THERMAL CONDUCTION IN GRANULAR MEDIA:

FROM INTERFACE, TOPOLOGY

TO EFFECTIVE PROPERTY

By

Weijing Dai

A thesis submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Civil Engineering

Faculty of Engineering and Information Technologies

University of Sydney

2019

Under the Supervision of

Dr Yixiang Gan

School of Civil Engineering

University of Sydney

Sydney, Australia

and

Dr Dorian A. H. Hanaor

Institute for Materials Science and Technology

Technische Universität Berlin

Berlin, Germany

i

Statement of Originality

This is to certify that to the best of my knowledge; the content of this thesis is my own work.

This thesis has not been submitted for any degree or other purposes.

I certify that the intellectual content of this thesis is the product of my own work and that all

the assistance received in preparing this thesis and sources have been acknowledged.

Weijing Dai

31/08/2018

ii

Abstract

Granular media are particulate substance featured by their unique discrete structure, which

are commonly seen in daily life and extensively used in industry. Differently from those

continuum materials whose properties are mostly defined by their chemical formulas and

status, granular media further require clarification about the effects of their topology on their

properties. Therefore, effective properties are used to emphasise this distinction in measuring

and describing granular media. In this study, we focus on the effective thermal conductivity

of generalised gas-filled granular media, which is highly related to energy technologies and

advanced fabrication processes. With particularly concentration on the topological transitions

in vibrated granular media, how the topology influences the effective thermal conductivity is

explored.

Aiming at revealing the mechanisms governing the heat conduction of granular media, a

bottom-up consequence scheme is employed in this study by decomposing the macroscale

phenomena into grain-scale interactions. Under such scheme, the objectives of this work are

further divided into (1) investigating the heat conduction mechanisms at inter-grain contact

interfaces and (2) integrating the thermal contact units based on the topology of granular

media. To accomplish the former investigation, the finite element analysis is implemented to

model the gas-solid thermal interaction contributed by the Smoluschowski effect that gives

rise to coupling dependence of gas pressure and grain size. With a systematic study on the

heat conduction of individual units, the later objective is tackled by introducing the grain-

scale thermal interaction into discrete element methods. With the combination of these cross-

scale studies, a numerical framework is established. Furthermore, the thermal measurement

system based on transient plane source techniques is applied to experimentally characterise

correlations between the effective thermal conductivity and external mechanical loading.

These experimental results as well as available literature data are used to quantitatively verify

the proposed numerical method.

In order to figure out the topological influence on the effective thermal conductivity, the

discrete element method is further employed to examine the mesoscale behaviours of agitated

granular media. The grain-scale structural characterisation unravels the topological

transitions in vibration. Granular crystallisation, a process prompting the disorder-to-order

transition, is identified as the major phenomenon and its boundary dependent mechanisms are

iii

proposed. Moreover, the topological influence on the effective thermal conductivity can be

assessed with respect to the crystallisation, i.e., the degree of structure order, of granular

media.

With the fundamental research in this thesis on the heat conduction mechanisms and the

granular crystallisation, the effective thermal conductivity is studied in a full range of scales

from individual grains to bulk media. In summary, we demonstrate and experimentally

validate a multiscale framework to solve the thermal problems in granular media that can also

be applied to other effective conduction properties.

iv

Acknowledgement

The journey has been more than three years since I first landed on Australia and started my

PhD program to explore new fields, both in academic and life. During this wonderful

expedition, I have received countless supports from people I work with as well as live with.

As I am approaching the final stage of my PhD program, I would like to use this moment to

acknowledge those fantastic people.

First, I would express my greatest thanks to my supervisor, Dr. Yixiang Gan without whom

this work would be impossible to complete. His creative thoughts and critical comments

always have the power to elevate the quality of the research in this work. No matter how bold

my idea is, he continuously encourages me to have a go without fear to face any failure,

which has established the confidence in my mind to tackle future challenges. Beyond the

academic, I also benefit from the style he manages the group and personal life, not only by

his words, but also through his conduct. Under his leadership, the group is never a boring

production line of papers and PhDs, but a piece of nutritious soil fertilising new perspectives.

I would also like to appreciate my co-supervisor Dr. Dorian Hanaor. His rigorousness and

wisdom are always supportive regardless to distance even after his move to Germany. His

endeavour, solidity and optimism in facing hardship create an excellent character to follow.

I would also like to thank A/Prof. Gwénaëlle Proust, who gave me the opportunity to become

a tutor to practise my communication and presentation skills. My appreciation will go beyond

Eurasia for Dr Joerg Reimann, who is a great representative for German efficiency and

professionalism, and Dr Claudio Ferrero (R.I.P), whose passion and dedication about

research will always inspire me. Further, I appreciate the efforts from all academic and

administrative staff in School of Civil Engineering. Their fabulous jobs have built such

inspiring environment of sustainable development.

Additionally, I want to send my gratitude to my group mates and research students I have

spent most of the time with. These memorable friendships make the PhD experience

enjoyable. In particular, I would like to mention Dr Chongpu Zhai, Dr Mingchao Liu, Mrs.

Ruoyu Wang, Ms. Shuoqi Li, Mr. Zhang Shi, Mr. Pengyu Huang, Ms. Sophie Liu and two

dear friends from Germany, Dr Simone Pupeschi, Mrs Marigrazia Mascardini.

v

Finally, my heartfelt thanks are delivered to my beloved family, my father Mr. Guosheng Dai

and mother Mrs Shuilian Li, as well as my girlfriend, Ms. Yi Zhang. I am always indebted for

their unconditional support and company that make everything possible and delightful. They

are definitely the driving force sustaining me to become a better man.

vi

Publications and Awards

The following publications have been produced as a result of this PhD study:

Journal papers:

1. Dai, W., Pupeschi, S., Hanaor, D., Gan, Y., (2017). Influence of gas pressure on the

effective thermal conductivity of ceramic breeder pebble beds. Fusion Engineering

and Design, 118, 45-51.

2. Dai, W., Reimann, J., Hanaor, D., Ferrero, C., Gan, Y., (2018). Modes of wall

induced granular crystallisation in vibrational packing (under revision).

arXiv: 1805.07865 [cond-mat.dis-nn].

3. Dai, W., Hanaor, D., Gan, Y., (2018). The effects of packing structure on the effective

thermal conductivity of granular media: A grain scale investigation (under revision).

arXiv: 1809.01379 [cond-mat.soft].

4. Cui, G., Liu, M., Dai, W., Gan, Y., (2018). Pore-scale Modelling of Gravity-driven

Drainage in Disordered Porous Media (under revision).

Refereed conference papers:

1. Dai, W., Gan, Y., Hanaor, D., (2015). Effective thermal conductivity of submicron

powders: A numerical study. The 2nd Australasian Conference on Computational

Mechanics (ACCM2015, Nov 30-Dec 1), Brisbane, Australia.

2. Dai, W., Gan, Y., (2017). Measurement of effective thermal conductivity of

compacted granular media by the transient plane source technique. The 8th

International Conference on Micromechanics of Granular Media (P&G2017, 3-7

Jul), Montpellier, France.

The following awards have been received during the course of this study:

1. Postgraduate Research Support Scheme, 2016, 2017, The University of Sydney.

2. Annie B Wilson 2nd Prize, Civil engineering Poster Presentation, 2015, The

University of Sydney.

3. Postgraduate Scholarship in Civil Engineering, 2015-2018, The University of

Sydney.

vii

Authorship Attribution Statement

viii

ix

x

xi

Content

Statement of Originality ........................................................................................................... i

Abstract .................................................................................................................................... ii

Acknowledgement ................................................................................................................... iv

Publications and Awards ........................................................................................................ vi

Authorship Attribution Statement ...................................................................................... vii

Content ..................................................................................................................................... xi

List of Figures ....................................................................................................................... xiii

List of Tables ....................................................................................................................... xvii

List of Symbols ................................................................................................................... xviii

Chapter 1 Introduction............................................................................................................ 1

1.1 General background ................................................................................................... 1

1.2 Research methodology and objectives ....................................................................... 3

1.3 Thesis outline ............................................................................................................. 4

Chapter 2 Literature Review .................................................................................................. 7

2.1 Analytical model ........................................................................................................ 7

2.1.1 Structural models ................................................................................................... 7

2.1.2 Representative geometry methods ......................................................................... 9

2.1.3 Unit cell methods ................................................................................................. 10

2.1.4 Challenges for analytical models ......................................................................... 11

2.2 Discrete element methods ........................................................................................ 12

2.2.1 Simulation principles ........................................................................................... 13

2.2.2 Recent development ............................................................................................. 15

2.2.3 Application on heat transfer of granular media ................................................... 16

2.3 Experimental characterisation .................................................................................. 17

2.3.1 Steady-state and transient-state methods ............................................................. 17

2.3.2 Transient plane source technique ......................................................................... 19

2.3.3 Potential techniques for nano-grain granular media ............................................ 20

xii

2.3.4 Versatility requirement of granular media ........................................................... 21

2.4 Effective properties vs. Topologies ......................................................................... 21

2.4.1 Topological characterisation ................................................................................ 21

2.4.2 Topological transitions in dynamic processes ..................................................... 23

2.4.3 Effective properties determined by topology ....................................................... 25

Chapter 3 Thermal Interaction at Gas-Solid Interface ...................................................... 27

Chapter 4 Conductivity Measurement Using Transient Techniques................................ 35

4.1 A proposal of planar 3ω sensor for nano-grain granular media ............................... 35

4.1.1 Nano-grain granular media .................................................................................. 35

4.1.2 Theoretical principles of 3ω method .................................................................... 36

4.1.3 Measurement system design and sensor fabrication ............................................ 38

4.1.4 Limitation and suggestions .................................................................................. 40

4.2 Transient plane source techniques ........................................................................... 41

4.2.1 Implementation of the HotDisk thermal measurement system ............................ 41

4.2.2 Comparison between the experimental observation and the numerical result ..... 46

Chapter 5 Topological Transition by Vibration ................................................................. 49

Chapter 6 Discrete Element Simulation with Interstitial Gas Phase ................................ 81

Chapter 7 Conclusions ......................................................................................................... 113

7.1 Major contributions ................................................................................................ 113

7.2 Future perspectives ................................................................................................ 114

Bibliography ......................................................................................................................... 117

Appendix I – C++ codes of heat transfer model implemented in LIGGGHTS.............. 133

AI.1 Grain-grain heat transfer ....................................................................................... 133

AI.2 concurrently Grain-wall heat transfer ................................................................... 143

Appendix II – Mathematica scripts for topology characterisation ................................. 149

xiii

List of Figures

Figure 1-1 Decomposition of heat conduction of granular media into sub-scales. ................... 3

Figure 3-1 Helium thermal conductivity predicted by the Gusarov’s model, the temperature

jump model and the Kistler’s model. ....................................................................................... 29

Figure 3-2 Heat conductance coefficient of helium by the truncated Gusarov model

(truncated G.) at 293 K with Helium pressure varying. ........................................................... 30

Figure 3-3 Schematic drawing of contact unit model: (a) two hemispheres in contact without

mechanical loading; (b) The contact under a constant mechanical loading; (c) The FEM

model with temperature profile with an arbitrary unit. Here, ∆𝑇 , ∆𝑥 , ℎg denote the

temperature difference, particle deformation, and heat flux through the gas phase,

respectively. ............................................................................................................................. 31

Figure 3-4 Size distribution of the pebbles. ............................................................................. 31

Figure 3-5 Influence of external mechanical loading on the contact unit of 𝐷 = 350 µm, and

the helium gas pressure is 120 kPa. ......................................................................................... 31

Figure 3-6 Gas pressure related size dependency of the contact units. ................................... 32

Figure 3-7 Effective thermal conductivity predicted by the Gusarov’s model, the SZB model,

the modified Batchelor’s model, and the proposed framework. .............................................. 32

Figure 3-8 Effective thermal conductivity comparison between model predictions and

experiments data from: (a) S. Pupeschi et al. [31] and (b) Abou-Sena, et al [1]. ................... 33

Figure 4-1 Typical configurations of the 3ω methods: (a) Conventional two-dimension

heating by line-source, (b) one-dimension heating by plate-source, and (c) bi-side two-

dimension heating by line-source. ........................................................................................... 38

Figure 4-2 schematic drawings of the 3ω measurement system and the 3ω sensor pattern. ... 39

Figure 4-3 Brief procedures designed for the sensor fabrication in the clockwise sequence

indicated by the U-shape arrow. .............................................................................................. 40

Figure 4-4 Preliminary implement of the fabricated sensors and the measurement circuit. .... 40

Figure 4-5 The simple measurement kit including two rings and a free weights supporter.

(Corresponding to Figure 1 in the attached paper) .................................................................. 43

Figure 4-6 The assembly of the experimental kit. (a) #5465 sensor, the diameter is 3.189 mm;

(b) #5501 sensor, the diameter is 6.403 mm; (c) another assembly for applying mechanical

loading. (Corresponding to Figure 2 in the attached paper) .................................................... 44

xiv

Figure 4-7 The effective thermal conductivity of glass beds consisting of beads in different

diameter range. The x coordinates of the points corresponding to the mid-point of the range.

(Corresponding to Figure 3 in the attached paper) .................................................................. 44

Figure 4-8 The measured effective thermal conductivity of beds versus packing factors and

the calculated Zehner-Schlunder-Bauer conductivity of the same beds. (Corresponding to

Figure 4 in the attached paper) ................................................................................................. 44

Figure 4-9 The effective thermal conductivity versus the external mechanical loadings for

glass beds (Corresponding to Figure 5 in the attached paper) ................................................. 44

Figure 4-10 The effective thermal conductivity versus the external mechanical loadings for

steel beds. (Corresponding to Figure 6 in the attached paper) ................................................. 45

Figure 5-1 Overall evolution of each granular medium subjected to vibrations of different

amplitude. The histograms (colouring with transparency) in the left column show

distributions of the Voronoi cell packing fraction of the initial state and two final relaxed

states after vibration (𝐴=0.1𝑑 and 𝐴=0.2𝑑) with legends giving the corresponding overall

packing fraction 𝛾 and structural index 𝐹6. The corresponding right column plots demonstrate

the time variation of the overall at transient states during vibration........................................ 58

Figure 5-2 Evolutional phase diagram of the crystallisation in granular media of different

height-to-diameter ratio. Each S6 density mapping consists of central plane slices of three

axes, the colour indicated by coarse grained S6 suggests disorder in violet direction and

crystallisation in reddish direction. In the phase diagrams three phases are marked by

background filling, (1) dual-modes cooperating; (2) single-mode prevailing; and (3) sole-

mode dictating which are approximately determined by the competition between cylindrical

and bottom modes. ................................................................................................................... 61

Figure 5-3 Evolution of 𝐹6 for particles groups separated by position. CW – first layer near

the cylindrical wall, BW – first layer near the bottom wall and Core – the bulk particles. ..... 63

Figure 5-4 (a) A similar trend in the 𝑆6 evolution is observed in all cases. The 𝑆6 accumulates

between 10 and 12 in the final state, while the peak right shifts as D increases. (b) Typical

granular temperature evolution (D50) in the small amplitude vibration scenario. (c) Typical

granular temperature evolution (D40) in the large amplitude vibration scenario. (d) Granular

temperature evolution in the relatively strong fluidisation case (D60). (b) and (c) have the

same legend shown in (d) with HC – highly crystallised, LL – liquid like, MO – moderately

ordered, AVE – averaged. ........................................................................................................ 65

xv

Figure 5-5 Smoothed probability density histograms of the crystallised structures appearing

in the granular media during vibration for D30 and D60. The (𝑊4local, 𝑄6

local) coordinates are

used to characterise the structure types. The intersects of pairs of dashed lines in orange and

green are the coordinates of the FCC and HCP, respectively. ................................................. 68

Figure 5-6 Rupture in the HCP structure near the cylindrical wall in D30. (a) Blue particles

are distorted HCP particles ( 0.465 ≤ 𝑄6local ≤ 0.505, 𝑊4

local ≥ 0.08 ) while yellow particles

are rupture particles ( 0.465 ≤ 𝑄6local ≤ 0.505, |𝑊4

local| ≤ 0.02 ) with size scaled by 0.5 for

visibility. (b) Typical rupture section with Particle 2 being the rupture particle. P1, P2, P3 are

the 12-neighbour configurations of Particle 1, 2, 3, respectively. ........................................... 70

Figure 5-7 Density distributions of crystallised structures at the final relaxed state. The right

column displays the corresponding packing structures with particles dyed according to the

(𝑊4local, 𝑄6

local) coordinates in red – HCP, blue – FCC, green – surface hexagon and yellow –

others. The diameters of the particles are rescaled for visualisation purposes. ....................... 71

Figure 5-8 Top – Evolution of the structural index 𝐹6 of the frictionless and frictional

granular media. The top two dashed lines serve as the extension for the final states in the

simulation and the middle one labelled with Exp. C represents the final state of the

experiment performed with a vibration intensity 𝛤 = 2 (Reimann et al., 2017). Bottom – 𝑆6

spatial distributions of the labelled states in the simulated evolution and the experimental

result. Friction coefficient (𝜇) and amplitude (𝐴) values for the simulations are displayed in

the legend. ................................................................................................................................ 73

Figure 5-9 Top – 𝑆6 distributions of the final state in the experiment Exp. C in (Reimann et

al., 2017) and the transient states in the simulations. Particles dyed according to the (𝑊4local,

𝑄6local) coordinates are displayed as insets in the 𝑆6 distribution in red – HCP, blue – FCC and

green – surface hexagon. Bottom – The corresponding ( 𝑊4local , 𝑄6

local ) coordinates

distributions. Friction (𝜇), amplitude (𝐴) and duration (𝑡) parameters for the simulations are

displayed in the legend............................................................................................................. 75

Figure 5-10 Top – 𝑆6 distributions of the final state in the experiment Exp. D in (Reimann et

al., 2017) and the simulations along with the selected 𝑆6 spatial distributions in the insets.

Bottom – The corresponding (𝑊4local, 𝑄6

local) coordinates distributions and the packing of dyed

particles. Friction (𝜇), amplitude (𝐴) and duration (𝑡) values for the simulations are displayed

in the legend. ............................................................................................................................ 76

Figure 6-1 Separated (left) and contacting (right) pairs of grains. .......................................... 88

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Figure 6-2 Separated (left) and contacting (right) geometries coloured with steady state

temperature field in the finite element simulation. .................................................................. 90

Figure 6-3 (a) and (b) compare the heat flow between the finite element simulation (filled

circles) and the original Batchelor & O’Brien model (solid lines), coloured according to 𝛼. (c)

and (d) plot 𝜒 for each 𝛼 and 𝑟cont/𝑟c pair as well as 𝛼 and ℎ/𝑟c pair with joining lines,

respectively. (e) gives the correlation between 𝜒 and 𝛼 and (f) compares the computed heat

flow between the modified model and the simulated result. ................................................... 91

Figure 6-4 Top – Experimental configuration. Bottom – Comparison between simulation

results and the experimental measurement in this work (left) and previously published studies

(right). ...................................................................................................................................... 96

Figure 6-5 𝜑V (– left) and 𝑆6 (– right) increase as the duration of vibration is extended,

noticing that the horizontal axis is arbitrary. ........................................................................... 98

Figure 6-6 Evolution of 𝑆6 spatial patterns in D40 with vibration duration (from left to right).

.................................................................................................................................................. 99

Figure 6-7 (a) The variation of 𝛼 with helium gas pressure in SiO2-He granular media

consisting of 2.3-mm-diameter SiO2 grains at room temperature. (b) and (c) show the

correlations of (𝑘eff, 𝜑V) and (𝑘eff , 𝑆6) in the minimum 𝛼 (≈ 1000), respectively. (d) and (e)

show similar correlations of the maximum 𝛼 (< 10). ........................................................... 100

Figure 6-8 Spatial evolution of 𝑘𝑧𝑧/𝑘s (top row), 𝜃𝑧 (middle) and 𝐻flow (bottom) of the D40

granular media. ...................................................................................................................... 102

Figure 6-9 Left – (𝑆6 , 𝑘𝑧𝑧 ), middle – (𝑆6 , 𝜃𝑧 ) and right – (𝜑V , 𝑘𝑧𝑧 ) plots of D30, D40 and

D50 (from top to bottom respectively), created to according to the mapping operation, in the

scenario of 𝛼 = 100. .............................................................................................................. 104

Figure 6-10 (𝑆6, 𝑘zz) and (𝑆6, 𝜃𝑧) plots of well-ordered packing structure belonging to three

types of granular media in different scenarios. (a) and (b) – 𝛼 ≈ 1000, (c) and (d) – 𝛼 ≈ 10.

................................................................................................................................................ 105

Figure 6-11 The variation of 𝑘eff against the change of packing fraction in artificial media

with disorder and well-order packing structures in 𝛼 = 1000 (left) and 𝛼 = 10 (right)

scenarios. ................................................................................................................................ 106

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List of Tables

Table 2-1 Structural models (heat flowing in the horizontal direction) ..................................... 8

Table 2-2 Representative geometry models (heat flowing in the vertical direction) ................. 9

Table 2-3 Two steady-state measurement schemes ................................................................. 18

Table 2-4 Three transient-state measurement techniques ........................................................ 19

Table 2-5 Comparison of different topological descriptors ..................................................... 23

Table 3-1 Li4SiO4 properties (T=293K) .................................................................................. 31

Table 5-1 Parameters for DEM simulations ............................................................................ 54

Table 6-1 Gas properties used in this work ............................................................................. 95

Table 6-2 Geometry parameters of granular media ................................................................. 97

xviii

List of Symbols

Symbol Meaning 𝑎 Deformation factor in ZSB model [/] 𝑏 Half width of sensors [m] 𝑐 Specific heat [J/(kg⋅K)] 𝑑 Dimension parameter in structural models [/] 𝑒 Restitution coefficient [/] g Gravitational acceleration [m/s2] ℎ Heat transfer coefficient [W/K] �̃� Thermal conductivity parameter in structural models [W/(m⋅K)] 𝑘 Thermal conductivity of continuum media [W/(m⋅K)] 𝑘eff Effective thermal conductivity of granular media [W/(m⋅K)] 𝑚 Mass [kg] 𝑞−1 Thermal penetration depth in the 3ω methods [m] 𝑟 Radius of a medium [m] 𝑡 Time [s] ∆𝑡 Time interval [s] 𝑣 Relative velocity [m/s] 𝐴 Cross-section area [m2] 𝐵 Shape factor in ZSB model [/] 𝐶 Volumetric heat capacity [J/(m3⋅K)] 𝐸 Elastic constant [N/m] 𝐺 Shear modulus [N/m2] 𝐻 Heat flow [W] 𝐼 Moment of inertia [kg⋅m2] 𝐿 Axial length [m] 𝑃 Heating power [W] 𝑅 Electrical resistance [𝛺] 𝑆 Contact stiffness[N/m] 𝑇 Temperature [K] ∆𝑇 Temperature difference [K] 𝑌 Young’s modulus [N/m2] 𝛼 Temperature coefficient of electrical resistance [/] 𝛽 Restitution constant[/] 𝛾 Volume fraction [/]

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Symbol Meaning λ Thermal conductivity ratio of solid phase over fluid/gas phase [/] 𝜑 Porosity [/] 𝛿𝑎 Deformation factor in the Lumped-cubic model [/] 𝛿𝑐 Porosity factor in the Lumped-cubic model [/] 𝜇 Friction coefficient [/] 𝜁 viscoelastic damping constant [kg] 𝛿 overlap distance [m] 𝜐 Poisson ratio [/] 𝜏 Characteristic time in transient plane source techniques [/] 𝜔 Angular frequency [Hz] 𝜅 Thermal diffusivity [m2/s] 𝒓 Rotational radius [m] 𝒙 Position vector [/] �̇�, �̈� Velocity vector [m/s] and acceleration vector [m/s2] 𝑭 Force vector [N] 𝑴 Torque [N⋅m] �̇� Rotational acceleration [rad/s2] Subscript f Fluid phase 𝑖 Phase 𝑖𝑗 Between phase 𝑖 and phase 𝑗 n Normal component nc Normal component in contact force s Solid phase t Tangential component tc Tangential component of contact force *The symbols in this list may be reused in the papers and manuscripts adapted in this thesis but have been specifically illustrated in the corresponding sections.

