Molecular Dynamics Simulation of the Thermodynamic Properties of Water and Atomistic Fluids

Tesfaye M. Yigzawe

Dissertation submitted in fulf llment for the degree of Doctor of Philosophy

Centre for Molecular Simulation

Swinburne University of Technology Melbourne, Australia

2012 Abstract

An alternative method of calculating the thermodynamic quantities as an average of the appropriate microscopic dynamical functions over the molecular dynamics ensemble is adopted. Pressure, heat capacities, compressibilities, isothermal pressure coefficient, Joule-Thomson coefficient, speed of sound at zero frequency and thermal expansion coefficient in a molecular dynamic ensemble were calculated for Lennard-Jones f uid, Weeks-Chandler-Anderson potential and MCYna water. Using the appropriate Lennard-Jones constants, the above mentioned thermodynamic quantities of supercritical argon and krypton are calculated. The simulation results are compared with experimental and/or previous simulation results. The effects of system size, cutoff radius and simulation time on the thermodynamic state variables of Lennard-Jones f uid and Weeks-Chandler-Anderson potential are studied. The Ewald sum is employed to calculate the long range Columbic interaction in MCYna water. Response functions (such as heat capacities, thermal expansion coefficient, compressibilities) in Lennard-Jones f uid, argon and krypton diverge when the critical point is approached. MCYna water potential predicted the pressure, isochoric heat capacity, speed of sound, adiabatic compressibility and thermal expansion coefficient with a very good agreement with experiment and previous simulations.

i Acknowledgment

First and for most I would like to thank my supervisor Professor Richard Sadus for his constant guidance, encouragement and insight full advices generally without whom I would not be able to complete my studies. I would also thank Professor Billy Todd and Professor Feng Wang for their constant support and encouragement.

I would like to thank Swinburne University of Technology for fnancial support through a Swinburne University Postgraduate Research Award scholarship (SUPRA). This work also received computational time from Victorian Partnership for Advanced Computing (VPAC) and Multi-modal Australian ScienceS Imaging and Visualization Environment (MASSIVE).

I would also thank my friends at Centre for Molecular Simulation (CMS), my facebook friends around the world. Special thanks to Geni, Kumar, Bill and Sergio for lighting up my days.

Finally, I would like to thank my mother who gave her life to me and my brothers. My f nal acknowledgment goes to three of my brothers (Dani, Asme and Abu).

ii Declaration

I hereby declare that the thesis entitled “Molecular Dynamics Simulation of the Thermodynamic Properties of Water and Atomistic Fluids” is my own work. To the best of my knowledge, it contains no materials previously published by other persons except where reference is made in the text of the thesis.

Tesfaye Mekonen Yigzawe

September 2012

iii Notation

SPC simple point charge SPC/E extendedsimplepointcharge SPC/Fw fexiblesimplepointcharge SPC/Fd fexiblesimplepointchargebyDangetal. TIP3P transferable intermolecular potential three point

TIP4P transferable intermolecular potential four point TIP4P/2005 transferable intermolecular potential four point 2005 TIP5P transferable intermolecular potential f ve point MCY Matsuoka-Clementi-Yoshimine water model

MCYL Matsuoka-Clementi-Yoshimine-Lie f exible water model MCYna MCY withnon-additivetermswater model BNS Ben-NaimandStillingermodel ST2 StillingerandRahmanmodel

PPC polarisablepointchargemodel GCPM Gaussian charge polarizable model NveD NadaandvanderEerdensixsitemodel LJ Lennard-Jones WCA Weeks-Chandler-Anderson

iv v

MD moleculardynamics MCMonte Carlo

CFT conventional f uctuation theory TMD temperatureofmaximumdensity TM meltingtemperature RDF radialdistributionfunction

EOS equationofstate Exp. experiment Ref. reference fcc face centered cubic

IAPWS international association for the properties of water and steam ∆t simulation step length t total simulation time

23 k Boltzmann constant (1.380662 10− J/K) B × 1 1 R universal gas constant (8.31441 J mol− K− )

23 1 N Avogadro number (6.022045 10 mol− ) A × 10 Å 10− m

9 ns nanosecond(10− s)

9 nm nano meter (10− m)

12 ps picosecond(10− s) Ω phase space volume ω phase space density

Ωmn derivatives of ω with respect to energy and volume S Entropy vi

T temperature

TC critical temperature τ ratio of temperature to the critical temperature ρ density

ρC critical density p pressure

pC critical pressure ∆p diﬀerence in pressure P total linear momentum G constant related to the initial position of the center of mass

m mass of a single atom/molecule M totalmassofthesystem V volume N number of molecules/atoms U potential energy per particle

∆U diﬀerence in the potential energy of LJ and WCA potentials E internal energy

rij relative position of particles i and j

ui j potential energy between particles i and j r position of the particle v velocity of the particle a acceleration of the particle

fi j force between particles i and j L length of the side of simulation box vii

AA absolute average q charge l1 bond length, between H and O l2 the distance between oxygen and the dummy charge site θ bond angle, between hydrogen-oxygen-hydrogen φ angle between hydrogen-oxygen-dummy site

σ Lennard-Jones length constant ǫ Lennard-Jones energy constant

Cv isochoric heat capacity

∆Cv Cv diﬀerence between the LJ and WCA potentials

Cp isobaric heat capacity

∆Cp Cp diﬀerence between the LJ and WCA potentials

βT isothermal compressibility

∆βT βT diﬀerence between the LJ and WCA potentials

βS adiabatic compressibility

∆βS βS diﬀerence between the LJ and WCA potentials

µJT Joule-Thomson coeﬃcient

∆µJT µJT diﬀerence between the LJ and WCA potentials

γv isothermal pressure coeﬃcient

∆γv γv diﬀerence between the LJ and WCA potentials

αp thermal expansion coeﬃcient

∆αp αp diﬀerence between the LJ and WCA potentials

ω0 speed of sound at zero frequency

∆ω0 ω0 diﬀerence between the LJ and WCA potentials Contents

Abstract i

Acknowledgment ii

Declaration iii

Notation iv

List of Figures xiii

List of Tables xv

1 Introduction 1

2 Water Potentials 5 2.1 Introduction...... 5 2.2 Whyallthesemodelsforwater? ...... 6 2.3 Watermodels,criteria...... 9

2.3.1 Theinteractionbetweenmolecules...... 11 2.3.2 Bondfexibility...... 13 2.3.3 Valuesusedforparametrization ...... 14

viii CONTENTS ix

2.3.4 Polarization...... 16

2.3.5 Thechargedistribution...... 18 2.3.5.1 Three-sitemodels ...... 19 2.3.5.2 Four-sitemodels ...... 19 2.3.5.3 Five-sitemodels ...... 20

2.3.5.4 Six-sitemodels ...... 22 2.4 Reviewofwatermodels...... 24 2.4.1 Rigidwatermodels...... 24 2.4.1.1 TIP3PandSPC ...... 24

2.4.1.2 SPC/E...... 25 2.4.1.3 TIP4P/2005 ...... 26 2.4.1.4 PPC ...... 27 2.4.1.5 GCPM...... 28 2.4.1.6 MCY...... 28

2.4.2 Flexiblewatermodels ...... 31 2.4.2.1 SPC/Fw...... 32 2.4.2.2 SPC/Fd...... 33 2.4.2.3 MCYL...... 33

2.5 MCYnawatermodel...... 34 2.6 Improvingwatermodels ...... 38

3 Calculation of Thermodynamic Properties from Molecular Simulation 41 3.1 Introduction...... 41 3.2 NVEPG Ensemble ...... 42 3.3 Thermodynamic quantities in the NVEPG ensemble ...... 50 x CONTENTS

3.3.1 Heatcapacities ...... 51

3.3.2 Isothermal pressure coeﬃcient...... 52 3.3.3 Compressibilities ...... 53 3.3.4 Speedofsoundatzerofrequency ...... 55 3.3.5 Joule-Thomson coeﬃcient ...... 56

3.4 Calculation of thermodynamic quantities in an NVEPG ensemble 58 3.5 Thermodynamicquantitiesfromthefuctuationtheory ...... 69 3.6 Thermodynamicspropertiesnearthecriticalpoint ...... 71

4 Simulation Details 75 4.1 Introduction...... 75 4.2 Integrators...... 77

4.2.1 Gearpredictor ...... 77 4.2.2 Leapfrog ...... 79 4.3 Periodic boundaries, cutoﬀ radiusandsimulationtime...... 80 4.4 Ewaldsummation...... 81

4.5 SimulationsofLJandWCApotentials ...... 85 4.5.1 LJpotential...... 88 4.5.2 WCApotential ...... 89 4.6 MCYnapotential ...... 90

5 Thermodynamic Properties of LJ and WCA f uids and Noble Gases 93 5.1 Introduction...... 93

5.2 ThermodynamicpropertiesofLJandWCApotentials ...... 96 5.2.1 Energyandpressure ...... 103 5.2.2 Isochoricandisobaricheatcapacities ...... 111 CONTENTS xi

5.2.3 Isothermal pressure coeﬃcient...... 117

5.2.4 Thermal expansion coeﬃcient ...... 119 5.2.5 Speedofsound ...... 121 5.2.6 Isothermalandadiabaticcompressibilities ...... 124 5.2.7 Joule-Thomson coeﬃcient ...... 126

5.3 Thermodynamic properties of supercritical argon and krypton . . . 130 5.3.1 Energyandpressure ...... 131 5.3.2 Heatcapacities ...... 135 5.3.3 Speedofsound ...... 138

5.3.4 Joule-Thomson coeﬃcient ...... 140 5.3.5 Compressibilities ...... 143 5.4 Summary ...... 146

6 Thermodynamic Properties of MCYna Water 149 6.1 Introduction...... 149 6.2 Thermodynamicpropertiesofwater ...... 150

6.3 Simulationresultsanddiscussion ...... 153 6.3.1 Pressure...... 154 6.3.2 Heatcapacities ...... 155 6.3.3 Isothermal pressure coeﬃcient...... 158 6.3.4 Compressibilities ...... 159

6.3.5 Joule-Thomson coeﬃcient ...... 161 6.3.6 Speedofsound ...... 163 6.3.7 Thermal expansion coeﬃcient ...... 164 6.4 Summary ...... 165 xii CONTENTS

7 Conclusions and Recommendations 166

7.1 Conclusion ...... 166 7.2 Recommendations...... 168

Bibliography 170

Appendices

A Simulation data for the WCA potential 207

B Simulation data for the LJ f uid 218

C Simulation data for the MCYna water potential 230 List of Figures

2.1 Simplerepresentationofwater ...... 6 2.2 Tetrahedralshapeofwater ...... 8

2.3 Hydrogenbondinginwater...... 12 2.4 Schematicrepresentationofwatermodels ...... 23 2.5 SchematicrepresentationofMCYwatermodel ...... 29 2.6 Schematicrepresentationofthreebodyinteraction...... 36

4.1 LJ,WCAandBuckinghampotentials ...... 88

5.1 UandpinaWCApotential,diﬀerentsystemsize ...... 98 5.2 Uand p inWCA potentials,diﬀerentsimulationtime ...... 99 5.3 UandpcomparisoninLJfuid ...... 102 5.4 UinWCAandLJpotentialsasafunctionofdensity ...... 106

5.5 ∆Uasafunctionofdensityandtemperature...... 107 5.6 p(LJ)and ∆pasafunctionofdensity...... 109 5.7 ∆pasafunctionoftemperature...... 110

5.8 Cv(LJ) and ∆Cv asafunctionofdensity ...... 113

5.9 Cp(LJ) and ∆Cp asafunctionofdensity ...... 115

5.10 γv inLJandWCAasafunctionoftemperature ...... 119

xiii xiv LIST OF FIGURES

5.11 ∆αp asafunctionofdensity...... 121

5.12 ω0(LJ) and ∆ω0 asafunctionofdensity ...... 123

5.13 ∆βS and ∆βT asafunctionofdensity ...... 125

5.14 µJT(LJ) and ∆µJT asafunctionofdensity...... 127 5.15 Internalenergyinsupercriticalargonandkrypton ...... 132

5.16 Pressureinargonandkrypton ...... 134 5.17 Heatcapacitiesinsupercriticalargon ...... 137 5.18 Speedofsoundinargonandkrypton ...... 139 5.19 Joule-Thomson coeﬃcientinargonandkrypton ...... 142

5.20 Adiabaticandisothermalcompressibilitiesinargon ...... 145

6.1 Pressureofwater ...... 154 6.2 Isochoricheatcapacityofwater ...... 156 6.3 Isobaricheatcapacityofwater ...... 157 6.4 Isothermal pressure coeﬃcientofwater ...... 159

6.5 Isothermalcompressibilityofwater ...... 160 6.6 Adiabaticcompressibilityofwater ...... 161 6.7 Joule-Thomson coeﬃcientofwater...... 162 6.8 Speedofsoundinwater...... 163

6.9 Thermal expansion coeﬃcientofwater...... 164 List of Tables

2.1 Summaryofpropertiesusedformodelparametrization ...... 15 2.2 ConstantsofMCYmodel...... 29

2.3 Parameters for diﬀerentwatermodels...... 37

3.1 Thermodynamicsquantitiesin terms of phase space function . . . 57 3.2 Valuesforsomeofmultinomialcombinations...... 65 3.3 Valuesforphasespacefunctionelements ...... 67 3.4 Fluctuationformulasinthermodynamics ...... 71

4.1 Summaryofreducedthermodynamicquantities ...... 76 4.2 TripleandcriticalpointsoftheLJfuid ...... 86

5.1 Absoluteaverage in Cv and Cp ...... 116 5.2 LocusoftheinversioncurveinLJfuid ...... 129

5.3 Constantsofargonandkrypton...... 130 5.4 Locusoftheinversioncurveinargonandkrypton ...... 141

5.5 Absolute average in βS and βT ...... 144

6.1 Experimentaltripleandcriticalpointsofwater ...... 150

6.2 Thermodynamicpropertiesofwatermodels ...... 151

xv xvi LIST OF TABLES

A.1 PressureinWCApotential...... 208

A.2 IsochoricheatcapacityinWCApotential...... 209 A.3 IsobaricheatcapacityinWCApotential...... 210 A.4 IsothermalcompressibilityinWCApotential...... 211 A.5 AdiabaticcompressibilityinWCApotential...... 212

A.6 SpeedofsoundinWCApotential...... 213 A.7 Joule-Thomson coefficientinWCApotential...... 214 A.8 Isothermal pressure coefficientWCApotential...... 215 A.9 Thermal expansion coefficientinWCApotential...... 216

A.10PotentialenergyinWCApotential...... 217

B.1 PressureinLJfuid...... 219 B.2 IsochoricheatcapacityinLJfuid...... 220 B.3 IsobaricheatcapacityinLJfuid...... 221 B.4 IsothermalcompressibilityinLJfuid...... 222

B.5 AdiabaticcompressibilityinLJfuid...... 223 B.6 SpeedofsoundinLJfuid...... 224 B.7 Joule-Thomson coeﬃcientinLJfuid...... 225 B.8 Isothermal pressure coeﬃcientinLJfuid...... 226

B.9 Thermal expansion coeﬃcientLJfuid...... 227 B.10PotentialenergyinLJfuid...... 228 B.11TotalenergyinLJfuid...... 229

C.1 ThermodynamicquantitiesinMCYnawater ...... 231 Chapter 1

Introduction

The complete thermodynamic information about an equilibrium system is contained in its fundamental equation of state. Once the fundamental equation of state is known, every thermodynamic state variable mentioned above can be calculated [1]. Current modeling of f uid properties focus on either the traditional f uctuation method [2–5] or equation of state models [6–8] to achieve accurate and reliable results over wide range of conditions.

An alternative method [9–13] of calculating the thermodynamic quantities as an average of the appropriate microscopic dynamical functions over the molecular dynamics ensemble is adopted. Pressure, heat capacities, compressibilities, isothermal pressure coefficient, Joule-Thomson coefficient, speed of sound at zero frequency and thermal expansion coefficient were calculated for Lennard- Jones f uid [14], Weeks-Chandler-Anderson potential [15] and MCYna water [16]. Using the appropriate Lennard-Jones constants, the above mentioned thermodynamic quantities of supercritical argon and krypton are calculated. The simulation results are compared with experimental and/or previous simulation results.

1 2 Introduction

Water and Lennard-Jones f uids [14] are the two most important f uids that we need to understand the thermodynamic properties such as pressure, heat capacities, compressibilities, Joule-Thomson coefficient, speed of sound, isothermal pressure coefficient and thermal expansion coefficient. Water is the only f uid medium at which chemical reaction in biological cells takes place, to understand these biological processes we need to have a very good understanding of the medium at which those process took place. Water is the only inorganic liquid which occurs naturally and also it is the only liquid which exists in all three states of matter.

Water has many interesting properties some of which are anomalous compared to other f uids. Most of the anomalous properties of water are crucial to the existence of life [17]. For example, ice f oats on water and density attains a maximum value when the temperature is at 277.134 K. It has high heat capacity, melting and boiling temperature. Unlike any other liquid isothermal compressibility of water decreases with temperature until it reaches its minima at 317 K [18]. Warmer water freezes faster than cold water, i. e. the Mpemba eﬀect [19]. The presence of these anomalous properties (there are more than sixty anomalies [20]), hydrogen bonding, polarization, angle bending, bond stretching and multibody interaction made modeling of water really diﬃcult. As reviewed in Chapter Two, there are many possible potentials for water. The focus of this work is the MCYna potential [16] because recent work [16,21,22] indicates that it could be used to improve the prediction of water potentials. The

MCYna potential is an ab-initio based MCY [23] water model with a non additive terms. Though highly studied, simple and ideal thermodynamic properties of Lennard- 3

Jones potential should be studied for the following reasons: it describes the property of noble gases, used in the simulation of most water models and give an easy way to compare new simulation methods with experiment. To compare the results of the Lennard-Jones simulation we have also simulated Weeks-Chandler- Anderson potential [15]. Studying Lennard-Jones f uids will serve the following purpose. First, to check the validity of the method that we will use for the calculation of thermodynamic quantities of water and more complex mixtures with a very simple f uid which we know very well. Second, to see how the dispersion term used in some water models behave by its own and see the accuracy of long range approximation. Third, to investigating its behaviour near the critical point which is an important tool to study the strength and limitations of a proposed technique in a more realistic situations [24]. We studied the WCA potential for the following three reasons. The f rst and foremost is to study the Lennard-Jones potential without the attractive term. This allows us to isolate the role of attractive interaction on the thermodynamic quantities of fuids. Second is to study the eﬀect of system size, cutoﬀ radius and simulation time on the thermodynamic quantities of an NVEPG ensemble. Last but not least, our work will be a reference in the future as there are no enough simulation results in the literature on thermodynamic state variables of a system of particles interacting with Weeks-Chandler-Anderson potential, data given in Appendix A. Generally speaking, studying Lennard-Jones and Weeks-Chandler- Anderson potentials is a proxy for real f uids.

There should be an accurate intermolecular potential to calculate all the thermodynamic variables from molecular simulations but few molecules have a real intermolecular potential such as helium [25] and argon [26]. The potential used 4 Introduction in the simulation of water and the review of diﬀerent water models are given in

Chapter Two. The description of molecular dynamic ensemble (NVEPG) [27] and the calculation of the thermodynamics quantities in this ensemble are given in Chapter Three. The simulation detail for all systems and the Lennard-Jones and Weeks-Chandler-Anderson potentials are given in Chapter Four. The thermodynamic quantities below and above the critical point in Lennard-Jones f uid, Weeks-Chandler-Anderson potential, super critical argon and krypton are given in Chapter Five. We have tested the advantage in increasing the system size to improve the statistics as claimed by Ahmed and Sadus [28] and also we have tested the advantage of increased simulation time and cutoﬀ radius to improve the simulation results near the critical point. The simulation and experimental results of water are given in Chapter Six. Finally, conclusion and recommendations are given in Chapter Seven. Thermodynamic quantities from WCA and LJ potentials in a system consisting of 2000 particles are given in Appendix A and B respectively. The simulation results of MCYna water in a system consisting of 500 water molecules at a density of 0.998 gm/cm3 is given in Appendix C. Chapter 2

Water Potentials

2.1 Introduction

In this chapter we will examine diﬀerent models of liquid water and review the progress in the development of water potentials from the Ben-Naim-Stillinger model [29] to the most recent six site model [30]. We will also highlight the strength and weakness of various water models, discuss their suitability for thermodynamic properties, and the role of both polarizable and f exible water models. In particular, the potential of MCYna [16,21] will be examined in detail and why it might be better than other alternatives for the calculation of thermodynamics properties will be discussed. Finally we will point out the weakness of water models and discuss what to improve so as to make the models predict more characteristics in diﬀerent phases accurately.

5 6 Water Potentials

2.2 Why all these models for water?

The structure of water is one of the 125 problems selected by Science magazine to be the most important question confronting researchers now and in the future [31]. A recent review by Guillot [32] found 46 distinct water models. One may ask why all these models? A comprehensive molecular theory for water is needed for the following reasons. First, it is a major constituent of our planet’s surface which we need to understand [33]. Second, it shows anomalies both in pure form and as a solvent [20,33–36]. Third, water exhibits one of the most complex phase diagram (plot of pressure versus temperature ), having f fteen diﬀerent solid structures [36].

Fourth, it is the only f uid medium capable of supporting biochemical processes [29]. Fifth, it has industrial application, such as supercritical water oxidation [37]. Six, it has an active role in molecular biology not only as a scaﬀold [38]. The availability of structural data of water from neutron scattering at ambient and supercritical conditions [39–42] contributed to the search of a better water model. The schematic representation of water is given in Figure 2.1.

Figure 2.1: Simple representation of water

Although water is one of the most studied molecules on earth [17,20,32,43,44] Why all these models for water? 7 our understanding of its thermodynamic and anomalous properties are inadequate.

Considerable eﬀort has gone into trying to understand the ways in which water is involved in processes like protein folding and stability. In contrast, there has been much less focus on trying to identify the specif c molecular characteristics of water that ‘nature’ exploits, and that evolution has capitalized upon [18].

The key to understanding the normal and anomalous properties of water in its different phases is to have a model which ref ects its true nature, i.e., a model which is f exible, polarizable, and considers multibody effects. If we try to include all the above mentioned characteristics of a ‘real’ water in a single model it will be computationally expensive. On the other hand, if we exclude some or most of those characteristics could the model be able to represent ‘real’ water? There is always a difference between the ‘real’ water and the simulated water, what matters is how different the two are and the purpose of the model at which it is built for. The main reason for having all these different models is the inability of a single model to describe its properties, which are results of either its high degree of hydrogen bonding and strong intermolecular interaction or its tetrahedral shape. The high degree of hydrogen bonding (shown in Figure 2.3) and strong intermolecular interaction cause the large heat capacity, the low solubility of inert solute and hydrophobic interaction. Perhaps its tetrahedral shape (shown in Fig- ure 2.2) is the cause of all the anomalies such as negative temperature dependence of the volume, the large negative entropy of solvation of inert solute, temperature dependence of density near freezing temperature, temperature dependence of isothermal compressibility, high boiling temperature and the large number of phases of ice [18,20,35,45]. Molecules which have four electron groups around their central atom, such as 8 Water Potentials

ammonia (NH3) and water have a tetrahedral shape with a bond angle of about

109.5◦ (shown in the appendix A of Ben-Naim [45]). The ammonia molecule has three bond groups and one lone pair, and the water molecules have two bond groups and two lone pairs. Tetrahedral conf guration is an arrangement with two positive and two negative charges [46]. In a tetrahedral conf guration the posi- tively charged end of the molecule is more orientationally constrained than in the negative lone-pair region, allowing both trigonal and tetrahedral local structures and enabling hydrogen bonding. Finney [18] described the tetrahedral geometry of the local order of water molecule to be the central point in understanding the water anomalies.

Figure 2.2: Schematic representation of the tetrahedral shape of water (source [47])

Diﬀerent researchers try to understand and explain the normal and anomalous properties of water from a diﬀerent perspective, which will lead to having a wider variety of model potentials and charge distributions. To mention some, Cho et al. [48] reported that understanding the density anomaly is the key to understanding the remaining anomalies of water. On the other hand, Stillinger [34],

Finney [18], Ben-Naim [45], Eisenberg and Kauzmann [36] and others point to Water models, criteria 9 the tehtraherality of water and hydrogen bond as the cause of those anomalies.

At lower temperature the eﬀect of hydrogen bonding becomes dominant [49]. To study the eﬀect caused by hydrogen bond in water, Poole et al. [50] proposed an extension of the van der Waals equation to include the network of hydrogen bond.

2.3 Water models, criteria

Molecular Dynamics (MD) and Monte Carlo (MC) simulations of aqueous solutions with explicit representation of the water molecule depend critically on the availability of water models that provide an accurate representation of the liquid. This can be rapidly evaluated, and are compatible with the force feld for the solutes [51]. Water is f exible and polarizable, however most of the models in the literature are rigid and non polarizable [20,32,52,53]. The main reasons for the wide spread use of rigid and non polarizable models are simplicity [54] and absence of experimental data for parametrization, especially on the many-body structure [55]. The absence of a polarization term and non bonded interactions, such as bond stretching and angle bending, are the main reasons limiting our ability to reproduce the experimental results of thermodynamic quantities and anomalous behaviours of all diﬀerent physical states of water and so large number of water models valid in a region which they are parametrized. The following are the criteria that most researchers use to choose the appropriate model for water. The model should be computationally economical, simple and physically justif able, estimate ‘most’ of the experimental properties of the real water with acceptable accuracy and transferable over a wide range of ther- 10 Water Potentials modynamic conditions and environmental particularities [56–58]. In choosing a model for simulation there are two competing issues, f rst the model should be computationally viable so we make the model simple, rigid, non polarizable and avoid many body interaction. The second issue is, if we do so, can we obtain a realistic representation of water from the model. In our review of diﬀerent water models we understand that there should be a compromise between these two competing issues, however with the ever increasing availability of computational facilities more properties of water should be included in the parametrization of the potential.

