thermal conduction in granular media: from...
TRANSCRIPT
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
xvi
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
xvii
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 [/]
xix
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.
66
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.
69
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.
72
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.
116
117
Bibliography
Abou-Sena, A., A. Ying and M. Abdou (2007). "Experimental measurements of the
effective thermal conductivity of a lithium titanate (Li2TiO3) pebbles-packed bed." Journal
of Materials Processing Technology 181(1-3): 206-212.
Abyzov, A. M., A. V. Goryunov and F. M. Shakhov (2013). "Effective thermal
conductivity of disperse materials. I. Compliance of common models with experimental
data." International Journal of Heat and Mass Transfer 67: 752-767.
Adjaoud, O. and K. Albe (2016). "Interfaces and interphases in nanoglasses: Surface
segregation effects and their implications on structural properties." Acta Materialia 113: 284-
292.
Alexander, K., S. Thorsten and K. Carolin (2014). "Evaporation model for beam based
additive manufacturing using free surface lattice Boltzmann methods." Journal of Physics D:
Applied Physics 47(27): 275303.
Algis Dˇziugys, B. P. (2001). "An approach to simulate the motion of spherical and non-
spherical fuel particles in combustion chambers." Granular Matter 3: 231-265.
An, X., R. Yang, K. Dong and A. Yu (2011). "DEM study of crystallization of monosized
spheres under mechanical vibrations." Computer Physics Communications 182(9): 1989-
1994.
An, X. Z., R. Y. Yang, R. P. Zou and A. B. Yu (2008). "Effect of vibration condition and
inter-particle frictions on the packing of uniform spheres." Powder Technology 188(2): 102-
109.
Assael, M. J., K. D. Antoniadis and W. A. Wakeham (2010). "Historical Evolution of the
Transient Hot-Wire Technique." International Journal of Thermophysics 31(6): 1051-1072.
Aste, T. (2005). "Variations around disordered close packing." Journal of Physics:
Condensed Matter 17(24): S2361.
Aste, T., T. D. Matteo, M. Saadatfar, T. J. Senden, S. Matthias and L. S. Harry (2007).
"An invariant distribution in static granular media." EPL (Europhysics Letters) 79(2): 24003.
Aste, T., M. Saadatfar and T. J. Senden (2006). "Local and global relations between the
number of contacts and density in monodisperse sphere packs." Journal of Statistical
Mechanics: Theory and Experiment 2006(07): P07010.
Bahrami, M., J. R. Culham, M. M. Yovanovich and G. E. Schneider (2004). "Thermal
Contact Resistance of Nonconforming Rough Surfaces, Part 1: Contact Mechanics Model."
Journal of Thermophysics and Heat Transfer 18(2): 209-217.
118
Bahrami, M., J. R. Culham, M. M. Yovanovich and G. E. Schneider (2004). "Thermal
Contact Resistance of Nonconforming Rough Surfaces, Part 2: Thermal Model." Journal of
Thermophysics and Heat Transfer 18(2): 218-227.
Bahrami, M., M. M. Yovanovich and J. R. Culham (2005). "A Compact Model for
Spherical Rough Contacts." Journal of Tribology 127(4): 884-889.
Bahrami, M., M. M. Yovanovich and J. R. Culham (2006). "Effective thermal
conductivity of rough spherical packed beds." International Journal of Heat and Mass
Transfer 49(19-20): 3691-3701.
Balakrishnan, A. R. and D. C. T. Pei (1979). "Heat Transfer in Gas-Solid Packed Bed
Systems. A critical review." Industrial & Engineering Chemistry Process Design and
Development 18(1): 30-40.
Barker, G. C. and A. Mehta (1993). "Transient phenomena, self-diffusion, and
orientational effects in vibrated powders." Physical Review E 47(1): 184-188.
Batchelor, G. K. and R. W. O'Brien (1977). "Thermal and Electrical Conduction Through
a Granular Material." The Royal Society Proceding A 355(21): 313-333.
Bauer, M. L. and P. M. Norris (2014). "General bidirectional thermal characterization via
the 3omega technique." Rev Sci Instrum 85(6): 064903.
Bauer, R. and E. Schlunder (1978). "Effective Radial Thermal-Conductivity of Packings
in Gas-Flow. 2. Thermal-Conductivity of Packing Fraction without Gas-Flow." International
Chemical Engineering 18(2): 189-204.
Bernal, J. D. and J. Mason (1960). "Packing of Spheres: Co-ordination of Randomly
Packed Spheres." Nature 188(4754): 910-911.
Boutreux, T. and P. G. de Geennes (1997). "Compaction of granular mixtures: a free
volume model." Physica A: Statistical Mechanics and its Applications 244(1): 59-67.
Cahill, D. G. (1990). "Thermal conductivity measurement from 30 to 750 K: the 3ω
method." Review of Scientific Instruments 61(2): 802.
Cahill, D. G. and R. O. Pohl (1987). "Thermal conductivity of amorphous solids above
the plateau." Physical Review B 35(8): 4067-4073.
Campbell, C. S. (2006). "Granular material flows – An overview." Powder Technology
162(3): 208-229.
