Applications Of The Virial Equation Of State To ...

Applications Of The Virial Equation

Of State To Determining The

Structure And Phase Behaviour Of

Fluids

A thesis submitted to the University of

Manchester for the degree of Doctor of

Philosophy in the Faculty of Engineering

and Physical Sciences

2016

Tom Bourne

School of Chemical Engineering and

Analytical Science

Contents

List of Figures 9

Abstract 10

1 Introduction 12

1.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Wider context of the project . . . . . . . . . . . . . . . . . . 13

1.3 Aims of this investigation . . . . . . . . . . . . . . . . . . . . 15

1.4 Structure of thesis . . . . . . . . . . . . . . . . . . . . . . . . 18

2 The virial equation of state 20

2.1 The canonical and grand canonical ensembles . . . . . . . . . 20

2.2 The virial expansion . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Existing applications of the virial expansion . . . . . . . . . . 22

2.4 Relating virial coefficients to particle interactions . . . . . . . 23

2.5 Obtaining the virial coefficients for a system . . . . . . . . . . 25

2.6 Mayer sampling Monte Carlo (MSMC) . . . . . . . . . . . . . 29

3 Applicability of the virial equation of state to a system 35

3.1 Convergence of the virial expansion . . . . . . . . . . . . . . . 35

3.2 Improving the performance of the virial expansion for systems

with a single fluid state . . . . . . . . . . . . . . . . . . . . . 38

3.3 Improving the performance of the virial expansion when sep-

arate liquid and vapour phases can exist . . . . . . . . . . . . 40

4 Describing the structure of a fluid 45

4.1 The Ornstein-Zernike equation . . . . . . . . . . . . . . . . . 45

4.2 Solving the O-Z equation using integral equation theory . . . 47

4.3 Using a virial expansion to describe the structure of a fluid . 49

5 Predicting the structure and formation of an ordered phase 52

5.1 Modelling the Helmholtz energy of a fluid . . . . . . . . . . . 52

5.2 Obtaining the excess contribution to the Helmholtz energy . . 54

5.3 Describing the solid-fluid phase transition of a fluid . . . . . . 57

5.4 Investigating the solid-fluid phase transition in terms of a

virial expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2

5.5 Improving the model of a solid-fluid phase transition . . . . . 61

6 Obtaining the structure of fluids governed by inverse power

potentials 63

6.1 The inverse power potential . . . . . . . . . . . . . . . . . . . 63

6.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.3 The structure of fluids governed by inverse-power potentials . 65

6.3.1 n = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.3.2 n = 6 and n = 9 . . . . . . . . . . . . . . . . . . . . . 68

6.3.3 n = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.3.4 n = 50 and hard spheres . . . . . . . . . . . . . . . . . 74

6.3.5 Asymptotic behaviour of the RDF and DCF for the

n = 12 case . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4 Analysis and interpretation of results . . . . . . . . . . . . . . 79

7 Extending the investigation to study the structure of a Lennard-

Jones fluid 81

7.1 The Lennard-Jones potential . . . . . . . . . . . . . . . . . . 81

7.2 The DCF of a Lennard-Jones fluid . . . . . . . . . . . . . . . 82

7.3 Perturbing the LJ potential . . . . . . . . . . . . . . . . . . . 85

7.4 Investigating the divergence of the LJ virials . . . . . . . . . 88

7.5 Re-summation of the virial expansion of the attractive com-

ponent of a Lennard-Jones potential . . . . . . . . . . . . . . 92

7.5.1 Re-summation with forced asymptotic behaviour . . . 93

7.5.2 Pade approximant schemes . . . . . . . . . . . . . . . 96

7.5.3 Asymptotically consistent approximation method . . . 98

7.5.4 Approximation using extrapolated high-order coeffi-

cients . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.6 Comparison of re-summation schemes . . . . . . . . . . . . . 101


8 Improving the convergence of the virial expansion in sys-

tems governed by a ‘square-shoulder’ potential form 106

8.1 The dissipative particle dynamics potential . . . . . . . . . . 106

8.2 The penetrative square well potential . . . . . . . . . . . . . . 108

8.3 Virial expansion for fluids governed by ‘square-shoulder’ po-

tentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3

8.3.1 DPD potential form . . . . . . . . . . . . . . . . . . . 110

8.3.2 PSW potential form . . . . . . . . . . . . . . . . . . . 112

8.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.5 Re-summation of the virial expansion for a fluid governed by

a DPD potential . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.5.1 Re-summation with an asymptotic limit of Z ∼ ρ∗

imposed . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.5.2 Re-summation to an asymptotic limit including mean-

field approximation . . . . . . . . . . . . . . . . . . . . 119

8.6 Re-summation of the virial expansion for a fluid governed by

a PSW potential . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.6.1 Re-summation with an asymptotic limit of Z ∼ ρ∗

imposed . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.6.2 Re-summation to an asymptotic limit including mean-

field approximation . . . . . . . . . . . . . . . . . . . . 125

8.7 Extrapolation of virial coefficients . . . . . . . . . . . . . . . . 128


9 Using the virial expansion to describe transition to an or-

dered phase 132

9.1 The hard-sphere model . . . . . . . . . . . . . . . . . . . . . . 132

9.2 The crystal lattice . . . . . . . . . . . . . . . . . . . . . . . . 133

9.3 Predicting the Helmholtz energy profile via stability analysis 134

9.4 Calculating and minimising the Helmholtz energy using a

virial expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.5 Calculating the thermodynamic properties of the system . . . 139

9.6 Searching for the point of phase coexistence . . . . . . . . . . 142

9.7 Application of Parsons-Lee theory to the model . . . . . . . . 144

9.8 Confirmation of results using the analytical derivative of the

Helmholtz energy . . . . . . . . . . . . . . . . . . . . . . . . . 147


10 Conclusions and future challenges 151

10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

10.2 Areas of possible further study and remaining challenges . . . 153

4

Appendix A Tabulated virial coefficients for the Lennard-Jones

fluid 156

Appendix B Tabulated virial coefficients for a fluid governed

by several types of DPD potential 157

Appendix C Tabulated virial coefficients for a fluid governed

by several types of PSW potential 158

Bibliography 160

Word count: 44953

5

List of Figures

2.1 Diagrammatic representation of virials up to B4 . . . . . . . . 27

3.1 Pressure-density plots for a Lennard-Jones fluid . . . . . . . . 37

3.2 Plot of polylogarithmic function Li aT(eb/T ρ) up to tenth order 43

4.1 RDF for a 12th-order inverse power potential from Rogers-

Young approximation . . . . . . . . . . . . . . . . . . . . . . 50

6.1 Low-density DCF expansions when n = 4 . . . . . . . . . . . 66

6.2 Higher-density DCF expansions when n = 4 . . . . . . . . . . 66

6.3 Low-density TCF expansions when n = 4 . . . . . . . . . . . 67

6.4 Higher-density TCF expansions when n = 4 . . . . . . . . . . 67



6.7 High-density DCF expansions when n = 6 . . . . . . . . . . . 70

6.8 High-density DCF expansions when n = 9 . . . . . . . . . . . 70


6.10 High-density DCF expansions when n = 12 . . . . . . . . . . 72

6.11 High-temperature DCF expansions when n = 12 . . . . . . . 73

6.12 Low-density TCF expansions when n = 12 . . . . . . . . . . . 73

6.13 High-density TCF expansions when n = 12 . . . . . . . . . . 74

6.14 DCF expansion when n = 50 . . . . . . . . . . . . . . . . . . 75

6.15 DCF expansions obtained using the hard sphere potential . . 75

6.16 TCF expansion when n = 50 . . . . . . . . . . . . . . . . . . 76

6.17 TCF expansions obtained using the hard sphere potential . . 77

6.18 RDF expansions with proposed density dependence of the

form g(r12ρ∗1/3) for n = 12 . . . . . . . . . . . . . . . . . . . 78

6.19 The mean squared average of the DCF of various inverse-

power fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.1 Sketch of the Lennard-Jones potential . . . . . . . . . . . . . 82

7.2 Pressure-density diagram given by the Kolafa-Nezbeda equa-

tion of state for an LJ fluid . . . . . . . . . . . . . . . . . . . 82

7.3 Virial expansion of an LJ fluid at T ∗ = 1.0 . . . . . . . . . . . 83

7.4 Virial expansion of an LJ fluid at T ∗ = 1.6 . . . . . . . . . . . 84

7.5 DCF of an LJ fluid at T ∗ = 1.0 and ρ∗ = 0.32 . . . . . . . . . 85

7.6 DCF of an LJ fluid at T ∗ = 1.6 and ρ∗ = 0.32 . . . . . . . . . 85

6

7.7 DCF of the repulsive contribution to an LJ fluid at T ∗ = 1

and ρ∗ = 0.32 . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.8 DCF of the attractive contribution to an LJ fluid at T ∗ = 1

and ρ∗ = 0.32 . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.9 The HTA for a Lennard-Jones fluid . . . . . . . . . . . . . . . 90

7.10 Comparison of compressibility from re-summation and HTA

with K-N equation of state . . . . . . . . . . . . . . . . . . . 92

7.11 Re-summation of an alternative form of the virial expansion

at T ∗ = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


at T ∗ = 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


with corrected leading-order behaviour . . . . . . . . . . . . . 96

7.14 Compressibility of an LJ fluid given by Pade approximants at

T ∗ = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.15 Compressibility of an LJ fluid given by Pade approximants at

T ∗ = 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.16 Re-summation of the virial expansion using ACAs at T ∗ = 1.0 99

7.17 Re-summation of the Lennard Jones virial expansion using

ACAs at T ∗ = 1.6 . . . . . . . . . . . . . . . . . . . . . . . . 99

7.18 Re-sumation using extrapolated virial coefficients at T ∗ = 1.0

and T ∗ = 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.19 Comparison of re-summation schemes at a sub-critical tem-

perature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.20 Comparison of re-summation schemes at a super-critical tem-

perature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.1 Sketch of DPD potentials . . . . . . . . . . . . . . . . . . . . 108

8.2 Sketch of PSW potentials . . . . . . . . . . . . . . . . . . . . 109

8.3 Virial expansion to 5th order for a DPD potential with C1 =

30 and C2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.4 Virial expansion to 5th order for DPD potentials with an

attractive component . . . . . . . . . . . . . . . . . . . . . . . 111

8.5 Virial expansion to 5th order for a DPD potential with C1 =

30 and C2 = 3.9505 . . . . . . . . . . . . . . . . . . . . . . . 111

8.6 Virial expansion to 5th order for a PSW potential with ∆σ = 0

and ǫaǫr

= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7

8.7 Virial expansion to 5th order for a PSW potential with ∆σ = 1

2

and ǫaǫr

= 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.8 Virial expansion to 5th order for PSW potentials close to and

beyond the Ruelle stability limit . . . . . . . . . . . . . . . . 114

8.9 Phase diagram for a fluid governed by a PSW potential. . . . 115

8.10 Re-summed virial expansion to 5th order for a DPD potential

with C1 = 30 and C2 = 0 . . . . . . . . . . . . . . . . . . . . 116


with C1 = 30 and C2 = 3 . . . . . . . . . . . . . . . . . . . . 117


with C1 = 30 and C2 = 3.5 . . . . . . . . . . . . . . . . . . . 118


with C1 = 30 and C2 = 3.9505 . . . . . . . . . . . . . . . . . . 118

8.14 Re-summed virial expansion to 5th order with fixed leading-

order behaviour for a DPD potential with C1 = 30 and C2 = 0 120


order behaviour for a DPD potential with C1 = 30 and C2 = 3 120


order behaviour for a DPD potential with C1 = 30 and C2 = 3.5121


order behaviour for a DPD potential with C1 = 30 and C2 =

3.9505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8.18 Re-summed virial expansion to 5th order for a PSW potential

with ∆σ = 0 and ǫa

ǫr= 0 . . . . . . . . . . . . . . . . . . . . . . 122


with ∆σ = 1

2 and ǫaǫr

= 18 . . . . . . . . . . . . . . . . . . . . . 123


with ∆σ = 3

4 and ǫaǫr

= 16 . . . . . . . . . . . . . . . . . . . . . 123


with ∆σ = 1 and ǫa

ǫr= 1 . . . . . . . . . . . . . . . . . . . . . . 124


order behaviour for a PSW potential with ∆σ = 0 and ǫa

ǫr= 0 . 126


order behaviour for a PSW potential with ∆σ = 1

2 and ǫaǫr

= 18 126


order behaviour for a PSW potential with ∆σ = 3

4 and ǫaǫr

= 16 127

8


order behaviour for a PSW potential with ∆σ = 1 and ǫa

ǫr= 1 . 127

8.26 Change in incremental ratio of virial coefficients with increas-

ing order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.27 Extrapolated virials up to B10 for several square-shoulder po-

tentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.28 Virial expansions for DPD potentials with extrapolated coef-

ficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8.29 Virial expansions for PSW potentials with extrapolated coef-

ficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9.1 The three types of crystal to be studied in this project . . . 133

9.2 Second order free energy profiles for an FCC lattice . . . . . . 138

9.3 Second order free energy profiles for an BCC lattice . . . . . 138

9.4 Second order free energy profiles for an HCP lattice . . . . . 139

9.5 Coexistence of inhomogeneous and isotropic phases in an FCC

lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.6 Coexistence of inhomogeneous and isotropic phases in an BCC

lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.7 Coexistence of inhomogeneous and isotropic phases in an HCP

lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.8 Helmholtz energy profile for an FCC lattice using Parsons-Lee

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.9 Helmholtz energy profile for a BCC lattice using Parsons-Lee

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.10 Helmholtz energy profile for an HCP lattice using Parsons-

Lee model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

9.11 Pressure-chemical potential phase diagram predicted by Parsons-

Lee theory for an FCC lattice . . . . . . . . . . . . . . . . . . 146


Lee theory for a BCC lattice . . . . . . . . . . . . . . . . . . 147


Lee theory for an HCP lattice . . . . . . . . . . . . . . . . . . 147

9.14 Derivative of the free energy profile for η = 1 . . . . . . . . . 149

9

Abstract

This work considers the extent to which the virial expansion can

describe the structure and phase behaviour of several model fluids.

These are the hard-sphere fluid, inverse-power potential fluids, the

Lennard-Jones fluid and two kinds of ‘square-shoulder’ potential.

The first novel contribution to knowledge that this work makes is in

using virials to obtain the direct correlation function of a hard-core

inverse-power potential fluid at densities close to freezing. Predicted

radial distribution functions for the fluid at these densities are found

that agree well with integral equation theory and simulation data.

For softer-core potentials, a convergent direct correlation function is

obtained at densities up to those at which a convergent virial

expansion is known to exist.

The study then extends to a Lennard-Jones fluid. At super-critical

temperatures, a convergent direct correlation function is found as

before. However, at sub-critical temperatures, the direct correlation

function is found to diverge at all points for densities below criticality.

Several recently-proposed re-summations of the pressure virial

expansion are studied to improve its convergence at high densities.

Some promise is shown in improving the accuracy of the virial

expansion at high densities, but a re-summed virial expansion is

found to be unable to fully capture the true behaviour of the system

at densities close to criticality.

A second novel contribution to knowledge is made by the reporting of

virial coefficient data for several dissipative particle dynamics and

penetrative square well potential forms. This is used to study the

effect of re-summing the virial expansion for these systems in order to

improve its convergence at high densities. The virial expansions of

these potentials are found to perform increasingly poorly in the

proximity of a vapour-liquid phase transition. This is in agreement

with the results of investigating the Lennard-Jones fluid.

Thirdly, this investigation considers the whether the virial expansion

can describe the freezing of a hard sphere fluid and therefore predict

the entire phase diagram for this system. This is investigated using a

virial expansion to model the excess contribution to the Helmholtz

energy functional. The virial expansion is not found to be able to

accurately the point of phase transition, most likely due to questions

remaining over the choice of a Gaussian basis set to describe lattice.

10

Declaration

The author declares that no portion of the work referred to in the thesishas been submitted in support of an application for another degree orqualification at this or any other university or other institute of learning.

Copyright statement

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11

Chapter 1

1 Introduction

1.1 Problem statement

This thesis describes the work done during an investigation into several

possible applications of the virial equation of state. The work focusses on

obtaining a convergent form of this equation and using it to find the

structure of a fluid and to predict the phase behaviour of a system.

An equation of state provides a mathematical relationship between

macroscopic thermodynamic properties such as pressure, temperature and

volume. The virial equation of state gives the pressure of a system as an

infinite power series in powers of density expanded about the ideal gas

limit, with each power multiplied by a coefficient based on the temperature

and the particle interactions taking place in the system. The virial

expansion provides a powerful, systematic and exact equation of state

which has the potential to have input into practical equations of state and

to be used in coarse-graining processes. Until recent advances in

computational power, these virial coefficients have proved extremely

difficult to calculate accurately beyond the lowest orders for most potential

forms. This has led to little being understood about the convergence of the

series and therefore the usefulness of the equation has been relatively

limited.

This project considers three main research questions. The first of these

concerns the convergence of the virial expansion and its ability to predict a

gas-liquid phase transition. The radius of convergence of the virial

expansion and its extension through re-summation techniques will be

considered. This would result in an increased range of densities for which

the virial equation of state is valid.

The direct correlation function of a fluid is a further property that can be

described by a virial expansion. The second aim of this project is to

predict the structure of a fluid at densities up to the solid phase, including

the liquid-solid transition behaviour if possible.

Relatively little is also known about the applicability of the virial

12

Chapter 1

expansion to an ordered solid phase. The final aim of this project is to use

a virial expansion to investigate the prediction of coexistence between the

solid and fluid phases of a system of hard spheres and present any evidence

for the most stable type of crystal that will form.

1.2 Wider context of the project

Theoretical modelling is important in representing physical phenomena in

situations where either laboratory experiments or computer simulations

are impractical or not possible at all. As well as this, theoretical methods

are useful in providing confirmation of results obtained via these other

methods. More recently, the application of results obtained from

theoretical studies and simulation to large-scale process design has been

investigated more closely (for example by Gani et al. (2002) and Adjiman

et al. (2014)). This provides a further potentially useful application of the

theoretical modelling of thermodynamic systems.

Knowledge of the thermodynamic behaviour of the system allows the

transfer of energy between a system and its surroundings and the

relationships between its macroscopic properties to be understood.

Statistical mechanics provides a set of tools with which to relate this

thermodynamic behaviour to that of the particles within the system using

probability theory.

An equation of state is a fundamentally useful thermodynamic relationship

since it allows the thermodynamic properties of a system to be calculated.

It is therefore important to obtain the most reliable and accurate equation

possible for a system. The virial expansion is an equation of state of

particular interest since theoretically it provides a mechanism by which

macroscopic thermodynamic properties can be derived rigorously from

knowledge of microscopic particle interactions through a comparatively

simple mathematical expression.

The virial equation of state itself takes the form of an expansion in powers

of the number density, ρ about the point at which compressibility Z is

equal to one (Mason and Spurling, 1969):

13

Chapter 1

Z =p

kBρT= 1 +B2(T )ρ+B3(T )ρ

2 + . . . (1.1)

= 1 +

∞∑

i=1

Bi+1(T )ρi. (1.2)

Here, p represents the pressure, kB represents Boltzmann’s constant and T

represents the absolute temperature. The coefficients Bn are known as the

virial coefficients and are functions of temperature and the inter-particle

potential which governs the system.

Although many equations of state exist, the virial equation of state is a

rare example of a thermodynamic model for a fluid that can be derived

rigorously from the interactions between particles described by the

potential function. It is also a convenient and systematic mathematical

expression, providing a bridge between the macroscopic and microscopic

scales.

With recent increases in computational power has come the ability to

obtain increasingly accurate expressions for the virial expansion of a fluid.

Therefore the virial expansion is able to describe an increasingly wide

range of microscopic-scale behaviour. Examples of this include analysis of:

mixtures of different types of particles (Schultz and Kofke, 2009b), density

inhomogeneities (Bellemans, 1962a,b, 1963), flexible molecules (Shaul

et al., 2011a; Caracciolo et al., 2006a,b), non-pairwise interactions

(Benjamin et al., 2007a; Hellmann and Bich, 2011; Jager et al., 2011; Kim

et al., 2012), electrostatic effects (Joslin, 1982) and nuclear quantum

effects (Garberoglio and Harvey, 2009, 2011; Garberoglio, 2012; Shaul

et al., 2012b,a).

Although, as pointed out by Schultz and Kofke (2015a), the capability of

the virial expansion of dealing with the thermodynamic behaviour of a

system is less well-understood, the virial expansion can theoretically be

modified to obtain information about a wide range of macroscopic

properties of a system. As well as pressure or compressibility, examples of

this include the use of the leading order concentration virial coefficient to

14

Chapter 1

provide information about the intrinsic conductivity of a fluid by Garboczi

and Douglas (1996) and the study of the dielectric constant by Ishihara

and Hanks (1962).

The virial equation of state is theoretically valid at all bulk fluid densities

and so is of particular use in describing the clustering of particles. Since

the thermodynamic properties of a system can theoretically be calculated

via a virial expansion, it is possible to investigate the ability of the virial

expansion to describe the phase transitions of a system. So long as the

virial coefficients can be obtained, this is true for any kind of interaction

between particles. The conditions at which a phase transition occurs are of

particular interest, since it is in these regions of the phase diagram that

the most drastic changes to the physical properties of the system occur,

and therefore much of the most interesting thermodynamic behaviour

takes place here.

However, the question of the range of densities over which the virial

expansion converges remains unsolved. Since an expansion of enumerated

coefficients represents an approximation to an infinite series, this question

is vitally important to the applicability of the expansion as an equation of

state for a given system.

1.3 Aims of this investigation

In order to further existing knowledge of the convergence behaviour of the

virial expansion and its ability to describe phase transitions, three

closely-linked investigations are described here. These investigations study

systems consisting of both one fluid phase and separate liquid and vapour

phases through the use of a variety of potential functions to define the

inter-particle interactions.

The n-th virial coefficient contains information about the interactions

between n particles in the system. This means that the number of possible

interactions to be included will increase rapidly with the order of the

expansion (i.e. the number of particles n whose behaviour is represented).

Due to the computational difficulty of obtaining an accurate value for

15

Chapter 1

these higher-order coefficients, relatively few virial coefficients have been

enumerated for any potential forms beyond the simplest. This in turn

means that in many cases, the virial expansion of a fluid either converges

extremely slowly or is divergent, and thus the usefulness of the virial

equation of state is limited.

The first aim of this work is therefore to obtain reliable values for the virial

coefficients of several systems and use them to study the convergence of

the virial expansion under different conditions. This will be done through

the application of re-summation techniques in order to extend the radius of

convergence of the system and thus reduce the number of virial coefficients

that are required to obtain a convergent series. Conclusions will then be

drawn on the reliability of the virial expansion and its capability in

describing the structure and phase transitions of the system.

The structure of a fluid (i.e. the behaviour of particles in relation to one

another) can be described by the direct correlation function and the radial

distribution function (Ornstein and Zernike, 1914). However, although it

may be deduced experimentally, the direct correlation function is not easy

to calculate directly either theoretically or by simulation methods. Also,

current theoretical methods frequently fail in regions of the phase diagram

close to a phase transition, which is the region in which the most

interesting thermodynamic behaviour often occurs. An alternative method

is to employ a virial expansion, which can be used to give a convenient

expression for the direct correlation function.

Obtaining an accurate radial distribution function at high densities is of

particular interest due to its applications to the study of liquid state

theory and applications to statistical associating fluid theory (SAFT). In

the first part of this investigation, the correlation structures of a system

using the virial expansion as a starting point is developed. This seeks to

accurately and reliably calculate the direct correlation function and thus

the radial distribution function of a fluid over an even broader range of

densities than has previously been seen.

This investigation also seeks to obtain new, higher-order virial coefficients

16

Chapter 1

for two types of ‘square-shoulder’ potential. These are the dissipative

particle dynamics potential and the penetrative square well potential.

Depending on the parameters selected, such systems can form either a

single fluid phase or separate liquid and gas phases. Sufficient virial

coefficients will be obtained and used in conjunction with newly-developed

techniques in order to seek a re-summed virial equation of state which is

valid across a wider density range than has previously been possible. In

both of these investigations, both systems forming one fluid phase and

systems forming separate liquid and gas phases are studied.

The final part of this investigation focusses on the virial expansion of a

hard sphere fluid. The commonly accepted behaviour of the hard sphere

phase diagram is summarised by Robles et al. (2014). Although the virial

expansion has been found to be capable of describing the vapour phase,

relatively little is known about the ability of this equation to describe the

development and structure of an ordered crystal phase. An accurate

analysis of the phase transitions of a system of hard spheres is particularly

useful since due to the strong short-range repulsion effects represented by

the model, hard spheres are well-suited as a reference system for a vast

range of more complicated systems.

A number of powerful and efficient methods currently exist to predict the

freezing transition, as is discussed in more detail in Chapter 5. Although it

would be inappropriate to consider the virial expansion as a practical

alternative to these methods, this investigation considers the question of

whether it is possible to describe the entire phase diagram of a hard sphere

fluid by means of the virial expansion.

The final topic studied in this investigation is the coexistence of a fluid

phase and an ordered crystal phase. This is done by using the virial

expansion to calculate the Helmholtz energy of a system in both the fluid

and solid phases and then searching for the point at which both phases

coexist. Several types of ordered phase are studied in order to ascertain

which is predicted by the virial expansion to be the most

thermodynamically stable crystal formation.

