriers This is of particular interest because the Arrhenius slope of the activation barrier is linked
to the glass transition and fragility The glass transition temperature is known to be related to the
set of activation barriers as it corresponds to a breakdown of ergodicity What is found here is that
the fragility is the ldquoconversion factorrdquo to account for the slope of the configurational entropy
(which as previously identified is related to the distribution of basins) This means that the barriers
themselves do not matter until the material is close to the breakdown of ergodicity and thus the
equilibrium liquid thermodynamics and kinetics can be described without explicit information
about the saddle points in the landscape
Understanding the role of the topography in the properties of glasses and liquids lead to
the most important insights derived from the AG model Since the configurational entropy is the
same for both stress and structural relaxation simple viscous parameters allow us to understand
fundamental thermodynamic quantities of the system without needing to map computationally
expensive energyenthalpy landscapes The process of converting experimentally accessible
properties to a usable landscape for thermodynamic calculations is what we are calling the toy
landscape model To show the utility of toy landscapes we can calculate the driving force for glass
relaxation This information can give insights into the physics of relaxation by dynamically
100
calculating the Gibbs free energy of the equilibrium liquid at any temperature which defines a
driving force for glass relaxation To do this calculation of the driving force for glass relaxation
we consider the ergodicity parameter ( x ) defined by the Mauro-Allan-Potuzak (MAP) model of
glass relaxation as the ratio between the glass entropy and the equilibrium liquid entropy
liquiss dglaS x S= (94)
To get information about the enthalpy we consider an instantaneous Boltzmann distribution on the
landscape given by an equilibrium distribution at some elevated temperature fT
1
exp
exp
iglass i i
fii
f
HH H g
kTHg
kT
= minus
minus
(95)
This formalism in conjunction with TLM gives us the ability to explore the temperature
and compositional dependence of the driving force for relaxation The enthalpy of the liquid is
calculated from an equilibrium distribution on the landscape The driving force ( ) is defined as
( ) ( )( ) ( )1iliquglas ds f l quii dH H TS xT T = minus + minus (96)
For our purposes this gives a nice approximation for the driving force improving on previous
approximations given by ( fT Tminus )[16] [42] TLM does not include a kinetic feature to calculate
the occupational probability in each basin however when coupled to a kinetic model such as the
MAP model[44] a complete description of the underlying thermodynamics of glass relaxation can
be considered For this calculation of the driving force the kinetics are calculated through the MAP
model but this is not necessarily a requirement In Figure 23 the compositional and temperature
dependence of the driving force is shown for three glasses with identical glass transitions and
varying distributions of basins (as listed in Table 1) The calculations are performed using the
RelaxPy script and MAP model with the stretching exponent fixed at the Phillips value of 37
101
explanations of which are available elsewhere[42] The systems were quenched from 700 K to 300
K at a rate of 10 Kmin Figure 23 clearly shows how simply changing the total number of basins
creates a larger driving force for glass relaxation This simple change of the number of basins
drastically increases the entropic contribution and due to the loss of ergodicity explains the increase
of the driving force of relaxation The kinetics could be calculated through any relaxation model or
directly from the landscape if some barrier approximations are made
Table 4 A table of parameter values for the three example glasses used in Figure 23 The
distribution of underlying inherent structure energies and the glass transition temperature (500 K)
were kept the same while the total number of basins were allowed to vary
Sample Total Basins [-] Distribution of States [log eV] 1
1493g
mT
minus
[K]
Fragility [-]
1 100 02 837 40
2 500 02 537 31
3 1000 02 466 28
102
Figure 23 The driving forces for different example glasses calculated using a combination of the
MAP model RelaxPy and the toy landscape model The parameters for each glass can be found in
Table 4
57 Toy Landscapes for the Design of Glasses and Glass Ceramics
The concept of a toy landscape can go even further building a method to predict the
dynamics of a glass system based on simple inputs This is possible because in the previous section
we have related the distribution of basins on an energy landscape to the experimental viscosity
which allows for thermodynamic insights Building on this we can assume that the barriers (H )
for transitioning between states is equal to that of the barrier mean for the viscous relaxation
ln10gH mkT= (97)
This means that the only key factor needed to then predict the dynamics of a landscape using the
metabasin approach is the vibrational frequency However the relaxation time at the glass transition
103
is known to be a constant of 100s and as such we can choose a vibrational frequency that reproduces
this fixed point This gives an entire description of relaxation without any need for fictive
temperature An example of the results predicted through this method are shown in Figure 24 This
prediction only required knowledge of the glass transition temperature and the fragility and from
this a pure prediction of relaxation is made
104
105
Figure 24 The predicted outputs from the toy landscape method showing predictions of the enthalpy
and entropy under a standard quench for barium disilicate This prediction does not require fictive
temperature or any such assumptions about the evolution of the non-equilibrium behavior
This method is also not limited to predicting the relaxation of glasses but can also capture
the crystallization It is known from the earlier work on nucleation presented in Chapter 3 that a
simple two state model was able to capture the underlying physics of the system Thus if we know
the free energy of the crystal and barrier to crystallization we can find predict key crystallization
106
phenomena By fitting the heat flow peak magnitude and location this gives enough information to
infer the barrier and the enthalpy of the crystal The degeneracy (and thus the entropy) of the crystal
can then be calculated through knowledge of the liquidus temperature
ex(
p)cry SCL l
cry
lk
G TH
T
minus=
(98)
This then gives everything for the model presented in Chapter 3 from only 1 calorimetry
experiment the glass transition the fragility and the liquidus temperature To predict nucleation a
series of estimates were done running the toy landscape at different temperatures until the
temperature with the fastest crystallization rate was found This was then assumed to be the
maximum rate of nucleation The value of the interfacial energy was then chosen in such a way so
that the predicted curve reproduces the peak at the same temperature This gives all the parameters
needed for a prediction using CNT It is also possible to make a prediction of the growth rate of
crystals using this expression
2
1 exp6
GU
kT
a kT
minus minus
=
(99)
The results for these estimates are shown in Figure 25 The reason why growth is more accurate
compared to nucleation is that the kinetic factor of growth is governed by viscosity while nucleation
is governed by a kinetic factor that includes an assumed vibrational frequency
These methods have a plethora of assumptions built in such as that the vibrational
frequency of basins undergoing relaxation is equal to the vibrational frequency when undergoing
crystallization however as more information is gained about a particular system (such as more
crystallization peak as a function of temperature) the information could be built into the system to
improve crystallization predictions If the peak size is known as a function of scan ratetemperature
even the enthalpy as a function of temperature could be known This provides a whole system for
the real estimates of crystallization can be improved
107
Figure 25 The prediction for nucleation and growth from the 5 parameters The volume in
nucleation is assumed to be on the order of one cubic angstrom while the a parameter in growth is
assumed to be around one nm both are in good agreement for estimates in literature The values
for the orange points are taken from these works[128] [167]
57 Discussion
The impact of the Adam-Gibbs model on glass science is hard to overstate and it has
become foundational to ideas that will enable the design of next generation glasses The Adam-
Gibbs equation still has more to offer the glass community in terms of fundamental understanding
and practical applications In this work we have shown that the Adam-Gibbs relationship is
consistent with both shear and bulk viscosities Also we have validated the main assumption of the
MYEGA model giving insights into the scaling of configurational entropy from the underlying
enthalpy landscape Using this knowledge of the relationship between the Adam-Gibbs model with
the underlying enthalpy landscapes we have developed a simplified ldquotoy landscaperdquo approach for
modeling glass relaxation The toy landscape is constructed numerically by understanding the
distribution of inherent structures and transition points in the landscape and then relating these
distributions to experimentally measured parameters of the MYEGA viscosity model This
1 00
1600
1400
1200
1000
00
600
400
200
000
0000 110000 130000 1 0000 1 0000
(m
s)
Temperature
Growth PredictionCassar et al
0
2
4
6
10
12
14
00 1100 1300 1 00
log 1
0(I) n
ucl
ei s
m3
Temperature
Predicted RateRodrigues et al
4
3
2
1
0
00 1000 1100 1200 1300
log
10(I m
ax(I))
nuclei s
m3
Temperature elvin
108
simplified landscape model can then be used to calculate the underlying thermodynamics and
kinetics of the system and represents a practical application of energy landscapes to understand
glass relaxation nucleation and other relevant thermodynamic properties
This also closes our discussion on understanding the thermal history effects on glass What
we have shown is that the dynamics of a glass system can be understood through energy landscapes
and fundamentally the same insights can be recreated through lsquoToy Landscapesrsquo This approach
allows fundamentally for the complex 3N dimensional landscape to be summarized and abbreviated
into a smaller landscape that can easily be implemented for any glass However this new
parameterization relies on knowing some key properties that are only due to the location in the
compositional phase space and for a complete picture of the system to be created we must also
create a sufficient approximation for the compositional phase space shown in the next 2 chapters
Chapter 6
Enabling the Prediction of Glass Properties
So far models have been created that describe the very nature of the thermal history effects on
glasses However as an input to these models we have to know about the compositional dependence
of key parameters namely the glass transition and the fragility In this chapter we extend
compositional models to other key properties needed for the design of glasses Here these key
properties that have not been readily predictable are approached using both machine learning and
TCT models The TCT models are focused on surface reactivity Youngrsquos modulus and ionic
conductivity The ML addresses CTE fragility melting point as well as Youngrsquos modulus
61 Controlling Surface Reactivity
Glass surfaces especially their interactions with water[168]ndash[171] are of the utmost importance to
nuclear waste glass cover glass and many other modern applications[106] It is generally accepted
that the outermost surface of oxide glasses readily reacts with water molecules where water
molecules chemisorb on an as-formed glass surface and populate it with surface hydroxyls Recent
models attempt to quantitatively explain the change in properties observed on hydroxylated glass
surfaces[77] [168] [172] [173] and most agree that the surface readily becomes hydrated because
it is more stable to have a bonded hydroxyl group than to have dangling bonds In glasswater
interactions hydrolysis and diffusion lead to two forms of adsorbed water chemisorbed and
physisorbed[171] [174]ndash[176] Both affect the network differently and are controlled by different
processes physisorption by hydrogen bonding[175] and chemisorption by reaction with the surface
to form bonded hydroxyl groups
110
equiv Si minus O minus Si equiv +H2O rarr equiv Si minus OH + OH minus Si equiv
Although binding energy studies have been performed using molecular dynamics simulations[177]
a specific study of the surface reactivity in tandem has not been conducted Recent advancements
in reactive force field modeling allow the direct observation of both diffusion and surface
reactivity[38] [178]ndash[180] and several studies have found surface reactivity constants consistent
with those reported experimentally[173] [178] [181]
Recently TCT has been used to predict various bulk properties of glassy systems such as
Vickers hardness and fragility[17] [182] [183] Based on the TCT description of the glass network
glassy materials have been postulated to have a so-called lsquointermediate phasersquo in which the atomic
structure of a material will self-organize so as to be isostatically constrained (n=3) as explored
extensively in various works[65]ndash[67] [133] [184] [185] In this case it has been shown that such
materials exhibit anomalous behavior in certain properties such as an high hardness and lower free
energy[43] [67] [77]
TCT has also been used to understand the impact of water on bulk glass structure and
properties Potter et al[168] found that by accounting for the impact of chemisorbed (dissociated)
water on glass network connectivity (viz breaking Si-O-Si bonds) and the impact of physisorbed
(molecular) water on strain of the Si-O-Si bond angle the glass transition temperature (Tg) could
be predicted This model could be further expanded to include other properties (such as
modulus[139] or dissolution kinetics[77]) predicted by TCT such as the work done by Liu et
al[186] on calcium-silicate-hydrate gel An extensive study relating topological constraints[6]
[17] [57] and surface energy was performed by Yu et al[172] specifically focusing on the
transition from hydrophilic to hydrophobic behavior on silica surfaces Their work used reactive
molecular dynamics to model the change in surface energy of a silicate glass and then correlate it
with the number of surface constraints present with the work focusing primarily on the global
average of the surface
111
To investigate the effects of glass network topology on the surface reactivity we model
the hydration of a silicate glass was modelled using molecular dynamics simulations according to
the following procedures Initially bulk sodium silicate glasses were simulated (150 atoms with a
molar composition of 70SiO2middot30Na2O) using the Teter potential[187] For a bulk glass this pair
potential is known to accurately simulate sodium silicate glasses and has been extensively studied
to investigate various properties including structural features transport of sodium ions and
vibrational density of states[188] The size of the system is initially set to achieve the
experimentally measured density of 2466 gcm3 [189] A total of 35 Si-atoms 85 O-atoms and 30
Na-atoms were randomly inserted within a periodic cubic box of a = 12686 Å and the initial
configuration was energy-minimized to avoid any overlaps prior to the glass formation The glass
was held for 05 ns with a constant number of atoms volume and energy (a microcanonical or
NVE ensemble) Since a thermal fluctuation of ~400 K (from 2000 K to 2400 K) was observed
during the microcanonical run the glass was allowed to evolve further at 2400 K for 05 ns with a
