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SOLIDSOLIDSOLIDSOLID

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MOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATION

SOLIDSOLIDSOLIDSOLID----LIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIA

AND AND AND AND


Centre for Molecular Simulation Centre for Molecular Simulation Centre for Molecular Simulation Centre for Molecular Simulation

Swinburne University of TechnologySwinburne University of TechnologySwinburne University of TechnologySwinburne University of Technology


LIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIA

AND AND AND AND SHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITY

Alauddin AhmedAlauddin AhmedAlauddin AhmedAlauddin Ahmed


Doctor of PhilosophyDoctor of PhilosophyDoctor of PhilosophyDoctor of Philosophy





SHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITY


DissertationDissertationDissertationDissertation

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ii

AbstractAbstractAbstractAbstract

The objectives of this thesis are to study the solid-liquid phase equilibria and

the shear viscosity for both bounded and unbounded intermolecular potentials.

A variety of molecular simulation techniques are used to calculate solid-liquid

coexistence whereas nonequilibrium molecular dynamics algorithm is used to

examine the shear viscosity.

The solid-liquid coexistence properties are calculated using the GWTS

algorithm which is self starting, independent of particle exchange mechanism

and does not rely upon a prior free energy equation of state. A combination of

equilibrium and nonequilibrium molecular dynamics simulation algorithms

constructs the framework for the GWTS algorithm. It has been demonstrated

by the solid-liquid phase coexistence data reported for 12-6 Lennard-Jones fluid

that the GWTS algorithm is capable of calculating solid-liquid coexistence

properties with comparable accuracy irrespective of temperature, density and

pressure range. The solid-liquid phase transition is found not to be very

sensitive to the choice of truncation and shifting schemes except close to the

vicinity of triple point where the coexistence properties vary significantly

compared to the entire melting line.

The effects of repulsive component �� on the solid-liquid coexistence properties of � � 6 Lennard-Jones potentials are reported. The estimated triple points of

iii

� � 6 Lennard-Jones potentials are reported for the first time. Scaling

relationships for the triple point pressures and temperatures have been

established with respect to �. The solid-liquid phase coexistence data for purely

repulsive Weeks-Chandler-Andersen system are presented from very low to high

temperatures and pressures. It has been observed that the WCA potential

approaches zero-temperature limit which is in contrast to the 12-6 Lennard-

Jones case. The data presented in this thesis also demonstrate that the GWTS

algorithm can also generate the phase diagram of bounded Gaussian core model

potential in presence of re-entrant melting scenario. The solid-liquid coexistence

properties are also reported for the state points closer to the common point. It

has been demonstrated how the GWTS algorithm is capable of calculating solid-

liquid coexistence properties for intermolecular potentials complex than 12-6

Lennard-Jones potential.

For the first time, the strain rate dependent shear viscosity data are reported

for the Gaussian core model fluid. It has been demonstrated that in the

reentrant melting region shear viscosity decreases with increasing density at

constant temperature whereas viscosity increases with increasing temperature.

This behaviour has been found to be consistent with viscosity measurements of

cationic surfactant solution and attributed to the “infinite-density ideal-gas

limit” of the Gaussian core potential.

iv

Extensive nonequilibrium molecular simulation data are reported for a wide

range of temperature, pressure, density, shear viscosity and strain rate. A

nonequilibrium steady-state equation of state has been developed from the

strain rate dependent pressure and energy data. A generic viscosity model has

also been developed to establish connections between the shear viscosity and the

steady state variables such as temperature, pressure, density and strain rate via

the nonequilibrium equation of state. The generic viscosity model has been

compared with recommended experimental data.

v

AcknowledgementAcknowledgementAcknowledgementAcknowledgement

I would like to acknowledge and thank my supervisors Prof. Richard J Sadus

and Prof. Billy D Todd for their positive direction and continuing support. In

particular, I am grateful to my principal supervisor Prof. Richard Sadus for his

invaluable encouragement, thought provoking criticism and remarkable patience

to achieve quality. Both of my supervisors gifted me the best attributes of a

researcher through their scientific knowledge, professional attitude and fantastic

interpersonal skill. I would also like to thank Prof. Feng Wang for her advice

and encouragement. My special thanks go to Angelica for her warm reception

and hot food at the end-of-year parties.

I would like to acknowledge Prof. Peter Mausbach for his enthusiastic

encouragement during his visit at CMS. I have learned many interesting things

from him in the course of our collaborative work. I am thankful to Jeffrey R

Errington of University at Buffalo for providing me the data of his published

work on shifted force Lennard-Jones potential.

I would like to thank my wife and best friend, Fermina, for her support and

encouragement without which this thesis, I believe, would not have been

written.

vi

I am grateful to my mother and father, whose love continues to encourage me,

as it has always done. I would like to dedicate this work to my loving mother.

I gratefully acknowledge A/Prof. Peter Daivis at RMIT for providing me

fruitful feedback whenever it was necessary for me. Special thanks go to Dr.

Ming Liu and Dr. Jesper S Hansen for their encouragement and thoughtful

discussions. I also thank Dr. Federico Frascoli, Dr. Tomas A Hunt, Dr. Jarek

Bosko and Dr. Alex Bosowski for their friendly behaviour.

I also like to acknowledge the support I have received from my fellow colleagues

at CMS. My special thanks go to the staff of FICT and Swinburne Research for

their continuing support with the highest professional standard possible.

I also acknowledge Swinburne University of Technology for providing me

financial support through a Swinburne University Postgraduate Research

Award (SUPRA). The Australian Partnership for Advanced Computing

generously provided an allocation of computing time to perform the simulations.

vii

DeclarationDeclarationDeclarationDeclaration

I hereby declare that the thesis entitled “Molecular Simulation of Solid-Liquid

Phase Equilibria and Shear Viscosity”, and submitted in fulfilment of the

requirements for the Degree of Doctor of Philosophy in the Faculty of

Information and Communication Technologies of Swinburne University of

Technology, is my own work and that it contains no material which has been

accepted for the award to the candidate of any other degree or diploma, except

where due reference is made in the text of the thesis. To the best of my

knowledge and belief, it contains no material previously published or written by

another person except where due reference is made in the text of the thesis.

Alauddin Ahmed

20 September 2010.

viii

Publications from this TPublications from this TPublications from this TPublications from this Thesishesishesishesis

JJJJournal Pournal Pournal Pournal Publicationsublicationsublicationsublications

The following papers have been published from part of this work:

Ahmed, A. and R. J. Sadus (2009) "Phase diagram of the Weeks-Chandler-

Andersen potential from very low to high temperatures and pressures",

Phys. Rev. E 80808080, 061101.

Ahmed, A. and R. J. Sadus (2009) "Solid-liquid equilibria and triple points of n-

6 Lennard-Jones fluids", J. Chem. Phys. 131131131131, 174504.

Ahmed, A. , P. Mausbach and R. J. Sadus (2009) "Strain rate dependent shear

viscosity of the Gaussian core model fluid", J. Chem. Phys. 131131131131, 224511.

Mausbach, P., A. Ahmed and R. J. Sadus (2009) "Solid-liquid phase equilibria

of the Gaussian core model fluid ", J. Chem. Phys. 131131131131, 184507.

Ahmed, A. and R. J. Sadus (2009) "Nonequilibrium equation of state for

Lennard-Jones fluids and the calculation of strain-rate dependent shear

viscosity ", AIChE J. (in press).

ix

Ahmed, A. , P. Mausbach and R. J. Sadus (2010) "Pressure and energy

behavior of the Gaussian core model fluid under shear", Phys. Rev. E.

82828282, 011201.

Ahmed, A. and R. J. Sadus (2010) "Effect of potential truncations and shifts on

the solid-liquid phase coexistance of Lennard-Jones fluids", J. Chem.

Phys. (in press).

ConfConfConfConference Proceedings Perence Proceedings Perence Proceedings Perence Proceedings Publicationublicationublicationublication

The following paper has been published from this work in the following

conference proceedings:

Ahmed, A. and R. J. Sadus (2008) "Shear viscosity along the freezing line of

Weeks-Chandler-Andersen fluid ", XXII ICTAM (The International

Congress of Theoretical and Applied Mechanics), 25-29 August 2008,

Adelaide, Australia.

x

ContentsContentsContentsContents

AbstractAbstractAbstractAbstract ........................................................................................................................................................................................................................................................................................................................................................................................................................................................ iiiiiiii

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements ........................................................................................................................................................................................................................................................................................................................................................................................ vvvv

DeclarationDeclarationDeclarationDeclaration ................................................................................................................................................................................................................................................................................................................................................................................................................................ viiviiviivii

Publications from this ThesisPublications from this ThesisPublications from this ThesisPublications from this Thesis ............................................................................................................................................................................................................................................................................................................ viiiviiiviiiviii

ContentsContentsContentsContents ........................................................................................................................................................................................................................................................................................................................................................................................................................................................ xxxx

List of FiguresList of FiguresList of FiguresList of Figures .................................................................................................................................................................................................................................................................................................................................................................................................... xviixviixviixvii

List of TablesList of TablesList of TablesList of Tables .................................................................................................................................................................................................................................................................................................................................................................................................... xxixxixxixxixxxx

NotationNotationNotationNotation ................................................................................................................................................................................................................................................................................................................................................................................................................................ xxxixxxixxxixxxiiiii

CCCCHAPTERHAPTERHAPTERHAPTER 1: 1: 1: 1: IIIINTRODUCTIONNTRODUCTIONNTRODUCTIONNTRODUCTION .................................................................................................................................................................................................................................................................................... 33338888

1.1 Motivations ........................................................................................ 38

1.2 Aims ................................................................................................... 42

1.3 Background and Current Progress ..................................................... 44

1.3.1 Solid-Liquid Phase Equilibria .............................................. 44

(i) Thermodynamic Integration .................................................... 45

(ii) Gibbs Ensemble....................................................................... 46

(iii) Gibbs-Duhem Integration ...................................................... 46

(iv) Direct Methods ....................................................................... 47

xi

(v) Density of States Method ........................................................ 49

(vi) Phase-Switch Monte Carlo Methods ...................................... 50

(vii) Molecular Dynamics Methods ............................................... 51

(viii) GWTS Method ..................................................................... 51

1.3.2 Validation of Solid-Liquid Phase Equilibria Data ............... 53

(i) Solid-Liquid Phase Coexistence from GWTS Algorithm and Its

Reliability ..................................................................................... 53

(ii) Effects of Truncation and Shifting Schemes on Solid-Liquid

Coexistence .................................................................................. 54

1.3.3 Solid-Liquid Phase Equilibria of Lennard-Jones Family of

Potentials ..................................................................................... 56

1.3.4 Phase Diagram of the Weeks-Chandler-Andersen Potential

...................................................................................................... 58

1.3.5 Phase Diagram of the Gaussian Core Model Fluid .............. 61

1.3.6 Strain Rate Dependent Shear Viscosity of the Gaussian Core

Bounded Potential ........................................................................ 62

1.3.7 Equation of State and Viscosity Modelling .......................... 63

(i) Steady State Equation of State ................................................ 63

(ii) Generic Viscosity Model ........................................................ 64

1.4 Organisation of the Dissertation ....................................................... 66

xii

CCCCHAPTERHAPTERHAPTERHAPTER 2: 2: 2: 2: MMMMOLECULAROLECULAROLECULAROLECULAR SSSSIMULATIONIMULATIONIMULATIONIMULATION ........................................................................................................................................................................................................ 69696969

2.1 Rationale for Molecular Simulation .................................................. 70

2.2 Intermolecular Potentials .................................................................. 71

2.2.1 Lennard-Jones Family of Potentials .................................... 72

2.2.2 Truncation and Shifting Schemes ....................................... 74

(i) Truncated Lennard-Jones Potential ....................................... 74

(iii) Truncated and Shifted Lennard-Jones Potential .................. 75

(iv) Shifted-Force Lennard-Jones Potential ............................... 75

2.2.3 Weeks-Chandler-Andersen Potential .................................... 75

2.2.4 Gaussian-Core Model Potential ........................................... 76

2.3 Reduced Unit Formalism .................................................................. 77

2.4 Molecular Dynamics .......................................................................... 78

2.4.1 Equations of Motion ............................................................ 81

2.4.2 Initial Lattice Configuration ................................................. 82

2.4.3 Initial Random Velocity ....................................................... 82

2.4.4 Force Calculation ................................................................. 83

2.4.5 Periodic Boundary Conditions (PBC) .................................. 83

2.5 Nonequilibrium Molecular Dynamics Simulation ............................... 86

2.5.1 Lees-Edwards Periodic Boundary Condition ........................ 86

2.5.2 The sllod Equations of Motion ............................................. 89

2.5.3 Gear Predictor-Corrector Integration Scheme ..................... 93

2.6 Algorithm to Study Solid-Liquid Phase Equilibria ............................ 98

2.6.1 GWTS Algorithm ............................................................... 100

xiii

(i) Fundamentals of the GWTS Algorithm ................................ 100

(ii) Calculating the Solid-Liquid Phase Coexistence ................... 104

2.6.2 Gibbs-Duhem Integration ................................................... 108

(i) Thermodynamic Basis of GDI Algorithm .............................. 108

(ii) Numerical Techniques in GDI Simulation Design ................ 110

CCCCHAPTERHAPTERHAPTERHAPTER 3: 3: 3: 3: VVVVALIDATION OFALIDATION OFALIDATION OFALIDATION OF SSSSOLIDOLIDOLIDOLID----LLLLIQUIDIQUIDIQUIDIQUID PPPPHASEHASEHASEHASE EEEEQUILIBRIA QUILIBRIA QUILIBRIA QUILIBRIA

DATADATADATADATA .................................................................................................................................................................................................................................................................................................................................................................................................................................................... 111111117777

3.1 Simulation Details ............................................................................ 117

3.1.1 Technical Details of the GWTS Simulation ....................... 117

3.1.2 Technical Details of the GDI Simulation ........................... 118

3.1.3 Calculation of Properties for Full LJ Potential.................. 119

3.2 Comparison of Lennard-Jones Solid-Liquid Phase Coexistence Data

............................................................................................................... 119

3.2.1 Data Collection ................................................................... 120

3.2.2 Data Analysis ..................................................................... 121

3.3 Finite Size Effect on Lennard-Jones Solid-Liquid Coexistence ....... 128

3.4 Solid-Liquid Phase Coexistence from the GWTS Algorithm and Its

Reliability .............................................................................................. 129

3.5 Independent Validation of the GWTS Algorithm............................133

3.6 The Effects of Potential Truncation and Shifting Schemes on Solid-

Liquid Coexistence ................................................................................ 134

xiv

CCCCHAPTERHAPTERHAPTERHAPTER 4: 4: 4: 4: SSSSOLIDOLIDOLIDOLID----LLLLIQUIDIQUIDIQUIDIQUID PPPPHASEHASEHASEHASE EEEEQUILIBRIAQUILIBRIAQUILIBRIAQUILIBRIA OFOFOFOF THETHETHETHE LENNARDLENNARDLENNARDLENNARD----

JONES FAMILY OF POTENTIALSJONES FAMILY OF POTENTIALSJONES FAMILY OF POTENTIALSJONES FAMILY OF POTENTIALS .................................................................................................................................................................................................................................................... 141414146666


4.2 Analysis of the n-Variation of the Solid-Liquid Coexistence ........... 148

4.3 Temperature Dependence of Coexistence Pressure and Densities ... 156

4.4 Estimation of the Triple Point ......................................................... 159

4.5 Melting and Freezing Rules ............................................................. 162

CCCCHAPTERHAPTERHAPTERHAPTER 5555: : : : PPPPHASE HASE HASE HASE DDDDIAGRAM OF THEIAGRAM OF THEIAGRAM OF THEIAGRAM OF THE WWWWEEKSEEKSEEKSEEKS----CCCCHANDLERHANDLERHANDLERHANDLER----

AAAANDERSENNDERSENNDERSENNDERSEN PPPPOTENTIALOTENTIALOTENTIALOTENTIAL .................................................................................................................................................................................................................................................................................................................... 161616169999


5.1.1 Simulations at Low and Intermediate Temperatures ......... 171

5.1.2 Simulations at High Temperatures ..................................... 173

5.1.3 Finite Size Effects ............................................................... 174

5.2 Solid-Liquid Coexistence .................................................................. 176

5.3 Low and High Temperature Limits .................................................. 184

5.4 Temperature Dependence of Coexistence Pressure and Densities ... 185

5.5 Comparison with Equation of State Calculations ............................ 186

5.6 Melting and Freezing Rules ............................................................. 188

5.7 Entropy of Fusion ............................................................................ 192

5.8 Volume Discontinuity at Phase Transition ...................................... 194

xv

CCCCHAPTERHAPTERHAPTERHAPTER 6: 6: 6: 6: PPPPHASEHASEHASEHASE DDDDIAGRAM OF THE IAGRAM OF THE IAGRAM OF THE IAGRAM OF THE GGGGAUSSIAN AUSSIAN AUSSIAN AUSSIAN CCCCOREOREOREORE MMMMODELODELODELODEL

FFFFLUIDLUIDLUIDLUID ................................................................................................................................................................................................................................................................................................................................................................................................................................................ 191919198888


6.2 System Size Analysis ........................................................................ 199

6.3 Low-Density Side of the Solid Region .............................................. 200

6.4 High-Density Side of the Solid Region ............................................. 203

6.5 The GCM Phase Diagram ................................................................ 206

CCCCHAPTERHAPTERHAPTERHAPTER 7: 7: 7: 7: SSSSTRAINTRAINTRAINTRAIN----RRRRATEATEATEATE DDDDEPENDENTEPENDENTEPENDENTEPENDENT SSSSHEARHEARHEARHEAR VVVVISCOSITY OF THEISCOSITY OF THEISCOSITY OF THEISCOSITY OF THE

GGGGAUSSIAN AUSSIAN AUSSIAN AUSSIAN CCCCOREOREOREORE BBBBOUNDEDOUNDEDOUNDEDOUNDED PPPPOTENTIALOTENTIALOTENTIALOTENTIAL ............................................................................................................................................................................................ 222210101010


7.2 Maximum Safe Strain-Rates ............................................................ 212

7.3 Shear Viscosity: Strain-Rate Behaviour ........................................... 216

7.4 Fitting Simulation Data ................................................................... 224

7.5 Zero-Shear Viscosities ...................................................................... 224

CCCCHAPTER HAPTER HAPTER HAPTER 8: 8: 8: 8: SSSSTEADYTEADYTEADYTEADY SSSSTATETATETATETATE EEEEQUATION OF STATE ANDQUATION OF STATE ANDQUATION OF STATE ANDQUATION OF STATE AND VVVVISCOSITYISCOSITYISCOSITYISCOSITY

MMMMODELLINGODELLINGODELLINGODELLING .................................................................................................................................................................................................................................................................................................................................................................................................... 222222227777

8.1 Steady State Equation of State ........................................................ 228

8.1.1 Simulation Details .............................................................. 228

8.1.2 Development of the Equation of State ............................... 230

8.1.3 Data Accumulation and Parameter Estimation ................. 234

8.1.4 Accuracy of the Proposed Steady-State EOS ..................... 237

8.2 Development of Generic Viscosity Model ........................................ 243

xvi

8.3 Connection between EOS and Generic Viscosity Model .................. 248

8.3.1 When Strain Rate is the Generic Variable ....................... 248

8.3.2 When Density is the Generic Variable .............................. 248

8.3.3 Pressure Dependent Shear Viscosity ................................. 248

8.3.4 Density Dependent Shear Viscosity .................................. 251

8.4 Experimental Verification of the Model for Strain Rate Dependent

Viscosity ................................................................................................. 254

8.5 Experimental Verification of the Model for Zero-Shear Viscosity ... 256

8.5.1 Monatomic Real Fluid ........................................................ 256

8.5.2 Complex Real Fluid ............................................................ 257

(i) Water ..................................................................................... 257

(ii) Carbon Dioxide ..................................................................... 264

(iii) Light Alkanes ....................................................................... 265

(iv) Hydrocarbons ....................................................................... 265

CCCCHAPTERHAPTERHAPTERHAPTER 9: 9: 9: 9: CCCCONCLUSIONSONCLUSIONSONCLUSIONSONCLUSIONS ........................................................................................................................................................................................................................................................................................ 262626267777

APPENDIX A:APPENDIX A:APPENDIX A:APPENDIX A: DATA FROMDATA FROMDATA FROMDATA FROM CHAPTER 3CHAPTER 3CHAPTER 3CHAPTER 3…………………………………....274…………………………………....274…………………………………....274…………………………………....274

APPENDIX B:APPENDIX B:APPENDIX B:APPENDIX B: DATA DATA DATA DATA FFFFROMROMROMROM CHAPTER 8CHAPTER 8CHAPTER 8CHAPTER 8…………………………………....275…………………………………....275…………………………………....275…………………………………....275

APPENDIX APPENDIX APPENDIX APPENDIX CCCC: D: D: D: DERIVATIONSERIVATIONSERIVATIONSERIVATIONS FROM CHAPTER FROM CHAPTER FROM CHAPTER FROM CHAPTER 8.................…………....2788.................…………....2788.................…………....2788.................…………....278

REFERENCESREFERENCESREFERENCESREFERENCES………………………………………………………………………281………………………………………………………………………281………………………………………………………………………281………………………………………………………………………281

xvii

List of FiguresList of FiguresList of FiguresList of Figures

Figure 2.1Figure 2.1Figure 2.1Figure 2.1 Comparison of n-6 Lennard-Jones pair potentials, where from top to

bottom � � 12, 11, 10, 9, 8, and 7………………………………………………..73

Figure 2.2Figure 2.2Figure 2.2Figure 2.2 Periodic boundary conditions in a three dimensional view. The

orange colour box is the central simulation box. All other boxes are the images

of the original simulation box. The particles move in and out as shown with

arrows………………………………………………………………………………84

Figure 2.3Figure 2.3Figure 2.3Figure 2.3 Lees-Edwards periodic boundary conditions for planar Couette flow

in a three dimensional view while the motion of the image cells defines the

strain rate for the flow. The pink colour boxes are taken to be stationary. The

indigo colour boxes are moving in the positive � direction with a velocity which

equals box length multiplied by the strain rate. The light blue colour boxes are

also moving with the same velocity but in the negative � direction…………..87

Figure 2.4Figure 2.4Figure 2.4Figure 2.4 Planner Couette flow geometry………………………………………90

Figure 2.5 Figure 2.5 Figure 2.5 Figure 2.5 Schematic views of the essential components of the GWTS algorithm.

Arrows are showing the next steps to follow in the algorithm. Blue colour

represents liquid and red colour represents solid…..........................................103

Figure 3.1Figure 3.1Figure 3.1Figure 3.1 Temperatures covered by different authors in their simulations of the

solid-liquid equilibria of 12-6 Lennard-Jones fluid. Shown are temperatures

studied by Hansen and Verlet (1969) (�, H & V), Agrawal and Kofke (1995c)

(�, A & K), Barroso and Ferreira (2002) (�, B & F), Mastny and de Pablo

(2007) (�, M & P), Morris and Song (2002) (�, M & S), Errington (2004) (⊳,

E) and McNeil-Watson and Wilding (2006) (�, M &W)……………………..121

Figure 3.2Figure 3.2Figure 3.2Figure 3.2 Solid-liquid phase coexistence pressure as a function of temperature

compiled from literature. Coexistence pressure as a function of temperature for

the temperature range (a) � 0.60 � 1.0, (b) � 1.0 � 2.0, (c) � 2.0 � 5.0

and (d) � 0.65 � 5.0 (most commonly used temperature range) calculated by

Agrawal and Kofke (1995c) (�), Barroso and Ferreira (2002) (�), Morris and

xviii

Song (2002) (�), Errington (2004) (⊳), McNeil-Watson and Wilding (2006)

(�), Mastny and de Pablo (2007) (�) and Hansen and Verlet (1969) (�)..125

Figure 3.3Figure 3.3Figure 3.3Figure 3.3 Comparison of (a) liquid and (b) solid densities as a function of

temperature at solid-liquid coexistence in the temperature range T = 0.65-1.10.

Data shown are from Agrawal and Kofke (1995c) (�), Barroso and Ferreira

(2002) (�), Morris and Song (2002) (�), Errington (2004) (⊳), McNeil-Watson

and Wilding (2006) (�), Mastny and de Pablo (2007) (�) and Hansen and

Verlet (1969) (�)………………………………………………………………...126

Figure 3.4Figure 3.4Figure 3.4Figure 3.4 Solid-liquid coexistence densities in the � � plane. Data shown are

from Agrawal and Kofke (1995c) (�), Barroso and Ferreira (2002) (�), Morris

and Song (2002) (�), Errington (2004) (⊳), McNeil-Watson and Wilding (2006)

(�), Mastny and de Pablo (2007) (�) and Hansen and Verlet (1969) (�). In all

cases solid lines are used to guide the symbols for the overall view of freezing

and melting lines............................................................................................127

Figure 3.5Figure 3.5Figure 3.5Figure 3.5 Comparison of the solid-liquid coexistence (a) pressure, (b) liquid

densities and (c) solid densities for the 12-6 Lennard-Jones potential calculated

in this work (�) with data from Agrawal and Kofke (1995c) (�) and Hansen

and Verlet (1969) (*). The errors are approximately equal to the symbol

size……………….............................................................................................130



in this work (Ο) with data from Mastny and de Pablo (2007) (�) and McNeil-

Watson and Wilding (2006) (�). The errors are approximately equal to the

symbol size………………………………………………………………………..131

Figure 3.7Figure 3.7Figure 3.7Figure 3.7 Solid-liquid coexistence pressures of 12-6 Lennard-Jones systems as a

function of cutoff radius. Shown are truncated (�), truncated-shifted (�) and

shifted-force (Ο) Lennard-Jones systems for temperatures (a) T � 1.0 and (b) T � 2.74………………………………………………………………………...137

Figure 3.8Figure 3.8Figure 3.8Figure 3.8 Potential energy as a function of cutoff radius for the liquid phase of

12-6 Lennard-Jones systems at solid-liquid coexistence. Shown are truncated

xix

(�), truncated-shifted (�) and shifted-force (Ο) Lennard-Jones systems for

temperatures (a) T � 1.0 and (b) T � 2.74……………………………………138

Figure 3.9Figure 3.9Figure 3.9Figure 3.9 Potential energy as a function of cutoff radius for the solid phase of

Lennard-Jones systems at solid-liquid coexistence. Shown are truncated (�),

truncated-shifted (�) and shifted-force (Ο) Lennard-Jones systems for

temperatures (a) T � 1.0 and (b) T � 2.74……………………………………139

Figure 3.10Figure 3.10Figure 3.10Figure 3.10 Comparison with full Lennard-Jones potential. Melting line of (a)

truncated-shifted LJ (�) and (b) shifted force LJ (Ο) at cutoff 2.5. In both

cases a comparison is made with the full LJ potential obtained in this work (×)

and reported by Agrawal and Kofke (1995c) (—)……………………………141

Figure 3.11Figure 3.11Figure 3.11Figure 3.11 Pressure variation of shifted force LJ (Ο) with respect to truncated-

shifted LJ potential (�). The melting line pressure for full LJ (—) potential,

obtained in this work, is also reported for comparison……………………… 142

Figure 3.12Figure 3.12Figure 3.12Figure 3.12 Melting line pressure variation of truncated and shifted LJ potential

as a function of cutoff radius. Shown are cutoff radius 2.5 (*) and cutoff radius

6.5 (�). The melting line pressure for full LJ (—) potential, calculated in this

work, is also reported for comparison…………………………………………142

Figure 3.13Figure 3.13Figure 3.13Figure 3.13 Phase diagram of shifted force LJ in � � plane. Shown are the

freezing (�) and melting (�) lines of LJ potential with cutoff radius 6.5 and

freezing (Ο) and melting (�) lines of LJ potential with cutoff radius 2.5. A

comparison is shown with the full LJ freezing line (—) and melting line (---)

obtained in this work……………………………………………………………144

Figure 4.1Figure 4.1Figure 4.1Figure 4.1 Solid-liquid coexistence (a) pressure (�), (b) liquid (�) and solid (O)

densities as functions of � at T = 2.74…………………………………………150

Figure 4.2Figure 4.2Figure 4.2Figure 4.2 Complete density-temperature phase diagrams of n-6 Lennard-Jones

potentials. Shown are (a) n = 12 (�, guided by a dashed line), 10 (�, guided

by a dotted line), 8 (O, guided by a solid line); and (b) n = 7 (�, guided by a

solid line), 9 (�, guided by a dotted line), 11 (∆, guided by a dashed line). The

vapour-liquid coexistence data are from (Kiyohara et al., 1996, Okumura and

xx

Yonezawa, 2000). Freezing and melting lines and triple points are from this

work………………………………………………………………………………151

Figure 4.3Figure 4.3Figure 4.3Figure 4.3 The solid-liquid coexistence pressure of n-6 Lennard-Jones potentials

calculated in this work as a function of reciprocal temperature on a log scale. (a)

n = 12 (∆, guided by solid line), 10 (�, guided by dotted line), and 8 (�,

guided by dashed line); and (b) n = 11 (�, guided by solid line), 9 (�, guided

by dotted line), and 7 (*, guided by dashed line)……………………………..153

Figure 4.4Figure 4.4Figure 4.4Figure 4.4 (a) Relative density difference (r.d.d) and (b) fractional density

difference (f.d.d) of n-6 Lennard-Jones potentials at T = 1.0 (�) and T = 2.74

(O)………………………………………………………………………………....154

Figure 4.5Figure 4.5Figure 4.5Figure 4.5 Triple point properties of n-6 Lennard-Jones potentials as a function

of 1/n. Shown are (a) triple point temperatures (�), (b) pressures (�) and (c)

liquid (�) and solid (�) phase densities. The lines represent the least-squares

fit of the data given by Eq. (4.7)………………………………………………..163

Figure 4.6Figure 4.6Figure 4.6Figure 4.6 Comparison of the liquid phase radial distribution functions for a 12-

6 Lennard-Jones potential (solid line) and a 7-6 Lennard-Jones potential

(dashed line) in the liquid phase at � 2.74 and ρ � 1.00……………………164

Figure 4.7Figure 4.7Figure 4.7Figure 4.7 Comparison of the (a) first maxima and the (b) first minima at the

freezing point for n-6 Lennard-Jones fluids, where n = 7 (solid line), 9 (dashed

line) and 12 (dotted line). T = 2.74 and ρ = 1.339, 1.218 and 1.116 for n = 7, 9

and 12, respectively……………………………………………………………….165


densities and (c) solid densities for the WCA potential calculated in this work

(�) with data from de Kuijper et al. (1990) (). The errors are approximately

equal to the symbol size………………………………………………………….178

xxi

Figure 5.2Figure 5.2Figure 5.2Figure 5.2 Comparison of the relative percentage difference of pressures (�),

liquid densities (�) and solid densities (�) as a function of temperature (de

Kuijper et al., 1990)………………………………………………………………179

Figure 5.3Figure 5.3Figure 5.3Figure 5.3 The solid-liquid coexistence (a) pressure (�), (b) freezing (upper)

and melting line (lower) density (�) as a function of temperature………….180

Figure 5.4Figure 5.4Figure 5.4Figure 5.4 Comparison of (a) r.d.d and (b) f.d.c WCA data as a function of

temperature obtained in this work (�) with the LJ (Ο) data from Agrawal and

Kofke (1995c)……………………………………………………………………192

Figure 5.5Figure 5.5Figure 5.5Figure 5.5 Comparison of (a) the overall pressure-temperature and the (b)

pressure-low temperature behavior of the WCA fluid calculated in this work

(�) with literature data (Agrawal and Kofke, 1995c) for LJ potential (---). The

LJ data were supplemented by calculations using Eq. (1) from van der Hoef

(2000)……………………………………………………………………………183

Figure 5.6Figure 5.6Figure 5.6Figure 5.6 Comparison of the solid-liquid coexistence pressure (�) of WCA

potential with 12-6 LJ potential (---) (Agrawal and Kofke, 1995c) as a function

of reciprocal temperature. …………………………………………………….185

Figure 5.7Figure 5.7Figure 5.7Figure 5.7 Comparison of molecular simulation data for the compressibility

factors of the WCA fluid obtained in this work (�) with calculations using the

Heyes and Okumura (Ο), Verlet and Weis EOS (�) and Kolafa and Nezbeda

EOS (�) equations of state. The solid line represents calculations of the

reparametrized Heyes and Okumura equation reported here…………………189

Figure 5.8Figure 5.8Figure 5.8Figure 5.8 (a) Comparison of radial distribution functions at T = 1.0 for the WCA fluid at freezing (solid line) and melting (dashed line) points. (b) A

typical structure factor curve for the WCA fluid at a freezing point �� 0.98, � 1.15�…………………………………………………………………………….193

xxii

Figure 5.9Figure 5.9Figure 5.9Figure 5.9 Comparison of entropy of fusion obtained in this work (�) for the

WCA fluid with the data for the Lennard-Jones potential (Ο) (Agrawal and

Kofke, 1995c)…………………………………………………………………........194

Figure 5.10Figure 5.10Figure 5.10Figure 5.10 Comparison of volume change, |��|, calculated in this work (�) for

the WCA fluid with the data for the 12-6 Lennard-Jones potential (Ο) (Agrawal

and Kofke, 1995c)…………………………………………………………………195

Figure 6.1Figure 6.1Figure 6.1Figure 6.1 Low-density side of the GCM solid state at T = 0.006. (a) Pressure

as a function of strain-rate at different constant densities. Shown are results for

densities of 0.1296 (�), 0.1297 (�), 0.1298 (�), 0.1299(�), 0.1300 (),

0.1301(⊳), 0.1302 (�), 0.1303(�), 0.1304 (�), 0.1305( ). Entry into the two-

phase solid-liquid region is clearly seen by the sudden drop in pressure at zero

strain-rate. (b) Pressure as a function of density for different strain-rates �� 0.0 (�), �� 0.001 (), �� 0.002 (�), all in the stable liquid state and its

metastable extension, and �� 0.0 (�) in the stable solid state and its

metastable extension. The symbols fp and mp refer to the freezing and the

melting point, respectively. A dashed arrow marks the jump in the equilibrium

pressure ……………………………………………………………………………201

FFFFigure 6.2igure 6.2igure 6.2igure 6.2 High-density side of the GCM solid state at T = 0.004. (a) Pressure

as a function of strain-rate at different densities. Shown are results for the

densities of 0.5424 (�), 0.5425 (�), 0.5426 (�), 0.5427 (�), 0.5428 (),


phase solid-liquid region is clearly seen by the sudden jump in pressure at zero

strain-rate. (b) Pressure as a function of density for different strain-rates �� 0.0 (�), �� 0.001 (), �� 0.002 (�), all in the stable liquid state and its

metastable extension and �� 0.0 (�) in the stable solid state and its



pressure……………………………………………………………………………202

Figure 6.3Figure 6.3Figure 6.3Figure 6.3 Phase diagram of the GCM fluid showing the freezing () and

melting lines (�) obtained in this work. The fps (�) reported by Prestipino et

xxiii

al. (2005) and freezing thresholds (�) predicted by the Hansen-Verlet rule

(Saija et al., 2006) are also illustrated…………………………………………205

Figure 6.4Figure 6.4Figure 6.4Figure 6.4 Comparison of the relative percentage difference in freezing (�) and

melting � densities on the low-density side and freezing (Ο) and melting (�)

densities on the high-density side at different temperatures obtained in this

work �ρ�� with data reported in Prestipino et al. (2005) � ρ!"#�$�%�&'�…207

Figure 6.5Figure 6.5Figure 6.5Figure 6.5 The solid-liquid density gap ∆�)* � +�) � �*+ on the low-density (�)

and high-density (�) sides of the GCM phase diagram as a function of

temperature………………………………………………………………………..208

Figure 7.1Figure 7.1Figure 7.1Figure 7.1 Phase diagram of the GCM fluid showing the state points (�)

covered by the NEMD simulations reported in this work…………………….211

Figure 7Figure 7Figure 7Figure 7.2.2.2.2 Strain-rate dependent internal energy per particle as a function of

strain-rate for different constant temperatures (as indicated on the lines) at a

density of ρ � 0.1. The sharp drop after the increase in energy indicates the

occurrence of the string phase…………………………………………………...213

Figure 7.3Figure 7.3Figure 7.3Figure 7.3 Shear viscosity isochors as a function of strain-rates for (a) T =

0.015, (b) T = 0.02, (c) T = 0.025 and (d) T = 0.03. The isochors were

obtained for � � 0.1 (�), 0.2 (�), 0.3 (�), 0.4 (�) and 1.0 (). Note the

anomalous behaviour at ρ , 0.3. The lines are for guidance only……………215

Figure 7.4Figure 7.4Figure 7.4Figure 7.4 Shear viscosity isochors as a function of strain-rates for (a) T = 0.06,

(b) T = 0.08, (c) T = 0.1 and (d) T = 0.3. The isochors were obtained for ρ =

0.1 (�), 0.2 (�), 0.3 (�) and 0.4 (�). The lines are for guidance only…….218

FigureFigureFigureFigure 7.57.57.57.5 (a) Shear viscosity at � � 1.0 versus strain-rate for various

temperatures as indicated. The lines are for guidance only. (b) Shear viscosity

as a function of temperature for four different densities……………………. ..219

xxiv

Figure 7.6Figure 7.6Figure 7.6Figure 7.6 Shear viscosity as a function of strain-rate at T = 0.015 and � �0.01. The lines indicate the fit to the simulation data (�) using Eq. (7.1) with . = 1/2 (dashed line) and 0.75 (solid ine)…………………………………222

Figure Figure Figure Figure 7.77.77.77.7 Comparison of the relative percentage difference of zero-shear

viscosities obtained from this work with Green-Kubo (GK) calculations

(Mausbach and May, 2009) as a function of temperature and densities of ρ �

0.1 (�), 0.2 (�), 0.3 (�), 0.4 (�) and 1.0 ()……………………………222

Figure 8.1Figure 8.1Figure 8.1Figure 8.1 Illustration of the range of state points ��, , �� for which NEMD

simulations were performed to obtain data for the steady-state equation of

state. Data were obtained for a total of 660 state points…………………..229

Figure 8.2Figure 8.2Figure 8.2Figure 8.2 (a) Comparison of equilibrium molecular simulation pressure data

(Ο) for Lennard-Jones fluid at � � 0.73 with values from Eq. (8.2) (solid line)

and (b) the corresponding relative percentage difference (�)……………..238

Figure 8.3Figure 8.3Figure 8.3Figure 8.3 Nonequilibrium steady-state contributions to (a) p0� and (b) 12� for a Lennard-Jones fluid as a function of temperature at four different densities..239

Figure 8.4Figure 8.4Figure 8.4Figure 8.4 Comparison of the relative percentage difference of steady-state

compressibility obtained from this work with the values calculated from Eq.

(8.1) as a function of strain-rate (a) for the temperature range T = 0.70 - 1.75

and (b) for the density range 0.73 - 0.95. Shown are (a) � � 0.73 (�), 0.8442

(�), 0.895 (�), 0.95 (�); (b) T = 0.7 (�), 0.90 (�), 1.10 (�), 1.35 (�), 1.75

(⊳)……………………………………………………………………………….241



(8.1) as a function of density. Shown are �T, γ� � � (0.75, 0.1) (�); (0.90, 0.3)(�);

(1.05, 0.5) (�); (1.20, 0.7) (�); (1.35, 0.9) (⊳); (1.75,1.1) (�)……………242



xxv

(8.1) as a function of temperature. Shown are �ρ, γ� � � (0.73, 0.2) (�); (0.8442,

0.4) (�); (0.895, 0.6) (�); (0.95, 0.8) (�)……………………………………242

Figure 8.7Figure 8.7Figure 8.7Figure 8.7 Shear viscosity for a 12-6 Lennard-Jones fluid as a function of

nonequilibrium steady-state compressibility obtained from NEMD simulation

(Ο) reported here and values obtained from Eq. (8.10) (solid lines) for T �0.70 � 1.75 and (a) ρ � 0.73, γ� � 0.1, (4�0, 5� � 1.5242, η�0, W� � -2.4179, Y =

-10.5966) and (b) ρ � 0.895, γ� � 0.4, (�4�0, 5� � 2.3128, η�0, W� � 0.6770, Y =

-0.3253)…………………………………………………………………………..246

Figure 8.8Figure 8.8Figure 8.8Figure 8.8 Comparison of the shear viscosity for a 12-6 Lennard-Jones fluid as a

function of nonequilibrium steady-state compressibility obtained from NEMD

simulations (Ο) at ρ � 0.8 reported here with values obtained from Eq. (8.6)

(solid lines) for �� = 0.3-1.1 at (a) � 1.0 � η�0, 5� � 3.4143, η�0, W� � -0.1224,

Y = -0.0760� and (b) T = 1.20 �η�0, W� � 3.24193, η�0, W� � -0.1120, Y = -

0.0879)…………………………………………………………………………..247

Figure 8.9Figure 8.9Figure 8.9Figure 8.9 Comparison of shear viscosity simulation data (Ο) reported here for

the 12-6 Lennard-Jones fluid as a function of pressure at four different densities

and constant strain-rates of (a) 0.3 �η�0, γ� � � 1.4378, 4�0, γ� � � -0.0335, Y = -

0.2925�, (b) 0.5 � η�0, γ� � � 2.6617, η�0, γ� � � -0.0653, Y = -0.0536�, (c) 0.7 �η�0, γ� � � 2.9511, η�0, γ� � � 0.0030, Y = -1.45� and (d) 1.0 �η�0, γ� � � 2.9294,

η�0, γ� � � 0.0499, Y = -0.0621� with values obtained from Eq. (8.15) (solid lines).

The data cover the temperature range of T = 0.70 to T = 1.75…………..249

Figure 8.10Figure 8.10Figure 8.10Figure 8.10 Comparison of shear viscosity simulation data (Ο) reported here for

the 12-6 Lennard-Jones fluid at T = 1.0 as a function of density with values

obtained from Eq. (8.17) (solid lines). Results are shown for strain-rates of 0.2 � η�0, �� 25.9563, η�0, γ� � � -72.1191, Y = -1.4613�, 0.4 �η�0, γ� � � 15.0339, η�0, γ� � � -43.1501, Y = -1.5513�, 0.6 � η�0, γ� � � 10.7986, η�0, γ� � � -31.5572, Y =

-1.6155� and 0.8 � η�0, γ� � � 8.7105, η�0, γ� � � -25.6701, Y = -1.6591�………250

Figure 8.11Figure 8.11Figure 8.11Figure 8.11 Comparison of experimental shear viscosities of squalane (�) at T � 20: and various strain rates and densities with values obtained from Eq.

xxvi

(8.13) (solid lines). Results are shown for strain-rates of 223s-1 (4�0, �� =4.17 ×

1010 cPcm3/g, 4�0, �� = -8.55×1010 cPcm3/g, Y = -1.025 cm3/g), 890s-1 (4�0, �� =

1.92 × 1010 cPcm3/g, 4�0, �� = -3.96×1010 cPcm3/g, Y = -1.029 cm3/g) and

1780s-1 (4�0, �� = 1.14 × 1010 cPcm3/g, 4�0, �� = -2.36×1010 cPcm3/g, Y = -1.03

cm3/g). In all cases the AAD is less than 1%............................................252

Figure 8.12Figure 8.12Figure 8.12Figure 8.12 Comparison of experimental shear viscosities of squalane (�) at T � 20: as a function of pressure obtained from Eq. (8.11) (solid lines).