Carslaw, H. S. and J. C. Jaeger (1959). Conduction of Heat in Solids, Oxford University
Press.
Carson, J. K. (2006). "Review of effective thermal conductivity models for foods."
International Journal of Refrigeration 29(6): 958-967.
119
Carson, J. K., S. J. Lovatt, D. J. Tanner and A. C. Cleland (2005). "Thermal conductivity
bounds for isotropic, porous materials." International Journal of Heat and Mass Transfer
48(11): 2150-2158.
Carson, J. K. and J. P. Sekhon (2010). "Simple determination of the thermal conductivity
of the solid phase of particulate materials." International Communications in Heat and Mass
Transfer 37(9): 1226-1229.
Carvente, O. and J. C. Ruiz-Suárez (2005). "Crystallization of Confined Non-Brownian
Spheres by Vibrational Annealing." Physical Review Letters 95(1): 018001.
Carvente, O. and J. C. Ruiz-Suárez (2008). "Self-assembling of dry and cohesive non-
Brownian spheres." Physical Review E 78(1): 011302.
Chen, L., X. Ma, X. Cheng, K. Jiang, K. Huang and S. Liu (2015). "Theoretical modeling
of the effective thermal conductivity of the binary pebble beds for the CFETR-WCCB
blanket." Fusion Engineering and Design 101: 148-153.
Chen, N., D. V. Louzguine-Luzgin and K. Yao (2017). "A new class of non-crystalline
materials: Nanogranular metallic glasses." Journal of Alloys and Compounds 707: 371-378.
Cheng, G. J. and A. B. Yu (2013). "Particle Scale Evaluation of the Effective Thermal
Conductivity from the Structure of a Packed Bed: Radiation Heat Transfer." Industrial &
Engineering Chemistry Research 52(34): 12202-12211.
Cheng, G. J., A. B. Yu and P. Zulli (1999). "Evaluation of effective thermal conductivity
from the structure of a packed bed." Chemical Engineering Science 54: 4199-4209.
Chu, K. W., B. Wang, A. B. Yu and A. Vince (2009). "CFD-DEM modelling of
multiphase flow in dense medium cyclones." Powder Technology 193(3): 235-247.
Cortis, A., Y. Chen, H. Scher and B. Berkowitz (2004). "Quantitative characterization of
pore-scale disorder effects on transport in ``homogeneous'' granular media." Physical Review
E 70(4): 041108.
Cundall, P. A. and O. D. L. Strack (1979). "A discrete numerical model for granular
assemblies." Géotechnique 29(1): 47-65.
Dai, W., S. Pupeschi, D. Hanaor and Y. Gan (2017). "Influence of gas pressure on the
effective thermal conductivity of ceramic breeder pebble beds." Fusion Engineering and
Design 118: 45-51.
Dai, W., J. Reimann, D. Hanaor, C. Ferrero and Y. Gan (2018). "Modes of wall induced
granular crystallisation in vibrational packing." arXiv:1805.07865 [cond-mat.dis-nn].
120
Daniels, K. E. and R. P. Behringer (2005). "Hysteresis and Competition between Disorder
and Crystallization in Sheared and Vibrated Granular Flow." Physical Review Letters 94(16):
168001.
de Gennes, P.-G. (2000). "Tapping of Granular Packs: A Model Based on Local Two-
Level Systems." Journal of Colloid and Interface Science 226(1): 1-4.
de Gennes, P. G. (1999). "Granular matter: a tentative view." Reviews of Modern Physics
71(2): S374-S382.
Di Renzo, A. and F. P. Di Maio (2005). "An improved integral non-linear model for the
contact of particles in distinct element simulations." Chemical Engineering Science 60(5):
1303-1312.
Dijksman, J. A. and M. v. Hecke (2009). "The role of tap duration for the steady-state
density of vibrated granular media." EPL (Europhysics Letters) 88(4): 44001.
Donev, A., F. H. Stillinger, P. M. Chaikin and S. Torquato (2004). "Unusually Dense
Crystal Packings of Ellipsoids." Physical Review Letters 92(25): 255506.
Dong, K., C. Wang and A. Yu (2015). "A novel method based on orientation
discretization for discrete element modeling of non-spherical particles." Chemical
Engineering Science 126: 500-516.
dShinde, D. P., A. Mehta and G. C. Barker (2014). "Shaking-induced crystallization of
dense sphere packings." Physical Review E 89(2): 022204.
Duran, J. (2000). Sands, Powders, and Grains-An Introduction to the Physics of Granular
Materials, Springer New York.
Elsahati, M., K. Clarke and R. Richards (2016). "Thermal conductivity of copper and
silica nanoparticle packed beds." International Communications in Heat and Mass Transfer
71: 96-100.
Elsahati, M. O. and R. F. Richards (2017). "Effect of moisture on nanoparticle packed
beds." International Journal of Heat and Mass Transfer 112: 171-184.
Enoeda, Mikio, Ohara, Yosihiro, Roux, Nicole, Ying, Alice, Pizza, Giovanni, Malang and
Siegfried (2001). Effective thermal conductivity measurement of the candidate ceramic
breeder pebble beds by the hot wire method. La Grange Park, IL, ETATS-UNIS, American
Nuclear Society: 890.