constant number of atoms volume and temperature (a canonical or NVT ensemble) Within the
same ensemble the melted system was then cooled with a constant cooling rate of 05 Kps and
after the temperature had reached 300 K it was equilibrated for 1 ns Finally a constant pressure
of 1 atm was applied to the system (NPT ensemble) for another 1 ns to ensure no significant
fluctuations of the density during the equilibration process Three different sodium silicate glasses
of the same nominal composition were constructed via the aforementioned procedure using
different initial atomic positions These repeated simulations compensate for the limitation of small
sample size by allowing us to capture the statistical behavior of glass surfaces during the glass-
water reactions The final densities of the sodium silicate are presented in Table 5 These final
structures were used as starting configurations for the glass-water reactions MD simulations with
the Teter potential creating three glass networks were carried out using the Large-scale
AtomicMolecular Massively Parallel Simulator (LAMMPS) package[30]
112
Table 5 Density of the simulated sodium silicate glasses after relaxation at 300K
Simulation box dimension a (cubic) [Å] Density [gcm3]
Run1 12979 2303
Run2 12630 2499
Run3 12737 2437
Initial dimension a=12686 Å
Experimental density (70SiO2middot30Na2O mol) 2466 gcm3
113
After the bulk sodium silicate glasses were obtained a reactive potential was employed to model
the glass surface and subsequent glass-water interface as the system of interest for this study
required characterization of the reactive processes during the simulation of surface phenomena In
this study all reactive MD simulations are performed with the NaSiOH parameterization using
the ReaxFF reactive force field framework15 The ReaxFF parameters are trained using a first-
principles data set that describes water interaction at the sodium silicate glass-water interface
Further details of the ReaxFF methodology and its potential forms can be found in earlier
publications by van Duin et al[38] [179] [181] [190]
114
Figure 26 Representative example of the initial non-hydrated sodium silicate used in the hydration
models Color scheme Si atom (ivory) O atom (red) and Na atom (blue) The z-axis is elongated
to allow space for an insert of water
115
As shown in Figure 26 a free surface was first created by expanding the c parameters of each of
the equilibrated bulk glasses This process results in a vacuum region above and below the two
surfaces of the glass slab along the z-direction These exposed surfaces were relaxed at 300 K for
100 ps with the ReaxFF reactive force field Following the relaxation of both top and bottom
surfaces the vacuum region was filled with water molecules The number of water molecules that
are inserted in the vacuum region was controlled to have a density of ~099 gcm3 In addition all
water molecules were kept at least 2 Å from the uppermost and lowermost atoms of the glass
surface to prevent any initial close contact with the surface Glass-water reaction simulations were
carried out at 300 K with the configurations shown in Table 6 for 500 ps in the NVT ensemble
From these runs trajectories at every 100 ps were obtained for the surface reactivity analyses
Figure 27 shows the initial and final positions of glass surface reaction with water at 0 and 500 ps
respectively
116
Figure 27 Initial (a) and final (b) states of the waterglass interface Note that only the top surface
in contact with water is shown here
a) b)
117
Table 6 System configurations for sodium silicate glass-water reactions
Simulation cell [Å3] Number of water molecules
Run1 12979 times 12979 times 3894 146
Run2 12630 times 12630 times 3789 134
Run3 12737 times 12737 times 3821 138
118
The hydrated surfaces of sodium silicate glasses from each run were used to investigate the
adsorption behavior of a water molecule Since the time evolution of surface reactivity is of key
interest the binding energy of a water molecule to the hydrated surface was calculated at 0 100
200 300 400 ps In order to evaluate the local heterogeneity of reactivity imposed by varied glass
surface structures each surface was divided into a 10 by 10 grid and the binding energy at each site
was mapped across the grid The water molecule position in the z-direction from top and bottom
surface was maintained to be 20 Å from the outermost atom during the energy calculation The
binding energies (Eb) were calculated as below where the energy of a surface with adsorbed water
(Esw) was taken with respect to the surface energy (Es) and the energy of molecular water (Ew)
b s wswE E E E= minus minus (100)
A negative binding energy would indicate that water adsorption to the surface site at the
corresponding grid is thermodynamically favorable The binding energy map may then be used to
locate where water binding would be most stable based on the ReaxFF calculations From three
independent glass-water reaction boxes a total of six hydrated surfaces are obtained (a top and
bottom surface from each run) increasing the statistical reliability of this analysis in the ensemble
despite the small overall system size of any one box The binding energy mapping process is shown
in Figure 28
119
Figure 28 (Top) Schematic of water binding energy calculations (Bottom) An example of the
electronic-structure DFT calculation using semilocal exchange-correlation functionals finding the
binding energy of a water lsquopixelrsquo to the surface
~20 Å
surface
10times10 grid
120
Similar calculations are carried out at an electronic structure level using the DFT framework as
implemented within the Vienna Ab-initio Simulation Package (VASP) software Total energies are
computed from the static geometries obtained from the reacted ReaxFF structures using the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional with projector augmented wave
(PAW) pseudopotentials to describe the ion cores A kinetic energy cutoff of 500 eV was used for
the expansion of the plane waves Due to the large non-crystalline structure the Brillouin zone is
only sampled at the Γ-point with 01 eV of Methfessel-Paxton smearing to help electronic
convergence Due to the computational cost associated with these DFT simulations the results were
primarily used to elucidate the electronic interactions between the water molecule and the glass
surface
To count the topological constraints in the structure generated from the ReaxFF MD
simulations an algorithm was developed to systematically exclude all non-bridging oxygens
alkali and hydrogens from the structure The number of surface rigid constraints (in the pixel)
around each network-forming atom was then calculated an example contour plot generated using
this procedure is shown in Figure 29 The surfaces were then converted to a 10times10 pixel grid and
the average coordination of network-forming atoms in each pixel calculated The depth of each
pixel taken was to be 4 Aring below the surface
121
Figure 29 Example contour surface showing the average coordination per atom on the glass
surface for the first run at 300 ps
122
Figure 30 shows the results from the MD binding energy studies where a clear transition in the
region around 30 constraintsatom is shown This region is isostatic because the number of
constraints is equal to the number of translational degrees of freedom and the width highlighted in
the figure is evidence for the existence of an intermediate phase In order to confirm that the binding
energies had indeed converged a larger system of 1500 atoms was simulated Good quantitative
agreement was found between the 150- and 1500-atom systems In this work the isostatic behavior
occurs over a wide range in which certain silicon network sites dramatically increase the binding
sites (binding energy approaching 0) The state or structure with nc outside of the intermediate range
will readily interact with water Only those in the isostatic region are likely to remain unaffected
after the interaction with water thus they are the rate limiting species in dissolution of silica glass
123
Figure 30 ReaxFF MD-derived water binding energies plotted versus the number of constraints for
surface atoms at the local pixel Results show a distinct maximum in which there is a near
hydrophilic-hydrophobic transition of the surface The error bars represent the standard deviation
A second system with 1500 atoms was performed to show convergence of the ReaxFF MD results
⟨119889119867minus119874119866119897119886119904119904⟩ = 345 Aring
⟨119889119867minus119874119866119897119886119904119904⟩ = 248 Aring
124
The intermediate phase result is intuitive due to the implicit stability that comes with having an
isostatic phase which is both energetically favorable and stress-free The larger negative binding
energies in the over-constrained regions arise from large differences between the free energies of
the chemically bonded state (chemisorbed) and unbound state (water in near proximity to the
surface but not bonded) in these regions in this region the chemical reaction with the water
molecule allows the glass surface to reduce the number of incompatible constraints which in turn
alleviates local stress in the network In other words a local region undergoes an alleviation of
stress when Q4 units convert to Q3 units The larger negative binding energy values in the under-
constrained regions can trace their origin to the sites available for bonding with the water molecule
Their high number of degrees of freedom allows for the facile reaction of water ie the only energy
cost is for the oxygen to dissociate from the network The isostatic network within the intermediate
phase is able to preserve its structural integrity because the energy barrier to deform the network is
high and there is no localized stress to create an additional driving force for reaction with water
Indeed the isostatic network is the most energetically favorable arrangement of a non-crystalline
structure This intermediate phase is governed by the fluctuations of the topology that arise with
the minimization of stresses These stresses change the local topology (within the range of the
fluctuations) so that they become isostatic[68] This leads to a flat region where there is a constant
binding energy at the surface Though it appears to be an intermediate phase it is important to note
that this may instead be simply a manifestation of an isostatic threshold (a peak in binding energies
rather than a plateau)
The average number of surface constraints has been shown in previous work to be largely
controlled by the annealing time[172] The surface constraints may then be used to predict surface
reactivity However it is worth noting that the local number of constraints is not constant in
time[17] Furthermore Potter et al[168] showed that molecular water in the network can radically
shift the free energy of the γ constraint Therefore the number of surface constraints is dynamic
125
during the influx of water Shifting constraint energy will also alter the fragility which in turn alters
the diffusion activation enthalpy In the future this may provide a path for developing more durable
glasses ndash currently it shows that a homogeneous intermediate phase (isostatic) surface should be
targeted to achieve maximum chemical durability
62 Elastic Modulus Prediction
Tailoring the elastic properties of glass is important in the design of new advanced compositions to
control the stiffness and damage resistance of a variety of glass products[2] [191]ndash[193] Elastic
moduli are defined based on the ratio of the stress exerted upon a material to the resulting strain
While various elastic moduli can be defined (eg Youngrsquos modulus shear modulus bulk modulus
and Poissonrsquos ratio) for an isotropic material such as glass only two of these quantities are
mutually independent[1] [191] [194]
Previous attempts to model elastic moduli have focused on either computationally costly
molecular dynamics simulations or empirical fitting methods The work by Makishima and
Mackenzie[195] offered a possible solution for predicting elastic moduli based on the dissociation
energy per unit volume of the glass however topological changes in the glass network are ignored
and the model cannot account for the temperature dependence of modulus With MD simulations
elastic moduli can be obtained by applying a stress and measuring the resulting strain on the
simulation cell assuming that accurate interatomic potentials are available[2] Machine learning
has also been applied to model elastic moduli using experimentally measured composition-property
databases with a high predictive ability being achieved[2] [196] Previous analytical modeling
techniques related to the topology of the glass-forming network have shown good qualitative
agreement with compositional trends in modulus but have a lack of quantitative accuracy[64]
[197]ndash[199]
126
It is common in the glass community to assume that Youngrsquos modulus may scale linearly
with hardness While this is clearly an oversimplification let us begin with previously derived
models for glass hardness as a potential starting point since glass hardness has received
considerable attention in the context of topological constraint theory [85] [192] [200] Several
models have been proposed to explain the origin of glass hardness all of which have a linear form
( ) vv
dHH n x n
dn= minus (101)
where Hv is the Vickers hardness x is chemical composition nrsquo is the critical level of constraints
needed for the substance to provide mechanical resistance to the indenter in three dimensions and
n is a measure of the constraint rigidity Recent work on glass hardness[192] has shown that n can
most accurately be defined in terms of either the density of rigid constraints (or alternatively the
density of rigid angular constraints) given by
)( )
( )( )
(c An xn
x Nx
M x
= (102)
where nc is either the number of constraints per atom or number of angular constraints per atom ρ
is the density of the composition M is the molar mass and NA is Avogadrorsquos number Since
hardness and elastic modulus are often considered to be correlated one might surmise that the
elastic modulus might also be some function of angular constraint density or total constraint
density However such an approach is not able to give quantitatively accurate predictions of
modulus
Here we propose an improved model of Youngrsquos modulus based on the free energy density of the
topological constraints
int
( ) ( ))
(
(
) i constr
c i
a s
Ai in x q T
M x
x NF F
=
= (103)
where qn(T) is the onset function for each associated constraint to give temperature dependence to
127
the predictions and is determined by the free energy ∆Fn The calculated values of Δ119865c are then
used to calculate Youngrsquos modulus (E) by
( )c c
c
FdE
EF
xd
F
= minus (104)
In this new model the Youngrsquos modulus is controlled by the free energy parameters related to the
strength of each constraint Each parameter in this model captures a key physical phenomenon with
each parameter corresponding to a physical property Moreover we demonstrate that a common
set of parameters can simultaneously capture both the temperature and compositional dependence
of modulus as well as predict other properties such as the glass transition temperature and fragility
To demonstrate the validity of this temperature-dependent constraint model Youngrsquos
modulus data are collected from literature for lithium borate[195] sodium borate[195] and
germanium selenide glass systems[201] Additionally the sodium phosphosilicate[202] system
was experimentally determined following the procedure described by Zheng et al[183] using room
temperature resonant ultrasound spectroscopy on prisms of dimensions 10 mm times 8 mm times 6 mm
Analytical topological constraint models have already been published for each of these
systems[17] [57] [202]
The proposed models for glass elasticity (the models previously proposed for hardness
based on constraint density and angular constraint density as well as the new model for modulus
based on the free energy density of constraints) were optimized in each case leaving the intercept