1

Chapter 1 Introduction

1.1 General background

Granular media, derived from the word granule, are a category of substances composed by

particulate constituents that are usually named as grains. From breakfast cereals on tables to

rock piles in gardens, people impossibly live without connections to granular media.

Similarly, they are handled by nearly all industries including not only conventional sectors

like crop storage, coal combustion and building foundation but also cutting-edge technologies

such as tritium breeding beds in fusion reactors and raw material powders in laser sintering.

If water is excluded, granular media are thought as the most manipulated materials in

industry (Richard et al., 2005) and probably even in daily life. In spite of this ubiquitousness,

the underlying mechanisms determining their properties and explaining their behaviours are

non-intuitive. Compared with their counterpart, the materials of their grains but in a

continuum form, whose macro characteristics are defined by the chemical formulas and

statuses at atomic scale, the observed phenomena emerging in granular media are additionally

but strongly influenced by their distinct discrete structure, the spatial arrangement of those

grains (Wang and Pan, 2008; Zhu et al., 2008). This mesoscale identity contributes to their

extensive usage, because such unique discrete nature brings about superior flowability,

deformability and permeability in granular media (Duran, 2000). However, this nature

inevitably weakens the validity of mechanisms and models based on continuum substances

(de Gennes, 1999; Zhao et al., 2016) and complicates the establishment of principles

governing their properties and behaviours (Zhao et al., 2016). Solution to these puzzles

requires appropriate integration of the grain-scale interactions and topological characteristics

of granular media. Aiming at partly resolving this matter, this work concentrates on heat

conduction in granular media, a non-negligible issue faced by every industry, especially in

this era of energy technologies.

Fundamentally, the steady state heat conduction in continuum media is formulated by the

Fourier’s law as 𝐻HeatFlow = 𝑘𝐴∆𝑇/𝐿 , where 𝐻HeatFlow and ∆𝑇 are considered as the

boundary conditions, the total heat flow and temperature difference, respectively; and 𝐴 as

well as 𝐿 are the geometrical parameters, the cross-sectional area and conductive length,

2

respectively. Then, 𝑘 is the only inherent property of the concerned media, thermal

conductivity of a particular material. Therefore, identifying thermal conductivity 𝑘 is

recognised as the essence to evaluate heat conduction. Equivalently, the Fourier’s law is hold

in granular media by replacing 𝑘 with the so-called effective thermal conductivity 𝑘eff .

However, different from the theoretically well-defined 𝑘 of continuum media, 𝑘eff of granular

media should be treat as a derived coefficient measuring the overall participation of various

heat transfer mechanisms. In practice, 𝑘eff can be experimentally assessed by various

techniques verified in continuum media (Presley and Christensen, 1997), but technical

limitation and labour intensity make it almost impossible to manually obtain 𝑘eff of every

species in granular media. So, reliable predictive methods based on proper evaluation of the

heat conduction are desired. For an apt solution to tackle such demand in granular media, two

key questions need to be deliberated: (1) what are the involving heat transfer mechanisms? (2)

how do those mechanisms propagate through granular media? Correspondingly, the answers

to the former question should elucidate the thermal interactions at grain-scale; and the

influence of mesoscale topological characteristics ought to be clarified in sorting out the latter

one.

For general binary solid-gas/fluid granular media, seven major thermal mechanisms are

summarised (Yagi and Kunii, 1957): (1) heat conduction in solid, (2) heat transfer through

contact surfaces of solid, (3) heat transfer through gas/fluid film near contact, (4) heat

radiation between solid surfaces in solid-gas cases and from solid surfaces into nearby liquids

in solid-fluid cases under high temperature condition ( 𝑇 > 450K) , (5) convective heat

transfer between solid and mobile gas/fluid, (6) natural convection in solids and fluids of high

Rayleigh number, (7) heat transfer through macroscopic flowing gas/fluid. Although some of

the above mechanisms are relied on thermal convection or radiation, they are yielded inside

the space occupied by granular media and so still considered. The following characteristics of

granular media that vary the manifestation of those heat transfer mechanisms are figured out

according to the scale of their dimension. In a sub-grain region, surface characteristics, e.g.,

roughness, affect heat transfer across grains by varying solid contact area (Bahrami et al.,

2005; Zhai et al., 2016). With increasing the length scale, sizes and shapes of grains (Presley

and Craddock, 2006; Dai et al., 2017) are important due to the geometry-dependency

occurring in contacts formation and some heat transfer mechanisms. Further expanding the

dimension to enclose entire media, topological characteristics that create different

propagation pathways for heat conduction, such as spatial distribution of pores,

3

order/disorder packing structure of grains and granular network of contacts, are necessary to

examine. In this study, the most common stagnant fluid/gas-filled granular media are used as

the starting point to establish frameworks to investigate the effective thermal conductivity.

Also, the evolution of granular packing structure is focussed on and efforts are contributed to

unravel correlations between those topology characteristics and heat conduction in granular

media.

1.2 Research methodology and objectives

It has been demonstrated that heat conduction in granular media is a multiscale problem, as

shown in Figure 1-1. To differentiate the involved scales, a bottom-up consequence scheme is

thereby deployed. In this scheme, the most bottom elements are considered as the individual

contact units formed by two adjacent grains, and granular media are treated as assemblies of

those elements. Accordingly, two distinct scales are identified, the grain-scale and the

topology-scale. For the grain-scale, finite element analysis is used to simulate how the

thermal mechanisms participate in the contact units, because the high precision of finite

element analysis ensures the accuracy of simulated result and the relatively simple

geometries of those contact units only bring tolerable computational burden. Regarding the

topology-scale, a discrete element method is instead applied in consideration of the elevated

computational requirement resulted from complicated geometries. To bridge these two scales,

a model that represents the findings from the grain-scale analysis is derived and integrated

into the topology-scale resolution. Hence, the heat conduction in granular media is

investigated thoroughly with neglecting neither interfacial interactions nor topological

variations, and the effective thermal conductivity can be extracted exactly. Further, by

evaluating the consistency between the numerically obtained effective thermal conductivity

and the experimentally measured result, the validity of the proposed numerical framework

can be examined.

Figure 1-1 Decomposition of heat conduction of granular media into sub-scales.

4

In order to successfully carry out research based on the methodology stated above, a series of

research objectives are required to be achieved.

1. First of all, heat transfer of individual contact units must be systematically

characterised with necessary thermal interactions enveloped. Specifically, the gas-

solid thermal interaction is essential in this work.

2. Another equivalent objective is to reveal topological evolution in granular media

which can be accomplished by agitating granular media and performing adequate

characterisation on packing structure consecutively.

3. Further, numerical frameworks incorporated with the heat transfer model in

accordance with the first achievement need to be established and validated. Based on

this, how the topology of granular media determines the heat conduction can be

addressed through the combination of the preceding findings.

4. Last but not the least, reusable and reliable measurement protocols must be

established to extend compatibility of available techniques for granular media.

1.3 Thesis outline

This thesis is organised in the following structure. Chapter 2 provides literature review on a

broad scope covering: (1) relevant theoretical and experimental works on the heat conduction

of granular media; (2) overview on the discrete element method; (3) particular perspective on

topology evolutions and their influences on granular media. Concentrated literature reviews

about particular subjects in this thesis are embedded in the following chapters. Chapter 3 is

adapted from a published article, in which a numerical framework combining finite element

analysis and semi-analytical solution is presented to estimate 𝑘eff for tritium breeding beds in

fusion blankets. Gas pressure and grain size dependency introduced by the Smoluschowski

effect is extensively discussed. Chapter 4 includes an article in EPJ Web of Conference series

discussing the application of transient plane source techniques to measure 𝑘eff of granular

media, and the size effect and loading dependency are observed concurrently in the

application. In addition, a measurement technique based on the 3ω methods is proposed for

nano-grain granular media, and a prototype sensor is designed and fabricated by semi-

conductor technologies. Chapter 5 reproduces an under-revision manuscript that explores the

topological disorder-to-order transitions in granular media undergoing vibration. Geometrical

influences on the formation of crystallisation patterns are identified. Chapter 6 demonstrates a

discrete element framework integrated with semi-analytical heat transfer solution of contact

5

units. The solution revises the original Batchelor & O’Brien model with an empirical

function derived from the finite element analysis. Granular media generated in Chapter 5 are

then utilised to further investigate the influence of topological variance on heat conduction in

granular media. This content has been formatted into a manuscript that is under revision.

Conclusions are drawn in Chapter 7 and potential research is also suggested. In addition,

codes and scripts for the implementation of the modified heat conduction module as well as

data processing are attached in Appendices.

6

7

Chapter 2 Literature Review

In this chapter, relevant theoretical and experimental works about heat conduction and

topology evolution of granular media in the literature are reviewed. Different approaches to

theoretically interpret heat conduction are presented in the first section. Next, since discrete

element methods are the most widely used numerical methods in the research about granular

media, the principles and applications of this type of methods are introduced briefly. On the

other side, the experimental techniques developed for characterising the effective thermal

conductivity of granular media are exhibited and categorised according their measurement

principles in the third section. Lastly, the studies about the resolution to characterise and

understand granular topology and its variation are discussed.

2.1 Analytical model

Broadly speaking, establishing an analytical model for a particular property or phenomenon

involves physically identifying influential factors, theoretically analysing relations between

these factors, and mathematically combining these relations. With respect to the heat

conduction in static granular media, 𝑘eff is determined by thermal conductivity of each

component in granular media and their volume fraction, primarily. Further, size distributions

and spatial arrangements of individual components as well as shapes and surface condition of

solid grains also affect 𝑘eff extensively. Externally, mechanical compression, temperature and

flow field of fluid/gas phases are necessary to be concerned depending on specific scenarios.

Ultimately, 𝑘eff is expected to express as a function include all those factors (Tsotsas and

Martin, 1987). However, either some factors are difficult to quantify physically, or some

relations are complicated to derive mathematically. In this regard, analytical models are

usually defined under particular assumptions, so they can be convenient to implement in

appropriate scenarios.

2.1.1 Structural models

In the consideration of simplicity and ease in practical usage, the structural models (Wang et

al., 2006), originally for bi-phase composites, are formulated with only taking the intrinsic

thermal conductivity of individual phases 𝑘1 and 𝑘2 as well as their volume fraction 𝛾1 and 𝛾2

as the input variables to calculated 𝑘eff (Carson et al., 2005; Carson, 2006; Wang et al., 2006;

8

Wang et al., 2008; Carson and Sekhon, 2010). The serial, parallel, Maxwell-Eucken model

and effective medium theory (EMT) models belong to this class but hold different

assumptions to abstract structures of given media. The serial and parallel models are designed

for layered structures; the Maxwell-Eucken model is based on dilute dispersed structure in

which no interference occurs between the dispersed phases; and the EMT assumes

completely random structure (Wang et al., 2006; Wang et al., 2008). A generalised formula

of the structural models has been proposed to describe 𝑘eff of 𝑚 multi-phase media (Wang et

al., 2006) as,

𝑘eff =

∑ 𝑘𝑖 𝛾𝑖 𝑑𝑖 �̃�

(𝑑𝑖−1) �̃�+𝑘𝑖𝑚𝑖=1

∑ 𝛾𝑖 𝑑𝑖 �̃�

(𝑑𝑖−1) �̃�+𝑘𝑖𝑚𝑖=1

, (2-1)

where 𝑘𝑖 and 𝛾𝑖 are the thermal conductivity and volume fraction of phase 𝑖, respectively,

and 𝑑𝑖 as well as �̃� are parameters selected according to the abstracted structure as listed in

Table 2-1. Further improvement has been achieved by combining different structural models

together to describe media of more complex structures (Wang et al., 2006; Wang et al., 2008;

Carson and Sekhon, 2010).

Table 2-1 Structural models (heat flowing in the horizontal direction)

Serial Parallel Maxwell-Eucken EMT

𝑑𝑖 = 1 or �̃� → 0

𝑑𝑖 → ∞ or �̃� → 𝑘𝑖

𝑑𝑖 → 3 and �̃� → 𝑘continuous

𝑑𝑖 → 3 and �̃� → 𝑘eff

The Maxwell-Eucken and the EMT models can be applied in granular media with proper

topological approximation (Tsotsas and Martin, 1987). The serial and parallel models are

usually treated as the lower and upper bounds (Balakrishnan and Pei, 1979) as initial tests for

developing theories and experiments, and a combined model of the serial and parallel models

has also been used to estimate 𝑘eff of granular media (Yagi and Kunii, 1957).

9

2.1.2 Representative geometry methods

The representative geometry methods are brought forward to specifically calculate 𝑘eff for

granular media. Rather than making structural abstraction like the structural models, these

methods employ particular geometries to represent granular media according to the porosity,

grain size and compression of these media.

Table 2-2 Representative geometry models (heat flowing in the vertical direction)

ZSB

𝑘eff𝑘f

=1 − √1 − 𝜑 + 2√1−𝜑

1−𝜆𝐵[ (1−λ)𝐵(1−λ𝐵)2 ln 1

λ𝐵−𝐵+12 − 𝐵−1

1−λ𝐵]

λ = 𝑘s𝑘f

, and 𝐵 is shape factor determined by 𝜑.

Contact-ZSB

𝑘eff𝑘f

=1 − √1 − 𝜑 + √1−𝜑𝑎

(1 − 1(1+𝑎𝐵)2) +

2√1−𝜑[1−𝜆𝐵+(1−𝜆)𝑎𝐵] ( (1−𝜆) (1+𝑎)𝐵

[1−𝜆𝐵+(1−𝜆)𝑎𝐵]2 × ln 1+𝑎𝐵(1+𝑎)𝜆𝐵

−

𝐵+1+2𝑎𝐵2(1+𝑎𝐵)2 − 𝐵−1

[1−𝜆𝐵+(1−𝜆)𝑎𝐵] (1+𝑎𝐵))

𝑎 is deformed factor indicating contact size.

Lumped-Cubic

𝑘eff𝑘f

= (1 − 𝛿𝑎2 − 2𝛿𝑎𝛿𝑐 + 2 𝛿𝑐𝛿𝑎

2) + 𝛿𝑎2𝛿𝑐

2

𝜆+

𝛿𝑎2−𝛿𝑎

2𝛿𝑐2

1−𝛿𝑎+𝜆𝛿𝑎+ 2(𝛿𝑎𝛿𝑐−𝛿𝑎

2𝛿𝑐)1−𝛿𝑎𝛿𝑐+𝜆𝛿𝑎𝛿𝑐

𝛿𝑎 is deformed factor, and 𝛿𝑐 is porosity factor.

Both are determined by 𝜑.

*The green parts and blue parts indicate the solid and fluid phases, respectively.

The Zehner-Schlunder-Bauer (ZSB) model (Zehner and Schlunder, 1970; Bauer and

Schlunder, 1978) constructs a cylindrical geometry consisting of two point-connecting

identical bodies, whose shapes are determined by the porosity 𝜑 of granular media, to

represent the solid phase. The first expression of ZSB model includes some empirical

parameters for calibration by experiment data. But this expression neglects the grain-grain

contacts in granular media and thus underestimates 𝑘eff . In order to include the flattened

contact surfaces due to compression into expression, a so called “contact-ZSB” model creates

a contact region between those two solid phase bodies and adds an parameter to stand for the

10

size of this region (Hsu et al., 1994). Later refinement of this type of model has introduced

the influence of binary-sized grains (Chen et al., 2015). Inspired by ZSB model, a lumped-

parameter model (Hsu et al., 1995) uses a cubic-like geometry to derive relatively simpler

formulations. This “lumped-cubic” model considers the size of contact surface merely

depending on the volume fraction of grains and explicitly use experiment data to fit the size.

The expressions of these models are summarised in Table 2-2 and only parts of the cross-

sections of the corresponding representative geometries are shown due to their rotational and

axial symmetry.

2.1.3 Unit cell methods

The previous two types of methods are established on hypothetical representation of granular

media without comprehensive analysis of heat transfer mechanisms at different scales of the

media. To overcome this disadvantage, the unit cell methods have been frequently used to

resolve such multiscale problems. Contact units made by two adjacent hemispheres are

commonly regarded as the most basic unit cells in these methods, which are considered as the

grain-scale interfaces of granular media. The identified heat transfer mechanisms are

formulated according to the geometry of these contact units. For purely solid and gas heat

conduction, the solution has been analytically acquired in the Batchelor-O’Brien (BOB)

model (Batchelor and O'Brien, 1977) but its overestimation becomes severe if the thermal

conductivity ratio between solid grains and fluid/gas phase is small. So, additional

compensation is necessary to improve its accuracy (Sasanka Kanuparthi, 2008; Yun and

Evans, 2010). By numerical calculation, another quantitative solution of the Gusarov model

(Gusarov and Kruth, 2005; Gusarov and Kovalev, 2009) is provided particularly for granular

media of fine grains, in which the gas thermal conductivity cannot be treated as constant due

to the Smoluschowski effect. Similar phenomenon is also captured by the BOB solution but

with additionally considering a feature size of grains (Moscardini et al., 2018). Based on

similar contact units, a taped-contact model (Cheng et al., 1999) is derived with further

consideration about the local porosity of individual grains, and thermal radiation can also be

plugged into this model (Cheng and Yu, 2013). Starting from a finer scale, the thermal

contact resistance between two contacting rough surfaces can be incorporated into the contact

units (Bahrami et al., 2004; Bahrami et al., 2004; Bahrami et al., 2006). Besides, by wrapping

up the thermal contact resistance, the Smoluschowski effect, and the thermal radiation

together, the Weidenfeld-Kalman (WK) model (Weidenfeld et al., 2004) and the SLH model

11

(Slavin et al., 2002) present comparable all-inclusive solution but of more complexity.

Differently, the former one uses similar contact units as the aforementioned models, while the

latter constructs a close-packed contact unit cell.

With knowing the heat conduction coefficient of contact unit cells, 𝑘eff of granular media can

be obtained by appropriately integrating all these contact units. The first method is to assume

a statistically homogeneous granular media and use its average contact number. Analytically,

the BOB model and the Gusarov model derive similar expression to calculate a multiplicator

according to the average contact number and volume fraction of solid grains. The Slavin

model directly considers the average contact number during formulating, so no further

integration is needed. The WK model employs empirical augmentation factors which imply

the magnification contributed by the existence of multiple contacts. Note that all these

models will require topological characteristics to complete the theoretical predictions. If the

exact topologies of granular media are known, the Kirchhoff's circuit laws can be applied to

numerically compute the steady-state temperature profiles of the given granular media

(Cheng et al., 1999; Peeketi et al., 2018), by which 𝑘eff can be further calculated.

In addition, other mathematical correlations without constructing unit cells and integrating

these units have been suggested (Huetter et al., 2008; van Antwerpen et al., 2010). But these

correlations are based on analysing heat transfer mechanisms and granular topologies

similarly at the scale of unit cells. So, they are considered as complementary methods in this

unit cell method category. In those methods, empirical equations are usually derived to

estimate 𝑘eff according to the representative topological characteristics, e.g., the mean grain

size, mean pore size, and volume fraction of components.

2.1.4 Challenges for analytical models

From the structural models to the representative geometry methods and to the unit cell

methods, with the intake of fundamental heat transfer mechanisms and granular topologies,

the resulted formulas become increasingly complicated. Establishing analytical models

always falls into the fight to capture physical detail without drastically increasing complexity.

One common challenge faced by all types of the analytical models is how to accurately and

concisely represent the structure of granular media. Using the volume fraction is the most

straightforward way, but the physical relationships between the volume fraction and those

derived empirical parameters are unclear. Despite the theoretical difficulty in building those

12

relationships, research based on numerical methods and experimental characterisation would

be useful to provide quantitative understanding. As the structural irregularity of granular

media blows up with the inclusion of poly-dispersity, random arrangement, non-sphericity,

etc., how to mathematically describe the influence of such factors is awaiting resolution.

Approximating representative simple structures according to the realistic irregular structures

is one common way, but physical insights are required to make solid assumption underlying

this approximation.

Another challenge mainly related to the unit cell methods is to mathematically solve the heat

transfer models. In order to enhance the accuracy, various heat transfer mechanisms are

necessary taken into account. To reduce the increasing troubles in the mathematical

derivation, one can theoretically or experimentally determine how a specific heat transfer

mechanism should be considered in specified condition. For instance, thermal radiation on

𝑘eff of a given granular media is evaluated by considering the positions of neighbour grains

surrounding individual grains. With respect to the properties of grains, e.g., thermal

conductivity, emissivity and size, as well as the temperature of the given media, the influence

of thermal radiation has been addressed quantitatively, where increment in these

characteristics are demonstrated to enhance the significance of thermal radiation (Cheng and

Yu, 2013). In addition, detailed examination about the view factor between two radiative

grains, one of the most essential parameters in thermal radiation, is conducted to identified

appropriate models for calculation about the view factor (Feng and Han, 2012; Wu et al.,

2016). On the other hand, on can also determine empirical relationships of simple functions

to bear some of the mathematical burden. Admittedly, reliable experiments are usually

laborious, hard to realise, or absent, so the implementation of appropriate numerical methods

is advantageous in helping to directly provide useful quantitative results. In this consideration,

combination these analytical models with numerical methods can precisely and effectively

solve heat transfer problems of granular media with tolerable computational bother.

2.2 Discrete element methods

Since computational theories and technologies develop rapidly, numerical simulation has

become prevalent in scientific research. Compared with analytical models, although

numerical simulation may lack the beauty of simplicity and straightforwardness, it can

provide in-situ information about definite processes and statuses. Discrete element methods

(DEM) have been most widely used due to their capability to tackle the discreteness of

13

granular media, since they were first introduced to investigate the kinetic behaviours of

granular assemblies (Cundall and Strack, 1979). By implementing proper mechanistic laws to

govern the interactions between grains, the behaviours of granular media in responding to

certain agitation are able to simulate as the propagation of such agitation from the nearest

grains to the outmost ones. In general, DEM assume grains to be spherical and rigid, and

iteratively update status, such as kinematic and temperature, of grains according to grain

properties and the defined laws. Fundamentally, the accurateness of those laws describing

grain-grain interactions is crucial to achieve trustable simulation. Concurrently, the time

interval for such explicit iteration equivalently decides whether simulation is realistic. In spite

of popularity, development about DEM is rather demanded in problems like non-spherical

grains, deformable grains, and grain breakage.