Fan, Z. (1996). "A microstructural approach to the effective transport properties of
multiphase composites." Philosophical Magazine A 73(6): 1663-1684.
121
Feng, Y. T. and K. Han (2012). "An accurate evaluation of geometric view factors for
modelling radiative heat transfer in randomly packed beds of equally sized spheres."
International Journal of Heat and Mass Transfer 55(23–24): 6374-6383.
Ferraro, A., A. Sufian and A. R. Russell (2017). "Analytical derivation of water retention
for random monodisperse granular media." Acta Geotechnica 12(6): 1319-1328.
Francois, N., M. Saadatfar, R. Cruikshank and A. Sheppard (2013). "Geometrical
Frustration in Amorphous and Partially Crystallized Packings of Spheres." Physical Review
Letters 111(14): 148001.
Gan, Y. and M. Kamlah (2010). "Discrete element modelling of pebble beds: With
application to uniaxial compression tests of ceramic breeder pebble beds." Journal of the
Mechanics and Physics of Solids 58(2): 129-144.
Gan, Y., M. Kamlah and J. Reimann (2010). "Computer simulation of packing structure
in pebble beds." Fusion Engineering and Design 85(10): 1782-1787.
Gan, Y., F. Maggi, G. Buscarnera and I. Einav (2013). "A particle–water based model for
water retention hysteresis." Géotechnique Letters 3(4): 152-161.
Ganeriwala, R. and T. I. Zohdi (2016). "A coupled discrete element-finite difference
model of selective laser sintering." Granular Matter 18(2): 1-15.
Garrett, D. and H. Ban (2011). "Compressive pressure dependent anisotropic effective
thermal conductivity of granular beds." Granular Matter 13(5): 685-696.
Georgalli, G. A. and M. A. Reuter (2006). "Modelling the co-ordination number of a
packed bed of spheres with distributed sizes using a CT scanner." Minerals Engineering 19(3):
246-255.
Golde, S., T. Palberg and H. J. Schope (2016). "Correlation between dynamical and
structural heterogeneities in colloidal hard-sphere suspensions." Nat Phys 12(7): 712-717.
Goodrich, C. P., A. J. Liu and S. R. Nagel (2014). "Solids between the mechanical
extremes of order and disorder." Nat Phys 10(8): 578-581.
Grimaldi, C. (2014). "Theory of percolation and tunneling regimes in nanogranular metal
films." Physical Review B 89(21): 214201.
Gusarov, A. V. and E. P. Kovalev (2009). "Model of thermal conductivity in powder
beds." Physical Review B 80(2): 024202(024201)-024202(024212).
Gusarov, A. V. and J. P. Kruth (2005). "Modelling of radiation transfer in metallic
powders at laser treatment." International Journal of Heat and Mass Transfer 48(16): 3423-
3434.
122
Gustafsson, S. E. (1991). "Transient plane source techniques for thermal conductivity and
thermal diffusivity measurements of solid materials." Review of Scientific Instruments 62(3):
797.
Gustavsson, M., E. Karawacki and S. E. Gustafsson (1994). "Thermal conductivity,
thermal diffusivity, and specific heat of thin samples from transient measurements with hot
disk sensors." Review of Scientific Instruments 65(12): 3856.
Hall, R. O. A. and D. G. Martin (1981). "The thermal conductivity of powder beds. a
model, some measurements on UO2 vibro-compacted microspheres, and their correlation."
Journal of Nuclear Materials 101: 172-183.
Hao, T. (2015). "Tap density equations of granular powders based on the rate process
theory and the free volume concept." Soft Matter 11(8): 1554-1561.
He, Y. (2005). "Rapid thermal conductivity measurement with a hot disk sensor."
Thermochimica Acta 436(1-2): 122-129.
He, Y., W. Peng, T. Wang and S. Yan (2014). "DEM Study of Wet Cohesive Particles in
the Presence of Liquid Bridges in a Gas Fluidized Bed." Mathematical Problems in
Engineering 2014: 14.
Healy, J. J., J. J. de Groot and J. Kestin (1976). "The theory of the transient hot-wire
method for measuring thermal conductivity." Physica B+C 82(2): 392-408.
Heitkam, S., W. Drenckhan and J. Fröhlich (2012). "Packing Spheres Tightly: Influence
of Mechanical Stability on Close-Packed Sphere Structures." Physical Review Letters
108(14): 148302.
Höhner, D., S. Wirtz and V. Scherer (2015). "A study on the influence of particle shape
on the mechanical interactions of granular media in a hopper using the Discrete Element
Method." Powder Technology 278: 286-305.
Hsiau, S. S., L. S. Lu and C. H. Tai (2008). "Experimental investigations of granular
temperature in vertical vibrated beds." Powder Technology 182(2): 202-210.
Hsu, C. T., P. Cheng and I. W. Wong (1994). "Modified Zehner-Schlunder models for
stagnant thermal conductivity of porous media." International Journal of Heat and Mass
Transfer 37(17): 2751-2760.
