Three-body effects on the phase behaviour of noble gases ... · In this work the phase behaviour of...
177
Three-Body Effects on the Phase Behaviour of Noble Gases from Molecular Simulation Liping Wang Dissertation Submitted in fulfilment of requirements for the degree of Doctor of Philosophy Centre for Molecular Simulation Faculty of Information and Communication Technologies Swinburne University of Technology 2005
Transcript of Three-body effects on the phase behaviour of noble gases ... · In this work the phase behaviour of...
Three-Body Effects on the Phase Behaviour of Noble Gases from
Molecular Simulation
Liping Wang
Dissertation Submitted in fulfilment of requirements for the degree of
Doctor of Philosophy
Centre for Molecular Simulation Faculty of Information and Communication Technologies
Swinburne University of Technology
2005
Abstract In this work the phase behaviour of noble gases is studied comprehensively by different
molecular simulation methods using different intermolecular potentials. The aim is to
investigate three-body effects on the phase behaviour of noble gases. A true two-body
potential model (Barker-Fisher-Watts potential) and the three-body potential model
(Axilrod-Teller term) have been used.
The results obtained from the two-body BFW potential with the three-body Axilrod-
Teller potential included for the vapour-liquid and solid-liquid phase equilibrium
properties of pure noble gases are compared with the calculations using the Lennard-
Jones potential with different suggested parameter values. The results have been
compared with experimental data and the best parameter values for simulating the
thermodynamic properties of noble gases are found.
Three-body effects on the phase behaviour of noble gases are reported for a large range
of density, temperature and pressure. Simple relationships have been found between
two-body and three-body potential energies for pure fluids and solids. Three-body
effects on the vapour-liquid phase equilibrium properties of argon, krypton, xenon and
argon-krypton systems are studied by the Gibbs-Duhem integration Monte Carlo
method. Three-body effects on the solid-liquid phase equilibrium properties of argon,
krypton and xenon are investigated by non-equilibrium and equilibrium molecular
dynamics techniques. All the calculations have been compared with experimental data,
which show that three-body interactions play an important role in the overall
interatomic interactions of noble gases.
i
Acknowledgements I would like to thank my supervisor Prof. Richard Sadus for his patience,
encouragement and support in the process of doing the project, which has broadened my
view and has developed my scientific knowledge and skills. I appreciated very much the
opportunity of working with him.
Many thanks go to A/Prof. Billy Todd, A/Prof. Feng Wang, Dr. Ming Liu, Jaroslaw
Bosko and all the other research fellows and PhD students for their warm
encouragement and help.
Special thanks go to my best friend Robert.
I thank the International Postgraduate Research Scholarship and the Faculty of
Information and Communication Technologies Scholarship of Swinburne University of
Technology. Thanks also go to the Australian Partnership for Advanced Computing
who provided an allocation of computing time to perform simulations.
I would like to express my gratitude to my family for their constant encouragement and
support.
ii
Declaration
I hereby declare that the thesis entitled “Three-Body Effects on the Phase Behaviour of
Noble Gases from Molecular Simulation” 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.
Liping Wang December, 2005
iii
Table of Contents Chapter 1: Molecular Simulation of Phase Equilibria ...…………………………….1 1.1 Aims ……………………………………………………………………….......1 1.2 Phase Equilibria of Pure Fluids and Mixtures …………………………….......3 1.3 Progress in Molecular Simulation of Phase Equilibria ………………………10 Chapter 2: Molecular Simulation Theories and Techniques ………………………17 2.1 Intermolecular Potentials …………………………………………………….17 2.1.1 Lennard-Jones Potential ……………………………………………….20 2.1.2 Two-body Potentials …………………………………………………...22 2.1.3 Many-body Interactions …………………………………………..........31 2.1.4 Relationship between Two-body and Three-body Interactions………...38 2.1.5 Combining Rules……………………………………………………….39 2.2 Phase Equilibria Simulation Techniques …………………………………….40 2.2.1 Metropolis Monte Carlo Method ………………………………………40 2.2.2 Gibbs Ensemble Monte Carlo Simulation …………………………......42 2.2.3 Gibbs-Duhem Integration Method ……………………………………..48 2.2.4 Molecular Dynamics and Non-equilibrium Molecular Dynamics .........54 Chapter 3: Phase Equilibrium Properties of Noble Gases – Lennard-Jones Calculation ………………………….…………………..62 3.1 Vapour-liquid Phase Equilibrium Properties …………………………..…….63 3.2 Solid-liquid Phase Equilibrium Properties ………………………………......69
iv
Chapter 4: Three-body Effects on Phase Equilibrium Properties of Noble Gases …………………………………………………………………………….73 4.1 Three-body Effects on Vapour, Liquid and Solid ………………………........73 4.2 Investigation of Relationship between Two-body and Three-body Potential Energies ………………………………………………………………………91 4.3 Three-body Effects on Vapour-liquid Equilibrium Properties of Pure Fluids .........................................................................................................................102 4.4 Three-body Effects on Vapour-liquid Equilibrium Properties of Mixtures ……………………………………………………………………………….109 4.5 Three-body Effects on Solid-liquid Equilibrium Properties of Noble Gases ………………………………………………………………………….........119 Chapter 5: Conclusions and Recommendations ………………………………......125 Appendix 1: Forces and Long-range Corrections Used in this Work ……….......128 Appendix 2: Tables of the Simulation Results Reported in Chapters 3 and 4. …………………………………………………………………………...131 References ……………………………………………………………………………151
v
Table of Symbols
Abbreviations
AT Axilrod-Teller potential
BFW Barker-Fisher-Watts intermolecular potential
Fcc Face-centred cubic
LJ Lennard-Jones potential
MD Molecular Dynamics
MC Monte Carlo
NEMD Non-equilibrium molecular dynamics
NPT Ensemble where number of particles, pressure and temperature are
kept constant
NVT Ensemble where number of particles, volume and temperature are
kept constant
Subscripts and superscripts
* Reduced units
V Vapour phase
L Liquid phase
S Solid phase
Latin alphabet
E Potential energy
k Boltzmann’s constant
N Number of particles
P Pressure
R Molar gas constant
T Temperature
V Volume
x,y,z Cartesian coordinates
vi
Greek alphabet
σ Atomic radius
ε Energy per particle or depth of potential well
ν Non-additive coefficient
ρ Numeric density
vii
List of Tables Table 2.1 Parameter values suggested for the Lennard-Jones potential ……………….22 Table 2.2 Parameters of the Barker-Pompe, Bobetic-Barker and Barker-Fisher-Watts (BFW) potentials……………………………………………………………………… 27 Table 2.3 Parameters of the two-body potentials for krypton and xenon………………29 Table 2.4 Coefficients for the three-body potentials. ………………………………….37 Table 2.5 Contribution of three-body interactions to the crystalline energy of noble gases ……………………………………………………………………………..37 Table 3.1 Vapour-liquid coexistence data obtained from the Gibbs-Duhem integration simulation using the Lennard-Jones potential………………………………………...131 Table 3.2 Solid-Liquid coexistence properties obtained from the Gibbs-Duhem simulations using the Lennard-Jones potential ……………………………………….132 Table 4.1 Three-body effects on phase behaviour of argon at …………...133 * 0.9914T = Table 4.2 Three-body effects on phase behaviour of argon at …………...134 * 1.2678T = Table 4.3 Vapour-liquid coexistence properties of argon obtained by Gibbs-Duhem Integration method simulations using BFW potential………………………………...135 Table 4.4 Vapour-liquid coexistence properties of krypton obtained by Gibbs-Duhem Integration method simulations using BFW potential. ……………………………….135 Table 4.5 Vapour-liquid coexistence properties of xenon obtained by Gibbs-Duhem Integration method simulations using BFW potential………………………………...135 Table 4.6 Vapour-liquid phase equilibria properties of argon from Gibbs-Duhem integration simulation using the two-body BFW potential + three-body potentials (DDD+DDQ+DQQ+QQQ+DDD4). …………………………………………………136 Table 4.7 Vapour-liquid phase equilibria properties of argon from Gibbs-Duhem integration simulation using the two-body BFW potential + AT term ………………………………………………………………………………………...137 Table 4.8 Vapour-liquid phase equilibrium properties of argon obtained from Gibbs-Duhem integration simulations using the relationship between 2–body and 3-body potentials ………………………………………………………………………….......138
viii
Table 4.9 Vapour-liquid phase equilibrium properties of krypton obtained from Gibbs-Duhem integration simulations using the relationship between 2–body and 3-body potentials. ……………………………………………………………………………..138 Table 4.10 Vapour-liquid phase equilibrium properties of xenon obtained from Gibbs-Duhem integration simulations using the relationship between 2–body and 3-body potentials…………….…………………………………………………………….......138 Table 4.11 Initial conditions for BFW calculations ………………………………….139 Table 4.12 Initial conditions for BFW+AT calculations……………………………...139 Table 4.13 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=177.38K using the two-body potential. ……………………………......140 Table 4.14 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=163.15K using the two-body potential…………………………………140 Table 4.15 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=158.15K using the two-body potential. ……………………………......141 Table 4.16 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=153.15K using the two-body potential. ……………………………….141 Table 4.17 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=148.15K using the two-body potential. ……………………………......142 Table 4.18 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=143.15K using the two-body potential. ……………………………......142 Table 4.19 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T = 177.38K using BFW+AT potentials. …………………………………143 Table 4.20 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=163.15K using BFW + AT potentials…………………………….…….144 Table 4.21 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=158.15K using BFW + AT potentials ………………………………….145 Table 4.22 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=153.15K using BFW + AT potentials ………………………………….146 Table 4.23 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=148.15K using BFW + AT potentials. …………………………………147 Table 4.24 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T=143.15K using BFW + AT potentials ………………………………….148 Table 4.25 Three-body effects on solid-liquid phase equilibrium properties of argon ………………………………………………………………………………………...149
ix
Table 4.26 Three-body effects on solid-liquid phase equilibrium properties of krypton ………………………………………………………………………………………...149 Table 4.27 Three-body effects on solid-liquid phase equilibrium properties of xenon ………………………………………………………………………………………...150
x
List of Figures
Figure 1.1 Pure fluid p-T phase diagram ………………………………………….…….5 Figure 1.2 Pure fluid T ρ− phase diagram…………………………………….………..6 Figure 1.3 Vapour-liquid phase diagram of binary mixture …….. ……………………. 8 Figure 2.1 Comparisons between LJ and BFW potential ……………………….........30 Figure 2.2 Triplet configuration of atom ………………………………….........35 , , .i j k Figure 2.3 the Gibbs ensemble Monte Carlo simulation method …..………………….44 Figure 2.4 Pressure as a function of strain rate at different densities and constant temperature …………………………………………………………………60 1.00T ∗ = Figure 3.1 Comparison of experiment (æ) with calculation using Lennard-Jones potential with different potential parameter values suggested by Hogervorst (ó), Horton (ç), Clifford (ò), Rodriguex (õ), and Walton (ô) for the vapour-liquid coexistence density of argon. …………………………………… …………………………………66 Figure 3.2 Comparison of experiment (æ) with calculation using Lennard-Jones potential with different potential parameter values suggested by Hogervorst (ó), Horton (ç), Clifford (ò), Rodriguex (õ), and Walton (ô) for the vapour-liquid coexistence density of krypton. ……………………………………. ………………………………66 Figure 3.3 Comparison of experiment (æ) with calculation using Lennard-Jones potential with different potential parameter values suggested by Hogervorst (ó), Horton (ç), Clifford (ò), Rodriguex (õ), and Walton (ô) for the vapour-liquid coexistence density of xenon. ..............................................................................…...........................67 Figure 3.4 Comparison of experiment (æ) with calculation using Lennard-Jones potential with the potential parameter values suggested by Horton (ç) for the vapour-liquid coexistence pressure of argon, krypton and xenon. …………………………….67 Figure 3.5 Comparison of experiment (æ) with calculation using Lennard-Jones potential with the potential parameter values suggested by Horton (ç) for the vapour-liquid coexistence latent heat of argon, krypton and xenon. ………………………......68 Figure 3.6 Vapour-liquid phase equilibrium potential energy of LJ system as a function of temperature …………………………………………………………….....................68 Figure 3.7 Comparison of experiment (æ) with calculation using Lennard-Jones potential with the potential parameter values suggested by Horton (ç) for the solid-liquid coexistence density of argon, krypton and xenon. .....................................…......71
xi
Figure 3.8 Comparison of experiment (æ) with calculation using Lennard-Jones potential with the potential parameter values suggested by Horton (ç) for the solid-liquid coexistence pressure of argon, krypton and xenon. …………………………….71 Figure 3.9 Comparison of experiment (æ) with calculation using Lennard-Jones potential with the potential parameter values suggested by Horton (ç) for the solid-liquid coexistence latent heat of argon, krypton and xenon ……………………...……72 Figure 3.10 Solid-liquid phase equilibrium potential energy of LJ system as a function of temperature ……………………………………………………………………......72 Figure 4.1a Comparison of two-body potential energies calculated for argon with different system size ( ó108, æ256, ç500 and ò864 atoms) at different reduced densities ……………………………………………………………………………......79 Figure 4.1b Relative percentage difference between two-body potential energy of different system size ( ó108, æ256, and ç500) and that of 864 atom system of argon at different reduced densities…...………………………………….……………………...79 Figure 4.2a Comparison of three-body potential energies calculated for argon with different system size ( ó108, æ256, ç500 and ò864 atoms) at different reduced densities ……………………………………………………………………………......80 Figure 4.2b Relative percentage difference between total-body potential energy of different system size ( ó108, æ256, and ç500) and that of 864 atom system of argon at different reduced densities……………………………………………………...………80 Figure 4.3a Comparison of total potential energies calculated for argon with different system size (ó108, æ256, ç500 and ò864 atoms) at different reduced densities……...81 Figure 4.3b Relative percentage difference between three-body potential energy of different system size ( ó108, æ256, and ç500) and that of 864 atom system of argon at different reduced densities …...……………………………………………...…………81 Figure 4.4 Comparison of potential energies (á 2E∗ , à 3E∗ and æ( 2 3E E E∗ ∗= + ∗ )) calculated for 500 atom system of argon at different reduced densities…….…………82 Figure 4.5 Comparison of pressures (æ totalP∗ , à 2P∗ , ç 3P∗ and ó ) calculated for 500 atom system of argon at different reduced densities……………………………….......82
kP∗
Figure 4.6 the total pressure ( 2 3total kP P P P∗ ∗ ∗ ∗= + + ) (-è-) and ( ) (-â-) calculated from simulations for 500 atom system of argon between the liquid and vapour coexistence densities. The pressures display “van der Waals” loops in the two-phase vapour/liquid region. The equilibrium coexisting pressures for the two cases (dotted and solid lines) were obtained from Gibbs Ensemble simulations…………………………83
2 kP P∗ + ∗
Figure 4.7 Comparison of pressures calculated for 500 atom system of argon between the liquid and vapour coexistence densities…………………………………... ………83
xii
Figure 4.8 Comparisons of the various components to the pressures of argon at different reduced densities at ……………………………………..………………...84 0.9914T ∗ = Figure 4.9 Comparisons of pressures calculated from simulations for 500 atom system of argon at different reduced densities at ………………………................84 0.9914T ∗ = Figure 4.10 the configuration of argon at * 0.91ρ = …………………………………......85 Figure 4.11 the configuration of argon at * 0.92ρ = . ……………………………………85 Figure 4.12 Two-body potential energies as a function of density at different temperatures. Results are shown for both subcritical temperatures ( (ò),
(à)) and supercritical temperatures ( (æ), (á), (ç))...………………………………………………………………………….88
0.9T ∗ =0.9914T ∗ = 1.2678T ∗ = 1.4168T ∗ =2.0T ∗ =
Figure 4.13 Three-body potential energies as a function of density at different temperatures. Results are shown for both subcritical temperatures ( (ò),
(à)) and supercritical temperatures ( (æ), (á), (ç))…..……………………………………………………………………......88
0.9T ∗ =0.9914T ∗ = 1.2678T ∗ = 1.4168T ∗ =2.0T ∗ =
Figure 4.14 Total potential energies as a function of density at different temperatures. Results are shown for both subcritical temperatures ( (ò), (à)) and supercritical temperatures ( (æ), (á), (ç))…..............89
0.9T ∗ = 0.9914T ∗ =1.2678T ∗ = 1.4168T ∗ = 2.0T ∗ =
Figure 4.15 Two-body pressures as a function of density at different temperatures. Results are shown for both subcritical temperatures ( (ò), (à)) and supercritical temperatures ( (æ), (á), (ç))..…………89
0.9T ∗ = 0.9914T ∗ =1.2678T ∗ = 1.4168T ∗ = 2.0T ∗ =
Figure 4.16 Three-body pressures as a function of density at different temperatures Results are shown for both subcritical temperatures ( (ò), (à)) and supercritical temperatures ( (æ), (á), (ç)). ………………………………………………………………………………………….90
0.9T ∗ = 0.9914T ∗ =1.2678T ∗ = 1.4168T ∗ = 2.0T ∗ =
Figure 4.17 Total pressures as a function of density at different temperatures. Results are shown for both subcritical temperatures ( (ò), (à)) and supercritical temperatures ( (æ), (á), (ç)). ………………………………………………………………………………………….90
0.9T ∗ = 0.9914T ∗ =1.2678T ∗ = 1.4168T ∗ = 2.0T ∗ =
Figure 4.18 the ratio of three-body and two-body potential energies of argon with different system size (ó108, æ256, ç500 and ò864 atoms) at different reduced densities …………………………………………………………………………….....................95 Figure 4.19a simple relationship between two-body and three-body potential energies of liquid argon (reduced density ranging from 0.45 to 0.75) with different system size (ó108, ø200, æ256, ç500 and ò864 atoms) ………………………………………......96 Figure 4.19b coefficient values as a function of atom numbers of liquid argon (reduced density ranging from 0.45 to 0.75) ……………………………………….....................96
xiii
Figure 4.20a simple relationship between two-body and three-body potential energies of liquid argon (reduced density ranging from 1.1 to 1.4) with different system size (ó108, æ256, ç500 and ò864 atoms) …………………………………………………97 Figure 4.20b coefficient values as a function of atom numbers of solid argon (reduced density ranging from 1.1 to 1.3) ……………………………………………………….97 Figure 4.21 the ratio of three-body and two-body energies calculated from simulations at different reduced densities for argon ( (æ)), krypton ( (ç))and xenon ( (ò)) ……………………………………………………………...100
0.9914T ∗ = 1.0T ∗ =0.9252T ∗ =
Figure 4.22 the ratio of three-body and two-body energies of argon at different temperatures. Results are shown for both subcritical temperatures ( (ò),
(à)) and supercritical temperatures ( (æ), (á), (ç))………………………………………………………………………......100
0.9T ∗ =0.9914T ∗ = 1.2678T ∗ = 1.4168T ∗ =2.0T ∗ =
Figure 4.23 a simple relationship between two-body and three-body potential energies of pure fluids. Results are shown for both subcritical temperatures ( (ò),
(à)) and supercritical temperatures ( (æ), (á))........…………………………………………………………...…......101
0.9T ∗ =0.9914T ∗ = 1.2678T ∗ =1.4168T ∗ =
Figure 4.24 a simple relationship between two-body and three-body potential energies of pure solids Results are shown for both subcritical temperatures ( (ò),
(à)) and supercritical temperatures ( (æ), (á))…. 0.9T ∗ =
0.9914T ∗ = 1.2678T ∗ = 1.4168T ∗ =………………………………………………………………………………………...101 Figure 4.25 Comparison of experiment (æ) with calculation using BFW potential (Gibbs (à), Gibbs-Duhem(á)) and BFW+AT (Gibbs (ó), Gibbs-Duhem(ò)) for the vapour-liquid coexistence density of argon…………………………………………...105 Figure 4.26 Comparison of the contribution of the various three-body terms to the configurational energy of the liquid phase of argon ………………………………….106 Figure 4.27 Comparison of the contribution of the various three-body terms to the pressure of the liquid phase of argon …………………………………………………106 Figure 4.28 Comparison of the calculated vapour-liquid phase coexistence density of argon using the ( 4BFW DDD DDQ DQQ QQQ DDD+ + + + + ) potentials (ò), the ( ) potentials (æ) and the simple relationship between two-body and three-
body potential energy (
BFW DDD+2
3 6
23v EE ρεσ
= − ) (ç), respectively………………….................107
Figure 4.29 Comparison of experiment (æ) with calculation using BFW potential (Gibbs (à), Gibbs-Duhem(á)) and BFW+AT (Gibbs (ó), Gibbs-Duhem(ò)) for the vapour-liquid coexistence density of krypton. ……………………………………….108
xiv
Figure 4.30 Comparison of experiment (æ) with calculation using BFW potential (Gibbs (à), Gibbs-Duhem(á)) and BFW+AT (Gibbs (ó), Gibbs-Duhem(ò)) for the vapour-liquid coexistence density of xenon. …………………………………………108 Figure 4.31 Isothermal vapour-liquid phase diagram of the system argon + krypton at 117.38K. …………………………………………………………………………….113 Figure 4.32 Isothermal vapour-liquid phase diagram of the system argon + krypton at 163.15K. ……………………………………………………………………………...113 Figure 4.33 Isothermal vapour-liquid phase diagram of the system argon + krypton at 158.15K. ……………………………………………………………………………...114 Figure 4.34 Isothermal vapour-liquid phase diagram of the system argon + krypton at 153.15K. …………………………………………………………………………......114 Figure 4.35 Isothermal vapour-liquid phase diagram of the system argon + krypton at 148.15K. …………………………………………………………………………......115 Figure 4.36 Isothermal vapour-liquid phase diagram of the system argon + krypton at 143.15K. …………………………………………………………………………......115 Figure 4.37 Comparisons of isothermal vapour-liquid phase diagram of the system argon + krypton at 163.15K. ……………………………………………………….....118 Figure 4.38 Comparisons of isothermal vapour-liquid phase diagram of the system argon + krypton at 148.15K. ………………………………………………………….118 Figure 4.39 Comparison of solid-liquid phase coexistence densities of argon calculated by NEMD and MD simulations with experimental data. …………….………………122 Figure 4.40 Comparison of solid-liquid phase coexistence densities of krypton calculated by NEMD and MD simulations with experimental data. …………….......122 Figure 4.41 Comparison of solid-liquid phase coexistence densities of xenon calculated by NEMD and MD simulations with experimental data. …………….………………123 Figure 4.429 comparison of solid-liquid phase coexistence pressures of argon calculated by NEMD and MD simulations with experimental data. ……….…….......................123 Figure 4.43 Comparison of solid-liquid phase coexistence pressures of krypton calculated by NEMD and MD simulations with experimental data. …………………124 Figure 4.44 Comparison of solid-liquid phase coexistence pressures of xenon calculated by NEMD and MD simulations with experimental data. ……………………………124
xv
Chapter 1
Molecular Simulation of Phase Equilibria 1.1 Aims Knowledge of phase equilibria is very important for the understanding of various
phenomena occurring in nature and industrial processes (Gubbins and Quirke 1996).
Over the last century many experimental efforts (Horton 1976; Laird and Howat 1990)
have been made to study phase equilibrium properties of pure fluids and mixtures,
which have also been investigated in theoretical methods such as the integral equations
theories (Wu and Chiew 2001) and equations of state (Wei and Sadus 2000). Recently,
with the development of computer techniques and knowledge of the intermolecular
interactions, molecular simulation (Allen and Tildesley 1987; Panagiotopoulos 1987b;
Kofke 1993a) has become an effective tool for investigating phase equilibria properties
of fluids and materials.
We all have the experience in our every day life that matter exists in three different
phases: solid, liquid and gas. In particular ranges of temperature and density or pressure,
one phase or two phases can be kept stable. The reason why this occurs is due to the
intermolecular interactions. Generally two-body intermolecular interactions dominate
the total intermolecular interactions, but many reports (Barker et al. 1968; Murphy and
Barker 1971; Sadus and Prausnitz 1996; Marcelli and Sadus 2001) have shown that
three-body interactions play an important role in the phase equilibrium properties of
pure fluids and mixtures, especially at high density and pressure.
It is a well-known fact that three-body forces should be taken into account in order to
very accurately describe some experimental properties of dense simple fluids properly.
Furthermore, it has been shown that a true two-body potential (Barker-Fisher-Watts
1
potential) (Barker 1976), plus three-body interaction of the Axilrod-Teller type
(Axilrod and Teller 1943), is sufficient to reproduce accurately the thermodynamics of
argon, krypton, and xenon throughout a wide range of temperatures and densities. The
effect of three-body interactions on the vapour-liquid equilibria of pure noble gases has
been studied (Marcelli and Sadus 1999). However, little work has been done for
mixtures (Marcelli and Sadus 2001) and solid-liquid equilibria have not been
investigated. In this thesis we intend to fill this gap using four powerful techniques:
Gibbs-Duhem integration method (Kofke 1993b), the Gibbs ensemble Monte Carlo
(GEMC) method (Panagiotopoulos 1987b), molecular dynamics (MD) and non-
equilibrium molecular dynamics (NEMD) (Evans and Morriss 1990), which are used on
three different potential models, Lennard-Jones potential, Barker-Fisher-Watts potential
(Barker 1976) and the Axilrod-Teller (Axilrod and Teller 1943) three-body potential.
Considering the aim of studying the phase equilibria properties of pure fluids and
mixtures, in this chapter, a brief introduction of the theory of phase equilibria will be
given. It includes one-component and two-component system phase diagram
explanations. A review of progress in molecular simulation of phase equilibria is also
provided.
The simulation techniques and accurate potential models are crucial to the study of
phase equilibria of pure fluids and mixtures. In Chapter 2, the potential models used in
molecular simulations will be introduced. The Lennard-Jones potential, Barker-Fisher-
Watts potential and Axilrod-Teller three-body potentials will be discussed in detail. A
brief introduction of the simulation techniques for the study of phase equilibria will also
be given in this section. Four main techniques used in the work are explained in detail,
which include the Gibbs ensemble Monte Carlo simulation, the Gibbs-Duhem
integration method, molecular dynamics and non-equilibrium molecular dynamics.
The main work will be presented in Chapter 3 and 4. In Chapter 3, the vapour-liquid
and solid-liquid phase equilibria properties of argon, krypton and xenon will be studied
using the Gibbs-Duhem integration method for a pair wise Lennard-Jones potential with
different potential parameter values.
2
In Chapter 4, three-body effects on phase behaviour of noble gas will be investigated
comprehensively. The three-body effects over a wide range of densities including
vapour, liquid and solid will be explored and simple relationships between two-body
and three-body potential energies will be investigated. The three-body effects on
vapour-liquid phase equilibrium properties of pure fluids and mixtures will be studied
and the three-body effects on solid-liquid phase equilibrium properties of pure fluids
will also be presented in this chapter. For the sake of clarity, the numerical results from
Chapters 3 & 4 are given in the Appendix.
Finally, the conclusions and recommendations for the future work will be made in
Chapter 5.
1.2 Phase Equilibria of Pure Fluids and Mixtures
Thermodynamic Definition of Phase Equilibrium The state of equilibrium can be defined as the following (Malanowski and Anderko
1992):
“An isolated system is in the state of thermodynamic equilibrium if measurable changes
of thermodynamic parameters do not occur in this system.”
This definition points to the empirical character of the thermodynamics of phase
equilibria. It is known that thermodynamic processes tend to the state of equilibrium in
an asymptotic way. The process of arriving at equilibrium is a nonreversible one and
only external forces can stop or reverse the process of approaching the state of
equilibrium. In general, the requirement for phase coexistence is satisfied when the
temperature, pressure and chemical potential of every component are equal in all
phases.
Phase Rule To deduce the law of equilibrium, Gibbs (Findlay 1951) regarded a system as
possessing only three independently variable factors: temperature, pressure, and the
3
concentration of the components of the system. The theory he published now known as
the phase rule, by which the conditions of equilibrium were defined as a relationship
between the number of the phases and the components of the system.
The phase rule may be stated as the following (Findlay 1951):
A system consisting of C components can exist in C+2 phases only when the
temperature, total pressure of the system, and concentration of each phase have fixed
and definite values. If there are components in C+ 1 phase, only one of the factors may
be arbitrarily fixed, and if there are only C phases, two of the varying factors may be
arbitrarily fixed. This rule can be summarised in the form of equation
2C P R= − + − F (1.1)
where is the number of degrees of freedom of the system, C the number of
components and the number of phases. R stands for additional restrictions, eg.
stoichiometric requirements or the two conditions for critical points.
F
P
From the phase rule it can be seen easily that the greater the number of the phases, the
fewer are the degrees of freedom.
One-component System Phase Diagrams According to the phase rule, the degree of freedom is 2 when one component exists in
only one phase, and this must be the maximum degree of freedom possible. All systems
of one component can therefore be defined by at most two variable factors and the
equilibrium conditions can be represented by the phase diagrams shown in Figure 1.1
and 1.2. They are typical one-component system phase diagrams that describe the
existence of different phases under different physical conditions such as temperature,
pressure and volume or density. The precise shape of the p-V-T phase diagram depends
on the particular substance under investigation, but its general shape is broadly similar
for many different substances. It is often useful to project the p-V-T surface on to the p-
T plane and T-V or T- ρ plane to study phase equilibria of pure fluids.
4
V
LS
Fusion curve
Vaporization curve
Critical point
Triple point
Sublimation curve
Temperature
Pres
sure
V
LS
Fusion curve
Vaporization curve
Critical point
Triple point
Sublimation curve
Temperature
Pres
sure
Figure 1.1 Pure fluid p-T phase diagram
For certain ranges of the independent variables (any two of p, V, and T), the substance
exists wholly as a solid (S), a liquid (L), or as a vapour (V). The regions corresponding
to these single phases are indicated in Fig 1.1. In accordance with our everyday
experience we see that the vapour phase exists at high volumes and temperatures, the
solid when the temperature and volume are low, and the liquid phase when the
temperature and volume have intermediate values. Within other ranges of p, V, and T,
two phases are present in equilibrium. Only at the particular values, are all three phases
(S-L-V) coexisting in equilibrium. The point where this occurs is called the triple point.
The line connecting the triple point to the critical point records how the boiling
temperature (the temperature at which the liquid-gas phases transition occurs) varies
with pressure. It is usually called the vapour pressure (or vaporization) curve. The line
showing the pressure dependence of the temperature where solid and liquid coexist is
called the fusion curve, and the line describing how the pressure changes with the
temperature where solid and vapour coexist is called the sublimation curve.
5
Tem
per
atu
re
Liquid-Vapour Coexistence
Freezing Liquid
Melting Solid
density
V+L
L+S
Cp
Figure 1.2 Pure fluid T ρ− phase diagram
In the meantime, the system consisting of solid and liquid (S-L), solid and vapour (S-
phase is vapour.
bove the critical temperature, we can not observe the phase coexistence. One can
V), or liquid and vapour (L-V) is also illustrated in Fig 1.2. When two phases are at
phase equilibrium, they have the same temperature, pressure and chemical potential.
Additionally, at temperatures below the critical temperature two fluid phases may
coexist in equilibrium. The denser phase is liquid and the less dense
A
continuously pass from low-temperature gas to low-temperature liquid by heating above
the critical temperature. Between the liquid freezing line and the solid melting line, both
solid and liquid phases coexist together.
6
Two-component System Phase Diagrams
a
ent systems, the vapour-liquid phase equilibria behaviour is different
from their pure fluids. When a gas dissolves in a liquid, a two-component system will
exist in two phases: vapour and liquid phases. If the temperature is fixed, the system
becomes isothermally invariant, while the composition of the solution still changes with
the pressure. The direction of variation can be predicted from the principle of Le
Chatelier (Findlay 1951). Since the act of solution of a gas in a liquid is necessarily
accompanied by a decrease of volume, the effect of pressure will always be to increase
the solubility of a gas in liquid.
Figure 1.3 is a typical vapour-liquid phase diagram of a two-component mixture. This
diagram shows how the liquid and vapour compositions of a mixture of component A
and B change with pressure. In this figure, is the mole fraction of component B and p
is the total pressure of the system. Here the temperature of the liquid mixture is kept
constant. When the vapour is progressively removed, the composition of the liquid will
in general alter as the process goes on. This process is known as isothermal distillation.
In the lower pressure area the mixture exists in the single pure vapour phase and in
higher pressure area it exists in the pure liquid phase. In the certain area between the
lower and higher pressures the system is in an equilibrium state where vapour and liquid
phases coex
According to the phase rule, in systems of two components, not only may there be
change of pressure and temperature, as in the case of pure fluids, but the concentration
of the components in the different phases m y also change.
For the discussion of the equilibria occurring in two-component systems, there are
several classifications such as two liquid phases only; liquid and vapour phases only;
solid and liquid phases only: solid, liquid and gas phases coexist and so on. In this work
we pay strict attention to vapour-liquid equilibria involving mixtures.
For two compon
x
ist.
7
Mole fraction
p
L
V
L+V
c a d e b fA B
p1
p2
Figure 1.3 Vapour-Liquid phase diagram of binary mixture (Findlay 1951)
ponent B than the
the more volatile component, the residual
with falling vapour pressure. Suppose
e residual liquid has the composition c: it will then be
given off an equilibrium vapour of com
to p2. If the vapour removed has been condensed to a liquid, the liquid distillate will
obviously have a composition somewhere between b and d, say e. If e is submitted to
For instance, a mixture of the composition a will have a total pressure of p1, this
pressure can be produced in a closed space by the liquid remaining essentially
unchanged if the vapour space is small and the volume of the liquid large. Under the
condition of the fixed temperature and the pressure p1, this mixture exists in the
equilibrium state where the liquid composition is a, and the vapour composition is b. It
will be noted that the vapour b is much richer in the more volatile com
liquid. If now the pressure is reduced below p1, the liquid will continue to give off
vapour and, since this vapour is richer in
liquid will become richer in the less volatile A, and its composition will move along the
curve of vapour pressure towards pure A
distillation is continued until th
position d with the total vapour pressure falling
8
isothermal distillation in the above manner it will at first give rise to a vapour having
e composition f, still richer in thth e more volatile component. Obviously, continued
action of the distillate in this manner will lead eventually to a distillate of pure B.
rediction of Phase Equilibrium Properties rom a qualitative point of view, changes of equilibrium or the general direction of the
quilibrium curve can be predicted by means of the principle of Le Chatelier, which can
e stated as follows: if an attempt is made to change the pressure, temperature, or
, in
ch a manner as to diminish the magnitude of the alteration in the factor which is
aried. This principle of Le Chatelier is very important because it applies to all systems
uantitative point of view, the equilibrium curve can be predicted by the
ermodynamic equation
fr
Continued removal of the vapour from the residual liquid would also lead to a residue of
pure A. In this way, complete separation of the mixture into its two components is
theoretically possible.
PF
e
b
concentration of a system in equilibrium, then the equilibrium will shift, if possible
su
v
and changes of the condition of equilibrium (Findlay 1951).
From q
th
( )2 1
dP qT v v
=−
(1.2) dT
where q is the transformation heat, 2v and 1v are the specific volumes of the two
phases, T is the absolute temperature.
The above equation enables one to calculate only the slope of the curve of the pressure.
However, if an initial point is given, it is possible to calculate the pressures along the
equilibrium curve. This observation is the basis of the Gibbs-Duhem integration
simulation technique that will be discussed in the next chapter.
9
1.3 Progress in Molecular Simulation of Phase Equilibria
The calculation of phase equilibria for macroscopic systems from knowledge of the
intermolecular interactions is one of the central goals of statistical mechanics (Barker
1976).
Generally there are two approaches for the calculation of the phase equilibria of pure
fluids and mixtures: direct and indirect methods.
Indirect simulation methods (Sadus 2002) are mainly based on the calculations of the
chemical potential (or free energy). For a series of state points, the chemical potentials
are calculated, and a coexistence point is determined by the intersection of two phase’s
branches in the pressure-chemical potential projection diagram. This approach is tedious
and time-consuming. In this situation, a number of other indirect methods have been
developed to avoid this problem. These include thermodynamic integration (Hansen and
Verlet 1969), grand-canonical Monte Carlo (Adams 1976, 1979), and probability ratio
methods (Torrie 1977). Since the 1980s, researchers have tried to calculate the free
nergy for dense fluids with the application of the Widom potential distribution
e because they are obtained from a simulation with a small total number
f molecules. However, these calculations provide detailed information on the structure
e
theorem (Widom 1963), which can be implemented in the NVT ensemble (Shing 1982),
the NVE ensemble (Powles et al. 1982; Romano and Singer 1982), and the NPT
ensemble (Shing and Chung 1987) as well.
In the direct simulation methods, thermophysical properties of both phases are
determined simultaneously. A two-phase system is set up in a small box and the
simulations are slow to equilibrate because of the presence of a real surface (Chapela et
al. 1987; Heffelfinger et al. 1987). The long-range corrections to the thermodynamic
properties are difficult to estimate in the inhomogeneous systems and it is not clear that
the densities of the two phases close to the interface are those of the macroscopic
coexisting phas
o
and properties of the interfacial region (Panagiotopoulos and Quirke 1988).
10
Since the 1980s, the application of molecular simulation to phase equilibria has made
significant progress with the development of the Gibbs ensemble Monte Carlo method
(Panagiotopoulos 1987b). It is one of the commonly used methods for the direct
alculation of fluid phase equilibria, where the problem of obtaining reliable chemical
ell (Vega et al. 1992), Yukawa (Rudisill and Cummings 1989) etc. The
ixtures studied with the Gibbs ensemble include Lennard-Jones binary mixtures (Van
liquid-liquid phase equilibria. At high densities, the
article transfer step in the Gibbs ensemble Monte Carlo simulation method has a very
c
potentials and uncertainties caused by interfaces are eliminated by representing the
coexisting phases as separate simulation boxes. The Gibbs ensemble has been used to
calculate phase equilibria in pure fluids and mixtures successfully (Panagiotopoulos
1987b; Panagiotopoulos and Quirke 1988; Panagiotopoulos 1989, 1992, 2000).
The Gibbs ensemble method has been extended to study inhomogeneous systems
(Panagiotopoulos 1987a) and it is particularly suited to study phase equilibria in multi-
component mixtures (Panagiotopoulos and Quirke 1988). The ternary mixture equilibria
have been investigated (Tsong et al. 1995). This method has also been applied to the
study of more than two phases at equilibrium (Lopes and Tildesley 1997), which is
based on setting up a simulation with as many boxes as the maximum number of phases
expected to be present. Kristóf and Liszi (Kristóf and Liszi 1997, 1998) have proposed
an implementation of the Gibbs ensemble in which the total enthalpy, pressure and
number of particles in the total system are kept constant. Molecular dynamics versions
of the Gibbs ensemble algorithm are also available (Kotelyanskii and Hentschke 1996;
Sadus 2002).
The Gibbs ensemble has been widely used to study phase equilibria in conjunction with
several intermolecular potentials including the Lennard-Jones (Panagiotopoulos 1987b),
square-w
m
Leeuwen et al. 1991), hard-sphere binary mixtures (Amar 1989) and hydrocarbon
mixtures (de Pablo 1989).
It should be noted that most of the applications of the Gibbs ensemble method are
restricted to either vapour-liquid or
p
low probability of acceptance. Hence, practical application of the Gibbs ensemble
Monte Carlo technique to solid-liquid equilibrium is not very successful. Direct
techniques for simulation of equilibria involving solids have also been developed
11
(Crawford and Daniels 1968; Barker and Klein 1973; Crawford et al. 1976; Tsang and
Tang 1978; Bobetic and Barker 1983; Morris et al. 1994; Errington 2004; Yoo et al.
004). One of the most widely used simulation method for solid-liquid equilibrium is
a (Lisal et al. 1997) (Lisal and Vacek 1996a), solid-liquid equilibria (Lisal and
acek 1997) for pure fluids and vapour-liquid equilibria (Lisal and Vacek 1996b),
l other advanced simulation
chniques based on the Gibbs ensemble method have been developed in order to deal
n this case, a combination of configurational
ias methods with the Gibbs ensemble has been proposed (Mooij et al. 1992). A
ing ion pairs. It is clear that attempted transfers into or out of such a clustered
ystem need to take into account the presence of clusters in order to lead to a reasonable
2
the Gibbs-Duhem integration method proposed by Kofke (Kofke 1993a, b; Kofke and
Agrawal 1995). This method combines elements of thermodynamic integration with the
Gibbs ensemble method; but it avoids the particle insertion or deletion in the Gibbs
ensemble technique. Starting at a state point for which the two phases are known to be
in equilibrium, the Gibbs-Duhem integration method can be used to trace out the phase
diagram directly and efficiently. It has been applied successfully to study vapour-liquid
equilibri
V
solid-liquid equilibria (Hitchcock and Hall 1999) (Lamm and Hall 2001, 2002) for
mixtures. Polymers have also been investigated by this method (Escobedo and de Pablo
1997).