There are f ve main differences between various water models [44, 59]. First, bond f exibility, naturally water is f exible but for practical and physical reasons most models are rigid. Second, polarization, as a result of an induced and/or external electric f eld there is always polarization in water but depending on the quantities to be predicted and for a computational reason most models do not include it. Third, values used for parametrization, different models use different quantities for parametrization depending on the availability of experimental data and ones intention in using the model. Fourth, the interaction between molecules, depending on the intermolecular and/or intramolecular interaction to be considered in the system different water models will have a unique interaction potential. Fifth, the charge distribution and position of Lennard-Jones interaction site, different water models put the negative charge at different positions with respect to the position of oxygen atom. As a result of which we will have different interaction potentials, bond length and bond angle for different water models. Testing of water models, from the original Ben-Naim and Stillinger (BNS) model [29] to the most recent six-site model [30] indicate that they will fail in at Water models, criteria 11 least two or more of the criteria set. In the process of improving the prediction of models many modif cation have been made. Generally speaking, each of the models are in a very good agreement with experiment at least for the values at which they are parametrized but there is no one single water model capable of describing its normal and anomalous properties in different phases [43, 45]. We will discuss the above mentioned differences of water models in detail.

2.3.1 The interaction between molecules

The interaction between molecules def ne the properties of a molecular system, which means it is important that the description of the interaction captures the correct and suﬃcient physical features for the application of interest [21, 60]. Water has a slightly negative end and a slightly positive end. It can interact with itself and form a highly organized inter-molecular network. The positive hydrogen end of one molecule interact favorably with the negative lone pair of another water molecule as shown in Figure 2.3. The result is hydrogen bonding, via weak electrostatic attraction. As each one of the water molecules can form four hydrogen bonds, an elaborate network of molecules is formed. An ideal interaction potential should be derived from ab initio quantum calculations [62] which predict reliably all known experimental data for all phases of water and intrinsically duplicate the instantaneous charge densities of water [63]. Such potentials do not exist for the following two reasons: ab initio methods are not accurate enough (because of limited number of basis sets and the approximation on the theory), and the interaction potentials are not pairwise additive.

Any model expected to predict properties of liquid water correctly must be either 12 Water Potentials

Figure 2.3: Hydrogen bonding in water showing the formation of hydrogen bonds (source [61]) non-pairadditive or it must use an effective pair potential that includes polarizability [64,65]. If one is committed to the use of an additive total interaction, the contributing pair function must be viewed as an “effective pair interaction”, which deviates signif cantly from pure pair potential [66–68]. The main reason that pure potentials cannot reproduce condensed-state properties for polar molecules is that such potentials neglect the effect of polarizability beyond the level of pair interactions. In water and in other polar liquids-there is a considerable average polarization, leading to a cooperative strengthening of intermolecular bonding. Thus, effective pair potentials invariably exhibit large dipole moments than the isolated molecules have and produce second virial coefficients larger than the experimental ones [69]. The effective pair potentials using f xed point charges which, leads to an enhanced dipole moment (greater than 1.85D) cannot be further improved [70,71]. The net effect of three-molecule, four-molecule, nonadditivity includes strengthening and shortening of the hydro- Water models, criteria 13 gen bonds, and perhaps slightly enhanced tetrahedrality in structure [66].

The choice of the molecular model of water sets the microscopical length scale on which the interactions need to be modeled. At this level of detail, the atoms and molecules still obey classical mechanics, and atom interactions can be described by potential functions [43]. For most problems it is not necessary to describe the system in terms of wave functions, although the technique for wave function propagation have been successfully applied in simulating liquid water by Laaso- nen et al. [72] and Matsuoka et al. [23] to mention some.

2.3.2 Bond f exibility

A review of water models revealed that most models are rigid despite the fact that water molecule is f exible. Making water rigid neglects the intramolecular degrees of freedom which in turn will affect the validity of the results from the simulation. To get an acceptable result we need to have a f exible water model but for a technical and physical reasons most of the water models are rigid. If we include the internal vibrations, which has the reorientation time of 2 ps at ambient conditions [73], in our model we need to use a much smaller time step when integrating the equations of motion this will take a very long time for a picosecond simulation, and also one can argue that internal vibrations are quantum mechanical in nature and cannot strictly be incorporated into a classical molecular dynamics simulation [53]. However, some researchers have developed a f exible model with promising results which could urge future models to consider including the f exibility of water molecules in the simulation. For example, Wu et al. [74] found that to improve 14 Water Potentials the prediction of diffusion coefficient and dielectric constant one should use f exible water model. Recently Raabe and Sadus [75–77] using the DL POLY [78] molecular simulation package reported that introducing bond f exibility has an observable effect on the prediction of the vapor-liquid coexistence curve, dielectric constant and pressure-temperature-density behaviour. Lie and Clementi [79] predicted the radial distribution function successfully and calculated thermodynamic variables using f exible Matsuoka-Clementi-Yoshimine-Lie (MCYL) model.

2.3.3 Values used for parametrization

The validity of molecular simulation depends only on the availability of reliable data [80]. Generally each model is developed to ft well with one or more particular physical structure or parameter in order to predict other values of interest. Therefore, it comes as no surprise when a model developed to f t certain parameters give good compliance with these same parameters and the thermophysical parameters close to those used in f tting the models.

Different models use different quantities for parametrization which can be found either from experiments [39–42,81–84] or quantum chemistry [85,86]. For some physical parameters such as the dipole moment, there is no agreement over which value to use. Most of the models use density and pressure of liquid water, density of ice, vaporization enthalpy and temperature of maximum density for parametrization. An extended list of parameters used in the parametrization of different water models is given in Table 2.1. Also the values of the parameters used in each of the models are given in Table 2.3. Water models, criteria 15 [87] [87] [57] [23] [79] [21] [29] [66] [54] [30] [64] [69] [74] [89] [90] [88] Ref. erent water models. ff ient ient c c ffi ffi potential potential potential potential b initio b initio b initio b initio a a a a u ff ummary of properties used for parametrization of di S exagonal ice and Vaporization correction energy aporization correction energy and density of water t ambient conditions Not applicable - Water vapor and second virial coe Water vapor and second virial coe Vaporization energy, density maximum of liquid water anda density of water Melting point of ice and densities of ice and water near the melting point Ground vibrational state frequency Not applicable - Internal energy and pressure of water Vaporization enthalpy and liquid density of waterTemperature at of ambient maximum condition density, phase diagram, meltingh temperature of Not applicable - Not applicable - Properties used for parametrization Energy and pressure of liquid water Vaporization enthalpy at room temperature, pressure ofv liquid water, Vaporization enthalpy and liquid density of waterBulk at di ambient condition sionanddielectricconstant,equilibriumbondlengthandangle Table 2.1: 2 / F E F / / / MCYna BNS ST2 TIP5P NveD SPC d PPC GCPM TIP4P TIP4P MCY 005 MCYL Model SPC SPC TIP3P SPC w 16 Water Potentials

The parameters determine the validity of the model when used in diﬀerent phases and thermodynamic conditions and also its ability to reproduce a range of experimental data [91]. In ab initio based models, such as Matsuoka-Clementi- Yoshimine (MCY) [23], there is no parametrization. Instead, the ab initio approach uses values from quantum mechanical calculations obtained by solving either the Schr¨odinger equation or Hartree-Fock approximation. However, the accuracy of force f eld calculated in ab initio is limited by unavoidable approxi- mations in the level of theory (i.e., truncation), neglect of intramolecular degree of freedom (if the model is rigid) and incompleteness of basis sets [35,92].

2.3.4 Polarization

Polarization is def ned as an induced dipole moment per unit f eld strength, when the molecule is placed in a uniform electric f eld [93]. The polarity of a molecule is a measure of the symmetry in the distribution of the charged particles. Molec- ular polarization may be electronic (caused by the redistribution of electrons), geometric (caused by changes in the bond length and angles) and/or orientational (caused by the rotation of the whole molecule) [94,95]. The total polarizability of a molecule is, as are all molecular properties, expressible as a sum of atomic contributions [96]. Polarizable potentials approximate the eﬀect of multi-body interactions that arise because the induced dipole of each molecule generates an electric f eld that aﬀect all other molecules . The total number of the positive and negative charges in water are equal so the water molecule is electrically neutral. However, the distribution of charges is not spherically symmetric, therefore water molecule as a whole is polar [45]. Water models, criteria 17

Most properties of water predicted by simple water models are in good agreement with the experimental values [54,87,97,98] at least for the values at which the model is parametrized. Why do we need to consider a very expensive polarizable models, such as Matsuoka-Clementi-Yoshimine with non-additive terms (MCYna) [16], if the simple and computationally less expensive models are work- ing well? The SPC/E model [69] using the polarization correction gives improved values for diffusion coefficient and radial distribution (O-O), which emphasizes the need to have a polarization term in the potential to predict those values that we cannot predict with the simple models. Li et al. [16] found that polarization is the main nonadditive inf uence resulting in good predictions of radial distribution function (RDF), dielectric constants and dipole moments. Svishchev et al. [99] asserted that in order to calculate the static and dynamic properties of liquid water from supercooled to near-critical conditions there is a need to consider polarization in the model. Polarizable potentials also approximate the effect of multibody interaction by using f uctuating charges or f exible geometries [16]. From quantum chemical studies of water it is found out that non-additive contribution are of great importance. For example, non polarizable models can not describe simultaneously the vapor phase and condensed phase with the same degree of accuracy [44]. About 10% of the total intermolecular interaction energy in a water trimer may arise from three-body interaction [100,101]. Dyer et al. [102] reported that the use of an explicitly polarizable solute improves agreement between experiment and simulation of the solubility of simple nonpolar solutes in water. Though it is computationally expensive, attempts have been made to incorporate polarization in the water model [63,90,103–106]. Sprik and Klein [107] considered polarizability as an explicit degree of freedom with an artif cial inertial 18 Water Potentials mass. Chialvo and Cummings [90,108] showed that the polarization energy accounts for 40 to 57% of the total conf guration internal energy of water. Recently Li et al. [16] reported that the energy contribution from the polarizable term was approximately 30% of the overall energy. In most models of water the polarization effect is either ignored or if considered it will be in an effective dipole moment [109,110]. The effective dipole approach, with suitably chosen effective moments, may yield a good approximation to the correct equilibrium properties. However it cannot be expected to a priori account equally well for the dynamics and the dielectric properties of a polarizable dipole system [111]. Stern et al. [105] described the three different technique of making an empirical model to be polarizable. The f rst method is the f uctuating charge model [112,113]. The second technique is the Thole type dipole polarizability model [114,115] and the third one is charge response kernel model [116–118]. Different authors [22,101,111,119] described how the polarization term in a molecular dynamics simulation can be calculated.

2.3.5 The charge distribution

Almost all water models put one positive charge on the hydrogen atom but diﬀer in the location of the negative charge(s) and Lennard-Jones interaction site. The cost of evaluating a pair potential is proportional to the square of the number of interaction sites each molecule has and so a model with more sites will be computationally expensive [43]. In order to mimic the tetrahedral shape of water and describe the dipole and quadrapole of water molecule, water models with more than three interaction sites have been proposed. Water models, criteria 19

Based on the charge distribution, upon which some of the water models are named after, there are diﬀerent water models. There are three, four, f ve and six site models where each site is occupied by either the charge of hydrogen, oxygen atom and/or the Lennard-Jones interaction site. We will describe the potential, values used for parametrization and brief y examine failure and success of selected models from each group.

2.3.5.1 Three-site models

This group of water models consists of three charge sites, the smallest possible number of charges for the water molecule. Two positive charges are situated at the hydrogen atoms and one negative charge situated at the oxygen atom as shown in Figure 2.4.a and Figure 2.4.b. The charges fall on the center of mass of each atom so that there will not be any reconstruction of charge centers and the redistribution of force and torque. Generally, the second neighbor peak in the O-O radial distribution function which is sensitive to the computed density will disappear for a three site models with an improved density prediction [87]. Since these models are computationally less expensive they are the one used for biological systems simulation, where a large number of solvatingwater molecules are often needed at which the addition of a single interaction site into the biosimulation system can lead to a drastic increase in simulation time which could reach as high as 50% [74,120].

2.3.5.2 Four-site models

The basic philosophyof having a four site water model is a need to have a structure which has a tetrahedral coordination. The four site water model consists of four 20 Water Potentials sites out of which there are three charge sites, two positive charges situated at the hydrogen atom and a negative charge situated at the dummy site (called the M site), and one chargeless oxygen site. The M site carrying the negative charge is not located at the oxygen atom but on H-O-H bisector at a distance of ‘l2’ from the oxygen atom as shown in Figure 2.4.c and given in Table 2.3. Bernal and Fowler

[121] proposed the four site geometry water model as early as 1933. More recently Jorgensen et al. [87] developed transferable intermolecular potential (TIP) four point (TIP4P) water model. The diﬃculty in using four site models, apart from the greater computational time compared with three site models is the existence of the massless M site. The reason for better density results of TIP4P is associated with the bent optimal water dimer structure, this lead to the subsequent addition of sites in the water model which promotes a more tetrahedral network in the liquid, which is ref ected in the second peak in the TIP4P oxygen-oxygen radial distribution function [122].

The TIP4P model predicts the temperature of maximum density better than the three site models, near -13 ◦C [122,123].

2.3.5.3 Five-site models

Due to their signif cance in separating the position of the negative charge from the mass center of oxygen, we will brief y consider the Ben-Naim and Stillinger model (BNS) [29] and the ST2 model developed by Stillinger and Rahman [66]. Both are the predecessors of the fve site model. BNS has many simplifying ap- proximations, however it provides the basis for subsequent water models. It is a rigid model with pair-wise additive potential and three body interaction. In this model instead of having point dipoles, there are four point charges of magnitude Water models, criteria 21

ηe, where the value of η is chosen to be 0.17. Two of the +ηe may be identif ed ± as water molecule protons partly shielded by electron cloud. The remaining two charges, ηe, represent crudely the unshielded pairs of valence-shell electron in − the molecule. The four charges are placed at the vertices of a regular tetrahedron whose centre is presumed coincident with the oxygen nucleus. The distance from this center to each of the four charges has been chosen to be 1.0 Å. ST2 was developed as a result of BNS unsatisfactory results in the liquid state of water [109]. ST2 is the same as BNS except that some of the parameters are changed (for example the value of charge and Lennard-Jones constants) and the distance between the negative charges and the oxygen nucleus is reduced to 0.8 Å. Both BNS and ST2 potential have a Lennard-Jones central potential acting between the oxygens. ST2 is more tetrahedral than TIP4P [122] and has a stiﬀer potential than the three and four site models, and has a more bound optimal dimer. The use of a cubic scaling function to dampen the short-range electrostatic interaction yields an overly structured oxygen-oxygen RDF [87] and the density maximum is at 27 ◦C [66]. The transferable intermolecular potential f ve point (TIP5P) potential is the modif cation of ST2. Mahoney and Jorgensen [54] developed this model with the intention of eliminating the scaling function, improved density results including a correct temperature of maximum density with out sacrif cing performance for other structural or thermodynamic properties in comparison to the TIP4P model. The two negative charges are located symmetrically along the lone-pair (L) direction with an intervening angle of 109.47◦ at a distance of 0.7 Å from the oxygen mass centre as shown in the Figure 2.4.d. A positive charge is placed on each of the hydrogen atom forming H-O-H angle of 104.52◦. There is no charge on 22 Water Potentials oxygen, and the interaction between diﬀerent oxygen atoms is obtained using the

Lennard-Jones potential. TIP5P water potential successfully predicted energy, density and temperature of maximum density [124] at the expense of a very long simulation run (billions of steps) and by forcing tetrahedral arrangement for hydrogen-bonded pairs to be more attractive than for real water, which leads to a very high f rst peak in H-H radial distribution function. The main problems of this model are that the isobaric heat capacity is too high and the density increases too rapidly with increasing pressure or decreasing temperature above temperature of maximum density. The model also predicted a higher dielectric constant.

2.3.5.4 Six-site models

Nada and van der Eerden [30] developed a six-site model (NvdE) for simulating ice and water near the melting point. A positivecharge is placed on each hydrogen (H) site and a negative charge on each lone-pair (L) site, similar to the TIP5P model, as shown in the Figure 2.4.d. A negative charge is also placed on a site M, which is located on the H-O-H bisector, as is the case for the TIP4P model shown in Figure 2.4.c. A point of diﬀerence from either the TIP4P or TIP5P models is that the Lennard-Jones interaction acts not only on the oxygen (O) site but also on the hydrogen (H) site. The melting point of ice, and densities of ice and water were used for parametrization near the melting point. This model predicted melting temperature in the range of 16 ◦C and 21 ◦C [125], but later the melting temperature is found to be at 16 ◦C [126]. The six-site water model is computationally very expensive and is only capable of reproducing the structure of water and ice in a temperature range very close to the melting temperature, Water models, criteria 23 which is not our region of interest.

All the above discussed water models are tetrahedral. Kiss and Baranyai [55] found out that simple models such as SPC/E and TIP3P exhibit too much of a planar shape which in turn will hinder their prediction of water properties. For our study we will use a four site polarizable, rigid and ab inito based water model, developed by Matsuoka et al. [23] and modif ed by Li et al. (MCYna) [16,21].

Figure 2.4: Schematic representation of water models: ‘a’ and ‘b’ represent three site, ‘c’ represent four site and ‘d’ represent f ve site. θ is the bond angle, φ is the angle between the negative charge and hydrogen atom, l1 is the bond length and l2 is the distance between lone-pair and oxygen atom center of mass (source [20]). 24 Water Potentials

2.4 Review of water models

We will examine different water models which will represent the majority of the models developed so far. Here in our review of different water models we will focus on ambient conditions and we don’t consider quantum effects. We have divided the water models under consideration into two categories as rigid and f exible. Properties used for the parametrization of each of the models is given in Table 2.1.

2.4.1 Rigid water models

Molecular dynamics simulations will be computationally economical if the model ignores the internal degrees of freedom. We will investigate the structure, success and failure of some of water models with a rigid structure such as TIP3P, SPC, SPC/E, TIP4P/2005, PPC and MCY.

2.4.1.1 TIP3P and SPC

Since both TIP3P and SPC have a similar geometry and number of sites we would like to see both models together. A transferable intermolecular potential three point (TIP3P) model was proposed by Jorgensen et al. [87] to improve the energy and density for liquid water. In this model the negative charge is located on the oxygen atom and the positive charge on the hydrogen atom with a bond angle of 104.52◦. The parameters of the model (the Lennard-Jones constants and the charge on the hydrogen atom) were obtained by reproducing the vaporization enthalpy and liquid density of water at ambient conditions. This model is commonly used in biological molecules for its computational eﬃciency [44]. Review of water models 25

With the same geometry as TIP3P model and with parameters chosen to reproduce both the energy and pressure of liquid water at ambient conditions Berendsen et al. [64] proposed a simple point charge water model (SPC). The SPC model is a simple yet reliable model for intermolecular potential of water. The model averages many-body interaction (included in the effective potential) and as in other three site models the Lennard-Jones site is on the oxygen atom. Because of its small number of sites this model is computationally less expensive and easy to incorporate in protein-water potential. This model has the capability of predicting large number of properties in an acceptable range, such as dipole moment [69] but fails to predict the density, radial distribution function (but better than the more expensive f ve site ST2 model) [82] and diffusion coefficient [84]. The model fails to predict the correct energy, which leads to the reparametrization and inclusion of polarization energy and increase of the charge.

2.4.1.2 SPC/E

The extended simple point charge (SPC/E) model, is possibly the most popular water model. It is similar to SPC and TIP3P in both the charge distributionand the values used for its parametrization. The model has a polarization term included because there is an induced permanent dipole moment. The correction term is

[44, 69] 2 Epol (µ µgas) = − (2.1) N 2αp where µ is the dipole moment of the molecule, µgas is the dipole moment of the molecule in the gas phase and αp is the polariazability of water molecule. The correction increases the atomic charges on hydrogen and oxygen site. i.e., the 26 Water Potentials values used for the charges of hydrogen and oxygen is slightly higher than the one used in SPC and TIP3P. SPC/E reproduces the vaporization enthalpy of real water when a polarization energy correction is included [69]. This model is capable of predicting challenging features such as the critical behaviour, pair-correlation and dielectric constant reasonably well [32]. Recently Kiss and Baranyai [55], and Vega et al. [127] reported that the above predictions are caused by fortuitous cancellation error rather than inherent improvement at the accuracy of intermolecular interaction. The SPC/E water model predicted a very high diﬀusion coeﬃcient and specif c heat capacity [128]. SPC/E failed to predict the phase diagram since its negative charge is on the oxygen. In order to improve the phase diagram the negative charge should be shifted towards the hydrogen which prompts to the choice of the four site water model than the three site [57].

SPC and SPC/E models have tetrahedral angle of 109.47◦, in order to have a tetrahedral arrangement in the water molecules, instead of the experimental bond angle of 104.5◦. Neither SPC nor SPC/E showed temperature of maximum density in the range between 15 ◦C and 25 ◦C as did most water models [129]. Baez and

Clancy [128] found the temperature of maximum density in SPC/E at -38 ◦C and also Jorgensen and Madura [122] indicated that the temperature of maximum density for SPC model to be between -37.5 ◦C and -50 ◦C, far below the experimental value of 4 ◦C.

2.4.1.3 TIP4P/2005

TIP4P/2005 is arguably the best non polarizable and rigid model which is capable of describing the different properties of water [44]. It is similar to TIP4P, in Review of water models 27 charge distribution, reparametrized by Abascal and Vega [57] with an intention of producing a general and computationally simple water model. Properties used for the parametrization of TIP4P/2005 model are temperature of maximum density, phase diagram, melting temperature of hexagonal ice and polarization correction. The model predicts a large number of properties in different phases such as phase diagram [127], density and temperature of maximum density at 5 ◦C as a result of which the model will give a low melting temperature [130]. The model is also able to provide a good description of the vapor-liquid equilibria [131] and the surface tension [132]. However, the model fails to predict the dielectric constant and also the model misses one of the natural characteristic of water, namely polarization. The success of this model could be attributed to the heavy parametrization and it will not be surprising if it become successful in predicting a large number of anomalous and thermodynamic properties (such as compressibility, coefficient of thermal expansion) [44].

2.4.1.4 Polarizable point charge (PPC)

The polarizable point-charge (PPC) model is a three site ab inito model, with out any parametrization, of water molecule in an applied electric f eld [89]. Kusalik and Svishchev [89] developed this model in order to overcome the limitations of simple water models caused by the absence of polarization. The model is intended to work in diﬀerent phases of water, from supercooled to the critical point of water. This model predicts the experimental value of temperature of maximum density at 4 ◦C, the oxygen-oxygen radial distribution function [99] and the critical point [133]. The model is capable of predicting the most challenging properties of water. The success of this model is based on the presence of polarization and 28 Water Potentials absence of parametrization.

2.4.1.5 Gaussian charge polarizable model (GCPM)

The Gaussian charge polarizable model (GCPM) [90] is a modif cation of the SPC [64] potential based self-consistent point-dipole polarizability model (SCPDP) [108] developed by Chialvo and Cummings [90]. To improve the short-range polarization behaviour of the SCPDP model the Gaussian distribution (smeared charge) is used to represent the partial charge on the water molecule centered at the sites described by SPC [90,134]. There is a dipole polarizability and the Buck- ingham potential (Eq. (4.21)) is used in place of Lennard-Jones oxygen-oxygen interaction potential.

This model describes accurately the pressure and conf gurational energy of water at ambient conditions [90]. However, the model decreased the values of dipole moment and polarization energy. Paricaud et al. [134] conf rmed that with some more parametrization this model is capable of predicting dielectric, structural, vapor-liquid equilibria and transport properties of water over the entire f uid region.

2.4.1.6 Matsuoka-Clementi-Yoshimine (MCY)

MCY is a four site water model developed by Matsuoka, Clementi and Yoshimine [23] with the intention of obtaining a quantitative accurate description of pair potential function for two water molecules. The model is based on conf guration- interaction (CI) method of ab initio calculation of potential surface for water dimer. There is no parametrization in this model. The schematic representation of MCY model charge distribution is shown in Figure 2.5. Review of water models 29

Figure 2.5: Schematic representation of MCY water model (source [23]). Points ‘M’, ‘H’ and ‘O’ represent the position of the dummy, hydrogen and oxygen sites respectively.

The potential used in the MCY water dimer interaction is

2 2 1 1 1 1 4q u2 = q + + + + r13 r14 r23 r24 ! r78

2 1 1 1 1 2q + + + + a1exp( b1r56) − r18 r28 r37 r47 ! − (2.2) + a [exp( b r ) + exp( b r ) + exp( b r ) + exp( b r )] 2 − 2 13 − 2 14 − 2 23 − 2 24 + a [exp( b r ) + exp( b r ) + exp( b r ) + exp( b r )] 3 − 3 16 − 3 26 − 3 35 − 3 45 a [exp( b r ) + exp( b r ) + exp( b r ) + exp( b r )] − 4 − 4 16 − 4 26 − 4 35 − 4 45 where q is a charge on hydrogen atom, ri j is the distance between two sites at i and j, and ai and bi to be determined from the ab inito calculation. The values of ai and bi used in MCY model are given in Table 2.2.