Hsu, C. T., P. Cheng and K. W. Wong (1995). "A Lumped-Parameter Model for Stagnant
Thermal Conductivity of Spatially Periodic Porous Media." Journal of Heat Transfer 117(2):
264-269.
Hu, X. J., R. Prasher and K. Lofgreen (2007). "Ultralow thermal conductivity of
nanoparticle packed bed." Applied Physics Letters 91(20): 203113.
123
Huetter, E. S., N. I. Koemle, G. Kargl and E. Kaufmann (2008). "Determination of the
effective thermal conductivity of granular materials under varying pressure conditions."
Journal of Geophysical Research 113(E12).
Ioannidou, K., F.-J. Ulm, P. Levitz, E. Del Gado and R. J.-M. Pellenq (2017). "Nano-
granular texture of cement hydrates." EPJ Web Conf. 140: 15027.
Jacquot, A., F. Vollmer, B. Bayer, M. Jaegle, D. G. Ebling and H. Böttner (2010).
"Thermal Conductivity Measurements on Challenging Samples by the 3 Omega Method."
Journal of Electronic Materials 39(9): 1621-1626.
Jaeger, H. M., C.-h. Liu and S. R. Nagel (1989). "Relaxation at the Angle of Repose."
Physical Review Letters 62(1): 40-43.
Jiang, M., H.-S. Yu and S. Leroueil (2007). "A simple and efficient approach to capturing
bonding effect in naturally microstructured sands by discrete element method." International
Journal for Numerical Methods in Engineering 69(6): 1158-1193.
Jiménez-Herrera, N., G. K. P. Barrios and L. M. Tavares (2018). "Comparison of
breakage models in DEM in simulating impact on particle beds." Advanced Powder
Technology 29(3): 692-706.
Kawasaki, T. and H. Tanaka (2010). "Formation of a crystal nucleus from liquid."
Proceedings of the National Academy of Sciences 107(32): 14036-14041.
Kloss, C., C. Goniva, A. Hager, S. Amberger and S. Pirker (2012). "Models, algorithms
and validation for opensource DEM and CFD-DEM." Progress in Computational Fluid
Dynamics 12(2-3): 140-152.
Klumov, B. A., S. A. Khrapak and G. E. Morfill (2011). "Structural properties of dense
hard sphere packings." Physical Review B 83(18): 184105.
Knight, J. B., C. G. Fandrich, C. N. Lau, H. M. Jaeger and S. R. Nagel (1995). "Density
relaxation in a vibrated granular material." Physical Review E 51(5): 3957-3963.
Körner, C., A. Bauereiß, F. Osmanlic, A. Klassen, M. Markl and A. Rai (2014).
Simulation of selective beam melting on the powder scale: mechanisms and process
strategies. Bremen, Germany, Materials Science and Technology of Additive Manufacturing
- ISEMP/Airbus.
Kumar, V. S. and V. Kumaran (2006). "Bond-orientational analysis of hard-disk and
hard-sphere structures." The Journal of Chemical Physics 124(20): 204508.
Lash, M. H., M. V. Fedorchak, J. J. McCarthy and S. R. Little (2015). "Scaling up self-
assembly: bottom-up approaches to macroscopic particle organization." Soft Matter 11(28):
5597-5609.
124
Lee, S. M. (2009). "Thermal conductivity measurement of fluids using the 3omega
method." Rev Sci Instrum 80(2): 024901.
Lee, S. M. and D. G. Cahill (1997). "Heat transport in thin dielectric films." Journal of
Applied Physics 81(6): 2590.
Levin, Y. (2000). "Crystallization of hard spheres under gravity." Physica A: Statistical
Mechanics and its Applications 287(1): 100-104.
Li, D. and D. Liang (2016). "Experimental study on the effective thermal conductivity of
methane hydrate-bearing sand." International Journal of Heat and Mass Transfer 92: 8-14.
Li, J., Y. Cao, C. Xia, B. Kou, X. Xiao, K. Fezzaa and Y. Wang (2014). "Similarity of
wet granular packing to gels." Nat Commun 5.
Li, Y., Y. Xu and C. Thornton (2005). "A comparison of discrete element simulations and
experiments for ‘sandpiles’ composed of spherical particles." Powder Technology 160(3):
219-228.
Lim, E. W. C. (2016). "Density segregation of dry and wet granular mixtures in vibrated
beds." Advanced Powder Technology 27(6): 2478-2488.
Lo Frano, R., D. Aquaro, S. Pupeschi and M. Moscardini (2014). "Thermo-mechanical
test rig for experimental evaluation of thermal conductivity of ceramic pebble beds." Fusion
Engineering and Design 89(7–8): 1309-1313.
Lubner, S. D., J. Choi, G. Wehmeyer, B. Waag, V. Mishra, H. Natesan, J. C. Bischof and
C. Dames (2015). "Reusable bi-directional 3omega sensor to measure thermal conductivity of
100-mum thick biological tissues." Rev Sci Instrum 86(1): 014905.
