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SOLIDSOLIDSOLIDSOLID
Submitted in Fulfilment of Submitted in Fulfilment of Submitted in Fulfilment of Submitted in Fulfilment of
MOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATION
SOLIDSOLIDSOLIDSOLID----LIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIA
AND AND AND AND
Submitted in Fulfilment of Submitted in Fulfilment of Submitted in Fulfilment of Submitted in Fulfilment of
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
MOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATION
LIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIA
AND AND AND AND SHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITY
Alauddin AhmedAlauddin AhmedAlauddin AhmedAlauddin Ahmed
Submitted in Fulfilment of Submitted in Fulfilment of Submitted in Fulfilment of Submitted in Fulfilment of
Doctor of PhilosophyDoctor of PhilosophyDoctor of PhilosophyDoctor of Philosophy
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
MOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATION
LIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIA
SHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITY
Alauddin AhmedAlauddin AhmedAlauddin AhmedAlauddin Ahmed
DissertationDissertationDissertationDissertation
Submitted in Fulfilment of Submitted in Fulfilment of Submitted in Fulfilment of Submitted in Fulfilment of the the the the Requirements for the DRequirements for the DRequirements for the DRequirements for the D
Doctor of PhilosophyDoctor of PhilosophyDoctor of PhilosophyDoctor of Philosophy
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
2020202010101010
MOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATION
LIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIA
SHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITY
Alauddin AhmedAlauddin AhmedAlauddin AhmedAlauddin Ahmed
DissertationDissertationDissertationDissertation
Requirements for the DRequirements for the DRequirements for the DRequirements for the D
Doctor of PhilosophyDoctor of PhilosophyDoctor of PhilosophyDoctor of Philosophy
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
10101010
MOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATIONMOLECULAR SIMULATION OFOFOFOF
LIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIA
SHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITYSHEAR VISCOSITY
Requirements for the DRequirements for the DRequirements for the DRequirements for the D
Doctor of PhilosophyDoctor of PhilosophyDoctor of PhilosophyDoctor of Philosophy
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
OFOFOFOF
LIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIALIQUID PHASE EQUILIBRIA
Requirements for the DRequirements for the DRequirements for the DRequirements for the Degree ofegree ofegree ofegree of
Swinburne University of TechnologySwinburne University of TechnologySwinburne University of TechnologySwinburne University of Technology
egree ofegree ofegree ofegree of
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
4.1 Simulation Details ............................................................................ 147
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 Simulation Details ............................................................................ 170
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
6.1 Simulation Details ............................................................................ 199
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
7.1 Simulation Details ............................................................................ 211
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
Figure 3.6Figure 3.6Figure 3.6Figure 3.6 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 (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
Figure 5.1Figure 5.1Figure 5.1Figure 5.1 Comparison of the solid-liquid coexistence (a) pressure, (b) liquid
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 (),
0.5429(⊳), 0.5430 (�), 0.5431(�), 0.5432 (�), 0.5433( ). Entry into the two-
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
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……………………………………………………………………………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
Figure 8.5Figure 8.5Figure 8.5Figure 8.5 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 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
Figure 8.6Figure 8.6Figure 8.6Figure 8.6 Comparison of the relative percentage difference of steady-state
compressibility obtained from this work with the values calculated from Eq.
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
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…….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-
Edwards periodic boundary condition
b� Jd)Ze[ velocity of the i-th particle after the move in Lees-
Edwards periodic boundary condition
b� Jfe)G[e velocity of the i-th particle before the move in Lees-
Edwards periodic boundary condition
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
Molecular Simulation
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
Molecular Simulation
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).
Molecular Simulation
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
Molecular Simulation
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)
Molecular Simulation
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)
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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
Molecular Simulation
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.
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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
Molecular Simulation
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.
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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:
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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
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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)
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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.
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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
Molecular Simulation
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:
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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.
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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
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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 �-
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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
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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
Molecular Simulation
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^ (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
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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),
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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
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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)
Molecular Simulation
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:
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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
Molecular Simulation
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
Molecular Simulation
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.