and slope (119889119864
119889119899
119889119864
119889119880119888) as adjustable parameters The constraint onset temperatures were also optimized
for the free energy density model Figure 31 and Figure 32 show the results for each model in the
phosphosilicate and sodium borate systems respectively
128
Figure 31 The Youngrsquos modulus prediction and experimentally determined values for
03Na2Omiddot07(ySiO2middot(1-y)P2O5) glasses The root mean square error (RMSE) values of the model
predictions are 641 GPa for constraint density 313 GPa for free energy density and 774 GPa
for angular constraint density
129
Figure 32 The Youngrsquos modulus residuals for different prediction methods sorted from minimum
to maximum error The free energy density model gives the most accurate results The constraint
density has a RMSE of 61 GPa the angular density has a RMSE of 20 GPa and the energy density
has a RMSE of 59 GPa
130
We have also evaluated the predictive ability of the model in terms of the temperature dependence
of the Youngrsquos modulus sing the 10 Na2Omiddot90 B2O3 data reported by Jaccani and
Huang[191] the model and experimental predictions of the temperature dependence of the Youngrsquos
modulus are plotted in Figure 33 Here the constraint onset temperatures are determined from the
room temperature modulus data The number of escape attempts was optimized since this controls
the width of each transition and is thermal history dependent
131
Figure 33 (A) Temperature dependence of the Youngrsquos modulus from theory and experiment for
10 Na2O 90 B2O3 Using the previously fitted onset temperatures the only free parameters are
then the vibrational frequency and the heating time in which their product was fitted to be 14000
Where each dip in the modulus corresponds to a constraint no longer being rigid as heated through
each onset The onsets were fitted from the compositional dependence and only the width of the
transition was fit which may account for the discrepancy around the inflection The data was fit
using a least-squares method and the resultant fit is shown as the calculated method The fit has an
R2 of 094 (B) The contribution from each constraint to the overall modulus
It is also possible to validate the model by comparing the free parameters fitted to the Youngrsquos
modulus data vs that of previously reported data as shown in Table 7 Results for two additional
systems are plotted in Figure 34 to show the general validity of the free energy density model
132
Table 7 Fitted values from this analysis compared to those reported in the literature The disparity
between the constraints evaluated with molecular dynamics most likely come from the speed in
which the samples are quenched
Value Fitted Value (K) Literature (K) Method Citation
Silicate Onset 2212 1986 MD (Potter et
al[168])
Silicate Onset 818 1600 MD (Potter et
al[168])
Silicate Onset 450-500 810 MD (Potter et
al[168])
Borate Onset 921 Not Reported Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 715 740-760 Fitting Parameter
(Mauro Gupta and
Loucks[69])
Borate Onset 393 328 Fitting Parameter
(Mauro Gupta and
Loucks[69])
133
Figure 34 The Youngrsquos modulus prediction (using the same fitting method described in Fig 3) and
experimentally determined values for (A) zGemiddot(100-z)Se with an R2 of 093 and (B) xLi2Omiddot(100-
x)B2O3 glasses with an R2 of 0986
(
A)
(
B)
134
In the previous work of Zheng et al[192] concerning hardness of glass the authors write
ldquoIt should be noted that each bond constraint corresponds to a certain energy since different kind
of bond has different bonding energyhellip and thus the constraint density also represents an energy
per unit volume In other words hardness is correlated to the energy per unit volume Our findings
for the borosilicate and phosphosilicate systems are further evidence in support of this argument
since both the total constraint density and angular constraint density approaches give better
prediction of glass hardness compared to models based on number of atomic constraintsrdquo
which makes it clear that the prediction of the hardnessmodulus should be closely related to the
onset temperatures (ie free energies) of the associated constraints The model that was previously
proposed could be extended to modulus if all constraints had equal amounts of potential energy
but due to the drastic change in strengths between the constraints the prediction fails
Bauchy et al[85] showed also that hardness in the calcium-silicate-hydrate system is
controlled by the angular constraints which led to the development of angular constraint density
as the governing control for hardness When the analysis of the density of angular constraints is
considered through the free energy view it becomes apparent why this method works effectively
for some systems Models for predicting the hardness and elastic modulus of glass attempt to
explicitly connect the rigid bond energy to the macroscopic properties of the system The
assumption for the hardness models is that the energies for breaking each type of constraint are all
approximately equal and hence only the number of the constraints matters The argument can then
be extended for elasticity since elastic modulus is a bulk material property with the approximation
of all constraints being equal strength the number of constraints per volume should be related To
correct for this approximation when a weighted sum of free energies is used the estimation
becomes significantly more accurate Using the density of the glass a precise free energy per
volume of the rigid constraints can be calculated The knowledge that the energy of the bonds is
tied to the elastic modulus has been widely known but had not been quantified nor placed within
the context of topological constraint theory[203]
135
63 Ionic Conductivity
Ionic conductivity 120590 is related to the number of charge carriers n and the mobility of the
carriers 120583 as they diffuse through a network by[1]
Ze n = (105)
where Ze is the charge of the conducting species Since the charge-carrying ion is constant within
a given glass family the ionic conductivity depends only on the mobility and the concentration of
the charge carriers present However it is not typically feasible to measure 120583 and n independently
of each other and thus it is unclear which variable scales in an Arrhenius fashion thereby
controlling the ionic conductivity of a glass In the weak electrolyte model by Ravaine and
Souquet[204]ndash[207] the mobility of the charge carriers is considered constant while the
concentration scales in an Arrhenius function Conversely the strong electrolyte model assumes a
constant number of charge carriers while the mobility follows an Arrhenius function[208] [209]
Recent MD simulations from Welch et al[210] support the weak electrolyte interpretation for alkali
silicate glasses concluding that ionic conductivity is dictated by the concentration of charge
carriers Previous reports in the literature also supporting this view have proposed several
hypotheses to explain this conclusion[208] [209] [211]ndash[213] however further exploration is
needed
The simplest form of the temperature dependence of free ions is given by
0 exp an nE
kT
= minus
(106)
Here n0 is the total possible number of mobile ions and Ea is the activation barrier for an ion to
diffuse It is often assumed that the activation enthalpy is related to the energy needed to deform
the network temporarily to allow for ion motion[193] [209] [210] [212] however it is our
hypothesis that there is a coupling between the ion hopping event and the relaxation of the local
136
network allows for a permanent relaxation or deformation along ion channels This means that
there is some cooperative relaxation of the network along the diffusion path in conjunction with an
ionic hopping event These cooperative relaxations are discussed explicitly in the Adam-Gibbs
formalism[53] for describing the relationship between viscosity and configurational entropy Sc
expc
B
TS
=
(107)
where B is an activation barrier is the viscosity and is the viscosity of the liquid in the limit
of infinite temperature In the high-temperature (low-viscosity) liquid state there is a well-defined
relationship between the viscosity and diffusion (Stokes-Einstein relation) However this
relationship breaks down in the low-temperature (high-viscosity) glassy phase
Herein we present a new model for predictions for the compositional dependence of ionic
conductivity We can write the activation free energy barrier Eac for an ionic hopping event as
a c c c cE T S= minus (108)
Here is an enthalpic barrier for the ion to hop to a neighboring site CT is the configurational
temperature (a value that describes the distribution of energies of the ions) and cS is the entropy
of the activation barriers These quantities are associated with configurational changes in the glass
indicated by the subscript c In the weak electrolyte model all charge carriers have the same
mobility and hence the entropic effects are dominant This indicates a small value of an
assumption that will be validated by results later in this work leading to a simplified approximation
for the activation energy at temperatures below the glass transition but suffiently high temperatures
such that
a c c cE T S (109)
In the case of a nonequilibrium glass at low temperature the system becomes trapped in a localized
137
region of the energy landscape known as a ldquometabasinrdquo with slow inter-metabasin transitions[156]
[161] The MAP model of the nonequilibrium viscosity of glass relies on this partial confinement
within metabasins where variation of the activation barrier for relaxation (H) is related to the
fragility of the system by[44]
1
ln10g
dm
dH kT= (110)
Here m is the fragility index (as proposed by Angell[58]) and Tg is the glass transition temperature
Splitting the differential in Eq (110) and integrating we can write
ln10gmH kT= (111)
This relationship was also proposed by Moynihan et al[214] for the activation enthalpy of
relaxation of a glass[214] [215] Setting the Arrhenius form of the barrier described in Eq (111)
equal to the Adam-Gibbs equation (Eq (107)) we write
exp exp c
H B
kT TS
=
(112)
Solving Eq (112) for the configurational entropy and inserting the result into Eq (109) we obtain
the simple relationship
2
ln10
ca
g
T BE
k T m (113)
Dividing the above expression with that of a reference state r and assuming that B and is cT are
constant with respect to compositional variation we have
g r ra
a r g
T mE
TE m= (114)
This new equation is the first to relate the activation free energy for ionic hopping in a glass
to the glass transition temperature and fragility of its corresponding supercooled liquid It also
138
predicts a coupling between ionic diffusion and the cooperative rearrangements described by the
Adam-Gibbs relation This possibility was first proposed by Ngai and Martin[216] who showed a
correlation between the product of the Kohlrausch stretching exponent ( ) and the activation
barrier for relaxation with the activation barrier for ionic conductivity This implies that the
diffusion in glasses is not governed by an elastic component but instead controlled by the
surrounding cooperative rearrangements ie the -relaxation of the glass[14] in good qualitative
agreement with previous results shown by Potuzak et al[217] Further evidence comes from the
polymer community where the decoupling of relaxation and conductivity is reported with work
showing that the decoupling of viscosity and diffusion varies with the fragility and the glass
transition temperature[218] This is a general relationship which we demonstrate later to be valid
for binary alkali borate alkali silicates and alkali phosphate glasses as well as validating the
atomistic diffusion method using computational techniques
In order to test this proposed relationship between relaxation and diffusion through
simulation a glass with the composition 10 Na2O-90 B2O3 (mol ) was synthesized in molecular
dynamics with 1150 atoms using the potentials of Wang et al[162] The system was quenched at a
rate of 1 Kps from randomly placed atoms at 2500 K in the NPT ensemble to room temperature
as shown in Figure 35[210] To find the transition point energy of the diffusing ion a nudged elastic
band (NEB) calculation was performed An alkali ion was chosen and its closest alkali neighbor
was moved to another location so that the NEB calculation could be performed between those two
sites All calculations were carried out using the LAMMPS software package Eleven reaction
coordinates were used in the NEB calculation and the results are shown in Figure 36
139
Figure 35 The structure for the initial minimum energy configuration showing the boron (blue)
network with interconnecting oxygens (red) and the interstitial sodium ions (yellow)
Figure 36 The energy barrier between the two sites Oxygen is blue boron is red and sodium is
ivory The barrier is overestimated compared to experimental data this could be from several
sources of error such as potential fitting thermal history fluctuations or sampling too few
transitions The line is drawn as a guide to the eye
140
Although the transition point energy is key for determining the activation barrier for diffusion here
we are also interested in the dynamics of the atoms around the mobile ion Since we have
hypothesized here that the ion diffusion is dependent on the ability for the local region to relax and
deform instead of elastically straining we expect a local deformation (relaxation) along the ion
path Figure 37 shows that along the ion diffusion tunnel there is indeed a permanent displacement
of the atoms
141
Figure 37 Snapshots of the NEB calculation On the top is the total network as a function of
reaction coordinates The middle shows the local deformation around the ion of any atom that
moves in between inherent structure mandating a relaxation force The color shows the degree of
deformation
142
In the final configuration of Figure 37 the permanent deformation required to lower the
energy of the system is clearly seen If it were purely an elastic response to dilate a pathway the
displacement between two inherent structures should be confined to changes with only the mobile
ion however this is not observed Instead along the pathway of the ion movement the network
forming atoms also deform and change local positions to minimize the energy implying a net
relaxation around the ion ldquopathwaysrdquo To apply this and show the validity of Eq (114) the sodium
borate lithium borate lithium phosphate and sodium silicate systems were considered with the
model results compared to experimental values in Figure 38 To confirm this model literature values
for activation barriers have been used The experimental activation barriers for the borates are taken
from the work of Martin[213] while the fragilities and the glass transition temperatures are from
Nemilov[219] The viscosity curve of the silicate system was fit with the Mauro-Yue-Ellison-
Gupta-Allan (MYEGA) equation of viscosity[55] using the data from Poole[220] and Knoche et
al[221] with the activation barriers from Martinsen[222] Lithium phosphate activation data was
taken from Ngai and Martin[216] with the glass transition and fragility taken from Hermansen et
al and predicted values from topological constraint theory[223]
143
Figure 38 Different network formers and the prediction of the activation barrier from our model
compared with activation barriers from literature (A) Sodium silicate predictions and experimental
values[222] the error is calculated from the error in the fragility when fitting the data (B) Lithium
phosphate activation energy[216] predicted with topological constraint theory and compared with
the experimental values (C) Predictions over two different systems of alkali borates[213] sodium
and lithium with a reported R2 of 097
144
It is worth noting that this model is predicting a non-equilibrium kinetic property using
equilibrium viscosity parameters a connection that has been demonstrated in the past for viscosity