17

Chapter 1

1.4 Structure of thesis

As an equation of state, the virial expansion is applicable to an extremely

wide range of situations. Recent computational advances have meant that

the pace of development and breadth of research in this area is continually

increasing. This means that a full review of recent developments in the

calculation and application of the virial coefficients (such as that by

Masters (2008)) lies beyond the scope of this thesis. To this end, the thesis

focusses on the following areas of existing knowledge and new research.

Firstly, in Chapter 2, the problem of calculating higher-order virial

coefficients accurately is addressed and the existing methods for doing so

summarised. Following on from this in Chapter 3, the issue of the

convergence of the virial expansion is considered, along with existing tools

for improving it in order to increase the usefulness of the virial equation of

state.

Existing literature concerning the application of the virial expansion to the

structure of fluids is then reviewed in Chapters 4 and 5. This focusses on

relating the virial expansion to the correlation structure of a fluid and

using a virial expansion to obtain the Helmholtz energy of a system.

Alternative means of achieving this and their results are also presented.

After reviewing existing knowledge, new results obtained in this

investigation are presented. These results begin by describing the

application of the virial equation of state to describing the correlation

structure of both a fluid defined by a purely repulsive potential and by a

potential containing an attractive term are presented in Chapters 6 and 7.

Following this, the improvement of the convergence of the virial expansion

for ‘square-shoulder’ potentials in Chapter 8 through re-summation

techniques is investigated. Lastly, an attempt to predict the freezing

transition of a hard-sphere fluid into an ordered crystal phase is discussed

in Chapter 9.

In Chapter 10, the thesis discusses what can be learned from these

investigations and what new contributions to knowledge have been made.

Suggestions are also made for further work to extend this project in order

18

Chapter 1

to meet any remaining challenges.

19

Chapter 2

2 The virial equation of state

This section will introduce the virial equation of state and the existing

methods for obtaining the associated coefficients.

2.1 The canonical and grand canonical ensembles

An ensemble is a theoretical collection of a large number of copies of the

same system of a large number of particles, known as microstates. Each

microstate represents one possible state in which the system could exist,

and so the ensemble describes the probability distribution of the

microscopic states of the system.

A canonical ensemble represents a system of given volume V which is

assumed to contain a constant number of particles N and to be in an

infinite heat bath, resulting in a constant temperature T .

The partition function Z(N,V, T ) describes the statistical properties of a

system and is based on the temperature and each of the energy

‘microstates’ that are available to the system. Defining the total energy of

the system in a state s as Es, an expression for the canonical partition

function Z is:

Z(N,V, T ) =∑

s

e−βEs , (2.1)

where β = 1/kBT and kB is the Boltzmann constant. This expression

treats the energy states as a set of discrete states in which the system may

exist. However, in the classical limit, the position and momentum of

particles within the system varies continuously, resulting in an uncountable

number of possible energy states in the set of all possible values of s. To

account for this, Z can be written as the following integral for a system of

N identical particles:

Z(N,V, T ) =1

N !h3N

∫

exp[−βH(p1 . . . pN , r1 . . . rN )] dNpN dNrN . (2.2)

In this integral, H represents the Hamiltonian depending on the values of

20

Chapter 2

momentum p and position r of each particle in the system. N ! is the

number of ways in which a system of N particles can be permuted among

themselves. Thus dividing by N ! ensures that each possible microstate is

only counted once since the particles are indistinguishable from one

another. Also present here is the term h3N , where h is Planck’s constant,

with the same units as pr. This means that when integrating over N

three-dimensional variables, the result for Z is rendered dimensionless.

The grand canonical ensemble represents the possible states of an isolated

system of fixed volume in thermodynamic equilibrium with an infinite

reservoir of particles. The macroscopic properties that define this system

are the chemical potential µ, the system volume V and the system

temperature T . This means that the system is able to exchange both

energy and particles with its environment. The grand canonical partition

function Ξ can be written as the following weighted sum of canonical

partition functions over different numbers of particles N :

Ξ(µ, V, T ) =∞∑

N=0

eµN/kBTZ(N,V, T ). (2.3)

2.2 The virial expansion

At sufficiently low densities, the interactive forces between particles in a

fluid may be neglected and the behaviour of the fluid may be described as

follows:

pV = NkBT. (2.4)

This equation is the ideal gas law and relates the pressure p, volume V ,

number of particles N and temperature T . In terms of the particle density

ρ = N/V , the ideal gas law can be rewritten as:

p = βρ. (2.5)

As the density of the fluid increases, the behaviour of the fluid deviates

from the ideal limit. To account for this, a series of corrections for

interactions between particles within the fluid are introduced. This leads

to the virial expansion form of the ideal gas law which introduces the virial

21

Chapter 2

coefficients Bi(T ) (Thiessen, 1885). These are functions of temperature

and an inter-particle potential function and account for the changes in free

energy due to the interactions between particles. The virial equation of

state (Kamerlingh Onnes, 1901; Mason and Spurling, 1969) is:

Z =p

kBρT= 1 +B2(T )ρ+B3(T )ρ

2 + . . . (2.6)

= 1 +∞∑

i=1

Bi+1(T )ρi. (2.7)

This equation represents a relationship between the macroscopic

thermodynamic properties of the system and particle interactions. The

first term on the right-hand side represents the case when there are no

collisions or interactions, B2(T )ρ represents the effect of two interacting

particles and so on. As ρ approaches zero, the leading term dominates the

right hand side of the equation and the ideal gas law is recovered.

2.3 Existing applications of the virial expansion

As well as the applications that are to be investigated in this project, there

are a range of other applications of the virial expansion. Firstly, the virial

expansion is an equation of state and can therefore be used to theoretically

predict PV T data for a system. Virial coefficients can also be used to

predict unknown parameters in other equations of state, as in the case of

Carnahan and Starling (1969).

With relatively few virial coefficients, Benjamin et al. (2006) have

successfully predicted gas phase molecular clustering of particles in water

models. It is also possible to use a short virial series to describe the

extraction of solids from supercritical solvents (Boublık, 2001; Tomberli

et al., 2001).

As the particles being studied become more complicated, more degrees of

freedom are introduced due to vibration effects, orientation, flexibility,

types of existing molecular interaction etc. This means that it becomes

increasingly difficult to accurately calculate the virial coefficients and

22

Chapter 2

therefore it is common that only the second or third virial can be easily

obtained.

Nevertheless, this information can still be useful. Hirschfelder et al. (1954)

have been able to use second and third virial data to predict the

Joule-Thomson inversion curve for a molecular fluid. Due to its

temperature dependence, the second virial coefficient can also be used to

give detailed information on the pairwise interactions in a system through

comparison with experimental data. The second virial coefficient is also

used as a predictor in the characterisation of the growth of protein crystals

(George et al., 1997). In this case, the second virial is measured

experimentally using dynamic light scattering. The advantage of this is

that the saturation of the protein is not increased by the measurement

technique. This minimises the likelihood of an undesirable alteration to

the amount of particle interactions and thus minimises the risk of

unwanted aggregation or poor crystal formation.

2.4 Relating virial coefficients to particle interactions

Consider a one-component, classical fluid of spherically symmetrical

particles with pairwise interactions between particles. These interactions

are represented by a potential u(rij), where rij = |ri − rj|. Mayer and

Mayer (1977) give an expression to describe the behaviour of the potential

u between the particles positioned at ri and rj, known as a Mayer

function:

fij = e−βu(rij) − 1. (2.8)

The virial expansion can be expressed in terms of these functions via the

following derivation. In a system where particle interactions are present, it

is not normally possible to calculate an exact value of the partition

function for that system and so it is not possible to directly evaluate the

thermodynamic properties of that system. However, the partition function

Z for the system can be factorised by writing:

Z = QNZ0. (2.9)

23

Chapter 2

Here, Z0 is the corresponding partition function for a system of

non-interacting particles and QN is the configurational integral for a

system of N particles. An expression for this configurational integral is:

QN =1

V N

∫

exp

−βN∑

i=1,i<j

u(rij)

drN . (2.10)

The exponential term in this expression can be rewritten in terms of Mayer

functions as:

exp

−β∑

i=1,i<j

u(rij)

=N∏

i=1,i<j

exp [−βu(rij)] =N∏

i=1,i<j

(1 + fij). (2.11)

This gives rise to the name ‘cluster’ expansions, since each term in the

expression on the right-hand side represents an increasingly large cluster of

particles. At low densities, only pairwise interactions between particles

need be considered. This means that this product can be approximated to:

N∏

i=1,i<j

(1 + fij) ≈ 1 +N∑

i=1,i<j

fij. (2.12)

This allows QN to be rewritten in terms of Mayer functions:

QN =1

V N

∫

N∑

i=1,i<j

(1 + fij)

drN . (2.13)

The sum in this integral is over all possible pairs of particles, and so

contains N(N − 1)/2 terms. As N → ∞, this tends to N2/2. As all

particles are assumed to be identical, each integral will be the same so

without loss of generality, Q can be rewritten as:

QN = 1 +N2

2V

∫

fij drj . (2.14)

Now, the partition function for the system can be obtained by subsituting

the new expression for the configurational integral back into equation (2.9):

24

Chapter 2

Z = Z0

(

1 +N2

2V

∫

fij drN

)

. (2.15)

The Helmholtz energy for the system is found from taking the logarithm of

the partition function:

A = −kBT logZ0 − kBT logQN . (2.16)

This results in the following expression for the Helmholtz energy:

A = Aid − kBTN2

2V

∫∫

f12 dr1 + . . . (2.17)

The pressure is found from differentiating the Helmholtz energy:

p = kBTN

V

[

1− N

2V

∫∫

f12 dr1 + . . .

]

. (2.18)

Hence it can be seen that the effect of the potential function u(rij) is to

modify the pressure from the ideal gas value. Although this provides the

correct result, more rigorous derivations of the virial series, based on the

grand canonical ensemble, may be found in the literature (McQuarrie,

1973). Equating the expression derived here with the virial equation of

state (2.7), expressions for the virial coefficients in terms of Mayer

functions can be obtained by comparing like powers of the number density.

For example, the second and third virial coefficients can be written in

terms of cluster integrals as:

B2 = −(

1

2V

)∫∫

f12 dr1dr2, (2.19)

B3 = −(

1

3V

)∫∫∫

f12f13f23 dr3dr2dr1. (2.20)

2.5 Obtaining the virial coefficients for a system

Mayer functions therefore allow the virial coefficients to be related to

particle interactions via a series of cluster integrals (neglecting any existing

non-additive interactions). At each order, the nth coefficient is expressed

in terms of cluster integrals over the positions of n particles to represent

the interaction of pairs of particles.

25

Chapter 2

Coefficients greater than B3 are expressed as sums of cluster integrals.

Generally, each increase in the order of the coefficient sought brings a more

complicated sum of cluster integrals to be calculated and also requires

integration over a further set of spatial variables, as well as any necessary

internal or orientational variables, thus rapidly increasing the dimension of

the cluster integrals.

The number of cluster integrals increases rapidly and non-linearly with n,

making calculation of higher order virial coefficients for even the simplest

potential function an extremely difficult and computationally expensive

task. This means that for all but the simplest systems, relatively few virial

coefficients have been accurately calculated. This in turn means that

comparatively little is known so far about the convergence properties of

the virial equation of state or its behaviour in relation to phase transitions.

Cluster integrals and therefore virial coefficients can also be represented

diagrammatically using graphs as described by Hansen and McDonald

(2006) and McQuarrie (1973). A graph consists of a diagram made up of a

number of circles (vertices) and straight lines (connections). Each vertex

represents the position of one particle and is labelled accordingly from 1 to

n. A white vertex indicates a root point, which corresponds to a position

held constant in the integration. A black vertex represents a field point,

which corresponds to a variable of integration. Each connection links a pair

of vertices and represents the Mayer function between those two particles.

Each cluster integral in the expression for a virial coefficient is represented

by a sum of biconnected graphs. This provides a convenient shorthand

method of expressing the integrals to be solved and any multiplying

coefficients. A biconnected graph is one which contains no articulation

points. An articulation point is a vertex whose removal would cause the

graph to become disconnected, i.e. split into two or more separate sections.

This means that if a vertex were to be removed from a biconnected graph,

the graph would remain connected.

Each graph in a sum has an assigned value equal to the value of the cluster

26

Chapter 2

it represents multiplied by a combinatorial factor. This combinatorial

factor is represented by the coefficients multiplying each contribution to

the sum and is equal to the symmetry number of that graph. The

symmetry number of an object is the number of different but equivalent

ways in which it can be observed (i.e. the order of its symmetry group).

This allows unlabelled diagrams to be manipulated more easily in the

solution of the integrals.

To demonstrate this system, an example showing graphs up to the fourth

order virial coefficient B4 is shown in Figure 2.1. Recall that the black

vertices correspond to the dummy variables in the integrations. This

means that altering the labels of these vertices will not change the value of

the integral. Therefore when a graph would usually have only one white

vertex, the labelling applied is irrelevant and only the way in which the

vertices are connected matters. Such graphs are topologically equivalent.

B2 = − 1

2!V

∫∫

f12 dr1 dr2 = ,

B3 = − 2

3!V

∫∫∫

[f12f13f23] dr1 dr2 dr3 = ,

B4 = − 3

4!V

∫∫∫∫

f12f23f34f14[3 + 6f13 + f13f24] dr1 dr2 dr3 dr4

= + + .

Figure 2.1: All graphs contributing to virial coefficients up to B4 weightedby their respective symmetry numbers.

For a system of spherically symmetrical particles such as those used

throughout this investigation, the virial coefficients can be expressed in

terms of cluster integrals using the following generalisation:

Bn =1− n

n!

∫

. . .

∫

FB(rn) dr1 . . . drn−1. (2.21)

27

Chapter 2

In these expressions, FB(rn) represents the sum of all biconnected graphs

on n vertices. For a pairwise additive potential, it can be written as a

product of Mayer functions:

FB(rn) =

∑

G

∏

i,j∈G

fij, (2.22)

where G is the subset of all possible graphs with n vertices which consists

of all biconnected graphs. With increasing n, the number of these graphs

and the dimensionality of the integrals they represent increases rapidly for

even the simplest of systems. Over n vertices, Wheatley (2013) points out

that there are 2n(n−1)/2 possible graphs and at higher orders, most of these

are biconnected.

This diagrammatic method has proved a useful and powerful tool for

systematically analysing cluster integrals in general and specifically in the

case of using them to obtain higher order virial coefficients. Much of the

early work in this area has concentrated on the hard sphere potential, since

this is amongst the simplest possible geometries and minimises the number

of variables which must be introduced.

Ree and Hoover (1964) introduce a modified Mayer function, f∗ij:

f∗ij ≡ fij + 1. (2.23)

This function is represented by a ‘wiggly’ line rather than a straight line

connecting two vertices and is introduced to a diagram whenever there is a

pair of unconnected vertices. The effect of this is to reduce the number of

topologically distinct biconnected diagrams and hence improve the

efficiency of any calculation making use of them. To illustrate this

improvement, Malijevsky and Kolafa (2008) tabulate the number of

diagrams that must be analysed for each order of virial coefficient in the

case of hard spheres. Implementing this method allowed Ree and Hoover

(1967) to obtain up to the 7th order virial coefficients for hard spheres and

hard disks, as well as improving existing results for lower order coefficients.

More recently, methods have been developed to make use of these

diagrams in order to calculate higher order virial coefficients for hard

28

Chapter 2

spheres and hard disks. van Rensburg (1993) describes a computational

scheme for calculating the 8th order coefficient using Ree-Hoover diagrams.

Labık et al. (2005) build on this work to suggest a more computationally

efficient method of analysing these diagrams. This has enabled them to

obtain estimated values for the 9th and 10th order coefficients which have

since been improved upon by Clisby and McCoy (2006).

A further alternative method for calculating cluster integrals is described

by Wheatley (2013) and has provided to date virial coefficients up to 12th

order. This approach begins for the nth coefficient by calculating the value

of sum the 2n(n−1)/2 graphs to give the total number of possible diagrams.

Then, the contribution to this sum made by disconnected graphs is

removed. Next, all graphs containing an articulation point (i.e. all singly

connected graphs) are removed, leaving the total number of biconnected

graphs. To do this, the properties of the sets of singly connected and

unconnected graphs are used to allow a recursion method which

systematically eliminates the contributions to the total sum from all

members of those sets. This leaves a sum of contributions from

biconnected diagrams only. An advantage of this method is that it requires

only 3nn operations and 2nn storage.

2.6 Mayer sampling Monte Carlo (MSMC)

A new alternative method for the evaluation of cluster integrals has been

introduced by Singh and Kofke (2004) which is theoretically able to

calculate a cluster integral and can be applied to any potential function for

which the Mayer function can be calculated.

The method relies on a simulation performed only on a number of particles

equal to the order of the cluster integral to be solved. Configurations of

particles are generated by a Metropolis Monte Carlo (MC) scheme using

importance sampling based on the magnitude of the interactions

represented by the cluster integral. Rather than using this to directly

evaluate the cluster integral, the ratio of the required cluster integral to a

known reference integral is sought.

29

Chapter 2

Singh and Kofke (2004) suggest a formula for obtaining a cluster integral

which is based on umbrella sampling, as described by Frenkel and Smit

(2002). This is a type of sampling useful in situations where the energy

landscape is such that there are large potential energy barriers between

different regions of configurational space. Therefore MC sampling may be

unable to sample a sufficient range of configurations to achieve an accurate

result. Umbrella sampling overcomes this problem by sampling parts of the

configurational space accessible to both the system of interest and a

reference system. To do this in a single simulation, the Markov chain

governing the sampling must be modified through the introduction of a

weighting function π(rN ). This means that the probability of being at a

point in the configuration space rN is now proportional to π(rN ).

Using these ideas, an equation can be written to describe a sampling

formula:

Γ = Γ0Γ

Γ0= Γ0

〈γ/π〉π〈γ0/π〉π

. (2.24)

It is necessary that π(rN ) is chosen such that it has some overlap in

configuration spaces with both systems, otherwise either the numerator or

denominator of this expression will be close to zero. The subscript 0

denotes the given quantity for a chosen reference system, while the

subscript π indicates that the integral is weighted by a normalised

probability distribution, i.e. for some property M :

〈M〉π ≡∫

Mπ drN∫

π drN. (2.25)

In equation (2.24), Γ is the cluster integral to be evaluated, in this case

Bn, and Γ0 is the cluster integral of equivalent order of a chosen reference

system. γ(rn) is a function of position in configurational space and is

defined as the integrand (or sum of integrands) of the cluster integral Γ.

The value of π(rn) is chosen by Singh and Kofke (2004) to be equal to the

absolute value of the sum of all integrands of the required cluster integral

with their appropriate weights, i.e. π = |γ|. A benefit of defining γ and π

30

Chapter 2

in this way is that the numerator in the above equation reduces to an

average of the sign of the overall sum, since all contributions to the

numerator are ±1. The angled brackets indicate that an ensemble average

is taken over all orientations and configurations of the particles being

modelled.

Singh and Kofke (2004) suggest two possible methods for choosing a

reference system. Firstly, a reference system of a different potential, ideally

defining a relatively well-understood system, can be selected. A choice such

as hard spheres of a known diameter is found to be advantageous since it

has no temperature dependence and its virial coefficients had already been

found to a higher order than other potentials. The second recommended

choice for a reference system is one governed by the same potential as that

which is being investigated, but with the system given by different

products of Mayer functions. This allows the method to be applied to

more complicated integrals by relating them to smaller, simpler ones.

Choosing the former type of reference system and π as described here

means that (2.24) provides the required virial coefficient. Often, the

solution of cluster integral sums suffers from errors introduced in the

cancellation of large terms to give a relatively small result. This is not the

case here, since the problem manifests in an alternative way. Singh and

Kofke (2004) argue that in situations where this cancellation would cause

errors to occur, the numerator of (2.24) becomes very small. The required

number of samples is inversely proportional to the square of this average,

leading to a larger number of samples being required and hence a more

computationally expensive calculation.

For each MC trial, a randomly chosen number of the total number of

particles are perturbed by displacing and/or rotating them. The trial is

then accepted with a probability of (1,min(πold, πnew)). The results of

these trials are then used to compute the averages required in equation

(2.24). Trials can be split into batches, with the fraction of trials accepted

calculated periodically. Allen and Tildesley (1988) provide a method of

varying the maximum possible displacement in order to control this

acceptance ratio. This method has been successfully used to calculate the

31

Chapter 2

virial coefficients for the Lennard-Jones potential up to sixth order, using a

hard sphere potential as a reference system.

While investigating various models for water, Benjamin et al. (2006) found

this method to be less successful. This is because equation (2.24) requires

that the configurational space of the reference system chosen be a subset of

that of the target system. However, the water particles were found to

strongly prefer their own energetic wells when sampled at low

temperatures and therefore many configurations of particles important to

the reference system were not being sampled, causing the direct sampling

method shown in equation (2.24) to fail. Since this sampling is carried out

using a reference system governed by a hard sphere potential, it can be

concluded that the configurational space of the hard sphere reference

system is not a subset of the configurational space of the water potentials,

but instead that there are regions of overlap between the two spaces.

To overcome this problem, Benjamin et al. (2006) suggest an improvement

to the sampling equation given in (2.24). Rather than a direct sampling

technique in which the system of interest is perturbed into the reference

system, the improved sampling equation is based on overlap sampling.

This requires that separate simulations of both systems are carried out and

a perturbation is performed into an overlapping region. The new overlap

equation is (Benjamin et al., 2006):

Γ = Γ0〈γ/π〉π/〈γOS/π〉π

〈γ0/π0〉π0/〈γOS/π0〉π0

. (2.26)

As explained by Benjamin et al. (2007a), the quantity measured during

both simulations is the ratio of the average value of each cluster integral to

the average overlap for that system. They define the ‘overlap function’

γOS, which represents the regions of configurational space that are of

importance to both the targeted system (in their case various model

potentials for a water molecule) and the chosen reference system (hard

spheres):

γOS =|γ0||γ|

α|γ0|+ |γ| . (2.27)

Here, α, is a chosen optimisation parameter. Bennett (1976) defines this

32

Chapter 2

parameter as:

∑

reference

γOS

π0= α

∑

target

γOS

π, (2.28)

where the subscripts ‘reference’ and ‘target’ refer to the respective systems.

This equation assumes that the statistical error is equal in both systems

for a given number of samples. However, in MSMC, the target system is

often more difficult to sample and therefore contains a greater degree of

statistical error, invalidating this assumption. Sampling the target system

more often leads to a decreased value of α and hence increases the error in

〈γOS/π0〉. Benjamin et al. (2007b) suggest generalising the above

expression to:

〈γOS/π0〉 = α〈γOS/π〉. (2.29)

In this case, the same value of α is still obtained when the target system is

sampled more often than the reference system with no increase in

statistical error.

The MSMC method has recently been modified by Schultz and Kofke

(2014) and Feng et al. (2015). To further improve performance, the

recursive counting method developed by Wheatley (2013) has been

incorporated into the sampling scheme. This introduces a loss of precision

in the case of configurations in which the particles are far apart and FB is

close to zero. To overcome this, a truncation is used, setting FB to 0 when

it is smaller than a desired cut-off value of ∼ 10−12. This increase in

efficiency has enabled virial coefficients up to n = 16 to be obtained for the

Lennard-Jones fluid.

The MSMC method has also been used successfully to obtain virial

coefficients for many other potentials, including a Lennard-Jones fluid

(Schultz and Kofke, 2009a), the Gaussian charge polarisable model

(GCPM) for water molecules (Benjamin et al., 2007a), flexible molecules

such as linear alkanes and Lennard-Jones fluid mixtures (Schultz and

Kofke, 2009b), methanol (Shaul et al., 2011a) and high-dimensional hard

sphere models (Zhang and Pettitt, 2014). Overall, it can be concluded that

this is an increasingly well-tested and reliable method and therefore a good

33

Chapter 2

choice for calculating the virial coefficients required for this project.

34

Chapter 3

3 Applicability of the virial equation of state to a

system

This section will consider how successfully the virial equation may be

applied to a system of spherically symmetrical particles and under what

circumstances its application is limited due to poor convergence of the

virial expansion. Suggested methods of overcoming these problems and

improving the radius of convergence of the virial expansion will be

presented.

3.1 Convergence of the virial expansion

A series Sn of n terms is the summation of a sequence of numbers, each of

which is generated in turn by the same formula or algorithm. A series is

said to be convergent if the sequence of its partial sums of the first n terms

for increasing values of n is convergent. Formally, this means that there

exists some limit ϕ such that for any sufficiently small number ε > 0 there

exists an integer N such that for all n ≥ N , |Sn − ϕ| ≤ ε. A series that is

not convergent is said to diverge. The radius of convergence of a series is

the radius of the largest disc in which the series is convergent, so is either a

positive real number or infinite.

Since the virial expansion (and related expansions such as that for the

direct correlation function) is an infinite series, any finite calculation of the

series is necessarily an approximation. Hence, the question of whether the

series is convergent and if so within what radius is an important

consideration in the study of the virial expansion.