Table 2.2: Values used in MCY model, all in a.u unless specif ed (source [23]). The α and β (dimensionless) values are from [21, 36].

a1 a2 a3 a4 b1 b2 1734.196 1.061887 2.319395 0.436006 2.726696 1.460975

2 3 b3 b4 l2 q α (Å ) β 1.567367 1.181792 0.505783 0.514783 1.44 0.557503 30 Water Potentials

MCY predicted a reasonable f rst peak in radial distribution function but the second neighbor peak is too short (which resembles a high temperature behaviour), energy of the liquid is small, high dielectric constant and the pressure is too high. Lie and Clementi [79] argued that MCY potential is too repulsive, which only al- low a computation of a small number of conf gurations that will be used for f tting and conclude that this compact conf gurations is the main reason for high pressure prediction. Different modif cations have been proposed to improve the prediction of water properties using MCY model. Berendsen et al. [64] and Impey et al. [135] suggested to scale the energy by a factor of 1.14 to get a better result. Wojcik and Clementi [136] found that the many-body interactions make up as much as 15% of the internal energy and can affect the liquid structure. This is a very big amount of energy to be ignored, future models should incorporate multibody interaction. In line with this Li et al. [16], Niesar et al. [137] and Barnes et al. [138] showed that introducing a polarization term will improve the model’s prediction. The NCC water model by Niesar et al. [137] is a modif cation of MCY model with an explicit incorporation of the many-body effects due to polarization. NCC is capable of accurately predicting a wide spectrum of static and dynamic properties of liquid water. NCC failed to predict the f rst neighborhood peak in the O-O RDF. Niesar et al. [137] recommended the use of an accurate dispersion energy term and an extended basis set to improve the failed properties. Li et al. [16] claimed that the added multibody interaction in NCC could lead to over counting. Later we will discuss the most recent modifcation of MCY model by Li et al. [16]. Review of water models 31

2.4.2 Flexible water models

Naturally water is a f exible but for a computational ease and considering molecular vibration to be a quantum effect most of the water models are rigid. The diffusion constant (or relaxation time) and the static dielectric constant are ex- tremely sensitive to the equilibrium bond length and equilibrium bond angle respectively [74,120]. In most of water models these properties are poorly predicted. The self diffusion constant and static dielectric constant are very important since they are directly related to the solvent dynamics and solvent-mediated electrostatic interactions. Lots of works are being done to justify the need of having a fexible water model. First we will see some of the justifcations for not including fexible in water models, apart from computational effectiveness. Waheed and Edholm [139] claimed that the atomic vibrations to be of quantum mechanical which results in about 65% overestimation of heat capacity at ordinary temperature when used in a classical model. Lie and Clementi [79] claimed that most f exible water models do not properly describe the dependence of the charge in dipole moment on molecular geometry. Therefore, Smith and Haymet [140] and Zhu et al. [103] suggested that geometric f exibility should be included only in polarizable models. However, Raabe and Sadus [75] found out that introducing internal vibration (intramolecular degree of freedom) into SPC has an observable effect on vapor-liquid coexistence curve. More recently, Raabe and Sadus [77] found that introducing bond f exibility signif cantly improves the prediction of both dielectric constants and pressure–temperature–density behaviour. In a model which will serve as a solvent and used in biological simulation 32 Water Potentials more properties such as density, compressibility and viscosity should be predicted including those mentioned above. As a result of their computational effectiveness three site models are in constant use for biological system simulation, most of the f exible water models are also three site. There is evidence which urge us to have a f exible water model in order to predict properties of water which will be affected by the intramolecular vibration [35].

2.4.2.1 SPC/Fw

Wu et al. [74] developed a f exible SPC model (SPC/Fw) with an intention of having a model with an improved dynamic and electrostatic properties of water without damaging other properties. It is a f exible SPC model derived by optimizing bulk diffusion and dielectric constants to the experimental value via the equilibrium bond length and angle. The bond angle and bond length (is responsible for the improvement of self diffusion constant) are slightly higher than the values used in SPC. This model has three internal degrees of freedom: two from bond stretching (O-H) and one from angle bending (H-O-H angle). The intramolecular interaction is represented by simple harmonic potential. This model predicted the diffusion constant, dielectric constant and predicted a set of thermodynamic properties of liquid water. The model is capable of predicting saturation densities and critical point [75]. More recently Raabe and Sadus [76] showed that this model to be capable of predicting the diffusion coef- f cient at ambient and supercritical temperatures over a wide range of pressure. Review of water models 33

2.4.2.2 SPC/Fd

SPC/Fd is a fexible water model developed by Dang and Pettitt [88] based on Urey-Bradley [141] modif cation of the intramolecular and the Jorgensen [142] and Berendsen et al. [64] intermolecular interaction. In SPC/Fd potential model there is additional interaction term for H-H in the harmonic potential (from the Urey-Bradley modif cation) compared with the intramolecular potential developed by Wu et al. [74]. Dang and Pettitt [88] developed this model with an intention of having a classical, simple and f exible water model which will predict most of the properties of water and serve as a solvent. The experimental ground vibrational state frequencies for the water monomer using the classical normal modes [36] is used to f t the force constants. The gas-phase monomer transition frequency is used and so the three force constants (for O-H, H-H and H-O-H) were chosen to reproduce the three observed frequencies for the water monomer. The model does not show second neighbor peak in O-O which is a result of structural insensitivity to intramolecular motion of water by the original TIP3P model [88].

2.4.2.3 Matsuoka-Clementi-Yoshimine-Lie (MCYL)

Lie and Clementi [79] developed the MCYL water model which is an analytical continuation of the MCY [23] conf guration interaction potential for rigid water- water interaction. MCYL include the intramolecular vibration with an intention of calculating the static and dynamic properties of liquid water. The potential is modif ed in such a way that, if there is no deformation of the two interacting water molecules the potential will be the original MCY potential. The MCY potential contains M-site extending this potential to a f exible model must include a speci- 34 Water Potentials f cation of how M changes with the deformation of water molecule, it is assumed that M-site to reside on the H-O-H bisector. Calculations for internal energy, heat capacity and pressure showed improvement compared with MCY. Because of the inherited problems from MCY model, MCYL predicted very high heat capacity and pressure compared with experiment .

2.5 MCYnawatermodel

MCYna is an extension of the MCY potential to include non-additive contribution from three body interaction and polarization [16]. The MCYna potential uses no empirical parameter other than the atomic mass, electronic charge so can be considered as a truly ab initio potential [79]. The potential of MCYna water model is,

u(r) = u2 + u3(dis) + upol (2.3)

where u(r) is the intermolecular potential, u2 is the additive dimer potential, u3(dis) is three body nonadditive term and upol is the polarizable term (i.e., the interaction between charge and dipole site). The total conf gurational energy U for a system consisting of N water molecules is N N

U = u2(ri, r j) + u3(ri, r j, rk) + upol (2.4) Xi< j iX< j

1 N u = µind.E0 (2.5) pol −2 i i Xi=1

0 ind where, Ei is the electrostatic f eld of surrounding charges, and µi is the induced dipole at site i given by N q r 0 = j i j Ei 3 (2.6) j=X1, j,i ri j

ind and µi is the induced dipole at site i given by

N µind = αβ = αβ 0 + µind i Ei Ei Ti j j  (2.7)  j=X1, j,i      where α is the polarizability, β is the scaling term (the values are given in Ta- ble 2.2) and Ti j is the dipole tensor given by

1 = ′ 2 . Ti j 5 [3ri jri j ri j] (2.8) 4πǫ0ri j −

The three-body interaction is assumed only between oxygen atoms, because of the electron poor feature of hydrogen. The value for Axilrod-Teller coeﬃ- cient is the value for Argon (5/9), since there is no known value for oxygen. The three body nonadditive interaction will be calculated using the Axilrod-Teller triple dipole term [145,146]

ν(1 + 3cosθ cosθ cosθ ) , , = i j k , u3(ri r j rk) 3 (2.9) (ri jrikr jk)

where ν is the Axilrod-Teller coeﬃcient, θi, θ j and θk are the inside angles of the 36 Water Potentials

triangle formed by three atoms donated by i, j and k, and ri j, rik and r jk are the three side length of the triangle shown in Figure 2.6.

Figure 2.6: Schematic representation of three body interaction.

We will use MCYna water model in the calculation of thermodynamic quantities using Lustig [9–11] and Meier and Kabelac [13] statistical method (given in

Chapter Three). For the following reasons we choose the MCYna water model: it is ab initio based, has nonadditive term (it has been established that the nonadditive contribution is roughly 15% of the binding energy of ice crystal [100]), did predict dielectric constant, dipole moment, O-O correlation. The MCY potential is a true pair potential not an eﬀective one, i.e., it does not have three-and higher body energy terms included in an average way [147]. It also predict the contribution of hydrogen bond better than other models [16]. Li et al. [21] reported that the contribution of induction to the overall energy is 30%, an amount which should not be ignored. Drawbacks of MCYna water model are it cannot be easily extended to mixtures and it is computationally expensive and rigid. The values of the parameters used in each of the models are given in Table 2.3. MCYna water model 37 [64] [87] [87] [57] [23] [79] [21] [29] [66] [54] [30] [69] [74] [88] [89] [90] Ref. mol) / – – – ( 0.6500 0.6364 0.6480 0.7749 0.3028 0.3169 0.6694 0.7100 0.6500 0.6364 0.6500 0.6000 0.5869 ǫ kJ ( σ Å) 3.1660 3.1506 3.1536 3.1589 2.8700 2.8700 2.8700 2.8200 3.1000 3.1200 3.1100 3.1660 3.1506 3.1660 3.2340 3.5140 are shown in Figure 2.4. ) ◦ 2 ( l – – – – – φ 52.26 52.26 52.26 52.26 52.26 52.26 52.26 111.00 109.28 109.47 109.47 and 1 l ) φ, ◦ , ( θ θ 109.47 104.52 108.00 109.47 104.52 109.47 106.00 104.52 104.52 104.52 104.52 104.52 104.52 109.28 109.47 104.52 ( – – – – – 2 l Å) 0.8892 0.0600 0.2500 0.1500 0.1546 0.2676 0.2677 0.2677 1.0000 0.8000 0.7000 ( erent water models. 1 ﬀ l Å) 1.0000 0.9572 0.9430 0.9572 0.9572 0.9572 0.9572 0.9572 1.0000 0.9572 0.9572 0.9800 1.0000 0.9600 1.0000 0.9572 0.6113 0.47700 0.41000 0.42380 0.41700 0.41700 0.41000 0.51700 0.52000 0.55640 0.71748 0.71740 0.71740 0.19000 0.24357 0.24100 q on H (e) arameters for di P 4 5 5 5 6 3 3 4 4 4 4 4 3 3 3 3 sites Table 2.3: 2 / F E F / / / MCYna BNS ST2 TIP5P NveD SPC d PPC GCPM TIP4P TIP4P MCY 005 MCYL Model SPC SPC TIP3P SPC w 38 Water Potentials

2.6 Improving water models

Most of the models considered fail in predicting either one or more of the following properties. Temperature of maximum density (TMD). In order to understand the hydrophobic eﬀect of water we need to have a correct description of density at all temperature [148]. At around 4 ◦C water will have maximum density. Predicting the density of water especially temperature of maximum density is the most challenging and important one.

The Second peak of the oxygen-oxygen pair correlation gOO(r) function. The pair distribution function provides an insight into the liquid structure [3]. The Second peak of the oxygen-oxygen pair correlation function gOO(r) at room temperature (around 4.5 Å) is the signature of the hydrogen-bond network [70]. Three site models (both rigid and f exible) do not predict second peak of the oxygen-oxygen pair correlation at all [87, 88]. Phase diagram/liquid-vapor coexistence curve. Almost all of the models considered ignore one or more of water’s properties or restrict the degree of freedom for computational and other reasons. This strategy of ignoring certain properties (parametrization) could give the needed result with the available computational resource in the intended region, however the model does not work in a region far a way from it is intended region. With a single model. Dielectric constant. Water’s ability to dissolve electrolytes plays a greater role in chemical reaction in cells [149], for this purpose we need to have a water model capable of predicting the dielectric constant. Unfortunately most of water models which will be used for biosimulation are not able to predicted the experimental Improving water models 39 value of the dielectric constant [120] for a detailed work on dielectric constant of water by different authors refer to the work of Ellison et al. [150]. Recently, Raabe and Sadus [77] improved the prediction of dielectric constant by using a f exible water model. Li et al. [16] found out that incorporating multi-body interaction will improve dielectric constant. Wu et al. found out that [74] equilibrium bond and angle affect dielectric constant. Critical temperature. Most water model do not predict the critical temperature and its behaviour near it [45,151]. Calculating the thermodynamic quantities near or at the critical point is challenging and at times impossible (for order parameters) which is a result f nite size and cutoff radius of the system. Though it is impossible to get the full phase diagram of water with a single model, it will be worth trying to get as many properties as possible from each of the models. One or more of the following could be implemented so as to improve the prediction of models.

Polarization. 30% of the total energy is contributed by the polarization term [16,90,108], this should not be ignored. Incorporating more terms in the model has an expensive computational cost but it is possible to use a simple relationship between two- and three-body interactions without signif cant additional computational cost as Wang and Sadus [152] recommended. Svishchev et al. [99] recommended incorporation of electronic polarization in water models for it to work in diﬀerent phases. Multibody interaction. Three body interaction has an eﬀect on vapor-liquid [80,

152] and solid-liquid [153] phase behaviour of f uids. Marcelli and Sadus [154] found that vapor–liquid equilibria are aﬀected substantially by three-body interactions. In particular, the subcritical liquid-phase densities are predicted accurately. 40 Water Potentials

More recently Wang and Sadus [152] showed that the addition of three-body interactions provides near perfect agreement with experiment for the vapor branch of the coexistence curve while simultaneously improving the agreement with experiment on the liquid branch. Leder [155] found out that incorporating three- body contribution in the calculation of thermodynamic properties of argon gives a very good agreement with experiment from near critical point to twice the critical density. Calculating the thermodynamic quantities with three body interaction included may improve the results. Replacing the Lennard-Jones potential. In some water molecule models there is

Lennard-Jones interaction site which accounts for the size of the molecules. Its repulsive nature at short distances ensures the structure does not completely col- lapse due to the electrostatic interactions [156]. If we replace the Lennard-Jones contribution by other realistic potentials, as recommended by Errington and Pana- giotopoulos [90, 134, 157]. In doing so the model will lose the computational eﬃciency of Lennard-Jones potential and the model might not be convenient to be used in conjunction with standard force f elds for organic and biomolecular systems [54]. Finally if possible the model should be ab inito based otherwise the properties for parametrization should be chosen carefully. Polarization should be included in water model, as it is conf rmed that it will contributes about 30% of the total interaction energy. Multibody interaction should also be included in the model so that it will work in diﬀerent states of water. In order not to restrict its degree of freedom and get better results from the model as recommended by Wu et al. [74] and Raabe and Sadus [75–77] the water model should be f exible. Chapter 3

Calculation of Thermodynamic Properties from Molecular Simulation

3.1 Introduction

In the classical molecular dynamics simulation it is possible to calculate only a few thermodynamic quantities, such as pressure and temperature, directly from a simulation. Despite the fact that the equation of state and f uctuation formulas fail at the critical point, until recently, to calculate the remaining thermodynamic state variables one need to use either f uctuation theory [3,4,54,158] or solve equation of state [159–162]. Lustig [9–12] formulated a direct method of calculating different thermodynamic quantities. Meier et al. [163–165], Lustig [166], Mausbach and Sadus [167] and Morsali et al. [168] showed that it is possible to calculate the transport coeﬃcients for Lennard-Jones f uid and the thermodynamic state vari-

41 42 Calculation of Thermodynamic Properties from Molecular Simulation ables for Gaussian-Core model [110] and nitrogen directly from the molecular dynamics simulation using the Lustig [9] and Meier and Kabelac [13] statistical formalism. It is the same approach as Mausbach and Sadus [167], Ray and Zhang [27], Lustig [9,166] and Pearson et al. [169] that we will follow to calculate the thermodynamic state variables in MCYna water [16], Lennard-Jones f uid, Weeks-Chandler-Anderson potential, and supercritical argon and krypton in a molecular dynamics ensemble. This chapter is organized in such a way that f rst we will see the molecular dynamics ensemble [27,170,171], followed by a brief revision of the thermodynamic state variables of interest. A detailed account on the statistical mechanical formalism in NVEPG ensemble developed by Ray and Zhang [27], Lustig [9], Meier and Kabelac [13] and Meier [172] will be given. We will review the f uctuation formulas of the thermodynamic quantities of interest. Finally we will explore the behaviour of the thermodynamic quantities near the critical point.

3.2 NVEPG Ensemble

For a molecular dynamics simulation one needs to properly def ne the ensemble. The specifc type of ensemble ( be it NVE, NVT or NPT) that we use for the calculation of thermodynamic quantities neither affects the result nor restrict the quantities to be calculated [166]. In principle, it is possible to calculate all the thermodynamic quantities such as heat capacities, compressibilities, isothermal pressure coefficient and Joule-Thomson coefficient from the T, V-derivatives of internal energy and pressure in any ensemble [173,174]. It is also possible to transform from one ensemble to the other [169,170]. In the thermodynamic limit NVEPG Ensemble 43 and in a region far away from the phase transition the averages from different ensembles should be the same [166,170,175]. A microcanonical ensemble is an isolated system conf ned in a constant volume, which does not exchange energy and molecules with its surrounding or any other system [169,176]. This kind of ensemble is central to the analysis of molecular dynamics trajectories computed for isolated system [177]. In an isolated system it is not possible to make any measurement or observation, which makes direct application of the statistical mechanics postulates (i.e., ergodicity and a priori probability) difficult. The common procedure to avoid this difficulty is either to introduce entropy [177,178] or to derive the properties from other ensembles [170]. For example, making use of the fact that a microcanonical ensemble is a degen- erate canonical ensemble we can calculate the quantities from canonical ensemble [178] and transfer those quantities [179]. In addition to the inability to make any direct measurement, the mathematical difficulty, i.e. the incorporation of delta and step functions which are not convenient for deriving statistical formulas, and ambiguity in the entropy def nition prohibited the development and wide use of microcanonical ensemble [169]. However ambiguous, we will use the entropy method of calculating thermodynamic quantities in a microcanonical ensemble. In the microcanonical ensemble every system has N molecules, a volume V, and an energy E and E + ∆E [180]. A molecular dynamics ensemble of isolated systems is a microcanonical ensemble with constant momentum-NVEP [3, 171], i.e., if the force between particles are conservative, the total linear momentum of the system P is constant. However, Ray and Zhang [27] showed that molecular dynamics ensemble to be NVEPG where G is a constant related to the initial 44 Calculation of Thermodynamic Properties from Molecular Simulation position of the center of mass def ned as

N N N G(t) = p t m r (t) = Pt m r (t) (3.1) i − i i − i i Xi=1 Xi=1 Xi=1

where ri, pi and mi correspond to the position, momentum and mass of particle i respectively and t is time. The above equation and the fact that the total linear momentum in a molecular dynamics ensemble is constant implies that

N miri(0) dG i=1 = 0 and G(0) = M P , (3.2) dt − N mi i=1 P where M is the total mass of the system. The constant of motion G is associated with the Galilean boost (transformation between inertial reference frame that have inf nitesimally different velocities) and it is a generator of inf nitesimal boosts like E is the generator of inf nitesimal time translation and P is the generator of inf nitesimal spatial translation. The introduction of the constant G into the NVEP ensemble simplif es the mathematics and correct the problem with the pressure and related quantities. In a molecular dynamics ensemble, here after NVEPG ensemble, there are no restriction on the order of thermodynamic derivatives and any measurable quantity is directly accessible [9], however as a result of the restriction on the momentum the degree of freedom will be one less that it will be if that restriction was not there [171]. The equipartition equation will be changed slightly and the temperature is a factor of N/(N 1), larger than the unconstrained microcanonical ensem- − 1 ble temperature. There will be a correction term of the order of O(N− ) which oc- NVEPG Ensemble 45 curs as a result of the molecular dynamics ensemble inability to mimic the microcanonical ensemble [171,181]. These two changes in the equipartition equation really does not affect the system if it consists of 100 or more molecules [171,177]. Shirts et al. [182] showed that, as a result of conservation of total linear momentum the equipartition theorem breaks down for a f nite, hard-sphere system when particles have different mass in molecular dynamics simulation. All the derivation given here, from the def nition of entropy to the calculation of thermodynamic variables, are based on the formulations reported by Meier and Kabelac [13], Meier [172], Lustig [9,10,12], Cagin and Ray [171], Pearson et al.

[169] and Ray and Zhang [27] unless otherwise stated. In preparation, for the calculation of thermodynamic quantities from entropy, we will def ne volume of phase space, density of phase space, entropy and enthalpy in a molecular dynamics ensemble. Consider Ω(N, V, E, P, G) as the volume in phase-space occupied by the molecular dynamics ensemble between E and E+∆E. The phase space volume is def ned as

1 N Ω(N, V, E, P, G) = Θ (E H) δ P p C " −  − i N Xi=1 H(p,r) E   ≤   (3.3)  N N δ G t p + m r drNdpN ×  − i i i Xi=1 Xi=1     where CN is a constant, H(p, r) is the Hamiltonian (total energy) of the system given as a sum of kinetic (K) and potential (U) energies, Θ is the Heaviside step function, δ is the Dirac-δ function, dpN and drN are the volume elements of phase space given as 46 Calculation of Thermodynamic Properties from Molecular Simulation

N dp = dp1 dp2 dp3 dpN · · ··· (3.4) drN = dr dr dr dr . 1 · 2 · 3 · ·· N Wood et al. [183] modif ed the expression for the phase space volume in order to avoid the appearance of time in it, making use of the def nition that G is related to the center of mass

1 N Ω(N, V, E, P, G) = Θ (E H) δ P p C " −  − i N Xi=1 H(p,r) E   ≤   (3.5)   N δ r r drNdpN ×  cm − i Xi=1     where r = G(0)/M is the position of the center of mass of the system and cm − ri should be the “inf nite checkerboard” position rather than the position of the particle i in the simulation cell for G to be continuous. This approach will give the same expression for the thermodynamic state variables [184]. The phase space density, ω(N, V, E, P, G), is the density of states of the system at the energy E [4,177], is def ned as

∂Ω ω = (3.6) ∂E !V therefore, the phase space density in a molecular dynamics ensemble will be

1 N ω(N, V, E, P, G) = δ (E H) δ P p C " −  − i N Xi=1 H(p,r)=E     (3.7)  N  N δ G t p + m r drNdpN. ×  − i i i Xi=1 Xi=1     NVEPG Ensemble 47

The phase space volume includes all points of phase space where the total energy of the system is lower or equal to the prescribed macroscopic energy, while the phase space density is obtained by integration over points for which the total energy is equal to the macroscopic energy, where the desired values of E, P and G are picked by Θ and δ functions.

An ensemble average of an arbitrary dynamical function A(rN, pN) can be calculated as [9, 172]

A = A rN, pN Ω(N, V, E, P, G) drNdpN h i Z 1 N = A rN, pN Θ(E H) δ P p ωC " −  − i (3.8) N Xi=1    N  N δ G t p + m r drNdpN. ×  − i i i Xi=1 Xi=1     in all the above equations the integration over momentum extends from - to + ∞ ∞ and the integration over the coordinates covers the volume of the system. For a given f uid at equilibrium two independent variables implicitly specify all other equilibrium properties. For an isolated system internal energy and volume will specify all the thermodynamic properties [173,185,186]. The connection between thermodynamic quantities and the molecular dynamics ensemble is via entropy. In all our simulations we will use molecular dynamics ensemble where the fundamental equation of state is entropy, S = S (N, V, E, P, G). Entropy is an extensive state function related to the number of states. There are two equally acceptable def nitions of entropy

S (N, V, E, P, G) = kB ln Ω(N, V, E, P, G), (3.9) 48 Calculation of Thermodynamic Properties from Molecular Simulation and

S (N, V, E, P, G) = kB ln ω(N, V, E, P, G), (3.10)

where kB is Boltzmann constant. For brevity we will use S , Ω and ω for S(N,V,E,P,G), Ω(N, V, E, P, G) and

ω(N, V, E, P, G) respectively. The above def nitions of entropy relate thermodynamics with quantum mechanics, it is also a bridge between microscopic states and macroscopic thermodynamics [185,186] and satisfy the f rst and second laws of thermodynamics [180]. The above two def nitions of entropy are in agreement to the order of 1/N [169,180]. Lustig [9] and Pearson et al. [169] examined the correct expression of entropy between the two def nitions but were unable to reach on a clear distinction as to which expression is correct. However, it is found out that both def nitions give equal thermodynamic results for a large system where the thermodynamic limit condition [9] is valid, namely

ln Ω ln ω lim = lim . (3.11) N N →∞ N →∞ N

If the system consists of small number of molecules, the thermodynamic limit condition does not work and so the two def nitions of entropy will give diﬀerent thermodynamic quantities [187]. In this work and in almost all references mentioned here the phase space volume def nition of entropy is chosen. Calculating entropy from the phase space volume follows from the adiabatic invariance of the phase space volume [171]. This gives expressions for temperature and pressure which are consistent with the equipartition and virial theorem respectively. On top of this, simulation results using the phase-space volume for the entropy calculation by Meier and Kabelac [13], Lustig [166], and Mausbach and Sadus [167] NVEPG Ensemble 49 predict the thermodynamic quantities in a better agreement than the traditional method of calculation. However, Baker and Johnson [188] used the phase space density def nition of entropy for the calculation of thermodynamic quantities and reached at the same results as in the other def nition of entropy. In a molecular dynamics ensemble temperature and pressure are not def ned explicitly. In this ensemble both can be calculated from the entropy of the system. Any thermodynamic quantity which can be immediately expressed as E and V derivative of S will be fundamental quantity from which all other thermodynamic variables will be constructed [169]. An inf nitesimal change in entropy caused by an infnitesimal change in either internal energy or volume or both can be calculated as [1] ∂S ∂S dS (E, V) = dE + dV (3.12) ∂E !V ∂V !E rewriting the above equation gives the f rst law of thermodynamics

TdS = dE + pdV (3.13) where T and p are temperature and pressure def ned as

1 ∂S ∂S and p T . (3.14) T ≡ ∂E !V ≡ ∂V !E

Finally, the enthalpy (S,p) of the system is def ned as H

(S, p) E(S, V) + V p. (3.15) H ≡ 50 Calculation of Thermodynamic Properties from Molecular Simulation

3.3 Thermodynamic quantities in the NVEPG

ensemble

We review the general def nition of some of thermodynamic state variables such as pressure, heat capacities, compressibilities, thermal expansion coefficient, isothermal pressure coefficient, speed of sound and Joule-Thomson coefficient in a molecular dynamics ensemble. It is possible to calculate the above mentioned thermodynamic state variables by solving either an equation of state [189–193], standard f uctuation formulas [5,54,179,187,194–196] or directly from the molecular dynamics simulation [9,13,27,166,167]. Recently, Mausbach and Sadus [167] showed that it is possible to use the statistical mechanical method developed by Lustig [9], Ray and Zhang [27], Cagin and Ray [171] and Meier and Ka- belac [13] to calculate a complete thermodynamic properties of a Gaussian core model f uid [110]. We will follow the same technique to calculate the complete thermodynamic properties of water, Lennard-Jones f uids and Weeks-Chandler- Anderson potential directly from the molecular dynamics simulation. Consider an isolated system consisting of N particles, volume V and internal energy E. In order to calculate the thermodynamic quantities let us def ne the phase space function Ωmn [9]

1 ∂m+nΩ Ω . (3.16) mn ≡ ω ∂Em∂Vn

For all calculations involving entropy we will use Eq. (3.9) the def nition with phase space volume. All the calculated thermodynamic state variables will be in terms of phase space function. Using Eq. (3.14) temperature and pressure of the Thermodynamic quantities in the NVEPG ensemble 51 system in terms of phase space function are

1 ∂S ∂ k = = (k ln Ω) = B , (3.17) B Ω T ∂E !V ∂E 00 and ∂S 1 ∂Ω p = T = k T = Ω (3.18) B Ω 01 ∂V !E ∂V where Ω00 and Ω01 are calculated from Eq. (3.16) and are given in Table 3.3.