and relaxation models[61] In this work we are applying the same principles to the diffusion of
ions in a network The diffusion and viscosity in the case of equilibrium are known to be intimately
linked using the Stokes-Einstein expression (which breaks down for highly cooperative
rearrangements near and below the glass transition temperature[224]) Here we are no longer
connecting the two properties explicitly but instead connecting the activation barrier for diffusive
hopping to the two key parameters governing the viscous flow of the supercooled liquid state
The failure of the Stokes-Einstein expression near the glass transition is a well-known
problem and has been studied extensively for purposes of predicting crystallization rates[167] This
occurs because there is a breakdown of the ergodicity of a system around the glass transition In
the MAP model[44] the viscosity is controlled by the ergodic parameter x and is given by
min( )
max( )
f
p
f
T Tx
T T
=
(115)
Here the exponent p is related to the sharpness of the ergodic to non-ergodic transition Using the
ergodic parameter the overall viscosity ( ) is expressed as a linear combination of equilibrium
viscosity ( eq ) and non-equilibrium viscosity ( ne ) such that
10 10 10log ) log ( ) (1 ) log ( )( f eq f ne fx T T x T TT T = + minus (116)
By combining the Stokes-Einstein expression and the model proposed in this work we can define
the diffusion of an ion through the network as
( )10 10 10 0log ( 1 log exp ) l6 )
og(
b af
f
kx
T ED T T D
a T Tx
kT
minus + minus =
(117)
Here a is an empirical constant related to the size of the diffusing species k is Boltzmannrsquos
145
constant aE is the activation barrier calculated with Eq (114) and D0 is the limit of diffusion as
the temperature approaches infinity D0 should be chosen such that at fT T= the diffusion is
continuous This can be justified because the preexponential factor is related to the entropy of the
system and the fictive temperature is the temperature at which the configurational entropy of the
transition barriers becomes nearly constant Only one parameter is left unknown a since the
viscosity and fictive temperature can be calculated with an arbitrary thermal history using a
relaxation modeling tool such as RelaxPy[42] Implicit in RelaxPy are the assumptions pertaining
to the viscosity made by Guo et al[61] A similar model was first proposed by Cassar et al[167]
for modeling diffusion during crystallization although their model used an empirical hyperbolic
tangent function to approximate the ergodic factor In contrast the current model can be considered
explicitly as a function of thermal history and composition
64 Machine Learning Expansion
Though the TCT models presented here for different properties are immensely useful they
do have limitations the greatest of which is that intimate knowledge of the glassy structure is
required to parameterize these models This is due to the fact that the landscape is fundamentally
informed by the underlying structure of the network and as such to predict the dynamics of the
landscape information on the structure must be incorporated in some form However there is an
alternate method in which the physical origin is not considered in the development of the model
machine learning Machine learning comes in many forms and has become crucial to the study of
materials in the modern era There is a plethora of machine learning techniques however in this
paper we are focused on four key methods linear fitting (LF) methods random forest (RF)
symbolic regression (SR) and neural networks (NN)[3] [80] [225] [226] These methods are
146
explored extensively in literature and the exact underlying mathematics is beyond the scope of this
work In this work these methods have been used to create models for number of constraints (LF)
glass stability (RF) fragility (NN) melting point (NN) the infinite limit of viscosity (SR) Youngrsquos
modulus (NN) and coefficient of thermal expansion (NN)
To enable a linear fitting method for the number of constraints a simple steepest descent
algorithm was used to find the number of constraints due to each component There is no direct
experimental way to measure the number of constraints so instead a technique leveraging some
other property must be used to approximate the number of constraints In this case hardness is
related to the number of constraints through
25v cH A n= minus (118)
In which A is an empirical scaling parameter that is dependent on load indenter geometry and
glass family In this linear approach to determining constraints cn can be given by
0
compon
k
c k k
ents
cn x n=
= (119)
Combining this expression with a hardness database grouped by composition (in this case Sciglass)
allows for an approximate number of constraints of the system but due to A being a variable there
must be a fixed constraint in the system and every set of data is given a unqiue value of A The
fixed constraint in this system is that silicon atoms are four coordinated The value of A is chosen
such that the error between the prediction and the hardness is minimized The total process is as
follows
Choose a set of c kn
1 Calculate number of constraints for all compositions
2 Iterate over every dataset in Sciglass with gt 1 datapoints and scale by a factor of A
3 Calculate total error for steepest descent algorithm
4 Descent error slope
5 Repeat until error has been minimized
147
This then leads to an estimate for the number of constraints for every atomic species which can
then be used to estimate important properties such as chemical durability hardness and
qualitatively can predict glass stability
Two other techniques that are used are RF and SR When implemented a grid search for
the hyperparameters were used with a testing set used to determine the efficiency of each set of
hyperparameters Additionally each model was trained repeatedly and the lowest error was taken
The RF method was used to predict glass stability while undergoing a novel processing technique
called alkali-proton substitution (APS) The SR method was used to understand the compositional
dependence of the infinite temperature viscosity limit used in 3-parameter viscosity models To do
this the Sciglass database was taken (as implemented in GlassPy[227]) and sets of data that
included more than eight datapoints were fit to the MYEGA equation If the RMSE error of the fit
was less than 001 Pa s the fragility was greater than 17 and the value of the log infinite limit was
293 5minus it was accepted into the final database Each of these criteria were added due to the well-
known fact that this database is full of error and imprecise measurements The symbolic regression
was then performed through the grid search and the equation that performed best and reappeared is
given as
10loggT
mA B = + (120)
With A and B being parameters that varied depending on the hyper-parameters the infinite limit
threshold and the accuracy threshold used when creating the database We are currently in the
process of refining values for A and B is ongoing
This method allows for an easy prediction of the infinite limit so if the glass transition and
fragility are predicted through some other means (such as ML or TCT) a better prediction of the
third parameter is known without just accepting -293 as the mean Plots of the infinite limit vs the
148
fragility glass transition and Tgm showing the clear dependance of the infinite limit on the term
Tgm are shown in Figure 39
149
Figure 39 The dependence of infinite viscosity limit for other viscous parameters (Top left) shows
the distribution of the infinite temperature limit in the database after limits exerted on the system
(Top right) The distribution of the infinite viscosity limit vs the glass transition (Bottom left) The
relationship between fragility and infinite temperature limit of viscosity (Bottom right) The infinite
temperature limit of viscosity vs the key metric predicted by SR
150
NN are perhaps the most widely discussed ML tool applied in literature This makes sense
given they are universal function approximators and are quite easy to train given sufficient data
The most difficult part of creating a NN is adjusting the hyperparameters so that good accuracy can
be found without causing overfitting To avoid overfitting the data is divided into a testing set then
a training set (just as for RF and SR) but instead of doing a grid search for the hyperparameters we
used gradient boosted regression trees to find the optimal set of parameters This consists of
creating an initial tree then evaluating it based on a small set of data and finding the minimum
point then running a new set with perturbations around that point to create another tree This is
repeated until the value converges to a small error In this work the value being determined is the
lowest error of 100 sequentially trained neural nets with a set of hyperparameters This is
particularly useful because there is no concern for the type of parameters (string int floats) that
determine the value at the end so one can do a mixed parameter optimization that includes every
conceivable variable The values being optimized are given by
bull Learning rate [10-5 10-1] (float on logarithmic scale)
bull Decay rate for Adam Optimization [10-6 10-2] (float on logarithmic scale)
bull Hidden layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull Output layer activation function lsquorelursquo lsquoselursquo lsquolinearrsquo lsquotanhrsquo lsquosigmoidrsquo (string)
bull The number of nodes in each layer [81024] (int)
bull Batch size [1256] (int)
bull Patience [5250] (int)
In Table 8 we have listed the hyper-parameters that were used for the fragility melting point
Youngrsquos modulus and co-efficient of thermal expansion (CTE)
151
Table 8 Hyperparameters for different neural networks after hyper-optimizations
Model Fragility Melting
Temperature
Youngrsquos
Modulus
CTE
Learning Rate 0035 0018 0027 00091
Decay Rate 000293 000095 000161 0002
Hidden Layer RELU RELU RELU RELU
Output Layer Linear Linear SELU RELU
Number of Hidden
Layers
2 1 1 1
Number of Nodes 235 506 514 470
Batch Size 4 118 211 125
Patience 25 250 23 22
RMSE 5 [-] 805 [K] 68 [GPa] 96 x 10-7 [1 K]
152
Chapter 7
Designing Green Glasses for the 21st Century
Thus far the work presented in this dissertation has enabled a new set of design tools for
everything from crystallization glass relaxation and optimizing over the ideal compositional space
we can use them for designing new glasses Building on these models and insights new applications
of glass are enabled specifically three glasses an ion conducting glass for batteries a proton
conducting glass for hydrogen fuel cells and a commodity glass for consumers All three are
currently undergoing testing but experimental results have been reported back as of yet Overall
both the hydrogen and ion conducting glasses have been designed thorough a combination of ML
and underlying physical models and as such hare reported herein
71 Glass Electrolytes
An emerging high-interest application of glass is solid state batteries Glasses offer a
possible solution to the growing energy crisis however to realize this new paradigm new research
methods are needed The main barrier for any new batteries to be commercialized is a high
conductivity at room temperature (gt 10-3 S cm) high stability ease of processing and ultimately
an alternative must be cheaper than current liquid state Li-ion batteries glass is an attractive
candidate due to the innate stability a simple processing technique the infinite variability and the
relatively low cost of production In addition the structure of a glass is a liquid-like structure which
may encourage ion migration like that seen in liquids Despite all of these advantages only a few
compositions have realized the requirements but further research is needed to find optimal
compositions
153
To optimize the composition for activation barriers we must understand the relationship
between conductivity and structure Structural effects can be propagated in two different methods
based on the Arrhrenius expression for conductivity
0 exp aE
kT
minus =
(121)
The two methods are through 0 (the infinite temperature conductivity) and the activation barrier
( aE ) The pre-exponential factor is approximately a constant for simliar compositions while the
activation barrier varies dramatically over small compositional spaces making it the larger of the
two concerns Many models have been presented to predict the compositional dependence of the
activation barrier the Anderson-Stuart (AS) model the Christensen-Martin-Anderson-Stuart
(CMAS) model the weak electrolyte (WE) model Kohlrausch exponent model (KEM) from Ngai
et al[216] and there is the model proposed in this dissertation which we will call the Wilkinson
viscous cooperative conductivity model (WVCC)[8] [88] [204] [207]ndash[209]
To optimize machine learning (ML) is a powerful technique that is implemented The best
method to train the ML is to use the direct relationship between the compositions and the activation
barrier but this is not currently feasible due to a lack of central depository or database for this
information To make the prediction of the activation barrier we need use some model that is reliant
on commonly accesible parameters This rules out all models for this type of analysis except since
WVCC since the other models require some fitting parameters not accessible Each models required
parameters fitted values (please note that the models typically include approximations for these
values) and caveats are shown in Table 9
154
Table 9 A table with some ionic conductivity models and the parameters needed for them as well
as the disadvantages for each These are not the only models but are representative of those
commonly used in literature
Model Required Parameters Fit Parameters Caveats
AS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
lsquoMadelungrsquo Constant
Covalency Parameter
Doorway Radius1
The Madelung
Constant is not smooth
as a function of
composition and the
fitting parameters scale
the activation energy
non-proportionally
CMAS Shear Modulus
Charge of Anion
Charge of Carrier
Radii of Anion
Radii of Carrier
Dielectric Permittivity
lsquoMadelungrsquo Constant
Doorway Radius1
Jump Distance of Ion1
The same issues as AS
but in addition the
dielectric permittivity
lacks a database as
well
WE Charge Concentration Equilibrium Coefficient1 The equilibrium
coefficient drastically
changes the prediction
non-linearly
KEM Stretching Exponent Proportionality
Coefficient
There is not enough
data available to know
the compositional
dependence of the
stretching exponent
WVCC Glass Transition
Fragility Index
Proportionality
Coefficient
Due to the
proportionality
constant the activation
barrier can only be
known in a local
composition range
1 There are ways to approximate this value based on additional data
155
The WVCC model predicts that the activation barrier for ionic conductivity is given by
a
g
EA
mT= (122)
in which A is a proportionality constant This means that to predict the glass behavior we donrsquot
need to know things about the glass but instead merely about the liquid state where the viscosity is
readily available Leveraging this technique if the viscosity information as a function of
compositions is available then the local glass with the lowest activation barrier can be found by
minimizing 1 gT m To access this information multiple options are available such as topological
constraint theory but in this work we will leverage ML since it is easily applied to a large
compositional space
To get the viscosity we will use neural networks (NN) A well trained NN for the glass
transition is readily available and as such will be used for our prediction of gT In the previous
section there was presented a trained fragility NN which enables a complete optimization of the
local composition based on WVCC To generate a database of fragility values that the NN was
trained on SciGlass viscosity values were fit with MYEGA model The data had to fit following
criteria
1 The total number of data points for fitting had to be greater than 5 from the same literature