Due to the difficulty in obtaining virial coefficients, in most cases a virial

expansion containing only a few terms can be calculated. If the virial

expansion for a system converges poorly or slowly, this could give rise to

an inaccurate description of the fluid behaviour when examined over a

large range of fluid densities. Since the virial expansion is a power series

expansion in powers of density, as density increases the behaviour of the

expansion will become increasingly dominated by the last term in the

series. This frequently gives incorrect asymptotic behaviour, since the

35

Chapter 3

series can only accurately represent Z(ρ) within its radius of convergence.

This problem is discussed by Barlow et al. (2012) in the context of hard-

and soft-core inverse power potentials. It is found that as the spheres

become less strongly repulsive (‘softer’), the virial expansion behaves

increasingly poorly with increasing density.

Whether or not a series has converged (and if so within what radius) can

be determined in two ways. Sometimes, it is possible to determine

convergence by observation, but when this is not possible a number of

mathematical convergence criteria exist. Two such criteria are

D’Alembert’s criterion and Cauchy’s criterion. These are of particular

convenience in the case of the virial expansion since they require only one

series to analyse and are not limited to a series with a large number of

terms known.

The D’Alembert criterion supposes that there exists x such that:

limn→∞

∣

∣

∣

∣

an+1

an

∣

∣

∣

∣

= x, (3.1)

where an and an+1 are consecutive terms in a series. Then if x < 1, the

series is convergent. If x > 1 the series is divergent and if x = 1 then the

test is inconclusive. The Cauchy criterion supposes that there exists x such

that:

lim supn→∞

n√

|an| = x, (3.2)

where an is again the nth term of a series. Then the same conclusions for

the value of x apply as for the D’Alembert criterion.

To demonstrate the usefulness of these tests, consider Figure 3.1. This

shows the change in pressure with density at temperatures above and

below the critical temperature of a Lennard-Jones fluid. The virial

coefficients used are generated using MSMC calculations by Schultz and

Kofke (2009a). In this case, an analytical result exists due to the

Kolafa-Nezbeda equation of state (Kolafa and Nezbeda, 1994).

It can be observed that under both sub- and super-critical conditions, the

36

Chapter 3

curves converge at very low densities and diverge at very high densities.

However, it is not clear by observation at what point the curves begin to

diverge. Therefore, convergence tests are useful to establish a more exact

idea of the radius of convergence of the virial expansion. In the sub-critical

case, both D’Alembert’s and Cauchy’s criteria conclude that the curves

converge up to densities of around 0.6ρ∗c . However, in the super-critical

case the radius of convergence is greater, at around 1.15ρ∗c . In both cases,

the curves converge strongly to the KN equation of state within these

density limits.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

KN EOS

p*

−20

−15

−10

−5

0

5

10

ρ*/ρc

*

0 0.5 1 1.5 2 2.5 3

−0.1

−0.05

0

0.05

0.1

0 0.2 0.4 0.6 0.8 1

*

0

/

.

6

h

S

*

0

/

.

6

hO

h

S SE6 * *E6 0 0E6 /

S

SES0

SES.

SES8

SES

SE*

S SES6 SE* SE*6 SE0 SE06

Figure 3.1: Pressure-density plot obtained from the virial expansion for aLennard-Jones fluid at temperatures of 1.0 (left) and 1.6 LJ units (right).These reduced units are defined in Chapter 6. Each curve represents a pres-sure virial expansion up to the stated order of powers of density.

Systems governed by a potential modelling only a repulsive inter-particle

force can form only one fluid phase and exhibit no vapour-liquid phase

transition. It can be observed that although the virial expansion converges

relatively well for some such systems (such as hard-sphere potentials), this

is not the case when an attractive component is included in the potential

function and so separate vapour and liquid phases can exist. In such a

case, the virial expansion converges better at low densities and at

super-critical temperatures. However, as the density approaches the

critical density, the virial expansion frequently diverges. This behaviour is

demonstrated by the data presented in Figure 3.1.

37

Chapter 3

3.2 Improving the performance of the virial expansion for

systems with a single fluid state

One possible technique for improving the radius of convergence of the

virial expansion is to construct some form of approximant for Z (or p).

These are polynomial functions which share some behaviour with the virial

expansion. As density tends to zero, the approximant must tend to the

ideal fluid. Also, the Taylor expansion of the approximant about ρ = 0 (i.e.

the Maclaurin expansion) must match the virial expansion to ith order.

However, there are an infinite number of possibilities when choosing such

an approximant. In order to correctly represent the equation of state, some

knowledge of the behaviour of the fluid at large densities is required. This

allows an approximant to be chosen which gives the correct asymptotic

fluid behaviour at both low and high densities. If the radius of convergence

of the virial expansion is known, Baker (1990) shows that an approximant

can be used to continue the series analytically beyond this radius. This

can lead to an accurate and generalised equation of state that is valid for

the entire fluid phase for a system.

Baker and Gammel (1961) used a symmetric Pade approximation scheme

in order to obtain the correct behaviour of a system where the asymptotic

behaviour is finite. However, this approach would be unsuitable for

treating a virial expansion, since it would not be possible to use a

truncated power series when the independent variable (in this case density)

became large.

A similar approach has been developed by Barlow et al. (2012) based on

an asymmetric Pade approximation scheme. This allows the required

asymptotic behaviour for the density to be enforced and was successfully

used to re-sum a virial expansion for fluids governed by an inverse-power

potential.

For a virial expansion to order ρJ−1, either [(M + k)/(M − 1− k)](α

2k+1) or

[(M + k)/(M − k)](α2k

) approximants are constructed depending on

38

Chapter 3

whether J is even or odd. These approximants take the following form:

Z =

[

N1 +N2ρ+ · · ·+NM+1+kρM+k

1 +D2ρ+ · · · +DM−kρM−1−k

]( α2k+1

)

, J even,M = J/2, (3.3)

Z =

[

N1 +N2ρ+ · · ·+NM+1+kρM+k

1 +D2ρ+ · · · +DM−k+1ρM−k

]( α2k

)

, J odd,M = (J − 1)/2. (3.4)

To give the correct ideal behaviour as ρ→ 0, N1 is set to 1. The variable k

is an integer greater than or equal to −M and defines the degree of

asymmetry between the numerator and denominator, while α is chosen to

give the correct asymptotic behaviour for the chosen system. This results

in an equation of state which matches the correct behaviour of the system

at both low and high densities. The sets of coefficients Ni and Di are

found in terms of the virial coefficients through Taylor expanding these

expressions and comparing them to the virial expansion at each order of ρ.

This method has the advantage of being adaptable to other models

provided the asymptotic behaviour is known and has produced results for

hard- and soft-sphere models which agree well with simulation data.

There have been a number of mathematical investigations into the

convergence of the virial expansion of a known system. Florindo and Bassi

(2014) demonstrate a method of estimating higher-order virial coefficients

for the hard sphere model based on Levin’s transformations, which are

designed to improve the performance of a slowly converging or strongly

divergent series (Levin, 1972; Baram and Luban, 1979). The resulting

estimated virial coefficients at 9th and 10th order are within 5% of those

obtained by Wheatley (2013).

The convergence of the virial expansion of mixtures of different types of

particle is considered by Jansen et al. (2014) through application of the

Lagrange-Good inversion formula, which generalises the Taylor expansion

of the inverse function of an analytic function to several variables. A

mathematical proof is provided to demonstrate that the virial expansion

converges fully at low densities, thus supporting the results of

previously-discussed investigations.

39

Chapter 3

The problem of obtaining a single expression which recreates the correct

behaviour in both the asymptotic limit and as the independent variable

tends towards zero is also considered by Muthukumar and Nickel (1984).

Rather than a density expansion, they consider the problem of describing

the mean square end-to-end length of a polymer chain in terms of a

defined dimensionless parameter. To re-sum this series and obtain the

correct behaviour for the chain length, a Borel summation is employed

which results in a relationship that is valid at all values of the

dimensionless parameter being obtained. The accuracy of these results is

confirmed independently by Des Cloiseaux et al. (1985).

3.3 Improving the performance of the virial expansion

when separate liquid and vapour phases can exist

As noted earlier, the convergence behaviour of the virial expansion of a

system capable of undergoing a vapour-liquid phase transition is less

well-understood. It has been claimed by Mayer and Mayer (1977) that the

virial expansion is not valid at pressures higher than those associated with

the spinodal (the limit of local stability with respect to small fluctuations).

This means that when a phase transition exists, a virial expansion cannot

describe the van der Waals’ loop since the virial equation of state cannot

describe a fully unstable phase. This basis is used by Schultz and Kofke

(2015b) to argue that the binodal may not play any part in determining

the point at which a phase transition occurs.

Recent work by Schultz and Kofke (2015b) and Ushcats (2012, 2013a)

seeks to investigate the region of the phase diagram between the binodal

and the spinodal pressures in order to predict the point at which a phase

transition will occur. This contends that the virial equation of state should

be able to account for this phase transition and in order to correctly do so,

the virial expansion may require a volume dependence, which is not

normally present when virial coefficients are expressed in terms of

infinite-volume cluster integrals.

Subsequent studies and comparisons with simulation work to identify the

point at which a phase transition from vapour to liquid occurs for a

40

Chapter 3

Lennard-Jones fluid (Ushcats, 2013a,b) have confirmed these limitations

and the observations shown earlier in Figure 3.1. The divergence observed

when a potential function has an attractive component has been attributed

by Ushcats (2013a) to the simplification of the cluster integrals.

Schultz and Kofke (2015b) hypothesise that the virial equation of state will

remain valid at densities up to the spinodal. In order to test this theory

and attempt to describe the vapour-liquid phase transition, the

convergence of the virial expansion for a Lennard-Jones fluid is studied

using extrapolated virial coefficients to infinite order.

To do this, the following relations for an extrapolated virial coefficient of

nth order, Ben, are suggested by Ushcats (2014) in (3.5) and Schultz and

Kofke (2015b) in (3.6):

Ben(T ) = − 0.01

(n− 1)(n − 2)

(

2.2

T

)3.51(n−1)

, (3.5)

Ben(T ) ∝ exp

(−aψ(n + 1)− γa+ bn

T

)

,

≈ C(T )n−a/T expbn/T . (3.6)

In this expression, γ is Euler’s constant and ψ is the digamma function.

The coefficients a and b are determined by considering the incremental

ratio of consecutive virial coefficients and fitting a curve to this ratio

plotted against n at several sub-critical temperatures and fitting to the

following approximate relationship:

T logBn(T )

Bn−1(T )= −a

n+ b. (3.7)

Based on this extrapolation, a closed-form equation of state can be

generated through summing the virial series:

∞∑

n=1

n−a/T expbn/T ρn = Li aT(expb/T ρ), (3.8)

where Lis(z) is the polylogarithm. The general form of this function can

be written as an expansion:

41

Chapter 3

Lis(z) =

∞∑

k=1

zk

ks= z +

z2

2s+z3

3s+ . . . , (3.9)

which reduces to the logarithm if s = 1. This form of the function is

defined on the complex plane within the unit disk, since the radius of

convergence this series is 1. It has a real argument within this radius,

which can be extended by analytic continuation. Otherwise, the imaginary

part of the argument outside the unit disk is given by Wood (1992). This

expansion is taken to be the principal branch since the function is

multivalued for some values of s. When the function diverges, branch point

singularities exist (Ablowitz and Fokas, 2003).

Since Schultz and Kofke (2015b) found that equation (3.6) gives a poor

estimate of lower-order coefficients, a re-summed equation of state is

written by introducing accurate values of Bn obtained by an alternative

method, such as MSMC:

βP e(ρ, T ) =

nmax(T )∑

n=1

(Bn(T )−Ben(T ))ρ

n

+ C(T )Li aT(expb/T ρ). (3.10)

An upper bound of the radius of convergence of this equation of state is

ρ = expb/T . This is in accordance with the polylogarithm having a branch

point singularity at z = 1. Schultz and Kofke (2015b) conclude that as the

order of the virial expansion used increases, the spinodal densities converge

to this point and therefore the point at which the virial series diverges is

the spinodal. However, this method cannot quantitatively describe a

vapour-liquid phase transition.

A further issue concerns the polylogarithmic function given in equation

(3.9), which is plotted up to tenth order in Figure 3.2. It can be observed

from the plot that this function itself is divergent at densities well below

the critical density for a Lennard-Jones fluid, which raises questions over

the convergence properties of equation (3.10).

Work in this area has also been carried out by Trokhymchuk et al. (2015)

42

Chapter 3

to 2nd order

to 3rd order

to 4th order

to 5th order

to 6th order

to 7th order

to 8th order

to 9th order

to 10th order

Li a/T (ebn/Tρ*)

0

10

20

30

40

ρ*0 0.1 0.2 0.3 0.4

Figure 3.2: Plot of Li aT(eb/T ρ) up to tenth order. The function converges at

densities up to ρ∗ = 0.1 but diverges rapidly beyond that point.

through the study of the virial expansion of a hard-core attractive Yukawa

fluid. The potential function governing the fluid was split into two separate

terms. Rather than splitting the potential into a term responsible for the

short-range repulsive forces and another responsible for the long-range

attractive forces, an alternative form of perturbation is used (Melnyk et al.,

2009). This is based on an ‘augmented’ version of van der Waals’ theory in

which one term represents the full interaction energy between neighbouring

particles (the full excluded volume energy) while the other represents the

weaker long-range attractive forces between particles further apart. This is

used to produce an augmented virial equation of state from which

thermodynamic properties are calculated. Although this gives results for

pressure close to those obtained from simulation at low densities, at higher

densities there is still a discrepancy. This means that although this method

shows some improvement in the prediction of the high density behaviour of

the fluid, it seems unable to fully predict a vapour-liquid phase transition.

The findings discussed in this chapter demonstrate that it is possible to

improve the performance of the virial expansion by improving the rate at

which it converges and increasing the radius in which this can take place.

This is especially true in systems of only one fluid phase and there are

43

Chapter 3

several methodologies that are able to achieve this accurately. However,

there are still some questions concerning the convergence of the virial

expansion in systems with separate liquid and vapour phases. As yet the

literature is unclear on the rigour of the claim that the radius of

convergence is the spinodal density despite the claims of Ushcats (2014)

and Schultz and Kofke (2015b) to be able to prove this and the plausibility

arguments offered by Mayer and Mayer (1977). If this contention is true,

then this does raise important issues concerning the validity of approaches

based on the van der Waals equation of state, in which a single

mathematical function describes both the vapour and liquid phases.

44

Chapter 4

4 Describing the structure of a fluid

Knowledge of the structure of a fluid, in particular the direct correlation

and radial distribution functions, is desirable since these functions are not

only fundamental to liquid state theory (Hansen and McDonald, 2006),

but can also provide useful information as to the behaviour of the

Helmholtz energy when local variations in density are present. This

enables the calculation of the stability limits of various phases and the

approximation of Helmholtz energy functionals (Haymet, 1987). This

section will describe how the correlation structure of a fluid is defined and

will consider possible methods of investigating it.

4.1 The Ornstein-Zernike equation

The correlation of particles within a fluid is a measure of the order within

the structure of that fluid and describes the effect that particles exert on

one another within the fluid. It has been shown by Ornstein and Zernike

(1914) that the total correlation between two identical particles labelled 1

and 2 in the fluid can be split into a direct contribution and an indirect

contribution.

The direct contribution describes the influence exerted on particle 2 by

particle 1. The indirect contribution to the total is thought of as the effect

of particle 1 on a neighbouring particle, labelled 3 without loss of

generality, which in turn exerts its own direct and indirect effects on

particle 2. These correlation effects are expressed as functions of the

distance between the particles in question.

These effects may be described mathematically in terms of correlation

functions. It is first convenient to define the total correlation function,

h(r12), in terms of the radial distribution function g(r12) for a particle

positioned at the origin:

h(r12) = g(r12)− 1. (4.1)

The radial distribution function is a ratio of the local density (over a scale

σ) to the average fluid density ρ. It effectively defines the probability of

45

Chapter 4

there being a given particle occupying a given spatial position. This means

that g(r12) can be used to find the variations of density within a fluid as a

function of distance from a chosen origin. It should be noted that the

number of particles in a known volume of fluid 4πr212dr12 can be written as:

dN = ρg(r12)4πr212 dr12, (4.2)

so it can be seen that at large distances (as r → ∞), g(r12) → 1.

The Ornstein-Zernike (OZ) equation relates the total correlation function

to a further function, the direct correlation function c(r12) :

h(r12) = c(r12) + ρ

∫

c(r12)h(r23) dr3, (4.3)

h(r12) = c(r12) + e(r12), (4.4)

where r3 represents the position of the third particle. Defining the distance

vector between two particles as rij = ri − rj and using the convolution

theorem allows the Fourier transforms of h(r12) and c(r12) to be defined,

written as h(k) and c(k) respectively. The form of the forward and

backward transform is shown below, using the total correlation function as

an example:

h(k) =

∫

e−ik.r12h(r12) d3r12, (4.5)

h(r12) =

(

1

(2π)3

)∫

eik.r12 h(k) d3k. (4.6)

Now, the OZ equation can be rewritten in terms of these transforms in

k-space:

h(k) = c(k) + ρh(k)c(k). (4.7)

Hence, an algorithm to obtain e(r12) from c(r12) can be constructed by

first obtaining the Fourier transform:

e(k) =c(k)

1− ρc(k)− c(k), (4.8)

46

Chapter 4

and then back-transforming from e(k) to e(r12). This is one possible

method by which information about the direct and total correlation

functions can be obtained. Alternatively, simulation techniques can be used

to measure g(r12) and then a result for c(r12) can be obtained using this.

The correlation structure and RDF of a fluid can be used to find the

mechanical thermodynamic properties of the fluid through equations such

as the pressure and compressibility equations, as shown by McQuarrie

(1973). These equations link information about the correlation structure to

the pressure via the compressibility. In terms of the RDF, the

compressibility equation is:

kBT

(

∂p

∂ρ

)

= 1 + ρ

∫

V[g(r12)− 1] dr2, (4.9)

and the pressure equation is:

p = ρkBT − ρ2

6

∫

Vr12g(r12)

du(r12)

dr12dr2. (4.10)

4.2 Solving the O-Z equation using integral equation theory

There is an alternative method for using equation (4.4) to find both the

total correlation function and the direct correlation function. It can be

observed that in this equation, there are two functions, the DCF and the

function representing indirect contribution. This means that the OZ

equation cannot by itself be directly solved for c(r12). Therefore, further

information is required before equation (4.4) can be solved.

This information takes the form of a closure relation. This is a second

equation to describe the total correlation function which introduces

information about the shape of the inter-particle potential u(r12). A wide

range of closures for a variety of types of interaction potential have been

suggested over recent decades.

In general, for a system of spherically symmetrical particles, it holds that:

g(r12) = exp[−βu(r12) + e(r12) +B(r12)], (4.11)

47

Chapter 4

where e(r12) is the indirect contribution to the total correlation function:

e(r12) = h(r12)− c(r12). (4.12)

The bridge function, represented by B(r12), is a function of inter-particle

distance. Although it can be represented via diagrammatic expansions, no

convenient closed-form solution exists. Closure relations are based on the

introduction of approximations for this function. There have been a great

many closure relations suggested over the last century for application to a

wide range of systems. As such only very few that are most useful to

systems of hard spherical particles will be discussed below. Malijevsky and

Kolafa (2008) provide a brief overview of the categories of closure that

have been suggested for similar systems.

The simplest closure relation is the hyper-netted chain closure, which

results from setting B(r12) = 0:

g(r12) = exp[−βu(r12)] exp[e(r12)]. (4.13)

This has been found to be a more accurate approximation for systems of

soft spheres (i.e. where some degree of overlapping of particles is

permitted). An alternative approximation is the Percus-Yevick equation

(1958):

g(r12) = exp[−βu(r12)](1 + e(r12)). (4.14)

This expression has the advantage that for systems of hard spherical

particles, an analytical solution exists. A drawback of both the

hyper-netted chain and Percus-Yevick approximations is that they are not

self-consistent. This means that when these approximations are used with

the OZ equation to obtain the total correlation function, the results for

pressure obtained from the pressure equation and compressibility equation

are not equal.

Although Malijevsky and Kolafa (2008) summarise a large number of

works that suggest self-consistent closure relations, of particular interest to

this investigation is the approximation given by Rogers and Young (1984),

for use with inverse power potentials of the form 1/rn. This approximation

48

Chapter 4

makes use of both the hyper-netted chain and Percus-Yevick

approximations and introduces a ‘thermodynamic consistency’ parameter

which varies with fluid density:

g(r12) = exp[−βu(r12)](

1 +exp[e(r12)ξ(r12)− 1]

ξ(r12)

)

, (4.15)

where ξ(r12) is a mixing function containing the thermodynamic

consistency parameter αRY :

ξ(r12) = 1− exp[αRY r12]. (4.16)

This parameter αRY is chosen in order to provide the best possible

consistency in the results of the pressure and compressibility equations.

An example of the application of the Rogers-Young approximation for a

12th-order inverse power potential is shown in Figure 4.1. In this case,

inter-particle distance is represented by x. This compares their

thermodynamically consistent approximation (labelled TC) with

pre-existing Monte Carlo data (labelled MC) taken from Hansen and Schiff

(1973).

4.3 Using a virial expansion to describe the structure of a

fluid

It has been shown so far that fluid structure can be found through

knowledge of the DCF. The required calculations can be done in one of

two ways. Firstly, the DCF can be obtained via simulation. Since there

appears to be no direct method of doing this, usually the two-particle

distribution function is found first, allowing equation (4.4) to then be used.

Secondly, the DCF can be found theoretically via integral equation theory,

using some form of closure approximation with (4.4) as described above.

Apart from the Percus-Yevick result for hard spheres (1958), these must be

solved numerically. However, these methods can fail to give a real solution

in some areas of the phase diagram, particularly near to a phase transition

and hence can be limited in their usefulness. An example of this is

presented for the HNC closure approximation by Belloni (1993).

A third approach is to use a virial expansion. One way to improve a

49

Chapter 4

Figure 4.1: Radial distribution function obtained via the Rogers-Young clo-sure approximation for a 12th-order inverse power potential and comparedwith MC data provided by Hansen and Schiff (1973). This figure is adaptedfrom Rogers and Young (1984).

closure approximation is to employ a virial expansion of the bridge

function B(r12) to provide a result for the direct correlation function via

integral equations. This has been achieved for hard spheres up to sixth

order by Kolafa and Labık (2006) using a range of diagrammatic methods.

MSMC has been used by Shaul et al. (2011b) to calculate corrections to

the virial expansions of the HNC and PY closures up to fifth order in the

case of a Lennard-Jones fluid. However, in this investigation, a simpler

idea of directly studying the virial expansion of the DCF is explored.

As with the virial expansion, the DCF c(r12) can be expressed

diagrammatically as an expansion of products of cluster integrals (Hansen

and McDonald, 2006):

50

Chapter 4

c(r12) = r1 r2+

r1 r2

+

r1 r2

+

r1 r2

+

r1 r2

+

r1 r2

+

r1 r2

+

r1 r2

+

r1 r2

. . . , (4.17)

where the black and white circles and connecting vertices represent

clusters of Mayer functions in the same manner as described previously.

The direct correlation function c(r12) can be obtained by considering a

density expansion:

c(r12) =

∞∑

n=2

nBn(r12)ρn−2, (4.18)

where the coefficient Bn(r12) is related to the nth-order virial coefficient by:

Bn =1

V (4π)2

∫

Bn(r12) dr1 dr2. (4.19)

This method has been used by Dennison et al. (2009) to successfully

calculate the direct correlation function for both systems of hard spheres

and hard spheroids. The application of this to this investigation is

explained in Chapter 6.

51

Chapter 5

5 Predicting the structure and formation of an

ordered phase

This chapter discusses the hard sphere model and considers the transition

from a hard sphere fluid to a solid crystal phase. The equations used to

find the Helmholtz energy are presented and the most widely-used

methods of solving them are discussed. Following this, the question is

considered through the application of a virial expansion.

5.1 Modelling the Helmholtz energy of a fluid

An inhomogeneous fluid of hard, spherically symmetrical particles can be

described in terms of density functionals, as demonstrated by Tarazona

et al. (2008). For any chosen temperature and form of the potential u(rij),

there is a unique free energy A[ρ], which is a functional of the density

distribution within the system ρ(r) but not a functional of any external

potential.