2.2.1 Simulation principles

In principle, DEM discretises a continuous procedure into a series of steps and tracks the

simulated grains by solving corresponding models with a specified increment of time. For a

step 𝑡, the kinetic status of individual grains 𝑖 is described by (Kloss et al., 2012)

𝑚𝑖�̈�𝑖 = 𝑭n, 𝑖 + 𝑭 t, 𝑖 + 𝑭f, 𝑖 + 𝑭b, 𝑖, (2-2)

𝐼𝑖�̇�𝑖 = 𝒓t, 𝑖 × 𝑭t, 𝑖 + 𝑴𝑖, (2-3)

where, respectively, 𝑚𝑖 and 𝐼𝑖 are the mass and moment of inertia; �̈�𝑖 and �̇�𝑖 are the

translational and rotational acceleration; 𝑭n, 𝑖 and 𝑭 t, 𝑖 are the total normal and tangential

force exerted by other grains; 𝑭f, 𝑖 is the force applied by surrounding fluid phase; 𝑭b, 𝑖 is the

body force due to external fields such as gravity and electric; 𝒓𝑖, t is the rotational radius; 𝑴𝑖

is the additional torque due to the non-sphericity of grains. For a step (𝑡 + ∆𝑡) after the above

step 𝑡, the updated status of grain 𝑖 follows a Verlet velocity integration (Verlet, 1967) as

𝒙𝑖(𝑡 + ∆𝑡) = 𝒙𝑖(𝑡) + �̇�𝑖(𝑡)∆𝑡 + 12

�̈�𝑖(𝑡)∆𝑡2, (2-4)

�̇�𝑖(𝑡 + ∆𝑡) = �̇�(𝑡) + 12

[�̈�𝑖(𝑡) + �̈�𝑖(𝑡 + ∆𝑡)]∆𝑡2, (2-5)

in which 𝒙𝑖 is the position and �̇�𝑖 is the velocity. Rotational degree of freedom can be treated

in a similar way (Gan and Kamlah, 2010). The majority events in granular media are contacts

and collisions that form force chains in static state and transfer kinetic energy in dynamic

state. Therefore, the formulation of contact forces is the foundation of DEM. One of the most

14

widely used model is the Hertz-Mindlin-Deresiewicz model (Di Renzo and Di Maio, 2005)

for elastic frictional contacts. In this model, the normal contact force 𝑭nc𝑖𝑗 and tangential

contact force 𝑭tc𝑖𝑗 are formulated as

𝑭nc𝑖𝑗 = 𝐸n

𝑖𝑗 𝛿n𝑖𝑗 − 𝜁n

𝑖𝑗 𝑣n𝑖𝑗, (2-6)

𝑭tc𝑖𝑗 = 𝐸t

𝑖𝑗 𝛿t𝑖𝑗 − 𝜁t

𝑖𝑗 𝑣t𝑖𝑗. (2-7)

Here, 𝐸n𝑖𝑗 and 𝜁n

𝑖𝑗 (𝐸t𝑖𝑗 and 𝜁t

𝑖𝑗 ) are elastic constant and viscoelastic damping constant for

normal (tangential) contact; 𝛿n𝑖𝑗 (𝛿t

𝑖𝑗) and 𝑣n𝑖𝑗 (𝑣t

𝑖𝑗) are overlap distance and relative velocity

in the normal (tangential) direction. In addition, 𝑭tc𝑖𝑗 is limited by the Coulomb friction limit,

𝑭tc𝑖𝑗 ≤ 𝜇𝑡 𝑭n

𝑖𝑗, in which 𝜇𝑡 is the friction coefficient. Those four quantities can be calculated

according to material properties by the equations shown below:

𝐸n

𝑖𝑗 = 43

𝑌𝑖𝑗∗ √𝑟𝑖𝑗

∗ 𝛿n𝑖𝑗, and 𝜁n

𝑖𝑗 = −2√56

𝛽√𝑆n𝑖𝑗 𝑚𝑖𝑗

∗ ; (2-8)

𝐸t

𝑖𝑗 = 8𝐺𝑖𝑗∗ √𝑟𝑖𝑗

∗ 𝛿n𝑖𝑗, and 𝜁t

𝑖𝑗 = −2√56

𝛽√𝑆t𝑖,𝑗 𝑚𝑖𝑗

∗ ; (2-9)

𝑆n

𝑖𝑗 = 2𝑌𝑖𝑗∗ √𝑟𝑖𝑗

∗ 𝛿n𝑖𝑗, and 𝑆t

𝑖𝑗 = 8𝐺𝑖𝑗∗ √𝑟𝑖𝑗

∗ 𝛿n𝑖𝑗. (2-10)

These derived material properties can be calculated by the primary material properties.

𝛽 = ln 𝑒√ln2𝑒+𝜋2

, (2-11)

1𝑌𝑖𝑗

∗ = (1−𝜐𝑖2)2

𝑌𝑖+

(1−𝜐𝑗2)2

𝑌𝑗, (2-12)

1𝐺𝑖𝑗

∗ = 2(2−𝜐𝑖)(1+𝜐𝑖)𝑌𝑖

+ 2(2−𝜐𝑗)(1+𝜐𝑗)𝑌𝑗

, (2-13)

1𝑟𝑖,𝑗

∗ = 1𝑟𝑖

+ 1𝑟𝑗

, (2-14)

1𝑚𝑖𝑗

∗ = 1𝑚𝑖

+ 1𝑚𝑗

, (2-15)

according to grain radius 𝑟, grain mass 𝑚, Young’s modulus of grains 𝑌, Poisson ratio of

grains 𝜐 and coefficient of restitution 𝑒. The 𝑭nc𝑖𝑗 in Eqn. (2-6) and 𝑭tc

𝑖𝑗 in Eqn. (2-9) are the

basic components in 𝑭n, 𝑖 in Eqn. (2-2) and 𝑭t, 𝑖 in Eqn. (2-3) respectively, but non-contact

15

force such as the capillary force and the van der Waals force can also be included if required.

Further, the selection of time increment for DEM simulation is of equivalent importance. The

Cundall time (Cundall and Strack, 1979; Gan and Kamlah, 2010), Rayleigh time (Li et al.,

2005), Hertz time (Otsubo et al., 2017) and other criteria (Otsubo et al., 2017) are proposed to

achieve realistic simulation. Using the minimum of these time increments is commonly

adopted in running DEM simulation.

2.2.2 Recent development

Besides the development of accurate contact models, the implementation of capillary forces,

the most significant non-contact forces (Zhu and Yu, 2006), is also widely made to extend the

capability of DEM to simulate wet granular media (Gan et al., 2013; He et al., 2014; Lim,

2016; Tsunazawa et al., 2016; Washino et al., 2016). When the size of simulated grains is

fine, the van der Waals force (Parteli et al., 2014) and electrostatic interactions (Yang et al.,

2015) are necessary to consider because of their attractive interaction. With the usage of

temperature depending attractive interactions, DEM have also been extended to study the

sintering in granular media (Luding et al., 2005; Steuben et al., 2016). Additionally, grains in

DEM simulation can be bonded for the purpose to study rock and soil mechanics (Potyondy

and Cundall, 2004; Jiang et al., 2007). Furthermore, breakage of grains in granular media is

possible to explored by grain replacement methods or bond breakage methods (Jiménez-

Herrera et al., 2018). Not only spherical grains are simulated, but also ellipsoid (Zhou et al.,

2011), cubic (Wu et al., 2017), and other non-spherical (Algis Dˇziugys, 2001; Dong et al.,

2015; Höhner et al., 2015) grains are compatible in DEM.

Although the capability of DEM grows extensively by those efforts, there are still various

problems that cannot be tackled by pure DEM simulation. Therefore, coupling DEM with

other numerical methods is gaining popularity. One of the most significant combinations is

CFD-DEM coupling (Zhu and Yu, 2006; Chu et al., 2009; Kloss et al., 2012) to resolve 𝑭f, 𝑖

in Eqn. (2-2). In such grain-fluid coupling, the geometry and kinetic status of grains

simulated by DEM are passed into a CFD solver. By meshing the identical simulation domain

according to this communication, the CFD solver calculates the surrounding fluid fields

around those grains, derives 𝑭f, 𝑖 on each grain, and then sends this information back to DEM.

Finally, DEM takes the 𝑭f, 𝑖 into Eqn. (2-2), continues iteration to next step, and repeats the

exchange process. Other numerical methods, e.g., the Lattice Boltzmann methods (Alexander

et al., 2014; Körner et al., 2014; Markl and Körner, 2016) and the finite element analysis

16

(Ganeriwala and Zohdi, 2016; Ma et al., 2016), often utilise DEM to produce base particulate

geometries as the input to simulate other processes like grain melting and fusion (Körner et

al., 2014; Markl and Körner, 2016).

2.2.3 Application on heat transfer of granular media

Regarding the concentration of this thesis, heat transfer can be added along with the iteration

of force interactions (Kloss et al., 2012). The temperature evolution of each grain is able to

calculate by

𝑚𝑖𝑐𝑖∆𝑇∆𝑡

= ∑ 𝐻𝑖𝑗𝑗 + 𝐻𝑖, src, (2-16)

𝐻𝑖𝑗 = ℎ𝑖𝑗 (𝑇𝑗 − 𝑇𝑖), (2-17)

where 𝑇𝑖(𝑗) is the temperature of grain 𝑖 (𝑗); 𝑐𝑖 is the specific heat capacity of grain 𝑖; ℎ𝑖𝑗 is

the heat transfer coefficient between two adjacent grains 𝑖 and 𝑗 ; 𝐻𝑖𝑗 is the heat flow

transferring from grain 𝑗 to grain 𝑖 and 𝐻𝑖, src is the heat source in grain 𝑖. By totalling all heat

flows from every adjacent grain of one particular grain 𝑖 to get ∑ 𝐻𝑖𝑗𝑗 , temperature variation

∆𝑇 of the considered grain after ∆𝑡 can be acquired.

As is shown by Eqn. (2-16) and Eqn. (2-17), the most crucial parameter to determine is ℎ𝑖𝑗.

In general, this parameter depends on the contact area or separating distance between two

adjacent grains. The most straightforward way is to use those relationship derived in the unit

cell methods of the analytical models indicated in the section 2.1.3. Since the computational

efforts for the calculation of those heat transfer models become a less limiting factor in DEM,

there is a higher demand about the accuracy of the models. For the stagnant granular media,

the questions focus on how to interpret the solid-gas/fluid interactions (Batchelor and O'Brien,

1977; Cheng et al., 1999), because these interactions are usually distance-depending, giving

difficulty in mathematical derivation. Consequently, Empirical coefficients (Yun and Evans,

2010; Moscardini et al., 2018) are always used in those models to resolve this issue, while

thorough evaluation of the coefficients awaits further investigation theoretically or

experimentally.

Because the position of individual grains in simulated granular media are known explicitly in

DEM, thermal radiation can be considered accurately with this advantage. By applying the

Voronoi tessellation onto the packing structures, the grain-scale view factor (Feng and Han,

17

2012; Wu et al., 2016) and the surrounding temperature field (Cheng and Yu, 2013) of

individual grains can be computed locally, ensuring the appropriate implementation of these

theoretical principles. However, computational demand increases with the improvement in

the sophistication of applied models. Therefore, a balance between acceptable precision and

tolerable computation becomes the practical issue in this topic and other similar ones (Qi and

Wright, 2018; Wu et al., 2018).

As shown in previous session, the CFD-DEM coupling method is preferred in solving the

granular media of moving gas/fluid phase. With the velocity fields of fluid/gas surrounding

individual grains computed by the CFD solvers, the thermal convection can be specifically

resolved (Zhou et al., 2010), which is the distinct problem in the heat transfer of fluidised

granular media. It has been demonstrated that convective heat transfer becomes the dominant

mechanism with the increase in fluidisation of granular media, which indicates that the

differentiated application fields of pure DEM and CFD-DEM coupling. Nevertheless, Like

the scenario of those stagnant granular media, determination of the coefficients used in the

convective heat transfer models is crucial. Since the CFD is a mesh-based computational

method, the additional question is to establish suitable meshing schemes to meet the

efficiency and precision requirement (Wu et al., 2018).

2.3 Experimental characterisation

To validate theoretical discoveries or numerical frameworks, reliable experimental

characterisation is indispensable. Generally, heating components to create temperature

gradient in particular media and sensing components to probe status variation induced by that

temperature gradient are the common foundations of thermal characterisation techniques.

Regarding to granular media, their unstable geometry and low 𝑘eff are necessary to handle

properly. Therefore, minimal requirement about sample geometry and good thermal shielding

are the two prerequisites to satisfy in establishment of accurate measurement systems.

2.3.1 Steady-state and transient-state methods

Steady-state methods and transient-state methods are two major categories of thermal

measurement, and both of them have been employed onto granular media. The former one

creates a stable temperature field in a given medium by a constant heating source. According

to the geometry of the medium, its 𝑘eff can be calculated based on the Fourier’s law (Carslaw

18

and Jaeger, 1959). The second one also applies a constant heating source but on the contrary

records a temperature versus time profile. By solving the transient heat transfer models

(Carslaw and Jaeger, 1959), 𝑘eff and of the measured granular medium can be obtained.

Table 2-3 gives two commonly used steady-state measurement schemes for granular media.

A variety of practical setups have been realised by using different heating or sensing methods,

including conventional thermocouples (Hall and Martin, 1981; Abou-Sena et al., 2007) and

water bath (Tsotsas and Schlünder, 1991), as well as advanced calorimeters in the radial flow

schemes and photometry in the axial flow schemes (Presley and Christensen, 1997). Besides,

commercialised measurement systems are also available (Abyzov et al., 2013). The steady-

state methods are straightforward in calculating 𝑘eff with little mathematical derivation, but

there are some disadvantages. First, the measurement time is likely to be long to reach the

steady states because of the relatively low 𝑘eff of granular media. Second, thermal loss into

sample holders and equipment and thermal convection into environment become severe

during long measurement processes.

Table 2-3 Two steady-state measurement schemes

Axial flow Radial flow

𝑘eff = 𝑃source𝐴

× 𝐿𝑇2−𝑇1

𝑘eff = 𝑃source2𝜋𝐿

× ln 𝑟2/𝑟1𝑇2−𝑇1

Compared with the steady-state methods the transient-state methods can relieve the concern

of long measuring time and excess heat loss to the surroundings. However, the transient-state

methods usually need to solve differential equations or fit numerical models. The thermal

probe methods (Woodside and Messmer, 1961; Lo Frano et al., 2014; Pupeschi et al., 2017),

the hot wire methods (Healy et al., 1976; Tavman, 1996; Presley and Christensen, 1997;

Enoeda et al., 2001; Assael et al., 2010; Merckx et al., 2012; Wei et al., 2016) and the

transient plane source techniques (Gustavsson et al., 1994; He, 2005; Garrett and Ban, 2011;

19

Mo et al., 2015; Li and Liang, 2016) are three common types of transient-state methods

employed for granular media. In the thermal probe methods, a thermally conductive probe

with both a heating element and a sensing element encapsulated is required to fabricate. The

hot wire methods are similar to the thermal probe methods, but the heating and sensing is

achieved by only one single metallic wire, e.g., a thin platinum wire. Joule-heating generated

by the metallic wire works as the thermal source and the temperature dependent resistance of

the corresponding metal is used as the sensing principle in the hot wire methods. Both two

methods are based on line-source heating assumption and governed by similar equations,

while the thermal probe methods are popular in bulky samples and the hot wire methods are

preferred in granular media of relatively small volume. The transient plane source techniques

instead are realised under a plane-source heating assumption. In this type of setup, a metallic

coil is usually used to mimic a plane source and act as a temperature sensor. By varying the

size of the plane source, this type of techniques has a better compatibility with respect to

sample volume. Table 2-4 displays the schematic drawings of the measurement setups for

these three types of techniques.

Table 2-4 Three transient-state measurement techniques

Thermal probe Hot wire Plane source

2.3.2 Transient plane source technique

The commercialised HotDisk system based on the transient plane source technique is utilised

in this work under the consideration about instrument availability. In this particular system, a

hot disk sensor is produced by encapsulating a spiral nickel coil inside a thin layer of Kapton

polymer (Gustavsson et al., 1994). Such Kapton shell provides electrical insulation and

mechanical protection for the heating and sensing coil. During individual measurement, a

20

constant direct current (DC) to generate joule heating, and the system continuously records

the electrical resistance 𝑅 of the sensor against the time 𝑡. According to the pre-characterised

temperature coefficient of electrical resistance 𝛼 of the sensor, the expression

𝑅 = 𝑅0[1 + 𝛼∆𝑇(𝑡)], (2-18)

is used to obtain temperature increase ∆𝑇 with the known initial electrical resistance 𝑅0 of the

sensor. With the thermal diffusivity of the measured sample 𝜅 and the radius of the sensor 𝑏,

the temperature increment ∆𝑇 of the sensor is further expressed as a function of the

characteristics time ratio 𝜏 = √𝜅𝑡/𝑏 as

∆𝑇(𝜏) = 𝑃𝜋 3 2⁄ 𝑏 𝑘

𝐷(𝜏), (2-19)

where 𝑃 is heating power, 𝑘 is thermal conductivity and 𝐷(𝜏) represents a non-dimensional

time variable resulted from a complex function of 𝜏 (Gustafsson, 1991). By numerical

analysis on this complicated heat transfer model, the optimised linearity between ∆𝑇 and

𝐷(𝜏) is yielded, and thus, 𝜅 and 𝑘 are obtained at the same time.

2.3.3 Potential techniques for nano-grain granular media

With the development of material fabrication technologies, nano-sized particles are more

common nowadays. As a result, nano-grain granular media will be easier to produce with

their properties modified by nano-engineering. Nano-grain granular media are shown to be

promising in thermal insulation application (Prasher, 2006). Therefore, it is of importance to

establish appropriate measurement to evaluate 𝑘eff of nano-grain granular media. Due to

limitation in bulk production of nano-sized particles, nano-grain granular media are often

restricted to small volume. Consequently, creating and sensing very small temperature

variation become challenge in this scenario. A steady-state method has been achieved in the

axial flow scheme (Hu et al., 2007) as shown in Table 2-3, which first evidences the ultra-low

thermal conductivity of nano-grain granular media. Using calorimeters, another steady-state

technique has been implemented to study the influences of moisture (Elsahati et al., 2016;

Elsahati and Richards, 2017) on heat transfer of nano-grain granular media. The HotDisk

system is also able to measure nano-grain granular media by employing a very small sensor.

But these techniques are likely to face difficulty when the volume of granular media is more

limited. To facilitate measurement in this scenario, the 3 method (Cahill, 1990) is of

potential. This method has been used for nano-film (Lee and Cahill, 1997), bio-tissue

21

(Lubner et al., 2015), liquid (Lee, 2009) and gases (Schiffres and Malen, 2011). The most

important advantage of the 3 method is that the heating can be restricted within a tiny

volume by appropriate configuration (Jacquot et al., 2010). Such advantage makes it a

promising technique for nano-grain granular media.

2.3.4 Versatility requirement of granular media

Due to the discrete nature of granular media, the effective thermal conductivity is possibly

related to applied external loads, interstitial gases, topologies, etc., so the key in developing

an appropriate thermal measurement system for granular media is to ensure versatility to

handle these influential factors. Building a system from scratch is possible but involves great

technical challenges. Alternatively, designing customised tools that are compatible for

existing systems is more cost-effective.

2.4 Effective properties vs. Topologies

Analogical to atomic structure determining properties in materials science, the effective

properties of granular media are also the result of the topologies, but the principles for this

subject is far from completeness. The first question is how to describe the topologies of

granular media both locally and globally. In general, packing structure of grains and spatial

distribution of pores (gas/fluid phase) constitute the topologies of granular media. Either

treating them separately or considering them collectively can bring different descriptors of

topologies. With topologies characterised, the second question about which descriptor to use

for a particular effective property also remains unclear. Answers to these two questions are

the foundations for those principles.

2.4.1 Topological characterisation

The most historical descriptors of the topology of granular media can be recognised as the

macroscopic packing fraction, which is defined as the volume fraction of grains in granular

media (Scott, 1960). With advanced mathematical algorithms, individual packing fraction at

grain scale can be derived by employing the Voronoi geometrical tessellation (Aste, 2005;

Rycroft, 2009). In the opposite way, a representative pore distribution can be achieved by the

Delaunay tessellation (Aste, 2005; Francois et al., 2013; Li et al., 2014; Ferraro et al., 2017).

Further, granular media can be transformed into spatial distributions of polyhedrons by other

delicate tessellation methods. Examination about the aforementioned distributions has

22

revealed the local topological origins for global topological changes (Aste, 2005; Aste et al.,

2006; Aste et al., 2007; Sufian et al., 2015; Saadatfar et al., 2017).

The contact-based packing structure of grains is of primary interests in practical application

of granular media, so efforts have been contributed to obtain better understandings. The

coordination number (Bernal and Mason, 1960; Saadatfar et al., 2005), which quantifies the

number of grains contacting or nearly contacting a specific grain, is considered as the

fundamental characteristic because it determines potential grain-grain interactions (Batchelor

and O'Brien, 1977; Gan and Kamlah, 2010). The radial distribution function (Scott, 1962;

Mason, 1968) gives a distribution of the distance between two grains, and the peaks of such

distributions are strong indicators for the regular packing structure, such as face-centred-

cubic (FCC) and hexagonal-close-packing (HCP) (Reimann et al., 2015; Reimann et al.,

2017). More sophisticatedly, bond orientation parameters can also be applied to quantify the

rotational symmetry of the neighbour configuration of individual grains (Steinhardt et al.,

1983; Kumar and Kumaran, 2006; Klumov et al., 2011; Tanaka, 2012).

All these methods are firstly created to characterise mono-dispersed granular media. To

include poly-dispersity into topological characterisation, more efforts are required. The

Voronoi and Delaunay tessellations (van der Linden et al., 2018) can be accomplished by

mathematically adjusting parameters accounting for the poly-dispersity. Studies have also

been attempted to reveal the change of the coordination number in respect to granular media

of binary (Yu and Standish, 1993; Pinson et al., 1998; Gan et al., 2010), ternary (Yu and

Standish, 1993; Zou et al., 2003; Yi et al., 2011) and continuous distributions (Suzuki and

Oshima, 1983; Georgalli and Reuter, 2006). While, the bond orientation parameters are very

sensitive to polydispersity and incompatible with grains of large size contrast (Russo and

Tanaka, 2012). Not only the poly-dispersity, but also the non-sphericity has been involved in

topological research. Grains of cubic shape (Wu et al., 2017), ellipsoids (Donev et al., 2004;

Zhou et al., 2011) and cylinder (Tangri et al., 2017) have been used to find out the

corresponding densest packing structure. Challenges arise when applying the aforementioned

methods onto those non-spherical grains, and specific methods for different types of grains

are required. Alignment of grains (Shi and Ma, 2013; Tamás et al., 2016) has been utilised in

characterising granular media consisting of gains with high aspect ratios and identified as the

major topological phenomenon.

23

As shown in Table 2-5, each method has its suitable applications as well as limitations.