source
2 The root-mean-square-error of the fit had to be less than 001 Pa s
3 The infinite temperature value of viscosity had to be within 3 orders of magnitude of the
accepted logarithmic value (-293 log(Pa s))
An additional neural network recently created by Cassar[228] was also used for an independent
comparison Cassarrsquos NN was based on a hybrid physical ML work and may offer better insights
when extrapolating far from experimental datapoints In this study to prove this concept we will
only focus on P2O5 B2O3 Al2O3 and Li2O
156
This gives an entire method to predict glass battery candidates This technique will find all
of the local minimum but due to the fact that there is a proportionality constant it is unknown which
local minimum is the absolute minimum This means that the composition with the lowest
activation barrier in the space is identified but another technique is needed to investigate which one
of the identified compositions is best The method follows
1 Randomize an initial glass candidate with the content of being normalized
2 Run the glass candidate through the NN to find values of fragility and the glass transition
3 Perturb the composition to find the gradient
4 Step down the gradient
5 If the new gradient is 0 this is a local minimum and a possible glass candidate return to 1
Otherwise return to step 3
This method cannot find global optima because there is no guarantee that the
proportionality constant will stay constant across wide compositional spaces To narrow it down
an additional technique to compare the predicted values is needed (such as another activation
barrier model molecular dynamics simulations a regression algorithm or simply melt the proposed
compositions) To further pair down in this work a k-means algorithm was used along with the
assumption that the proportionality constant was slowly changing over compositional space This
allowed us to take the best glass in each grouping predicted by k-means The glasses we have
predicted (using both the separate predictor NN and the method by Cassar et al[228]) are listed
below in Table 10 while an example prediction for 1000 gmT (which should scale with the
activation barrier) in the Na2O ndash B2O3 ndash SiO2 system is shown in Figure 40 The glasses are currently
undergoing melting at Coe College and Iowa State It is worth noting that composition D is a well-
known composition that is known to exhibit criteria close to the criteria for conductivity If this
method shows that these compositions are better than other compositions in the family found in
literature then it can be expanded to include a wider range of component and hopefully finding a
universal candidate for glass batteries
157
Table 10 The predicted compositions based on the optimization scheme proposed
Compositions Li2O Al2O3 P2O5 B2O3
A- This work NN 55 31 0 14
B- This work NN 44 1 5 50
C- Cassar 60 26 2 13
D- Cassar 50 0 0 50
Figure 40 The prediction of the scaling of the activation barriers for a common sodium borosilicate
system
72 Hydrogen Fuel Cell Glasses
An emerging high-interest application of glass that is proton-conducting intermediate
temperature fuel cells[7] [229] A fuel cell in short is a device that turns chemical fuel into
electricity without the use of combustion It is often described as a chemical battery due to the
similarity of a fuel construction with batteries One of the crucial components to a working
intermediate range fuel cell is an electrolyte with a high proton migration The electrolyte must be
stable over a wide range of temperatures (up to 673 K) and a proton conductivity greater than 10-2
S cm-1 at the operating temperature[7] Previously phosphate glasses near the metaphosphate
compositions were used since they carried the appropriate number of residual protons for
conductivity studies However this restricted the phase space to only n-2 dimensions drastically
reducing the degrees of freedom of the problem and artificially restricting possible solutions In
2013[230] to access the full phase space a new method was developed where a sodium phosphate
glass could have all the sodium ions replaced with protons This allows high concentrations of
protons to be achieved with a variety of starting compositions A previous report[230] has
summarized successful samples these data are used as a training set for this study with additional
data on compositions that have failed taken from internal theses
To choose the candidate material for each model a cost function is defined The cost function
is an analytical function whose inputs are the fraction of each oxide component and the output is a
value that rates the composition The cost function consists as of many properties that are of interest
In the application of oxide electrolytes for an intermediate range fuel cell the two main properties
to worry about are the ionic conductivity at high temperatures (we will choose 473 K due to this
being the low end in which these materials are considered) and the relative stability of each phase
159
This is especially important since the material will undergo the APS (alkali-proton substitution) in
which the sodium will be replaced with protons in the bulk of the material The cost function used
in this work is given by
473 1( ) log 000T K Sx = minus= minus (123)
In which S is either a 0 or 1 (representing a crystal or a glass respectively) and is the
ionic conductivity The function is arbitrary and could be expanded to include as many terms are
as needed in the goal of this glass The stability is heavily weighted because it is more important
than the conductivity and any sample that is not stable should not be considered In order to predict
each of these samples a hybrid physicalempirical approach is used For predicting the ionic
conductivity of a proton-swapped glass we began with an observed relationship between the glass
transition temperature and the conductivity at 473 K This empirical relationship is shown in Figure
41 The relationship although empirical can provide some insights into the physics of proton
transferred glasses (no universality is claimed in this work) The activation barrier for these glasses
has widely remained unchanged as noted in a previous work due to the fact that the activation
barrier is simply to disassociate the proton with the non-bridging oxygen (NBO) and once the
proton is free interacts rarely with the network until it is rebounded to said network Thus the pre-
factor can be understood as being related to the degeneracy of proton conduction pathways When
the network has lower configurational entropy there are fewer pathways for the proton to travel in
this case it is known that the configurational entropy is then the dominating effect for the glass
transition[53]
The relationship between the glass transition temperature and the configurational entropy
can be expanded through the Stokes-Einstein relationship evaluated at the glass transition
22
1273 10
6 10g
g
T T g
kTnZeD nZe
anT
minus
= = = (124)
160
In which a is approximately the size of a proton and n is the number of charge carriers (which is
approximately a constant according to the weak electrolyte theory) This determines the intercept
of the glassy form proton conductivity and then the activation barrier (slope) is found to be
approximately the same for all proton conducting glasses meaning that the intercept is the dominant
effect on the behavior of the glass and the only variable controlling the intercept is the glass
transition
Leveraging the configurational entropyrsquos relationship with the glass transition along with
topological constraint theory we can write an expression for the glass transition and as such gain
predictive power for proton conductivity To predict the glass transition temperature (Tg) we start
with the Adam Gibbs model of viscosity[53] Topological constraint theory then states that the
degrees of freedom of the network is proportional to the configurational entropy[23] [56] [57]
Using the liquid state definition of the glass transition (1210 = Pa s) and the well-known infinite
temperature of viscosity we can then write
1
1493 lng
BT
kf=
(125)
Where f is the degrees of freedom Letting the constants be equal to A and rewriting the expression
in terms of constraints we arrive at a predictive formula for the glass transition[109]
3
g
x c x
x
AT
m n=
minus (126)
Where nc is the number of constraints provided by each component at the glass transition associated
with each component x and their molar fraction mx The value of the constraints was then linearly
parameterized to the glass transition data from literature[7] [230] [231] This is not the most
explicit approach to counting constraints however it is the most convenient when considering
large phase spaces being optimized over This glass transition temperature is then converted to
conductivity using the empirical translation shown in Figure 41
161
The other term in our cost function is that of stability The question of what forms glass is
a notorious question that goes back to the first serious days of research into materials It has been
shown that no one metric is a good predictor and no metric is universal However one must be used
to rule out bad compositions from the start To do this a Random Forest method is implemented
Though this may fail in edge cases it will at least help identify the right area to explore A
comprehensive review of random forest methods and machine learning methods for prediction of
glass properties can be found elsewhere[3] [4] [225] The amount of information previously
obtained about what forms glass in the compositional family we will be working on is not enough
to use random forest methods on just the fraction of each phase In order to circumvent this issue
we chose four physical parameters that would be used as features for the predictions
bull Total mol oxide modifier This was chosen because it is well known that modifiers break
up the network and most glass forming theories are related to the network percentages We
considered the oxide versions of H Na Ba Sr and Ca as the modifiers
bull Total mol network former Similarly it is well known the network influences glass
forming We considered B Ge and P as the network forming cations
bull Mean cation charge Though there has been previous work on this topic it is rarely
considered as a lsquonormalrsquo predictor of glass forming capability It is included here because
these glasses are mainly invert and as such a higher field strength will increase the cohesive
nature of the network
bull Entropy of mixing This is an important parameter to consider with so many components
it may be stabilized by the entropy of mixing The entropy of mixing (Sm) was calculated
using the Gibbs entropy
The Gibbs entropy is given by
lnm x
x
xS k m m= minus (127)
The random forest model consisted of 100 trees with a maximum depth of 3 The results of both
the random forest model the topological predictions the optimization and the resultant glass are
shown in the results section The optimization was limited to the range of each component that has
been explored experimentally as well as an additional constraint of at least 5 mol rare earth oxide
and at least 5 mol of boron oxide or germania this is justified elsewhere[231]
162
Figure 41 The relationship between the glass transition and the proton conductivity This is justified
two ways one through the relationship of the entropy of diffusion and glass formation (the Adam-
Gibbs model) and through the fact that water is known to depress the glass transition
163
To explain the results we will first start with the models listed to confirm their validity
The first model is the prediction of the glass transition temperature Figure 42 shows the accuracy
of the glass transition prediction as well as the relative contribution of each component to the glass
transition In this figure any component that has a number of constraints greater then PO52 will
increase the glass transition and as such decrease the conductivity Thus it is preferable to construct
a glass with only components whose constraints at Tg are less than PO52 as long as the glass
remains stable Interestingly whether the data includes only the glass transitions of protonated glass
or all glasses determines the role of La Barium and sodium both decrease the glass transition
playing the role of the more traditional modifier however they do provide some rigidity to the
network Protons have a net negative effect which we believe to be a result of the APS process
where they are introduced as the sodium is leaving so this negative effect corresponds to a breaking
of the rigidity that exists It is also interesting to note that adding a boron phase to the sample
increases the glass transition though not as much as some of smaller alkaline earth samples
assumingly due to their increased field strength The worst network former appears to be GeO2
To calibrate the accuracy of the random forest model a confusion matrix is shown It shows
that the RF always predict that a glass will be a glass however there is some error when it forms a
crystal with 23 of the time being misidentified Though random forest methods are empirical we
can derive some physical meaning by looking at the relative importance of each feature From the
random forest study of stability we are able to rank the relative importance of each feature used
The mean cation charge mixing entropy and percent network formers are all of considerable
importance while the percent modifiers is less so
Once these models were developed and checked the reliability of the cost function is
confirmed After which the minimization of the cost function was performed and one glass was
chosen as optimal Boron was found to be preferential in the glass but was limited since the
compositional space was bound by previous samples (an additional sample is undergoing
164
characterization where the boron content was slightly increased) The optimized sample when
synthesized had some visible nucleation so the optimization while redone with limiting the
component maxima to just the limits of those where compositions in which APS was successfully
performed This resulted in the OP composition The three samples currently are undergoing
experimental characterization are listed in Table 11
165
Table 11 The compositions synthesized in this work These compositions were predicted by
minimizing the cost function described in Eq (123) OP is the variant that was melted after OP
partially crystallized B-OP appeared to have surface nucleation in some spots but was cut and
removed before APS treatment
Name PO52 NaO12 BaO LaO32 GeO2 BO32
OP 44 36 10 5 3 2
OP 49 36 5 5 3 2
B-VOP 44 36 10 5 1 4
166
Figure 42 (A) The glass transition prediction vs the experimental values showing a good
correlation (B) The confusion matrix of the random forest method used to determine the glass
forming region Over top the constraints at the glass transition provided by each oxide species is
listed Since the objective is to decrease Tg while staying in the glass forming region we will
attempt to minimize use of elements that increase the glass transition temperature (nc gt 17)
167
Chapter 8
Conclusions
The rapid rate of new information in the current scientific climate and the infinite
variability of glass both stand as both unique challenges and opportunities In this dissertation we
have divided the influence of energy landscapes on the effects of properties into two independent
spaces the compositional and the thermal history dimensions By building models that succinctly
and accurately describe the dynamics of different hyper-coordinate (composition thermal history
crystallinity) changes the feasibility of designing new glasses for the challenges of the 21st century
is obtainable
Before designing new models the current state-of-the-art models need to be implemented
and understood To reach this end two softwares based on previous work were created RelaxPy
and ExplorerPy RelaxPy is an implementation of the MAP model and though the MAP model is
powerful when predicting the dynamics of the glass RelaxPy ultimately showed that fictive