Considering a grand canonical system, the equilibrium density distribution

in the vicinity of some external potential v(r) that minimises the grand

potential energy density functional Ω[ρ] for any chosen function ρ(r) is:

Ω[ρ] ≡ A[ρ] +

∫

ρ(r)(v(r)− µ) dr. (5.1)

Here, µ is the chemical potential of the system. The minimising condition

on Ω can then be expressed using the Euler-Lagrange equation:

δΩ[ρ]

δρ(r)≡ δA[ρ]

δρ(r)+ v(r)− µ = 0. (5.2)

The free energy functional of a system of hard spheres may be split into an

ideal contribution and an excess contribution. This expresses the overall

Helmholtz energy in terms of a perturbation to the ideal case which takes

into account the changes in energy due to the existence of particle

interactions in a real fluid. To first obtain an expression for the ideal

contribution to the functional, Tarazona et al. (2008) begin by considering

the grand canonical partition function Ξ:

52

Chapter 5

Ξ = e−βΩ =∑

N

eβµN

N !Λ3N

∫ N∏

i=1

e−βUNdri, (5.3)

where:

UN (r1 . . . rN ) =∑

ij

u(rij) +∑

i

v(ri). (5.4)

The equilibrium density distribution can also be calculated from the grand

canonical partition function:

ρ(r) =1

Ξ=

∑

N

eβµN

(N − 1)!Λ3N

∫ N∏

i=1

e−βUN δ(r − ri) dri. (5.5)

In the case of an ideal fluid, there are no interactions between particles, i.e.

u(rij) = 0. Inputting this into (5.3) results in:

Ξid =∞∑

N=0

(eβµ∫

e−βv(r)dr)N

N !Λ3N. (5.6)

From this, (5.3) also allows the grand potential energy at equilibrium to be

calculated:

Ωid ≡ − 1

βln(Ξid) =

eβµ

β

∫

e−βv(r)dr. (5.7)

The equilibrium density distribution in the ideal case can now be

calculated from equations (5.5) and (5.7) and is:

ρid(r) = e−β(µ−v(r)). (5.8)

Rearranging the two expressions for Ωid and ρid(r) leads to an expression

for the free energy functional for an ideal gas:

βAid[ρ] =

∫

ρ(r)[ln(ρ(r)Λ3)− 1] dr, (5.9)

and Hansen and McDonald (2006) show that through functional

differentiation:

δAid[ρ]

δρ(r)=

1

βln(ρ(r)Λ3). (5.10)

53

Chapter 5

This result directly links the Euler-Lagrange equation (5.2) with the

equilibrium density.

Having obtained an expression for the ideal contribution to the free energy

functional, it remains to seek an expression for the excess contribution,

Θ([ρ]; r), to enable the total free energy functional to be calculated:

βA[ρ] = βAid[ρ] + βAex[ρ],

=

∫

ρ(r)[ln(ρ(r)Λ3)− 1] + Θ([ρ]; r)dr. (5.11)

The following sections will examine these two terms more closely and

discuss methods of solving them to obtain an expression for the overall

Helmholtz energy of the fluid.

5.2 Obtaining the excess contribution to the Helmholtz

energy

In recent decades, increases in available computational power has enabled

the development of several powerful and efficient methods for

approximating the excess contribution to the Helmholtz energy. Density

functional theory (DFT) is an important computational modelling method

in these investigations. This theory uses functions of functions, known as

functionals, to determine the properties of the many-body system. When

applied to a statistical mechanical system, the Helmholtz energy is a

function of the local density, which itself is a function of the local position

of the particles.

DFT was originally developed to investigate the electronic structure

(especially the ground state) of many-body systems. The two theorems

enabling this were originally developed by Hohenberg and Kohn (1964).

The first of these shows that the ground state of properties of the

many-body system of electrons are uniquely determined by an electron

density that depends only on three spatial coordinates. The second

theorem defines an energy functional for the system and demonstrates that

54

Chapter 5

there exists a ground state electron density that minimises this functional.

This theory has been adapted for use in statistical mechanical systems by

Evans (1979). In this case, there is a density which is a function of position

within an inhomogeneous fluid that uniquely defines the Helmholtz energy

of a system. Minimising this density enables the equilibrium density

function and true Helmholtz energy of the system to be obtained.

In the case of the statistical mechanical system, a lemma is required in

order for the Hohenberg-Kohn theorems to be applied. Firstly, the

‘classical trace’ is defined to be the following operator:

Tr · · · ≡∞∑

N=0

1

h3NN !

∫∫

. . . drNpN , (5.12)

where p is the three-dimensional momentum. Then, for the functional of a

normalised phase-space probability density Ω[f ],

Ω[f ] = Trf(H−Nµ+ kBT log f), (5.13)

then:

Ω[f ] ≥ Ω[f0], (5.14)

where H is the classical energy Hamiltonian and f0 is the equilibrium

probability density. The first Hohenberg-Kohn theorem states that for a

given system volume, temperature and chemical potential, the intrinsic

free energy functional Ω[ρ(1)(r)]:

Ω[ρ(1)(r)] = Trf0(K + V + kBT log f0), (5.15)

is a unique functional of the equilibrium single particle density ρ(1)(r).

Here, K represents the kinetic energy of the system.

The second theorem states that for an arbitrary single particle density

a(r), the functional

Ω[a(r)] +

∫

a(r)φ(r)dr − µ

∫

a(r) dr, (5.16)

55

Chapter 5

is minimised when a(r) is equal to the equilibrium single-particle density

ρ(1)(r). Here, φ(r) is some external potential. Mathematical proofs for the

above lemma and theorems are given by Hansen and McDonald (2006).

One of the most widely-used and advanced techniques for approximating

the density functional of a hard-sphere fluid is fundamental measure theory

(FMT), which was first developed by Rosenfeld (1989). In FMT, the

‘fundamental measure’ chosen is the shape of one particle, rather than the

excluded volume between two. Hence, the density functional for the

Helmholtz energy is represented by a function of the packing fraction and

weighted densities based on particle geometries. The equations describing

the use of these properties to represent the excess Helmholtz energy are

presented with a fuller explanation by Tarazona et al. (2008).

Advantages of FMT over earlier methods, such as weighted-density

approximation, include the fact that the functional involves fundamental

information about the geometric shape of the hard particles and the use of

the packing fraction as a measure of the non-local Helmholtz energy. The

packing fraction is the fraction of the total system volume that is occupied

by particles. This packing fraction is represented as the probability that

for a random configuration of particles, a given point in space is occupied

by a hard sphere. Hence, this gives information on the limits of the density

functional more efficiently and accurately than earlier methods could.

A shortcoming of early FMT theory when applied to a highly

inhomogeneous system such as a hard-sphere crystal lattice is that the

density weightings chosen often lead to divergent results. However,

Rosenfeld et al. (1997) addressed this problem by suggesting improved

density weightings, resulting in a much improved approximation for the

Helmholtz energy of the inhomogeneous solid phase. Since the original

development of FMT, there have been a wide range of proposed variations

and improvements in order to produce increasingly accurate results for a

wide range of systems. Fuller historical reviews of this work and

summaries of more recent developments of the theory are presented by

Lowen (2002) and Roth (2010).

56

Chapter 5

There are many possible ways of separating the overall excess contribution

into a sum of local contributions and so there are many choices for

approximating the expression Θ([ρ]; r) which can result in a similar final

answer. One solution for systems with pairwise interactions is to make use

of the virial expansion to show (Hansen and McDonald, 2006):

βAex[ρ] ≈ −1

2

∫∫

ρ(r1)ρ(r2)f12 dr2 dr1 + . . . , (5.17)

where subscripts refer to labelled particles. This allows an expression for

the full Helmholtz energy functional in terms of cluster integrals to be

written:

A[ρ] = Aid[ρ] +Aex[ρ], (5.18)

βA =

∫

ρ(r1)[ln(ρ(r1)Λ3)− 1] dr1 −

1

2

∫∫

ρ(r1)ρ(r2)f(r12) dr1 dr2 + . . .

(5.19)

5.3 Describing the solid-fluid phase transition of a fluid

A system of solid, spherically symmetrical particles will undergo a phase

transition from an isotropic liquid phase to a crystal lattice. In recent

years, an increase in the computational power available, as well as

developments in modelling and simulation techniques have allowed a range

of studies into the prediction of a freezing transition of a fluid through

analysis of the Helmholtz energy to take place.

Some of the earliest simulations of the solid-fluid phase transition have

been carried out by Alder and Wainwright (1957). This work uses

molecular dynamics simulations of various sizes of system and is able to

predict the freezing of hard spheres into an FCC lattice. Robles et al.

(2014) use MC simulations to analyse a number of equations of state for

the hard sphere fluid. Their results are in agreement with Alder and

Wainwright (1957) that the freezing transition takes place at a packing

fraction of around 0.492.

Bharadwaj et al. (2013) approach the freezing of systems governed by

inverse power potentials through a density functional based on correlation

57

Chapter 5

functions. They conclude that for soft-core potentials, a BCC crystal is

favoured but for hard spheres, they also agree that an FCC crystal lattice

is formed.

The earliest applications of DFT to the question of the freezing transition

were developed by Ramarkrishnan and Yussouff (1979) and refined by

Haymet and Oxtoby (1981). This involved considering both the Helmholtz

energy functional of the crystal and of the fluid phase. In the case of the

crystal, A[ρ] is a unique functional of ρ(r), whereas in the fluid phase A is

a function of the average fluid density. Minimising the density functional

results in an expression in terms of the direct pair correlation function.

It is also possible to study the freezing transition of a system through

experimental work. A recent example of this is demonstrated by Franke

et al. (2013), who use dynamic light scattering experiments to analyse the

freezing transition of a colloidal suspension of hard spherical particles.

It can be concluded that a range of well-understood methods exist to

calculate the excess contribution to the Helmholtz energy functional in

order to predict a freezing transition. The rest of this chapter will consider

the question of the convergence of the virial expansion in the case of a

system of hard spheres and whether it can be used to find the excess

contribution to the Helmholtz energy functional and solve equation (5.19).

This would mean that a virial expansion would theoretically be able to

predict all regions of the phase diagram for a system of hard spheres.

5.4 Investigating the solid-fluid phase transition in terms of

a virial expansion

It has been demonstrated by Onsager (1949) that the transition from an

isotropic fluid to a nematic liquid crystal phase can be predicted using a

virial expansion as the starting point for calculating the Helmholtz energy.

For elongated particles, only low order virials are important, but as the

particle approaches a spherical shape, higher order virial coefficients are

required.

58

Chapter 5

This means that for any system, the Helmholtz energy calculations must

include at least the second virial coefficient and higher orders for systems

of spherical particles. Re-summations of these higher order contributions

have been proposed by Tjipto-Margo and Evans (1990, 1991) for various

hard convex bodies. This results in conclusions in agreement with those

found by the simulation. This idea has been extended to the re-summation

of the two particle DCF by Samborski et al. (1994).

The most thermodynamically stable state is that given by the density that

minimises the grand potential functional. At high pressures, this will

correspond to the solid phase while at low pressures it corresponds to a

fluid phase. The point at which the grand potential functional is equal in

the solid and fluid phases is the point at which a phase transition occurs.

This point can be obtained through the unconstrained minimisation of the

grand potential function. This is equivalent to the minimisation of the

Helmholtz energy functional in a system of constant volume and

temperature with the following constraint:

∫

V

ρ(r) dr = N. (5.20)

In other words, the density integrated over the whole volume of the system

must give the total number of particles in the system. Using the Helmholtz

energy profile found in equation (5.19), the point of coexistence between

both phases may be calculated. In order to do this, the Helmholtz energy

is minimised with respect to the density ρ since this is the variational

parameter in a system at constant N,V and T .

The ideal contribution to equation (5.11) is a functional of ρ and may be

evaluated numerically at a local density. In an infinite crystal lattice, the

local density profile is periodic and at a given point is related to the

position of the occupied sites in the lattice relative to that point. As well

as this, the local density must include contributions from all neighbouring

sites that are sufficiently close to the point of interest. A widely-used

method of representing this density field at a given point in the lattice ρ(r)

is as a sum of Gaussian functions φ:

59

Chapter 5

ρ(r) =∑

i=1

φ(r −Ri), (5.21)

where Ri are the positions of the lattice sites.

The excess contribution Θ([ρ]; r) is a function of position and a functional

of ρ and is expressed as a volume integral. An approximation of the

contribution to the excess Helmholtz energy from a second order cluster

integral is suggested by Dong and Evans (2006), although this is not

achieved via a virial expansion. For a fluid freezing into a crystal lattice,

this contribution can be approximated as a sum over lattice vectors, i.e.:

∫∫

V

ρ(r1)ρ(r2)f12 dr1 dr2 = −1

2Nsω, (5.22)

where Ns is the number of lattice sites (with positions denoted Rj) and:

ω =∑

j

ω(bj), (5.23)

ω(bj) =

√

2σ

πl2e−b2j/2l

2

bj

1∫

0

−e−x2σ2/2l2 sinh

(

xσbjl2

)

dx. (5.24)

Here, bj = |Rj −R1|, where the lattice site R1 is taken to be an origin.

Solving the integral in this expression analytically gives the following

result:

ω(bj) = −1

2

[

erf

(

σ − bj

l√2

)

+ erf

(

σ + bj

l√2

)

+l

bj

√

2

π

(

e−(bj+σ)2/2l2 − e−(bj−σ)2/2l2)

]

. (5.25)

Equation (5.25) is an expression for the excess contribution to the

Helmholtz energy that can be solved analytically. This method has been

found to provide a good agreement with DFT approximations for the

freezing transition of spherically symmetrical particles into a face-centred

cubic crystal lattice, but no freezing transition in the case of a

60

Chapter 5

body-centred cubic lattice. This finding contradicts the earlier results of

Alexander and McTague (1978), who concluded that if there was a local

stability limit (spinodal point) associated with the phase transition then

the body-centred cubic crystal would be the most stable form. However,

the results of Dong and Evans (2006) are in agreement with those of Groh

and Mulder (1999), who conclude through stability analysis that the

face-centred cubic crystal is the most stable form into which hard spherical

particles freeze.

5.5 Improving the model of a solid-fluid phase transition

The methods presented so far for modelling a freezing transition in a fluid

of hard spheres do not account for the fact that only a finite range of

packing fractions are possible for a given type of particle and lattice due to

the close packing limit. This means that it is theoretically possible for a

minimum Helmholtz energy to exist for a given system at a density

corresponding with a packing fraction exceeding the close packing limit.

This is because the methods suggested so far describe an Onsager crystal

(1949) which cannot itself adequately represent the existence of the close

packing limit for a given lattice.

The relationship between density ρ and packing fraction η can be found

from:

η =NcellVparticle

Vcell. (5.26)

Here, Ncell is the number of particles per unit cell, Vparticle is the volume of

one particle and Vcell is the volume of one unit of the crystal lattice which

is formed. In the case of hard, spherically-symmetrical particles, this

equation becomes:

η =ρπσ3

6, (5.27)

where σ is the radius of a particle.

One way of accounting for this issue is to employ the rescaling suggested

by Parsons (1979) and Lee (1987, 1988). They construct a modified form

61

Chapter 5

of equation (5.19) in which the excess contribution to the Helmholtz

energy is weighted by a function of the packing fraction, F (η):

βA =

∫

V

ρ(r)(ln(ρ(r))− 1) dr − F (η)

∫∫

V

ρ(r1)ρ(r2)f12 dr1 dr2. (5.28)

The function F (η) is defined as:

F (η) =(4− 3η)

4(1 − η)2. (5.29)

The effect of this function is to provide a correction to the excess Helmholtz

energy term based on the value of η chosen, and so only physically realistic

values of η can now be considered. This weighting of the excess Helmholtz

energy has been found to improve the quality of results obtained at larger

packing fractions and therefore higher densities. For an isotropic hard

sphere fluid, equation (5.28) simplifies to the Carnahan-Starling equation

of state for hard spheres (Carnahan and Starling, 1969).

This chapter has discussed the question of predicting the freezing

transition of a hard-sphere fluid and presented a range of possible methods

for finding this phase transition. Although other more powerful methods

for doing so exist, it has been shown that the question can theoretically be

considered through the application of a virial expansion.

62

Chapter 6

6 Obtaining the structure of fluids governed by

inverse power potentials

This chapter presents the results obtained for the fluid structure from a

virial expansion method for several potential functions and discusses their

usefulness and reliability.

6.1 The inverse power potential

For this work, a series of pairwise-additive, spherically symmetrical and

purely repulsive potentials are examined. They take the following general

form:

u(r12) = ǫ

(

σ

r12

)n

, (6.1)

where ǫ, σ and n are positive parameters which represent the energy, size

and ‘hardness’ of the particles respectively. Since the potential is purely

repulsive, it gives rise to one single fluid phase with no vapour-liquid

transition due to the lack of any long-range attractive component. When

n→ ∞, this potential tends to the same form as the hard sphere potential.

Due to the simplicity of this model (it can be represented with one

effective parameter, ǫσn) it has been used in studies of phase transition

and thermodynamic behaviour for simple fluids as summarised by Tan

et al. (2011). Other examples of the application of this form of potential

include the studies of microgel properties by Pyett and Richtering (2005)

and soft-sphere glasses by Heyes et al. (2009).

Throughout this Chapter and Chapters 7 - 9, density and temperature will

be defined in dimensionless units as ρ∗ = ρσ∗3 and T ∗ = TkB/ǫ

respectively. In addition, without loss of generality, the distance between

two particles r12 is henceforth measured in units of σ∗, which is defined as

σ∗ = σ(βǫ)1n . This means that the virial coefficients no longer contain a

temperature dependence as described by Wheatley (2005).

63

Chapter 6

6.2 Methodology

Initially, the direct correlation function of fluids governed by potentials

with a range of values of n has been initially obtained via the calculation

of virial coefficients for each system. This was made possible through the

use of the Etomica software library (Schultz and Kofke, 2015a). The

MSMC technique is used to achieve this, with the contribution from each

generated configuration accumulated and binned into a histogram. The

bins are then re-weighted in the same manner as when calculating a virial

coefficient at the end of the sampling calculation.

The values of n are chosen to represent a wide variation in the type of

system. n = 4 represents a very soft (less strongly repulsive) potential,

with a freezing density of ρ∗ ≈ 5.6 (Hoover et al., 1971). n = 12 represents

a much harder (more strongly repulsive) potential with a freezing density

of ρ∗ ≈ 1.14. For n = 6 and 9, the freezing densities are ρ∗ ≈ 2.3 and 1.4

respectively (Tan et al., 2011).

A series of curves representing each increasing order of c(r12) from f(r12)

up to a sixth order expansion is presented in each case. These calculations

are first carried out at a density comparable with the critical density of a

Lennard-Jones fluid as found by Singh and Kofke (2004) of ρ∗c = 0.32. The

calculations are then repeated at higher densities closer to the respective

freezing densities of each system for n = 4, 6, 9 and 12. In the latter case,

data is obtained at a temperature of 1.6 reduced units, representing a

system which is comparable with a supercritical Lennard-Jones system.

Data for much harder-core potentials is obtained by setting n = 50 and by

allowing n to tend to infinity, resulting in a hard sphere system, confirming

the other findings and providing a comparison with the work of Dennison

et al. (2009).

The total correlation function, h(r12), is obtained by performing Fourier

transforms on the OZ equation to obtain a value of e(r12) from c(r12) as

discussed in Chapter 4. This provides a series of curves for h(r12)

representing each order of the expansion in the same manner as for the

direct correlation function. In turn, this then gives information on both

h(r12) and the radial distribution function g(r12) as in (4.1).

64

Chapter 6

The results obtained via virial coefficients are compared with results

obtained from integral equation theory and simulation via use of the

Rogers-Young closure approximation as shown in (4.15) and an NVT

simulation of 2048 particles.

6.3 The structure of fluids governed by inverse-power

potentials

6.3.1 n = 4

Figure 6.1 shows the direct correlation function for a system with n = 4.

At a low density of ρ∗ = 0.32, the curves appear to converge slowly, despite

the density being far below that of the freezing transition. This is in line

with the behaviour observed by Barlow et al. (2012) for the convergence of

the pressure virial expansions of this potential. A small increase in density

leads to even slower convergence at small values of r12, whilst at higher

values of r12, no convergence is evident. These trends also occur at a

higher density of 0.4 (as shown in Fgiure 6.2), where similarly slow

convergence at low values of r12 and little evidence of any convergence at

longer distances can be observed. Both values of density shown here are

far below the value of the freezing density in this case.

Relatively slow convergence to simulation and integral equation theory

data is again observed for the total correlation function, as shown in

Figures 6.3 and 6.4 for ρ∗ = 0.32 and ρ∗ = 0.4 respectively and T ∗ = 1.0.

65

Chapter 6

f(r12)1st order2nd order3rd order4th order5th order6th orderRogers-Young

c(r

12)

−3

−2

−1

0

r12

0 1 2 3 4 5

−1

−0.8

−0.6

−0.4

−0.2

0

1 1.2 1.4 1.6 1.8 2

Figure 6.1: Convergence of the direct correlation function for n = 4 withρ∗ = 0.32 and T ∗ = 1.0.


c(r

12)

−4

−3

−2

−1

0

r12

0 1 2 3 4 5 6 7


66

Chapter 6

f(r12)1st order2nd order3rd order4th order5th order6th orderRogers-YoungSimulation

h(r

12)

−1.25

−1

−0.75

−0.25

0

r12

0 1 2 3 4 5 6 7

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

1 1.2 1.4 1.6

Figure 6.3: Convergence of the total correlation function for n = 4 withρ∗ = 0.32 and T ∗ = 1.0.


h(r

12)

−1.25

−1

−0.75

−0.25

0

0.25

r12

0 1 2 3 4 5 6 7

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

1 1.2 1.4 1.6 1.8 2


67

Chapter 6

6.3.2 n = 6 and n = 9

These potentials give rise to systems with stronger short-range repulsive

forces than the case where n = 4. At densities far below the freezing

transition, rapid convergence can now be seen as shown in Figures 6.5 and

6.6.

Convergence is only observed up to a limit, in agreement with the findings

of Tan et al. (2011). This is demonstrated by Figures 6.7 and 6.8, which

show data obtained nearer to the respective freezing densities of both

systems. However, convergence is only observed for densities up to around

0.7 in the case of n = 6 and 0.85 in the case of n = 9, representing roughly

30% and 60% of the respective freezing densities.

68

Chapter 6


c(r

12)

−4

−3

−2

−1

0

r12

0 1 2 3 4 5 6 7



c(r

12)

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

r12

0 1 2 3 4 5 6 7


69

Chapter 6


c(r

12)

−15

−12.5

−10

−7.5

−5

−2.5

0

2.5

r12

0 1 2 3 4 5 6 7

−1

−0.75

−0.5

−0.25

0

0.25

0.5

1 1.25 1.75 2 2.25



c(r

12)

−30

−25

−20

−10

−5

0

r12

0 1 2 3 4 5 6 7

−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.8 1 1.2 1.4 1.6 1.8 2


70

Chapter 6

6.3.3 n = 12

For a system of this type, the virial expansion is known to converge at

densities very close to freezing (Tan et al., 2011). It can be seen that the

direct correlation functions obtained converge extremely well at ρ∗ = 0.32

(Figure 6.9) and as the density approaches the point of solid-fluid

transition for n = 12, convergence is still generally very good but weakens

at longer ranges (Figure 6.10).

Data is also obtained at a higher temperature in this case to represent the

repulsive contribution to a Lennard-Jones potential under both sub- and

super-critical conditions. A more rapid convergence is observed at the

increased temperature, as demonstrated in Figure 6.11.

Rapid convergence to simulation and integral equation theory data can

also be seen for the total correlation function at a low density (Figure

6.12), whereas poorer convergence is seen at a density closer to the phase

transition (Figure 6.13). In this case, oscillating curves are observed at

longer ranges. This is possibly suggestive of these virial expansions picking

up a solid-fluid instability.

71

Chapter 6


c(r

12)

−4

−3

−2

−1

0

r12

0 1 2 3 4 5 6 7


f(r12)1st order2nd order3rd order4th order5th order6th order

c(r

12)

−50

−40

−30

−20

−10

0

r12

0 0.5 1 1.5 2 2.5 3

−1.5

−1

−0.5

0

0.5

0.8 1 1.2 1.4 1.6 1.8 2


72

Chapter 6


c(r

12)

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

r12

0 0.5 1 1.5 2 2.5 3



h(r

12)

−1.25

−1

−0.75

−0.25

0

0.25

r12

0 1 2 3 4 5 6 7


73

Chapter 6

f(r12)2nd order4th order6th orderSimulation

h(r

12)

−15

−10

−5

0

5

10

r12

0 1 2 3 4 5 6 7


Convergence has generally been rapid at ρ∗ = 0.32 in all the results

presented so far. However, as the density is increased towards the freezing

density, it can be seen in all cases that there is a density beyond which

obtaining a convergent expansion becomes problematic. This is shown by

the extremely slow convergence (if at all) at low values of r12 and the

frequent lack of any apparent convergence at values of r12 of 1 and above.

6.3.4 n = 50 and hard spheres

Results for the direct correlation function have also been obtained using an

extremely strongly repulsive potential with n = 50. The curves shown in

Figure 6.14 support the findings above for a harder-core potential. At

densities below the phase transition, convergence is excellent and the

curves show very similar behaviour to a true hard sphere potential (Figure

6.15).

The total correlation functions for these systems were also obtained and

are shown in Figures 6.16 and 6.17. These results are similar to those

presented by Dennison et al. (2009), taking into account that only a sixth

74

Chapter 6


c(r

12)

−4

−3

−2

−1

0

r12

0 0.5 1 1.5 2 2.5 3



c(r

12)

−60

−50

−40

−20

−10

0

r12

0 0.5 1 1.5 2 2.5 3

−6

−4

−2

0

2

0.6 0.8 1 1.2 1.4

Figure 6.15: Convergence of the direct correlation function for a hard spherefluid with ρ∗ = 0.32.

75

Chapter 6


h(r

12)

−1.25

−1

−0.75

−0.5

0

0.25

0.5

r12

0 1 2 3 4 5 6


order expansion could be used, rather than 8th as in the previous study.

This is because it has proven impossible to obtain sufficiently accurate

data for B7 and above due to the limitations of the available

computational facilities.

76

Chapter 6


h(r

12)

−1

−0.5

0

0.5

r12

0 1 2 3 4 5 6

Figure 6.17: Convergence of the total correlation function for a hard spherefluid with ρ∗ = 0.32.