Results are shown for strain-rates of 223s-1 (4�0, �� = 1.94 × 1017 cP/Pa, 4�0, �� = -4.99×1017 cP/Pa, Y = -1.3×109 Pa-1), 890s-1 (4�0, �� = 7.13 × 1016

cP/Pa, 4�0, �� = -1.96×1017 cP/Pa, Y= -1.407×109 Pa-1) and 1780s-1 (4�0, �� =

3.96 × 1016 cP/Pa, 4�0, �� = -1.10×1017 cP/Pa, Y = -1.448×109 Pa-1). In all cases

the AAD is less than 1%.............................................................................253

Figure 8.13Figure 8.13Figure 8.13Figure 8.13 Zero-shear viscosity isotherms of monatomic fluids as a function of

pressure. (a) For neon, isotherms presented are T = 26 K (�), 100 K (�), 500

K ( �), 1000 K (�), 1300 K (⊳). (b) For argon, isotherms presented are T =

90 K (�), 500 K (�), 1000 K ( �), 1300 K (�). (c) For krypton, isotherms

presented are T = 120 K (�), 500 K (�), 1000 K ( �), 1300 K (�). (d) For

xenon, isotherms presented are T = 170 K (�), 500 K (�), 1000 K ( �), 1300

K (�). In all cases solid lines represent the model the model of Eq. (8.13) with

the fitting parameters and statistics illustrated in Table B.1 (Appendix B)..255

Figure 8.14Figure 8.14Figure 8.14Figure 8.14 Zero-shear viscosity of water as a function of pressure (in the range

from 40 to 100 MPa) at 273 K (�) and 773 K (Ο). Experimental data taken

from Watanabe and Dooley (2003). The convex and concave behavior of water

viscosity towards the pressure axis can be seen from these experimental

data………………….. ……………………………………………………………258

Figure 8.15Figure 8.15Figure 8.15Figure 8.15 Zero-shear viscosity isotherm of water as a function of pressure (in

the range from 0.5 to 100 MPa) at 373 K (Ο) and the least squares fit (—) of

the model (Eq. (8.13)). Experimental data taken from Watanabe and Dooley

(2003). Any exponential or quadratic type viscosity model cannot fit this

viscosity behavior of water………..……………………………………………..258

xxvii

Figure 8.16Figure 8.16Figure 8.16Figure 8.16 Zero-shear viscosity isotherms of water as a function of pressure in

the high pressure region. Shown are the isotherms for (a) T = 273 K (�), 298

K (�), 323 K ( �), 348 K(�), 373 K (⊳), 423 (�), 573 (); (b) T = 523 K

(�), 573 K (�), 623 K ( �), 648 K(�); (c) T = 673 K (�), 698 K (�), 723 K

( �), 748 K (�), 773 K (⊳); (d) T = 823 K (�), 873 K (�), 923 K ( �), 973

K (�), 1023 K (⊳), 1073 (�). In all cases solid lines represent the model of Eq.

(8.13) with the fitting parameters and statistics illustrated in Table B.2

(Appendix B)……………………………………………………………………259

Figure 8.17Figure 8.17Figure 8.17Figure 8.17 Zero-shear viscosity isotherms of water as a function of pressure in

the low pressure region. Shown are the isotherms for (a) T = 523 K (�), 573 K

(�), 623 K ( �), 648 K (�); (b) T = 673 K (�), 698 K (�), 723 K ( �), 748

K (�), 773 K (⊳). In all cases solid lines represent the model of Eq. (8.13) with

the fitting parameters and statistics illustrated in Table B.2 (Appendix

B)…………………………………………………………………………………260

Figure 8.18Figure 8.18Figure 8.18Figure 8.18 Zero-shear viscosity isotherms of Carbon dioxide as a function of

pressure. Shown are the isotherms for (a) T = 1100 K (�), 1200 K (�), 1300

K ( �), 1400 K(�), 1500 K (⊳); (b) T = 580 K (�), 600 K (�), 620 K ( �),

640 K(�), 660 (), 680 (⊳), 700 (�), 800 (�), 900 (�); (c) T = 400 K (�),

420 K (�), 440 K ( �), 460 K (�), 480 K (⊳), 500 K (�), 520 K (), 540 K

( ), 560 K (�) ; (d) T = 220 K (�), 240 K (�), 260 K ( �), 280 K (�), 300

K (⊳), 320 (�), 340 K (), 360 K ( ), 380 K (�). In all cases solid lines

represent the model of Eq. (8.13) with the fitting parameters and statistics

illustrated in Table B.2 (Appendix B)………………………………………….261

Figure 8.19Figure 8.19Figure 8.19Figure 8.19 Zero-shear viscosity isotherms (Set-I) of hydrocarbons as a function

of pressure. (a) For n-C12, isotherms presented are T = 310.78 K (�), 333 K

(�), 352.44 K ( �), 371.89 K(�), 388.0 K (⊳), 408 (�); (b) For n-C15,

isotherms presented are T = 311.93 K (�), 334.15 K (�), 353.59 K ( �), 373.4

K(�), 389.15 (⊳), 409.15 (�); (c) For n-C18, isotherms presented are T = 333

K (�), 352.44 K (�), 371.89 K ( �), 388 K (�), 408 K (⊳); (d) For cis-

Decahydro-napthalene, isotherms presented are T = 288.56 K (�), 310.78 K

xxviii

(�), 333 K ( �), 352 K (�), 371.89 K (⊳). In all cases solid lines represent the

model of Eq. (8.13) with the fitting parameters and statistics illustrated in

Table B.2 (Appendix B)………………………………………………………262

Figure 8.20Figure 8.20Figure 8.20Figure 8.20 Zero-shear viscosity isotherms (Set-II) of hydrocarbons as a function

of pressure. (a) For 7-n-Hexyltridecane, isotherms presented are T = 310.78 K

(�), 333 K (�), 371.89 K ( �); (b) For 9-n-Octylheptadecane, isotherms

presented are T = 310.78 K (�), 333 K (�), 352.44 K ( �), 371.89 K(�), 388

(⊳); (c) For 11-n-Decylheneicosane, isotherms presented are T = 310.78 K (�),

333 K (�), 371.89 K ( �), 408 K (�); (d) For 13-n-Dodecylhexacosane,

isotherms presented are T = 310.78 K (�), 334 K (�), 371.89 K ( �), 408 K

(�). In all cases solid lines represent the model of Eq. (8.13) with the fitting

parameters and statistics illustrated in Table B.2 (Appendix B)………..263

xxix

List of TablesList of TablesList of TablesList of Tables

Table 2.1Table 2.1Table 2.1Table 2.1 The relationship of reduced unit with real unit in terms of Lennard-

Jones σ and ε parameters……………………………………………………......78

Table 3.1Table 3.1Table 3.1Table 3.1 Sources of solid-liquid phase equilibria data of 12-6 Lennard-Jones

fluid……………………………………………………………………………....122

Table 3.2Table 3.2Table 3.2Table 3.2 Common temperatures found in literature to validate 12-6 LJ solid-

liquid phase coexistence properties……………………………………………123

Table 3.3Table 3.3Table 3.3Table 3.3 System size dependencies of the solid-liquid coexistence properties of

12-6 Lennard-Jones fluid at T = 1.0 obtained using the GWTS algorithm..128

Table3.4Table3.4Table3.4Table3.4 Solid-liquid coexistence properties calculated from the GWTS

algorithm. Gibbs free energy is calculated via Lennard-Jones equation of state of

Johnson et al. (1993).....................................................................................134

TableTableTableTable 4.14.14.14.1 Molecular simulation data for the solid-liquid coexistence properties of

n-6 Lennard-Jones fluids. The statistical uncertainty is given in brackets…149

Table Table Table Table 4.24.24.24.2 Melting line shifts of n-6 Lennard-Jones potentials with respect to 12-

6 Lennard-Jones potential along the temperature axes form three and four

parameters Simon-Glatzel equations. Values in brackets are errors………..157

TableTableTableTable 4.34.34.34.3 Parameters for the scaling behaviour (Eq. (4.4)) of pressure as a

function of inverse temperature for n-6 nLennard-Jones potentials. Errors are

given in parenthesis……………………………………………………………..158

Table 4.4Table 4.4Table 4.4Table 4.4 Parameters for the polynomial fit (Eq. 4.6) for the coexisting liquid

and solid densities for n-6 Lennard-Jones potential………………………….159

Table 4.5Table 4.5Table 4.5Table 4.5 Comparison of triple point properties for the 12-6 Lennard-Jones fluid

obtained from molecular simulation studies. Errors are given in brackets…160

Table 4.6Table 4.6Table 4.6Table 4.6 Estimated triple point properties for n-6 Lennard-Jones potentials.162

Table 4.7Table 4.7Table 4.7Table 4.7 Summary of parameters for melting and freezing rules for n-6

Lennard-Jones potentials at T = 1.0 and the melting or freezing densities….164

xxx

Table Table Table Table 5.15.15.15.1 Coexistence pressures and melting and freezing densities of a WCA

fluid at T � 1.0 for different number of particles………………………………173

Table 5.2Table 5.2Table 5.2Table 5.2 Solid-liquid phase coexistence properties of the WCA potential at low

to intermediate temperatures. Values in brackets represent the uncertainty in

the last digit……………………………………………………………………175

Table 5.3Table 5.3Table 5.3Table 5.3 Solid-liquid phase coexistence properties of the WCA potential at high

temperatures. Values in parentheses represent the uncertainty in the last digit.

………………………………………………………………………………….177

Table 5.4Table 5.4Table 5.4Table 5.4 Comparison of WCA solid-liquid coexistence data at T = 1.0…179

Table 5.5Table 5.5Table 5.5Table 5.5 Invariants of the Lindemann (1910), Raveché et al. (1974) and

Hansen and Verlet (1969) melting or freezing rules as a function of coexistence

temperature…………………………………………………………………….190

Table 5.6Table 5.6Table 5.6Table 5.6 Parameters of Simon’s equation and van der Putten’s relation both

for WCA and 12-6 LJ potentials obtained from the least square fit of solid-

liquid coexistence pressure data and volume jump data as a function of

temperature, respectively……………………………………………………….195

Table 6.1Table 6.1Table 6.1Table 6.1 System size dependency of the freezing density of the GCM fluid at T

= 0.006 obtained using the GWTS algorithm………………………………..200

Table 6.2Table 6.2Table 6.2Table 6.2 Freezing and melting densities for the low-density and high-density

sides of the solid state of the GCM fluid obtained using the GWTS

algorithm…………………………………………………………………………204

Table 7.1Table 7.1Table 7.1Table 7.1 Maximum safe strain-rates at different densities and temperatures.

These strain-rates avoid string phases and shear-induced ordering. For state

points marked with a ‘minus’ the drop in the internal energy profiles occurs at

strain-rates higher than a dimensionless value of 9.0. This situation occurs for

all densities with temperatures greater than T = 0.30………………………..214

xxxi

Table 7.2Table 7.2Table 7.2Table 7.2 Parameters of temperature dependent viscosity model of GC fluid.

Errors are in the brackets………………………………………………………..221

Table 8.1Table 8.1Table 8.1Table 8.1 Values of =>, p0� and α appearing in Eq. (8.2) for three different

densities and a range of temperatures. The values were obtained from a least-

squares fit of NEMD simulation data for a range of strain-rates (detailed in the

text). The statistical uncertainty in the last digit is given in brackets……235

Table 8.2Table 8.2Table 8.2Table 8.2 Values of E>D'&E, E0� and α appearing in Eq. (8.2) for three different



text). The statistical uncertainty in the last digit is given in brackets…….236

Table 8.3Table 8.3Table 8.3Table 8.3 Parameters for the nonequilibrium steady-state equation of state

regressed from the simulation data of this work…………………………….237

Table A.1Table A.1Table A.1Table A.1 Solid-liquid coexistence properties of full Lennard-Jones potential

obtained in this work using the GDI algorithm starting with the coexistence

properties obtained from GWTS algorithm at T = 2.74 (Appendix A)…..274

Table B.1Table B.1Table B.1Table B.1 Pressure dependent viscosity model parameters for monatomic real

fluids and the relevant statistics (Appendix B)……………………………….276

Table B.2Table B.2Table B.2Table B.2 Pressure dependent viscosity model parameters for complex molecular

fluid and the relevant statistics (Appendix B)…………………………………277

xxxii

NotationNotationNotationNotation

AbbreviationsAbbreviationsAbbreviationsAbbreviations

AAD Average Absolute Deviation

A&K Agrawal and Kofke

B&F Barroso and Ferreira

E Errington

EMD Equilibrium Molecular Dynamics

EOS Equation of State

EXEDOS Extended Ensemble Density-of-State Monte Carlo Method

fcc face-centered-cubic

f.d.c fractional density change

fp freezing point

G Ratio of the first maximum to the nonzero first minimum

GC Gaussian Core

GCM Gaussian Core Model

GWTS Ge, Wu, Todd and Sadus

GDI Gibbs-Duhem Integration

HS Hard Sphere

LJ Lennard-Jones potential

MC Monte Carlo

MD Molecular Dynamics

M&P Mastny and de Pablo

xxxiii

mp melting point

M&S Morris and Song

M&W McNeil-Watson and Wilding

NEMD Non-Equilibrium Molecular Dynamics

NpT Isothermal Isobaric Ensemble

NVE Canonical Ensemble

NVT Isothermal Isochoric Ensemble

R2 Squared Correlation Coefficient

RMS Raveché-Mountain-Streett

r.d.d relative density difference

WCA Weeks-Chandler-Anderson potential

Symbols Symbols Symbols Symbols ---- Latin alphabetLatin alphabetLatin alphabetLatin alphabet

1 potential energy per particle

1FGH) configurational energy 1>FGH) equilibrium contribution of configurational energy

I force acting between atoms

IJ the vector force exerted on atom K IJL the vector force exerted by M on atom K NJ probability of i-th atom in the velocity distribution

OP Boltzmann constant

QR length of the simulation box in the O-th direction

QS length of the simulation box in the �-direction

xxxiv

QT length of the simulation box in the U-direction

QV length of the simulation box in the z-direction

W number of time steps

X mass of a an atom/particle

XJ mass of the i-th atom

Y number of atoms/particles

� integer ranging from 7 to 12.

= hydrostatic pressure

=Z[ triple point pressure \ overall momentum of a system

] pressure tensor

]ST xy element of the pressure tensor

]SS xx element of the pressure tensor

]TT xx element of the pressure tensor

]VV xx element of the pressure tensor

^_ generalized momentum in N-dimension

^ peculiar momentum

=T U-component of ^

=SJ �-component of ^ for particle i

=TJ U-component of ^ for particle i

� rate of change of momentum

^J momentum of the i-th atom

`_ generalized coordinate in N-dimension

xxxv

a distance between two atoms

aJL magnitude of the vector distance between atoms K and M bJL pair separation vector aSJL

�-component of the pair separation vector bJL aTJL

U-component of the pair separation vector bJL aVJL

c-component of the pair separation vector bJL aSJ position of the i-th atom along the � coordinate

aTJ position of the i-th atom along the U coordinate

aVJ position of the i-th atom along the c coordinate a�SJ velocity of the i-th atom along the � coordinate

a�TJ velocity of the i-th atom along the y coordinate

a�VJ velocity of the i-th atom along the z coordinate

b position vector of an atom

aT U-component of b bJ position vector of the i-th atom

bJd)Ze[ position of the i-th particle after the move in Lees-

Edwards periodic boundary condition

bJfe)G[e position of the i-th particle before the move in Lees-


b� Jd)Ze[ velocity of the i-th particle after the move in Lees-


b� Jfe)G[e velocity of the i-th particle before the move in Lees-


xxxvi

b� velocity vector of an atom

temperature

Z[ triple point temperature

g time

∆g integration time step

h interaction potential energy

hJL interaction energy between particles K and M hZ truncated interaction potential energy hZij truncated-shifted interaction potential energy hZij) shifted force interaction potential energy

k volume of the simulation box

l local fluid velocity

mS �-component of the local fluid velocity l

n measurable physical quantity

Symbols Symbols Symbols Symbols ---- Greek alphabetGreek alphabetGreek alphabetGreek alphabet

. �, U, c o characteristic energy parameter of LJ

4 shear viscosity

p thermostatting multiplier/coefficient

�� strain rate

q real number from the division of 22/7

� number density

�r freezing point

xxxvii

�j melting point

�rJs,Z[ triple point liquid side density �jGr,Z[ triple point solid side density t characteristic length parameter of LJ

u simulation time

uv relaxation/correlation time

w correlation length

38

Chapter 1Chapter 1Chapter 1Chapter 1 Introduction Introduction Introduction Introduction

Molecular simulation techniques based on postulates of statistical mechanics are

critical tools for the justification of theoretical development and experimental

outcomes. For several decades, molecular simulation has played a major role in

physical and chemical research in providing exact data for comparison to the

results of statistical mechanical theories (Allen and Tildesley, 1987). “One of the

fascinating attributes of computer simulation is the possibility of obtaining

computer generated experimental data for systems that do not occur in nature

yet are amenable to theoretical analysis” (Evans et al., 1984). Molecular

simulation techniques based on linear and nonlinear statistical mechanics are

playing a crucial role in the modelling of fluid phase equilibria and

thermophysical properties of chemical process design. These methods provide an

intermediate layer between direct experimental measurements and engineering

models (Sheng et al., 1995).

1.1 1.1 1.1 1.1 MotivationsMotivationsMotivationsMotivations

This work is motivated by the following three aspects:

(i) Simulation algorithms that work well for the liquid-vapour

phase transitions are not always adequate for the study of solid-liquid phase

Introduction

39

transitions. For example, Gibbs-Ensemble Monte Carlo simulation is usually

limited to vapour-liquid phase transitions because of the insufficient acceptance

probability of particle transfer between liquid and solid phases. The Gibbs-

Duhem integration (GDI) (Agrawal and Kofke, 1995c; Kofke, 1993a) algorithm

can calculate the solid-liquid phase transition but it requires at least one known

starting point. Not being a self starting the GDI method cannot be used for the

study of solid-liquid phase transitions for systems without known coexistence

temperature, pressure, and densities. Moreover, with an erroneous initial

condition GDI method also propagates the error in a systematic way (Agrawal

and Kofke, 1995a; Kofke, 1993a). To address the challenge of simulating solid-

liquid phase transitions several other simulation algorithms have also been

developed in last fifteen years (Agrawal and Kofke, 1995c; Hansen, 1970;

Hansen and Verlet, 1969; Raveche et al., 1974; Streett et al., 1974; Ladd and

Woodcock, 1977; Ladd and Woodcock, 1978a; Chokappa and Clancy, 1987a;

Chokappa and Clancy, 1987b; Hsu and Mou, 1992; Barroso and Ferreira, 2002;

Morris and Song, 2002; Ge et al., 2003b; Errington, 2004; Mastny and de Pablo,

2005; Mastny and de Pablo, 2007; McNeil-Watson and Wilding, 2006). Most of

them have only been used with the 12-6 Lennard-Jones potential. The

versatility of these methods was not being tested for different model potentials.

The Monte Carlo methods often suffer sufficient sampling problems with the

change of model potentials (Kiyohara et al., 1996).

Introduction

40

It is widely believed that repulsive part of the intermolecular potentials plays a

significant role in the solid-liquid phase transitions (Kirkwood, 1952; Alder and

Wainwright, 1962; Longuet-Higgins and Widom, 1964; Hansen and Schiff, 1973;

Stillinger, 1976; Khrapak and Morfill, 2009). Simple models such as hard-sphere

potential, inverse potential and 12-6 Lennard-Jones potential are no way

capable of revealing the complete picture of the solid-liquid phase equilibria for

varying repulsive components. But such studies on solid-liquid equilibria are

rarely found in the literature. The absence of solid-liquid data can be partly

attributed to the difficulty of particle insertion between dense phases common

to many molecular simulation techniques. This difficulty has been recently

eliminated by a molecular dynamics algorithm that combines aspects of

equilibrium and nonequilibrium simulation techniques (Ge et al., 2003b).

(ii) Most of the theoretical and simulation studies on non-

Newtonian behavior of fluids have largely focused on unbounded potentials.

However, recently bounded potentials such as the Gaussian core model (GCM)

have become a focus of attention because they appear to be useful in the

description of soft condensed matter physics (Likos, 2001). Recent studies

(Mausbach and May, 2006; Mausbach and May, 2009) indicate that the

equilibrium transport properties of the GCM fluid show anomalous behaviour at

some state points. For example, at constant temperature but increasing density,

the diffusivity increased and the shear viscosity decreased, violating the Stokes-

Introduction

41

Einstein relation (May and Mausbach, 2007). The occurrence of equilibrium

transport anomalies in the GCM fluid gives rise to the question of whether or

not the strain-rate dependent shear viscosity is also anomalous.

(iii) Conventional equilibrium equations of state cannot be used

for either nonequilibrium or nonequilibrium steady-state processes. However, it

is feasible to formulate an equation of state specifically for nonequilibrium

steady-states. Evans and Hanley (Evans and Hanley, 1980a; Evans and Hanley,

1980b; Hanley and Evans, 1982; Evans, 1983; Evans et al., 1984; Romig and

Hanley, 1986) have used molecular simulation data to devise equations of state

for both the pressure and energy of a steady-state fluid. In contrast to equations

of state based on extended irreversible thermodynamics (EIT) (Jou et al., 2001)

principles, which use a quadratic correction to the energy, these simulation-

based equations use a strain-rate exponent of α = 3/2. The α = 3/2 exponent is

consistent with both mode-coupling theory (Evans et al., 1984) and simulation

data at the triple point of the Lennard-Jones fluid. However, nonequilibrium

molecular dynamics (NEMD) simulation data (Ge et al., 2003a; Marcelli et al.,

2003a; Marcelli et al., 2001; Ge et al., 2001) for the pressure of a fluid under

shear away from the triple-point indicates that the value of α varies

continuously from ~1.2 to ~2 depending both on temperature and density. A

steady-state equation of state applicable to wide ranges of temperatures,

densities and strain rates is required.

Introduction

42

Shear viscosity is a complex function of pressure, temperature, density (or

volume), molecular structure (or degree of branching), molecular weight, and

strain rate (if external field is applied). For zero-shear case, temperature and

pressure dependent viscosity behavior (So and Klaus, 1980; Fresco et al., 1969)

is well known. In contrast, for applied strain rates, pressure dependence of shear

viscosity is rarely studied in literature. Although shear viscosity as a function of

strain rate is widely studied, few attempts have been made to correlate pressure,

temperature and density with strain rates.

1.2 Aims1.2 Aims1.2 Aims1.2 Aims

The first aim of this dissertation is to identify an appropriate algorithm capable

of producing benchmark data and handling complex potentials. There remain

wide discrepancies in the solid-liquid phase equilibria data of 12-6 Lennard-

Jones potential which is one of the most studied potentials in molecular

simulations. A Chapter (Chapter 3) is devoted finding the most probable causes

behind the variation of data in literature and providing benchmark data set

with rigorous analysis. The effects of system size and various truncation and

shifting schemes on the solid-liquid phase equilibria are extensively investigated

for the sake of benchmarking 12-6 Lennard-Jones solid-liquid coexistence data.

In this dissertation, two different classes of potential models will be used to

study the solid-liquid equilibria. The first category of the potential is so called

Introduction

43

“unbounded” and the second category is so called “bounded”. The potentials

chosen are as follows:

(i) � � 6 Lennard-Jones family of potentials, where � represents the repulsive part of the potential and can take any value between 7

and 12 whereas ‘6’ represents the attractive part of the potential.

This is an unbounded potential.

(ii) Weeks-Chandler-Andersen (WCA) potential which is a variant of

12-6 Lennard-Jones potential and also an unbounded potential. In

contrast to 12-6 Lennard-Jones potential, it is of purely repulsive

in nature.

(iii) Gaussian core model (GCM) potential is a bounded potential and

allows particles to overlap. In contrast to this potential,

unbounded potentials allow atoms to reorganise themselves in a

crystal structure, during solidification, without overlapping.

Three Chapters (Chapters 2-5) will be devoted for examining the solid-liquid

phase behavior of the above mentioned fluids in conjunction with an analysis of

the efficiency and versatility of the algorithm adopted in this dissertation

compared to other reported works. Different thermophysical properties will also

be studied for these fluids either providing analytical expressions or validating

the melting and freezing rules commonly proposed for 12-6 Lennard-Jones

potential.

Introduction

44

The second aim of this dissertation is determining the shear viscosity of the

Gaussian core model fluid using nonequilibrium molecular dynamics simulation

algorithm. This is the first reported work on the sheared flow of a bounded

potential. The similarities and contrasts of the shear viscosity behavior of GCM

fluid with 12-6 Lennard-Jones potential will be discussed. A comprehensive data

analysis will also be presented along with a comparison to Green–Kubo

Calculations.

The last aim of this dissertation is developing a nonequilibrium equation of

state at its steady state and designing a generic model to predict shear viscosity.

The ability of the generic viscosity model will be tested compared to

experimental data for both zero-shear viscosity and strain rate dependent shear

viscosity. All predictions from the generic viscosity model will be validated via

nonequilibrium steady state EOS.

1.3 Background 1.3 Background 1.3 Background 1.3 Background and Cand Cand Cand Current urrent urrent urrent PPPProgress rogress rogress rogress

1.3.1 1.3.1 1.3.1 1.3.1 SSSSolidolidolidolid----LLLLiquid iquid iquid iquid PPPPhase hase hase hase EEEEquilibriaquilibriaquilibriaquilibria

The review presented in this Section is organised on the basis of algorithms used

for the study of solid-liquid phase equilibria.

Introduction

45

(i) Thermodynamic Integration(i) Thermodynamic Integration(i) Thermodynamic Integration(i) Thermodynamic Integration

The implementation of a thermodynamic integration technique for the

determination solid-liquid phase coexistence has been the most commonly

adopted approach after Hoover and Ree (1968). Hansen and Verlet (1969) first

reported solid-liquid coexistence points at four temperatures with the

corresponding pressures, solid phase densities at melting and liquid phase

densities at freezing. They included a phase transition in the thermodynamics

integration path to calculate the free energy of the solid phase and evaluated

the liquid phase free energy independently. They also estimated the triple point

from these data combining results from vapour-liquid coexistence. Subsequently,

Hansen (1970) published similar results for high temperature regime and

compared them with the results of 12th power soft-sphere potential and made

prediction that at very high temperature Lennard-Jones system will behave like

12th power soft-sphere potential. In that study Hansen considered the attractive

part as a perturbation to the exact Monte Carlo simulation of the repulsive

part. On the freezing line of Hansen and Verlet (1969), Street and Mountain

(1974) and Raveché et al. (1974) repeated the simulations at two temperatures

and found a 20% deviation in pressure. They calculated solid-phase free energy

for 256 particles through the interpolation of a van der Waals loop between

metastable crystal and fluid phases. The most widely used variant of this

approach is the method of Frenkel and Ladd (1984), which utilizes an Einstein

crystal as the reference system. A weakness of this approach is the possibility of

encountering singularities along the common path. In addition, it requires

Introduction

46

rigorous searching method for coexistence parameters is to achieve the condition

of mutual equilibrium in chemical potential.

(ii) Gibbs Ensemble Method(ii) Gibbs Ensemble Method(ii) Gibbs Ensemble Method(ii) Gibbs Ensemble Method

Gibbs Ensemble Monte Carlo (Panagiotopoulos, 1987; Sadus, 1999) eliminated

the need for a physical interface. It uses particle interchange to equilibrate

chemical potential, volume exchange for pressure equilibrium and displacements

for temperature equilibrium. However, the acceptance ratio of particle

interchange for solid-liquid phase transition is too low to achieve chemical

potential equality and so exhibit phase transition. Roughly 1 out of 4000

attempts is successful to accommodate a particle from liquid phase to solid

phase due to the compact organization of solid crystal structure.

(iii) Gibbs(iii) Gibbs(iii) Gibbs(iii) Gibbs----Duhem Duhem Duhem Duhem IIIIntentententegration gration gration gration

In 1993, Kofke introduced the Gibbs-Duhem integration method (Kofke, 1993a;

Kofke, 1993b). In this method one can simulate two phases simultaneously as in

case of Gibbs ensemble method but without a physical interface and particle

transfer between the phases. A Gibbs-Duhem equation describes the mutual

dependence of the state variables in an individual thermodynamic phase (van 't

Hof et al., 2006). Gibbs-Duhem integration facilitates the numerical integration

of the Clapeyron equation. Agrawal and Kofke (1995c) carried out simulation

series starting from temperatures of 2.74 (approximately twice the critical

Introduction

47

point), 1.15 and 0.75 (close to the triple point). They applied Gibbs-Duhem

integration to track the melting point as the potential of interaction was first

changed from purely repulsive to the repulsive part of the LJ potential energy

function, and then to the complete Lennard-Jones energy function. They used

half a box length as the cutoff radius and added standard long range tail

corrections to the pressure and energy for two system sizes. Finite size effects

were not considered during the simulations as they have used only 236 and 932

particles. However, Errington (2004) showed that their coexistence pressures on

the melting line were lower than those of Agrawal and Kofke (1995c). The

slightly inaccurate reference point adopted by Agrawal and Kofke is thought to

be the cause behind the discrepancy of the data generated (Mastny and de

Pablo, 2005; Mastny and de Pablo, 2007). The main limitation of the GDI

method is that known coexistence properties are needed to initiate numerical

integration (van 't Hof et al., 2006). A common feature of both the

thermodynamics and Gibbs-Duhem integration techniques is the requirement to

connect the system of interest to a reference state. In many cases, this

requirement can be difficult or impossible to satisfy, rendering these methods

ineffective.

(iv) (iv) (iv) (iv) Direct Methods Direct Methods Direct Methods Direct Methods

The direct method appears as an alternative to the indirect method and

simulates the coexistence directly with an explicit interface. Though the first

approach of its kind failed (Ladd and Woodcock, 1977; Ladd and Woodcock,

Introduction

48

1978a) due to small system size and short simulation times, melting

temperature have been calculated directly with MD simulations (Morris et al.,

1994; Belonoshko et al., 2000) assuming that both phases of the system will

evolve towards the equilibrium melting point. Morris and Song (2002) carried

out a large scale molecular dynamics simulation to locate solid-liquid equilibria

using a potential which was truncated and splined to zero at cutoff radius. They

carried out simulations on 2000 and 16000 particles using cutoff radius 2.1t, 4.2t and 8.0t. They found no appreciable finite size effects for their method

and found 4% deviation for the melting temperature at 2.1t compared to the

cutoffs 4.2t and 8.0t whereas the difference was less than 1% between 4.2t

and 8.0t. They had to manually adjust box sizes to calculate the hydrostatic

pressure. Also in direct methods one has to construct a simulation cell where an

explicit interface between the coexisting phases will exist. Following are the

limitations of direct methods:

• The system is not necessarily under hydrostatic pressure for a given

geometry. Manual adjustment necessary for calculating the

hydrostatic pressure.

• Construction of simulation cell where an explicit interface between

the coexisting phases will exist. Such an interface can be difficult to

stabilize. Quantify the contribution of such interface to the bulk

phase is difficult.

Introduction

49

• Presence of the interface can produce stress in the system, even when

the box size is optimized for the bulk crystal and liquid densities.

(v) Density of (v) Density of (v) Density of (v) Density of States MStates MStates MStates Methodethodethodethod

A many-replica, multidimensional density-of-states method (Mastny and de

Pablo, 2005) has been proposed for direct simulation of solid-liquid phase

transitions. This method provides a direct estimate of the relative density of

states (energy and volume space relevant to both phases) and thus the relative

energy within these regions, which is subsequently used to determine portions of

the melting curve over wide ranges of pressure and temperature. Mastny and de

Pablo (2005) used a Monte Carlo sampling method with order parameters of

potential energy and volume. They simulated 500 particles with cutoff radius

2.5t but neither finite-size nor cutoff effects were investigated. They noted that

their results are sensitive to the methods used for matching the free energy

functions in the energy and volume space between the equilibrium solid and

liquid regions. Recently, Mastny and de Pablo (2007) have used the relative free

energies from the thermodynamic integrations and corrected for large cutoff

radius as references to connect the Lennard-Jones fluid equation of state of van

der Hoef (2000) with the Lennard-Jones fluid equation of state of Johnson et al.

(1993). Upon applying the references the two equations of state were used to

identify the melting curve. They have used an extended-ensemble density-of-

states method to determine free energy changes for each phase as a continuous

Introduction

50

function of the cutoff radius and the resulting melting temperatures exhibited

oscillatory behaviour with the increase of cutoff radius.

(vi) Phase(vi) Phase(vi) Phase(vi) Phase----Switch Monte Carlo MSwitch Monte Carlo MSwitch Monte Carlo MSwitch Monte Carlo Methodethodethodethod

Errington (2004) carried out simulations on 108-, 256- and 500-particles to

examine the finite-size effect and adopted two cutoff schemes to locate the

coexistence of face centered cubic (f.c.c) crystal and liquid phases of Lennard-

Jones system. Firstly, he used standard long range correction beyond half the

box length both for the liquid and crystal phase and then applied perfect lattice

correction beyond half the box length only for the solid phase. He did not find

any scaling relationship of coexistence pressure with either type of corrections to

extrapolate his results to infinite system-size. McNeil-Watson and Wilding

(2006) also used the phase switch Monte Carlo method described by Errington

to predict freezing properties for a system interaction with LJ potential. They

then reweighted their data to obtain a portion of the melting line. Their results

were quantitatively similar to those of Errington. They explained that the

reason they did not see a 1/Y scaling of the melting temperature was due to the

fact that the choice of a “fluctuating” cutoff at one-half the box length

introduces a coupling between the cutoff and system size. Mastny and de Pablo

(2007) pointed out that it is unclear whether a cutoff radius that changed with

system size had a significant effect. Mastny and de Pablo (2007) also argued

that one would not expect different systems to have any predictable scaling

behaviour.

Introduction

51

(vii) Molecular Dynamic M(vii) Molecular Dynamic M(vii) Molecular Dynamic M(vii) Molecular Dynamic Methods ethods ethods ethods

The melting and crystallisation of the Lennard-Jones system is analysed in

detail (Chokappa and Clancy, 1987a; Chokappa and Clancy, 1987b) by

Chokappa and Clancy and Nosé and Yonezawa (1985). Both studies are

performed with the NpT MD simulation. Chokappa and Clancy (1987b)

determined the mechanical stability of liquids and solids. They demonstrated

the hysteresis loops enclosing liquid, solid, superheated solid and supercooled

liquid through enthalpy-temperature phase diagram.

(viii) (viii) (viii) (viii) GWTS GWTS GWTS GWTS MethodMethodMethodMethod

Ge et al. (2003b) used two entirely different approaches to calculate the solid-

liquid coexistence of 12-6 Lennard-Jones fluid. The first method is based on a

scaling relationship (Ge et al., 2003a): . � y � z{ � |�{ where y � 3.67 }0.04 ; z � 0.69 } 0.03; | � 3.35 } 0.03 ; { is a reduced temperature and �{ is a reduced density. This scaling relationship has been obtained from

nonequilibrium molecular dynamics simulations. Assuming . � 1 one can

calculate melting density for a given temperature within the range 0.687 ~ ~1.26. Ge et al. (2003b) reported a melting density for { � 1 and found it

consistent with the literature. The main limitation of this method is the

determination of exact value of . for the entire range of melting curve. There is

no known theoretical method which can produce accurate and reliable value for

.. However, one solution of this problem is to find the exact extrapolated value

Introduction

52

of . at the melting transition. But we have pointed out the following constraints

in getting extrapolated values:

i) At high pressures and temperatures the nonequilibrium molecular

dynamics simulations will exceed the stability criteria at higher strain

rates. This will necessarily limit the observation of desired scaling

relationship.

ii) Kuksin et al. (2007) found that at a given temperature and density

Lennard-Jones fluid may find itself in two significantly different

metastable phase states, namely disordered and crystalline ones. It is

not clearly defined so far what scaling relationship will be applicable

for solid-like (ordered) and liquid-like (disordered) metastable states.

Moreover, using this method one can only determine the melting density.

Without the solid-liquid coexistence pressure and solid and liquid enthalpies a

comprehensive study on solid-liquid coexistence properties is impossible.

Ge et al. (2003b) also proposed a second method to calculate the solid-liquid

coexistence which is independent of the above mentioned scaling behavior and

combines the best elements from equilibrium and nonequilibrium molecular

dynamics simulation techniques. This is the method we have used in this work

to determine the solid-liquid phase equilibria and will be discussed in details in

Chapter 2. This method has been designated as “GWTS” (after the first letter

Introduction

53

of the surnames of the authors) algorithm in the literature (Mausbach et al.,

2009). Recently, this method has been successfully used to study the effect of

three-body interactions on the solid-liquid phase boundaries of argon, krypton,

and xenon (Wang and Sadus, 2006); to calculate the solid-liquid coexistence and

the triple points of � � 6 Lennard-Jones potential (Ahmed and Sadus, 2009b);

to generate the phase diagram of Weeks-Chandler-Andersen potential for very

low to high temperatures and pressures (Ahmed and Sadus, 2009a); to

determine the solid-liquid phase equilibria of Gaussian core bounded potential

(Mausbach et al., 2009).

1.3.2 1.3.2 1.3.2 1.3.2 Validation of Validation of Validation of Validation of SSSSolidolidolidolid----LLLLiquid iquid iquid iquid PPPPhase hase hase hase EEEEquilibria quilibria quilibria quilibria DDDDataataataata

(i)(i)(i)(i) SSSSolidolidolidolid----LLLLiquid iquid iquid iquid PPPPhase hase hase hase CCCCoexistenceoexistenceoexistenceoexistence from GWTS Algorithm and Its from GWTS Algorithm and Its from GWTS Algorithm and Its from GWTS Algorithm and Its

ReliabilityReliabilityReliabilityReliability

The 12-6 Lennard-Jones potential is often used as part of a force field of

molecular system and also used as part of a molecular mechanics potential

(Sadus, 1999). Complex molecular liquids like methane (Saager and Fischer,

1990), organic liquids (Parsafar et al., 1999), and fullerenes (Caccamo, 1996) are

also studied successfully with LJ potential. In the literature, 12-6 Lennard-Jones

simulation data vary from 12% to 30% (Mastny and de Pablo, 2007). Without

any systematic analysis of the data, it is impossible to figure out the actual

causes behind the differences found in the literature.

Introduction

54

(ii)(ii)(ii)(ii) EffectEffectEffectEffectssss of Tof Tof Tof Truncation and runcation and runcation and runcation and SSSShifting hifting hifting hifting SSSSchemes on chemes on chemes on chemes on SolidSolidSolidSolid----LLLLiquid iquid iquid iquid

CCCCoexistenceoexistenceoexistenceoexistence

Thermodynamic properties of phase transitions, glass transitions, liquid state

study, and amorphus state properties, many-body effects, correlation studies,

rheological behavior, interfacial properties characteristics of bulk system, critical

phenomenon and even the flow behavior of confined systems have been studied

with different versions of Lennard-Jones potential. The 12-6 Lennard-Jones

potential models can be categorized on the basis of potential truncation and

shifting schemes adopted. The most commonly used truncation scheme is fixing

the cutoff radius of spherically symmetric Lennard-Jones potential. The

immediate benefit of using a truncation scheme is the significant reduction of

computation time which is supplemented by the availability of long range

corrections. Even with the tail corrections for cutoff radii aF , 2.5t melting

temperatures could fluctuate (Mastny and de Pablo, 2007) up to 2%. A study

on the effect of the cutoff on thermodynamics properties showed that the

normal long range corrections (Vogelsang and Hoheisel, 1985) are only exact

near the triple point for aF � 4.0. They have used Baxter’s continuation method

(Baxter, 1970) to calculate the truncation effect on the low pressure.

The vapour-liquid phase equilibria of LJ potential with cutoff radii 2, 2.5 and

5.0 have established the fact that the details of the truncation significantly

change the shape of the liquid-vapour phase diagrams (Finn and Monson, 1989;

Introduction

55

Finn and Monson, 1990; Panagiotopoulos, 1994). Smit (1992) calculated the

vapour-liquid phase diagram of truncated and shifted Lennard-Jones potential

with cutoff radius aF � 2.5. They have also found a noticeable difference in the

shape of the respective phase diagrams. The vapour-liquid phase diagram of

shifted-force (Powles et al., 1982; Errington et al., 2002) 12-6 Lennard-Jones

potential also varies profoundly. Critical parameters for truncated, truncated-

shifted, and long-range corrected Lennard-Jones system varies quite significantly

with cutoff radius (Shi and Johnson, 2001). From the solid-liquid phase diagram

of shifted force LJ potential Errington et al. (2003) calculated the triple point

thermodynamics properties and showed that it varies significantly. Recently,

Mastny et al. (2007) calculated the Gibbs free energy of both the solid and

liquid phases as a function of cutoff radius aF � 2.5 at � 0.77 and = � 1.0

using the Extended Ensemble Density-of-States Monte Carlo Method

(EXEDOS) (Kim et al., 2002) simulation. They found that the effect of the

cutoff radius is more pronounced on the solid phase than on the liquid phase

and calculated the effect of cutoff radius on the melting temperature. Most of

the solid-liquid coexistence studies have considered only a single cutoff radius

and also applied long range corrections. We are not aware of any study that

compared the thermodynamics properties at solid-liquid coexistence of truncated

Lennard-Jones (tLJ) potential, truncated and shifted Lennard-Jones (tsLJ)

potential and truncated and shifted-force (tsfLJ) potential in their original

forms. It is, therefore, unclear what the true effect of potential cutoff and

shifting schemes are on the melting line properties of 12-6 LJ system. In this

Introduction

56

dissertation (Chapter 3), we have presented a comprehensive study of various

truncation and shifting schemes on the melting line.

1.3.1.3.1.3.1.3.3333 SolidSolidSolidSolid----Liquid Phase Equilibria of the LennardLiquid Phase Equilibria of the LennardLiquid Phase Equilibria of the LennardLiquid Phase Equilibria of the Lennard----JonJonJonJones es es es

Family of PotentialsFamily of PotentialsFamily of PotentialsFamily of Potentials

A molecular level understanding of the freezing and melting transitions can be

acquired using simple molecular models constructed from the concept of

interatomic and intermolecular interactions. A short-range repulsive potential

part, a long-range attractive potential part, and model parameters are the key

elements of any intermolecular potential. Even though all the models are not

designed to retrieve the actual physics behind a real molecular system they can

indeed provide valuable information about the nature of interaction among the

atoms or molecules depending on the shape, size, and closeness of interacting

particles. In the case of two-body interaction models the potential energy and

the force between the particles are governed by the distance between atoms.

The short range repulsive contributions and the long range attractive

interactions are usually modelled applying a functional relationship between

distance, potential energy, and force. The most common functional relationships

are of the form of inverse-power, exponential, or a combination of other

functional forms. The best known widely used molecular model is the 12-6

Lennard-Jones potential where ‘12’ represent the short range repulsive part and

‘6’ represent the long range attractive part of the potential for the inverse

Introduction

57

distance between them. The attractive part, repulsive part, and the combined

effect of both of these on the solid-liquid coexistence properties are of

considerable current interest (Lowen, 1994; Monson and Kofke, 2000).

Many successful theories have been developed on the observation that the

structure of dense fluid is dominated by the steep repulsive interaction between

the atoms or molecules (Frenkel and McTague, 1980; Weeks et al., 1971;

Longuet-Higgins and Widom, 1964; Barker and Henderson, 1967). The melting

temperatures are strongly influenced by the interatomic repulsive forces while

the attractive interatomic forces are a very weak function of the orientation and

shape of the atoms. For modelling larger molecules or monomers of polymer

chains one needs, in most cases, a potential whose repulsive part is often softer

than the standard 6-12 LJ potential (Meyer et al., 2000). A soft core potential

model can be obtained by replacing the exponent ‘12’ by a smaller integer which

is greater than ‘6’. Though a number of studies (Meyer et al., 2000; Okumura

and Yonezawa, 2000; Charpentier and Jakse, 2005; Kiyohara et al., 1996) have

been carried out for � � 6 Lennard-Jones potential for vapour-liquid system no

such study has ever been conducted for solid-liquid phase transitions. In a liquid

phase, the possibilities of arrangement of the atoms and molecules, which are

dependent on their shape and orientation, will thus determine the

characteristics of the solid phase to a great degree (density and symmetry in

particular). They will also influence the value of the freezing and the melting

Introduction

58

temperature since the characteristics of transition points are strongly influenced

by the interatomic repulsive forces.

It has been found that the varying steepness of purely repulsive potential has a

profound effect on the transport properties (Gordon, 2006; Heyes and Powles,

1998; Heyes et al., 2004), liquid-vapour phase coexistence, and the critical points

(Okumura and Yonezawa, 2000). In this dissertation (Chapter 4), we attempt to

calculate the solid-liquid coexistence properties as a function of the repulsive

term �. This will provide a global perspective of the link between solid-liquid

phase behavior and molecular interactions. In particular we will demonstrate

how physical properties and the melting rules vary with the variation of

repulsive potential contribution for a fixed attractive potential.

1.3.1.3.1.3.1.3.4 4 4 4 Phase DPhase DPhase DPhase Diagram iagram iagram iagram of the Weeksof the Weeksof the Weeksof the Weeks----ChandlerChandlerChandlerChandler----Andersen Andersen Andersen Andersen

PPPPotentialotentialotentialotential

The Week-Chandler-Andersen (Weeks et al., 1971) (WCA) interaction potential

acts as the building block of many modelled complex molecular structures

(Kröger, 2005; Kröger, 2004) and behaves as a generic fluid in many rheological

studies and polymer simulations (Rapaport, 2004; Kroger et al., 1993). Due to

its short range of interaction and its smooth cutoff the WCA potential is quite

popular in equilibrium and nonequilibrium molecular dynamics (MD and

NEMD) computer simulation studies (Hess et al., 1998). Because of its

Introduction

59

simplicity WCA potential can be used as nonbonded potential, bonded

potential, solvent-solute potential, and solvent-solvent potential. In many liquid

state theories WCA is the governing potential (Tang, 2002; Weeks et al., 1971;

Chandler et al., 1983). Since the WCA potential is softer than hard-sphere (HS)

potential and harder than 12-6 Lennard-Jones potential and consists of both

attractive part and repulsive part. The absence of liquid-vapour phase

transition, critical point and triple point makes WCA potential essentially

different from LJ potential.