Luding, S., K. Manetsberger and J. Müllers (2005). "A discrete model for long time
sintering." Journal of the Mechanics and Physics of Solids 53(2): 455-491.
Ma, G., W. Zhou, X.-L. Chang and M.-X. Chen (2016). "A hybrid approach for modeling
of breakable granular materials using combined finite-discrete element method." Granular
Matter 18(1): 1-17.
Marchal, P., C. Hanotin, L. J. Michot and S. K. de Richter (2013). "Two-state model to
describe the rheological behavior of vibrated granular matter." Physical Review E 88(1):
012207.
Marchal, P., N. Smirani and L. Choplin (2009). "Rheology of dense-phase vibrated
powders and molecular analogies." Journal of Rheology 53(1): 1-29.
Markl, M. and C. Körner (2016). "Multiscale Modeling of Powder Bed–Based Additive
Manufacturing." Annual Review of Materials Research 46(1): 93-123.
125
Martin, C. L., D. Bouvard and S. Shima (2003). "Study of particle rearrangement during
powder compaction by the Discrete Element Method." Journal of the Mechanics and Physics
of Solids 51(4): 667-693.
Marty, G. and O. Dauchot (2005). "Subdiffusion and Cage Effect in a Sheared Granular
Material." Physical Review Letters 94(1): 015701.
Mason, G. (1968). "Radial Distribution Functions from Small Packings of Spheres."
Nature 217(5130): 733-735.
Mathonnet, J. E., P. Sornay, M. Nicolas and B. Dalloz-Dubrujeaud (2017). "Compaction
of noncohesive and cohesive granular materials under vibrations: Experiments and stochastic
model." Physical Review E 95(4): 042904.
Mehta, A. and G. C. Barker (1991). "Vibrated powders: A microscopic approach."
Physical Review Letters 67(3): 394-397.
Merckx, B., P. Dudoignon, J. P. Garnier and D. Marchand (2012). "Simplified Transient
Hot-Wire Method for Effective Thermal Conductivity Measurement in Geo Materials:
Microstructure and Saturation Effect." Advances in Civil Engineering 2012: 1-10.
Mo, J., D. Garrett and H. Ban (2015). "Anisotropic Effective Thermal Conductivity of
Particle Beds Under Uniaxial Compression." International Journal of Thermophysics.
Moscardini, M., Y. Gan, S. Pupeschi and M. Kamlah (2018). "Discrete element method
for effective thermal conductivity of packed pebbles accounting for the Smoluchowski
effect." Fusion Engineering and Design 127: 192-201.
Nowak, E. R., J. B. Knight, E. Ben-Naim, H. M. Jaeger and S. R. Nagel (1998). "Density
fluctuations in vibrated granular materials." Physical Review E 57(2): 1971-1982.
Nowak, E. R., J. B. Knight, M. L. Povinelli, H. M. Jaeger and S. R. Nagel (1997).
"Reversibility and irreversibility in the packing of vibrated granular material." Powder
Technology 94(1): 79-83.
Otsubo, M., C. O'Sullivan and T. Shire (2017). "Empirical assessment of the critical time
increment in explicit particulate discrete element method simulations." Computers and
Geotechnics 86: 67-79.
Panaitescu, A. and A. Kudrolli (2014). "Epitaxial growth of ordered and disordered
granular sphere packings." Physical Review E 90(3): 032203.
Parteli, E. J. R., J. Schmidt, C. Blümel, K.-E. Wirth, W. Peukert and T. Pöschel (2014).
"Attractive particle interaction forces and packing density of fine glass powders." Scientific
Reports 4: 6227.
126
Peeketi, A. R., M. Moscardini, A. Vijayan, Y. Gan, M. Kamlah and R. K. Annabattula
(2018). "Effective thermal conductivity of a compacted pebble bed in a stagnant gaseous
environment: An analytical approach together with DEM." Fusion Engineering and Design
130: 80-88.
Philippe, P. and D. Bideau (2002). "Compaction dynamics of a granular medium under
vertical tapping." EPL (Europhysics Letters) 60(5): 677.
Pica Ciamarra, M., A. Coniglio, D. De Martino and M. Nicodemi (2007). "Shear- and
vibration-induced order-disorder transitions in granular media." The European Physical
Journal E 24(4): 411-415.
Pinson, D., R. P. Zou, A. B. Yu, P. Zulli and M. J. McCarthy (1998). "Coordination
number of binary mixtures of spheres." Journal of Physics D: Applied Physics 31(4): 457.
Potyondy, D. O. and P. A. Cundall (2004). "A bonded-particle model for rock."
International Journal of Rock Mechanics and Mining Sciences 41(8): 1329-1364.
Pouliquen, O., M. Belzons and M. Nicolas (2003). "Fluctuating Particle Motion during
Shear Induced Granular Compaction." Physical Review Letters 91(1): 014301.
Pouliquen, O., M. Nicolas and P. D. Weidman (1997). "Crystallization of non-Brownian
Spheres under Horizontal Shaking." Physical Review Letters 79(19): 3640-3643.
Prasher, R. (2006). "Ultralow thermal conductivity of a packed bed of crystalline
nanoparticles: A theoretical study." Physical Review B 74(16): 165413.