3.3.1 Heat capacities

Calculating the heat capacity of water is one of the key goals for diﬀerent simulations as it show anomaly and is necessary to understand the solvation [20]. Heat capacity is the amount of hear needed to raise the temperature of a material by one degree. The measurement could be conducted either at constant volume or pressure. The isochoric heat capacity (Cv) using second law of thermodynamics and def nition of temperature is def ned as

1 ∂E ∂S ∂2S − = = = . (3.19) Cv T 2 ∂T !V ∂T !V " ∂E !V #

The heat capacity at constant volume in terms of phase space function is

1 ∂T 1 ∂ Ω 1 = = = (Ω10 Ω00Ω20) (3.20) Cv ∂E !V kB ∂E "ω# kB −

where the values of Ω10 and Ω20 are given in Table 3.3. The isobaric heat capacity

(Cp) which denotes the temperature variation of enthalpy is def ned as, 52 Calculation of Thermodynamic Properties from Molecular Simulation

∂ ∂S Cp = H = T . (3.21) ∂T !p ∂T !p

3.3.2 Isothermal pressure coeﬃcient

Isothermal pressure coeﬃcient, whose negative value indicates the anomalous behaviour of density in phase space, is def ned as

∂p γv = . (3.22) ∂T !V

It can also can be calculated from the thermodynamic identity

∂γ ρ ∂C T v = v . (3.23) ∂T !v −v ∂ρ !T

Rewriting Eq. (3.22) gives

∂p ∂p ∂E ∂p γv = = = Cv , (3.24) ∂T !V ∂E !V ∂T !V ∂E !V where

∂p ∂ 1 ∂Ω = = Ω20Ω01 + Ω11 (3.25) ∂E !V ∂E ω ∂V !V − substituting the values of Eqs. (3.20) and (3.25) into Eq. (3.24) gives

Ω Ω Ω γ = k 11 − 01 20 (3.26) v B Ω Ω Ω 10 − 00 20 where the value of Ω11 is given in Table 3.3. Thermodynamic quantities in the NVEPG ensemble 53

3.3.3 Compressibilities

Isothermal (βT) and isentropic (βS) compressibilities measure the changes in the volume of the system (i.e., measure the response to the volume to a pressure stim- ulus) when heated by maintaining the internal energy at constant temperature and entropy respectively. Isothermal compressibility increases when a liquid is super- heated [197]. Isothermal and adiabatic compressibilities are def ned as,

1 ∂p ∂p βT− = V = ρ (3.27) − ∂V !T ∂ρ !T and

1 ∂p βS− = V . (3.28) − ∂V !S

There is a relation between compressibilities and the heat capacities so that if the compressibilities and one of the heat capacities are known the remaining heat capacity can be calculated using the following relation, and vice versa

Cp β = T (3.29) Cv βS or we can use the following relation to calculate isothermal compressibility from adiabatic compressibility, isochoric heat capacity and isothermal pressure coeﬃ- cient at a given temperature and volume

2 1 1 TVγv βT− = βS− . (3.30) − Cv

To calculate the right hand side of Eq. (3.28), Equation (A.17) from Callen [1] is used 54 Calculation of Thermodynamic Properties from Molecular Simulation

∂p ∂p ∂p ∂S = + (3.31) ∂V !E ∂V !S ∂S !V ∂V !E using the def nition of pressure Eq. (3.14) and rearranging terms give

∂p ∂p p ∂p ∂E = + . (3.32) ∂V !E ∂V !S T ∂E !V ∂S !V

Again inserting Eq. (3.14) and rearranging terms gives

1 β− ∂p ∂p ∂p S = = p (3.33) − V ∂V !S ∂V !E − ∂E !V solving for the f rst term on the right gives

∂p 1 ∂2Ω 1 ∂Ω 1 ∂2Ω = + = Ω Ω Ω . 2 ( 02 11 01) (3.34) ∂V !E −ω ∂V∂E ω ∂V ω ∂V −

Ω 1 where 02 is given in Table 3.3. Solving for βS− after inserting Eqs. (3.34), (3.18) and (3.25) into Eq. (3.33) gives

1 β− = V[Ω (2Ω Ω Ω ) Ω ]. (3.35) S 01 11 − 01 20 − 02

Substituting Eqs. (3.35),(3.20), (3.26) into Eq. (3.30) the thermal expansion coef- f cient in terms of the phase space function will be

Ω Ω Ω Ω Ω Ω2 1 01(2 11 01 20) 00 11 β− = V − − Ω . (3.36) T " 1 Ω Ω − 02# − 00 20

Here, after showing the expression of isothermal compressibility and isothermal pressure coeﬃcient, is the right time to introduce thermal expansion coeﬃ- Thermodynamic quantities in the NVEPG ensemble 55

cient. The thermal expansion coeﬃcient (αp) is the measure of the tendency of matter to change in volume in response to a change in temperature keeping the pressure constant. Thermal expansion coeﬃcient is def ned as

1 ∂V αp = = βTγv. (3.37) V ∂T !p

3.3.4 Speed of sound at zero frequency

Due to the high accuracy of the experimental data for the speed of sound in diﬀer- ent liquids, this property provides a very sensitive test in the course of developing an equation of state [190] and so validity of simulations. The thermodynamic speed of sound is related to the propagation of an adiabatic pressure wave. The zero frequency speed of sound is related to the zero frequency bulk modulus, it is a derivative of the pressure with respect to mass density at constant entropy [172]. The speed of sound at zero frequency is def ned as

2 ∂p V ∂p ω0 = = , ∂ρS !m −ρm ∂ρS !S 2 (3.38) V ∂p V 1 = = βS− − M ∂V !S M where ρm is mass density. Using Eq. (3.35) speed of sound in terms of the phase space function can be written as

V2 ω2 = [Ω (2Ω Ω Ω ) Ω ] . (3.39) 0 M 01 11 − 01 20 − 02 56 Calculation of Thermodynamic Properties from Molecular Simulation

3.3.5 Joule-Thomson coeﬃcient

The Joule-Thomson eﬀect describes the temperature change of a gas or liquid when it is forced through a valve or porous plug while kept insulated so that no heat is exchanged with the environment [4]. The Joule-Thomson coeﬃcient

(µJT ) is a measure of how much the temperature of a f uid changes as pressure is changed at constant enthalpy [140]. The Joule-Thomson def ned as

∂T 1 ∂ V µJT = H = (Tαp 1) (3.40) ≡ ∂p ! −Cp ∂p !T Cp − H

Depending on state conditions the Joule-Thomson coefficient may be positive, negative or zero [198, 199]. If µJT is positive, reduction in pressure causes reduction in temperature. This will happen at lower initial pressure (this could have a malfunction or catastrophic effect by liquefying or solidifying a gas in a pipe or machine which transport or work in gas). If the coefficient is negative, reduction in pressure causes increase in temperature. This will happen if the initial pressure is high. Therefore, temperature increases with increasing pressure for an isenthalpic process, reaches a maximum point and then starts to decrease with increasing pressure. The temperature corresponding to this maximum point

(i. e., the crossover from heating to cooling) at which µJT = 0 is referred as the “inversion point”. The inversion curve is the locus of these inversion points on a T-p graph [140, 198, 200]. There are two temperatures at which µJT = 0. The f rst one is in the lower density supercritical f uid part of the phase diagram and the other point is when the f uid approaches the liquid state as p 0 [201]. The → inversion curve for diﬀerent simple f uids is given in [201–203].

For an ideal gas, µJT is always equal to zero. Implying that ideal gases neither Thermodynamic quantities in the NVEPG ensemble 57 warm nor cool upon being expanded at constant enthalpy. A Joule-Thomson process is commonlyusedto cool or liquefy gases and its coeﬃcient is the indicator of whether the throttling process produces cooling or heating. Only when the Joule- Thomson coeﬃcient for the given gas at the given temperature is greater than zero can the gas be liquef ed at that temperature by the Linde cycle [198,204,205].

In other words, a gas must be below its inversion temperature to be liquef ed by the Linde cycle. The experimental data of the inversion curve of Joule-Thomson is scarce since it demands precise measurement of volumetric and caloric properties [206] and also it covers large area of the T-p curve, which extends to f ve-fold of the critical temperature and about twelve-fold of the critical pressure [202]. Fundamental thermodynamic quantities are calculated in terms of phase space function. The remaining thermodynamic quantities, isothermal compressibility

(βT), isobaric heat capacity (Cp), thermal expansion coeﬃcient (αp), speed of sound (ω0) and Joule-Thomson coeﬃcient (µJT ) can be calculated from Eqs. (3.30), (3.29), (3.37), (3.38) and (3.40) respectively. The results of all fundamental thermodynamic quantities are summarized in Table 3.1.

Table 3.1: Summary of thermodynamic quantities in terms of phase space function.

Thermodynamic quantities Expression in terms of Ωmn

T Ω00/kB

p Ω01

1 C k [Ω Ω Ω ]− v B 01 − 00 20 1 β− V [Ω (2Ω Ω Ω ) Ω ] S 01 11 − 01 20 − 02 γ k (Ω Ω Ω ) / (Ω Ω Ω ) v B 11 − 01 20 10 − 00 20 58 Calculation of Thermodynamic Properties from Molecular Simulation

3.4 Calculation of thermodynamic quantities in an

NVEPG ensemble

The thermodynamic state variables are calculated in terms of the phase space functions Ωmn. The explicit expression of Ωmn given by Eq. (3.16) are related with the ensemble averages of instantaneous phase variables. We will calculate the relation between Ωmn and the ensemble average of kinetic and potential energies as part of an eﬀort to get the thermodynamic state variables directly from the simulation. Ray and Zhang [27] introduced the Jacobi coordinates so as to simplify the calculation of the phase space function by eliminating the Hamiltonian from phase space volume [207]. The Jacobi coordinates are the generalizations of the Carte- sian coordinate. The f rst N-1 Jacobi coordinates are internal coordinates whereas the Nth coordinate is the center of mass coordinate of the system of particles. For

N particles the Jacobi coordinates Rα and Pα are def ned as [208]

α miri i=1 Rα = P rα+ , 1 α N, r + = 0, α − 1 ≤ ≤ N 1 mi i=1 P α α mα+1 M1 (3.41) Pα = p p , 1 α N 1, α+1 i − α+1 α+1 ≤ ≤ − M1 Xi=1 M1 N

PN = pi, Xi=1 where j M j = m , i j, 1 i, j, k N. (3.42) i k ≤ ≤ ≤ Xk=i In Jacobi coordinates the kinetic energy of the system which depend only on the momentum coordinates becomes Calculation of thermodynamic quantities in an NVEPG ensemble 59

N N 1 p2 − P2 P2 K = i = α + N , (3.43) 2m 2µα 2µ Xi=1 i Xα=1 N where the reduced masses µα are def ned by

α mα+ M µ = 1 1 , α , µ = N. α α+1 1 N 1 N M1 (3.44) M1 ≤ ≤ −

Eq. (3.43) gives the kinetic energy in terms of the internal kinetic energy plus the kinetic energy of the center of mass. It is a common practice in classical mechanics to separate the center of mass in order to reduce the multi body problem to the motion of one particle with mass µα [207]. The reduced masses satisfy the following product relation

N N 1 mi − i=1 (3.45) µα = Q µ Yα=1 N where µN = M is the total mass of the system. The expression for the phase space volume, after substituting the Jacobi coordinates and kinetic energy into Eq. (3.3), will be

N 1 − 2 2 1 Pα PN Ω = Θ E U δ (P PN) C "  − 2µα − 2µ −  − N Xα=1 N   (3.46)   δ (G P t + MR ) dRNdPN × − N N where dRN and dPN have the same expression as Eq. (3.4) except that it is a Jacobi coordinate. Using the property of δ-functions 60 Calculation of Thermodynamic Properties from Molecular Simulation

∞ f (x ) = δ(x x ) f (x)dx (3.47) 0 Z − 0

−∞ integrating over PN gives

N 1 2 − 2 1 P Pα N N 1 Ω = Θ E U δ (G Pt + MRN) dR dP − . (3.48) C "  − 2M − 2µα −  − N Xα=1     Assuming that the potential depends only on the relative coordinates of the particles and using the same Dirac delta function property as above, integrating over the center of mass coordinate RN gives

N 1 2 − 2 1 P Pα N 1 N 1 Ω = Θ E U dR − dP − . (3.49) C "  − 2M − 2µα −  N Xα=1     To calculate the remaining Jacobi momenta and position integrals, and express the phase space function as an average of dynamic functions we need to separate the momenta and position components of the integral. To accomplish this,

Laplace transform phase space volume with respect to energy, do the momentum integral and again inverse transform it to Jacobi coordinates. We will present the steps taken in Laplace transforming and inverse Laplace transforming of phase space volume by Pearson et al. [169] and Meier and Kabelac [13]. The Laplace transform of the phase space volume using Laplace transform for Heaviside step function [209]

x Γ(x + 1) sH (E H) Θ(E H) exp− , when H 0 (3.50) − − −→ sx+1 ≥ is Calculation of thermodynamic quantities in an NVEPG ensemble 61

N 1 2 − 2 1 1 P Pα N 1 N 1 E Ω = exp s U + + dP − dR − (3.51) L { } C " s −  2M 2µα  N   Xα=1        separating the above equation into position and momentum parts

2 1 1 P N 1 E Ω = exp s U + dR − L { } CN Z s (− 2M !) + ∞ ∞ 2 2 2 (3.52) P1 P2 PN 1 exp s + − dP1 dP2 dPN 1. × Z · ·· Z (− 2µ1 2µ2 ··· 2µN 1 !) · ·· · − − −∞ −∞

In Laplace space evaluating the N-1 Jacobi momenta of phase space volume using ∞ 3/2 aX2 π e− dX = (3.53) Z a −∞ and Eq. (3.45) gives

3(N 1) + 2− 1 2 1 1 P N 1 E Ω = exp s U + dR − (3.54) L { } C0 Z s ! (− 2M !) where 1 N − 3(N 1)/2 3/2 C = C M (2π) − m . (3.55) 0 N  i  Yi=1     The inverse Laplace of phase space in Jacobi coordinate, using the correspon- dence [210] x x 1 1 (E H) − exp( sH) − Θ(E H), (3.56) s ! − −→ Γ(x) − will be 62 Calculation of Thermodynamic Properties from Molecular Simulation

3(N 1) 1 1 P2 2− Ω = E U C Γ[3(N 1)/2 + 1] − 2M − ! 0 Z (3.57) − 2 P N 1 Θ E U dR − . × − 2M − ! Now we have a phase space with position Jacobi coordinate as the variable of integration where the limit of integration depend on the volume. Introducing a scaled Jacobi coordinate

R = V1/3R′ , (3.58) will remove the dependence of limit of integration on volume of the system [179]. The phase space volume in a scaled Jacobi coordinates, with a limit of integration ranging between zero and one independent of volume becomes

3(N 1) − 1 1 P2 2 Ω = E U Γ / + − − ! C0 [3(N 1) 2 1] Z 2M (3.59) − 2 P N 1 N 1 Θ E U V − dR′ − . × − 2M − !

Using the property of gamma function [210]

xΓ(x) = Γ(x + 1), (3.60) the phase space density in a scaled Jacobi coordinates will be

3(N 1) − 1 1 1 P2 2 − ω = E U C Γ[3(N 1)/2] − 2M − ! 0 Z (3.61) − 2 P N 1 N 1 δ E U V − dR′ − . × − 2M − !

Starting from Eq. (3.59) the phase space function can be calculated. As de- Calculation of thermodynamic quantities in an NVEPG ensemble 63 f ned in Eq. (3.16) the general equation for the phase space function is derived by taking the mth derivative with respect to energy and the nth derivative with respect to volume and dividing this by the phase-space density (Eq. (3.61)) [172]

3(N 1) N 1 − 3N/2 n V − (2π) 2 m N 1 3(1 N) Ω = ( 1)m − ( 1)n mn C Γ + 3/2 ( V Nn − 2 ! − 0 [3(N 1)/2 1]µN m − 3(N 1) − m 2 2 − 2 P P ′ N 1 (1 N)n E U Θ E U dR − × − Z − 2µN − ! − 2µN − ! n n i i n n i n i N − 1 m+l 3(1 N) + (1 δ ) ( 1) − (1 N) − ( 1) − − 0n i! − − V Nn i − 2 ! Xi=1 − Xl=1 m+l 3(N 1) − m l k (i,l) P2 2 − − max P2 Θ ′N 1 E U  cilk ilk E U dR −  , × Z − 2µN − ! W − 2µN − !   Xk=1           (3.62) where (x) = x(x + 1)(x + 2) (x + n 1) with (x) = 1 is Pochhammer symbol, n ··· − 0 δ δ = = δ = , n i j is the Kronecker delta, i j 1 if i j and i j 0 if i j, i is binomial distribution, c is the multinomial coeﬃcient and is related with the product ilk Wilk of the diﬀerential of the system potential with respect to the volume.

Molecular dynamics simulations are carried out at P = 0, i.e. the system is at rest, the phase space volume is then

1 1 3(N 1)/2 N 1 N 1 Ω = (E U) − Θ (E U) V − dR′ − , (3.63) C Γ[3(N 1)/2 + 1] Z − − 0 − the very nature of the kinetic energy makes Θ(K = E U) in the above equation − to be always equal to one. Thus, the simplifed expression for the phase space density is

1 1 3(N 1)/2 1 N 1 N 1 ω = (E U) − − δ (E U) V − dR′ − . (3.64) C Γ[3(N 1)/2] Z − − 0 − 64 Calculation of Thermodynamic Properties from Molecular Simulation

If a phase variable ‘A’ depend only on the N spatial coordinates rN Eq. (3.8) { } can be simplifed in the same way as the phase space volume and density, this condition is necessary in getting rid of the center of mass coordinates using the above mentioned property of Dirac-δ function, Eq. (3.47). From Eq. (3.62) it is possible to conclude that, any phase variable required to calculate thermodynamic state variables depends only on powers of the kinetic energy of the system and on the conf gurational energy or volume derivatives of it [172]. The latter two quantities depend only on the coordinates. Since the total energy of the system is the sum of the kinetic and conf gurational energy and is constant in the NVEPG ensemble, it follows that the kinetic energy also depends only on the coordinates. Using scaled Jacobi coordinates, an ensemble average of phase variables is

1 1 3(N 1)/2 1 N 1 N 1 A = A (E U) − − Θ (E U) V − dR′ − . (3.65) h i ωC Γ[3(N 1)/2] Z − − 0 −

By comparing the results of the diﬀerentiation of phase space volume with respect to E and V in Eq. (3.62) with the general formula for an arbitrary ensemble average (Eq. (3.65)) the expressions for the corresponding phase space functions

Ω are found. Ω corresponds to A and A is to be identif ed by the comparison. mn mn h i Comparing Eq. (3.62) with Eq. (3.65) gives the following expression for the phase space function in a molecular dynamics ensemble

n N 1 m 2 3(N 1) n (m 1) Ω = ( 1) − ( 1) ( [N 1]) K− − mn V Nn − 3(N 1) − 2 ! − − − nh i − m n n i n n i N − 1 2 + (1 + δ0n) ( 1) − ( [N 1])n i i! − − − − V Nn i 3(N 1) (3.66) Xi=1 − − i kmax(i,l) m+l 3(N 1) (m+l 1) ( 1) − K− − c × − − 2 ! *  ilkWilk+ Xl=1 m+l  Xk=1      Calculation of thermodynamic quantities in an NVEPG ensemble 65

where kmax(i, l) is the number of terms with the same number of factors cilk is a multinomial coeﬃcient, which is the number of ways of partitioning a set of i = a + 2a + + ia of diﬀerent objects into a subsets containing k objects for 1 2 ··· i k k = 1, 2, , i [210]. In other words i is decomposed into l distinct summands of ··· positive integers with out regard to others, given as

i! cilk = (3.67) (1!)a1 a !(2!)a2 a ! (i!)ai a ! 1 2 ·· · i and is the kth product made up of l factors, where each factor is a derivative Wilk given as [10]

∂mU ∂nU = (3.68) Wilk ∂Vm ∂Vn ··· so that the total order of derivatives is i. Some of the values for c and are ilk Wilk given in Table 3.2

Table 3.2: Values for some of multinomial combinations.

i l k c number of ways to partition ilk Wilk 1 1 1 1 1 1 = 1 ∂U × − ∂V 2 2 1 1 1 1 0 + 2 1 = 2 ∂ U × × − ∂V2 2 2 2 1 1 1 2 + 2 0 = 2 ∂U × × ∂V

If we substitute m = 0 and n = 2 into Eq. (3.66) we will get the following expression for Ω02 66 Calculation of Thermodynamic Properties from Molecular Simulation

2 2 i 2(N 2) 2 2 i N − 2 Ω02 = − K + ( 1) − (1 N)2 i 3V2 h i ( i! − − − V 3(N 1) Xi=1 − (3.69) i kmax(i,l) l 3(N 1) (l 1) ( 1) − K− − c × − −  ilkWilk  = 2 !l * = + Xl 1  Xk 1      for simplicity let us divide the second expression of the right hand side equation = Ω1 = Ω2 into two, i 1 ( 02) and i 2 ( 02)

2(N 1) 2(N 1) ∂U Ω1 = − c = − (3.70) 02 V h 111W111i − V * ∂V + and

2 3(N 1) 3(N 1) 3(N 1) Ω2 = − c − 1 − c 02 3(N 1) ( 2 h 211W211i − 2 − 2 ! h 221W221i) − 2 2 ∂ U 3(N 1) 1 ∂U = 1 − K− − * ∂V2 + − − 2 !* ∂V ! + (3.71) substituting Eqs. (3.70) and (3.71) into Eq. (3.69) gives

2 N 2 N 1 ∂U ∂2U Ω = − K 2 − 02 3V V h i − V * ∂V + − * ∂V2 + 2 (3.72) 3(N 1) 1 ∂U 1 − K− . − − 2 !* ∂V ! +

In order to calculate Ωmn, w can also make a direct comparison between the derivatives of the phase space volume (Eq. (3.63)) with respect to the independent variables E and V and the ensemble average of phase space variable Eq. (3.65). Since there are no complicated multinomial expressions this method is easier than the one we have seen earlier. For a comparison we have calculated Ω02 Calculation of thermodynamic quantities in an NVEPG ensemble 67

2 2 1 ∂ Ω 3(N 1)/2 1 ∂ U N 1 N 1 Ω = = (E U) − − Θ(E U) V − dR′ − 02 ω ∂V2 (− Z − ∂V2 ! −

2(N 1) 3(N 1)/2 1 ∂U N 1 N 1 − (E U) − − Θ(E U) V − dR′ − − V Z − ∂V ! − 2 3(N 1) 3(N 1)/2 2 ∂U N 1 N 1 1 − (E U) − − Θ(E U) V − dR′ − − − 2 ! Z − ∂V ! −

2(N 2) 3(N 1)/2 N 1 N 1 − (E U) − Θ(E U) V − dR′ − /ωC Γ[3(N 1)/2]. − 3V2 Z − − ) 0 − (3.73)

Comparing Eq. (3.73) with Eq. (3.65) gives the same result as Eq. (3.72). The values of Ωmn calculated from either Eq. (3.66) or by comparison of Ω with respect to the derivative of E and V are given in Table 3.3.

Table 3.3: Def nition of phase space function in terms of phase space volume and ensemble averages [9, 13, 27, 169, 171, 172].

Ωmn Def nition Ensemble average of phase space function Ω 1 ∂Ω 10 ω ∂E 1 Ω Ω 2 00 ω 3(N 1) K − h i 1 ∂Ω N 1 ∂U Ω − k T 01 ω ∂V V B − ∂V D E 1 ∂2Ω 3(N 1) 1 Ω 1 − K− 20 ω ∂E2 − − 2 D E 1 ∂2Ω N 1 3(N 1) 1 ∂U Ω − + 1 − K− 11 ω ∂E∂V V − 2 ∂V D E 2 1 ∂2Ω 2 N 2 3(N 1) ∂U ∂2U 2(N 1) ∂U Ω − 1 − 02 ω ∂V2 3V V− K 1 2 K− ∂V ∂V2 V ∂V h i − − h i − D E − D E

It is possible to calculate the ensemble average of kinetic energy directly from MD simulation. Now, the diﬀerential of potential with respect to the volume completes the calculation. We will show this calculation as outlined by Lustig [10,12] and Meier and Kabelac [13]. The result we have so far is valid for any kind of conservative intermolecular potential. Here, to calculate ∂U/∂V and higher 68 Calculation of Thermodynamic Properties from Molecular Simulation order terms let us consider a pairwise additive potential. For a system consisting of N particles the pairwise additive intermolecular potential is given by

N 1 N − U = u(ri j) (3.74) Xi=1 Xj>i

where u is the pair potential energy and ri j is the distance between particle i and j. If the system is contained in a cubic box of volume V the distance between the particles implicitly depend on the volume of the system. To remove this depen-

′ = 1/3 ′ dence we will introduce the scaled distance ri j with a transformation ri j V ri j.

