Microstructure of a soft glass Béla Joós Matthew L. Wallace Michael Plischke (SFU)

Microstructure of

a soft glass

Béla Joós

Matthew L. WallaceMichael Plischke (SFU)

Queen's CSE Colloquium, October 2007

Glass is a phase of matter

• Glasses are ubiquitous in nature • A glass is a phase such as the solid or gaseous, or liquid

phases as opposed to a type of material• It is a disordered phase, amorphous, like a liquid frozen in

time


Phase Transitions

• In nature there are various kinds of transitions:

First order: ex: solid -> liquid, jump in physical observables such as volume, or energy

Continuous: ex: Gelling transition as an example of percolation transition(the gel is rigid, i.e. resists shear)


Mechanical vs entropic rigidity

• Triangular lattice: geometric percolation at p=pc (0.349), rigidity percolation p= pr > pc (pr = 0.66) .

• Multiple connectivity required for mechanical rigidity


The Glass transition

• The glass-maker’s viewpoint:

at TG viscosity= 1012 Pa s

• A continuous transition characterized by a divergence in viscosity

• As to what really happens microscopically, there is really no consensus. There are a number of competing pictures

Conference: Mechanical Behaviour of Glassy Materials (UBC, July 2007)


The three viewpoints

1. A transition to an ideal zero entropy state

2. A dynamical transition resulting from the jamming of particles together

3. Not a transition but a cross-over where there is a rapid change in viscosity (critical slowing down)


Some facts to illustrate the issues

• Heat capacity: heat transferred into object as its temperature is raised

• In experiments: T raised by increments ΔT during time Δt

• Drop in Cp, critical slowing down

The three viewpoints have common features (slowing down), but very different views of the glass. How to distinguish them?


The challenges

• As T decreases, slowing down in the system, increasing run times to simulate anything (also an issue experimentally)

• Configuration space very complex often represented as an energy landscape

• Glasses age: they continuously evolve

Glasses evolve towards lower energy states: consequently longer relaxation times

Bouchaud (2000)

Have to find clever ways to characterize the glass


Our perspective

• Model: a short chain polymer melt (10 monomers) (e.g. plastic)

• The glass transition and the onset of rigidity

• Shearing the glass: the elastic and plastic regimes

• Microstructure of the deformed glass: displacements, stresses,


Molecular Dynamics of a Polymer Glass

• Polymer “melt” of ~1000 particles with chains of length 10.

• LJ interactions between all particles• + FENE potential between nearest

neighbours in a chain (Kremer and Grest, 1990)

• Competing length scales prevent crystallization

FENE

L-J

L-J

L-J

L-J


Approaching the Glass Transition

• Instead of approaching the final states along isobars by lowering T (very high cooling rates)

• We propose an isothermal compression method (blue curves) for better exploration of phase space

• System gets “stuck” in wells of lower potential energy

• Below TG, the system is closer to equilibrium (less aging)

P

T

Initial state

TG

Final States


• Equilibrate in the NVT ensemble with Brownian dynamics as a thermostat

• Apply a steady compression rate of 0.015

• Final volume realized in the NPT ensemble with a damped-force algorithm

)(2

2

tWdt

dxm

x

U

dt

xdm i

i

i

ii

external “piston” force regulates pressure

Numerical algorithms


The glass transition temperature TG

Φ: Packing Fraction

• At TG, there is kinetic arrest, the liquid can no longer change configurations

(expt. time scale issue). TG determined by a change in the volume density.

• We obtain TG = 0.465 + 0.005

• But we cannot assume

TG to be the rigidity onset: the viscosity does not diverge at TG.


Outline

• Our way of preparing the polymer melt near the glass transition: pressure quench at constant temperature to improve statistics

• Onset of rigidity in the glass: a new angle on the glass transition

• Deforming the glass below the rigidity transition: the elastic and plastic regime

• Macroscopic signatures• Changes in the microstructure• What is learned, what needs to be learned.


Rigidity of Mechanical Structures


Onset of mechanical rigidity

Triangular lattice: geometric percolation at p=pc (0.349), rigidity percolation p= pr > pc (pr = 0.66) .