Combination of different methods for particular questions are generally favourable.

Regarding the subjects in this thesis, the bond orientation parameters and Voronoi tessellation

are employed to investigate both dynamic and static phenomena of granular media. To carry

out the investigation, the discrete element methods are extensively used since the grain

positions can be easily extracted for topological characterisation. Further, experimental data

obtained from X-ray computational tomography techniques are utilised to validate the

corresponding numerical research about the static phenomena, which serves as the added

weight to the validity of the discoveries from the dynamic phenomenon research despite of

the unavailable dynamic X-ray experiments.

Table 2-5 Comparison of different topological descriptors

Descriptor Applications Limitations

Voronoi

tessellation

• Grain-scale packing fraction

differentiation;

• Poly-dispersed characterisation;

• Densification identification.

Not suitable in dynamic

state

Delaunay

tessellation

• Pore-scale properties

characterisation;

• Compatible in poly-dispersed

granular media.

Not suitable in dynamic

state

Coordination

number

• Contact identification;

• Contact network investigation

Changeable according to

characterisation criteria.

Radial distribution

function

• Structure identification;

• Densification identification.

Not represented by

individual quantities.

Bond orientation

parameters

• Packing order evaluation;

• Dynamic state investigation;

• Structure identification.

Not suitable for poly-

dispersed granular media

2.4.2 Topological transitions in dynamic processes

The variety of the granular topology has been widely demonstrated by the characterisation

methods above. How various topologies are generated and transited remains elusive. A series

24

of agitation, such as vibrating, tapping, shaking and shearing, has been widely utilised to

trigger topological transitions in granular media (Mehta and Barker, 1991; Pouliquen et al.,

1997; Philippe and Bideau, 2002; Pouliquen et al., 2003). The first three processes are

normal-collision dominating, while the last one mostly introduces much stronger tangential

motion. The relaxation of packing fraction is the most significant experimental observation in

these processes (Jaeger et al., 1989; Knight et al., 1995; Nowak et al., 1997; Nowak et al.,

1998; Pouliquen et al., 2003; Tsai et al., 2003). Depending on the strength of the agitation,

different temporal relaxation profiles (Nowak et al., 1997; Philippe and Bideau, 2002;

Richard et al., 2003; Daniels and Behringer, 2005; Richard et al., 2005; Dijksman and Hecke,

2009) are identified. Unless excessively agitated, granular media generally exhibit collective

compaction after such processes. To understand mechanisms behind these phenomena, the

dynamic rearrangement of grains, such as caging effect, position fluctuation and activated

diffusion (Barker and Mehta, 1993; Martin et al., 2003; Marty and Dauchot, 2005; Yu et al.,

2006; An et al., 2008; Richter et al., 2009; An et al., 2011; Royer and Chaikin, 2015), are

examined. Preservation and reconstruction of original contact topologies by analysing the

change of coordination number are found as the compaction mechanisms in the weak and

strong perturbation, respectively (Yu et al., 2006). By utilising size distributions of pores in

granular media the compaction has been described by statistics methods (Boutreux and de

Geennes, 1997; de Gennes, 2000; Hao, 2015; Mathonnet et al., 2017; Rondet et al., 2017)

which has been extended to explain rheology of granular media (Campbell, 2006; Marchal et

al., 2009; Marchal et al., 2013).

Further dedicated characterisation on topologies has revealed that disorder-to-order

transitions and their reverses (Pica Ciamarra et al., 2007; Panaitescu and Kudrolli, 2014; Lash

et al., 2015; Royer and Chaikin, 2015; Saadatfar et al., 2017) are the essential procedures

bridging the motion of individual grains to the collective behaviours of entire granular media.

The disorder-to-order transitions, commonly called as crystallisation, are usually observed

when granular media of initial random topologies are agitated (Pouliquen et al., 1997; Levin,

2000; Tsai et al., 2003; Carvente and Ruiz-Suárez, 2005; Daniels and Behringer, 2005; An et

al., 2011; dShinde et al., 2014). This granular crystallisation brings about periodic densified

topologies that are identifiable by the bond orientation parameters, the radial distribution

function and the Voronoi tessellation. However, the underlying driving force of this

phenomenon remains elusive. The improvement of mechanical stability (Heitkam et al.,

2012), energy-transferring efficiency (Carvente and Ruiz-Suárez, 2008; Dai et al., 2018) and

25

dynamic uniformity (Hsiau et al., 2008; Tai and Hsiau, 2009; Dai et al., 2018) have been

proposed as the reasons for the preferential crystallisation in those granular media. The

reversed transitions can be realised by perturbing highly ordered granular media (Panaitescu

and Kudrolli, 2014; Saadatfar et al., 2017). In addition, approaches used to investigate

crystallisation in other systems, e.g., liquid (Steinhardt et al., 1983; Kawasaki and Tanaka,

2010; Russo and Tanaka, 2012), colloidal suspension (Tan et al., 2014; Golde et al., 2016)

and glass (Sanz et al., 2014; Yanagishima et al., 2017), can be modified to prompt research

about topological transitions in granular media because similarities have been identified

between those systems and granular media (Li et al., 2014). It will be of great significance to

address the uniqueness and resemblance of granular crystallisation when compared with

similar phenomena in other systems. Inspired by this, this thesis will contribute to the

understanding of dynamic processes in the granular crystallisation.

2.4.3 Effective properties determined by topology

From a practical perspective, the research towards insights of the topological characteristics

is fundamentally related to a better understanding about the origins of the effective properties

of granular media. Mechanical properties has been demonstrated to correlate with topological

order degree of granular media (Goodrich et al., 2014) by application of the jamming theory

(Torquato and Stillinger, 2010). Both thermal conduction (Watson L. Vargas, 2002; Wang

and Pan, 2008) and electrical conduction in granular media are also determined by the

packing structure of grains (Batchelor and O'Brien, 1977; Fan, 1996), with respect to contact

networks and coordination numbers. Not only the packing structure, but also the spatial

distributions of pores are analysed (Ferraro et al., 2017) to study the properties of fluid phase

in granular media that contributes to knowledge about the water retention in soil mechanics.

Additionally, the pore space characterisation is also used to study the permeability of

granular media (Cortis et al., 2004; van der Linden et al., 2016). Compared to the abundant

utility of granular media, understanding about the correlations between effective properties

and topology is still insufficient. Therefore, more efforts are demanded to improve the

integrity of this subject. Motivated by the principles in material science, it is intuitive to think

about that the effective properties of granular media are originated from the local topology of

individual grains in the corresponding granular media. However, limited efforts have been

devoted to unveiling the masks on it. To fill this gap, this thesis focuses on investigating the

relationship between the effective thermal conductivity and the multiscale topological

26

characteristics of granular media by further employing the discrete element methods and the

findings of the topological transitions.

27

Chapter 3 Thermal Interaction at Gas-Solid Interface

Granular media are intrinsically heterogeneous, consisting of different materials as well as

phases. The most widely seen granular media are in the form of solid-gas mixture. Within the

scope of this work, heat conduction in such granular media of static state is contributed by (1)

heat transfer through solid phase, (2) heat transfer through gas phase, and (3) heat transfer

through solid contacts. The main content of this chapter is based on the following published

journal paper:

Dai, W., Pupeschi, S., Hanaor, D., Gan, Y., (2017). Influence of gas pressure on the

effective thermal conductivity of ceramic breeder pebble beds. Fusion Engineering

and Design, 118, 45-51.

This paper mainly addresses the second issue in the granular media made by sub-millimetre

grains in the context of pebble beds used in nuclear fusion reactors to breed tritium. In this

scenario, heat transfer through gas phase is strongly influenced by the geometrical dimension

of the region near the contacts due to the presence of the Smoluschowski effect. Because the

pressure of gas phase in this application is adjustable, the resulted effective thermal

conductivity 𝑘eff is variable. By finite element analysis, the size dependence of heat transfer

is proved and extensively discussed. A numerical method to estimate 𝑘eff of pebble beds with

known grain size distributions is put forward and quantitatively verified. This paper

demonstrates the applicability of the finite element analysis for investigating the thermal

interactions at grain-scale, which is one of the foundations in the bottom-up scheme. Such

method will be further incorporated in Chapter 6 to provide a correction to improve the

accuracy of conventional heat transfer model.

28

29

Figure 3-1 Helium thermal conductivity predicted by the Gusarov’s model, the temperature

jump model and the Kistler’s model.

30

Figure 3-2 Heat conductance coefficient of helium by the truncated Gusarov model

(truncated G.) at 293 K with Helium pressure varying.

31

Figure 3-3 Schematic drawing of contact unit model: (a) two hemispheres in contact without

mechanical loading; (b) The contact under a constant mechanical loading; (c) The FEM

model with temperature profile with an arbitrary unit. Here, ∆𝑇 , ∆𝑥 , ℎg denote the

temperature difference, particle deformation, and heat flux through the gas phase,

respectively.

Figure 3-4 Size distribution of the pebbles.

Figure 3-5 Influence of external mechanical loading on the contact unit of 𝐷 = 350 µm, and

the helium gas pressure is 120 kPa.

Table 3-1 Li4SiO4 properties (T=293K)

32

Figure 3-6 Gas pressure related size dependency of the contact units.

Figure 3-7 Effective thermal conductivity predicted by the Gusarov’s model, the SZB model,

the modified Batchelor’s model, and the proposed framework.

33

Figure 3-8 Effective thermal conductivity comparison between model predictions and

experiments data from: (a) S. Pupeschi et al. [31] and (b) Abou-Sena, et al [1].

34

35

Chapter 4 Conductivity Measurement Using Transient Techniques

Thermal measurement on granular media is usually challenging, because the effective

thermal conductivity 𝑘eff is relatively low, making steady-state techniques become time

consuming and sensitive to environmental interferences. Therefore, reliable transient-state

measurement gains more prevalence in this field. In the first section, a transient technique

based on the 3ω method is proposed for measuring 𝑘eff of nano-grain granular media. The

reusable 3ω sensor has been successfully designed and produced by semiconductor

fabrication technologies. The implementation of this sensor is retarded by the unavailability

of some essential equipment and the highly technical difficulty, because it would be costly in

the current circumstance. Nevertheless, the necessary technical requirement is further

discussed for realising its function in future. In order to experimentally measure effective

thermal conductivity of granular media, we have switched to another alternative transient

technique to ensure the comprehensiveness of this thesis. In Section 4.2, based on

standardised equipment (HotDisk system) of the transient plane source techniques, a

measurement protocol with the particularly designed holder for granular media is established.

The size-effect on 𝑘eff proved in Chapter 3 is observed by this method. The sensitivity to

mechanical loading of 𝑘eff is tested in granular media of different grain materials. Discussion

about these two observed phenomena are extended in Section 4.3. In later Chapter 6, this

protocol will be further applied to verify the numerical framework.

4.1 A proposal of planar 3ω sensor for nano-grain granular media

4.1.1 Nano-grain granular media

Traditionally, a lower bound of grain size of 1 micrometre is defined for granular media.

However, with development of nano-technologies, particles of size in sub-micro regions

become more approachable. As a result, granular media comprising grains of diameter below

1 μm are gradually attracting researchers’ attentions (Takahashi et al., 2001; Tai et al., 2006;

Vandamme and Ulm, 2009; Yuan et al., 2011; Grimaldi, 2014; Adjaoud and Albe, 2016;

Chen et al., 2017; Ioannidou et al., 2017) . The nano-grain granular media, or called as nano-

granular materials, usually possesses different mechanical characteristics (Yuan et al., 2011)

due to stronger van der Waals interaction introduced by fine grain size. Meanwhile, it has

36

also been proved that micromechanics at nanoscale are the origins of macroscopic behaviours

of granular media (Tai et al., 2006; Vandamme and Ulm, 2009). Beyond the fundamental

research about mechanics, huge potential of nano-grain granular media is revealed both

theoretically and experimentally in electrical percolation (Grimaldi, 2014), radiation

shielding (Takahashi et al., 2001), and thermal insulation (Prasher, 2006; Hu et al., 2007).

Therefore, an initial touch on heat transfer of nano-grain granular media is attempted by this

work as well.

Heat transfer in conventional granular media is majorly explained by the mechanisms

established in continuum media. Yet, because of abundant nano-interfaces in nano-grain

granular media, heat transfer through bulk solid/gas phases is far from determinative (Prasher,

2006). Nano-scale heat transfer theories point out that great thermal resistance of those nano-

interfaces makes nano-grain granular media become excellent thermal insulators of ultra-low

thermal conductivity (Hu et al., 2007). Despite promising, nano-grain granular media are still

facing critical challenges such as bulk production, safe manipulation and accurate

measurement. In terms of thermal application, reliable techniques to characterise thermal

properties of nano-grain granular media are not only essential for demonstration but also

useful to study other phenomena (Elsahati and Richards, 2017). Due to difficulty in

production and process of nano-grain granular media, good technique candidates should have

little requirement on volume and shape of samples. In literature, the guard-heated calorimeter

technique (Elsahati et al., 2016) and a axial flow steady-state method have been achieved for

this purpose. To provide an alternative solution, a measurement system based on 3ω method

is proposed here, which is suitable for extreme small volume measurement.

4.1.2 Theoretical principles of 3ω method

The 3ω method is a type of transient-state techniques for characterising thermal conductivity.

The uniqueness of these techniques is their alternating current (AC) source heating and

frequency domain sensing. Therefore, the 3ω method is considered as a frequency domain

quasi-static method. In principle, regarding to joule-heating in a metallic sensor, the AC

current 𝐼𝜔(𝜔) of which 𝜔 is the angular frequency generates a heating power fluctuating at

2𝜔, which also makes the sensor temperature modulate at 2𝜔 but with a time lag. Eventually,

this temperature modulation induces a small but detectable voltage signal fluctuating at 3𝜔

(Cahill and Pohl, 1987; Cahill, 1990). Theoretically, this small voltage of 3𝜔 is proved to

have a relation with source voltage of 1𝜔 and temperature modulation of 2𝜔 as

37

∆𝑇2𝜔(𝜔) = 2𝑉3𝜔(𝜔)𝛼𝑉1𝜔(𝜔)

, (4-1)

where ∆𝑇2𝜔 is the 2𝜔 temperature modulation amplitude, 𝑉3𝜔 and 𝑉1𝜔 are the fluctuating

voltage amplitudes measured at 1𝜔 and 3𝜔, respectively, and 𝛼 is the temperature coefficient

of the sensor. Conventionally, the 3ω method is based on a setup of a metal strip of width 2𝑏

and length 𝑙 attaching to the surface of a large medium, assuming a line-source heating the

semi-infinite medium as shown in Figure 4-1 (a). A typical the isotropic heating profile in the

plane perpendicular to the metal sensor (Carslaw and Jaeger, 1959) is solved as

∆𝑇(𝑟) = 𝑃𝜋𝑙𝑘

𝐾0(𝑞𝑟), (4-2)

where ∆𝑇(𝑟) is the temperature fluctuation at position 𝑟 = (𝑥2 + 𝑦2)1 2⁄ , 𝑃 is the effective

power of the AC current source, 𝑘 is the thermal conductivity of the measured medium, and

𝐾0 is the zeroth order of the modified Bessel function of the second kind. The thermal

penetration depth 𝑞−1 in a medium of thermal diffusivity 𝜅 defines the characteristics length

of a thermal wave with 2𝜔 frequency by 𝑞−1 = (𝜅 2𝜔𝑖⁄ )1/2. Here, 𝑞−1 quantifies the most

significant region affected by the thermal wave, so the semi-infinite medium assumption is

validated if this region is smaller than the measured medium. In accordance with the half

width 𝑏 of the sensor, the quasi-static fluctuating temperature amplitude at 𝑟 = 𝑏 in

frequency domain is approached as (Cahill, 1990)

∆𝑇2𝜔(𝜔) = 𝑃𝜋𝑙𝑘

(12

ln κ𝑏2 + 𝛿 − 1

2ln(2𝜔) − 𝑖𝜋

4), (4-3)

where 𝑘 and 𝜅 and the thermal conductivity and thermal diffusivity of the medium and 𝛿 is a

constant. According to this formula, the ∆𝑇2𝜔(𝜔) is linear to ln (2𝜔) and the 𝑘 can be

extracted from the slope of the linearity between ∆𝑇2𝜔(𝜔)and ln 2𝜔,

𝑘 = 𝑃𝜋𝑙

𝑑[∆𝑇2𝜔(𝜔)]𝑑[ln(2𝜔)]

. (4-4)

The above solution is relied on an assumption that 𝑞−1 ≫ 𝑏 . On the other hand, in the

opposite limit 𝑞−1 ≪ 𝑏 , a one-dimensional heating profile as shown in Figure 4-1 (b)

(Carslaw and Jaeger, 1959) can be expressed as

∆𝑇2𝜔(𝜔) = 𝑃2𝑙𝑏

( 1√2𝜔𝑘𝐶

) 𝑒−𝜋𝑖4 , (4-5)

38

where 𝐶 is the volumetric heat capacity of the medium and 𝑘 = 𝜅𝐶 . 𝑘 can be derived by

fitting the equation with known 𝐶. By varying the heating power 𝑃, 𝑙 and 𝑏, 𝐶 can also be

fitted as well.

(a) (b) (c)

Figure 4-1 Typical configurations of the 3ω methods: (a) Conventional two-dimension

heating by line-source, (b) one-dimension heating by plate-source, and (c) bi-side two-

dimension heating by line-source.

In those conventional semi-infinite configurations, measurement is always conducted in

vacuum to eliminate heat loss to surrounding environment. However, such metal strips are

hard to fabricate onto the surface of granular media. Therefore, a bi-side 3ω method (Bauer

and Norris, 2014) as shown in Figure 4-1 (c) is capable for this requirement. Rather than only

attaching to one medium, the sensor in the bi-side 3ω method is sandwiched by two media. In

this way, individual heating profiles are created in both media but the temperature

fluctuations of both sides adjacent to the sensor are considered to be identical, ∆𝑇1 = ∆𝑇2.

With the above assumption satisfied, the quasi-static temperature fluctuation amplitude of the

sensor can be formulated as

∆𝑇1 = ∆𝑇2 = 𝑃𝜋𝑙(𝑘1+𝑘2)

𝑓(ln 2𝜔), (4-6)

and ∆𝑇1 = ∆𝑇2 = 𝑃2𝑙𝑏

( 1√2𝜔𝑘1𝐶1

+ 1√2𝜔𝑘2𝐶2

) 𝑒−𝜋𝑖4 . (4-7)

According to the Eqn. (4-6) and Eqn. (4-7), by knowing the thermal properties of either side,

the other side can be easily characterised. In these approaches, granular media can be

measured by loaded onto the surface of a well-characterised medium integrated with

particular sensors.

4.1.3 Measurement system design and sensor fabrication

The schematic drawing of the measurement system is shown in Figure 4-2. In general, a

probe station is deployed to form electrical connection from the 3ω sensor to external circuits

39

and equipment; a data process circuit is fabricated to acquire and refine rough signals

detected by probes connecting to the sensor; a lock-in amplifier is used to record electrical

signals of particular frequency; a DC power source is used to heat the 3ω sensor and enable

the data process circuit. The 3ω sensor is placed in the probe station and granular media

samples are loaded onto the sensor surface. The layout of the in-house 3ω sensor is zoomed

and shown in the right insert of Figure 4-2. Five types of sensors are fabricated onto one

single chip because the optimal dimension of sensor size is likely to be varied according to

specific measurement scenarios.

Figure 4-2 schematic drawings of the 3ω measurement system and the 3ω sensor pattern.

The 3ω sensors are fabricated by semi-conductor fabrication technologies according to the

following procedures that are sequenced in Figure 4-3: (1) A silicon wafer is diced into 5 cm

× 5 cm square which used as the substrate carrying the sensors. (2) A 200 nm thick SiO2 film

is produced on the top of the silicon chip by thermal oxidation. This non-conducting SiO2

film serves as the insulation layer to prevent electrical leakage into the silicon chip. After that,

the pattern of the sensors is transferred on to the chips by the combination of a positive mask

with the exact same layout and negative photoresist. (3) In this step, the surface of the chip is

firstly covered with a layer of negative photoresist by a spin-coating method. (4) Next, this

layer of negative photoresist is selectively exposed to intense UV light with the sensor pattern

shielded by the mask. During this procedure, the exposed part becomes polymerised and

extremely hard to dissolve. (5) Then, the chip is immersed into a developer solution to

dissolve the unexposed parts, leaving the remaining photoresist of a pattern negative to the

sensors. (6) With the patterning procedure done, the chip is sent to electron beam physical

vapour deposition (EBPVD) equipment to form metal connection. In the consideration of

physical and chemical stability, a layer of 200 nm Pt film is deposited onto the surface of the

chip by a vaporised Pt phase produced by high energy electron beam. (7) To remove the

40

unwanted photoresist, a lith-off procedure is conducted in which a liquid resist stripper

chemically alters adhesion of the photoresist to the chip. Finally, the patterned Pt sensors are

fabricated.

Figure 4-3 Brief procedures designed for the sensor fabrication in the clockwise sequence

indicated by the U-shape arrow.

4.1.4 Limitation and suggestions

Because of a lack of accessible probe stations, wire bonding was used to attempt the

electrical connection between the sensors on chip and the external electrical circuit in the

original design of the system as Figure 4-4. However, due to the fragility of those bonding

wires, it is very difficult to achieved stable connection in this scenario. Although soldering is

considered as an alternative method, the metal films of those sensors are too thin to suffer

thermal stress and likely detached from the chip.

Figure 4-4 Preliminary implement of the fabricated sensors and the measurement circuit.

41

It is unfortunate that the integration between the 3ω sensors and the external circuit was

unable to implement at this scenario. But this conceptual design for the sensor and

measurement can be realised if a capable probe station is available. The following steps are

suggested to practically implement the 3ω measurement system:

• First of all, the temperature coefficient of resistance of the sensors on chip must be

characterised, which is the primary factor influencing the accuracy of the 3ω method.

Without any measured samples, the sensors are ready to measure thermal conductivity

of the silicon wafer as well as the thermal SiO2 layer. So, the validation can be done

by directly measuring their thermal conductivity based on the single side principles.

• Further, since the metallic sensors are required to be insulated from any conductive

materials, the current design is only suitable for non-conductive granular media. To

measure electrically conductive granular media, additional insulating layer is

necessary to deposit on the surface of the chip. It is suggested that atomic layer

deposition (ALD) of Al2O3 is good for this purpose; because it can be fabricated in

very thin thickness (tens of nanometres) but maintain excellent insulating properties.

Such advantages introduce as minimum influence as possible to the thermal transport

in the measurement, which is demanded owing to the low effective thermal

conductivity of granular media.

4.2 Transient plane source techniques

4.2.1 Implementation of the HotDisk thermal measurement system

Despite the incompleteness of the previous section due to the technical issues, a ready-to-use

commercial thermal measurement system, named HotDisk, is selected to further carry out the

experimental work in this thesis. To make the HotDisk system compatible with granular

media, a customised sample holder has been designed and fabricated. The content of this

section is based on the following published peer-reviewed conference paper:

Dai, W., Gan, Y., (2017). Measurement of effective thermal conductivity of compacted

granular media by the transient plane source technique. The 8th International

Conference on Micromechanics of Granular Media (P&G2017, 3-7 Jul), Montpellier,

France.