6.3.5 Asymptotic behaviour of the RDF and DCF for the

n = 12 case

A further question in the prediction of fluid structure is that of the

behaviour of the direct correlation function at very high densities.

Empirical data has been studied by Rosenfeld and Baram (1981); Song

and Mason (1991) and Heyes et al. (2009), who found that the internal

energy of a system governed by an inverse-power potential with n = 12 has

the following form:

βU

N∼ any + bny

14 + cn, (6.2)

where an, bn and cn are constants evaluated through fitting a curve to

simulation data and y = βρ∗n3 . This leads to the scaling suggested by

Barlow et al. (2012) for the compressibility of Z ∼ ρ∗1/3. This information

can be used to propose a rescaling for c(r12). At sufficiently high densities,

the system will form a solid phase. Considering the bulk modulus of this

phase, a change in the pressure should lead to a change in the system

volume and therefore the internal energy, but the coordination number of

the crystal will remain the same. Now, considering the compressibility

equation:

77

Chapter 6

Z =ρ∗

3

∫

V

4πr212

(

r12du(r12)

dr12

)

g(r12; ρ∗) dr12, (6.3)

a scaling for the RDF can be proposed that as the asymptotic limit is

approached, g(r12, ρ∗) ∼ g(r12ρ

∗ 13 ). A plot of the RDF re-scaled in this

way is demonstrated for the case of n = 12 in Figure 6.18. The expected

‘collapsing’ of the curves is not observed even at the initial peak of g(r12),

despite RDFs of this form generally converging, suggesting that the

density dependence of the RDF is likely to be of a more complicated form.

This is supported by the fact that the pressure equation given by equation

(4.10) contains no dependency on n:

p = ρ∗kBT − ρ∗2

6

∫

Vr12g(r12)

du(r12)

dr12dr2, (6.4)

therefore this re-scaling does not result in a thermodynamically consistent

relation.

ρ* = 0.1ρ* = 0.2ρ* = 0.3ρ* = 0.4ρ* = 0.5ρ* = 0.6ρ* = 0.7

g(r

12 ρ

*1/3)

0

0.5

1

1.5

2

r12 ρ*1/3

0 2 4 6 8 10

Figure 6.18: RDF expansions with proposed density dependence of the formg(r12ρ

∗1/3) for n = 12.

Further information regarding the possible re-scaling of g(r12) can be

found by examining the mean squared average of the DCF, < r212 >:

78

Chapter 6

< r212 >=

∫

V

r212c(r12)4πr212 dr12

∫

V

c(r12)4πr212 dr12. (6.5)

Figure 6.19 uses the RY closure approximation to show the change in

< r212 > found from this relation with increasing density on a logarithmic

scale. It can be seen that that there is no obvious linear dependence on the

density and therefore no obvious scaling coefficient.

n = 4

n = 12

ln <r

12

2>

−1

−0.5

0

0.5

1

1.5

2

ln ρ*

−3 −2.5 −2 −1.5 −1 −0.5 0

Figure 6.19: Calculation of ln < r212 > using the RY approximation forvarious n values.

6.4 Analysis and interpretation of results

The results presented in this chapter demonstrate that in some cases it is

possible to obtain a convergent DCF for fluids governed by an

inverse-power potential function using a virial expansion. For a softer-core

potential, this is possible up to a density limit similar to that defined by

Tan et al. (2011) and based on the convergence properties of the virial

expansion itself. For a harder-core potential (for at least n = 12 and

above), this is possible at all densities up to the freezing density for the

fluid.

79

Chapter 6

These curves have been successfully used to obtain TCF curves which at

low densities agree very well with those obtained by simulation and

integral equation theory at all values of n which have been studied here.

At higher densities, convergence tends to take place more slowly at low

values of r12 and in those cases when convergence is not apparent, it tends

to break down at values of r12 > 1 first.

Overall, it has been shown that it is possible to obtain reliable results for

the structure of a fluid using a virial expansion. The techniques

demonstrated in this chapter have the potential to be useful in any case

where the virial coefficients are known and the virial expansion is found to

be convergent. They constitute a fast and efficient way of calculating the

RDF over a wide range of densities, which can then be input into an

equation of state. One such application of this is to statistical associating

fluid theory (SAFT), which requires knowledge of the radial distribution

function at contact in the chain and association terms (McCabe and

Galindo, 2010).

80

Chapter 7

7 Extending the investigation to study the

structure of a Lennard-Jones fluid

This chapter presents the results of investigating the structure of a

Lennard-Jones fluid using a virial expansion.

7.1 The Lennard-Jones potential

A potential approximating the interaction between two solid spherical

particles to include attractive as well as repulsive forces was presented by

Lennard-Jones (1924) and defined as:

u(r12) = 4ǫ

[

(

σ

r12

)12

−(

σ

r12

)6]

. (7.1)

Here, σ is the distance at which the potential is equal to zero. The depth

of the potential well, ǫ, represents a measure of the strength of attraction

between the two particles. The first term in parentheses represents the

short range Pauli repulsion due to electron orbitals overlapping, and the

second term represents the attractive component due to

dispersive-attractive forces. A sketch of the potential function is shown in

Figure 7.1. The blue dotted line here represents the repulsive contribution

to the potential and the red dotted line the attractive contribution. The

purple line is the sum of these two contributions, giving the shape of the

full Lennard-Jones potential.

An analytical equation of state for a fluid defined by this potential is given

by Kolafa and Nezbeda (1994), which provides expressions for several

thermodynamic properties. Figure 7.2 below shows the pressure variation

with density.

81

Chapter 7

u(r

12)

r12

Figure 7.1: Sketch of the Lennard-Jones potential as a function of inter-particle distance r12.

T* = 1.0

T* = 1.3

T* = 1.6

p*

−2.5

0

2.5

5

7.5

10

12.5

15

ρ*/ρc

*

0 0.5 1 1.5 2 2.5 3 3.5

Figure 7.2: Pressure variation with density for a Lennard-Jones fluid ac-cording to Kolafa-Nezbeda equation of state at sub-, super- and critical tem-peratures.

7.2 The DCF of a Lennard-Jones fluid

For a Lennard-Jones fluid, the pressure virials have been calculated to 8th

order by Schultz and Kofke (2009a). These values have been confirmed

82

Chapter 7

during this investigation and are summarised in Appendix A. It should be

noted that at sub-critical temperatures the virial coefficients begin to

increase in size by roughly one order of magnitude at each increase in order

of coefficient. This agrees with the previous reports of Schultz and Kofke

(2009a) and is supported by the idea that in the liquid state, the long-range

attractive forces (i.e. those more likely to include a larger number of

particles) would make a larger contribution to the overall virial coefficient.

The resulting expansions to 6th order are shown in Figures 7.3 and 7.4 for

temperatures of T ∗ = 1.0 and 1.6 respectively. These curves show that no

van der Waals’ loops are shown and the virial expansion begins to diverge

at a sub-critical density below the critical temperature. Above the critical

temperature, the expansion converges quite rapidly. Here and throughout

the rest of this chapter, the density is presented as the ratio ρ∗/ρ∗c , where

ρ∗c is the critical density for a Lennard-Jones fluid and takes a value of 0.32

reduced units.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

KN EOS

p*

−20

−15

−10

−5

0

5

10

ρ*/ρc

*

0 0.5 1 1.5 2 2.5 3

−0.1

−0.05

0

0.05

0.1

0 0.2 0.4 0.6 0.8 1

Figure 7.3: The virial expansion to 6th order at T ∗ = 1.0 using the data ofSchultz and Kofke (2009a).

The DCF of the Lennard-Jones potential was obtained via the virial

expansion using MSMC in the same manner as for inverse power potentials

in the previous chapter. This has been carried out at the critical density of

83

Chapter 7

*

0

/

.

6

h

S

*

0

/

.

6

hO

h

S SE6 * *E6 0 0E6 /

S

SES0

SES.

SES8

SES

SE*

S SES6 SE* SE*6 SE0 SE06

Figure 7.4: The virial expansion to 6th order at T ∗ = 1.6 using the data ofSchultz and Kofke (2009a).

ρ∗ = 0.32 and temperatures above and below the critical point. Figures 7.5

and 7.6 show that as for the virial expansion of pressure, there is good

convergence up to 6th order at super-critical conditions. However, this is

not observed at sub-critical conditions, where the curves diverge over all

ranges of r12.

84

Chapter 7


c(r

12)

−200

−100

0

100

200

r12

0 0.5 1 1.5 2 2.5

−6

−4

−2

0

2

4

6

0.8 0.9 1 1.1

Figure 7.5: The DCF for an LJ potential up to 6th order under sub-criticalconditions at T ∗ = 1 and ρ∗ = 0.32.


c(r

12)

−4

−3

−2

−1

0

1

2

r12

0 0.5 1 1.5 2 2.5

Figure 7.6: The DCF for an LJ potential up to 6th order under super-criticalconditions at T ∗ = 1.6 and ρ∗ = 0.32.

7.3 Perturbing the LJ potential

In order to investigate this divergence further, a perturbation scheme can

be applied to the Lennard-Jones potential and split it into separate terms

85

Chapter 7

representing the repulsive and attractive contributions,

u(r12) = urep(r12) + uatt(r12). There are many possible choices of the

manner in which the Lennard-Jones potential is split to obtain urep(r12)

and uatt(r12). The MK perturbation (McQuarrie and Katz, 1966) takes the

form:

urep(r12) = 4ǫ

(

σ

r12

)12

, (7.2)

uatt(r12) = −4ǫ

(

σ

r12

)6

. (7.3)

However, the equation of state resulting from this perturbation has been

found to work well only at temperatures above T ∗ ≈ 3.0 (Hansen and

McDonald, 2006). At temperatures closer to the critical temperature of a

Lennard-Jones fluid, a better agreement with simulation data is provided

by the perturbation scheme suggested by Barker and Henderson (1967):

urep(r12) =

u(r12), r12 < σ

0, r12 > σ,(7.4)

uatt(r12) =

0, r12 < σ

u(r12), r12 > σ,(7.5)

This splits the full potential into the part that is positive and the part that

is negative. A further improvement is suggested by Andersen et al. (1971):

urep(r12) =

u(r12) + ǫ, r12 < rmin

0, r12 > rmin,(7.6)

uatt(r12) =

−ǫ, r12 < rmin

u(r12), r12 > rmin,(7.7)

where rmin is the value of r12 at which the potential is minimised. The MK

perturbation scheme will be useful since it confers an advantage over the

others described here, despite its relatively poor performance at

temperatures as low as considered in this investigation. The MK

86

Chapter 7

perturbation scheme treats both the repulsive and attractive components

as effectively inverse-power potentials, meaning that the asymptotic

behaviour of the pressure (and therefore compressibility) at high density is

known. This is advantageous in the application of re-summation schemes

in order to improve the convergence of a virial expansion obtained from

the perturbed potentials.

The repulsive term in the MK perturbation has been treated in the same

way as the full LJ potential to produce a set of DCF curves to each order,

as shown in Figure 7.7. As well as this, a DCF ∆c(r12) has been calculated

from the virial expansion of the difference between the LJ virials and those

generated from this repulsive potential, to represent the effect of only the

attractive contribution to the LJ potential. This is shown in Figure 7.8.


crep(r

12)

−7

−6

−5

−4

−3

−2

−1

0

r12

0 0.5 1 1.5 2 2.5

Figure 7.7: The DCF for the repulsive contribution to an LJ fluid up to 6thorder at T ∗ = 1 and ρ∗ = 0.32.

The purely repulsive potential leads to rapid convergence in a similar

manner to the 12th-order inverse power potential. This finding also

supports the conclusion reached in the previous chapter about the use of a

virial expansion to predict the structure of a fluid with a purely repulsive

potential.

87

Chapter 7


Δc(r

12)

−200

−100

0

100

200

r12

0 0.5 1 1.5 2 2.5 3

−6

−4

−2

0

2

4

6

0.8 0.9 1 1.1

Figure 7.8: The DCF for the attractive contribution to an LJ fluid up to 6thorder at T ∗ = 1 and ρ∗ = 0.32.

The DCF curves of these perturbations clearly show that the source of the

divergence lies in the attractive part of the potential and is evident across

the entire range of r12. This means that there is likely to be no convenient

perturbation theory that provides a plausible solution to the problem since

such methods (e.g. the random phase approximation as described by

Hansen and McDonald (2006)) generally work on the basis of separating

the attractive contribution to the potential and treating it as a

perturbation.

7.4 Investigating the divergence of the LJ virials

To consider the phenomena of divergence in the correlation functions seen

in the previous section, it is first necessary to study the pressure virial

expansion of the Lennard-Jones fluid in greater depth. As the virial

expansion itself has been shown to be divergent at sub-critical

temperatures, an alternative approximation is required.

The ‘high temperature approximation’ (HTA) (Hansen and McDonald,

2006) provides a guide as to the behaviour of the fluid. This is a first order

88

Chapter 7

WCA perturbation theory in which the attractive contribution to the

potential is treated as an expansion parameter. The Helmholtz energy can

be written in terms of the fluctuations in the potential energy WN as

demonstrated by Zwanzig (1954):

βA = βA0 + β〈WN 〉0 −1

2β2

(

〈W 2N 〉0 − 〈WN 〉20

)

+ . . . , (7.8)

where the subscript ‘0’ refers to a reference system. Higher order terms in

this expansion contain higher powers of fluctuations. At higher pressures,

the effects of these fluctuations are increasingly quenched and so their

importance decreases. This is because the particles in the system are

generally closer together and therefore the structure of the fluid is

determined mainly by the repulsive forces between the particles. As in

WCA perturbation theory, the RDF of the perturbed system is

approximately equal to that of the reference system. Thus through an

appropriate choice of reference system, the RDF of the perturbed system

and thus the thermodynamic properties of that system can be obtained.

This relationship provides the basis of many practical equations of state,

such as van der Waals’ and statistical associating fluid theory (SAFT).

A HTA has been successfully used at high densities and a range of

temperatures for a Lennard-Jones fluid by Cuadros et al. (1996) to

calculate Helmholtz energies. The HTA can relate density to the

Helmholtz energy, A through the following expansion:

βA = βArep +Nρ∗

2

∫

grep(r12)uatt(r12) dr12,

βA

N− βArep

N= ∆A. (7.9)

For the RDF of the reference system grep(r12), it is convenient to choose a

12th-order inverse power potential as studied in the previous chapter due

to the similarity in behaviour between the DCFs obtained from this and

the repulsive part of the MK perturbation. The potential function

uatt(r12) refers to the attractive contribution to the Lennard-Jones

potential and the RDF can be calculated via a suitable closure

approximation, in this case the Rogers-Young closure.

89

Chapter 7

The investigation will now consider if it is possible to obtain ∆A as

calculated via equation (7.9) from a re-summed virial expansion. ∆A can

be thought of as the difference in Helmholtz energy per particle between

the Lennard-Jones fluid and the fluid governed by the 12th-order inverse

power potential. Figure 7.9 shows the change in ∆A as the system density

increases above criticality.

Result from HTA

Fitted curve

ΔA

0

0.1

0.2

0.3

0.4

0.5

ρ*/ρc

*

0 0.2 0.4 0.6 0.8

Result from HTA

Fitted curve

ΔA

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

ρ*/ρc

*

0 0.2 0.4 0.6 0.8

Figure 7.9: The change in Helmholtz energy (found using the HTA) with den-sity for a Lennard-Jones fluid at sub-critical (left) and super-critical (right)temperatures.

Examining these curves reveals that a polynomial of the form

∆A = aρ∗ + bρ∗2 can be fitted extremely well. In the sub-critical case

(T ∗ = 1.0), a = 0.332 and b = 0.542. In the super-critical case (T ∗ = 1.6),

a = 0.208 and b = 0.338.

The information contained in these graphs can be used to show a predicted

pressure-density phase diagram for a fluid governed by the attractive part

of the Lennard-Jones potential. This can then be compared with curves

that result from applying various re-summation techniques in order to

obtain a convergent expansion. If this can be achieved, it should be

possible to use this re-summation to improve the convergence of the DCF

for a Lennard-Jones fluid and continue to obtain information about the

fluid structure in the same way as in the previous chapter.

90

Chapter 7

In order to usefully compare the information in Figure 7.9 to that obtained

from re-summing the virial expansion, first the HTA must be used to

develop an expression for the pressure in terms of the density.

The following thermodynamic relationship connects the pressure P to the

Helmholtz energy A:

P = −(

∂A

∂V

)

T

= −(

∂A

∂ρ

)

T

(

∂ρ

∂V

)

N

= −(

∂A

∂ρ

)

T

N

V 2

= −(

∂(A/N)

∂ρ

)

ρ2. (7.10)

Using this relationship and the expression obtained from curve fitting, an

approximate expression for the compressibility in terms of density can be

obtained:

Z − Zrep = ∆Z = −[a+ 2bρ∗]ρ∗. (7.11)

The validity of this expression can be checked by calculating the total

compressibility Z and comparing the results with the Kolafa-Nezbeda

equation of state. The total of the compressibility obtained by this fit can

be added to a compressibility representing the repulsive contribution, Zrep.

This is obtained from a 12-th order inverse-power potential, since the virial

expansion in this case is known to be convergent at densities beyond the

critical density of a Lennard-Jones fluid. The results of this are shown in

Figure 7.10.

91

Chapter 7

Zrep

ΔZ from HTAZ = Zrep + ΔZK-N EOS

Z

−4

−2

0

2

4

ρ*/ρc

*

0 0.5 1 1.5 2

Zrep

ΔZ from HTAZ = Zrep + ΔZK-N EOS

Z

−4

−2

0

2

4

ρ*/ρc

*

0 0.5 1 1.5 2

Figure 7.10: Comparison of the attractive contribution to the compressibilityobtained from HTA analysis and a repulsive contribution obtained from avirial expansion of an inverse-power potential with n = 12. The left graphshows the sub-critical case T ∗ = 1.0 and the right graph shows the super-critical case T ∗ = 1.6

7.5 Re-summation of the virial expansion of the attractive

component of a Lennard-Jones potential

As discussed in Chapter 3, there are an infinite number of ways in which a

virial expansion can be re-summed. Based on the result obtained in

equation (7.11), an expression is required that results in a value of

∆Z = Z − Zrep obtained from the virial coefficients representing the

contribution to the compressibility due to the attractive part of the

potential function, which tends to 0 as ρ∗ → 0 and tends to ρ∗2 as ρ∗ → ∞.

To make use of the high temperature expansion, an expression for the

contribution to the compressibility of a Lennard-Jones fluid from the

long-range attractive forces is required. Since both the virial coefficients of

the Lennard Jones fluid and a fluid governed by the potential:

u(r12) = 4ǫ(σ

r

)12, (7.12)

are readily available using MSMC, a virial series approximating the

contribution to the compressibility of the fluid due to the attractive

component of the potential can be defined:

92

Chapter 7

∆Z = (1−1)+(B2−Brep2 )ρ∗+(B3−Brep

3 )ρ∗2+· · ·+(Bn−Brepn )ρ∗n−1. (7.13)

Here, the superscript ‘rep’ refers to virial coefficients obtained from the

purely repulsive component of the Lennard-Jones potential. Several

different types of re-summation have been considered, although it is not

possible to carry out a fully conclusive study here. Results are presented

for the compressibility obtained through considering both the repulsive

and attractive contributions to the Lennard-Jones potential. This allows

the suitability of suggested expansions as re-summation schemes in general

to be checked, since a purely repulsive potential of the form ˜r−12 has

already been found to be convergent.

The curves presented in the following section give values of Z plotted

against values of ρ∗/ρ∗c , obtained for increasing orders of density from first

up to fifth or sixth. This is done for temperatures of both 1.0 and 1.6

reduced units since this part of the investigation is concerned with the

performance of the re-summation schemes as the density approaches

criticality and whether a convergent re-summation of the expansion is

obtained under sub-critical conditions. Investigating the super-critical

phase alongside this allows the quality of the methods themselves to be

checked. This allows greater certainty to be placed in any conclusions that

are drawn concerning the behaviour of the virial expansion in the

sub-critical phase.

7.5.1 Re-summation with forced asymptotic behaviour

Firstly, the virial expansion has been written in an alternative form in

order to force it to fit the above criteria. This is written as:

Z = 1 + ρ∗B2

(

1 +B3

B2ρ∗ +

B4

B2ρ∗2 + · · ·+ Bn

B2ρ∗(n−2)

)

, (7.14)

= 1 + ρ∗B2(1 +D1ρ∗ +D2ρ

∗2 + · · · +Dmρ∗m)

1m . (7.15)

Here, Dm = Bn/B2 (where m = n+ 2) represents coefficients that are

93

Chapter 7

functions of the virial coefficients. These are obtained by Taylor expansion

of the re-summed virial series and comparing the coefficients of like powers

of ρ∗. The results of this re-summation are shown up to 6th order in ρ∗.

Figure 7.11 shows that convergence can be observed for densities up to

around 0.5ρ∗c for both contributions to Z in the sub-critical case. However,

beyond this point the curves diverge, meaning the re-summation scheme

performs increasingly poorly at high densities. Figure 7.12 shows that

similar behaviour is observed in the super-critical case.

to 2nd order

to 3rd order

to 4th order

to 5th order

to 6th order

Zrep

0

2

4

6

8

ρ*/ρc

*

0 0.5 1 1.5 2

to 1st orderto 2nd orderto 3rd order

to 4th order

to 5th order

HTA

ΔZ

−10

−8

−6

−4

−2

0

ρ*/ρc

*

0 0.5 1 1.5 2

Figure 7.11: Re-summation of an alternative form of the virial expansion atT ∗ = 1.0 representing both repulsive (left) and attractive (right) contribu-tions to the Lennard-Jones potential.

In both cases, the radius of convergence of the virial expansion has been

improved. However, even when convergent, the expansions do not converge

to values close to the result suggested by the HTA. This can also be seen

in the leading-order coefficients of density B2D1mm , which are not similar to

the value of the ρ∗2 coefficient given by the HTA at any point. To improve

the accuracy of this re-summation, the information provided by the HTA

can be incorporated into the approximation scheme. At sufficiently high

densities, it is required that:

∆Z = −2bρ∗2 ≈ 1 + ρ∗B2(Dmρ∗m)

1m . (7.16)

In order to achieve this, Dm is set equal to (−2b/B2)m. The results of this

further correction are shown in Figure 7.13.

94

Chapter 7

to 2nd orderto 3rd order

to 4th order

to 5th order

to 6th order

Zrep

1

2

3

4

5

6

ρ*/ρc

*

0 0.5 1 1.5 2

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

HTA

ΔZ

−5

−4

−3

−2

−1

0

ρ*/ρc

*

0 0.5 1 1.5 2

Figure 7.12: Re-summation of an alternative form of the virial expansion atT ∗ = 1.6 representing both repulsive (left) and attractive (right) contribu-tions to the Lennard-Jones potential.

These curves show some improvement in the accuracy of the scheme, but

there is still some degree of error observable. The radius of convergence

has not been improved any further.

95

Chapter 7

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

HTA

ΔZ

−40

−30

−10

0

ρ*/ρc

*

0 1 2 3 4 5

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

HTA

ΔZ

−15

−12.5

−10

−5

−2.5

0

ρ*/ρc

*

0 1 2 3 4 5

Figure 7.13: Re-summation of an alternative form of the virial expansion atT ∗ = 1.0 (left) and T ∗ = 1.6 (right). In this case, the leading-order term ofthe re-summation has been set equal to that of the HTA.

7.5.2 Pade approximant schemes

An alternative re-summation scheme is to use Pade approximants. This is

a technique developed to improve the convergence of an infinite series

using a rational function as a best approximation. The following general

expression for compressibility in terms of a Pade approximant of order

[m/n] is used:

Z =D0 +D1ρ

∗ +D2ρ∗2 + · · · +Dmρ

∗m

1 + E1ρ∗ + E2ρ∗2 + · · · +Enρ∗n, (7.17)

where Di and Ei represent coefficients to be found. This approximant is

set equal to the virial expansion and the unknown coefficients computed

systematically. To preserve the correct behaviour as ρ∗ → 0, the first term

in the numerator is set equal to 1.

Results are presented here for all approximants that are available with

m+ n ≤ 5, where a re-summation has been possible. New curves for the

compressibility found using both the repulsive and attractive contributions

to the Lennard-Jones potential are shown in Figure 7.14 for the

sub-critical case and 7.15 for the super-critical case.

In both cases, there is generally a stronger agreement between the

96

Chapter 7

[1/1] Pade[2/1] Pade[1/2] Pade[3/1] Pade[2/2] Pade[1/3] Pade[4/1] Pade[3/2] Pade[1/4] Pade

Zrep

0

2

4

6

8

10

ρ*/ρc

*

0 0.25 0.5 0.75 1 1.25 1.5

[1/1] Pade[2/1] Pade[1/2] Pade[2/2] Pade[1/3] Pade[1/4] PadeHTA

ΔZ

−7

−6

−5

−4

−3

−2

−1

0

ρ*/ρc

*

0 0.25 0.5 0.75 1 1.25

Figure 7.14: Pade approximants of representing compressibility obtainedfrom both repulsive (left) and attractive (right) contributions to the Lennard-Jones potential at T ∗ = 1.0.