′ The scaled distance ri j do not depend on the volume of the system

dri j 1 2/3 ri j = V− r′ = . (3.75) dV 3 i j 3V

Applying the chain rule, the f rst derivative of the potential energy is then

N 1 N N 1 N ∂U − ∂u ∂ri j 1 − ∂u = = r . (3.76) ∂V ∂r ∂V 3V i j ∂r Xi=1 Xj>i i j Xi=1 Xj>i i j

The second volume derivative of the potential energy using chain rule

N N 1 ∂2U ∂ 1 − ∂u = r , ∂V2 ∂V 3V i j ∂r   Xi=1 Xj>i i j     N N 1  N N 1 1 − ∂u  1 − ri j ∂u ∂ ∂u ∂ri j = ri j + + ri j , (3.77) −3V2 ∂r 3V "3V ∂r ∂r ∂r ! ∂V # Xi=1 Xj>i i j Xi=1 Xj>i i j i j i j N 1 N 1 − ∂u ∂2u = 2r + r2 . 9V2 − i j ∂r i j ∂ 2  Xi=1 Xj>i i j ri j     Generally the nth volume derivative of the potential energy is Thermodynamicquantitiesfromthefuctuationtheory 69

N 1 N n ∂nU 1 − ∂ku = a rk , (3.78) ∂Vn 3nVn nk i j ∂ k Xi=1 Xj>i Xk=1 ri j where ank is constructed from the recursive relation

ank = an 1,k 1 + ( 3(n 1) + k)an 1,k, − − − − − n 1 − (3.79) a = ( 3l + 1), a = 1. n1 − nn Yl=0

Finally, the results of this chapter are valid for any system of classical particles contained in a cubic box, interacting through a pairwise potential which depends only on their relative position in a molecular dynamics ensemble.

3.5 Thermodynamic quantities from the f uctuation

theory

The conventional thermodynamic f uctuation theory (CFT) developed by Ein- stein [211] express the time-independent probability distribution for the state of a f uctuating system in terms of thermodynamic quantities. The CFT does not consider the local correlation and so it fails in a very interesting region-near critical points [194,195]. Results from the f uctuation formulas were less well con- verged [54]. Thermodynamic quantities which are not def ned for a single quantum state, such as temperature and entropy can not be calculated from f uctuation [178]. In a molecular dynamic ensemble energy is constant, calculating the f uctuation of energy around its mean value is not possible. However Lado [212] and 70 Calculation of Thermodynamic Properties from Molecular Simulation

Ray and Graben [179] independently calculated the f uctuation in kinetic and potential energies within the molecular dynamics ensemble. Hill [178], Allen and Tildesley [3] and Lebowitz et al. [170] transformed thermodynamic quantities calculated in one ensemble to the other. The “ensemble correction” the diﬀerence among various ensemble averages is of the order of 1/N for an intensive variable in a f nite system [213]. Ray and Graben [179] and Cheung [187] derived the f uctuation formula for thermodynamics properties in a molecular dynamics ensemble. Equilibrium and dynamical properties of a system can be obtained as time averages. The time average of dynamical function ‘A’ in phase space is [3,170,172, 214–216]

t0+τ 1 A(t) = A(rN(t), pN(t)) dt (3.80) h i τ Z t0 where τ is the period. It is one of the basic postulates of statistical mechanics that the time average must be equal to the ensemble average at equilibrium. Both Eqs. (3.8) and (3.80) must give the same result at equilibrium. We can choose either the time average or the ensemble average for the calculation of thermodynamic variables. For its convenience to implement in the simulation and the ability to work near the critical point the ensemble average which uses the statistical formalism is preferable. The root-mean-square deviation of ‘A’ is

(δA)2 = A2 A 2 (3.81) h i h i − h i where δA is the instantaneous diﬀerence between ‘A’ and its time average A . h i Thermodynamics properties near the critical point 71

Table 3.4 summarizes calculation of the thermodynamic quantities from the root- mean-square f uctuation formulas.

Table 3.4: Thermodynamic variables calculation from f uctuation formulas (Source [3,4, 158, 214, 217]).

Variables Fluctuation formulas

T 1 p ∂H and 1 q ∂H kB k ∂pk kB k ∂qk D E 2 D1 E 3NkB 2 (δU) − C 1 h i2 v 2 3N(kBT) − C (δH)2 /(k T) p h i B γ 2C 1/V δpδU /(N(k T)2) /3 v v − h i B β 2Nk T/3V + F /V + p V δp2 /(k T) S B h i h i − h i B β (δV)2 /(k T V ) T h i B h i α ( VH V H )/(k T 2 V ) p h i − h ih i B h i

where pk and qk are the generalized momentum and position coordinates, F is the free energy.

3.6 Thermodynamics properties near the critical

point

The liquid-vapor coexistence curve does not exist forever, as the equilibrium line between liquid and solid state appears to do, but rather that it terminates in a point - critical point [45,151,218]. The fact that the vapor pressure curve terminates in a critical point means that one can convert a liquid to a gas continuously, without crossing the phase transition line. In this sense the distinction between the liquid and gaseous phases is almost non-existent [219]. This is associated with the phe- 72 Calculation of Thermodynamic Properties from Molecular Simulation nomenon of critical opalescence, a milky appearance of the liquid due to density f uctuations at all possible wavelengths. At or very near the critical point water will be “Milky” and opaque [151]. It is possible to liquefy a gas by applying a pressure only if its temperature is less than the critical value [151,220]. The critical points of argon, krypton and water are given in Table 5.3 and 6.1. Near the critical point it is not possible to have systematic and direct calculation of thermodynamic properties of realistic molecular models, as result of which most of the simulations avoid the region [60,221–225]. Thermodynamic phase transition is thermally driven phase transition of a thermodynamic system from one phase or state of matter to another [226,227] at the thermodynamic limit [228]. During phase transition there is a jump in a suppos- edly continuous functions such as density at the thermodynamic limit [229]. There is no general agreement as to what causes the singularity in thermodynamic phase transition [230–233]. At low temperature there is a very big diﬀerence between the liquid and gas densities but as the critical temperature is approached this density diﬀerence approaches to zero. A quantity which is non-zero below the critical temperature and zero above, which signify the onset of order at the phase transition, is referred as order parameter [226]. Inside the coexistence curve (on both side of the critical point) there will be a phase change with out change in order parameter for example density in f uids [219]. The potential energy at the critical point will have a f at minima [1]. In this situation the basic assumption of statistical mechanics (i.e., the ergodic hypothesis which states that the ensemble and time averages give the same result) does not hold true. The system will not be able to explore all the phase space. There is a priori knowledge of the particle that will make it to choose some path and not others. Thermodynamics properties near the critical point 73

In the case of a pure compound, the critical coordinates may be obtained by assuming the following scaling law ρ ρ = A(T T )β where ρ and ρ are G − L − C G L the densities of the coexisting vapor and liquid phases respectively and A is a constant, β = 0.325 is a universal critical exponent [234,235]. Molecular simulation cannot be used in the close vicinity of the critical point, near the critical point all the particles will be highly correlated as a result of which the system size will be small. Indeed, the characteristic size of density or energy f uctuations increases when the critical point is approached, and it would be necessary to simulate systems of excessively large size that are incompatible with reasonable computing times [236]. The isothermal compressibility of a f uid given in Eq. (3.27), and hence the f at portion of the isotherm corresponds to an isothermal compressibility which diverges to inf nity as the critical point is approached. Of course an inf nite value

1 of ρ− (∂ρ/∂p)T means that the response of the density to a very small pressure f uctuation is inf nite. Thus we might expect that this divergence in the isothermal compressibility is connected to the huge density f uctuation which is associated with critical opalescence. The value of ρ ρ (where ρ is the density at the critical point) is positive if − C C the f uid is liquidand negativeif the f uid isin gas phase [231]. There are two pairs of response functions (response of density or entropy to changes in temperature or pressure) which are very sensitive to the critical point (actually they will diverge at the critical point [1,197]. The molar heat capacity, the same is true for isothermal

′ compressibility, diverges at the critical point with exponents α above TC and α 74 Calculation of Thermodynamic Properties from Molecular Simulation

below TC

α (T T )− if T > T  − C C Cv  (3.82) ∼   α′ (TC T)− if T < TC  −  and, 

γ (T T )− if T > T  − C C βT  (3.83) ∼   γ′ (TC T)− if T < TC  −  where the critical exponent α and γ determined by evaluation of the renormalization group theory (α = 0.110 0.003 and γ =) [237]. Another response function is ± coeﬃcient of thermal expansion αp. The response functions are not independent to each other

β (C C ) = TVα2 and C (β β ) = TVα2. (3.84) T p − v p p T − S p

In order to that our f uid system to be thermally and mechanically stable the specif c heat and compressibility should be positive for all temperature. Eq. (3.84) implies that C C and β β (3.85) p ≥ v T ≥ S for all temperatures. In particular as T T we shall see that C C and → C p ≫ v β β . In Eq. (3.85) the equality holds true when either T = 0 or α = 0. T ≫ S p Chapter 4

Simulation Details

4.1 Introduction

This chapter outlines all the simulation details necessary to understand the simulation conducted for the Lennard-Jones f uid [14], Weeks-Chandler-Anderson [15] and MCYna water [16] potentials. All simulations were conducted in an NVEPG ensemble, described in Chapter Three. Since our systems are isolated it is not possible to use an external thermostats so that we will use a molecular thermostat to maintain the temperature of the systems. Initial conf guration in the simulations are face centered cubic lattice structure with a periodic boundary condition. This chapter is organized in such a way that frst we will present the integrators, cutoﬀ radius and Ewald summation followed by the potential and the simulation details of the LJ and WCA potentials. Finally we will present the simulation details of the MCYna water potential. All simulation were performed on Swinburne’s GREEN and VPAC’s TANGO and MASSIVE supercomputers. The calculation for the MCYna water are reported in real units. However as the calcu-

75 76 Simulation Details lations for both LJ and WCA potentials represent theoretical f uids, it is convenient to represent the results in terms of a system of reduced units. The advantage of using reduced units is that the results can be mapped onto any real system if the values of ǫ and σ are later determined. The reduction scheme of the thermodynamic quantities are summarized in Table 4.1.

Table 4.1: Summary of reduced thermodynamic quantities used for the LJ and WCA potentials [3, 167, 238].

Thermodynamic quantities Def nition of reduction unit

3 Number density ρ∗ = ρσ

Temperature T ∗ = kBT/ǫ

Energy u∗ = u/ǫ

3 Pressure p∗ = pσ /ǫ = Heat capacities C∗p,V Cp,V /kB = 3 Compressibilities βT∗ ,S βT,S ǫ/σ ffi = 3 Isothermal pressure coe cient γV∗ γV σ /kB ffi = Thermal expansion coe cient α∗P αPǫ/kB = √ Speed of sound ω0∗ ω0 m/ǫ ffi = 3 Joule-Thomson coe cient µ∗JT µJT kB/σ

2 Time t∗ = t ǫ/mσ p

The mass and the LJ constants used for the calculation of thermodynamic quantities of the LJ f uid and WCA potential are reduced to m = 1.0, σ = 1.0 and

ǫ/kB = 1.0. Here after the superscript asterisk will be omitted. Results of LJ f uid and WCA potentials (presented in Chapter Five and the Appendix A and B) are in a reduced units. However, the results of argon and krypton (presented in Chapter Integrators 77

Five) and water (presented in Chapter Six and Appendix C) are in SI units.

4.2 Integrators

In this section we will present two of the integrators used in the simulation. The position, velocity and acceleration of the classical particles can be calculated from Newton’s second using the appropriate integrators. For a classical particle the equation of motion is

fi(t) = mi ai(t) (4.1)

th where fi(t) is the net external force acting on the i particle of mass mi and ai is the acceleration. Since we will consider a system consisting of a single molecule of equal mass we will drop the subscript i here after. for no specif c reason we have used both Gear predictor and leap-frog integrators. In the molecular dynamics simulation for the LJ and WCA potentials we used the Gear predictor-corrector integrator scheme. The Leap-frog integrator is used in molecular dynamics simulation of water.

4.2.1 Gear predictor

The Gear predictor-corrector method [94,214,239] is a three step process which include predicting, calculating and correcting of position, velocity and acceleration. First, new positions, velocities and accelerations are predicted according to Taylor expansion 78 Simulation Details

∂r ∆t2 ∂2r ∆t3 ∂3r r(t + ∆t) = r(t) + ∆t + + + .... ∂t 2! ∂t2 3! ∂t3 ∂v ∆t2 ∂2v v(t + ∆t) = v(t) + ∆t + + .... (4.2) ∂t 2! ∂t2 ∂a a(t + ∆t) = a(t) + ∆t + .... ∂t The truncation in these expansions does not yield correct trajectories, which means that the values ‘predicted’ by the above equation must be ’corrected’ to yield the actual path. Second, the forces are evaluated at the new position and the acceleration ac(t + ∆t) will be calculated from from Eq. (4.1). Finally these accelerations are compared with the acceleration that are predicted from Eq. (4.2). The diﬀerence between the predicted and calculated accelerations

∆a(t + ∆t) = a (t + ∆t) a(t + ∆t) (4.3) C − is then used to correct the positions and velocities in the correction step

rC(t + ∆t) = r(t + ∆t) + c0∆a(t + ∆t)

vC(t + ∆t) = v(t + ∆t) + c1∆a(t + ∆t) (4.4)

aC(t + ∆t)/2 = a(t + ∆t)/2 + c2∆a(t + ∆t)

where ci (i = 0, 1, 2) is a constant (Gear corrector coeﬃcients) which depend on the nature of the time derivative denoted by i. The values of ci’s are given in [214,239]. A detailed implementation of this integration technique in a molecular dynamics simulation is described in [214]. Integrators 79

4.2.2 Leapfrog

The Leap frog integration technique [240] is used for integration of the equation of motion of water. This integration algorithm evaluates the velocities at half-integral time steps and uses velocities to compute the new positions

∆t r(t) r(t ∆t) v t − − (4.5) − 2 ! ≡ ∆t and ∆t r(t + ∆t) r(t) v t + − (4.6) 2 ! ≡ ∆t

From Eq. (4.6) we can calculate the late position of the particle

∆t r(t + ∆t) = r(t) + v t + ∆t (4.7) 2 !

The later position of the particle r(t +∆t) from the velocity-Verlet algorithm [241] is f(t) r(t + ∆t) = 2r(t) r(t ∆t) + ∆t2. (4.8) − − m

Comparing Eqs. (4.7) and (4.8) gives the velocity at half integral time

∆t ∆t f(t) v t + = v t + ∆t. (4.9) 2 ! − 2 ! m

The main advantage of this algorithm is that numerical imprecision is reduced because it uses diﬀerences between smaller quantities. The implementation of this algorithm in MD simulation is illustrated in [214]. Velocities are not def ned at the same time as the positions as a consequence kinetic and potential energies are 80 Simulation Details also not def ned at the same time and hence we cannot directly compute the total energy in the leap-frog scheme.

4.3 Periodic boundaries, cutoﬀ radius and simula-

tion time

To avoid surface effects and the limited volume of the system, periodic boundary conditions are required [3]. This involves surrounding the simulation box with periodic images. When particles leave the volume of the simulation box, they are replaced by their periodic image at the opposite side. The maximum cutoff value when using periodic boundaries is half of the box length. However it is computationally expedient to use smaller values. The value used will affect the accuracy of the simulation and should be investigated beforehand. The full values to the intermolecular potential are recovered using long range correction [3]. Before deciding the appropriate cutoff radius we run test simulations. To check the effect of cutoff radius on the thermodynamic behaviour of the LJ fuid we used two cutoff values, half box length and 6.5σ, which are longer than the usual 2.5σ [236,238,242]. Noise in the molecular dynamics simulation are mainly due to the application of a cutoff in the interaction. Based on the test simulations we have decided to use a longer cutoff radius for simulations above and below the critical point in a LJ potential, 6.5σ. In MCYna water the cutoff radius of the MCY potential and polarization is half box length and 0.81. The length of the simulation time plays a paramount role in the accuracy of the simulation results. The effect of the total simulation time, which is a result Ewald summation 81 of either single simulation step length ∆t, total number of steps or both, on the simulation results in LJ and WCA potentials is examined. The values of ∆t for water simulations should be longer than the natural vibrational frequency [243]. We choose a simulation time which will give a reasonable stability in the system, ∆t = 0.002 ps. Equilibration is especially important when the initial conf guration is a lattice, and the state point of interest is in the liquid region of the phase diagram [3]. Depending on the nature of the fuid we have used at least 60% of the total simulation time for equilibration.

4.4 Ewald summation

In order to achieve a macroscopic behaviour for investigated quantities in computer experiment based on the observation of f nite systems, it is necessary to reduce the inf uence of surface eﬀects to a minimum. This is especially important for polar systems with the long-range nature of interactions. Long range interactions such as Coulombic or dipolar interaction in molecular dynamic simulations have a very long inf uence further than half the simulation box. Incorporating long range interactions in simulation is a complex problem. The simple way out for it could be using a very large simulation box, however using a very large simulation box is computationally expensive.

For a polar molecules minimizing surface eﬀect by truncating the potential is not a solution rather we have to use either the reaction feld [3,214,243,244], Ewald summation [3, 214, 245, 246] or Particle-Particle and Particle-Mesh [214, 247,248] techniques.

The advantage of Ewald sum over other methods is that Coulombic systems 82 Simulation Details can be simulated with a “full” Hamiltonian, without having to restore to arbitrary truncations of the intermolecular potential [245]. In addition to this, the ability to produce correct results, computational eﬃciency and following trend makes the Ewald sum to be our preferred choice to calculate long range interaction of water. The Ewald sum is adequate for phase diagram calculation since it can be used not only for the f uid phase but also for the solid phase [44]. Comparison of diﬀerent truncation methods is given in the work of Linse and Anderson [249] and in Sadus [214]. Detailed implementation of Ewald sum in a molecular dynamics simulation is given in Sadus [214], de Pablo et al. [245,250] and Adams [246].

The following discussion about Ewald summation is from [214]. The electrostatic interaction potential between N molecules is

N N 1 qiq j U = (4.10) 2 r Xi=1 Xj=i i j where the half is to avoid over counting. If the simulation box is cubic, there will be six periodic images of the central particle at a distance L with coordinates (0,0,L), (0,0,-L), (0,L,0), (0,-L,0), (L,0,0) and (-L,0,0). Modifying Eq. (4.10) to calculate the contribution of the Coulombic interaction from all images in the surrounding boxes and the charges in the central box is

6 N N 1 qiq j U = (4.11) 2 r + r nboXx=1 Xi=1 Xj=1 | i j box| where rbox is the coordinates of the six images to the central box. Following the same procedure it is possible to construct fve periodic images for each of these new boxes. A sphere of simulation boxes is constructed by continual repetition of Ewald summation 83 this process. in general, the energy of interaction of an ion and all of its periodic images can be obtained from

N N 1 ∞ qiq j U = (4.12) 2 r + n Xn=0 Xi=1 Xj=1 | i j | where the summation over n is taken over all lattice points, n = (nxL, nyL, nzL) where nx, ny, nz are integers. Eq. (4.12) is the Ewald sum. The above equation converges very slowly. The solution is to divide the above equation into two series of which converge more rapidly. The assumption to do so is to assume that each charge is surrounded by a neutralizing charge distribution of equal magnitude but of opposite sign, the typical charge is Gaussian charge distribution.

q α3 ρ (r) = i exp( α2 r) (4.13) i π3/2 −

The new summation is often referred to as the ‘real space’ summation. The real space energy is

N N 1 ∞ qiq j erfc(α ri j + n ) Ureal s = | | (4.14) − 2 r + n Xi=1 Xj=1 Xn=0 | i j | where erfc is the complementary error function given as [210]

2 ∞ erfc(x) = 1 erf(x) = exp( t2) dt (4.15) − √π Zx − where erf(x) is the error function (where erf( x) = erf(x) which implies that − − erfc( x) = 1+erf(x) = 2 erfc(x)), α is chosen such that only terms corresponding − − to n = 0 survive, i.e., the interaction which involve charges in the central box. In 84 Simulation Details this work we use α = 2.75 as reported by Li [21].

N N 2 1 ∞ 4πqiq j k Urecip s = exp cos(k.ri j) (4.16) − 2 k2L3 4α2 ! Xk,0 Xi=1 Xj=1 where k = 2πn/L2 are the reciprocal vectors. The summation of Gaussian functions in real space also includes the interaction of each Gaussian with itself. The eﬀect of this interaction on the energy is:

α N U = q2 (4.17) Gauss − √π i Xk=1 this potential does not contribute to the phase space function Ωmn (Eq. (3.16)) terms since it does not depend on the position of the particle (Eqs. (3.76) and (3.77)). The medium surrounding the sphere of simulation boxes must also be considered in the calculation because the sphere can interact with its surrounding. No correction is required if the surrounding medium is a good conductor characterized by inf nite relative permittivity (ǫ = ). However, if the surrounding 0 ∞ medium is vacuum (ǫ0 = 1), the following correction applies.

2 2π N Ucorr = qiri (4.18) 3L3 Xk=1

The f nal expression for the Coulombic interaction, Eq. (4.10) is

U = Ureal s + Urecip s + UGauss + Ucorr. (4.19) − −

The potential drawback of the Ewald sum method is that it is a model for an inf nitely periodic system which is used to simulate a periodic system, naturally Simulations of LJ and WCA potentials 85 all the systems are non periodic [215].

4.5 Simulations of the Lennard-Jones and Weeks-

Chandler-Anderson potentials

The MCYna water potential was previously described in Chapter Two, Eq (2.3). Here the potential and simulation details of each of the LJ and WCA potential are given below. The LJ potential is a prototypical model potential for simple f uids. It is one of the simplest models capable of reproducing thermodynamical behaviour of classical f uids, gases, and model dispersion and overlap interactions in molecular models [251]. It is particularly accurate for noble gas atoms and is a good approximation at long and short distances for neutral atoms and molecules. The LJ potential has the following mathematical form [14],

σ 12 σ 6 ui j = 4ǫ (4.20)  ri j ! − ri j !      where ǫ and σ are the depth of the potential well and the f nite distance at which the inter-particle potential is zero respectively and ri j is the distance between two particles i and j. The constants of LJ potential can be obtained from second virial coeﬃcients [252,253], viscosity data for gases [254,255] or from the position of the critical point [256,257]. The later having a deviation as high as 20% on the value of ǫ [254,256]. The positions of the critical and triple point of LJ f uid are given in Table 4.2. In Table 4.2 the subscripts ‘t’ and ‘c’ represent triple and critical points respectively. 86 Simulation Details

Table 4.2: Triple and critical points of the LJ f uid (source [225, 251, 252, 258]).

Properties Tt ρt (liquid) ρt (solid) Tc ρc pc Values 0.68 0.85 0.96 1.312 0.316 0.13

12 The repulsive term (r− ) describes Pauli repulsion at short ranges due to over- lapping electron orbitals. At intermediate distances it is signif cantly attractive but non-directional and competes with the directional attractive electrostatic interac-

6 tions. The long-range attractive term (r− ) represents van der Waals or dispersion force which can be ignored at high temperature [258]. The functional form of the attractive term has a clear physical justif cation but the repulsive term has no theoretical justif cation. It is used because it approximates the Pauli repulsion well, and is more convenient due to the relative computational eﬃciency of calculating

12 6 r− as the square of r− , and there are suggestions that an exponent of 9 or 10 gives better results [259]. The LJ potential is an approximation to Buckingham potential [260], which has an exponential repulsive part, given as

Br 6 u = Ae− i j Cr− (4.21) i j − i j where A, B and C are constants . The above equation can be rewritten as [259]

α σ 6 6 (1 ri j/σ) ui j = ǫ e − (4.22) α 6 − α 6 ri j !   − −    where ǫ and σ are the same LJ constants used in Eq.( 4.20), and α is a free parameter which can be considered as a f tting constant. Choosing α value of 12 gives a long-range behaviour identical to the LJ potential, while a value of 13.772 Simulations of LJ and WCA potentials 87 reproduces the LJ force constant at the equilibrium distance. The Buckingham potential diverges to inf nity when the distance between the two non bonded atoms becomes small. This characteristic makes the utilization of this potential difficult in simulations. The other potential that we used in our simulations is WCA potential [15] which is a modifcation of the LJ potential with only the repulsive part, short range and having a smooth cutoff. This potential treats the attractive term in LJ as a perturbation [261]. Due to the repulsive nature of the potential there are no triple and critical points in WCA potential. Ahmed and Sadus [28] claimed that system size has an effect on thermodynamic quantities during solid-liquid phase transition in a system interacting with WCA potential. The application of this potential is described in Ahmed and Sadus [28], and Hess et al. [261]. WCA potential has the following mathematical form

12 6 σ σ 1/6 4ǫ + ǫ, if ri j 2 σ  ri j ri j u = − ≤ (4.23) i j   1/6 0, if ri j > 2 σ    where ǫ and σ are the LJ constants. The potential profle of the LJ, WCA and

Buckingham (for α = 12) potentials are shown in Figure 4.1. 88 Simulation Details

Figure 4.1: WCA (solid line), LJ (dot) and Buckingham potentials (red dash) versus distance in reduced unit.

4.5.1 LJ potential

The effects of system size and cutoff radius on the thermodynamic quantities of particles interacting via LJ potential, Eq. (4.20), both above and below the critical point is studied. We conducted test simulations in systems with different simulation setup. To study the effect of system size we considered two systems consisting of 1000 and 2000 particles, and to study the effect of cutoff radius on the simulation result two longer than usual radii were used at 6.5σ and half box length. The standard long range corrections [3,214] were applied. The total simulation time of the test simulation run is 10,000,000 steps, i.e., a total simulation Simulations of LJ and WCA potentials 89 time of 20ps with a time step of ∆t = 2 f s. The simulations used 8,000,000 steps for equilibration. The temperature and densities range of our simulations were 0.7 T 2.624 ≤ ≤ and 0.1 ρ 1.0 respectively. The temperatures used in the simulation are ≤ ≤ multiples of the reduced critical temperature of the LJ fuid, TC = 1.312. To simplify the discussion, we denote these multiples using the symbol τ = T/TC, where for example τ = 1.05 means that T=1.05 T = 1.3776. × C To decide the best simulation setup below and above the critical point, pressure and potential from all the simulation setups are compared in Chapter Five. The simulation results from a setup with better prediction will be analyzed in Chapter Five. All the simulation results are give in Appendix B. The results of the LJ f uid will be used to calculate the thermodynamic quantities of argon and krypton using the appropriate LJ constants given in Table 5.3.

4.5.2 WCA potential

The effects of system size and total simulation time on the thermodynamic quantities of particles interacting via WCA potential, Eq. (4.23), is studied. We conducted test simulations in systems with different simulation setup. To study the effect of system size we considered two systems consisting of 1000 and 2000 particles. To study the effect of total simulation time we run the simulation for 10,000,000 steps (with ∆t = 2 f s) and 6,000,000 steps (with ∆t = 1 f s). The simulations used 8,000,000 and 5,000,000 steps for equilibration respectively. The simulation start from a face centered cubic (fcc) lattice.