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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.
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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.
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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
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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.
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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
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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.
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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.
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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
Molecular Simulation
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.
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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)
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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
Molecular Simulation
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
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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
Molecular Simulation
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
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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.
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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
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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 Ǝ 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)
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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.
Validation of Solid-Liquid Phase Equilibria Data
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 Ǝ 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,
Validation of Solid-Liquid Phase Equilibria Data
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.
Validation of Solid-Liquid Phase Equilibria Data
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).
Validation of Solid-Liquid Phase Equilibria Data
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).
Validation of Solid-Liquid Phase Equilibria Data
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
5.68 1.024 0.936 Hansen and Verlet (1969)
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)
32.2 1.179 1.113 Hansen and Verlet (1969)
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)
Validation of Solid-Liquid Phase Equilibria Data
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
Validation of Solid-Liquid Phase Equilibria Data
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) (�).
Validation of Solid-Liquid Phase Equilibria Data
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) (�).
Validation of Solid-Liquid Phase Equilibria Data
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).
Validation of Solid-Liquid Phase Equilibria Data
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
CCCCoexistenceoexistenceoexistenceoexistence
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
Validation of Solid-Liquid Phase Equilibria Data
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
Validation of Solid-Liquid Phase Equilibria Data
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
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.
Validation of Solid-Liquid Phase Equilibria Data
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
Figure 3.6 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 (Ο) with data from Mastny and de Pablo (2007) (�) and McNeil-
Watson and Wilding (2006) (�). The errors are approximately equal to the
symbol size.
Validation of Solid-Liquid Phase Equilibria Data
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-
Validation of Solid-Liquid Phase Equilibria Data
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
Validation of Solid-Liquid Phase Equilibria Data
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
Validation of Solid-Liquid Phase Equilibria Data
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
Validation of Solid-Liquid Phase Equilibria Data
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.
Validation of Solid-Liquid Phase Equilibria Data
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.
Validation of Solid-Liquid Phase Equilibria Data
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.
Validation of Solid-Liquid Phase Equilibria Data
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.
Validation of Solid-Liquid Phase Equilibria Data
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.
Validation of Solid-Liquid Phase Equilibria Data
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) (—).
Validation of Solid-Liquid Phase Equilibria Data
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.
Validation of Solid-Liquid Phase Equilibria Data
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
Validation of Solid-Liquid Phase Equilibria Data
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.
Validation of Solid-Liquid Phase Equilibria Data
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
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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
CCCCoexistenceoexistenceoexistenceoexistence
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
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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.
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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.
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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).
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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).
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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.
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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��.���� (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��.���� (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.
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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)
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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.
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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).
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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.
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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).
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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.
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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:
Solid-Liquid Phase Equilibria of the Lennard-Jones Family of Potentials
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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%,
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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).
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Figure 5.1 Comparison of the solid-liquid coexistence (a) pressure, (b) liquid
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.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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).
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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��.
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
Phase Diagram of the Weeks-Chandler-Andersen Potential
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).
Phase Diagram of the Weeks-Chandler-Andersen Potential
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
6.1 6.1 6.1 6.1 Simulation DetailsSimulation DetailsSimulation DetailsSimulation Details
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).
Phase Diagram of the Gaussian Core Model Fluid
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
Phase Diagram of the Gaussian Core Model Fluid
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 (),
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 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
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.
Phase Diagram of the Gaussian Core Model Fluid
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 (),
0.5429(⊳), 0.5430 (�), 0.5431(�), 0.5432 (�), 0.5433( ). Entry into the two-
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
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.
Phase Diagram of the Gaussian Core Model Fluid
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
Phase Diagram of the Gaussian Core Model Fluid
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
Phase Diagram of the Gaussian Core Model Fluid
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.
Phase Diagram of the Gaussian Core Model Fluid
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.