[1/1] Pade

[2/1] Pade

[1/2] Pade

[3/1] Pade

[2/2] Pade

[1/3] Pade

[4/1] Pade

[3/2] Pade

[1/4] Pade

Zrep

0

2

4

6

8

10

ρ*/ρc

*

0 0.25 0.5 0.75 1 1.25 1.5

[1/1] Pade

[2/1] Pade

[1/2] Pade

[3/1] Pade

[2/2] Pade

[1/3] Pade

[4/1] Pade

[3/2] Pade

[2/3] Pade

[1/4] Pade

HTA

ΔZ

−7

−6

−5

−4

−3

−2

−1

0

ρ*/ρc

*

0 0.25 0.5 0.75 1 1.25 1.5 1.75

Figure 7.15: Pade approximants of representing compressibility obtainedfrom both repulsive (left) and attractive (right) contributions to the Lennard-Jones potential at T ∗ = 1.6.

97

Chapter 7

approximants representing the repulsive contribution to Z at densities

below around 0.5ρ∗c . However, at the sub-critical temperature, the

approximants representing the attractive component do not agree closely

and there is no evidence of convergence to the HTA. Therefore this

re-summation scheme does not accurately represent the attractive

contribution to the virial equation of state in this case.

In the case of the super-critical system, the approximants agree more

strongly and to densities of around 0.5ρ∗c . However, the re-summation

scheme still appears unable to approach the prediction of the HTA in most

cases. The exception to this in both the sub- and super-critical cases is the

[2/2] approximant, which gives a result that agrees quite well with the

prediction of the HTA. This outcome is expected since Pade approximants

closest to the case when m = n are generally found to give the most

accurate result (Baker and Gammel, 1961). While this re-summation

shows some promise, further work is required in order to accurately

describe the sub-critical fluid.

7.5.3 Asymptotically consistent approximation method

A series of asymptotically consistent approximants are obtained in the

same way as is described in Chapter 3 to attempt to improve on the result

obtained from a Pade approximation. The equations used to generate

these approximants are (Barlow et al., 2012):

Z =

[

N1 +N2ρ∗ + · · ·+NM+1+kρ

∗M+k

1 +D2ρ∗ + · · ·+DM−kρ∗M−1−k

]( α2k+1

)

, J even,M = J/2, (7.18)

Z =

[

N1 +N2ρ∗ + · · ·+NM+1+kρ

∗M+k

1 +D2ρ∗ + · · ·+DM−k+1ρ∗M−k

]( α2k

)

, J odd,M = (J − 1)/2.

(7.19)

All approximants possible for a fifth-order virial expansion have been

obtained, using values of k = 0 and k = 1. The attractive component

behaves like ρ∗2 as ρ∗ → ∞, so α = 2. The repulsive component requires

α = 4 to correctly represent the required high-density behaviour. The

98

Chapter 7

results produced by using ACAs are shown in Figure 7.16 for T ∗ = 1.0 and

in Figure 7.17 for T ∗ = 1.6.

[2/0]2

α

[3/0]3

α

[3/1]2

α

[4/1]3

α

[3/0]α

[4/1]α

Zrep

0

5

10

15

20

ρ*/ρc

*

0 0.5 1 1.5 2

[2/0]2

α

[3/0]3

α

[3/1]2

α

[4/1]3

α

[3/0]α

[4/1]α

HTA

ΔZ

−30

−20

−10

0

10

20

ρ*/ρc

*

0 0.5 1 1.5 2

Figure 7.16: Application of the ACA method to a virial expansion represent-ing both the Zrep (left) and ∆Z (right) contributions to the Lennard-Jonesvirial expansion with T ∗ = 1.0.

[2/0]2

α

[3/0]3

α

[3/1]2

α

[4/1]3

α

[3/0]α

[4/1]α

Zrep

0

2.5

5

7.5

10

12.5

15

ρ*/ρc

*

0 0.5 1 1.5 2

[2/0]2

α

[3/0]3

α

[3/1]2

α

[4/1]3

α

[3/0]α

[4/1]α

HTA

ΔZ

−30

−20

−10

0

10

20

ρ*/ρc

*

0 0.5 1 1.5 2

Figure 7.17: Application of the ACA method to a virial expansion represent-ing both the Zrep (left) and ∆Z (right) contributions to the Lennard-Jonesvirial expansion with T ∗ = 1.6.

In both cases, the higher-order approximants representing Zrep converge at

densities beyond criticality. In the sub-critical case, little agreement is

evident among the approximants representing ∆Z. However, at high

density the [3/1]α2 approximant gives an excellent agreement with the

99

Chapter 7

HTA, despite its poor performance at low density. For the super-critical

case, several approximants agree well with the HTA at low densities, but

at higher densities the [2/0]α2 approximant gives the best agreement.

7.5.4 Approximation using extrapolated high-order coefficients

The final method of re-summation is that which has been recently

proposed by Schultz and Kofke (2015b) and discussed in Chapter 3. In this

case, rather than applying the re-summation to the entire virial expansion,

it is applied only to ∆Z. As presented in Chapter 3, the equation for the

re-summation of the virial expansion is:

βP e(ρ∗, T ) =

nmax(T )∑

n=1

(Bn(T )−Ben(T ))ρ

∗n

+C(T )Li aT(expb/T ρ∗). (7.20)

In this case, the value of the proportionality constant C(T ) is determined

to match the highest order virial available, (B6 −Brep6 ). The values of a

and b are taken to be the same as those used by Schultz and Kofke

(2015b), 4.06 and 3.01 respectively.

The results of this re-summation to each order of ρ∗ (i.e. increasing values

of nmax) for the attractive contribution to the compressibility are shown in

Figure 7.18 for T ∗ = 1.0 and T ∗ = 1.6.

In both the sub- and super-critical cases, this method gives the most

strongly convergent result for a re-summed virial expansion, but does not

converge to the HTA.

100

Chapter 7

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

HTA

ΔZ

−5

−4

−3

−2

−1

0

ρ*/ρc

*

0 0.5 1 1.5 2 2.5

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

HTA

ΔZ

−2.5

−2

−1.5

−1

−0.5

0

ρ*/ρc

*

0 0.25 0.5 0.75 1 1.25 1.5 1.75

Figure 7.18: Re-summation scheme proposed by Schultz and Kofke (2015b)applied to the attractive contribution to the compressibility at T ∗ = 1.0 (left)

and T ∗ = 1.6 (right). The spinodal densities are e−bT = 0.049 and 0.15

respectively, according to Schultz and Kofke (2015b).

7.6 Comparison of re-summation schemes

Each of these schemes show some degree of improvement in the

convergence of the series over the original virial expansion for the attractive

contribution to the compressibility ∆Z to fifth order. Figures 7.19 and

7.20 compare the highest order curves produced from re-summation with

forced leading-order behaviour and extrapolating virial coefficients with

the most promising Pade approximant and ACA results with the original

virial expansion and the HTA for both the sub- and super-critical case.

Since it was noted earlier that a HTA based on the MK perturbation

scheme may not give the most accurate result possible at the temperatures

used here, the KN equation of state can also be used to approximate the

behaviour of ∆Z. This can be made by subtracting Zrep from the result

given by the KN equation of state. As the density increases above

criticality, there is increasingly good agreement between this result and the

HTA and the asymptotic behaviour appears to be similar.

Considering first the sub-critical case, it can be concluded that at low

densities (less than around 0.5ρ∗c), the extrapolated virials and the [2/2]

Pade show promising agreement with the HTA and the result obtained

101

Chapter 7

HTAVirial expansion of (Z - Zrep)ACA with [3/1]3

α

Extrapolated virials[2/2] PadeAlternative schemeKN EOS - Zrep

ΔZ

−30

−25

−20

−15

−10

−5

0

5

ρ*/ρc

*

0 0.5 1 1.5 2

Figure 7.19: Comparison at T ∗ = 1.0 with a 5th-order virial expansion of∆Z of: the HTA; alternative form of virial expansion; [2/2] Pade approxi-mant; ACA with [3/1]

α2 and approximation using extrapolated higher-order

coefficients.

HTAVirial expansion of ( Z - Zrep)ACA with [2/0]

3

α

Extrapolated virials[3/1] PadeAlternative schemeKN EOS - Zrep

ΔZ

−10

−8

−6

−4

−2

0

ρ*/ρc

*

0 0.5 1 1.5 2

Figure 7.20: Comparison at T ∗ = 1.6 with a 5th-order virial expansion of∆Z of: the HTA; alternative form of virial expansion; [3/1] Pade approxi-mant; ACA with [2/0]

α2 and approximation using extrapolated higher-order

coefficients.

102

Chapter 7

from the KN equation. However, these re-summations begin to perform

more poorly at increased densities. The [3/1]α2 ACA scheme does not

appear to provide a re-summed result at densities below 0.5ρ∗c and is the

least effective re-summation at this point. However, above criticality, it

shows very good agreement with the asymptotic behaviour of the HTA and

appears to be approaching closer agreement with the result obtained from

the KN equation. At the critical point itself, no re-summation shows

complete agreement with the result obtained from the KN equation,

although the extrapolated virial coefficients and [2/2] Pade approximant

are closest.

In the super-critical case, each re-summation scheme generally shows much

closer agreement with the asymptotic behaviour of the HTA at high

densities, extending well beyond criticality. The best-performing of these

appears to be the ACA with [2/0]α2 , which shows very close agreement

with the attractive contribution to the KN equation at all densities.


In some cases, the results presented show some promising agreement

between a re-summed form of the virial expansion and the asymptotic

behaviour of the curves obtained from the HTA and the attractive

contribution to the KN equation of state. However, generally these

re-summations still do not agree well with the behaviour of the HTA in the

region of the critical density at sub-critical temperatures.

It has been established in the previous chapter that the virial expansion is

able to describe a convergent function grep(r12). This means that the

expression for ∆A and thus ∆Z found via the HTA appear to be derivable

from a virial expansion, implying that there is an issue concerning the

virial expansion of ∆Z. If the HTA does give the high density limit of ∆Z,

then neither the direct summation of the virial series nor the various

re-summation techniques attempted in this chapter are able to reproduce

this behaviour accurately in the region close to the critical density.

It is possible that the re-summation schemes proposed so far and used here

103

Chapter 7

may be insufficient to capture the true behaviour of the virial expansion at

densities close to criticality. However, this seems unlikely since the

methods used in this investigation are among the most recently developed

and advanced available and also perform well when applied to the

super-critical case. It should also be noted that due to limitations in the

accuracy of computing high-order virial expansions, it is likely that the

series would show divergence at very high densities as the accuracy of the

virial coefficients themselves is lost.

One remaining question is that of the radius of convergence of the virial

expansion of a Lennard-Jones fluid. A finite radius of convergence is

indicative of the presence of a singularity in the complex plane. The

existence of such a singularity in this case is supported by the fact that

re-summation of the virial expansion has shown evidence of convergence up

to a limit below the critical density. This could be indicative of the

existence of a discontinuity on the complex plane at a density below the

critical density.

Although a standard Pade approximant assumes that any singularity is in

the form of a pole in the complex plane, the ACA method assumes that in

fact any singularity is instead a branch point (Santos and Lopez de Haro,

2009). A branch point is a point at which the argument of the

approximant representing ∆Z is zero and therefore its derivative with

respect to ρ∗ is undefined. Since the virial series is an expansion around

∆Z = 0, the radius of convergence is determined by the first value of ρ∗ at

which the derivative of ∆Z is undefined.

The findings of this investigation seem to indicate that in the case of a

fluid governed by a Lennard-Jones potential, a convergent virial expansion

can be obtained at super-critical temperatures and in some cases at

sub-critical temperatures for low densities. However, a convergent virial

expansion is not obtained at all densities up to the freezing transition even

through means of a re-summation scheme. Therefore it may not be

possible to describe the behaviour of the fluid at all densities up to the

freezing transition through a single expression for Z. Instead, equations of

state may need to be developed separately for sub- and super-critical

104

Chapter 7

conditions in order to adequately describe the gas-liquid phase transition.

This conclusion is in agreement with the recent work of Schultz and Kofke

(2015b) and Ushcats (2014), which also suggest that a single equation of

state for a fluid based on the virial expansion may not be possible.

105

Chapter 8

8 Improving the convergence of the virial

expansion in systems governed by a

‘square-shoulder’ potential form

This chapter will introduce the two types of ‘square-shoulder’ potential.

These kinds of potential differ from the Lennard-Jones potential in the

respect that they are described by a function u(r12) that is finite when

r12 = 0. This chapter will also present the methods used to improve the

convergence of the virial expansion for each system and discuss any

similarities with the results of Chapter 7 that arise from the results of this

investigation. All results in this chapter are obtained at T ∗ = 1.0.

8.1 The dissipative particle dynamics potential

Dissipative particle dynamics (DPD) is a technique that aims to improve

the speed of the simulation of rheological properties of complex fluids

(Espanol and Warren, 1995; Groot and Warren, 1997). To do this, a

coarse-graining technique is introduced with small groups of molecules

constituting ‘fluid particles’ on which simulations are performed and

dissipative forces are introduced. A form of the potential governing this

system can be defined in terms of the coefficients C1 and C2, which

represent the effects of the temperature and the forces between the

particles. The particular potential used in this investigation takes the

following form:

βu(r12) =

C1(rc − r12)2 − C2(r

′c − r12)

2, 0 < r12 ≤ rc,

−C2(r′c − r12)

2, rc < r12 ≤ r′c,

0, otherwise.

(8.1)

The critical properties of a system governed by a potential of this type

depend on the choice of parameters C1 and C2. marsh and Yeomans

(1997) suggest relations for the critical temperature and density for these

systems. The potential obeys the following stability criterion for many

body systems (Fisher and Ruelle, 1966):

106

Chapter 8

∫

βu(r12) dr12 ≥ 0, (8.2)

which holds when:

C2 ≤ C1(rcr′c)5. (8.3)

A system satisfies this Ruelle stability criterion if it has a potential energy

which has a fixed lower bound. This means that the energy per particle

has a lower bound in the thermodynamic limit. Thus the additivity of

extensive thermodynamic properties in the system holds within the Ruelle

stability limit.

The coefficient C1 is set to 30 and the coefficient C2 is varied. When

C2 = 0, the potential becomes purely repulsive and thus no gas-liquid

transition can exist. The case when C2 = 3.9505 represents the maximum

value of C2 that satisfies the Ruelle stability condition. Therefore

calculations are carried out at C2 = 3, 3.5 and C2 = 3.9505 in order to

show the effects of adding an increasingly strong long-range attractive

component to the potential. In this investigation, rc = 1 and r′c =32 . This

means that the quantity rc represents the radius of a fluid particle, since

σ = 1. The shape of several kinds of DPD potential with these rc and r′c

values are illustrated in the sketch in Figure 8.1.

The mean-field approximation for a fluid can be calculated to determine

analytically the approximate behaviour of the compressibility Z at high

densities and so give the curve to which an expansion should be expected

to converge. A mean-field approximation is designed to study the

behaviour of a large, many-body system through a simpler model. Thus, a

many-body problem can be reduced to a one-body problem by reducing

the contribution of all other bodies to a single averaged effect using a

correctly chosen external field. Following the argument given by Hansen

and McDonald (2006) discussing the mean-field theory of vapour-liquid

coexistence, in the case of a fluid governed by a DPD potential, the mean

field approximation to second order can be written as:

107

Chapter 8

C1 = 30, C2 = 0

C1 = 30, C2 = 2

C1 = 30, C2 = 3

C1 = 30, C2 = 4

C1 = 30, C2 = 5

u(r

12)

−5

0

5

10

15

20

25

30

r12

0 0.25 0.5 0.75 1 1.25 1.5

Figure 8.1: A sketch of the shape of the DPD potential for various values ofthe coefficients C1 and C2.

Z = 1 + β1

2ρ∗

∫ ∞

04πr212βu(r12) dr12, (8.4)

= 1 + βπρ∗[

C1

15− C2

2

]

. (8.5)

8.2 The penetrative square well potential

The penetrative square well (PSW) potential is the second type of

potential that is investigated here. This is an extension of the square well

potential that reduces the infinite short range repulsion to a finite value.

Such a potential function is defined as:

u(r12) =

ǫr, r12 ≤ σ,

−ǫa, σ < r12 ≤ σ +∆,

0, r12 > σ +∆.

(8.6)

In this definition, ǫr and ǫa represent two positive energies accounting for

the repulsive and attractive components of the potential respectively. ∆ is

the width of the attractive square well and σ accounts for the width of the

108

Chapter 8

repulsive barrier. As ǫr → ∞, the square well potential is recovered and for

ǫa = 0 or ∆ = 0, the penetrable-sphere model is recovered.

The potential can be defined in terms of two ratios. ǫa/ǫr is the

penetrability ratio, a measure of the penetrability of a particle. The ratio

∆/σ is a dimensionless measure of the size of the particles in the system.

Figure 8.2 shows a PSW potential for a variety of size and penetrability

ratios.

=======

===

=====

(2

Figure 8.2: A sketch of the shape of the PSW potential for several differentpenetrability and size ratios.

For a PSW fluid, the mean field approximation can again be calculated

using equation (8.4). In this case, the approximation gives:

Z = 1 +2

3ρ∗βπ

[

1 +ǫaǫr

− ǫaǫr

(

1 +∆

σ

)]

. (8.7)

For PSW potentials of the form described here, the Ruelle stability

condition can be written as:

ǫaǫr

≤ 1(

1 + ∆σ

)3 − 1. (8.8)

109

Chapter 8

8.3 Virial expansion for fluids governed by

‘square-shoulder’ potentials

Virial coefficients for up to fifth order in density for several variants of

both potential functions have been obtained using the MSMC method

described in Chapter 2. These coefficients and the error associated with

each are tabulated in Appendices B and C. Plots of the virial expansion at

increasing orders of density can then be found using this data.

8.3.1 DPD potential form

Figures 8.3, 8.4 and 8.5 show the behaviour of the virial expansion for a

fluid governed by a DPD potential.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Z

0

5

10

15

20

25

ρ*

0 0.5 1 1.5 2 2.5 3

Figure 8.3: Virial expansion to fifth order for a fluid governed by a DPDpotential with C1 = 30 and C2 = 0. Convergence tests show that the curvesconverge at densities below ρ∗ ≈ 0.7.

The graphs show that for a potential with no attractive component, both

the D’Alembert and Cauchy tests indicate that convergence is good up to

around ρ∗ = 0.7. As C2 increases below the Ruelle stability limit, the

expansion converges well at low densities, but performs increasingly poorly

at higher densities approaching the region of phase transition. In the case

110

Chapter 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Z

0

5

10

15

20

25

ρ*

0 0.5 1 1.5 2 2.5 3

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Z

0

5

10

15

20

25

ρ*

0 0.5 1 1.5 2 2.5 3

Figure 8.4: Virial expansion to fifth order for a fluid governed by a DPDpotential with C1 = 30, C2 = 3 (left) and C1 = 30, C2 = 3.5 (right).Convergence tests show that the curves converge at densities below ρ∗ ≈ 0.5.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Z

−10

−7.5

−5

0

2.5

5

ρ*

0 0.5 1 1.5 2 2.5 3

Figure 8.5: Virial expansion to fifth order for a fluid governed by a DPDpotential with C1 = 30 and C2 = 3.9505. Convergence tests show that thecurves diverge at densities above ρ∗ ≈ 0.35.

111

Chapter 8

of the Ruelle stability limit being approached, C2 = 3.9505, the expansion

performs relatively poorly at low densities and extremely poorly at higher

densities.

8.3.2 PSW potential form

Figure 8.6 shows the case of a potential with both width and penetrability

ratios set to zero, giving rise to one single fluid phase. In contrast, Figure

8.7 shows the case of a potential with width ratio 12 and penetrability ratio

18 , which gives rise to a fluid with separate liquid and vapour phases.

Figures 8.6 and 8.7 show that the virial expansion is only convergent at

relatively low densities.

Figures 8.6, 8.7 and 8.8 show the virial expansion of a fluid governed by a

PSW potential at each order up to fifth. For this potential form, the case

where only a single fluid phase exists is given by setting both the ǫaǫr

and ∆σ

to 0, as shown in Figure 8.6. To introduce an attractive component to the

potential, the case where ∆σ = 1

2 and ǫaǫr

= 18 has been chosen (Figure 8.7).

Under these conditions, the critical density has been calculated by Fantoni

et al. (2011) to be ρ∗c = 0.302. In order to show the effects close to and

beyond the Ruelle stability limit, the cases where ∆σ = 3

4 and ǫaǫr

= 16 and

both ratios are set equal to 1 are also considered and shown in Figure 8.8.

From these graphs, it can be seen that at high densities, the virial

expansion is divergent in all cases. For a purely repulsive potential form,

convergence is evident up to a density of around 0.5. However, when∆σ = 1

2 and ǫaǫr

= 18 , convergence is only evident up to a density of around

half this value. For the unstable case, the virial expansion is divergent at

all densities.

In general, for both types of potential form, the virial expansion is only

convergent at relatively low densities and therefore is unable to accurately

predict vapour-liquid phase transition or the high-density behaviour of the

fluid. To improve the convergence behaviour of the equation, a

re-summation scheme is required.

112

Chapter 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th orderZ

−5

0

5

10

15

ρ*

0 1 2 3 4 5

1

1.25

1.5

1.75

2

2.25

2.5

0 0.2 0.4 0.6 0.8 1

Figure 8.6: Virial expansion to fifth order for a fluid governed by a PSWpotential with ∆

σ = 0 and ǫaǫr

= 0. This is equivalent to a system with onlyone fluid phase.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Z

0

1

2

3

4

5

ρ* /ρc

*

0 2 4 6 8


σ = 12 and ǫa

ǫr= 1

8 . This is equivalent to a system where agas-liquid phase transition is possible.

113

Chapter 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Z

0

1

2

3

4

5

ρ*

0 2 4 6 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Z

−100

−80

−60

−40

−20

0

ρ*

0 0.5 1 1.5 2 2.5


σ = 34 and ǫa

ǫr= 1

6 (left) and with ∆σ = 1 and ǫa

ǫr= 1 (right).

This is equivalent to systems close to and beyond the Ruelle stability limit.

8.4 Methodology

To improve the convergence behaviour of the virial expansion, a

re-summation scheme is required. Using arguments developed in the study

of liquid state theory, it is assumed that as the density ρ∗ tends to ∞, the

pressure of the system p will behave roughly like ρ∗2 and therefore the

compressibility Z like ρ∗. This assumption is supported by the work of

Fantoni et al. (2011) in the case of a fluid governed by a PSW potential.

The phase diagram they obtained for this type of system is shown in

Figure 8.9.

As ρ∗ → 0, the virial expansion should tend to the ideal gas and so Z → 1.

Imposing these constraints on a series expansion of unknown coefficients

Di gives:

Z = 1 +B2ρ∗ +B3ρ

∗2 + · · ·+Bnρ∗n−1, (8.9)

=(

1 +D2ρ∗ +D3ρ

∗2 + · · · +Dnρ∗n−1

)1

n−1 . (8.10)

As in the Chapter 7, this expression can be Taylor expanded and the

coefficients of like powers of ρ∗ compared to obtain expressions for Di in

terms of known virial coefficients.

114

Chapter 8

Figure 8.9: The phase diagram for a PSW potential adapted from that ob-tained by Fantoni et al. (2011). These results are taken from NPT MonteCarlo simulations for ǫa

ǫr= 1

8 and ǫaǫr

= 115 .

This re-summation scheme can be extended further to include information

given by the mean field approximation. Since this approximation is valid

at high densities, it can be approximated that when ρ∗ is large:

Z ∼ (Dnρ∗n−1)

1n−1 ≈ β

1

2ρ∗

∫ ∞

04πr212βu(r12) dr12 =Mρ∗, (8.11)

where M is a constant. This means that the leading-order term in

equation (8.10) can be fixed so that:

Z =(

1 +D2ρ∗ +D3ρ

∗2 + · · · +Dn−1ρ∗n−2 + (Mρ∗)n−1

)1

n−1 , (8.12)

which can also be dealt with using the Taylor expansion technique

described above. This re-summation should force the virial expansion to

adopt the same asymptotic behaviour as the mean field approximation.

115

Chapter 8

8.5 Re-summation of the virial expansion for a fluid

governed by a DPD potential

8.5.1 Re-summation with an asymptotic limit of Z ∼ ρ∗ imposed

Figures 8.10 to 8.13 show the re-summation scheme in Equation (8.10).

Figure 8.10 shows that by fifth order the re-summation converges to a

curve close to the mean-field approximation when no attractive forces are

present. Both the D’Alembert and Cauchy convergence tests support this

observation. However, as attractive forces are introduced, the series

converges more slowly in general and no longer converges to the mean-field

approximation, although still appears to converge at densities up to

criticality. This is demonstrated in Figure 8.11 for a system with C2 = 3.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Mean field result

ρ*

Z-1

0

0.5

1

1.5

2

2.5

3

3.5

ρ*0 1 2 3 4 5

Figure 8.10: Re-summation of a virial expansion to fifth order for a fluidgoverned by a DPD potential with C1 = 30 and C2 = 0. Convergence testsshow that the series converges to a result close to the mean-field approxima-tion.

116

Chapter 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Mean field resultρ*

Z-1

0

0.25

0.5

0.75

1

1.25

1.5

1.75

ρ*0 0.5 1 1.5 2

Figure 8.11: Re-summation of a virial to fifth order for a fluid governed bya DPD potential with C1 = 30 and C2 = 3. The series converges rapidly atlow densities to a result less than the mean-field approximation.