Apart from the Gibbs-Duhem integration method, severa
te
with the limitation of the Monte Carlo Gibbs ensemble method that the particle transfer
step has a very low probability of acceptance for highly non-spherical, multi-segment,
or strongly interacting molecules.
For multisegment molecules, the simple particle transfer move of the Gibbs ensemble
becomes impractical due to steric overlap. I
b
rotational insertion bias method for the Gibbs ensemble applicable to dense phase of
structured particles such as water has been described as well (Cracknell et al. 1990).
For ionic fluids, at low reduced temperatures for which phase separation is observed,
the low-density phase consists almost exclusively of clusters of ions, the dominant
species be
s
probability of acceptance of the transfer moves. The problem of biased pair insertions
and removals in ionic systems has been addressed (Orkoulas and Panagiotopoulos
1994).
12
The Gibbs ensemble simulation has also been advanced to deal with chemical reactions.
An ensemble where the number of atoms, rather than the number of molecules is kept
constant during the simulation has been introduced (Shaw 1991). More recently, a
eneralized framework for handling chemical reactions in canonical or Gibbs ensemble
alculations.
phase coexistence point is determined at the intersection of the vapour and liquid
be implemented in combination
ith the Gibbs ensemble or Gibbs-Duhem integrations (Escobedo and de Pablo 1997).
e theory of histograms has been developed (Salsburg et al. 1959) and the earlier
imulation work has also been done (McDonald and Singer 1967). The histogram re-
os e
ynamics (NEMD) has mainly been confined
the liquid phase. Recently it has been demonstrated (Ge et al. 2003b) that the NEMD
g
simulations has been developed (Smith and Triska 1994).
Additionally, a NPT + test particle method (Moller and Fischer 1990) has been used to
study vapour-liquid equilibria. The method is based on chemical potential c
A
branches of the chemical potential versus pressure diagram. The method has roughly the
same range of applicability and limitations as the Gibbs ensemble method, but requires
multiple simulations per coexistence point.
The pseudo-ensembles method is another alternative path to investigate phase equilibria
of fluids. This method provides a significant flexibility in determinations of phase
equilibria under different external constraints and can
w
This technique has been applied to an expanded grand canonical ensemble (Escobedo
1998).
A histogram re-weighting method can also been used to study phase equilibria. Since
1959 th
s
weighting method provides the free energy and phase behaviour with excellent accuracy
and can be used in the vicinity of critical points. This method has been employed very
successfully for phase equilibria evaluations (Kiyohara et al. 1998; Panagiotopoul t
al. 1998).
Traditionally, Non-equilibrium molecular d
to
technique, in conjunction with standard (NVT) equilibrium molecular dynamics (MD),
can be used to determine the solid-liquid phase coexistence at equilibrium.
13
In summary, the Gibbs ensemble Monte Carlo simulation method is currently one of the
most popular techniques for studying phase behaviour of fluids, and more advanced
Monte Carlo techniques based on the Gibbs ensemble are being developed. Molecular
dynamics simulation is also becoming an alternative promising route to investigate
phase equilibria. In comparison with other techniques, the Gibbs-Duhem integration
method is possibly the most computationally efficient, particularly for obtaining
multiple state points. Therefore, it is used in this work.
Another issue in the calculation of phase equilibria of pure fluids and their mixtures is
the intermolecular potentials (Allen and Tildesley 1993). A sufficient knowledge of
termolecular interactions is necessary for the simulation of a system because the
using a
ombination of knowledge from theory (quantum mechanics, electrostatics) and
the Lennard-Jones potential is not
true two-body potential. It is an effective intermolecular potential which incorporates
in
simulation accuracy depends mostly on the reliability of the intermolecular potentials
used.
The interaction between particles is commonly evaluated from a suitable potential
function. In general, intermolecular potentials can be classified as ab initio potentials
and semi-empirical potentials. Since computers are still too slow to calculate reliable
ab initio potentials, most workers use semi-empirical potentials developed by
c
experimental data (Panagiotopoulos 2000).
Computer simulation methods are typically implemented assuming that intermolecular
interactions are confined to pairs of molecules. The most commonly used pair
interaction model is the Lennard-Jones (LJ) potential, which can estimate well the
effective pair potential between normal molecular (i.e. non-polar, non-ionic etc.) fluids
at normal conditions. It is important to emphasise that
a
many-body interactions in an average way. Therefore no conclusions regarding two-
body interaction can be obtained from Lennard-Jones calculations. There is a large body
of simulation studies for a variety of different systems based on the LJ potential. The
phase equilibria of the LJ fluids have been predicted (Kalyuzhnyi and Cummings 1996;
Kotelyanskii and Hentschke 1996; Lisal et al. 1997; Errington 2004) and its binary
mixture phase properties have also been investigated (Hitchcock and Hall 1999; Lamm
and Hall 2001; Monica and Hall 2001; Lamm and Hall 2002).
14
It is well known that two-body interactions dominate other multi-body interaction and
pair-wise potentials alone are often sufficiently accurate. However, from a theoretical
perspective, it is also known that some aspects of intermolecular interactions cannot be
accurate enough without considering three or multi-body interactions. Strictly, pair-wise
additivity only applies to electrostatic, magnetic, and short-range penetration
interactions. To some extent, many-body effects will be present in induction, dispersion,
resonance, exchange, repulsion, and charge transfer interactions (Sadus 2002).
Modern developments in experimental and computational technology have recently
allowed a great advance in exploring many-body effects through computer simulations.
theory, all possible n-body interactions must be considered to study many-body
ince the 1940s, the effects of three-body interactions have been reported. The early
ork was done using the potential of Axilrod and Teller (Axilrod and Teller 1943).
arker et al. (Barker et al. 1971) demonstrated that the Axilrod-Teller potential makes a
gnificant contribution (5%-10%) to the overall energy of liquid argon. Recently,
arcelli and Sadus (Marcelli and Sadus 1999, 2000) reported good results for the
f the vapour-liquid equilibria of the pure substances argon, krypton, and
enon using accurate two-body potentials such as the Barker-Fisher-Watts (BFW)
otential (Barker et al. 1971) plus three-body contributions. The simple relationship
etween two-body and three-body interactions has also been proposed (Marcelli and
molecular dynamics simulation (Marcelli et
. 2001). Unlike the Lennard-Jones potential, the BFW potentials provide genuine
presentations of the contribution of only two-body interactions. On this basis, the
apour-liquid phase coexistence of the binary argon-krypton mixture using BFW
otential and AT term was studied as well (Marcelli and Sadus 2001). More recent work
lated to three-body interaction effect on phase equilibria uses an pair
otential with the Axilrod-Teller three-body term to study thermodynamic properties of
In
interactions. However, in practice the contribution from interactions other than
molecular pairs and triplets is likely to be extremely small. Generally including only
two-body and three-body interactions is assumed as an excellent approximation for
many-body interactions (Barker 1976; Sadus 2002).
S
w
B
si
M
prediction o
x
p
b
Sadus 2000) and used for nonequilibrium
al
re
v
p
ab initiore
p
15
argon, krypton and xenon pure fluids and argon-krypton, neon-argon binary mixtures
asrabad and Deiters 2003).
lthough three-body effects have been studied, most of work has been confined within
e vapour-liquid phase equilibria of pure fluids. In this work we also investigate the
ree-body effects on the solid-liquid phase equilibria of pure fluids and the vapour-
mixtures at different temperatures.
(N
A
th
th
liquid phase properties of
16
Chapter 2
Molecular Simulation Theories and Techniques 2
.1 Intermolecular Potentials
A key question about the use of any
accurate for the application of interest. The validity of
u accuracy of the equations used for the
rmolecular potentials.
An overview of intermolecular potentials will be given in this section and the
is known that the properties of solids, liquids and gases can be understood in terms of
in ematical points
urrounded by forces which are alternately attractive and repulsive (Barker 1976). With
l phenomena including phase changes can be
xplained.
ted pair
molecular simulation is whether the intermolecular
potential model is sufficiently
any sim lation will rest on the suitability and
inte
intermolecular potentials used to study argon, krypton and xenon will be discussed in
detail.
It
teratomic or intermolecular forces. Atoms can be considered as math
s
these ideas a wide range of physica
e
The interaction between atoms can be expressed in terms of potential energy. We define
the potential energy ( )u r of two atoms a distance r apart as the energy required to
bring one atom up from infinity to a distance r from the second one. The total molecular
potential energy of a system of N particles can be written as a sum of isolaNU
potentials, plus a sum of three-body correction term, plus a sum of four-body correction
terms, etc.
17
( ) ( )2 3, ,ij i j k
, , ...N ij ij ik jku r u r r r= + +∑ ∑ (2.1)
le hard spheres, i.e.
U
where are particle separations.
The leading term in this expansion 2u is a pair wise potential that is the most important
term in the total potential. Many pair wise potentials have been developed and applied
to atoms. The simplest approximation is the hard-sphere potential that treats atoms as
impenetrab
ijr
( )0
ru r
rσσ
∞ ≤⎧= ⎨ ≥⎩
(2.2)
where σ is the hard-sphere diameter.
The first m c lder and
Wainwright (Alder and Wainwright 1957), and their results were in a good agreement
ulations reported by Wood and Jacobson (Wood and Jacobson
t-sphere potential is a simple yet more realistic alternative to the hard-sphere
ole ular dynamics simulation of hard spheres was reported by A
with Monte Carlo sim
1957). The disadvantage of the hard-sphere potential model is that it cannot predict the
properties of real fluids due to the absence of a term of attractive interactions.
The sof
potential. It is not infinite at interatomic separations less than the sphere diameter. It
may be expressed in the following equation:
( )0
n
ru r rr
σε σ
σ
⎧ ⎛ ⎞ ≤⎪ ⎜ ⎟= ⎨ ⎝ ⎠⎪ ≥⎩
(2.3)
where n is an empirical constant and ε is a measure of the strength of intermolecular
determine the effect of repulsion in fluids and solid-liquid equilibria (Hoover et al.
1972; Hansen and Schiff 1973; Laird and Haymet 1992). However it can not be used to
interactions. Many simulation studies have been reported using soft-sphere potentials to
18
study vapour-liquid or liquid-liquid equilibria for the same reason as the hard-sphere
potential.
The simplest intermolecular potential that allows us to study the properties of liquids is
the square-well potential. This potential is combined with attractive term so that it can
e used to calculate the properties of liquids, and it can be expressed in the following b
form:
( )0
u r ε∞⎧
⎪= −⎨⎪⎩
r
rr
σσ λσ
λσ
≤< ≤
≥ (2.4)
where λ is some multiple of the hard-sphere diameter and ε is a measure of the
ttractive interaction. The properties of the square-well fluid have been investigated
mings 1996). The Yukawa potential
as also been used to study the properties of a charged hard sphere binary fluid (de
a
widely (Haile 1992).
There are also other pair potentials based on the concept of hard sphere + attractive term
such as the Sutherland potential and Yukawa potential (Hoover et al. 1972; Kalyuzhnyi
and Cummings 1996; Rosenfield 1996). The phase diagram of fluids with the Yakawa
potential has been calculated (Kalyuzhnyi and Cum
h
Carvalho and Evans 1997) in conjunction with the restrictive primitive model.
The next term in the expansion of Eq.(2.1) is the three-body potential 3u . The leading
contribution to this term is the Axilrod-Teller term that is triple-dipole (DDD)
interactions. There is empirical evidence that the influence of higher order dispersion
terms (DDQ, DQQ, QQQ,…) (Q means quadrupole here) and short-range contribution
to bulk fluid properties fortuitously cancels (Anta et al. 1997; Marcelli and Sadus 1999;
Lotrich and Szalewicz 2000; Bukowski and Szalewicz 2001). The DDD term is
generally considered to be a reasonably accurate effective three-body potential in the
fluid region.
19
Very little is known about the influence of four-body and the other higher-order terms
Eq. (2.1) on phase properties of fluids. Johnson and Spurling (Johnson and Spurling
l investigation of four-body interaction. It is generally believed that the
fluence of higher-order terms on fluid properties is negligible (Hoef and Madden
1
ld curacy
accuracy of the intermolecular potential used. The Car-Parrinello method (Car and
method combines m
ed and the equations of motion are derived
for the nuclear coordinates
this work we study phase equilibria properties of the noble gases argon, krypton and
r
s
ost widely used
alistic potential. The purpose of these calculations is to identify the accuracy of the
in
1974) investigated the effect of four-body interactions on the fourth virial coefficient
using the intermolecular potential (Bade 1958), which is a rare example of the
theoretica
in
999).
It shou be noted that the ac of conventional simulations is limited to the
Parrinello 1985) can be an alternative way to provide an accurate calculation. This
olecular dynamics with some elements of electronic-structure
theory. A new Lagrangian function is form
and electronic orbitals.
In
xenon. The remaining stable noble gases, helium and neon, will not be conside ed
because of uncertaintie arising from quantum effects. We used the Lennard-Jones
potential (LJ) to study the phase equilibria of noble gases, since it is a m
re
Lennard-Jones potential. It should be noted that we pay more attention to the three-body
potentials since they are the focus of our investigation. In order to study the three-body
interaction effects, true two-body potentials are needed. Several accurate pair potentials
are available for argon, krypton and xenon (Maitland et al. 1981). We selected the
Barker-Fisher-Watts potential (BFW) because of its well-known accuracy and the
availability of intermolecular potential parameters for argon, krypton and xenon (Sadus
2002).
2.1.1 Lennard-Jones Potential
In contrast to the hard sphere and soft sphere potentials, a more realistic description of
intermolecular interaction is given by the Lennard-Jones (LJ) potential. The general
form is:
20
( ) n mm nu r x xn m n m
ε − −⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎢ ⎥− −⎝ ⎠ ⎝ ⎠⎣ ⎦ (2.5)
where n and m are constants, / mx r r= , and mr is the separation where the energy is
minimum. The most common form of the Lennard-Jones potential is obtained when
= . Then the for d by the equation: m and 12n = m can be expresse6
12 6
( ) 4u rr rσ σε
⎡ ⎤⎞ ⎛ ⎞⎥⎟ ⎜ ⎟
⎛= −⎠ ⎝ ⎠ ⎥⎦
(2.6)
w h tance b two particles,
⎢⎜⎝⎢⎣
here r is t e dis etween ε is the well-d f poten depth o tial an σ
is the distance at which . It i ost c ly us ractio l in
molecular simulations.
In Eq. (2.6) the term els the n b parti h d s at
s nce two les are t ver e to e r, th ronic
the energy increases dramatically. The
rm contributes the attraction part which is dominating at large distances. This
rm is caused by the van der Waals dispersion forces. The parameters
u is zero s the m ommon ed inte n mode
121/ r mod repulsio etween cles whic ominate
hort dista s. When partic brough y clos ach othe e elect
clouds surrounding the particles overlap and
te 61/ r
ε and σte are
data. Some representative values of ε and σdetermined via fitting to experimental for
dynam
mulations (Wood and
acobson 1957; Errington 2004).
hanics
Ar, Kr and Xe are given in Table 2.1.
The Lennard-Jones potential is probably the most widely used potential for molecular
simulation. The first molecular ics simulation of a LJ fluid was reported by
Rahman (Rahman 1964), where the properties of the noble gas argon was studied. The
LJ potential has also been investigated by Monte Carlo si
J
The Lennard-Jones potential is often used as part of a larger potential for molecular
systems. For example, it is used frequently to calculate the interactions between atoms
that constitute a model of a many-atom molecule such as polymer. It is also used
commonly as part of a molecular mec potential. The appeal of the Lennard-Jones
21
potential is that it combines a realistic description of the intermolecular interaction
separation with computational simplicity (Sadus 2002).
Table 2.1 Parameter values suggested for the Lennard-Jones potential
Ar Kr Xe Reference
/ kε (K) σ (Å) / kε (K) σ (Å) / kε (K) σ (Å)
Hogervorst (1971) 135 3.36 193 3.57 256 3.92
Horton (1976) 119.8 3.405 164.4 3.638 231.1 3.961
Clifford (1977) 141.2 3.336 191.4 3.575 257.4 3.924
Rodriguex (1978) 118 3.41 164.91 3.631 227.17 3.988
Walton(1982)
123.2 3.40 166.67 3.68 224.64 4.07
2.1.2 Two-body Potentials
Generally there are two approaches to determining a suitable equation for the pair
he first is a direct calculation based on quantum mechanics. The
second is to build a sem
theoretical k xperimental property data.
The
use input constant but electron
theoretical chem odynamic
ds without parame ed to experimental data
ab initiopotential. T
iempirical equation that is based on a combination of partial
nowledge and fitting to e
ab approach is the most desirable one because ab initio pair potentials do not
charge, Planck’s constant etc, and one of the goals of
istry and physics is to determine quantitatively bulk therm
initio
properties of gases and liqui ters fitt
(Malijevsky and Malijevsky 2003). Ab initio potentials have been proposed for helium
(Janzen and Aziz 1997), argon (Leonhard and Deiters 2000), krypton (Nasrabad and
Deiters 2003), xenon (Fass et al. 2000), and neon (Leonhard and Deiters 2000).
22
It should be noted that the ab initio calculations are still strongly limited by available
computer power since the difficulty of the calculations rises very rapidly with the
number of electrons. But with the rapid development of computer technologies and
numerical methods, the ab initio approach will become a more and more promising
route to study phase equilibria and materials.
Historically, the empirical approach was used with the parameters of the potential being
btained from experimental data such as second virial coefficient, viscosities, molecular
(Sadus
002).
he qualitative form of the potential energy function from the interaction of a pair of
o the
(i) the experimental quantity must depend reasonably sensitively on some
principle quantum mechanical calculations could be used to determine the whole of
the potential function by solving the Schrödinger equation. But in practice it is not easy
to get accurate results in the attractive region of the potential. The reason is that the
attractive forces are due primarily to electron correlation effects, which are very
o
beam cross sections etc. Calculations regarding the accuracy of a pair potential were
made by comparing the properties predicted by the potential with experiment
2
T
noble gas atoms is well known. Different kinds of experiments provide information on
different regions of this function, and it is important that a particular experimental
quantity can provide useful informati n on interatomic potentials, for which there
are three conditions that must be satisfied (Barker 1976):
feature of the potential
(ii) it must be possible to measure the quantity with sufficient accuracy
(iii) adequate theories and computational procedures must exist to permit
calculation of the experimental quantity from a given potential or of the
potential of the experimental quantity
In order to understand deeply about two-body potentials we will refer to the interatomic
potentials for inert gases from experimental data (Barker 1976) and give a brief
description of the kinds of experimental information that are available and the features
of the potential function to which they are sensitive.
In
23
difficult to incorporate with sufficient accuracy into the calculation. For the repulsive
region of the potential, quantum mechanics calculations could pr vide results of
reasonab
o
le accuracy (Murrell 1976).
A antum mechanical
erturbation theory can be used to derive an asymptotic expression of the form,
0r C r C r C r= − − − (2.7)
where is the interatomic distance, the coefficients etc. can be related to properties
f the isolated atoms. In particular can be expressed in terms of dipole oscillator
tial for large separations
arker 1976).
is
ow preferable to determine the potential in the repulsive region from viscosity data, for
ilute gas properties that include the second virial coefficient
t large distances, where electronic overlap can be neglected, qu
p
6 8 1
6 8 10( ) / / / ...u
r 6C
o 6
strengths and approximate values of 8C and 10C have been derived in this way. This is
undoubtedly the most accurate way to determine the poten
C
(B
Molecular beam scattering experiments have been used to determine the repulsive
potential in a narrow range of distances. These measurements were shown to contain
similar information to the high-temperature viscosity measurements. Therefore, it
n
which the measurements seem to be more reliable (Barker 1976).
Spectroscopic data on inert gas dimers (Tanaka et al. 1970) has been used to determine
vibrational level spacings. The lowest level spacings could provide a good estimate of
the curvature at the minimum of the potential, while the higher level spacings could
provide information of the anharmonic and detailed shape of the potential curve (Barker
1976).
D B and gas transport
teraction. The second coefficient properties depend only on the pair in B in the equation
of state
21 ...pV B CRT V V
= + + + (2.8)
24
is related directly to the pair potential by the equation (Allen and Tildesley 1987).
( )
2
0
2 13
u rkTNB e r drπ −∞ ⎛ ⎞
= −⎜ ⎟⎝ ⎠
determine the radial distribution function by X-ray
nd neutron diffraction to describe the local density of atoms around a given central
ll the thermodynamic properties of the solid phase depend more or less strongly on the
l. The crystal structure, as the most obvious solid state property, provides
ve
a d structure should be more stable than the face-centred cubic
tructures, in which all the inert gases except helium crystallize. In order to determine
the relative stability, a further ca
nd it indicates that the use of many-body potentials favours the cubic structure.
Bne of the aims at this work is to investigate three-body effects on phase behaviour of
with the three-body term was used to study vapour-liquid phase coexistence
roperties (Sadus 1996; Sadus and Prausnitz 1996). Using only the Lennard-Jones
otential could get much better agreement with experiment. The main reason is that
∫ (2.9)
Viscosity is the most accurately measured transport property of gases. The thermal
diffusion ratio and thermal conductivity also contain the information related to the form
of the potential but it is difficult to measure them accurately.
Liquid state properties provide the information for a potential function by comparing
with the results studied by molecular simulations or the thermodynamic perturbation
theory. It is also possible to ( )g r
a
atom.
A
potentia
additional useful information. Most calculations using potentials ha shown that the
hexagon l close–packe
s
lculation for both crystal structures has been performed
a
arker-Fisher-Watts Potential O
noble gases. Choosing a true two-body potential is an important condition to study
three-body effects accurately. It has been investigated that the calculation deviated
appreciably from the experimental data when the Lennard-Jones potential in
conjunction
p
p
25
Lennard-Jones potential is an effective potential and already includes many-body
teractions. In the literature, several pair potentials for argon were presented (Sadus
002). A highly accurate interatomic potential Aziz-Slaman potential for argon (Aziz
interaction base to study three-body effects of
argon (Ba er et al. 19 pot o-body potential as it was
derived only by properties depending on two- ilar
analytical expressions are also available for other noble gases such as krypton and
xenon (Barker et al. 1974).
A detailed d scussion about the BFW potentials for argon, krypton an will be
given here.
Argon
The BFW potential is a linear combination of the Barker-Pompe ( obetic-
Barker ( potentials:
in
2
1993; Anta et al. 1997) has been studied. In this work we choose the Barker-Fisher–
Watts potential (BFW) as a two-body
rk 71). The BFW ential is a true tw
body interactions. Furthermore, sim
i d xenon
BPu ) and B
BBu )
( ) ( ) ( )P r 2 0.75 0.25BB Bu r u= + (2.10)
where the po Barker-Pompe and Bobetic-Barker have the following form:
)
u r
tentials of
( ) ( ) (5 2
2 62
01 expi j
i ji j
Cu r A x
xε α
δ+
= =
⎡ ⎤= − −⎡ ⎤⎢ ⎥⎣ ⎦ +⎣ ⎦
∑ ∑ (2.11)
the above equation,
6+ 0
1 x−
/ mx r r= where is the intermolecular separation at which the In m
potential has a minimum value. The
r
σ term is the value where the potential is zero and
it is usually defined as the atomic diameter. The other parameters are summarised in
Table 2.2.
26
Table 2.2 Parameters of the Barker-Pompe, Bobetic-Barker and Barker-Fisher-Watts
(BFW) potentials (Barker et al. 1971).
Barker-Pompe Bobetic-Barker Barker-Fisher-Watts
( )/ k Kε 147.70 140.235 142.095
( )Åσ 3.7560 3.7630 3.3605
( )Åmr 3.341 3.3666 3.7612
0A 0.2349 0.29214 0.27783
-4.7735 -4.41458 -4.50431
-10.2194 -7.70182 -8.331215
1.0698 1.11976 1.10727
0.1642 0.171551 0.16971325
0.013611
1A
2A
3A -5.2905 -31.9293 -25.2696
4A 0.0 -136.026 -102.0195
5A 0.0 -151.0 -113.25
6C
8C
10C 0.0132 0.013748
α 12.5 12.5 12.5
δ 0.01 0.01 0.01
Barker et al. (Barker et al. 1971) used the following experimental data to determine the
otential of argon:
h h-
3. the known long-range coefficients of
p
1. ig energy molecular beam data;
2. the zero-temperature and pressure lattice spacing, energy and Debye parameter,
derived from specific heat measurement of solid argon; 6r− , 8r− and 10r− ;
4. second virial coefficients;
5. the liquid-phase pressure at one temperature and density.
27
This potential was shown to be consistent with available information on pair
interactions (second virial coefficients, gas-transport properties, molecular-beam
scattering, and known long-rang interaction coefficients). It also gave approximately
correct values for third virial coefficients as well as for the lattice spacing and energy of
crystalline argon at 0 Ko , provided that the three-body interaction (Axilrod and Teller
1943) was included (Barker et al. 1968).
Krypton and xenon
The molecular-specific nature of the BFW potential is illustrated by attempts to use Eq.
(2.11) for other oble gases such as krypto d xenon. Barker et al. (Barker et al. 1971)
reported that modifications to Eq. (2. were required to obt n optimal
representation f r these larger noble gase r krypton and xenon, th termined a
potential of the form:
(2.12)
where is identical to Eq. (2.11) and
n n an
11) ain a
o s. Fo ey de
( ) ( ) ( )2 0 1u r u r u r= +
( )0u r ( )1u r is given by
1
0 1
P x Q x xu r( ) ( ) ( ) ( )4 5
1
1 1 exp ' 1 x
x
ε α⎧ ⎡ ⎤− + − >⎡ ⎤⎪ ⎣ ⎦⎣ ⎦= ⎨⎪
−
≤⎩ (2.13)
where 'α , and are additional parameters obtained by fitting data for differential
scattering cross-sections. We used Eqs. (2.12) and (2.13) to calculate the properties of
eters summarized in Table 2.3.
P Q
krypton and xenon with the param
28
Table 2.3 Parameters of the two-body potentials for krypton and xenon (Barker et al.
1974).
krypton xenon
( )/ k Kε 201.9 281.0
( )Åσ 3.573 3.890
( )Åmr 4.0067 4.3623
-4.78686 -4.8169
0.1660
0.0143 0.0323
0A 0.23526 0.2402
1
2A -9.2 -10.9
3A -8.0 -25.0
4A -30.0 -50.7
5A -205.8 -200.0
6C 1.0632 1.0544
8C 0.1701
A
10
P -9.0 59.3
Q 68.67 71.1
C
α 12.5 12.5
'α 12.5 12.5
δ 0.01 0.01
The experimental data used to derive these potentials were:
1. lattice spacing and cohesive energy of the crystal at 0 K;
2. bulk modulus and Debye parameter at 0 K;
3. lower vibrational level spacings derived from spectroscopic data;
4. gas viscosity data;
;
6. second virial coefficients.
5. differential scattering crossing sections
29
It should b
experiment
experimen
The DiffeThe LJ and
e noted that the determination of a potential of xenon consistent with all
al data has proved unexpectedly difficult due to a real inconsistency in the
tal data (Barker 1976).
rence between LJ and BFW Potentials BFW potentials are compared in Figure 2.1.
Figure 2.1 Comparisons between LJ and BFW potential
From the a
Lennard-Jo -body
potential since it was derived from the experimental data such as second virial
which are only associated with two-body
inte
evaluated actions. The main
fluence of this many-body contribution is to reduce the depth of the intermolecular
well compared with only two-body interactions.
bove plot we can see that the Barker-Fisher-Watts potential is different from
nes potential. The basic reason is that the BFW potential is a true two
coefficients and collision cross sections,
ractions. While LJ potential is calculated using “effective” parameters that are often
from bulk experimental data including many-body inter
in
30
In order to investigate many-body effects on the phase equilibria properties of fluids,
the true-two body potentials have to be chosen. A comparison of experiments with
simulations using “effective” two-body potentials can not lead to a definite conclusion
regarding the accuracy of the two-body potential and the many-body effect on the
studying properties.
2.1 Gen
met
tho d another is those which study microscopic
roperties.
In the first
(i) tructures and binding energies: The measurement
of crystal structures of the rare gases (excluding helium) is one of the first
Crystal binding energies have also been measured and showed a deviation
ion of a liquid or dense gas, which describe the local
density of molecules around a given central molecule. These experiments are
l to pe
.3 Many-body Interactions
erally many-body interaction effect can be studied by experimental and theoretical
hods (Elrod and Saykally 1994). There are two categories of experiment. One is
se that explore macroscopic properties an
p
category, the experiments include:
Measurements of crystal s
experiments that indicate the possible effects of many-body forces. The
results were not in agreement with predictions from pair wise additive
potentials. X-ray diffraction reveals a face centred structure for the noble
gases, while the pair potentials predict a hexagonal close-packed structure.
The inclusion of three-body terms in the potential gives a better result.
from pair additivity of about 10% (Elrod and Saykally 1994). These all
indicate the existence and importance of the many-body interactions.
(ii) Measurements of the structure of liquids: It is also possible by X-ray and
neutron scattering techniques (Egelstaff 1988) to determine the radial
distribution funct
difficu t rform with a high level of accuracy. But there are still some
examples dealing with many-body interaction studies (Barker 1976).
31
(iii) Measurements of the absorption and scattering of light by dense media: The
observation of an absorption spectrum due to a pure rare gas medium would
be direct information on many-body forces since the two-body induced
dipole moment in such systems is exactly zero by symmetry (Guillot 1989;
Guillot et al. 1989).
(iv) Measurements of other macroscopic properties, which are sensitive to many-
body forces, such as viscosity, thermal conductivity and diffusion. But it is
more difficult to investigate many-body effects using these properties.
The second category experiments include:
(i) Virial coefficient determination: The third virial coefficient C depends on
both pair and triplet potentials, according to the equation
( ) [ ]{ }
[ ]{ } [ ]{ } [ ]
determine the three-body interactions.
ique: It has played an important role in the
pair potential energy interfaces, have also been applied to
{ }
2
12 13 23 123 2 3
2
12 13 23 2 3
exp / exp / 13
exp / 1 exp / 1 exp / 13
NC u u u kT u kT dr dr
N u kT u kT u kT dr dr
= − − + + − −⎡ ⎤⎣ ⎦
− − − − − − −
∫∫
∫∫(2.14)
Barker and Pompe (Barker and Pompe 1968) found that values of C
calculated neglecting all three-body interactions differed from experimental values
for argon by as much as 50%, but that values calculated including the Axilrod-
Teller triple-dipole interaction were within about 10% of the experimental values
which are probably uncertain by almost as much as this. Therefore it would be very
important to measure the third virial coefficient sufficiently accurately in order to
(ii) Molecular beam scattering techn
determination of
the many-body problems.
(iii) The spectroscopy of van der Waals molecules: Finally, it has recently
become a very important tool in the study of many-body forces. The study of
van der Waals models is ideal for a systematic examination of many-body
32
terms because it will provide extremely accurate data (~1 ppm accuracy)
(Elrod and Saykally 1994) when coupled with high-resolution spectroscopy.
Many-body interaction can be also investigated in a theoretical way. Basically
theoretical methods are classified into two categories: quantum mechanics and statistical
echanics. Quantum mechanical calculations give more direct information on many-
ince the 1940s quantum mechanical methods have been applied to the study
ion for the diatomic molecule constituted by the pair of atoms. In
ractice it is only for helium that calculations of this kind have an accuracy
ap e the attractive forces are due
rimarily to electron correlation effects, which are very difficult to incorporate with
ed the most useful theoretical insight into
e effects of many-body interactions. an example is a simultaneous implementation of
the Møller-Plesset perturbation theory (MPPT) and intermolecular MPPT (Chalasinsky
and Szczesniak 1988 expansion that allows the
position of the interaction energy into the usual terms of the theory of
olecular forces: exchange, induction, electrostatic, and dispersion. More recently
m
body interactions. Statistical mechanical simulations require microscopic intermolecular
potential functions that can be obtained from quantum mechanical calculations or
semiempirical formulations.
ab initioS
of many-body interactions. There are two approaches to the ab initio calculations:
supermolecular and perturbation theory techniques.
The supermolecular method is relatively easy to apply but calculations must be made
carefully and accurately because the van der Waals interaction energy is much smaller
than the energies of the monomers. In principle the whole potential energy function
could be calculated directly by a sufficiently accurate solution of the electronic
Schrödinger equat
ab initiop
proaching that of other methods. This is becaus
p
sufficient accuracy into the calculations.
The perturbation theory technique has provid
th
). This method used double perturbation
decom
interm
symmetry-adapted perturbation theory (Lotrich and Szalewicz 1997) has been
developed to study many-body effects.
33
The second theoretical method to investigate many-body interaction is statistical
mechanics which involves two kinds of simulation techniques: Monte Carlo and
molecular dynamics. Both of these methods have found widespread use in the study of
any-body forces through comparing simulation results based on the microscopic
an
e measured by condensed phase experiments.
hree-body Interactions
Three-body interaction includes dispersion term and repulsive term. The effect of three-
ody dispersion is well defined whereas it is difficult to formulate an intermolecular
potential for three-body repulsion. Indeed, some commonly used three-body repulsion
potentials are tied to the LJ potential. The relative importance of repulsion fo
gases has been reported (Sadus 2002). The repulsive term completely dominates the
xilrod-Teller term at small interatomic separations. For argon, krypton and xenon, the
for
elium and neon, repulsion is important at considerably larger distances. Recently
L reported the importance of the three-body exchange
teraction in dense rare-gas solid.
of molecular properties. The issue
bout the relative importance between the repulsive term and the Axilrod-Teller term
this work, we focus on the phase equilibria properties of noble gases. Work on three-
b n of atoms
adus 2002). Various contributions to three-body dispersion interactions can be
pote
and Teller 1943).
m
potential energy function and the related macroscopic equilibrium properties which c
b
T
b
r noble
A
dominance of repulsion commences at a similar interatomic separation, whereas
h
oubeyre (Loubeyre 1988) has also
in
It should be noted that in many circumstances, the empirical pair potential with Axilrod-
Teller term alone can provide a good description
a
strongly depends on the reference pair potential used.
In
ody interaction effects has been confined largely to the dispersion interactio
(S
envisaged arising from instantaneous dipole (D), quadrupole (Q), octupole (O) and
hexadecapole (H) moments of a triplet of atoms. In principle, the dispersion is the sum
of these various combinations of multipole moments (Bell 1970). The triple-dipole
ntial can be evaluated from the formula proposed by Axilrod and Teller (Axilrod
34
( )( )3
1 cos cos cosDDD i j kv θ θ θ+uDDD
ij ik jkr r r= (2.15)
where the angles and interm separations r triangular co on of
atoms ( see figure 2.2 ), and where
olecular efer to a nfigurati
DDDv is the non-additive coefficient which can be
estimated from o served oscil trengths.
b lator s
rik
i
θj
θ
rij
rjk
θk
i
j
of atom
The contribution of the AT potential can be either positive or negative depending on the
orientation of the three atoms. The potential is positive for an acute triangular
arran ent of a ereas it is negative for linear ge ries. Th
can b pected to n overa sive con on in a close-packed s in
e liquid phase. The terms indicate that the magnitude of the potential is very
ensitive to intermolecular separation.
k
Figure 2.2 Triplet configuration , , .i j k
gem toms wh near omet e potential
e ex make a ll repul tributi olid and3r−th
s
Bell (Bell 1970) has derived the other multipolar non-additive third-order potentials:
35
( )( )( )43
39cos 25cos3 6cos 3 5cos 2
16DDQ
DDQ k k
ij jk ik
ur r r
i j k
νθ θ θ θ θ⎡ ⎤= × − + − +⎣ ⎦ (2.16)
( )( ) ( )( )
5
3 cos 5cos3 20cos 1 3cos 215
64DQQ
DQQ
jk ij ik
ur r r ( )4 70cos 2 cos
i i j k i
j k i
θ θ θ θ θν ⎡ ⎤+ + − −=
θ θ θ⎢ ⎥×⎢ ⎥+ −⎣ ⎦
(2.17)
( ) ( ) ( ) ( )5
27 220cos cos cos 490cos 2 cos 2 cos 215175 cos 2 cos 2 cos 2128
i j k i j kQQQQQQ
i j j k k iij ik jk
ur r r
θ θ θ θ θ θνθ θ θ θ θ θ
− + +⎡ ⎤⎢ ⎥= ×
⎡ ⎤+ − + − + −⎢ ⎥⎣ ⎦⎣ ⎦ (2.18)
the fourth-order triple-dipole term can be evaluated from (Doran and Zucker 1971)
( ) ( ) ( )22 2
44
1 cos1 cos 1 cos45 ji kDDDDDD
θ6 6 664
ik ij ij jk ik jkr r r r r r
θ θν ⎡ ⎤++ +⎢ ⎥= + +⎢ ⎥
(2.19) u⎣ ⎦
where Eq. (2.15), (2.16), (2.17) and (2.18) represent the effect of dipole-dipole-
quadrupole, dipole-quadrupole-quadrupole, quadrupole-quadrupole-quadrupole and the
forth-order triple-dipole interactions, respectively.
The non-additive coefficient DDDv can be calculated approxima ely from the 6C
dispersion coefficient and po
t
la ity datrizabil aα :
634DDD
C α= (2.20) v
Sadus (Sadus 2002) has investigated the relative importance of DDDv with respect to
molecular size. The result shows that the non-additive dispersion coefficient increases
progressively with molecular size relative to the two-body dispersion coefficientC . 6
DDDv can also be obtained directly from experimental data by analysing dipole oscillator
strength distributions (Kumar and Meath 1985). Values of three-body interaction
coefficient for noble gases are given in Table 2.4.
36
Table 2.4 Coefficients for the three-body potentials.
Argon Krypton Xenon
( ). .DDDv a u 517.4 1554 5603 a
( ). . bDDQv a u 687.5 2272 9448
( ). . bDQQv a u 2687 9648 45770
( )b. .QQQv a u 10639 41478 222049
( )4 . . cDDDv a u -10570 -48465 -284560
a from ref (Leonard and Barker 1975)
98)
from ref. (Bade 1958)
teractions to the crystalline energy of noble
1971; Barker et al. 1972)
b from ref. (Hoef and Madden. 19
c
The relative importance of the above various three-body interaction terms has also been
studied (Doran and Zucker 1971; Barker et al. 1972). The results are listed in Table 2.5.
Table 2.5 Contribution of three-body in
gases (Doran and Zucker
( / )E J mol
Atom
Ne
DDD DDQ QQD QQQ 4DDD
62.4 15.5 2.7 0.2 -4.8
Ar 579.6 173.8 37.0 3.5 -126.7
Kr 1004.0 220.1 34.3 2.4 -281.4
Xe 1597.9 373.6 62.0 4.6 -636.6
From Table 2.5 we
and the fourth-order triple-dipole contribution nearly cancels the
contribution from
orth-order triple-dipole contribution is much larger than the combined contributions
from all the other higher multipole terms.
can see that the leading contribution is from the third-order triple-
dipole term
the other higher multipole terms. It should be noted that for xenon,
the f
37
2.1.4 Relationship between Two-body and Three-bod
Interactions
y
ccording to the previous discussion it can be seen that there are many contributions to
alone can be an
A
three-body interactions but the triple-dipole term of Axilrod and Teller
excellent approximation. Even with this simplification, the three-body calculations are
still tedious and time-consuming. Recently Monte Carlo simulations (Marcelli and
Sadus 2000) have been reported that there is a simple and accurate relationship between
the two-body ( E ) and three-body ( E ) configurational energies of a fluid: 2 3
23 6
23v EE ρεσ
= − (2.21)
where v is the non-additive coefficient, ε is the characteristic depth of the pair
potential, σ is the characteristic molecular diameter used in the pair potential, and
/N Vρ = is the nu density obtained by dividing the number of molecules ( N ) b
the volume (V ). The si
mber y
gnificance of this r tionship is that it allows us to obtain an
intermolecular potential ( ) solely in terms of pair contribu )
and well-known intermolecu
ela
accurate overall u tions ( 2u
lar parameters:
2 6
23
vu u 1 ρεσ
⎛ ⎞= −⎜ ⎟⎝ ⎠
(2.22)
Therefore, the effect of three-body interactions can be incorporated into a simulation
involving pair interactions wit
this approach with a full two-body plus three-body calculation indicates that there is no
ignificant loss of accuracy (Marcelli and Sadus 2000).
t of the Eq. (2.21) potential yields:
hout any additional computational cost. Comparison of
s
The Eq. (2.22) is a density-dependent potential and care must be taken when calculating
the pressure. Applying the formula proposed by Smit et al. (Smit et al. 1992) to the
density-dependent par
38
( ) ( ) 22 2
26 61 1
1 2 23 9 3
N Nij ij
ij iji j i jij ij
du r du rv vV dr V dr
ρ ρεσ εσ< = < =
(2.23) . .P r r E= − + −∑ ∑
i
f the pressure calculated using the effective intermolecular potential compared with the
full two-body + three-body calculation
e calculation
2.1.5 Combining Rules
ecular potentials mentioned previously were developed originally for pure
uids, but they can be applied directly to binary mixtures by assuming suitable
co eters. In general if we denote the energy-
ke parameters
where the angle brackets represent ensemble averages, 2E is the two-body energy per
particle and the rema ning terms are defined as above. There is a very good agreement
o
s (Marcelli et al. 2001; Marcelli et al. 2004)
According to the above discussion, the Eq. (2.21) can be used as an accurate alternative
to th s for both energy and pressure.