42

43

Figure 4-5 The simple measurement kit including two rings and a free weights supporter.

(Corresponding to Figure 1 in the attached paper)

44

Figure 4-6 The assembly of the experimental kit. (a) #5465 sensor, the diameter is 3.189 mm;

(b) #5501 sensor, the diameter is 6.403 mm; (c) another assembly for applying mechanical

loading. (Corresponding to Figure 2 in the attached paper)

Figure 4-7 The effective thermal conductivity of glass beds consisting of beads in different

diameter range. The x coordinates of the points corresponding to the mid-point of the range.

(Corresponding to Figure 3 in the attached paper)

Figure 4-8 The measured effective thermal conductivity of beds versus packing factors and

the calculated Zehner-Schlunder-Bauer conductivity of the same beds. (Corresponding to

Figure 4 in the attached paper)

Figure 4-9 The effective thermal conductivity versus the external mechanical loadings for

glass beds (Corresponding to Figure 5 in the attached paper)

45

Figure 4-10 The effective thermal conductivity versus the external mechanical loadings for

steel beds. (Corresponding to Figure 6 in the attached paper)

46

4.2.2 Comparison between the experimental observation and the numerical result

It is demonstrated in Figure 4-7 of the Section 4.2.1 that the effective thermal conductivity of

granular media consisting of glass beads is positively correlated to the size of the beads in air

environment. Although the experiments were not conducted in the pressure-controlled air

atmosphere, the observed correlation is still the reflection of the Smoluschowski effect that is

extensively studied in Chapter 3. Recalling Figure 3-6 of the Chapter 3, it not only

emphasises that the effective thermal conductivity of Li4SiO4 is positively influenced by the

filling helium pressure, but also stressed that the increase of particle size enhances the

effective thermal conductivity by vertical comparison. For the universality of Smoluschowski

effect, it will certainly introduce reduction in the thermal conductivity of air when the pore

size decreases with the shrinking of the bead size. As a result, the effective thermal

conductivity of granular media filled with air will become smaller. Although the numerical

work is not the exact simulation of the same glass beads combined with air, the thermal

conductivity ratio of the solid phase to the gas phase is of the similar order (atmospheric air

vs. glass and 105 Pa Helium vs. Li4SiO4), and the resulted relative enhancement is

comparable, around 10% increase with the size changing from 0.3 mm to 0.6 mm in both

cases. Thus, we conclude that the Smoluschowski effect is one of the major reasons

contributing to the positive correlation between the effective thermal conductivity and the

bead size as stated in the Section 4.2.1, a quantitative agreement between the experimental

observation and the numerical result.

Another experimental observation in the Section 4.2.1 is about the influence of the

mechanical loading on the effective thermal conductivity. According to the experiments

conducted in the Section 4.2.1, the influence varies depending on the materials of the beads

in the granular media. When the thermal conductivity of the beads is low, i.e. glass beads, the

effective thermal conductivity is insignificantly affected by the mechanical loading, giving

around 5% enhancement in the corresponding range of increasing mechanical loading as

shown by Figure 4-9. On the other hand, beads of high thermal conductivity, e.g., steel beads,

strengthens the influence of mechanical loading, about 25% in the identical loading range.

The former one of these two scenarios validates the assumption of using the point-contact

model in the FEM simulation performed in Chapter 3 because of the similar thermal

conductivity ratios, although the latter case reminds that this assumption is not suitable if the

thermal conductivity ratio is relatively large. Therefore, the experimental observation verifies

47

the assumption made in Chapter 3 with the supporting evidence and confirms the validity of

the numerical result.

48

49

Chapter 5 Topological Transition by Vibration

Following the bottom-up research scheme, the next step is to solve the meso-scale problems

of granular media. In static state, the major problem is how to describe packing structure of

grains in granular media beyond the macroscopic packing fraction. While, in dynamic state,

due to rearrangement of grains by external agitation, topological evolution is promoted and

creates various patterns. So how to characterise this evolution and interpret the mechanisms

of pattern formation is indispensable in developing fundamental theories for granular media.

Thus, a finalised manuscript, currently under review, about granular crystallisation induced

by mechanical vibration is delivered in this chapter to present efforts to extend understanding

on these topics:

Dai, W., Reimann, J., Hanaor, D., Ferrero, C., Gan, Y., (2018). Modes of wall

induced granular crystallisation in vibrational packing (under review).

arXiv: 1805.07865 [cond-mat.dis-nn].

The disorder-to-order transitions and the influence of geometrical boundary on these

transitions are thoroughly studied. Packing structure of those granular media undergoing

vibration is also quantitatively characterised from macro-scale down towards grain-scale.

Besides, this investigation into the meso-scale structural problems is of equivalent

significance as other chapters in the context of the current thesis, which provide fruitful

granular packing structures for expanding research. These packing structures offer the

topologies to systematically assemble the grain-scale units of which thermal interactions are

deliberated in Chapter 3. Further, this source of packing structure will be fed into Chapter 6

to demonstrate how these meso-scale characteristics of granular media determine the

effective thermal conductivity. Thereby, with the outcome of this chapter, the grain-scale heat

transfer models can be appropriately applied to solve the heat conduction in granular media.

50

51

52

53

54

Table 5-1 Parameters for DEM simulations

55

56

57

58

Figure 5-1 Overall evolution of each granular medium subjected to vibrations of different

amplitude. The histograms (colouring with transparency) in the left column show

distributions of the Voronoi cell packing fraction of the initial state and two final relaxed

states after vibration (𝐴=0.1𝑑 and 𝐴=0.2𝑑) with legends giving the corresponding overall

packing fraction 𝛾 and structural index 𝐹6. The corresponding right column plots demonstrate

the time variation of the overall at transient states during vibration

59

60

61

Figure 5-2 Evolutional phase diagram of the crystallisation in granular media of different

height-to-diameter ratio. Each 𝑆6 density mapping consists of central plane slices of three

axes, the colour indicated by coarse grained 𝑆6 suggests disorder in violet direction and

crystallisation in reddish direction. In the phase diagrams three phases are marked by

background filling, (1) dual-modes cooperating; (2) single-mode prevailing; and (3) sole-

mode dictating which are approximately determined by the competition between cylindrical

and bottom modes.

62

63

Figure 5-3 Evolution of 𝐹6 for particles groups separated by position. CW – first layer near

the cylindrical wall, BW – first layer near the bottom wall and Core – the bulk particles.

64

65

Figure 5-4 (a) A similar trend in the 𝑆6 evolution is observed in all cases. The 𝑆6 accumulates

between 10 and 12 in the final state, while the peak right shifts as D increases. (b) Typical

granular temperature evolution (D50) in the small amplitude vibration scenario. (c) Typical

granular temperature evolution (D40) in the large amplitude vibration scenario. (d) Granular

temperature evolution in the relatively strong fluidisation case (D60). (b) and (c) have the

same legend shown in (d) with HC – highly crystallised, LL – liquid like, MO – moderately

ordered, AVE – averaged.

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67

68

Figure 5-5 Smoothed probability density histograms of the crystallised structures appearing

in the granular media during vibration for D30 and D60. The (𝑊4local, 𝑄6

local) coordinates are

used to characterise the structure types. The intersects of pairs of dashed lines in orange and

green are the coordinates of the FCC and HCP, respectively.

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70

Figure 5-6 Rupture in the HCP structure near the cylindrical wall in D30. (a) Blue particles

are distorted HCP particles ( 0.465 ≤ 𝑄6local ≤ 0.505, 𝑊4

local ≥ 0.08 ) while yellow particles

are rupture particles ( 0.465 ≤ 𝑄6local ≤ 0.505, |𝑊4

local| ≤ 0.02 ) with size scaled by 0.5 for

visibility. (b) Typical rupture section with Particle 2 being the rupture particle. P1, P2, P3 are

the 12-neighbour configurations of Particle 1, 2, 3, respectively.

71

Figure 5-7 Density distributions of crystallised structures at the final relaxed state. The right

column displays the corresponding packing structures with particles dyed according to the

(𝑊4local, 𝑄6

local) coordinates in red – HCP, blue – FCC, green – surface hexagon and yellow –

others. The diameters of the particles are rescaled for visualisation purposes.

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73

Figure 5-8 Top – Evolution of the structural index 𝐹6 of the frictionless and frictional

granular media. The top two dashed lines serve as the extension for the final states in the

simulation and the middle one labelled with Exp. C represents the final state of the

experiment performed with a vibration intensity 𝛤 = 2 (Reimann et al., 2017). Bottom –𝑆6

spatial distributions of the labelled states in the simulated evolution and the experimental

result. Friction coefficient (𝜇) and amplitude (𝐴) values for the simulations are displayed in

the legend.

74

75

Figure 5-9 Top – 𝑆6 distributions of the final state in the experiment Exp. C in (Reimann et

al., 2017) and the transient states in the simulations. Particles dyed according to the (𝑊4local,

𝑄6local) coordinates are displayed as insets in the 𝑆6 distribution in red – HCP, blue – FCC and

green – surface hexagon. Bottom – The corresponding (𝑊4local, 𝑄6

local) coordinates

distributions. Friction (𝜇), amplitude (𝐴) and duration (𝑡) parameters for the simulations are

displayed in the legend.

76

Figure 5-10 Top – 𝑆6 distributions of the final state in the experiment Exp. D in (Reimann et

al., 2017) and the simulations along with the selected 𝑆6 spatial distributions in the insets.

Bottom – The corresponding (𝑊4local, 𝑄6

local) coordinates distributions and the packing of dyed

particles. Friction (𝜇), amplitude (𝐴) and duration (𝑡) values for the simulations are displayed

in the legend.

77

78

79

80

81

Chapter 6 Discrete Element Simulation with Interstitial Gas Phase

This chapter serves as the integration procedure to establish an apt solution tackling the heat

conduction in granular media and further to utilise the outcomes from previous chapters to

extend the research. In this chapter, a numerical framework to simulate heat conduction in

stagnant solid-gas/fluid granular media is realised in an open source platform based on

discrete element methods. Not only the granular media extracted from Chapter 5 are

simulated to evaluate their effective thermal conductivity, but also the grain-scale

mechanisms influencing the heat conduction of those media are particularly analysed. It is

found that ordered packing structure can enhance the effective thermal conductivity, which is

further explained by the influence of the grain-scale variations of packing structure on the

local thermal conductivity of individual grains. With the conclusions of Chapter 6, the

bottom-up research scheme completely covers multi-scale problems of heat conduction in

granular media.

The main content of this Chapter is based on the following finalised manuscript:

Dai, W., Hanaor, D., Gan, Y., (2018). The effects of packing structure on the effective

thermal conductivity of granular media: A grain scale investigation (under review).

arXiv: 1809.01379 [cond-mat.soft].

82

83

84

85

86

87

88

Figure 6-1 Separated (left) and contacting (right) pairs of grains.

89

90

Figure 6-2 Separated (left) and contacting (right) geometries coloured with steady state

temperature field in the finite element simulation.

91

Figure 6-3 (a) and (b) compare the heat flow between the finite element simulation (filled

circles) and the original Batchelor & O’Brien model (solid lines), coloured according to 𝛼. (c)

and (d) plot 𝜒 for each 𝛼 and 𝑟cont/𝑟c pair as well as 𝛼 and ℎ/𝑟c pair with joining lines,

respectively. (e) gives the correlation between 𝜒 and 𝛼 and (f) compares the computed heat

flow between the modified model and the simulated result.

92

93

94

95

Table 6-1 Gas properties used in this work

96

Figure 6-4 Top – Experimental configuration. Bottom – Comparison between simulation

results and the experimental measurement in this work (left) and previously published studies

(right).

97

Table 6-2 Geometry parameters of granular media

98

Figure 6-5 𝜑V (– left) and 𝑆6 (– right) increase as the duration of vibration is extended,

noticing that the horizontal axis is arbitrary.

99

Figure 6-6 Evolution of 𝑆6 spatial patterns in D40 with vibration duration (from left to right).

100

Figure 6-7 (a) The variation of 𝛼 with helium gas pressure in SiO2-He granular media

consisting of 2.3-mm-diameter SiO2 grains at room temperature. (b) and (c) show the

correlations of (𝑘eff , 𝜑V) and (𝑘eff , 𝑆6) in the minimum 𝛼 (≈ 1000), respectively. (d) and (e)

show similar correlations of the maximum 𝛼 (< 10).

101

102

Figure 6-8 Spatial evolution of 𝑘𝑧𝑧/𝑘s (top row), 𝜃 (middle) and 𝐻flow (bottom) of the D40

granular media.

103

104

Figure 6-9 Left – (𝑆6 , 𝑘𝑧𝑧 ), middle – (𝑆6 , 𝜃𝑧 ) and right – (𝜑V , 𝑘𝑧𝑧 ) plots of D30, D40 and

D50 (from top to bottom respectively), created to according to the mapping operation, in the

scenario of 𝛼 = 100.

105

Figure 6-10 (𝑆6, 𝑘𝑧𝑧) and (𝑆6, 𝜃) plots of well-ordered packing structure belonging to three

types of granular media in different scenarios. (a) and (b) – 𝛼 ≈ 1000, (c) and (d) – 𝛼 ≈ 10.

106

Figure 6-11 The variation of 𝑘eff against the change of packing fraction in artificial media

with disorder and well-order packing structures in 𝛼 = 1000 (left) and 𝛼 = 10 (right)

scenarios.

107

108

109

110

111

112

113

Chapter 7 Conclusions

7.1 Major contributions

A cross-scale investigation on the heat conduction of granular media has been achieved in

this work. The interfacial thermal interactions at grain-scale, topological characteristics and

transitions at meso-scale, and effective property at macro-scale are extensively studied and

the integration of different scales is also realised.

• A finite element analysis based numerical method has been applied to study the grain-

grain thermal interaction. Grain-scale heat conduction is accurately simulated, which

has been combined with an analytical description of granular media to estimate the

effective thermal conductivity, capturing the Smoluschowski effect. Further, this

numerical method is further used to improve its accuracy and applicability of the

conventional heat conduction model, in particular reproducing the dependency on

grain size and interstitial gas pressure.

• A numerical platform based on the discrete element method has been employed to

simulate the topological transitions of granular media subjected to vibration. The

crystallisation phenomenon and the corresponding evolution of topological

characteristics are thoroughly examined. Particularly, the geometrical boundary

influence is investigated, which introduces different topological patterns.

• To develop the capability of this platform for simulating bi-phase heat conduction of

granular media, the modified Batchelor and O’Brien model is implemented.

Integrated with the results regarding the topology, grain-scale interplays between heat

conduction and topology characteristics have been identified, and the underlying

mechanism for enhancing effective thermal conductivity are generalised.

• Last but not the least, the HotDisk measurement system is customised for the

characterisation on effective thermal conductivity of granular media, which provides

validation about the numerical results. Further, the mechanical and size dependency

of the heat conduction is also observed by this technique.

At the end, a complete loop going through heat conduction problems in different scales of

granular media has been established in this work. This approach has demonstrated its

114

potential for granular media, which can be adapted for investigation on other issues of

granular media, especially the transport phenomena.

7.2 Future perspectives

According to the findings of this work, the following perspectives are suggested for future

research.

• The numerical finite element analytics to investigate heat transfer phenomena at

grain-scale can be extended to incorporate other heat transfer activities, such as

thermal radiation, thermal convection, thermal contact resistance and poly-dispersity.

Not only can this provide more accurate correlations for the contact unit model used

in this work, but also other models that require empirically fitted parameters will also

benefit from this investigation, e.g., establishing links between these parameters and

measurable physical quantities.

• On the other side of discrete element methods, the basic heat transfer model can be

extended to incorporate thermal radiation and thermal expansion, which will improve

the accuracy of the presented heat transfer simulation. Further beyond the scope of bi-

phase granular media, how to simulate heat conduction in unsaturated granular media,

which is a typical tri-phase problem, is awaiting to resolve. Two main tasks are

necessary to tackle in order to achieve an appropriate solution. At the grain-scale, how

to describe the influence of encased fluid phase due to capillary effect should be

formulated first. The more challenging part is to generate realistic spatial distribution

of the fluid phase in the simulated granular media, rather than just by imposing an

average quantity calculated by global characteristics.

• Apart from the heat transfer problems, complicated topology puzzles of granular

media remain substantial. Though mono-dispersed granular media has been

extensively investigated in this work, poly-dispersed granular media are hardly

touched. The biggest challenge is to find appropriate topological characterisation

methods to quantitatively describe the poly-dispersed structure. The coordination

number and the Voronoi/Delaunay tessellation are quite robust in the application onto

poly-dispersed granular media. But it is difficult to construct similar measure like the

bond orientation order to quantify the symmetrical characteristics of poly-dispersed

granular media. Therefore, the crystallisation phenomenon has been scarcely explored

in combination of poly-dispersity. Even in the mono-dispersed granular media, it still

115

requires further investigation to clarify how the dynamic behaviours of granular

media, e.g., granular convection, relate with crystallisation and other topological

transitions.

• Equivalently important as the understanding of the mechanisms under topological

behaviours and characteristics of granular media, learning the methods to manipulate

the granular topologies will have great practical impacts. Though systematic

formulations to govern those topological phenomena have not been established yet,

useful means to modify and control the topologies of granular media are possible to

generalise by appropriate simulation and experiment. It has been pointed out in this

work that the boundary geometry strongly influences the topologies. Such influence

can be further studied with respect to the roughness and curviness of the boundary

regions.

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Appendix I – C++ codes of heat transfer model implemented in

LIGGGHTS

The implemented codes are developed based on the original source codes of LIGGGHTS

(version LIGGGHTS-PUBLIC 3.8.0). Only the modified parts are presented, and other

unchanged parts are denoted as … in the following text.

AI.1 Grain-grain heat transfer

This code is originated from the code of fix_heat_gran_conduction.cpp in LIGGGHTS

source code.

In order to use this code, a new version of the fix heat/gran and fix heat/gran/conduction

commands is used in the input scripting for LIGGGHTS with the template as following:

fix ID group-ID heat/gran (or heat/gran/conduction) initial_temperature T contact_area c

gas_type g gas_pressure p solid_atomic_solid s

• T – the numeric value to set the initial temperature of grains;

• c – string value from overlap, constant, projection and hertz to determine which

method to calculate contact area;

• g – string value from helium and air to select the gas phase or constant to define an

artificial uniform continuum phase;

• p – numeric value to set gas pressure for the gas phase or thermal conductivity for the

corresponding artificial uniform continuum phase;

• s – numeric value to set the molar mass of solid phase for the calculation of reduced

thermal conductivity due to the Smoluschowski effect if g is set for helium and air.

Detail can be referred from:

https://www.cfdem.com/media/DEM/docu/fix_heat_gran_conduction.html

#include "fix_heat_gran_conduction.h" ... using namespace FixConst; // modes for conduction contact area calaculation // same as in fix_wall_gran.cpp

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enum{ CONDUCTION_CONTACT_AREA_OVERLAP, CONDUCTION_CONTACT_AREA_CONSTANT, CONDUCTION_CONTACT_AREA_PROJECTION, CONDUCTION_CONTACT_AREA_HERTZ, HELIUM,AIR,CONSTANT}; /* --------------------------------------------------------------------- */ FixHeatGranCond::FixHeatGranCond(class LAMMPS *lmp, int narg, char **arg) : FixHeatGran(lmp, narg, arg), fix_conductivity_(0), conductivity_(0), store_contact_data_(false), fix_conduction_contact_area_(0), fix_n_conduction_contacts_(0), fix_wall_heattransfer_coeff_(0), fix_wall_temperature_(0), conduction_contact_area_(0), n_conduction_contacts_(0), wall_heattransfer_coeff_(0), wall_temp_(0), area_calculation_mode_(CONDUCTION_CONTACT_AREA_OVERLAP), fixed_contact_area_(0.), area_correction_flag_(0), gas_pressure_(0.), solid_atomic_mass_(1.), gas_type_(HELIUM), deltan_ratio_(0) { iarg_ = 5; bool hasargs = true; while(iarg_ < narg && hasargs) { hasargs = false; if(strcmp(arg[iarg_],"contact_area") == 0) { if(strcmp(arg[iarg_+1],"overlap") == 0) area_calculation_mode_ = CONDUCTION_CONTACT_AREA_OVERLAP; else if(strcmp(arg[iarg_+1],"hertz") == 0) area_calculation_mode_ = CONDUCTION_CONTACT_AREA_HERTZ; else if(strcmp(arg[iarg_+1],"projection") == 0) area_calculation_mode_ = CONDUCTION_CONTACT_AREA_PROJECTION; else if(strcmp(arg[iarg_+1],"constant") == 0) { if (iarg_+3 > narg) error->fix_error(FLERR,this,"not enough arguments for keyword

'contact_area constant'"); area_calculation_mode_ = CONDUCTION_CONTACT_AREA_CONSTANT; fixed_contact_area_ = force->numeric(FLERR,arg[iarg_+2]); if (fixed_contact_area_ <= 0.) error->fix_error(FLERR,this,"'contact_area constant' value must

be > 0"); iarg_++;

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} else error->fix_error(FLERR,this,"expecting 'overlap', 'projection' ,

'hertz' or 'constant' after 'contact_area'"); iarg_ += 2; hasargs = true; } else if(strcmp(arg[iarg_],"area_correction") == 0) { if (iarg_+2 > narg)

error->fix_error(FLERR,this,"not enough arguments for keyword 'area_correction'");

if(strcmp(arg[iarg_+1],"yes") == 0) area_correction_flag_ = 1; else if(strcmp(arg[iarg_+1],"no") == 0) area_correction_flag_ = 0; else

error->fix_error(FLERR,this,"expecting 'yes' or 'no' after 'area_correction'");

iarg_ += 2; hasargs = true; }

else if(strcmp(arg[iarg_],"store_contact_data") == 0) {...} else if(strcmp(arg[iarg_],"gas_type") == 0) { if (iarg_+2 > narg)

error->fix_error(FLERR,this,"not enough arguments for keyword 'gas_type'");

if(strcmp(arg[iarg_+1],"helium") == 0) gas_type_ = HELIUM; else if(strcmp(arg[iarg_+1],"air") == 0) gas_type_ = AIR;

else if(strcmp(arg[iarg_+1],"constant") == 0) gas_type_ = CONSTANT; else error->fix_error(FLERR,this,"expecting 'helium', 'air' or

'constant' after 'gas type'"); iarg_ += 2; hasargs = true; } else if(strcmp(arg[iarg_],"gas_pressure") == 0.) { if (iarg_+2 > narg)

error->fix_error(FLERR,this,"not enough arguments for keyword 'gas_pressure'");

gas_pressure_ = force->numeric(FLERR,arg[iarg_+1]); if (gas_pressure_ < 0.) error->fix_error(FLERR,this,"'gas pressure' value must be > 0"); iarg_ += 2; hasargs = true; } else if(strcmp(arg[iarg_],"solid_atomic_mass") == 0) { if (iarg_+2 > narg)

error->fix_error(FLERR,this,"not enough arguments for keyword 'solid_atomic_mass'");

solid_atomic_mass_ = force->numeric(FLERR,arg[iarg_+1]); if (solid_atomic_mass_ <= 0.)