Presley, M. A. and P. R. Christensen (1997). "Thermal conductivity measurements of
particulate materials 1. A review." Journal of Geophysical Research: Planets 102(E3): 6535-
6549.
Presley, M. A. and P. R. Christensen (1997). "Thermal conductivity measurements of
particulate materials 2. Results." Journal of Geophysical Research: Planets 102(E3): 6551-
6566.
Presley, M. A. and R. A. Craddock (2006). "Thermal conductivity measurements of
particulate materials: 3. Natural samples and mixtures of particle sizes." Journal of
Geophysical Research: Planets 111(E9): n/a-n/a.
Pupeschi, S., R. Knitter and M. Kamlah (2017). "Effective thermal conductivity of
advanced ceramic breeder pebble beds." Fusion Engineering and Design 116: 73-80.
Qi, F. and M. M. Wright (2018). "Particle scale modeling of heat transfer in granular
flows in a double screw reactor." Powder Technology 335: 18-34.
Reimann, J., E. Brun, C. Ferrero and J. Vicente (2015). "Pebble bed structures in the
vicinity of concave and convex walls." Fusion Engineering and Design 98-99: 1855-1858.
127
Reimann, J., J. Vicente, E. Brun, C. Ferrero, Y. Gan and A. Rack (2017). "X-ray
tomography investigations of mono-sized sphere packing structures in cylindrical
containers." Powder Technology 318(Supplement C): 471-483.
Richard, P., M. Nicodemi, R. Delannay, P. Ribiere and D. Bideau (2005). "Slow
relaxation and compaction of granular systems." Nat Mater 4(2): 121-128.
Richard, P., P. Philippe, F. Barbe, S. Bourlès, X. Thibault and D. Bideau (2003).
"Analysis by x-ray microtomography of a granular packing undergoing compaction."
Physical Review E 68(2): 020301.
Richter, S. K. d., G. L. Caër and R. Delannay (2009). "Heterogeneous dynamics of a
granular pack under vertical tapping." EPL (Europhysics Letters) 85(5): 58004.
Rondet, E., M. Delalonde, S. Chuetor and T. Ruiz (2017). "Modeling of granular
material's packing: Equivalence between vibrated solicitations and consolidation." Powder
Technology 310: 287-294.
Royer, J. R. and P. M. Chaikin (2015). "Precisely cyclic sand: Self-organization of
periodically sheared frictional grains." Proceedings of the National Academy of Sciences
112(1): 49-53.
Russo, J. and H. Tanaka (2012). "The microscopic pathway to crystallization in
supercooled liquids." Scientific Reports 2: 505.
Rycroft, C. H. (2009). "VORO++: A three-dimensional Voronoi cell library in C++."
Chaos: An Interdisciplinary Journal of Nonlinear Science 19(4): 041111.
Saadatfar, M., A. Kabla, T. Senden and T. Aste (2005). The geometry and the number of
contacts of monodisperse sphere packs using X-ray tomography. Powders and Grains 2005-
Proceedings of the 5th International Conference on Micromechanics of Granular Media.
Saadatfar, M., H. Takeuchi, V. Robins, N. Francois and Y. Hiraoka (2017). "Pore
configuration landscape of granular crystallization." Nature Communications 8: 15082.
Sanz, E., C. Valeriani, E. Zaccarelli, W. C. K. Poon, M. E. Cates and P. N. Pusey (2014).
"Avalanches mediate crystallization in a hard-sphere glass." Proceedings of the National
Academy of Sciences 111(1): 75-80.
Sasanka Kanuparthi, G. S., Thomas Siegmund, and Bahgat Sammakia (2008). "An
Efficient Network Model for Determining the Effective Thermal Conductivity of Particulate
Thermal Interface Materials." IEEE TRANSACTIONS ON COMPONENTS AND
PACKAGING TECHNOLOGIES 31(3): 11.
128
Schiffres, S. N. and J. A. Malen (2011). "Improved 3-omega measurement of thermal
conductivity in liquid, gases, and powders using a metal-coated optical fiber." Rev Sci
Instrum 82(6): 064903.
Scott, G. D. (1960). "Packing of Spheres: Packing of Equal Spheres." Nature 188(4754):
908-909.
Scott, G. D. (1962). "Radial Distribution of the Random Close Packing of Equal
Spheres." Nature 194(4832): 956-957.
Shi, X.-q. and Y.-q. Ma (2013). "Topological structure dynamics revealing collective
evolution in active nematics." Nature Communications 4: 3013.
Slavin, A. J., V. Arcas, C. A. Greenhalgh, E. R. Irvine and D. B. Marshall (2002).
"Theoretical model for the thermal conductivity of a packed bed of solid spheroids in the
presence of a static gas, with no adjustable parameters except at low pressure and
temperature." International Journal of Heat and Mass Transfer 45(20): 4151-4161.
Steinhardt, P. J., D. R. Nelson and M. Ronchetti (1983). "Bond-orientational order in
liquids and glasses." Physical Review B 28(2): 784-805.