The intermol
fl
mbining rules for the intermolecular param
ν and εli by the symbol X, the cross potential parameters of interacting
:
pairs and triplets can be calculated from
3ijk iii jjj kkkX X X X= (2.24)
(2.25) ij ii jjX X X=
In general if we denote all the remaining parameters such as 6, ,A Cσ etc by the symbol
Y, the cross potential parameters on interacting pairs can be calculated from:
3iii jjj kkk
ijk
Y Y Y+ +Y = (2.26)
2ii jj
ij
Y YY
+= (2.27)
39
ε and σ In the argon-krypton mixture simulations the potential parameters of argon
were used to obtain reduced quantities in the standard way. It is important to stress that
ese commonly used combining rules do not have physical rationale.
ar dynamics and non-equilibrium molecular
ynamics.
th
2.2 Phase Equilibria Simulation Techniques The main simulation techniques used in this work are described in detail in this section,
which includes the Metropolis Monte Carlo method, the Gibbs ensemble, the Gibbs-
Duhem integration method, molecul
d
2.2.1 Metropolis Monte Carlo Method
The Monte Carlo method is a stochastic strategy that relies on probability, and it may be
used to study molecular systems through statistical mechanics. The average of any
physical property ( )NA r can be obtained by calculating the following integral on the
N particles in the system:
( ) ( ) ( )N N N NA A dρ r r (2.28)
here
= ∫r r
W ρ is the probability of obtaining configuration , and it can be evaluated by the
otential energy
Nr
p of the configuration: E
( )( )
exp( )
exp
NN
N N
E
E d
βρ
β
⎡ ⎤−⎣ ⎦=⎡ ⎤−⎣ ⎦∫
rr
r r (2.29)
The Metropolis Monte Carlo technique (Metropolis et al. 1953) is a convenient way to
calculate the above equation. This method is a computational approach for generating a
set of n configurations of the system. The Metropolis method algorithm can be
escribed as follows: d
40
Step (1): pick a configuration ir (this initial configuration can be any configuration of
the system, but usually the atoms are positioned in a crystalline lattice sites).
Step (2): generate a trial configuration tr (usually a configuration similar to ir ). At this
step it has to be decided whether to accept or reject the trial configuration. The
transition probability ( )t iπ →r r can be expressed as:
( ) ( ) ( )t i t i t iaccπ α→ = → × →r r r r r r (2.30)
where ( )t iα →r r is a transition matrix that indicates the probability to perform the trial
move, and ( )t iacc →r r is the probability of accepting the trial move. In the Metropolis
method ( )t iα →r r is chosen to be a symmetric matrix that
eans t i→r r . The average number of accepted moves from to is
ibrium. The Metropolis
1 ( ) ( )t i
t i
ifa
ifρ ρ ρ ρ
ρ ρ<
→
( ) ( )i tα α→ =r r ir trm
exactly cancelled by the number of reverse moves at equil
method used the following choice to decide acceptance.
( )( ) t i
i tcc⎧
=/ ( ) ( ) ( )
⎨ ≥⎩
r rr r
(2.31) r r
r r
The above equation indicates that if ( ) ( )t iρ ρ≥r r , this trial move will be accepted,
otherwise a probability ratio
( ) / ( )t iR
ρ ρ= r r (2.32)
has to be ca generated,
nd if this random number is less than the probability ratio the trial move will be
mber
lculated. In this case a random number between 0 and 1 will be
a
accepted.
Step (3): go to step (2) replacing. Step (3) is repeated n times, where n is a sufficiently
large nu .
The Metropolis method can be used for any ensemble. According to the ensemble under
study, several trial moves can be chosen, and the acceptance criteria depends on the
41
partition of the ensemble considered. In the following section the trial moves and the
relative acceptance criteria are given for the Gibbs ensemble simulation.
2.2.2 Gibbs Ensemble Monte Carlo Simulation
Since its introduction some years ago (Panagiotopoulos 1987b), the Gibbs ensemble
Monte Carlo technique has proved to be one of the most efficient ways of directly
simulating two existing phases in equilibrium. This method is based on si aneous
compositions. The two regions represent equilibr
mult
calculations in two distinct physical regions of generally different densities and
ium phases, coupled indirectly via
article transfers and volume changes.
e great advantage (Sadus 2002) of the Gibbs method over the conventional
spon
itions and then construct the coexistence line. The Gibbs
nsemble Monte Carlo methodology provides a direct and efficient route to the phase
co ulations of moderate accuracy. A single Gibbs
nsemble simulation gives a point on the phase envelope of a system. The method is
he prediction of thermodynamic and transport properties of fluids and their mixtures is
one of the main aims of resea
he Gibbs ensemble method is one of the most powerful techniques used to realize this
o
p
Th
techniques to study phase coexistence is that, in the Gibbs method, the system
taneously “finds” the densities and compositions of the coexisting phases. Hence,
there is no need to compute the relevant chemical potentials as a function of pressure at
a number of different compos
e
existence properties of fluids, for calc
e
now commonly used for obtaining phase equilibria of fluids, because of its simplicity
and speed.
The Gibbs Ensemble Method T
rch in applied thermodynamics and statistical mechanics.
T
goal. The methodology for determination of phase equilibria using the Gibbs ensemble
method was described in detail for pure fluids (Panagiot poulos 1987b) and mixtures
(Panagiotopoulos and Quirke 1988) by Panagiotopoulos.
42
The thermodynamic requirements for phase coexistence are that each region should be
in internal equilibrium, and that temperature, pressure and the chemical potentials of all
omponents should be the same in two phases. System temperature in Gibbs ensemble
Monte Carlo simulations is specified in advance. The remaining three conditions are
satisfied by performi
that ensures internal equilibrium with each region, an equal and opposite change in the
s, and random transfers of
olecules that satisfy equality of the chemical potentials in the two regions.
The acceptance
originally derived from fluctuation theory. For pure component systems, the phase rule
requires that only one intensive variable (in this case system temperature) can be
multi-component systems pressure can be specified in
dvance, with the total system being considered at constant NpT.
he NVT Gibbs Ensemble Simulation for Pure Fluids
c
ng three types of perturbations, a random displacement of particles
volume of the two regions that results in equality of pressure
m
criteria (Panagiotopoulos 1987b) for the Gibbs ensemble were
independently specified when two phases coexist. The pressure is obtained from the
simulation. By contrast, for
a
T
For pure fluids, we performed NVT ensemble simulation, where the temperature ( )T ,
the total number of particles ( )I IIN n n= + and the total volume ( )I IIV V V= + of the
two boxes I and II , are fixed.
A convenient method to generate trial configurations is to perform a simulation in
cycles. One cycle consi rticle in one
of the randomly chosen boxes, one attempt to change the volume of the subsystems, and
attempts to exchange particles between the boxes. These three different moves
sts of N attempts to displace a randomly selected pa
N
performed in Gibbs ensemble simulation have been showed in the Figure 2.3.
43
displacements Volu Particle transfers
Figure 2.3 the Gibbs ensemble Monte Carlo simulation m
me changes
ethod
lume change in
mpts to exchange particles require
ome care. To ensure that detailed balance is obeyed, it is important to first select at
tation
here the different trial moves are performed in a fixed order.
The implementation of trial moves for particle displacement and vo
Gibbs ensemble simulations is very similar to that of the corresponding trial moves in a
normal NVT or NpT simulation. However, the atte
s
random from which box a particle will be removed and subsequently select at random in
this box. An alternative would be to first select one particle at random from all N
particles and then try to move this particle to the other simulation box however, in that
case, acceptance rule has to be replaced by a slightly different one (Rull et al. 1995).
The method we used to generate trial configurations is the original implemen
w
For the particle displacement moves, the new configuration is generated from the old
one, displacing randomly a particle in the selected box. The old coordinates of the
randomly selected particles are changed adding random numbers between
max−∆ and max∆ , where max∆ is the maximum displacement allowed:
44
( )( )
max
max
2 1
2 1new old
new old
new
x x rand
y y rand
z
← + × − × ∆
← + × − × ∆
←
(2.33)
( ) max2 1oldz rand+ × − × ∆
ion in the interval. We chose
in order to have about 50% acceptance rate. The potential energies with the particle
Here rand is a random number from a uniform distribut
∆max
in the old position and in the new position respectively are calculated and the
acceptance criterion applied.
( ) [ ]{ }(min 1;expacc old new→ = − )/new oldu u kT− (2.34)
ral, if the move is rejected, the
x to be expanded is randomly chosen. If the
Where u is the total potential energy in each box. In gene
old configuration is kept.
For the volume fluctuation move, the bo
chosen box is expanded by V∆ , another one will be compressed by - in order to V∆
keep the total volume constant. The quantity V∆ is given by:
( )min ;I IIV V V Vζ∆ = × ∆ × (2.35) max
Where ζ is a uniformly distributed random number in the range [0, 1] and is the
aximum fractional volume change allowed. Typically,
maxV∆
maxV∆m is chosen in order to
ould guarantee that equilibrium is achieved
llowing
uantities are calculated:
have at least 50% acceptance rate, which sh
efficiently. However, this does not have any theoretical justification. The fo
q
3
ii new
iold
RatV
,i I II= (2.36)
and used to scale the coordinates of the particles in each box:
V=
45
new old
new old
new old
x x Ray y R
tat
Rat
← ×← ×← ×
(2.37)
The potentia
ac
z z
l energies relative to the old and new coordinates are calculated and the
ceptance criterion applied.
( ) ( ) }( ) {( ) ( ) { }
exp /min 1;
I In N nI IInew new newV V u kT
acc old new−⎛ −⎜→ =
exp /I In N nI IIold old oldV V u kT
−
⎞⎟
⎜ ⎟−⎝ ⎠ (2.38)
( )
0.5
0.5new
For the exchange particle move, one of the two boxes is randomly chosen where a
randomly selected particle is cancelled from and inserted into another box. The new
coordinates of the particle are randomly assigned in the following way (the centre of the
simulation box is the origin of the coordinates):
( )( )0.5
newx rand L
rand L
← − ×
←
← − ×
y rand L− × L Boxlength=
newz
(2.39)
lacement, in order to a
the particles is required. The probability of accepting the new configuration is given by:
As with the particle disp pply the acceptance criterion in the
exchange move only the potential energy between the exchanged particle and the rest of
( ) ( ) ( )( ) ( )
exp /min 1; I new
I
n V V u kTold new
⎛ ⎞− −⎜ ⎟→ =
I
acc1 exp /I oldN n V u kT⎜ ⎟− +
(2.40)
pT Gibbs Ensemble for Binary Mixtures
hod is the ease of extension to mixtures. In the
ase of a ensemble for binary mixtures, the total number of particles, temperature
of the total
er of particles of both
−⎝ ⎠
NOne of the strengths of the Gibbs met
NPTc
and pressure are kept constant. The total number of particles is the sum
numb species. The acceptance criterion for the particle
46
displacement is the same as in the Gibbs method for one component. The volume NVT
fluctuation acceptance criterion becomes:
( )( ) ( ) ( ){ }
( ) ( ) ( )
exp /I IIn nI II I IIV V u P V V k⎛ ⎡ ⎤− + ∆ + ∆min 1;
exp /I II
new new new
n nI IIold old old
Told new
V V u kT
⎞⎣ ⎦⎜ ⎟→ = ⎜ ⎟⎜ − ⎟
acc
⎝ ⎠
(2.41)
e it is possible that I
constant. And for the particle exchange step, the criterion and now
presents the number of particles of the species being interchanged.
due to the use
and periodic boundary conditions.
W e transition, the equilibrium
ly determined by
averag is approaching the
critical point of the phase transition, the occasional abnormal fluctuations in the
pro ties of the coexisting phases app
the coexisting phases has been pointed out by Smit (Smit et al. 1989; Smit 1996).
rce for a Gibbs ensemble simulation to remain in a state with two stable
gions of different density is the free energy penalty for formation of interface within
ach of the two regions. When the system is close to a critical point, the penalty for
changes of the identity of the
be determined to high accuracy because the density probability function
enerally is quite noisy. However, finite size scaling may provide an alternative way to
st rovsky and Freed 1989; De et al.
002).
Where in this cas since the total volume is not I IV V∆ ≠ −∆
In II In N n= −
re
Approaching Critical Points by Gibbs Ensemble Simulations
Approaching critical points the Gibbs ensemble simulation is complicated
of a finite system
hen the system is away from the critical point of the phas
densities and compositions of the coexisting phase can be simp
ing the observations after equilibration. But when the system
per ear. The presence of “drift” in the properties of
The driving fo
re
e
formation of interfaces is small, and there are frequent ex
two boxes. In such cases, the only way to obtain estimates of the coexisting densities is
to obtain the density probability function. It should be noted that the coexisting densities
cannot
g
udy the phase transition close to critical point (Nemi
2
47
2.2.3 Gibbs-Duhem Integration Method The Gibbs-Duhem simulation method proposed by Kofke (Kofke 1993b) is an efficient
means study p ase equilibria properties of fluids and even solids by molecular
simulation because it do
to h
es not require particle insertions and removals and is applicable
transitions involving solids.
Gibbs-Duhem Integration Simulation Algorithm the Gibbs-Duhem integration method, each simulation yields one coexistence point.
A
l point to start the simulation
. Perform NPT simulations of both phases simultaneously at the estimated conditions
d d adjust pressure and temperature by
corrector during the process of simulation
d dp
to
In
brief description of the algorithm is as follows:
a. Get initia
b. Estimate pressure or temperature from predictor
c
. Estimate new slop from simulation data an
e. Calculate simulation average value after phase equilibrium obtained
f. Repeat for the next state point.
Gibbs-Duhem Integration Simulation
The Gibbs-Duhem equation for pure substance (Kofke 1993a) can be written as
( )d hβµ β βν= + , (2.42)
here µ is the chemical potential, h the molar enthalpy, ν the molar volume, p the
ressure, and 1/ kTβ = , with the Boltzmann constant and the absolute k Tp
temperature. By writing Eq. (2.42) for two coexisting phases, phase I and phase II, and
by equating the right-hand sides, the Clapeyron equation results in
lnd p ( , )h f pd pσ
ββ β ν
∆= − =⎟ ∆⎠
(2.43) ⎛ ⎞⎜⎝
48
in Eq. (2.43), II Ih h h∆ = − is the difference in mo ar enthalpies of the coexisting phases,
and II I
l
ν ν ν∆ = − the difference in molar volumes of coexisting phases; the subscript
σ indicates that the derivative is taken along the saturation line. Eq. (2.43) is a first-
order nonlinear differential equat
the temperature for two phases to remain in coexistence.
.
iven an initial condition, i.e., the pressure, temperature and
ion that describes how the pressure must change with
The starting point for the simulation is a known coexistence point that can be obtained
by performing a conventional Gibbs ensemble simulation or from experimental data
G ( , )f pβ at one
coexistence point, Eq. (2.43) can be solved numerically by a predictor-corrector
method. We applied the Adams predictor-corrector (Kofke 1993a) to calculate the
pressure.
P 1 1 2 3(55 59 37 9 )24i i i i i iy y f f f fβ
+ − − −
∆= + − + − (2.44)
C 1 1 1 2(9 19 5 )24i i i i i iy y f f f fβ
+ + − −
∆= + + − + (2.45)
In Eqs. (2.44) and (2.45), y = ln p , f for ( , )f pβ , P stands for the predictor, C for the
corrector, and β∆ is the step in the β . The Adams algorithm requires four prior
simulations. We performed the start-up as follows: the pressure at the first simulation
point was predicted by the trapezoid predic o rector
P 1 0 0y y f
tor-c r
β= + ∆ (2.46)
C 1 0 1 02y y f f( )β∆
= + + (2.47)
Then, the midpoint predictor-corrector
P 2 0 12y y fβ= + ∆ (2.48)
C 2 0 2 1 0( 4 )3
y y f f fβ∆= + + + (2.49)
49
was used to determine the pressure at the second point. Finally, the midpoint predictor
P 3 1 22y y fβ= + ∆ (2.50)
With the Adams co
rrector
C 3 2 3 2 1 024(9 19 5 )y y f f f fβ∆
= + + − + (2.51)
w he quantities needed to evaluate
simultaneous but
in nt phases.
In each NPT Monte Carlo sim moves performed:
1. attem
min (2.52)
as used to compute the pressure at the third point. T
the right-hand side of the Clapeyron equation were obtained from
dependent NPT Monte Carlo simulations of the two differe
ulation, there are two different
pt particle move accepted with probability of
( )1,exp Uβ− ∆⎡ ⎤⎣ ⎦
2. attempt volume fluctuation accepted with probability of
( )( )min 1,exp U P Vβ⎡ ⎤− ∆
above formulas, and
+ ∆ − ∆⎣ ⎦ (2.53)
In the U∆ V∆ are the change in internal energy and volume
spectively.
the
rm:
re
Gibbs-Duhem Integration Method for Mixtures Gibbs-Duhem integration method has been extended to study mixtures (Mehta and
Kofke 1994). The Gibbs –Duhem equation for a binary mixture may be written in
fo
1 1 2 2ln ln lnrx d f h d Zd p x d fβ= + − (2.54)
50
here p is the pressure, and Z is the compressibility factor, /pv RT , where v is the
molar volume; 1x is the mole fraction of species 1, with fugacity 1f , rh is the residual
molar enthalpy, defined as the enthalpy above an ideal gas at the same tem erature;
nally
p
fi 1/ RTβ = , where R is the gas constant and is the a perature. The T bsolute tem
fugacity fraction of species 2 is
22
1 2
ff f+
(2.55) ξ =
2ξ varies from zero to unity as the mixture composition goes from pure species 1 to
pure species 2.
According to the definition of 2ξ , the Gibbs-Duhem equation can be written as
( ) ( )2 2
1 2 2ln lnrxd f f h d Zd p d
2 21ξβ ξ
ξ ξ−
+ = + − −
(2.56)
Eq. (2.54) is designated as the “osmotic” form of the Gibbs-Duhem equation, while Eq.
(2.56) is referred to as the “semigrand” form.
Clapeyron type formulae are developed by considering in the intensive “field” variables
that maintain equilibrium between coexisting phases. In particular, changes that
maintain coexistence at constant temperature are governed by a Clapeyron differential
equation, which may take the following form:
1 2 1 2
2 1 1,
/ /g l l g
g l l g
x xpf x Z x Z
β σ
φ φ⎛ ⎞ −∂=⎜ ⎟∂ −
he subscript
⎝ ⎠ (2.57)
σ indicates a change along the saturation line, and the fugacity T
coefficient ( )2 2 2/f pxφ = . This equation prescribes simulation in an osmotic ensemble,
for which the independent variables include temperature, pressure, and species-2
fugacity.
51
An equivalent development may be advanced from the semigrand form of the Gibbs-
Duhem equation. In this case, we consider variations in temperature, pressure and
fugacity fraction that keep changes in the sum ( )1 2f f+ equal between phases.
Variations at constant temperature obey
( )( )2 2
2 2 2,
ln1
l g
l g
x xpZ Zβ σ
ξ ξ ξ⎛ ⎞ −∂
=⎜ ⎟∂ − −⎝ ⎠ (2.58)
This equation specifies simulations in a semigrand ensemble, which has independent
variables of temperature, pressure, fugacity fraction, and total number of molecules N.
Each method for evaluating coexistence diagrams of mixtures has certain advantages
depending on the system under study. The semigrand method (Mehta and Kofke 1994)
never requires particle insertions, but it does demand that particles are able to change
species identities. The osmotic approach does not require such identity changes, but it
must be possible to insert particles of one of the species. The semigrand thus is best
suited for mixtures of species that are not too dissimilar, although each component may
be complicated enough to preclude insertion. With the osmotic method, the species may
be quite dissimilar, but as long as one of them is insertable, the method can be applied.
We are studying the argon-krypton system, so we choose the semigrand method. In the
igrand implementation, integration proceeds according to Eq. (2.58) and the Adams sem
predictor-corrector applied to calculate the pressure. Simulations are conducted in the
isobaric semigrand ensemble, for whichT , P , gN or lN and 2ξ are the independent
variables. The two simulated systems that represent the coexisting phases are
independent but for their comm and on choices of T , P 2ξ ; the total number of
articles for each phase,p gN and , may be selected f r convenience, and they do not
composition in each phase is however not fixed, and
ractions in each phase are then computed as
verages.
lN o
change during the simulation, the
as a result the molecules must sample species identities, just as they sample positions
within the simulation box. The mole f
a
52
Henry’s constants are required to connect the integration to the pure-component limits.
r o For the semigrand simulations, only the ati of the solute Henry’s constant to the
solvent fugacity is required. The ratio can be determined in a pure solvent simulation by
performing trial identity changes: one of the solvent molecules is converted into a solute
molecule and the “exchange energy” u∆ is noted. The particle is converted back before
e simulation proceeds. The ensemble average of the Boltzmann factor of the exchange
en
th
ergy gives the desired ratio
0
exp( )j
i
uH
β= − ∆ (2.59) f
pt particle move accepted with probability of
In the semigrand ensemble simulation, there are three different Monte Carlo moves to
be performed.
1 attem
( )min 1,exp Uβ− ∆⎡ ⎤⎣ ⎦ (2.60)
(min 1,exp Uβ⎡ − ∆ + ∆⎣ ⎦
3. attempt identity exchange accepted with probability of
2 attempt volume fluctuation accepted with probability of
( ) )P V ⎤− ∆ (2.61)
2
21ξ
ξ
⎡ ⎤⎞⎛ ⎞min 1,exp lnU mβ
⎛− ∆ +⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠
(2.62)
for identity change from species 2 to species 1, and otherwise.
⎝ ⎠⎣ ⎦
1m = − 1m = +
In the above formulae, and VU∆ ∆ are the change in internal energy and volume
respectively, accompanying the move. The fugacity fraction of component 2 is 2ξ and
in indicates the minimum of the two arguments. Other variables are as defined in the
ite natural to compare the Gibbs-Duhem technique to the Gibbs ensemble
ethod. However, the two methods should be viewed as complementary rather than
competing approaches to the problem of coexistence evaluation in model systems.
m
text.
Comparison between Gibbs Ensemble and Gibbs-Duhem Integration
Method
It is qu
m
53
One of the appealing features of Gibbs ensemble technique is its simplicity. This
important feature is not lost with the Gibbs-Duhem method. The absence of particle
exchange between the phases eliminates the bookkeeping needed to monitor the
positions of a variable number of particles in each simulation volume. Due to the same
ason, there are many difficult problems to which the Gibbs ensemble can not be
ap tegration may. On the other hand, Gibbs-Duhem
ethod is not self-starting but requires an initial point. From this initial point, the entire
ain limitation of the Gibbs-Duhem integration method is that it is non-self
tarting. For liquid-liquid and vapour-liquid properties, the initial starting point can be
obtained from Gibbs ensemble simulations. However, as discussed in Section 1.3, this is
not possible for solid-liquid equilibrium. To overcome this problem, we performed non-
equilibrium molecular dynamics (NEM
simulations to study solid-liquid phase coexistence properties.
olecular dynamics provides an alternative path to study phase equilibria of fluids.
d onditions applied,
on-equilibrium molecular dynamics approach falls into two categories: homogeneous
and inhomogeneous methods. Homogeneous methods employ periodic boundaries
while inhomogeneous methods may not involve periodic boundary conditions.
Generally homogenous methods are p
make all particles experience the same environment. The disadvantage of homogenous
methods is that the equations of motions of particles must be altered artificially (Sadus
002).
re
plied, and to which Gibbs-Duhem in
m
phase diagram can in principle be determined.
2.2.4 Molecular Dynamics and Non-equilibrium Molecular
Dynamics
The m
s
D) combined with molecular dynamics (MD)
M
Unlike Monte Carlo methods that rely on transition probabilities, molecular dynamics
obtain phase properties by solving the equations of motions of the molecules (Allen and
Tildesley 1987). In addition to equilibrium properties, molecular dynamics can also be
used to study non-equilibrium systems. According to the boun ary c
n
referred because of using periodic boundaries that
2
54
Recently several NEMD algorithms (Evans and Morriss 1990) have been discussed
ased on the statistical mechanics of non-equilibrium liquids. In practice the most
is exact for arbitrar
non-Newtonian systems (Evans and Morriss 1990). In SLLOD method, an external
brium. In order to maintain
ermodynamic fluxes or gradients in the system and prevent relaxation to the
quilibrium state, the external force field need work on the system continuously, which
ill lead to the system over-heating. In this case, thermostat algorithms have been
pplied to remove the heat and maintain the system at a fixed temperature. The
ommonly used thermostat methods are the Nosé-Hoover thermostat (Hoover 1985)and
e Gaussian thermostat (Evans et al. 1983).
he main task of the SLLOD algorithm is to solve isokinetic SLLOD equations of
otion, which will be used in conjunction with suitable periodic boundary conditions
uch as the Lees-Edwards moving periodic boundary conditions (Lees and Edwards
972). In principle, the equations of motion can be dealt with by any standard finite-
ifference algorithm. In practice, most commonly used algorithms are time-consuming
omputationally due to the multiple force evaluation requirements. Recently some
nite-difference algorithms have been developed to apply to molecular dynamics.
1967), where the
b
efficient NEMD algorithms are DOLLS tensor method (Hoover et al. 1980) and the
SLLOD algorithm (Evans and Morriss 1984). It has been shown that the SLLOD
algorithm ily large strain rates and it can be applied to nonlinear,
force field is applied to perturb the system away equili
th
e
w
a
c
th
T
m
s
1
d
c
fi
Generally there are two kinds of algorithms: predictor and predictor-corrector methods.
the most widely used predictor method is the Verlet algorithm (Verlet
molecular coordinates are updated from quantities from previous step or in the current
step. In contrast, the most widely used predictor-corrector method is the Gear algorithm
(Gear 1971), where the new molecular coordinates will be predicted and later will be
corrected using the equations of motion.
The SLLOD equations of motion and the Gear predictor-corrector method will be
discussed in detail because part of this work is to use the SLLOD method combined
with equilibrium NVT molecular dynamics to study solid-liquid phase equilibria of
noble gases.
55
SLLOD Equations of Motion For a system of N particles, the force experienced by a particle during displacement is
given by Newton’s equation of motion:
i i im=F r&& (2.63)
where iF is the force acting on the particle i , im is its mass, ir is its position and ir&& is its
acceleration. This is a second order differential equation. In practice, this equation is
often expressed as two first order differential equations by time derivative of the particle
position or time derivative of the momenta.
i ii
ddt m
= =r pr&
i
(2.64)
ii i
ddt
= =pp F& (2.65)
where p is the momentum of the particle i . i
In contrast to the above clas
motion can be expressed in the following forms due to the external force worked on the
.
sic equations of motion, the isokinetic SLLOD equations of
system
.ii i
i
ddt m
= = + ∇irr r u& (2.66) p
.i i i iddt
α= = − ∇ −ipp F p u p& (2.67)
where ( ,0,0)xu u and xu= yγ= is the velocity field corresponding to planar Couette
ow with a strain rate γ . αfl is the Gaussian isokinetic thermostating constant given by:
1
2
( )N
i xi yii
N
i1i
p p
p
γα =
−=
∑
∑
iFp (2.68)
=
56
The thermostating term α ip in Eq. (2.67) enables the system to reach a steady state by
removing the dissipative heat produced by the driving shearing force.
The SLLOD equations of motion must be implemented with compatible periodic
nditions that are widely used in the study of planar Couette flow. The
etails about the Lees-Edward boundary conditions have been described by different
boundary conditions (Evans and Morriss 1990). In this work, we use the Lees-Edward
boundary co
d
researchers (Lees and Edwards 1972; Allen and Tildesley 1987; Sadus 2002).
Gear Predictor-corrector Method To solve the first order SLLOD equations of motion (Eqs. (2.66) and (2.67)), proper
predictor-corrector algorithms should be performed. Generally the evolution of particle
coordinates or any time-dependent property can be estimated from a Taylor series
expansion. Taking the coordinate vector as an example, the Talylor series expansion is:
2 3 4
2 3 4
1 1 1( ) ( ) ...2! 3! 4!
d d d dt t t tdt dt dt dt
+ ∆ = + ∆ + + + +r r r rr r (2.69)
where t∆ is the time step. It should be noted that the accuracy of the values obtained
from the above equation will depend partly on the extent of truncation. For the Gear
predictor-corrector method, the above equation can not be truncated below the third
term because the SLLOD algorithm depends on the evaluation of accelerations in the
corrector step. In my work, we used a fourth-order Gear predictor-corrector algorithm
due to its efficiency and accuracy. According to the Taylor series expansion, the
predicted coordinates of the particle i can be written as the following using a matrix
form:
00
11
22
33
( )1 1 1 1 1( )( )0 1 2 3 4( )( )0 0 1 3 6( )( )0 0 0 1 4( )( )0 0 0 0 1( )
pii
pii
pii
pii
p
tt ttt ttt ttt ttt t
⎛ ⎞
44 ii
+ ∆ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟+ ∆⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟=+ ∆⎜ ⎟ ⎜ ⎟⎜ ⎟
+ ∆⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ ∆ ⎝ ⎠⎝ ⎠⎝ ⎠
rrrrrrrrrr
(2.70)
57
After the predicted values are calculated, Lees-Edwards periodic boundary conditions
are app ied. The relativl e distances between pair particles are recalculated and are used
determine the forces performed on each particle. Finally the corrected values are
alculated according to the equation (2.71). The derivatives of are:
2
33 3
44 4
( ) ( )))
( ) ( )( ) ( )
c pi ic p
i
i rc pi r
ct t t tccct t t tct t t t
⎞ ⎛ ⎞+ ∆ + ∆ ⎛ ⎞⎟ ⎜ ⎟ ⎜ ⎟
⎟ ⎜ ⎟⎟ ⎜ ⎟+ ∆⎟ ⎜ ⎟
+ ∆ + ∆⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎟+ ∆ + ∆ ⎝ ⎠⎝ ⎠⎠
r r
rr r
(2.71)
to
irc
00 0
11 1
2 2
( ) (( ) (
i ic pi rc p
t t t tt t t t+ ∆ + ∆⎜ ⎟ ⎜
⎜ ⎟ ⎜=+ ∆ + ∆⎜ ⎟ ⎜
r rr r
⎛⎜
⎜⎝r r
where: 1 ( )ii i i
i
pr y tm
γ∆ = − + ∆r (2.72)
re , , , and are the corrector coefficients that depend on the order of the
fferential equation being solved (Gear 1971; Allen and Tildesley 1987). In this
ork
he 0c 1c 2c 3c 4c
di
w , 0 251/ 720c = , 1 1c = , 2 11/12c = , 3 1/ 3c = and 4 1/ 24c = . Equivalently Eq. (2.67)
an also be solved for the momentum using the Gear predictor-corrector method. ipc
58
Method for Obtaining Solid-liquid Phase Equilibrium Properties by Non-
equilibrium Molecular Dynamics and Molecular Dynamics
Prediction of solid-liquid equilibria can be achieved directly by thermodynamic
integration method. but it is tedious, time-consuming and difficult because it involves
calculation of the equation of state and requires accurate evaluation of free energy for
liquids and solids at temperatures of interest (Hansen and Verlet 1969; Meijer et al.
1990; Baez and Clancy 1995). It is also problematic to determine the solid-liquid
transition by conventional simulation techniques because of the computational
challenges posed by two dense phases. The widely used Gibbs ensemble simulation
lgorithm (Panagiotopoulos 1987b) has proven to be a successful technique to
etermine vapour-liquid and liquid-liquid phase equilibria. However, it is not practical
r solid-liquid phase equilibrium study due to the difficulty of inserting particles into
tegration
ain
isadvantage of Gibbs-Duhem integration is that it requires prior knowledge of at least
points to start the algorithm. The success in predicting the phase
t th
termined
se curve.
a
d
fo
the solid phase. This limitation has been overcome by the Gibbs-Duhem in
technique (Kofke 1993b) that does not involve particle exchange. But the m
d
one pair of coexistence
boundary largely depends on the accuracy of this starting point.
Recent work (Ge et al. 2003b) has demonstrated that the NEMD technique, in
conjunction with standard (NVT) equilibrium MD, can be used to determine the solid-
liquid phase coexistence at equilibrium. Two alternative methods have been reported.
One is to locate the solid density according o e observation that the scaling exponent
of the pressure or energy of a shearing Lennard-Jones liquid is about 1.0 at the solid
phase. The liquid density is de by constructing a tie line between the coexisting
solid phase point and the liquid pha
The pressure and energy of a shearing simple liquid can be expressed as the following
equations according to the work by Ge et al. (Ge et al. 2001, 2003a).
0P P a αγ∗ ∗ ∗= + & (2.73)
0E E b αγ∗ ∗ ∗= + & (2.74)
( ),T A BT Cα ρ ρ∗ ∗ ∗ ∗= + − (2.75)
59
where 0P∗ , 0E∗ are the equilibrium reduced values of pressure and energy. γ ∗& , ρ∗ and
T ∗ are the reduced strain rate, reduced density and temperature, respectively. A , B , C
are c ants andonst α is an exponent that can be expressed as a linear function of density
nd temperature. For the liquid system, the typical values of are in the range of 1.2~2, a α
where α decreases as a function of density. The system enters into solid ph e ase when th
value of α falls to about 1.0,
We pay more attention to the second method that is more accurate compared with the
first one. The phase equilibrium liquid density can be obtained by observing the change
in pressure as a function of strain rate and density, and the solid density can be
determined straightforward from a line tied to the solid curve.
In the Figure 2.4 (Ge et al. 2003b) the changes of pressure with varying strain rate over
a range of densities at 1.00T ∗ = have been illustrated.
Figure 2.4 Pressure as a function of strain rate at different densities and constant
temperature 1.00T ∗ =
60
In this figure, the densities ( )ρ∗ from bottom to top curves are 0.74, 0.76, 0.78, 0.80,
0.82, 0.8442, 0.88, 0.92, 0.94, 0.96, 0.98 and 1.0 respectively. All these results were
obtained form the equilibrium and non-equilibrium molecular dynamics simulations
starting from an initial face centred cubic (FCC) lattice configuration. By observing the
points at 0γ =& , we can see that there is a sharp drop in pressure when the system enters
the liquid-solid coexisting area from the pure liquid region. In this way the liquid phase
density of solid-liquid phase equilibrium can be determined. In principle, the liquid
phase density can be obtained only by performing equilibrium molecular dynamics
simulations and noticing where the pressure drop occurs. But in practice non-
quilibr molecular dynam cs simulations can make the liquid phase density to be
ma e
e ium i
determined more efficiently and accurately. The pressure drop observation work only by
equilibrium molecular dynamics is tedious because it would require many equilibrium
MD simulations and each separated in density by a very s ll incr mental amount ρ∆ .
Furthermore, the transition point may be easily missed due to choosing too large a value
of ρ∆ when the system is considered to be still in liquid region but in fact it has already
entered liquid-solid coexistence area. According to Figure 2.4, the system can be easily
ightforward to identify
elting point. For any liquid density, extending an isobaric tie line from the liquid
judged to be still in the pure liquid phase or enter the two-phase metastable region by
observing the difference in pressure at the shear rate between 0 and 0.1. If there is large
discontinuity the system has entered solid-liquid coexisting region or pure solid,
otherwise the system is still in pure liquid phase.
Once the freezing transition point has been determined it is stra
the m
branch to the solid branch obtained from equilibrium molecular dynamics simulations,
the point of intersection gives the corresponding density of the melting transition.
It should be noted that the transition densities determined by this method are accurate to
within ρ∆ , and generally 0.001ρ∆ = ± .
61
Chapter 3
hase Equilibrium Properties of Noble
ses – Lennard-Jones Calculation
Phase equilibria properties of noble gases have been widely investigated by molecular
simulation techniques with different potential models (Barker and Klein 1973; Moller
and Fischer 1990; Leonhard and Deiters 2000; Chen et al. 2001). In this chapter, the
vapour-liquid and solid-liquid phase equilibria properties of argon, krypton and xenon
are studied using the Gibbs-Duhem integration method for the Lennard-Jones potential
with different potential parameter values. One of the aims is to explore the validity of
different Lennard-Jones potential parameter values to fit the experimental data on phase
quilibrium properties. A further aim is to observe the differences in phase equilibrium
potential has been re-examined by several workers in order to obtain
reliable parameter values for this model, which is widely used in the calculation of
phase equilibrium properties of fluids. The most commonly used potential parameter
values determined from second virial coefficients at temperatures above 273K (Klein
and Venables 1976). The new parameter values for a few simple gases from their
viscosity measurements have been presented (Hogervorst 1971; Clifford et al. 1977).
The potential parameters have also been examined for pure substances from the p-V-T
behaviour of binary mixtures containing the pure substance as a component in different
proportions (Rodriguez 1978). Walton has also provided suggested potential parameters
(Walton 1982). These parameter values, listed in Table 2.1, will be used in the present
P
Ga
e
properties using Lennard-Jones potential and the true two-body Barker-Fisher-Watts
potential (see Chapter 4).
The Lennard-Jones
62
simulation work to test for their ability to fit the experimental data on phase equilibrium
properties of argon, krypton and xenon over a wide temperature range.
3.1 Vapour-liquid Phase Equilibrium Properties
Simulation details The Gibbs-Duhem integration Monte Carlo simulations were performed to study
vapour-liquid phase equilibria properties of pure noble gases. An adequate system size
of 512 atoms was used. The initial point for the Gibbs-Duhem integrations is from the
data of Sadus and Prausnitz (Sadus and Prausnitz 1996). From this initial point, the
simulations were performed in two directions by increasing and decreasing
temperatures. The step is calculated by 1/ *Tβ = ( *T
step
is reduced temperature). When the
temperature decreased, we chose the simulation 0.05β∆ = . When the temperature
increased, we chose the smaller step 0.02β∆ = − to get better results due to the difficulty
in simulating close to critical points.
The simulations were performed in cycles, and the total 512 atoms were partitioned
etween two boxes to simulate two different phases. The equilibration period was
10,000 cycles and a further 10,000 cycles were used to accumulate the averages. The
calculations were truncated at intermolecular separations greater than half the box
length, and appropriate long-range corrections were used to obtain the full contribution
of pair interactions to energy and pressure. The values of various properties of the
system such as the coexistence temperatures, densities, energies, and latent heat were
obtained.
Results and discussion The results of the Gibbs-Duhem integration Monte Carlo simulations for vapour-liquid
phase equilibria properties are reported in Table 3.1 (Appendix). Tabulated in columns
1-9 respectively, are vapour-liquid phase coexistence temperature, the set pressure,
b
63
vapour-phase pressure, density, energy, liquid-phase pressure, density, energy, and the
latent heat for the coexisting vapour-liquid phases. The values in brackets represent the
uncertainty of the last digit.
The reduced unit was adopted according to the normal convention for simulation in all
the following work. The reduced temperature is obtained by * /T T , densityε= 3*ρ ρσ= ,
pressure 3* /P Pσ ε= , energy * /E E ε= , and molar enthalpy *H /H ε= .