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error->fix_error(FLERR,this,"'solid atomic mass' value must be > 0"); iarg_ += 2; hasargs = true; } else if(strcmp(style,"heat/gran/conduction") == 0) error->fix_error(FLERR,this,"unknown keyword"); }

if(CONDUCTION_CONTACT_AREA_OVERLAP != area_calculation_mode_ && 1 == area_correction_flag_)

error->fix_error(FLERR,this,"can use 'area_correction' only for 'contact_area = overlap'");

/* --------------------------------------------------------------------- */ FixHeatGranCond::~FixHeatGranCond() {...} ... /* --------------------------------------------------------------------- */ void FixHeatGranCond::post_force(int vflag) { if(history_flag == 0 && CONDUCTION_CONTACT_AREA_OVERLAP == area_calculation_mode_) post_force_eval<0,CONDUCTION_CONTACT_AREA_OVERLAP>(vflag,0); if(history_flag == 1 && CONDUCTION_CONTACT_AREA_OVERLAP == area_calculation_mode_) post_force_eval<1,CONDUCTION_CONTACT_AREA_OVERLAP>(vflag,0); if(history_flag == 0 && CONDUCTION_CONTACT_AREA_CONSTANT == area_calculation_mode_) post_force_eval<0,CONDUCTION_CONTACT_AREA_CONSTANT>(vflag,0); if(history_flag == 1 && CONDUCTION_CONTACT_AREA_CONSTANT == area_calculation_mode_) post_force_eval<1,CONDUCTION_CONTACT_AREA_CONSTANT>(vflag,0); if(history_flag == 0 && CONDUCTION_CONTACT_AREA_HERTZ == area_calculation_mode_) post_force_eval<0,CONDUCTION_CONTACT_AREA_HERTZ>(vflag,0); if(history_flag == 1 && CONDUCTION_CONTACT_AREA_HERTZ == area_calculation_mode_) post_force_eval<1,CONDUCTION_CONTACT_AREA_HERTZ>(vflag,0); if(history_flag == 0 && CONDUCTION_CONTACT_AREA_PROJECTION == area_calculation_mode_) post_force_eval<0,CONDUCTION_CONTACT_AREA_PROJECTION>(vflag,0); if(history_flag == 1 && CONDUCTION_CONTACT_AREA_PROJECTION == area_calculation_mode_) post_force_eval<1,CONDUCTION_CONTACT_AREA_PROJECTION>(vflag,0); } /* ---------------------------------------------------------------------- */ void FixHeatGranCond::cpl_evaluate(ComputePairGranLocal *caller)

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{ if(caller != cpl) error->all(FLERR,"Illegal situation in FixHeatGranCond::cpl_evaluate"); if(history_flag == 0 && CONDUCTION_CONTACT_AREA_OVERLAP == area_calculation_mode_) post_force_eval<0,CONDUCTION_CONTACT_AREA_OVERLAP>(0,1); if(history_flag == 1 && CONDUCTION_CONTACT_AREA_OVERLAP == area_calculation_mode_) post_force_eval<1,CONDUCTION_CONTACT_AREA_OVERLAP>(0,1); if(history_flag == 0 && CONDUCTION_CONTACT_AREA_CONSTANT == area_calculation_mode_) post_force_eval<0,CONDUCTION_CONTACT_AREA_CONSTANT>(0,1); if(history_flag == 1 && CONDUCTION_CONTACT_AREA_CONSTANT == area_calculation_mode_) post_force_eval<1,CONDUCTION_CONTACT_AREA_CONSTANT>(0,1); if(history_flag == 0 && CONDUCTION_CONTACT_AREA_HERTZ == area_calculation_mode_) post_force_eval<0,CONDUCTION_CONTACT_AREA_HERTZ>(0,1); if(history_flag == 1 && CONDUCTION_CONTACT_AREA_HERTZ == area_calculation_mode_) post_force_eval<1,CONDUCTION_CONTACT_AREA_HERTZ>(0,1); if(history_flag == 0 && CONDUCTION_CONTACT_AREA_PROJECTION == area_calculation_mode_) post_force_eval<0,CONDUCTION_CONTACT_AREA_PROJECTION>(0,1); if(history_flag == 1 && CONDUCTION_CONTACT_AREA_PROJECTION == area_calculation_mode_) post_force_eval<1,CONDUCTION_CONTACT_AREA_PROJECTION>(0,1); } /* -------------------------------------------------------------------- */ template <int HISTFLAG,int CONTACTAREA> void FixHeatGranCond::post_force_eval(int vflag,int cpl_flag) { double contactArea,delta_n,flux,dirFlux[3]; int i,j,ii,jj,inum,jnum; double xtmp,ytmp,ztmp,delx,dely,delz; double radi,radj,radsum,rsq,r,tcoi,tcoj,radij,dij,reff,hci,hcj,hcont,asg; double rcont,heff; double deltaT = 0.; double Kn,mT,dm,cl,beta,ac,rKn,cL,mg; /* Smoluchowski effect */ const double kb = 1.38e-23; /* Boltzman constant in J/K */ int *ilist,*jlist,*numneigh,**firstneigh; int *contact_flag,**first_contact_flag; double eij = 0.5; /* heat transfer criterion */ double xij = 0.; /* effective radius of equivalent cylinder */ double tcog,tcogs; /* gas thermal conductivity */ int newton_pair = force->newton_pair; if (strcmp(force->pair_style,"hybrid")==0)

error->warning(FLERR,"Fix heat/gran/conduction implementation may not be valid for pair style hybrid");

if (strcmp(force->pair_style,"hybrid/overlay")==0) error->warning(FLERR,"Fix heat/gran/conduction implementation may not be valid for pair style hybrid/overlay");

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inum = pair_gran->list->inum; ilist = pair_gran->list->ilist; numneigh = pair_gran->list->numneigh; firstneigh = pair_gran->list->firstneigh; if(HISTFLAG) first_contact_flag = pair_gran->listgranhistory->firstneigh; double *radius = atom->radius; double **x = atom->x; int *type = atom->type; int nlocal = atom->nlocal; int *mask = atom->mask; updatePtrs(); if(store_contact_data_) { fix_conduction_contact_area_->set_all(0.); fix_n_conduction_contacts_->set_all(0.); } // vacuum if(gas_pressure_ == 0.) { // loop over neighbors of my atoms for (ii = 0; ii < inum; ii++) {...} } else { for (ii = 0; ii < inum; ii++) { i = ilist[ii]; xtmp = x[i][0]; ytmp = x[i][1]; ztmp = x[i][2]; radi = radius[i]; mT = Temp[i]; jlist = firstneigh[i]; jnum = numneigh[i]; if(HISTFLAG) contact_flag = first_contact_flag[i]; for (jj = 0; jj < jnum; jj++) { j = jlist[jj]; j &= NEIGHMASK; if (!(mask[i] & groupbit) && !(mask[j] & groupbit)) continue; if(!HISTFLAG) { delx = xtmp - x[j][0]; dely = ytmp - x[j][1]; delz = ztmp - x[j][2]; rsq = delx*delx + dely*dely + delz*delz; radj = radius[j]; radsum = radi + radj;

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radij = 2. * radi*radj / (radi + radj); r = sqrt(rsq); dij = r - radsum; } if ((HISTFLAG && contact_flag[jj]) || (!HISTFLAG && (dij < eij*radij))) { //contact if(HISTFLAG) { delx = xtmp - x[j][0]; dely = ytmp - x[j][1]; delz = ztmp - x[j][2]; rsq = delx*delx + dely*dely + delz*delz; radj = radius[j]; radsum = radi + radj; radij = 2.*radi*radj/(radi + radj); r = sqrt(rsq); dij = r - radsum; if(dij >= eij*radij) continue; } tcoi = conductivity_[type[i]-1]; tcoj = conductivity_[type[j]-1]; if(gas_type_==CONSTANT) { rKn = std::min(radi,radj); tcog = gas_pressure_; asg = tcoi/tcog; if(dij >= 0.) { if ( asg*asg*dij/radij < 0.01)

hcont = M_PI*tcog*radij* log(asg*asg); else if ( asg*asg*dij/radij > 100.)

hcont = M_PI*tcog*radij* log(1. + radij/dij); else hcont = std::min( M_PI*tcog*radij* log (asg*asg),

M_PI*tcog*radij* log(1. + radij/dij)); contactArea = 0.; } else { if(CONTACTAREA == CONDUCTION_CONTACT_AREA_OVERLAP) { if(area_correction_flag_) { delta_n = radsum - r; delta_n *= deltan_ratio_[type[i]-1][type[j]-1]; r = radsum - delta_n; } if (r < fmax(radi, radj)) // one sphere is inside the other { // set contact area to area of smaller sphere contactArea = fmin(radi,radj); contactArea *= contactArea * M_PI;

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} else //contact area of the two spheres contactArea = - M_PI/4.0 * ( (r-radi-radj)*(r+radi-radj)

*(r-radi+radj)*(r+radi+radj) )/(r*r); } else if (CONTACTAREA == CONDUCTION_CONTACT_AREA_HERTZ) contactArea = M_PI * radi*radj/(radi+radj) * (radsum-r); else if (CONTACTAREA == CONDUCTION_CONTACT_AREA_CONSTANT) contactArea = fixed_contact_area_; else if (CONTACTAREA == CONDUCTION_CONTACT_AREA_PROJECTION) { double rmax = std::max(radi,radj); contactArea = M_PI*rmax*rmax; } rcont = sqrt (contactArea/M_PI); if (asg*rcont/radij > 100.)

hcont = M_PI*tcog*radij*(2.*asg*rcont/radij/M_PI – 2.*log (asg*rcont/radij) + log (asg*asg));

else if (asg*rcont/radij < 1.) hcont = M_PI*tcog*radij*(0.22*(asg*rcont/radij)* (asg*rcont/radij)-0.05*(asg*rcont/radij)*(asg*rcont/radij) + log (asg*asg));

else hcont = M_PI*tcog*radij*(((2.*100./M_PI - 2.*log (100.) ) - 0.17) / 99. * ((asg*rcont/radij) - 1.) + (0.17 + log (asg*asg)));

} } else { /* Smoluchowski effect */ mT = (Temp[i]+Temp[j])/2.; if(gas_type_ == HELIUM) { tcog = 3.366e-3*pow(mT+273.,0.668); dm = 2.15e-10; mg = 4.; } if(gas_type_ == AIR) { tcog = 0.0241 - 1e-11*pow(mT,3) - 4e-8*mT*mT + 8e-5*mT; dm = 3.66e-10; mg = 28.96; } if(dij >= 0.) { rKn = std::min(radi,radj); cL = dij + radi*((asin(rKn/radi) - rKn/radi)/asin(rKn/radi)) +

radj*((asin(rKn/radj) - rKn/radj)/asin(rKn/radj)); Kn = (kb*(mT + 273.)/(sqrt(2.)*M_PI*dm*dm*gas_pressure_))/cL; ac = 2.4*(mg/solid_atomic_mass_)/((1. + mg/solid_atomic_mass_)*(1.

141

+ mg/solid_atomic_mass_)); beta = (2. - ac)/ac; tcogs = tcog/(1. + 2.*beta*Kn); asg = tcoi/tcogs; if ( asg*asg*dij/radij < 0.01)

hcont = M_PI*tcogs*radij* log(asg*asg); else if ( asg*asg*dij/radij > 100.)

hcont = M_PI*tcogs*radij* log(1. + radij/dij); else

hcont = std::min( M_PI*tcogs*radij* log (asg*asg), M_PI*tcogs*radij* log(1. + radij/dij));

contactArea = 0.; } else { if(CONTACTAREA == CONDUCTION_CONTACT_AREA_OVERLAP) { if(area_correction_flag_) { delta_n = radsum - r; delta_n *= deltan_ratio_[type[i]-1][type[j]-1]; r = radsum - delta_n; } if (r < fmax(radi, radj)) // one sphere is inside the other { // set contact area to area of smaller sphere contactArea = fmin(radi,radj); contactArea *= contactArea * M_PI; } else //contact area of the two spheres contactArea = - M_PI/4.0 * ( (r-radi-radj)*(r+radi-radj)*

(r-radi+radj)*(r+radi+radj) )/(r*r); } else if (CONTACTAREA == CONDUCTION_CONTACT_AREA_HERTZ) contactArea = M_PI * radi*radj/(radi+radj) * (radsum-r); else if (CONTACTAREA == CONDUCTION_CONTACT_AREA_CONSTANT) contactArea = fixed_contact_area_; else if (CONTACTAREA == CONDUCTION_CONTACT_AREA_PROJECTION) { double rmax = std::max(radi,radj); contactArea = M_PI*rmax*rmax; } rcont = sqrt (contactArea/M_PI); rKn = std::min(radi,radj); cL = radi*(asin(rKn/radi) - asin(rcont/radi) - rKn/radi +

rcont/radi)/(asin(rKn/radi) - asin(rcont/radi)) + radj*(asin(rKn/radj) - asin(rcont/radj) - rKn/radj + rcont/radj)/(asin(rKn/radj) - asin(rcont/radj));

Kn = (kb*(mT + 273.)/(sqrt(2.)*M_PI*dm*dm*gas_pressure_))/cL; ac = 2.4*(mg/solid_atomic_mass_)/((1. + mg/solid_atomic_mass_)*

(1. + mg/solid_atomic_mass_)); beta = (2. - ac)/ac; tcogs = tcog/(1. + 2.*beta*Kn); asg = tcoi/tcogs;

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if (asg*rcont/radij > 100.) hcont = M_PI*tcogs*radij*(2.*asg*rcont/radij/M_PI - 2.* log (asg*rcont/radij) + log (asg*asg));

else if (asg*rcont/radij < 1.) hcont = M_PI*tcogs*radij*(0.22*(asg*rcont/radij)* (asg*rcont/radij)-0.05*(asg*rcont/radij)* (asg*rcont/radij) + log (asg*asg));

else hcont = M_PI*tcogs*radij*(((2.*100./M_PI - 2.*log (100.) )

- 0.17) / 99. * ((asg*rcont/radij) - 1.) + (0.17 + log (asg*asg)));

} }

if (tcoi < SMALL_FIX_HEAT_GRAN || tcoj < SMALL_FIX_HEAT_GRAN) heff = 0.;

xij = 1.3121*pow(asg,-0.19); reff = xij*rKn; hci = M_PI*tcoi*reff*reff/radi; hcj = M_PI*tcoj*reff*reff/radj; heff = 1./(1./hci + 1./hcj + 1./hcont); flux = (Temp[j]-Temp[i])*heff; dirFlux[0] = flux*delx; dirFlux[1] = flux*dely; dirFlux[2] = flux*delz; ... } } } } ... } ... void FixHeatGranCond::unregister_compute_pair_local(ComputePairGranLocal *ptr){...}

143

AI.2 concurrently Grain-wall heat transfer

This code is originated from the code of fix_wall_gran.cpp in LIGGGHTS source code.

The new version of the fix wall/gran command has an identical additional template as the fix

heat/gran and fix heat/gran/conduction commands. Detail can be referred in Section 7.3 as

well as https://www.cfdem.com/media/DEM/docu/fix_wall_gran.html.

#include <cmath> ... const double SMALL = 1e-12; // modes for conduction contact area calaculation // same as in fix_heat_gran_conduction.cpp enum{ CONDUCTION_CONTACT_AREA_OVERLAP, CONDUCTION_CONTACT_AREA_CONSTANT, CONDUCTION_CONTACT_AREA_PROJECTION, CONDUCTION_CONTACT_AREA_HERTZ, HELIUM, AIR, CONSTANT }; /* -------------------------------------------------------- */ FixWallGran::FixWallGran(LAMMPS *lmp, int narg, char **arg) : Fix(lmp, narg, arg), impl(NULL), fix_sum_normal_force_(NULL) { ... area_calculation_mode_ = CONDUCTION_CONTACT_AREA_OVERLAP; gas_type_ = HELIUM; gas_pressure_ = 0.; solid_atomic_mass_ = 1.; ... while(iarg_ < narg && hasargs) { ... else if (strcmp(arg[iarg_],"temperature") == 0) {...} else if(strcmp(arg[iarg_],"contact_area") == 0)

{ if(strcmp(arg[iarg_+1],"overlap") == 0) area_calculation_mode_ = CONDUCTION_CONTACT_AREA_OVERLAP; else if(strcmp(arg[iarg_+1],"projection") == 0) area_calculation_mode_ = CONDUCTION_CONTACT_AREA_PROJECTION; else if(strcmp(arg[iarg_+1],"hertz") == 0) area_calculation_mode_ = CONDUCTION_CONTACT_AREA_HERTZ;

144

else if(strcmp(arg[iarg_+1],"constant") == 0) { if (iarg_+3 > narg) error->fix_error(FLERR,this,"not enough arguments for

keyword 'contact_area constant'"); area_calculation_mode_ = CONDUCTION_CONTACT_AREA_CONSTANT; fixed_contact_area_ = force->numeric(FLERR,arg[iarg_+2]); if (fixed_contact_area_ <= 0.) error->fix_error(FLERR,this,"'contact_area constant' value

must be > 0"); iarg_++; } else

error->fix_error(FLERR,this,"expecting 'overlap', 'hertz', 'projection' or 'constant' after 'contact_area'");

iarg_ += 2; hasargs = true; } else if(strcmp(arg[iarg_],"gas_type") == 0) { if (iarg_+2 > narg)

error->fix_error(FLERR,this,"not enough arguments for keyword 'gas_type'");

if(strcmp(arg[iarg_+1],"helium") == 0) gas_type_ = HELIUM; else if(strcmp(arg[iarg_+1],"air") == 0) gas_type_ = AIR; else if(strcmp(arg[iarg_+1],"constant") == 0)

gas_type_ = CONSTANT; else error->fix_error(FLERR,this,"expecting 'helium', 'air' or

'constant' after 'gas type'"); iarg_ += 2; hasargs = true; } else if(strcmp(arg[iarg_],"gas_pressure") == 0) { if (iarg_+2 > narg)

error->fix_error(FLERR,this,"not enough arguments for keyword 'gas_pressure'");

gas_pressure_ = force->numeric(FLERR,arg[iarg_+1]); if (gas_pressure_ < 0.) error->fix_error(FLERR,this,"'gas pressure' value must

be >= 0"); iarg_ += 2; hasargs = true; } else if(strcmp(arg[iarg_],"solid_atomic_mass") == 0) { if (iarg_+2 > narg)

error->fix_error (FLERR,this,"not enough arguments for keyword 'solid_atomic_mass'");

solid_atomic_mass_ = force->numeric(FLERR,arg[iarg_+1]); if (solid_atomic_mass_ <= 0.) error->fix_error(FLERR,this,"'solid atomic mass' value must

145

be > 0"); iarg_ += 2; hasargs = true; } } ... } /* ------------------------------------------------------------------- */ void FixWallGran::post_create() {...} ... void FixWallGran::wall_temperature_unique(bool &has_temp,bool &temp_unique, double &temperature_unique) {...} /* --------------------------------------------------------------------*/ void FixWallGran::addHeatFlux(TriMesh *mesh,int ip, const double ri, double delta_n, double area_ratio) { //r is the distance between the sphere center and wallTemp_wall double tcop, tcowall, hs, heff, r, rcont, asg; double hcont = -0.; double Acont=-0.; double reff_wall = 2.*ri; double eij = 0.5; /* heat transfer criterion */ double xij = 0.; /* effective radius of equivalent cylinder */ double reff; double tcog,tcogs; /* gas thermal conductivity */ double Kn,cl,beta,ac,rKn,cL; /* Smoluchowski effect */ double dm = -1.; double mg = -1.; double mT = -1.; int itype = atom->type[ip]; if(mesh) { ScalarContainer<double> *temp_ptr = mesh->prop().getGlobalProperty

< ScalarContainer<double> >("Temp"); if (!temp_ptr) return; Temp_wall = (*temp_ptr)(0); } double *Temp_p = fppa_T->vector_atom; double *heatflux = fppa_hf->vector_atom; tcop = th_cond[itype-1]; //types start at 1, array at 0 tcowall = th_cond[atom_type_wall_-1]; mT = (Temp_wall + Temp_p[ip])/2;

146

if ((fabs(tcop) < SMALL) || (fabs(tcowall) < SMALL)) heff = 0.; else { if(gas_pressure_ == 0.) { if(delta_n <= 0) heff = 0.; else { if(CONDUCTION_CONTACT_AREA_OVERLAP == area_calculation_mode_) { if(deltan_ratio)

delta_n *= deltan_ratio[itype-1][atom_type_wall_-1]; r = ri - delta_n; Acont = (ri*ri-r*r)*M_PI*area_ratio; //contact area sphere-wall } else if (CONDUCTION_CONTACT_AREA_CONSTANT == area_calculation_mode_) Acont = fixed_contact_area_; else if (CONDUCTION_CONTACT_AREA_HERTZ == area_calculation_mode_) Acont = M_PI*ri*delta_n; else if (CONDUCTION_CONTACT_AREA_PROJECTION == area_calculation_mode_) Acont = M_PI*ri*ri; hcont = 4.*tcop*tcowall/(tcop+tcowall)*sqrt(Acont/M_PI); heff = hcont; } } else

{ if(gas_type_ == CONSTANT)

{ tcog = gas_pressure_; asg = tcop/tcog; if(-delta_n > eij*reff_wall) hcont = 0.; else if(delta_n <= 0 && -delta_n <= eij*reff_wall) {

if ( -asg*asg*delta_n/reff_wall < 0.01) hcont = M_PI*tcog*reff_wall*log(asg*asg);

else if ( -asg*asg*delta_n/reff_wall > 100) hcont = M_PI*tcog*reff_wall*log(1 + reff_wall/-delta_n);

else hcont = std::min(M_PI*tcog*reff_wall*log(asg*asg), M_PI*tcog*reff_wall*log(1 + reff_wall/-delta_n));

} else { if(CONDUCTION_CONTACT_AREA_OVERLAP == area_calculation_mode_) { if(deltan_ratio) delta_n *= deltan_ratio[itype-1][atom_type_wall_-1]; r = ri - delta_n; Acont = (ri*ri-r*r)*M_PI*area_ratio; //contact area sphere-wall } else if (CONDUCTION_CONTACT_AREA_CONSTANT == area_calculation_mode_) Acont = fixed_contact_area_;

147

else if (CONDUCTION_CONTACT_AREA_HERTZ == area_calculation_mode_) Acont = M_PI*ri*delta_n; else if (CONDUCTION_CONTACT_AREA_PROJECTION == area_calculation_mode_) Acont = M_PI*ri*ri; rcont = sqrt(Acont/M_PI); if (asg*rcont/reff_wall > 100) hcont = M_PI*tcog*reff_wall*(2*asg*rcont/reff_wall/M_PI –

2*log(asg*rcont/reff_wall) + log(asg*asg)); else if (asg*rcont/reff_wall < 1)

hcont = M_PI*tcog*reff_wall*(0.22*(asg*rcont/reff_wall)* (asg*rcont/reff_wall) - 0.05*(asg*rcont/reff_wall)* (asg*rcont/reff_wall) + log(asg*asg));

else hcont = M_PI*tcog*reff_wall*(((2*100/M_PI - 2*log(100) ) - 0.17) / 99 * ((asg*rcont/reff_wall) - 1) + (0.17 + log(asg*asg)));