Further increasing the strength of the attractive forces results in the

breakdown of the convergence of the re-summed series at a point before

the phase transition is reached. This is demonstrated by the lack of

convergence demonstrated by the re-summations given in Figures 8.12 and

8.13.

117

Chapter 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Mean field result

ρ*

Z-1

−1.5

−1

−0.5

0

0.5

1

ρ*0 0.25 0.5 0.75 1 1.25 1.5

to 2nd order

to 3rd order

to 4th orderto 5th order

to 6th order

Mean field result

p

−40

−20

0

20

40

ρ*

0 1 2 3 4 5

Figure 8.12: Re-summation of a virial expansion to fifth order for a fluidgoverned by a DPD potential with C1 = 30 and C2 = 3.5. The left graphshows compressibility and the right graph shows the pressure-density diagramfor this fluid. The series does not appear to converge and no van der Waals’loops are observed.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Mean field result

ρ*

Z-1

−3

−2

−1

0

1

2

ρ*0 0.25 0.5 0.75 1 1.25 1.5

Figure 8.13: Re-summation of a virial expansion to fifth order for a fluidgoverned by a DPD potential with C1 = 30 and C2 = 3.9505. The seriesonly appears to converge up to around ρ∗ = 0.3. Beyond this point there-summation scheme performs poorly.

118

Chapter 8

8.5.2 Re-summation to an asymptotic limit including

mean-field approximation

Figures 8.14 to 8.17 show the effect of forcing the asymptotic behaviour of

the virial expansion to match that of the mean field result.

In the case of a purely repulsive potential in Figure 8.14, convergence is

evident at all densities and the result obtained from the re-summed virial

expansion agrees reasonably well with the mean field approximation. As

attractive forces are introduced, the convergent results no longer agree well

with the mean field approximation and eventually convergence is no longer

evident at high densities. At the Ruelle stability limit, divergence is

observed.

119

Chapter 8

to 2nd order

to 3rd order

to 4th order

to 5th order

to 6th order

Mean field result

ρ*

Z-1

0

0.5

1

1.5

2

2.5

3

3.5

ρ*0 1 2 3 4 5

Figure 8.14: Re-summation of a virial expansion to fifth order with fixedleading-order behaviour for a fluid governed by a DPD potential with C1 =30 and C2 = 0. The curves converge to a result close to the mean fieldapproximation at high densities.

to 2nd order

to 3rd order

to 4th order

to 5th order

to 6th order

Mean field result

ρ*

Z-1

0

0.5

1

1.5

2

ρ*0 0.5 1 1.5 2 2.5

Figure 8.15: Re-summation of a virial expansion to fifth order with fixedleading-order behaviour for a fluid governed by a DPD potential with C1 = 30and C2 = 3. Although the curves converge at high densities, agreement withthe mean field approximation is poor.

120

Chapter 8

to 2nd order

to 3rd order

to 4th order

to 5th order

to 6th order

Mean field result

ρ*

Z-1

−3

−2

−1

0

1

ρ*0 0.5 1 1.5 2 2.5

Figure 8.16: Re-summation of a virial expansion to fifth order with fixedleading-order behaviour for a fluid governed by a DPD potential with C1 = 30and C2 = 3.5. Divergence is evident at densities greater than around 0.4.

to 2nd order

to 3rd order

to 4th order

to 5th order

to 6th order

Mean field result

ρ*

Z-1

−4

−3

−2

−1

0

1

2

3

ρ*0 0.5 1 1.5 2 2.5

Figure 8.17: Re-summation of a virial expansion to fifth order with fixedleading-order behaviour for a fluid governed by a DPD potential with C1 = 30and C2 = 3.9505. Divergence is evident at all densities.

121

Chapter 8

8.6 Re-summation of the virial expansion for a fluid

governed by a PSW potential

8.6.1 Re-summation with an asymptotic limit of Z ∼ ρ∗ imposed

The virial expansion for a PSW potential can be re-summed in the same

way as has been done for a DPD potential. This is shown in figures 8.18 to

8.21.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Mean field result

ρ*

Z-1

0.5

1

1.5

2

2.5

ρ*0 1 2 3 4 5

Figure 8.18: Re-summation of a virial expansion to fifth order for a fluidgoverned by a PSW potential with ∆

σ = 0 and ǫaǫr

= 0. Rapid convergencecan be observed.

Figure 8.18 shows that at all densities studied, the re-summation scheme

results in a strong convergence for a system with a single fluid phase.

Figure 8.19 shows that this is also the case up to densities of around 0.8

when fairly weak attractive forces are introduced in a system with ∆σ = 1

2

and ǫaǫr

= 18 . However, at higher densities the re-summation scheme

performs poorly. In the case when stronger attractive forces exist shown in

Figure 8.21, convergence is not apparent even at low densities and

divergence is observed. At this point, the negative gradient of the mean

field approximation is indicative of the system being beyond the gas-liquid

transition at all densities. In all cases, the re-summed virial expansion

shows poor agreement with the mean field approximation.

122

Chapter 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Mean field result

ρ*

Z-1

0

0.2

0.4

0.6

0.8

ρ* /ρc*

0 0.5 1 1.5 2 2.5 3

to 2nd order

to 3rd order

to 4th order

to 5th order

to 6th order

Mean field result

p

0

0.5

1

1.5

2

2.5

ρ*/ ρ*c

0 0.5 1 1.5 2 2.5 3


σ = 12 and ǫa

ǫr= 1

8 . Convergence isevident up to a densities of around 0.8, but beyond this the re-summationperforms poorly. The left graph shows compressibility, while the right graphshows the pressure-density diagram. No van der Waals’ loops are observed.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Mean field result

ρ*

Z-1

0

0.1

0.2

0.3

0.4

0.5

0.6

ρ*0 0.5 1 1.5 2


σ = 34 and ǫa

ǫr= 1

6 . The curves aredivergent.

123

Chapter 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Mean field result

ρ*

Z-1

−30

−25

−20

−15

−10

−5

0

5

ρ*0 0.5 1 1.5 2 2.5


σ = 1 and ǫaǫr

= 1. Convergence is notobserved and the mean field approximation has a negative gradient, which isindicative of an unstable system.

124

Chapter 8

8.6.2 Re-summation to an asymptotic limit including

mean-field approximation

Figures 8.22 to 8.25 show the effect of forcing the asymptotic behaviour of

the virial expansion to match that of the mean field result.

In the case of a purely repulsive PSW potential, the virial expansion is

convergent at high densities and agrees reasonably well with the mean field

approximation. Introducing an attractive component to the potential

results in poor agreement with the mean field result and as the strength of

the attraction increases, a decreased radius of convergence of the

re-summed expansion. Close to and beyond the Ruelle stability limit, the

curves become increasingly divergent even when re-summed.

125

Chapter 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order


Z-1

1

1.25

1.5

1.75

2

2.25

2.5

2.75

ρ*0 1 2 3 4 5

Figure 8.22: Re-summation of a virial expansion to fifth order with fixedleading-order behaviour for a fluid governed by a PSW potential with ∆

σ = 0and ǫa

ǫr= 0. Curves to fourth order converge at high densities to a value

close to the mean field approximation.

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order

Mean field result

ρ*

Z-1

1.1

1.2

1.3

1.4

1.5

ρ*/ρ*c

0 2 4 6 8


σ = 12

and ǫaǫr

= 18 . Curves converge at densities below criticality but agree porrly

with the mean field approximation.

126

Chapter 8

to 1st order

to 2nd order

to 3rd order

to 4th order

to 5th order


Z-1

0.25

0.5

0.75

1

1.25

1.5

ρ*0 0.5 1 1.5 2 2.5


σ = 34

and ǫaǫr

= 16 . Convergence is evident at densities up to ρ∗ ≈ 0.4, but at high

densities the curves diverge.

u

*

-

−

0

h

au

−l

-l

*l

ul

l

ul

*l

-l

h

l * − 6


σ = 1and ǫa

ǫr= 1. The curves are divergent and agree poorly with the mean field

approximation at all densities.

127

Chapter 8

8.7 Extrapolation of virial coefficients

The method recently proposed by Schultz and Kofke (2015b) and

described in Chapter 3 can be used to extend the range of known virial

coefficients up to theoretically infinite order. The potentials selected for

analysis here describe both systems where a single fluid phase forms and

systems where a vapour-liquid phase transition is expected for both a DPD

and a PSW potential form. The change in natural logarithm of the

incremental ratio of coefficients with increasing order of coefficient is

plotted in Figure 8.26 and the following approximate relationship is seen:

T lnBn

Bn−1= −a

n+ b. (8.13)

C1 = 30, C1 = 0

C1 = 30, C1 = 3.9505

εa/εr =0, Δ/σ = 0

εa/εr = 81, Δ/σ = 0.5

ln (B

n/B

n-1)

−6

−5

−4

−3

−2

−1

0

1

1/n

0 0.1 0.2 0.3 0.4 0.5

Figure 8.26: The change in incremental ratio of virial coefficients with in-creasing order n. The dashed lines represent the fitted lines with the equation− a

n + b.

The coefficients a and b are found through curve fitting. These values are

tabulated in Table 8.1 for several potentials.

Equation (8.13) can be used to define the following expression for an

extrapolated virial coefficient Ben:

Ben ≈ C(T )n−a/T exp [bn/T ]. (8.14)

128

Chapter 8

a b

DPD with C1 = 30, C2 = 0 −3.580 −1.269

DPD with C1 = 30, C2 = 3.9505 −16.728 −5.795

PSW with ǫaǫr

= 0, ∆σ = 0 −4.711 −2.162

PSW with ǫaǫr

= 18 ,

∆σ = 1

2 −9.721 −4.276

Table 8.1: Coefficients a and b for extrapolating higher order virials. Alldata is shown at T ∗ = 1.0. Values are given to 3 decimal places.

Due to limitations in the available computational power, the coefficient B6

is the highest which can be confidently used to calculate the constant of

proportionality C in order to extrapolate higher order coefficients. Despite

this limitation, higher order virial coefficients can be plotted for all of the

potentials tabulated here. Virial coefficients up to B10 are shown in Figure

8.27.

DPD with C1 = 30, C2 = 0

DPD with C1 = 30, C2 = 3.9505

PSW with εa/εr = 0, Δ/σ = 0

PSW with εa/εr = 81, Δ/σ = 0.5

Bn

−2

−1

0

1

2

n

0 2 4 6 8 10

Figure 8.27: Extrapolated virials for several types of square-shoulder potentialup to B10.

These virial coefficients can be used to plot virial expansions for the

compressibility Z as usual. The results of this are shown in Figures 8.28

and 8.29.

For the purely repulsive DPD potential, convergence can be seen up to

129

Chapter 8

−

0

1

*

.

*

s

*

us

u*

−s

6

s si* u ui* − −i*

−

1

0

*

.

u*

us

*

s

*

6

s si* u ui* − −i*

Figure 8.28: Virial expansions for DPD potentials with extrapolated coeffi-cients up to sixth order in density at T ∗ = 1.0. The forms of the potentialshown are when C1 = 30, C2 = 0 (left) and C1 = 30, C2 = 3.9505 (right).

−

.

0

7

1

7

−i7

s

−i7

7

*i7

us

6

s si7 u ui7 − −i7

−

.

0

7

1

7

−i7

s

−i7

7

*i7

us

6l

6

s − 0 1 /

Figure 8.29: Virial expansions for PSW potentials with extrapolated coeffi-cients up to sixth order in density at T ∗ = 1.0. The forms of the potentialshown are when ǫa

ǫr= 0, ∆

σ = 0 (left) and ǫaǫr

= 18 ,

∆σ = 1

2 (right).

130

Chapter 8

around ρ∗ = 1.0. In all other cases, convergence is only evident at lower

densities. At high densities, the curves are divergent for every kind of

potential analysed here. As well as this, there is generally poor agreement

with the mean field approximation.


The results presented in this chapter show that it is possible to use the

MSMC technique to obtain virial coefficients for up to sixth order with

relatively small degrees of error. It has then been possible to use these

results to re-sum a virial expansion for both types of square-shoulder

potential to improve the convergence of the expansion.

From these findings, it can be generally concluded that it is possible to

improve the convergence of the virial expansion in a system with only one

fluid phase. However, as the strength of the long-range attractions between

particles is increased, the proposed re-summation scheme is increasingly

ineffective and is unable to describe a gas-liquid phase transition. This is

evidenced by the fact that even though the convergence of the series is

improved in some cases at higher densities, no van der Waals’ loops are

observed in plots of density against pressure. This indicates that a

vapour-liquid phase transition is not predicted by the virial expansion.

When an increasingly strong attractive component is added to either kind

of potential form, there is increasingly poor agreement with the mean field

approximation at all densities. This raises questions over the quality of the

mean field approximation in this case.

These findings provide corroborative evidence that the conclusions drawn

in Chapters 6 and 7 are reliable. This is because it appears that the

re-summmed virial expansions considered here are unable to accurately

describe a gas-liquid phase transition when one is known to exist. This

phenomena appears to be common to many fluids governed by potentials

that consist of repulsive and attractive components and is not limited to

only the Lennard-Jones potential.

131

Chapter 9

9 Using the virial expansion to describe

transition to an ordered phase

This chapter presents and discusses the results of investigating how a virial

expansion may be used to predict phase transition from a fluid to a crystal

phase in a hard sphere system.

9.1 The hard-sphere model

The hard sphere model considers particles to be hard, impenetrable

spheres of diameter σ and is defined in terms of the pairwise interaction

potential u(r12) between two spheres:

u(r12) =

0, r12 ≥ σ,

∞, r12 < σ.(9.1)

Using this potential, the Mayer function can be written as:

f12 =

0, r12 ≥ σ,

−1, r12 < σ.(9.2)

This model therefore provides a simple geometry while still giving a good

approximation of the strong short-range repulsive forces exhibited by

spherical particles. The lack of an attractive component to the potential

results in the model predicting one fluid phase rather than both liquid and

vapour phases. The hard sphere model is important since despite its

apparent simplicity, it contains sufficient thermodynamic richness that it is

widely used as a reference system for other more complicated models, as

discussed by Mulero (2008). Due to advances in the development of

colloidal suspensions, the behaviour of a hard sphere system can also be

studied increasingly accurately using experimental methods, as well as

theoretical and simulation techniques.

132

Chapter 9

9.2 The crystal lattice

A crystal lattice is a set of an infinite number of arranged points in space

which are related by translational symmetry. This investigation considers

three types of crystal lattice: the face-centred cubic (FCC), body-centred

cubic (BCC) and hexagonal close-packed (HCP) lattices, shown in Figure

9.1. Each of these crystals can be split into a number of unit cells, the

smallest repeating units that makes up each crystal lattice.

A crystal lattice can be categorised using its physical properties. Firstly,

the number of particles per unit cell, Ncell, is the number of particles per

unit cell in the lattice. For example, this value is 4 for an FCC lattice, but

only 2 for a BCC lattice. Since the lattice site at each unit cell vertex is

shared by all neighbouring cells, the particle at each occupied site belongs

equally to all of the cells.

A second important property based on the size of the unit cell. In the

HCP crystal, the key quantity is ca , which is the ratio of the distance

between vertical hexagonal layers of sites and the horizontal distance

between sites within a hexagonal lattice. For cubic lattices, the important

parameter is d, the distance between two nearest neighbour lattice sites in

a unit cell. This information along with the particle radius σ determines

the packing fraction in a particular system.

Figure 9.1: A sketch of each of the three types of crystal lattice to be modelled:BCC, FCC and HCP.

Both the FCC and BCC crystals are examples of a Bravais lattice. This

means that they can be represented as an infinite array of discrete points

that can be described by a set of primitive vectors ai that span the array.

133

Chapter 9

So for a lattice site at R:

R = la1 +ma2 + na3, (9.3)

where l,m, n are integers. For an FCC lattice, the unit vectors can be

written as: a1 = d(0, 1, 1),a2 = d(1, 0, 1),a3 = d(1, 1, 0); for a BCC lattice:

a1 = d(1, 0, 0),a2 = d(0, 1, 0),a3 = d(1, 1, 1).

The HCP lattice consists of two inter-penetrating hexagonal Bravais

lattices, but is not itself a Bravais lattice. The unit vectors in this case can

be written as: a1 = c(0, 0, 1),a2 =a2 (1,

√3, 0),a3 =

a2 (−1,

√3, 0). This

crystal is based on particles at two sites, one at the origin (0, 0, 0) and one

at (2/3a1 + 1/3a2 + 1/2a3). The HCP lattice and FCC lattices share the

property of admitting the highest possible packing fraction η. The packing

fraction is a measure of the amount of space occupied by particles within a

lattice and can reach a maximum value of π/(3√2) for an FCC or HCP

lattice (Hansen and McDonald, 2006).

9.3 Predicting the Helmholtz energy profile via stability

analysis

Recalling from Chapter 5, the expression for the Helmholtz energy profile

is given in terms of an ideal and excess contribution to second order by:

A[ρ] = Aid[ρ] +Aex[ρ], (9.4)

βA =

∫

ρ(r1)(ln[Λ3ρ(r1)]− 1) dr1

− 1

2

∫∫

ρ(r1)ρ(r2)f12 dr1 dr2 + . . . (9.5)

The hard sphere model contains no dependence on temperature, which

means that the system being investigated here is athermal. Therefore the

de Broglie thermal wavelength Λ plays no role in the phase equilibrium

since it is equal in both the fluid and crystal phases. For this reason, it is

assumed equal to 1 and henceforth neglected.

134

Chapter 9

A stability analysis can be used to approximate the instability density so

that the accuracy of the numerical results obtained in this investigation

can be better understood. Consider first the grand potential:

βΩ =

∫

ρ(r1)(ln[Λ3ρ(r1)]− 1) dr1

− 1

2

∫∫

ρ(r1)ρ(r2)f12 dr1 dr2 − µ

∫

ρ(r1) dr1, (9.6)

and its functional derivative with respect to ρ(r1):

δβΩ

δρ(r1)= ln[Λ3ρ(r1)]−

1

2

∫

ρ(r1)ρ(r2)f12 dr2 − µ. (9.7)

Now, both the density ρ(r1) and the chemical potential µ can be written

in terms of perturbations from the exact solutions in terms of the

parameter ǫ, where ǫ << 1:

ρ(r1) = ρ0 + ǫρ1(r1) + . . . , (9.8)

µ = µ0 + ǫµ1 + . . . (9.9)

Setting the expression for the derivative of the grand potential equal to

zero and substituting these expansions into it leads to the results that

µ = µ0 and:

ρ1(r1) = ρ0

∫

ρ1(r2)f12 dr2. (9.10)

Taking the Fourier transform of this integral gives:

ρ1(k) = ρ0f(k)ρ1(k), (9.11)

which leads to the conclusion that either ρ0f(k) = 1 or ρ1(k) = 0. This

provides an instability condition, since at densities above ρ0 = 1/f(k), the

isotropic phase will become unstable and a phase transition will occur.

This condition can now be used to evaluate the coexistence point for a

system. Let the perturbation that is applied be of the form:

ǫρ(r1) = eik.r1 , (9.12)

135

Chapter 9

Substituting this expression into equation (9.10) results in:

ρ

∫

[f12e−ik.r2 ] dr2 = −4πρσ3

(

sin kσ − kσ cos kσ

(σk)3

)

= 1. (9.13)

The minimum value of the density ρ obtained from this equation is the

instability density. This minimum corresponds to a particular value of k.

This instability density can be used to find the packing fraction at which

the Helmholtz energy is minimised. In the case of the FCC crystal, this

occurs at a packing fraction of η ≈ 1.46. This approximation is close to

reported values such as those of Groh and Mulder (1999, 2000), which

place the corresponding packing fraction for the FCC crystal in the range

1.2 to 1.5.

9.4 Calculating and minimising the Helmholtz energy using

a virial expansion

The density profile over the crystal lattice is periodic in nature. Each site

Ri within the lattice is described by a Gaussian trial function of width l,

representing the density profile over that particular site, with the optimum

value of l taken to be that which minimises A. An expression for the

density in terms of spatial position can be described by summing Gaussian

functions over Ns neighbouring sites Ri. Here, the density profile is

represented by a sum of Gaussian functions similar to that used by Dong

and Evans (2006) and Verma and Ford (2008):

ρ(r1) =

Ns∑

i

ρ(ri)

=

Ns∑

i

1

(π3/2)l3e−(Ri−ri)2/2l2 . (9.14)

This expression provides a density profile that can now be used to find a

numerical solution for the integral for the ideal contribution to the

Helmholtz energy in (9.5).

To calculate the ideal part of this equation the integral must be solved

136

Chapter 9

numerically. This is done here using the trapezium rule in three

dimensions with a sufficiently small step size, i.e. d/nsteps, to ensure an

accurate result. Over a unit cell of side length d, 100 steps are used in each

direction, effectively splitting the three-dimensional integral over Vcell into

106 small cubes, each with a side length of 0.01d. The size of a unit cell

varies with η as shown in the definition given in equation (5.26). In the

case of the HCP crystal, the ratio c/a is used to determine the step size in

place of d. Once again, 100 steps are used in each direction.

Equation (9.14) is used to calculate the density at each step in the unit

cell, taking into account the effect of nearest neighbour lattice sites. The

grid size and cut-off distance have been varied through a trial and error

method and summation over ten neighbouring sites in each direction has

been found to give a convergent result to eight decimal places for the value

of the contribution to the overall sum. This leads to a result at each ‘step’

within the unit cell which can then be summed to give the value of the

integral of ρ(ln(ρ)− 1). The accuracy of this approximation can be

confirmed by first calculating the integral of the density over the volume of

one unit cell and ensuring that the programme returned the number of

lattice sites per cell, accurate to six decimal places. This test was

successful in the case of all three types of lattice.

To calculate the excess free energy, the method of Dong and Evans (2006)

as set out in Section 5.4. To recap, the value of ω(bj) at each position bj

within the cell is calculated and summed to give an expression for −12Nsω.

The terms to be summed are given by:

ω(bj) = −1

2

[

erf

(

σ − bj

l√2

)

+ erf

(

σ + bj

l√2

)

+l

bj

√

2

π

(

e−(bj+σ)2/2l2 − e−(bj−σ)2/2l2)

]

. (9.15)

This results in an expression for the excess free energy as a function of l/d.

In this case, accuracy could be ensured by checking that at large values of

l, value of the excess energy tends to η/Ncell. This method is repeated over

a range of packing fractions. The total Helmholtz energies for FCC, BCC

137

Chapter 9

and HCP crystals are shown below in Figures 9.2, 9.3 and 9.4.

η = 0.1

η = 0.2

η = 0.3

η = 0.4

η = 0.5

η = 0.6

η = 0.7

η = 0.8

η = 0.9

η = 1.0

η = 1.1

η = 1.2

η = 1.4

η = 1.5

β A[ρ]

−2

0

2

4

6

8

l/d

0 0.5 1 1.5 2 2.5

Figure 9.2: Unweighted free energy βA for an FCC lattice with increasingratio l/d for increasing values of packing fraction η.

η = 0.1η = 0.2η = 0.3η = 0.4η = 0.5η = 0.6η = 0.7

η = 0.8η = 0.9η = 1.0η = 1.1η = 1.2η = 1.4η = 1.5

β A[ρ]

−1

0

1

2

3

4

l/d

0 0.5 1 1.5 2 2.5

Figure 9.3: Unweighted free energy βA for an BCC lattice with increasingratio l/d for increasing values of packing fraction η.

In the case of each of the three lattice types, the Helmholtz energy profiles

138

Chapter 9

η = 0.1η = 0.2η = 0.3η = 0.4η = 0.5η = 0.6η = 0.7

η = 0.8η = 0.9η = 1.0η = 1.1η = 1.2η = 1.4η = 1.5

β A[ρ]

−15

−10

−5

0

5

10

l/(c/a)

0 0.5 1 1.5 2 2.5

Figure 9.4: Unweighted free energy βA for an HCP lattice with increasingratio l/d for increasing values of packing fraction η.

presented above do not have any obvious minima for almost all values of η

which have been studied. Although a cubic spline interpolation method

does yield a numerical minimum in each case, it is always extremely

shallow and therefore very little confidence can be placed in this being the

true minimum value of βA rather than a numerical artefact.

9.5 Calculating the thermodynamic properties of the

system

Expressions for pressure p and chemical potential µ can be developed using

the thermodynamic relationships between free energies. Firstly:

(

∂A

∂V

)

N,T

= −p, (9.16)

(

∂A

∂N

)

V,T

= µ. (9.17)

Using the definitions of enthalpy H, Gibbs free energy G and entropy S, it

is simple to show that for a system of N particles:

139

Chapter 9

pV = H(S, p)− U(S, V ) = H(S, p)− (A(V, T ) + TS),

= G(T, p)−A(V, T ). (9.18)

Now, in terms of the grand potential Ω:

βΩ = −βpV,= β[A−G]. (9.19)

This free energy relationship can be expressed in terms of density

functionals and the second virial coefficient:

βΩ = −βpV =

∫

V

ρ(r1)(ln[ρ(r1)]− 1) dr1 −1

2

∫∫

V

ρ(r1)ρ(r2)f12 dr1 dr2

− βµ

∫

V

ρ(r1) dr1. (9.20)

As before, the de Broglie thermal wavelength is set to 1. Now, settingδβΩδρ(r) = 0 and using functional differentiation (Hansen and McDonald,

2006) leads to an expression for the chemical potential:

βµ = ln[ρ(r1)] +

∫

V

f12ρ(r2) dr2. (9.21)

Rearranging this expression, multiplying by ρ(r1) and integrating with

respect to r1 results in:

∫

V

ρ(r1)[ln ρ(r1)− βµ] dr1 =

∫∫

V

f12ρ(r1)ρ(r2) dr1 dr2. (9.22)

Equaton (9.22) can be used to find expressions for both the chemical

potential and the pressure in terms of density functionals. Rearranging

this expression, it can be observed that:

140

Chapter 9

βµ

∫

V

ρ(r1) dr1 =

∫

V

ρ(r1)[ln ρ(r1)] dr1 −∫∫

V

f12ρ(r1)ρ(r2) dr1 dr2,

βµN =

∫

V

ρ(r1)[ln ρ(r1)] dr1 −∫∫

V

f12ρ(r1)ρ(r2) dr1 dr2.