In order to compare the calculations with experimental data, the results with reduced
units have been converted to experimental units according to the Lennard-Jones
potential parameter values listed in Table 2.1. Comparisons of the vapour-liquid phase
equilibria properties of argon, krypton and xenon obtained from the simulations with the
experimental data (Vargaftik et al. 1996) are shown in Figure 3.1-3.5. In these figures,
the solid circles represent the experimental data and the others are the results computed
using the Gibbs-Duhem integration method.
Comparisons of the calculated V-L coexistence densities of argon, krypton and xenon
with the experimental data are shown in Figures 3.1, 3.2, 3.3 respectively. It is observed
that the best agreement with the experimental data is found when the potential
parameters of Horton (Klein and Venables 1976) are employed to compute the vapour-
liquid coexistence densities of argon, krypton and xenon. For argon and krypton, the
simulation results using the parameter values provided by Rodriguez (Rodriguez 1978)
ng the same parameter values can not give a good agreement over a
suggested by
Hogervorst (Hogervorst 1971) and Clifford (Clifford et al. 1977) can not be employed
simulate vapour-liquid phase equilibrium properties of argon, krypton and xenon
apour and liquid branches have a large deviation compared with
ults wit real units for the other
phase equilibrium properties of argon, krypton and xenon.
can give a very good agreement with the experimental data. However, for xenon, the
calculations usi
lower temperature range for the liquid branch. The parameter values
to
properly as both the v
the experimental data. The calculations using the parameter values provided by Walton
(Walton 1982) can only give a good agreement at lower temperatures for argon and
higher temperatures for krypton. The liquid phase calculation for xenon has a large
deviation from the experimental data. Based on the above analysis, we chose the
parameter values suggested by Horton to obtain res h
64
Comparison of the calculated V-L coexistence pressures of argon, krypton and xenon,
based on the potential parameter values suggested by Horton, with the experimental
data has been described in Figure 3.4. It can be seen that ln P and 1/Tβ = have a good
linear relationship and the calculations give a very good agreement for all the three pure
fluids. The simulation can estimate the triple points and can also give a further
prediction over the critical point.
Comparison of the calculated V-L coexistence latent heat of argon, krypton and xenon,
based on the potential parameters suggested by Horton, with the experimental data is
howed in Figure 3.5. The simulation can give an excellent agreement with
n. For argon and xenon, the latent heat values are a little bit
vapour phase energy
ecreases and the liquid phase energy increases. The changes are smooth and steady. It
simulate vapour-liquid phase equilibrium properties of argon,
rypton and xenon when the potential parameter values are chosen properly.
s
experimental data for krypto
larger than the experimental data, but the agreement is still acceptable. The simulation
provides more data at lower temperatures than the experiments.
Changes in the vapour-liquid phase equilibrium potential energy with temperature are
illustrated in Figure 3.6. With the increase of temperature, the
d
can be predicted that the vapour-liquid phase energy will be equal to the liquid-phase
energy when the critical point is reached. To the best of our knowledge, experimental
data is not available for comparison with the calculations.
The above discussion indicates that the Lennard-Jones potential is an efficient potential
and it can be used to
k
65
180
0 200 400 600 800 1000 1200 1400 160070
170
90
100
110
120
130
140
150
160
T(K)
80
ρ (kg/m3)
Figure 3.1 Comparison of experiment (æ) with calculation using the Lennard-Jones
potential with different potential parameter values suggested by Hogervorst (ó), Horton
), Clifford (ò), Rodriguex (õ), and Walton (ô) for the vapour-liquid coexistence (ç
density of argon.
220
240
260
80
100
120
140
160
180
200
0 400 800 1200 1600 2000 2400 2800
T(K
)
tion using the Lennard-Jones
potential with different potential parameter values suggested by Hogervorst (ó), Horton
(ç), Clifford (ò), Rodriguex (õ), and Walton (ô) for the vapour-liquid coexistence
density of krypton.
ρ (kg/m3)
Figure 3.2 Comparison of experiment (æ) with calcula
66
0 500 1000 1500 2000 2500 3000
140
160
180
200
220
240
260
280
300
320
340
T(K
)
ρ (kg/m3)
Figure 3.3 Comparison of experiment (æ) with calculation using the Lennard-Jones
potential with different potential parameter values suggested by Hogervorst (ó), Horton
(ç), Clifford (ò), Rodriguex (õ), and Walton (ô) for the vapour-liquid coexistence
density of xenon.
8
9
0.004 0.006 0.008 0.010 0.012 0.014
3
4
5
6
7
lnP(
MP
a)
β=1/T(K)
Xe Kr Ar
Figure 3.4 Comparison of experiment (æ) with calculation using the Lennard-Jones
potential with the potential parameter values suggested by Horton (ç) for the vapour-
liquid coexistence pressure of argon, krypton and xenon.
67
50 100 150 200 250 300
0
20
40
60
80
100
120
140
160
180
∆H(k
J/kg
)
T(K)
Ar Kr Xe
Figure 3.5 Comparison of experiment (æ) with calculation using the Lennard-Jones
ce latent heat of argon, krypton and xenon.
0
potential with the potential parameter values suggested by Horton (ç) for the vapour-
liquid coexisten
-2
-1
-5
-4
-3
0.6 0.7 0.8 0.9 1.0 1.1-7
-6
E*
T*
Figure 3.6 vapour-liquid phase equilibrium potential energy of LJ system as a function
of temperature
68
3.2 Solid-liquid Phase Equilibrium Properties Simulation details The Gibbs-Duhem integration Monte Carlo simulations were performed to study solid-
quid phase equilibria properties of noble gases. The initial point for the Gibbs-Duhem
. From
this initial point, the simulations were carried out by decreasing temperatures. The
li
integrations is from the data of Agrawal and Kofke (Agrawal and Kofke 1995)
temperature change per step was calculated by 1/ *Tβ = , and 0.05β∆ = .
The simulations were performed in cycles, and the total 932 atoms were partitioned
between two boxes to simulate two different phase: 432 atoms in liquid phase box and
500 atoms in solid phase box to make sure that the simulation in the solid phase box
starts from a FCC lattice. The equilibration period was 10,000 cycles and a further
10,000 cycles was used to accumulate the averages. The calculations were truncated at
termolecular separations greater than half the box length, and appropriate long-range
to obtain the full contribution of pair interactions to energy and
pressure. The values of various properties of the system such as the solid-liquid
e Lennard-Jones potential, we chose the parameter values suggested by
1976) to study the solid-liquid phase equilibria properties
f argon, krypton and xenon. The results with real units are compared with the
d and Daniels 1968) and the comparisons have been showed
in the Figures 3.7-3.9.
in
corrections were used
coexistence temperatures, densities, energies, and latent heat were obtained.
Results and discussion The results of the Gibbs-Duhem integration Monte Carlo simulations for solid-liquid
phase equilibria properties are summarized in Table 3.2 (Appendix). Tabulated in
columns 1-9 respectively, are solid-liquid phase coexistence temperature, the set
pressure, liquid-phase pressure, density, energy, solid-phase pressure, density, energy,
and the heat latent for the coexisting vapour-liquid phases.
According to the above discussion of the vapour-liquid phase equilibrium properties
calculated by th
Horton (Klein and Venables
o
experimental data (Crawfor
69
Comparison of the calculated solid-liquid coexistence densities of argon, krypton and
xenon with the experimental data is illustrated in Figure 3.7. The simulation results are
in very good agreement with experiment for all the three different gases and also give
more information for high temperatures.
Comparison of the calculated solid-liquid coexistence pressures of argon, krypton and
g krypton and xenon.
an be seen that the Lennard-Jones potential
arameter values provided by Horton can be used to simulate the vapour-liquid and
xenon with the experimental data is showed in Figure 3.8. The simulation has good
agreement with experimental data with the increase of temperature starting from triple
points and also provides further information for high temperatures and pressures.
The calculated solid-liquid coexistence latent heat of argon, krypton and xenon is
reported in Figure 3.9. It can be seen that with the increase of temperature starting from
triple points the latent heat is going higher. The simulation has a good agreement with
experimental data for argon and gives a further prediction for higher temperatures. The
experimental data for krypton and xenon is not available, but our calculation may
provide more information for studyin
The variation of solid-liquid coexistence potential energy changes with temperature is
plotted in Figure 3.10. Both the liquid and solid phase energies increase with increasing
temperatures, which is in contrast with the situation of vapour-liquid phase equilibrium
where the vapour phase energies decrease with the increase of temperatures. The
changes are smooth and steady.
According to the above analysis, it c
p
solid-liquid phase equilibrium properties of argon, krypton and xenon successfully. All
these calculations and discussion can also give a valuable comparison for the work
regarding three-body effects on phase behaviour of noble gases in the next chapter.
70
700
800
1500 2000 2500 3000 3500 4000 45000
100
200
300
400
500
Kr
Xe
600
T(K)
ρ (kg/m3)
Ar
Figure 3.7 Comparison of experiment (æ) with calculation using the Lennard-Jones
potential with the potential parameter values suggested by Horton (ç) for the solid-
liquid coexistence density of argon, krypton and xenon.
2000
2200
100 200 300 400 500 600 7000
200
400
600
800
1000
1200
1400
ArKr
Xe
1600
1800
P(G
Pa)
T(K)
Figure 3.8 Comparison of experiment (æ) with calculation using the Lennard-Jones
potential with the potential parameter values suggested by Horton (ç) for the solid-
liquid coexistence pressure of argon, krypton and xenon.
71
90
100 200 300 400 500 600 70010
20
30
40
50
60
70
80
Ar
∆H
(kJ/
kg)
T(K)
KrXe
Figure 3.9 Comparison of experiment (æ) with calculation using the Lennard-Jones
potential with the potential parameter values suggested by Horton (ç) for the solid-
liquid coexistence latent heat of argon, krypton and xenon.
0.5 1.0 1.5 2.
-7
-6
-5
-4
-3
0 2.5 3.0-8
E*
T*
Figure 3.10 Solid-liquid phase equilibrium potential energy of LJ system as a function
of temperature
72
Chapter 4
Three-body Effects on Phase Equilibrium
wide range of densities
cluding vapour, liquid and solid are explored using NVT molecular dynamics
all energy in the calculations of configuration energies. In order to study
ree-body effects on phase behaviour of noble gases, a series of NVT molecular
ynamics simulations were performed using a real two-body potential (BFW) with
) included.
Properties of Noble Gases
In this chapter, we focus on the study of the three-body effects on phase equilibrium
properties. In Section 4.1, the three-body effects over a
in
simulations for a real two-body potential (Barker-Fisher-Watts potential) and the three-
body interactions (Axilrod-Teller term). In Section 4.2 simple relationships between
two-body and three-body potentials are investigated and the empirical expressions that
link the two-body and three-body potential energies are reported. In Section 4.3 and 4.4
the three-body effects on vapour-liquid phase equilibrium properties of both pure noble
gases and argon-krypton mixtures are studied using the Gibbs-Duhem integration
method. In Section 4.5 the three-body effects on solid-liquid coexistence properties are
examined using Non-equilibrium and equilibrium molecular dynamics simulations.
4.1 Three-body effects on vapour, liquid and solid It is well established that the physical properties of fluids are determined
overwhelmingly by pair interactions. However, it is also well-know that three-body
interactions can make a small but significant contribution to some properties of fluids. It
has been reported that three-body interactions make a contribution of typically 5%-10%
to the over
th
d
three-body interactions (AT term
73
Simulation details
he NVT simulations were performed using molecular dynamics for different system
izes: 108, 256, 500, and 864 atoms respectively at different reduced densities ranging
om 0.05 to 1.3 for argon. For krypton and xenon, only the 500 atoms system size was
vestigated. The starting structure is a face centred cubic (FCC) lattice. The equations
f motion were integrated by a 4th order Gear predictor-corrector scheme with a reduced
tegration time step of 0.001. The first 50,000 time steps of each trajectory were used
equilibrate the system, and a further 200,000 time steps were carried out to calculate
verage values. Periodic boundary conditions were applied. The BFW two-body
otentials were truncated at half the box length and long-range corrections were used to
cover the full contribution to the intermolecular potential, while the three-body
teractions of AT term were assumed to be zero at separations greater than a quarter of
e box length.
Results and discussion
differences of two-body potential energies are presented in Figure 4.1b. It can be
observed that the deviation exists at all densities, particularly at both low and high
densities. Even so, the calculation of 500 atom system is very close to that of 864 atom
system.
The system size effect on three-body potential energies is illustrated in Figure 4.2a, and
relative percentage differences of three-body potential energies are shown in Figure
4.2b. Contrasting two-body potential energy calculation, the system size effect is much
greater for the three-body potential energy calculation. It is particularly significant for
the 108 atom system. The relative percentage differences of both two-body and three-
body potential energies are very clear at low and high densities. The main reason is that
at low density, the two-body or three-body potential energy values are very small and
em size.
T
s
fr
in
o
in
to
a
p
re
in
th
The system size effect on two-body potential energies is shown in Figure 4.1a, in which
there is a discontinuity when the solid-liquid transition occurs. Relative percentage
only a small fluctuation will lead to a large relative percentage difference. While at high
density, the potential energy value is big, especially for three-body potential energy that
is much affected by the syst
74
The system size effect on the total potential energies is presented in Figure 4.3a, in
deviation is very small over the liquid region. The interesting thing is that the relative
difference of total potential energy is not as large as either two-body or three-body
potential cases at medium and high densities. This indicates that the two-body and
three-body potential energy fluctuations with system size to some extent cancel each
other when the density is not very low.
From the above investigation, it can be said that there are finite size effects at all
densities, particularly at both low and high densities. Increasing the system size, the
system finite size effect becomes smaller. When the size is over 500 atoms, the relative
percentage differences are very small for the two-body, three-body and total potential
energy calculations, particularly when the density is at medium region. Therefore, a
system size of 500 atoms may represent a reasonable compromise between system size
effects and the large computational cost involved in performing three-body calculation.
rgon at
which the system size effect is difficult to notice. The relative percentage differences of
the total potential energies are shown in Figure 4.3b. It can be observed that the
Due to the similarity of argon, krypton and xenon, only 500 atom systems of krypton
and xenon are used to perform the similar simulations. The three-body effects on
vapour, liquid and solid of a * 0.9914T = with densities ranging from 0.0 to 1.3 are
summarized in Table 4.1 (Appendix).
ity. The
tal energy and two-body potential energy has the similar trend shape, which shows
that two-body potential dominates the total potential. The total energy or two-body
energy falls with the increase of density before . It is easily seen that there is a
big gap when the density is changed from 0.91 and 0.92, where the solid and liquid
phase transition happens. After the range of equilibrium state, the total energy and two-
body potential energy increase with the rise of density.
The two-body, three-body, kinetic and total potential pressures change with densities
are presented in Figure 4.5. Over the range of vapour and liquid (reduced
density
The two-body, three-body and total potential energies change with densities are
presented in Figure 4.4. The three-body energy increases with increasing dens
to
0.91ρ∗ =
* 0.03 0.70ρ = − ), all the types of pressures fluctuate around a very small value and
75
do not change much. As the dense liquid and solid density are approaching, the three-
body pressure and the kinetic pressure increase smoothly and slowly, but the two-body
and total pressure change dramatically especially in the range of solid. This shows again
that two-body interaction dominates the total interactions.
A vapour-liquid transition occurs at the temperature as shown in Figure
4.6, where the total pressure as a functi vapour-liquid
phase equilibrium (
0.9914T ∗ =
on of density over the range of
* 0.03 0.55ρ = − ) has been plotted. The vapour-liquid phase transition
is evident from a “van der Waals loop”. The pressure increases with the increase of
vapour density. When the density goes up to about 0.15, the pressure starts to fall down,
where the phase transition between vapour to liquid happens. When the density goes
down to the lowest pressure at about 0.45, the pressure rises again until it goes up to the
pressure when the transition happens, the density is about 0.55, where the vapour-liquid
ent with the theory of van der Waals loop.
aals isotherm develops a loop which extends
through the full, equilibrium two-phase region and represents an isothermal, real
vapour and liquid densities
can be determined from the van der Waals loop by choosing a pressure which cuts the
It should be noted that the “van der Waals” loop is observed both in analytical equations
of states and molecular simulations. It represents a metastable extension of the liquid
and vapour branches inside of the two-phase region. The van der Waals loop is not
observed in real systems which display a discontinuity at the equilibrium pressure. The
equilibrium pressure can be obtained by using a “Maxwell’s rule”. At equilibrium, an
isobar passing through the van der Waals loop will result in two regions, above and
below the isobar, which must be of equal area. In this work, the equilibrium pressures
were determined independently using the NVT-Gibbs Ensemble and the pressures are
illustrated in Figure 4.6 using solid line (total potential pressure) and dotted line (2body
+ kinetic term). If a sufficiently large number of particles are used in the simulation, the
equilibrium state finishes and it will enter into pure liquid. It can be seen that the
pressure does change over the range of vapour-liquid zone and the pressure has a fall
and rise process, which has a good agreem
Below the critical point, the van der W
analytical contribution of the equation of state through the coexistence curve or phase
boundary (Fisher and Zinn 1998). In principle the coexisting
loop in equal areas.
76
van der Waals loop vanishes (Yamamoto et al. 1995), which is consistent with
experiment. Although the van der Waals loop is an artefact, it serves the useful purpose
of signalling the presence of a phase transition.
It should be noted that when Gibbs ensemble simulations are performed for two-body
plus kinetic term only, the equilibria liquid density changes substantially, but the
pressure is only slightly lowered. When a MD NVT calculation is performed for two-
body and kinetic term, the Van der Waals loop occurs at a substantially lower pressure
than the full (2body + 3body + kinetic term) potential calculation. In view of the small
change in equilibria pressure observed in the Gibbs ensemble simulations, it is apparent
that the Maxwell equal area rule cannot be used in this case to determine the pressure.
Indeed, it would indicate a substantial lower, even negative pressure. This illustrates the
fact that the van der Waals loop is an artefact and that the Maxwell rule does not
necessarily apply to all simulation results.
ty, and the value is always positive. While the two-
body pressure falls down with the increase of density and always is negative. It is
ic at vapour-liquid equilibria.
Figure 4.8 illustrates combinations of the kinetic pressure with either two-body
interactions or three-body interactions. It is apparent that it is the contribution of two-
body plus kinetic interactions that gives rise to a van der Waals loop and hence a
vapour-liquid transition. The effect of three-body interactions is to reduce the depth of
the van der Waals loop resulting in a vapour-liquid transition of higher pressures and
different densities. Indeed, in some circumstances, the effect of three-body interactions
may be to eliminate a vapour-liquid phase transition totally.
Figure 4.9 shows the effect of various components of pressure on the solid-liquid
transition. When the density goes from 0.91 to 0.92, the total pressure and
We show more interest in the relationship between the different pressures over the range
of vapour-liquid phase transition zone. Figure 4.7 illustrates the various components to
the pressure. We can see that both the three-body pressure and kinetic pressure go up
smoothly with the increase of densi
apparent that none of the kinetic, two-body and three-body contribution to pressure
results in the van der Waals loop characterist
*totalP , *
2P
77
( * *2 KP P+
me
where the atom
equilibrium
for solid-
) has a very sharp drop, where the liquid-solid transition happens. It should be
noted that the regions on the solid curve in the close proximity to this drop is a
tastable extension before their pressure rises up to the transition pressure. The
configurations before and after the drop are shown in the Figure 4.10 and 4.11
respectively. Before dropping the configuration is more like liquid whose atoms
distribute more randomly, while after dropping the configuration is more solid like
s distribute as FCC lattice. In common with vapour-liquid equilibria
(Figure 4.8), the combination of kinetic with three-body contributions alone is not
sufficient to observe a phase transition. However, two-body interactions alone are
sufficient to observe a solid-liquid transition. This is a direct contrast to vapour-liquid
which needs the contribution of both kinetic and two-body components.
The effect of three-body interactions is to increase the coexistence pressure. Comparing
Figure 4.8 with 4.9, it is apparent that the increase in pressure is much more substantial
liquid equilibrium whereas the change in density is relatively minor.
78
0.0 0.2 0.4 0.6 0.8 1.0 1.2-7
-6
-5
-4
-3
-2
-1
0
E*2
ρ∗
Figure 4.1a Comparison of two-body potential energies calculated for argon with
ifferent system size (ó108, æ256, ç500 and ò864 atoms) at different reduced densities d
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-10
-5
0
5
10
15
100(
E* 2(
864)-E
* 2)/E* 2(
864)
ρ∗
Figure 4.1b Relative percentage difference between two-body potential energy of
different system size (ó108, æ256, and ç500) and that of 864 atom system of argon at
different reduced densities
79
0.9
1.0
0.6
0.7
0.8
0.5
0.0 0.2 0.4 0.6 0.8 1.0 1.20.0
0.1
0.2
0.3
0.4
E*
Figure 4.2a Com
different system size ( 108, 256, 500 and 864 atoms) at different reduced densities
3
ρ∗
parison of three-body potential energies calculated for argon with
ó æ ç ò
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-10
0
10
20
30
40
50
60
70
100(
E* 3(86
4)-E
* 3)/E* 3(
864)
ρ∗
Figure 4.2b Relative percentage difference between three-body potential energy of
different system size (ó108, æ256, and ç500) and that of 864 atom system of argon at
different reduced densities
80
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-6
-5
-4
-3
-1
0
-2
E*total
ρ*
Figure 4.3a Comparison of total potential energies calculated for argon with different
system size (ó108, æ256, ç500 and ò864 atoms) t d a ifferent reduced densities
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-6
-4
-2
0
2
4
6
8
10
100(
E* to
tal(8
64)-E
* tota
l)/E* to
tal(8
64)
ρ∗
Figure 4.3b Relative percentage difference between total potential energy of different
system size (ó108, æ256, and ç500) and that of 864 atom system of argon at different
reduced densities
81
0.0 0.2 0.4 0.6 0.8 1.0 1.2-7
-6
-5
-4
-3
-2
-1
0
1
2
E*
ρ
∗
Figure 4.4
Comparison of potential energies (á 2∗ , à 3E∗ and æ(E 2 3E E∗ = E∗ ∗+ ))
calculated for 500 atom system of argon at different reduced densities.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0
10
20
30
40
P*
ρ∗
Figure 4.5 Comparison of pressures (æ totalP∗ , à 2P∗ , ç 3P∗ and ó calculated for 500
atom system of argon at different reduced densities
kP∗ )
82
0.0 0.1 0.2 0.3 0.4 0.5 0.6
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
P*total
ρ∗
Figure 4.6 the total pressure ( 2 3total kP P P P∗ ∗ ∗ ∗= + + ) (-è-) and ( 2 kP P∗ ∗+ ) (-â-) calculated
from simulations for 500 atom system of argon between the liquid and vapour
coexistence densities. The ressu “van der Wp res display aals” loops in the two-phase
vapour/liquid region. The equilibrium coexisting pressures for the two cases (dotted and
solid lines) were obtained from Gibbs Ensemble simulations.
0.0 0.1 0.2 0.3 0.4 0.5 0.6-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
P*
ρ∗
P*3
P*total
P*2
P*K
Figure 4.7 Comparison of pressures calculated for 500 atom sys between
the liquid and vapour coexis ence d
tem of argon
t ensities.
83
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
P*K+P*
2
P*
ρ∗
Figure 4.8 Comparisons of the var
P*K+P*
3
P*total
ious components to the pressures of argon at different
reduced densities at . 0.9914T ∗ =
0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150
2
4
6
8
10
12
14
P*
ρ∗
P*total
P*2+P*
K
P*3
P*2
P*3+P*
K
Figure 4.9 Comparisons of pressures calculated from s for 500 atom system
of argon at different reduced densities at 0T ∗ =
imulations
. .9914
84
0 2 4 6
2
4
6
8
0
8
Y A
xis
X Axis
Figure 4.10 the configuration of argon at * 0.91ρ = .
0 2 4 6 8
0
2
4
6
8
Y A
xis
X Axis
Figure 4.11 the configuration of argon at * 0.92ρ = .
85
Three-body Effects on the Phase Behaviour of Argon at Different
Temperatures In this part we explore how three-body interactions effect on the phase behaviour of
similarity of argon, krypton and xenon, we take argon as an example.
noble gases at high temperatures both below and above the critical point. Due to the
500 atoms of argon were chosen and NVT
rmed by molec
of , 1.2678, 1.4168 and 2.0000 (the critical point of argonT = ). All
the other simulation details are the same as the situation fo
According to the above investigation,
simulations were perfo ular dynamics at different temperatures
* 0.9000T = * 1.0616c
r * 0.9914T = .
The three-body effects on phase behaviour of argon with densities ranging from 0.05 to
1.3 at are summarized in Table 4.2 (Appendix). Here we list only the
as the calcula
ilar. And the simulations for
* 1.2678T =
calculations for * 1.267T = tions for temperatures above the critical point 8
* 0.9000T =are very sim is similar to the one for
t lower density (
* 0.9914T = .
The two-body potential energy, three-body potential energy and total potential energy
variation with density at the different temperatures are shown in Figure 4.12, 4.13 and
4.14, respectively. It can be observed that the three-body potential energy changes are
almost independent of temperature except in the vapour-liquid phase equilibrium area,
where the three-body potential energies are higher and the rise is more obvious when
the temperature below the critical point is lower. When the temperature is above the
critical point, the three-body potential energies are independent of temperature over all
the phase areas from pure vapour to pure solid. In contrast to the three-body potential
energy, the two-body and the total potential energy changes with density are
dramatically affected by temperature when phase transitions occur. It can be noticed
that a * 0.8ρ < ), the two-body and the total potential energy changes
ilibrium area. The two-body and the total potential energies are
duced when the vapour-liquid phase coexistence exists, which is in contrast to the
three-body potential energies going up when the vapour-liquid coexistence occurs.
with densities are almost along a straight line when the temperature is above the critical
point, while there is a curve when the temperature is below the critical point due to the
vapour-liquid phase equ
re
86
Between before and after the solid-liquid phase transition occurs, the two-body and total
otential energy has a big drop, and they are greatly affected by temperatures. It also
hases are not much affected by temperatures, which makes possible for us
explore a relatively steady relationship between two-body and three-body potential
ery similar to the one for energies. Although when the transition occurs
om vapour to liquid, the three-body pressures have a small but noticeable increase.
p
can be observed that the two-body and the total potential energy in pure vapour and
pure liquid p
to
energies in these areas.
The two-body, three-body pressures and total pressures changes with density at the
different temperatures are shown in Figure 4.15, 4.16 and 4.17, respectively. The
situation is v
fr
The three-body pressure changes with densities are nearly independent of temperatures
while the two-body and the total pressures are greatly affected by temperature,
especially at dense fluids and solids. In the Figure 4.16 the van der Waals loop in the
two-phase vapour/liquid region has been shown when temperatures ( 0.9000T ∗ = and
0.9914) are below the critical point temperature ( 1.0616T ∗ = ).
87
0.0 0.2 0.4 0.6 0.8 1.0 1.2-7
-6
-5
-4
-3
-2
-1
0
E*2
ρ*
Figure 4.12 Two-body potential energies as a function of density at different
temperatures. Results are shown for both subcritical temperatures ( ò),
à)) and supercritical temperatures ( æ), á),
ç)).
0.9T ∗ = (
0.9914T ∗ = ( 1.2678T ∗ = ( 1.4168T ∗ = (
2.0T ∗ = (
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
E*3
ρ*
Figure 4.13 Three-body potential energies as a function of density at different
temperatures. Results are shown for both subcritical temperatures ( ò),
à)) and supercritical temperatures ( æ), á),
ç)).
0.9T ∗ = (
0.9914T ∗ = ( 1.2678T ∗ = ( 1.4168T ∗ = (
2.0T ∗ = (
88
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-6
-5
-4
-3
-2
-1
0
E*total
ρ*
Figure 4.14 Total potential energies as a function of density at different temperatures.
Results are shown for both subcritical temperatures ( ò), à)) and
supercritical temperatures ( æ), á), ç)).
45
0.9T ∗ = ( 0.9914T ∗ = (
1.2678T ∗ = ( 1.4168T ∗ = ( 2.0T ∗ = (
35
40
0.0 0.2 0.4 0.6 0.8 1.0 1.2-5
0
5
10
15
20
25
30
P*2
ρ*
Figure 4.15 Two-body pressures as a function of density at different temperatures.
Results are shown for both subcritical temperatures ( ò), à)) and
supercritical temperatures ( æ), á), ç)).
0.9T ∗ = ( 0.9914T ∗ = (
1.2678T ∗ = ( 1.4168T ∗ = ( 2.0T ∗ = (
89
0.0 0.2 0.4 0.6 0.8 1.0 1.20.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
P*3
ρ*
Figure 4.16 Three-body pressures as a function of density at different temperatures.
Results are shown for both subcritical temperatures ( ò), à)) and 0.9T ∗ = ( 0.9914T ∗ = (
supercritical temperatures ( 1.2678T ∗ = (æ), 1.4168T ∗ = (á), 2.0T ∗ = (ç)).
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0
10
40
50
0.08
0.12
0.16
20
30
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-0.08
-0.04
0.00
0.04
igure 4.17 Total pressures as a function of density at different temperatures. Results
P*total
ρ*
F
are shown for both subcritical temperatures ( 0.9T ∗ = (ò), 0.9914T ∗ = (à)) and
supercritical temperatures ( 1.2678T ∗ = (æ), 1.4168T ∗ = (á), 2.0T ∗ = (ç)).
90
4.2 Investigation of Relationship between -body
and Three-body Potential Energies
Two
is well known that many-body interaction calculations are very time-consuming. The
al. 1992; Marcelli et
l. 2001) has demonstrated that density dependent pair potentials seemed to be a
practical way to account for three-body or higher body interactions. In this part, the
lationship between two-body and three-body potential energy is investigated. This
tational
enalty caused by three-body interaction calculations.
ystem Size Effects on the Relationship between Two-body and Three-body Potential Energies
the two-body, three-
ody and total potentials, especially when the density is very low or very high. In the
he ratio of three-body and two-body potential energies of argon with different system
izes has been presented in Figure 4.18. It can be observed that system size does have
n effect on the relationship between two-body and three-body potential energies. When
e system is very small e.g. 108 atoms, the ratio of three-body and two-body potential
nergies of fluid argon (reduced density ranging from 0.2 to 0.8) is virtually
dependent of density. However, the absolute ratio (as the ratio is negative) increases
ith the increase of system size. When the system size exceeds 500 atoms, the variation
f the ratio with density becomes independent of system size.
It
introduction of the three-body potential increases the computing time of a normal two-
body potential simulation by a factor of ten. Previous work (Smit et
a
re
investigation is significant as it makes possible to estimate the three-body effects with
sufficient accuracy only from two-body potential calculations without the compu
p
S
From the previous study of the three-body effects on the phase behaviour of noble gases
in Section 4.1, we know that system size does have an effect on
b
following section, the system size effect on the relationship between two-body and
three-body potential energies is investigated.
T
s
a
th
e
in
w
o
91
In order to further investigate the system effect on the relationship between two-body
nd three-body potential energies, we study the liquid and solid argon respectively.
he variation of
a
* *3 2/E ET with density for liquid argon is presented in Figure 4.19a. It has
been analysed that for each different system size, the ratio changes with argon density
according to a linear relationship, which can be expressed by the following equations:
108 atom system *
*3* *2
0.38EE
ρα
= − (4.1)
200 atom system *
*3* *2
0.65EE
ρα
= − (4.2)
256 atom system *
*3* *2
0.72EE
ρα
= − (4.3)
500 atom system *
*3* *2
0.83EE
ρα
= − (4.4)
*
864 atom system *3* *2
0.85EE
ρα
It should be noted that for the system of 200 atoms, the results that we used here are
taken from the Gibbs ensemble simulations preformed by Marcelli and Sadus (Marcelli
and Sadus 2000). In their work, 500 atoms were employed but distributed between the
two boxes which made up the Gibbs ensemble. The three-body potential energy
averages in the liquid phase involved typically 200 atoms. It should also be noted that
the densities studied in the Gibbs ensemble were reported with an error range. In
contrast, calculations presented here are for a fixed density. The simulations were run
for a much longer period, of which the first 50,000 time steps were used to equilibrate
the system, a further 200,000 time steps were then employed to calculate averages.
However, in the Gibbs ensemble simulations, only 1500 steps were run to equilibrate
the system and a further 1500 steps for the average calculation purpose. Consequently
the error in the calculation of this work is very small (see Table 4.1 Appendix).
= − (4.5)
92
From Eq. (4.1) to (4.5), it can been observed that for pure liquid argon, the simple
relationship between two-body and three-body potential energies can be presented as
g formula: the followin
*
*3* *2
EE
λρα
= − (4.6)
The coefficient λ value changes from 0.38 to 0.85 when the system size increases from
108 to 864.
Figure 4.19b shows the variation of λ with system size. It is clear that the coefficient
value increases with the system size when the atom number is less than 500. However,
when the system size is larger than 500 atoms, the coefficient value is going to be stable
at about 0.85. It should be noted that this is only valid when the density is ranging from
0.45 to 0.75.
body potent s argon is pr . It has been
observed that for each different system size, the ratio changes with argon density
In contrast to liquid argon, the ratio of solid argon changes not only with the density but
also with the square of the density. In Figure 4.20a, the ratio of three-body and two-
ial energy changes with density of olid esented
according to the second order polynomial relationship, which can be expressed as the
following equations.
*
108 atom system 23* *2
3.7 3.8EE
ρ ρα
∗ ∗= − (4.7)
256 atom system *3 4.3E 2
* *2
4.7E
ρ ρα
∗ ∗− (4.8) =
*
500 atom system 23* *2
5.57 6.0EE
ρ ρα
∗ ∗= − (4.9)
864 atom system *
23* *2
5.58 6.0EE
ρ ρα
∗ ∗= − (4.10)
93
It can been observed that for pure solid argon, the simple relationship between two-
body and three-body potential energies can be presented as the following formula:
*
23* *2
a bE
Eλ ρ λ ρ
α∗ ∗= − (4.11)
In Figure 4.20b, the variation of aλ and bλ with system size for solid argon is given. It
is obvious that both coefficient value aλ and bλ increase with the system size when the
atom number is less than 500. Once the system size is larger than 500 atoms, both
coefficient values are going to be stable ( aλ is about 5.5, bλ is about 6.0). It should be
noted that this is only valid when the reduced density is ranging from 1.1 to 1.3.
94
0.0 0.2 .4 0.8 1.0 1.2 1.4
-2.8
0 0.6
-2.4
-0.8
-0.4
0.0
-2.0
-1.6
-1.2
E* 3/(
E* 2α∗ )
ρ∗
Figure 4.18 the ratio of three-body and two-body potential energies of argon with
different system size (ó108, æ256, ç500 and ò864 atoms) at different reduced densities
95
0.45 0.50 0.55 0.60 0.65 0.70 0.75
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
E* 3/(E* 2α
∗ )
ρ∗
ple relationship between two-body and three-body
Figure 4.19a Sim potential energies
of liquid argon (reduced density ranging from 0.45 to 0.75) with different system size
(ó108, ø200, æ256, ç500 and ò864 atoms)
0.9
0 100 200 300 400 500 600 700 800 9000.3
0.4
0.5
0.6
0.7
0.8
atom numbe
λ
r
igure 4.19b Coefficient values as a function of atom numbers of liquid argon (reduced
ensity ranging from 0.45 to 0.75)
F
d
96
0.4
0.8
1.0 1.1 1.2 1.3 1.4
-3.2
-2.8
-2.4
-2.0
-1.6
-1.2
-0.8
-0.4
0.0
ρ∗
E* 3/(
E* 2α∗ )
Figure 4.20a Simple relationship between two-body and three-body potential energies
of liquid argon (reduced density ranging from 1.1 to 1.4) with different system size
(ó108, æ256, ç500 and ò864 atoms)
100 200 300 400 500 600 700 800 9003.5
4.0
4.5
5.0
5.5
6.0
6.5
atom number
λa
λb
λ
Figure 4.20b Coefficient values as a function of atom numbers of solid argon (reduced
density ranging from 1.1 to 1.3)
97
Temperature Effects on the Relationship between Two-body and Three-body Potential Energies
According to the above analysis, it can be observed again that a system size of 500
atoms can provide a very good compromise between system size effects and the large
computational cost in studying three-body potential. Therefore, temperature effects on
the relationship between two-body and three-body potential energies were investigated
with the 500 atom systems.
The ratio of three-body and two-body energies calculated from simulations at different
reduced densities for 500 atoms of argon, krypton and xenon is presented in Figure
4.21. It can be seen that the curves of argon, krypton and xenon almost overlap, which
means that the simple relationship between three-body and two-body interactions holds
for the three different systems. So the following discussion we take argon as an
example.
otential ener
F er
ensities
The ratio of three-body and two-body p gies of argon as a function of
reduced number density at different temperatures is shown in igure 4.22. At low
d ( * 0.8ρ < ), the ratio is not much affected at low to medium temperatures.
ion is observed for high super critical temperature
). Due to the transition from vapour to liquid when the temperatures are below
e critical point, the ratio curves have an “S” shape. In contrast, when the temperature
is above the critical point, the ratio change with density is almost linear.
To investigate a relationship between two-body and three-body potential energies of
pure fluids at different temperatures, the validity of the Eq. (4.6) :
However, a significant deviat
( 2.0T ∗ =
th
**3
* *2
EE
λρα
= −
with 0.85λ = at different temperatures is examined in Figure 4.23. It is apparent that
the relationship remains valid at low and moderate temperatures for pure fluids.
98
To investigate a relationship between two-body and three-body potential energies of
pure solids at different temperatures, the validity of the Eq.(4.11):
*
23* *2
a bE
Eλ ρ λ ρ
α∗ ∗= −
with 5.5aλ = and 6.0bλ = is examined in Figure 4.24. It can be observed that this is a
reasonable approximation for pure solid at different temperatures.
According to the above discussion, the relationship between two-body and three-body
potential energy for pure fluids and solids has been found, which will benefit the further
study of three-body effects on the phase behaviour of pure fluids and solids.
99
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-3.5
0.0
-2.5
-2.0
-1.5
-1.0
-0.5
T*=0.9914 Ar T*=1.0000 Kr T*=0.9252 Xe
E* 3/(E* 2α
∗ )
-3.0
ρ∗
Figure 4.21 the ratio of three-body and two-body energies calculated from simulations
at different reduced densities for argon ( 0.9914T ∗ = (æ)), krypton ( 1.0T ∗ = (ç))and
xenon ( 0.9252T ∗ = (ò)).
-4.5
-4.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-5.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
E* 3/(E
* 2α)
ρ*
Figure 4.22 the ratio of three-body and two-body energies of argon at different
temperatures. Results are shown for both subcritical temperatures ( 0.9T ∗ = (ò),
0.9914T ∗ = (à)) and supercritical temperatures ( 1.2678T ∗ = (æ), 1.4168T ∗ = (á),
2.0T ∗ = (ç)).
100
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.7
-0.6
-0.5
-0.3
-0.1
-0.2
-0.4
E* 3/(E
* 2α)
f pure fluids. Results are shown for both subcritical temperatures ( ò),
1
ρ*
Figure 4.23 a simple relationship between two-body and three-body potential energies
0.9T ∗ = (o
0.9914T ∗ = (à)) and supercritical temperatures ( 1.2678T ∗ = (æ), 1.4168T ∗ = (á)).
2
-3
1.0 1.1 1.2 1.3 1.4-5
-4
-2
-1
0
E* 3/(E
* 2α)
ρ*
Figure 4.24 a simple relationship between two-body and three-body potential energies
of pure solids Results are shown for both subcritical temperatures ( 0.9T ∗ = (ò),
0.9914T ∗ = (à)) and supercritical temperatures ( 1.2678T ∗ = (æ), 1.4168T ∗ = (á)).
101
4.3 Three-body effects on vapour-liquid equilibrium
properties of pure fluids In this part the three-body effects on vapour-liquid phase coexistence properties of
argon, krypton and xenon are investigated using the Gibbs-Duhem integration Monte
Carlo simulation. For the true two-body potential we used Barker-Fisher-watts potential
proposed by Barker et al (Barker et al. 1971; Barker et al. 1974) and for the three-body
teractions, we took consideration of the third-order , and DDD , DDQ DQQ , QQQin
the fourth-order DDD terms (see Section 2.1.4). The simple relationship between two-
body and three-body potential energies (Marcelli and Sadus 2000) has also been tested
by applying it to the simulation of vapour-liquid phase properties of argon.