temperature is insufficient at capturing the underlying physics and is intensive to parameterize
ExplorerPy was created to standardize the approach to mapping energy landscapes Energy
landscapes served as the key method of understanding the deeper dynamics of glasses glass-
ceramics and liquids
The thermal history dependence of glass with respect to both relaxation and crystallization
has been incorporated into the existence of new models called ldquotoy landscapesrdquo The toy landscapes
have built upon the physics of previous models such as the MAP model however due to the lower
parametrization cost and the built-in increased physicality toy landscapes pose as a tool to increase
our understanding and speed-up the rate of new glass discoveries This tool can deal with the
complexity of the higher dimensional spaces due to crystallization and relaxation without
assumptions concerning fictive temperature
168
The compositional dimensions are the remaining dimensions that must be optimized when
designing a glass or glass-ceramic Building on previous work models have been developed that
enable predictions of Youngrsquos modulus surface reactivity and ionic conductivity However there
are additional properties that are needed and for those we have proposed novel machine learning
approaches This dissertation has not enabled the design of glass for every application However
it lays the groundwork and approaches to design glasses for societyrsquos growing needs Presented as
well are the methods used to design glasses that could satisfy the requirements for solid state glass
electrolytes and hydrogen fuel cells All of these techniques together promise to be a powerful new
framework to build the glasses of the future
169
References
[1] A K Varshneya and J C Mauro Fundamentals of Inorganic Glasses 3rd ed Elsevier
2019
[2] J C Mauro A Tandia K D Vargheese Y Z Mauro and M M Smedskjaer
ldquoAccelerating the Design of Functional Glasses through Modelingrdquo Chem Mater vol
28 no 12 pp 4267ndash4277 2016 doi 101021acschemmater6b01054
[3] J C Mauro ldquoDecoding the glass genomerdquo Curr Opin Solid State Mater Sci pp 1ndash7
2017 doi 101016jcossms201709001
[4] H Liu Z Fu Yang X Xu and M Bauchy ldquoMachine learning for glass science and
engineering A reviewrdquo J Non Cryst Solids no March p 119419 2019 doi
101016jjnoncrysol201904039
[5] M M Smedskjaer C Hermansen and R E Youngman ldquoTopological engineering of
glasses using temperature-dependent constraintsrdquo MRS Bull vol 42 no 01 pp 29ndash33
2017 doi 101557mrs2016299
[6] J C Mauro and M M Smedskjaer ldquoStatistical mechanics of glassrdquo J Non Cryst Solids
vol 396ndash397 pp 41ndash53 2014 doi 101016jjnoncrysol201404009
[7] T Omata et al ldquoProton transport properties of proton-conducting phosphate glasses at
their glass transition temperaturesrdquo Phys Chem Chem Phys vol 21 no 20 pp 10744ndash
10749 2019 doi 101039c9cp01502g
[8] Z A Grady C J Wilkinson C A Randall and J C Mauro ldquoEmerging Role of Non-
crystalline Electrolytes in Solid-State Battery Researchrdquo Front Energy Res vol 8 p
218 2020 doi 103389fenrg202000218
[9] R C Welch et al ldquoDynamics of glass relaxation at room temperaturerdquo Phys Rev Lett
170
vol 110 no 26 p 265901 2013 doi 101103PhysRevLett110265901
[10] M Montazerian E D Zanotto and J C Mauro ldquoModel-driven design of bioactive
glasses from molecular dynamics through machine learningrdquo Int Mater Rev vol 0 no
0 pp 1ndash25 2019 doi 1010800950660820191694779
[11] E D Zanotto and J C Mauro ldquoThe glassy state of matter Its definition and ultimate
faterdquo J Non Cryst Solids vol 471 pp 490ndash495 2017 doi
101016jjnoncrysol201705019
[12] R G Palmer ldquoBroken ergodicityrdquo Adv Phys vol 31 no 6 pp 669ndash735 1982 doi
10108000018738200101438
[13] J C Mauro P Gupta and R J Loucks ldquoContinuously broken ergodicityrdquo J Chem
Phys vol 126 no 18 p 184511 2007 doi 10106312731774
[14] M Micoulaut ldquoRelaxation and physical aging in network glasses A reviewrdquo Reports
Prog Phys vol 79 no 6 p 066504 2016 doi 1010880034-4885796066504
[15] D J Wales and H A Scheraga ldquoGlobal optimization of clusters crystals and
biomoleculesrdquo Science (80- ) vol 285 no 5432 pp 1368ndash1372 1999 doi
101126science28554321368
[16] J C Mauro R J Loucks and P Gupta ldquoFictive temperature and the glassy staterdquo J
Am Ceram Soc vol 92 no 1 pp 75ndash86 2009 doi 101111j1551-2916200802851x
[17] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 2009 doi 10106313152432
[18] F H Stillinger and T A Weber ldquoHidden structure in liquidsrdquo Phys Rev A vol 25 no
2 pp 978ndash989 1982 doi 101103PhysRevA25978
[19] F H Stillinger ldquoSupercooled liquids glass transitions and the auzmann paradoxrdquo J
Chem Phys vol 88 no 12 pp 7818ndash7825 1988 doi 1010631454295
171
[20] S Neelamraju C Oligschleger and J C Schoumln ldquoThe threshold algorithm Description of
the methodology and new developmentsrdquo J Chem Phys vol 147 no 15 p 152713
2017 doi 10106314985912
[21] D Prada-Gracia J Goacutemez-Gardentildees P Echenique and F Falo ldquoExploring the free
energy landscape From dynamics to networks and backrdquo PLoS Comput Biol vol 5 no
6 p e1000415 2009 doi 101371journalpcbi1000415
[22] E Pollak A Auerbach and P Talknerz ldquoObservations on rate theory for rugged energy
landscapesrdquo Biophys J vol 95 no 9 pp 4258ndash4265 2008 doi
101529biophysj108136358
[23] G G Naumis ldquoEnergy landscape and rigidityrdquo Phys Rev E vol 71 p 026114 2005
doi 101103PhysRevE71026114
[24] S J Benkovic G G Hammes and S Hammes-Schiffer ldquoFree-energy landscape of
enzyme catalysisrdquo Biochemistry vol 47 no 11 pp 3317ndash3321 2008 doi
101021bi800049z
[25] M Roca B Messer D Hilvert and A Warshel ldquoOn the relationship between folding and
chemical landscapes in enzyme catalysisrdquo Proc Natl Acad Sci U S A vol 105 no 37
pp 13877ndash13882 2008 doi 101073pnas0803405105
[26] P Li G Henkelman J A eith and J Johnson ldquoElucidation of aqueous solvent-
mediated hydrogen-transfer reactions by ab initio molecular dynamics and nudged elastic-
band studies of NaBH4 hydrolysisrdquo J Phys Chem C vol 118 no 37 pp 21385ndash21399
2014 doi 101021jp507872d
[27] S P Niblett M Biedermann D J Wales and V De Souza ldquoPathways for diffusion in
the potential energy landscape of the network glass former SiO2rdquo J Chem Phys vol
147 no 15 p 152726 2017 doi 10106315005924
[28] Y Yu M Wang M M Smedskjaer J C Mauro G Sant and M Bauchy
172
ldquoThermometer Effect Origin of the Mixed Alkali Effect in Glass Relaxationrdquo Phys Rev
Lett vol 119 no 9 pp 1ndash5 2017 doi 101103PhysRevLett119095501
[29] Y Yu M Wang D Zhang B Wang G Sant and M Bauchy ldquoStretched Exponential
Relaxation of Glasses at Low Temperaturerdquo Phys Rev Lett vol 115 no 16 p 165901
2015 doi 101103PhysRevLett115165901
[30] Steve Plimpton ldquoFast Parallel Algorithms for Short-Range Molecular Dynamicsrdquo J
Comput Phys vol 117 no 1 pp 1ndash19 1995 doi
httpsdoiorg101006jcph19951039
[31] N Mousseau and G T Barkema ldquoTraveling through potential energy landscapes of
disordered materials The activation-relaxation techniquerdquo Phys Rev E vol 57 no 2
pp 2419ndash2424 1998 doi 101103PhysRevE572419
[32] F El-Mellouhi N Mousseau and L J Lewis ldquo inetic activation-relaxation technique
An off-lattice self-learning kinetic Monte Carlo algorithmrdquo Phys Rev B vol 78 p
153202 2008 doi 101103PhysRevB78153202
[33] D J Wales ldquoDiscrete path samplingrdquo Mol Phys vol 100 no 20 pp 3285ndash3305 2002
doi 10108000268970210162691
[34] G Henkelman and H Joacutensson ldquoImproved tangent estimate in the nudged elastic band
method for finding minimum energy paths and saddle pointsrdquo J Chem Phys vol 113
no 10 pp 9978ndash9985 2000 doi 10106314961868
[35] D Sheppard P Xiao W Chemelewski D D Johnson and G Henkelman ldquoA
generalized solid-state nudged elastic band methodrdquo J Chem Phys vol 136 p 074103
2012 doi 10106313684549
[36] A Pedone G Malavasi M C Menziani A N Cormack A V and N York ldquoA New
Self-Consistent Empirical Interatomic Potential Model for Oxides Silicates and Silica-
Based Glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
173
[37] A Takada C R A Catlow and G D Price ldquoComputer modelling of B2O3 II Molecular
dynamics simulations of vitreous structuresrdquo J Phys Condens Matter vol 7 no 46 pp
8693ndash8722 1995 doi 1010880953-8984746004
[38] A C T Van Duin S Dasgupta F Lorant and W A Goddard ldquoReaxFF A reactive
force field for hydrocarbonsrdquo J Phys Chem A vol 105 no 41 pp 9396ndash9409 2001
doi 101021jp004368u
[39] J C Mauro and A Varshneya ldquoModel interaction potentials for selenium from ab initio
molecular simulationsrdquo Phys Rev B vol 71 p 214105 2005 doi
101103PhysRevB71214105
[40] J C Mauro R J Loucks J Balakrishnan and S Raghavan ldquoMonte Carlo method for
computing density of states and quench probability of potential energy and enthalpy
landscapesrdquo J Chem Phys vol 126 no 19 2007 doi 10106312733674
[41] Y Z Mauro C J Wilkinson and J C Mauro ldquo ineticPy A tool to calculate long-time
kinetics in energy landscapes with broken ergodicityrdquo SoftwareX vol 11 p 100393
2020 doi 101016jsoftx2019100393
[42] C J Wilkinson Y Z Mauro and J C Mauro ldquoRelaxPy Python code for modeling of
glass relaxation behaviorrdquo SoftwareX vol 7 pp 255ndash258 2018 doi
101016jsoftx201807008
[43] J C Mauro ldquoTopological constraint theory of glassrdquo Am Ceram Soc Bull vol 90 no
4 pp 31ndash37 2011 doi 101039c3ee40810h
[44] J C Mauro D C Allan and M Potuzak ldquoNonequilibrium viscosity of glassrdquo Phys Rev
B vol 80 p 094204 2009 doi 101103PhysRevB80094204
[45] M Cardona R V Chamberlin and W Marx ldquoThe history of the stretched exponential
functionrdquo Ann der Phys vol 16 no 12 pp 842ndash845 2007 doi
101002andp200710269
174
[46] P Grassberger and I Procaccia ldquoThe long time properties of diffusion in a medium with
static trapsrdquo J Chem Phys vol 77 no 1982 pp 6281ndash6284 1982 doi
1010631443832
[47] J C Phillips ldquoMicroscopic aspects of Stretched Exponential Relaxation (SER) in
homogeneous molecular and network glasses and polymersrdquo J Non Cryst Solids vol
357 no 22ndash23 pp 3853ndash3865 2011 doi 101016jjnoncrysol201108001
[48] J C Phillips ldquo ohlrausch explained The solution to a problem that is 1 0 years oldrdquo J
Stat Phys vol 77 no 3ndash4 pp 945ndash947 1994 doi 101007BF02179472
[49] A Q Tool and C G Eichlin ldquoVariations caused in the heating curves of glass caused by
heat treatmentrdquo J Am Ceram Soc vol 14 pp 276ndash308 1931
[50] A Q Tool ldquoRelation Between Inelastic Deformability and Thermal Expansion of Glass in
Its Annealing Rangerdquo J Am Ceram Soc vol 29 no 9 pp 240ndash253 1946 doi
101111j1151-29161946tb11592x
[51] H N Ritland ldquoLimitations of the Fictive Temperature Conceptrdquo J Am Ceram Soc vol
39 no 12 pp 403ndash406 Dec 1956 doi 101111j1151-29161956tb15613x
[52] O S Narayanaswamy ldquoA model of structural relaxation in glassrdquo J Am Ceram Soc
vol 54 no 10 pp 491ndash498 1971 doi 101111j1151-29161971tb12186x
[53] G Adam and J Gibbs ldquoOn the Temperature Dependence of Cooperative Relaxation
Properties in Glass‐Forming Liquidsrdquo J Chem Phys vol 43 no 1 pp 139ndash146 1965
doi 10106311696442
[54] Q Zheng J C Mauro A J Ellison M Potuzak and Y Yue ldquo niversality of the high-
temperature viscosity limit of silicate liquidsrdquo Phys Rev B vol 83 no 21 p 212202
2011 doi 101103PhysRevB83212202
[55] J C Mauro Y Yue A J Ellison P Gupta and D C Allan ldquoViscosity of glass-
forming liquidsrdquo Proc Natl Acad Sci vol 106 no 47 pp 19780ndash19784 2009 doi
175
101073pnas0911705106
[56] G G Naumis ldquoGlass transition phenomenology and flexibility An approach using the
energy landscape formalismrdquo J Non Cryst Solids vol 352 no 42-49 SPEC ISS pp
4865ndash4870 2006 doi 101016jjnoncrysol200601160
[57] P Gupta and J C Mauro ldquoComposition dependence of glass transition temperature
and fragility I A topological model incorporating temperature-dependent constraintsrdquo J
Chem Phys vol 130 no 9 p 094503 2009 doi 10106313077168
[58] C A Angell ldquoRelaxation in liquids polymers and plastic crystals - strongfragile patterns
and problemsrdquo J Non Cryst Solids vol 13 no 31 pp 131ndash133 1991 doi
1010160022-3093(91)90266-9
[59] G S Fulcher ldquoAnalysis of Recent Measurements of the Viscosity of Glassesrdquo J Am
Ceram Soc vol 75 no 5 pp 1043ndash1055 1992 doi 101111j1151-
29161992tb05536x
[60] I Avramov ldquoViscosity in disordered mediardquo J Non Cryst Solids vol 351 no 40ndash42
pp 3163ndash3173 Oct 2005 doi 101016JJNONCRYSOL200508021
[61] X Guo J C Mauro D C Allan and M M Smedskjaer ldquoPredictive model for the
composition dependence of glassy dynamicsrdquo J Am Ceram Soc vol 101 pp 1169ndash
1179 2018 doi 101111jace15272
[62] Doss C J Wilkinson Y Yang H Lee L Huang and J C Mauro ldquoMaxwell
relaxation time for nonexponential α-relaxation phenomena in glassy systemsrdquo J Am
Ceram Soc vol 103 no 6 pp 3590ndash3599 2020 doi 101111jace17051
[63] P Gupta and A R Cooper ldquoTopologically disordered networks of rigid polytopesrdquo J
Non Cryst Solids 1990 doi 1010160022-3093(90)90768-H
[64] J C Phillips and M F Thorpe ldquoConstraint theory vector percolation and glass
formationrdquo Solid State Commun vol 53 no 8 pp 699ndash702 1985 doi 1010160038-
176
1098(85)90381-3
[65] X Feng W Bresser and P Boolchand ldquoDirect Evidence for Stiffness Threshold in
Chalcogenide Glassesrdquo Phys Rev Lett vol 78 no 23 pp 4422ndash4425 1997 doi
101103PhysRevLett784422
[66] W Bresser P Boolchand and P Suranyi ldquoRigidity Percolation and Molecular Clustering
in Network Glassesrdquo Phys Rev Lett vol 56 no 23 pp 2493ndash2496 1986 doi
101103PhysRevLett562493
[67] Y Vaills T Qu M Micoulaut F Chaimbault and P Boolchand ldquoDirect evidence of
rigidity loss and self-organization in silicate glassesrdquo J Phys Condens Matter vol 17
no 32 pp 4889ndash4896 2005 doi 1010880953-89841732003
[68] A irchner and J C Mauro ldquoStatistical Mechanical Model of the Self-Organized
Intermediate Phase in Glass-Forming Systems With Adaptable Network Topologiesrdquo
Front Mater vol 6 p 11 2019 doi 103389fmats201900011
[69] J C Mauro P Gupta and R J Loucks ldquoComposition dependence of glass transition
temperature and fragility II A topological model of alkali borate liquidsrdquo J Chem Phys
vol 130 no 23 p 234503 Jun 2009 doi 10106313152432
[70] D R Cassar A C P L F de Carvalho and E D Zanotto ldquoPredicting glass transition
temperatures using neural networksrdquo Acta Mater vol 159 pp 249ndash256 Oct 2018 doi
101016JACTAMAT201808022
[71] J C Mauro and R J Loucks ldquoSelenium glass transition A model based on the enthalpy
landscape approach and nonequilibrium statistical mechanicsrdquo Phys Rev B vol 76 no
17 p 174202 2007 doi 101103PhysRevB76174202
[72] C J Wilkinson et al ldquoEnergy Landscape Modeling of Crystal Nucleationrdquo Nat Comput
Mater p Submitted 2020
[73] M E Mc enzie and J C Mauro ldquoHybrid Monte Carlo technique for modeling of crystal
177
nucleation and application to lithium disilicate glass-ceramicsrdquo Comput Mater Sci vol
149 no January pp 202ndash207 2018 doi 101016jcommatsci201803034
[74] M E McKenzie et al ldquoImplicit glass model for simulation of crystal nucleation for glass-
ceramicsrdquo npj Comput Mater vol 4 no 1 pp 1ndash7 2018 doi 101038s41524-018-
0116-5
[75] E D Zanotto and P F James ldquoExperimental tests of the classical nucleation theory for
glassesrdquo J Non Cryst Solids vol 74 no 2ndash3 pp 373ndash394 1985 doi 1010160022-
3093(85)90080-8
[76] Y Yu N M A Krishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[77] I Pignatelli A Kumar M Bauchy and G Sant ldquoTopological control on silicatesrsquo
dissolution kineticsrdquo Langmuir vol 32 pp 4434ndash4439 2016 doi
101021acslangmuir6b00359
[78] C J Wilkinson et al ldquoTopological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phaserdquo J Phys Chem Lett vol 10 pp
3955ndash3960 2019 doi 101021acsjpclett9b01275
[79] Y Zhang A Li B Deng and Hughes ldquoData-driven predictive models for chemical