} } else { /* Smoluchowski effect */ if(gas_type_ == HELIUM) { tcog = 3.366e-3*pow(mT + 273,0.668); dm = 2.15e-10; mg = 4; } else { tcog = 0.0241 - 1e-11*pow(mT,3) - 4e-8*mT*mT + 8e-5*mT; dm = 3.66e-10; mg = 28.96; } if(-delta_n > eij*reff_wall) hcont = 0.; else if(delta_n <= 0 && -delta_n <= eij*reff_wall) { cL = -delta_n + ri*(M_PI/2 - 1)/(M_PI/2); Kn = (kb*(mT + 273)/(sqrt(2)*M_PI*dm*dm*gas_pressure_))/cL; ac = 2.4*(mg/solid_atomic_mass_)/((1 + mg/solid_atomic_mass_)

*(1 + mg/solid_atomic_mass_)); beta = (2 - ac)/ac; tcogs = tcog/(1 + 2*beta*Kn); asg = tcop/tcogs; if ( -asg*asg*delta_n/reff_wall < 0.01)

hcont = M_PI*tcogs*reff_wall*log(asg*asg); else if ( -asg*asg*delta_n/reff_wall > 100)

hcont = M_PI*tcogs*reff_wall*log(1 + reff_wall/-delta_n); else

hcont = std::min(M_PI*tcogs*reff_wall*log(asg*asg), M_PI*tcogs*reff_wall*log(1 + reff_wall/-delta_n));

} else { if(CONDUCTION_CONTACT_AREA_OVERLAP == area_calculation_mode_)

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{ if(deltan_ratio) delta_n *= deltan_ratio[itype-1][atom_type_wall_-1]; r = ri - delta_n; Acont = (ri*ri-r*r)*M_PI*area_ratio; //contact area sphere-wall } else if (CONDUCTION_CONTACT_AREA_CONSTANT == area_calculation_mode_) Acont = fixed_contact_area_; else if (CONDUCTION_CONTACT_AREA_HERTZ == area_calculation_mode_) Acont = M_PI*ri*delta_n; else if (CONDUCTION_CONTACT_AREA_PROJECTION == area_calculation_mode_) Acont = M_PI*ri*ri; rcont = sqrt(Acont/M_PI); cL = ri*(M_PI/2 - asin(rcont/ri) - 1 + rcont/ri)/(M_PI/2 –

asin(rcont/ri)); Kn = (kb*(mT + 273)/(sqrt(2.)*M_PI*dm*dm*gas_pressure_))/cL; ac = 2.4*(mg/solid_atomic_mass_)/((1 + mg/solid_atomic_mass_)*(1 +

mg/solid_atomic_mass_)); beta = (2 - ac)/ac; tcogs = tcog/(1 + 2*beta*Kn); asg = tcop/tcogs; if (asg*rcont/reff_wall > 100)

hcont = M_PI*tcogs*reff_wall*(2*asg*rcont/reff_wall/M_PI – 2*log(asg*rcont/reff_wall) + log(asg*asg));

else if (asg*rcont/reff_wall < 1) hcont = M_PI*tcogs*reff_wall*(0.22*(asg*rcont/reff_wall) *

(asg*rcont/reff_wall) - 0.05*(asg*rcont/reff_wall)* (asg*rcont/reff_wall) + log(asg*asg));

else hcont = M_PI*tcogs*reff_wall*(((2*100/M_PI - 2*log(100) ) –

0.17) / 99 * ((asg*rcont/reff_wall) - 1) + (0.17 + log(asg*asg)));

} }

xij = 1.3121*pow(asg,-0.19); hs = M_PI*tcop*xij*xij*ri; if (hcont == 0.) heff = 0.; else heff = 1/(1/hs+1/hcont); } } if(computeflag_) { double hf = (Temp_wall-Temp_p[ip]) * heff; heatflux[ip] += hf; Q_add += hf * update->dt; if(fppa_htcw) fppa_htcw->vector_atom[ip] = heff; } if(cwl_ && addflag_) cwl_->add_heat_wall(ip,(Temp_wall-Temp_p[ip]) * heff); }

149

Appendix II – Mathematica scripts for topology characterisation

This script is used to calculate the bond orientation order parameters of grains in granular

media according to their position coordinates

SetDirectory "My\\file\\Directory\\" ;$overall OpenAppend "Overall.dat", BinaryFormat True ;WriteString $overall,

"Packing factor\t Solid Number\t F6\t xMSD\t yMSD\t zMSD\t MSDtotal\txMSDstep\t yMSDstep\t zMSDstep\t MSDstep\t \n" ;

Definition of the radial distribution functionRDFunction

Compile Data, Real, 2 , NumG, Integer , diameter, Real ,Rho, Real ,

Module i, j, dR, Rmax, dVol, dx, x, xbin, NumBin ,dR 0.01 diameter; Rmax 6.0 diameter;x Table i, i, dR, Rmax, dR ;NumBin Dimensions x 1 ;x x diameter;xbin Table 0.0, i, dR, Rmax, dR ;DoDo

dx Norm Data i Data j ;If dx Rmax, xbin Ceiling dx dR 2.0; dx 0 ;

dx 0,j, i 1, NumG ,

i, 1, NumG ;DodVol 4 3 i 1 ^3 i^3 dR^3;xbin i xbin i dVol Rho NumG,i, 1, NumBin ;

Transpose x, xbin , CompilationTarget "C" ;

Prepare varibale for Wigner 3 j symbolSol6 x1, x2, x3 .

Solve x1 x2 x3 0 && Abs x1 6 && Abs x2 6 && Abs x3 6,x1, x2, x3 , Integers ;

Sol6Num Dimensions Sol6 1 ;Sol4

x1, x2, x3 .Solve x1 x2 x3 0 && Abs x1 4 && Abs x2 4 && Abs x3 4,

x1, x2, x3 , Integers ;Sol4Num Dimensions Sol4 1 ;

the number of closest particle to calculate bond orientation, 12 for general useNeighNum 12;xyz0 Import "Initial coordiante", "Table" 10 ;; 5009, 3 ;; 5 ;xyzO xyz0;BondOrient step :Module , FileNo ToString step ;

Input particle coordinate

xyzData Import "Selected snapshot coordinate", "Table" 10 ;; 5009,3 ;; 5 ;

xyzNum Dimensions xyzData 1 ;

Mean square displacementxMoveT Table xyzData i, 1 xyz0 i, 1 ^2, i, xyzNum ;yMoveT Table xyzData i, 2 xyz0 i, 2 ^2, i, xyzNum ;zMoveT Table xyzData i, 3 xyz0 i, 3 ^2, i, xyzNum ;dMoveT Table xMoveT i yMoveT i zMoveT i , i, xyzNum ;SDtotal Transpose xMoveT, yMoveT, zMoveT, dMoveT ;MSDtotal Mean SDtotal ;Export "SDtotal " FileNo ".dat", SDtotal, "Table" ;xMoveS Table xyzData i, 1 xyzO i, 1 ^2, i, xyzNum ;yMoveS Table xyzData i, 2 xyzO i, 2 ^2, i, xyzNum ;zMoveS Table xyzData i, 3 xyzO i, 3 ^2, i, xyzNum ;dMoveS Table xMoveS i yMoveS i zMoveS i , i, xyzNum ;SDstep Transpose xMoveS, yMoveS, zMoveS, dMoveS ;MSDstep Mean SDstep ;Export "SDstep " FileNo ".dat", SDstep, "Table" ;

Packing factordiameter 0.0023; Rcylinder 0.03; maxco Max Transpose xyzData ;minco Min Transpose xyzData ;Hcylinder maxco 3 minco 3 diameter;PF diameter^3 6 xyzNum Rcylinder^2 Hcylinder ;

Neighbour list

NeighPTableFlattenTable Cases Nearest xyzData, xyzData i , NeighNum 1 ,

Except xyzData i , i, xyzNum , 1 ;Neighbour vector list

NeighVTable Table Table xyzData i NeighPTable i, j , j, NeighNum ,i, xyzNum ;Neighbour index list

NeighIndexTable Flatten Table Position xyzData, NeighPTable i, j , j, NeighNum ,

i, xyzNum ;Export "NeighIndex " FileNo ".dat", NeighIndex, "Table" ;

xyz spherical coordiantionNeighSpheCoor ToSphericalCoordinates NeighVTable . Indeterminate 0.0;

calculate individual spherical harmonic Q6m & Q4m

q6mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 6, m, NeighSpheCoor i, j, 2 ,NeighSpheCoor i, j, 3 , j, NeighNum , m, 6, 6 , i, xyzNum ;

q6sum Table Sum Norm q6m i, j ^2 , j, 13 , i, xyzNum ;q4mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 4, n, NeighSpheCoor i, j 2 ,NeighSpheCoor i, j 3 , j, NeighNum , n, 4, 4 ,

i, xyzNum ;q4sum Table Sum Norm q4m i, j ^2 , j, 9 , i, xyzNum ;

local bond orientationQ6local Flatten Table N Sqrt 4 2 6 1 q6sum i , i, xyzNum ;Q4local Flatten Table N Sqrt 4 2 4 1 q4sum i , i, xyzNum ;

local Wigner 3 j symbol

w6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

q6m j, Sol6 i, 1 7 q6m j, Sol6 i, 2 7q6m j, Sol6 i, 3 7 , i, Sol6Num q6sum j ^ 3 2 ,

j, xyzNum ;

w4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

q4m j, Sol4 i, 1 5 q4m j, Sol4 i, 2 5q4m j, Sol4 i, 3 5 , i, Sol4Num q4sum j ^ 3 2 ,

j, xyzNum ;BOOlocal Transpose Q4local, Q6local, w4, w6 ;Export "BOOlocal " FileNo ".dat", BOOlocal, "Table" ;

Averaged bond order parameters

Aq6mTableFlattenTable 1 NeighNum

q6m i, m Sum q6m NeighIndex i, j , m , j, NeighNum ,m, 13 , i, xyzNum ;

Aq6sum Table Sum Norm Aq6m i, j ^2 , j, 13 , i, xyzNum ;

Aq4mTableFlattenTable 1 NeighNum

q4m i, m Sum q4m NeighIndex i j , m , j, NeighNum ,m, 9 , i, xyzNum ;

Aq4sum Table Sum Norm Aq4m i, j ^2 , j, 9 , i, xyzNum ;AQ6 Flatten Table N Sqrt 4 2 6 1 Aq6sum i , i, xyzNum ;AQ4 Flatten Table N Sqrt 4 2 4 1 Aq4sum i , i, xyzNum ;

Average Wigner 3 j symbol

Aw6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

Aq6m j, Sol6 i, 1 7 Aq6m j, Sol6 i, 2 7Aq6m j, Sol6 i, 3 7 , i, Sol6Num Aq6sum j ^ 3 2 ,

j, xyzNum ;

Aw4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

Aq4m j, Sol4 i, 1 5 Aq4m j, Sol4 i, 2 5Aq4m j, Sol4 i, 3 5 , i, Sol4Num Aq4sum j ^ 3 2 ,

j, xyzNum ;BOOaverage Transpose AQ4, AQ6, Aw4, Aw6 ;Export "BOOaverage " FileNo ".dat", BOOaverage, "Table" ;

solidlike and liquidlikeCosSimi a , b : N Re a Norm a .Conjugate b Norm b ;q6c 0.7;

q6Table Flatten Table CosSimi q6m i , q6m NeighIndex i, j ,

j, NeighNum , i, xyzNum ;S6 Flatten Table Sum q6 i, j , j, NeighNum , i, xyzNum ;

SorLFlatten Table UnitStep Sum UnitStep q6 i, j q6c , j, NeighNum 7 ,

i, xyzNum ;SolidNum Sum SorL i , i, xyzNum ;

order and disorder

f6Flatten Table N 1 NeighNum Sum UnitStep q6 i, j q6c , j, NeighNum ,

i, xyzNum ;F6 Mean f6 ;Solidliquid Transpose S6, SorL, f6 ;Export "Solid Liquid " FileNo ".dat", Solidliquid, "Table" ;

Calculate radial distribution functionRDFlist RDFunction xyzData, xyzNum, diameter, PF ;Export "RDFlist " FileNo ".dat", RDFlist, "Table" ;

Overall

Overall PF, SolidNum, F6, MSDtotal 1 , MSDtotal 2 ,MSDtotal 3 , MSDtotal 4 , MSDstep 1 , MSDstep 2 , MSDstep 3 ,MSDstep 4 ;

Export $overall, Overall, "Table" ;

WriteString $overall, "\n" ;

xyzO xyzData;

Do BondOrient i , i, step1, step2, intervalClose $overall ;

150

SetDirectory "My\\file\\Directory\\" ;$overall OpenAppend "Overall.dat", BinaryFormat True ;WriteString $overall,

"Packing factor\t Solid Number\t F6\t xMSD\t yMSD\t zMSD\t MSDtotal\txMSDstep\t yMSDstep\t zMSDstep\t MSDstep\t \n" ;

Definition of the radial distribution functionRDFunction

Compile Data, Real, 2 , NumG, Integer , diameter, Real ,Rho, Real ,

Module i, j, dR, Rmax, dVol, dx, x, xbin, NumBin ,dR 0.01 diameter; Rmax 6.0 diameter;x Table i, i, dR, Rmax, dR ;NumBin Dimensions x 1 ;x x diameter;xbin Table 0.0, i, dR, Rmax, dR ;DoDo

dx Norm Data i Data j ;If dx Rmax, xbin Ceiling dx dR 2.0; dx 0 ;

dx 0,j, i 1, NumG ,

i, 1, NumG ;DodVol 4 3 i 1 ^3 i^3 dR^3;xbin i xbin i dVol Rho NumG,i, 1, NumBin ;

Transpose x, xbin , CompilationTarget "C" ;

Prepare varibale for Wigner 3 j symbolSol6 x1, x2, x3 .

Solve x1 x2 x3 0 && Abs x1 6 && Abs x2 6 && Abs x3 6,x1, x2, x3 , Integers ;

Sol6Num Dimensions Sol6 1 ;Sol4

x1, x2, x3 .Solve x1 x2 x3 0 && Abs x1 4 && Abs x2 4 && Abs x3 4,

x1, x2, x3 , Integers ;Sol4Num Dimensions Sol4 1 ;

the number of closest particle to calculate bond orientation, 12 for general useNeighNum 12;xyz0 Import "Initial coordiante", "Table" 10 ;; 5009, 3 ;; 5 ;xyzO xyz0;BondOrient step :Module , FileNo ToString step ;

Input particle coordinate

xyzData Import "Selected snapshot coordinate", "Table" 10 ;; 5009,3 ;; 5 ;

xyzNum Dimensions xyzData 1 ;

Mean square displacementxMoveT Table xyzData i, 1 xyz0 i, 1 ^2, i, xyzNum ;yMoveT Table xyzData i, 2 xyz0 i, 2 ^2, i, xyzNum ;zMoveT Table xyzData i, 3 xyz0 i, 3 ^2, i, xyzNum ;dMoveT Table xMoveT i yMoveT i zMoveT i , i, xyzNum ;SDtotal Transpose xMoveT, yMoveT, zMoveT, dMoveT ;MSDtotal Mean SDtotal ;Export "SDtotal " FileNo ".dat", SDtotal, "Table" ;xMoveS Table xyzData i, 1 xyzO i, 1 ^2, i, xyzNum ;yMoveS Table xyzData i, 2 xyzO i, 2 ^2, i, xyzNum ;zMoveS Table xyzData i, 3 xyzO i, 3 ^2, i, xyzNum ;dMoveS Table xMoveS i yMoveS i zMoveS i , i, xyzNum ;SDstep Transpose xMoveS, yMoveS, zMoveS, dMoveS ;MSDstep Mean SDstep ;Export "SDstep " FileNo ".dat", SDstep, "Table" ;

Packing factordiameter 0.0023; Rcylinder 0.03; maxco Max Transpose xyzData ;minco Min Transpose xyzData ;Hcylinder maxco 3 minco 3 diameter;PF diameter^3 6 xyzNum Rcylinder^2 Hcylinder ;

Neighbour list

NeighPTableFlattenTable Cases Nearest xyzData, xyzData i , NeighNum 1 ,

Except xyzData i , i, xyzNum , 1 ;Neighbour vector list

NeighVTable Table Table xyzData i NeighPTable i, j , j, NeighNum ,i, xyzNum ;Neighbour index list

NeighIndexTable Flatten Table Position xyzData, NeighPTable i, j , j, NeighNum ,

i, xyzNum ;Export "NeighIndex " FileNo ".dat", NeighIndex, "Table" ;

xyz spherical coordiantionNeighSpheCoor ToSphericalCoordinates NeighVTable . Indeterminate 0.0;

calculate individual spherical harmonic Q6m & Q4m

q6mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 6, m, NeighSpheCoor i, j, 2 ,NeighSpheCoor i, j, 3 , j, NeighNum , m, 6, 6 , i, xyzNum ;

q6sum Table Sum Norm q6m i, j ^2 , j, 13 , i, xyzNum ;q4mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 4, n, NeighSpheCoor i, j 2 ,NeighSpheCoor i, j 3 , j, NeighNum , n, 4, 4 ,

i, xyzNum ;q4sum Table Sum Norm q4m i, j ^2 , j, 9 , i, xyzNum ;

local bond orientationQ6local Flatten Table N Sqrt 4 2 6 1 q6sum i , i, xyzNum ;Q4local Flatten Table N Sqrt 4 2 4 1 q4sum i , i, xyzNum ;

local Wigner 3 j symbol

w6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

q6m j, Sol6 i, 1 7 q6m j, Sol6 i, 2 7q6m j, Sol6 i, 3 7 , i, Sol6Num q6sum j ^ 3 2 ,

j, xyzNum ;

w4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

q4m j, Sol4 i, 1 5 q4m j, Sol4 i, 2 5q4m j, Sol4 i, 3 5 , i, Sol4Num q4sum j ^ 3 2 ,

j, xyzNum ;BOOlocal Transpose Q4local, Q6local, w4, w6 ;Export "BOOlocal " FileNo ".dat", BOOlocal, "Table" ;

Averaged bond order parameters

Aq6mTableFlattenTable 1 NeighNum

q6m i, m Sum q6m NeighIndex i, j , m , j, NeighNum ,m, 13 , i, xyzNum ;

Aq6sum Table Sum Norm Aq6m i, j ^2 , j, 13 , i, xyzNum ;

Aq4mTableFlattenTable 1 NeighNum

q4m i, m Sum q4m NeighIndex i j , m , j, NeighNum ,m, 9 , i, xyzNum ;

Aq4sum Table Sum Norm Aq4m i, j ^2 , j, 9 , i, xyzNum ;AQ6 Flatten Table N Sqrt 4 2 6 1 Aq6sum i , i, xyzNum ;AQ4 Flatten Table N Sqrt 4 2 4 1 Aq4sum i , i, xyzNum ;

Average Wigner 3 j symbol

Aw6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

Aq6m j, Sol6 i, 1 7 Aq6m j, Sol6 i, 2 7Aq6m j, Sol6 i, 3 7 , i, Sol6Num Aq6sum j ^ 3 2 ,

j, xyzNum ;

Aw4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

Aq4m j, Sol4 i, 1 5 Aq4m j, Sol4 i, 2 5Aq4m j, Sol4 i, 3 5 , i, Sol4Num Aq4sum j ^ 3 2 ,

j, xyzNum ;BOOaverage Transpose AQ4, AQ6, Aw4, Aw6 ;Export "BOOaverage " FileNo ".dat", BOOaverage, "Table" ;

solidlike and liquidlikeCosSimi a , b : N Re a Norm a .Conjugate b Norm b ;q6c 0.7;

q6Table Flatten Table CosSimi q6m i , q6m NeighIndex i, j ,

j, NeighNum , i, xyzNum ;S6 Flatten Table Sum q6 i, j , j, NeighNum , i, xyzNum ;

SorLFlatten Table UnitStep Sum UnitStep q6 i, j q6c , j, NeighNum 7 ,

i, xyzNum ;SolidNum Sum SorL i , i, xyzNum ;

order and disorder

f6Flatten Table N 1 NeighNum Sum UnitStep q6 i, j q6c , j, NeighNum ,

i, xyzNum ;F6 Mean f6 ;Solidliquid Transpose S6, SorL, f6 ;Export "Solid Liquid " FileNo ".dat", Solidliquid, "Table" ;

Calculate radial distribution functionRDFlist RDFunction xyzData, xyzNum, diameter, PF ;Export "RDFlist " FileNo ".dat", RDFlist, "Table" ;

Overall

Overall PF, SolidNum, F6, MSDtotal 1 , MSDtotal 2 ,MSDtotal 3 , MSDtotal 4 , MSDstep 1 , MSDstep 2 , MSDstep 3 ,MSDstep 4 ;

Export $overall, Overall, "Table" ;

WriteString $overall, "\n" ;

xyzO xyzData;

Do BondOrient i , i, step1, step2, intervalClose $overall ;

151

SetDirectory "My\\file\\Directory\\" ;$overall OpenAppend "Overall.dat", BinaryFormat True ;WriteString $overall,

"Packing factor\t Solid Number\t F6\t xMSD\t yMSD\t zMSD\t MSDtotal\txMSDstep\t yMSDstep\t zMSDstep\t MSDstep\t \n" ;

Definition of the radial distribution functionRDFunction

Compile Data, Real, 2 , NumG, Integer , diameter, Real ,Rho, Real ,

Module i, j, dR, Rmax, dVol, dx, x, xbin, NumBin ,dR 0.01 diameter; Rmax 6.0 diameter;x Table i, i, dR, Rmax, dR ;NumBin Dimensions x 1 ;x x diameter;xbin Table 0.0, i, dR, Rmax, dR ;DoDo

dx Norm Data i Data j ;If dx Rmax, xbin Ceiling dx dR 2.0; dx 0 ;

dx 0,j, i 1, NumG ,

i, 1, NumG ;DodVol 4 3 i 1 ^3 i^3 dR^3;xbin i xbin i dVol Rho NumG,i, 1, NumBin ;

Transpose x, xbin , CompilationTarget "C" ;

Prepare varibale for Wigner 3 j symbolSol6 x1, x2, x3 .