Simulation details The Gibbs-Duhem integration Monte Carlo simulations were performed to study
vapour-liquid phase equilibria properties of argon, krypton and xenon. An adequate
system size of 512 atoms was used. The initial points for the Gibbs-Duhem integrations
are from the Gibbs ensemble Monte Carlo simulations (Marcelli and Sadus 1999).
Starting form this initial point, the simulations were performed with increasing
temperatures. The temperature change between states was calculated by 1/ *Tβ =
and 0.05β∆ = − .
The simulations were performed in cycles, and the total 512 atoms were partitioned
between two boxes to simulate two different phases. For the simulations using only
BFW potential, the equilibration period was 10,000 cycles and a further 10,000 cycles
was used to accumulate the averages. for the simulations through two-body with three-
body ( 4BFW DDD DDQ DQQ QQQ DDD+ + + + + ) potential or two-body with
Axilrod-Teller term only ( BFW DDD+ ) , considering computing time we chose 1500
ycles for the equilibration period and a further 1500 cycles to accumulate the averages. c
Each cycle is a standard NPT Monte Carlo simulation for each phase that involves
randomly chosen 512 attempted displacements or an attempted volume change.
102
The standard periodic boundary conditions were applied. The calculations of BFW
potential were truncated at intermolecular separations greater than half the box length,
and appropriate long-range corrections were used to obtain the full contribution of pair
teractions to energy and pressure. The simulations for the three-body interactions were
uncated at intermolecular separations greater than a quarter of box length according to
e previous work (Marcelli and Sadus 1999).
esults and discussion
he results of Gibbs-Duhem integration Monte Carlo simulations for vapour-liquid
phase equilibrium properties of argon, krypton and xenon are reported in Tables 4.3-
4.10 (Appendix). The uncertainties in the ensemble averages for density and energy
reported in the tables were calculated by dividing the post-equilibrium results into ten
blocks. The estimated errors represent the standard deviations of the section averages.
An error estimate for the molar enthalpy difference between ten phases cannot be
estimated in this way because it is the average of the entire post-equilibrium simulation.
The phase equilibrium properties obtained from argon using the BFW potential are
summarized in Table 4.3 (Appendix), for the
in
tr
th
RT
( 4BFW DDD DDQ DQQ QQQ DDD+ + + + +
the potential ( BFW DDD+ ) simulations in Table 4.7 (A
) calculations in Table 4.6 (Appendix),
ppendix), and the results using
the simple relationship between two-body and three-body potential energies are listed in
Table 4.8.
4.30. The Figure 4.28 gives a comparison of
simulation results of argon for (
Comparisons of simulation results with experiments of argon, krypton and xenon are
given in Figure 4.25, 4.26, 4.27, 4.29 and
4BFW DDD DDQ DQQ QQQ DDD+ + + + + ),
) and for the simple relationship between two-body and three-body
potential energies.
The simulation results for the vapour-liquid phase equilibrium envelope of argon and
those obtained by the Gibbs ensemble Monte Carlo simulations using BFW potential
( BFW DDD+
103
with and without ( ) included are compared with
the experimental data in Figure 4.25. It is clear that only using the BFW potential can
estimate the vapour branch of the coexistence curve properly but it can not predict the
liquid phase coexistence density of argon adequately. This contrasts with calculations
using the Lennard-Jones potential (Chapter 3) which generally yields a good agreement
with experiment for liquid densities. The main reason is that Lennard-Jones potential is
an effective potential which includes many-body interactions. The comparison with
experiment of argon in figure 4.25 indicates that the genuine two-body potential BFW
cannot predict the liquid phase densities of argon properly. It also shows that the
addition of the three-body terms to the BFW potential results in a good agreement of
theory with experimental data for both vapour and liquid branches.
The contributions of the various multipole terms to the three-body interactions of argon
are listed in Table 4.6 (Appendix). The three-body effect on the vapour phase is quite
the configuration potential energy and pressure of the liquid
phase of argon are illustrated in Figure 4.26 and 4.27, respectively. It can be seen that
. The oth d-order multipole terms ( and )
contributions are relatively small and their sums (
4DDD DDQ DQQ QQQ DDD+ + + +
small but it is very apparent on the liquid phase. Comparisons of the contributions of the
various three-body terms to
the triple-dipole term ( DDD ) makes the dominant contribution to the three-body
interactions er thir DDQ , DQQ QQQ
DDQ DQQ QQQu u u+ + ) or
( DDQ DQQ QQQP P P+ +
term ( 4DDD
) are similar to the contributions from the fourth-order triple-dipole
) but with the opposite sign. So they largely cancel. Therefore, the
Axilrod-Teller term alone can be a very good approximation of three-body dispersion
interactions, which is consistent with earlier work.
A comparison of simulations obtained using the potential ( ), the potential BFW DDD+
( 4BFW DDD DDQ DQQ QQQ DDD+ + + + + ) and using the simple relationship
between two-body and three-body interactions is showed in the Figure 4.28. It proves
again that the Axilrod-Teller (DDD) alone is an excellent approximation of three-body
dispersion interaction. The Figure 4.28 also indicates that the simple relationship
between two-body and three-body interactions is a good alternative to simulation three-
body effects without further computing consumptions.
104
We applied the simple relationship between two-body and three-body potential energies
to the systems of krypton and xenon. The simulation results of krypton and xenon are
listed in Tables 4.9 and 4.10 (Appendix) espectively. Comparisons of simulation
. From these two figures, it can be seen that only the BFW potential can not
otential
130
140
r
results with experiment for krypton and xenon are illustrated in Figure 4.29 and 4.30
respectively
predict the liquid phase density adequately but can estimate the vapour phase properly.
They also indicate that the addition of the three-body interaction to the BFW p
can improve considerably the agreement between theory and experiment for krypton
and xenon. The situations are the same for argon.
150
160
0 200 400 600 800 1000 1200 140090
100
110
120
K)
ρ (kg/m3)
lation usi
T(
Figure 4.25 Comparison of experiment (æ) with calcu ng BFW potential
(Gibbs (à), Gibbs-Duhem(á)) and BFW+AT (Gibbs (ó), Gibbs-Duhem(ò)) for the
vapour-liquid coexistence density of argon.
105
0.16
0.18
0.20
0.0
0.10
0.12
0.14
8
0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
-0.04
-0.02
0.00
0.02
0.04
0.06
E*
ρ∗
DDD
DDQ
DQQ
DDD4QQQ
igure 4.26 Comparison of the contribution of the various three-body terms to the
onfigurational energy of the liquid phase of argon
0.15
0.20
0.25
0.30
0.35
0.40
F
c
0.05
0.10
0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80-0.10
-0.05
0.00
P*
ρ*
DDD
DDQ
DQQ
QQQ
DDD4
Figure 4.27 Comparison of the contribution of the various three-body terms to the
pressure of the liquid phase of argon
106
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
T*
ρ∗
Figure 4.28 Comparison of the calculated vapour-liquid phase coexistence density of
argon using the ( BFW D+ 4DD DDQ DQQ QQQ DDD+ + + + ) potentials (ò), the
( BFW DDD+ ) potentials (æ) and the simple relationship between two-body and three-
body potential energy ( 23 6
23v EE ρεσ
= − ) (ç), respectively.
107
190
200
180
150
160
170
0 500 1000 1500 2000 2500140
210
220
T(K)
ρ (kg/m3)
Figure 4.29 Comparison of experiment (æ) with calculation using BFW potential
(Gibbs (à), Gibbs-Duhem(á)) and BFW+AT (Gibbs (ó), Gibbs-Duhem(ò)) for the
vapour-liquid coexistence density of krypton.
300
320
0 500 1000 1500 2000 2500 3000
200
220
240
280
260
K)T(
ρ (kg/m3)
Figure 4.30 Comparison of experiment (æ) with calculation using BFW potential
(Gibbs (à), Gibbs-Duhem(á)) and BFW+AT (Gibbs (ó), Gibbs-Duhem(ò)) for the
vapour-liquid coexistence density of xenon.
108
4.4 Three-body effects on the vapour-liquid
equilibrium properties of mixtures In this part the effects of three-body interactions on the vapour-liquid phase equilibrium
properties of argon-krypton system at different temperatures are explored by means of
the Gibbs-Duhem integration Monte Carlo simulation method. We find that using
Barker-Fisher-Watts (BFW) potential alone is neither sufficient nor efficient to simulate
the vapour-liquid phase equilibria properties of argon-krypton mixtures. But the
combination of BFW true two-body interaction with the standard Axilrod-Teller triple-
ipole potential leads to the excellent prediction of the experimental coexistence curve.
imulation details
he Gibbs ensemble Monte Carlo simulations are performed for a system of 500 atoms.
were performed in cycles consisting typically of 512 attempted
d
The simple relationship between two-body and three-body potential energies of pure
noble gases has been extended to argon-krypton mixtures. Calculations using this
simple relationship indicate that it is valid for mixtures as well.
SIn this work, a convenient choice for the initial coexistence condition for argon-krypton
mixtures is the vapour-liquid equilibria condition for either of the pure components. We
take pure krypton as a starting point. The vapour-liquid coexistence data for single
component krypton have been obtained via the Gibbs ensemble Monte Carlo
simulations. We tried six different temperatures with and without three-body
interactions.
T
1500 cycles were used for equilibration and a further 1500 cycles were used to calculate
ensemble averages. In each cycle a standard NPT Monte Carlo was carried out, which
includes 500 attempted displacements, an attempted volume change and 500 atom
exchange attempts. It should be noted that we referred to the experimental pressure of
krypton due to the large fluctuations in the pressure.
Due to the similarity of argon and krypton atoms we chose semigrand ensemble
simulations which were implemented for a system of 512 atoms. The ratio of fugacity
and Henry constant for initial conditions is obtained by performing NPT Monte Carlo
simulations, which
109
displacements, an attempted volume change and 512 attempts “fake” identity exchange.
1 and 4.12
ppendix).
10000 cycles were used for equilibration and a further 10000 cycles was used to
accumulate ensemble averages. The initial conditions at different temperatures using the
potential BFW with and without three-body potential are listed in Table 4.1
(A
The Gibbs-Duhem integration Monte Carlo simulations were as follows:
Starting from the initial point, 20 simulations were performed according to Calperyon
equations. The step size is 0.05ξ∆ = . In each step, there are 10000 cycles, including
8000 cycles for equilibration and a further 2000 cycles to accumulate ensemble
verages. NPT Monte Carlo simulations were performed to give the information
required for the predictor-corrector algorithm. Each cycle involved 512 attempted
displacements, or an attempted volume change or 512 attempts identity exchange. These
three different moves were chosen at random.
Other details of the simulations are similar to the ones of pure noble gases using the
BFW potential with and without three-body interactions.
The normal conventions for reduced units and mixing rules were applied (see Chapter
2).
Results and discussion
In this section, we present the results of the Gibbs-Duhem integration calculations of the
vapour-liquid phase behaviour for argon-krypton mixtures using the BFW potential
We chose argon-krypton mixture systems to study the role of the three-body interaction
d their accurate intermolecular
potentials are available.
a
with and without 3-body interactions included.
in the vapour-liquid phase equilibrium properties of mixture. The main reason is that the
systems provide a rare example of a binary mixture, an
110
The results of the Gibbs-Duhem integration simulations at different temperatures using
the BFW potential with and without three-body interactions are reported in Tables 4.13-
4.23 (Appendix), where the reduced units are based on the BFW potential parameter
values of krypton.
In the above tables, denotes the mole fraction of argon in the liquid (L) and vapour
(V) phases. The uncertainties in ensemble averages for pressure, density, temperature,
mole fraction reported in the tables were calculated by dividing the post-equilibrium
results into ten blocks. The total pressure is the sum of both kinetic and configurational
terms. The estimated errors represent the standard deviations of the section averages.
Comparisons of simulations with experiments for the pressure-composition behaviour
of argon-krypton at different temperatures are illustrated in Figure 4.31-4.36. It can be
seen that the liquid branch curve is almost a straight line over a wide range of
phase branches have a large deviation from experimental data. Inclusion of the three-
ment deteriorates when close to
e binary critical point, which is caused by the limitation of simulation techniques that
there will be large fluctuations due to the small size simulation box.
The above observation is also confirmed by the recent work of Nasrabad et al.
(Nasrabad et al. 2004). They use a different but accurate ab initio pair potential with
and without the Axilrod-Teller three-body corrections. Like our work, their simulations
without the Axilrod-Teller term can predict the trend of the phase behaviour of argon-
krypton system, but the phase composition does not agree well with the experimental
data. After adding the three-body corrections, the calculations can give a reasonable
agreement with experimental data. Three-body effects on argon-krypton mixtures at
temperatures 117.38K, 163.15K and 158.15K have been investigated by Nasrabad et al.
and us. Our work of using BFW two-body potential only is not as good as their results
using ab initio only. But the BFW+AT term (DDD) gives a better match to experiment
than the ab initio + three-body interactions (DDD). The ab initio with the three-body
potential inclusion overestimates the pressure at most compositions of vapour branch.
x
compositions. It is obvious that the simulations using only the BFW potential can not
predict the phase behaviour of argon-krypton mixtures, and both vapour and liquid
body AT term (DDD) leads to a reasonable agreement with experiments at low and
moderate pressure. However, at high pressures the agree
th
111
Nasrabad et al. also performed the simulation at a temperature of 193.15K, where the
calculations using only ab initio potential show large deviations from experiment for
both vapour and liquid branches. The addition of three-body interactions improves the
agreement but still underestimates the pressure at most compositions of liquid branch.
We did not perform the simulation for the temperature of 193.15K. However, we have
investigated three-body effects on the phase equilibria of argon-krypton mixtures at the
other three different temperatures of 148.15K, 143.15K and 153.15K, where the
inclusion of three-body interactions gives a good match to experiment for both vapour
and liquid branches.
It should be mentioned that Marcelli and Sadus (Marcelli and Sadus 2001) investigated
three-body interactions on the phase equilibria of argon-krypton mixtures only at one
temperature 163.15K, which showed that the composition of coexisting vapour and
liquid phases is relatively unaffected by three-body interactions. This result was due to
hort simulation runs (total 3000 steps including 1500 steps for averaging calculation) s
and small system sizes (500 atoms for two phases).
112
60
70
80
90
0 10 20 30 40 50 60 70 80 90
20
30
40
50
experiment 2+3 body simulation 2 body simualtion
atm
T=177.38K
P/
xAr%
Figure 4.31 Isothermal vapour-liquid phase diagram of the system argon + krypton at
177.38K.
70
0 10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
experiment 2+3 body simulation 2 body simualtion
P/at
m
xAr%
T=163.15K
Figure 4.32 Isothermal vapour-liquid phase diagram of the system argon + krypton at
63.15K. 1
113
30
40
50
60
0 10 20 30 40 50 60 70 80 90 1000
10
experiment 2+3 body simulation 2 body simualtion
P/a
tm
T=158.15
20
xAr%
Figure 4.33 Isothermal vapour-liquid phase diagram of the system argon + krypton at
158.15K.
60
0 10 20 30 40 50 60 70 80 90 100 1100
10
20
30
40
50 experiment 2+3 body simulation 2 body simulation
P/a
tm
xAr%
T=153.15K
al vapour-liquid phase diagram of the system argon + krypton at Figure 4.34 Isotherm
153.15K.
114
0 10 20 30 40 50 60 70 80 90 100 1100
5
10
15
20
25
30
35
40
45
50
experiment 2+3 body simulation 2 body simualtion
T=148.15K
P/a
tm
xAr%
Figure 4.35 Isothermal vapour-liquid phase diagram of the system argon + krypton at
148.15K.
0 10 20 30 40 50 60 70 80 90 100 1100
experiment 2+3 body simulation
40
T=143.15K35
10
15
20
30 2
25
5
body simualtion
P/at
m
igure 4.36 Isothermal vapour-liquid phase diagram of the system argon + krypton at
1
xAr%
F
43.15K.
115
Simulating Phase Equilibria of Mixture Using A Simple
and Three-body Potentials
is well known that the two-body calculation is a relatively routine but the addition of
the three-body calculation is computationally expens
the Section 4.2 we have demonstrated that the simple relationship between two-body
and three-body potential energies (Marcelli and Sadus 2000) could be used to predict
e vapour-liquid phase equilibrium properties of argon, krypton and xenon efficiently.
In this part we will extend this simple relationship to apply to the prediction of vapour-
liquid phase equilibrium properti
Relationship between Two-body It
ive.
In
th
es of mixtures. The relationship may be expressed as:
( )2 23 2 1 111 1 2 112 1 2 221 2 2223
E x x x x x xρ ν ν ν ν∗ ∗ ∗ ∗ ∗ ∗ ∗= − + + + 2E (4.12)
where
ix is molar fraction of component is three-body potential coefficient for
the three different components . The above equation is based on the implicit
assumption that the radial distribution funct identical,
and they both explicitly contain a contribution from interactions between dissimilar
molecules.
To test the accuracy of this relationship, we performed Gibbs-Duhem integration Monte
arlo simulations for argon-krypton mixtures at
i , ijkv
, ,i j k
ions of the component species are
C 163.15T K= and . In the
simulation the contribution of three-body interactions is determined by the Eq. (4.12)
and all the other simulation details are the same as the previous description of mixtures.
omparisons of experiment with simulations for the BFW with the three-body potential
and for the simple relationship between two-body and three-body potential energies are
illustrated in Figure 4.37 and 4.38. It can be seen that the calculations using the Eq.
(4.12) are almost the same as the results obtained from the full two-body and three-body
calculations. The comparisons indicate that Eq. (4.12) is an accurate estimate of three-
y contribution for mixtures from two-body calculation alone, which will
ecrease the computing time efficiently.
148.15T K=
C
body energ
d
116
It should be noted that the validity of this simple relationship should be the same as the
o s within the vapour-liquid phase
equilibrium area.
ne for pure fluids. It is only effective for the system
117
10
20
30
40
50
60
0 20 40 60 80
experiment 2+3 body simulation simple relationship
P(at
m)
x Ar%
p f ou p g t tem
20
35
he sysFigure 4.37 Com arisons o isothermal vap r-liquid hase dia ram of
argon + krypton at 163.15K
0 20 40 60 80 100
5
10
15
25
30
40
45
50
experiment 2+3 body simulation simple relationship
P(a
x Ar%
F .3 parisons of isotherm u p ag th tem
argon + krypton at 148.15K
tm)
igure 4 8 Com al vapo r-liquid hase di ram of e sys
118
4.5 Three-body Effects on Solid-liquid Equilibrium
Properties of Noble Gases
Non-equilibrium sim
technique in studying the tra ph a id s orr 0).
T nal MD in ne e l ha is r w the
N tec c ed ta NV ui m ar ics
(M de e t d- ha is t e ium m has
been discussed in detail in Chapter 2.
Hanson and Verlet (Hansen and Verlet 1969) reported the solid-liquid phase
equilibrium properties of the Lennard-Jones potential system using a thermodynamic
in on d f irs W de ent of comp d s ion
te es av b di nt bo te (K and
A 1 nd ul sal a 7 t a 3b)
m . In or iv a to bo cts so uid
phase equilibrium pr s o , an on te se are
a real two-body potential (Barker-Fisher-watts potential) with or without the Axilrod-
eller three-body interactions included. The results are compared with the experimental
ata (Horton 1976).
imulation details
Simulation has been performed by both NEMD and MD. The only difference is that
n we ed tion M she wa s 0 the
wing e onl etai e NE imul
simulations were performed plyin stand LO tion otion
lanar te fl e SL equ for a omp atom wing
stream eloc in t irec d co a
molecular dynamics (NEMD) ulation has proven to be a powerful
nsport enomen of liqu s (Evan and M iss 199
raditio ly, NE is ma ly confi d to th iquid p se. In th chapte e use
EMD hnique ombin with s ndard ( T) eq librium olecul dynam
D) to termin he soli liquid p se coex tence a quilibr . This ethod
tegrati metho or the f t time. ith the velopm uter an imulat
chniqu they h e also een stu ed rece ly by th Mon Carlo ofke
grawal 995) a molec ar dynamics (Li and V cek 199 ; Ge e l. 200
ethods this w k we g e more ttention three- dy effe on the lid-liq
opertie f argon krypton d xen . The po ntials u d here
T
d
S
whe perform simula s using D, the ar rate s set a . So in
follo we giv y the d ls of th MD s ations.
The by ap g the ard SL D equa s of m
for p Couet ow. Th LOD ations one-c onent ic flo
with ing v ity v x he x -d tion an nstant str in rate xdν wer=γ& e
ied. T atio mot ere ted order Gear predictor-
dy
appl he equ ns of ion w integra by a 4th
119
*t t 2/ mεcorrector s with a reduced (cheme integration time step σ= ) of 0.0 non-
librium lation trajector ypic n fo 00 teps first
00 tim of jec e us ake e sy eaches a steady
. A f 200 e s ere o ca ave lues very
0 time the r ere d o
inal re
ults a cus
e the soli d ph uili prop n
ied(La Ev 196 loss Fer 991 o a gram
; Boe . 2 ecau estig of s ubst ike ases
provid tter u anding of the f g an g p es. g of
heavie ga h as on a non mo orta these
ents f rysta ere m e are not . So f the
ious in tion one olec mula chn nd f the
lation sed nna es p l (T d T 78). little
has ne in -bod cts o -liq se rium
erties e ga eci r the r me Ba Kl arker
Klein hav ied three-body effects on solid-liquid phase equilibrium
roperties of argon, but they only calculated the properties using the BFW potential
ith the AT three-body corrections, and they did not perform calculations using the
only and using BFW plus three-body interactions. Furthermore, their simulations were
ement nly on Ac to iou is ork
e begi of th ter si cts a ap he tom
er is an 5 stu hre effe ha vio oble
espe or d luid lid gon,
ton an sy e in to p mor te c ons
e-bod s on iqu e eq m of argon, krypton and xenon have
liste abl -4.2 pen spe Th lts een
01. A
equi simu y is t ally ru r 250,0 time s . The
50,0 e steps each tra tory ar ed to m sure th stem r
state urther ,000 tim teps w used t lculate rage va and e
5,00 steps esults w printe ut. The last printing out result is regarded as
the f sult.
Res nd dis sion Sinc 1960s d-liqui ase eq brium erties of noble gases have bee
stud hr and ersole 2; Sch er and rante 1 ; Nard nd Bil
hler et al1995 001) b se inv ations imple s ances l noble g
can e a be nderst reezin d meltin roperti Studyin
the r noble ses suc krypt nd xe is even re imp nt as
elem orm c ls wh quantu ffects hardly iceable me o
prev vestiga was d by m ular si tion te iques a most o
simu work u the Le rd-Jon otentia sang an ang 19 Very
work been do to exam e three y effe n solid uid pha equilib
prop of nobl ses, esp ally fo heavie mbers. rker and ein (B
and 1973) e stud
p
w
BFW potential only. So there was no comparison between the properties using BFW
impl ed for o 108 arg atoms. cording the prev s analys of our w
at th nning is chap , system ze effe re very parent w n the a
numb less th 00 for dying t e-body cts on p se beha ur of n
gases cially f enser f s or so s. In our work we use 500 atoms as ar
kryp d xenon stem siz order rovide e accura alculati .
Thre y effect solid-l id phas uilibriu
been d in T es 4.25 7 (Ap dix) re ctively. e resu have b
120
compared per dat on nd son se
ities a sure ustr Fig 9-4 ec
the r nd we e th hree nte alm not
t the l hase y, b can cant ge ity olid
e. Th t is eem e a of dy on hase
viour e fl d s th h
tial i ons ate se n o ste pu d to
. The re g urs am de ith out ody
action
ng ar an le, dy tion app gi ood
ment liqu e d art at h pe The ent
s goo t tures near the triple point ood ent for liquid
e prop tra h th qua eme e has h of
apour qu showed in F .39 lid den sing
pote y d re xpe . Th on ree term
oves t me di cy per ill pa ly at
r temp n rip . Th con the tio g LJ
tial w e v d a nt peri or k and , the
ituation is similar.
he above observation suggests that the BFW potential including Axilrod-Teller term
ed to be considered.
arin ula t res (Figur .4 b that
o-b tera o t ur ad f t ody
ive r p r ri s e at ree-
effe pr n . ar an
le, te r 1 th o e in ions
ses ssu .
with ex imental a (Hort 1976) a compari of pha equilibrium
dens nd pres s are ill ated in ures 4.3 .44, resp tively.
From esults a figures can se at the t -body i ractions ost do
affec iquid p densit ut they signifi ly chan the dens of the s
phas is resul in agr ent th nalysis three-bo effects the p
beha of pur uids an olids at e beginning of this chapter. T e two-body
poten nteracti domin the pha transitio f the sy m from re liqui
solid pressu ap occ at the s e liquid nsity w or with three-b
inter s.
Taki gon as examp two-bo calcula only ears to ve a g
agree for the id phas ensity, p icularly igh tem ratures. agreem
is les d at low empera . The g agreem
phas erty con sts wit e inade te agre nt for th liquid p e branc
the v -liquid e ilibrium igure 4 . The so phase sities u
BFW ntial onl o not ag e with e riment e additi of the th -body
impr he agree nt but a screpan with ex iment st remains rticular
lowe eratures ear the t le point is is in trast to calcula ns usin
poten hich giv ery goo greeme with ex ment. F rypton xenon
s
T
does not adequately describe the solid-liquid coexistence density. Other effects such as
the other three-body or higher body interactions may ne
Comp g calc ted and experimen al pressu e 4.42-4 4), we o serve
the tw ody in ctions al ne underestimate he press e. The dition o hree-b
term g s a nea erfect ag eement with expe ment. It hould b noted th the th
body ct on essure at high temperature is of sig ificance Using gon as
examp at the mperatu e of 20 .32 K, e additi n of thr e-body teract
increa the pre re by 71 49 GPa.
121
220
80
100
120
140
160
180
200 2 bo dy 2+3 body experiment
T(K)
1400 1500 1600 1700 1800 1900
ρ (kg/m3)
Figure 4. pa id o e es n ated
MD D x t
0
200
00
50
00
39 com rison of solid-liqu phase c existenc densiti of argo calcul
by NE and M simulations with e perimen al data.
4
2400 2600 2800 3000 3200 3400
100
experiment3 2 body 2+3 body
3
250)(K
15
T
(kg/
Figure 4.40 comparison of solid-liquid phase coexistence densities of krypton calculated
by NEMD and MD simulations with experimental data.
m3)ρ
122
3000 3200 3400 3600 3800 4000
160
200
2
280
320
400
experiment360 2 body
3 bodyT(
K)
40
ρ (kg/m
Figure 4.41 comparison of solid-liquid phase coexistence dens f e ed
by NEMD and MD simulations with experimental data.
120 14 0 2 220
100
2
300
400
500
700
3)
ities o x non calculat
2 body600 2+3 body experimen
00
0
80 100 0 16 180 00
t
P(G
Pa)
T(K)
Figure 4.42 Com i s d-liqu ph e x ssu s of argon calcul d
by NEMD and MD simulations with experimental data.
par son of oli id as coe istence pre re ate
123
30 50
0
40
800
10
1200
experiment 2 body 2+
00
600
200
0
100 150 200 250 0 3
3 bodyP(
GPa
)
T(K)
Figure . o a o o solid- qui p
calculated by NEMD and MD simulations with experimental data.
4 43 C mp ris n f li d hase coexistence pressures of krypton
150 200 250 300 350 400-100
0
100
200
300
400
500
600
700
800
experiment 2 body 2+3 body
4.4 pa s s of xenon calculated
MD D o x t
P(G
Pa)
T(K)
Figure 4 Com rison of olid-liquid phase coexistence pressure
by NE and M simulati ns with e perimen al data.
124
Chap
bo cts h f h ide of
y, atu p e g ork. Simple
nsh twe o h l of pure fluids, solids
ix hav . i e r o arlo
s on the vapour-liquid
hase equilibrium of pure fluids and mixtures. We also use non-equilibrium molecular
ody interactions play an important role in the phase behaviour of noble
gases. To the best of our knowledge, this is the first time of investigating the three-body
on ho e se ds also
th exa f s o o cts s uid
properties.
er y t y o a i o s, wo-
ote od e e Fis atts
ial. om o l dy ial the
-liquid phase equilibrium properties
of argon, krypton and xenon have been studied using the Lennard-Jones potential by
means of the Gibbs-Duhem ra onte Ca thod. The results indicate that
only Lennard-Jones potential can be used to simulate the phase behaviour of noble
gases successfully because the Lennard-Jones pair potential not only reflects the
con on also effectively includes contributions
from y-b te n
ter 5
Conclusions and Recommendations
Three- dy effe on the phase be aviour o noble gases wit in a w range
densit temper re and ressure have be n investi ated in this w
relatio ips be en two-b dy and t ree-body potentia energies
and m tures e been presented The G bbs-Duh m integ ation M nte C
simulation programs have been written to study three-body effect
p
dynamics combined with molecular dynamics to explore the three-body effects on the
solid-liquid phase equilibrium properties of noble gases. All the calculations indicate
that three-b
effects the w le phase state prop rties ranging from pure ga s to soli , and
one of e rare mples o the inve tigation f three-b dy effe on the olid-liq
phase equilibrium
In ord to stud hree-bod effects n the ph se behav our of n ble gase true t
body p ntial m els have to be used. In this work w used th Barker- her-W
potent To c pare the calculati ns using this rea two-bo potent with
effective pair potential, the vapour-liquid and solid
integ tion M rlo me
tributi from two-body interactions, but
man ody in ractio s.
125
NVT molecular dynamics simulations have been performed to investigate three-body
from vapour to solid phases. It
has been found that system s ze affects the three-body effects greatly when the system is
les 500 s. The ratio of two-body and three-body potential energy at different
tem ures so be plore results indicate that for pure fluids and solids
tem ure d ot affect the rati h, however the phase transi vapour-
liquid and solid-liquid) happen, the reatly af ected by temperature. Simple
relationships between two-body and three-body potential energy for pure fluids and
olids have been determined, respectively. The validity of the relationship equation
eeds to be tested in future work, which will be significant because calculations
cluding three-body interactions are very expensive, especially for the study of
roperty of solids.
hree-body effects on the vapour-liquid phase equilibrium properties of argon, krypton
nd xenon have been studied by the Gibbs-Duhem integration Monte Carlo simulation.
he results are compared with experimental data and the other work (Marcelli and
adus 1999). Comparison shows that three-body interactions do not affect the vapour
hase coexisting density much. However, the BFW potential can not predict the liquid
hase coexisting density accurately. Inclusion of three-body corrections leads to a good
greement with experimental data. Marcelli and Sadus have compared their calculation
ith the simulation using the Aziz-Slaman potential (Anta et al. 1997), which shows
at Aziz-Slaman potential is very similar to the BFW potential. Both of them can not
redict the liquid phase density adequately.
tion has been performed to study
three-body effects on the vapour-liquid phase equilibrium properties of argon-krypton
mixtures. The calculation indicates that the BFW potential can only predict the trend of
th e u o n bu ha positions have a big
de n pe l cl t - rrections can give a
go ee it im a e he o ra .
20 in ti o th r r ur tio s a
be red of u si re ip
be tw a e e g p id eq m
effects on the phase behaviour of noble gases ranging
i
s than atom
perat has al en ex d. The
perat oes n o muc when tions (
ratio is g f
s
n
in
p
T
a
T
S
p
p
a
w
th
p
The Gibbs-Duhem integration Monte Carlo simula
e phas behavio r of arg n-krypto system t the p se com
viatio with ex rimenta data. In usion of he three body co
od agr ment w h exper ental dat . Compar d with t other w rk (Nas bad et al
04) us g ab ini o pair p tential wi the Axil od-Telle term, o calcula n give
tter p iction the liq id phase coexisting compo tions. A simple lationsh
tween o-body nd thre -body pot ntial ener y for va our-liqu phase uilibriu
126
ha e r e rg t m tes is
si ela a x
Three-body effects on the solid-liquid phase equilibrium properties of argon, krypton
and xenon have also been investigated by non equilibrium molecular dynamics
combined with molecular dynamics simulation. The results indicate that three-body
interactions nearly do not affect the liquid phase coexisting density but do affect the
solid branch density. The BFW potential alone can not be used to simulate the
coexisting pressure accurately and inclusion three-body corrections can lead to a good
agreement with experimental data. Compared with previous other work (Barker and
Klein 1973), our calculation should provide a more accurate result because they only
use a 108 atom system size that is not large enough to study three-body interactions
according to the analysis of system size effects in our work.
It should be noted that in this work we did not study three-body short range repulsive
interactions. Our calculations indicate that just inclusion of Axilrod-Teller term can lead
a gr w ri a o
work (Bukowski and Szalewicz 2001). However, some other work showed that three-
body exchange interaction has an important role in dense noble-gas solids (Loubeyre
1987, 1988), and three-body exchange interactions may contribute significantly in the
short-distance range (Polymeropoulos et al. 1984). Therefore further investigation of
three-body effects on dense noble gases in this direction is still necessary.
We also found three-body calculations are very expensive. It is known that accounting
for three-body interactions requires normally at least one order of magnitude more
computing time than simple pair calculations (Sadus 2002). So time saving algorithms
have to be developed to speed the calculation and make it more accurate and efficient.
We have studied three-body effects on the vapour-liquid, solid-liquid phase equilibrium
properties of pure fluids and the vapour-liquid phase equilibrium properties of mixtures.
Three-body effects on liquid-liquid phase equilibrium of pure fluids and mixtures, solid-
liquid phase equilibrium properties of mixtures need to be investigated in the future
work.
s been xtended to mixtu es. The t st with a on-kryp on syste indica that th
mple r tionship is also v lid for mi tures.
good a eement ith expe mental d ta, which is also c nfirmed by the very recent
127
Appendix 1:
Forces and long-range corrections for the potentials used in this work
Lennard-Jones potential
12 6
( ) 4u rr rσ σε
⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
(A1.1)
12 6
224r rσ σε
⎡ ⎤⎛ ⎞ ⎛ ⎞= 2 −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
rFr
(A1.2)
9 3
8 13 3lrc
c c
NEr r
πρ σ σ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
(A1.3)
9 3216 2
3 3lrcc c
Pr r
πρ σ σ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
(A1.4)
Barker-Fisher-Watts potential
( ) ( ) ( )5 2
2 62 6
0 0
1 exp 1i ji j
i j
Cu r A x x
xε α
δ+
+= =
⎡ ⎤= − − −⎡ ⎤⎢ ⎥⎣ ⎦ +⎣ ⎦
∑ ∑ (A1.5)
( )( ) ( ) ( )( ) ( )( )
2 55 21 2 6
22 60 0
2 6exp 1 1 1
ji i j
i ji j m
j C xA x i x x
rrxε α α
δ
+− +
+= =
⎡ ⎤+⎢ ⎥= − − − − − +⎢ ⎥+⎣ ⎦∑ ∑ rF (A1.6)
(A1.7)
5 23
2 60 0
2lrc m i ji j
E Nr I Jπρ ε += =
⎡ ⎤= −⎢ ⎥
⎣ ⎦∑ ∑
232
3lrc mP rπρ= − I (A1.8)
128
in Eq.(A1.7) and (A1.8):
( )( )( )( )( )
0 0 2
1 1 3 2
2 2 4 3 2
3 3 5 4 3 2
4 4 6 5 4 3 2
5 5 7 6 5 4 3 2
2
3 3
4 6 4
5 10 10 5
I A DI A D D
I A D D D
I A D D D D
I A D D D D D
I A D D D D D D
=
= −
= − +
= − + −
= − + − +
= − + − + −
22 2
3 23 2 3
4 3 24 2 3 4
5 4 3 25 2 3 4 5
6 5 4 3 26 2 3 4 5 6
7 6 57 2
2 2
3 6 6
4 12 24 24
5 20 60 120 120
6 30 120 360 720 720
7 42 210
D Q R R
D Q R R R
D Q R R R R
D Q R R R R R
D Q R R R R R R
D Q R R R
α α
α α α
α α α α
α α α α α
α α α α α α
α α
⎛ ⎞= − +⎜ ⎟⎝ ⎠⎛ ⎞= − + −⎜ ⎟⎝ ⎠⎛ ⎞= − + − +⎜ ⎟⎝ ⎠⎛ ⎞= − + − + +⎜ ⎟⎝ ⎠⎛ ⎞= − + − + − +⎜ ⎟⎝ ⎠
= − + − 4 3 23 4 5 6
840 2520 5040 5040R R R Rα α α α α
⎛ ⎞+ + + +⎜ ⎟⎝ ⎠7
1 exp 1
c
m
c
m
rRr
rQr
αα
=
⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
3
6 6
8 8 5
10 10 7
1 123 3
1
5
1
7
c
m
c
m
c
m
rr
J C arctg
J Crr
J Crr
πδ δ δ
⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪⎢ ⎥⎜ ⎟⎪ ⎪⎢ ⎥⎝ ⎠= −⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭⎧ ⎫⎪ ⎪⎪ ⎪= ⎨ ⎬
⎛ ⎞⎪ ⎪⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
⎧ ⎫⎪ ⎪⎪ ⎪= ⎨ ⎬
⎛ ⎞⎪ ⎪⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
129
3 5 22 6
2 62 60 0
1 exp 1 3i
jc c ci i jj
i j im m m c
m
Cr r rI A Ir r r r
r
ε α ε
δ
+++
= =
⎧ ⎫⎪ ⎪
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ ⎪= − − − − − −⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎛ ⎞⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎣ ⎦⎪ ⎪+ ⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
∑ ∑ ∑ 5 2
0 0jJ
= =∑
Axilrod-Teller three-body potential
( )( )3
1 cos cos cosDDD i j kDDD
ij ik jk
vu
r r r
θ θ θ+= (A.19)
4 3 3 5 5 6 5 6 5 2 3 5
2 5 3 4 5 4 5 6 3 6 3 4 3 3
3 551 1 18 8 8 8
1 3 3 5 5 68 8 8 8 8 8
ij jkx ikijk
ij ij ik jk ik jk ij jk ij ik ij ik jk
ij ik jk ij ik jk ik ik jk ij ik jk ij ik jk ij ik jk
x rrFr r r r r r r r r r r r r
r r r r r r r r r r r r r r r r r r
ν⎡ −
= − − + + −⎢⎢⎣
⎤− − − − − + ⎥
⎥⎦
(A.20)
Similar expressions exist for the other terms of the triplet potential.