Despite its important role in liquid state theories and molecular simulation,

relatively few data are available for solid-liquid equilibria for the WCA potential

(Hess et al., 1998; de Kuijper et al., 1990). In contrast, there are extensive

simulation data (Barroso and Ferreira, 2002; Morris and Song, 2002; Ge et al.,

2003b; Errington, 2004; Mastny and de Pablo, 2005; Mastny and de Pablo,

2007; McNeil-Watson and Wilding, 2006) for the solid-liquid coexistence of 12-6

Lennard-Jones fluids. Two previous investigations of WCA solid-liquid

coexistence have been performed for either a single state point or for a limited

temperature range (Hess et al., 1998; de Kuijper et al., 1990). de Kuijper et al.

(1990) obtained the WCA melting line from Monte Carlo simulations, whereas

Hess et al. (1998) approximately located the solid-liquid phase coexistence for

one temperature using canonical (NVT) and isothermal-isobaric (NpT)

molecular dynamics algorithms. The freezing point densities and pressures

obtained for the two MD simulation methods showed discrepancies of 5% and

Introduction

60

18.5%, respectively, whereas the different simulations were in good agreement

for the melting point properties. Although the freezing point densities and

pressures from both MC and NVT MD simulations are in agreement, there is a

discrepancy of 5.2% and 15.5% for the melting point density and pressure,

respectively. These discrepancies are somewhat surprising because, unlike the

Lennard-Jones potential, solid-liquid coexistence for the WCA potential is not

affected by cutoff errors that can contribute as much as 10% to the properties

(Mastny and de Pablo, 2005; Mastny and de Pablo, 2007). Without the cutoff

potential the system size effect is thought to be the only systematic source of

errors in case of WCA phase coexistence study.

In this dissertation (Chapter 5), we report data for the phase diagram of WCA

system using the GWTS (Ge et al., 2003b) and the GDI (Kofke, 1993a; Kofke,

1993b) techniques. The conjecture (Nelson and Halperin, 1979; Hoover and Ree,

1968) of abnormal behaviour at low temperatures is investigated by examining

the melting behaviour at very low temperatures. We also trace the melting line

of the WCA potential to very high temperatures to test the hypothesis that it

approaches a 12-th power soft-sphere asymptote. Three empirical expressions for

the solid-liquid coexistence pressure, freezing density and melting density are

reported.

Introduction

61

1.3.51.3.51.3.51.3.5 Phase Diagram of the Gaussian Core Model FPhase Diagram of the Gaussian Core Model FPhase Diagram of the Gaussian Core Model FPhase Diagram of the Gaussian Core Model Fluidluidluidluid

The GCM fluid is well known for its virtues to qualitatively imitate the

anomalies of complex molecular fluids and their solutions. This potential

exhibits a density anomaly (Stillinger and Stillinger, 1997) as well as other

water-like anomalies (Mausbach and May, 2006) associated with re-entrant

melting behaviour. Furthermore, it shows a structural order anomaly

(Krekelberg et al., 2009). The GC potential is shown to be describing effective

interactions of micellar aggregates of ionic surfactants suggested by Baeurle et

al. (2004).

In the past, different approaches have been applied to observe the topology of

the GCM phase diagram (Lang et al., 2000; Stillinger and Stillinger, 1997;

Giaquinta and Saija, 2005). Currently, the most accurate simulation results

were reported by Prestipino et al. (2005) using Monte Carlo simulations in

conjunction with calculations of the solid free energies. Compared with the

results reported by Prestipino et al. (2005), the one-phase entropy criterion

(Giaquinta and Saija, 2005) underestimates *dS by approximately 30%. In

contrast, the approach used by Lang et al. (2000) yields a value of *dS that is approximately 10% higher than reported by Prestipino et al. (2005).

Furthermore, the calculations reported by Prestipino et al. (2005) lead to a

partially modified phase diagram for the face centered cubic-body centered cubic

(fcc-bcc) solid transition compared with previous calculations (Stillinger and

Introduction

62

Stillinger, 1997; Lang et al., 2000). To the best of our knowledge, the successful

use of direct coexistence methods for GCM-like fluids has not been reported. In

view of these considerations, the GCM fluid provides a severe test for the

GWTS algorithm. It should be noted that because the GWTS algorithm uses

liquid state simulation methods, it can not be used to determine solid-solid

transitions.

1.3.61.3.61.3.61.3.6 StrainStrainStrainStrain----Rate DRate DRate DRate Dependent ependent ependent ependent Shear VShear VShear VShear Viscosity of the iscosity of the iscosity of the iscosity of the

Gaussian Gaussian Gaussian Gaussian Core Bounded PCore Bounded PCore Bounded PCore Bounded Potentialotentialotentialotential

The viscoelastic behaviour of non-equilibrium fluids is of significant theoretical

and industrial interest (Onuki, 1997; Barnes et al., 1989). It has been

experimentally determined that many fluids display shear thinning, which is

characterized by a decrease in viscosity with increasing strain-rate (Barnes et

al., 1989). In contrast, some complex fluids, such as colloidal suspensions show

shear thickening, i.e., their shear viscosities increase with increasing strain-rate

(Barnes et al., 1989). Theoretical studies of flows under shear have largely

focused on unbounded interaction potentials, such as hard spheres or the

Lennard-Jones potential (Todd and Daivis, 2007). However, during the last

decade bounded potentials such as the Gaussian core model (GCM) have proved

useful in the field of soft condensed matter physics (Likos, 2001).

Introduction

63

The absence of any detailed internal structure in the GCM means that it is

difficult to deduce substance-specific behaviour. In this dissertation, we report

non-equilibrium molecular dynamics (NEMD) calculations for the shear

viscosity behaviour of GCM fluids at different strain-rates and state points. To

the best of our knowledge no non-equilibrium studies have been reported using

potentials, such as the GCM, which permit particle overlap.

1.3.7 1.3.7 1.3.7 1.3.7 Equation of Equation of Equation of Equation of SSSState and tate and tate and tate and VVVViscosity iscosity iscosity iscosity MMMModellingodellingodellingodelling

(i) Steady S(i) Steady S(i) Steady S(i) Steady State tate tate tate EEEEquation of quation of quation of quation of SSSState tate tate tate

An equation of state typically provides an analytical relationship between the

pressure (p), volume (V), temperature (T), and in the case of mixtures,

composition (x) of a fluid. The prediction of fluid properties at thermodynamic

equilibrium has been greatly facilitated by improvements in equations of state

(Wei and Sadus, 2000) that increasingly incorporate many of the underlying

subtleties of intermolecular interactions. In particular, equations of state

developed in conjunction with molecular simulation data (Wei and Sadus, 2000)

have proved valuable in predicting phenomena that are not easily

experimentally accessible. However, many interesting thermodynamic processes

(Jou et al., 1988; Trepagnier et al., 2004) never attain thermodynamic

equilibrium. Instead, some nonequilibrium phenomena, such as viscosity at

constant strain rate (Evans and Morriss, 2008), eventually attain a

nonequilibrium steady-state. In this dissertation, we make use of this insight to

Introduction

64

formulate a steady-state equation of state for the pressure of a Lennard-Jones

fluid as a function of density, temperature and strain-rate. NEMD simulations

for shear viscosity are reported for a wide range of temperatures, densities and

strain-rates. These extensive data are analysed to obtain the parameters of the

equation of state.

(ii)(ii)(ii)(ii) Generic Viscosity MGeneric Viscosity MGeneric Viscosity MGeneric Viscosity Modelodelodelodel

The non-Newtonian viscosity is usually studied as a function of strain rate. But

during the application of applied external field in the real experiments or in the

NEMD simulations the influence of temperature, density, and pressure are quite

significant. The involvement of the four variables made it complex to model the

complete behavior. It is common to study the shear viscosity as a function of

strain rate. Recently, McCabe et al. (2001) have demonstrated qualitatively the

pressure (or equivalently density) dependence of non-Newtonian viscosity for 9-

octlyheptadecane. From NEMD simulation data they have found that at higher

pressure 9-octlyheptadecane first showed shear thinning behavior at lower strain

rates. For alkane molecules, the pressure dependence of viscosity is commonly

treated simply by using Barus’s equation (McCabe et al., 2001; Coy, 1998),

which indicates that the viscosity decreases with increase in pressure. For

multigrade oils, shear viscosity as a function of strain rate and pressure, 4�� , =�, is usually described by a four parameter relation (Coy, 1998). The similar

functional relationships for viscosity such as 4�� , �, 4�� , ��, and 4�� , , �, =� are

Introduction

65

rarely found in the literature. The dependency of shear viscosity on strain rate,

pressure, density and temperature is not only essential for modeling and

simulation but also necessary for the development of new theories and cost

optimization in experiments. Such unified relationships are rarely seen in

current literature. This limitation is largely attributed to the complexity arises

from four variables. With the definition of steady state compressibility factor

and nonequilibrium steady state equation of state it is possible to model

viscosity as a function of pressure, density, temperature and shear rate.

Although in literature several attempts of modelling zero-shear viscosity as a

function of pressure can be found, the nature of complexity of the problem made

the progress of development slow. In fact, a robust pressure dependent viscosity

model must address the following points: (i) the model must fit, at least

individually, the experimental data for ��4/�=�� , 0 and ��4/�=�� 0; (ii) can

reproduce linear, quadratic and exponential parts of the viscosity curves; (iii)

needs a least number of model parameters; (iv) the quality of reproducibility

measured in terms of average absolute deviation, maximum deviation, and bias.

If a theoretical or empirical model compromise the number of model parameters

it could be used to reproduce the experimental data with variable slope ��4/�=��. One such example is the friction theory (Quinones-Cisneros et al., 2000)

with 18 adjustable parameters which is also limited by the range of pressures

and temperatures.

Introduction

66

1.4 Organisation of the 1.4 Organisation of the 1.4 Organisation of the 1.4 Organisation of the DDDDissertationissertationissertationissertation

In Chapter 2, equilibrium and nonequilibrium molecular dynamics simulation

algorithms will be explained in details. The algorithms adopted in this work for

the study of solid-liquid equilibria also introduced in this Chapter. The

Lennard-Jones � � 6 family of potentials, variants of truncated and shifted LJ

potentials, Weeks-Chandler-Andersen potential and Gaussian core models

potentials will also be presented in this Chapter.

Chapter 3 will be devoted to present the results of solid-liquid phase transitions

obtained as a part of this work. The benchmarking problem of Lennard-Jones

simulation data will be investigated along with supplementary accurate

simulation data from the algorithm adopted in this study. The solid-liquid

phase transitions of truncated, truncated-shifted and shifted force LJ potentials

will also be investigated comprehensively.

Chapter 4 will present the solid-liquid phase coexistence data of � � 6 Lennard-

Jones potential. Triple point properties and their scaling relationships with �

will also be discussed. The simulation data will be verified using a number of

melting rules and new empirical models and parameters will also be introduced

in this Chapter.

Introduction

67

Chapter 5 will describe the solid-liquid phase transitions of Weeks-Chandler-

Andersen potential from very low to high temperatures and pressures. Low and

high temperature limits of the WCA phase diagram will be analysed. The

compressibility factors of the WCA fluid will be compared with the available

equation of state calculations and, if necessary, will be modified accordingly.

The simulation data will be verified using a number of melting rules and new

empirical models and parameters will also be introduced in this Chapter.

Entropy of fusion for the WCA will also be estimated.

Chapter 6 will be devoted to the study of solid-liquid phase diagram of

Gaussian core model potential. The particular emphasis will be given to the

high precision data on the low and high density sides of the Gaussian core phase

envelope. The approach of the re-entrant melting line close to the common

point of the phase envelope will be studied extensively. The solid-liquid

coexistence data obtained from simulations will be compared with the free

energy based calculations and freezing rule.

In Chapter 7 we will calculate the shear viscosities of Gaussian core model using

nonequilibrium molecular dynamics technique. The onset of Non-Newtonian

shear viscosity will be discussed as a function of strain rate considering the fact

of re-entrant melting. The shear viscosity behavior will also be discussed for

varying temperatures and densities. The strain rate dependent viscosity data

will be tested against a suitable model and with the mode-coupling theory. The

Introduction

68

zero-shear viscosities will be estimated form the NEMD data and will be

compared with the Green-Kubo calculations.

Chapter 8 will outline the design and development procedure of a

nonequilibrium steady state equation of state and a generic viscosity model. It

will be shown, how the steady state equation state can be coupled with the

generic viscosity model to predict shear viscosity as functions of strain rate,

temperature, pressure and density. Complete statistics will be presented to

verify the steady state equation of state and generic viscosity model against

simulation data. Finally, in this Chapter, all the models will be verified using

experimental data.

Finally, the conclusions and recommendations for future work will be made in

Chapter 9.

69

Chapter 2Chapter 2Chapter 2Chapter 2 Molecular SimulationMolecular SimulationMolecular SimulationMolecular Simulation

The research findings presented in this dissertation are obtained from molecular

simulation techniques. A variety of simulation algorithms is used for the

purpose. The choices of the algorithms are made by the guidance of problems

under consideration to obtain accurate simulation data. One of the objectives of

this dissertation, outlined in Chapter 1, is to study solid-liquid phase equilibria

for systems capable of modelling complex molecular system. Limitations of

current algorithms, reviewed in Chapter 1, tempted us to choose an algorithm

which is self starting and free from particle transfer between phases. One such

algorithm is the GWTS algorithm which combines the elements of both

equilibrium and nonequilibrium molecular dynamics simulation technics. For the

ease of drawing entire solid-liquid phase diagrams, without compromising

significant amount of computation time, we have also used the Gibbs-Duhem

integration technique starting from the coexistence values obtained through the

GWTS algorithm. Hence the essentials aspects of the GDI algorithm are also

discussed in this Chapter.

In Section 2.1, various intermolecular potentials used in this dissertation are

introduced. Reduced units are presented in Section 2.2. In Section 2.3, essential

components of molecular dynamics simulation algorithm are discussed and the

distinct features of nonequilibrium simulation technique are elaborated in

Molecular Simulation

70

Section 2.4. In Section 2.5, the integrated approach of the GWTS and the GDI

algorithms are discussed.

2.1 2.1 2.1 2.1 Rationale Rationale Rationale Rationale forforforfor Molecular SMolecular SMolecular SMolecular Simulationimulationimulationimulation

Experiments search for meaningful patterns in nature and theories model these

patterns into mathematical language which provides the predictive laws of

nature (Nakano et al., 1999). Molecular simulation attempts to closely imitate

experiments on a real system using model potentials. In other words, molecular

simulation uses microscopic properties of a system to calculate macroscopic

variables. Wilding pointed out the following features of molecular simulations

compared to experiments (Wilding, 2001):

• As in real experiments, we have to prepare equilibrated samples under

desired thermodynamic conditions.

• As in real experiments, we can measure the physical properties of the

sample.

• Many different algorithms can reproduce the same physical properties

within the limit of uncertainties.

• Because the simulator has access to complete information about the state

of the model system, there are fewer restrictions on which properties can

be measured. Accordingly, information and insight can be gleaned from a


71

simulation that is not only easily obtainable by experiments. For

example, a comparison of the model’s phase behavior can be useful in

helping to refine the model parameters.

• Simulations can be used as test bed for theories.

The distinctive advantages of computer simulations over real experiments are

that materials can be studied that are too expensive, too complicated, or too

dangerous to be tackled by real experiments (Wilding, 2001).

The choice of algorithm is usually determined by factors such as desired

thermodynamic conditions, expected thermophysical properties, computational

efficiency, reproducibility of the experimental data, minimization of statistical

fluctuation and ease of use.

2.2 2.2 2.2 2.2 InterInterInterIntermolecularmolecularmolecularmolecular PPPPotentialotentialotentialotentialssss

Intermolecular interactions involved in solid-liquid coexistence are of

considerable scientific interest (Monson and Kofke, 2000). Theories of solid-

liquid coexistence are commonly based on the observation that the structure of

dense fluids is dominated by steep repulsive interaction between the atoms or

molecules (Frenkel and McTague, 1980; Barker and Henderson, 1967; Weeks et

al., 1971; Longuet-Higgins and Widom, 1964). Melting temperatures are

strongly influenced by interatomic repulsive forces. The 12-6 Lennard-Jones


72

potential is adequate for atomic fluids, whereas modelling the behaviour of

molecules or monomers of polymer chains usually requires a potential with a

softer repulsive part (Meyer et al., 2000). In this Section a set of such

intermolecular potentials is introduced and is researched in coming Chapters.

2.22.22.22.2.1.1.1.1 LennardLennardLennardLennard----Jones Family of PJones Family of PJones Family of PJones Family of Potentialsotentialsotentialsotentials

The n-6 Lennard-Jones family of potential is:

h�a� � o � �� 6� ��6�� Hi�� ta�H � �ta�� (2.1)

where t is the atomic diameter and � is the well depth. We will consider

potentials with values of n ranging from 7 to 12. The smaller the index n, the

wider the attractive part and weaker the repulsive force as depicted in the Fig.

2.1. It should be noted that attributing n and ‘6’ contributions to repulsion and

attraction, respectively is only a convenient approximation. The continuous

nature of the potential with respect to interatomic separation (r) means that it

is impossible to isolate either purely repulsive or purely attractive contributions.

As n approaches infinity, the leading coefficient of Eq. (2.1) approaches ε and

the n-6 Lennard-Jones potential reaches the limiting case of a “hard-sphere +

attractive term” potential (Charpentier and Jakse, 2005).


73

0.9 1.2 1.5 1.8 2.1 2.4

-1

0

1

2

u(r)/εεεε

r/σσσσ

Figure 2.1 Comparison of n-6 Lennard-Jones pair potentials, where from top to

bottom � � 12, 11, 10, 9, 8, and 7.

In case of � � 12 and X � 6 the n-6 Lennard-Jones potential takes the form of

12-6 Lennard-Jones potential and can be written as

h�a� � 4o ��ta�� ta�� (2.2)

where ε and σ are the characteristic energy and distance parameters,

respectively. Thus a simple soft core potential can be obtained by replacing the

“12” exponent in the 12-6 Lennard-Jones potential by a smaller integer. It has

been found that varying the value of the exponent and thereby the steepness of

the main repulsive branch of the potential significantly affects vapour-liquid

equilibria (Charpentier and Jakse, 2005; Meyer et al., 2000; Okumura and

Yonezawa, 2000; Kiyohara et al., 1996), the critical point (Okumura and


74

Yonezawa, 2000) and transport properties (Heyes and Powles, 1998; Heyes et

al., 2004; Gordon, 2006). In Chapter 4, we will study the effect of varying � for

the solid-liquid phase transition and the triple points.

2.22.22.22.2.2 .2 .2 .2 Truncation and Truncation and Truncation and Truncation and Shifting SShifting SShifting SShifting Schemeschemeschemeschemes

Variants of Lennard-Jones potential using different truncation and shifting

schemes are widely used in molecular simulations of pure and multicomponent

systems. In case of molecular dynamics simulations potential shifting schemes

are also suggested by many authors. A detailed discussion on these schemes can

be found elsewhere (Sadus, 1999; Allen and Tildesley, 1987; Frenkel and Smit,

2001; Smit, 1992). The phase transition is very sensitive to the details of the

implementation of the intermolecular potential. Surface tension in the vicinity of

critical temperature is one of the examples of sensitivity of simulation data on

the liquid-vapour coexistence. The effect of truncation and shifting schemes on

vapour-liquid phase equilibria is well known. But such studies for solid-liquid

phase equilibria are rarely seen in the literature. In Chapter 3, we will present a

detailed study of solid-liquid equilibria employing following truncation and

shifting schemes.

(i)(i)(i)(i) Truncated LennardTruncated LennardTruncated LennardTruncated Lennard----Jones Jones Jones Jones PPPPotentialotentialotentialotential

hZ�a� � �h�a� a ~ aF0 a , aF �, (2.3)


75

where aF is the radial distance of the potential cutoff and hZ is the truncated potential.

(ii)(ii)(ii)(ii) TruncTruncTruncTruncated and Shifted Lennardated and Shifted Lennardated and Shifted Lennardated and Shifted Lennard----Jones PJones PJones PJones Potentialotentialotentialotential

hZij�a� � �h�a� � h�aF� a ~ aF0 a , aF �, (2.4)

where hZij is the truncated and shifted potential.

(iii)(iii)(iii)(iii) ShiftedShiftedShiftedShifted----Force Force Force Force LennardLennardLennardLennard----Jones PJones PJones PJones Potentialotentialotentialotential

hZij)�a� � �h�a� � h�aF� � �a � aF�h� �aF� a ~ aF0 a , aF �, (2.5)

where hZij) is the shifted-force potential, h� is the first derivative of the full Lennard-Jones potential.

2.22.22.22.2.3 .3 .3 .3 WeeksWeeksWeeksWeeks----ChandlerChandlerChandlerChandler----Andersen PAndersen PAndersen PAndersen Potentialotentialotentialotential

The Weeks-Chandler-Andersen (Weeks et al., 1971) (WCA) is the Lennard-

Jones potential truncated at the minimum potential energy at a distance

aJL � 2�/�t on the length scale and shifted upwards by the amount o on the energy scale such that both the energy and force are zero at or beyond the cut-

off distance:

h�a� � � 4o ��ta�� ta�� o, a ~ 2�/�t0, a , 2�/�t� (2.6)


76

where o and t are the characteristic energy and distance parameters,

respectively. Eq. (2.6) is a purely repulsive potential. WCA potential is

commonly used as part of a broader model for molecular fluids such polymers

(Kröger, 2005; Kröger, 2004) and dendrimers (Bosko et al., 2004a; Bosko et al.,

2005). It can be applied to both bonded and non-bonded interactions and it

forms the basis of many liquid state theories (Sadus, 1999; Chandler et al., 1983;

Tang, 2002). The utility of the potential is that it provides a simpler alternative

to the Lennard-Jones potential that is more realistic than a crude hard-sphere

potential. It possesses many of the physical attributes of the Lennard-Jones

system. However, a key difference is that WCA potential is limited to solid-

liquid equilibria.

2.22.22.22.2.4.4.4.4 GaussianGaussianGaussianGaussian----Core MCore MCore MCore Modelodelodelodel Potential Potential Potential Potential

Gaussian core potential (Berne and Pechukas, 1972) is given by

h�a� � o ��= �� at�� (2.7)

where t is the length scale and o is the energy scale of the model. A feature of

Eq. (2.7) is that the particles can overlap, that is, a ~ t, without catastrophic

consequences for the simulation. In a number of studies the GCM is used as an

effective potential to explain aspects of soft condensed matter. For example, the

effective interaction between self-avoiding polymer coils, dispersed in a good

solvent, can be described by the GCM (Lang et al., 2000; Louis et al., 2000).

Additionally, the GCM has been applied (Baeurle and Kroener, 2004) to micelle


77

aggregates to reproduce results from calorimetric experiments of aqueous

suspension of the ionic surfactant sodium. Simulation of such aggregates built

up from several thousands of molecules become rapidly intractable if a detailed

description on an atomic level is retained. It is therefore tempting to consider

such aggregates as “soft” particles where the detailed interaction is replaced by

an effective interaction between the soft particles. We will use GCM potential

to calculate the solid-liquid coexistence and shear viscosity in Chapters 6 and 7,

respectively.

2.2.2.2.3333 Reduced Reduced Reduced Reduced UUUUnit nit nit nit FFFFormalism ormalism ormalism ormalism

Unless otherwise stated, all quantities in this dissertation will be expressed in

the conventional (Allen and Tildesley, 1987; Sadus, 1999) reduced forms relative

to the depth �o� and size �t� parameters of Lennard-Jones potential and are

given in Table 2.1. The asterisk superscript and the prefix “reduced” will be

omitted in the rest of the dissertation. It is to be noted that in Chapter 8, we

have used both reduced and real units where the appropriate SI units are the

companion of real units in all cases.


78

Table 2.1 The relationship of reduced unit with real unit in terms of Lennard-

Jones � and � parameters.

Quantity Symbol Relation to real units

Reduced length a{ a{ � at

Reduced temperature { { � OPo

Reduced pressure ={ ={ � =t�o

Reduced density �{ �{ � �t� Reduced energy 1{ 1{ � 1o

Reduced time g{ g{ � g� oXt� Reduced strain rate �� { �� { � ��t�Xo

Reduced viscosity 4{ 4{ � 4t�√Xo

2.2.2.2.4444 Molecular DMolecular DMolecular DMolecular Dynamics ynamics ynamics ynamics

It has been first realized by Ge et al. (2003b) that solid-liquid phase coexistence

of model (Section 2.2) fluids can be determined via equilibrium molecular

dynamics technique in conjunction with a nonequilibrium molecular dynamics

simulation algorithm. One of the main goals of this dissertation is to determine

the solid-liquid phase equilibria for both unbounded and bounded potentials and

the essential elements of the molecular dynamics algorithm are discussed in this

Section. However, some of the necessary components of EMD and NEMD

algorithms are similar and these are also elaborated in this Section. In the

subsequent Section (Section 2.5), the distinctive features of the NEMD

algorithm are presented. The integration scheme adopted in this dissertation


79

relates equations of motion with appropriate boundary conditions and thus is

introduced in Section 2.5 with necessary modifications.

The general idea behind MD is that if one allows a system of particles to evolve

in time literally infinitely, that system will eventually pass through all possible

configurations (atomic system) or conformations (molecular system). In MD

simulations the time average of the measurable physical quantity X is given by

�n�ZJ*e � lim�� 1u � n�^_�g�, `_�g��g � 1W n�^_, `_�¡Z¢�

�Z¢>

(2.8)

Where u is the simulation time, M is the number of time steps in the

simulation, and n�^_�g�, `_�g�� is the instantaneous value of X at time t when

`_ and ^_ are the generalised coordinates and momenta, respectively. In

practice, n is calculated from the atomic trajectories which are the signature of

relative atomic positions and momenta. The dynamics of the atoms in a

trajectory dictates by the two-body instantaneous forces readily available as the

gradient of a potential energy function. The numerical values of position and

momenta of all particles can be calculated from two key classical mechanics

formalism. In Lagrangian (aka Newtonian) formalism position and momenta of

N particles can be found by numerically solving 3N second-order differential

equations and in Hamiltonian formalism the same information can be gathered

by solving 6N first-order differential equations.


80

The essential assumptions of any MD simulation are based on the definition of

time-correlation function. If n��g� is a time dependent quantity at time g and n��g� is another related quantity at some latter time g, then average of the product of n� and n� over some equilibrium ensemble is the time-correlation

function (Zwanzig, 1965). This is the simplest definition of time-correlation

function and a rigours definition is out of scope of this dissertation and can be

found elsewhere (Zwanzig, 1965; Zwanzig, 1964). The assumptions on which any

molecular dynamics simulation runs are:

(i) Simulation must be longer than the relaxation time, £¤ (aka correlation time)

of the system and can be estimated from the rate of decay of time-correlation

functions (Haile, 1997).

(ii) Correlation length, ¥ of the spatial correlation function of the system must

be converged and well below the simulation box length. ¥ can be estimated from

the rate of decay of time-correlation function (Ma, 1985; Hansen and McDonald,

1986).

(iii) Unavoidable surface effects must be appropriately minimized (Allen and

Tildesley, 1987).

The essential components of any molecular dynamics simulation are described

sequentially in the following subsections:


81

2.4.12.4.12.4.12.4.1 Equations of MEquations of MEquations of MEquations of Motionotionotionotion

Molecular dynamics is the numerical way of solving N-body problem which is

apparently impossible analytically. In MD, a set of coordinates, ¦aSJ, aTJ , aVJ§ where K � 1 … Y, governed by a model interaction potential controls the

dynamics of a system (consisting of N atoms) via Newton’s differential

(discretized) equations of motion and readily provides atomic trajectories

(positions, bJ�g� and momenta, ^J�g�) when integrated numerically with respect

to time, g.

The trajectory of a typical atom at time g, �b�g�, b� �g�� , evolves from a given

initial atomic positions and velocities, �b�0�, b� �0��, by integrating the set of first-order differential equations derived from Hamiltonian formulation of

mechanics (Goldstein, 1980):

�b� � *� � I© (2.9)

An alternative set of a second-order differential equations derivable from

Lagrangian formulation of mechanics (Goldstein, 1980) could be used in stead of

Eqs. (2.9). However, because of the nonlinearity of second-order differential

equations, simulations carried out in this work employed Eqs. (2.9). In

computer simulations, discretized sequence of states is used instead of continuos

trajectory �b�g�, b� �g�� for g � 0. That discretized sequence of states forward in

time can be obtained by applying appropriate finite-difference integration

scheme (will be discussed in details in Section 2.5.3). The timestepping


82

mechanism of finite-difference scheme must be initiated by using appropriate

initial conditions given by the coordinates and velocities of all particles at g � 0.

2.4.22.4.22.4.22.4.2 Initial Lattice CInitial Lattice CInitial Lattice CInitial Lattice Configurationonfigurationonfigurationonfiguration

The time evaluation of the MD trajectory passing through all possible points in

the phase space is independent of the choice of initial configuration for a

simulation of adequate time duration. Thus in our simulations we have adopted

the popular face centered cubic (f.c.c) lattice (Kittel, 2005; Dekker, 1969)

configuration as initial configuration at g � 0.

2.4.32.4.32.4.32.4.3 Initial Random VInitial Random VInitial Random VInitial Random Velocityelocityelocityelocity

The initial distribution of velocities are usually determined from a random

distribution with the magnitudes conforming to the required temperature and

corrected so there is no overall momentum:

\ � X�b� �_J¢� � 0 (2.10)

The velocities are often chosen randomly from Maxwell-Boltzmann distribution

(or a Gaussian function with suitable scaling) at given temperature which gives

the probability that an atom K has a velocity, a�SJ , a�TJ , a�VJ in the �, U, c-directions respectively at a temperature :

NJ�a�ªJ� � « XJ2qOP¬�/� ��= � XJa�ªJ�2OP® (2.11)


83

where . � �, U, c. The temperature can be calculated from the velocities using

the relation

� 13Y |N�|2X�_

�¢� (2.12)

2.4.42.4.42.4.42.4.4 Force CForce CForce CForce Calculationalculationalculationalculation

The force (Eq. (2.9)) needed to integrate the equations of motion is derivable

from intermolecular potential (Section 2.2) of interest. If h¯aJL° is the pair

potential specific to pair �K, M�, aJL being the magnitude of the vector distance

between atoms K and M, the force on the particle K: IJ � � �h¯aJL°�bJ

_L¢� � IJL

_L¢� (2.13)

where IJL is the vector force exerted by M on atom K . MD program intends to

take the advantage of the symmetry imposed by the fact that IJL � �ILJ (Newton’s third law).

2.4.52.4.52.4.52.4.5 Periodic Boundary Periodic Boundary Periodic Boundary Periodic Boundary CCCConditionsonditionsonditionsonditions (PBC)(PBC)(PBC)(PBC)

If the atoms of the system under consideration contained in a rectangular

simulation box, the periodic boundary conditions utilize replicas of this

simulation box to form an infinite lattice.


84

Figure 2.2 Periodic boundary conditions in a three dimensional view. The

orange colour box is the central simulation box. All other boxes are the images

of the original simulation box. The particles move in and out as shown with

arrows.

The application of PBC allows us to simulate equilibrium bulk solid and liquid

thermodynamic properties with a manageable number of atoms by eliminating

surface effects (Born and von Karman, 1912). The basic idea behind the PBC is

that if an atom moves in the original simulation box, all its images move in a

concerted manner by the same amount and in the same fashion. The

computational advantage of this method is that we need to keep track of the

original image only as representative of all other images. As the simulation

evolves, atoms can move through the boundary of the simulation cells. When


85

this happens, an image atom from one of the neighbouring cell enters to replace

the lost particle. The situation is visualized in Fig. 2.2. As a result of applying

PBC the number of interacting pairs increases enormously. This is because of

each particle in the simulation box not only interacts with other particles in the

box but also with their images. This problem can be handled by choosing a

finite range potential within the criteria of minimum image convention. The

essence of the minimum image criteria is that it allows only the nearest

neighbours of particle images to interact. In practice, the mechanism of doing so

is to use the potential in a finite range such that the interaction of two distant

particles at or beyond a finite length can be neglected. This maximum length

must be equal to or less than the half of the box length used in the simulation.

The periodic boundary condition algorithm with minimum image convention

could be implemented by considering an imaginary box around the atom of

interest which interacts only with other atoms within the imaginary box. If

QS , QT , QV are edge lengths of the imaginary box and aSJL , aTJL , aVJL are the

components of the pair separation vector bJL, one must apply the following

condition during computation:

�� QS2 ~ aSJL ~ QS2� QT2 ~ aTJL ~ QT2� QV2 ~ aVJL ~ QV2 ±²

³² (2.14)

This could be written in the program as follows:


86

�aSJL µ aSJL � ¶·Y aSJLQS ® � QSaTJL µ aTJL � ¶·Y ¸aTJL

QT ¹ � QTaVJL µ aVJL � ¶·Y aVJLQV ® � QV ±²

²³²² (2.15)

where ¶·Y�y� is a function that returns the nearest integer to y. In practice,

the calculation of image distances can be simplified by using reduced units

(Allen and Tildesley, 1987).

2.5 2.5 2.5 2.5 NonequilibriumNonequilibriumNonequilibriumNonequilibrium Molecular Dynamics SMolecular Dynamics SMolecular Dynamics SMolecular Dynamics Simulationimulationimulationimulation

Nonequilibrium molecular dynamics is a variant of conventional equilibrium

molecular dynamics for systems far from equilibria. In the study of transport

properties NEMD is orders of magnitude more efficient than EMD (Evans and

Morriss, 2008). The essential components of the NEMD algorithm are discussed

in the following sub-sections.

2.2.2.2.5555....1111 LeesLeesLeesLees----Edwards Periodic Boundary CEdwards Periodic Boundary CEdwards Periodic Boundary CEdwards Periodic Boundary Conditiononditiononditionondition

Figure 2.3 shows Lees-Edwards periodic boundary conditions (Lees and

Edwards, 1972; Evans and Morriss, 2008) in a three dimensional view. It

illustrates a way of adapting periodic boundary conditions to planar Couette

flow (see Fig. 2.4) and can be seen as a modified version of fixed orthogonal

periodic boundary condition described in Section 2.4.5 and shown in Fig. 2.2.


87

Figure 2.3 Lees-Edwards periodic boundary conditions for planar Couette flow

in a three dimensional view while the motion of the image cells defines the

strain rate for the flow. The pink colour boxes are taken to be stationary. The

indigo colour boxes are moving in the positive º direction with a velocity which

equals box length multiplied by the strain rate. The light blue colour boxes are

also moving with the same velocity but in the negative º direction.

The perpendicular height of a typical cell remains fixed so that the shearing

deformation occur isochorically. If a particle exits a cell through a top face, it is

replaced by its periodic image which enters at the bottom face. This image will

be positioned according to the current angle of the slewing lattice vector. Thus

the Fig. 2.3 depicts the repositioning of nonorthogonal lattice vectors as a


88

function of time. The �-component of its velocity will be the old velocity minus

the strain rate multiplied by the perpendicular height of the cell. Its peculiar

velocity is unchanged.

In the Lees-Edwards boundary conditions, the movement of the imaged particles

can be viewed in three distinct ways (Sadus, 1999). In the case of U boundary

moving Couette-flow geometry, if the particle exits through the top face of the

simulation box, its position and velocity are calculated from

�bJd)Ze[ � ¯bJfe)G[e � »�� QRg°X¼� QRb� Jd)Ze[ � b� Jfe)G[e � »�� QR © (2.16)

where bJfe)G[e and bJd)Ze[

are the positions and b� Jfe)G[e and b� Jd)Ze[

are the

velocities of the particle K before and after the move, respectively, and QR is the length of the simulation box in the direction O � �, U, c. Whereas, if the particle

exits through the bottom face of the box:

�bJd)Ze[ � ¯bJfe)G[e � »�� QRg°X¼� QRb� Jd)Ze[ � b� Jfe)G[e � »�� QR © (2.17)

If the particle exits through either face parallel to the U axis, the position vector

after the move is obtained from:

bJd)Ze[ � �bJ� X¼� QR (2.18)

To implement the time varying Lees-Edwards periodic boundary conditions in

computer programs in lieu of original fixed orthogonal periodic boundary

condition (Section 2.4.5) the Eq. (2.15) must be modified. Since the U-

boundaries of our planner Couette flow geometry are in motion, the �-


89

component of Eq. (2.15) is modified to accommodate the effect of sheared flow

in the following way

�aSJL µ aSJL � ½1�\¾ � ¶·Y aSJLQS ® � QS

aSJL µ aSJL � ¶·Y aSJLQS ® � QSaTJL µ aTJL � ¶·Y ¸aTJL

QT ¹ � QTaVJL µ aVJL � ¶·Y aVJLQV ® � QV ±²

²²²³²²²² (2.19)

with

½1�\¾ � X¼�� Qy=¿�� gKX��; QS�, (2.20)

where X¼��y; z� is a function that returns the remainder of the division of z

into y.

2.2.2.2.5555....2222 The The The The sllodsllodsllodsllod Equations of MEquations of MEquations of MEquations of Motionotionotionotion

Analytically successful infinite-dimensional Hamiltonian thermal reservoirs are

not suitable for computer simulations because of its essential finite degrees of

freedom. Early nonequilibrium molecular dynamics computer simulations

employed stochastic models of heat baths which were partly considered to be

inefficient. Since heat is generated during the run of sllod algorithm, a

thermostat should be added to the equations of motion to remove this heat from

the system. We employed the Gaussian constraint thermostat assuming a linear

velocity profile (Evans et al., 1983; Evans, 1983; Hoover et al., 1982). The


90

Gaussian thermostatted sllod algorithm provides an apparently simple model of

shearing nonequilibrium steady states (Evans et al., 1989). The sllod method is

known to applicable not only in the linear but also in the nonlinear region of the

shear flow when a thermostat is not introduced (Evans and Morriss, 1984b). In

the limit of zero shear rates it is known that the Gaussian thermostatted sllod

equations of motion generate the Green-Kubo relation for the shear viscosity

(Evans and Morriss, 1984a; Evans, 1986).

Figure 2.4 Planner Couette flow geometry.

X

Y


91

NEMD is a numerical technique which explains nonlinear statistical phenomena

using a special machinery of fluid flow geometry. The computational set up for

this algorithm is a steady state planar couette flow in a homogeneous system

and is depicted schematically in Fig. 2.4. The steaming velocity of this flow is

determined by the sheared algorithm. The sllod equations of motions are the

modified version of Newton’ equation of motion for sheared geometry. At time

g , 0Á they are equivalent with a linear shift applied to the initial velocities of

the particles along the applied shear.

Let us consider a nonequilibrium stationary state of a fluid driven by an

external strain rate

�� mS�U (2.21)

which is the gradient of the � component of the local velocity l of the fluid in

the U direction and coupled to a thermostat to assure a stationary state.

The equations of motion of the particles in such a system are the so called sllod

equations (Evans and Morriss, 2008):

� b� � X � »��aT� � I � »��=T � p^Â (2.22)

where X is the mass of the fluid; b� is the velocity; » is the unit vector in the �

direction; I is the intermolecular force; aT is the U component of the position b; =T is the U component of the peculiar momentum ^ which is the component of

momentum in excess of that caused by the strain-rate and is given by


92

� � Xb� � »XmS�b� (2.23)

with respect to the local fluid velocity

mS�b� � ��aT (2.24)

The Gaussian thermostat multiplier p can be determined using Gauss’ principle

of least constraint (Evans and Morriss, 2008) and is given by

p � ∑ ¯IJ · ^J � ��=SJ=TJ°_J¢� ∑ =»ÅÆ»¢Ç (2.25)

The total energy E and pressure tensor ] in terms of peculiar momentum are

obtained via:

1 � ^J�2X_

J¢� � 12 hJL_J,L (2.26)

]k � ^J�X_

J¢� � 12 bJLIJL_J,L (2.27)

where aJL � aJ � aL, IJL is the force on K due to M and hJL is the intermolecular

interaction energy between the particles. The second summation of the right

hand side of Eq. (2.26) represents the configurational energy ¯1FGH)°. In the nonlinear regime, where the local thermodynamic equilibrium hypothesis is not

valid, the equations for ] and E are consistent with the macroscopic

conservation equations of hydrodynamics (Irving and Kirkwood, 1950).

In principle, for a system at mechanical equilibrium, the off-diagonal elements of

the pressure tensor should be zero, whereas the on-diagonal elements should all

be equal and related to ambient external hydrostatics pressure (Brown and

Neyertz, 1995),


93

= � �]SS � ]TT � ]VV�3 (2.28)

The hydrostatic pressure is thus one third the trace of the pressure tensor and is

given by

= � 13 tr�]� (2.29)

The shear viscosity �4� is calculated from a component of the pressure tensor

¯]ST° and the strain-rate �� via the relationship: 4 � � 1�� ]ST� (2.30)

The implementation of sllod algorithm requires that the Lees-Edwards periodic

boundary conditions (Lees and Edwards, 1972) are already in use.

2.2.2.2.5555....3333 Gear PredictorGear PredictorGear PredictorGear Predictor----Corrector Integration Corrector Integration Corrector Integration Corrector Integration SSSSchemechemechemecheme

An integrator advances the trajectory of particles over small time increments.

The key features of an ideal MD integrator would be (Allen and Tildesley,

1987): (i) less expensive force calculation; (ii) zero error accumulation during

the progression of trajectory; (iii) energy and momentum conservation; (iv)

time-reversible trajectory of particles motion; (v) during sampling the integrator

will preserve the volume of the phase space, that is, the integrator will be

symplectic.

A family of integrator algorithms (Allen and Tildesley, 1987) were evolved in

time based on accuracy, fastness and coupling with other algorithms. Since the

symplectic nature of integrator and the time reversibility of simulation


94

trajectory are not the essential characteristics of the integration algorithms, the

remaining problems are the conservation of energy and momentum along with

the accuracy. Due to the complex and chaotic nature of N-particle dynamics one

has to compromise with the absolute accuracy of the integration. Finally, it

appears that a good integrator algorithms must conserve energy and momentum

for a moderately long run. This eventually related with the choice of finite-

difference method adopted for the numerical integration.

To solve Newtons equations of motion (Eq. (2.9)) and to solve the so called

sllod (Section 2.5.2) equations of motion (Eq. (2.22)), we used a five-value Gear

predictor-corrector method (Allen and Tildesley, 1987; Evans and Morriss, 2008;

Gear, 1971) for its efficiency and accuracy. Despite its 4th order accuracy it

requires only first derivatives of the intermolecular potential which is calculated

once per time-step. Other alternative integration methods are leap frog method

(Verlet, 1967) and Runge-Kutta (Gear, 1971) method. However, the Runge-

Kutta method is computationally very expensive.