Multiple connectivity required for mechanical rigidity

in disordered systems


Entropic rigidity

At T>0 K, rigidity sets in at the onset of geometric percolation,

through the creation of an entropic

spring

Plischke and Joos, PRL 1998

Moukarzel and Duxbury, PRE 1999


The entropic spring

RNa

TkB2

3force =

It is a Gaussian spring (zero equilibrium length) whose strength is proportional to the temperature T


The onset of rigidity in melts

With permanent crosslinks, at a fixed temperature:

Well defined point of onset of the entropic rigidity : It is geometric percolation pc where there is a diverging length scale (such as in rubber)


Rigidity in melts without crosslinks

• Not clear where the onset is

• Is it at TG that we have percolating regions of “jammed” or immobile particles that can carry the strain?

Wallace, Joos, Plischke, PRE 2004


Calculating the shear viscosity

• Using the intrinsic fluctuations in the system:

The shear viscosity equals:


Viscosity diverges at onset of rigidity

Empirical models of :• VFT (Vogel-Fulcher-

Tamann)

(T0 associated with an “ideal” glass state)

T0 = 0.41 + 0.02 Tc=0.422 + 0.006

• dynamical scaling (Colby, 2000)

0

exp~TT

Eact

kT

E

T

TT

C

C exp~9

measured to T=0.49 > TG=0.465 extrapolation required


Calculating the shear modulus

Two ways: • Applying a finite affine deformation

• Using the intrinsic fluctuations in the system driven by temperature to obtain its shear strength, as the limit to ∞ of G(t) called Geq

where


Geq or extrapolating G(t) to infinity

Power law fit of tail:

G(t) = Geq + A t-

G'eq = G(t=150)

Geq = G(t=)


The shear modulus : Geq vs s

s (=0.1) < < Geq

These µ’s are the response of the system to the finite deformation and not the shear modulus of the deformed relaxed system


The shear modulus G'eq , Geq , and μs

G'eq : short time(t=150)

Geq : extrapolatedto infinity*

μs : applied shear

Rigidity onset at T1 =0.44 < TG = 0.465

* using distribution of energy barriers observed during first t=150


Meaning of T1: the onset of rigidity

T1

T0 (0.41) and Tc (0.422) gave extrapolated values for the onset of rigidity. Measurement of stopped at 0.49 (TG = 0.465)

T1 = 0.44 is the onset ofGeq and s, and the cusp in CP, the heat capacity(is it the appearance of floppy modes with rising T ?)


Issues on rigidity in the polymer glass

•TG is the temperature at which the melt stops flowing. It is not a point of divergence of the viscosity (For glass makers: s= 1012 Pa ·s or = s / G = 400 s for SiO2

In simulations: s= 107 or = s / G = 105

(simulations 103, unit of time: 2 ps)(issues of time scale and aging)

•Comparison with gelation due to permanent crosslinks: no clearly defined length scale, but there could be a dynamical one

•Onset of rigidity: divergence of viscosity, onset of shear modulus, cusp in heat capacity (disappearance of floppy modes)


Polymer glass under deformation

• Glasses are heterogeneous

• What happens to the glass when deformed: a lot of questions from aging, mechanical properties, and thermal properties

• Which properties are we interested in this study? We will focus on the microstructure as a first step in understanding the effect of deformation on the properties of the glass.

Main message: deformation reduces heterogeneity


Properties of the deformed “rigid” glassy system

• Glassy system just below a temperature T1 (“rigidity threshold”): very little cooperative movement (except at long timescales)

• Previous study: examining mechanical properties of a polymer glass (e.g. shear modulus) across TG .

TG

T1 TMCSamples used to investigate effects of shear (present work)

Wallace and Joos, PRL 2006


Plastic and elastic deformations

• Glassy systems have a clear yield strain

• What specific local dynamical and structural changes occur?