130
Appendix 2: Tables of the Simulation Results Reported in Chapter 3 and 4 Table 3.1 Vapour-liquid coexistence data obtained from the Gibbs-Duhem integration
simulation using the Lennard-Jones potential
T ∗ setP∗ VP∗ Vρ∗ VE∗ LP∗ Lρ∗ LE∗ H ∗∆
0.9500 0.0180 0.01823(3) 0.0220(0) -0.213(1) -0.008(20) 0.725(0) -5.091(2) 5.6708
0.9069 0.0128 0.01289(9) 0.0158(1) -0.157(2) 0.013(18) 0.749(7) -5.29(6) 5.9241
0.8676 0.0092 0.00914(7) 0.0114(1) -0.117(2) 0.009(28) 0.768(7) -5.45(6) 6.0850
0.8315 0.0065 0.00654(5) 0.0084(1) -0.090(1) 0.008(20) 0.785(8) -5.60(6) 6.2584
0.7983 0.0046 0.00467(4) 0.00615(5) -0.068(1) 0.014(14) 0.804(4) -5.76(3) 6.4315
0.7677 0.0033 0.00330(2) 0.00447(3) -0.051(1) 0.008(16) 0.817(5) -5.87(4) 6.5471
0.7393 0.0023 0.00234(2) 0.00326(3) -0.037(8) -0.002(16) 0.825(6) -5.94(5) 6.6454
0.7129 0.0016 0.00166(1) 0.00238(1) -0.0283(6) -0.006(35) 0.840(3) -6.08(2) 6.7143
0.6884 0.0012 0.00117(6) 0.00173(8) -0.0212(5) 0.002(26) 0.851(2) -6.18(2) 6.8056
0.6655 0.0008 0.00082(5) 0.00126(9) -0.0161(4) -0.023(29) 0.854(4) -6.21(3) 6.8744
0.6441 0.0006 0.00058(0)) 0.00091(8) -0.0122(4) 0.013(20) 0.867(5) -6.32(4) 6.9350
0.9500 0.0180 0.01824(3) 0.0220(0) -0.213(1) -0.008(20) 0.725(0) -5.09(2) 5.6708
0.9684 0.0206 0.0206(1) 0.0246(2) -0.237(3) 0.020(23) 0.722(6) -5.07(5) 5.6217
0.9875 0.0236 0.0237(2) 0.0284(3) -0.269(4) 0.020(35) 0.705(7) -4.94(6) 5.4972
1.0074 0.0270 0.0271(2) 0.0323(3) -0.303(4) 0.011(30) 0.691(6) -4.82(4) 5.3463
1.0281 0.0309 0.0310(2) 0.0372(3) -0.343(4) 0.033(17) 0.684(11) -4.77(8) 5.2246
1.0497 0.0354 0.0356(3) 0.0424(8) -0.39(1) 0.045(25) 0.675(11) -4.69(8) 5.0942
1.0722 0.0406 0.0409(7) 0.0493(13) -0.44(1) 0.048(20) 0.661(13) -4.58(10) 4.9082
1.0957 0.0466 0.0470(5) 0.0578(10) -0.51(1) 0.048(14) 0.649(11) -4.49(8) 4.6684
1.1203 0.0536 0.0537(10) 0.0666(25) -0.58(2) 0.062(20) 0.633(16) -4.37(11) 4.4961
1.1460 0.0616 0.0616(10) 0.0774(23) -0.67(2) 0.070(17) 0.609(11) -4.19(7) 4.1957
1.1728 0.0711 0.0709(16) 0.0930(63) -0.79(2) 0.070(16) 0.590(17) -4.05(12) 3.9164
1.2010 0.0822 0.0817(10) 0.1140(66) -0.95(6) 0.082(10) 0.570(15) -3.91(9) 3.5407
1.2306 0.1030 0.1020(14) 0.5354(188) -3.66(11) 0.100(15) 0.542(17) -3.72(10) 1.1719
1.2616 0.1521 0.1532(20) 0.5562(142) -3.79(8) 0.163(17) 0.556(14) -3.78(10) -0.0131
131
Table 3.2 Solid-Liquid coexistence properties obtained from the Gibbs-Duhem
simulations using the Lennard-Jones potential
T ∗ setP∗ LP∗ Lρ∗ LE∗ SP∗ Sρ∗ SE∗ H ∗∆
2.7397 36.900 36.67(24) 1.144 -3.31(4) 36.83(13) 1.211 -4.43(2) 2.9020
2.4096 29.8095 29.89(10) 1.114(2) -4.00(4) 29.83(15) 1.182(1) -5.09(3) 2.6074
2.1505 24.5002 24.57(8) 1.088(1) -4.52(1) 24.51(16) 1.156(2) -5.57(2) 2.3764
1.9418 20.3624 20.35(11) 1.064(1) -4.92(2) 20.38(16) 1.134(1) -5.93(2) 2.1891
1.7699 17.0711 17.05(12) 1.045(1) -5.25(2) 17.04(9) 1.115(2) -6.21(2) 2.0227
1.6260 14.4563 14.43(9) 1.027(2) -5.48(3) 14.46(9) 1.099(2) -6.43(2) 1.8840
1.5038 12.3002 12.30(9) 1.008(2) -5.62(2) 12.31(7) 1.084(2) -6.60(2) 1.7501
1.3986 10.4884 10.47(10) 0.996(1) -5.81(2) 10.43(11) 1.069(1) -6.72(1) 1.6650
1.3072 8.9689 8.99(7) 0.980(3) -5.88(2) 8.99(4) 1.057(1) -6.82(1) 1.5412
1.2270 7.6722 7.67(5) 0.967(3) -5.97(3) 7.68(6) 1.045(2) -6.90(2) 1.4573
1.1561 6.5448 6.56(5) 0.956(3) -6.05(3) 6.53(5) 1.036(1) -6.98(1) 1.4208
1.0929 5.5648 5.54(4) 0.943(2) -6.09(2) 5.57(5) 1.027(2) -7.04(2) 1.3412
1.0363 4.7099 4.73(4) 0.936(2) -6.16(2) 4.71(5) 1.019(2) -7.09(1) 1.3189
0.9852 3.9608 3.99(3) 0.922(2) -6.15(2) 3.95(5) 1.010(2) -7.11(2) 1.2832
0.9390 3.3004 3.32(5) 0.918(2) -6.23(2) 3.32(7) 1.005(2) -7.16(1) 1.2595
0.8969 2.7119 2.73(4) 0.907(3) -6.22(3) 2.73(4) 0.997(1) -7.18(1) 1.2333
0.8584 2.1871 2.17(5) 0.892(3) -6.17(3) 2.17(4) 0.990(1) -7.20(1) 1.2286
0.8231 1.7147 1.73(2) 0.891(3) -6.24(3) 1.69(3) 0.983(1) -7.20(1) 1.1773
0.7905 1.2901 1.30(4) 0.878(3) -6.19(2) 1.28(2) 0.979(1) -7.23(2) 1.1641
0.7605 0.9104 0.92(1) 0.871(4) -6.19(3) 0.92(4) 0.975(2) -7.24(2) 1.1532
0.7326 0.5676 0.57(3) 0.862(3) -6.18(3) 0.54(3) 0.970(1) -7.26(2) 1.1235
0.7067 0.2440 0.26(3) 0.855(5) -6.16(4) 0.24(4) 0.966(2) -7.27(1) 1.1214
0.6826 0.0000 0.00(2) 0.851(3) -6.17(3) 0.00(3) 0.964(2) -7.28(2) 1.1896
132
Table 4.1 Three-body effects on phase behaviour of argon at * * 0.9914T =
ρ∗ P∗ *2P 3P∗ *
KP E ∗ 2E∗ 3E∗
0.03 0.0260 -0.0038 0.00006 0.0297 -0.2352 -0.2358 0.0006 0.05 0.0394 -0.0104 0.0003 0.0496 -0.3968 -0.3986 0.0018 0.07 0.0498 -0.0203 0.0007 0.0694 -0.5578 -0.5613 0.0035 0.10 0.0605 -0.0410 0.0023 0.0991 -0.8135 -0.8213 0.0078 0.12 0.0645 -0.0585 0.0040 0.1190 -0.9770 -0.9882 0.0112 0.13 0.0667 -0.0671 0.0050 0.1289 -1.0434 -1.0562 0.0127 0.15 0.0693 -0.0867 0.0073 0.1487 -1.1864 -1.2026 0.0162 0.18 0.0686 -0.1221 0.0123 0.1785 -1.4147 -1.4375 0.0228 0.20 0.0671 -0.1478 0.0167 0.1983 -1.5669 -1.5947 0.0278 0.25 0.0628 -0.2132 0.0282 0.2479 -1.8539 -1.8915 0.0376 0.30 0.0566 -0.2846 0.0438 0.2974 -2.1349 -2.1836 0.0487 0.35 0.0483 -0.3585 0.0598 0.3470 -2.3481 -2.4051 0.0570 0.40 0.0347 -0.4422 0.0803 0.3966 -2.5770 -2.6440 0.0670 0.43 0.0281 -0.4909 0.0927 0.4263 -2.6946 -2.7665 0.0719 0.45 0.0269 -0.5223 0.1030 0.4461 -2.7858 -2.8622 0.0764 0.47 0.0281 -0.5525 0.1146 0.4660 -2.8818 -2.9631 0.0813 0.50 0.0360 -0.5926 0.1330 0.4957 -3.0213 -3.1100 0.0887 0.52 0.0414 -0.6206 0.1465 0.5155 -3.1155 -3.2094 0.0939 0.54 0.0548 -0.6422 0.1616 0.5354 -3.2159 -3.3156 0.0998 0.545 0.0610 -0.6449 0.1656 0.5403 -3.2406 -3.3418 0.1013 0.55 0.0683 -0.6470 0.1700 0.5453 -3.2680 -3.3710 0.1030 0.60 0.1609 -0.6525 0.2187 0.5948 -3.5370 -3.6585 0.1215 0.65 0.3553 -0.5694 0.2804 0.6444 -3.8121 -3.9558 0.1438 0.70 0.6813 -0.3702 0.3575 0.6940 -4.0872 -4.2575 0.1703 0.75 1.2056 0.0091 0.4528 0.7436 -4.3479 -4.5492 0.2013 0.80 1.9847 0.6233 0.5682 0.7931 -4.5831 -4.8199 0.2368 0.82 2.3739 0.9401 0.6209 0.8130 -4.6702 -4.9226 0.2524 0.85 3.0906 1.5399 0.7080 0.8427 -4.7855 -5.0632 0.2777 0.88 3.9553 2.2788 0.8040 0.8724 -4.8811 -5.1857 0.3046 0.90 4.6253 2.8592 0.8739 0.8923 -4.9338 -5.2575 0.3237 0.91 4.9902 3.1775 0.9106 0.9022 -4.9567 -5.2903 0.3336 0.92 2.8478 0.9956 0.9401 0.9121 -5.4667 -5.8074 0.3406 0.93 3.0484 1.1476 0.9788 0.9220 -5.5255 -5.8763 0.3508 0.94 3.3010 1.3500 1.0191 0.9319 -5.5764 -5.9378 0.3614 0.96 3.9052 1.8498 1.1037 0.9517 -5.6639 -6.0472 0.3832 0.98 4.6307 2.4653 1.1939 0.9716 -5.7350 -6.1411 0.4061 1.00 5.5029 3.2216 1.2899 0.9914 -5.7854 -6.2154 0.4300 1.02 6.5179 4.1146 1.3920 1.0112 -5.8163 -6.2712 0.4549 1.05 8.3256 5.7277 1.5570 1.0410 -5.8236 -6.3179 0.4943 1.07 9.7334 6.9970 1.6756 1.0610 -5.8012 -6.3232 0.5220 1.10 12.179 9.2229 1.8664 1.0905 -5.7218 -6.2874 0.5656 1.13 15.0566 11.8620 2.0743 1.1203 -5.5845 -6.1964 0.6119 1.15 17.2267 13.8642 2.2224 1.1401 -5.4588 -6.1030 0.6442 1.18 20.8861 17.2558 2.4604 1.1699 -5.2168 -5.9119 0.6950 1.20 23.6143 19.7951 2.6296 1.1897 -5.0168 -5.7473 0.7304 1.22 26.5852 22.5680 2.8077 1.2095 -4.7848 -5.5520 0.7671 1.25 31.5120 27.1806 3.0922 1.2393 -4.3748 -5.1994 0.8246 1.27 35.1273 30.5743 3.2939 1.2591 -4.0582 -4.9227 0.8645 1.30 41.0657 36.1616 3.6153 1.2888 -3.5165 -4.4435 0.9270
133
Table 4.2 Three-body effects on phase behaviour of argon at * 1.2678T = *
ρ∗ P∗ *2P 3P∗ *
KP E ∗ 2E∗ 3E∗ 0.03 0.0350 -0.0030 0.00004 0.0380 -0.2072 -0.2076 0.0005 0.05 0.0555 -0.0081 0.0002 0.0634 -0.3424 -0.3436 0.0013 0.07 0.0737 -0.0155 0.0005 0.0887 -0.4740 -0.4765 0.0025 0.10 0.0974 -0.0309 0.0015 0.1268 -0.6669 -0.6718 0.0048 0.12 0.1112 -0.0434 0.0025 0.1521 -0.7976 -0.8044 0.0069 0.15 0.1292 -0.0657 0.0047 0.1902 -0.9848 -0.9952 0.0105 0.18 0.1452 -0.0907 0.0078 0.2282 -1.1610 -1.1753 0.0144 0.20 0.1536 -0.1105 0.0105 0.2536 -1.2849 -1.3024 0.0176 0.25 0.1774 -0.1585 0.0189 0.3170 -1.5489 -1.5741 0.0252 0.30 0.1976 -0.2138 0.0311 0.3803 -1.8246 -1.8591 0.0345 0.35 0.2231 -0.2675 0.0469 0.4437 -2.0836 -2.1282 0.0446 0.40 0.2596 -0.3142 0.0667 0.5071 -2.3377 -2.3933 0.0556 0.45 0.3103 -0.3528 0.0926 0.5705 -2.5974 -2.6660 0.0686 0.50 0.3962 -0.3622 0.1245 0.6339 -2.8551 -2.9381 0.0830 0.55 0.5414 -0.3216 0.1657 0.6973 -3.1208 -3.2213 0.1004 0.60 0.7504 -0.2268 0.2165 0.7607 -3.3851 -3.5054 0.1203 0.65 1.0802 -0.0248 0.2810 0.8241 -3.6499 -3.7939 0.1441 0.70 1.5746 0.3272 0.3599 0.8875 -3.8989 -4.0702 0.1714 0.75 2.2764 0.8686 0.4570 0.9509 -4.1308 -4.3339 0.2031 0.80 3.2707 1.6821 0.5743 1.0143 -4.3292 -4.5685 0.2393 0.82 3.7460 2.0788 0.6276 1.0396 -4.4006 -4.6557 0.2551 0.85 4.5888 2.7962 0.7150 1.0776 -4.4912 -4.7716 0.2804 0.88 5.5824 3.6550 0.8118 1.1157 -4.5638 -4.8713 0.3075 0.90 6.3553 4.3319 0.8824 1.1410 -4.5981 -4.9249 0.3268 0.92 7.2093 5.0853 0.9575 1.1664 -4.6219 -4.9688 0.3469 0.94 8.1624 5.9330 1.0376 1.1917 -4.6329 -5.0008 0.3679 0.95 8.6580 6.3740 1.0797 1.2044 -4.6370 -5.0158 0.3788 0.96 9.1950 6.8551 1.1228 1.2171 -4.6343 -5.0242 0.3898 0.97 6.3806 3.9927 1.1581 1.2298 -5.3120 -5.7100 0.3980 0.98 6.7540 4.3076 1.2039 1.2424 -5.3473 -5.7567 0.4095 0.99 7.1634 4.6570 1.2513 1.2551 -5.3779 -5.7992 0.4213 1.00 7.6156 5.0477 1.3001 1.2678 -5.4019 -5.8353 0.4334 1.01 8.1050 5.4742 1.3503 1.2805 -5.4205 -5.8662 0.4456 1.02 8.6469 5.9515 1.4022 1.2932 -5.4314 -5.8896 0.4583 1.03 9.2064 6.4449 1.4556 1.3058 -5.4407 -5.9117 0.4711 1.04 9.8132 6.9841 1.5106 1.3185 -5.4429 -5.9271 0.4842 1.05 10.4603 7.5618 1.5673 1.3312 -5.4400 -5.9375 0.4976 1.06 11.1480 8.1786 1.6255 1.3439 -5.4309 -5.9421 0.5112 1.07 11.8836 8.8413 1.6858 1.3566 -5.4158 -5.9410 0.5252 1.08 12.6639 9.5472 1.7475 1.3692 -5.3939 -5.9333 0.5393 1.10 14.3550 11.0842 1.8761 1.3946 -5.3328 -5.9013 0.5685 1.12 16.2399 12.8077 2.0122 1.4199 -5.2455 -5.8443 0.5988 1.15 19.4358 15.7466 2.2313 1.4580 -5.0655 -5.7123 0.6467 1.18 23.1208 19.1566 2.4682 1.4960 -4.8196 -5.5168 0.6972 1.20 25.8606 21.7024 2.6369 1.5213 -4.6181 -5.3506 0.7324 1.22 28.8403 24.4792 2.8144 1.5467 -4.3849 -5.1538 0.7689 1.26 35.5585 30.7642 3.1969 1.5974 -3.8183 -4.6641 0.8457 1.30 43.3523 38.0854 3.6187 1.6481 -3.1121 -4.0400 0.9278
134
Table 4.3 Vapour-liquid coexistence properties of Argon obtained by Gibbs-Duhem
Integration method simulations using BFW potential.
T ∗ setP∗ VP∗ Vρ∗ VE∗ LP∗ Lρ∗ LE∗ H ∗∆
0.7000 0.0040 0.00400(5) 0.0060(0) -0.0650(7) -0.04(4) 0.806(0) -5.188(6) 5.7847 0.7202 0.0051 0.00515(5) 0.00760(8) -0.080(1) 0.02(4) 0.799(3) -5.12(1) 5.7183 0.7415 0.0066 0.00656(5) 0.00954(8) -0.098(1) 0.009(20) 0.789(5) -5.04(3) 5.6318 0.7642 0.0084 0.00840(4) 0.01205(9) -0.120(2) 0.014(19) 0.777(4) -4.94(3) 5.4909 0.7883 0.0107 0.01062(4) 0.0149(1) -0.146(1) 0.003(25) 0.760(8) -4.82(5) 5.4186 0.8140 0.0136 0.01376(8) 0.0193(1) -0.184(7) 0.013(30) 0.743(10) -4.69(7) 5.2422 0.8414 0.0174 0.0174(2) 0.0241(3) -0.225(4) 0.009(23) 0.728(9) -4.57(5) 5.1127 0.8707 0.0223 0.0221(2) 0.0305(4) -0.276(4) 0.033(17) 0.713(13) -4.46(8) 4.9315 0.9021 0.0285 0.0288(3) 0.0402(9) -0.35(1) 0.036(17) 0.696(13) -4.32(8) 4.6662 0.9358 0.0367 0.0366(4) 0.052(1) -0.45(1) 0.032(26) 0.676(11) -4.19(4) 4.3753 0.9722 0.0473 0.0468(6) 0.068(1) -0.57(1) 0.053(25) 0.652(8) -4.01(12) 4.0562 1.0116 0.0619 0.0619(12) 0.103(7) -0.85(8) 0.073(24) 0.620(19) -3.79(8) 3.4996 1.0542 0.0909 0.092(13) 0.581(21) -3.56(1) 0.095(14) 0.589(15) -3.60(9) 1.4927 1.1006 0.1651 0.172(17) 0.579(18) -3.50(6) 0.170(9) 0.575(16) -3.49(5) 0.0173
Table 4.4 Vapour-liquid coexistence properties of Krypton obtained by Gibbs-Duhem
Integration method simulations using BFW potential.
T ∗ setP∗ VP∗ Vρ∗ VE∗ LP∗ Lρ∗ LE∗ H ∗∆
0.7000 0.0050 0.00463(1) 0.0070(0) -0.0734(5) -0.044(26) 0.800(0) -5.054(5) 5.6882 0.7202 0.0063 0.00634(5) 0.0095(1) -0.098(2) 0.011(25) 0.795(3) -4.99(2) 5.5196 0.7415 0.0080 0.0081(5) 0.0119(1) -0.117(1) 0.009(15) 0.783(7) -4.90(5) 5.4715 0.7642 0.0102 0.0102(1) 0.0150(1) -0.145(2) 0.002(27) 0.774(8) -4.84(5) 5.3747 0.7883 0.0130 0.0130(1) 0.0189(2) -0.179(2) 0.018(25) 0.761(9) -4.73(6) 5.2204 0.8140 0.0166 0.0167(2) 0.0241(5) -0.222(7) 0.026(20) 0.740(9) -4.57(5) 5.0078 0.8414 0.0213 0.0213(1) 0.0307(4) -0.278(2) 0.023(22) 0.729(11) -4.49(7) 4.8756 0.8707 0.0272 0.0271(3) 0.0392(6) -0.349(10) 0.032(19) 0.708(7) -4.33(4) 4.6742 0.9021 0.0349 0.0353(4) 0.0523(4) -0.446(14) 0.042(22) 0.687(10) -4.18(6) 4.3730 0.9358 0.0452 0.0458(3) 0.0716(22) -0.606(24) 0.050(16) 0.663(13) -4.02(7) 3.9844 0.9722 0.0594 0.0593(11) 0.1073(112) -0.866(109) 0.072(16) 0.634(9) -3.82(6) 3.3889 1.0116 0.0882 0.0788(184) 0.6071(100) -3.634(63) 0.100(14) 0.612(6) -3.66(3) 1.4563 1.0542 0.1614 0.1600(101) 0.5993(101) -3.565(33) 0.152((16) 0.590(15) -3.51(9) 0.0220
Table 4.5 Vapour-liquid coexistence properties of Xenon obtained by Gibbs-Duhem
Integration method simulations using BFW potential.
T ∗ setP∗ VP∗ Vρ∗ VE∗ LP∗ Lρ∗ LE∗ H ∗∆
0.7000 0.0040 0.00400(0) 0.0060(0) -0.0632(3) -0.011(55) 0.801(0) -5.075(7) 5.6738 0.7202 0.0051 0.00510(4) 0.0075(1) -0.078(1) -0.002(19) 0.791(7) -4.99(6) 5.5992 0.7415 0.0065 0.00650(6) 0.0094(1) -0.094(2) 0.016(13) 0.783(8) -4.93(6) 5.5030 0.7642 0.0082 0.00820(5) 0.0117(1) -0.114(2) 0.005(27) 0.771(9) -4.83(6) 5.4238 0.7883 0.0104 0.01046(7) 0.0146(2) -0.139(3) -0.006(22) 0.748(4) -4.66(3) 5.2792 0.8140 0.0132 0.0132(1) 0.0182(2) -0.169(3) 0.010(18) 0.744(8) -4.62(5) 5.1361 0.8414 0.0167 0.0167(1) 0.0229(3) -0.207(3) 0.010(12) 0.719(6) -4.43(4) 4.9909 0.8707 0.0212 0.0211(1) 0.0288(3) -0.251(2) 0.021(22) 0.706(5) -4.33(4) 4.7811 0.9021 0.0268 0.0270(2) 0.0364(6) -0.312(6) 0.026(10) 0.687(8) -4.20(5) 4.5989 0.9358 0.0339 0.0342(4) 0.0467(11) -0.390(11) 0.039(30) 0.661(10) -4.01(6) 4.3727 0.9722 0.0428 0.0431(5) 0.0587(10) -0.471(9) 0.049(14) 0.634(16) -3.84(10) 4.0682 1.0116 0.0542 0.0549(8) 0.0765(16) -0.599(21) 0.059(19) 0.594(27) -3.57(15) 3.6889 1.0542 0.0681 0.0681(12) 0.0958(35) -0.715(41) 0.077(11) 0.531(32) -3.20(17) 3.0327 1.1006 0.0820 0.0819(15) 0.1146(35) -0.832(35) 0.083(35) 0.117(3) -0.85(3) 1.1148
135
∆Η
EL D
DD
4
EL Q
EL D
EL D
DQ
EL D
DD
EL 2body
EL Total
PL D
DD
4
PL Q
PL D
PL D
DQ
PL D
DD
PL 2body
PL Total
ρL
EV
DD
D4 10 -3
EVQ
QQ 10
-5
EVD
QQ 10
-4
EVD
DQ 10
-3
EV
DD
D 10-3
EV
2body
EV
Total
PV D
DD
4 10-4
PV Q
QQ 10
-6
PV D
10-5
PV
DD
Q 10-4
PV D
DD
10-3
PV
2body
PV
Total
ρV
Pset
T
5.1493
-0.0200(2)
0.00065(!)
0.00820(7)
0.0451(3)
0.162(1)
-4.75(1)
-4.55(1)
-0.0593(4)
0.00239(2)
0.0264(2)
0.1230(4)
0.361(2)
-0.988(69)
0.021(70)
0.742(0)
-0.027(3)
0.040(12)
0.053(15)
0.032(8)
0.131(30)
-0.0969(3)
-0.0967(3) -0.010(2)
0.019(5)
0.022(6)
0.011(3)
0.0037(8)
-0.0005(1)
0.0066(1)
0.0095(0)
0.0067
07500
5.0654
-0.0189(5)
0.00059(3)
0.0075(3)
0.042(1)
0.152(5)
-4.58(8)
-4.40(7)
-0.054(3)
0.0021(1)
0.0234(1)
0.111(6)
0.330(1)
-0.952(32)
0.020(45)
0.721(11)
-0.040(5)
0.055(17)
0.075(21)
0.047(11)
0.192(39)
-0.123(4)
-0.123(4) -0.020(2)
0.035(10)
0.041(11)
0.022(4)
0.0073(14)
-0.0008(1)
0.0089(1)
0.0125(3)
0.0088
0.7792
4.9781
-0.0182(2)
0.00055(1)
0.0071(1)
0.0396(9)
0.146(2)
-4.47(5)
-4.29(4)
-0.052(1)
0.00195(7)
0.0217(8)
0.103(3)
0.310(9)
-0.950(29)
0.007(29)
0.707(7)
-0.068(14)
0.103(32)
0.136(40)
0.082(22)
0.331(82)
-0.157(7)
-0.156(7) -0.044(9)
0.083(26)
0.095(28)
0.048(13)
0.0160(41)
-0.0014(1)
0.0117(3)
0.0161(4)
0.0116
0.8108
4.6849
-0.0176(3)
0.00051(1)
0.0066(2)
0.037(1)
0.139(3)
-4.32(5)
-4.16(5)
-0.048(1)
0.00176(7)
0.0197(8)
0.094(3)
0.286(9)
-0.906(31)
0.027(30)
0.686(6)
-0.092(9)
0.117(21)
0.161(28)
0.102(15)
0.434(54)
-0.189(7)
-0.188(7) -0.077(8)
0.123(22)
0.146(24)
0.078(11)
0.0272(33)
-0.0022(3)
0.0153(2)
0.0206(5)
0.0153
0.8451
4.5345
-0.0165(5)
0.00045(2)
0.0059(3)
0.034(1)
0.128(5)
-4.10(10)
-3.95(9)
-0.043(2)
0.0015(1)
0.017(3)
0.080(6)
0.251(16)
-0.858(39)
0.028(34)
0.656(17)
-0.145(24)
0.185(36)
0.253(48)
0.159(28)
0.684(11)
-0.239(14)
-0.238(14) -0.159(34)
0.254(62)
0.300(71)
0.160(36)
0.0562(23)
-0.0036(5)
0.0203(5)
0.0269(11)
0.0201
0.8824
4.1985
-0.0153(4)
0.00040(2)
0.0052(2)
0.030(1)
0.117(4)
-3.91(10)
-3.77(10)
-0.039(2)
0.0013(1)
0.014(1)
0.070(5)
0.221(15)
-0.828(39)
0.022(37)
0.631(20)
-0.213(23)
0.269(43)
0.366(16)
0.230(28)
0.991(10)
-0.293(11)
-0.292(11) -0.297(39)
0.471(81)
0.554(87)
0.295(41)
0.104(13)
-0.0054(5)
0.0267(7)
0.0345(11)
0.0264
0.9231
3.8999
-0.0128(3)
0.00027(1)
0.0036(1)
0.022(1)
0.089(3)
-3.30(7)
-3.19(7)
-0.027(1)
0.00071(5)
0.0082(6)
0.042(3)
0.141(9)
-0.658(22)
0.017(28)
0.527(14)
-0.319(51)
0.390(88)
0.534(11)
0.340(67)
0.148(26)
-0.356(22)
-0.354(18) -0.561(115)
0.862(232)
1.021(265)
0.549(134)
0.196(44)
-0.0079(13)
0.0341(90)
0.0431(18)
0.0341
0.9677
3.4438
-0.0110(5)
0.00019(1)
0.0026(2)
0.016(1)
0.069(5)
-2.83(13)
-2.75(12)
-0.020(1)
0.00044(6)
0.0051(7)
0.027(3)
0.093(12)
-0.527(43)
0.034(13)
0.448(25)
-0.424(42)
0.502(69)
0.688(89)
0.440(51)
0.196(20)
-0.417(21)
-0.415(21) -0.920(138)
0.136(25)
1.618(285)
0.876(143)
0.318(48)
-0.0116(18)
0.0432(13)
0.0535(27)
0.0434
1.0169
1.8649
-0.00056(6)
0.000006(1)
0.0008(1)
0.0006(1)
0.0026(3)
-0.48(2)
-0.48(2)
-0.00014(2)
0.000002(0)
0.000025(4)
0.00013(2)
0.00048(7)
-0.015(1)
0.0526(1)
0.063(3)
-0.619(71)
0.729(98)
0.995(12)
0.637(76)
0.283(32)
-0.50026)
-0.497(31) -1.647(279)
2.426(440)
2.872(508)
1.554(266)
0.564(94)
-0.0164(26)
0.0547(13)
0.0658(21)
0.0533
1.0714
Table 4.6 Vapour-liquid phase equilibria properties of argon from G
ibbs-Duhem
integration simulation using the tw
o-body BFW
potential + three-body potentials (DD
D+D
DQ
+DQ
Q+Q
+DD
D4).
136
∆Η
EL D
DD
EL 2body
EL Total
PL D
DD
PL 2body
PL Total
ρL
EV
DD
D 10-3
EV 2body
EV Total
PV
DD
D 10
-3
PV 2body
PV Total
ρV
Pset
T
5.1822
0.1622(7)
-4.74(1)
-4.58(1)
0.361(1)
-0.991(51)
-0.073(52)
0.742(0)
0.11(1)
-0.095(2)
-0.095(2)
0.0033(6)
-0.00051(2)
0.00662(2)
0.0095(25)
0.007
0.75
5.1134
0.160(3)
-4.68(6)
-4.52(6)
0.353(12)
-0.907(44)
0.017(48)
0.734(9)
0.19(3)
-0.123(6)
-0.123(6)
0.0073(15)
-0.0009(1)
0.0089(1)
0.0126(2)
0.009
0.7792
4.9687
0.154(4)
-4.58(7)
-4.43(7)
0.334(13)
-0.925(46)
-0.004(37)
0.724(9)
0.27(4)
-0.149(7)
-0.149(7)
0.0131(24)
-0.0012(1)
0.01180(1)
0.0161(3)
0.012
0.8108
4.8665
0.137(5)
-4.27(9)
-4.13(9)
0.281(16)
-0.850(42)
0.007(37)
0.682(13)
0.43(9)
-0.193(10)
-0.192(10)
0.0278(69)
-0.0023(3)
0.0155(3)
0.0211(6)
0.016
0.8451
4.6072
0.133(6)
-4.20(11)
-4.07(11)
0.270(20)
-0.819(30)
0.045(45)
0.674(17)
0.70(7)
-0.242(7)
-0.242(7)
0.0584(68)
-0.0036(4)
0.0206(3)
0.0274(5)
0.021
0.8824
4.3605
0.122(4)
-3.99(7)
-3.87(7)
0.238(12)
-0.774(47)
0.060(54)
0.646(11)
1.09(13)
-0.306(13)
-0.305(13)
0.1207(17)
-0.0061(6)
0.0276(7)
0.0364(13)
0.027
0.9231
4.0891
0.110(3)
-3.75(5)
-3.64(5)
0.200(8)
-0.760(36)
0.029(37)
0.608(10)
1.73(34)
-0.389(32)
-0.387(32)
0.2528(68)
-0.0100(16)
0.0364(15)
0.0476(29)
0.037
0.9677
3.5692
0.885(9)
-3.28(20)
-3.19(20)
0.143(24)
-0.638(58)
0.045(34)
0.532(37)
2.85(30)
-0.499(17)
-0.496(18)
0.5406(67)
-0.0159(14)
0.0486(12)
0.0629(20)
0.048
1.0169
2.7278
0.028(14)
-1.67(49)
-1.64(47)
0.025(26)
-0.215(111)
0.081(9)
0.252(87)
4.37(51)
-0.628(36)
-0.623(36)
1.096(17)
-0.0271(43)
0.0625(18)
0.0827(46)
0.061
1.0714
Table 4.7 Vapour-liquid phase equilibria properties of argon from
Gibbs-D
uhem integration sim
ulation using the two-body B
FW potential + A
T term
137
Table 4.8 Vapour-liquid phase equilibrium properties of argon obtained from Gibbs-
Duhem integration simulations using the relationship between 2–body and 3-body
potentials.
T ∗ setP∗ VP∗ Vρ∗ VE∗ LP∗ Lρ∗ LE∗ H ∗∆
0.7500 0.0067 0.00662(5) 0.0095(0) -0.097(1) -0.036(84) 0.742(0) -4.55(1) 5.1490
0.7792 0.0088 0.0088(2) 0.0124(4) -0.123(7) 0.003(30) 0.738(10) -4.51(7) 5.0694
0.8108 0.0117 0.0117(2) 0.0162(4) -0.157(8) 0.028(56) 0.719(16) -4.37(11) 4.9980
0.8451 0.0154 0.0155(4) 0.0209(8) -0.194(10) 0.006(38) 0.696(9) -4.22(6) 4.8112
0.8824 0.0203 0.0203(5) 0.0270(8) -0.240(11) 0.039(53) 0.676(9) -4.06(5) 4.5775
0.9231 0.0264 0.0264(8) 0.0343(17) -0.299(21) 0.006(53) 0.637(10) -3.82(6) 4.3858
0.9677 0.0345 0.0345(6) 0.0441(9) -0.364(16) 0.043(26) 0.596(11) -3.57(8) 4.0253
1.0169 0.0449 0.0446(13) 0.0573(27) -0.462(29) 0.047(21) 0.523(16) -3.12(9) 3.6676
1.0714 0.0567 0.0571(21) 0.0705(39) -0.534(33) 0.059(5) 0.0760(14) -0.58(11) 2.1344
Table 4.9 Vapour-liquid phase equilibrium properties of krypton obtained from Gibbs-
Duhem integration simulations using the relationship between 2–body and 3-body
potentials.
T ∗ setP∗ VP∗ Vρ∗ VE∗ LP∗ Lρ∗ LE∗ H ∗∆
0.7500 0.0074 0.00729(4) 0.0105) -0.103(3) -0.108(66) 0.712(0) -4.26(1) 4.8515
0.7792 0.0097 0.0097(1) 0.0136(2) -0.129(4) 0.028(38) 0.720(10) -4.27(7) 4.9099
0.8108 0.0127 0.0128(1) 0.0177(2) -0.168(10) -0.027(32) 0.679(16) -4.01(9) 4.7582
0.8451 0.0166 0.0165(2) 0.0221(6) -0.197(7) -0.013(40) 0.662(14) -3.90(6) 4.5720
0.8824 0.0217 0.0217(4) 0.0288(10) -0.249(12) 0.026(24) 0.644(18) -3.77(9) 4.3201
0.9231 0.0281 0.0278(4) 0.0363(9) -0.304(14) 0.012(35) 0.605(15) -3.54(9) 3.9792
0.9677 0.0359 0.0364(9) 0.0472(15) -0.377(11) 0.030(22) 0.486(19) -2.94(8) 3.2168
1.0169 0.0435 0.0441(15) 0.0544(29) -0.419(24) 0.044(9) 0.055(2) -0.42(1) 2.0730
1.0714 0.0498 0.0501(12) 0.0578(17) -0.427(17) 0.050(1) 0.059(2) -0.44(2) -0.0105
Table 4.10 Vapour-liquid phase equilibrium properties of Xenon obtained from Gibbs-
Duhem integration simulations using the relationship between 2–body and 3-body
potentials.