Steuben, J. C., A. P. Iliopoulos and J. G. Michopoulos (2016). "Discrete element
modeling of particle-based additive manufacturing processes." Computer Methods in Applied
Mechanics and Engineering 305: 537-561.
Sufian, A., A. R. Russell, A. J. Whittle and M. Saadatfar (2015). "Pore shapes, volume
distribution and orientations in monodisperse granular assemblies." Granular Matter 17(6):
727-742.
Suzuki, M. and T. Oshima (1983). "Estimation of the Co-ordination number in a Multi-
Component Mixture of Spheres." Powder Technology 35(2): 159-166.
Tai, K., F.-J. Ulm and C. Ortiz (2006). "Nanogranular Origins of the Strength of Bone."
Nano Letters 6(11): 2520-2525.
Tai, S.-C. and S.-S. Hsiau (2009). "The flow regime during the crystallization state and
convection state on a vibrating granular bed." Advanced Powder Technology 20(4): 335-349.
Takahashi, H., K. Watanabe, K. Hoshino, H. Hoshiya, N. Yoshida, H. Kimura, K.
Nakamoto, Y. Hamakawa and T. Kawabe (2001). "Nano-granular metal/insulator multilayer
for reader shielding materials." IEEE Transactions on Magnetics 37(4): 1758-1760.
Tamás, B., S. Ellák, S. Balázs, W. Sandra, M. Pascal, R. Georg and S. Ralf (2016).
"Packing, alignment and flow of shape-anisotropic grains in a 3D silo experiment." New
Journal of Physics 18(9): 093017.
129
Tan, P., N. Xu and L. Xu (2014). "Visualizing kinetic pathways of homogeneous
nucleation in colloidal crystallization." Nat Phys 10(1): 73-79.
Tanaka, H. (2012). "Bond orientational order in liquids: Towards a unified description of
water-like anomalies, liquid-liquid transition, glass transition, and crystallization." The
European Physical Journal E 35(10): 113.
Tangri, H., Y. Guo and J. S. Curtis (2017). "Packing of cylindrical particles: DEM
simulations and experimental measurements." Powder Technology 317: 72-82.
Tavman, I. H. (1996). "EFFECTIVE THERMAL CONDUCTIVITY OF GRANULAR
POROUS MATERIALS." International Communications in Heat and Mass Transfer 23(2):
169-176.
Torquato, S. and F. H. Stillinger (2010). "Jammed hard-particle packings: From Kepler to
Bernal and beyond." Reviews of Modern Physics 82(3): 2633-2672.
Tsai, J. C., G. A. Voth and J. P. Gollub (2003). "Internal Granular Dynamics, Shear-
Induced Crystallization, and Compaction Steps." Physical Review Letters 91(6): 064301.
Tsotsas, E. and H. Martin (1987). "Thermal conductivity of packed beds- A review."
Chemical Engineering Process 22: 19-37.
Tsotsas, E. and E.-U. Schlünder (1991). "Impact of particle size dispersity on thermal
conductivity of packed beds: Measurement, numerical simulation, prediction." Chemical
Engineering & Technology 14(6): 421-427.
Tsunazawa, Y., D. Fujihashi, S. Fukui, M. Sakai and C. Tokoro (2016). "Contact force
model including the liquid-bridge force for wet-particle simulation using the discrete element
method." Advanced Powder Technology 27(2): 652-660.
van Antwerpen, W., C. G. du Toit and P. G. Rousseau (2010). "A review of correlations
to model the packing structure and effective thermal conductivity in packed beds of mono-
sized spherical particles." Nuclear Engineering and Design 240(7): 1803-1818.
van der Linden, J. H., G. A. Narsilio and A. Tordesillas (2016). "Machine learning
framework for analysis of transport through complex networks in porous, granular media: A
focus on permeability." Physical Review E 94(2): 022904.
van der Linden, J. H., A. Sufian, G. A. Narsilio, A. R. Russell and A. Tordesillas (2018).
"A computational geometry approach to pore network construction for granular packings."
Computers & Geosciences 112: 133-143.
Vandamme, M. and F.-J. Ulm (2009). "Nanogranular origin of concrete creep."
Proceedings of the National Academy of Sciences 106(26): 10552-10557.
130
Verlet, L. (1967). "Computer "Experiments" on Classical Fluids. I. Thermodynamical
Properties of Lennard-Jones Molecules." Physical Review 159(1): 98-103.
Wang, J., J. K. Carson, M. F. North and D. J. Cleland (2006). "A new approach to
modelling the effective thermal conductivity of heterogeneous materials." International
Journal of Heat and Mass Transfer 49(17–18): 3075-3083.
Wang, J., J. K. Carson, M. F. North and D. J. Cleland (2008). "A new structural model of
effective thermal conductivity for heterogeneous materials with co-continuous phases."
International Journal of Heat and Mass Transfer 51(9–10): 2389-2397.
Wang, M. and N. Pan (2008). "Predictions of effective physical properties of complex
multiphase materials." Materials Science and Engineering: R: Reports 63(1): 1-30.