The total simulation time for a system consisting of 2000 particles interacting 90 Simulation Details with WCA potential is 297 cpu hours. In contrast to this the simulation time for virtually the same system consisting of 1000 particles is 100 cpu hours on MASSIVE supercomputer. The total simulation time for system which run for six million steps (12 ns) is about 100 cpu hours where as the system which run for ten million steps (20 ns) is about 170 cpu hours.

4.6 MCYna potential

The MCYna potential [21], Eq. (2.3), in an NVEPG ensemble has been used in a molecular-dynamics simulation study of liquid water at a density of 55.371 dm3/mol (i.e., 0.998 g/cm3 or 100 molecules per (nm)3) and temperature between

298 - 645 K. The system consisted of 500 water molecules conf ned to a cubic box and subject to periodic boundary conditions. There are electrostatic, three body and polarization interactions in MCYna model, the induced dipole moment is calculated using conjugate gradient method. Ewald summation (with a convergence parameter of α = 5.0/L) for the long-range Columbic interactions. The real-space cutoff for the Ewald sum was L/2 where L is the box length, the reciprocal-space cutoff was 5/2L, and the screening parameter was set to 5.6/L. A spherical cutoff radius equal to half the box length, 12.331 Å, is used in evaluating potentials and forces. The forces due to the the intermolecular interaction and Ewald summation were evaluated and the Newtonian equations of motion were then solved by leap-frog algorithm for each atom. The simulation start from a face centered cubic (fcc) lattice. Positions and velocities of all the atoms were collected every 10 time steps for later analysis of the static and dynamic properties of the liquid water.

The centre of mass of water molecules are placed on a fcc crystal lattice as the MCYna potential 91 initial conf guration. The basic cubic length of the fcc crystal is determined by the number density corresponding to simulation conditions. We set the initial number of molecules and density (in fact, number of molecules per (nm)3) from that the volume will be calculated. The box length is adjusted according to the density of water and the box length for the ensemble are determined as 2.4631 nm when the density of water is 100 molecules per nm3. In order to determine the challenging thermodynamic quantities of water, such as the maximum density, the simulation run should be very long [52]. The simulation run is for 500,000 steps with a single

15 time step of 2 f s (where the total simulation time is 500000 2 10− s = 1 ns) × × out of which we have used 400,000 steps for equilibration. The potential energy is monitored during the equilibration. In all the equilibration simulation, the system temperature is scaled at every step using molecular thermostat, molecular kinetic energy. Equilibrium is assumed to be achieved when the potential energy is signif cantly stabilized during the equilibration. After equilibration is reached the molecular thermostat will be switched off. This simulation takes about 100 hrs on a f ve node Swinburne’s green supercomputer. The electrostatic interaction between point charges are evaluated with Ewald technique. The SHAKE algorithm [214,262] is applied to constrain the molecular structure which enables moderate convergence speed and accurate constraint on water molecules. In this model there is one polarizable site on the negative charge center of each water molecule. Intramolecular interaction is ignored, for simplicity, which lead that the induced dipole does not interact with the partial charge on the same water molecule. Gas phase polarizability coefficient (1.44 Å3) [36] gives a larger value for dipole moment which shows the incompatibility experimental value of 92 Simulation Details gas with the polarization term in liquid water, to improve the induced dipole, scale the polarizability coefficient by a factor of 0.557503 and so the polarizability coefficient becomes 0.802804 Å3. This givesa dipole moment of 2.9 D, with 0.9D attributed to induction interaction. The three-body interaction is assumed only between oxygen atoms, because of the electron poor feature of hydrogen. The value for Axilrod-Teller coefficient is the value for argon (5/9), since there is no known value for oxygen. At every dynamical step, we compute the electric f eld at site i produced by the f xed charges in the system using Eq. (2.6). The calculated electric f led is then used Eq. (2.7) to generate the initial estimate of the induced dipole moments [263]. The initial estimate of the induced dipoles and the calculated electric f eld are used to compute the total electric f eld Ei. Chapter 5

Thermodynamic properties of Lennard-Jones and Weeks-Chandler-Anderson f uids and Noble Gases

5.1 Introduction

We present calculations for a wide range of thermodynamic state points. For the case of the LJ f uid, these calculations are both above and below the critical point. We have used results from LJ simulation to calculate the thermodynamic quantities of selected supercritical noble gases, namely argon and krypton. By comparing results with the experimental data and earlier simulation work in the literature we will check the validity of our simulations. Results from repulsive

WCA potential, which does not have a critical point, is also used as a basis for

93 94 Thermodynamic Properties of LJ and WCA fuids and Noble Gases comparison.

Mastny and de Pablo [251] investigated the effects of parametrized potentials, fnite size and fnite cutoff radius (beyond which there is no particle correlation [166]) in calculating the thermodynamic properties of LJ f uid along the melting curve in a thermodynamic limit. Smit and Frenkel [264] and Smit [265] showed that the phase diagram (position of the critical point) depends largely on the details of the truncation of the potential. It is shown that, for many bulk properties the nature of the truncation makes little difference, however for the liquid-vapor coexistence curve a small change in it makes a very large difference [264,266,267]. Morris and Song [268] found out that there is no appreciable effect from f nite size and f nite cutoff on the calculation of melting temperature of LJ fuid. Blas et al. [269] studied the effect of truncation on thermodynamic and interfacial properties of f exible LJ chain and found that all the calculated properties have dependence on the cutoff distance to a different level. Vogt et al. [267] recommended a larger cutoff length than the usual 2.5σ. Ahmed and Sadus [270] investigated the effect of potential truncation and shifts on the solid-liquid phase coexistence of LJ f uids and found that the truncation has an effect in the vicinity of triple point and on the liquid phase.

Mausbach et al. [271] showed that simulations around the phase transitions might be sensitive to the system size of a f uid. Sadus [272] found that the incorporation of dipole has increased the critical temperature and predicted the critical temperature of LJ f uid to be less than 1.3. Varied simulation setups used different truncation length, position of critical point, ensembles and system size making it difficult to get a simulation result to compare with. Adams [225] used a simulation setup of larger system consisting of 6000 par- Introduction 95 ticles and longer cutoff radius of 4σ near the critical point in LJ f uid and water.

We have investigated the effect of f nite size and f nite cutoff radius on a system of particles interacting with LJ and WCA potentials. To asses the effect of truncation on different thermodynamical properties of a LJ f uid above the critical point we have used two different truncation lengths at 6.5σ and half box length, both of which are longer than the usual cutoff radius of 2.5σ, the later depending on the number of particles. Errington [273] tried to f nd a scaling relation between two sets of data calculated considering either the f nite size effect or the fnite cutoff radius to develop a scaling relation that can be used to calculate an inf nite system size. This chapter is organized in such a away that f rst we will discuss the simulation results of the LJ and WCA potentials and compare the results from [60,161, 166,221,242,261]. Using the appropriate values of the LJ constants for argon and krypton (given in Table 5.3) we will calculate the thermodynamic quantities of argon and krypton from the LJ f uid simulation. Finally, we will compare the simulation results of argon and krypton with experimental [221–223,253,274,275] and equation of state [237,274] results. Note that all values in LJ f uid and WCA potential are dimensionless (see Chapter Four for the relations) whereas the results for argon and krypton are in SI unit. 96 Thermodynamic Properties of LJ and WCA fuids and Noble Gases

5.2 Thermodynamic properties of LJ and WCA po-

tentials

We examine the effect of system size, cutoff radius and length of simulation time on thermodynamic properties of WCA potential and LJ f uid prior to presenting the simulation results. Based on this analysis we will decide the optimum simulation setup, i.e., number of particles in the system, length of cutoff radius and length of total simulation time for different potentials. First, to study the effect of system size on thermodynamic quantities of the WCA potential we have compared the potential energy and pressure of two systems consisting of 1000 and 2000 particles. Figure 5.1a and 5.1b respectively illustrates the potential energy and pressure of two different systems consisting of 1000 and 2000 particles. The temperatures are given as a multiple of the LJ critical temperature τ as discussed in Chapter Four. As shown in Figure 5.1 (data given in Appendix A), the potential energy and pressure calculated in both systems are the same at each density in the entire simulation region. The potential energy and pressure of system of particles interacting with WCA potential are smooth functions of density. Based on the premises that the thermodynamic variables of the

LJ potential vary with the system size [251,268], Ahmed et al. [28] expected the thermodynamic variables will be affected by the system size the same was as in LJ potential. To study the effect of length of single simulation step (∆t) and total simulation time on the thermodynamic properties of particles interacting with the WCA potential we considered three different simulation setups where each system consists of 1000 particles and τ = 1.05 with a varied number of simulation steps and ThermodynamicpropertiesofLJandWCApotentials 97

∆t. The difference in total simulation time is a result of difference in number of simulation steps, ∆t or both. The frst two systems are setup to run for a total of six million steps with ∆t = 1 f s or 2 f s. The third system is setup to run for ten million steps with ∆t = 2 f s. Figures 5.2a and 5.2b respectively illustrate the potential energy and pressure of the three different simulation setups with a varying simulation time. In order to get a clear view of each of the values of the potential energy and pressure the values of each in two of the systems were shifted by adding or subtracting 0.1 in the potential and 0.01 in the pressure. As shown in Figure 5.2a the value of the potential in a simulation setup which runs for 20 ns (i. e., 10 million steps and ∆t = 2 f s) is rescaled by adding 0.1 in the simulated value. Whereas, the values of the potential in a simulation setup which runs for 6 ns (i. e., 6 million steps and ∆t = 1 f s) is rescaled by subtracting 0.1 from the simulated result. Onto the same simulation setups a value of 0.01 is used to rescale the pressure of the system, shown in Figure 5.2b. Had there not be any rescaling of the potential energy and pressure, as shown in the respective f gures, the potential energy and pressure calculated in the three simulation setups are the same throughout the simulation. Which we can draw a conclusion that the neither number of simulation steps nor the single simulation steps (generally the total simulation time) affect the simulation time. However, the total simulation time should give enough time for equilibration. 98 Thermodynamic Properties of LJ and WCA fuids and Noble Gases

Figure 5.1: Potential energy (a) and pressure (b) as a function of density in systems interacting with the WCA potential, where τ = 1.10(), 1.25(), 1.40(N), 1.60(H), 1.70() and 1.90(). Results for N = 1000 are represented by the symbols, whereas the solid lines represent calculations for N = 2000. ThermodynamicpropertiesofLJandWCApotentials 99

Figure 5.2: Potential energy (a) and pressure (b) as a function of density in a system consisting of 1000 particles interacting with the WCA potential at τ = 1.05 where the total simulation time is 6 ns(), 12 ns() and 20(N) ns. The values of 6 ns and 20 ns are shifted by the values given in the f gure for clarity. 100 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

In Figure 5.1 and 5.2, we have shown that systems interacting with the WCA potential with larger number of particles or longer simulation time neither give a better simulation result nor a different one than the alternative simulation setups. In deciding the simulation setup (as discussed in Chapter Two) the prominent factor is the efficiency of simulation in terms of the simulation time and validity of the results. It could be wiser to do a simulation on a system with fewer particles, and of course shorter simulation time, if neither the system size nor simulation time does not affect the results. However, from the statistical point of view it is important to make the system as large as possible, for this reason and another reason to be explained next we will present simulation results of a system consisting of 2000 particles which run for 12 ns (six million steps with ∆t = 2 f s) in an NVEPG ensemble. Simulation data of a system consisting of 2000 particles are given in Appendix A. To test the effect of system size and cutoff radius on the thermodynamic quantities of a LJ fuid four different simulation setups are arranged. As explained in Chapter Four, the different simulation setups will have either 1000 or 2000 particles with a cutoff radius of either 6.5 or half box length. Results of the thermodynamic quantities from the simulation setup which show a relative stability along the isotherm in the simulated region, computationally cheap and statistically reliable will be presented later. For the purpose of comparison we will show results of potential and pressure from these four simulation setups. All the simulation data are give in Appendix B.

Potential energy and pressure of LJ f uid as a function of density at different constant temperatures from a different simulation setups are illustrated in Fig- ure 5.3. As shown in Figure 5.3a, the potential energy in all simulation setups ThermodynamicpropertiesofLJandWCApotentials 101 along the different isotherms are almost the same, the different values from different simulation setups and temperature are indistinguishable from each other. Similar to the potential energy, the isothermal pressure coefficient from different simulation setups are indistinguishable. The system pressure, shown in Fig- ure 5.3b, along the isotherms for different simulation setups is the same. Values of the Joule-Thomson coefficient, adiabatic compressibility and speed of sound along the isotherm from different simulation setups are also comparable. Results for the potential energy and pressure (Figure 5.3) implies that number of particles in the system and cutoff radius does not affect the potential, pressure and their derivative thermodynamic quantities such as speed of sound, adiabatic compressibility and isothermal pressure coefficient. Our simulation result is a recon- f rmation to Adams [225] result, the truncation length on the LJ potential neither affect the pressure nor make a signif cant change on isochoric heat capacity and compressibility.

Near the critical point most of the response functions, such as the isochoric heat capacity, f uctuates along the isotherm attaining maxima and minima in all the simulation setups. As discussed in Chapter Three, response functions will diverge when the state points of the system are very close to the critical point.

The f uctuation of heat capacity along the isotherm is common for all response functions in all the simulation setups near the critical point and is conf rmed experimentally [276], by previous simulations [242, 277] and by EOS [160,191]. The simulation setup to predict these f uctuating and worse diverging response functions should be statistically reliable, i.e., should have large system size and longer simulation time. 102 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Figure 5.3: (a) Potential energy as a function of density at different constant temperatures for the LJ f uid. Results are for a system consisting of 1000 and 2000 atoms with a cutoff radius of 6.5σ and L/2 at τ = 1.05, 1.10 and 1.15 and (b) pressure as a function of density at different constant temperatures for the LJ f uid. Results are shown for 1000 and 6.5σ (), 1000 and L/2 (), 2000 and 6.5σ (N) and 2000 and L/2 (). The temperature is shown in the f gure for clarity. ThermodynamicpropertiesofLJandWCApotentials 103

Apart from getting a longer simulation time for larger systems, neither the system size nor the cutoff radius (so long as it is fairly long) affect most of the calculated thermodynamic quantities in a region away from the critical point. However, to minimize the f nite size effect (in simulations fairly close to the critical point) and to improve the statistics of the result we prefer to present results from a system with large number of particles, longer cutoff radius and simulation time. We present simulation results of a system consisting of 2000 LJ particles with a cutoff radius of 6.5 which run for 20 ns in an NVEPG ensemble. To be compatible with this simulation setup we used 2000 particles in a system of particles interacting with the WCA potential. We have checked the effect of system size, cutoff radius and simulation time on LJ and WCA potentials. We also decided the simulation setups to be presented, here after. We will examine the contribution of the attractive part of LJ potential on the thermodynamical properties of the f uid. To achieve this we deduct the thermodynamic quantities of WCA from those values calculated in the LJ f uid.

5.2.1 Energy and pressure

The potential energy as a function of density at diﬀerent constant temperatures from WCA and LJ potentials is illustrated in Figure 5.4a and 5.4b respectively.

The interaction potential energies of both potentials depend on the density and temperature. The potential energy in WCA potential is repulsive, increase linearly in the lower and middle density region (0.1 ρ 0.4) and increases monoton- ≤ ≤ ically faster in the higher density region along the isotherm. As shown in Fig- ure 5.4b the attractive potential energy in LJ f uid increases linearly with density 104 Thermodynamic Properties of LJ and WCA f uids and Noble Gases along the isotherm and the potential in the lower density region from different isotherms is the same. In the entire simulation region the potential energy calculated from both potentials is continuous. At each state point, independent simulations were performed for the WCA and LJ potentials. To illustrate the difference in the thermodynamic properties we represent the data by subtracting the WCA values from the LJ values, for example the difference in the potential energy is represented as ∆u = u(LJ) - u(WCA). In the content of the discussion below the ‘∆’ symbol in front of any thermodynamic quantity refers to this type of difference. In doing so, we are in effect isolating the contribution of the negative part of the LJ potential i. e., regions where u < 0. We will refer to this as the ‘attractive’ contribution of the LJ potential. Therefore, apart from having a convenient way of representing the data for the two potentials, it also allows us to approximate the relative inf uence of attractive and repulsive interactions. These inferences are only approximate because the negative part of the LJ potential has also repulsive characteristics, particularly at small separation. Furthermore, there is a small discontinuity between the WCA potential and the negative LJ well at u(r) = 0 [156]. Recently Morsali et al. [168] reported the repulsive and attractive contribution to heat capacity by performing separate simulation using the WCA separation of attractive and repulsive contributions. Although such approach allows the isola- tion of purely repulsive and attractive contribution, it neglects the transition region connecting positive and negative values of the LJ potential.

∆u (i.e., attractive part of the LJ potential) as a function of density and temperature at different constant temperatures and densities is illustrated in Figures 5.5. As shown in Figure 5.5a, the attractive potential energy increases linearly with ThermodynamicpropertiesofLJandWCApotentials 105 density along the isotherm and the values from different isotherms are almost the same which implies that the attractive part of the LJ potential does not depend on the temperature of the system. Comparing Figure 5.4b and 5.5a, the attractive part of the LJ potential in the higher density region is higher than in the lower density region and the net attractive potential of LJ fuid increased by 0.5 at all points from the LJ potential. Though small the effect of attractive term in LJ potential is central to the near critical behaviour. The attractive part of LJ potential decreases almost linearly with temperature along the isochore, Figure 5.5b. The decrease in the attractive potential along the isochore is insignif cant compared to the increase along the isotherm. In the absence of the WCA repulsive term contribution the potential energy of the LJ f uid is almost independent of temperature, Figure 5.4a. 106 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Figure 5.4: Potential energy as a function of density at diﬀerent constant temperatures for (a) WCA and (b) LJ f uid, where τ = 1.17(), 1.25(), 1.40(N), 1.60(H), 1.80(), 2.0() and the solid lines in WCA potential (Figure 5.4a) are polynomial f t to the data and are for guidance only. ThermodynamicpropertiesofLJandWCApotentials 107

Figure 5.5: ∆U as a function of (a) density at diﬀerent constant temperatures where τ = 1.17(), 1.25(), 1.40(N), 1.60(H), 1.80() and 2.0() (b) temperature at diﬀerent densities where ρ = 0.1(), 0.2(), 0.25(N), 0.3(H), 0.35(), 0.4() and the solid lines are linear f ts to the data and are for guidance only. 108 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Pressure is a continuous function of density and sensitive to the shape of the potential [267]. Pressure calculated for the LJ f uid and ∆p as a function of density at diﬀerent constant temperatures are illustrated in Figures 5.6. As shown in Figure 5.6a, pressure calculated for the LJ f uid is continuous and increases monotonically along the isotherm in the supercritical region, however for a density less than and equal to 1.3 (data is given in Appendix Table A.1) the simulation predicted a negative pressure which indicates that the f uid is in a metastable state and related with the virial coeﬃcient of the fuid. As a result of the repulsive nature of the potential, shown in Figure 5.4a, in the entire simulation region the pressure calculated from WCA potential is higher than the pressure in the LJ f uid, Fig- ure 5.6b. Similar to the potential energy, the pressure by the repulsive part of the LJ potential does not depend on the temperature of the system. Comparing results from Figure 5.6a and 5.6b it will be clearly seen that the pressure from Weeks- Chandler-Andersen potential is twice or more than the pressure from LJ f uid at lower temperature and high density. The pressure from the attractive part of the LJ f uid (i.e., ∆p) is insensitive to the temperature (Figure 5.6b). ThermodynamicpropertiesofLJandWCApotentials 109

0.0

(b)

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

-3.5

-4.0

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

Figure 5.6: (a) Pressure calculated in the LJ f uid and (b) ∆p as a function of density at diﬀerent constant temperatures where τ = 1.17(), 1.25(), 1.40(N), 1.60(H), 1.80() and 2.0(). The solid lines in Figure 5.6a are polynomial f ts to the data and are for guidance only. 110 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

∆p as a function of temperature at diﬀerent constant densities is illustrated in Figures 5.7. As shown in Figure 5.7, the diﬀerence between pressures (i. e., ∆p) is constant along the isotherm. In both WCA and LJ f uid pressure increases linearly along the isochor consistent with an ideal gas law (p ∝ T) with a varying slope proportional to the density of the system. It is shown that denser systems to be more pressurized.

Figure 5.7: ∆p as a function of temperature at diﬀerent constant densities where ρ = 0.1(), 0.2(), 0.25(N), 0.3(H), 0.35(),0.4() and the solid lines are linear f ts to the data and are for guidance only. ThermodynamicpropertiesofLJandWCApotentials 111

5.2.2 Isochoric and isobaric heat capacities

Isochoric heat capacity calculated by Hess et al. [261], using f uctuation formula, and from our simulation (not shown, data given Appendix Table A.2) for a system of particles interacting with WCA potential increase monotonically along the isotherm with density showing no peak or minima observed in the LJ f uid. This observation implies that the extremum values of heat capacity, perhaps more generally the critical behaviours, in the LJ f uid are results of the attractive term in the potential.

Heat capacities behave erratically near the critical density. This erratic behaviour and divergence of the heat capacities near the critical values, shown in Eq. (3.82), is a result of fnite size effect and has been experimentally conf rmed [276] and also shown by previous simulations [242,278]. Isochoric heat capacities show maxima and minima when plotted as a function of density for a range of temperatures above the critical point [242]. Isochoric heat capacity which can be expressed using the power law (Eq. (3.82)) predicts a weak divergence at the critical point [237]. The f uctuation in heat capacities along the isothermal curve near the critical point can be improved by increasing the system size but can not be avoided. However, it should be remembered that increasing the system size increase the computational time by the square of the number of particles. Isochoric heat capacity of LJ f uid, from our simulation and Freasier et al. [242] which has a cutoff radius of 2.5σ and using the f uctuation formula, as a function of density at different constant temperature is illustrated in Figure 5.8a. In Figure 5.8a, isochoric heat capacity shows a maximum and minimum values as the density increases along the isotherm as predicted by Gregorowicz et al. 112 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

[6]. Fluctuation of isochoric heat capacity along the isotherm near the critical density is observed in both f gures. The f uctuation of isochoric heat capacity along the isotherm disappears and f nally becomes a smooth increasing function of density at higher temperatures. The peak in the heat capacity indicates the presence of a critical point at or near that point. As a result of the diﬀerence in the system size and cutoﬀ length in our simulation and Freasier et al. [242] simulation the peak and the magnitude of f uctuation in the isochoric heat capacity along the isotherm varies signif cantly at temperatures either below or above the critical temperature, Figure 5.8a. However, heat capacity calculated at higher temperatures are comparable.

∆Cv as a function of density at diﬀerent constant temperatures is illustrated in Figures 5.8b. Subtracting WCA values greatly reduces the heat capacity, showing that most of the contribution is from the repulsive interaction. Nonetheless, the attractive interaction is required to observe the maximum in the heat capacity.

At high temperature where the repulsive part of the LJ potential dominates, comparing Figure 5.8a and 5.8b (data is given in Appendix Table A.2 and B.2), it is clearly seen that the isochoric heat capacity from both potentials is comparable. ThermodynamicpropertiesofLJandWCApotentials 113

Figure 5.8: (a) Cv as a function of density at constant temperature from our simulation and Freasier et al. [242] in a LJ f uid where τ = 1.3(), 1.4(), 1.6(N) and 1.8(). The open symbols are for Freasier et al. and corresponding solid symbols and broken lines are from our simulation and (b) ∆Cv as a function of density at diﬀerent constant temperatures where τ = 1.17(), 1.25(), 1.40(N), 1.60(H), 1.80(), 2.0() and the solid lines are polynomial f ts to the respective data and are for guidance only. 114 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Isobaric heat capacity of LJ fuid as a function density at diﬀerent constant temperatures is shown in Figure 5.9a. Isobaric heat capacity f uctuates along the isotherm near the critical point, the magnitude of the f uctuations along the isotherm becomes smaller at higher temperatures away from the critical point. Isobaric heat capacity attained maximum value near the critical density along the isotherm and diverges, shown in Eq. (3.85), when moved very close to the critical point (not shown in the f gure but the data is given in Appendix Table B.3). Unlike the isochoric heat capacity isobaric heat capacity does not shown minima in the simulation region.

∆Cp as a function of density at diﬀerent constant temperatures is illustrated in Figure 5.9b. Since the WCA potential does not have critical point the isobaric heat capacity does not show those unusual f uctuating behaviours along the isotherm that we have observed in the LJ f uid near the critical point. The diﬀerence in the isobaric heat capacities from the two potentials is dominated by the values from the LJ f uid and will have the same shape and comparable values with isobaric heat capacity of the LJ f uid. The disparity between the isobaric heat capacities from the two potentials is signif cant near the critical point without as much reduction as seen in the isochoric heat capacity. The attractive part of the potential not only provide the peak but also dominate the whole value of the isobaric heat capacity. ThermodynamicpropertiesofLJandWCApotentials 115

Figure 5.9: (a) Cp(LJ) f uid and (b) ∆Cp as a function of density at diﬀerent constant temperature where τ = 1.17(), 1.25(), 1.40(N), 1.60(H), 1.80(), 2.0() and the solid lines are polynomial f ts to the data and are for guidance only. 116 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

To quantify what has been discussed so far, we have calculated the absolute average (AA) of isochoric and isobaric heat capacities between the simulation results of LJ and WCA potentials. The absolute average of a thermodynamic quantity ‘χ’ isdefnedas 1 N AA = χ , (5.1) N | i| Xi=1 where χi is the ith term and N is the total number of values of χ. The absolute average of heat capacities calculated using Eq. (5.1) is given in Table 5.1.

Table 5.1: Absolute average values of various ratios of heat capacities (χ)inLJandWCA potentials.