T ∗ setP∗ VP∗ Vρ∗ VE∗ LP∗ Lρ∗ LE∗ H ∗∆
0.7500 0.0075 0.00755(1) 0.0109(0) -0.1082(8) -0.098(26) 0.706(0) -4.198(6) 4.7670
0.7792 0.0098 0.0098(1) 0.0139(1) -0.133(2) 0.019(14) 0.717(6) -4.24(4) 4.7983
0.8108 0.0128 0.0128(1) 0.0175(1) -0.163(3) 0.022(21) 0.689(11) -4.06(6) 4.5835
0.8451 0.0166 0.0166(1) 0.0224(2) -0.201(3) 0.008(10) 0.661(9) -3.87(5) 4.3891
0.8824 0.0215 0.0215(1) 0.0283(3) -0.245(4) 0.016(21) 0.626(13) -3.67(7) 4.0909
0.9231 0.0276 0.0277(4) 0.0361(9) -0.300(9) 0.027(17) 0.587(26) -3.43(12) 3.8566
0.9677 0.0349 0.0350(2) 0.0446(5) -0.358(7) 0.040(7) 0.203(161) -1.36(98) 3.2132
1.0169 0.0376 0.0388(6) 0.0463(13) -0.359(12) 0.039(1) 0.046(1) -0.36(1) 0.0032
1.0714 0.0330 0.0332(2) 0.0354(3) -0.263(2) 0.033(1) 0.035(1) -0.26(1) -0.0075
138
Table 4.11 Initial conditions for BFW calculations
(K) T *T Vρ∗ Lρ∗ P∗ / ( ) / (fugacity henry L fugacity henry V )
177.38 0.8786 0.0349 0.6976 0.0247 0.8871 0.2368
163.15 0.8081 0.0212 0.7459 0.0148 0.9175 0.1844
158.15 0.7833 0.0141 0.7661 0.0100 0.9408 0.1703
153.15 0.7585 0.0112 0.7714 0.0078 0.9511 0.1633
148.15 0.7338 0.0074 0.7893 0.0051 0.9645 0.1529
143.15 0.7090 0.0054 0.7942 0.0037 0.9746 0.1288
Table 4.12 Initial conditions for BFW+AT calculations
(K) T Vρ∗ Lρ∗
P∗ / ( ) / (fugacity henry L fugacity henry V )
177.38 0.0500 0.6585 0.0335 0.8442 0.2765
163.15 0.0294 0.7039 0.0194 0.8921 0.2286
158.15 0.0239 0.7163 0.0158 0.9060 0.2135
153.15 0.0189 0.7334 0.0125 0.9221 0.1945
148.15 0.0151 0.7432 0.0098 0.9319 0.1677
143.15 0.0118 0.7619 0.0076 0.9405 0.1639
139
Table 4.13 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=177.38K using the two-body potential. *
setP ArVx Ar
Lx *VP *
LP *Vρ *
Lρ *VE *
LE
0.0247 0.00(0) 0.00(0) 0.0252(1) 0.022(0) 0.035(0) 0.703(8) -0.30(0) -4.28(6) 0.0258 0.044(3) 0.012(1) 0.0259(4) 0.005(11) 0.036(1) 0.695(1) -0.30(2) -4.23(3) 0.0269 0.091(4) 0.024(1) 0.0268(7) 0.055(68) 0.037(2) 0.700(13) -0.31(2) -4.23(8) 0.0283 0.135(5) 0.039(2) 0.0292(9) 0.028(11) 0.041(2) 0.686(0) -0.32(3) -4.17(2) 0.0298 0.180(7) 0.053(2) 0.0311(19) -0.002(81) 0.044(3) 0.693(8) -0.34(3) -4.13(5) 0.0314 0.226(5) 0.071(4) 0.0317(11) 0.054(77) 0.045(4) 0.701(15) -0.33(4) -4.12(9) 0.0332 0.266(3) 0.088(3) 0.0335(16) -0.065(83) 0.049(4) 0.699(0) -0.36(2) -4.14(2) 0.0352 0.312(6) 0.107(5) 0.0347(16) 0.120(126) 0.050(4) 0.723(10) -0.35(3) -4.21(7) 0.0375 0.361(6) 0.133(4) 0.0376(9) 0.094(83) 0.054(2) 0.713(6) -0.36(2) -4.08(4) 0.0401 0.406(8) 0.167(6) 0.0393(27) -0.043(94) 0.057(6) 0.665(0) -0.37(5) -3.81(3) 0.0431 0.448(3) 0.191(5) 0.0440(18) 0.054(108) 0.067(4) 0.699(4) -0.43(3) -3.87(2) 0.0466 0.500(6) 0.221(6) 0.0465(23) 0.043(61) 0.071(6) 0.707(10) -0.43(3) -3.87(5) 0.0507 0.542(5) 0.276(6) 0.0522(26) 0.053(70) 0.081(5) 0.673(16) -0.47(4) -3.56(8) 0.0558 0.594(6) 0.323(9) 0.0565(21) 0.024(71) 0.088(3) 0.661(23) -0.49(4) -3.42(13) 0.0618 0.639(8) 0.378(7) 0.0632(37) -0.040(52) 0.104(8) 0.654(0) -0.56(5) -3.34(2) 0.0692 0.695(9) 0.450(8) 0.0662(36) -0.037(90) 0.107(11) 0.654(0) -0.56(9) -3.20(3) 0.0791 0.729(12) 0.521(9) 0.0836(71) -0.023(74) 0.160(23) 0.654(0) -0.79(12) -3.13(2) 0.0931 0.666(33) 0.618(6) 0.0538(80) -0.022(56) 0.517(92) 0.654(0) -2.29(43) -3.00(3) 0.1195 0.779(10) 0.726(4) 0.1119(21) 0.059(38) 0.478(22) 0.654(0) -1.95(11) -2.78(1) 0.1654 0.876(4) 0.854(4) 0.1521(35) 0.076(76) 0.505(31) 0.654(0) -1.98(9) -2.57(1)
Table 4.14 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=163.15K using the two-body potential. Ar
Vx ArLx
*setP
*VP
*LP
*Vρ
*Lρ
*VE
*LE
0.0148 0.0(0) 0.0(0) 0.0149(1) 0.019(26) 0.021(0) 0.740(5) -0.19(0) -4.57(3) 0.0155 0.045(2) 0.009(1) 0.0153(6) 0.013(99) 0.022(1) 0.743(7) -0.20(1) -4.59(5) 0.0162 0.090(3) 0.018(1) 0.0163(7) 0.201(82) 0.023(1) 0.740(0) -0.20(1) -4.58(2) 0.0170 0.139(5) 0.031(1) 0.0172(5) 0.025(94) 0.024(1) 0.749(0) -0.20(1) -4.60(1) 0.0179 0.185(4) 0.042(3) 0.0184(8) 0.038(90) 0.026(1) 0.749(0) -0.22(2) -4.62(1) 0.0190 0.229(5) 0.057(3) 0.0191(9) 0.071(99) 0.027(1) 0.749(0) -0.21(2) -4.58(3) 0.0201 0.278(9) 0.070(3) 0.0202(6) 0.027(98) 0.030(1) 0.744(10) -0.23(2) -4.44(7) 0.0214 0.321(6) 0.090(3) 0.0219(5) 0.044(99) 0.032(2) 0.758(0) -0.24(2) -4.60(2) 0.0229 0.372(5) 0.109(3) 0.0224(1) 0.045(39) 0.032(1) 0.758(0) -0.24(1) -4.54(2) 0.0245 0.420(6) 0.126(5) 0.0244(1) 0.041(80) 0.036(2) 0.754(6) -0.25(1) -4.39(5) 0.0264 0.468(5) 0.157(6) 0.0265(9) 0.037(51) 0.039(2) 0.741(9) -0.26(2) -4.24(6) 0.0287 0.513(6) 0.187(7) 0.0285(9) 0.163(90) 0.043(2) 0.737(0) -0.28(2) -4.24(3) 0.0314 0.564(5) 0.219(14) 0.0310(18) 0.165(74) 0.047(3) 0.737(0) -0.28(2) -4.19(5) 0.0347 0.609(7) 0.257(5) 0.0353(19) 0.139(75) 0.055(4) 0.737(0) -0.33(3) -4.10(2) 0.0387 0.653(4) 0.309(11) 0.0395(33) 0.102(99) 0.064(5) 0.737(0) -0.38(4) -3.99(3) 0.0438 0.707(5) 0.378(9) 0.0440(22) 0.025(85) 0.070(6) 0.737(0) -0.38(3) -3.83(4) 0.0503 0.751(5) 0.455(10) 0.0511(40) 0.001(68) 0.093(6) 0.737(0) -0.49(4) -3.65(3) 0.0596 0.807(6) 0.552(11) 0.0591(47) 0.039(98) 0.108(11) 0.737(0) -0.53(6) -3.49(3) 0.0735 0.849(7) 0.664(7) 0.0737(96) 0.052(75) 0.159(19) 0.737(0) -0.81(11) -3.2(2) 0.0976 0.825(12) 0.819(2) 0.0773(42) 0.096(62) 0.685(33) 0.737(0) -2.69(15) -2.99(22)
140
Table 4.15 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=158.15K using the two-body potential. Ar
Vx ArLx *
setP *VP *
LP *Vρ *
Lρ *VE *
LE
0.0100 0.0(0) 0.0(0) 0.0101(1) 0.017(2) 0.014(0) 0.759(11) -0.13(0) -4.72(8) 0.0105 0.047(2) 0.009(1) 0.0104(3) 0.002(84) 0.015(1) 0.755(8) -0.13(1) -4.66(6) 0.0110 0.095(3) 0.019(1) 0.0110(3) 0.110(99) 0.015(6) 0.760(3) -0.14(1) -4.66(2) 0.0115 0.140(6) 0.028(2) 0.0118(6) -0.296(99) 0.017(1) 0.731(0) -0.14(1) -4.54(2) 0.0121 0.188(3) 0.041(2) 0.0123(5) -0.244(96) 0.017(1) 0.731(0) -0.14(1) -4.54(1) 0.0128 0.237(6) 0.051(1) 0.0124(5) -0.370(72) 0.018(1) 0.731(0) -0.15(9) -4.56(2) 0.0136 0.283(4) 0.065(2) 0.0138(4) -0.418(75) 0.020(1) 0.731(0) -0.16(1) -4.59(2) 0.0145 0.332(7) 0.080(2) 0.0147(5) -0.506(91) 0.021(1) 0.731(0) -0.16(1) -4.57(2) 0.0154 0.381(4) 0.099(5) 0.0155(3) -0.261(99) 0.022(1) 0.731(0) -0.16(1) -4.41(3) 0.0165 0.429(3) 0.123(5) 0.0164(4) -0.201(91) 0.024(1) 0.731(0) -0.16(1) -4.32(3) 0.0178 0.476(3) 0.143(7) 0.0180(6) -0.284(99) 0.027(1) 0.731(0) -0.18(1) -4.30(3) 0.0194 0.525(4) 0.171(6) 0.0197(7) -0.295(84) 0.029(1) 0.731(0) -0.19(1) -4.29(2) 0.0211 0.573(6) 0.204(5) 0.0214(2) -0.394(76) 0.032(2) 0.731(0) -0.20(2) -4.27(2) 0.0233 0.620(5) 0.244(8) 0.0235(8) -0.307(99) 0.035(1) 0.731(0) -0.21(1) -4.12(3) 0.0258 0.668(5) 0.295(7) 0.0268(11) -0.197(97) 0.042(2) 0.731(0) -0.25(3) -4.06(2) 0.0291 0.720(4) 0.363(8) 0.0299(12) -0.223(88) 0.046(3) 0.731(0) -0.26(3) -3.89(2) 0.0333 0.769(4) 0.432(7) 0.0334(9) -0.214(51) 0.053(2) 0.731(0) -0.28(2) -3.77(3) 0.0387 0.826(5) 0.528(6) 0.0388(24) -0.201(80) 0.062(5) 0.731(0) -0.31(2) -3.60(2) 0.0458 0.879(4) 0.647(8) 0.0447(22) -0.127(80) 0.074(2) 0.731(0) -0.35(2) -3.33(2) 0.0557 0.937(4) 0.806(5) 0.0555(25) -0.116(61) 0.099(9) 0.731(0) -0.45(3) -3.08(1)
Table 4.16 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=153.15K using the two-body potential. Ar
Vx ArLx
*setP
*VP
*LP
*Vρ
*Lρ
*VE
*LE
0.0078 0.0(0) 0.0(0) 0.0078(1) 0.009(27) 0.0112(1) 0.767(8) -0.110(3) -4.78(6) 0.0081 0.047(2) 0.008(1) 0.0080(2) -0.151(142) 0.0115(3) 0.768(0) -0.108((9) -4.85(2) 0.0085 0.095(2) 0.015(1) 0.0085(3) -0.164(145) 0.0122(4) 0.768(0) -0.112(7) -4.83(2) 0.0090 0.142(2) 0.025(2) 0.0092(3) -0.156(135) 0.0134(5) 0.768(0) -0.122(7) -4.87(3) 0.0095 0.191(4) 0.036(3) 0.0095(3) -0.175(139) 0.0137(7) 0.768(0) -0.113(12) -4.81(4) 0.0100 0.240(5) 0.047(3) 0.0101(3) -0.289(109) 0.0147(6) 0.764(0) -0.117(11) -4.60(3) 0.0106 0.288(4) 0.060(3) 0.0107(3) -0.120(132) 0.0154(6) 0.764(0) -0.122(8) -4.66(2) 0.0113 0.337(4) 0.074(3) 0.0113(4) -0.173(110) 0.0165(9) 0.762(0) -0.122(9) -4.64(1) 0.0121 0.382(5) 0.093(3) 0.0122(4) 0.240(94) 0.0176(8) 0.797(0) -0.132(7) -4.76(2) 0.0130 0.431(5) 0.113(5) 0.0132(5) 0.142(109) 0.0194(10) 0.786(0) -0.140(13) -4.66(2) 0.0140 0.479(6) 0.129(7) 0.0140(4) 0.015(185) 0.0205(8) 0.778(1) -0.146(7) -4.57(4) 0.0152 0.531(3) 0.157(7) 0.0151(4) -0.024(85) 0.0221(9) 0.768(9) -0.145(12) -4.43(7) 0.0167 0.579(5) 0.183(7) 0.0169(4) -0.052(84) 0.0248(9) 0.774(11) -0.157(7) -4.43(7) 0.0185 0.626(4) 0.218(8) 0.0189(10) 0.001(10) 0.0281(16) 0.794(0) -0.173(10) -4.50(3) 0.02061 0.678(4) 0.278(8) 0.0206(5) 0.02(86) 0.0309(11) 0.773(4) -0.183(12) -4.21(2) 0.0233 0.725(3) 0.329(5) 0.0238(10) -0.204(59) 0.0366(22) 0.740(0) -0.202(13) -3.95(2) 0.0268 0.776(5) 0.410(5) 0.0271(7) 0.048(46) 0.0423(19) 0.759(10) -0.228(20) -3.82(5) 0.0313 0.831(4) 0.496(13) 0.0313(18) -0.056(97) 0.0493(32) 0.767(1) -0.255(20) -3.74(4) 0.0374 0.882(3) 0.619(7) 0.0377(11) 0.003(70) 0.0619(30) 0.768(1) -0.307(20) -3.49(2) 0.0462 0.939(3) 0.789(7) 0.0452(23) 0.089(88) 0.0791(84) 0.768(1) -0.370(40) -3.14(2)
141
Table 4.17 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=148.15K using the two-body potential. Ar
Vx ArLx *
setP *VP *
LP *Vρ *
Lρ *VE *
LE
0.0051 0.0(0) 0.0(0) 0.0051()) 0.007(25) 0.0074(1) 0.780(10) -0.074(2) -4.89(7) 0.0053 0.049(3) 0.006(1) 0.0053(1) -0.303(131) 0.0076(2) 0.778(0) -0.074(7) -4.99(2) 0.0056 0.096(2) 0.014(1) 0.0056(1) -0.253(130) 0.0082(6) 0.778(0) -0.076(6) -4.90(2) 0.0059 0.144(3) 0.023(1) 0.0059(2) -0.227(82) 0.0085(3) 0.778(0) -0.082(6) -4.91(1) 0.0062 0.190(3) 0.031(2) 0.0063(1) -0.313(89) 0.0092(3) 0.778(0) -0.081(6) -4.95(3) 0.0066 0.245(4) 0.044(2) 0.0065(2) 0.086(121) 0.0095(3) 0.800(9) -0.081(5) -4.92(7) 0.0070 0.288(6) 0.053(3) 0.0070(1) -0.238(154) 0.0100(2) 0.761(9) -0.082(4) -4.69(4) 0.0074 0.338(3) 0.070(5) 0.0075(2) 0.030(90) 0.0109(4) 0.791(0) -0.083(5) -4.85(3) 0.0080 0.389(4) 0.081(4) 0.0080(3) -0.182(125) 0.0116(6) 0.791(0) -0.090(8) -4.88(3) 0.0085 0.437(4) 0.098(5) 0.0085(2) -0.050(110) 0.0122(5) 0.786(8) -0.092(8) -4.71(5) 0.0092 0.487(6) 0.124(6) 0.0092(3) -0.215(73) 0.0133(5) 0.762(0) -0.091(6) -4.52(3) 0.0100 0.538(7) 0.143(6) 0.0101(3) 0.031(67) 0.0149(5) 0.795(6) -0.099(7) -4.64(6) 0.0110 0.585(4) 0.176(9) 0.0110(3) 0.134(135) 0.0162(5) 0.807(0) -0.102(6) -4.68(4) 0.0121 0.632(4) 0.214(7) 0.0122(4) 0.164(114) 0.0181(6) 0.807(0) -0.116(8) -4.58(3) 0.0135 0.681(3) 0.253(8) 0.0135(3) 0.098(78) 0.0199(6) 0.807(0) -0.120(4) -4.50(3) 0.0153 0.731(5) 0.306(10) 0.0155(4) 0.086(99) 0.0236(10) 0.804(0) -0.140(11) -4.33(3) 0.0176 0.782(5) 0.371(14) 0.0177(5) -0.016(140) 0.0273(13) 0.792(0) -0.151(13) -4.15(5) 0.0207 0.835(5) 0.471(12) 0.0211(7) 0.056(115) 0.0329(16) 0.783(5) -0.172(13) -3.84(3) 0.0248 0.888(4) 0.596(8) 0.0248(11) 0.014(37) 0.0396(27) 0.773(15) -0.19919) -3.54(9) 0.0308 0.941(3) 0.775(6) 0.0307(15) -0.078(70) 0.0506(29) 0.761(0) -0.249(17) -3.21(2)
Table 4.18 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=143.15K using the two-body potential. Ar
Vx ArLx
*setP
*VP
*LP
*Vρ
*Lρ
*VE
*LE
0.0037 0.0(0) 0.0(0) 0.0037(0) -0.010(20) 0.0054(0) 0.794(9) -0.055(1) -5.00(7) 0.0039 0.048(3) 0.006(1) 0.0039(1) -0.055(90) 0.0058(2) 0.799(0) -0.057(3) -5.07(3) 0.0041 0.098(3) 0.012(2) 0.0041(1) -0.154(99) 0.0060(2) 0.808(0) -0.059(5) -5.17(2) 0.0043 0.144(3) 0.023(1) 0.0042(1) 0.224(97) 0.0062(2) 0.808(0) -0.058(4) -5.04(1) 0.0045 0.195(3) 0.029(2) 0.0045(1) 0.124(90) 0.0067(2) 0.817(0) -0.061(4) -5.09(2) 0.0048 0.242(7) 0.038(2) 0.0048(1) -0.017(70) 0.0072(1) 0.805(7) -0.061(2) -4.99(6) 0.0051 0.293(4) 0.049(3) 0.0051(1) -0.225(85) 0.0076(3) 0.788(0) -0.065(7) -4.93(2) 0.0054 0.342(6) 0.060(4) 0.0054(1) -0.358(99) 0.0081(2) 0.788(2) -0.064(4) -4.98(1) 0.0058 0.389(4) 0.074(5) 0.0058(1) -0.067(90) 0.0086(2) 0.801(6) -0.066(5) -4.89(6) 0.0062 0.440(3) 0.091(3) 0.0062(1) -0.171(99) 0.0092(2) 0.798(0) -0.069(5) -4.88(1) 0.0067 0.490(4) 0.108(5) 0.0067(1) -0.198(97) 0.0100(3) 0.797(0) -0.072(6) -4.87(2) 0.0073 0.537(5) 0.130(5) 0.0075(3) 0.061(98) 0.0112(5) 0.809(9) -0.076(6) -4.78(5) 0.0080 0.591(4) 0.154(5) 0.0081(4) 0.064(96) 0.0121(6) 0.831(0) -0.083(6) -4.96(3) 0.0089 0.638(5) 0.189(10) 0.0089(4) 0.124(98) 0.0135(8) 0.831(0) -0.085(6) -4.84(5) 0.0100 0.688(3) 0.239(9) 0.0101(4) 0.304(99) 0.0153(8) 0.831(0) -0.094(9) -4.64(4) 0.0114 0.737(4) 0.280(8) 0.0116(4) 0.007(75) 0.0178(9) 0.798(5) -0.107(8) -4.36(6) 0.0131 0.782(3) 0.347(12) 0.0133(8) -0.103(96) 0.0207(16) 0.797(0) -0.116(9) -4.24(4) 0.0155 0.839(4) 0.439(12) 0.0154(4) 0.023(68) 0.0242(8) 0.800(9) -0.129(5) -4.02(10) 0.0190 0.890(5) 0.562(12) 0.0195(7) 0.045(90) 0.0308(17) 0.820(0) -0.157(9) -3.89(4) 0.0241 0.944(3) 0.746(7) 0.0237(11) 0.120(84) 0.0391(19) 0.820(0) -0.189(9) -3.49(2)
142
Table 4.19 Molecular simulation results for the vapour-liquid equilibria of argon-krypton at T = 177.38K using BFW+AT potentials.
*P 0.0335 0.0350 0.0367 0.0384 0.0404
( )Vx Ar 0.0(0) 0.042(2) 0.082(4) 0.124(4) 0.166(4)
( ) 0.0(0) 0.013(1) 0.029(2) 0.047(1) 0.062(2) 0.034(0) 0.034(1) 0.036(2) 0.039(2) 0.040(2)
Lx Ar * totalVP
* -0.012(1) -0.014(3) -0.015(5) -0.015(4) -0.018(8) 2V bodyP
* 0.00046(7) 0.00049(17) 0.00072(29) 0.00080(28) 0.00077(39) 3V bodyP
* 0.037(17) -0.008(18) -0.058(71) 0.0009(47) 0.031(106) L totalP
* -0.825(13) -0.829(10) -0.865(71) -0.796(41) -0.787(103) 2L bodyP
* 0.332(14) 0.272(5) 0.247(4) 0.242(10) 0.279(12) 3L bodyP
* 0.051(2) 0.053(4) 0.058(7) 0.062(7) 0.066(10) Vρ * 0.658(10) 0.647(1) 0.636(3) 0.631(10) 0.667(9) Lρ * -0.435(23) -0.442(57) -0.489(72) -0.508(69) -0.494(83)
2V bodyE * 0.0028(3) 0.0029(7) 0.0039(1) 0.0040(10) 0.0036(12)
3V bodyE * -4.03(6) -3.96(2) -3.82(2) -3.76(6) -3.95(6)
2L bodyE * 0.143(3) 0.140(2) 0.129(2) 0.127(3) 0.139(4)
3L bodyE *P
0.0426 0.0451 0.0479 0.0512 0.0550
( )Vx Ar 0.212(4) 0.256(6) 0.299(5) 0.322(5) 0.386(13)
( ) 0.080(3) 0.105(1) 0.128(5) 0.153(5) 0.192(6) 0.042(1) 0.045(2) 0.047(2) 0.057(8) 0.056(4)
Lx Ar * totalVP
* -0.017(4) -0.021(6) -0.018(3) -0.049(10) -0.034(15) 2V bodyP
* 0.00085(22) 0.00100(61) 0.00098(17) 0.00305(68) 0.0019(12) 3V bodyP
* 0.0226(11) 0.0633(64) 0.0238(62) 0.065(82) 0.056(68) L totalP
* -0.745(109) -0.753(47) -0.789(58) -0.758(91) -0.686(70) 2L bodyP
* 0.289(3) 0.250(13) 0.245(18) 0.252(9) 0.211(6) 3L bodyP
* 0.067(4) 0.075(9) 0.074(5) 0.118(4) 0.102(18) Vρ * 0.679(0) 0.644(15) 0.648(15) 0.664(8) 0.618(4) Lρ * -0.503(51) -0.528(87) -0.524(32) -0.793(65) -0.629(13)
2V bodyE * 0.0038(4) 0.0042(6) 0.0043(6) 0.0086(18) 0.0057(25)
3V bodyE * -3.99(2) -3.76(7) -3.72(10) -3.74(5) -3.46(4)
2L bodyE * 0.142(1) 0.129(4) 0.126(6) 0.126(3) 0.113(3)
3L bodyE *P
0.0593 0.0641 0.0700 0.0771 0.0867
( )Vx Ar 0.423(14) 0.469(9) 0.504(14) 0.564(11) 0.375(9)
( ) 0.224(7) 0.257(9) 0.298(8) 0.359(8) 0.420(9) 0.060(5) 0.066(5) 0.069(6) 0.072(4) 0.039(93)
Lx Ar * totalVP
* -0.037(14) -0.045(13) -0.055(15) -0.053(13) -0.634(81) 2V bodyP
* 0.0024(13) 0.0032(8) 0.0047(19) 0.0035(14) 0.1267(88) 3V bodyP
* 0.055(26) 0.084(56) 0.069(36) 0.094(44) 0.079(36) L totalP
* -0.690(45) -0.712(54) -0.704(34) -0.634(42) -0.610(40) 2L bodyP
* 0.203(21) 0.223(7) 0.208(15) 0.181(18) 0.158(16) 3L bodyP
* 0.109(24) 0.123(17) 0.136(14) 0.139(16) 0.697(8) Vρ * 0.617(24) 0.653(5) 0.643(15) 0.622(20) 0.604(23) Lρ * -0.683(12) -0.750(79) -0.860(12) -0.766(10) -3.503(80)
2V bodyE * 0.0069(28) 0.0084(13) 0.0113(34) 0.0082(24) 0.0604(35)
3V bodyE * -3.38(14) -3.49(4) -3.38(10) -3.15(12) -2.96(12)
2L bodyE *
3L bodyE 0.109(7) 0.113(3) 0.107(5) 0.096(6) 0.087(5)
143
Table 4.20 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=163.15K using BFW + AT potentials.
*P 0.0194 0.0202 0.0212 0.0222 0.0234 0.0247
( )Vx Ar 0.0(0) 0.045(3) 0.088(2) 0.134(2) 0.176(3) 0.230(4)
( ) 0.0(0) 0.010(1) 0.024(1) 0.035(1) 0.051(2) 0.066(5) 0.0196(3) 0.0202(6) 0.0214(6) 0.0227(10) 0.0234(10) 0.0246(14)
Lx Ar * totalVP
* -0.0045(6) -0.0046(15) -0.0044(6) -0.0042(8) -0.0057(10) -0.0053(9) 2V bodyP
* 0.00015(2) 0.00016(3) 0.00015(5) 0.00021(2) 0.00016(5) 0.00015(4) 3V bodyP
* 0.027(22) -0.064(78) -0.006(64) -0.011(78) -0.067(99) -0.031(86) L totalP
* -0.878(26) -0.912(76) -0.891(32) -0.863(77) -0.906(101) -0.827(84) 2L bodyP
* 0.340(15) 0.338(4) 0.328(2) 0.348(2) 0.326(2) 0.330(4) 3L bodyP
* 0.029(1) 0.031(2) 0.032(1) 0.033(1) 0.036(2) 0.037(2) Vρ * 0.701(11) 0.704(0) 0.698(15) 0.715(0) 0.705(0) 0.705(0) Lρ * -0.276(13) -0.278(31) -0.288(21) -0.277(15) -0.301(28) -0.288(28)
2V bodyE * 0.0012(2) 0.0012(2) 0.0015(5) 0.0012(2) 0.0015(4) 0.0013(3)
3V bodyE * -4.33(7) -4.34(1) -4.26(11) -4.35(1) -4.25(2) -4.23(2)
2L bodyE *
3L bodyE
0.161(4) 0.160(2) 0.156(7) 0.162(1) 0.154(2) 0.156(2)
*P 0.0261 0.0277 0.0296 0.0316 0.0341 0.0372
( )Vx Ar 0.272(5) 0.315(5) 0.363(5) 0.407(4) 0.455(5) 0.497(6)
( ) 0.083(2) 0.105(5) 0.128(5) 0.147(6) 0.180(5) 0.216(8) 0.0263(12) 0.0276(13) 0.0296(16) 0.0322(16) 0.0349(14) 0.0377(22)
Lx Ar * totalVP
* -0.0069(13) -0.0070(17) -0.0080(23) -0.0093(20) -0.0095(21) -0.0140(31) 2V bodyP
* 0.00017(4) 0.00021(7) 0.00026(7) 0.00030(7) 0.00034(7) 0.00052(13) 3V bodyP
* -0.180(67) -0.176(103) -0.129(122) -0.297(123) -0.191(72) -0.168(75) L totalP
* -0.886(64) -0.854(99) -0.806(120) -0.876(123) -0.838(70) -0.800(74) 2L bodyP
* 0.326(4) 0.317(4) 0.309(2) 0.309(5) 0.295(4) 0.288(2) 3L bodyP
* 0.041(2) 0.043(3) 0.046(4) 0.0511(3) 0.055(4) 0.063(5) Vρ * 0.705(0) 0.705(0) 0.705(0) 0.705(0) 0.705(0) 0.705(0) Lρ * -0.301(22) -0.312(32) -0.334(33) -0.352(30) -0.357(29) -0.411(37)
2V bodyE * 0.0014(2) 0.0016(4) 0.0018(3) 0.0019(3) 0.0020(3) 0.0026(5)
3V bodyE * -4.21(2) -4.15(2) -4.08(2) -4.07(4) -3.99(2) -3.91(2)
2L bodyE *
3L bodyE
0.154(2) 0.150(2) 0.146(1) 0.146(2) 0.140(1) 0.136(1)
*P 0.0405 0.0447 0.0498 0.0565 0.0654 0.0819
( )Vx Ar 0.544(7) 0.588(7) 0.643(10) 0.686(9) 0.729(15) 0.584(9)
( ) 0.251(8) 0.305(10) 0.348(5) 0.417(6) 0.492(6) 0.586(10) 0.0398(23) 0.0452(18) 0.0481(22) 0.0582(43) 0.0656(33) -0.101(125)
Lx Ar * totalVP
* -0.0134(32) -0.0221(40) -0.0210(50) -0.0319(71) -0.0606(285) -0.640(119) 2V bodyP
* 0.00055(20) 0.00094(24) 0.00112(55) 0.00190(76) 0.0039(21) 0.0953(61) 3V bodyP
* -0.198(94) -0.087(97) -0.075(42) -0.030(71) 0.043(72) 0.063(96) L totalP
* -0.813(92) -0.678(97) -0.750(43) -0.736(69) -0.684(71) -0.613(96) 2L bodyP
* 0.281(5) 0.276(4) 0.257(6) 0.240(4) 0.231(4) 0.210(2) 3L bodyP
* 0.065(6) 0.082(5) 0.084(7) 0.109(10) 0.151(34) 0.658(0) Vρ * 0.705(0) 0.705(0) 0.705(0) 0.705(0) 0.705(0) 0.705(0) Lρ * -0.409(46) -0.493(43) -0.501(85) -0.610(76) -0.746(148) -2.982(26)
2V bodyE * 0.0027(7) 0.0037(8) 0.0042(18) 0.0056(17) 0.0078(27) 0.0482(30)
3V bodyE * -3.85(3) -3.75(4) -3.65(3) -3.51(2) -3.39(2) -3.20(3)
2L bodyE *
3L bodyE 0.133(2) 0.130(4) 0.122(2) 0.114(2) 0.109(2) 0.099(1)
144
Table 4.21 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=158.15K using BFW + AT potentials
*P 0.0157 0.0165 0.0172 0.0181 0.0190 0.0201
( )Vx Ar 0.0(0) 0.045(3) 0.092(4) 0.138((3) 0.178(3) 0.227(6)
( )Lx Ar *
0.0(0) 0.010(1) 0.021(1) 0.034(1) 0.047(3) 0.059(1) 0.0157(1) 0.0168(7) 0.0175(6) 0.0182(4) 0.0195(11) 0.0208(10)
totalVP * -0.0029(2) -0.0030(9) -0.0034(8) -0.0036(9) -0.0043(11) -0.0040(9) 2V bodyP * 0.000061(9) 0.000080(28) 0.000067(20) 0.0000086(23) 0.000094(31) 0.000101(36) 3V bodyP * 0.029(22) -0.166(63) 0.013(87) -0.017(49) 0.051(52) -0.021(53) L totalP
* -0.901(24) -1.023(61) -0.901(84) -0.912(46) -0.862(37) -0.945(46) 2L bodyP * 0.367(16) 0.342(4) 0.379(3) 0.342(12) 0.351(16) 0.3561(15) 3L bodyP * 0.024(0) 0.025(1) 0.027(1) 0.028(1) 0.030(2) 0.032(1) Vρ * 0.719(10) 0.707(0) 0.732(0) 0.708(8) 0.718(10) 0.727(10) Lρ * -0.227(8) -0.235(27) -0.233(22) -0.244(20) -0.250(25) -0.254(20) 2V bodyE * 0.0008(1) 0.0010(3) 0.0008(2) 0.0010(2) 0.0010(3) 0.0010(9) 3V bodyE * -4.47(7) -4.38(1) -4.51(1) -4.34(5) -4.35(7) -4.39(6) 2L bodyE *
3L bodyE
0.170(5) 0.161(2) 0.173(1) 0.161(4) 0.162(5) 0.163(4)
*P 0.0213 0.0226 0.0241 0.0258 0.0278 0.0301
( )Vx Ar 0.277(6) 0.321(3) 0.367(3) 0.411(3) 0.460(5) 0.508(8)
( )Lx Ar *
0.078(3) 0.100(3) 0.121(4) 0.139(6) 0.165(5) 0.194(6) 0.0212(5) 0.0231(7) 0.0246(9) 0.0265(17) 0.0284(11) 0.0308(12)
totalVP * -0.0041(4) -0.0052(10) -0.0065(11) -0.0067(13) -0.0074(18) -0.0088(16) 2V bodyP * 0.000091(16) 0.000130(24) 0.000182(37) 0.000210(93) 0.000226(65) 0.000264(91) 3V bodyP * 0.013(75) 0.087(79) 0.036(70) -0.029(108) 0.035(88) -0.027(98) L totalP
* -0.891(57) -0.806(77) -0.832(68) -0.881(100) -0.847(75) -0.871(97) 2L bodyP * 0.341(15) 0.337(7) 0.315(17) 0.338(3) 0.333(10) 0.328(4) 3L bodyP * 0.032(0) 0.036(1) 0.039(2) 0.042(3) 0.045(2) 0.050(2) Vρ * 0.720(9) 0.711(0) 0.706(12) 0.726(0) 0.728(6) 0.731(0) Lρ * -0.248(10) -0.272(21) -0.296(17) -0.304(31) -0.314(26) -0.323(33) 2V bodyE * 0.0009(1) 0.0011(1) 0.0015(3) 0.0016(5) 0.0016(4) 0.0017(5) 3V bodyE * -4.30(6) -4.23(3) -4.12(9) -4.23(2) -4.18(4) -4.13(3) 2L bodyE *
3L bodyE 0.158(5) 0.158(3) 0.148(5) 0.155(1) 0.152(3) 0.149(2)
*P 0.0329 0.0363 0.0404 0.0457 0.0525 0.0666
( )Vx Ar 0.553(9) 0.604(5) 0.650(6) 0.704(7) 0.739(13) 0.528(15)
( )Lx Ar *
0.230(7) 0.281(11) 0.329(13) 0.404(5) 0.466(8) 0.556(9) 0.0345(15) 0.0365(13) 0.0396(25) 0.0456(17) 0.0547(31) 0.0459(105)
totalVP * -0.0113(29) -0.0126(20) -0.0147(29) -0.0177(52) -0.0350(99) -0.6531(1142) 2V bodyP * 0.00039(11) 0.00040(9) 0.000058(18) 0.000072(28) 0.00194(83) 0.13524(151) 3V bodyP * 0.313(80) 0.420(99) 0.380(136) -0.131(55) 0.133(70) 0.105(53) L totalP
* -0.622(80) -0.477(103) -0.507(137) -0.838(51) -0.699(70) -0.695(55) 2L bodyP * 0.256(1) 0.256(4) 0.256(2) 0.208(2) 0.141(2) 0.139(3) 3L bodyP * 0.058(4) 0.062(3) 0.068(5) 0.080(7) 0.112(14) 0.760(30) Vρ * 0.772(0) 0.772(0) 0.772(0) 0.667(1) 0.734(0) 0.734(0) Lρ * -0.367(3) -0.371(29) -0.405(40) -0.435(54) -0.598(100) -3.581(178) 2V bodyE * 0.0021(5) 0.0021(3) 0.0026(6) 0.0028(8) 0.0056(17) 0.0590(44) 3V bodyE * -4.31(2) -4.19(4) -4.09(4) -3.36(2) -3.63(2) -3.46(2) 2L bodyE *
3L bodyE 0.199(1) 0.194(1) 0.187(1) 0.104(1) 0.107(1) 0.098(1)
145
Table 4.22 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=153.15K using BFW + AT potentials *P
0.0125 0.0130 0.0136 0.0143 0.0151 0.0160 0.0169
( )Vx Ar 0.0(0) 0.046(1) 0.091(2) 0.138(3) 0.185(2) 0.235(5) 0.276(5)
( )Lx Ar*
0.0(0) 0.009(1) 0.019(2) 0.031(2) 0.043(2) 0.053(2) 0.069(3)
0.0125(1) 0.0132(4) 0.0135(5) 0.0142(11) 0.0153(5) 0.0160(7) 0.0175(10) totalVP
* -0.0018(1) -0.0023(5) -0.0020(4) -0.0022(5) -0.0028(4) -0.0026(5) -0.0033(8) 2V bodyP
* 0.000031(4) 0.000036(9) 0.000038(13) 0.000048(25) 0.000047(14) 0.000050(12) 0.000073(26) 3V bodyP
* 0.013(16) -0.166(73) 0.022(85) -0.071(123) -0.003(68) -0.009(74) -0.120(91) L totalP
* -0.931(18) -0.996(72) -0.902(77) -0.954(120) -0.893(66) -0.945(59) 1.006(91) 2L bodyP
* 0.389(15) 0.349(4) 0.372(15) 0.371(3) 0.387(3) 0.377(14) 0.369(2) 3L bodyP
* 0.019(0) 0.020(1) 0.020(1) 0.021(1) 0.024(1) 0.024(1) 0.027(2) Vρ * 0.734(9) 0.710(0) 0.729(10) 0.727(0) 0.741(0) 0.736(8) 0.738(0) Lρ * -0.182(5) -0.189(15) -0.184(15) -0.193(30) -0.201(14) -0.202(14) -0.219(25)
2V bodyE * 0.00054(7) 0.00057(15) 0.00060(21) 0.00071(36) 0.00064(18) 0.00066(18) 0.00086(29)
3V bodyE * -4.58(6) -4.41(1) -4.49(7) -4.49(2) -4.53(1) -4.49(5) -4.46(2)
2L bodyE *
3L bodyE 0.177(4) 0.164(4) 0.170(4) 0.170(1) 0.174(1) 0.171(4) 0.167(1)
*P
0.0180 0.0192 0.0207 0.0223 0.0242 0.0264 0.0292
( )Vx Ar 0.325(4) 0.372(6) 0.417(5) 0.467(3) 0.511(7) 0.561(6) 0.611(6)
( )Lx Ar *
0.088(4) 0.105(5) 0.127(4) 0.153(8) 0.180(7) 0.222(4) 0.266(7)
0.0182(12) 0.0192(5) 0.0211(11) 0.0222(12) 0.0250(14) 0.0269(12) 0.0297(12) totalVP
* -0.0034(10) -0.0039(10) -0.0043(10) -0.0040(5) -0.0061(17) -0.0062(6) -0.0077(22) 2V bodyP
* 0.000074(30) 0.000087(26) 0.000097(24) 0.000097(25) 0.000162(60) 0.000167(46) 0.000214(62)3V bodyP
* -0.008(110) -0.125(131) -0.059(65) 0.010(171) 0.232(80) 0.004(56) -0.031(77) L totalP
* -0.927(100) -0.941(132) -0.922(61) -0.898(150) -0.755(81) -0.862(59) -0.822(78) 2L bodyP
* 0.365(3) 0.363(2) 0.357(5) 0.348(19) 0.399(3) 0.315(10) 0.300(3) 3L bodyP
* 0.028(2) 0.030(1) 0.033(2) 0.034(2) 0.041(3) 0.043(2) 0.049(4) Vρ * 0.738(0) 0.739(0) 0.739(0) 0.738(14) 0.776(1) 0.727(7) 0.725(0) Lρ * -0.221(21) -0.231(20) -0.243(18) -0.236(20) -0.280(31) -0.280(23) -0.296(30)
2V bodyE * 0.00082(22) 0.00093(24) 0.00094(20) 0.00092(21) 0.00127(37) 0.00125(29) 0.00140(33)
3V bodyE * -4.41(3) -4.39(3) -4.34(2) -4.28(7) -4.45(3) -4.05(6) -3.95(7)
2L bodyE *
3L bodyE 0.164(1) 0.163(1) 0.161(2) 0.157(5) 0.172(1) 0.144(3) 0.138(1)
*P
0.0326 0.705(4) 0.755(8) 0.804(6) 0.846(8) 0.794(8)
( )Vx Ar 0.658(4) 0.365(12) 0.440(7) 0.538(8) 0.658(9) 0.811(7)
( )Lx Ar*
0.314(8) 0.0378(20) 0.0431(21) 0.0499(15) 0.0690(74) 0.4226(1310)
0.0330(21) -0.0147(32) -0.0177(44) -0.0289(52) -0.0648(134) -0.3055(1284) totalVP
* -0.0100(24) 0.00055(16) 0.00082(31) 0.00132(35) 0.0051(17) -0.5921(33) 2V bodyP
* 0.00034(13) 0.175(114) 0.051(45) 0.100(71) 0.066(68) 0.116(97) 3V bodyP
* -0.042(91) -0.660(114) -0.793(38) -0.694(72) -0.694(84) -0.634(97 L totalP
* -0.812(90) 0.255(2) 0.280(15) 0.263(3) 0.229(4) 0.198(1) 2L bodyP
* -0.822(3) 0.068(5) 0.079(6) 0.102(7) 0.169(20) 0.817(0) 3L bodyP
* 0.056(5) 0.761(0) 0.745(11) 0.749(0) 0.742(0) 0.742(0) Vρ * 0.721(0) -0.395(41) -0.444(53) -0.530(52) -0.815(101) -3.299(42) Lρ * -0.341(38) 0.00258(57) 0.00332(10) 0.00420(93) 0.00958(22) 0.08247(13)
2V bodyE * 0.00195(57) -3.98(4) -3.72(7) -3.54(3) -3.28(3) -3.00(2)
3V bodyE * -3.87(9) 0.179(1) 0.125(5) 0.117(1) 0.103(1) 0.089(1)
2L bodyE *
3L bodyE 0.141(1) 0.705(4) 0.755(8) 0.804(6) 0.846(8) 0.794(8)
146
Table 4.23 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=148.15K using BFW + AT potentials.