If bJ be the position of particle K, the predictor step uses a Taylor series to

obtain an estimate of the new positions and momenta one time step, represented

byΔg, later:

b»�g � Δg� � b»�g� � �Δg �b»�g �b»�Z� � ËΔg�2! ��bJ�g� Í[»�Z� � ËΔg�3! ��bJ�g� ÍbÎ�Z�� ËΔgÏ4! �Ïb»�gÏ ÍbÎ�Z� �. .. (2.31)


95

Consider the predictor step for the first derivative by differentiating with

respect to g

b� »�g � Δg� � b� »�g� � �Δg �b� »�g �b»�Z� � ËΔg�2! ��b� J�g� Í[»�Z�� ËΔg�3! ��b� J�g� ÍbÎ�Z� �. .. (2.32)

Similarly for other derivatives:

bÐ »�g � Δg� � bÐ »�g� � �Δg �bÐ »�g �b»�Z� � ËΔg�2! ��bÐ J�g� Í[»�Z� �. .. (2.33)

bÑ»�g � Δg� � bÑ»�g� � �Δg �bÑ»�g �b»�Z� �. .. (2.34)

b»�Ò��g � Δg� � bJ�Ï��g� (2.35)

It is convenient to define successive scaled time derivatives of the form

bJHÓ � ÔZÕH! ÖÕbÖZÕ for 1 ~ � ~ 4 (2.36)

The advantage of using time scaled coordinate is that the predictor step of the

Gear algorithm can be expressed in terms of Pascal Triangle matrix which is

easy to apply (Sadus, 1999) and the corrector coefficients are also quoted by

assuming that time scaled variables are used. The predicted value of the

function is simply the sum

�bJÓ � bJ�g� � bJ�g� � bJ�g� � bJ�g� � bJ�g�bJ�Ó �g� � bJ��g� � 2bJ��g� � 3bJ��g� � 4bJÏ�g�bJ�Ó �g� � bJ��g� � 3bJ��g� � 6bJÏ�g�bJ�Ó �g� � bJ��g� � 4bJÏ�g�bJÏÓ �g� � bJÏ�g� ±²

³² (2.37)

In matrix notation:


96

×ØØÙbJÓ�g � ∆g�bJ�Ó �g � ∆g�bJ�Ó �g � ∆g�bJ�Ó �g � ∆g�bJÏÓ �g � ∆g�Ú

ÛÛÜ �

×ØÙ

1 1 1 1 10 1 2 3 40 0 1 3 60 0 0 1 40 0 0 0 1ÚÛÜ

×ØÙ

bJ�g�bJ��g�bJ��g�bJ��g�bJÏ�g�ÚÛÜ (2.38)

Equivalently for the momentum ^J, we have

×ØØÙ^JÓ�g � ∆g�^J�Ó �g � ∆g�^J�Ó �g � ∆g�^J�Ó �g � ∆g�^JÏÓ �g � ∆g�Ú

ÛÛÜ �

×ØÙ

1 1 1 1 10 1 2 3 40 0 1 3 60 0 0 1 40 0 0 0 1ÚÛÜ

×ØÙ

^J�g�^J��g�^J��g�^J��g�^JÏ�g�ÚÛÜ (2.39)

The values predicted from the Taylor series are not exactly the real values. The

correction vector .F must be chosen on the basis of accuracy and stability

requirements (Berendsen and van Gunsteren, 1986; Berendsen et al., 1984). The

corrector takes the form

×ØØÙ

bJF�g � ∆g�bJ�F �g � ∆g�bJ�F �g � ∆g�bJ�F �g � ∆g�bJÏF �g � ∆g�ÚÛÛÜ �

×ØØÙbJÓ�g � ∆g�bJ�Ó �g � ∆g�bJ�Ó �g � ∆g�bJ�Ó �g � ∆g�bJÏÓ �g � ∆g�Ú

ÛÛÜ �

×ØÙ

OO�O�O�OÏÚÛÜ ∆bJ (2.40)

×ØØÙ

^JF�g � ∆g�^J�F �g � ∆g�^J�F �g � ∆g�^J�F �g � ∆g�^JÏF �g � ∆g�ÚÛÛÜ �

×ØØÙ^JÓ�g � ∆g�^J�Ó �g � ∆g�^J�Ó �g � ∆g�^J�Ó �g � ∆g�^JÏÓ �g � ∆g�Ú

ÛÛÜ �

×ØÙ

OO�O�O�OÏÚÛÜ ∆^J (2.41)

The values of the corrector coefficients depend upon the order of the differential

equation being solved and independent of the detailed of the equation of motion.

Since in our simulation codes we have implemented the first-order equation of

motion the coefficients are


97

.F �×ØÙ

OO�O�O�OÏÚÛÜ �

×ØÙ

251/720111/121/31/24 ÚÛÜ (2.42)

The calculations of ΔbJ and Δ^J depends on the detailed of the equations of motion, that means, they are different for equilibrium and nonequilibrium

molecular dynamics. For first order equation of motion the usual definition of

ΔbJ and Δ^J are Δb» � b�D � b�% (2.43)

Δ^» � ^�D � ^�% (2.44)

Where b� and ^�% are the predicted first derivatives according to Eq. (2.38) and

Eq. (2.39). b�D and ^�D are the corrected first derivatives obtained by substituting bÓ and ^Ó into the equations of motion. For equilibrium molecular dynamics:

ΔbJ � Ý��J � =JSΔgU�J � =JTΔgc�J � =JVΔgÞ (2.45)

Δ^J � Ý=�JS � �ßJS � .F=JS�Δg=�JT � �ßJT � .F=JT�Δg=�JV � �ßJV � .F=JV�Δg Þ (2.46)

In Couette flow geometry (Fig. 2.4) when the predicted values are calculated,

Lees-Edwards periodic boundary conditions (Section 2.5.1) are applied to

reintroduce particles into the simulation box, which may have cross the

boundaries. The relative distances between pairs of particles are first calculated

and then used to determine the forces acting on each atom. Finally, sllod Eqs.

(2.22) are used in the corrector step to calculate the corrected values of bJ and


98

^J and their derivatives. For sheared flow Eq. (2.45) and Eq. (2.46) are modified

successively in the following way:

ΔbJ � Ý��J � ¯=JS � ��aJT°ΔgU�J � =JTΔgc�J � =JVΔg Þ (2.47)

Δ^J � Ý=�JS � �ßJS � .F=JS � ��=JT�Δg=�JT � �ßJT � .F=JT�Δg=�JV � �ßJV � .F=JV�Δg Þ (2.48)

2.62.62.62.6 Algorithms Algorithms Algorithms Algorithms to Sto Sto Sto Study tudy tudy tudy SolidSolidSolidSolid----Liquid Phase Liquid Phase Liquid Phase Liquid Phase

EEEEquilibriaquilibriaquilibriaquilibria

Solid-liquid phase transition is one the most challenging algorithmic problems in

molecular simulations for about half a century. Solid-liquid phase equilibria are

complicated by several factors:

1) First order phase transition between the solid and liquid phases prevents

the use of methods that rely on transitioning smoothly between the two

phase using standard simulations.

2) Particle exchanges in high density liquid and solid phase typically have

low acceptance probabilities.

3) Conventional procedures for inserting or deleting molecules alter the free

energy of the crystalline phase.


99

4) Multiple stable or metastable crystal structures (polymorphs) can exist at

a given state point for a compound, each with a different free energy.

5) Multi atom molecular species and structured molecular systems.

Many existing algorithms (Ladd and Woodcock, 1977; Ladd and Woodcock,

1978; Hansen, 1970; Hansen and Verlet, 1969; Raveche et al., 1974; Streett et

al., 1974; Chokappa and Clancy, 1987a; Chokappa and Clancy, 1987b; Hsu and

Mou, 1992; Agrawal and Kofke, 1995c; Barroso and Ferreira, 2002; Morris and

Song, 2002; Ge et al., 2003b; Errington, 2004; Mastny and de Pablo, 2005;

Mastny and de Pablo, 2007; McNeil-Watson and Wilding, 2006) have been

mainly applied using the Lennard-Jones potential. We note that there are

considerable inconsistencies in the results reported for the Lennard-Jones solid-

liquid phase transition (Mastny and de Pablo, 2007). A detailed discussion on

benchmarking problem of Lennard-Jones data is presented in Chapter 3. Thus

we have chosen GWTS algorithm which is self starting and free from particle

transfer. The objectives of using this algorithm are two fold:

(i) To provide accurate simulation data for solid-liquid coexistence

data.

(ii) To test the algorithm for more complex potentials than Lennard-

Jones potential.


100

2.2.2.2.6666.1 GWTS A.1 GWTS A.1 GWTS A.1 GWTS Algorithmlgorithmlgorithmlgorithm

(i) Fundamentals of GWTS (i) Fundamentals of GWTS (i) Fundamentals of GWTS (i) Fundamentals of GWTS AAAAlgorithmlgorithmlgorithmlgorithm

The algorithm developed by Ge et al. (2003b) is completely based on following

observations made on a model fluid obtainable via MD and NEMD algorithms:

• In the vicinity of solid-liquid phase transition, a few hundred

thousand time steps are sufficient to equilibrate model system event

at strain rates of the order of 10i� in reduced units. In contrast, millions more time steps are required to equilibrate a model system

near two -phase solid/liquid region.

• In the high density two phase region a significant speed up of

equilibration process can be achievable by applying strain rates. This

process can be further accelerated by introducing higher strain rates

within the stability region of the fluid flow.

• Metastable phase can be easily determined by applying small strain

rates which force the solid to yield. In contrast, conventional MC and

MD simulations demand very long runs to determine metastable

points.

• Accurate determination of the metastable points lessens the risk of

over-running the phase transition in determining the freezing line.


101

• In the two-phase liquid/solid freezing transition a sharp and

distinguishable pressure discontinuity is observed in the pressure

versus strain rate curves.

This algorithm does not apply NEMD technique solely to determine solid-liquid

phase equilibria rather it facilities the EMD technique to the system under

shear approaches steady state several order faster than the conventional

equilibrium molecular dynamics simulation. In contrast to the EMD algorithm

the approaching steady state at NEMD simulation is independent of solid-liquid

phase boundary. Although the original basis of the GWTS algorithm is

empirical an ad hoc theoretical basis of this algorithm could be as follows.

Since the pressure is calculated from one third the trace of the pressure tensor

for a homogeneous nonequilibrium liquid the key to understanding why the

discontinuity occurs at freezing density can be easily explained via the physical

origin of pressure tensor. Two commonly used methods of calculating pressure

tensor (Irving and Kirkwood, 1950; Todd et al., 1995) involve mass and

momentum continuity equation of hydrodynamics. For a single phase system

the pressure tensor is the linear sum of the kinetic and potential components.

When the liquid phase transformed into a two-phase solid/liquid region, that is,

the liquid freezes the kinetic contribution of the pressure tensor does not depend

on the redistribution of atoms across the solid-liquid phase boundary but


102

depends on the applied external field (i.e., strain rate) and on the relative

position of the atoms. Since in this case atoms remain on either side of the

phase boundary the rate of change of influx of fluid through the boundary will

be zero. Thus the contribution of applied external field dominates and the

pressure jump is observed when the fluid enters into solid-liquid phase boundary

or in other words the system freezes. To discover the physical origin of pressure

drop during melting and to locate the melting point the usual formulation of

Irving-Kirkwood pressure tensor (Irving and Kirkwood, 1950) must be modified

for two phase system considering different degrees of freedom. In practice, since

nonequilibrium molecular dynamics simulation algorithm calculate pressure

from the symmetric pressure tensors, a slight change in this behavior can be

realised applying a small strain rate. The mechanical stability of the sheared

system is governed by the temperature and density.


103

Figure 2.5 Schematic views of the essential components of GWTS algorithm.

Arrows are showing the next steps to follow in the algorithm. Blue colour

represents liquid and red colour represents solid.

à

NEMD

EMD

Post Processing

Post Processing

Freezing point

^

à

^

^

à

^

à

Melting

Post Processing

á 0.1 0.2

^

àâ»`

àãäâ

å�

å� � á

å� � á. Ç

å� � á. Å

StepStepStepStep----1 &21 &21 &21 &2 StepStepStepStep----3333

StepStepStepStep----4444 StepStepStepStep----5555

StepStepStepStep----6666 StepStepStepStep----6666


104

For a given temperature and strain rate, the increase in density influences the

stability and hence two phase region can easily be identified by a pressure jump

in NEMD simulation from its EMD counterpart.

(ii) Calculating the Solid(ii) Calculating the Solid(ii) Calculating the Solid(ii) Calculating the Solid----Liquid Phase CLiquid Phase CLiquid Phase CLiquid Phase Coexistenceoexistenceoexistenceoexistence

The heart of the GWTS algorithm is determining the pressure difference from

equilibrium and nonequilibrium molecular dynamics simulations at a give state

point to make decision whether the state point in question indicates phase

transition or not. Theoretical basis and numerical methods of isothermal-

isochoric (NVT) EMD and NEMD algorithms have already been discussed in

Sections 2.4 and 2.5. Following are the key considerations prior to

implementation of the GWTS algorithm:

Selection of the DenSelection of the DenSelection of the DenSelection of the Density Rsity Rsity Rsity Range:ange:ange:ange: At a given temperature, the choice of

density range to run simulations is crucial for saving significant

amount of computation time. If at least a single melting point density

is known for the system, it is a trivial matter to make the choice

about density range. But if the solid-liquid phase coexistence is not

known at all, for the system, one has to carry out several test runs to

find the approximate solid-liquid phase transition by observing

discontinuity in the density-pressure isotherm generated by test EMD

simulations.


105

Choice of the Strain RChoice of the Strain RChoice of the Strain RChoice of the Strain Rates:ates:ates:ates: In principle, a single non-zero strain rate

is sufficient to locate the freezing point given that the choice of strain

rate is consistent with low Reynolds’s number condition and stability

criteria of the fluid. Stable fluid flow in the Couette flow geometry

may not be known a priori for the potential model under

consideration. To find the appropriate strain rate range suitable for

the potential at hand one can conduct some test runs. Starting

density for such simulations must be in the liquid phase or otherwise

the tests will be failed. For any suitable temperature and density in

the liquid phase one can run several short NEMD simulations (4-5 is

recommended) to see the variation of pressure as a function of strain

rate. If the pressures as a function of strain rate (obtained from

NEMD runs) do not differ from the zero-strain rate pressure (from

EMD run), the choice of strain rates are acceptable for the final

production run. The strain rates chosen for the GWTS algorithm

should be greater than zero and must be well below the notorious

“string phase” (Erpenbeck, 1984; Woodcock, 1985; Evans and

Morriss, 1986). For example, in case of Lennard-Jones system strain

rates 0.1, 0.2 are sufficient whereas for low temperature WCA system

these values are 0.01 and 0.02. For bounded Gaussian core potential,

suitable strain rates are found to be 0.001 and 0.002.


106

Let us assume that for a given temperature at solid-liquid coexistence a

liquid density �r (freezing point) and a solid density �j (melting point) exist

within a set of possible densities æ��, ��, … , �Rç. GWST algorithm then can be

implemented according to the following scheme as depicted schematically in

Fig. 2.5:

StepStepStepStep----1:1:1:1: At temperature and density �R, run one EMD (where

��> � 0 ) simulation and two NEMD simulations for strain rates �� and �� . StepStepStepStep----2:2:2:2: For each density of the set of densities æ��, ��, … , �Rç, repeat Step-1.

StepStepStepStep----3:3:3:3: From the data obtained from Step-2, either calculate the

pressure differences to observe the pressure jump, necessary to locate

the freezing point, or draw a strain rate versus pressure graph to see

at which density pressure jumps while it goes from ��> to ��. It is to be noted that for some fluids pressure drops may also be possible (see

Fig. 6.2(a) in Chapter 6). For a given density (in liquid phase),

pressure variation for strain rates �� and �� is almost constant (i.e.,

linear on the �� = plot). Let the pressure jump is observed for

density �r, which is the desired freezing point density at

temperature . The difference ∆� � �r � �ri� is critical for the

accuracy of phase transition. In most of the cases, a suitable choice is

∆� � 0.01 (in reduced units) which is accurate up to two decimal


107

places. However, to improve the precision of the data one can repeat

Step-3 in between the densities �r and �ri�. The accuracy of the

algorithm can be improved from two to three decimal places just

dividing ∆� by 10. That means, one needs to repeat Step–3 for 10

more densities (in steps of 0.001 reduced unit) in between �r and �ri�. In this way one can ensure the accuracy of the simulations according

to the desired level.

StepStepStepStep----4:4:4:4: Now generate a density-pressure isotherm from the EMD

simulations on densities æ��, ��, … , �Rç. This isotherm will not be

continuous and will show two distinct curves: one for the liquid phase

and the other one for solid phase.

StepStepStepStep----5:5:5:5: Mark the freezing density �r on the density-pressure isotherm

for .

StepStepStepStep----6:6:6:6: Draw a tie line from the freezing point of the liquid phase

density line on to the solid phase and the point where tie line

intersects the solid phase density line is the desired melting point

density �j.

Thus one can draw a complete solid-liquid phase diagram (also known as

melting line) repeating Steps 1-6 for any chosen set of temperatures

æ�, �, … , Jç, where K is any suitable number according to choice.


108

2.2.2.2.6666....2222 GibbsGibbsGibbsGibbs----Duhem IDuhem IDuhem IDuhem Integrationntegrationntegrationntegration

Gibbs-Duhem integration (GDI) method is commonly used to trace a complete

phase diagram including vapour, liquid and solid phases from a predetermined

initial coexistence point. This method implies numerical integration of so called

monovariant Clapeyron equation that combines the Gibbs-Duhem equations of

coexistence phases.

(i)(i)(i)(i) ThermodynamiThermodynamiThermodynamiThermodynamic Bc Bc Bc Basis of GDIasis of GDIasis of GDIasis of GDI AAAAlgorithmlgorithmlgorithmlgorithm

A Gibbs-Duhem equation describes the mutual dependence of state variables in

a pure phase via the relation (Denbigh, 1971; Kofke, 1993b)

��éê� � ë�é � ém�\ (249)

where ê is the chemical potential, ë is the molar enthalpy defined by ë � h �\/� with � is the density, m is the molar volume, \ is the pressure, and é is

the reciprocal of temperature defined by é � 1/OP where OP is the Boltzmann

constant and T is the absolute temperature. For three phases of matter Eq.

(2.49) can be written as

Vapour: �¯éìdÓêìdÓ° � ëìdÓ�éìdÓ � éìdÓmìdÓ�\ìdÓ (2.50)

Liquid: �¯érJsêrJs° � ërJs�érJs � érJsmrJs�\rJs (2.51)

Solid: ��éjGrêjGr� � ëjGr�éjGr � éjGrmjGr�\jGr (2.52)


109

where the subscripts “vap”, “liq” and “sol” represent vapour, liquid, solid

phases, respectively. For solid-liquid phase equilibria the following conditions

must be satisfied:

�éjGr � érJs � éjGrirJsêjGr � êrJs � êjGrirJs\jGr � \rJs � \jGrirJs í (2.53)

At solid-liquid phase equilibria, equating (Eq. (2.51)) and (Eq. (2.52)) one can

get

�\jGrirJs�éjGrirJs � � ∆ëjGrirJséjGrirJs∆mjGrirJs (2.54)

where ∆ëjGrirJs � ëjGr � ërJs and ∆mjGrirJs � mjGr � mrJs. This is the famous

Clapeyron equation and is a simple first order nonlinear equation that describes

how pressure changes with temperature when two phases are in equilibrium.

Kofke (1993a) has used the Clapeyron equation to trace phase coexistence line

numerically. In computer simulations, it is customary to rewrite the equation in

terms of compressibility factor î � é=m [21]:

�\jGrirJs�éjGrirJs � � ∆ëjGrirJsîjGr � îrJs (2.55)

where îjGr and îrJs are being the compressibilities of solid and liquid phases

respectively. In principle, this form (Eq. (2.55)) of Clapeyron equation and its

variant (Eq. (2.58) below) could be used to calculate the solid-liquid and solid-

vapour phase equilibria. But for the ease of computer simulations and stable

numerical techniques slightly modified versions are used for these purposes.

Agrawal and Kofke (1995c) found that at low temperatures (a typical value for

Lennard-Jones potential is ~ 2.74) the freezing and melting line can be


110

extended up to the triple point value if they swap the choice of dependent and

independent field variables such that the Clapeyron equation yields

�éjGrirJs� Q� \jGrirJs®rGï� � � îjGr � îrJs∆ëjGrirJs ®rGï� (2.56)

Similarly, for solid-vapour phase equilibria the choice of field variables modify

the usual Clapeyron equation in the form

�ìdÓijGr�\ìdÓijGr � � ¯mìdÓ � mjGr°∆ëìdÓijGr (2.57)

(ii) (ii) (ii) (ii) Numerical TNumerical TNumerical TNumerical Techechechechniques in GDI Simulation Dniques in GDI Simulation Dniques in GDI Simulation Dniques in GDI Simulation Designesignesignesign

The correct choice of the integration path, integration method and

understanding of error estimation and stability checks are the key elements in

the successful design and performance of GDI algorithm. In GDI algorithms,

following sequence of computational techniques are required for the best

approximation of the integration results:

(a)(a)(a)(a) Suitable TransformaSuitable TransformaSuitable TransformaSuitable Transformattttion of the Clapeyron Eion of the Clapeyron Eion of the Clapeyron Eion of the Clapeyron Equation quation quation quation

The first step towards a correct GDI simulation is to achieve a stable

integration path without accumulating the errors. The integration path along

the phase coexistence line on a known plane is largely set by the definition of

the problem (Kofke, 1999). The precise and much interesting simpler shape of

the coexistence line can be obtained by virtue of adaptable transformation of

the thermodynamic variables involved in the Clapeyron equation. As long as the


111

right-hand side of the Clapeyron equation (Eq. (2.55)) does not vary

significantly from the ideal linear dependency with independent variable, these

transformations can sometimes ensure accuracy, stability and precision of the

integration.

The Clapeyron equation is numerically attractive because the right hand side of

Eq. (2.55) is almost constant. For computational purpose Eq. (2.55) can be

expresses as (Kofke, 1993b):

�¯Q�\jGrirJs°�éjGrirJs � � ∆ëjGrirJsîjGr � îrJs � NH¯\jGrirJs, éjGrirJs° (2.58)

In conventional numerical integration the evaluation of the slope

NH¯\jGrirJs , éjGrirJs° yields its exact value at GDI cycle �, where � � 1,2, … . But

in GDI algorithm this slope is determined with molecular simulation and the

longer the simulation proceeds, the better the estimate of the slope gets.

(b)(b)(b)(b) Initial ConInitial ConInitial ConInitial Conditionditionditiondition

It is essential to have a known coexistence point for the starting of integration

(Eq. (2.58)). The saturation pressure and temperature from any standard

simulation is particularly recommended. The use of Gibbs-Ensemble simulation

results to initiate GDI is common for vapour-liquid phase equilibria given that

the state point is moderately far from critical point. It is because near critical

point Gibbs-Ensemble Monte Carlo simulation data is not enough reliable. For


112

the solid-liquid phase equilibria Agrawal and Kofke (1995c) used soft-sphere

phase coexistence data as a starting point of their simulation. There is an active

discussion on this choice of solid-liquid phase coexistence (Errington, 2004;

Mastny and de Pablo, 2005; Mastny and de Pablo, 2007). Recently we have

successfully used a starting point from NEMD-MD simulation (Ahmed and

Sadus, 2009a). The last option of choosing initial point is from a very accurate

theory. For vapour-solid equilibria the GDI usually starts from triple point

value.

(c)(c)(c)(c) Pressure or Temperature Estimation from PredictorPressure or Temperature Estimation from PredictorPressure or Temperature Estimation from PredictorPressure or Temperature Estimation from Predictor----CCCCorrectororrectororrectororrector

Given a starting phase coexistence point NH¯\jGrirJs , éjGrirJs° Eq. (2.58) can be

solved numerically by a predictor-corrector method. We applied the Adams

predictor-corrector (Kofke, 1993b; Press et al., 1992) scheme to calculate the

pressure. The Adams algorithm requires a sequence of four prior predictor-

corrector simulations to run rest of the simulations. The Adams algorithm can

be described mathematically adopting a symbolic scheme where P represents

“Predictor”, C represents “Corrector”, ∆é represents steps in é and U � Q�\.

First Point (PFirst Point (PFirst Point (PFirst Point (Pressureressureressureressure from 1from 1from 1from 1stststst GDI CGDI CGDI CGDI Cycleycleycleycle): ): ): ): The pressure at the first simulation

point was predicted by the trapezoid predictor-corrector:

P U� � U> � ∆éN> (2.59)

C U� � U> � ∆é2 �N� � N>� (2.60)

Here N> is calculated from the start up value and is calculated via


113

N> � � ∆ëé\∆m

where ∆ë is the difference of molar enthalpies in liquid �ërJs � hrJs � \/�rJs� and solid �ëjGr � hjGr � \/�jGr� states and is given by ∆ë � ërJs � ëjGr and similarly ∆m is the difference of volumes in liquid �mrJs� and solid �mjGr� phases and is given by ∆m � mrJs � mjGr. It is to be noted that for a single GDI

simulation N> must be kept constant since this is calculated from the given

values to start the GDI simulation.

Second PSecond PSecond PSecond Pointointointoint (Pressure from 2nd GDI C(Pressure from 2nd GDI C(Pressure from 2nd GDI C(Pressure from 2nd GDI Cycle)ycle)ycle)ycle): : : : The pressure at this is calculated

via midpoint predictor-corrector of the form:

P U� � U> � 2∆éN� (2.61)

C U� � U> � ∆é3 �N� � 4N� � N>� (2.62)

where N� is the estimation from the simulation in progress after second GDI

cycle and can be determined from the running averages of the enthalpy and

volume.

Third PThird PThird PThird Pointointointoint (Pressure from 3rd GDI C(Pressure from 3rd GDI C(Pressure from 3rd GDI C(Pressure from 3rd GDI Cycle)ycle)ycle)ycle): : : :

P U� � U� � 2∆éN� (2.63)

C U� � U� � ∆é24 �9N� � 19N��5N� � N>� (2.64)

where N� is the estimation from the simulation in progress after third GDI cycle

and can be determined from the running averages of the enthalpy and volume.


114

Fourth Point (PFourth Point (PFourth Point (PFourth Point (Pressure for 4ressure for 4ressure for 4ressure for 4thththth & Rest of the GDI C& Rest of the GDI C& Rest of the GDI C& Rest of the GDI Cycles): ycles): ycles): ycles):

P UJÁ� � UJ � ∆é24 �55NJ � 59NJi� � 37NJi� � 9NJi�� (2.65)

C UJÁ� � UJ � ∆é24 �9NJÁ� � 19NJ � 5NJi� � NJi�� (2.66)

(d)(d)(d)(d) A Complete GDI Simulation at Several Different TA Complete GDI Simulation at Several Different TA Complete GDI Simulation at Several Different TA Complete GDI Simulation at Several Different Teeeemperaturemperaturemperaturemperaturessss

StepStepStepStep----1:1:1:1: Determine the solid-liquid coexistence pressure�\>�, liquid density ¯�>rJs° and solid density ��>jGr� at inverse temperature �é> � 1/�� using GWTS (Section 2.6.1) algorithm.

StepStepStepStep----2:2:2:2: Starting from f.c.c lattice, perform the NVT MC/MD (in this

dissertation MC has been used) simulations of the liquid and solid phases at

temperature éjZd[Z and densities �rJs_jZd[Z and �jGr_jZd[Z obtained from Step-

1. This part is used to calculate the molar enthalpies of both liquid and

solid phases.

StepStepStepStep----3:3:3:3: Divide the total number of GDI simulation cycles (production) in

blocks (typical number blocks is 10 or 8). In practice, for each incremental

temperature each block average is used to calculate the pressure, enthalpies

and liquid and solid densities. The number of GDI blocks is usually

determined by the formula:

GDI blocks �n� � |éjZd[Z � éeHÖ|∆é (2.67)

where éjZd[Z and éeHÖ are the starting and the terminating temperature of

the GDI simulations. Thus the sequence of éH , where


115

� � start, 1,2, … , k, … , end,

will be

�é� � éjZ[dZ } ∆é� � � � � � � �éR � éR}� } ∆é� � � � � � � �éeHÖ � éeHÖi� ù ∆é±²³

²

Here ‘+’ and ‘�’ signs are used for decreasing and increasing temperatures,

respectively. The choice of ∆é and its sign must be made on the basis of

integration path chosen.

StepStepStepStep----4444: : : : At each éH, in both liquid and solid phases simultaneous but

independent Y\ Monte Carlo simulations must be performed to obtain the

quantities needed to evaluate the right-hand side of the Clapeyron equation

(Eq. (2.58)). In each Y\ Monte Carlo simulation the following two moves

are performed:

ParticleParticleParticleParticle DisplacementDisplacementDisplacementDisplacement: : : : The attempted particle move is accepted with

probability of

XK�ú1, ��=��é∆û�ü (2.68)

where ∆û is the change in internal energy.

Volume CVolume CVolume CVolume Change: hange: hange: hange: The attempted volume fluctuation is accepted with

probability of

XK�ú1, ��=��é�∆û � ∆\� � ∆k�ü (2.69)


116

where ∆k is the volume change.

Following are the essential steps in simultaneous NPT simulations for

calculating solid-liquid coexistence properties using Clapeyron equation:

(i) Calculate the predictor pressure using Eq. (2.59).

(ii) Before the production run, few thousands of MC cycles (typically

10000) are needed for the equilibration of liquid and solid phase simulations.

After that all the accumulators must set to zero to initiate production run.

(iii) In the production run, calculate the liquid and solid enthalpies and

liquid and solid densities using the right hand side of Eq. (2.58). Then

calculate the corrector pressure via Eq. (2.60).

(iv) Evaluate the predictor and corrector pressures according to the

appropriate equations: Eqs. (2.61)-(2.66).

117

Chapter 3Chapter 3Chapter 3Chapter 3 Validation of SolidValidation of SolidValidation of SolidValidation of Solid----Liquid Liquid Liquid Liquid

Phase Equilibria DPhase Equilibria DPhase Equilibria DPhase Equilibria Dataataataata

Since the GWTS algorithm (Chapter 2) is (a) free from the complexity arises

from the particle transfer in a condensed phase (b) independent of starting point

and (c) is not limited by the choice of the free energy equation of state, it is

desirable to thoroughly test its ability to predict solid-liquid coexistence. An

obvious starting point would be to compare it to results in the literature for the

12-6 Lennard-Jones potential. However, there are considerable inconsistencies in

the available data. Here we address this issue.

3.1 Simulation Details3.1 Simulation Details3.1 Simulation Details3.1 Simulation Details

3.1.1 Technical Details of 3.1.1 Technical Details of 3.1.1 Technical Details of 3.1.1 Technical Details of the the the the GWTS SimulationGWTS SimulationGWTS SimulationGWTS Simulation

The solid-liquid coexistence properties were determined employing the GWTS

algorithm discussed in Section 2.6.1. The initial configuration in all the

simulations was a face centered cubic (f.c.c) lattice structure. The equations of

motion were integrated with a five-value Gear predictor corrector scheme

(Section 2.5.3). All simulation trajectories were typically run for 2 � 10ý time

steps. The first 5 � 10Ï time steps of each trajectory were used either to

equilibrate zero-shearing field equilibrium molecular dynamics or to achieve

Validation of Solid-Liquid Phase Equilibria Data

118

nonequilibrium steady state after the shearing field is switched on. The rest of

the time steps in each trajectory were used to accumulate the average values of

thermodynamic variables with average standard deviations. All the data

presented in this paper were average over run of 5-10 independent simulations.

A system size of 4000 Lennard-Jones particles was used for all the simulations

reported in this Chapter.

3.1.2 Technical Details of 3.1.2 Technical Details of 3.1.2 Technical Details of 3.1.2 Technical Details of the the the the GDI SimulationGDI SimulationGDI SimulationGDI Simulation

The solid-liquid coexistence was determined at high temperatures starting from

an initial point obtained from the GWTS algorithm described in the previous

Chapter (Section 2.6.1). The Clapeyron equation used in the evaluation of GDI

(Section 2.6.2) series was shown to be related to the stability of the integration

at a given temperature (Agrawal and Kofke, 1995c; Kofke, 1993b). Agrawal and

Kofke (Agrawal and Kofke, 1995c; Kofke, 1993b) demonstrated that for

Lennard-Jones potential two different versions of Clapeyron equation must be

used below and above the temperature � 2.74 to maintain the stability of

integration. Without any rigorous testing of the GDI series for LJ potential, we

have chosen this temperature as the starting point of our high temperature GDI

simulation. At the beginning of the simulation 932 atoms were distributed

between boxes represent solid and liquid phases. The box in liquid phase

contains 432 atoms while the box in solid phase contains 500 atoms in the face-

centred-cubic (f.c.c) lattice structure. The simulations were performed in cycles.


119

A simulation period of 10,000 was used to accumulate the simulation averages

followed by a period of equilibration cycles 10,000. In all the series of

simulations, temperature change per step in decreasing direction was, ∆é �0.01, where é is the reciprocal of the temperature. We have also tested three

different series with different ∆é and found no difference in the results.

3.1.3 Calculation of Properties for Full LJ Potential 3.1.3 Calculation of Properties for Full LJ Potential 3.1.3 Calculation of Properties for Full LJ Potential 3.1.3 Calculation of Properties for Full LJ Potential

Whenever necessary the pressure and energy for full LJ potential were

calculated using standard procedure (Sadus, 1999; Allen and Tildesley, 1987).

The 12-6 LJ potential were truncated at half of a box length and appropriate

long range correction terms were evaluated to recover the contribution to

pressure and energy of the full potential. The same procedure was adopted both

for the GWTS and the GDI algorithms. The GDI calculations for full LJ

potential were initiated with the coexistence pressure and liquid and solid

densities at a chosen temperature obtained from full LJ calculations via GWTS

algorithm.

3.3.3.3.2222 Comparison ofComparison ofComparison ofComparison of LennardLennardLennardLennard----Jones Jones Jones Jones SSSSolidolidolidolid----LLLLiquid iquid iquid iquid PPPPhase hase hase hase

CCCCoexistence oexistence oexistence oexistence DDDDataataataata

Inspite of the rigorous simulation techniques (Agrawal and Kofke, 1995c;

Barroso and Ferreira, 2002; Morris and Song, 2002; Ge et al., 2003b; Errington,


120

2004; Mastny and de Pablo, 2005; Mastny and de Pablo, 2007; McNeil-Watson

and Wilding, 2006) available for the study of solid-liquid phase equilibria of 12-6

Lennard-Jones (Lennard-Jones, 1931) fluid, a definitive standard for comparison

of the simulation data has not been established. Literature survey (Chapter 1)

shows that for a given temperature, the coexistence pressure, liquid density and

solid density vary as much as 12%-30%. After so many attempts, the actual

causes behind these discrepancies have not been revealed. Now the question

arises how these results were benchmarked against one another. In this Section

we have addressed the benchmarking issue of 12-6 LJ solid-liquid coexistence

data via a systematic comparison and provided a set of accurate data.

3.3.3.3.2222.1. Data C.1. Data C.1. Data C.1. Data Collectionollectionollectionollection

We have carried out an extensive literature survey on the solid-liquid phase

coexistence properties of 12-6 LJ potential. In Table 3.1 we summarized all the

state points previously used in the molecular simulation study of solid-liquid

phase transition for 12-6 LJ fluid. Fig. 3.1 illustrates the temperature range

covered by different authors. In this figure, very high temperatures �10.96, 17.2, 30.44, 68.49, 273.97 studied by Agrawal et al. (1995c) and �10,20,50,100 studied by Hansen (1970) are not shown.


121

3.23.23.23.2.2 .2 .2 .2 Data AnalysisData AnalysisData AnalysisData Analysis

It is particularly note worthy that of the 91 temperatures surveyed from 11

published simulation results (Table 3.1), there are only 4 common temperatures

(Table 3.2).

0

1

2

3

4

5

6

T

SourceH & V A & K B & F M & PM & S E M & W

Figure 3.1 Temperatures covered by different authors in their simulations of the

solid-liquid equilibria of 12-6 Lennard-Jones fluid. Shown are temperatures

studied by Hansen and Verlet (1969) (�, H & V), Agrawal and Kofke (1995c)

(�, A & K), Barroso and Ferreira (2002) (�, B & F), Mastny and de Pablo

(2007) (�, M & P), Morris and Song (2002) (�, M & S), Errington (2004) (⊳,

E) and McNeil-Watson and Wilding (2006) (�, M & W).


122

Table 3.1 Sources of solid-liquid phase equilibria data for 12-6 Lennard-Jones

fluid.

Source Method/Ensemble Y Cutoff

Mastny and de Pablo

(2007)

Direct method of thermodynamic

integration (NpT)

256, 500, 864,

2048

2.5-6.0

McNeil-Watson and

Wilding (2006)

Phase-switch Monte Carlo method

(NpT)

32, 108, 256,

500

L/2

Mastny and de Pablo

(2005)

Direct method of thermodynamic

integration (NpT)

500 2.5

Errington (2004)

Phase-switch Monte Carlo method

(NVT)

108,256, 500 L/2

Ge et al. (2003b) Equilibrium and nonequilibrium

molecular dynamics (NVT)

500 3.5

Morris and Song (2002)

Direct method of molecular

dynamics (NpT)

2000,4000, 000,

16000

2.1,

4.0, 8.0

Barroso and Ferreira

(2002)

Absolute Helmholtz free energy

calculation

108, 256,343,

500, 729, 864

L/2

Agrawal and Kofke

(1995c)

Gibbs-Duhem integration 236, 932 L/2

Hsu and Mou (1992) Molecular dynamics 864 2.5

Shifted

Chokappa and Clancy

(1987a)

Molecular dynamics (NpT) 108, 256, 500,

864, 1372,

2048, 2916,

4000

2.5

Ladd and Woodcock

(1977)

Molecular dynamics 2916, 4000

Raveche et al. (1974) NVT Monte Carlo

simulation (NVT)

108 and 256 L/2

Hansen and Verlet

(1969)

Thermodynamic

integration (NVT)

864 2.5

a EXEDOS: Extended Ensemble Density-of-States Monte Carlo Method (Kim et al., 2002).


123

Table 3.2 Common temperatures found in literature to validate 12-6 LJ solid-

liquid phase coexistence properties.

= �jGrJÖ �rJs Source

0.70 0.0 0.96 0.85 Hansen and Verlet (1969)

0.75

0.67 0.973 0.875 Hansen and Verlet (1969)

0.554 0.9652 0.8566 Errington (2004)

1.15


6.21 1.031 0.95 Barroso and Ferreira

(2002)

2

21.1 1.137 1.067

Barroso and Ferreira

(2002)

20.2 1.1277 1.0581 Errington (2004)


26.3 1.131 1.066 Streett et al. (1974)

36.9 1.211 1.144 Agrawal and Kofke (1995c)

0.84-0.88 0.593 --- --- Nose and Yonezawa (1985)

0.82-0.83 0.593 --- --- Chokappa and Clancy

(1987a)


124

To reveal the detailed picture of the literature data so far obtained from

molecular simulations, we have divided the popular temperature range in three

different regions namely, low temperature region � � 0.65 � 1.0�, medium

temperature region � � 1.0 � 2.0� and high temperature region � � 2.0 � 7.0�. A comparison of literature data is shown in the = � plane (Figure 3.2). It is

observed that the qualitative and quantitative trend of the melting line pressure

of Agrawal and Kofke (1995c), Barroso and Ferreira (2002) and Morris and

Song (2002) are almost identical. In contrast, the pressure of the melting curve

is lower for McNeil-Watson and Wilding (2006). Although the data from

Mastny and de Pablo (2007) varies slightly, it follows the main trend of

pressure. Errington’s (2004) single data point below � 1.0 shows a slightly

higher pressure than McNeil-Watson and Wilding (2006). In contrast to the

solid-liquid coexistence pressure, the scenario is different for the freezing and

melting densities: these densities vary considerably for lower temperatures and

higher temperatures.

Figure 3.3 shows the freezing and melting densities in the temperature

range 0.65 � � 1.10. The results of McNeil-Watson and Wilding (2006) are

significantly lower than the data of Agrawal and Kofke (1995c), Barroso and

Ferreira (2002), Morris and Song (2002) and Hansen and Verlet (1969).

Errington’s (2004) freezing and melting densities are also lower than the main

trend and close to the McNeil-Watson and Wilding (2006) data. However, the

data of Mastny and de Pablo (2007) is slightly lower than the main trend. The


125

effects of these variations can be easily understood from Fig. 3.4. Liquid side

densities at solid-liquid coexistences are difficult to determine than the solid

0.67 0.76 0.85 0.94 1.03-1

0

1

2

3

4

5

6

p

T

a

1.0 1.2 1.4 1.6 1.8 2.02

6

10

14

18

22

p

T

b

2.0 2.5 3.0 3.5 4.0 4.5 5.020

35

50

65

80

95

p

T

c

0.66 1.28 1.90 2.52 3.14 3.76 4.38 5.000

20

40

60

80

100

p

T

d

Figure 3.2 Solid-liquid phase coexistence pressure as a function of temperature

compiled from literatures. Coexistence pressure as a function of temperature for

the temperature range (a) þ � á.�á � Ç. á, (b) þ � Ç. á � Å. á, (c) þ � Å. á ��. á and (d) þ � á.�� . á (most commonly used temperature range)

calculated by Agrawal and Kofke (1995c) (�), Barroso and Ferreira (2002)

(�), Morris and Song (2002) (�), Errington (2004) (⊳), McNeil-Watson and

Wilding (2006) (�), Mastny and de Pablo (2007) (�) and Hansen and Verlet

(1969) (�).


126

0.65 0.80 0.95 1.100.83

0.87

0.91

0.95

ρρρρ

a

T

0.65 0.80 0.95 1.100.95

0.97

0.99

1.01

T

b

ρρρρ

Figure 3.3 Comparison of (a) liquid and (b) solid densities as a function of

temperature at solid-liquid coexistence in the temperature range T = 0.65-1.10.

Data shown are from Agrawal and Kofke (1995c) (�), Barroso and Ferreira

(2002) (�), Morris and Song (2002) (�), Errington (2004) (⊳), McNeil-Watson

and Wilding (2006) (�), Mastny and de Pablo (2007) (�) and Hansen and

Verlet (1969) (�).


127

0.65

0.80

0.95

1.10

1.25

0.82 0.86 0.90 0.94 0.98 1.02 1.06

ρρρρ

T

liquid solid

Figure 3.4 Solid-liquid coexistence densities in the à � þ plane. Data shown are

from Agrawal and Kofke (1995c) (�), Barroso and Ferreira (2002) (�), Morris

and Song (2002) (�), Errington (2004) (⊳), McNeil-Watson and Wilding (2006)

(�), Mastny and de Pablo (2007) (�) and Hansen and Verlet (1969) (�). In all

cases solid lines are used to guide the symbols for the overall view of freezing

and melting lines.

side densities (Fig. 3.4). The results of Hansen and Verlet (1969) and Agrawal

and Kofke (1995c) show more fluctuation on the freezing line. The main

implication of this density variation is the different melting temperatures (Fig.

3.4) and solid-liquid coexistence pressures (Fig. 3.3).


128

3.3.3.3.3333 FiniteFiniteFiniteFinite SSSSize ize ize ize EEEEffect on Lennardffect on Lennardffect on Lennardffect on Lennard----Jones Jones Jones Jones SolidSolidSolidSolid----liquid liquid liquid liquid


Errington (2004) examined finite-size effects with standard long-range tail

correction applying the phase-switch Monte Carlo method of Wilding and Bruce

(2000). Mastny and de Pablo (2007) were carried out simulations at � 0.77

and = � 1.0 for system sizes 256, 500, 864 and 2048. They inferred that using a

Table 3.3 System size dependencies of the solid-liquid coexistence properties of

12-6 Lennard-Jones fluid at þ � Ç. á obtained using the GWTS algorithm.

Y = �rJs �jGr 108 2.78385 0.9 0.9878

864 3.57273 0.91 1.0014

2048 3.9077 0.92 1.0072

4000 3.8979 0.92 1.011

13500 3.464 0.92 1.002

system of 256 or 108 particles for melting temperature predictions can result an

errors of 3% to 6% of the infinite-size melting temperature. Morris and Song

(2002) found little change (almost independent of system size) in melting

temperature (0.5263% decrease) and pressure (0.5020% increase) as the system

size is varied from 2000 to 16000 atoms (for large system size). To reduce


129

systematic errors from finite-size effects below 1%, at least 864 particles should

be used (Mastny and de Pablo, 2007). But most of the data found in literature

were obtained from simulations carried out with less than 1000 LJ atoms.

Therefore, we performed separate simulation runs to analyze the dependency of

the simulation results on the particle number. We calculated the solid-liquid

coexistence properties for system sizes of Y � 108, 864, 2048, 4000, 13500

particles (Table 3.3) at � 1.0. The average pressure, liquid density and solid

density calculated from Table 3.3 is 3.52 } 0.13,0.914 } 0.002 and 1.001 }0.002, respectively, within a 95% confidence interval. The effect of system size

does not appear to scale with 1/Y. Using Y � 2048 represents a reasonable

compromise between maintaining accuracy and minimizing computational effort.

3.3.3.3.4444 SSSSolidolidolidolid----LLLLiquid iquid iquid iquid Phase CPhase CPhase CPhase Coexistenceoexistenceoexistenceoexistence from from from from the the the the GWTGWTGWTGWTSSSS

Algorithm and Its ReliabilityAlgorithm and Its ReliabilityAlgorithm and Its ReliabilityAlgorithm and Its Reliability

We calculated solid-liquid equilibria for the 12-6 Lennard-Jones system at

various temperatures and compared the results to data available in literature

(Figure 3.5) (Agrawal and Kofke, 1995c; Hansen, 1970; Hansen and Verlet,

1969). Figure 3.5(a) shows that our results for the pressure-temperature

behaviour are in good agreement with previous studies. The only exception is

the pressure at T = 2.74 is somewhat higher than reported elsewhere (Agrawal

and Kofke, 1995c; Hansen, 1970; Hansen and Verlet, 1969). This discrepancy in

pressure reflects the fact that our results for both freezing and melting points


130

0 .5 1 .0 1 .5 2 .0 2 .5 3 .0

0

1 0

2 0

3 0

4 0

p

T

a

0 .5 1 .0 1 .5 2 .0 2 .5 3 .00 .8 5

0 .9 6

1 .0 7

1 .1 8

T

ρρρρliq

b

0 .5 1 .0 1 .5 2 .0 2 .5 3 .00 .95

1 .04

1 .13

1 .22

T

ρρρρ so l

c

Figure 3.5 Comparison of the solid-liquid coexistence (a) pressure, (b) liquid


in this work (�) with data from Agrawal and Kofke (1995c) (�) and Hansen

and Verlet (1969) (*). The errors are approximately equal to the symbol size.