τ Cp(WCA) Cp(WCA) Cv(WCA) Cv(WCA) Cp(LJ) ∆Cp Cv(LJ) ∆Cv 1.17 0.309 0.536 0.858 7.574

1.25 0.385 0.722 0.888 9.585

1.40 0.485 1.029 0.925 13.938

1.60 0.573 1.425 0.951 20.562

1.80 0.640 1.885 0.966 29.841

2.00 0.685 2.292 0.973 37.323

The absolute average of the ratio of heat capacities (both isochoric and isobaric) of WCA to LJ increases with temperature. This implies that heat capacities from WCA potential increases faster than heat capacities from LJ potential. we can also infer that the contribution of attractive potential from LJ potential is higher at lower temperatures. The ratio of absolute average of isochoric heat capacity i. e., Cv(WCA) is almost one, with a diﬀerence under 3% at a temperature Cv(LJ) twice the critical temperature of the LJ f uid. This implies that the value of the ThermodynamicpropertiesofLJandWCApotentials 117 isochoric heat capacity of the WCA potential to be comparable with the value of the LJ potential along the isochore. This is reconf rmed by a very high value of the ratio of Cv(WCA) to ∆Cv, which goes as high as 37 at a temperatures twice the critical temperature of the LJ f uid. From this value we can draw the conclusion that the isochoric heat capacity calculated in the attractive part of the LJ potential is very small compared with the value calculated from WCA and LJ potentials.

The absolute average of the ratio of Cp(WCA) to ∆Cp increases with temperature, however the increase is not comparable to that of Cv(WCA)/∆Cv. This small value of the absolute average of the ratio of Cp(WCA) to ∆Cp implies that the isobaric heat capacity calculated from WCA potential is very small compared to the value calculated from the LJ potential. We can infer that the contribution of the attractive potential in LJ potential towards the isobaric heat capacity to be dominant.

5.2.3 Isothermal pressure coeﬃcient

Employing the thermodynamic identity Eq. (3.23) and from the maximum and minimum behaviour of isochoric heat capacity beyond the critical point, the isothermal pressure coefficient will have a constant value at the extremum points of the isochoric heat capacity (Figure 5.8a). As a result of which, at the maximum and minimum positions of the isochoric heat capacity there will be two linear lines in isothermal pressure coefficient when drawn as a function of temperature. It is only between these two linear isochore lines that isothermal pressure coefficient increases with temperature. It decreases above the minimum and below the maximum values of isochoric heat capacity with temperature. 118 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Isothermal pressure coefficients from the LJ fuid and WCA potentials as a function of temperature at different constant densities are illustrated in Fig- ure 5.10. The isothermal pressure coefficient calculated from the LJ and WCA potentials are the same, with a difference comparable to the error committed in the calculation at higher density and lower temperature region. The similarities of the value of isothermal pressure coefficient to the two different potentials implies that it depends only on the repulsive part of the potentials. It is also shown that the isothermal pressure coefficients of both potentials are almost constant (with a negative gradient) along the isochore. Systems with high density showed higher isothermal pressure coefficient. When density is between 0.3 ρ 0.4 both isothermal pressure coefficients deceases linearly with temper- ≤ ≤ ature along the isochore at the same rate. However in a density region higher than 0.4 isothermal pressure coefficient calculated for the WCA potential has higher negative gradient along the isochore than the same value calculated for the LJ f uid. Neither the system size nor the cutoff radius of the system affect the value of the isothermal pressure coefficient in both LJ and WCA potentials (not shown, data given Appendix Table A.8 and B.8). In the simulated region, the isothermal pressure coefficient is positive, which implies that both potentials do not show the density anomaly in this region. ThermodynamicpropertiesofLJandWCApotentials 119

Figure 5.10: Isothermal pressure coeﬃcient as a function of temperature at diﬀerent constant densities for the LJ f uid (symbols) and WCA potential (solid line), where ρ = 0.1(), 0.2(), 0.25(N), 0.3(H), 0.35() and 0.4().

5.2.4 Thermal expansion coeﬃcient

Thermal expansion coefficient is the measure of the tendency of matter to change in volume in response to a change in temperature. Near the critical point, as in all response functions, thermal expansion coefficient showed f uctuations along the isotherm. At temperatures close to the critical value thermal expansion coefficient attained a maximum value and at a point very close to the critical point it will diverge. The divergence of thermal expansion is caused by its dependence on isobaric heat capacity, shownin Eq. (3.84) which in turn is a result of the f nite size of the system, when all the particles in the system are highly correlated beyond 120 Thermodynamic Properties of LJ and WCA f uids and Noble Gases the system size.

The thermal expansion coeﬃcient for the WCA potential decrease monotonically along the isochore with temperature (not shown, data given in Appendix Table A.9). The decrease in thermal expansion coeﬃcient is larger at lower densities than at higher density. Changing the volume of a system consisting of fewer particles by changing the temperature at lower temperature is easier than changing the volume of the same system by changing the temperature of the system at higher temperature.

∆αp as a function of density at different constant temperatures is illustrated in Figure 5.11. The value calculated from the LJ potential is higher than values calculated from the WCA potential as a result of which the difference in the thermal expansion coefficient depends mainly on the contribution from LJ potential. Sim- ilar to isobaric heat capacity, thermal expansion coefficient attained a peak value when calculated near the critical point, it diverges at a temperature close to the critical point and do not have minima. For a temperature close to the critical value, thermal expansion coefficient of the LJ f uid attained a maximum value when the density is between 0.2 and 0.4 (not shown, data given in Table B.9) and so did the difference between the thermal expansion coefficient between the two potentials.

Increasing the temperature minimizes the f uctuation along the isotherm and the peak height. ThermodynamicpropertiesofLJandWCApotentials 121

Figure 5.11: ∆αp as a function of density at diﬀerent constant temperatures where τ = 1.17(), 1.25(), 1.40(N), 1.60(H), 1.80(), 2.0() and the solid lines are polynomial f ts to the data and are for guidance only.

5.2.5 Speed of sound

The speed of sound calculated for the LJ fuid and ∆ω0 as a function of densities at diﬀerent constant temperature is illustrated in Figure 5.12. As shown in Figure 5.12a the speed of sound increases almost linearly with density along the isotherm without f uctuation near the critical point. The speed of sound attained a minimum value near the critical point. The power law expression for speed of sound at the critical point predicts the speed of sound to be zero [237]. In the two phase region the LJ f uid, i.e., below the critical point τ 1.3, pressure is ≤ not def ned (having a negative value) as a result of discontinuity in density which 122 Thermodynamic Properties of LJ and WCA f uids and Noble Gases implies that the speed of sound is not def ned (not shown, data given in Appendix

Table B.6) in that region. The speed of sound in a system of particles interacting with WCA potential increases linearly with density along the isotherm. Neither system size nor cutoﬀ radius aﬀect speed of sound in both potentials. Pressure calculated from WCA potential is higher than the value calculated from LJ potential, Figure 5.6b. The speed of sound depends on pressure (Eq. (3.38)), so that the speed of sound calculated for the WCA potential will be higher than the speed of sound calculated for the LJ f uid. As shown in Figure 5.12b, the speed of sound from the attractive potential energy (shown in Figure 5.5) decrease almost linearly in the lower to middle density region (0.1 ρ <0.4), with some ≤ f uctuations along the isotherm. Figure 5.12a indicates that in the absence of the WCA contribution, the speed of sound would have physically unrealistic negative values. This highlights the important role of repulsive, particularly at small sep- arations, on thermophysical properties of f uids. The attractive potential energy makes particles in a colder system to move faster than particles in hotter system at any given density. It attains a minimum value when the density is around 0.4 (i.e., when the LJ f uid becomes supercritical) and increases when the density increases further. ThermodynamicpropertiesofLJandWCApotentials 123

Figure 5.12: (a) Speed of sound in LJ f uid and (b) ∆ω0 as a function of density at diﬀerent constant temperatures where τ = 1.17(), 1.25(), 1.40(N), 1.60(H), 1.80(), 2.0() and the solid lines are polynomial f ts to the data and are for guidance only. 124 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

5.2.6 Isothermal and adiabatic compressibilities

Isothermal and adiabatic compressibilities calculated for particles interacting with the WCA potential are smoothly decreasing with density along the isotherm and are the same except there is a scaling factor (not show, data given in Appendix Ta- ble A.4 and A.5). Both the isothermal and adiabatic compressibilities from WCA potential are small compared to the values from LJ f uid at the same temperature. The adiabatic compressibility calculated for the LJ f uid decreases with density with a minimum f uctuation near the critical point.

∆βs and ∆βT as a function of density at diﬀerent constant temperatures are illustrated in Figures 5.13. As shown in Figure 5.13a, the adiabatic compressibility decreases with density monotonically in the lower and middle density region (ρ < 0.4). The decrease in compressibility becomes steady for a density beyond 0.4 and compressibility become zero when the density is 0.6 or more for all temperatures. Colder systems of lower density attained higher adiabatic compressibility. Isothermal compressibility is positive in one phase region and diverges when there is discontinuity in the density, shown in Eq. (3.27). Isothermal compressibility f uctuates near the critical point and at the critical point where density of the system is discontinuous. Eq. (3.83) implies that isothermal compressibility diverges. As shown in Figure 5.13b, the isothermal compressibility decreases with density showing f uctuations along the isotherm near the critical point. Increas- ing the temperature smooths the curve and it becomes a monotonically decreasing function of density. ThermodynamicpropertiesofLJandWCApotentials 125

Figure 5.13: ∆βs (a) and ∆βT (b) as a function of density at diﬀerent constant temperatures where τ = 1.17(), 1.25(), 1.40(N), 1.60(H), 1.80(), 2.0() and the solid lines are a polynomial f ts to the data and are for guidance only. 126 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

5.2.7 Joule-Thomson coeﬃcient

The Joule-Thomson coefficient calculated for the LJ f uid and ∆µJT as a function of density at different constant temperatures are illustrated in Figures 5.14. As shown in Figure 5.14a, the Joule-Thomson coefficient for the LJ f uid decreases with density along the isotherm, with some f uctuations near the critical point, attaining a negative value at higher densities. There will be an inversion curve

(locus of µ jt = 0) for the LJ f uid. There are two temperatures at which µJT = 0 the f rst one is in the lower density supercritical f uid part of the phase diagram and the other point when the f uid approaches the liquid state as p 0 [201]. In the region → where the Joule-Thomson coefficient of the LJ f uid is positive(0.1 ρ 0.6) the ≤ ≤ decrease in pressure has caused a decrease in temperature which occurs as a result of lower initial pressure. In the region where the Joule-Thomson coefficient is negative, in a region where ρ 0.6 for the LJ f uid and always in WCA potential ≥ (data given in Appendix Table A.7 and B.7), the decrease in pressure causes an increase in temperature, i.e., there will be heating on expansion. As shown in f gure 5.14b, the Joule-Thomson coefficient of the system of particles interacting with an attractive LJ potential decreases with density along the isotherm. Warmer systems will have the lowest Joule-Thomson coefficient. The Joule-Thomson coefficient from the WCA potential is negative (though increasing) in the entire simulation region. There will not be an inversion curve from a potential with only attractive or repulsive part. ThermodynamicpropertiesofLJandWCApotentials 127

Figure 5.14: (a) Joule-Thomson coeﬃcient calculated from LJ and (b) ∆µJT as a function of density at diﬀerent constant temperatures where τ = 1.17(), 1.25(), 1.40(N), 1.60(H), 1.80() and 2.0() and the solid lines are polynomial f ts to the data and are for guidance only. 128 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Colina and Müller [279] described the difficulty in f nding the inversion curve experimentally and the need and importance of simulation to calculate the inversion curve. Kioupis et al. [198] and Kioupis and Maginn [280] reported that the inversion curve can be accurately calculated using molecular dynamics. The inability of Heyes and Llaguno [201] to predict the inversion curve using molecular dynamics is a result of the difficulty in controlling the thermodynamic state points within the conventional molecular dynamics framework [198,280]. In Fig- ure 5.14a it is shown that the value of the Joule-Thomson coefficient changes sign from positive to negative when the density increase. The change in sign of the Joule-Thomson coefficient implies that it is possible to calculate the inversion curve in a LJ f uid using molecular dynamics. Near the inversion curve temperature is insensitive to large pressure difference [279], which makes the calculation of the locus of the inversion curve prone to larger errors. The inversion curve of several simple fuids are given in [203]. The inversion curve for the LJ fuid is extrapolated from our simulation results (Figure 5.14a). The procedure that we followed to calculate the inversion curve is that, f rst we calculated the density at which µJT is zero for each of the isotherms in Fig- ure 5.14a. This procedure is prone to error, in order to minimize the error there should be large number of data points where the Joule-Thomson coefficient changes sign. The change in sign of the Joule-Thomson coefficient takes place when the density is between 0.4 and 0.7. We then f nd the corresponding value of the pressure at that density from Figure 5.6a for each of the isotherms. This is done by frst fnding the best ft to each of the isotherms and then using a ftting equation to calculate the pressure at each of the densities (where µJT = 0). Now for each of the temperatures we will have a corresponding value of density and pressure ThermodynamicpropertiesofLJandWCApotentials 129 which will be the locus of the inversion curve. It should be noted that our calculation is a rough calculation as there are no enough points for its calculation and the result is only a partial locus of the inversion curve. To obtain the full inversion curve, as shown in [198,202,279,281,282] would require a very large number of simulation results at τ 2. The locus of the inversion curve for the LJ f uid from ≥ our simulation, Heyes and Llaguno [201], Colina and Müller [279] and Kioupis et al. [198] is given in Table 5.2.

Table 5.2: Locus of the inversion curve for the LJ f uid from Heyes and Llaguno [201], Colina and M¨uller [279], Kioupis et al. [198] and from our simulation.

Heyes et al. [201] Colina [279] Simulation Kioupis et al. [198]

τ p/pc ρ/ρc p/pc τ p/pc τ ρ/ρc Constraint Extended

0.85 1.28 1.84 1.53 0.89 4.16 0.99 2.03 p/pc τ p/pc τ 0.97 2.96 1.77 2.31 0.91 4.07 1 2.01 3.61 0.99 3.46 0.99 1.1 5.96 1.76 3.07 0.96 4.72 1.05 1.95 5.15 1.13 5.38 1.12 1.17 6.94 1.73 3.84 1.01 4.93 1.1 1.9 6.38 1.21 6.46 1.2 1.25 5.95 1.59 4.61 1.06 6.11 1.15 1.89 6.76 1.28 7.23 1.28 1.49 8.04 1.44 5.38 1.1 6.23 1.17 1.88 8.84 1.52 8.84 1.52 1.66 7.97 1.31 6.15 1.15 6.51 1.2 1.86 9.61 1.69 9.53 1.69 1.84 8.03 1.16 6.92 1.19 6.87 1.22 1.84 9.92 1.88 10.15 1.88 2.04 10.06 1.21 7.69 1.26 6.76 1.25 1.81 10.15 2.08 10.38 2.08 2.24 11 1.15 8.46 1.33 8.72 1.4 1.74 10.76 2.28 10.76 2.28 2.46 10.01 0.99 9.23 1.46 9.73 1.6 1.61 10.46 2.5 10.15 2.5 2.9 10.02 0.85 9.61 1.64 10.21 1.8 1.48 9.23 2.96 9.23 2.95 3.38 11.18 0.79 10 1.82 10.7 1.9 1.43 7.76 3.46 7.69 3.46 3.73 5.14 0.38 – – 10.36 2.0 1.34 6.53 3.73 6.53 3.72 3.92 8.05 0.54 – – – – – 2.31 4.57 5.38 4.00 5.26 8.07 0.41 – – – – – – – – – 130 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

5.3 Thermodynamic properties of supercritical ar-

gon and krypton

Wood and Parker [283], McDonald and Singer [284], and Verlet [285] established that the equilibrium properties of the system of LJ atoms to be very similar to those of argon and krypton if the appropriate values of LJ constants are used. The correct value of the LJ constants for the corresponding noble gases should f t to the second virial coeﬃcient [252,253], howeverthe f tting should not be at too low temperature. The values of LJ constants vary according to the value of the critical

2 points at which both ∂p and ∂ p are zero [275]. Applying the conversion ∂ρ T ∂ρ2 T scheme described in Chapter Four (Table 4.1), we calculated the corresponding thermodynamic quantities of argon and krypton from the LJ simulation results presented earlier. The critical points, LJ constants and mass of LJ f uid, argon and krypton used in converting the dimensionless thermodynamic LJ potential results to argon and krypton are given in Table 5.3.

Table 5.3: Triple and critical points, mass and LJ constants of argon and krypton.

3 Tt (K) Tc (K) ρc(mol/dm ) m(gm/mol) σ (Å) ǫ/kB (K) Ref.

Ar 83.8 150.68 13.407 39.948 3.3605 119.8 [286, 287]

Kr 115.7 209.48 10.85 83.798 3.644 171.0 [255, 288]

Due to the onset of macroscopic fuctuation and fnite size of the simulation system, simulation results showed deviation and in some thermodynamic quantities divergence in the region close to the critical point. We used results from the LJ simulation and constants from Table 5.3 to calculate the thermodynamic Thermodynamic properties of supercritical argon and krypton 131 quantities of supercritical argon and krypton. We compare the simulation results with [220–223,237,253,274,275,287,289,290] as appropriate.

5.3.1 Energy and pressure

The potential energy of supercritical argon and krypton is always continuous and attractive in the simulated region. The potential energy calculated in supercritical krypton is more attractive than the potential calculated in supercritical argon, which is attributed to the diﬀerence in electronic conf guration and charge. The attractive potential energy in both noble gases decreases almost linearly along the isochore with temperature (graph not given). We can also infer that denser systems has more attractive potential than the raref ed system at all temperatures. Total internal energy, from our simulation and [220, 223], as a function of temperature at diﬀerent constant densities in a supercritical argon and krypton is illustrated in Figure 5.15. The total internal energy in both noble gases is a linear increasing function of temperature. Denser systems have smaller total internal energy. Our simulation result is in total agreement with the results from [220,223]. As shown in Figure 5.15a total internal energy of supercritical argon is negative only at higher densities and lower temperature and positive at lower densities at all temperatures. However, in supercritical krypton the total internal energy is positive at all temperature for all densities considered. This implies that the main contributor to internal energy of krypton at all temperatures is the kinetic energy which is a result of the higher atomic weight. This observation is in agreement with the idea that at lower temperature the interaction potential energy contributes more than the kinetic energy to the total internal energy in an isolated system. 132 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Figure 5.15: Internal energy as a function of temperature at diﬀerent constant densities in a supercritical (a) argon, where ρ = 6.57(), 8.75(), 10.94(N), 13.13(H), 15.32() and 17.51() (b) krypton, where ρ = 5.15(), 6.87(), 8.58(N), 10.29(H),12.02() and 13.73(). Results from our simulation are represented by symbols, whereas the solid lines represent experimental results from [220, 223]. Thermodynamic properties of supercritical argon and krypton 133

Pressure, from our simulation and [220,237], as a function of temperature at different constant densities in a supercritical argon and krypton is illustrated in Figure 5.16. Pressure, in both supercritical argon and krypton, increases linearly along the isochor with temperature. The increase is more rapid in systems with higher density than a system with fewer particles at the same temperature. The supercritical argon and krypton showed an ideal gas behaviour (p T) in the ∝ simulated region. As shown Figure 5.16a, the pressure from our simulation is lower than the experimental data at higher densities whereas pressure calculated in supercritical krypton is higher than the experimental data (Figure 5.16b). The pressure at lower densities (ρ = 6.36–10.61 mol/dm3 in argon and ρ = 5.15–8.58 mol/dm3 in krypton) from our simulationis the same as the results from [237], where the difference is comparable to the error in calculation. As the density increase the difference between the two results increases. In most cases the disparity between the thermodynamic quantities from our simulation and reference data [220,222,237,253,275] is attributed to the difference in the values of LJ constant used in the conversion rather than an error in the simulation. 134 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Figure 5.16: Pressure as a function of temperature at diﬀerent constant densities in a supercritical (a) argon, where ρ = 6.57(), 8.75(), 10.94(N), 13.13(H), 15.32() and 17.51() (b) krypton, where ρ = 5.15(), 6.87(), 8.58(N), 10.29(H),12.02() and 13.73(). The solid lines are experimental data from [220,237] and the symbols are from our simulation. Thermodynamic properties of supercritical argon and krypton 135

5.3.2 Heat capacities

Unlike the other thermophysical properties of supercritical argon, the value of heat capacities will not be affected by the choice of the LJ constants when calculating it from the dimensionless heat capacities of LJ f uid. Any inconsistency, if there is any, between the simulation and reference data in the heat capacities is a result of either system size, cutoff radius, effect of the critical point or the statistical method used. Previously we showed that the value of the isochoric heat capacity from our simulations to be consistent with results from Freasier et al. [242] (Figure 5.8a).

This rules out a problem with the statistical method employed in the simulation. As discussed at the beginning of this chapter, the results of response functions in our simulation could be aﬀected by either the f nite system size or the cutoﬀ radius near the critical point. Isochoric and isobaric heat capacities, from our simulation and Stewart and

Jacobsen [275], as a function of temperature at diﬀerent constant densities in a supercritical argon are illustrated in Figure 5.17. As one should expect, heat capacities show some f uctuation near the critical point along the isochore. For this reason we plotted the values the heat capacity at higher temperature, far away from the critical point. Consistent with the prediction of Stewart and Jacobsen [275], Boda et al. [278], Michels et al. [223] and Estrada-Alexanders and Truslert [274] isochoric heat capacity decreases monotonically with temperature along the isochor, Figure 5.17a. However the simulation underpredicted the isochoric heat capacity consistently in all densities. Isobaric heat capacity is challenging to calculate near the critical point as it diverges at points very close to the critical point [278]. Isobaric heat capacity 136 Thermodynamic Properties of LJ and WCA f uids and Noble Gases showed a peak near the critical point and it diverges very close to the critical point. The isobaric heat capacity showed f uctuation along the isotherm when the density is between 7 to 18 mol/dm3 at which the f uctuation decreases at higher temperature (not shown, data given in Appendix Table B.3). As illustrated in Figure 5.17b at higher temperature (T 250 K) the isobaric heat capacity from ≥ our simulation is in a very good agreement with results from Stewart and Jacobsen [275]. Thermodynamic properties of supercritical argon and krypton 137

Figure 5.17: (a) Isochoric and (b) isobaric heat capacity as a function of temperature at diﬀerent constant densities in a supercritical argon, where ρ = 6.57(), 8.75() and 15.32(N). The solid symbols are from our simulation and dashed line with open symbols are from [275]. 138 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

5.3.3 Speed of sound

Speed of sound, from our simulation and [220,221], as a function of temperatures at different constant densities in a supercritical argon and krypton is illustrated in Figure 5.18. Speed of sound increase monotonically with temperature along the isochore. Our calculation of speed of sound is in good agreement with the results from [220,221] in the entire simulation region. The speed of sound at the lowest density shown in the respective f gures is in total agreement with the reference data. Speed of sound from our simulation is over predicted At higher density at higher density in supercritical krypton, Fig- ure 5.18b. The simulation results of the adiabatic compressibility (Figure 5.20a) is in very good agreement with the calculated value. From Eq. (3.38), we can conclude that the difference between the results at higher densities is a result of the difference in the masses used. It is shown that sound travels faster in a dense and hot system. The increase in speed of sound with temperature is faster in denser systems. Since speed of sound depend on pressure it does not exhibit bigger f uctuation along the isochore near the critical point as exhibited in heat capacities. Thermodynamic properties of supercritical argon and krypton 139

Figure 5.18: Speed of sound as a function of temperature at diﬀerent constant densities in a supercritical (a) argon, where ρ = 6.57(), 8.75(), 10.94(N), 13.13(H), 15.32() and 17.51() (b) krypton, where ρ = 5.15(), 6.87(), 8.58(N), 10.29(H),12.02() and 13.73(). The solid lines are from [220, 221] and the symbols are from our simulation. 140 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

5.3.4 Joule-Thomson coeﬃcient

The Joule-Thomson coefficient, from our simulation and [220], as a function of temperatures at different constant densities in a supercritical argon and krypton is illustrated in Figure 5.19. The Joule-Thomson coefficient decreases monotonically along the isochore with temperature in both argon and krypton. Our simulation predicts higher values than the results from [220] in supercritical argon in the lower density region. In our simulation, near the critical point it f uctuates and becomes smooth at higher temperature. The f uctuation near the critical point increases with density. The Joule-Thomson coefficient changes sign from positive to negative as the density increases in the LJ f uid, Figure 5.14a. The same will happen in supercritical argon and krypton (not shown in the respective f gures). The existence of the locus implies that both argon and krypton has an inversion curve, as a result of which both will be liquifed. From the locus of the inversion curve of the LJ f uid given in Table 5.2 we have calculated the corresponding values of the locus of the inversion curve of both argon and krypton. The locus of the inversion curve of supercritical argon and krypton are given in Table 5.4. Thermodynamic properties of supercritical argon and krypton 141

Table 5.4: Locus of the inversion curve in supercritical argon and krypton.

supercritical argon supercritical krypton T(K) p(MPa) ρ(mol/dm3) T(K) p(MPa) ρ(mol/dm3) 118.602 23.577 28.078 169.29 18.491 22.021 119.8 23.067 27.801 171 18.091 21.804 125.79 26.751 26.972 179.55 20.980 21.153 131.78 27.941 26.280 188.1 21.914 20.611 137.77 34.629 26.142 196.65 27.159 20.502 140.166 35.309 26.003 200.07 27.692 20.394 143.76 36.896 25.727 205.2 28.937 20.177 146.156 38.936 25.450 208.62 30.537 19.960 149.75 38.313 25.035 213.75 30.048 19.635 167.72 49.421 24.067 239.4 38.761 18.875 191.68 55.146 22.269 273.6 43.250 17.465 215.64 57.866 20.471 307.8 45.384 16.055 227.62 60.643 19.779 324.9 47.562 15.512 239.6 58.716 18.534 342 46.050 14.536 142 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Figure 5.19: Joule-Thomson coeﬃcient as a function of temperature at diﬀerent constant densities in a supercritical (a) argon, where ρ = 6.57(), 8.75(), 10.94(N), 13.13(H), 15.32() and 17.51() (b) krypton, where ρ = 5.15(), 6.87(), 8.58(N), 10.29(H),12.02() and 13.73(). The solid lines are from [220] and the symbols are from our simulation. Thermodynamic properties of supercritical argon and krypton 143

5.3.5 Compressibilities

Adiabatic and isothermal compressibilities of argon as a function temperature at diﬀerent constant densities is illustrated in Figure 5.20. Note that adiabatic compressibility is calculated from the experimental value of speed of sound using the statistical relation Eq. (3.38). Using the calculated value of adiabatic compressibility and heat capacities we calculated the isothermal compressibility from the statistical relation Eq. (3.29). Both compressibilities decrease monotonically with temperature. Colder systems with lower density has the highest compressibility.