Phase Diagram of the Gaussian Core Model Fluid
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
−×
Phase Diagram of the Gaussian Core Model Fluid
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
Phase Diagram of the Gaussian Core Model Fluid
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
7.1 7.1 7.1 7.1 Simulation DetailsSimulation DetailsSimulation DetailsSimulation Details
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
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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.
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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).
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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.
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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.
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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.
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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.
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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.
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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.
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded 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
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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
Strain-Rate Dependent Shear Viscosity of the Gaussian Core Bounded Potential
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.
Steady State Equation of State and Viscosity Modelling
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.
Steady State Equation of State and Viscosity Modelling
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:
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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)
Steady State Equation of State and Viscosity Modelling
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).
Steady State Equation of State and Viscosity Modelling
235
Table 8.1 Values ^á, ^å� and " 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.
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)
Steady State Equation of State and Viscosity Modelling
236
Table 8.2 Values of #á$ä��,#å� and " 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.
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)
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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.
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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
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) 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
−× & &
&
Steady State Equation of State and Viscosity Modelling
242
0.72 0.76 0.80 0.84 0.88 0.92 0.96-4
-2
0
2
4
ρρρρ
Figure 8.5 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 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
Figure 8.6 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 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
−× & &
&
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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)
Steady State Equation of State and Viscosity Modelling
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)
Steady State Equation of State and Viscosity Modelling
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).
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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.
Steady State Equation of State and Viscosity Modelling
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.
Steady State Equation of State and Viscosity Modelling
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.
Steady State Equation of State and Viscosity Modelling
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%.
Steady State Equation of State and Viscosity Modelling
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%.
Steady State Equation of State and Viscosity Modelling
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.
Steady State Equation of State and Viscosity Modelling
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).
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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.
Steady State Equation of State and Viscosity Modelling
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.
Steady State Equation of State and Viscosity Modelling
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).
Steady State Equation of State and Viscosity Modelling
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).
Steady State Equation of State and Viscosity Modelling
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).
Steady State Equation of State and Viscosity Modelling
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).
Steady State Equation of State and Viscosity Modelling
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).
Steady State Equation of State and Viscosity Modelling
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
Steady State Equation of State and Viscosity Modelling
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)
Steady State Equation of State and Viscosity Modelling
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.
Conclusions and Recommendations
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.
Conclusions and Recommendations
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
Conclusions and Recommendations
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.
Conclusions and Recommendations
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)
Conclusions and Recommendations
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)
Successive integration gives
� &> � «�4�=, �= ¬� � 4�0, � exp�&>=�4�=, � � 4�0, � � 4�0, ��exp�&>=� � 1�
&> ±²³² (C1.2)
Expanding the exponential and truncating after the first two terms leads, after
simplification to:
4�=, � � 4�0, � � 4�0, ��= � 0.5&>=�� (C1.3)
This means that the viscosity can be simply calculated from three adjustable
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)
Successive integration gives
� &Ó � «�4�=, ���= ¬2� � 4�0, �� � exp¯&Ó=°4�=, �� � � 4�0, �� � � 4�0, �� �¯exp¯&Ó=° � 1°
&Ó ±²³² (C2.2)
Expanding the exponential and truncating after the first two terms leads, after
simplification to:
4�=, �� � � 4�0, �� � � 4�0, �� �¯= � 0.5&Ó=�° (C2.3)
This means that the viscosity can be simply calculated from three adjustable
parameters 4�0, �� �, 4�0, �� � and &Ó which can be easily obtained by fitting Eq.
(C2.3) to the simulation results.
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)
Successive integration gives
� &! � «�4��, ���� ¬� � 4�0, �� � exp¯&!�°4��, �� � � 4�0, ��� � 4�0, �� �¯exp¯&!�° � 1°
&! ±²³² (C3.2)
Expanding the exponential and truncating after the first two terms leads, after
simplification to:
4��, �� � � 4�0, �� � � 4�0, �� �¯� � 0.5&!��° (C3.3)
This means that the viscosity can be simply calculated from three adjustable
parameters 4�0, ��, 4�0, �� and &! which can be easily obtained by fitting Eq.
(C3.3) to the simulation results.
281
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