0,00 0,05 0,10 0,15 0,20 0,25

0,55

0,60

0,65

0,70

0,75

0,80

0,85

0,90

P

Pressure variations in an NVT ensemble

Plastic


Decay of the shear stress after deformation

Shows both the initial stress and the subsequent decay in the system


Structural changes (1)

• Changes in the energy of the inherent structures (eIS) are relevant to subtle structural changes

• Initial decrease / increase in polymer bond length for elastic / plastic deformations

• Plastic deformations create a new “well” in the PEL – different from those explored by slow relaxations in a normal aging process

• In “relaxed”, deformed system, changes in the energy landscape are entirely due to L-J interactions

0,00 0,08 0,16 0,24

11,05

11,10

11,15

Total Energy

e IS

15,956

15,960FENE potential

0,00 0,08 0,16 0,24

-4,92

-4,86

-4,80LJ potential

Immediately after deformationAfter tw=103 time units


Local bond-orientational order parameter Q6

• Q6 measures subtle angular correlations (towards an FCC structure) between particles at long time tw after deformations

• We can resolve a clear increase in Q6 for elastic deformations, but limited impact on system dynamics


Diffusion

N

i

ii rtrNdt

d

1

2)0()(

1

6

1D

Effect of "caging" observed near the transition (T G = 0.465).

At TG, still possibility to rearrange under deformation.


Glasses are heterogeneous

Widmer-Cooper, Harrowel, Fynewever, PRL 2004

The propensity reveals more acurately the fast and slow regions than a single run

Propensity: Mean squared deviation of the displacements of a particle in different iso-configurations


Mobility and “sub-diffusion”

• Initially, plastic shear forces the creation of “mobile” regions of mobile particles

• Once the system is allowed to relax, cooperative re-arrangements remain possible

• Rearrangements from plastic deformations allow cage escape in more regions

• In the case of elastic deformations, new mobile particles can be created, but only temporarily

100 1000 10000

0,1

plastic

elastic / reference

beginning of sub-diffusion

<r

2 (t)>

t

100 10001E-4

1E-3

0,01

0,1

plastic

elastic / reference

fra

ctio

n o

f m

ob

ile p

art

icle

s


Heterogeneous dynamics

• The non-Gaussian parameter α2(t) measures deviations from Gaussian behavior

• Deviations from a Gaussian distribution become less apparent for plastic deformations

0,0 0,5 1,0 1,50,0

0,1

0,2

0,3

0,4

0,5

long times

short times

fixed mobile

P(

r)

r


Cooperative movement

• The dynamical heterogeneity is spatially correlated• The peak of α2(t) coincides with the beginning of sub-diffusive behavior – can

indicate a maximum in “mobile cluster” size

Snapshots of dynamically heterogeneous systems. Left: the clusters are localized. Right: as cluster size increases, significant large-scale relaxation is possible.


Effect of shear on the microstructure

• Based on changes in L-J potentials and the formation of larger mobile clusters, plastic deformations must induce substantial local reconfigurations


Fraction of nearest neighbours which

are the fastest 5% the slowest 5%

ε = 0, reference system, ε = 0.2, smaller domains of fast and slow particles


Fraction of n-n’s on the same chain

which are the fastest which are the slowest 5%

This means that the islands of fast particles are getting smaller


Average distance between fast particles

• Evidence of reduction in size of mobile regions and increase in size of jammed regions with increasing deformation

• Increasing jamming in elastic region, as seen in slowest particle

fast particles slow particles


Distances between particles

There is homogenization with applied deformation, most evident

with the fast particles


Glasses age!

Glasses evolve towards lower energy states: consequently longer relaxation times

N

jwjwj

wwq

trtrqiN

ttC

1

)(exp1

),(

Incoherent intermediate scattering function:

Bouchaud, 2000

Kob, 2000


On route to irreversible changes

Statistics of big jumpsshow accelerated equilibrium for large ε, but also that fast regions become smaller.

More stable glass, less aging?


Irreversible microstructural changes

Polymers shrink after deformation

Reduction in grain size or correlations in inhomogeneities


Conclusion (1)

• Real glasses versus glasses on the computer: time scales and a better grasp on the computer of the microstructure

• At the latest conference at UBC on glasses, there was a growing consensus that this is really an issue of critical slowing down, in other words not a real transition


Conclusion (2)

• We have presented attempts to characterize the effect of deformations on the structure of the glass that did not require huge computing times

• The net effect of deformations appears to be connected to general “jamming” phenomena, and what the deformations can do to un-jam the structure

• What they reveal is a more homogeneous glass with a smaller “grain” structure

• More studies are required (highly computer intensive)• Currently working on applying oscillating shear to the glass, and

monitoring the aging of the glasses prepared by shear deformation

Documents

Microstructure of a soft glass Béla Joós Matthew L. Wallace Michael Plischke (SFU)