Solve x1 x2 x3 0 && Abs x1 6 && Abs x2 6 && Abs x3 6,x1, x2, x3 , Integers ;

Sol6Num Dimensions Sol6 1 ;Sol4

x1, x2, x3 .Solve x1 x2 x3 0 && Abs x1 4 && Abs x2 4 && Abs x3 4,

x1, x2, x3 , Integers ;Sol4Num Dimensions Sol4 1 ;

the number of closest particle to calculate bond orientation, 12 for general useNeighNum 12;xyz0 Import "Initial coordiante", "Table" 10 ;; 5009, 3 ;; 5 ;xyzO xyz0;BondOrient step :Module , FileNo ToString step ;

Input particle coordinate

xyzData Import "Selected snapshot coordinate", "Table" 10 ;; 5009,3 ;; 5 ;

xyzNum Dimensions xyzData 1 ;

Mean square displacementxMoveT Table xyzData i, 1 xyz0 i, 1 ^2, i, xyzNum ;yMoveT Table xyzData i, 2 xyz0 i, 2 ^2, i, xyzNum ;zMoveT Table xyzData i, 3 xyz0 i, 3 ^2, i, xyzNum ;dMoveT Table xMoveT i yMoveT i zMoveT i , i, xyzNum ;SDtotal Transpose xMoveT, yMoveT, zMoveT, dMoveT ;MSDtotal Mean SDtotal ;Export "SDtotal " FileNo ".dat", SDtotal, "Table" ;xMoveS Table xyzData i, 1 xyzO i, 1 ^2, i, xyzNum ;yMoveS Table xyzData i, 2 xyzO i, 2 ^2, i, xyzNum ;zMoveS Table xyzData i, 3 xyzO i, 3 ^2, i, xyzNum ;dMoveS Table xMoveS i yMoveS i zMoveS i , i, xyzNum ;SDstep Transpose xMoveS, yMoveS, zMoveS, dMoveS ;MSDstep Mean SDstep ;Export "SDstep " FileNo ".dat", SDstep, "Table" ;

Packing factordiameter 0.0023; Rcylinder 0.03; maxco Max Transpose xyzData ;minco Min Transpose xyzData ;Hcylinder maxco 3 minco 3 diameter;PF diameter^3 6 xyzNum Rcylinder^2 Hcylinder ;

Neighbour list

NeighPTableFlattenTable Cases Nearest xyzData, xyzData i , NeighNum 1 ,

Except xyzData i , i, xyzNum , 1 ;Neighbour vector list

NeighVTable Table Table xyzData i NeighPTable i, j , j, NeighNum ,i, xyzNum ;Neighbour index list

NeighIndexTable Flatten Table Position xyzData, NeighPTable i, j , j, NeighNum ,

i, xyzNum ;Export "NeighIndex " FileNo ".dat", NeighIndex, "Table" ;

xyz spherical coordiantionNeighSpheCoor ToSphericalCoordinates NeighVTable . Indeterminate 0.0;

calculate individual spherical harmonic Q6m & Q4m

q6mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 6, m, NeighSpheCoor i, j, 2 ,NeighSpheCoor i, j, 3 , j, NeighNum , m, 6, 6 , i, xyzNum ;

q6sum Table Sum Norm q6m i, j ^2 , j, 13 , i, xyzNum ;q4mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 4, n, NeighSpheCoor i, j 2 ,NeighSpheCoor i, j 3 , j, NeighNum , n, 4, 4 ,

i, xyzNum ;q4sum Table Sum Norm q4m i, j ^2 , j, 9 , i, xyzNum ;

local bond orientationQ6local Flatten Table N Sqrt 4 2 6 1 q6sum i , i, xyzNum ;Q4local Flatten Table N Sqrt 4 2 4 1 q4sum i , i, xyzNum ;

local Wigner 3 j symbol

w6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

q6m j, Sol6 i, 1 7 q6m j, Sol6 i, 2 7q6m j, Sol6 i, 3 7 , i, Sol6Num q6sum j ^ 3 2 ,

j, xyzNum ;

w4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

q4m j, Sol4 i, 1 5 q4m j, Sol4 i, 2 5q4m j, Sol4 i, 3 5 , i, Sol4Num q4sum j ^ 3 2 ,

j, xyzNum ;BOOlocal Transpose Q4local, Q6local, w4, w6 ;Export "BOOlocal " FileNo ".dat", BOOlocal, "Table" ;

Averaged bond order parameters

Aq6mTableFlattenTable 1 NeighNum

q6m i, m Sum q6m NeighIndex i, j , m , j, NeighNum ,m, 13 , i, xyzNum ;

Aq6sum Table Sum Norm Aq6m i, j ^2 , j, 13 , i, xyzNum ;

Aq4mTableFlattenTable 1 NeighNum

q4m i, m Sum q4m NeighIndex i j , m , j, NeighNum ,m, 9 , i, xyzNum ;

Aq4sum Table Sum Norm Aq4m i, j ^2 , j, 9 , i, xyzNum ;AQ6 Flatten Table N Sqrt 4 2 6 1 Aq6sum i , i, xyzNum ;AQ4 Flatten Table N Sqrt 4 2 4 1 Aq4sum i , i, xyzNum ;

Average Wigner 3 j symbol

Aw6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

Aq6m j, Sol6 i, 1 7 Aq6m j, Sol6 i, 2 7Aq6m j, Sol6 i, 3 7 , i, Sol6Num Aq6sum j ^ 3 2 ,

j, xyzNum ;

Aw4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

Aq4m j, Sol4 i, 1 5 Aq4m j, Sol4 i, 2 5Aq4m j, Sol4 i, 3 5 , i, Sol4Num Aq4sum j ^ 3 2 ,

j, xyzNum ;BOOaverage Transpose AQ4, AQ6, Aw4, Aw6 ;Export "BOOaverage " FileNo ".dat", BOOaverage, "Table" ;

solidlike and liquidlikeCosSimi a , b : N Re a Norm a .Conjugate b Norm b ;q6c 0.7;

q6Table Flatten Table CosSimi q6m i , q6m NeighIndex i, j ,

j, NeighNum , i, xyzNum ;S6 Flatten Table Sum q6 i, j , j, NeighNum , i, xyzNum ;

SorLFlatten Table UnitStep Sum UnitStep q6 i, j q6c , j, NeighNum 7 ,

i, xyzNum ;SolidNum Sum SorL i , i, xyzNum ;

order and disorder

f6Flatten Table N 1 NeighNum Sum UnitStep q6 i, j q6c , j, NeighNum ,

i, xyzNum ;F6 Mean f6 ;Solidliquid Transpose S6, SorL, f6 ;Export "Solid Liquid " FileNo ".dat", Solidliquid, "Table" ;

Calculate radial distribution functionRDFlist RDFunction xyzData, xyzNum, diameter, PF ;Export "RDFlist " FileNo ".dat", RDFlist, "Table" ;

Overall

Overall PF, SolidNum, F6, MSDtotal 1 , MSDtotal 2 ,MSDtotal 3 , MSDtotal 4 , MSDstep 1 , MSDstep 2 , MSDstep 3 ,MSDstep 4 ;

Export $overall, Overall, "Table" ;

WriteString $overall, "\n" ;

xyzO xyzData;

Do BondOrient i , i, step1, step2, intervalClose $overall ;

152

SetDirectory "My\\file\\Directory\\" ;$overall OpenAppend "Overall.dat", BinaryFormat True ;WriteString $overall,

"Packing factor\t Solid Number\t F6\t xMSD\t yMSD\t zMSD\t MSDtotal\txMSDstep\t yMSDstep\t zMSDstep\t MSDstep\t \n" ;

Definition of the radial distribution functionRDFunction

Compile Data, Real, 2 , NumG, Integer , diameter, Real ,Rho, Real ,

Module i, j, dR, Rmax, dVol, dx, x, xbin, NumBin ,dR 0.01 diameter; Rmax 6.0 diameter;x Table i, i, dR, Rmax, dR ;NumBin Dimensions x 1 ;x x diameter;xbin Table 0.0, i, dR, Rmax, dR ;DoDo

dx Norm Data i Data j ;If dx Rmax, xbin Ceiling dx dR 2.0; dx 0 ;

dx 0,j, i 1, NumG ,

i, 1, NumG ;DodVol 4 3 i 1 ^3 i^3 dR^3;xbin i xbin i dVol Rho NumG,i, 1, NumBin ;

Transpose x, xbin , CompilationTarget "C" ;

Prepare varibale for Wigner 3 j symbolSol6 x1, x2, x3 .

Solve x1 x2 x3 0 && Abs x1 6 && Abs x2 6 && Abs x3 6,x1, x2, x3 , Integers ;

Sol6Num Dimensions Sol6 1 ;Sol4

x1, x2, x3 .Solve x1 x2 x3 0 && Abs x1 4 && Abs x2 4 && Abs x3 4,

x1, x2, x3 , Integers ;Sol4Num Dimensions Sol4 1 ;

the number of closest particle to calculate bond orientation, 12 for general useNeighNum 12;xyz0 Import "Initial coordiante", "Table" 10 ;; 5009, 3 ;; 5 ;xyzO xyz0;BondOrient step :Module , FileNo ToString step ;

Input particle coordinate

xyzData Import "Selected snapshot coordinate", "Table" 10 ;; 5009,3 ;; 5 ;

xyzNum Dimensions xyzData 1 ;

Mean square displacementxMoveT Table xyzData i, 1 xyz0 i, 1 ^2, i, xyzNum ;yMoveT Table xyzData i, 2 xyz0 i, 2 ^2, i, xyzNum ;zMoveT Table xyzData i, 3 xyz0 i, 3 ^2, i, xyzNum ;dMoveT Table xMoveT i yMoveT i zMoveT i , i, xyzNum ;SDtotal Transpose xMoveT, yMoveT, zMoveT, dMoveT ;MSDtotal Mean SDtotal ;Export "SDtotal " FileNo ".dat", SDtotal, "Table" ;xMoveS Table xyzData i, 1 xyzO i, 1 ^2, i, xyzNum ;yMoveS Table xyzData i, 2 xyzO i, 2 ^2, i, xyzNum ;zMoveS Table xyzData i, 3 xyzO i, 3 ^2, i, xyzNum ;dMoveS Table xMoveS i yMoveS i zMoveS i , i, xyzNum ;SDstep Transpose xMoveS, yMoveS, zMoveS, dMoveS ;MSDstep Mean SDstep ;Export "SDstep " FileNo ".dat", SDstep, "Table" ;

Packing factordiameter 0.0023; Rcylinder 0.03; maxco Max Transpose xyzData ;minco Min Transpose xyzData ;Hcylinder maxco 3 minco 3 diameter;PF diameter^3 6 xyzNum Rcylinder^2 Hcylinder ;

Neighbour list

NeighPTableFlattenTable Cases Nearest xyzData, xyzData i , NeighNum 1 ,

Except xyzData i , i, xyzNum , 1 ;Neighbour vector list

NeighVTable Table Table xyzData i NeighPTable i, j , j, NeighNum ,i, xyzNum ;Neighbour index list

NeighIndexTable Flatten Table Position xyzData, NeighPTable i, j , j, NeighNum ,

i, xyzNum ;Export "NeighIndex " FileNo ".dat", NeighIndex, "Table" ;

xyz spherical coordiantionNeighSpheCoor ToSphericalCoordinates NeighVTable . Indeterminate 0.0;

calculate individual spherical harmonic Q6m & Q4m

q6mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 6, m, NeighSpheCoor i, j, 2 ,NeighSpheCoor i, j, 3 , j, NeighNum , m, 6, 6 , i, xyzNum ;

q6sum Table Sum Norm q6m i, j ^2 , j, 13 , i, xyzNum ;q4mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 4, n, NeighSpheCoor i, j 2 ,NeighSpheCoor i, j 3 , j, NeighNum , n, 4, 4 ,

i, xyzNum ;q4sum Table Sum Norm q4m i, j ^2 , j, 9 , i, xyzNum ;

local bond orientationQ6local Flatten Table N Sqrt 4 2 6 1 q6sum i , i, xyzNum ;Q4local Flatten Table N Sqrt 4 2 4 1 q4sum i , i, xyzNum ;

local Wigner 3 j symbol

w6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

q6m j, Sol6 i, 1 7 q6m j, Sol6 i, 2 7q6m j, Sol6 i, 3 7 , i, Sol6Num q6sum j ^ 3 2 ,

j, xyzNum ;

w4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

q4m j, Sol4 i, 1 5 q4m j, Sol4 i, 2 5q4m j, Sol4 i, 3 5 , i, Sol4Num q4sum j ^ 3 2 ,

j, xyzNum ;BOOlocal Transpose Q4local, Q6local, w4, w6 ;Export "BOOlocal " FileNo ".dat", BOOlocal, "Table" ;

Averaged bond order parameters

Aq6mTableFlattenTable 1 NeighNum

q6m i, m Sum q6m NeighIndex i, j , m , j, NeighNum ,m, 13 , i, xyzNum ;

Aq6sum Table Sum Norm Aq6m i, j ^2 , j, 13 , i, xyzNum ;

Aq4mTableFlattenTable 1 NeighNum

q4m i, m Sum q4m NeighIndex i j , m , j, NeighNum ,m, 9 , i, xyzNum ;

Aq4sum Table Sum Norm Aq4m i, j ^2 , j, 9 , i, xyzNum ;AQ6 Flatten Table N Sqrt 4 2 6 1 Aq6sum i , i, xyzNum ;AQ4 Flatten Table N Sqrt 4 2 4 1 Aq4sum i , i, xyzNum ;

Average Wigner 3 j symbol

Aw6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

Aq6m j, Sol6 i, 1 7 Aq6m j, Sol6 i, 2 7Aq6m j, Sol6 i, 3 7 , i, Sol6Num Aq6sum j ^ 3 2 ,

j, xyzNum ;

Aw4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

Aq4m j, Sol4 i, 1 5 Aq4m j, Sol4 i, 2 5Aq4m j, Sol4 i, 3 5 , i, Sol4Num Aq4sum j ^ 3 2 ,

j, xyzNum ;BOOaverage Transpose AQ4, AQ6, Aw4, Aw6 ;Export "BOOaverage " FileNo ".dat", BOOaverage, "Table" ;

solidlike and liquidlikeCosSimi a , b : N Re a Norm a .Conjugate b Norm b ;q6c 0.7;

q6Table Flatten Table CosSimi q6m i , q6m NeighIndex i, j ,

j, NeighNum , i, xyzNum ;S6 Flatten Table Sum q6 i, j , j, NeighNum , i, xyzNum ;

SorLFlatten Table UnitStep Sum UnitStep q6 i, j q6c , j, NeighNum 7 ,

i, xyzNum ;SolidNum Sum SorL i , i, xyzNum ;

order and disorder

f6Flatten Table N 1 NeighNum Sum UnitStep q6 i, j q6c , j, NeighNum ,

i, xyzNum ;F6 Mean f6 ;Solidliquid Transpose S6, SorL, f6 ;Export "Solid Liquid " FileNo ".dat", Solidliquid, "Table" ;

Calculate radial distribution functionRDFlist RDFunction xyzData, xyzNum, diameter, PF ;Export "RDFlist " FileNo ".dat", RDFlist, "Table" ;

Overall

Overall PF, SolidNum, F6, MSDtotal 1 , MSDtotal 2 ,MSDtotal 3 , MSDtotal 4 , MSDstep 1 , MSDstep 2 , MSDstep 3 ,MSDstep 4 ;

Export $overall, Overall, "Table" ;

WriteString $overall, "\n" ;

xyzO xyzData;

Do BondOrient i , i, step1, step2, intervalClose $overall ;

153

SetDirectory "My\\file\\Directory\\" ;$overall OpenAppend "Overall.dat", BinaryFormat True ;WriteString $overall,

"Packing factor\t Solid Number\t F6\t xMSD\t yMSD\t zMSD\t MSDtotal\txMSDstep\t yMSDstep\t zMSDstep\t MSDstep\t \n" ;

Definition of the radial distribution functionRDFunction

Compile Data, Real, 2 , NumG, Integer , diameter, Real ,Rho, Real ,

Module i, j, dR, Rmax, dVol, dx, x, xbin, NumBin ,dR 0.01 diameter; Rmax 6.0 diameter;x Table i, i, dR, Rmax, dR ;NumBin Dimensions x 1 ;x x diameter;xbin Table 0.0, i, dR, Rmax, dR ;DoDo

dx Norm Data i Data j ;If dx Rmax, xbin Ceiling dx dR 2.0; dx 0 ;

dx 0,j, i 1, NumG ,

i, 1, NumG ;DodVol 4 3 i 1 ^3 i^3 dR^3;xbin i xbin i dVol Rho NumG,i, 1, NumBin ;

Transpose x, xbin , CompilationTarget "C" ;

Prepare varibale for Wigner 3 j symbolSol6 x1, x2, x3 .

Solve x1 x2 x3 0 && Abs x1 6 && Abs x2 6 && Abs x3 6,x1, x2, x3 , Integers ;

Sol6Num Dimensions Sol6 1 ;Sol4

x1, x2, x3 .Solve x1 x2 x3 0 && Abs x1 4 && Abs x2 4 && Abs x3 4,

x1, x2, x3 , Integers ;Sol4Num Dimensions Sol4 1 ;

the number of closest particle to calculate bond orientation, 12 for general useNeighNum 12;xyz0 Import "Initial coordiante", "Table" 10 ;; 5009, 3 ;; 5 ;xyzO xyz0;BondOrient step :Module , FileNo ToString step ;

Input particle coordinate

xyzData Import "Selected snapshot coordinate", "Table" 10 ;; 5009,3 ;; 5 ;

xyzNum Dimensions xyzData 1 ;

Mean square displacementxMoveT Table xyzData i, 1 xyz0 i, 1 ^2, i, xyzNum ;yMoveT Table xyzData i, 2 xyz0 i, 2 ^2, i, xyzNum ;zMoveT Table xyzData i, 3 xyz0 i, 3 ^2, i, xyzNum ;dMoveT Table xMoveT i yMoveT i zMoveT i , i, xyzNum ;SDtotal Transpose xMoveT, yMoveT, zMoveT, dMoveT ;MSDtotal Mean SDtotal ;Export "SDtotal " FileNo ".dat", SDtotal, "Table" ;xMoveS Table xyzData i, 1 xyzO i, 1 ^2, i, xyzNum ;yMoveS Table xyzData i, 2 xyzO i, 2 ^2, i, xyzNum ;zMoveS Table xyzData i, 3 xyzO i, 3 ^2, i, xyzNum ;dMoveS Table xMoveS i yMoveS i zMoveS i , i, xyzNum ;SDstep Transpose xMoveS, yMoveS, zMoveS, dMoveS ;MSDstep Mean SDstep ;Export "SDstep " FileNo ".dat", SDstep, "Table" ;

Packing factordiameter 0.0023; Rcylinder 0.03; maxco Max Transpose xyzData ;minco Min Transpose xyzData ;Hcylinder maxco 3 minco 3 diameter;PF diameter^3 6 xyzNum Rcylinder^2 Hcylinder ;

Neighbour list

NeighPTableFlattenTable Cases Nearest xyzData, xyzData i , NeighNum 1 ,

Except xyzData i , i, xyzNum , 1 ;Neighbour vector list

NeighVTable Table Table xyzData i NeighPTable i, j , j, NeighNum ,i, xyzNum ;Neighbour index list

NeighIndexTable Flatten Table Position xyzData, NeighPTable i, j , j, NeighNum ,

i, xyzNum ;Export "NeighIndex " FileNo ".dat", NeighIndex, "Table" ;

xyz spherical coordiantionNeighSpheCoor ToSphericalCoordinates NeighVTable . Indeterminate 0.0;

calculate individual spherical harmonic Q6m & Q4m

q6mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 6, m, NeighSpheCoor i, j, 2 ,NeighSpheCoor i, j, 3 , j, NeighNum , m, 6, 6 , i, xyzNum ;

q6sum Table Sum Norm q6m i, j ^2 , j, 13 , i, xyzNum ;q4mTableFlattenTable 1 NeighNum

Sum SphericalHarmonicY 4, n, NeighSpheCoor i, j 2 ,NeighSpheCoor i, j 3 , j, NeighNum , n, 4, 4 ,

i, xyzNum ;q4sum Table Sum Norm q4m i, j ^2 , j, 9 , i, xyzNum ;

local bond orientationQ6local Flatten Table N Sqrt 4 2 6 1 q6sum i , i, xyzNum ;Q4local Flatten Table N Sqrt 4 2 4 1 q4sum i , i, xyzNum ;

local Wigner 3 j symbol

w6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

q6m j, Sol6 i, 1 7 q6m j, Sol6 i, 2 7q6m j, Sol6 i, 3 7 , i, Sol6Num q6sum j ^ 3 2 ,

j, xyzNum ;

w4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

q4m j, Sol4 i, 1 5 q4m j, Sol4 i, 2 5q4m j, Sol4 i, 3 5 , i, Sol4Num q4sum j ^ 3 2 ,

j, xyzNum ;BOOlocal Transpose Q4local, Q6local, w4, w6 ;Export "BOOlocal " FileNo ".dat", BOOlocal, "Table" ;

Averaged bond order parameters

Aq6mTableFlattenTable 1 NeighNum

q6m i, m Sum q6m NeighIndex i, j , m , j, NeighNum ,m, 13 , i, xyzNum ;

Aq6sum Table Sum Norm Aq6m i, j ^2 , j, 13 , i, xyzNum ;

Aq4mTableFlattenTable 1 NeighNum

q4m i, m Sum q4m NeighIndex i j , m , j, NeighNum ,m, 9 , i, xyzNum ;

Aq4sum Table Sum Norm Aq4m i, j ^2 , j, 9 , i, xyzNum ;AQ6 Flatten Table N Sqrt 4 2 6 1 Aq6sum i , i, xyzNum ;AQ4 Flatten Table N Sqrt 4 2 4 1 Aq4sum i , i, xyzNum ;

Average Wigner 3 j symbol

Aw6TableNReSum ThreeJSymbol 6, Sol6 i, 1 , 6, Sol6 i, 2 , 6, Sol6 i, 3

Aq6m j, Sol6 i, 1 7 Aq6m j, Sol6 i, 2 7Aq6m j, Sol6 i, 3 7 , i, Sol6Num Aq6sum j ^ 3 2 ,

j, xyzNum ;

Aw4TableNReSum ThreeJSymbol 4, Sol4 i, 1 , 4, Sol4 i, 2 , 4, Sol4 i, 3

Aq4m j, Sol4 i, 1 5 Aq4m j, Sol4 i, 2 5Aq4m j, Sol4 i, 3 5 , i, Sol4Num Aq4sum j ^ 3 2 ,

j, xyzNum ;BOOaverage Transpose AQ4, AQ6, Aw4, Aw6 ;Export "BOOaverage " FileNo ".dat", BOOaverage, "Table" ;

solidlike and liquidlikeCosSimi a , b : N Re a Norm a .Conjugate b Norm b ;q6c 0.7;

q6Table Flatten Table CosSimi q6m i , q6m NeighIndex i, j ,

j, NeighNum , i, xyzNum ;S6 Flatten Table Sum q6 i, j , j, NeighNum , i, xyzNum ;

SorLFlatten Table UnitStep Sum UnitStep q6 i, j q6c , j, NeighNum 7 ,

i, xyzNum ;SolidNum Sum SorL i , i, xyzNum ;

order and disorder

f6Flatten Table N 1 NeighNum Sum UnitStep q6 i, j q6c , j, NeighNum ,

i, xyzNum ;F6 Mean f6 ;Solidliquid Transpose S6, SorL, f6 ;Export "Solid Liquid " FileNo ".dat", Solidliquid, "Table" ;

Calculate radial distribution functionRDFlist RDFunction xyzData, xyzNum, diameter, PF ;Export "RDFlist " FileNo ".dat", RDFlist, "Table" ;

Overall

Overall PF, SolidNum, F6, MSDtotal 1 , MSDtotal 2 ,MSDtotal 3 , MSDtotal 4 , MSDstep 1 , MSDstep 2 , MSDstep 3 ,MSDstep 4 ;

Export $overall, Overall, "Table" ;

WriteString $overall, "\n" ;

xyzO xyzData;

Do BondOrient i , i, step1, step2, intervalClose $overall ;

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