(9.23)

If equation (9.22) is substituted for the term including µ in equation

(9.20), then an expression for the pressure can be derived:

βΩ = −∫

V

ρ(r1) dr1 +1

2

∫∫

V

f12ρ(r1)ρ(r2) dr1 dr2,

βpV = N − 1

2

∫∫

V

f12ρ(r1)ρ(r2) dr1 dr2. (9.24)

It should be noted that the unconstrained minimisation of Ω is equivalent

to the constrained minimisation of A. Examining equations (9.23) and

(9.24) shows that to find values of µ and p, density functionals must be

solved. This can be done using the same methods as for the ideal and

excess contributions to the Helmholtz energy described earlier in this

chapter.

The phase coexistence point is the point at which the hard sphere fluid

undergoes a liquid-solid phase transition. At this point, the pressure and

chemical potential of both the solid crystal phase and the isotropic liquid

phase are equal. The isotropic thermodynamic properties can be obtained

directly from the virial equation of state:

βP = ρ+B2ρ2 + . . . , (9.25)

βµ = ln ρ+ 2B2ρ+ . . . (9.26)

Note that here the isotropic form of the second virial coefficient is used,

B2 =23πσ

3, i.e. half of the excluded volume of a hard sphere.

141

Chapter 9

9.6 Searching for the point of phase coexistence

The pressure and chemical potential of both the solid and the fluid phases

of the system are required to find the coexistence point. These have been

derived for both the inhomogeneous solid phase and the isotropic fluid

phase in the previous section.

Values of p and µ for a known l/d ratio can be calculated from these

expressions using similar techniques to the calculation of the free energy

profiles, since equations (9.24) and (9.23) contain similar terms to the

expression for Helmholtz energy used earlier. This is carried out at the

values of l/d corresponding to the minimised Helmholtz energy for each

type of crystal. Results obtained for the phase diagrams of both the

inhomogeneous and isotropic phases are shown in Figures 9.5, 9.6 and 9.7

for an FCC, BCC and HCP lattice respectively.

Inhomogenous phaseisotropic phase

βμ

−5

−2.5

0

2.5

5

7.5

βp

0 1 2 3 4 5

Figure 9.5: Pressure βp variation with chemical potential βµ for an FCClattice in both the inhomogeneous and isotropic phases.

In each case, it has been possible to identify an apparent coexistence point.

This is represented by the point at which the two curves intersect. This

intersection corresponds to an approximated point of phase coexistence,

since at this point the pressures and chemical potentials of both phases are

142

Chapter 9

Isotropic phaseInhomogeneous phase

βμ

−5

0

5

10

15

20

βp

0 2 4 6 8 10

Figure 9.6: Pressure βp variation with chemical potential βµ for an BCClattice in both the inhomogeneous and isotropic phases.


βμ

−5

−2.5

0

2.5

5

7.5

βp

0 1 2 3 4 5

Figure 9.7: Pressure βp variation with chemical potential βµ for an HCPlattice in both the inhomogeneous and isotropic phases.

equal. Based on the minimum Helmholtz energies calculated here, the

resulting solid packing fraction at coexistence is around 0.42 for the FCC

crystal, 0.28 for the BCC crystal and 0.47 for the HCP crystal.

143

Chapter 9

Although the order of stabilities of the possible solid crystal phases

appears to be generally as expected in these results, the solid phase

appears to be meta-stable compared to the fluid phase. This finding points

to some flaw in the model used. Although a solid-solid phase transition

would not normally be expected in a hard sphere system, it may be

possible in principle to search for one in this case. However, this

investigation focusses only on accurately identifying a fluid-solid transition,

which has so far not been possible.

9.7 Application of Parsons-Lee theory to the model

So far, it has been possible to obtain Helmholtz energy profiles at packing

fractions that are greater than the close packing limit in the case of all

three types of lattice and are therefore unphysical. Introducing a weighting

function to limit possible packing fractions to those below the close

packing limit of a given lattice type may improve the accuracy of the

Helmholtz energies calculated and therefore of the thermodynamic data

obtained. The function F (η) is introduced into the expressions for the

excess contribution to the Helmholtz energy that was presented earlier.

The general approach described in Section 9.4 is repeated here, with the

‘weighting function’ F (η) applied to the excess Helmholtz energy term. As

before, a minimum value of the Helmholtz energy can be obtained through

interpolation and so the thermodynamic properties of the lattice can again

be calculated. The free energy profiles obtained for each lattice type are

shown in Figures 9.8, 9.9 and 9.10 for an FCC, BCC and HCP lattice

respectively. Again, the minima of these curves are shown to be extremely

shallow, but it is still possible to obtain values of the minimum Helmholtz

energy at all values of η.

As before, only a very shallow minimum is evident. These results can now

be used to produce pressure-chemical potential phase diagrams for each

type of lattice in the same way as in the previous sections. Doing this

allows a new coexistence point to be observed, as shown in Figures 9.11,

9.12 and 9.13.

144

Chapter 9

η = 0.1η = 0.2η = 0.3η = 0.4

η = 0.5η = 0.6η = 0.7

β A[ρ]

−2.5

0

2.5

5

7.5

10

l/d

0 0.5 1 1.5 2 2.5

Figure 9.8: Helmholtz energy profile obtained for an FCC crystal lattice atpacking fractions up to the close packing limit of 0.74.

η = 0.1η = 0.2η = 0.3

η = 0.4η = 0.5η = 0.6

β A[ρ]

−1

0

1

2

3

l/d

0 0.5 1 1.5 2 2.5

Figure 9.9: Helmholtz energy profile obtained for a BCC crystal lattice atpacking fractions up to the close packing limit of 0.68.

The curves show that in general, the coexistence point predicted by the

Parsons-Lee form of the model produces an improvement in the results

obtained in all cases in that the coexistence points predicted now are closer

to those predicted by the stability analysis. However, these results are still

145

Chapter 9

η = 0.1η = 0.2η = 0.3η = 0.4

η = 0.5η = 0.6η = 0.7

β A[ρ]

−15

−10

−5

0

5

10

l/(c/a)

0 0.5 1 1.5 2 2.5

Figure 9.10: Helmholtz energy profile obtained for an HCP crystal lattice atpacking fractions up to the close packing limit of 0.74.


βμ

0

0.5

1

1.5

2

2.5

3

βp

0 0.5 1 1.5 2 2.5 3

Figure 9.11: Parsons-Lee predictions for pressure βp against chemical po-tential βµ for an FCC lattice for both the liquid and solid phases at packingfractions up to the close packing limit.

in themselves a poor approximation of the actual point of phase

coexistence for these types of lattice.

146

Chapter 9

Inhomogeneous phaseIsotropic phase

β μ

0

2.5

5

7.5

10

12.5

15

β p

0 2 4 6 8

Figure 9.12: Parsons-Lee predictions for pressure βp against chemical po-tential βµ for a BCC lattice for both the liquid and solid phases at packingfractions up to the close packing limit.


βμ

0

0.5

1

1.5

2

2.5

3

βp

0 0.5 1 1.5 2 2.5 3

Figure 9.13: Parsons-Lee predictions for pressure βp against chemical po-tential βµ for an HCP lattice for both the liquid and solid phases at packingfractions up to the close packing limit.

9.8 Confirmation of results using the analytical derivative

of the Helmholtz energy

The results obtained so far have been found through numerical methods

and are therefore approximations to the true result. To confirm the validity147

Chapter 9

of these results, an attempt can be made to minimise the Helmholtz energy

functional analytically at a given choice of packing fraction by calculating

the derivative of the expression in equation (9.5) with respect to l:

∂(βAid)

∂l=

∫

V

ln

[

Λ3Ns∑

i

1

π32 l3

e−(Ri−ri)2/2l2

]

×

Ns∑

i

1

π32

e−(Ri−ri)2/2l2(

− 3

l4− (Ri − ri)

2

l6

)

dr1.

(9.27)

Examining this expression, it can be seen that as l → ∞, ∂(βAid)∂l → 0

(after invoking L’Hopital’s Rule). Performing a similar calculation on the

excess contribution to the free energy:

∂(βAex)

∂l= −Ns

4

√

2

π

∑

j

[

e−(bj−σ)2

2l2

(

1

l2(bj − σ)− 1

bj− 1

bjl2(bj − σ)2

)

+ e−(bj+σ)2

2l2

(

1

l2(bj + σ) +

1

bj+

1

bj l2(bj + σ)2

)]

.

(9.28)

This expression behaves similarly to that above in that as l → ∞,∂(βAex)

∂l → 0. This means that overall it can be concluded that to second

order, l → ∞, ∂(βA)∂l → 0. The next piece of information required is

whether the overall derivative approaches ∞ from a positive or a negative

direction, as that will define whether or not a finite minimum point exists.

As an example, values for the derivative of βA when η = 1 for an FCC

lattice were plotted against increasing l using both the analytical

expressions given above and a two-step finite difference method in Figure

9.14.

Examining these curves, it appears that there is no non-infinite value of l

for which ∂(βAex)∂l = 0, since there is no clear point at which the curves

cross the horizontal axis.

148

Chapter 9

llAlll lldl

lll ll

c5

ch

y5

yh

5

h

5

yh

h hdc5 hd5 hd15 y ydc5 yd5 yd15

Figure 9.14: The derivative of the free energy profile to second order ateta = 1 obtained through both numerical and analytical methods for an FCClattice.


In the cases of each of the three types of lattice, it has not been possible to

accurately predict a freezing transition. This is evident in the case of the

FCC lattice, where the value of η ≈ 0.42 obtained for the coexistence point

is far below that predicted by the stability analysis, and applying the

Parsons-Lee model does little to improve this. Similarly large errors are

also present in the results obtained for the BCC and HCP lattices. Since

such a stability analysis is an analytical calculation and therefore does not

contain the inherent numerical errors of the approximation methods, this

finding casts doubt on the reliability of the results obtained via a virial

expansion.

Despite this lack of confidence in the prediction of the point of phase

coexistence, further analysis of the curves obtained here shows that some

reliable conclusions can still be drawn from the work. In general, at high

pressures, the chemical potential in the solid phase is less than that in the

fluid phase, as expected. Also, when comparing the thermodynamic

properties with the density profile of each lattice, the BCC lattice is found

149

Chapter 9

to be the least stable, in agreement with many recent studies. Similarly,

the FCC crystal is shown to be the most stable form, in agreement with the

most recent investigations in this area, including Groh and Mulder (1999).

The inaccuracies in the calculated coexistence points confirm the suspicion

that the minima obtained from the Helmholtz energy profiles are not true

minima, but are numerical artefacts. This is supported by the analytical

calculation of the derivative of the Helmholtz energy functional.

Questions therefore exist over the choice of trial function. Based on the

findings of this investigation, it appears that a second order model with a

Gaussian basis cannot accurately predict a stable solid phase for a hard

sphere system. Although a functional form based on a Gaussian function is

widely used to represent the density profile of a lattice site, the function is

non-analytic in nature as l → ∞. This could be leading to a poor

representation of Helmholtz energy profile and therefore a poor prediction

of the freezing transition of the hard-sphere fluid.

150

Chapter 10

10 Conclusions and future challenges

This chapter discusses the conclusions that can be drawn from the

investigations that have been documented in this thesis and how they

answer the questions posed at the start of the project. Possible areas of

future work and remaining challenges are also considered.

10.1 Conclusions

The aim of this work has been to investigate a number of applications of

the virial expansion to systems of spherically symmetrical particles. The

virial equation of state is a systematic and exact equation with a

potentially wide variety of uses if some inherent difficulties can be

overcome. The most notable of these difficulties are the high

computational demands of calculating virial coefficients and uncertainty

around the convergence of the expansion.

This work has sought to improve the understanding of the convergence of

the virial expansion and its performance in areas of the phase diagram

close to phase transitions. This has been done through the application of

re-summation schemes in order to increase the radius of convergence of the

series. Therefore the virial equation of state would require fewer

coefficients in order to provide sufficiently accurate results.

Chapter 6 considered the application of a virial expansion to calculating

the DCF of fluids governed by (purely repulsive) inverse-power potentials

in order to be able to describe the correlation structure of the fluid. In the

cases where the repulsive force between particles is weak, convergence of

the DCF virial expansion was found to be poor at all densities up to

freezing. As the strength of the repulsive force was increased, the virial

expansion performed better and a convergent DCF could be obtained up

to density limits similar to those found for the pressure virials by Tan

et al. (2011). For the n = 12 case, the virial expansion converges at

densities approaching the point of the fluid-solid phase transition.

The same investigation has been repeated for a Lennard-Jones potential in

151

Chapter 10

Chapter 7. In this case, the virial expansion of the DCF is found to be

convergent at high densities only under super-critical conditions. Under

sub-critical conditions, the expansion is divergent at liquid densities and

therefore not capable of predicting an accurate DCF under these

conditions. To attempt to improve the convergence of this expansion, a

perturbed form of the potential has been studied. The virial expansion of

the DCF derived from the repulsive component of the potential is generally

found to converge well under all conditions, mirroring the findings of

Chapter 6. However, an expansion derived from a representation of the

attractive component of the potential is shown to be divergent. To

investigate this further, re-summation schemes are applied to a perturbed

Lennard-Jones virial expansion. Some of these, most notably the ACA

scheme (Barlow et al., 2012), show promise in improving the convergence

of the series and obtaining the same asymptotic behaviour as that

predicted by the HTA at very high densities. However, none are able to

accurately predict the vapour-liquid phase transition.

Applying a re-summation scheme to the virial equation of state of fluids

governed by ‘square-shoulder’ potentials in Chapter 8 results in similar

findings although application of the re-summation scheme does provide at

least some improvement to convergence in all cases. In the case when a

single fluid phase exists, convergence to a result close to a mean field

approximation is generally rapid whereas in the case of a potential with an

attractive component, convergence is generally poor, especially at high

densities and in the region of the vapour-liquid phase transition where this

is known.

In Chapter 9, the application of the virial expansion to calculating the

Helmholtz energy of the hard-sphere crystal has been studied. This is

found to be possible and the results of this work agree with the general

understanding that the most stable form of lattice in this case is the FCC

crystal. However, the prediction of the point of the solid-liquid phase

transition is generally inaccurate when using a second order virial

expansion. The reason for this seems to be that the choice of a Gaussian

form for the trial function describing the density profile over a lattice site

is a poor one due to the non-analytic behaviour of the function in cases

152

Chapter 10

where the width becomes very large.

The accuracy of the prediction of a phase transition could possibly be

improved by improving the accuracy with which the Helmholtz energy is

calculated. For the ideal contribution, this could be done either by

reducing the step size used in the numerical calculation or by using a more

advanced quadrature method to numerically calculate the integral. To

improve the accuracy in calculating the excess contribution, higher order

terms in the virial expansion would be required, rather than limiting the

expansion to second order as in this investigation.

These findings demonstrate that the virial expansion generally performs

well in describing the thermodynamic behaviour of a system with only one

fluid phase when only strong, short-range repulsive forces are present.

However, the same is not true in the case of a system where separate liquid

and vapour phases exist and long-range attractive forces or weak repulsive

forces exist. Although the virial expansion gives convergent results in some

cases, it frequently exhibits divergence and so performs poorly.

Although this could be accounted for in part by errors in the calculation of

the virial coefficients themselves, it is much more likely to be the case that

the radius of convergence of the virial expansion is small in the cases when

it performs poorly. This seems to indicate that the virial expansion is

unable to describe the thermodynamic behaviour of the fluid at all

densities. Instead, separate equations of state may be required to fully

describe the behaviour of the fluid and the vapour-liquid phase transition

rather than working to improve the accuracy of a single expression so that

it is applicable at all points on the phase diagram.

10.2 Areas of possible further study and remaining

challenges

It is clear from the conclusions presented here that although some

applications of the virial expansion have proved to be successful, there are

several questions that remain to be fully answered. The most notable of

these is how best to use the equation to describe the vapour-liquid phase

153

Chapter 10

transition and whether this is possible. The convergence of the virial

expansion is known to be slow in many cases, so it is possible that with

improvements in computational power and re-summation schemes beyond

the limits of this work, the radius of convergence of the virial expansion

could be improved, making it better suited to describing a vapour-liquid

phase transition.

Generally, this research has been limited to systems based on

comparatively simple potential functions. In order to expand on the

investigations carried out in Chapters 6 and 7, similar investigations into

obtaining the DCF of a fluid could be carried out for a variety of other,

more complicated potential functions using the same techniques presented

here. As discussed in Chapter 3, there are also many other re-summation

schemes that could be applied to improve the performance of a virial

expansion. Using the MSMC scheme described in Chapter 2 it is possible,

with sufficient computational power, to obtain higher order coefficients for

the virial expansion and so help to increase the speed of convergence in

those cases where it is slow.

Some questions remain concerning the correct asymptotic behaviour of the

systems considered in this investigation. The study of the asymptotic

behaviour of the RDF and DCF that is discussed in Chapter 6 could be

developed further by posing a more complicated form for the density

dependence of the RDF. Chapter 7 discussed how the high-density

behaviour of the attractive contribution to the virial expansion was

approximated using an ansatz based on the MK perturbation scheme since

the correct asymptotic behaviour of the re-summed virial expansion is

known. This study could be extended through the use of a more advanced

perturbation scheme, such as the WCA method. This would require the

desired scaling of the perturbed Lennard-Jones potential to be established

empirically since the repulsive and attractive contributions would no

longer be power law relationships. Hansen and McDonald (2006)

demonstrates that this should result in improved agreement with data

obtained from the KN equation of state or simulation data at the low

temperatures used in this study. In Chapter 8, the asymptotic behaviour

of many of the re-summed potential functions did not agree well with the

154

Chapter 10

predictions of the mean field approximations for those potential functions.

Pressure data obtained either from integral equation theory or simulation

would be helpful in making certain of the correct behaviour of these

expansions at high densities.

A possible extension to the work described in Chapter 9 is to repeat the

investigation using an alternative form of trial function to the Gaussian,

that is better able to predict the stable solid phase of a hard sphere system.

Another possible reason for the poor numerical predictions could be due to

the limiting of the virial expansion to second order. At high densities close

to any point of fluid-solid phase transition, the particles in the fluid are

likely to be closer together and therefore higher order terms are likely to

play a comparatively greater role in the result. Increasing the number of

terms in the expansion describing the excess contribution to the Helmholtz

energy could therefore prove valuable and could be done using the MSMC

method. If reliable results could be obtained, some possible further areas

of study are anisotropic crystals and crystals containing empty lattice

sites. Lattices with these features have been considered by Tarazona

(2000) and Yamani and Oettel (2013) respectively, meaning that results

exist with which to compare those obtained using a virial expansion.

In spite of its inherent difficulties, several applications of the virial

expansion have been successfully demonstrated in this investigation and

have provided some new insight into answering the questions posed in

Chapter 1. The virial equation of state has been proven to be a powerful

and adaptable method for describing the thermodynamic behaviour of a

system of particles, although it appears to be limited in its ability to

adequately describe a vapour-liquid transition.

155

Appendix A Tabulated virial coefficients for the

Lennard-Jones fluid

n Bn at T ∗ = 1.0 Absolute error Bn at T ∗ = 1.6 Absolute error

2 −5.31508 9.03 × 10−4 −0.10665 7.18× 10−4

3 1.88234 3.20 × 10−3 2.27247 1.50× 10−3

4 −2.44596 0.045129 1.74458 0.011965 −53.72246 0.98630 −0.85458 0.156206 −592.29241 23.60099 −4.24365 2.42250

Table A.1: Virial coefficients up to B6 for a Lennard-Jones fluid at sub-and super-critical temperatures obtained using MSMC. Data is given to 5decimal places.

156

Appendix B Tabulated virial coefficients for a

fluid governed by several types of

DPD potential

Value of C2 : 0.0 1.0 2.0 3.0 3.5 3.9505 4.5

Value of B2 1.61387 1.24949 0.75372 0.05491 −0.39839 −0.88345 −1.60974Absolute error 0.00173 0.00217 0.00192 0.00152 0.00205 0.00271 0.00361

Value of B3 1.61232 1.07044 0.64929 0.50104 0.58413 0.75219 0.98390Absolute error 0.00396 0.00374 0.00273 0.00304 0.00487 0.00840 0.01683

Value of B4 1.14045 0.75164 0.46648 0.18315 0.07067 0.13339 0.71147Absolute error 0.00100 0.00058 0.00046 0.00095 0.00207 0.00440 0.01275

Value of B5 0.64831 0.46357 0.31428 0.17565 −0.01355 −0.33118 −0.41400Absolute error 0.00105 0.00062 0.00072 0.00261 0.00546 0.01497 0.05723

Value of B6 0.31596 0.24372 0.19198 0.14246 0.21213 0.01735 −1.47818Absolute error 0.00145 0.00113 0.00174 0.00666 0.02242 0.08582 0.47292

Table B.1: Virial coefficients for a DPD potential with C1 = 30 and T ∗ =1.0. The Ruelle stability limit is reached when C2 = 3.9505. Data is givento 5 decimal places.

Value of C2 : 0.0 2.0 2.6337 3.0

Value of B2 1.51326 0.44832 −0.49241 −0.10061Absolute error 0.00012 0.00012 0.00012 0.00010

Value of B3 1.40007 0.45917 0.51887 0.43345Absolute error 0.00027 0.00015 0.00029 0.00019


Value of B5 0.45345 0.18136 −0.06975 0.08560Absolute error 0.00051 0.00046 0.00352 0.00140



157

Value of C2 : 0.0 2.0 3.0 5.2675

Value of B2 1.67471 0.91568 0.33167 −1.99028Absolute error 0.00013 0.00014 0.00013 0.00033




Value of B6 0.41678 0.26883 0.19657 −3.33722Absolute error 0.00113 0.00119 0.00495 1.51867


Appendix C Tabulated virial coefficients for a

fluid governed by several types of

PSW potential

Value of ǫaǫr: 1 1

214

16

18 0

Value of B2 1.32404 1.32387 1.32403 1.32403 1.32399 1.32388Absolute error 0.00010 0.00010 0.00011 0.00012 0.00011 0.00011


Value of B4 −0.20391 −0.20393 −0.20417 −0.20420 −0.20403 −0.20417Absolute error 0.00009 0.00009 0.00009 0.00008 0.00009 0.00009



Table C.1: Virial coefficients for a PSW potential with ∆/σ = 0.0 andselected values of ǫa/ǫr at T ∗ = 1.0. Data is given to 5 decimal places.

158


214

16

18 0

Value of B2 −1.68872 0.18679 0.82566 1.00624 1.09053 1.32405Absolute error 0.00042 0.00016 0.00017 0.00017 0.00019 0.00011




Value of B6 0.11667 −0.05922 −0.06314 −0.03506 −0.01109 −7.15768Absolute error 0.00030 0.00291 0.00037 0.00020 0.00014 0.44505

Table C.2: Virial coefficients for a PSW potential with ∆/σ = 12 and selected

values of ǫa/ǫr at T ∗ = 1.0. Data is given to 5 decimal places.


214

16

18 0

Value of B2 −3.40945 −0.46285 0.54173 0.82420 0.95721 1.32385Absolute error 0.00057 0.00020 0.00018 0.00019 0.00019 0.00011




Value of B6 0.11623 0.08376 −0.09651 −0.08441 −0.06350 −708.87796Absolute error 0.00030 0.01471 0.00087 0.00041 0.00027 6.73971

Table C.3: Virial coefficients for a PSW potential with ∆/σ = 34 and selected

values of ǫa/ǫr at T ∗ = 1.0. Data is given to 5 decimal places.


214

16

18 0

Value of B2 −5.25602 −1.16022 0.23664 0.62925 0.81416 1.32387Absolute error 0.00062 0.00023 0.00018 0.00019 0.00019 0.00011

Value of B3 −2.43629 0.71129 0.38998 0.41508 0.45636 0.69239Absolute error 0.00545 0.00041 0.00017 0.00016 0.00016 0.00013

Value of B4 −53.43934 −0.45184 −0.04397 0.00168 −0.00967 −0.20416Absolute error 0.12244 0.00227 0.00029 0.00015 0.00010 0.00009


Value of B6 0.11666 0.71847 −0.09429 −0.11858 −0.10372 −14846.43638Absolute error 0.00028 0.06432 0.00198 0.00071 0.00046 68.59667

Table C.4: Virial coefficients for a PSW potential with ∆/σ = 1.0 andselected values of ǫa/ǫr at T ∗ = 1.0. Data is given to 5 decimal places.

159

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