Washino, K., K. Miyazaki, T. Tsuji and T. Tanaka (2016). "A new contact liquid
dispersion model for discrete particle simulation." Chemical Engineering Research and
Design 110: 123-130.
Watson L. Vargas, J. J. M. (2002). "Stress effects on the conductivity of particulate beds."
Chemical Engineering Science 57: 13.
Wei, G., L. Wang, C. Xu, X. Du and Y. Yang (2016). "Thermal conductivity
investigations of granular and powdered silica aerogels at different temperatures and
pressures." Energy and Buildings 118: 226-231.
Weidenfeld, G., Y. Weiss and H. Kalman (2004). "A theoretical model for effective
thermal conductivity (ETC) of particulate beds under compression." Granular Matter 6(2-3):
121-129.
Woodside, W. and J. H. Messmer (1961). "Thermal Conductivity of Porous Media. I.
Unconsolidated Sands." Journal of Applied Physics 32(9): 1688-1699.
Wu, H., N. Gui, X. Yang, J. Tu and S. Jiang (2016). "Effect of scale on the modeling of
radiation heat transfer in packed pebble beds." International Journal of Heat and Mass
Transfer 101: 562-569.
Wu, H., N. Gui, X. Yang, J. Tu and S. Jiang (2018). "Particle-Scale Investigation of
Thermal Radiation in Nuclear Packed Pebble Beds." Journal of Heat Transfer 140(9):
092002-092002-092007.
Wu, H., N. Gui, X. Yang, J. Tu and S. Jiang (2018). "A smoothed void fraction method
for CFD-DEM simulation of packed pebble beds with particle thermal radiation."
International Journal of Heat and Mass Transfer 118: 275-288.
Wu, Y., X. An and A. B. Yu (2017). "DEM simulation of cubical particle packing under
mechanical vibration." Powder Technology 314: 89-101.
131
Yagi, S. and D. Kunii (1957). "Studies on effective thermal conductivities in packed
beds." AIChE Journal 3(3): 373-381.
Yanagishima, T., J. Russo and H. Tanaka (2017). "Common mechanism of
thermodynamic and mechanical origin for ageing and crystallization of glasses." Nature
Communications 8: 15954.
Yang, J., C.-Y. Wu and M. Adams (2015). "DEM analysis of the effect of electrostatic
interaction on particle mixing for carrier-based dry powder inhaler formulations."
Particuology 23: 25-30.
Yi, L. Y., K. J. Dong, R. P. Zou and A. B. Yu (2011). "Coordination Number of the
Packing of Ternary Mixtures of Spheres: DEM Simulations versus Measurements." Industrial
& Engineering Chemistry Research 50(14): 8773-8785.
Yu, A. B., X. Z. An, R. P. Zou, R. Y. Yang and K. Kendall (2006). "Self-Assembly of
Particles for Densest Packing by Mechanical Vibration." Physical Review Letters 97(26):
265501.
Yu, A. B. and N. Standish (1993). "A study of the packing of particles with a mixture size
distribution." Powder Technology 76(2): 113-124.
Yuan, C.-N., Y.-F. Li, Y.-J. Sheng and H.-K. Tsao (2011). "Dry nanogranular materials."
Applied Physics Letters 98(14): 144102.
Yun, T. S. and T. M. Evans (2010). "Three-dimensional random network model for
thermal conductivity in particulate materials." Computers and Geotechnics 37(7-8): 991-998.
Zehner, P. and E. Schlunder (1970). "Thermal conductivity of packings at moderate
temperatures." Chemie Ingenieur Technik 42(14): 933-&.
Zhai, C., D. Hanaor, G. Proust, L. Brassart and Y. Gan (2016). "Interfacial electro-
mechanical behaviour at rough surfaces." Extreme Mechanics Letters 9: 422-429.
Zhao, J., M. Jiang, K. Soga and S. Luding (2016). "Micro origins for macro behavior in
granular media." Granular Matter 18(3): 1-5.
Zhou, Z.-Y., R.-P. Zou, D. Pinson and A.-B. Yu (2011). "Dynamic Simulation of the
Packing of Ellipsoidal Particles." Industrial & Engineering Chemistry Research 50(16):
9787-9798.
Zhou, Z. Y., A. B. Yu and P. Zulli (2010). "A new computational method for studying
heat transfer in fluid bed reactors." Powder Technology 197(1–2): 102-110.
Zhu, H. P. and A. B. Yu (2006). "A theoretical analysis of the force models in discrete
element method." Powder Technology 161(2): 122-129.
132
Zhu, H. P., Z. Y. Zhou, R. Y. Yang and A. B. Yu (2008). "Discrete particle simulation of
particulate systems: A review of major applications and findings." Chemical Engineering
Science 63(23): 5728-5770.
Zou, R. P., X. Bian, D. Pinson, R. Y. Yang, A. B. Yu and P. Zulli (2003). "Coordination
Number of Ternary Mixtures of Spheres†." Particle & Particle Systems Characterization
20(5): 335-341.
133
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
134
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_++;
135
} 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.)
136
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)
137
{ 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");
138
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;
139
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;
140
} 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;
142
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_)
148
{ 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 ;
154