131

0 .6 1 .2 1 .8 2 .4 3 .0

0

6

1 2

1 8

2 4

3 0

3 6

p

T

a

0 .6 1 .2 1 .8 2 .4 3 .00 .8 2

0 .9 0

0 .9 8

1 .0 6

1 .1 4

ρρρρ l i q

T

b

0 .6 1 .2 1 .8 2 .4 3 .00 .9 5

1 .0 0

1 .0 5

1 .1 0

1 .1 5

1 .2 0

ρρρρ s o l

T

c



in this work (Ο) with data from Mastny and de Pablo (2007) (�) and McNeil-

Watson and Wilding (2006) (�). The errors are approximately equal to the

symbol size.


132

occur at different densities (Figures 3.5(b) & 3.5(c)). Our densities are lower

than reported by Agrawal and Kofke (1995c) but higher than the results of

Hansen and Verlet (1969). Agrawal and Kofke’s pressure data is 12 % higher

then the data of Hansen and Verlet. The cause of this discrepancy is commonly

attributed to uncertainties in the starting point required by the Gibbs-Duhem

integration (GDI) method (Mastny and de Pablo, 2005; Mastny and de Pablo,

2007; McNeil-Watson and Wilding, 2006). The inverse 12th-power soft-sphere

initial condition needed to start the GDI procedure was p = 16.89T5/4 which is

higher than reported by Hoover et al. (1970), Hansen (1970), Cape and

Woodcock (1978). Any error in the initial condition for the GDI method will be

systematically (Agrawal and Kofke, 1995a; Kofke, 1993b) applied to all other

state points. The GWTS (Section 2.6.1) method used in this dissertation is free

from this uncertainty.

It is also instructive to benchmark the simulation data obtained via the GWTS

algorithm with some recent studies. We have compared the solid-liquid

coexistence pressure obtained in this work with the data from Mastny et al.

(2007) and McNeil-Watson et al. (2006) (Fig. 3.6). Figure 3.6(a) shows that the

Lennard-Jones pressure-temperature behavior obtained applying the GWTS

algorithm is in good agreement with the recent literature data (Mastny and de

Pablo, 2007; McNeil-Watson and Wilding, 2006). At low temperatures (T <

1.0) our freezing and melting densities are slightly higher than Mastny et al.

(2007) and at significantly higher values from McNeil-Watson et al. (McNeil-


133

Watson and Wilding, 2006) data. At temperatures T > 1.0, the overall trend of

our data is in very good agreement with both Mastny et al. (2007) and McNeil-

Watson et al. (2006). It is interesting to note that at temperature � 0.72019

the solid-liquid coexistence liquid and solid densities determined by McNeil-

Watson et al. (2006) are 0.8422 and 0.9587, respectively, which are well below

the triple point liquid and solid densities determined by Hansen-Verlet (1969),

Agrawal and Kofke (1995c), Barroso and Ferreira (2002) and Ahmed and Sadus

(2009b). This clearly indicates the algorithm used by McNeil-Watson and

Wilding (2006) underestimates the solid-liquid equilibria at low temperatures.

The discrepancy of our data and Mastny and de Pablo (2007) data can also be

explained through similar argument since at � 0.7793 they also obtained

significantly lower liquid (0.8699) and solid (0.9732) densities at solid-liquid

coexistence, which are also in the close vicinity of triple point densities. It

clearly indicates that even at low temperatures the GWTS algorithm is able to

locate solid-liquid phase coexistence with comparatively good accuracy. It

remains a challenge for the simulation community to develop a standardized

algorithm and methodology to explain the variation of the thermodynamic

variables in the solid-liquid phase coexistence of Lennard-Jones system.

3.53.53.53.5 Independent Validation of the GWTS AlgorithmIndependent Validation of the GWTS AlgorithmIndependent Validation of the GWTS AlgorithmIndependent Validation of the GWTS Algorithm

Since the equality of Gibbs free energies at phase coexistence is the essential

thermodynamic condition, we validate the accuracy of the GWTS algorithm via


134

an independent method. We have calculated reduced Gibbs free energy

corresponding to our solid-liquid coexistence point from the Lennard-Jones

equation of state of Johnson et al. (1993). The equality of Gibbs free energies at

solid-liquid coexistence for three different temperatures shown in Table 3.4

demonstrates the accuracy of the GWTS method.

Table 3.4 Solid-liquid coexistence properties calculated from the GWTS

algorithm. Gibbs free energy is calculated via Lennard-Jones equation of state of

Johnson et al. (1993).

Liquid phase Solid phase = �rJs 1rJs �rJs �jGr 1jGr �jGr 1.0 4.05(1) 0.923 -6.149(2) 2.14 1.008 -7.05(2) 2.12

1.5 11.2(1) 0.993 -5.63(3) 11.31 1.069 -6.57(3) 11.28

2.74 33.2(3) 1.116 -3.48(6) 35.69 1.181 -4.60(6) 35.60

3.63.63.63.6 The The The The EffectEffectEffectEffectssss of Potential Truncation and Shifting of Potential Truncation and Shifting of Potential Truncation and Shifting of Potential Truncation and Shifting

Schemes on SolidSchemes on SolidSchemes on SolidSchemes on Solid----Liquid Liquid Liquid Liquid CCCCoexistence oexistence oexistence oexistence

The solid-liquid coexistence properties were examined at two different

temperatures as the cut-off radius increased from 2.5t to 6.5t in steps of 0.5t.

Figure 3.7 shows the variation of pressure as a function of cut-off radius at

� 1.0 (Fig. 3.7(a)) and � 2.74 (Fig. 3.7(b)). In general, the pressure

decreases systematically with the increase of cut-off radius on the solid-liquid


135

coexistence. The truncated and truncated and shifted LJ potentials yield the

same pressure because the shift is a constant value (see Eq. (2.4)), which does

not affect the derivative used in calculating the virial contribution to pressure.

In contrast, the shifted-force potential yields considerably higher values of

pressure, particularly at small cut-off values. At all cut-off values, the

differences in pressures between the different LJ potentials are less noticeable at

� 2.74 than � 1, which reflects the greater relative contribution of kinetic

interactions at the higher temperature.

We note that our results for the sifted-force LJ potential with a cut-off radius

aF � 2.5t are slightly different to GDI data reported by Errington et al. (2003).

Our coexistence pressure, liquid density, and solid density are 5.38% , 1.84%

and 1.56% lower respectively, than their results. These discrepancies could be

largely attributed to finite-size effects and errors in choosing the original

reference point. We have validated our sifted-force LJ data with results reported

by Powles et al. (1982) for cut-off radius aF � 3t and obtained very good

agreement.

The use of potential truncations and shifts requires the addition of long-range

corrections to recover the full contribution to pressure. In contrast, Powles

(1984) did the exact opposite, i.e., the pressure was corrected from the full LJ

pressure to that of the truncated-shifted and shifted-force LJ pressure. This

transformation mechanism was verified via the equation of state of Nicolas et


136

al. (1979) and simulations on truncated-shifted and shifted-force LJ potentials.

In the same spirit, Johnson et al. (1993) rigorously derived mean-filed

corrections for truncated-shifted LJ potential and found remarkable accuracy of

these corrections at aF � 4t. However, they also found that the accuracy of the

mean-field corrections was compromised for lower cut-off radii in case of the

truncated-shifted LJ potential and could only produce reasonable results

for aF , 3t.

The configurational energy variation with respect to cut-off values is shown in

Figs. 3.8 and 3.9. In common with the pressure results, Figs. 3.8 and 3.9 show

that the configurational energy at solid-liquid coexistence depends both on the

cutoff radius and on the shift used. The energy variation is more prominent for

the lower cut-off values. At � 1 and at aF � 2.5t, the truncated-shifted and

shifted-force LJ potentials yield energies that are 9.34% and 21% higher,

respectively, than those observed for the truncated LJ potential. This gap

becomes progressively smaller at higher cut-off values.

In contrast to the results for the liquid phase (Fig. 3.8), the truncated LJ

configurational energies obtained for the solid phase (Fig. 3.9) are relatively

insensitive to the cut-off values. This result is also in contrast to the solid phase

energies obtained for the truncated-shifted LJ and shifted-force LJ potentials,

which are both dependent on the cut-off radius, particularly at low values.


137

2 3 4 5 6 74.2

4.8

5.4

6.0

p

rc

a

2 3 4 5 6 734

35

36

37

38

p

rc

b

Figure 3.7 Solid-liquid coexistence pressures of 12-6 Lennard-Jones systems as a

function of cut-off radius. Shown are truncated (�), truncated-shifted (�) and

shifted-force (Ο) Lennard-Jones systems for temperatures (a) þ � Ç. á and

(b) þ � Å.�Ò.


138

2 3 4 5 6 7-6.2

-5.8

-5.4

-5.0

-4.6

-4.2

Eliq

rc

a

2 3 4 5 6 7-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

Eliq

rc

b

Figure 3.8 Potential energy as a function of cut-off radius for the liquid phase of

12-6 Lennard-Jones systems at solid-liquid coexistence. Shown are truncated

(�), truncated-shifted (�) and shifted-force (Ο) Lennard-Jones systems for

temperatures (a) � 1.0 and (b) � 2.74.


139

2 3 4 5 6 7-7.2

-6.8

-6.4

-6.0

-5.6

-5.2

Esol

rc

a

2 3 4 5 6 7-4.8

-4.1

-3.4

-2.7

-2.0

Esol

rc

b

Figure 3.9 Potential energy as a function of cut-off radius for the solid phase of

Lennard-Jones systems at solid-liquid coexistence. Shown are truncated (�),

truncated-shifted (�) and shifted-force (Ο) Lennard-Jones systems for

temperatures (a) þ � Ç. á and (b) þ � Å.�Ò.


140

We found that, at any given temperature, the liquid and solid phase coexisting

densities only vary by approximately 10-2 - 10-3 depending on the truncations

and shifts used. This is in contrast to the significant potential dependencies

observed in the densities for vapour-liquid phase equilibria at different

temperatures. At low temperatures ( � 1.0� the density of the liquid phase varies by }0.002 depending on the cut-off radius. In contrast, at higher

temperatures � � 2.74� there is no noticeable dependency on the cut-off radius. The solid phase densities are almost insensitive to the cut-off value irrespective

of the temperature. This observation is in contrast to the work of Mastny and

de Pablo (2007), which reported that solid phase densities were more dependent

on the cut-off radius than liquid phase densities.

Since the melting temperature and pressure is related through the Clausius-

Clapeyron equation, a small change in temperature affects the pressure and vice

versa. As our data suggest a monotonic variation of pressure with respect to

cut-off distance, we can also expect a monotonic change of melting temperature

as a function of cut-off radius. In contrast to this observation, Mastny and de

Pablo (2007) found an oscillatory behavior of melting temperature with

increasing cut-off values. No theoretical justification was provided for such

aberrant behavior. In view of the fact that vapour-liquid equilibria pressure and

temperature also vary regularly as a function of cut-off radius, Mastny and de

Pablo’s (2007) observation may be an artefact of the simulation algorithm.


141

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.610-2

10-1

100

101

102

103

104

p

ββββ

a

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.610-2

10-1

100

101

102

103

104

p

ββββ

b

Figure 3.10 Comparison with full Lennard-Jones potential. Melting line of (a)

truncated-shifted LJ (�) and (b) shifted force LJ (Ο) at cut-off 2.5. In both

cases a comparison is made with the full LJ potential obtained in this work (×)

and reported by Agrawal and Kofke (1995c) (—).


142

2.6 2 .8 3 .0 3 .2 3 .4 3.6 3.8 4.032

37

42

47

52

57

62

67

p

T

Figure 3.11 Pressure variation of shifted force LJ (Ο) with respect to truncated-

shifted LJ (�). The melting line pressure for full LJ (—) potential, obtained in

this work, is also reported for comparison.

2 .6 2 .8 3 .0 3 .2 3 .4 3 .6 3 .8 4 .03 0

3 5

4 0

4 5

5 0

5 5

6 0

6 5

p

T

Figure 3.12 Melting line pressure variation of truncated and shifted LJ potential

as a function of cut-off radius. Shown are cut-off radius 2.5 (*) and cut-off

radius 6.5 (�). The melting line pressure for full LJ (—) potential, calculated in

this work, is also reported for comparison.


143

In common with the well-known system size dependency of the melting line

properties, the effect of cut-off radius may be algorithm dependent.

Figs. 3.10(a) and 3.10(b), compare of solid-liquid coexistence pressures for

truncated, truncated-shifted and shifted-force, (both at a cut-off radius of 2.5t�, and the full LJ potentials. It is apparent from this comparison that the

truncated-shifted and shifted-force LJ yield similar deviations from the LJ

potential as a function of temperature. Indeed, for either the truncated-shifted

or shifted-force LJ, deviations from the LJ pressure only become really

significant at very low temperatures, i.e., in the proximity of the triple point. It

is apparent that both the truncated-shifted and shifted-force LJ potentials

would predict a lower triple point temperature than the LJ potential.

The pressures obtained from the shifted-force and truncated LJ potentials at a

common cut-off radius �aF � 2.5t� are compared to the full LJ potential in Fig.

3.11. It is apparent from this comparison that either truncating or shifting the

potential considerably increases the coexistence pressure. In particular, the

shifted-force LJ calculations yield significant deviations from the full LJ

pressure.

Figure 3.12 illustrates the effect of the cut-off radius on the melting pressure

obtained at different temperatures using the truncated-shifted LJ potential. The

comparison with results obtained from the full LJ potential indicates that


144

1.10 1 .15 1 .20 1 .25 1 .30 1 .35 1 .40 1 .452 .5

3 .0

3 .5

4 .0

4 .5

5 .0

5 .5

6 .0

6 .5

T

ρρρρ

Figure 3.13 Phase diagram of shifted force LJ in à � þ plane. Shown are the

freezing (�) and melting (�) lines of LJ potential with cutoff radius 6.5 and

freezing (Ο) and melting (�) lines of LJ potential with cut-off radius 2.5. A

comparison is shown with the full LJ freezing line (—) and melting line (---)

obtained in this work.

choosing a small cut-off value �aF � 2.5t� consistently results in an increase in the melting pressure at all temperatures. A significantly higher cut-off value

�aF � 6.5t� yields good agreement with the LJ potential at low temperatures

but the pressures are slightly under predicted at , 3.2.

Figure 3.13 illustrates the effect of the cut-off radius on the temperature-density

behavior of the shifted-force LJ potential. When a small cut-off value is used

�aF � 2.5t� both the coexisting liquid and solid phase densities are increased at all temperatures. In contrast a cut-off value of aF � 6.5t results in densities that

are almost indistinguishable from the LJ potential.


145

In a summary, in this Chapter a detailed analysis has been presented for

determining the position of LJ melting line data obtained from the GWTS

algorithm. It is found that solid-liquid coexistence properties vary systematically

with potential truncations, shifts and cut-off radius. A cut-off radius of 6.5 is

recommended to achieve consistency between all methods. Potential truncation

and shifts have important consequences at low temperatures, particularly in the

vicinity of the triple point. The data suggests a regular variation of the melting

temperature as a function of cut-off radius, which contradicts the oscillatory

behavior of the melting temperature reported by Mastny and de Pablo (Mastny

and de Pablo, 2007).

146

Chapter 4Chapter 4Chapter 4Chapter 4 SolidSolidSolidSolid----Liquid Liquid Liquid Liquid Phase Phase Phase Phase Equilibria Equilibria Equilibria Equilibria

of of of of the the the the LennardLennardLennardLennard----Jones Jones Jones Jones Family of Family of Family of Family of

PotentialsPotentialsPotentialsPotentials

Many of the current simulation algorithms for the investigation of solid-liquid

phase coexistence have not either extensively verified or extended for handling

complexities beyond the 12-6 Lennard-Jones potential. In this Chapter

molecular dynamics simulations are reported for the solid-liquid coexistence

properties of n-6 Lennard-Jones fluids, where n = 12, 11, 10, 9, 8 and 7. In

Section 4.2, the complete phase behaviour for these systems has been obtained

by combining these data with vapour-liquid simulations. The influence of n on

the solid-liquid coexistence region is compared through relative density

difference and miscibility gap calculations. In Section 4.3, analytical expressions

for the coexistence pressure, liquid and solid densities as a function of

temperature have been determined. The triple point temperature, pressure and

liquid and solid densities are estimated in Section 4.4. The scaling behavior of

triple point temperature and pressure is also examined in this Section. In

Section 4.5, various melting and freezing rules are tested on the solid-liquid

coexistence lines of n-6 Lennard-Jones potentials.

Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials

147

4.1 4.1 4.1 4.1 Simulation DetailsSimulation DetailsSimulation DetailsSimulation Details

We have used the GWTS algorithm as described in Chapter 2 (Section 2.6.1) to

calculate the solid-liquid coexistence properties of the � � 6 Lennard-Jones

potential (Section 2.2.1). The initial configuration in all the simulations carried

out in this Chapter was a face centred cubic (f.c.c) lattice structure. The

isothermal isochoric NEMD simulations were performed by applying the

standard sllod equations (Eq. (2.22)) of motion for planner Couette flow coupled

with Lees-Edwards (Section 2.5.1) periodic boundary conditions. If the applied

strain rate is switched off the sllod algorithm behaves like a conventional

equilibrium molecular dynamics algorithm in the canonical ensemble (NVT).

The NVT EMD simulations were performed using conventional cubic periodic

boundary conditions (Section 2.4.5). A Gaussian thermostat multiplier (Eq.

(2.25)) was used to keep the kinetic temperature of the fluid constant. The

equations of motion were integrated with a five-value Gear predictor corrector

scheme (Section 2.5.3). The details of these techniques are given in Chapter 2.

The results presented in this Chapter are the ensemble averages for 5

independent simulations corresponding to different MD trajectories. The

simulation trajectories were typically run for 2 × 105 time steps of τ = 0.001.

The first 5 × 104 time steps of each trajectory were used either to equilibrate

zero-shearing field equilibrium molecular dynamics or to achieve non-equilibrium

steady state after the shearing field was switched on. The rest of the time steps


148

in each trajectory were used to accumulate the average values of

thermodynamic variables and standard deviations. A system size of 2048

Lennard-Jones particles was used for all the simulations with a cutoff distance

of 2.5σ. Conventional long-range corrections were used to recover the properties

of the full Lennard-Jones fluid. Simulations were performed for � 0.8, 0.9 and

2.74. These choices have been made to benchmark and compare the simulation

data with literature.

4.2 Analysis of the 4.2 Analysis of the 4.2 Analysis of the 4.2 Analysis of the nnnn----VVVVariation of the ariation of the ariation of the ariation of the SSSSolidolidolidolid----liquid liquid liquid liquid


The solid-liquid coexistence pressure as a function of n is shown in Figure 4.1(a)

for � 2.74. It indicates that there is an approximately linear inverse

relationship between pressure and �. Decreasing the value of �, causes an

increase in the coexistence pressure. Decreasing the value of n means the

distance at which atoms start to experience significant repulsive forces is

decreased. Therefore, higher pressures are required to overcome this increased

repulsion to form a solid phase. The coexisting solid and liquid densities for

different � values are illustrated in Figure 4.1(b). In common with the

coexistence pressure, decreasing the value of � causes both the liquid and solid

phase coexisting densities to increase. However, the relationship is not linear

and the difference between the liquid and solid densities decreases slightly with


149

decreasing n. These data and data for other temperatures are summarized in

Table 4.1.

Table 4.1 Molecular simulation data for the solid-liquid coexistence properties

of � � � Lennard-Jones fluids. The statistical uncertainty is given in brackets.

� = �rJs 1rJs �jGr 1jGr ∆ë

2.74

7 46.7(3) 1.339 -4.72(7) 1.391 -5.83(7) -2.41

8 42.6(3) 1.267 -4.01(7) 1.3206 -5.14(7) -2.48

9 39.9(3) 1.218 -3.66(6) 1.2781 -4.73(7) -2.61

10 37.2(3) 1.176 -3.53(6) 1.2416 -4.60(7) -2.73

11 35.5(3) 1.147 -3.45(6) 1.212 -4.55(7) -2.76

12 33.2(3) 1.116 -3.48(6) 1.1807 -4.60(6) -2.75

0.90

7 1.63(7) 0.965 -9.18(2) 1.031 -10.08(2) -1.01

8 2.17(7) 0.939 -8.04(2) 1.0119 -8.93(2) -1.05

9 2.40(8) 0.924 -7.31(2) 1.0025 -8.22(2) -1.11

10 2.63(9) 0.917 -6.83(2) 1 -7.76(2) -1.16

11 2.80(9) 0.913 -6.48(2) 1 -7.43(2) -1.21

12 2.7(1) 0.908 -6.21(1) 0.99935 -7.18(2) -1.24

0.80

7 0.39(6) 0.937 -9.22(1) 1.01191 -10.15(2) -0.96

8 0.82(6) 0.913 -8.07(1) 0.99178 -8.99(20 -0.99

9 1.10(7) 0.9 -7.34(1) 0.98928 -8.31(2) -1.07

10 1.41(7) 0.898 -6.88(1) 0.98428 -7.83(1) -1.08

11 1.43(8) 0.892 -6.53(1) 0.983 -7.49(2) -1.10

12 1.65(8) 0.891 -6.25(1) 0.983 -7.23(1) -1.15


150

6 7 8 9 10 11 12 1331

34

37

40

43

46

49

p

n

a

6 7 8 9 10 11 12 131.1

1.2

1.3

1.4

1.5

ρρρρ

n

b

Figure 4.1 Solid-liquid coexistence (a) pressure (�), (b) liquid (�) and solid (O)

densities as functions of n at T = 2.74.


151

0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.6

1.1

1.7

2.3

2.8

T

ρρρρ

a

0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.6

1.1

1.7

2.3

2.8

T

ρρρρ

b

Figure 4.2 Complete density-temperature phase diagrams of n-6 Lennard-Jones

potentials. Shown are (a) n = 12 (�, guided by a dashed line), 10 (�, guided

by a dotted line), 8 (O, guided by a solid line); and (b) n = 7 (�, guided by a

solid line), 9 (�, guided by a dotted line), 11 (∆, guided by a dashed line). The

vapour-liquid coexistence data are from (Kiyohara et al., 1996; Okumura and

Yonezawa, 2000). Freezing and melting lines and triple points are from this

work.


152

The data for the other temperatures show the same trend as � 2.74. The

energy and change in enthalpy are also given in Table 4.1 for the benefit of

completeness.

The temperature-density behaviour of the freezing and melting lines of n-6

Lennard-Jones potentials is illustrated in Figure 4.2. Vapour-liquid coexistence

data (Kiyohara et al., 1996; Okumura and Yonezawa, 2000) and triple point

data are also included to complete the phase diagram of the pure n-6 Lennard-

Jones fluids. Figure 4.2(a) represents the complete phase diagrams for n = 12,

10 and 8 and Figure 4.2(b) represents the complete phase diagrams for n = 11,

9 and 7.

It is well known (Okumura et al., 2000; Okumura and Yonezawa, 2000;

Kiyohara et al., 1996) that a decrease in the value of n increases the

temperature at the critical point, increasing the temperature range for two-

phase vapour-liquid coexistence. The main effect of decreasing n on solid-liquid

coexistence is to shift the melting and freezing curves to higher densities.

The variation in pressure for different n values with respect to temperature is

examined in Figures 4.3(a) and 4.3(b). At high temperatures, the pressure

decreases with increasing n. However, this trend is reversed at medium to low

temperatures, at which pressure increases with increasing n. The most notable


153

0.35 0.75 1.15 1.550.15

1

10

p

1/T

a

0.35 0 .75 1 .15 1 .550 .25

1

10

p

1/T

b

Figure 4.3 The solid-liquid coexistence pressure of n-6 Lennard-Jones potentials

calculated in this work as a function of reciprocal temperature on a log scale. (a)

n = 12 (∆, guided by solid line), 10 (�, guided by dotted line), and 8 (�,

guided by dashed line); and (b) n = 11 (�, guided by solid line), 9 (�, guided

by dotted line), and 7 (*, guided by dashed line).


154

7 8 9 10 11 120.035

0.050

0.065

0.080

0.095

n

r.d.d

a

7 8 9 10 11 120.04

0.05

0.06

0.07

0.07

0.08

0.09

f.d.d

n

b

Figure 4.4 (a) Relative density difference and (b) fractional density difference of

n-6 Lennard-Jones potentials at T = 1.0 (�) and T = 2.74 (O).


155

change occurs at very low temperatures, where an increase in the value of n

results in a sharp increase in pressure.

The solid-liquid coexistence region of the phase diagram is sensitive to the

nature of the interaction potential (Wang and Gast, 1999). The relative density

difference (r.d.d) and the fractional density change (f.d.c) at freezing (commonly

known as the miscibility gap) are two measures that can be used to quantify the

effect of the interaction potential on solid-liquid coexistence. The relative

density difference (r.d.d) is defined as (Hess et al., 1998) 2��jGr � �rJs�/��jGr ��rJs� where �jGr and �rJs are the solid and liquid coexistence densities of the system. The lower bound of the relative density difference is 0.037, which is the

approximate value for 12-inverse-power soft sphere systems (Hoover et al.,

1971). The upper bound is 0.098, which is the relative density difference for

hard spheres (Hoover and Ree, 1968). The n dependency of the r.d.d is shown in

Figure 4.4(a). We have also calculated the relative density difference of the 12-6

Lennard-Jones system from Agrawal and Kofke’s (Agrawal and Kofke, 1995c;

Kofke, 1993a) data, obtaining a value of 0.093. The miscibility gap or f.d.c is

defined as (Agrawal and Kofke, 1995c; Wang and Gast, 1999) ��jGr � �rJs�/�rJs and is shown in Fig. 4.4(b). It is evident that both f.d.c and r.d.d decrease with

decreasing � values. This means that decreasing n results in a smaller two-phase

region. Both metrics also decrease significantly with the increasing temperature.

Therefore, the size of the two-phase region is narrower at high temperatures

compared with low temperatures.


156

4.3 4.3 4.3 4.3 Temperature Temperature Temperature Temperature DDDDependence of Coexistence ependence of Coexistence ependence of Coexistence ependence of Coexistence Pressure Pressure Pressure Pressure

and Densitiesand Densitiesand Densitiesand Densities

The unique feature of � � 6 Lennard-Jones melting lines are that they shift

along the temperature axes with the variation of �. We have quantified the shift

of � � 6 Lennard-Jones melting lines along the temperature axes with respect to

the melting line of 12-6 Lennard-Jones potential using three (Simon and Glatzel,

1929) and four (Crawford and Daniels, 1971) parameters Simon-Glatzel

equations. To do that we have first fitted the 12-6 Lennard-Jones melting line

data with the three parameter Simon-Glatzel equation of the form:

= � y � zF (4.1)

where y, z, | are fitting parameters. We have obtained:

= � �4.8�1� � 8.9�1��.Ï�� (4.2)

Then we have used the following 4-parameter Simon-Glatzel equation to

calculate the shifts in the melting lines of � � 6 Lennard-Jones potentials for

� � 11,10,9,8,7 :

= � �4.8�1� � 8.9�1�� j��.Ï�� (4.3)

It is found that the shifts of the melting lines on the temperature axes decrease

with the increase of �-values and these data are collected in Table 4.2. These

shifts directly quantify the effects of the repulsive potentials on the melting

lines. It is expected that at sufficiently high temperatures � � 6 Lennard-Jones

potential models will behave as systems of aiH (inverse-power) soft spheres.


157

Table 4.2 Melting line shifts of � � � Lennard-Jones potentials with respect to

12-6 Lennard-Jones potential along the temperature axes form three and four

parameters Simon-Glatzel equations. Values in brackets are errors.

� of LJ � � 6 j 11 0.03(8)

10 0.05(8)

9 0.09(8)

8 0.13(8)

7 0.21(8)

At high temperatures repulsive force dominates and it is expected that the 12-6

Lennard-Jones potential must approach the scaling behaviour of the inverse

12th-power potential (Hoover et al., 1971). Using the special scaling properties

(Hoover et al., 1971; Hoover et al., 1970; Matsuda and Hiwatari, 1973) of the

inverse nth-power potential, Agrawal and Kofke (1995c) showed that

=��i� � éiý/Ï exp ¯��é�/�° ú16.89 � O�é � O�é�ü (4.4)

where é � 1/OP, 16.89 is the limiting soft sphere value of pβ5/4, D = 0.4759

was determined from soft-sphere simulation data and k1 and k2 are fitting

parameters. van der Hoef (2000) used Eq. (4.4) to reproduce the solid-liquid

coexistence data with very good accuracy. For soft-core systems, the equilibrium

melting temperature and pressure should satisfy Cn = pβα, where α = (3 +n)/n.

Morris et al. (2002) reported values of C12 = 16.89 ± 0.03 and C9 = 22.90 ± 0.03


158

for soft sphere potentials. We propose that Agrawal and Kofke’s (1995c) original

semi-empirical fit can be generalized for any n-6 Lennard-Jones potential:

=Hi� � éi��ÁH�/H exp��0.4759é�/��úO>�� O�é � O�é�ü (4.5)

The values of k0, k1 and k3 obtained from fitting our simulation data to Eq.

(4.5) are summarized in Table 4.3. Eq. (4.5) accurately reproduces the pressure-

temperature behaviour (0.8 ≥ T ≤ 2.74) as evident from a squared correlation

coefficient (R2) value of 0.99 for all n-6 Lennard-Jones potentials.

Table 4.3 Parameters for the scaling behaviour (Eq. (4.4)) of pressure as a

function of inverse temperature for � � � Lennard-Jones potentials. Errors are

given in parenthesis.

� � 6 k0 O� O� 12 1.36(1) -20.5(6) 3.8(4)

11 1.49(1) -23.6(9) 4.7(5)

10 1.61(2) -27 (1) 6.3(8)

9 1.82(4) -34 (2) 9(1)

8 1.93(2) -35(1) 8(1)

7 2.28(3) -48(2) 15(1)

van der Hoef (2000) fitted the freezing and melting densities for a 12-6 Lennard-

Jones via the following relationships involving β.

ρliq ==== β −−−−1/4 l0 ++++ l1β ++++ l2β

2 ++++ l3β3 ++++ l4β 4 ++++ l5β

5

ρsolid ==== β −−−−1/4 s0 ++++ s1β ++++ s2β2 ++++ s3β

3 ++++ s4β 4 ++++ s5β5

(4.6)


159

We found that simulation data for all of the n-6 Lennard-Jones potentials could

be accurately (R2 = 0.99) fitted to these equations. The values of the required

parameters are summarized in Table 4.4.

Table 4.4 Parameters for the polynomial fit (Eq. 4.6) for the coexisting liquid

and solid densities for n-6 Lennard-Jones potentials.

� Q> Q� Q� Q� QÏ Qý S0 s1 s2 s3 s4 s5

12 1.43985 1.31919 1.58685 1.82755 1.68777 1.86046 0.63619 4.49482 -12.9164 15.89534 -9.07314 1.97157

11 -1.3116 -0.19095 -1.68211 -2.66671 -1.5078 -1.89277 0.88257 3.1642 -9.78499 12.13758 -6.84999 1.45931

10 1.48422 -1.49835 1.98672 3.47735 1.07835 1.41498 1.97316 -3.80765 7.13875 -7.30466 3.79866 -0.78307

9 -1.01681 2.52647 -1.49615 -2.25097 -0.29379 -0.39337 1.84997 -2.43489 2.9722 -1.77939 0.41464 0

8 0.39561 -1.57908 0.68041 0.56061 0 0 1.74461 -1.58736 1.37156 -0.603 0.10872 0

7 -0.06812 0.35203 -0.14072 0 0 0 1.87403 -1.72006 1.1946 -0.29628 0 0

4.4.4.4.4444 Estimation of the Triple PointEstimation of the Triple PointEstimation of the Triple PointEstimation of the Triple Point

We have obtained estimates of the triple point by performing solid-liquid

equilibria simulations at low densities and, where necessary, slightly

extrapolating vapour-liquid data. The triple point liquid density and

temperature were identified by the intersection of the solid-liquid and vapour-

liquid coexistence data. The solid densities were estimated by extrapolating data

for the melting densities to the triple point temperature. We have determined

the triple point pressures from extrapolating our solid-liquid coexistence data for

T ≤ 0.8. The use of extrapolation means that the triple point values should only

be considered as reasonable approximations rather than accurate values.


160

Table 4.5 Comparison of triple point properties for the 12-6 Lennard-Jones fluid

obtained from molecular simulation studies. Errors are given in brackets

Source System

Size Z[ =Z[ �rJs,Z[ �jGr,Z[

Ladd and Woodcock

(1978a) 1500 0.67(1) -0.47(3) 0.818(4) 0.963(6)

Hansen and Verlet

(1969) 864 0.68(2) - 0.85(1) -

Agrawal and Kofke

(1995c) 236 0.698 0.0013 0.854 0.963

Agrawal and Kofke

(1995c) 932 0.687(4) 0.0011 0.850 0.960

This work 2048 0.661 0.0018 0.864 0.978

Triple point data in the literature are confined exclusively to the 12-6 Lennard-

Jones potential. The estimated triple point for the 12-6 Lennard-Jones potential

is compared with literature sources in Table 4.5. Our triple point temperature

differs by less that 4% from the values reported by either Hansen and Verlet

(1969) or Agrawal and Kofke (1995c). Indeed, it is well within the uncertainty

reported by Hansen and Verlet (1969). Our triple point densities are somewhat

higher than reported earlier (Agrawal and Kofke, 1995c; Hansen and Verlet,

1969). The triple point pressure is higher than reported elsewhere (Agrawal and

Kofke, 1995c), reflecting differences in both the estimated triple point

temperature and densities. We note that estimating the pressure is prone to

considerable uncertainties (Kofke, 1999) with early estimates yielding negative

values (Ladd and Woodcock, 1978b; Ladd and Woodcock, 1978a). The

differences between our calculations and that of Agrawal and Kofke can be


161

partly attributed to the effect of system size. Agrawal and Kofke observed a

1.6% decrease in temperature by increasing the system size from 236 to 932

atoms. In contrast, 2048 atoms were used for our simulations in the vicinity of

the triple point. The triple points for the remaining n-6 Lennard-Jones

potentials are summarised in Table 4.6. We were not able to reliably determine

the pressures for n = 7 and 8 because of the precipitous nature of the pressure

change close to the triple point. For the other n values a scaling relationship for

both triple point temperature (Figure 4.5(a)) and pressure (Figure 4.5(b)) with

respect to 1/n can be observed. In contrast, scaling behaviour is not apparent

for the densities (Figure 4.5(c)). These data can be adequately fitted by:

Ttr (n) ==== 2.10 / n ++++ 0.482

ptr (n) ==== 0.1104 / n −−−− 0.0073

(4.7)

From Eq. (4.7), the triple point temperature for the ∞-6 Lennard-Jones

potential is 0.482. The relatively small value of the intercept for the pressure

equation, suggests that the triple point pressure for the ∞-6 Lennard-Jones

potential is zero. This compares with a critical temperature of either 0.572 or

0.607 reported by Camp and Patey (2001) and Charpentier and Jakse (2005),

respectively and a critical pressure (Camp, 2003; Camp and Patey, 2001) of

0.079. These data are likely to be of value in calibrating “hard sphere +

attractive term” equations of state (Wei and Sadus, 2000).


162

Table 4.6 Estimated triple point properties for n-6 Lennard-Jones potentials.

� Z[ =Z[ �rJs,Z[ �jGr,Z[ ∞ 0.482 0 - -

12 0.661 0.0018 0.864 0.978

11 0.673 0.0028 0.867 0.982

10 0.689 0.0038 0.867 0.992

9 0.718 0.0049 0.883 1.000

8 0.748 - 0.899 1.028

7 0.782 - 0.932 1.050

4.5 Melting and Freezing Rules4.5 Melting and Freezing Rules4.5 Melting and Freezing Rules4.5 Melting and Freezing Rules

It has been observed that liquid freezing and solid melting follow certain

empirical rules. Most freezing rules involve the liquid structure as quantified by

the radial distribution function, whereas melting rules typically involve either

geometrical attributes or free energy calculations. In view of this, it is of interest

to examine the radial distribution functions for the n-6 Lennard-Jones fluids.

Figure 4.6 compares the radial distribution functions for the 12-6 Lennard-Jones

and 7-6 Lennard-Jones potentials at a common state point. It is apparent that

decreasing the value of n, results in higher maxima and lower minima, resulting

in narrower peaks.


163

0 . 0 8 0 0 . 0 9 6 0 . 1 1 2 0 . 1 2 8 0 . 1 4 40 . 6 2

0 . 6 8

0 . 7 4

0 . 8 0

T t r

1 / n

a

0 . 0 8 0 0 . 0 9 6 0 . 1 1 2 0 . 1 2 8 0 . 1 4 40 . 0 0 0

0 . 0 0 1

0 . 0 0 2

0 . 0 0 3

0 . 0 0 4

0 . 0 0 5

p t r

1 / n

b

0 . 0 8 0 0 . 0 9 6 0 . 1 1 2 0 . 1 2 8 0 . 1 4 40 . 8 5 0

0 . 9 0 5

0 . 9 6 0

1 . 0 1 5

1 . 0 7 0

1 / n

c

ρρρρ t r

Figure 4.5 Triple point properties of n-6 Lennard-Jones potentials as a function

of 1/n. Shown are (a) triple point temperatures (�), (b) pressures (�) and (c)

liquid (�) and solid (�) phase densities. The lines represent the least-squares

fit of the data given by Eq. (4.7).


164

0 1 2 3-0.0

0.5

1.0

1.5

2.0

2.5

3.0

g(r)

r

Figure 4.6 Comparison of the liquid phase radial distribution functions for a 12-

6 Lennard-Jones potential (solid line) and a 7-6 Lennard-Jones potential

(dashed line) in the liquid phase at þ � Å.� and à � Ç. á.

Table 4.7 Summary of parameters for melting and freezing rules for n-6

Lennard-Jones potentials at T = 1.0 and the melting or freezing densities.

n ·v¡ � �a*JH��a*dS� � const. Lindemann’s

Constant

Maximum in ��O>� 12 0.14 0.157 4.4

11 0.14 0.186 4.8

10 0.13 0.186 5.28

9 0.13 0.181 5.87

8 0.13 0.182 6.57

7 0.13 0.181 7.52


165

0.8 0.9 1.0 1.1 1.2 1.3-0.1

0.4

0.9

1.4

1.9

2.4

2.9

3.4

g(r)

r

a

1.0 1.2 1.4 1.6 1.8 2.0-0.1

0.4

0.9

1.4

1.9

2.4

2.9

3.4

g(r)

r

b

Figure 4.7 Comparison of the (a) first maxima and the (b) first minima at the

freezing point for n-6 Lennard-Jones fluids, where n = 7 (solid line), 9 (dashed

line) and 12 (dotted line). T = 2.74 and ρ = 1.339, 1.218 and 1.116 for n = 7, 9

and 12, respectively.


166

Figure 4.6 compares the radial distribution maxima (Figure 4.7(a)) and minima

(Figure 4.7(b)) for different n-6 Lennard-Jones fluids at the freezing point. It is

evident that the choice of n has a considerable influence of the structure of the

fluid at the freezing transition.

The Lindemann (1910), Simon-Glatzel (Crawford and Daniels, 1971; Simon and

Glatzel, 1929), and Ross (1969) are widely used examples of melting rules. The

most commonly used freezing rules are the Hansen-Verlet (Hansen, 1970;

Hansen and Verlet, 1969), Raveché-Mountain-Street (RMS) (Raveche et al.,

1974; Streett et al., 1974), and Giaquinta-Giunta (Giaquinta and Giunta, 1992)

rules. Agrawal and Kofke (1995c) concluded that many melting and freezing

rules were both temperature and density dependent. In contrast, the freezing

rule of RMS is almost invariant for the entire solid-liquid coexistence curve from

the triple point to the high temperature soft-sphere limit.

RMS (Raveche et al., 1974; Streett et al., 1974) observed that experimental

radial distribution function data generally obeyed the following relationship:

·v¡ � �a*JH�/�a*dS� � 0.2 (4.8)

where a*JH is the position of the first non-zero minimum of the pair

distribution, a*dS is the position of its first maximum. We calculated ·v¡ from

our simulation data for the n-6 Lennard-Jones potentials at a common

temperature of T = 1.0 (Table 4.7). It is apparent from Table 4.7 that IRMS is

largely invariant for all values of n. The difference in the value of IRMS (0.14) for


167

the 12-6 Lennard-Jones potential compared with the experimentally observed

value (0.2) partly reflects the limitation of the potential to fully represent the

properties of real fluids.

The Hansen-Verlet (Hansen, 1970; Hansen and Verlet, 1969) rule states that on

freezing the structure factor has a maximum value of ��O>� = 2.85. We have

obtained the structure factor via a Fourier transformation of the pair-correlation

function (Weeks et al., 1971). Hansen (1970) found that S(k0) changes with

increasing temperature for 864 particles. Agrawal and Kofke (1995c) also

observed a 10 % variation in this value with respect to temperature change.

They also found that Hansen-Verlet freezing rule also varied significantly with

the system size. For a system of 2048 Lennard-Jones (12-6) atoms at T = 2.74

we calculated the maximum structure factor to be 4.2. Values for the other n-6

Lennard-Jones potentials are summarized in Table 4.7. It is evident from the

data in Table 4.7 that the value of S(k0) depends on the value of n. This means

that the Hansen-Verlet freezing rule is not valid for n-6 Lennard-Jones

potentials.

The most commonly used model for predicting the melting line is the

Lindemann rule (Lindemann, 1910). The Lindemann ratio (L) is defined as the

root-mean-square displacement of particles in a crystalline solid about their

equilibrium lattice positions divided by the nearest neighbour distance (a). In a

MD simulation, L can be evaluated (Luo et al., 2005) via:


168

½ � ��aJ��Z � �aJ�Z��Jy (2.1)

Where �… �Z and �… �J denote ensemble averages over time and particles. The

Lindemann rule states that a solid melts if the root mean square displacement

of particles around their ideal position is approximately 10% of their nearest

neighbour distance, i.e., L ≈ 0.1. Many authors (Agrawal and Kofke, 1995c;

Hansen, 1970; Hansen and Verlet, 1969; Luo et al., 2005) have questioned the

quantitative prediction of Lindemann’s rule, although it is generally accepted as

being at least qualitatively correct. We have calculated Lindemann’s constant

for all n-6 Lennard-Jones potentials (Table 4.7) and as expected, L ≠ 0.1.

Nonetheless, L is close to being constant irrespective of the value of n, which

indicates that it is a valid indicator of the melting transition.

In a summary, in this Chapter, we have used the GWTS algorithm for

investigating solid-liquid phase coexistence of n-6 Lennard-Jones fluids as a

function of n. The data provide an insight into the role of intermolecular

repulsion on the solid-liquid transition. We have demonstrated how physical

properties and the melting rules vary with n. The data also allows us to

complete the phase diagrams of the n-6 Lennard-Jones fluids and estimate the

triple points.

169

Chapter 5Chapter 5Chapter 5Chapter 5 Phase DPhase DPhase DPhase Diagram iagram iagram iagram of the Weeksof the Weeksof the Weeksof the Weeks----

ChandlerChandlerChandlerChandler----Andersen PAndersen PAndersen PAndersen Potentialotentialotentialotential

In view of success of the GWTS algorithm to calculate the solid-liquid phase

coexistence of complex fluid described in Chapter 4, a comprehensive study of

the phase diagram of popular WCA potential is presented in this Chapter. In

this Chapter, causes behind the discrepancy in the literature data on WCA

solid-liquid coexistence properties are examined and resolved. Here the GWTS

and the GDI molecular simulation algorithms are used to determine the solid

liquid coexistence of the WCA fluid from low temperatures up to very high

temperatures. In Section 5.1, technical details of the GWTS and the GDI

algorithms, particular to this Chapter, are presented along with the system size

analyses. In Section 5.2, solid-liquid coexistence properties of WCA potential are

reported and validated from low to very high temperatures and pressures.

Relative density differences and fraction density changes are also measured in

this Section for the coexistence gaps and studied as a function of temperature.