durability of oxide glass under different chemical conditionsrdquo npj Mater Degrad vol 4
no 1 pp 1ndash11 2020 doi 101038s41529-020-0118-x
[80] N M Anoop Krishnan S Mangalathu M M Smedskjaer A Tandia H Burton and M
Bauchy ldquoPredicting the dissolution kinetics of silicate glasses using machine learningrdquo J
Non Cryst Solids vol 487 no February pp 37ndash45 2018 doi
101016jjnoncrysol201802023
[81] R M Potter J Hoffman and J Hadley ldquoAn update of the equation for predicting the
178
dissolution rate of glass fibers from their chemical compositionsrdquo Inhal Toxicol vol 29
no 4 pp 145ndash146 2017 doi 1010800895837820171321702
[82] B Deng ldquoMachine learning on density and elastic property of oxide glasses driven by
large datasetrdquo J Non Cryst Solids vol 529 no August 2019 p 119768 2020 doi
101016jjnoncrysol2019119768
[83] S Feller N Lower and M Affatigato ldquoDensity as a probe of oxide glass structurerdquo
Phys Chem Glas vol 42 no 3 pp 240ndash246 2001 Accessed Jul 19 2018 [Online]
Available
httpswwwingentaconnectcomcontentsgtpcg200100000042000000034203240
[84] Q Zheng and H Zeng ldquoProgress in modeling of glass properties using topological
constraint theoryrdquo Int J Appl Glas Sci vol 11 no 3 pp 432ndash441 2020 doi
101111ijag15105
[85] M Bauchy M Javad Abdolhosseini Qomi C Bichara F-J Ulm and R J-M Pellenq
ldquoRigidity Transition in Materials Hardness is Driven by Weak Atomic Constraintsrdquo
2015 doi 101103PhysRevLett114125502
[86] Yang X Xu B Yang B Cook H Ramos and M Bauchy ldquoPrediction of Silicate
Glassesrsquo Stiffness by High-Throughput Molecular Dynamics Simulations and Machine
Learningrdquo Accessed Mar 13 201 Online Available
httpsarxivorgpdf190109323pdf
[87] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological constraint model for
the elasticity of glass-forming systemsrdquo J Non Cryst Solids X vol 2 2019 doi
101016jnocx2019100019
[88] C J Wilkinson K Doss D R Cassar R S Welch C B Bragatto and J C Mauro
ldquoPredicting Ionic Diffusion in Glass from Its Relaxation Behaviorrdquo J Phys Chem B vol
124 no 6 pp 1099ndash1103 2020 doi 101021acsjpcb9b10645
179
[89] C Wilkinson and J C Mauro ldquoExplorerpy Mapping the energy landscapes of complex
materialsrdquo SoftwareX vol Submitted 2020
[90] J C Mauro R J Loucks and J Balakrishnan ldquoA simplified eigenvector-following
technique for locating transition points in an energy landscaperdquo J Phys Chem A vol
109 no 42 pp 9578ndash9583 2005 doi 101021jp053581t
[91] Roumlder and D J Wales ldquoAnalysis of the b to b-CR Transition in biquitinrdquo
Biochemistry vol 57 no 43 pp 6180ndash6186 2018 doi 101021acsbiochem8b00770
[92] J C Mauro R J Loucks and J Balakrishnan ldquoSplit-step eigenvector-following
technique for exploring enthalpy landscapes at absolute zerordquo J Phys Chem B vol 110
no 10 pp 5005ndash5011 2006 doi 101021jp056803w
[93] J C Mauro R J Loucks J Balakrishnan and A Varshneya ldquoMapping the potential
energy landscapes of selenium clustersrdquo J Non Cryst Solids vol 353 pp 1268ndash1273
2007 doi 101016jjnoncrysol200609062
[94] B W H van Beest G J Kramer and R A van Santen ldquoForce fields for silicas and
aluminophosphates based on ab initio calculationsrdquo Phys Rev Lett vol 64 no 16 pp
1955ndash1958 Apr 1990 doi 101103PhysRevLett641955
[95] Q Zheng and J C Mauro ldquoViscosity of glass-forming systemsrdquo J Am Ceram Soc vol
100 no 1 pp 6ndash25 2017 doi 101111jace14678
[96] J C Mauro and Y Z Mauro ldquoOn the Prony Series Representation of Stretched
Exponential Relaxationrdquo Physica A vol 506 pp 75ndash87 2018 doi
101016jphysa201804047
[97] E D Zanotto and D R Cassar ldquoThe microscopic origin of the extreme glass-forming
ability of Albite and B2O3rdquo Sci Rep vol 7 no February pp 1ndash13 2017 doi
101038srep43022
[98] P Pedevilla S J Cox B Slater and A Michaelides ldquoCan Ice-Like Structures Form on
180
Non-Ice-Like Substrates The Example of the K-feldspar Microclinerdquo J Phys Chem C
vol 120 no 12 pp 6704ndash6713 2016 doi 101021acsjpcc6b01155
[99] O Bjoumlrneholm et al ldquoWater at Interfacesrdquo Chem Rev vol 116 no 13 pp 7698ndash7726
2016 doi 101021acschemrev6b00045
[100] Reichelt ldquoNucleation and growth of thin filmsrdquo Vacuum vol 38 no 12 pp 1083ndash
1099 Jan 1988 doi 1010160042-207X(88)90004-8
[101] G H Beall ldquoDr S Donald (Don) Stookey (1 1 ndash2014) Pioneering researcher and
adventurerrdquo Front Mater vol 3 no August pp 1ndash8 2016 doi
103389fmats201600037
[102] J C Mauro C S Philip D J Vaughn and M S Pambianchi ldquoGlass science in the
nited States Current status and future directionsrdquo Int J Appl Glas Sci vol 5 no 1
pp 2ndash15 2014 doi 101111ijag12058
[103] J Deubener et al ldquo pdated definition of glass-ceramicsrdquo J Non Cryst Solids vol 501
no January pp 3ndash10 2018 doi 101016jjnoncrysol201801033
[104] X Hao ldquoA review on the dielectric materials for high energy-storage applicationrdquo J Adv
Dielectr vol 03 no 01 p 1330001 2013 doi 101142s2010135x13300016
[105] J Deubener G Helsch A Moiseev and H Bornhoumlft ldquoGlasses for solar energy
conversion systemsrdquo J Eur Ceram Soc vol 29 no 7 pp 1203ndash1210 2009 doi
101016jjeurceramsoc200808009
[106] G S Frankel et al ldquoA comparative review of the aqueous corrosion of glasses crystalline
ceramics and metalsrdquo npj Mater Degrad vol 2 no 1 p 15 2018 doi
101038s41529-018-0037-2
[107] G H Beall ldquoDesign and properties of glass-ceramicsrdquo Annu Rev Mater Sci vol 22
no 1 pp 91ndash119 1992 doi 101146annurevms22080192000515
[108] J Deubener ldquoConfigurational entropy and crystal nucleation of silicate glassesrdquo Phys
181
Chem Glas vol 45 pp 61ndash63 2004
[109] J C Mauro A J Ellison D C Allan and M M Smedskjaer ldquoTopological model for the
viscosity of multicomponent glass-forming liquidsrdquo Int J Appl Glas Sci vol 4 no 4
pp 408ndash413 2013 doi 101111ijag12009
[110] M M Smedskjaer J C Mauro S Sen and Y Yue ldquoQuantitative design of glassy
materials using temperature-dependent constraint theoryrdquo Chem Mater vol 22 no 18
pp 5358ndash5365 2010 doi 101021cm1016799
[111] Y T Sun H Y Bai M Z Li and W H Wang ldquoMachine Learning Approach for
Prediction and Understanding of Glass-Forming Abilityrdquo J Phys Chem Lett vol 8 no
14 pp 3434ndash3439 2017 doi 101021acsjpclett7b01046
[112] Q Zheng et al ldquo nderstanding Glass through Differential Scanning Calorimetryrdquo Chem
Rev vol 119 no 13 pp 7848ndash7939 2019 doi 101021acschemrev8b00510
[113] X Xia I Dutta J C Mauro B G Aitken and F elton ldquoTemperature dependence of
crystal nucleation in BaOmiddot2SiO2 and 5BaOmiddot8SiO2 glasses using differential thermal
analysisrdquo J Non Cryst Solids vol 459 pp 45ndash50 2017 doi
101016jjnoncrysol201612032
[114] V M Fokin A A Cabral R M C V Reis M L F Nascimento and E D Zanotto
ldquoCritical assessment of DTA-DSC methods for the study of nucleation kinetics in
glassesrdquo J Non Cryst Solids vol 356 no 6ndash8 pp 358ndash367 2010 doi
101016jjnoncrysol200911038
[115] D C Van Hoesen X Xia M E Mc enzie and F elton ldquoModeling nonisothermal
crystallization in a BaO∙2SiO2 glassrdquo J Am Ceram Soc no December 2019 pp 2471ndash
2482 2019 doi 101111jace16979
[116] S C C Prado J P Rino and E D Zanotto ldquoSuccessful test of the classical nucleation
theory by molecular dynamic simulations of BaSrdquo Comput Mater Sci vol 161 no
182
January pp 99ndash106 2019 doi 101016jcommatsci201901023
[117] A O Tipeev and E D Zanotto ldquoNucleation kinetics in supercooled Ni 0Ti 0 Computer
simulation data corroborate the validity of the Classical Nucleation Theoryrdquo Chem Phys
Lett vol 735 no August p 136749 2019 doi 101016jcplett2019136749
[118] K F Kelton and A L Greer Nucleation in Condensed Matter Applications in Materials
and Biology 1st ed Pergamon 2010
[119] V M Fokin E D Zanotto N S Yuritsyn and J W P Schmelzer ldquoHomogeneous
crystal nucleation in silicate glasses A 40 years perspectiverdquo J Non Cryst Solids vol
352 no 26ndash27 pp 2681ndash2714 2006 doi 101016jjnoncrysol200602074
[120] A M Rodrigues D R Cassar V M Fokin and E D Zanotto ldquoCrystal growth and
viscous flow in barium disilicate glassrdquo J Non Cryst Solids vol 479 pp 55ndash61 2018
doi 101016jjnoncrysol201710007
[121] A Pedone G Malavasi M C Menziani A N Cormack and U Segre ldquoA new selft-
consistent empirical interatomic potential model for oxides silicates and silica-based
glassesrdquo J Phys Chem B vol 110 pp 11780ndash11795 2006
[122] J J Maldonis A D Banadaki S Patala and P M Voyles ldquoShort-range order structure
motifs learned from an atomistic model of a Zr 0Cu4 Al metallic glassrdquo Acta Mater
vol 175 pp 35ndash45 2019 doi 101016jactamat201905002
[123] A irchner S Goyal M E Mc enzie J T Harris and J C Mauro ldquoStatistical
Description of the Thermodynamics of Glass-Forming Liquidsrdquo Physica A vol 559 p
125059 2020 doi 101016jphysa2020125059
[124] D R Cassar ldquoCrystallization Driving Force of Supercooled Oxide Liquidsrdquo Int J Appl
Glas Sci vol 7 no 3 pp 262ndash269 2016 doi 101111ijag12218
[125] V M Fokin A S Abyzov E D Zanotto D R Cassar A M Rodrigues and J W P
Schmelzer ldquoCrystal nucleation in glass-forming liquids Variation of the size of the
183
lsquostructural unitsrsquo with temperaturerdquo J Non Cryst Solids vol 447 pp 35ndash44 2016 doi
101016jjnoncrysol201605017
[126] M Lenoir A Grandjean Y Linard B Cochain and D R Neuville ldquoThe influence of
SiB substitution and of the nature of network-modifying cations on the properties and
structure of borosilicate glasses and meltsrdquo Chem Geol vol 256 no 3ndash4 pp 316ndash325
2008 doi 101016jchemgeo200807002
[127] P Gupta D R Cassar and E D Zanotto ldquoRole of dynamic heterogeneities in crystal
nucleation kinetics in an oxide supercooled liquidrdquo J Chem Phys vol 145 no 21 2016
doi 10106314964674
[128] A M Rodrigues ldquoDiffusion Processes Crystallization and Viscous Flow in Barium
Disilicate Glassrdquo Dissertation 2014
[129] E D Zanotto ldquoThe effects of amorphous phase sepatration on crystal nucleation in baria-
silica and lithia-silica glassesrdquo Dissertation 1982
[130] D Vargheese A Tandia and J C Mauro ldquoOrigin of dynamical heterogeneities in
calcium aluminosilicate liquidsrdquo J Chem Phys vol 132 no 19 p 194501 May 2010
doi 10106313429880
[131] K Deenamma Vargheese A Tandia and J C Mauro ldquoStatistics of modifier distributions
in mixed network glassesrdquo J Chem Phys vol 132 p 24507 2010 doi
10106313429880
[132] M Micoulaut ldquoThe Deep Effect of Topology on Glass Relaxationrdquo Physics (College
Park Md) vol 6 p 72 2013 doi 101103Physics672
[133] M Micoulaut ldquoRigidity and intermediate phases in glasses driven by speciationrdquo Phys
Rev B vol 74 p 184208 2006 doi 101103PhysRevB74184208
[134] M Kodama S Kojima S Feller and M Affatigato ldquoThe occurrence of minima in the
Borate anomaly anharmonicity and fragility in lithium borate glassesrdquo Phys Chem
184
Glas vol 46 no 2 pp 190ndash193 2005
[135] Y Fukawa et al ldquoVelocity of Sound and Elastic Properties of Li2O-B2O3 Glassesrdquo Jpn
J Appl Phys vol 34 p 2570 1995 doi httpsdoiorg101143JJAP342570
[136] M Reiner ldquoThe Deborah Numberrdquo Physics Today vol 17 no 1 p 62 1964 doi
10106313051374
[137] P K Gupta and J C Mauro ldquoThe laboratory glass transitionrdquo J Chem Phys vol 126
no 22 p 2240504 2007 doi 10106312738471
[138] R Richert and C A Angell ldquoDynamics of glass-forming liquids V On the link between
molecular dynamics and configurational entropyrdquo J Chem Phys vol 108 no 21 pp
9016ndash9026 1998 doi 1010631476348
[139] C J Wilkinson Q Zheng L Huang and J C Mauro ldquoTopological Constraint Model for
the Elasticity of Glass-Forming Systemsrdquo J Non Cryst Solids X vol 2 p 100019 2019
doi 101016jnocx2019100019
[140] S Feller S Bista A OrsquoDonovan-Zavada T Mullenbach M Franke and M Affatigato
ldquoPacking in alkali and alkaline earth borosilicate glass systemsrdquo Phys Chem Glas Eur
J Glas Sci Technol Part B vol 50 no 3 pp 224ndash228 2009
[141] P J Bray S Feller G E Jellison and Y H Yun ldquoB10 NMR studies of the structure of
borate glassesrdquo J Non Cryst Solids vol 38ndash39 no PART 1 pp 93ndash98 May 1980 doi
1010160022-3093(80)90400-7
[142] N Andersson and G L Comer ldquoRelativistic Fluid Dynamics Physics for Many Different
Scales Living Reviews in Relativityrdquo Living Rev Relativ vol 10 p 1 2007 Accessed
Feb 18 2019 [Online] Available httpwwwlivingreviewsorglrr-2007-
1httprelativitylivingreviewsorghttpwwwmathssotonacukstaffAnderssonhttpw
wwslueducollegesASphysicsprofscomerhtml
[143] P Ilg and H C Oumlttinger ldquoNonequilibrium relativistic thermodynamics in bulk viscous
185
cosmologyrdquo Phys Rev D - Part Fields Gravit Cosmol vol 61 no 2 p 023510 2000
doi 101103PhysRevD61023510
[144] P Romatschke ldquoRelativistic viscous fluid dynamics and non-equilibrium entropyrdquo Class
Quantum Gravity vol 27 no 2 p 025006 Jan 2010 doi 1010880264-
9381272025006
[145] D C Allan ldquoInverting the MYEGA equation for viscosityrdquo J Non Cryst Solids vol
358 no 2 pp 440ndash442 Jan 2012 doi 101016JJNONCRYSOL201109036
[146] M Potuzak R C Welch and J C Mauro ldquoTopological origin of stretched exponential
relaxation in glassrdquo J Chem Phys vol 135 no 21 p 214502 Dec 2011 doi
10106313664744
[147] L Ding M Thieme S Demouchy C unisch and B J P aus ldquoEffect of pressure and
temperature on viscosity of a borosilicate glassrdquo J Am Ceram Soc vol 101 no 9 pp
3936ndash3946 2018 doi 101111jace15588
[148] R Richert and M Richert ldquoDynamic heterogeneity spatially distributed stretched-
exponential patterns and transient dispersions in solvation dynamicsrdquo Phys Rev E vol
58 no 1 pp 779ndash784 1998 doi 101103PhysRevE58779
[149] P Richet ldquoViscosity and configurational entropy of silicate meltsrdquo Geochim Cosmochim
Acta vol 48 no 3 pp 471ndash483 Mar 1984 doi 1010160016-7037(84)90275-8
[150] P Gupta and J C Mauro ldquoTwo factors governing fragility Stretching exponent and
configurational entropyrdquo Phys Rev E vol 78 no 6 p 063501 2008 doi
101103PhysRevE78062501
[151] Z Zheng J C Mauro and D C Allan ldquoModeling of delayed elasticity in glassrdquo J Non
Cryst Solids vol 500 no August pp 432ndash442 2018 doi
101016jjnoncrysol201808033
[152] R Boumlhmer L Ngai C A Angell and D J Plazek ldquoNonexponential relaxations in
186
strong and fragile glass formersrdquo J Chem Phys vol 99 no 5 pp 4201ndash4209 1993
doi 1010631466117
[153] O Gulbiten J C Mauro and P Lucas ldquoRelaxation of enthalpy fluctuations during sub-
Tg annealing of glassy seleniumrdquo J Chem Phys vol 138 no 24 p 244504 2013 doi
10106314811488
[154] D Sidebottom R Bergman L Boumlrjesson and L M Torell ldquoTwo-step relaxation decay
in a strong glass formerrdquo Phys Rev Lett vol 71 no 14 pp 2260ndash2263 1993 doi
101103PhysRevLett712260
[155] G M Bartenev and V A Lomovskoi ldquoRelaxation time spectra and the peculiarities of
the process of boron anhydride glass transitionrdquo J Non Cryst Solids vol 146 no C pp
225ndash232 1992 doi 101016S0022-3093(05)80495-8
[156] A Heuer ldquoExploring the potential energy landscape of glass-forming systems From
inherent structures via metabasins to macroscopic transportrdquo J Phys Condens Matter
vol 20 no 37 p 373101 2008 doi 1010880953-89842037373101
[157] J C Phillips ldquoStretched exponential relaxation in molecular and electronic gasesrdquo
Reports Prog Phys vol 59 pp 1133ndash1207 1996
[158] P Richet ldquoResidual and configurational entropy Quantitative checks through applications
of Adam-Gibbs theory to the viscosity of silicate meltsrdquo J Non Cryst Solids vol 355
no 10ndash12 pp 628ndash635 2009 doi 101016jjnoncrysol200901027
[159] C J Wilkinson Doss G Palmer and J C Mauro ldquoThe relativistic glass transition A
thought experimentrdquo J Non-Crystalline Solids X vol 2 2019 doi
101016jnocx2019100018
[160] J C Mauro R J Loucks and S Sen ldquoHeat capacity enthalpy fluctuations and
configurational entropy in broken ergodic systemsrdquo J Chem Phys vol 133 no 16 pp
1ndash9 2010 doi 10106313499326
187
[161] J C Mauro R J Loucks and P Gupta ldquoMetabasin approach for computing the