*P
0.0098 0.0103 0.0108 0.0113 0.0119 0.0126 0.0134
( )Vx Ar 0.0(0) 0.045(1) 0.092(3) 0.139(4) 0.188(3) 0.233(4) 0.279(4)
( )Lx Ar*
0.0(0) 0.008(1) 0.018(1) 0.029(1) 0.040(4) 0.051(4) 0.066(3)
0.0098(1) 0.0103(4) 0.0109(4) 0.0112(2) 0.0121(4) 0.0129(5) 0.0132(4) totalVP
* -0.0012(1) -0.0015(3) -0.0014(3) -0.0015(3) -0.0015(3) -0.0019(4) -0.0017(4) 2V bodyP
* 0.000017(2) 0.000025(15) 0.000022(6) 0.000020(4) 0.000026(7) 0.000029(8) 0.000027(4) 3V bodyP
* 0.029(26) -0.075(119) -0.064(102) -0.061(99) -0.016(83) 0.023(62) 0.201(83) L totalP
* -0.930(23) -0.993(100) -0.920(101) -0.909(96) -0.957(83) -0.955(66) -0.743(85) 2L bodyP
* 0.412(11) 0.406(12) 0.406(4) 0.403(3) 0.395(2) 0.420(25) 0.452(3) 3L bodyP
* 0.015(0) 0.016(1) 0.017(1) 0.017(1) 0.018(1) 0.020(9) 0.020(1) Vρ * 0.746(7) 0.747(7) 0.747(0) 0.747(0) 0.746(12) 0.761(14) 0.783(0) Lρ * -0.150(3) -0.158(23) -0.160(13) -0.157(11) -0.164(12) -0.173(14) -0.168(11)
2V bodyE * 0.00036(3) 0.00051(29) 0.00043(11) 0.00038(1) 0.00045(10) 0.00046(12) 0.00044(6)
3V bodyE * -4.69(5) -4.67(6) -4.65(2) -4.62(1) -4.59(9) -4.68(11) -4.76(2)
2L bodyE *
3L bodyE 0.184(3) 0.181(4) 0.181(1) 0.179(1) 0.176(6) 0.184(7) 0.192(1)
*P
0.0142 0.0152 0.0163 0.0176 0.0192 0.0211 0.0233
( )Vx Ar 0.332(4) 0.377(6) 0.424(5) 0.473(4) 0.521(6) 0.569(4) 0.615(5)
( )Lx Ar*
0.082(4) 0.100(4) 0.120(7) 0.147(7) 0.172(7) 0.200(10) 0.248(6)
0.0145(3) 0.0157(4) 0.0169(8) 0.0178(6) 0.0186(9) 0.0213(6) 0.0242(11) totalVP
* -0.0022(5) -0.0026(6) -0.0032(9) -0.0029(6) -0.0033(11) -0.0042(8) -0.0057(9) 2V bodyP
* 0.000034(7) 0.000047(1) 0.000074(43) 0.000051(7) 0.000061(24) 0.000092(22) 0.000143(50) 3V bodyP
* -0.026(65) -0.019(57) -0.128(131) 0.017(113) -0.046(103) 0.122(101) 0.033(53) L totalP
* -0.956(65) -0.916(57) -0.973(131) -0.832(113) -0.851(107) -0.793(95) -0.836(54) 2L bodyP
* 0.383(27) 0.364(13) 0.348(3) 0.371(2) 0.362(5) 0.219(7) 0.325(10) 3L bodyP
* 0.023(1) 0.025(1) 0.027(1) 0.028(1) 0.030(2) 0.035(1) 0.041(2) Vρ * 0.747(16) 0.736(10) 0.733(0) 0.752(0) 0.752(0) 0.770(0) 0.741(6) Lρ * -0.177(8) -0.192(17) -0.212(28) -0.198(9) -0.205(25) -0.233(14) -0.259(23)
2V bodyE * 0.00048(10) 0.00061(16) 0.00087(46) 0.00059(75) 0.00066(24) 0.00086(19) 0.00113(33)
3V bodyE * -4.51(12) -4.39(7) -4.32(4) -4.38(3) -4.32(4) -4.41(4) -4.09(5)
2L bodyE *
3L bodyE 0.171(8) 0.165(3) 0.158(1) 0.164(1) 0.160(2) 0.198(12) 0.146(3)
*P
0.0261 0.0296 0.0345 0.0401 0.0501 0.0761
( )Vx Ar 0.668(6) 0.710(6) 0.763(6) 0.815(5) 0.860(6) 0.777(12)
( )Lx Ar*
0.289(9) 0.338(6) 0.411(10) 0.507(8) 0.630(9) 0.795(4)
0.0256(7) 0.0307(13) 0.0354(17) 0.0400(22) 0.0492(26) -0.0850(788) totalVP
* -0.0052(11) -0.0095(20) -0.0130(38) -0.0141(33) -0.0340(76) -0.7155(743) 2V bodyP
* 0.000122(33) 0.000278(79) 0.000441(144) 0.000507(92) 0.00182(77) 0.117(10) 3V bodyP
* 0.014(100) 0.122(56) -0.023(109) 0.078(72) 0.153(97) 0.090(97) L totalP
* -0.855(92) -0.795(61) -0.795(110) -0.721(72) -0.636(97) -0.654(97) 2L bodyP
* 0.328(14) 0.350(4) 0.323(3) 0.296(2) 0.274(5) 0.199(5) 3L bodyP
* 0.042(2) 0.054(3) 0.065(6) 0.073(4) 0.111(10) 0.770(22) Vρ * 0.751(10) 0.778(1) 0.776(0) 0.776(0) 0.783(0) 0.745(9) Lρ * -0.247(19) -0.321(34) -0.365(38) -0.384(23) -0.574(85) -3.207(136)
2V bodyE * 0.00096(23) 0.00165(37) 0.00216(50) 0.00227(29) 0.00532(20) 0.0506(33)
3V bodyE * -4.07(7) -4.12(2) -3.95(3) -3.74(2) -3.52(3) -3.02(4)
2L bodyE *
3L bodyE 0.145(4) 0.149(1) 0.139(1) 0.127(1) 0.116(2) 0.089(2)
147
Table 4.24 Molecular simulation results for the vapour-liquid equilibria of argon-
krypton at T=143.15K using BFW + AT potentials *P
0.0076 0.0079 0.0082 0.0087 0.0092 0.0097 0.0103
( )Vx Ar 0.0(0) 0.048(4) 0.094(1) 0.145(5) 0.187(6) 0.238(4) 0.285(4)
( )Lx Ar*
0.0(0) 0.008(1) 0.018(1) 0.026(2) 0.037(1) 0.046(2) 0.061(2)
0.0076(1) 0.0079(3) 0.0083(1) 0.0087(2) 0.0090(2) 0.0098(5) 0.0102(3) totalVP
* -0.00076(4) -0.00080(15) -0.00083(16) -0.00090(22) -0.00091(25) -0.00123(23) -0.00109(31) 2V bodyP
* 0.000008(1) 0.000011(3) 0.000009(2) 0.000011(3) 0.000012(3) 0.000015(5) 0.000018(7) 3V bodyP
* 0.022(23) -0.010(160) 0.206(56) 0.016(126) -0.038(77) -0.099(104) 0.008(53) L totalP
* -0.950(21) -0.957(163) -0.729(55) -0.908(118) -0.944(75) -1.007(101) -0.905(52) 2L bodyP
* 0.435(7) 0.442(1) 0.443(4) 0.428(6) 0.417(3) 0.417(5) 0.419(3) 3L bodyP
* 0.012(0) 0.012(1) 0.013(1) 0.014(0) 0.014(0) 0.015(1) 0.016(1) Vρ * 0.759(4) 0.767(0) 0.767(0) 0.759(0) 0.757(0) 0.760(0) 0.764(0) Lρ * -0.119(4) -0.126(11) -0.126(9) -0.126(11) -0.1289(8) -0.137(12) -0.134(10)
2V bodyE * 0.00023(4) 0.00029(8) 0.00023(4) 0.00026(8) 0.00027(9) 0.00031(9) 0.00036(15)
3V bodyE * -4.79(2) -4.82(3) -4.77(2) -4.74(1) -4.68(1) -4.69(2) -4.68(1)
2L bodyE *
3L bodyE 0.191(2) 0.192(1) 0.192(2) 0.187(2) 0.184(1) 0.183(2) 0.183(1)
*P
0.0110 0.0117 0.0126 0.0137 0.0149 0.0163 0.0180
( )Vx Ar 0.334(5) 0.380(5) 0.430(5) 0.476(5) 0.526(2) 0.569(5) 0.621(6)
( )Lx Ar*
0.077(1) 0.092(4) 0.110(5) 0.133(6) 0.156(6) 0.190(8) 0.223(9)
0.0109(2) 0.0123(4) 0.0126(4) 0.0136(5) 0.0152(8) 0.0163(5) 0.0181(6) totalVP
* -0.00135(51) -0.00160(26) -0.00163(42) -0.00203(32) -0.00205(38) -0.00245(70) -0.00332(10) 2V bodyP
* 0.000021(9) 0.000027(6) 0.000026(5) 0.000030(12) 0.000035(5) 0.000048(10) 0.000056(15) 3V bodyP
* 0.043(39) 0.052(94) 0.041(114) -0.029(73) -0.217(121) -0.162(111) 0.032(102) L totalP
* -0.901(30) -0.895(92) -0.867(112) -0.917(75) -0.999(123) -0.965(111) -0.882(96) 2L bodyP
* 0.406(1) 0.428(3) 0.406(4) 0.392(3) 0.355(5) 0.345(5) 0.372(15) 3L bodyP
* 0.017(1) 0.019(1) 0.020(1) 0.022(1) 0.024(1) 0.026(1) 0.030(2) Vρ * 0.759(9) 0.775(1) 0.764(0) 0.763(0) 0.747(0) 0.742(0) 0.768(10) Lρ * -0.145(14) -0.159(10) -0.155(8) -0.164(15) -0.170(9) -0.184(12) -0.193(17)
2V bodyE * 0.00040(18) 0.00045(9) 0.00043(9) 0.00045(16) 0.00048(8) 0.00060(10) 0.00060(12)
3V bodyE * -4.60(6) -4.67(2) -4.57(3) -4.50(3) -4.35(3) -4.25(3) -4.33(7)
2L bodyE *
3L bodyE 0.178(4) 0.184(1) 0.177(1) 0.171(1) 0.158(2) 0.155(2) 0.161(4)
*P
0.0202 0.0229 0.0265 0.0315 0.0385 0.0500
( )Vx Ar 0.672(3) 0.724(5) 0.769(5) 0.823(4) 0.871(4) 0.927(5)
( )Lx Ar*
0.271(15) 0.319(7) 0.391(3) 0.481(9) 0.612(11) 0.774(4)
0.0200(7) 0.0235(7) 0.0267(18) 0.0312(11) 0.0398(28) 0.0515(51) totalVP
* -0.00428(79) -0.0051(18) -0.0069(19) -0.0089(16) -0.0172(34) -0.0459(13) 2V bodyP
* 0.000079(12) 0.000113(35) 0.000191(11) 0.000255(50) 0.00067(18) 0.00257(77) 3V bodyP
* -0.044(142) -0.134(81) -0.161(34) -0.139(77) -0.040(110) 0.143(54) L totalP
* -0.843(146) -0.922(81) -0.887(34) -0.875(76) -0.796(109) -0.585(54) 2L bodyP
* 0.343(5) 0.329(5) 0.313(2) 0.285(4) 0.251(3) 0.255(2) 3L bodyP
* 0.034(1) 0.040(3) 0.047(4) 0.056(3) 0.079(7) 0.133(17) Vρ * 0.760(0) 0.760(0) 0.760(0) 0.760(0) 0.760(0) 0.792(0) Lρ * -0.212(11) -0.236(27) -0.269(33) -0.306(21) -0.413(37) -0.641(75)
2V bodyE * 0.00075(8) 0.00090(21) 0.00129(62) 0.00148(23) 0.00273(54) 0.00616(12)
3V bodyE * -4.17(6) -4.08(3) -3.94(1) -3.74(2) -3.46(3) -3.31(1)
2L bodyE *
3L bodyE 0.150(2) 0.144(2) 0.137(1) 0.125(1) 0.110(1) 0.141(1)
148
149
Table 4.25 three-body effects on solid-liquid phase equilibrium properties of argon *
T* *2Lρ *
2Sρ *2P *
3Lρ *3Sρ *
3P
0.6667 0.859 0.968 0.6448 0.860 0.956 1.2970
0.7609 0.878 0.976 1.6672 0.874 0.964 2.2740
0.8505 0.894 0.987 2.7083 0.889 0.973 3.3081
0.9500 0.908 0.996 3.8320 0.910 0.989 4.7172
0.9914 0.916 1.002 4.3726 0.917 0.995 5.2566
1.0500 0.928 1.010 5.2110 0.928 1.000 6.1056
1.1288 0.941 1.021 6.3171 0.939 1.012 7.1697
1.2678 0.970 1.045 8.6969 0.963 1.033 9.3784
1.4168 0.987 1.059 10.7257 0.991 1.057 12.1093
Table 4.26 three-body effects on solid-liquid phase equilibrium properties of krypton *
T* *2Lρ *
2Sρ *2P *
3Lρ *3Sρ *
3P
0.5750 0.827 0.951 -0.3933 0.829 0.934 0.3058
0.7000 0.852 0.956 0.7789 0.851 0.944 1.5217
0.8370 0.882 0.973 2.2954 0.885 0.966 3.3019
1.0000 0.918 1.000 4.5653 0.921 0.994 5.7115
1.1738 0.951 1.026 7.0942 0.951 1.018 8.287
1.2630 0.966 1.038 8.4488 0.964 1.03 9.6026
1.4512 0.998 1.066 11.6064 0.997 1.062 12.8523
1.6196 1.020 1.086 14.3178 1.021 1.081 15.9501
1.7930 1.043 1.106 17.2907 1.042 1.100 18.9673
150
Table 4.27 three-body effects on solid-liquid phase equilibrium properties of xenon *
T* *2Lρ *
2Sρ *2P *
3Lρ *3Sρ *
3P
0.5743 0.829 0.934 -0.7392 0.823 0.924 0.3319
0.6405 0.842 0.943 -0.031 0.842 0.935 0.8784
0.7117 0.858 0.957 0.8908 0.853 0.941 1.7337
0.7829 0.874 0.966 1.7007 0.872 0.954 2.6064
0.8541 0.888 0.976 2.6856 0.887 0.965 3.7321
0.9252 0.902 0.986 3.4959 0.899 0.974 4.5068
0.9964 0.913 0.995 4.3701 0.915 0.987 5.6279
1.0356 0.923 1.001 4.9744 0.920 0.991 6.2333
1.1477 0.940 1.012 6.6436 0.936 1.005 7.7804
1.2918 0.959 1.033 8.525 0.959 1.025 10.0271
* The calculation reported in these tables, the error range for density (if not fixed) is ± 0.001, for pressure is ± 0.0001, for two-body and total potential energy is ± 0.001 and for three-body potential is ± 0.0001.
151
References Adams, D. J. (1976) "Calculating low-temperature vapor line by Monte Carlo", Mol. Phys. 32, 647-657. Adams, D. J. (1979) "Calculating the high-temperature vapor line by Monte Carlo", Mol. Phys. 37, 211-221. Agrawal, R. and D. A. Kofke (1995) "Thermodynamic and structural properties of model systems at solid-liquid coexistence II. Melting and sublimation of the Lennard-Jones system", Mol. Phys. 85, 43-59. Alder, B. J. and T. E. Wainwright (1957) "Phase transition for a hard sphere system", J. Chem. Phys. 27, 1208-1209. Allen, M. P. and D. J. Tildesley (1987) "Computer Simulation of Liquids", Clarendon press:Oxford. Allen, M. P. and D. J. Tildesley (1993) "Computer simulation in chemical physics", Kluwer Academic Publishers. Amar, J. G. (1989) "Application of the Gibbs ensemble to the study of fluid-fluid phase equilibrium in a binary mixture of symmetric non-additive hard spheres." Mol. Phys. 67, 739-745. Anta, J. A., E. Lomba and M. Lombardero (1997) "Influence of three-body forces on the gas-liquid coexistence of simple fluids: the phase equilibrium of argon", Phys. Rev. E 55, 2707-2712. Axilrod, B. M. and E. Teller (1943) "Interaction of the van der Waals' type between three atoms." J. Chem. Phys. 11, 299-300. Aziz, R. A. (1993) "A highly accurate interatomic potential for argon", J. Chem. Phys. 99, 4518-4525. Bade, W. L. (1958) "Drude-model calculation of dispersion forces. III. the forth-order contribution", J. Chem. Phys. 28, 282-284. Baez, L. A. and P. Clancy (1995) "Phase equilibria in extended simple point charge ice-water systems." J. Chem. Phys. 103, 9744-9755. Barker, J. A. (1976) "Interatomic potentials for inert gases from experimental data", London, Academic. Barker, J. A., R. A. Fisher and R. O. Watts (1971) "Liquid argon: Monte Carlo and molecular dynamics calculations", Mol. Phys. 21, 657-673. Barker, J. A., D. Henderson and W. R. Smith (1968) "Three-body forces in dense systems", Phys. Rev. Lett. 21, 134-136.
152
Barker, J. A., C. H. J. Johnson and T. H. Spurling (1972) "On the cancellation of certain three- and four-body interactions in inert gases." Aust.J.Chem. 25, 1811-1812. Barker, J. A. and M. L. Klein (1973) "Monte Carlo Calculations for solid and liquid Argon", Phys. Rev. B 7, 4707-4712. Barker, J. A. and A. Pompe (1968) "Atomic interactions in argon", Aust.J.Chem. 21, 1683-1694. Barker, J. A., R. O. Watts, J. K. Lees, T. P. Schafer and Y. T. Lee (1974) "Intermolecular potential for krypton and xenon", J. Phys. Chem. 61, 3081-3089. Bell, R. J. (1970) "Multipolar expansion for the non-additive third-order interaction energy of three atoms", J. Phys. B: atom.molec.phys. 3, 751-762. Bobetic, M. V. and J. A. Barker (1983) "Solid-state properties of argon, krypton, and xenon near 0 K from an [n(r)]-6 potential", Phys. Rev. B 28, 7317-7320. Boehler, R., M. Ross, P. Soderlind and D. B. Boercker (2001) "High-Pressure melting curves of argon, krypton, and xenon: deviation from corresponding states theory", Phys. Rev. Lett. 86, 5731-5734. Bukowski, R. and K. Szalewicz (2001) "Complete ab initio three-body nonadditive potential in Monte Carlo simulations of vapor-liquid equilibria and pure phases of argon", J. Chem. Phys. 114, 9518-9531. Car, R. and M. Parrinello (1985) "Unified approach for molecular dynamics and density-functional theory", Phys. Rev. Lett. 55, 2471-2474. Chalasinsky, G. and M. M. Szczesniak (1988) "On the connection between the supermolecular Møller-Plesset treatment of the interaction energy and the perturbation theory of intermolecular forces." Mol. Phys. 63, 205-224. Chapela, G. A., S. E. Martinez-Casas and C. Varea (1987) "Square well orthobaric densities via spinodal decomposition", J. Chem. Phys. 86, 5683-5668. Chen, B., Siepmann and M. L. Klein (2001) "Direct Gibbs ensemble Monte Carlo simulations for solid-vapor phase equilibria: applications to Lennard-Jonesium and Carbon dioxide", J. Phys. Chem. B 105, 9840-9848. Clifford, A. A., G. P and N. Platts (1977) "Lennard-Jones 12-6 parameters for 10 small molecules", J. Chem. Soc. Faraday Trans 73, 381-382. Cracknell, R. F., D. Nicholson, N. G. Parsonage and H. Evans (1990) "Rotational insertion bias: a novel method for simulating dense phases of structured particles, with particular application to water", Mol. Phys. 71, 931-943. Crawford, R. K. and W. B. Daniels (1968) "Melting in argon at high temperatures", Phys. Rev. Lett. 21, 367-369.
153
Crawford, R. K., W. F. Lewis and W. B. Daniels (1976) "Thermodynamics of solid argon at high temperature", J. Phys. C: Solid St. Phys. 9, 1381-1404. de Carvalho, R. J. F. L. and R. Evans (1997) "The screened Coulomb (Yukawa) charged hard sphere binary fluid." Mol. Phys. 92, 211-228. de Pablo, J. J. (1989) "Phase equilibria for fluid mixtures from Monte Carlo simulation." Fluid Phase Equilibria 53, 177-189. De, S., S. Teitel, Y. Shapir and E. H. Chimowitz (2002) "Monte Carlo simulation of Fickian diffusion in the critical region", J. Chem. Phys. 116, 3012-3017. Doran, M. B. and I. J. Zucker (1971) "Higher order multipole three-body van der Waals interactions and stability of rare gas solids", J. Phys. C: Solid St. Phys. 4, 307-312. Egelstaff, P. A. (1988) "Experimental tests for many-body forces in noble gas fluids", Canadian Journal of Chemistry 66, 596-608. Elrod, M. J. and R. J. Saykally (1994) "Many-body effects in intermolecular forces", Chem. Rev. 94, 1975-1997. Errington, J. R. (2004) "Solid-liquid phase coexistence of the Lennard-Jones system through phase-switch Monte Carlo simulation", J. Chem. Phys. 120, 3130-3141. Escobedo, F. A. (1998) "Novel pseudoensembles for simulation of multicomponent phase equilibria." J. Chem. Phys. 108, 8761-8772. Escobedo, F. A. and J. J. de Pablo (1997) "Pseudo-ensemble simulations and Gibbs-Duhem integrations for polymers", J. Chem. Phys. 106, 2911-2923. Evans, D. J., W. G. Hoover, B. H. Failor, B. Moran and A. J. C. Ladd (1983) "Nonequilibrium molecular dynamics via Gauss's principle of least constraint", Phys. Rev. A 28, 1016-1021. Evans, D. J. and G. P. Morriss (1984) "Nonlinear-response theory for steady planar Couette flow", Phys.l Rev. A 30, 1528-1530. Evans, D. J. and G. P. Morriss (1990). Statistical Mechanics of Nonequilibrium Liquids. London, Academic Press. Fass, S., J. H. V. Lenthe and J. G. Sinijder (2000) "Regular approximated scalar relativistic correlated ab initio schemes: applications to rare gas dimers", Mol. Phys. 98, 1467-1472. Findlay, A. (1951) "The phase rule and its applications", New York, Dover Publications,Inc. Fisher, M. E. and S.-y. Zinn (1998) "The shape of the van der Waals loop and unversal critical ampltude ratios", J. Phys. A: Math. Gen., L629-L635.
154
Ge, J. L., G. Marcelli, B. D. Todd and R. J. Sadus (2001) "Energy and pressure of shearing fluids at different state points", Phys. Rev. E 64, 021201-5. Ge, J. L., G. Marcelli, B. D. Todd and R. J. Sadus (2003a) "Scaling behavior for the pressure and energy of shearing fluids", Phys. Rev. E 67, 061201-3. Ge, J. L., G. W. Wu, B. D. Todd and R. J. Sadus (2003b) "Equilibrium and nonequilibrium molecular dynamics methods for determining solid-liquid phase coexistence at equilibrium", J. Chem. Phys. 119, 11017-11023. Gear, C. W. (1971) "Numerical initial value problems in ordinary differential equations", London, Pretice Hall. Gubbins, K. E. and N. Quirke (1996) "Molecular simulation and industrial applications: methods, examples and prospects." Gordon and breach science publishers. Guillot, B. (1989) "Triple dipoles in the absorption spectra of dense rare gas mixtures. II. Long rang interactions", J. Chem. Phys. 91, 3456-3462. Guillot, B., R. D. Mountain and G. Birnbaum (1989) "Triple dipoles in the absorption spectra of dense rare gas mixtures. I. Short range interactions." J. Chem. Phys. 90, 650-662. Haile, J. M. (1992) "Molecular dynamics simulation. Elementary methods." New York, Wiley. Hansen, J. P. and D. Schiff (1973) "Influence of interatomic repulsion on the structure of liquids at melting", Mol. Phys. 25, 1281-1290. Hansen, J. P. and L. Verlet (1969) "Phase transitions of the Lennard-Jones system", Phys. Rev. 184, 151-161. Heffelfinger, G., F. Van Swol and K. E. Gubbins (1987) "Liquid-vapor coexistence in a cylindrical pore", Mol. Phys. 61, 1381-1390. Hitchcock, M. R. and C. K. Hall (1999) "Solid-liquid phase equilibrium for binary Lennard-Jones mixtures", J. Chem. Phys. 110, 11433-11444. Hoef, M. A. V. d. and P. A. Madden (1999) "Three-body dispersion contributions to the thermodynamic properties and effective pair interactions in liquid argon", J. Chem. Phys. 111, 1520-1526. Hoef, M. A. V. d. and P. A. Madden. (1998) "Novel simulation model for many-body multipole dispersion interactions." Mol. Phys. 94. Hogervorst, W. (1971) "Transport and equilibrium properties of simple gases and forces between like and unlike atoms", Physica 51, 77-89. Hoover, W. G. (1985) "Canonical dynamics: equilibrium phase-space distributions", Phys. Rev. A 31, 1659-1697.
155
Hoover, W. G., D. J. Evans, R. B. Hickman, A. J. C. Ladd, W. T. Ashurst and B. Moran (1980) "Lennard-Jones triple-point bulk and shear viscosity. Green-Kubo theory, Hamiltonian mechanics, and nonequilibrium molecular dynamics", Phys. Rev. A 22, 1690-1697. Hoover, W. G., D. A. Young and R. Grover (1972) "Statistical mechanics of phase diagrams. I. Inverse power potentials and the close-packed to body-centered cubic transition." J. Chem. Phys. 56, 2207-2210. Horton, G. K. (1976). "Rare gas solids". M. L. Klein and J. A. Venables. London, Academic Press. Janzen, A. R. and R. A. Aziz (1997) "An accurate potential energy curve for helium based on ab initio calculations", J. Chem. Phys. 107, 914-919. Johnson, C. H. J. and T. H. Spurling (1974) "Non-additivity of intermolecular forces on the fourth virial coefficient." Aust. J. Chem. 27, 241-247. Kalyuzhnyi, Y. V. and P. T. Cummings (1996) "Phase diagram for the Lennard-Jones fluid modelled by the hard-core Yukawa fluid." Mol. Phys. 87, 1459-1462. Kiyohara, K., K. E. Gubbins and A. Z. Panagiotopoulos (1998) "Phase coexistence properties of polarizable water models." Mol. Phys. 94, 803-808. Klein, M. L. and J. A. Venables (1976) "Rare gas solids", London,, Academic Press. Kofke, D. A. (1993a) "Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line", J. Chem. Phys. 98, 4149-4163. Kofke, D. A. (1993b) "Gibbs-Duhem integration: a new method for direct evaluation of phase coexistence by molecular simulation", Mol. Phys. 78, 1331-1336. Kofke, D. A. and R. Agrawal (1995) "Thermodynamics and structural properties of model systems at solid-fluid coexistence I. Fcc and bcc soft sphere", Mol. Phys. 85, 23-42. Kotelyanskii, M. J. and R. Hentschke (1996) "Gibbs-ensemble molecular dynamics: Liquid-gas equilibria for Lennard-Jones spheres and n-hexane", Mol. Simul. 17, 95-112. Kristóf, T. and J. Liszi (1997) "Application of a new Gibbs ensemble Monte Carlo method to site-site interaction model fluids." Mol. Phys. 90, 1031-1034. Kristóf, T. and J. Liszi (1998) "Alternative Gibbs ensemble Monte Carlo implementations: application in mixtures", Mol. Phys. 94, 519-525. Kumar, A. and W. J. Meath (1985) "Pseudo-spectral dipole oscillator strengths and dipole-dipole and triple-dipole dispersion energy coefficients for HF, HCl, HBr, He, Ne, Ar, Kr and Xe." Mol. Phys. 54, 823-833.
156
Lahr, P. H. and W. G. Eversole (1962) "Compression isotherms of argon, krypton and xenon through the freezing zone", Journal of Chemical And Engineering data 7, 42-47. Laird, B. B. and A. D. J. Haymet (1992) "Phase diagram for the inverse six power potential system from molecular dynamics computer simulation." Mol. Phys. 75, 71-80. Laird, D. G. and C. S. Howat (1990) "Vapor-Liquid phase equilibria and molar volumes of Butadiene-Acetonitrile system from 300K to 335K", Fluid Phase Equilibria 60, 173-190. Lamm, M. H. and C. K. Hall (2001) "Monte Carlo simulation of complete phase diagrams for binary Lennard-Jones mixtures", Fluid Phase Equilibria 182, 37-46. Lamm, M. H. and C. K. Hall (2002) "Equilibria between solid, liquid and vapor phase in binary Lennard-Jones mixtures", Fluid Phase Equilibria 194-197, 197-206. Lees, A. W. and S. F. Edwards (1972) "The computer study of transport processes under extreme conditions", J. Phys.: Condense Matter 5, 1921-1929. Leonard, P. J. and J. A. Barker (1975) "Dipole oscillator strengths and related quantities for inert gases", London, Academic. Leonhard, K. and U. K. Deiters (2000) "Monte Carlo simulations of neon and argon using ab initio potentials", Mol. Phys. 98, 1603-1616. Lisal, M., R. Budinsky and V. Vacek (1997) "Vapour-liquid equilibria for dipolar two-centre Lennard-Jones fluids by Gibbs-Duhem integration", Fluid Phase Equilibria 135, 193-207. Lisal, M. and V. Vacek (1996a) "Direct evaluation of vapour-liquid equilibria by molecualr dynamics using Gibbs-Duhem integration", Mol. Simul. 17, 27-39. Lisal, M. and V. Vacek (1996b) "Direct evaluation of vapour-liquid equilibria of mixtures by molecular dynamics using Gibbs-Duhem integration", Mol. Simul. 18, 75-99. Lisal, M. and V. Vacek (1997) "Direct evaluation of solid-liquid equilibria by molecular dynamics using Gibbs-Duhem integration", Mol. Simul. 19, 43-61. Lopes, J. N. C. and D. J. Tildesley (1997) "Multiphase equilibria using the Gibbs ensembel Monte Carlo method", Mol. Phys. 92, 187-195. Lotrich, V. F. and K. Szalewicz (1997) "Symmetry-adapted perturbation theory of three-body nonadditivity in Ar trimer." J. Chem. Phys. 106, 9688-9702. Lotrich, V. F. and K. Szalewicz (2000) "Perturbation theory of three-body exchange nonadditivity and application to helium trimer", J. Chem. Phys. 112, 112-121.
157
Loubeyre, P. (1987) "Three-body exchange interaction in dense helium", Phys. Rev. Lett. 58, 1857-1860. Loubeyre, P. (1988) "Three-body-exchange interaction in dense rare gases", Phys. Rev. B 37, 5432-5439. Maitland, G. C., M. Rigby, E. B. Smith and W. A. Wakeham (1981) "Intermolecular forces, their origin and determination", Clarendon:Oxford. Malanowski, S. and A. Anderko (1992) "Modelling Phase Equilibria : thermodynamic background and practical tools", New York, John Wiley&Sons ,Inc. Malijevsky, A. and A. Malijevsky (2003) "Monte Carlo simulations of the thermodynamic properties of argon, krypton and xenon in liquid and gas state using new ab initial pair potential", Mol. Phys. 101, 3335-3340. Marcelli, G. and R. J. Sadus (1999) "Molecular simulation of the phase behavior of noble gases using accurate two-body and three-body intermolecular potentials", J. Chem. Phys. 111, 1533-1540. Marcelli, G. and R. J. Sadus (2000) "A link between the two-body and three-body interaction energies of fluids from molecular simulation", J. Chem. Phys. 112, 6382-6385. Marcelli, G. and R. J. Sadus (2001) "Three-body interactions and the phase equilibria of mixtures", High Temperatures-High Pressures 33, 111-118. Marcelli, G., B. D. Todd and R. J. Sadus (2001) "On the relationship between two-body and three-body interactions from nonequilibrium molecular dynamics simulation", J. Chem. Phys. 115, 9410-9413. Marcelli, G., B. D. Todd and R. J. Sadus (2004) "Erratum:"On the realtionship between two-body and three-body interactions from nonequilibrium molecular dynamics simualtion" [J.Chem.Phys. 115, 9410 (2001)]", J. Chem. Phys. 120, 3043. McDonald, I. R. and K. Singer (1967) "Calculation of thermodynamic properties of liquid argon from Lennard-Jones parameters by a Monte Carlo method." Discuss. Faraday Soc. 43, 40-49. Mehta, M. and D. A. Kofke (1994) "Coexistence diagrams of mixtures by molecular simulation", Chemical Engineering Science 49, 2633-2645. Meijer, E. J., D. Frenkel, R. A. LeSar and A. J. C. Ladd (1990) "Location of melting point at 300 K of nitrogen by Monte Carlo simulation." J. Chem. Phys. 92, 7570-7575. Metropolis, N., A. W. Rosenbluth, A. N. Teller and E. Teller (1953) "Equation of state calculations by fast computing machines", J. Chem. Phys. 21, 1087-1092.
158
Moller, D. and J. Fischer (1990) "Vapour liquid equilibrium of a pure fluid from test particle method in combination with NpT molecular dynamics simulations." Mol. Phys. 69, 463-473. Monica, H. L. and C. K. Hall (2001) "Molecular simulation of complete phase diagrams for binary mixtures", Thermodynamics 47, 1664-1675. Mooij, G. C. A. M., D. Frenkel and B. Smit (1992) "Direct simulation of phase equilibria for chain molecules", J. Chem. Phys. 97, L255-L259. Morris, J. R., J. L. Wang, Horita.J. and Chan.C.T. (1994) "Melting line of aluminum from simulations of coexisting phase", Phys. Rev. B 49, 3109-3115. Murphy, R. D. and J. A. Barker (1971) "Three-Body interactions in liquid and solid Helium", Phys. Rev. A 3, 1037-1040. Murrell, J. N. (1976) "Short and intermediate range forces", London, Academic. Nardo, S. D. and J. H. Bilgram (1995) "Fluctuation during freezing and melting at the solid-liquid interface of xenon", Phys. Rev. B 51, 8012-8017. Nasrabad, A. E. and U. K. Deiters (2003) "Prediction of thermodynamic properties of krypton by Monte Carlo simulation using ab initio interaction potentials", J. Chem. Phys. 119, 947-952. Nasrabad, A. E., R. Laghaei and U. K. Deiters (2004) "Prediction of the thermophysical properties of pure neon, pure argon, and the binary mixtures neon-argon and argon-krypton by Monte Carlo simulation using ab initio potentials", J. Chem. Phys. 121, 6423-6434. Nemirovsky, A. M. and K. F. Freed (1989) "Finite-size scaling close to the critical point: renormalisation group and ε expansion", J. Phys. A: Math. Gen. 19, 591-597. Orkoulas, G. and A. Z. Panagiotopoulos (1994) "Free energy and phase equilibria for the restricted primitive model of ionic fluids from Monte Carlo Simulations", J. Chem. Phys. 101, 1452-1459. Panagiotopoulos, A. Z. (1987a) "Adsorption and capillary condensation of fluids in cylindrical pores by Monte Carlo simulation in the Gibbs ensemble." Mol. Phys. 62, 701. Panagiotopoulos, A. Z. (1987b) "Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble", Mol. Phys. 61, 813-826. Panagiotopoulos, A. Z. (1989) "Exact calculation of fluid-phase equilibria by Monte Carlo simulation in a new statistical ensemble", International Journal of Thermophysics 10, 447-457. Panagiotopoulos, A. Z. (1992) "Direct determination of fluid phase equilibria by simulation in the Gibbs ensemble : a review", Mol. Simul. 9, 1-23.
159
Panagiotopoulos, A. Z. (2000) "Monte Carlo methods for phase equilibria of fluids", J. Phys: Condens. Matter 12, R25-R52. Panagiotopoulos, A. Z. and N. Quirke (1988) "Phase equilibria by simulation in the Gibbs ensemble alternative derivation, generalization and application to mixture and memberane equilibria", Mol. Phys. 63, 527-545. Panagiotopoulos, A. Z., V. Wong and M. A. Floriano (1998) "Phase equilibria of lattice polymers form histogram reweighting Monte Carlo simulations." Macromolecules 31, 912-918. Polymeropoulos, E. E., J. Brickmann, L. Jansen and R. Block (1984) "Analysis of three-body potentials in systems of rare-gas atoms: Axilrod-Teller versus three-atom exchange interactions", Phys. Rev. A 30, 1593-1599. Powles, J. G., W. A. B. Evans and N. Quirke (1982) "Non-destructive molecular-dynamics simulation of the chemical-potential of a fluid", Mol. Phys. 46, 1347-1370. Rahman, A. (1964) "Correlations in the motion of atoms in liquid argon", Phys. Rev. 136, 405-411. Rodriguez, L. (1978) "Intermolecular Force parameters for some pure substances, derived from binary gas mixture data", Ind. Eng. Chem. Fundam 17, 228-230. Romano, S. and K. E. Singer (1982) "Calculation of the entropy of liquid chlorine and bromine by computer-simulation", Mol. Phys. 37, 1765-1772. Rosenfield, Y. (1996) "Ewald method for simulating Yukawa systems." Mol. Phys. 88, 1357-1363. Rudisill, E. N. and P. T. Cummings (1989) "Gibbs ensemble simulation of phase equilibrium in the hard core two-Yukawa fluid model for the Lennard-Jones fluid." Mol. Phys. 68, 629-635. Rull, L. F., G. Jackson and B. Smit (1995) "The condition of microscopic reversibility in the Gibbs-ensemble Monte Carlo simulation of phase equilibria." Mol. Phys. 85, 435-447. Sadus, R. J. (1996) "Monte Carlo simulation of vapour-liquid equilibria in "Lennard-Jones + three-body potential" binary fluid mixtures", Fluid Phase Equilibria 116, 289-295. Sadus, R. J. (2002) "Molecular simulation of fluids: theory, algorithms and object-orientation", Elsever. Sadus, R. J. and J. M. Prausnitz (1996) "Three-body interactions in fluids from molecular simulation: Vapor-liquid phase coexistence of argon", J. Chem. Phys. 104, 4784-4787.
160
Salsburg, Z. W., J. D. Jackson, W. Fickett and W. W. Wood (1959) "Application of the Monte Carlo method to the lattice-gas model. I. Two-dimensional triangular lattice." J. Chem. Phys. 30, 65-72. Schlosser, H. and J. Ferrante (1991) "Pressure dependence of the melting temperature of solids: Rare-gas solids", Phys. Rev. B 43, 13 305-13 308. Shaw, M. S. (1991) "Monte Carlo simulation of equilibrium chemical composition of molecular fluid mixtures in the Natoms PT ensemble", J. Chem. Phys. 94, 7550-7554. Shing, K. S. (1982) "The chemical-potential in dense fluids and fluid mixtures via computer-simulation", Mol. Phys. 46, 1109-1128. Shing, K. S. and S. T. Chung (1987) "Computer simulation methods for the calculation of solubility in supercritical extraction systems", J. Phys. Chem. 91, 1674-1681. Smit, B. (1996). "Advanced Monte Carlo techniques". "Understanding Molecular Simulation: From Algorithms to Applications". D. Frenkel and B. Smit, Academic Press. Smit, B., P. De Smedt and D. Frenkel (1989) "Computer simulations in the Gibbs ensemble." Mol. Phys. 68, 931-950. Smit, B., T. Hauschild and J. M. Prausnitz (1992) "Effect of density-dependent potential on phaser behavior of fluid." Mol. Phys. 77, 1021-1031. Smith, W. R. and B. Triska (1994) "The reaction ensemble method for the computer simulation of chemical and phase equilibria. I. Theory and basic examples." J. Chem. Phys. 100, 3019-3027. Tanaka, Y., G. Ciccotti and G. Renato (1970) "Absorption spectrum of the argon molecule in the vacuum-uv region." J. Chem. Phys. 53, 2012-2030. Torrie, G. M. (1977) "Monte Carlo study of a phase-separating liquid mixture by umbrella sampling", J. Chem. Phys. 66, 1402-1408. Tsang, T. and H. T. Tang (1978) "Solid-Liquid phase transition in argon", Phys. Rev. A 18, 2315-2320. Tsong, P. C., O. N. White, B. Y. Perigard, L. F. Vega and A. Z. Panagiotopoulos (1995) "Phase equilibria in ternary Lennard-Jones systems", Fluid Phase Equilibria 107, 31-43. Van Leeuwen, M. E., C. J. Peters, J. de Swaan Aron and A. Z. Panagiotopoulos (1991) "Investigation of the transition to liquid-liquid immiscibility for Lennard-Jones (12,6) systems using Gibbs ensemble molecular simulations", Fluid Phase Equilibria 66, 57-75.
161
Vargaftik, N. B., Y. K. Vinogradov and V. S. Yargin (1996) "Handbook of physical properties of liquids and gases: pure substances and mixtures", New York, Wallingford (UK), Begell house, Inc. Vega, L. F., E. de Miguel, L. F. Rull, G. Jackson and I. A. McLure (1992) "Phase equilibria and critical behavior of square-well fluids of variable width by Gibbs ensemble Monte Carlo simulation", J. Chem. Phys. 96, 2296-2305. Verlet, L. (1967) "Computer "experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules." Phys. Rev. 159, 98-103. Walton, A. J. (1982) "Three phases of matter", Clarendon press Oxford. Wei, Y. S. and R. J. Sadus (2000) "Equations of state for the calculation of fluid phase equilibria", American Institute of Chemical Engineers Journal 46, 169-196. Widom, B. (1963) "Some topics in the theory of fluids." J. Chem. Phys. 39, 2808-2812. Wood, W. W. and J. D. Jacobson (1957) "Monte Carlo equation of state of molecules interacting with the Lennard-Jones potential." J. Chem. Phys. 27, 720-733. Wu, N. and Y. C. Chiew (2001) "Multi-Density Integral Equation Theory for a Hard Sphere-Sticky Hard Sphere Heteronuclear Dimer Fluid: Thermodynamic and Structural Properties", J. Chem. Phys. 115, 6641-6652. Yamamoto, R., O. Kitao and K. Nakanishi (1995) "Can the 'van der Waals loop' vanish? II. Effect of domain size", Mol. Phys. 84, 757-768. Yoo, S., X. C. Zeng and J. R. Morris (2004) "The melting lines of model silicon calculated from coexisting solid-liquid phases", J. Chem. Phys. 120, 1654-1656.