In Section 5.3, low temperature and high temperature limits of WCA phase

diagram are estimated. Temperature dependence of coexistence pressure and

densities are examined in Section 5.4. In Section 5.5, suitable choice of WCA

equation of state is discussed via examination of the available models using

solid-liquid coexistence data. In Section 5.6, different melting and freezing

Phase Diagram of the Weeks-Chandler-Andersen Potential

170

hypotheses are tested based on simulation data of this Chapter. Entropy of

fusion is measured in Section 5.7 for the WCA system. In the last Section, phase

transition discontinuities are investigated via Simon-Glatzel and van der Putten

relations.

5.1 5.1 5.1 5.1 Simulation DSimulation DSimulation DSimulation Detailsetailsetailsetails

Since our simulations covered a wide range of temperatures and densities we

had to carefully choose the integration time step for different state points such

that the time step was small enough to solve the equations of motion correctly

and large enough to sample phase space adequately (Johnson et al., 1993). To

do that we have carried out short simulations to approximately locate solid-

liquid phase coexistence for selected temperatures using the same integration

time step 0.001. Thereafter, to determine the appropriate time step for any

given temperature, we performed NVE simulations at 10 random densities

around the approximate phase coexistence density to observe the conservation

of total energy. An order of 10-4 fluctuations in total energy ensures the correct

choice of time step (Johnson et al., 1993).

All simulation trajectories were typically run for 2 � 10ý time steps. The first

5 � 10Ï time steps of each trajectory were used either to equilibrate zero-

shearing field equilibrium molecular dynamics or to achieve non-equilibrium

steady state after the shearing field is switched on. The rest of the time steps in


171

each trajectory were used to accumulate the average values of thermodynamic

variables and standard deviations.

To obtain the most accurate results (as discussed below) for any given

temperature, different simulation algorithms were used for different ranges of

temperature. It is convenient to identify three different ranges of temperature.

The low temperature region is from � 0 to the Lennard-Jones triple point

temperature of T = 0.68. Intermediate temperatures are 0.68 < T < 2.74,

whereas T > 2.74 are high temperatures.

5.1.1 5.1.1 5.1.1 5.1.1 SimulationsSimulationsSimulationsSimulations at Low and Intermediate Tat Low and Intermediate Tat Low and Intermediate Tat Low and Intermediate Temperaturesemperaturesemperaturesemperatures

At low and intermediate temperatures, the solid-liquid phase coexistence

properties were obtained using the GWTS algorithm (Section 2.6.1). The initial

configuration in all the simulations was a face centered cubic (f.c.c) lattice

structure. The isothermal isochoric NEMD simulations were performed by

applying the standard sllod equations (Eq. (2.22)) of motion for planer Couette

flow (Fig. 2.4) coupled with Lees-Edwards (Section 2.5.1) periodic boundary

conditions. If the applied strain rate is switched off in sllod algorithm it behaves

like Newton’s equation of motions and NEMD converts to EMD. The NVT

EMD simulations were performed applying conventional cubic periodic

boundary conditions. In these molecular dynamics simulations a Gaussian

thermostat multiplier (Eq. (2.25)) was used to keep the kinetic temperature of


172

the fluid constant. The equations of motion were integrated with a five-value

Gear predictor-corrector scheme (Section 2.5.3). Details of all these algorithms

are given in Chapter 2.

We performed a wide range of simulations for locating the solid-liquid phase

transitions with a 0.01 increment in densities. For each state point we have

carried out three simulations, one for zero strain rate equilibrium molecular

dynamics and normally two for strain rates of 0.1 and 0.2. In principle, only the

difference between the 0 and 0.1 strain rate isothermal-isochoric simulation is

sufficient to detect the phase transitions. But to clarify the issue of solid-like

and liquid-like metastable states we performed additional simulations for the 0.2

strain rate. Details on this choice can be found in Section 2.6.1.

For temperatures 0.0001 � � 0.01 we used a density interval 0.01 and strain

rates of 0.01 and 0.02. At these very low temperatures, the higher strain rates of

0.1 and 0.2 did not exhibit the necessary pressure jump for locating the solid-

liquid phase transition (discussed in Section 2.6.1). We conducted test runs on

WCA liquid at low temperatures with different strain rates to check for strain

rate independent Couette flow behaviour. The melting point density is accurate

to within the limit of the density change, whereas the accuracy of the freezing

point density is 0.01.


173

5.1.2 5.1.2 5.1.2 5.1.2 Simulations at High TSimulations at High TSimulations at High TSimulations at High Temperaturesemperaturesemperaturesemperatures

The solid-liquid coexistence properties at high temperatures were obtained using

the GDI algorithm (Section 2.6.2). The starting point required by the GDI

method was obtained from the results obtained at intermediate temperatures

described above. The Clapeyron equation used in the evaluation of GDI series is

related to the stability of the integration at a given temperature

Table 5.1 Coexistence pressures and melting and freezing densities of a WCA

fluid at þ � Ç. á for different number of particles.

Y = �rJs �jGr 108 11.195 0.93 0.99

256 12.014 0.94 1.01

864 13.127 0.96 1.03

2048 12.577 0.95 1.016

4000 12.576 0.95 1.016

13500 13.097 0.96 1.03

32000 13.058 0.96 1.03

62500 13.061 0.96 1.03

(Kofke, 1993b; Kofke, 1993a). For the Lennard-Jones potential, it has been

reported (Agrawal and Kofke, 1995c) that two different versions of Clapeyron

equation must be used below and above T = 2.74 to maintain the stability of


174

the integration. The starting point of T = 2.74 was used for the high

temperature GDI simulations without any further rigorous testing of the GDI

series for the WCA potential.

At the beginning of the simulation 932 atoms were distributed between boxes to

represent solid and liquid phases. The box in the liquid phase contained 432

atoms while the box in solid phase contained 500 atoms in the face-centred-

cubic lattice structure. The simulations were performed in cycles. A simulation

period of 20,000 was used to accumulate the simulation averages followed by an

equilibration period of cycles 20,000. In all the GDI series of simulations, the

temperature change per step in the decreasing direction was, ∆é � 0.03, where

é is the reciprocal of the temperature.

5.1.3 5.1.3 5.1.3 5.1.3 Finite Size EFinite Size EFinite Size EFinite Size Effectsffectsffectsffects

The system size dependency of the solid-liquid phase coexistence pressure,

temperature and densities for WCA system are not known. For the Lennard-

Jones potential thermodynamic variables can vary from 3% to 6% depending on

the system size (Mastny and de Pablo, 2007, Morris and Song, 2002) and we

can also expect similar size effect for WCA system. To test for the effect of

system size on our results, we performed simulations at temperature T = 1.0 for

N = 108, 256, 864, 2048, 4000, 13500 and 62500 and the results are summarized

in Table 5.1. The maximum variation of the coexistence pressure is 14.3%,


175

whereas a maximum variation of 3.1% is observed for the coexistence densities.

The effect of system size does not appear to scale with 1/ N . Using N = 4000

represents a reasonable compromise between maintaining accuracy and

minimizing computational effort.

Table 5.2 Solid-liquid phase coexistence properties of the WCA potential at low

to intermediate temperatures. Values in brackets represent the uncertainty in

the last digit.

= �rJs 1rJs �jGr 1jGr ∆ë

0.001 0.0068(3) 0.65 6.5E-5(4) 0.7 5.9E-5(4) 0.00076

0.003 0.0217(7) 0.66 3.4E-4(1) 0.72 3.1E-4(1) 0.00278

0.006 0.045(1) 0.67 9.8E-4(4) 0.735 8.8E-4(4) 0.00607

0.009 0.069(1) 0.68 18.1E-4(7) 0.744 16.2E-4(6) 0.00904

0.01 0.077(1) 0.68 21.1E-4(8) 0.74 18.5E-4(7) 0.00954

0.05 0.439(6) 0.73 22.7E-3(5) 0.8 0.0201(5) 0.05534

0.1 0.94 (1) 0.76 0.062(1) 0.833 0.054(1) 0.11700

0.2 2.03(2) 0.80 0.166(2) 0.87 0.141(2) 0.22884

0.3 3.14 (3) 0.83 0.285(4) 0.90 0.248(4) 0.33166

0.4 4.37(4) 0.85 0.427(6) 0.92 0.364(6) 0.45545

0.5 5.60(5) 0.87 0.574(7) 0.94 0.492(7) 0.56269

0.6 6.95(5) 0.89 0.737(9) 0.96 0.633(9) 0.67325

0.7 8.42(6) 0.91 0.91(1) 0.98 0.78(1) 0.78951

0.8 9.60(7) 0.92 1.07(1) 0.99 0.92(1) 0.88071

0.9 11.28(8) 0.94 1.27(1) 1.00 1.07(1) 0.92090

1.0 12.57(9) 0.95 1.44(1) 1.016 1.24(1) 1.05865

1.15 15.5(1) 0.98 1.80(1) 1.05 1.57(1) 1.28860

2.00 30.4(1) 1.07 3.72(3) 1.14 3.31(3) 2.15879

2.74 45.1(2) 1.13 5.60(4) 1.20 5.04(5) 2.89380


176

5.2 5.2 5.2 5.2 SolidSolidSolidSolid----Liquid CLiquid CLiquid CLiquid Coexistenceoexistenceoexistenceoexistence

The WCA simulation data for low to intermediate temperatures and high

temperatures are summarized in Tables 5.2 and 5.3, respectively. Solid-liquid

coexistence data for the WCA potential have only been reported previously (de

Kuijper et al., 1990) at T = 0.5, 0.75, 1.0, 1.25, 1.5, 2.0 and 5.0. A comparison

of our data with literature data for the pressure and coexisting liquid and solid

densities is given in Fig. 5.1. The relative difference between our calculations

and literature data is quantified in Fig. 5.2. For T = 0.50 the reported pressure,

freezing density and melting density are approximately 13.1%, 3.3%, and 3.0%

higher, respectively than our results. For T = 1.25 our pressure is 1.9% higher

than reported elsewhere. The differences in pressure at T = 1.5 and T = 2.0 are

approximately 3% and 5.9%, respectively.

As discussed above, and illustrated in Table 5.4, there are considerable

discrepancies in the literature between the values previously reported by other

workers for T = 1.0. At this temperature, our liquid density is in good

agreement with the values reported for either NVT MD (Hess et al., 1998) or

MC (de Kuijper et al., 1990) simulations, which in turn is higher than NpT MD

(Hess et al., 1998) calculations. Our solid density also agrees reasonably well

with NVT MC data, which is higher than either NVT or NpT MD simulations.

The liquid phase pressure is in agreement with the either the NVT MC or MD

calculations, which is higher than the NpT MD simulations. In view of this, our


177

MD results clearly resolve the discrepancy in the literature in favour of the

NVT MC (de Kuijper et al., 1990) data.

Table 5.3 Solid-liquid phase coexistence properties of the WCA potential at high

temperatures. Values in parentheses represent the uncertainty in the last digit.

= �rJs 1rJs �jGr 1jGr Δë

3.63636 66.327 1.165(1) 7.21(2) 1.218(1) 6.31(1) 3.033

4.08163 77.605 1.198(1) 8.72(2) 1.252(1) 7.73(2) 3.376

4.65116 92.632 1.237(1) 10.65(2) 1.294(1) 9.53(3) 3.795

5.40540 113.476 1.286(1) 13.29(3) 1.342(1) 12.04(4) 4.364

6.45161 143.809 1.342(1) 17.08(3) 1.401(1) 15.62(3) 4.969

8.00000 191.135 1.415(2) 22.78(4) 1.475(1) 21.04(4) 5.973

10.52631 274.739 1.514(2) 32.46(7) 1.576(1) 30.29(5) 7.159

15.38461 450.978 1.662(2) 51.6(1) 1.729(1) 48.55(7) 9.311

28.57142 1021.973 1.924(5) 109.2(2) 2.000(3) 103.3(2) 23.769

The solid-liquid phase diagram is illustrated in Figure 5.3. It is apparent from

Figure 5.3(a) that the difference between liquid and solid densities is relatively

small at low temperatures but progressively increases with increasing

temperature.

The effect of intermolecular interactions on the two phases can be quantified in

terms of both the relative density difference (r.d.d) and the fractional density

change (f.d.c) at freezing (also known as miscibility gap).


178

0 .4 1 .6 2 .8 4 .0 5 .20

1 5

3 0

4 5

6 0

7 5

9 0

1 0 5

p

T

a

0 .4 1 .6 2 .8 4 .0 5 .20 .8 0

0 .9 1

1 .0 2

1 .1 3

1 .2 4

1 .3 5

ρρρρ l iq

T

b

0 .4 1 .6 2 .8 4 .0 5 .20 .9

1 .0

1 .1

1 .2

1 .3

1 .4

ρρρρ s o l

T

c


densities and (c) solid densities for the WCA potential calculated in this work

(�) with data from de Kuijper et al. (1990) (). The errors are approximately

equal to the symbol size.


179

0.4 1.6 2.8 4.0 5.2-15

-10

-5

0

5

10

15

T

Figure 5.2 Comparison of the relative percentage difference of pressures (�),

liquid densities (�) and solid densities (�) as a function of temperature (de

Kuijper et al., 1990).

Table 5.4 Comparison of WCA solid-liquid coexistence data at þ � Ç. á.

Method Y =rJs �rJs =jGr �jGr Source

NVT MC 500 12.60 0.952 12.60 1.023 de Kuijper et al. (1990)

NVT MD 2048 12.62 0.960 10.65 0.970 Hess et al. (1998)

NpT MD 8788 10.65 0.912 10.67 0.971 Hess et al. (1998)

GWTS 4000 12.57 0.950 12.57 1.016 This work

100 ×literature − simulation

literature


180

0 5 10 15 20 25 300

400

800

1200

p

T

a

0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.10

5

10

15

20

25

30

T

ρρρρ

b

Figure 5.3 The solid-liquid coexistence (a) pressure (�), (b) freezing (upper)

and melting line (lower) density (�) as a function of temperature.


181

The relative density difference (r.d.d) is defined as (Hess et al., 1998) �� 2��jGr � �rJs�/��jGr � �rJs� where �jGr and �rJs are the solid and liquid densities of the system at solid-liquid coexistence. The miscibility gap or f.d.c is defined

as (Agrawal and Kofke, 1995c) ��jGr � �rJs�/�rJs. The lower bound of the

relative density difference is 0.037 which corresponds approximately to 12-

inverse-power soft spheres (Hoover et al., 1971, Hoover and Ree, 1968) and the

upper bound value is 0.098, which is the value for hard spheres (Hoover and

Ree, 1968).

The temperature dependency of the r.d.d and f.d.c are illustrated in shown in

Fig. 5.4. It is evident that both metrics decreases with increasing temperature.

A comparison is also made in Fig. 5.4 with the 12-6 LJ potential. It is apparent

that the values obtained for the WCA are lower than the 12-6 LJ values in all

cases. The average relative density difference of 12-6 LJ system from Agrawal

and Kofke’s data is 0.093 compared with 0.060 obtained from our data, which

lies between the hard sphere and ai�� soft sphere values. For T = 1.0 Hess et

al. (1998) and de Kuijper et al. (1990) reported �� 0.063 and �� 0.0718,

respectively. This compares to �� 0.067 obtain from our data.


182

0 5 10 15 20 25 300.038

0.042

0.046

0.050

0.054

0.058

r.d.d

T

a

0 5 10 15 20 25 300.038

0.042

0.046

0.050

0.054

0.058

f.d.c

T

b

Figure 5.4 Comparison of (a) r.d.d and (b) f.d.c WCA data as a function of

temperature obtained in this work (�) with the LJ (Ο) data from Agrawal and

Kofke (1995c).


183

0 5 10 15 20 25 301.5

10

100

1000

p

T

a

Figure 5.5 Comparison of (a) the overall pressure-temperature and the (b)

pressure-low temperature behavior of the WCA fluid calculated in this work

(�) with literature data (Agrawal and Kofke, 1995c) for LJ potential (---). The

LJ data were supplemented by calculations using Eq. (1) from van der Hoef

(2000).

0.00 0.32 0.64 0.961E-8

1E-5

0.01

10

p

T

b


184

5.3 5.3 5.3 5.3 Low and Low and Low and Low and High Temperature LHigh Temperature LHigh Temperature LHigh Temperature Limitsimitsimitsimits

It is expected that at high temperatures the average kinetic energy will be such

that it would be impossible to distinguish between 12-6 LJ and WCA

interactions. de Kuijper et al. (1990) predicted that this would occur at T > 10.

We have calculated solid-liquid coexistence for temperatures up to � 28.57

and these data are compared with results for the Lennard-Jones potential in

Fig. 5.5(a). It is apparent from this comparison that the WCA and LJ results

begin to converge at high temperatures. Both the WCA and 12-6 LJ

calculations show convergence to a common asymptote (Agrawal and Kofke,

1995c) at 1/ � 0 which corresponds the 12th-power soft sphere limit.

In contrast to the high temperature behaviour, Fig. 5.5(b) indicates that the

WCA and 12-6 LJ fluids behaviour very differently as � 0. A clear divergence

in the pressure-temperature behaviour of the two potentials is evident for �1.0. There is an apparent discontinuity (Agrawal and Kofke, 1995c) in pressures

for the 12-6 LJ potential, which stops short of approaching T = 0. In contrast,

the WCA calculation approach T = 0. At � 0, we estimate that the pressure

is => �� 0.0068.


185

5.4 Temperature D5.4 Temperature D5.4 Temperature D5.4 Temperature Dependence of ependence of ependence of ependence of CCCCoexistence oexistence oexistence oexistence PPPPressure ressure ressure ressure

and and and and DDDDensitiesensitiesensitiesensities

The fact that the WCA fluid has the same (Fig. 5.6) soft-sphere high

temperature limiting behaviour as the 12-6 LJ potential means that it should be

possible to fit Eq. (4.4) (Section 4.3) to our data. However, unlike the 12-6 LJ

potential the WCA pressure is continuous until T = 0, which means a fit can be

made for the entire temperature range.

0 1 2 3 4 51.2

10

100

1000

p

1/T

Figure 5.6 Comparison of the solid-liquid coexistence pressure (�) of WCA

potential with 12-6 LJ potential (---) (Agrawal and Kofke, 1995c) as a function

of reciprocal temperature.


186

For the WCA, we found that the introduction of an additional parameter �O>� in Eq. (4.4) was required to account for these differences:

= �� éiý/Ï��=¯��é�/�°�16.89O> � O�é � O�é�� (5.1)

When O> � 2.34, O� � �59 and O� � 12 , Eq. (5.1) accurately reproduces the

pressure-temperature behaviour as evident from a squared correlation coefficient

(R2) value of 0.99.

In Section 4.3 we fitted our n-6 Lennard-Jones freezing and melting densities via

two polynomial relationships involving β (Eq. (4.6)) following the approach of

van der Hoef (2000). However, we fitted our WCA data via the following

relationships:

��rJs � éiýÏ��= «�0.51é�ý¬� �0.027 � 1.47é � 0.12é� � 1.4 � 10iÏé� � 1.94 � 10i�éÏ��jGr � éiýÏ��= «�0.51é�ý¬� �0.027 � 1.55é � 0.14é� � 1.1 � 10iÏé� � 2.1 � 10i�éÏ� ±²

³² (5.2)

In Eq. (5.2) we have introduced an exponential term to van der Hoef’s original

formulas to accommodate the low temperature behavior of the WCA fluid. We

found that the WCA simulation data could be accurately (R2 = 0.99) fitted to

these equations.

5.5 5.5 5.5 5.5 Comparison with Comparison with Comparison with Comparison with EEEEquation of quation of quation of quation of SSSState tate tate tate CCCCalculationsalculationsalculationsalculations

Attempts (Heyes and Okumura, 2006, Kolafa and Nezbeda, 1994, Verlet and

Weis, 1972) have been made to develop an equation of state for the WCA


187

potential. The usual stating point is the Carnahan-Starling equation (1969),

which provides an accurate description of the compressibility factor of hard

spheres (Z) in terms of the packing fraction of hard spheres (y = πρσ3/6):

î � 1 � U � yU� � zU��1 � U�� (5.3)

where y and z are adjustable parameters. When y � z � 1, Eq. (5.3) is the

original Carnahan-Starling equation. A WCA equation of state is obtained by

simply substituting the hard-sphere diameter with a temperature-dependent

formula:

( )1/ 6

1/ 6

1 0.5

0.5

2Heyes and Okumura

1

0.3837 1.068Verlet and Weis

0.4293 1

0.11117524 0.076383859

1.08014248 0.000693129 Kolafa and Nezbeda

0.063920968log

T

T

T

T T

T

T

σ

σ

σ − −

=

+

+ = + = −

+ + −

(5.4)

Figure 5.7 compares the compressibility factor predicted by these WCA

equations of state with our data for the freezing line at different temperatures.

It is apparent from this comparison that the Kolafa and Nezbeda (1994) and

Verlet and Weis (1972) equations fail to even qualitatively reproduce the

variation of the compressibility factor with respect to temperature. The Heyes

and Okumura (2006) equation yields qualitative agreement for the


188

compressibility factors at all temperatures. The average absolute deviation

between the compressibility factors predicted by the Heyes and Okumura

equation and our simulation data is 8.5%. In view of this reasonably good

quantitative agreement, we used our simulation data to revaluated the y and z

parameters of the Heyes and Okumura equation. Values of y � 17.22 and z �31.1 result in an average absolute deviation of approximately 0.38% and good

agreement at all temperatures (Fig. 5.7). This suggests that the reparametrized

Heyes and Okumura equation could have a role in equation of state

development for real fluids as an alternative to the Carnahan-Starling hard-

sphere term.

5.6 Melting and Freezing Rules5.6 Melting and Freezing Rules5.6 Melting and Freezing Rules5.6 Melting and Freezing Rules

A commonly used method to predict melting is the Lindemann (1910) rule

which states that a solid melts if the root mean square displacement of particles

around their ideal position is approximately 10% of their nearest neighbour

distance. Luo et al. (2005) reported that the Lindemann rule is valid for the

Lennard-Jones potential for a wide range of pressures. In view of this, it is

reasonable for it to also apply to WCA fluids.


189

0 5 10 15 20 25 3012

19

26

33

40

Z

T

Figure 5.7 Comparison of molecular simulation data for the compressibility

factors of the WCA fluid obtained in this work (�) with calculations using the

Heyes and Okumura (Ο), Verlet and Weis EOS (�) and Kolafa and Nezbeda

EOS (�) equations of state. The solid line represents calculations of the

reparametrized Heyes and Okumura equation reported here.


190

Ashcroft and Lekner (1966) and Ashcroft and Langreth (1967) showed that,

when viewed from the liquid side of the phase transition, Lindemann’s melting

rule and can be expressed as: L � σHS�T�3 Vm⁄ � const., where k* equal to the

volume of the liquid and t�� is the temperature-dependent hard-sphere

diameter. Heyes and Okumura (2006) found that good approximation for the

Table 5.5 Invariants of the Lindemann (1910), Raveché et al. (1974) and

Hansen and Verlet (1969) melting or freezing rules as a function of coexistence

temperature....

½ � t��/k* � � �a*dS�/�a*JH� ��O� 0.1 1.106 0.136 3.10

0.2 1.095 0.204 3.02

0.3 1.084 0.151 2.97

0.4 1.080 0.160 2.90

0.5 1.081 0.165 2.86

0.6 1.084 0.168 2.83

0.7 1.077 0.168 2.80

0.8 1.084 0.175 2.76

0.9 1.080 0.173 2.75

1 1.100 0.174 2.70

1.15 1.179 0.152 2.85

2 1.154 0.143 2.86

2.74 1.237 0.187 2.47


191

effective hard sphere diameter of the WCA potential is t� � 2�/�/¯1 � √°�/�.

Values of L obtained from our simulation data at various temperatures are

summarized in Table 5.5. It is evident that L is close to being constant for most

temperatures.

Raveché et al. (1974) proposed that the ratio of the first maximum to the

nonzero first minimum of the radial distribution function on the freezing line is

constant �� a*dS� �a*JH� �⁄ |¼�¿g. �. Figure 5.8 (a) compares the radial

distribution functions at a melting and freezing densities. We have calculated

values of � for the freezing densities at low and intermediate temperatures and

the results are summarized in Table 5.5. Although there is some variability in

the value of �, it can be used as a reasonable indicator of freezing.

We have also tested the Hansen and Verlet (1969) freezing rule, which says that

upon freezing the structure factor has a maximum value of ��O>� = 2.85. We

have obtained the structure factor via a Fourier transformation of the pair-

correlation function. An example of our calculations is illustrated in Fig. 5.8(b)

and the maximum values at the freezing density at various temperatures are

summarized in Table 5.5. Although the required value is not exactly obtained,

in most cases the deviation is relatively small.


192

5.7 Entropy of F5.7 Entropy of F5.7 Entropy of F5.7 Entropy of Fusionusionusionusion

The Clausius Clapeyron equation relates the slope to the entropy change and

the difference between the specific volumes of the solid and the liquid phases

through the relation: �=/� � ��/�k. A different form of Clapeyron equation

also relates the enthalpy change: �=/� � ��/�k. Comparing these forms of

Clausius-Clapeyron relations we have calculated the entropy change: �� /

at solid-liquid coexistence. The simulation data for the change in enthalpy

(Tables 5.2 and 5.3) allows us to calculate the entropy of fusion, i.e.,

∆S = ∆H / T (Fig. 5.9). It is evident that for T < 10, there is a steep increase

in ∆S, which reflects the high degree of order of the solid phase compared with

higher temperatures which approach a constant value. Fig. 5.9 also compares

∆� for the LJ and WCA potentials and it is evident that the WCA values are

lower at all temperatures. The average value of entropy change calculated was

∆� � 1.20 with a variation of about 14%, which compares with an average value

of 1.36 calculated from LJ data of Agrawal and Kofke (1995c). For a real

substance like aluminium the value of the calculated (Morris et al., 1994)

entropy change is ∆� � 1.2 compared to an experimental value of ∆� � 1.4.

We note that, because of the way the reduce constants are defined, the value of

the entropy of fusion can be transformed into real units by simply multiply the

reduced value by the Boltzmann constant. This means that the entropy of

fusion predicted by the WCA, Lennard-Jones and other such two-parameter


193

0 1 2 3 40

1

2

3

4

r

g(r)

a

5.5 9.0 12.5 16.00.30

0.95

1.60

2.25

2.90

r

S(k)

b

Figure 5.8 (a) Comparison of radial distribution functions at þ � Ç. á for the WCA fluid at freezing (solid line) and melting (dashed line) points. (b) A

typical structure factor curve for the WCA fluid at a freezing point �à � á.��,þ � Ç. Ç��.


194

0 5 10 15 20 25 300.82

0.86

0.90

0.94

0.98

1.02

1.06

1.10

��S

T

Figure 5.9 Comparison of entropy of fusion obtained in this work (�) for the

WCA fluid with the data for the Lennard-Jones potential (Ο) (Agrawal and

Kofke, 1995c).

potentials is independent of the value of the potential parameters. This provides

a convenient way of directly comparing different intermolecular potentials with

one another. More importantly, if experimental entropy of fusion data is

available for comparison, this insight allows us to quickly assess how accurately

an intermolecular potential is likely to predict the thermodynamic properties of

real fluids at any temperature.

5.8 5.8 5.8 5.8 Volume Discontinuity at Volume Discontinuity at Volume Discontinuity at Volume Discontinuity at the the the the Phase TPhase TPhase TPhase Transitionransitionransitionransition

The discontinuities of volume have been calculated for the WCA solid-liquid

phase coexistence using the limited number of data points for the temperature


195

0 5 10 15 20 25 300.01

0.04

0.07

0.10

0.13

0.16

|δδδδV|

T

Figure 5.10 Comparison of volume change, |��|, calculated in this work (�) for

the WCA fluid with the data for the 12-6 Lennard-Jones potential (Ο) (Agrawal

and Kofke, 1995c).

Table 5.6 Parameters of Simon’s equation and van der Putten’s relation both

for WCA and 12-6 LJ potentials obtained from the least-squares fit of solid-

liquid coexistence pressure data and volume jump data as a function of

temperature, respectively.

Equation Parameters WCA potential 12-6 LJ potential

Simon and Glatzel

(1929)

=> �� or =>�� -19.1(8) -11.2(8)

b 16.2(1) 14.30(7)

c 1.324(2) 1.274(9)

van der Putten et al.

(1986)

d 0.087(4) 0.0679(7)

e -0.49(8) 0.53(1)

f 1.48(3) 1.39(1)

Relation between exponents c and f f = c + 0.15705 f = c + 0.11906


196

range 0.5 ~ ~ 5. It is interesting to observe the behavior of discontinuities

along the melting line considering a wide range of data points for temperature

range 0.001 <T< 28.57 (Fig. 5.10). Using the value of the melting temperature

exponent of Simon and Glatzel (1929) equation van der Putten and co-workers

(van der Putten and Schouten, 1986; van der Putten and Schouten, 1987a; van

der Putten and Schouten, 1987b) has demonstrated that the absolute value of

volume change, |�k| during the solid-liquid transition will decrease with the

increase of temperature and approximates the shape of the melting line given by

the relation: |�k| � �/� � ��Fi� where e is approximately two-third of the

triple point temperature in case of 12-6 Lennard-Jones potential, c is the power

law exponent obtained from the Simon-Glatzel equation and d is the fitting

parameter. We have tested van der Putten equation by fitting with more recent

comprehensive data of 12-6 LJ system obtained from Agrawal and Kofke

(1995c) and found that their claim is true only for temperature range 0.5 ~ ~5. Then, without assuming any values for the parameter e and c, we

determined the values via a least-squares fit of the volume jump as a function of

temperature both for 12-6 LJ system and WCA system and the results were

summarized in Table 5.6. The sources of the discrepancy are clearly understood

from the wrong choice of Simon-Glatzel exponent for WCA system equal to the

12-6 LJ system and the lack of enough low temperature and high temperature

data considered in de Kuijper et al. (1990).


197

In a summary, in this Chapter, we have provided a comprehensive description

of the solid-liquid equilibria of the WCA fluid. We have also reported data for

the phase diagram using the GWTS and the GDI techniques. The conjecture of

abnormal behavior at low temperatures has been investigated by examining the

melting behavior at very low temperatures. We have also traced the melting line

of the WCA potential to very high temperatures to test the hypothesis that it

approaches a 12th-power soft-sphere asymptote. An improved WCA equation of

state and three empirical expressions for the solid-liquid coexistence pressure,

freezing density, and melting density have been reported. In this Chapter it has

been established that the GWTS algorithm can also efficiently simulate the

phase diagram of purely repulsive potential. It remains to test the capability of

this algorithm in calculating solid-liquid phase coexistence for bounded potential

and this problem will be addressed in the next Chapter.

198

Chapter 6Chapter 6Chapter 6Chapter 6 Phase Diagram of the Phase Diagram of the Phase Diagram of the Phase Diagram of the

Gaussian Core Model FGaussian Core Model FGaussian Core Model FGaussian Core Model Fluidluidluidluid

Determining solid-liquid phase transitions of the GCM fluid is a severe test for

the GWTS algorithm because the GC model has a very small range of densities

in which phase separation can occur and it has a complex re-entrant melting

scenario. The interaction potentials used in Chapters 3, 4 and 5, to calculate

solid-liquid phase coexistence, are unbounded potentials. In contrast, the

Gaussian core model potential adopted in this Chapter is bounded potential. In

this Chapter the solid-liquid phase equilibria of the Gaussian core model are

determined using the GWTS algorithm. This is the first reported use of the

GWTS algorithm for a fluid system displaying a re-entrant melting scenario. In

Section 6.3, system size effect on the GC phase envelope is studied. In Sections

6.3 and 6.4, the phase transitions of low-density side and the high-density side

are described, respectively. The complete GCM phase diagram is drawn in

Section 6.5 with a focus on the comparative analyses of the advantage of using

GWTS algorithm.

Phase Diagram of the Gaussian Core Model Fluid

199


As detailed in Chapter 2, we used the sllod algorithm (Section 2.5.2), with a

Gaussian thermostat (Eq. (2.25)) and 5-value Gear predictor-corrector scheme

(Section 2.5.3) with a time step of τ = 0.005 and a cut-off radius for the

potential of 3.2σ. We used three different strain-rates at �� = 0.0 (EMD

simulation), �� = 0.001 and �� = 0.002 (NEMD simulations). For each state point

(ρ, T, ��) simulation trajectories were obtained for a length of 8 × 105 τ. Periods

of 3 × 105 τ of each trajectory were used either to equilibrate zero-shearing field

EMD or to achieve non-equilibrium steady state after the shearing field was

switched on. The remaining time periods were used to accumulate the average

values of thermodynamic variables. We used 2048 GC particles for all

simulations reported in this work. Near the solid-liquid transition we used very

small density increments ∆ρ = 10-4 in order to sample the extremely small two-

phase liquid-solid region of the GCM with high accuracy.

6.2 6.2 6.2 6.2 System Size AnalysisSystem Size AnalysisSystem Size AnalysisSystem Size Analysis

Simulation of phase transitions might be sensitive to the system size of a fluid.

Therefore, we performed separate simulation runs to analyse the dependency of

the simulation results on the particle number. In particular, we calculated the

freezing point on the low-density side of the solid region of the GCM for a single

temperature at T = 0.006. We analysed the occurrence of the freezing point for

system sizes of N = 256, 864, 2048, 4000, 6912 and 10976 particles (Table 6.1).


200

For particle numbers N = 256 and 864 the freezing densities were determined

with greater uncertainties. The uncertainties for the freezing densities of particle

numbers N = 2048 to 10976 are moderate. The average freezing density

calculated from Table 6.1 is 0.129983 ± 0.000492 within the 95% confidence

interval. The freezing density for N = 2048 fits fairly well within the average

density and therefore we believe that this particle number is a very good choice

for the purpose of our study.

Table 6.1 System size dependency of the freezing density of the GCM fluid at T

= 0.006 obtained using the GWTS algorithm.

N �

256 0.1309

864 0.1291

2048 0.1299

4000 0.1299

6912 0.1304

10976 0.1297

6.3 6.3 6.3 6.3 LowLowLowLow----DDDDensity ensity ensity ensity SSSSide of the ide of the ide of the ide of the SSSSolid olid olid olid RRRRegionegionegionegion

On the low-density side of the solid state the GCM fluid behaves as a “normal”

liquid (Mausbach and May, 2006). In Fig. 6.1 we show a typical result of our

simulation in this density region for a temperature of T = 0.006. The strain-rate


201

0 0 .0 0 0 5 0 .0 0 1 0 0 .0 0 1 5 0 .0 0 2 00 .0 2 1 5

0 .0 2 1 9

0 .0 2 2 3

0 .0 2 2 7

γ &

p

a

0 .1 2 9 0 0 .1 2 9 8 0 .1 3 0 6 0 .1 3 1 4 0 .1 3 2 20 .0 2 1 5

0 .0 2 1 9

0 .0 2 2 3

0 .0 2 2 7

0 .0 2 3 1

f pm p

p

ρρρρ

b

Figure 6.1 Low-density side of the GCM solid state at T = 0.006. (a) Pressure

as a function of strain-rate at different constant densities. Shown are results for

densities of 0.1296 (�), 0.1297 (�), 0.1298 (�), 0.1299(�), 0.1300 (),


phase solid-liquid region is clearly seen by the sudden drop in pressure at zero

strain-rate. (b) Pressure as a function of density for different strain-rates å� � á. á (�), å� � á. ááÇ (), å� � á. ááÅ (�), all in the stable liquid state and

its metastable extension, and å� � á. á (�) in the stable solid state and its



pressure.


202

0 0 .0005 0 .0010 0 .0015 0 .00200 .7978

0 .7989

0 .8000

0 .8011

0 .8022

γ &

p

a

0 .5417 0 .5424 0 .5431 0 .54380 .7970

0 .7981

0 .7992

0 .8003

0 .8014

0 .8025

m pfp

p

ρρρρ

b

Figure 6.2 High-density side of the GCM solid state at T = 0.004. (a) Pressure

as a function of strain-rate at different densities. Shown are results for the

densities of 0.5424 (�), 0.5425 (�), 0.5426 (�), 0.5427 (�), 0.5428 (),


phase solid-liquid region is clearly seen by the sudden jump in pressure at zero

strain-rate. (b) Pressure as a function of density for different strain-rates å� � á. á (�), å� � á. ááÇ (), å� � á. ááÅ (�), all in the stable liquid state and

its metastable extension and å� � á. á (�) in the stable solid state and its



pressure.


203

dependent pressure is shown in Fig. 6.1(a) for a density range of ρ = 0.1296-

0.1305 in steps of ∆ρ = 10-4. Up to a density of 0.1299 the system is still in the

liquid state because the pressure is nearly constant for all three strain-rates.

Increasing the density to 0.13 results in a sudden drop of the pressure at zero

strain-rate. For densities ρ ≥ 0.13 the equilibrium pressures are even lower than

those for ρ ≤ 0.1296. This indicates the entry into the two-phase solid-liquid

region, i.e., the fp. To determine the mp we plot the results in the pressure-

density plane in Fig. 6.1(b). The curves for strain-rates at �� 0.0, �� 0.001

and �� 0.002 nearly lie on top of each other in the liquid branch. A dashed

arrow marks the drop in the equilibrium pressure starting at fp. The pressures

for strain-rates at �� 0.001 and 0.002 extend from the stable liquid branch into

the two-phase solid-liquid region. Drawing an isobaric line from fp to the solid

branch identifies mp. The construction at T = 0.006 yields densities of ρf =

0.1299 and ρm = 0.13134. On the low-density side we calculated transitions at T

= 0.002, 0.004, 0.006, 0.008 and 0.0089 and the results are summarized in Table

6.2.

6.4 High6.4 High6.4 High6.4 High----Density Side of the Solid RDensity Side of the Solid RDensity Side of the Solid RDensity Side of the Solid Regionegionegionegion

On the high-density side of the solid state, where overlapping of particles

becomes important, the GCM fluid displays re-entrant melting into the stable

liquid state. Contrary to the normal case, the liquid coexisting with the solid

has a higher density than the solid. In this region we have to reverse our


204

Table 6.2 Freezing and melting densities for the low-density and high-density

sides of the solid state of the GCM fluid obtained using the GWTS algorithm.

low-density side high-density side

T �) �* �* �) 0.002 0.0761 0.07817 0.72097 0.7215

0.004 0.1017 0.10357 0.54199 0.5429

0.006 0.1299 0.13134 0.43279 0.4338

0.008 0.1687 0.16960 0.33288 0.3337

0.0089 0.2124 0.21270 - -

0.009 - - 0.25626 0.2565

method in the sense that we have to start in the liquid phase at higher densities

and decrease the density in order to enter the two-phase solid-liquid region. In

Fig. 6.2 we show results for the high-density (re-entrant melting) region for a

temperature of T = 0.004. The strain-rate dependent pressure for densities

ranging from ρ = 0.5433 to 0.5424 is shown in Fig. 6.2(a). Down to a density of

0.5429 the system is still in the liquid phase. The interesting fact in this region

is that opposite to the low-density side, the equilibrium pressure jumps up to

higher values for densities ρ ≤ 0.5428. At a density of ρf = 0.5429 we find the fp.

The pressure-density projection of the results is shown in Fig. 6.2(b).

Again, the curves for strain-rates at �� 0.0, 0.001 and 0.002 nearly lie on top of each other in the liquid branch. A dashed arrow marks the jump of the


205

equilibrium pressure. An analogous construction of mp yields a melting density

of ρm = 0.54199. On the high-density side we calculated transitions at T =

0.002, 0.004, 0.006, 0.008 and 0.009 and the results are summarized in Table

6.2.

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.750.0015

0.0027

0.0039

0.0051

0.0063

0.0075

0.0087

0.0099

T

ρρρρ

Figure 6.3 Phase diagram of the GCM fluid showing the freezing () and

melting lines (�) obtained in this work. The fps (�) reported by Prestipino et

al. (2005) and freezing thresholds (�) predicted by the Hansen-Verlet rule

(Saija et al., 2006) are also illustrated.


206

6.5 The GCM Phase Diagram6.5 The GCM Phase Diagram6.5 The GCM Phase Diagram6.5 The GCM Phase Diagram

In Fig. 6.3 we show our results of solid-liquid phase coexistence at equilibrium

and compare them with the currently most accurate simulation results of

Prestipino et al. (2005). In addition, we also show freezing thresholds (Saija et

al., 2006) predicted by the Hansen-Verlet rule based on the height of the first

peak of the structure factor at freezing. Lang et al. (2000) established that the

phase boundaries of the GCM are well reproduced by the Hansen-Verlet

criterion. In general, the coexistence lines are double lines, but they cannot be

resolved on the scale of the figure because the solid-liquid density gap is too

small. On the low-density side our results are in very good agreement with those

of Prestipino et al. (2005). The solid region in our simulation is broader at

higher temperatures (T = 0.008).

This tendency continues on the high-density side where the melting and the

freezing lines are shifted slightly to higher densities, compared with those

obtained by Prestipino et al. (2005). At almost all temperatures studied, the

liquid phase is transformed into a bcc-solid. The only exception is the low-

density side at T = 0.002 where the liquid phase is transformed into a fcc solid.


207

0.0015 0.0027 0.0039 0.0051 0.0063 0.0075 0.0087 0.0099-6

-4

-2

0

2

4

6

T

Figure 6.4 Comparison of the relative percentage difference in freezing (�) and

melting �� densities on the low-density side and freezing (Ο) and melting (�)

densities on the high-density side at different temperatures obtained in this

work �àã»�� with data reported in Prestipino et al. (2005) �àPrestipino�.

Pr

Pr

( )100 estipino sim

estipino

ρ ρρ

−×


208

0.002 0.004 0.006 0.008 0.0100.2

0.6

1.0

1.4

1.8

2.2

T

Figure 6.5 The solid-liquid density gap ∆à�� +à� � à�+ on the low-density

(�) and high-density (�) sides of the GCM phase diagram as a function of

temperature.

Figure 6.4 provides a quantitative comparison of our coexistence densities to

those obtained from Prestipino et al. (2005). The comparison indicates that at

any temperature, the discrepancies between the two calculation methods are

typically less than 5%. Our results are in between the values reported by

Prestipino et al. (2005) and the predictions of the Hansen-Verlet freezing rule

(Saija et al., 2006).

103 × ∆ρ fm


209

In Fig. 6.5 we show the solid-liquid density gap �)* � +�) � �*+ on the low- and

the high-density side depending on temperature. The density gap is larger on

the low-density side. For both density sides the density difference decreases

when increasing the temperature for T ≥ 0.006. Extrapolating the density gaps

to temperatures higher than T = 0.009 suggests that the two-phase solid-liquid

region disappears completely for both density sides at a common point, as

predicted by Stillinger (1976), with a maximum freezing/melting temperature

Tmax. We located this maximum value at Tmax ≈ 0.00903 for ρmax ≈ 0.24265. This

compares with maximum values Tmax ≈ 0.00874 for ρmax ≈ 0.239 obtained by

Prestipino et al. (2005).

In a summary, in this Chapter, we have presented a precise phase envelope of

the GC potential using the GWTS algorithm. The results for the low-density

and high-density (reentrant melting) sides of the solid state are in good

agreement with those obtained by Monte Carlo simulations in conjunction with

calculations of the solid free energies. The common point on the Gaussian core

envelope, where equal-density solid and liquid phases are in coexistence, could

be determined with high precision.

210

Chapter 7Chapter 7Chapter 7Chapter 7 StrainStrainStrainStrain----Rate Dependent Shear Rate Dependent Shear Rate Dependent Shear Rate Dependent Shear

VVVViscosiiscosiiscosiiscosity of the ty of the ty of the ty of the Gaussian CGaussian CGaussian CGaussian Core ore ore ore Bounded Bounded Bounded Bounded

PPPPotentialotentialotentialotential

The success of using of GCM bounded potential, as an effective potential, to

explain anomalies observed in complex molecular fluids has made it attractive

potential. Since strain rate dependent shear viscosity of complex molecular fluid

is an important property, it is also desirable to calculate shear viscosity of GCM

fluid. In this Chapter nonequilibrium molecular dynamics simulations are

reported for the shear viscosity of the Gaussian core model fluid over a wide

range of densities, temperatures and strain-rates. The relevant simulation details

are discussed in Section 7.1. In Section 7.2, it is shown how the decision has

been made on the maximum safe strain rates for GCM fluid. In Section 7.3,

shear viscosity data is reported as a function of strain rate. A strain-rate

dependent viscosity model is obtained via fitting the simulation data in Section

7.4. In Section 7.5, zero-shear viscosities are estimated and compared with

Green-Kubo calculations.

Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential

211


We used nonequilibrium molecular dynamics simulation algorithm as discussed

in Chapter 2 to obtain the shear viscosity. The initial configuration in all the

simulations was a face centred cubic (f.c.c) lattice structure. The simulations

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10.00

0.01

0.02

0.03

0.04

0.05

0.06

T

ρρρρ

solid

Figure 7.1 Phase diagram of the GCM fluid showing the state points (�)

covered by the NEMD simulations reported in this work.

covered five isochors of densities ρ = 0.1, 0.2, 0.3, 0.4 and 1.0, temperatures

ranging from T = 0.015 to 3.0 and various strain-rates from �� = 0.005 to 9.0.

The phase state points for our NEMD simulations are shown in Fig. 7.1. The


212

solid-liquid coexistence lines in Fig. 7.1 are calculated in Chapter 6. Since our

simulations covered a wide range of temperatures and densities we had to

carefully choose the integration time step for different state points such that the

time step was small enough to solve the equations of motion correctly and large

enough to sample phase space adequately. The equations of motion were

integrated with a time step of τ = 0.001. The cutoff radius for the potential was

3.2σ. The ensemble averages are reported without any long-range corrections

because the potential rapidly goes to zero at larger separations.

For each state point (ρ, T, ��) simulation trajectories were typically run for 2 ×

106 time steps. The first 4 × 105 time steps of each trajectory were used either to

equilibrate zero-shearing field equilibrium molecular dynamics or to achieve non-

equilibrium steady state after the shearing field was switched on. The remaining

time steps in each trajectory were used to accumulate the average values of

thermodynamic variables standard deviations. A system size of 4000 GC

particles was used for all the simulations reported in this Chapter.

7.2 7.2 7.2 7.2 Maximum Safe StrainMaximum Safe StrainMaximum Safe StrainMaximum Safe Strain----RatesRatesRatesRates

A well-known limitation of the sllod algorithm (Section 2.5.2) coupled to a

Gaussian thermostat (Eq. (2.25)) is that it generates an artificial “string-phase”

at high strain-rates (Evans and Morriss, 1986, Woodcock, 1985, Erpenbeck,

1984). To avoid this problem and to also avoid possible shear-induced ordering


213

effects in our analysis, we estimated the maximum strain-rate that could be

safely used by analysing the strain-rate dependent internal energy per particle.

The formation of strings of particles or shear-induced ordering causes a clear

and easily detectable breakdown in the internal energy profiles. For example, in

Fig. 7.2 we show the strain-rate dependent internal energy per particle for

different temperatures at a density of � � 0.1. A similar trend can be observed

1E-4 1E-3 0.01 0.1 1 100.037

0.097

0.157

0.217

γ&

E

0.10

0.08

0.06

0.04

0.030.0250.020

0.015

Figure 7.2 Strain-rate dependent internal energy per particle as a function of

strain-rate for different constant temperatures (as indicated on the lines) at a

density of � � 0.1. The sharp drop after the increase in energy indicates the

occurrence of the string phase.


214

using viscosity data, but the drop in the viscosity profiles is less pronounced,

especially at higher strain-rates. Using this procedure, we estimated the

maximum safe strain-rates at different densities and temperatures. These data

are summarized in Table 7.1.

Table 7.1 Maximum safe strain-rates at different densities and temperatures.

These strain-rates avoid string phases and shear-induced ordering. For state

points without an entry the drop in the internal energy profiles occurs at strain-

rates higher than a dimensionless value of 9.0. This situation occurs for all

densities with temperatures greater than T = 0.30.

T � � 0.1 � � 0.2 � � 0.3 � � 0.4 � � 1.0

0.015 0.4 0.4 0.5 1.2 3.0

0.020 0.6 0.8 1.2 1.8 3.0

0.025 0.7 1.0 1.6 2.0 5.0

0.030 0.9 1.2 2.0 3.0 5.0

0.040 1.0 1.8 3.0 5.0 7.0

0.060 1.6 5.0 5.0 7.0 …

0.080 3.0 7.0 7.0 … …

0.100 5.0 … … … …

0.300 … … … … …

We note that an alternative method for the accurate detection of string phases

would be to observe the drop in the strain-rate configurational temperature

(Delhommelle, 2005).


215

0 . 0 1 0 . 1 1

0 . 0 2

0 . 0 4

0 . 0 6

0 . 0 80 . 1

γ &

η

T = 0 . 0 1 5 a

0 . 0 1 0 . 1 10 . 0 1

0 . 0 3

0 . 0 5

0 . 0 7

0 . 0 9

γ &

ηηηη

T = 0 . 0 2 b

0 . 0 1 0 . 1 10 . 0 1

0 . 0 3

0 . 0 5

0 . 0 7

0 . 0 9

γ &

ηηηη

T = 0 . 0 2 5 c

0 . 0 1 0 . 1 10 . 0 1

0 . 0 3

0 . 0 5

0 . 0 7

0 . 0 9

γ &

ηηηη

T = 0 . 0 3 0 d

Figure 7.3 Shear viscosity isochors as a function of strain-rates for (a) T =

0.015, (b) T = 0.02, (c) T = 0.025 and (d) T = 0.03. The isochors were

obtained for à � á. Ç (�), 0.2 (�), 0.3 (�), 0.4 (�) and 1.0 (). Note the

anomalous behaviour at à , 0.3. The lines are for guidance only.


216

There are also alternatives (Delhommelle et al., 2003) to the use of a Gaussian

thermostat which avoid the formation of the artificial string phases.

Nonetheless, we found that the energy-drop method was sufficiently reliable to

avoid string phases and no string phases were detected within the safe range of

strain-rates reported here.

7.3 Shear Viscosity7.3 Shear Viscosity7.3 Shear Viscosity7.3 Shear Viscosity:::: StrainStrainStrainStrain----Rate BehaviourRate BehaviourRate BehaviourRate Behaviour

The shear viscosity (Eq. (2.30)) as a function of strain-rate at different

temperatures and densities is illustrated in Fig. 7.3. In all cases we can observe

a transition from Newtonian (strain-rate independent) to non-Newtonian

(strain-rate dependent) behaviour. For a given temperature up to densities of

� � 0.3, the onset of this transition generally occurs at a lower strain-rate as the

density is increased. Similarly, for any given density, increasing the temperature

also generally reduces the strain-rate required to observe the transition between

Newtonian and non-Newtonian behaviour. The effect of temperature is

somewhat weaker than the effect of density. The above description is consistent

with the behaviour reported for the Lennard-Jones fluid. However, there is a

very noticeable exception. Normally, we would expect the shear viscosity

isochors (i.e., shear viscosity at constant density) to be progressively shifted to

higher viscosity values, with increasing densities. At T = 0.015 (Fig. 7.3(a)), the

ρ = 0.1, 0.2 and 0.3 isochors occur at progressively higher shear viscosities. This

trend is arrested at the ρ = 0.4 isochor which straddles the ρ = 0.3 isochor. The


217

ρ = 1.0 isochor commences at viscosities less than that observed for ρ = 0.2.

Furthermore, the onset of non-Newtonian behaviour does not occur until much

higher strain-rates. The non-Newtonian part of these anomalous isochors occurs

in the conventional density sequence relative to the non-Newtonian parts of the

other isochors. Very high strain-rates appear to restore normal behaviour in the

non-Newtonian region. Increasing the temperature to T = 0.020 (Fig. 7.3(b)),

0.025 (Fig. 7.3(c)) progressively removes the anomalous behaviour at all strain-

rates. The crossover point to normal behaviour is at a temperature of T = 0.030

(Fig. 7.3(d)). We performed additional calculations for temperatures up to T =

1.0. At these higher temperatures (Fig. 7.4), normal behaviour was observed.

This anomalous behaviour reflects an approaching solid state transition at low

temperatures and moderately high densities (around ρ ≈ 0.25) at which the

shear viscosity rises sharply (see Fig. 4a in Mausbach and May (2009)). Re-

entrant melting occurs on the high-density side of the solid region (Fig. 7.1),

which again results in a decrease in the shear viscosity for densities near the

melting density.

Anomalous shear behaviour ��4> ��⁄ �� 0 has also been reported (Mausbach

and May, 2009) in equilibrium calculations at temperatures up to approximately

T = 0.032, which coincides with the anomalous range of temperature observed

here.


218

0 . 0 2 0 . 1 1

0 . 0 0 3

0 . 0 1

γ &

ηηηη

T = 0 . 0 6 a

0 . 0 2 0 . 1 10 . 0 0 1 5

0 . 0 1

γ &

ηηηη

T = 0 . 0 8 b

0 . 0 2 0 . 1 10 . 0 0 1 5

0 . 0 1

γ &

ηηηη

T = 0 . 1 c

0 . 0 2 0 . 1 10 . 0 0 1 5

0 . 0 1

0 . 1

γ &

ηηηη

T = 0 . 3 0 d

Figure 7.4 Shear viscosity isochors as a function of strain-rates for (a) T = 0.06,

(b) T = 0.08, (c) T = 0.1 and (d) T = 0.3. The isochors were obtained for ρ =

0.1 (�), 0.2 (�), 0.3 (�) and 0.4 (�). The lines are for guidance only.


219

0.004 0.04 0.4 40.006

0.06

0.6

6

γ&

ηηηη

0.04

0.1

0.3

0.50.71.0

2.0

3.0

a

0 1 2 3 4 5 6-2

8

18

28

38

48

58

ηηηη

T

0.1

0.2

0.3

1.0b

Figure 7.5 (a) Shear viscosity at à � Ç. á vs strain rate for various temperatures

as indicated. The lines are for guidance only. (b) Shear viscosity as a function of

temperature for four different densities as indicated.


220

In this region the slope of the density dependent viscosity along an isotherm can

be characterized within three different stages (see Fig. 4b in Mausbach and May

(2009)). For ρ ≤ 0.3, the zero-shear viscosity increases with increasing the

density, which is typical for “normal” liquids. Thereafter, the zero-shear

viscosity passes through a maximum, followed by an anomalous decrease of 4> upon further compression. At higher densities, the zero-shear viscosity passes

through a minimum and increasing the density further at constant temperature

again causes an increase in 4>. The last situation coincides with a region of very

high particle overlap. For , 0.032, the anomaly disappears and 4> increases with increasing �.

Figure 7.5 illustrates the viscosity profiles at ρ = 1.0 for temperatures ranging

from T = 0.04 to 3.0 and strain-rates starting from �� 0.005. Generally, the

viscosity increases with temperature, which is contrary to the behavior of

normal dense liquids. However, at very high densities where penetration of GC

particles is dominant and the repelling force between the particles becomes very

small, the GCM system approaches the so called “infinite-density ideal-gas

limit” (Lang et al., 2000). Many thermodynamic and dynamic quantities

indicate (Mausbach and May, 2006) that the GCM system is approaching this

limit at a density of � � 1.0 and here, the system behaves like a dense gas

rather than a dense liquid. The unusual temperature dependence of the viscosity

shown in Fig. 7.5 reflects this peculiarity of the GCM and the viscosity behavior

at this density is in excellent agreement with equilibrium Green-Kubo


221

calculations (Mausbach and May, 2009). Thus the dependence of the viscosity

on temperature can be used as an invaluable source of information on

intermolecular force. We found that the temperature dependence of GC fluid

can be expressed as

4�� X�*� (7.1)

where X� and X� are density and strain rate dependent parameters and these

parameters are summarized in Table 7.2 for the temperature range � 0.015 �5.0 and densities obtained via the least-squares fit.

Table 7.2 Parameters of temperature dependent viscosity model of GC fluid.

Errors are in the brackets.

� X� X� 0.1 0.79(5) 1.55(4)

0.2 0.72(4) 1.94(4)

0.3 2.0(1) 1.51(3)

0.4 1.72(3) 1.70(1)

1.0 4.19(7) 1.58(1)

We note that there is recent experimental evidence (Kalur et al., 2005) for such

abnormal behaviour.


222

0.0025 0.01 0.10.01

0.015

0.02

0.025

γ&

ηηηη

Figure 7.6 Shear viscosity as a function of strain-rate at T = 0.015 and à �á. áÇ. The lines indicate the fit to the simulation data (�) using Eq. (7.1) with α � 1/2 (dashed line) and 0.75 (solid line).

0.010 0.025 0.040 0.055 0.070 0.085-18

-9

0

9

18

T

Figure 7.7 Comparison of the relative percentage difference of zero-shear

viscosities obtained from this work with Green-Kubo (GK) calculations

(Mausbach and May, 2009) as a function of temperature and densities of à �

0.1 (�), 0.2 (�), 0.3 (�), 0.4 (�) and 1.0 ().

100 ×(ηGK − ηNEMD )

ηGK


223

Kalur et al. (2005) measured the viscosity behaviour of cationic surfactant

solutions and observed an increase in viscosity with increasing temperature as

depicted in Figure 7.5(a). They attributed the anomaly to wormlike micelles. It

is unlikely that such phenomena could be predicted using a conventional

unbounded potential, which suggests that the GCM might have a useful role in

understanding this aspect of surfactant behaviour. It is also evident from Fig.

7.5 (a) that increasing the temperature causes an increase of the degree of shear

thinning, i.e., the crossover between the Newtonian and the non-Newtonian

regime is shifted to lower strain-rates. This is also consistent with experimental

data (Kalur et al., 2005) for cationic surfactant solutions.

NEMD simulation studies commonly suffer from the weakness that the quoted

statistical uncertainties become increasingly large in the zero-shear limit. This

means that the results cannot be applied directly to real fluids, which typically

experience strain-rates much lower than used in simulations. Nonetheless, as

evident from Figures 7.3, 7.4 and 7.5, reasonable statistical uncertainties are

obtained from the GCM at moderately low strain-rates. The reliability of the

calculations for low strain-rates improves with increasing density. The statistical

uncertainties for the CGM are lower than observed for unbounded potentials

such as the Lennard-Jones potential.


224

7.4 Fitting Simulation Data7.4 Fitting Simulation Data7.4 Fitting Simulation Data7.4 Fitting Simulation Data

As discussed in detail elsewhere (Bosko et al., 2004b; Ge et al. 2003a; Todd,

2005; Travis et al., 1998; Cross, 1965; Trozzi and Ciccotti, 1984), shear viscosity

data can be fitted to relatively simple relationships such as:

4 � 4> � 4�� ª (7.2)

where 4> is the zero shear viscosity and 4� is a coefficient, which depends on

temperature and density. At temperatures at or near the triple point of a

Lennard-Jones fluid, good agreement is obtained when α = ½, which is

consistent with mode-coupling theory. However, better overall agreement

(Todd, 2005) for other temperatures and densities can be obtained using other

values of α. Fig. 7.6 compares our simulation at T = 0.015 and ρ = 0.01, fitted

to Eq. (7.2), using α = ½ (η0 = 0.0245, η1 = 0.020) and a best fit value of α =

0.75 (η0 = 0.0230, η1 = 0.025). It is evident from this comparison that using a

value of ½ fails to adequately reproduce the simulation data, particularly at

moderate strain-rates, whereas a value of α = 0.75 gives good agreement for the

entire range of strain-rates. The value of α = ½ is also an inadequate choice for

other temperatures and densities (not shown).

7.5 Zero7.5 Zero7.5 Zero7.5 Zero----Shear ViscositiesShear ViscositiesShear ViscositiesShear Viscosities

In view of the relatively modest statistical uncertainties reported at low to

moderate strain-rates, it is reasonable to extrapolate the NEMD results to zero

strain-rate and thereby obtain the equilibrium or zero-shear viscosities. It is of


225

interest to compare these extrapolated values with equilibrium values obtained

elsewhere (Mausbach and May, 2009) from Green-Kubo calculations.

Fig. 7.7 compares Green-Kubo and extrapolated NEMD zero-shear viscosities 4> along isochors at ρ = 0.1, 0.2, 0.3. 0.4 and 1.0 as a function of temperature. The

comparison indicates that the discrepancies between NEMD zero shear

viscosities than Green-Kubo calculations for ρ ≤ 0.4 are typically less than 5%.

For ρ =1.0, the NEMD values are between 1 to 12% higher than the Green-

Kubo calculations. The zero-shear viscosity 4> shows a non-monotonic

dependence on density for certain state conditions, which is consistent with

equilibrium simulations (Mausbach and May, 2009).

In a summary, a transition from Newtonian and non-Newtonian behavior is

observed in all cases reported in this Chapter for sufficiently high strain rates.

On the high-density side of the solid region where re-entrant melting occurs, the

shear viscosity decreases significantly when the density is increased at constant

temperature and Newtonian behavior persists until very high strain rates. This

behavior, which is attributed to particle overlap, is in contrast to the monotonic

increase in shear viscosity with density observed for the Lennard-Jones

potential. Contrary to the behavior of normal fluids, the viscosity is observed to

increase with increasing temperatures at high densities. This reflects a

peculiarity of the GCM, namely the approach to the “infinite-density ideal-gas


226

limit”. This behavior is also consistent with viscosity measurements of cationic

surfactant solutions. In contrast to other potentials, the shear viscosities for the

Gaussian core potential at low to moderate strain rates are obtained with

modest statistical uncertainties. Zero shear viscosities extrapolated from the

nonequilibrium simulations are in good agreement with equilibrium Green-Kubo

calculations.

227

Chapter 8Chapter 8Chapter 8Chapter 8 Steady Steady Steady Steady State EState EState EState Equation of quation of quation of quation of

SSSState and tate and tate and tate and VVVViscosity iscosity iscosity iscosity MMMModellingodellingodellingodelling

Nonequilibrium steady state thermophysical properties are of significant

industrial and theoretical interests. A unified method of analysing steady-state

thermophysical properties is absent in the literature though plenty of

experimental and simulation data are available. In contrast to this, equations of

states are commonly used to analyze both experimental and simulation data

obtained from equilibrium thermophysical properties. In this Chapter a

nonequilibrium steady-state equation of state is developed with the help of

simulation data obtained from nonequilibrium molecular dynamics simulations.

Temperature dependent zero-shear viscosity models are well known. But

pressure, density and temperature dependent non-zero shear viscosity models

are rarely seen in the current literature.

In Section 8.1, simulation details are given for the construction of a steady-state

equation of state for Lennard-Jones fluid. A comprehensive design, test and

verification suite is also presented in this Section for the development of the

nonequilibrium EOS.

Steady State Equation of State and Viscosity Modelling

228

In Section 8.2, a viscosity model is developed with an introduction of a generic

variable. The distinctive feature of this model is that it can also be used in

conjunction with equations of state. The model can be applied to both shear

independent and shear dependent systems with the inclusion of all

thermodynamic variables. Pressure, temperature, density and strain rate

dependent viscosity models can be explicitly derived form the generic viscosity

model with simple mathematical manipulations. In Section 8.3, accuracy of the

newly developed (in Section 8.2) generic viscosity model, pressure dependent

viscosity model and density dependent viscosity model are tested against the

data obtained via the EOS developed in Section 8.1. A strain-rate dependent

shear-viscosity model is investigated with the squalane experimental data in

Section 8.4. With the availability of a large body of experimental data, the

pressure dependent zero-shear viscosity model is verified for monatomic and

complex fluids in Section 8.5. Argon, neon, krypton and xenon are the

representatives of monatomic fluids whereas water, carbon dioxide and

hydrocarbons are the representatives of complex molecular fluids.

8.1 Steady S8.1 Steady S8.1 Steady S8.1 Steady State tate tate tate EEEEquation of quation of quation of quation of SSSState tate tate tate

8.1.1 Simula8.1.1 Simula8.1.1 Simula8.1.1 Simulation tion tion tion DDDDetailsetailsetailsetails

The development of a nonequilibrium steady-state equation of state requires

extensive simulation data for a wide range of state points. We used equilibrium

and nonequilibrium molecular dynamics algorithms as discussed in Chapter 2.


229

The configurational energy per particle ¯1FGH)°, pressure and shear viscosity at different strain-rates were obtained for 660 different state points (Fig. 8.1)

employing 2048 Lennard-Jones particles. This involved performing simulations

at constant reduced densities of ρ = 0.73, 0.8442, 0.895 and 0.95 and a reduced

temperature range 0.70 ≤ T ≤ 1.75. At each density and temperature

simulations were performed for 11 strain-rates lie within 0.1 ~ �� ~ 1.1 at equal

intervals of 0.1. The strain-rates were chosen to be well below the “string

phase” region (Erpenbeck, 1984; Woodcock, 1985; Evans and Morriss, 1986).

0.700.73

0.770.81

0.840.88

0.910.95

0.2

0.4

0.6

0.8

1.0

0.6

1.01.41.8

γ&

T ρρρρ

Figure 8.1 Illustration of the range of state points �à, þ, å� � for which NEMD

simulations were performed to obtain data for the steady-state equation of

state. Data were obtained for a total of 660 state points.


230

A nonequilibrium simulation trajectory was typically run for 4 × 105 time steps.

To equilibrate the system, the NEMD trajectory was first run without a

shearing field. After the shearing field was switched on, the first 2 × 105 time

steps of the trajectory were ignored, and the fluid was allowed to relax to a

nonequilibrium steady-state. The statistical uncertainty in results depends on

the state point. It is particularly sensitive to the strain rate, with low strain-

rates associated with larger errors than high strain-rates. Typically, the

standard error in the reduced configurational energy is 10-5-10-4, compared with

a 10-3-10-2 error range for the reduced pressure. The range of standard errors in

the reduced viscosity is typically 10-2-10-1.

8.1.2 Development of the E8.1.2 Development of the E8.1.2 Development of the E8.1.2 Development of the Equation of quation of quation of quation of SSSStatetatetatetate

To develop the desired equation of state we first introduce the following

definition for the nonequilibrium steady-state compressibility factor:

î2� � =2� �� ª�

(8.1)

It should be noted that unlike its equilibrium counterpart when Eq. (8.1) is

expressed in terms of real units is not a dimensionless quantity. Following the

approach reported by Evans and Hanley (1980b), the pressure and the

configurational energy (Econf) at a given temperature, number density (ρ =

N/V) and strain-rate can be obtained as the sum of equilibrium ( &γ = 0) and

nonequilibrium ( &γ > 0) contributions:


231

� =�, �, �� =>��, � � =2� ��, �� ª1FGH)�, �, �� 1>FGH)��, � � 12� ��, �� ª© (8.2)

Where =>��, � and 1>FGH)��, � are the equilibrium pressure and equilibrium

configurational energy, respectively. The only difference between Eq. (8,2) and

similar equations reported previously is that the α term replaces a fixed value of

3/2. Although it is tempting to refer to =2� and 12� as nonequilibrium “pressure”

and “energy,” terms respectively, it is evident that they do not have the

corresponding dimensions for these quantities. Thus p0� and α of Eq. (8.1) can

be obtained via the least squares fit of either experimental or simulation data

using the Eqs. (8.2). The experimental or simulation data may be obtained from

strain rate dependent set up or strain rate independent set up for a range of

temperature, density and pressure.

To obtain the value of α, we used the relationship reported by Ge et al. (2003a)

from NEMD simulations of a Lennard-Jones fluid:

.��, � � y � z � |�, (8.3)

Where y � 3.67 } 0.04, z � 0.69 } 0.03 and | � 3.35 } 0.03.

The parameterisation of Eq. (8.3) is valid (Ge et al., 2003a) over a wide range

of densities and temperatures. Although these parameters were obtained

specifically for the Lennard-Jones potential, there is some recent evidence

(Desgranges and Delhommelle, 2009) to suggest that they may be independent


232

of the intermolecular potential. We note that theoretical considerations (Evans

and Hanley, 1981) require that the value of α should be less than 2.

To make use of Eq. (8.2), we need analytical relationships for the equilibrium

contributions of the Lennard-Jones potential. There are at least four (Johnson

et al., 1993; Kolafa and Nezbeda, 1994; Wang et al., 1996; Cuadros et al., 1996;

Adachi et al., 1988) accurate expressions for the calculation of the equilibrium

pressure for the Lennard-Jones potential. The equilibrium pressure =>�, �� can

be obtained by fitting simulation data to the modified Benedict-Webb-Rubin

(MBWR) equation proposed by Johnson et al. (1993) which is the most updated

version of the equation of state first proposed by Nicolas et al. (1979):

î � 1 � 1 ¶J��J � exp ��

J¢� ¾J��J�J¢� (8.4)

where δ typically equals 3 and each of the A and B terms is the sum of multiple

parameters (Nicolas et al., 1979). This version of the equation of state

accurately correlates pressure and internal energies from the triple point to

about 4.5 times the critical temperature over the entire fluid range. The

equilibrium configurational energy can be obtained from the following

thermodynamic relationship:

1>FGH)��, � � � = � «�=�¬!

® ��!

> (8.5)

We calculated 12� and =2� by using Eq. (8.2) in conjunction with our simulation

data for 660 state points (Figure 8.1). Values of =2� and 12� were determined


233

using least-squares fit of the sheared steady-state energy and pressure data

obtained from the NEMD simulations.

The attributes of Eq. (8.1) allows us to formulate a nonequilibrium equation of

state at its steady-state in the form of following polynomial which accurately fit

our simulation data:

î2� � n* NJL�J nJL�

L¢>�

J¢> (8.6)

where n � �i�/Ï. It has been reported (Hanley and Evans, 1982) that this

definition of n could be used to fit data for a soft sphere potential. Hanley and

Evans (1982) used a similar approach to fit their data for �� 1 and . � 2/3.

However, using �� 1 effectively means that neither �� nor . influenced the fit as

1 raised to any power remains unchanged. In contrast, our data covers values of

�� from 0.1 to 1.1, and the value of alpha is not constant but varies as given by

Eq. (8.3). Applying Eq. (8.6) to our data incorporates a strain-rate dependency

in the fit.

Combining equilibrium and steady-state contributions via Eq. (8.2) means that

the pressure of a nonequilibrium steady-state Lennard-Jones fluid experiencing

constant shear can be obtained from

= � � � ¶J��JÁ�

J¢�� exp �� ¾J��JÁ� � � NJL�� JnJÁ*�iL�

L¢>�

J¢>�

J¢�

(8.7)


234

It should be noted that because Eq. (8.7) was obtained from fitting our

simulation data it is free of any assumption concerning the thermodynamics of

the nonequilibrium steady-state.

8.1.38.1.38.1.38.1.3 Data Accumulation and Parameter EstimationData Accumulation and Parameter EstimationData Accumulation and Parameter EstimationData Accumulation and Parameter Estimation

The state point and strain-rate coverage of our simulation data used to obtain

nonequilibrium steady-state contributions is illustrated in Fig. 8.1. The

simulations were confined to the dense fluid region because at very low density

neither the energy nor pressure is strain-rate dependent. Equation (8.2) is valid

for the entire fluid region with the exception of state points close to the freezing

point. The pressure and energy data obtained by fitting the simulation data to

Eq. (8.2) are summarized in Tables 8.1 and 8.2, respectively. It should be noted

that the fits for energy and pressure should not be done independently.

A least-squares estimate of =2� was obtained for each state point ��, � over a range of strain-rates by using a multiple non-linear Levenberg-Marquardt

regression algorithm (Press et al., 1992).


235

Table 8.1 Values ^á, ^å� and " appearing in Eq. (8.2) for three different



text). The statistical uncertainty in the last digit is given in brackets.

T � � 0.73 � � 0.8442 � � 0.895

=> =2� . => =2� . => =2� .

0.7 -0.384(3) 8.431(4) 1.985(2) 0.806(8) 9.49(1) 1.910(4) 1.95(1) 10.15(1) 1.853(5)

0.722 -0.301(3) 8.430(4) 1.981(2) 0.943(7) 9.479(9) 1.912(4) 2.12(1) 10.12(1) 1.859(5)

0.75 -0.182(3) 8.418(4) 1.983(2) 1.122(9) 9.44(1) 1.918(5) 2.32(1) 10.07(1) 1.863(6)

0.8 0.015(3) 8.401(4) 1.988(2) 1.431(6) 9.393(8) 1.925(3) 2.67(1) 10.01(1) 1.874(6)

0.85 0.209(3) 8.392(4) 1.981(2) 1.733(6) 9.353(7) 1.935(3) 3.02(1) 9.95(1) 1.883(6)

0.9 0.410(3) 8.372(4) 1.991(2) 2.028(4) 9.303(5) 1.937(2) 3.36(1) 9.89(1) 1.890(6)

0.95 0.604(3) 8.362(4) 1.989(2) 2.319(4) 9.271(5) 1.940(2) 3.71(1) 9.83(1) 1.901(6)

1 0.794(3) 8.355(4) 1.993(2) 2.609(4) 9.233(4) 1.949(2) 4.04(1) 9.78(1) 1.908(6)

1.05 0.988(3) 8.338(4) 1.994(2) 2.889(1) 9.203(1) 1.954(7) 4.36(1) 9.74(1) 1.912(6)

1.1 1.169(3) 8.330(4) 1.992(2) 3.164(3) 9.167(4) 1.954(2) 4.69(1) 9.69(1) 1.921(6)

1.15 1.351(3) 8.328(4) 1.990(2) 3.448(5) 9.141(7) 1.965(3) 5.01(1) 9.64(1) 1.927(6)

1.2 1.539(3) 8.311(4) 1.997(2) 3.718(3) 9.104(4) 1.966(2) 5.33(1) 9.59(1) 1.935(6)

1.25 1.710(3) 8.313(4) 1.990(2) 3.982(3) 9.086(4) 1.967(2) 5.64(1) 9.57(1) 1.935(6)

1.35 2.065(3) 8.295(4) 1.990(2) 4.501(3) 9.042(4) 1.968(2) 6.25(1) 9.49(1) 1.947(6)

1.75 3.421(3) 8.257(4) 2.001(2) 6.492(3) 8.910(3) 1.985(1) 8.58(1) 9.28(1) 1.974(6)


236

Table 8.2 Values of #á$ä��,#å� and " appearing in Eq. (8.2) for three different



text). The statistical uncertainty in the last digit is given in brackets.

T � � 0.73 � � 0.8442 � � 0.895

1> 12� . 1> 12� . 1> 12� .

0.7 -4.9411(5) 0.1043(6) 1.59(2) -5.6651(7) 0.2226(7) 1.256(9) -5.933(1) 0.322(1) 1.08(1)

0.722 -4.9220(4) 0.1014(4) 1.58(1) -5.6404(6) 0.2177(6) 1.259(8) -5.903(1) 0.314(1) 1.09(1)

0.75 -4.8977(4) 0.0982(5) 1.61(2) -5.6086(9) 0.212(1) 1.26(1) -5.867(1) 0.305(1) 1.11(1)

0.8 -4.8579(70 0.0941(8) 1.61(3) -5.5515(6) 0.2006(6) 1.319(9) -5.804(1) 0.293(1) 1.14(1)

0.85 -4.8169(4) 0.0888(5) 1.68(2) -5.4971(4) 0.1924(5) 1.348(8) -5.740(1) 0.279(1) 1.17(1)

0.9 -4.7783(4) 0.0855(5) 1.71(2) -5.4432(7) 0.1830(8) 1.36(1) -5.678(1) 0.266(1) 1.19(1)

0.95 -4.7783(4) 0.0855(5) 1.71(2) -5.3912(8) 0.1758(9) 1.36(1) -5.615(1) 0.253(1) 1.24(1)

1 -4.7028(3) 0.0787(4) 1.77(2) -5.3379(7) 0.1674(7) 1.42(1) -5.556(1) 0.243(1) 1.25(1)

1.05 -4.6659(4) 0.0753(4) 1.78(2) -5.2868(4) 0.1616(5) 1.47(1) -5.498(1) 0.234(1) 1.28(1)

1.1 -4.6303(5) 0.0736(6) 1.81(3) -5.238(1) 0.155(1) 1.44(2) -5.4397(6) 0.2246(6) 1.314(8)

1.15 -4.5952(4) 0.0712(4) 1.79(2) -5.186(1) 0.148(1) 1.53(2) -5.3807(8) 0.2146(8) 1.36(1)

1.2 -4.5606(4) 0.0691(4) 1.77(2) -5.137(1) 0.142(1) 1.54(2) -5.324(1) 0.205(1) 1.38(1)

1.25 -4.5258(4) 0.0677(5) 1.84(3) -5.0895(7) 0.1383(8) 1.56(2) -5.2697(6) 0.1995(6) 1.39(1)

1.35 -4.4577(4) 0.0628(5) 1.88(3) -4.9946(5) 0.1288(6) 1.61(1) -5.159(1) 0.183(1) 1.45(2)

1.75 -4.2012(4) 0.0538(5) 1.94(4) -4.6353(4) 0.1021(5) 1.73(1) -4.7463(9) 0.141(1) 1.61(2)


237

The R-squared value is 0.97, which indicates that we have accounted for almost

all of the possible variability with the parameters given in the model. For each

value of =2� at a given T, ρ and α (Table 8.1), several values of î2� can be obtained from Eq. (8.1) corresponding to different values of �� . These data in

turn can be accurately fitted to Eq. (8.6) using the coefficients summarized in

Table 8.3. This means that Eq. (8.6) can be used over the entire range of

densities, temperatures, and strain-rates studied (Fig. 8.1). In contrast to our

work, Hanley and Evans (1982), assigned a value of zero to several of the NJL coefficients, which probably partly reflects the much more limited scope of their

simulation data.

Table 8.3 Parameters for the nonequilibrium steady-state equation of state

regressed from the simulation data of this work.

M � 0 M � 1 M � 2 M � 3 NGL 14.474498040195400 -6.647556155245270 3.917705698095090 -0.915316394378783 N�L -1.717513016301310 2.732677555831720 -0.446753567923021 -0.232780581631844 N�L 6.717727912940890 -13.095727298095300 6.374714825509740 -0.834708529404541 N�L -7.343084320883140 15.874167538948800 -10.100511777914100 2.135496485879440 X 0.870890115604038

8.1.4 Accuracy of the P8.1.4 Accuracy of the P8.1.4 Accuracy of the P8.1.4 Accuracy of the Proproproproposed Steadyosed Steadyosed Steadyosed Steady----SSSState EOState EOState EOState EOS

To check the validity of our fit we performed independent zero-shear rate

equilibrium molecular dynamics simulations at a density of � � 0.73 and various

temperatures. These simulation data are compared with the equilibrium

pressure => obtained from our least-squares fit (Fig. 8.2(a)). It is evident that


238

0.70 0.83 0.96 1.09 1.22 1.35-0.5

0.0

0.5

1.0

1.5

2.0

2.5

peq

T

a

0.70 0.83 0.96 1.09 1.22 1.35-4

-2

0

2

4

T

b

Figure 8.2 (a) Comparison of equilibrium molecular simulation pressure data

(Ο) for Lennard-Jones fluid at à � á.�% with values from Eq. (8.2) (solid line)

and (b) the corresponding relative percentage difference (�).

100 ×( psim − pcal )

psim


239

0.70 0.83 0.96 1.09 1.22 1.358.0

8.8

9.6

10.4

11.2

γ&p

T

a

ρ ρ ρ ρ = 0.73

ρ ρ ρ ρ = 0.8442

ρ ρ ρ ρ = 0.895

ρ ρ ρ ρ = 0.95

0.70 0.83 0.96 1.09 1.22 1.350.03

0.18

0.33

0.48

γ&E

T

b

ρρρρ = 0.73

ρρρρ = 0.8442

ρρρρ = 0.895

ρρρρ = 0.95

Figure 8.3 Nonequilibrium steady-state contributions to (a) ^å� and (b) #å� for a Lennard-Jones fluid as a function of temperature at four different densities.


240

the fit yields very good agreement with the simulation data. In most cases the

relative deviation between the calculated and simulated equilibrium pressure is

less than 2% (Fig. 8.2(b)). In view of this, we can be confident that the

estimates of =2� , 12� and . are quite reasonable.

Figure 8.3 illustrates the variation of =2� (Fig. 8.3(a)) and 12� (Fig. 8.3(b)) as a

function of temperature at various constant densities. It is apparent that both

quantities progressively decline with increasing temperature. Increasing the

density also increases the values of =2� and 12� for any given temperature.

However, the effect of density is most noticeable at relatively low temperatures.

Figure 8.4(a) compares the steady-state compressibilities calculated at different

strain-rates using Eq. (8.6) with simulation data at different temperatures. It is

apparent that irrespective of either the strain-rate or the temperature, Eq. (8.6)

can reproduce the simulation data to a typical accuracy of approximately 2%.

A similar comparison for the accuracy of Eq. (8.6) with respect to both strain-

rate and density is illustrated in Fig. 8.4(b). The quality of the agreement for

density is similar to that observed for temperature.

The accuracy of Eq. (8.6) for a given strain-rate and temperature at different

densities is examined in Fig 8.5. At both low and high densities there is a

tendency to overestimate î2� whereas the data is generally underestimated at

intermediate densities. However, the error is small resulting in a relative


241

0.0 0.2 0.4 0.6 0.8 1.0 1.2-4

-2

0

2

4

γ&

a

0.0 0.2 0.4 0.6 0.8 1.0 1.2-4

-2

0

2

4

γ&

b

Figure 8.4 Comparison of the relative percentage difference of steady-state


(8.1) as a function of strain-rate (a) for the temperature range T = 0.70 - 1.75

and (b) for the density range 0.73 - 0.95. Shown are (a) à � 0.73 (�), 0.8442

(�), 0.895 (�), 0.95 (�); (b) T = 0.7 (�), 0.90 (�), 1.10 (�), 1.35 (�), 1.75

(⊳).

( ( ) ( ))100

( )

Z sim Z cal

Z simγ γ

γ

−× & &

&

( ( ) ( ))100

( )

Z sim Z cal

Z simγ γ

γ

−× & &

&


242

0.72 0.76 0.80 0.84 0.88 0.92 0.96-4

-2

0

2

4

ρρρρ



(8.1) as a function of density. Shown are �þ, å� � � (0.75, 0.1) (�); (0.90, 0.3)(�);

(1.05, 0.5) (�); (1.20, 0.7) (�); (1.35, 0.9) (⊳); (1.75,1.1) (�).

0.70 0.83 0.96 1.09 1.22 1.35-4

-2

0

2

4

T



(8.1) as a function of temperature. Shown are �à, å� � � (0.73, 0.2) (�); (0.8442,

0.4) (�); (0.895, 0.6) (�); (0.95, 0.8) (�).

( ( ) ( ))100

( )

Z sim Z cal

Z simγ γ

γ

−× & &

&

( ( ) ( ))100

( )

Z sim Z cal

Z simγ γ

γ

−× & &

&


243

deviation of less than 2% in most cases. Figure 8.6 compares the ability of Eq.

(8.6) to reproduce the compressibility data for a given strain-rate and density at

various temperatures. For most temperatures the relative deviation is less than

2%. Within this small error range, over-estimates or under-estimates appear

equally likely irrespective of the temperature.

The analysis presented in Figures 8.4, 8.5 and 8.6 clearly indicates that the

steady-state equation of state can reproduce the simulation results over the

range of densities, temperatures and strain-rates covered by this study. The

quality of the fit for î2� is very close to the quality of agreement obtained for the

pressure for the equilibrium Lennard-Jones equation of state (see Fig. 4 in

Johnson et al. (1993)). This means that we can expect the nonequilibrium

equation of state to be of similar accuracy as its equilibrium counterpart.

8.2 8.2 8.2 8.2 Development of Generic Viscosity ModelDevelopment of Generic Viscosity ModelDevelopment of Generic Viscosity ModelDevelopment of Generic Viscosity Model

To model viscosity, we first define a generic variable W that can represent T,ρ,

p or �� . Following the approach used by Kapoor and Dass (2005), we assume

the ratio of the first and second derivatives of viscosity with respect to the

compressibility factor are a W-independent parameter (Y):

& � ��4�î2� , 5�î2�� ®

�4�î2� , 5�î2� ®

' (8.8)

Successive integration gives


244

� & � �4�î2� , 5�î2� ®

� 4�0, 5� exp¯&î2� °4¯î2� , 5° � 4�0, 5� � 4�0, 5�¯exp¯&î2� ° � 1°

& ±²³² (8.9)

Expanding the exponential and truncating after the first two terms leads, after

simplification to:

4¯î2� , 5° � 4�0, 5� � 4�0, 5�¯î2� � 0.5&î2��° (8.10)

This means that the viscosity can be simply calculated from three adjustable

parameters 4�0, 5�, 4�0, 5� and & which can be easily obtained by fitting Eq.

(8.10) to the simulation results. Replacing the implicit generic variable with the

explicit thermodynamic variable following relationships can be established:

When 5 � �� 4¯î2� , �� ° � 4�0, �� 4�0, ��¯î2� � 0.5&î2��° (8.11)

When 5 � � 4¯î2� , �° � 4�0, �� 4�0, ��¯î2� � 0.5&î2��° (8.12)

With the similar arguments (as Eq. (8.8) and (8.9)) we can also develop the

following viscosity models dependent on pressure and density:

(i)(i)(i)(i) Pressure dependent Pressure dependent Pressure dependent Pressure dependent zerozerozerozero----shear shear shear shear viscosityviscosityviscosityviscosity �å� � á� 4�=, � � 4�0, � � 4�0, ��= � 0.5&>=��, (8.13)

where &> is defined by: &> � ��4�=, �=� ®2� ¢> «�4�=, �= ¬2� ¢>' (8.14)


245

(ii)(ii)(ii)(ii) Pressure dependent Pressure dependent Pressure dependent Pressure dependent shear viscosityshear viscosityshear viscosityshear viscosity �å ( á� 4�=, �� 4�0, �� 4�0, �� ¯= � 0.5&Ó=�°, (8.15)

where &Ó is defined by: &Ó � ��4�=, ��=� ®2� «�4�=, ��= ¬2�' (8.16)

(iii)(iii)(iii)(iii) Density dependent Density dependent Density dependent Density dependent shear shear shear shear viscosityviscosityviscosityviscosity

4��, �� 4�0, �� 4�0, �� ¯� � 0.5&!��°, (8.17)

where &! is defined by:

&! � ��4��, �� ®2� «�4��, �� ¬2�' (8.18)


246

0.06 0.08 0.10 0.12 0.14 0.16 0.180.2

0.7

1.2

1.7

2.2

2.7

γ&

ηηηη

Z

a

0.90 1.45 2.00 2.55 3.102.3

2.6

2.9

3.2

3.5

3.8

γ&

ηηηη

Z

b

Figure 8.7 Shear viscosities for a 12-6 Lennard-Jones fluid as a function of

nonequilibrium steady-state compressibility obtained from NEMD simulation

(Ο) reported here and values obtained from Eq. (8.10) (solid lines) for þ �á.�á � Ç.�� and (a) à � á.�%, å� � á. Ç, ()�á,*� � 1.5242, )�á,*� � -2.4179, Y

= -10.5966) and (b) à � á.��, å� � á. Ò, ()�á,*� � 2.3128, )�á,*� � 0.6770, Y

= -0.3253).


247

0.3 2.2 4.1 6.0 7.9 9.8 11.7 13.62.4

2.7

3.0

3.3

3.6

3.9

γ&

ηηηη

Z

a

0.3 2.1 3.9 5.7 7.5 9.3 11.12.5

2.9

3.3

3.7

γ&

ηηηη

Z

b

Figure 8.8 Comparison of the shear viscosity for a 12-6 Lennard-Jones fluid as

a function of nonequilibrium steady-state compressibility obtained from NEMD

simulations (Ο) at à � á.� reported here with values obtained from Eq. (8.6)

(solid lines) for = 0.3-1.1 at (a) þ � Ç. á �)�á,*� � 3.4143, )�á,*� � -

0.1224, Y = -0.0760� and (b) T = 1.20 �)�á,*� � 3.24193, )�á,*� � -0.1120,

Y = -0.0879).

&γ


248

8.3 Connection between EOS and Generic Viscosity 8.3 Connection between EOS and Generic Viscosity 8.3 Connection between EOS and Generic Viscosity 8.3 Connection between EOS and Generic Viscosity

Model Model Model Model

8.3.1 When Strain Rate is the Generic Variable8.3.1 When Strain Rate is the Generic Variable8.3.1 When Strain Rate is the Generic Variable8.3.1 When Strain Rate is the Generic Variable

The ability of Eq. (8.10) to reproduce our shear viscosity data is illustrated in

Fig. 8.7. The comparison, which involves both different densities and strain-

rates, indicates that good agreement can be obtained for the full range of

compressibility values.

8.3.2 When Density is 8.3.2 When Density is 8.3.2 When Density is 8.3.2 When Density is the Generic Vthe Generic Vthe Generic Vthe Generic Variableariableariableariable

An analysis at a common density but different temperatures is given in Fig. 8.8,

which indicates that Eq. (8.10) can also accurately reproduce the temperature-

dependence of shear viscosity.

8.3.3 Pressure Dependent Shear V8.3.3 Pressure Dependent Shear V8.3.3 Pressure Dependent Shear V8.3.3 Pressure Dependent Shear Viscosity iscosity iscosity iscosity

Figure 8.9 compares the viscosity-pressure behaviour obtained from Eq. (8.15)

with simulation data at several different densities and strain-rates. We observe

that it is rare for the shear-dependent viscosity to be investigated either

experimentally or theoretically as a function of pressure. It is apparent that

there is good agreement between Eq. (8.15) and the simulation data. From Fig.