master equation dynamics of systems with broken ergodicityrdquo J Phys Chem A vol 111
no 32 pp 7957ndash7965 2007 doi 101021jp0731194
[162] M Wang N M Anoop Krishnan B Wang M M Smedskjaer J C Mauro and M
Bauchy ldquoA new transferable interatomic potential for molecular dynamics simulations of
borosilicate glassesrdquo J Non Cryst Solids vol 498 no December 2017 pp 294ndash304
2018 doi 101016jjnoncrysol201804063
[163] I Avramov and A Milchev ldquoEffect of disorder on diffusion and viscosity in condensed
systemsrdquo J Non Cryst Solids vol 104 no 2ndash3 pp 253ndash260 Sep 1988 doi
1010160022-3093(88)90396-1
[164] G Scherer Relaxation in Glass and Composites Krieger Publishing Company 1992
[165] L Huang J Nicholas J Kieffer and J Bass ldquoPolyamorphic transitions in vitreous B2O3
under pressurerdquo J Phys Condens Matter vol 20 no 7 2008 doi 1010880953-
8984207075107
[166] W Capps P B Macedo B OrsquoMeara and T A Litovitz ldquoTemperature Dependecne of
the High-Frequency Moduli of Vitreous B2O3rdquo J Chem Phys vol 45 p 3431 1966
doi 107868s0002337x1402002x
[167] D R Cassar A M Rodrigues M L F Nascimento and E D Zanotto ldquoThe diffusion
coefficient controlling crystal growth in a silicate glass-formerrdquo Int J Appl Glas Sci
vol 9 no 3 pp 373ndash382 Jul 2018 doi 101111ijag12319
[168] A R Potter C J Wilkinson S H im and J C Mauro ldquoEffect of Water on
Topological Constraints in Silica Glassrdquo Scr Mater vol 160 pp 48ndash52 2019 doi
101016jscriptamat201809041
[169] D B Asay and S H im ldquoEvolution of the adsorbed water layer structure on silicon
oxide at room temperaturerdquo J Phys Chem B vol 109 pp 16760ndash16763 2005 doi
188
101021jp053042o
[170] M Tomozawa ldquoWater diffusion oxygen vacancy annihilation and structural relaxation in
silica glassesrdquo J Non Cryst Solids vol 179 pp 162ndash169 1994 doi 1010160022-
3093(94)90693-9
[171] S Kapoor R E Youngman K Zakharchuk A Yaremchenko N J Smith and A Goel
ldquoStructural and Chemical Approach toward nderstanding the Aqueous Corrosion of
Sodium Aluminoborate Glassesrdquo J Phys Chem B vol 122 pp 10913ndash10927 2018 doi
101021acsjpcb8b06155
[172] Y Yu N M A rishnan M M Smedskjaer G Sant and M Bauchy ldquoThe hydrophilic-
to-hydrophobic transition in glassy silica is driven by the atomic topology of its surfacerdquo
J Chem Phys vol 148 p 74503 2018 doi 10106315010934
[173] Y Yu B Wang M Wang G Sant and M Bauchy ldquoReactive Molecular Dynamics
Simulations of Sodium Silicate Glasses mdash Toward an Improved Understanding of the
Structurerdquo Int J Appl Glas Sci vol 8 no 3 pp 276ndash284 2017 doi
101111ijag12248
[174] E Stolper ldquoThe speciation of water in silicate meltsrdquo Geochim Cosmochim Acta vol
46 no 12 pp 2609ndash2620 Dec 1982 doi 1010160016-7037(82)90381-7
[175] M G Mesko P A Schader and J E Shelby ldquoWater solubility and diffusion in sodium
silicate meltsrdquo Phys Chem Glas vol 43 no 6 pp 283ndash290 2002 doi
101016jsemradonc201010001
[176] M Tomozawa M Takata J Acocella E Bruce Watson and T Takamori ldquoThermal
properties of Na2Omiddot3SiO2 glasses with high water contentrdquo J Non Cryst Solids vol 56
no 1ndash3 pp 343ndash348 Jul 1983 doi 1010160022-3093(83)90491-X
[177] E A Leed and C G Pantano ldquoComputer modeling of water adsorption on silica and
silicate glass fracture surfacesrdquo J Non Cryst Solids vol 325 pp 48ndash60 2003 doi
189
101016S0022-3093(03)00361-2
[178] J C Fogarty H M Aktulga A Y Grama A C T van Duin and S A Pandit ldquoA
reactive molecular dynamics simulation of the silica-water interfacerdquo J Chem Phys vol
132 no 17 p 174704 2010 doi 10106313407433
[179] S H Hahn et al ldquoDevelopment of a ReaxFF Reactive Force Field for NaSiOx Water
Systems and Its Application to Sodium and Proton Self-Diffusionrdquo J Phys Chem C vol
122 no 34 pp 19613ndash19624 2018 doi 101021acsjpcc8b05852
[180] T S Mahadevan W Sun and J Du ldquoDevelopment of Water Reactive Potentials for
Sodium Silicate Glassesrdquo J Phys Chem B vol 123 pp 4452ndash4461 2019 doi
101021acsjpcb9b02216
[181] A C T van Duin A Strachan S Stewman Q Zhang X Xu and W A Goddard III
ldquoReaxFF SiO Reactive Force Field for Silicon and Silicon Oxide Systemsrdquo J Phys
Chem A vol 107 no 19 pp 3803ndash3811 2003 doi 101021jp0276303
[182] M M Smedskjaer J C Mauro R E Youngman C L Hogue M Potuzak and Y Yue
ldquoTopological principles of borosilicate glass chemistryrdquo J Phys Chem B vol 115 no
44 pp 12930ndash12946 2011 doi 101021jp208796b
[183] Q Zheng M Potuzak J C Mauro M M Smedskjaer R E Youngman and Y Yue
ldquoComposition-structure-property relationships in boroaluminosilicate glassesrdquo J Non
Cryst Solids vol 358 pp 993ndash1002 2012 doi 101016jjnoncrysol201201030
[184] K Rompicharla D I Novita P Chen P Boolchand M Micoulaut and W Huff
ldquoAbrupt boundaries of intermediate phases and space filling in oxide glassesrdquo J Phys
Condens Matter vol 20 p 202101 2008 doi 1010880953-89842020202101
[185] M Micoulaut and J C Phillips ldquoRings and rigidity transitions in network glassesrdquo Phys
Rev B vol 67 p 104204 2003 doi 101103PhysRevB67104204
[186] H Liu S Dong L Tang N M A rishnan G Sant and M Bauchy ldquoEffects of
190
polydispersity and disorder on the mechanical properties of hydrated silicate gelsrdquo J
Mech Phys Solids vol 122 pp 555ndash565 Jan 2019 doi 101016JJMPS201810003
[187] A Tilocca N H De Leeuw and A N Cormack ldquoShell-model molecular dynamics
calculations of modified silicate glassesrdquo Phys Rev B - Condens Matter Mater Phys
vol 73 p 104209 2006 doi 101103PhysRevB73104209
[188] J Du and A N Cormack ldquoThe medium range structure of sodium silicate glasses A
molecular dynamics simulationrdquo J Non Cryst Solids vol 349 no 1ndash3 pp 66ndash79 2004
doi 101016jjnoncrysol200408264
[189] N P Bansal and R H Doremus Handbook of glass properties Academic Press 1986
[190] S H Hahn and A C T van Duin ldquoSurface Reactivity and Leaching of a Sodium Silicate
Glass under an Aqueous Environment A ReaxFF Molecular Dynamics Studyrdquo J Phys
Chem C p acsjpcc9b02940 2019 doi 101021acsjpcc9b02940
[191] S P Jaccani and L Huang ldquo nderstanding Sodium Borate Glasses and Melts from Their
Elastic Response to Temperaturerdquo Int J Appl Glas Sci vol 7 no 4 pp 452ndash463 Dec
2016 doi 101111ijag12250
[192] Q Zheng Y Yue and J C Mauro ldquoDensity of topological constraints as a metric for
predicting glass hardnessrdquo Appl Phys Lett vol 111 no 1 p 011907 Jul 2017 doi
10106314991971
[193] J C Dyre ldquoElastic models for the non-Arrhenius relaxation time of glass-forming
liquidsrdquo in AIP Conference Proceedings 2006 vol 832 pp 113ndash117 doi
10106312204470
[194] Y T Cheng and C M Cheng ldquoRelationships between hardness elastic modulus and the
work of indentationrdquo Appl Phys Lett vol 73 no 5 pp 614ndash616 Jul 1998 doi
1010631121873
[195] A Makishima and J D Mackenzie ldquoDirect calculation of Youngrsquos modulus of glassrdquo J
191
Non Cryst Solids vol 12 no 1 pp 35ndash45 1973 doi 1010160022-3093(73)90053-7
[196] G Pilania C Wang X Jiang S Rajasekaran and R Ramprasad ldquoAccelerating materials
property predictions using machine learningrdquo Sci Rep vol 3 pp 1ndash6 2013 doi
101038srep02810
[197] H He and M Thorpe ldquoElastic properties of glassesrdquo Phys Rev Lett vol 54 no 19 pp
2107ndash2110 1985 Accessed Aug 29 2018 [Online] Available
httpsjournalsapsorgprlpdf101103PhysRevLett542107
[198] M Thorpe ldquoElastic properties of glassesrdquo Cambridge Arch 1985 doi 101557PROC-
61-49
[199] J C Phillips ldquoChemical Bonding Internal Surfaces and the Topology of Non-Crystalline
Solidsrdquo Phys status solidi vol 101 no 2 pp 473ndash479 Oct 1980 doi
101002pssb2221010204
[200] M M Smedskjaer J C Mauro and Y Yue ldquoPrediction of glass hardness using
temperature-dependent constraint theoryrdquo Phys Rev Lett vol 105 no 11 2010 doi
101103PhysRevLett105115503
[201] J D Musgraves J Hu and L Calvez Springer Handbook of Glass Springer US 2021
[202] C Hermansen X Guo R E Youngman J C Mauro M M Smedskjaer and Y Yue
ldquoStructure-topology-property correlations of sodium phosphosilicate glassesrdquo J Chem
Phys vol 143 no 6 p 064510 Aug 2015 doi 10106314928330
[203] L Wondraczek et al ldquoTowards ltrastrong Glassesrdquo Adv Mater vol 23 no 39 pp
4578ndash4586 Oct 2011 doi 101002adma201102795
[204] D Ravaine ldquoGlasses as solid electrolytesrdquo J Non Cryst Solids vol 38ndash39 no PART 1
pp 353ndash358 1980 doi 1010160022-3093(80)90444-5
[205] D Ravaine and J L Souquet ldquoA thermodynamic approach to ionic conductivity in oxide
glasses Part 1 Correlation of the ionic conductivity with the chemical potential of alkali
192
oxide in oxide glassesrdquo Phys Chem Glas vol 18 no 2 pp 27ndash31 1977
[206] J Swenson and L Boumlrjesson ldquoCorrelation between Free Volume and Ionic Conductivity
in Fast Ion Conducting Glassesrdquo Phys Rev Lett vol 77 no 17 pp 3569ndash3572 1996
doi 101103PhysRevLett773569
[207] C B Bragatto A C M Rodrigues and J L Souquet ldquoDissociation Equilibrium and
Charge Carrier Formation in AgI-AgPO3 Glassesrdquo J Phys Chem C vol 121 no 25 pp
13507ndash13514 2017 doi 101021acsjpcc7b02477
[208] S W Martin R Christensen G Olson J ieffer and W Wang ldquoNew Interpretation of
Na Ion Conduction in and the Structures and Properties of Sodium Borosilicate Mixed
Glass Former Glassesrdquo J Phys Chem C vol 140 pp 6343ndash6352 2019 doi
101021acsjpcc8b11735
[209] O L Anderson and D A Stuart ldquoCalculation of Activation Energy of Ionic Conductivity
in Silica Glasses by Classical Methodsrdquo J Am Ceram Soc vol 37 no 12 pp 573ndash580
1954 doi 101111j1151-29161954tb13991x
[210] R S Welch C J Wilkinson J C Mauro and C B Bragatto ldquoCharge Carrier Mobility
of Alkali Silicate Glasses Calculated by Molecular Dynamicsrdquo Front Mater vol 6 p
121 May 2019 doi 103389fmats201900121
[211] Y J Zhang ldquoEntropy and ionic conductivityrdquo Phys A Stat Mech its Appl vol 391 no
19 pp 4470ndash4475 2012 doi 101016jphysa201204021
[212] J C Dyre ldquoOn the mechanism of glass ionic conductivityrdquo J Non Cryst Solids vol 88
no 2ndash3 pp 271ndash280 1986 doi 101016S0022-3093(86)80030-8
[213] M Steve ldquoIonic Conduction in Phosphate Glassesrdquo JAmCeramSoc vol 74 no 8 pp
1767ndash1784 1991 doi 101111j1151-29161991tb07788x
[214] C T Moynihan A J Easteal J Wilder and J Tucker ldquoDependence of the Glass
Transition Temperature on Heating and Cooling Raterdquo J Phys Chem vol 78 no 26 pp
193
2673ndash2677 1974 Accessed Jul 13 2018 [Online] Available
httpspubsacsorgsharingguidelines
[215] Ito C T Moynihan and C A Angell ldquoThermodynamic determination of fragility in
liquids anda fragile-to-strong liquid transition inwaterrdquo Lett to Nat vol 398 no April
p 492 1999
[216] L Ngai and S W Martin ldquoCorrelation between the activation enthalpy and
Kohlrausch exponent for ionic conductivity in oxide glassesrdquo Phys Rev B vol 40 no
15 pp 10550ndash10556 1989 doi 101103PhysRevB4010550
[217] M Potuzak X Guo M M Smedskjaer and J C Mauro ldquoAre the dynamics of a glass
embedded in its elastic propertiesrdquo J Chem Phys vol 138 no 12 2013 doi
10106314730525
[218] A L Agapov and A P Sokolov ldquoDecoupling ionic conductivity from structural
relaxation A way to solid polymer electrolytesrdquo Macromolecules vol 44 no 11 pp
4410ndash4414 2011 doi 101021ma2001096
[219] S V Nemilov ldquoA Structural Investigation of Glasses in the B2O3ndashNa2O System by the
Viscosimetric Methodrdquo Izv Akad Nauk SSSR Neorg Mater vol 2 no 2 pp 349ndash359
1966
[220] J P Poole ldquoLow‐Temperature Viscosity of Alkali Silicate Glassesrdquo J Am Ceram Soc
vol 32 no 7 pp 230ndash233 1949 doi 101111j1151-29161949tb18952x
[221] R noche D B Dingwell F A Seifert and S L Webb ldquoNon-linear properties of
supercooled liquids in the system Na2O SiO2rdquo Chem Geol vol 116 pp 1ndash16 1994 doi
1010160009-2541(94)90154-6
[222] W Martinsen ldquoSelected properties of sodium silicate glasses and their structural
significancerdquo Dissertation 1969
[223] C Hermansen J C Mauro and Y Yue ldquoA model for phosphate glass topology
194
considering the modifying ion sub-networkrdquo J Chem Phys vol 140 no 15 2014 doi
10106314870764
[224] M L F Nascimento and E Dutra Zanotto ldquoDoes viscosity describe the kinetic barrier for
crystal growth from the liquidus to the glass transitionrdquo J Chem Phys vol 133 no 17
p 174701 2010 doi 10106313490793
[225] D Denisko and M M Hoffman ldquoClassification and interaction in random forestsrdquo Proc
Natl Acad Sci U S A vol 115 no 8 pp 1690ndash1692 2018 doi
101073pnas1800256115
[226] G Varney C Dema B E Gul C J Wilkinson and Akgun ldquo se of machine learning
in CARNA proton imagerrdquo in Progress in Biomedical Optics and Imaging - Proceedings
of SPIE 2019 vol 10948 doi 101117122512565
[227] ldquoMDLregSciGlass-6 MDL Information Systems San Leandro CArdquo 2003
[228] D R Cassar ldquoViscNet Neural Network for predicting the fragility index and the
temperature-dependency of viscosityrdquo pp 1ndash33 2020
[229] D J L Brett A Atkinson N P Brandon and S J Skinner ldquoIntermediate temperature
solid oxide fuel cellsrdquo Chem Soc Rev vol 37 no 8 pp 1568ndash1578 2008 doi
101039b612060c
[230] T Ishiyama S Suzuki J Nishii T Yamashita H awazoe and T Omata ldquoProton
conducting tungsten phosphate glass and its application in intermediate temperature fuel
cellsrdquo Solid State Ionics vol 262 pp 856ndash859 2014 doi 101016jssi201310055
[231] T Yamaguchi et al ldquoProton-conducting phosphate glass and its melt exhibiting high
electrical conductivity at intermediate temperaturesrdquo J Mater Chem A vol 6 no 46 pp
23628ndash23637 2018 doi 101039C8TA08162J
VITA
Collin Wilkinson was born in Mt Carroll IL He attended Coe College as an undergraduate and
studied glass resulting in a physics bachelorrsquos degree in 201 He worked with Dr Ugur Akgun
and Dr Steve Feller on modeling different key properties of glass propertiesresponses to external
stimuli Collin joined the group of Dr John Mauro at Pennsylvania State University in 2018
List of publications written by first author (or co-first author) Collin while at Penn State
1 CJ Wilkinson JC Mauro Explorerpy Mapping the Energy Landscapes of Complex
Materials SoftwareX Submitted
2 CJ Wilkinson DR Cassar AV DeCeanne KA Kirchner ME McKenzie ED Zanotto and
JC Mauro Energy Landscape Modeling of Crystal Nucleation NPJ Computational
Materialis Submitted
3 CJ Wilkinson K Doss O Gulbiten DC Allan JC Mauro Fragility and Temperature
Dependence of Stretched Exponential Relaxation in Glass-Forming Systems JACERS
Submitted
4 CJ Wilkinson JC Mauro Comment on ldquoThe Fragility of Alkali Silicate Glass Melts Part
of a niversal Topological Patterrdquo by DL Sidebottom JNCS (2020) ( 2 ) 11
5 CJ Wilkinson K Doss DR Cassar RS Welch CB Bragatto JC Mauro Predicting Ionic
Diffusion in Glass From its Relaxation Behavior Journal of Physical Chemistry B (2020)
124 (6) 1099-1103
6 Z Ding CJ Wilkinson J Zheng Y Lin H Liu J Shen SH Kim Y Yue J Rent CJ Mauro
Q Zheng Topological Understanding of the Mixed Alkaline Earth Effect in Glass JNCS
(2020) (526) 119696
7 CJ Wilkinson AR Potter RS Welch CB Bragatto Q Zheng M Bauchy M Affatigato SA
Feller JC Mauro Topological Origins of Mixed Alkali Effect in Glass Journal of Physical
Chemistry B (2019) 123 (34) 7482
8 CJ Wilkinson K Doss SH Hahn Nathan Keilbart AR Potter NJ Smith I Dabo ACT Van
Duin SH Kim JC Mauro Topological Control of Water Reactivity on Glass Surfaces
Evidence of a Chemically Stable Intermediate Phase JPCL (2019) 10 (14) 3955
9 CJ Wilkinson K Doss G Palmer JC Mauro The Relativistic Glass Transition A Through
Experiment JNCSX (2019) 100018
10 CJ Wilkinson Q Zheng L Huang JC Mauro Topological Constraint Model for the
Elasticity of Glass-Forming Systems JNCSX (2019) 100019
11 AR Potter CJ Wilkinson SH Kim JC Mauro Effect of Water on Toplogical Constraints
in Silica Glass Scripta Materialia (2019) (160) 48-52
12 CJ Wilkinson E Pakhomenko MR Jesuit AV DeCeanne B Hauke M Packard SA Feller
JC Mauro Topological Constraint Model of Alkali Tellurite Glasses JNCS (2018) (502)
172
13 CJ Wilkinson YZ Mauro JC Mauro RelaxPy Python code for Modeling of Glass
Relaxation Behavior SoftwareX (2018) (7) 255