8.9 it can be observed that, irrespective of the strain-rate the shear viscosity at


249

0 2 4 6 8 1 0 1 2 1 41 . 0

2 . 5

4 . 0

5 . 5

ηηηη

p

a

0 . 9 5

0 . 8 9 5

0 . 8 4 4 2

0 . 7 3

1 . 5 4 . 1 6 . 7 9 . 3 1 1 . 9 1 4 . 51 . 2

2 . 0

2 . 8

3 . 6

4 . 4

ηηηη

p

b

0 . 7 3

0 . 8 4 4 2

0 . 8 9 5

0 . 9 5

3 . 5 6 . 1 8 . 7 1 1 . 3 1 3 . 9 1 6 . 51

2

3

4

ηηηη

p

c 0 . 9 5

0 . 8 9 5

0 . 8 4 4 2

0 . 7 3

7 . 5 1 0 . 3 1 3 . 1 1 5 . 9 1 8 . 7 2 1 . 51 . 1

1 . 6

2 . 1

2 . 6

3 . 1

3 . 6

ηηηη

p

d 0 . 9 5

0 . 8 9 5

0 . 8 4 4 2

0 . 7 3

Figure 8.9 Comparison of shear viscosity simulation data (Ο) reported here for

the 12-6 Lennard-Jones fluid as a function of pressure at four different densities

and constant strain-rates of (a) 0.3 �)�á, å� � � 1.4378, )�á, å� � � -0.0335, Y = -

0.2925�, (b) 0.5 �)�á, å� � � 2.6617, )�á, å� � � -0.0653, Y = -0.0536�, (c) 0.7 �)�á, å� � � 2.9511, )�á, å� � � 0.0030, Y = -1.45� and (d) 1.0 �)�á, å� � � 2.9294,

)�á, å� � � 0.0499, Y = -0.0621� with values obtained from Eq. (8.15) (solid

lines). The data cover the temperature range of T = 0.70 to T = 1.75.


250

0 .6 9 0 .7 7 0 .8 5 0 .9 3 1 .0 11

2

3

4

5

6

ηηηη

ρρρρ

0 .2

0 .4

0 .60 .8

Figure 8.10 Comparison of shear viscosity simulation data (Ο) reported here for

the 12-6 Lennard-Jones fluid at T = 1.0 as a function of density with values

obtained from Eq. (8.17) (solid lines). Results are shown for strain-rates of 0.2 �)�á, å� � � 25.9563, )�á, å� � � -72.1191, Y = -1.4613�, 0.4 �)�á, å� � � 15.0339, )�á, å� � � -43.1501, Y = -1.5513�, 0.6 �)�á, å� � � 10.7986, )�á, å� � � -31.5572, Y

= -1.6155� and 0.8 �)�á, å� � � 8.7105, )�á, å� � � -25.6701, Y = -1.6591�.

a given pressure increases as the density is increased. For most densities, at low

strain-rates, the shear viscosity declines noticeably as pressure is increased.

However, as the strain-rate is increased, the rate of decline in the viscosity with

respect to increasing pressure progressively decreases. This means that at a

sufficiently high strain-rate, the shear viscosity is likely to be independent of

pressure. However, the data at � � 0.73, which shows a small increase in shear

viscosity with increasing pressure, appears to be an exception to this general

behaviour.


251

8.3.4 Density Dependent Shear V8.3.4 Density Dependent Shear V8.3.4 Density Dependent Shear V8.3.4 Density Dependent Shear Viscosity iscosity iscosity iscosity

The variation of shear viscosity with respect to density at constant temperature

is examined in Fig. 8.10, which indicates that Eq. (8.17) can be used to

reproduce the simulation data. The comparison involves data at different strain-

rates. At low density, the shear viscosity is very similar irrespective of the

strain-rate. However, a distinction in the shear viscosity begins to emerge at

moderate density and increases progressively as the density is increased. At any

moderate to high density, there is an inverse relationship between the shear

viscosity and the strain-rate. That is, the shear viscosity decreases with

increasing strain-rate.

It is apparent from the above comparisons that our shear viscosity model can

accurately reproduce the simulation data. The most common method for

reproducing strain-rate dependent shear viscosity data is to collapse the data

onto a single characteristic curve (Bird et al., 1987). It has been demonstrated

(Bair et al., 2002b) this approach can yield a good qualitative representation

between simulation and experimental data of the viscosity versus strain-rate

behaviour of squalane. The obvious disadvantage of this approach is that details

of both pressure and temperature dependence of shear-viscosity are lost. To the

best of our knowledge, accurate models for the strain-rate dependent shear

viscosity, as functions of either pressure or temperature have not been reported.


252

0.975 0.978 0.982 0.985 0.989 0.9922

4

6

8

10

12

14

1780 s-1

890 s-1

η η η η ,,,, 106cP

ρ ,ρ ,ρ ,ρ , g/cm3

223 s-1

Figure 8.11 Comparison of experimental shear viscosities of squalane (�) at

+ � Åá: and various strain rates and densities with values obtained from Eq.

(8.13) (solid lines). Results are shown for strain-rates of 223s-1 ()�á, å� �=4.17 ×

1010 cPcm3/g, )�á, å� � = -8.55×1010 cPcm3/g, Y = -1.025 cm3/g), 890s-1 ()�á, å� � = 1.92 × 1010 cPcm3/g, )�á, å� � = -3.96×1010 cPcm3/g, Y = -1.029 cm3/g) and

1780s-1 ()�á, å� � = 1.14 × 1010 cPcm3/g, )�á, å� � = -2.36×1010 cPcm3/g, Y = -1.03

cm3/g). In all cases the AAD is less than 1%.


253

0.79 0.82 0.85 0.88 0.91 0.94 0.972

4

6

8

10

12

14

1780 s-1

890 s-1

η,η,η,η, 101010106666cP

p, GPa

223 s-1

Figure 8.12 Comparison of experimental shear viscosities of squalane (�) at

+ � Åá: as a function of pressure obtained from Eq. (8.11) (solid lines).

Results are shown for strain-rates of 223s-1 ()�á, å� � = 1.94 × 1017 cP/Pa,

)�á, å� � = -4.99×1017 cP/Pa, Y = -1.3×109 Pa-1), 890s-1 ()�á, å� � = 7.13 × 1016

cP/Pa, )�á, å� � = -1.96×1017 cP/Pa, Y= -1.407×109 Pa-1) and 1780s-1 ()�á, å� � =

3.96 × 1016 cP/Pa, )�á, å� � = -1.10×1017 cP/Pa, Y = -1.448×109 Pa-1). In all cases

the AAD is less than 1%.


254

In contrast, as illustrated above, our approach accurately reproduces the

simulation data for the shear viscosity with respect to both temperature and

pressure. The agreement between our model and the simulation data is typically

within an absolute average deviation of less than 1%.

8.4 Experimental Verification of the Model for Strain 8.4 Experimental Verification of the Model for Strain 8.4 Experimental Verification of the Model for Strain 8.4 Experimental Verification of the Model for Strain

Rate Dependent ViscosityRate Dependent ViscosityRate Dependent ViscosityRate Dependent Viscosity

It would be highly desirable to compare our model with experimental data. In

many cases, there are insufficient experimental data for strain-rate dependent

shear viscosities at different temperatures, densities and pressures to obtain

reliable values of the parameters of our model. Typically experimental measures

focus on the effect of shear and as such they are conducted at a common

pressure. In contrast, the shear viscosity of squalane has been reported (Bair et

al., 2002a; Bair et al., 2002b) at different pressures and densities. We have

compared our model with these simulation data in Figs. 8.11 and 8.12. The

comparison indicates that the model accurately reproduces the experimental

viscosity-density and viscosity-pressure behaviour of squalane.


255

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 50 . 1

0 . 4

0 . 7

1 . 3 5

1 . 5 0

1 . 6 5

ηη ηη ×× ××1010 1010

66 66 , µ, µ , µ, µP

a.S

p , M P a

a

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 50 . 3

0 . 5

0 . 7

2 . 4

2 . 7

3 . 0

3 . 3

ηη ηη×× ×× 1

010 101066 66 , µ, µ , µ, µ

Pa.

S

p , M P a

b

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 50 . 2 5

0 . 5 0

0 . 7 5

3 . 6

3 . 8

4 . 0

4 . 2

ηη ηη ×× ××1010 1010

66 66 , µ, µ , µ, µP

a.S

p , M P a

c

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 50 . 3

0 . 6

0 . 84 . 5 0

4 . 7 5

5 . 0 0

5 . 2 5

ηη ηη×× ×× 1

010 101066 66 , µ, µ , µ, µ

Pa.

S

p , M P a

d

Figure 8.13 Zero-shear viscosity isotherms of monatomic fluids as a function of

pressure. (a) For neon, isotherms presented are T = 26 K (�), 100 K (�), 500

K ( �), 1000 K (�), 1300 K (⊳). (b) For argon, isotherms presented are T =

90 K (�), 500 K (�), 1000 K ( �), 1300 K (�). (c) For krypton, isotherms

presented are T = 120 K (�), 500 K (�), 1000 K ( �), 1300 K (�). (d) For

xenon, isotherms presented are T = 170 K (�), 500 K (�), 1000 K ( �), 1300

K (�). In all cases solid lines represent the model the model of Eq. (8.13) with

the fitting parameters and statistics illustrated in Table B.1 (Appendix B).


256

8.5 Experimental Verification of the Model for Zero8.5 Experimental Verification of the Model for Zero8.5 Experimental Verification of the Model for Zero8.5 Experimental Verification of the Model for Zero----

Shear ViscosityShear ViscosityShear ViscosityShear Viscosity

8.5.1 Monatomic Real Fluid8.5.1 Monatomic Real Fluid8.5.1 Monatomic Real Fluid8.5.1 Monatomic Real Fluidssss

The most commonly studied monatomic fluids are argon, neon, krypton, and

xenon. Although pressure is the most accessible experimental parameter, the

complicated representation of monatomic fluid viscosity as a function of

pressure made it difficult to model pressure dependent viscosity. At low

pressures viscosity increase with the increase of temperature while at sufficiently

high pressures the viscosity increases as the temperature decrease. This behavior

of the pressure dependent viscosity is manifested by the intersection of the

isotherms. The lower the temperatures, the lower the intersection pressures are.

The slope of the pressure-viscosity isotherms vary widely with the temperature

and in the vicinity of critical temperature, it changes from negative to positive.

This irregular nature of viscosity curves in the pressure-viscosity planes put

restrictions even in the modelling of simple monatomic liquids. Fig. 8.13 shows

pressure dependence of the shear viscosity of neon, argon, krypton, and xenon

for temperature range 26K -1300 K, pressure range 0.1-100 MPa, and viscosity

range 2.44 � 10� -3.18 � 10� µPa using data of Rabinovich et al. (1988). Fig.

8.13 shows that there is good agreement between experimental and model

calculations. Table B.1 (Appendix B) illustrates the estimated parameters and

relevant statistical analysis. It is found that our model can reproduce the


257

experimental results for neon, argon, krypton, and xenon with remarkable

agreement with the experimental results.

8.5.2 8.5.2 8.5.2 8.5.2 Complex Complex Complex Complex Real FReal FReal FReal Fluidluidluidluidssss

(i) (i) (i) (i) WaterWaterWaterWater

Current theories are not capable of fitting experimental viscosity data of water

from the liquid and vapour phases together. For our analysis we have used the

experimental data from IAPWS recommended viscosity data (Watanabe and

Dooley, 2003). Figures 8.14 - 8.17 show how differently viscosities of water vary

with pressure depending on the temperatures. For experimentally covered range

of temperatures and pressures three different behaviour of viscosity curve was

observed. At � 273 K and pressure range 40 � 100 MP the viscosity curve with respect to pressure is concave while it is convex for T = 773 K (Fig. 8.14).

At T = 473 K the increase of viscosity is almost proportional to the pressure as

can be observed from Fig. 8.15. Schmelzer et al. (2005) extensively analysed the

experimental viscosity of water as a function of pressure and found that the

dependence of the viscosity on pressure changes qualitatively with the increase

of temperature and a minimum occurs at � 298 K. They showed that the

slope ��4/�=�� of the viscosity curve 4 � 4�=, � const.� does not depend on the function and exclusively a function of temperature. In this study we have

not considered the low pressure viscosities.


258

35 45 55 65 75 85 95 10530

50

70

886

888

890

892

η,η,η,η,µµµµPas

p,MPa

Figure 8.14 Zero-shear viscosity of water as a function of pressure (in the range

from 40 to 100 MPa) at Å�% . (�) and ��% . (Ο). Experimental data taken

from Watanabe and Dooley (2003). The convex and concave behavior of water

viscosity towards the pressure axis can be seen from these experimental data.

0 15 30 45 60 75 90 105280

285

290

295

300

305

310

ηηηη,µµµµPaS

p, MPa

Figure 8.15 Zero-shear viscosity isotherm of water as a function of pressure (in

the range from 0.5 to 100 MPa) at 373 K (Ο) and the least squares fit (—) of

the model (Eq. (8.13)). Experimental data taken from Watanabe and Dooley

(2003). Any exponential or quadratic type viscosity model cannot fit this

viscosity behavior of water.


259

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 50

4 0 0

8 0 0

1 7 0 0

1 8 0 0

η , µη , µη , µη , µ P a . S

p , M P a

a

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 55 5

8 0

1 0 5

1 3 0

η , µη , µη , µη , µ P a . S

p , M P a

b

3 0 4 5 6 0 7 5 9 0 1 0 53 2

4 6

6 0

7 4

8 8

η , µη , µη , µη , µ P a . S

p , M P a

c

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 53 0

3 5

4 0

4 5

5 0

5 5

6 0

η , µη , µη , µη , µ P a . S

p , M P a

d

Figure 8.16 Zero-shear viscosity isotherms of water as a function of pressure in

the high pressure region. Shown are the isotherms for (a) T = 273 K (�), 298

K (�), 323 K ( �), 348 K(�), 373 K (⊳), 423 (�), 573 (); (b) T = 523 K

(�), 573 K (�), 623 K ( �), 648 K(�); (c) T = 673 K (�), 698 K (�), 723 K

( �), 748 K (�), 773 K (⊳); (d) T = 823 K (�), 873 K (�), 923 K ( �), 973

K (�), 1023 K (⊳), 1073 (�). In all cases solid lines represent the model of Eq.

(8.13) with the fitting parameters and statistics illustrated in Table B.2

(Appendix B).


260

0 3 6 9 12 15 18 2117

20

23

26

η, µη, µη, µη, µPa.S

p, MPa

a

0 4 8 12 16 20 24 28 3224

26

28

30

32

η, µη, µη, µη, µPa.S

p, MPa

b

Figure 8.17 Zero-shear viscosity isotherms of water as a function of pressure in

the low pressure region. Shown are the isotherms for (a) T = 523 K (�), 573 K

(�), 623 K ( �), 648 K (�); (b) T = 673 K (�), 698 K (�), 723 K ( �), 748

K (�), 773 K (⊳). In all cases solid lines represent the model of Eq. (8.13) with

the fitting parameters and statistics illustrated in Table B.2 (Appendix B).


261

0 5 1 0 1 5 2 0 2 5 3 04 3

4 5

4 7

4 9

5 1

5 3

5 5

5 7

η , µη , µη , µη , µ P a . S

p , M P a

a

0 4 0 8 0 1 2 0 1 6 0 2 0 0 2 4 0 2 8 0 3 2 02 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

p , M P a

η , µη , µη , µη , µ P a . S

b

0 4 0 8 0 1 2 0 1 6 0 2 0 0 2 4 0 2 8 0 3 2 00

5 0

1 0 0

1 5 0

2 0 0

2 5 0

η , µη , µη , µη , µ P a . S

p , M P a

c

0 4 0 8 0 1 2 0 1 6 0 2 0 0 2 4 0 2 8 0 3 2 00

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

η , µη , µη , µη , µ P a . S

p , M P a

d

Figure 8.18 Zero-shear viscosity isotherms of Carbon dioxide as a function of

pressure. Shown are the isotherms for (a) T = 1100 K (�), 1200 K (�), 1300

K ( �), 1400 K(�), 1500 K (⊳); (b) T = 580 K (�), 600 K (�), 620 K ( �),

640 K(�), 660 (), 680 (⊳), 700 (�), 800 (�), 900 (�); (c) T = 400 K (�),

420 K (�), 440 K ( �), 460 K (�), 480 K (⊳), 500 K (�), 520 K (), 540 K

( ), 560 K (�) ; (d) T = 220 K (�), 240 K (�), 260 K ( �), 280 K (�), 300

K (⊳), 320 (�), 340 K (), 360 K ( ), 380 K (�). In all cases solid lines

represent the model of Eq. (8.13) with the fitting parameters and statistics

illustrated in Table B.2 (Appendix B).


262

2 5 6 0 9 5 1 3 0 1 6 5 2 0 0 2 3 5 2 7 0 3 0 5 3 4 0 3 7 50

1 5 0 0

3 0 0 0

4 5 0 0

6 0 0 0

7 5 0 0

9 0 0 0

η , µη , µη , µη , µ P a . S

p , M P a

a

2 5 6 0 9 5 1 3 0 1 6 5 2 0 0 2 3 5 2 7 0 3 0 5 3 4 0 3 7 50

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

1 0 0 0 0

1 2 0 0 0

η , µη , µη , µη , µ P a . S

p , M P a

b

2 5 6 0 9 5 1 3 0 1 6 5 2 0 0 2 3 5 2 7 0 3 0 5 3 4 0 3 7 55 0 0

3 0 0 0

5 5 0 0

8 0 0 0

1 0 5 0 0

1 3 0 0 0

η , µη , µη , µη , µ P a . S

p , M P a

c

2 5 6 0 9 5 1 3 0 1 6 5 2 0 0 2 3 5 2 7 0 3 0 5 3 4 0 3 7 50

2 0 0 0 0

4 0 0 0 0

6 0 0 0 0

8 0 0 0 0

η , µη , µη , µη , µ P a . S

p , M P a

d

Figure 8.19 Zero-shear viscosity isotherms (Set-I) of hydrocarbons as a function

of pressure. (a) For n-C12, isotherms presented are T = 310.78 K (�), 333 K

(�), 352.44 K ( �), 371.89 K(�), 388.0 K (⊳), 408 (�); (b) For n-C15,

isotherms presented are T = 311.93 K (�), 334.15 K (�), 353.59 K ( �), 373.4

K(�), 389.15 (⊳), 409.15 (�); (c) For n-C18, isotherms presented are T = 333

K (�), 352.44 K (�), 371.89 K ( �), 388 K (�), 408 K (⊳); (d) For cis-

Decahydro-napthalene, isotherms presented are T = 288.56 K (�), 310.78 K

(�), 333 K ( �), 352 K (�), 371.89 K (⊳). In all cases solid lines represent the

model of Eq. (8.13) with the fitting parameters and statistics illustrated in

Table B.2 (Appendix B).


263

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 00

1 5 0 0 0

3 0 0 0 0

4 5 0 0 0

6 0 0 0 0

7 5 0 0 0

9 0 0 0 0

η , µη , µη , µη , µ P a . S

p , M P a

a

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 00

4 0 0 0 0

8 0 0 0 0

1 2 0 0 0 0

1 6 0 0 0 0

2 0 0 0 0 0

η , µη , µη , µη , µ P a . S

p , M P a

b

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 00

8 0 0 0 0

1 6 0 0 0 0

2 4 0 0 0 0

3 2 0 0 0 0

4 0 0 0 0 0

η , µη , µη , µη , µ P a . S

p , M P a

c

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 00

5 0 0 0 0

1 0 0 0 0 0

1 5 0 0 0 0

2 0 0 0 0 0

2 5 0 0 0 0

η , µη , µη , µη , µ P a . S

p , M P a

d

Figure 8.20 Zero-shear viscosity isotherms (Set-II) of hydrocarbons as a function

of pressure. (a) For 7-n-Hexyltridecane, isotherms presented are T = 310.78 K

(�), 333 K (�), 371.89 K ( �); (b) For 9-n-Octylheptadecane, isotherms

presented are T = 310.78 K (�), 333 K (�), 352.44 K ( �), 371.89 K(�), 388

(⊳); (c) For 11-n-Decylheneicosane, isotherms presented are T = 310.78 K (�),

333 K (�), 371.89 K ( �), 408 K (�); (d) For 13-n-Dodecylhexacosane,

isotherms presented are T = 310.78 K (�), 334 K (�), 371.89 K ( �), 408 K

(�). In all cases solid lines represent the model of Eq. (8.13) with the fitting

parameters and statistics illustrated in Table B.2 (Appendix B).


264

It is very difficult to find a single viscosity model that can address three

different viscosity curves with respect to pressure. Our viscosity model can fit

these three curves very well as indicated by the statistics of our fit given in

Table B.2 (Appendix B) and shown in Figs. 8.16 and 8.17. However, by virtue

of its simplicity our model cannot be used to analyse the viscosity curve near

the critical point for the entire range of experimentally studied pressure range at

a time. But if we divide the entire pressure range near critical points into low

pressure and high pressure parts our model fits very well. Since all the viscosity

models are valid for a certain range of temperatures and pressure, we can argue

that our model is more versatile than any other existing model and theories to

reproduce the experimental results with better and reliable statistics.

(ii) Carbon D(ii) Carbon D(ii) Carbon D(ii) Carbon Dioxideioxideioxideioxide

The qualitative shape of the pressure-viscosity curve of carbon dioxide also

changes with the increasing temperature. Unlike water the viscosity curve of

carbon dioxide change its shape at temperature � 1100 K. Below 1100 K the

viscosity curve is increasing in a convex shape for the range of pressure studied

and above this temperature the viscosity curve shape is concave. Quiones-

Cisneros et al. (2006), applying friction theory, have demonstrated that the

theory can reproduce the data obtained from the regression of Fenghour et al.

(1998) with AAD 0.21% for the range of temperatures 200 � 800 K and

pressures 0.1 � 300 MPa. To retain the simplicity of our model we have carried

out two different regression fits to the recommended carbon dioxide


265

experimental data of Fenghour et al. (1998). The reliability of the calculated

data from the model is shown in Fig. 8.18 and is illustrated in Table B.2

(Appendix B). Clearly the model demonstrates with very good reproducibility of

the experimental data. The estimated model parameters can also be found in

the columns 5, 6, and 7 of Table B.2 (Appendix B).

(iii) (iii) (iii) (iii) Light ALight ALight ALight Alkanelkanelkanelkanessss

We have chosen propane as one of the representatives of the light alkane group

(methane through octane). Table B.2 (Appendix B) indicates the quality of the

model predications compared to the recommended viscosity data provided by

Vogel et al. (1998). The model can reproduce experimental data with AAD of

0.025% for � 90K and AAD 0.0002% for � 200K. The calculated values are

better than reported by f-theory (Quinones-Cisneros et al., 2000) in which case

the AAD was 1.8 %.

(iv) (iv) (iv) (iv) HydrocarbonsHydrocarbonsHydrocarbonsHydrocarbons

We have used the data from Hogenboom et al. (1967) and Lowitz et al. (1959)

to test our model for hydrocarbons. Fig. 8.19 (Set-I) shows the experimental

data along with the model fits for � � ��, � � ��ý, � � �� and cis-Decahydro-

naphthalene (Lowitz et al., 1959). Table B.2 (Appendix B) gives the

corresponding model parameters and the statistics for the model. The

experimental data of Hogenboom et al. (1967) are shown in Fig. 8.20 (Set-II)


266

with the quality of model fit represented by the continuous lines. Our model

exhibits better fitting statistics than Kapoor and Das (2005).

In a summary, in this Chapter, an EOS has been developed, analysed and

tested. It has been found from comparison with simulation data that the

nonequilibrium contributions calculated from the steady-state EOS can be

obtained with a similar accuracy to the equilibrium contributions calculated

from the EOS. One of the utilities of such EOS is that it can be used in

conjunction with a generic viscosity model also developed in this Chapter.

Experimental verifications have been provided for the proposed model.

267

Chapter 9Chapter 9Chapter 9Chapter 9 ConclusionsConclusionsConclusionsConclusions and and and and

RecommendationsRecommendationsRecommendationsRecommendations

In this dissertation, we have attempted to provide a comprehensive

understanding of two distinct phenomena: the solid-liquid phase equilibria and

the shear viscosity for various model systems, both unbounded and bounded.

Thermophysical properties include shear viscosity and nonequilibrium equation

of state. The GWTS and GDI algorithms have been used for the simulations of

solid-liquid phase equilibria whereas nonequilibrium molecular dynamics

technique has been used for the calculation of thermophysical properties.

Validation is the most challenging part of any molecular simulation technique.

Since we have chosen the GWTS algorithm for the determination of solid-liquid

phase equilibria, we have carried out an extensive verification of the algorithm

on 12-6 Lennard-Jones potential. It has been demonstrated via extensive

analysis and comparison with literature that the GWTS algorithm (i) can

provide accurate estimation close to the triple point and (ii) can give better

results even at higher temperatures. The effects of system size and various

truncation and shifting schemes have been analysed to obtain benchmark data

for the 12-6 Lennard-Jones potential.

Conclusions and Recommendations

268

The effects of truncation and shifting schemes on the entire melting line of 12-6

Lennard-Jones fluid have been investigated via the GDI algorithm and it has

been found that the low temperature region, close to the vicinity of triple point,

is more sensitive to the truncation and shifting schemes. It has been found that

solid-liquid coexistence properties systematically vary with the details of

truncation and shifting scheme as well as with the cutoff radius. Before

comparing any solid-liquid coexistence data these considerations must be taken

into account. We believe that using the GWTS algorithm can also contribute to

the understanding of other unusual phase diagram topologies.

Using the GWTS algorithm, we have determined the solid-liquid coexistence

properties of fluids from the triple point to high pressures, interacting via n-6

Lennard-Jones potentials, where n = 12, 11, 10, 9, 8 and 7. By combining this

data with early vapour-liquid simulation, the complete phase behaviour for

these systems has been obtained. Analytical expressions for the coexistence

pressure liquid and solid densities as a function of temperature have been

determined which accurately reproduce the molecular simulation data. The

triple point temperature, pressure and liquid and solid densities have been

estimated. The triple point temperature and pressure scale with respect to 1/n,

resulting in a simple linear relationship that can be used to determine the

pressure and temperature for the limiting ∞ � 6 Lennard-Jones potential. Data

are obtained for the Raveché, Mountain and Streett and Lindemann melting

rules, which indicate that they are obeyed by the n-6 Lennard Jones potentials.


269

In contrast, it is demonstrated that the Hansen-Verlet crystallization rule is not

valid for n-6 Lennard-Jones potentials.

Given the success of GWTS algorithm in determining solid-liquid phase

transition for varying repulsive components of a family of Lennard-Jones

potentials, it has been used to determine the solid liquid coexistence of the

WCA fluid from low temperatures up to very high temperatures. At very high

temperatures, the coexistence pressure approaches the same 12-th power soft

sphere asymptote as the 12-6 Lennard Jones potential. However, in contrast to

the Lennard-Jones potential, which shows a discontinuity of pressure at low

temperatures, the coexistence pressure of the WCA potential approaches the

zero-temperature limit. Solid-liquid coexistence of the WCA potential

commences at densities close to the limiting packing fraction of hard spheres,

whereas the triple point is the commencement point for the Lennard-Jones fluid.

Three empirical relationships are determined to accurately reproduce the

coexistence pressure and both solid and liquid phase densities from near zero-

temperature up the very high temperatures. The simulation data are used to

reparametrized the Heyes and Okumura WCA equation of state, resulting in

considerably greater accuracy for the compressibility factor. The Lindemann

and Raveché et al. melting rules can be used to predict the onset of melting and

the Hansen and Verlet freezing rule can be applied for crystallization.


270

In view of the success of the GWTS algorithm in case of unbounded potentials,

we investigated its capability beyond unbounded potential. The GCM potential

is an important example of an unbounded potential. GCM has a very small

range of densities in which phase separation can occur and it has a complex re-

entrant melting scenario. The solid-liquid phase envelope of the GC potential

have been calculated using the GWTS algorithm more precisely than previously

possible. Our results are consistent with that of other work (Prestipino et al.,

2005). On the high-density side the solid-liquid coexisting line is slightly shifted

to higher densities compared with the results of Prestipino et al. (2005). The

common point, predicted by Stillinger (1976), where the crystal and its melt

have the same density, could be determined with high precision. The common

point on the GC envelope has not been resolved so far and a detailed analysis

will be of considerable interest.

NEMD simulations have been performed to calculate the shear viscosity of

GCM fluid under sheared flow. At low temperatures, shear viscosity isochors of

the GCM fluid as a function of strain rate, indicate anomalous behaviour. The

shear viscosity is lower than would be normally expected and the onset of shear

thinning is delayed until much higher strain-rates. The high strain-rate, non-

Newtonian region of the isochor appears to behave normally. Increasing the

temperature progressively reduces the anomaly, which is attributed to particle

overlap. At T ≥ 0.3, the viscosity isochor behaves normally at all strain-rates,

which is consistent with zero-shear viscosity results for the Gaussian core


271

potential. The GCM viscosity increases with temperature at high densities,

which is consistent with the behavior of dense gases when the GCM system is

approaching the so called infinite-density ideal-gas limit. The statistical

uncertainty in the viscosities reported for low strain-rates is considerably less

than can be obtained from the Lennard-Jones potential. Zero shear viscosities,

extrapolated from the NEMD data, are in generally good agreement with

equilibrium Green-Kubo calculations.

Extensive NEMD data for 12-6 Lennard-Jones fluids have been obtained for a

wide range of temperatures, density and strain-rates, which can be used to

deduce the nonequilibrium contributions to the energy and pressure of the fluid

under steady-state conditions. The nonequilibrium compressibility factor can be

accurately fitted to a polynomial function involving temperature, density and

strain-rate. Using this fit in conjunction with an equilibrium equation of state

yields a nonequilibrium steady-state equation of state for the 12-6 Lennard-

Jones potential. Comparison with simulation data indicates that the

nonequilibrium contributions can be obtained with similar accuracy to the

equilibrium contributions. Relationships for the shear viscosity as functions of

density and pressure have been obtained, which adequately reproduce the

simulation data. The isochoric shear viscosity as a function of pressure is shown

to be independent of strain-rate at sufficiently high strain-rates.


272

In view of the good results we have presented in this dissertation, the following

recommendations are possible for future work:

(i) GWTS algorithm could be extended for the study of solid-liquid

binary mixtures. Hitchcock and Hall (1999) calculated solid-liquid

phase diagrams for binary mixtures of 12-6 LJ spheres using Monte

Carlo and the GDI technique. However, the discrepancies of their

calculations with the experiments for the argon-krypton (Figure 4 in

Hitchcock and Hall (1999)) system are not known. An extension of

the GWTS algorithm could provide a reasonable answer for the

existing differences in literature and experiments.

(ii) Studies of solid-liquid phase equilibria for polymer chains are of both

theoretical and practical interests. For freely jointed chains the

GWTS algorithm can be easily extended for either with self avoiding

mechanism or for united atom models.

(iii) Since GCM potential acts as an effective potential, a combination of

this potential with other potentials such as WCA potential and LJ

potential could provide invaluable information about the phase and

shear viscosity behavior of complex molecular system. One such

example is of the form:

h�a� � h0�¡�a� � h �� , (9.1)


273

where h0�¡�a� is defined in Eq. (2.7) and h ��a� is defined in Eq.

(2.6). Eq. (9.1) can be thought as a special case of more general form

of the hard-core-soft-shoulder (Rascón et al., 1997) type potential

which is a combination of WCA potential and generalized exponential

model of the form:

h�a� � h01¡�a� � h ��a� , (9.2)

where h01¡�a� is defined as h01¡�a� � o��= 2� [

34�s5, (9.3)

where in Eq. (9.2) the value of 6 can be chosen (GCM is a special

case when q =2 ) arbitrarily and the ratio tj can be used as multiple

of t.

(iv) Nonequilibrium EOS can be extended for mixtures with the help of an

appropriate approximate theory such as the conformal solution theory

or corresponding states (Johnson et al., 1993, Roming and Hanley,

1986). Shear induced phase changes can be studied both for pure

fluid and mixtures using the nonequilibrium EOS.

(v) In conjunction with an appropriate conformal solution theory a

generic viscosity model could be an interesting tool study the

viscosity of mixtures.

274

AppAppAppAppendix Aendix Aendix Aendix A

Table A.1 Solid-liquid coexistence properties of full 12-6 Lennard-Jones

potential obtained in this work using the GDI algorithm starting with the

coexistence properties obtained from GWTS algorithm at T = 2.74. The

statistical uncertainty in the last digit is given in brackets.

= �rJs �jGr = �rJs �jGr 66.66667 2969(6) 2.040(3) 2.122(4) 3.508772 51.6(2) 1.192(1) 1.259(1)

40 1520(4) 2.001(1) 2.113(8) 3.389831 48.9(1) 1.184(1) 1.249(1)

28.57143 974(1) 1.925(2) 2.008(3) 3.278689 46.5(1) 1.174(1) 1.241(1)

22.22222 698(1) 1.819(2) 1.900(1) 3.174603 44.0(1) 1.166(2) 1.233(1)

18.18182 534(1) 1.738(2) 1.813(2) 3.076923 41.8(1) 1.158(2) 1.224(1)

15.38462 426(1) 1.670(2) 1.744(1) 2.985075 39.9(1) 1.151(2) 1.218(1)

13.33333 352(1) 1.617(2) 1.690(1) 2.898551 37.9(1) 1.144(3) 1.211(1)

11.76471 297.1(5) 1.569(2) 1.643(1) 2.816901 36.2(2) 1.136(2) 1.204(1)

10.52632 255.6(5) 1.532(1) 1.601(2) 2.739726 34.4(1) 1.127(1) 1.196(1)

9.52381 222.4(3) 1.496(2) 1.564(1) 2.290(5) 28.0966 1.085(1) 1.149(1)

8.695652 195.6(7) 1.462(2) 1.532(1) 2.065(5) 22.7861 1.065(1) 1.134(1)

8 174.4(3) 1.437(2) 1.504(1) 1.839(5) 18.4674 1.039(1) 1.108(1)

7.407407 156.5(3) 1.409(1) 1.478(1) 1.651(5) 14.9276 1.016(1) 1.088(1)

6.896552 141.6(4) 1.388(1) 1.456(1) 1.491(6) 12.0379 0.995(2) 1.068(1)

6.451613 128.6(3) 1.367(2) 1.434(1) 1.354(6) 9.6725 0.975(1) 1.051(1)

6.060606 117.8(4) 1.349(2) 1.416(1) 1.237(6) 7.7390 0.959(2) 1.036(1)

5.714286 107.9(2) 1.330(2) 1.397(1) 1.138(7) 6.1600 0.942(2) 1.021(1)

5.405405 99.4(2) 1.312(2) 1.379(1) 1.054(6) 4.8639 0.929(2) 1.011(1)

5.128205 92.0(3) 1.298(2) 1.364(1) 0.983(7) 3.8048 0.915(1) 1.001(1)

4.878049 85.5(2) 1.286(2) 1.350(1) 0.923(7) 2.9363 0.905(1) 0.994(1)

4.651163 79.7(2) 1.270(1) 1.336(1) 0.873(7) 2.2296 0.893(2) 0.986(1)

4.444444 74.4(3) 1.256(2) 1.323(1) 0.831(7) 1.6502 0.886(1) 0.979(1)

4.255319 69.6(1) 1.246(2) 1.311(1) 0.795(7) 1.1748 0.877(2) 0.975(1)

4.081633 65.3(1) 1.233(1) 1.299(1) 0.766(7) 0.7850 0.872(1) 0.971(1)

3.921569 61.5(1) 1.222(1) 1.288(1) 0.741(7) 0.4673 0.866(2) 0.967(1)

3.773585 58.0(2) 1.212(2) 1.278(1) 0.721(7) 0.2063 0.860(2) 0.965(1)

3.636364 54.7(1) 1.203(1) 1.268(1) 0.704(7) 0.0069

0.858(2)

0.963(1)

275

Appendix BAppendix BAppendix BAppendix B

The statistics presented in Tables B.1 and B.2 below for the evaluate of

performance are defined in the following way

DeviationJ=Xcalc,i-Xexp�or, sim�,iXexp,i

AAD � 1N

|Deviationi|N

i=1

MxD � Max|Deviationi| Bias � 1

N Deviationi

N

i=1

where N is the number of experimental points.

Appendix B

276

Tables of the Simulation Results Reported in Chapter Tables of the Simulation Results Reported in Chapter Tables of the Simulation Results Reported in Chapter Tables of the Simulation Results Reported in Chapter 8888

Table B.1 Pressure dependent viscosity model parameters for monatomic real fluids and the relevant statistics.

Fluid

T

(K) =-range (MPa)

4-range (µPa)

4�0, � (µPa/MPa)

4�0, � (µPa/MPa)

î

(MPa-1)

AAD

(%)

Max

Dev.

(%)

Bias

(%)

Source

Argon

90 1-20 2.44E+6-3.18E+6 2417150 (1321) 28755(322) 0.0343(3) 0.079 0.143 3.90E-05

Rabinovich et al. (1988)

1300

0.1-100

6.64E+5-7.14E+5

664400(583)

470 (32)

0.0012 (6)

0.055

0.764

1.81E-04

Neon

26

0.4-8 1.38E+6-1.63E+6 1380640(1380) 17319 (1381) 0.1946(3) 0.005 0.01 -1.28E-04

Rabinovich et al. (1988) 1300 0.1-100 8.38E+5-8.50E+5 119(56) 119 (56) 0.008 4(1) 0.002 0.005 -8.93E-08

Krypton

120

1-14 3.75E+6-4.10E+6 3721600(3463) 29805(1101) -0.0103(1) 0.0209 0.037 5.62E-05

Rabinovich et al. (1988) 1000 0.1-100 6.62E+5-8.20E+5 663374(1241) 1110(68) 0.0081(1) 0.020 0.088 7.07E-06

Xenon

170 1-22 4.53E+6-5.20E+6 4506180(8629) 33660(1914) -0.0049(1) 0.050 0.1 1.11E-04

Rabinovich et al. (1988) 500 0.1-100 3.72E+5-1.40E+6 328516(4067) 11225(220) -0.00018(4) 3.02 11.4 -1.83E-02

Appendix B

277

Table B.2 Pressure dependent viscosity model parameters for complex molecular fluid and the relevant statistics.

Fluid

T

(K) =-range (MPa)

4-range (µPa)

4�0, � (µPa/MPa)

4�0, � (µPa/MPa)

î

(MPa-1)

AAD

(%) Max Dev. (%)

Bias

(%)

Source

Water

273

0.1-100 1791-1651 1792.0(1) -2.653(9) -0.0095(1) 0.056 0.1313 -0.125

Watanabe and

Dooley (2003)

523 5-100 106.5-127.9 105.2(3) 0.26(1) -0.0030(2) 0.032 0.0762 -0.0001

684 0.1-20 23.45-25.85 23.57(8) -0.14(2) -0.1714(3) 0.617 1.0693 0.004

673 35-100 56.4-85 27(2) 1.05(7) -0.0085(1) 0.747 2.89988 0.002

773 0.5-30 28.64-31.68 28.64(9) -0.01(1) -0.7000(1) 0.142 0.3229 -0.032

1073 0.1-100 40.5-51 40.0(1) 0.081(8) 0.0061(1) 0.482 1.18145 0.003

Carbon dioxide

220 1-25 242.46-287.61 240.49(1) 1.988(3) -0.0042(2) 0.005 0.01 -0.015

Fenghour et al.

(1998)

400

0.1-30 19.7-45.08 19.6(2) 0.009(39) 6.333(1) 0.955 1.8274 0.033

35-300 51.02-222.94 29(1) 0.75(2) -0.0093(3) 1.50 7.7955 0.143

600

0.1-30 28-34.23 27.98(1) 0.042(1) 0.2619(1) 0.050 0.1224 0.001

35-300 36.09-127.75 22.5(4) 0.411(7) -0.0009(2) 0.718 1.5702 0.042

1000

0.1-30 41.26-43.44 41.253(3) 0.025(5) 0.1240(6) 0.010 0.0239 -0.0007

35-300 44.05-88.84 37.1(2) 0.179(3) 0.0002(6) 0.381 1.34795 -0.0009

1500 0.1-30 54.13-55.24 54.129(1) 0.016(2) 0.0812(5) 0.003 0.00861 -0.0005

Propane

90 0.01-40 7388-11950 7389(5) 88(1) 0.0143(1) 0.025 0.0594 0.009

Vogel et al. (1998) 200 0.05-100 289.1-562.6 289.03(3) 2.459(3) 0.0022(3) 0.0002 0.0735 0.0002

n-C12 408 40-360 580-3360 452(30) 0.35(2) 0.0714(5) 1.50 4.78 0.106

Lowitz et al.

(1959a)

n-C15 388 40-320 1060-6200 836(50) 4.8(6) 0.0152(5) 1.11 2.86 0.061

n-C18 371.89 40-320 1810-12820 1550(164) 5(2) 0.0360(2) 2.08 5.87 0.145

cis-Decahydro-naphthalene 310.78 80-360 5810-106000 28992(8849) -376(89) -0.0085(3) 17.2 57.5 1.47

278

Appendix CAppendix CAppendix CAppendix C

(C1) (C1) (C1) (C1) Derivation of pressure dependent Derivation of pressure dependent Derivation of pressure dependent Derivation of pressure dependent

zerozerozerozero----shear shear shear shear ((((å� � á) ) ) ) viscosity relation viscosity relation viscosity relation viscosity relation

(Eq(Eq(Eq(Eqssss. (8.13) . (8.13) . (8.13) . (8.13) ----(8.14))(8.14))(8.14))(8.14))

Following the approach used by Kapoor and Dass (2005) and generic viscosity

model developed in this Thesis (Chapter 8), we assume the ratio of the first and

second derivatives of viscosity with respect to the pressure are a pressure

independent parameter (&>): &> � ��4�=, �=� ®� «�4�=, �= ¬�' (C1.1)


� &> � «�4�=, �= ¬� � 4�0, � exp�&>=�4�=, � � 4�0, � � 4�0, ��exp�&>=� � 1�

&> ±²³² (C1.2)


simplification to:

4�=, � � 4�0, � � 4�0, ��= � 0.5&>=�� (C1.3)


parameters 4�0, �, 4�0, � and &> which can be easily obtained by fitting Eq.

(C1.3) to the simulation results.

279

(C2) (C2) (C2) (C2) Derivation of pressure dependent Derivation of pressure dependent Derivation of pressure dependent Derivation of pressure dependent

shear (shear (shear (shear (å ( á� ) viscosity relation (Eqs. ) viscosity relation (Eqs. ) viscosity relation (Eqs. ) viscosity relation (Eqs.

(8.15)(8.15)(8.15)(8.15)---- (8.16)) (8.16)) (8.16)) (8.16))

Following the approach used above in the calculation of pressure dependent

zero-shear viscosity model, we assume the ratio of the first and second

derivatives of viscosity with respect to the pressure are a pressure independent

parameter (&Ó): &Ó � ��4�=, ��=� ®2� «�4�=, ��= ¬2�' (C2.1)


� &Ó � «�4�=, ��= ¬2� � 4�0, �� exp¯&Ó=°4�=, �� 4�0, �� 4�0, �� ¯exp¯&Ó=° � 1°

&Ó ±²³² (C2.2)


simplification to:

4�=, �� 4�0, �� 4�0, �� ¯= � 0.5&Ó=�° (C2.3)


parameters 4�0, �� , 4�0, �� and &Ó which can be easily obtained by fitting Eq.


280

((((C3C3C3C3) Derivation of ) Derivation of ) Derivation of ) Derivation of densitydensitydensitydensity dependent dependent dependent dependent

shear (shear (shear (shear (å� ( á) viscosity relation (Eqs) viscosity relation (Eqs) viscosity relation (Eqs) viscosity relation (Eqs. . . .

(8.17(8.17(8.17(8.17) ) ) ) ----(8.18(8.18(8.18(8.18))))))))

Following the approaches used above in the calculation of pressure dependent

shear viscosity models, we assume the ratio of the first and second derivatives of

viscosity with respect to the density are a density independent parameter (&!):

&! � ��4��, �� ®2� «�4��, �� ¬2�' (C3.1)


� &! � «�4��, �� ¬� � 4�0, �� exp¯&!�°4��, �� 4�0, �� 4�0, �� ¯exp¯&!�° � 1°

&! ±²³² (C3.2)


simplification to:

4��, �� 4�0, �� 4�0, �� ¯� � 0.5&!��° (C3.3)


parameters 4�0, ��, 4�0, �� and &! which can be easily obtained by fitting Eq.


281

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