As shown in Figure 5.20a, the adiabatic compressibility of supercritical argon from our simulation is in total agreement with the calculated value in the entire simulation region. The value isothermal compressibility in a supercritical argon very far away (up to about 190 K) from the the critical point (at 150 K) from our simulation is very large compared with the calculated result. In Fig- ure 5.20b we only show values at higher temperature. To quantify the observation we have calculated the absolute average (AA) of adiabatic and isotherm compressibilities using Eq. (5.1). The absolute average of compressibilities calculated using Eq. (5.1) is given in Table 5.5. In Table 5.5 ‘Sim’ represents the simulation and ‘Exp’ are the experimental values of compressibilities calculated from other experimental thermodynamic quantities via Eqs. (3.29) and (3.38). Here ∆β = ∆β (Sim) ∆β (Exp). S,T S,T − S,T 144 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Table 5.5: Absolute average values of various ratios of compressibilities (χ) in supercritical argon.

χ T(K) βS (Exp) βs (Exp) βT (Exp) βT (Exp) βS (Sim) ∆βS βT (Sim) ∆βT 165.04 0.955 470.601 0.551 2.210 172.9 0.973 34.491 0.815 9.721 180.75 0.964 42.419 0.789 17.264 183.9 0.978 72.156 0.831 12.201 188.61 0.969 42.997 0.847 7.645 192.54 0.968 263.509 0.863 9.566 196.47 0.981 145.367 0.917 15.533 220.05 0.982 132.494 0.928 18.674 251.48 0.980 88.301 0.946 28.248 282.92 0.985 109.694 0.961 28.512 298.64 0.968 41.737 0.927 15.752 314.36 0.978 52.924 0.954 31.616

The values of the adiabatic compressibility in supercritical argon from our simulation is comparable with the calculated value at all temperatures. However, the values of isothermal compressibility from our simulation and the calculated are quite diﬀerent at lower temperature but far away from the critical point. The diﬀerence between the simulated and calculated values decrease as the temperature increases. Thermodynamic properties of supercritical argon and krypton 145

Figure 5.20: (a) Adiabatic and (b) isothermal compressibilities as a function of of temperature at diﬀerent constant densities in a supercritical argon where ρ = 6.57(), 8.75(), 10.94(N), 13.13(H), 15.32() and 17.51(). The solid lines are calculated from the reference data of Cv,Cp and ω0. 146 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

5.4 Summary

The effect of system size, cutoff radius and simulation time on the thermodynamic quantities of LJ f uid and WCA potential were investigated. Thermody- namic properties of system of particles interacting with WCA potential are not affected by system size, length of the cutoff radius and simulation time so long as these parameters are sufficiently large. None of these parameters signif cantly affected the values of the thermophysical quantities for the LJ f uid. By extension, we can conclude that the above mentioned parameters will have a little perhaps no effect on the thermophysical properties of atomistic f uids. However, in order to improve the statistics and reduce the effect of fnite size and cutoff effect near the critical point on the simulation results we have considered systems consisting of 2000 particles with a cutoff radius of 6.5σ in WCA potential and LJ f uid. The ‘attractive’ part of the LJ potential energy ∆u, calculated by subtracting the WCA potential from LJ potential, is independent of temperature (Figure 5.5a). It is also shown in Figure 5.6b that ∆p, the pressure in an attractive LJ potential, is independent of the temperature of the system. The main contributer to the isochoric heat capacity is the repulsive part, as a result of which the isochoric heat capacity calculated in LJ and WCA potentials is comparable. This makes the isochoric heat capacity due to the attractive part of the

LJ potential, ∆Cv, to be very small compared with its value from the LJ potential. Nonetheless, the attractive interaction is required to observe the maximum and the critical behaviour in the isochoric heat capacity. The isobaric heat capacity due to the attractive part of the LJ potential, ∆Cp, is comparable with the value from the LJ potential. The main contributer to the isobaric heat capacity is the attractive Summary 147 part of the potential. It is interesting to note that, even if both heat capacities are related to each other part of the potential energy mainly contributing to each of them is different. Our simulation results of argon and krypton are consistent with reference data at lower densities. The difference between simulation and experimental results of the thermodynamic quantities at higher densities and temperatures is not that signif cant. We need to emphasize that the difference in the thermodynamic quantities of our simulation and the reference data is mainly caused by the difference in the LJ constants, mass, the precision of any constant used, and numerical errors and rounding in the calculation. The heat capacities are the strictest test for the validity of the simulation results as there are no LJ constants involved in converting the value from LJ reduced form to the value for noble gases, which could be conf rmed by experimental results. The isochoric heat capacity from our simulation is underpredicted at higher temperature. However, the isobaric heat capacity from our simulation at higher densities is in a very good agreement with experiment. This leads to a very good agreement of Joule-Thomson coefficient with the experimental result.

We found the isothermal pressure coefficient γv from the LJ and WCA potential to be the same (Figure 5.10). This implies that that isothermal pressure coefficient depends only on the repulsive part of the potential in atomistic f uids. An ideal gas is characterized by a zero Joule-Thomson coefficient. Neither LJ nor WCA potentials are ideal gas potentials. There will not be an inversion curve for the WCA potential. It is possible to calculate the inversion curve of the LJ f uid, however we need to have a large amount of data. There will be an inversion curve only in a potential which has both attractive and repulsive parts. 148 Thermodynamic Properties of LJ and WCA f uids and Noble Gases

Sound travels faster in system where the particles interact with WCA potential than in the LJ f uid. The presence of attractive potential part in the LJ f uid makes vibration of particles harder than in the repulsive WCA potential. The speed of sound in WCA potential attained a maximum value in the middle density range relative to the speed of sound in LJ f uid (Figure 5.12b). Chapter 6

Thermodynamic Properties of MCYna Water

6.1 Introduction

We have implemented the statistical method (discussed in Chapter Three) to calculate the thermodynamic quantities of MCYna water, discussed in Chapter Two. The MCYna water model is for liquid water only and we do not consider the three body interaction, Eq. (2.6), in our simulation. Accordingly, the simulation is conducted in a temperature region between 298 - 645 K and a density of 0.998gm/cm3 (55.317 mol/dm3). All the simulation results are in SI units. The simulation setup for MCYna water is discussed in Chapter Four. The simulation run is for 500,000 steps with ∆t = 2 f s (where the total simulationtime is 1 ns)out of which we have used 400,000 steps for equilibration. After equilibration is reached the molecular thermostat will be switched oﬀ. As in the previous chapter, this chapter is organized in such away that, f rst we will present the positions of the triple and

149 150 Thermodynamic Properties of MCYna Water critical points and the values of the thermodynamic quantities calculated from different water models discussed previously. Second, the simulation results will be analyzed and compared with EOS [291] results followed by summary.

6.2 Thermodynamic properties of water

In the phase diagram there are two most important state points, the triple and critical points. The triple point is a point at which the three phase of state coexist. For water, at the triple point there will be steam, ice and liquid water. Whereas the critical point is a point at which there is no any distinction between the gas and liquid state of matter. Above the critical temperature a liquid cannot be formed by an increase in pressure, but with enough pressure a solid may be formed. The positions of the triple and critical points of water are given in Table 6.1.

Table 6.1: Experimental triple and critical points of water, source [190, 192, 235, 292– 294].

3 3 Properties Tt (K) pt (Pa) ρt (mol/dm ) Tc (K) pc (MPa) ρc (mol/dm ) Values 273.16 611.657 55.497 647.14 22.064 17.874 where the subscripts ‘t’ and ‘c’ represent triple and critical points respectively and the density at the triple point is for the liquid water. The thermodynamic properties obtained from diﬀerent intermolecular potentials are most commonly reported at 298 K and 0.1 MPa. A summary of such values of water models discussed in Chapter Two is given in Table 6.2. Depending on the availability, values presented here are from the original models or experiments as referred there. Thermodynamic properties of water 151 [57] [79] [66] [54] [30] [74] [99] [54] Ref. [291] [130] [128] [130] [134] [74, 88] [23, 295] this work [39,81,98,296] erent water models. ﬀ K) / 4 − 1 6.7 2.8 9.4 -6.9 2.63 2.54 2.57 7.51 5.14 8.56 4.98 5.08 ( p α0 ) s / m ( 0 3040 3130 1496.7 1498.4 1505.1 ω ) G / 1 ( 0.45 0.67 0.63 0.46 0.56 0.458 0.461 0.446 0.495 0.454 0.463 0.448 0.448 T β Pa m – / J 122 101 ( 74.63 75.338 84.516 77.613 70.291 80.751 79.077 92.885 75.770 114.516 116.148 p C olK) ) m / J ( 74.44 62.341 73.638 74.357 87.864 74.836 v C olK ) K – – – – – – 300.15 277.15 287.15 277.15 260.15 248.15 278.15 228.13 235.15 182.15 277.134 TMD ( omputed thermodynamic properties for liquid water at 298 K and 1 atm from di C 2 / F E F / / / Table 6.2: ST2 TIP5P NveD IAPWS-95 Exp. SPC d PPC GCPM TIP4P TIP4P MCY MCYL 005 MCYna Potential SPC SPC TIP3P SPC w 152 Thermodynamic Properties of MCYna Water

We presented only the fundamental thermodynamic quantities such as heat capacities, isothermal compressibility and thermal expansion coeﬃcient. The remaining quantities can be calculated using relations given in Chapter Three. It is apparent from Table 6.2 that there is a considerable range in the values predicted by the various potentials. For example, prediction of the TMD range from approximately 182 - 287 K compared with the experimental value of 277.134 K. The SPC model underpredicted the TMD, whereas good agreement is attained for PPC, TIP4P/2005 and TIP5P. In almost all cases, the isobaric heat capacity is overpredicted whereas good agreement is obtained for the SPC/E, TIP4P and

TIP4P/2005 potentials. In contrast most potentials are reasonably accurate for isothermal compressibility whereas there is consistently poor agreement for thermal expansion coefficient for all potentials except the six site model. It is interesting to note that the thermodynamic quantities calculated in MCYna potential are in a very good agreement with the experiment. The error in the thermodynamic quantities calculated from the MCYna water simulation is less than 1%. However, the model underpredicted isothermal compressibility and also our model overpredicted the isothermal pressure coefficient by about 2.3% at 298 K and 1 atm. The values of the response functions near the critical point are highly depen- dent on the density of water. It is known that, near the critical point the thermodynamic behaviours of water behave differently at higher and lower densities. For example, the isochoric heat capacity decreases monotonically with temperature even close to the critical point at higher densities. However, the isochoric heat capacity diverges at lower densities near the critical point. Simulation results and discussion 153

6.3 Simulation results and discussion

We will present the simulation results of the thermodynamic properties of water of density 0.998 gm/mol (55.371 mol/dm3) in a temperature range between 298 and 645 K. Depending on the availability of the result we would like to compare the simulation results with experimental [81,292,296] and equation of state [98,190,192,193,220,235,289,294,297,298] results. However, almost all the equation of state and experimental results of water are at isobaric condition. In order to compare our simulation with those results we need to convert them to constant volume or use other source which provide the thermodynamic quantities in a constant volume. For that purpose we have used the IAPWS-95 software package, developed by Wagner [291], to calculate the thermodynamic quantities from EOS in an isochoric condition. The IAPWS-95 software is a modif cation of the IAPS-84. The EOS used in IAPWS-95 are from Wagner and Prub [190]. Re- sults from the IAPWS-95 are as reliable as the experimental data which has been used and tested by different researchers for a very long time. We compared the simulation results with the outputs from IAPWS-95 software [291]. Note that it is only possible to calculate pressure, heat capacities, the Joule-Thomson coefficient and speed of sound from the IAPWS-95 software. The remaining thermodynamic quantities such as compressibilities, isothermal pressure coefficient and thermal expansion coefficient presented in this chapter are calculated from the previously mentioned quantities using the thermodynamic relation given in Chapter Three, Eqs. (3.29), (3.30), (3.38) and (3.40). 154 Thermodynamic Properties of MCYna Water

6.3.1 Pressure

The pressure of liquid water as a function of temperature from our simulation and equation of state [291] is illustrated in Figure 6.1. The pressure of water is an almost linearly increasing function of temperature. The pressure calculated from our simulation is in a very good agreement with the results from [291] in the entire simulation region. The pressure from our simulation at a temperature less than 290 K gives a negative result (not shown in Figure 6.1) which imply that water is in a metastable state [300].Negative pressure for metastable water is also reported by Poole et al. [299] and Stanley et al. [300]. The change in the sign of the pressure is a result of the presence of low-temperature phase transition in the metastable water [300].

Figure 6.1: Pressure of water as a function of temperature. The open symbols are from our simulation and solid line is from [291]. Simulation results and discussion 155

6.3.2 Heat capacities

Isochoric heat capacity of water as a function of temperature from our simulation and equation of state [291] is illustrated in Figure 6.2. The isochoric heat capacity is a monotonically decreasing function of temperature, even near the critical temperature, at higher densities. The comparison with the EOS results in Fig- ure 6.2 shows that the MCYna water potential can be used to accurately calculate isochoric heat capacity over a wide range of temperatures. At lower densities, the isochoric heat capacity of water decreases monotonically with temperature, however it increases very sharply when the temperature is close to the critical point [193]. The isochoric heat capacity of water diverges at the critical point, which signals change in phase. This behaviour is observed in our simulation (at a density 344.828 Kg/m3) and from the EOS calculation by Wagner et al. [190], Hill [301], Wyczalkowska et al. [302] and Saul et al. [193]. 156 Thermodynamic Properties of MCYna Water

Figure 6.2: Isochoric heat capacity of water as a function of temperature. The open symbols are from our simulation and solid line is from [291].

Isobaric heat capacity of water as a function of temperature from our simulation and equation of state [291] is illustrated in Figure 6.3. The values of the isobaric heat capacity deviates from the EOS results. The MCYna water model underpredicted the isobaric heat capacity of water. As shown in Chapter Three the isobaric heat capacity is calculated from the values of isochoric heat capacity and compressibilities, Eq.(3.29). An error in either of the thermodynamic quantities will be ref ected on the results of isobaric heat capacity. It is evident from Fig- ure 6.5 and 6.6 that the MCYna potential underpredicts both of these quantities, which account for isobaric heat capacity results. The average percentage error

C (EOS) C (Sim) p − p in calculating the isochoric heat capacity is 2.78%. It is apparent Cp(EOS) Simulation results and discussion 157 from the comparison of isobaric heat capacity (Table 6.2) calculated from other potentials that the MCYna potential yields the best agreement with experiment (deviation of less than 1%)at 298 K and 0.1 MPa. It is of interest to note, that with the exception of TIP3P all potentials overpredict isobaric heat capacity whereas the MCYna potential underpredicts it. Figure 6.3 indicates that this underpredic- tion widens with increasing temperature. There is no extensive data available for this behaviour from other potentials.

Figure 6.3: Isobaric heat capacity of water as a function of temperature. The open symbols are from our simulation and solid line is from [291]. 158 Thermodynamic Properties of MCYna Water

6.3.3 Isothermal pressure coeﬃcient

The isothermal pressure coefficient of water as a function of temperature from our simulation and equation of state [291] is illustrated in Figure 6.4. Isother- mal pressure coefficient calculated in MCYna water potential is comparable with the results from IAPWS-95 [291] in the entire simulation region. In the lower to middle temperature region (298-425 K) the agreement is appreciable. Isothermal pressure coefficient increase rapidly when the temperature is between 298 and 400 K, the increase becomes almost linear when the temperature is between 400 and

500 K and it becomes almost constant for a temperature beyond 500 K. Isother- mal pressure coeﬃcient becomes negative in phase-space region with anomalous density behavior [167]. Isothermal pressure coeﬃcient is positive in the entire simulation region, which implies that water (of the density 0.998 gm/mol) does not show anomaly in this temperature region. Simulation results and discussion 159

Figure 6.4: Isothermal pressure coeﬃcient of water as a function of temperature. The open symbols are from our simulation and solid line is from [291].

6.3.4 Compressibilities

Isothermal compressibility is positive in one phase region. For an ordinary liquid isothermal compressibility increases with temperature (it becomes less dense). Whereas isothermal compressibility of water at lower temperature decreases with temperature passes through a minimum (at 319 K and 1 bar [197,299]) and contin- ues to increase with temperature similar to ordinary liquid [18]. In our simulation the isothermal compressibility is a monotonically decreasing function of temperature. Isothermal compressibility of water as a function of temperature from our sim- 160 Thermodynamic Properties of MCYna Water ulation and the EOS [291] are illustrated in Figure 6.5. As described above the isothermal compressibility of water decreases until it attains the minimum value and increase thereafter if the pressure is 0.1 MPa. In our simulation and from the Wagner [291] the value decreases even after the minimum point since the simulation is in a highly pressurized container, shown in Figure 6.1. Isothermal compressibility predicted by the MCYna water potential is less than the values from IAPWS-95. As shown in Figure 6.5 the diﬀerence between the two results increase with temperature. It is this value that we have used to calculate the isobaric heat capacity which has caused that big disparity between the simulation and

EOS results.

Figure 6.5: Isothermal compressibility of water as a function of temperature. The open symbols are from our simulation and solid line is from [291]. Simulation results and discussion 161

Adiabatic compressibility of water as a function of temperature from our simulation and EOS [291] are illustrated in Figure 6.6. Adiabatic compressibility calculated from the MCYna water potential is in agreement with the result calculated from IAPWS-95 [291] in the entire simulation region. As shown in Fig- ure 6.6 adiabatic compressibility decreases with density. System having lower density attained higher adiabatic compressibility.

Figure 6.6: Adiabatic compressibility of water as a function of temperature. The open symbols are from our simulation and solid line is from [291].

6.3.5 Joule-Thomson coeﬃcient

The Joule-Thomson effect describes the temperature change of a gas or liquid when it is forced through a valve or porous plug while kept insulated so that no 162 Thermodynamic Properties of MCYna Water heat is exchanged with the environment. A system with the lowest initial temperature has the highest the Joule-Thomson coefficient. The Joule-Thomson coefficient decreases along the isotherm with density. The Joule-Thomson coefficient is negative in the entire simulation region which indicates that there is no inversion curve (locus of µJT = 0) in water at higher densities. The Joule-Thomson coefficient of water in the simulated region is negative. The Joule-Thomson coefficient water as a function of temperature from our simulation and equation of state [291] is illustrated in Figures 6.7. The MCYna water potential underpredicted the Joule-Thomson coefficient, where the disparity between the results increase with temperature. This is consistent with both the pattern shown in isobaric heat capacity (Figure 6.3) and Eq. (3.40).

Figure 6.7: Joule-Thomson coeﬃcient of water as a function of temperature. The open symbols are from our simulation and solid line is from [291]. Simulation results and discussion 163

6.3.6 Speed of sound

The speed of sound in water as a function of temperature from the simulation and equation of state [291] are illustrated in Figures 6.8. It is apparent from Figure 6.8 that the MCYna potential yields good agreement with results from EOS for temperature between 298 and 400 K. For T > 400 K the MCYna potential increasingly overpredicts the speed of sound. The error increases with increasing temperature but arguably the maximum discrepancy of 2.19% at 645 K is acceptable.

Figure 6.8: Speed of sound in water as a function of temperature. The open symbols are from our simulation and solid line is from [291]. 164 Thermodynamic Properties of MCYna Water

6.3.7 Thermal expansion coeﬃcient

Thermal expansion coeﬃcient is the measure of the tendency of matter to change in volume in response to a change in temperature. Thermal expansion coeﬃcient as a function of temperatures is illustrated in Figure 6.9. Thermal expansion coef- f cient increases almost linearly until it reach its peak value at 430 K, (the tendency of water to change volume reaches its peak when the temperature is around 400 to 450 K) and decreases. It is apparent from the comparison given in Figure 6.9 that the MCYna water potential mimics the behaviour from the EOS [291]. At all temperatures, the MCYna potential only slightly underpredicts the true value of

4 thermal expansion coeﬃcient. The maximum value of 6.629 10− /K is obtained × at 445 K.

Figure 6.9: Thermal expansion of water as a function of temperature. The open symbols are from our simulation and solid line is from [291]. Summary 165

6.4 Summary

Pressure, isochoric heat capacity, adiabatic compressibility and speed of sound predicted by the MCYna water potential are consistent with the results from IAPWS- 95 [291]. If there are some under and/or overprediction of the value of the calculated thermodynamic quantities by the model potential it will be in an acceptable error range of less than 2%. However the isobaric heat capacity and Joule- Thomson coeﬃcient are underpredicted by the model potential in the entire simulation region. Both thermodynamic quantities underpredicted are related to each other Eq. (3.29).

The isothermal compressibility is underpredicted by the model which causes the decrease in the isobaric heat capacity and Joule-Thomson coeﬃcient. It should

9 be noted that the compressibilities are calculated in the order of 10− . The errors accumulated might have manifested here. In the case of argon, the isochoric heat capacity was underpredicted in the higher temperature region (Figure 5.17a). However the isobaric heat capacity was consistent with the experimental result. Chapter 7

Conclusions and Recommendations

7.1 Conclusion

The effect of system size, cutoff radius and simulation time on the thermodynamic quantities of LJ f uid and WCA potential were investigated. We found that the above mentioned parameters will have a little perhaps no effect on the thermophysical properties of atomistic f uids. However, in order to improve the statistics and reduce the effect of fnite size and cutoff effect we considered systems consisting of 2000 particles with a cutoff radius of 6.5σ in WCA potential and LJ f uid. The ‘attractive’ part of the LJ potential energy ∆u is independent of temperature. It is also shown that ∆p is independent of the temperature of the system. We found that, isothermal pressure coefficient γv from the LJ and WCA potential to be the same. Implying that that isothermal pressure coefficient depends only on the repulsive part of the potential in atomistic f uids. The main contributer to the isochoric heat capacity is the repulsive part, as a

166 Conclusion 167 result of which the isochoric heat capacity calculated in LJ and WCA potentials is comparable. The attractive interaction is required to observe the maximum and the critical behaviour in the isochoric heat capacity. However, the main contributer to the isobaric heat capacity is the attractive part of the potential. It is interesting to note that, even if both heat capacities are related to each other part of the potential energy mainly contributing to each of them is different. Our simulation results of argon and krypton are consistent with reference data at lower densities. The difference between simulation and experimental results of the thermodynamic quantities at higher densities and temperatures is not that signif cant. We need to emphasize that the difference in the thermodynamic quantities of our simulation and the reference data is mainly caused by the difference in the LJ constants, mass, the precision of any constant used, and numerical errors and rounding in the calculation. The heat capacities are the strictest test for the validity of the simulation results as there are no LJ constants involved in converting the value from LJ reduced form to the value for noble gases, which could be con- f rmed by experimental results. The isochoric heat capacity from our simulation is underpredicted at higher temperature. However, the isobaric heat capacity from our simulation at higher densities is in a very good agreement with experiment.

An ideal gas is characterized by a zero Joule-Thomson coeﬃcient. Neither LJ nor WCA potentials are ideal gas potentials. There will not be an inversion curve for the WCA potential. It is possible to calculate the inversion curve of the LJ f uid, however we need to have a large amount of data. There will be an inversion curve only in a potential which has both attractive and repulsive parts. Sound travels faster in system where the particles interact with WCA potential than in the LJ f uid. The presence of attractive potential part in the LJ f uid makes 168 Conclusions and Recommendations vibration of particles harder than in the repulsive WCA potential. The speed of sound for the WCA potential attained a maximum value in the middle density range relative to the speed of sound in LJ f uid. Pressure, isochoric heat capacity, adiabatic compressibility and speed of sound predicted by the MCYna water potential are consistent with the results from the

IAPWS-95 [291]. If there are some under and/or overprediction of the value of the calculated thermodynamic quantities by the model potential it will be in an acceptable error range of less than 2%. However the isobaric heat capacity and Joule-Thomson coeﬃcient are underpredicted by the model potential in the entire simulation region. The isothermal compressibility is underpredicted by the MCYna water model which causes the decrease in the isobaric heat capacity and Joule-Thomson coef- f cient. It should be noted that the compressibilities are calculated in the order of

9 10− . Since these thermophysical quantities are related to each other, Eq. (3.29), at least the numerical error in calculating the very small values of the compressibility

9 (of the order of 10− ) will aﬀect the results of the heat capacities.

7.2 Recommendations

All the conclusions that we draw are based on a small range of temperature and density and so the conclusions are valid in that region. To make a general conclusion more simulations should be done for a wide range of temperatures and densities. If possible the model should be ab inito based otherwise the properties for parametrization should be chosen carefully. Polarization should be included in Recommendations 169 the water model, as it is conf rmed that it will contributes about 30% of the total interaction energy. Multibody interaction should also be included in the model so that it will work in diﬀerent states of water. In order not to restrict its degree of freedom and get better results from the model as recommended by Wu et al. [74] and Raabe and Sadus [75–77] the water model should be f exible.

The thermodynamics properties of water should be studied for diﬀerent densities so as to observe the critical behaviour and TMD of the MCYna water. The model should work for mixtures, some work should be done in this direction. It is not possible to measure any of the thermodynamic quantities at constant

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Simulation data for the WCA potential

Simulation results of WCA potential are given below. The simulation data of system of particles interacting with WCA potential consisting of 2000 particles with a cutoﬀ radius of half the box length given below. The results are given in reduced units, where the reduction scheme is given in Chapter Four.

207 208 Simulation data for the WCA potential 1.560 1.640 1.736 1.824 1.930 2.022 2.135 2.232 2.340 2.480 3.870 5.929 8.806 12.937 19.011 27.250 2.624 0.321 0.792 0.847 0.911 0.971 1.037 1.104 1.171 1.238 1.318 1.395 1.474 1.487 1.570 1.656 1.749 1.840 1.930 2.021 2.149 2.251 2.363 3.737 5.686 8.525 12.585 18.342 26.321 2.4928 0.305 0.755 0.807 0.866 0.926 0.986 1.049 1.115 1.181 1.256 1.334 1.411 1.417 1.503 1.576 1.667 1.749 1.856 1.943 2.037 2.130 2.235 3.587 5.427 8.202 12.066 17.646 25.660 2.3616 0.289 0.717 0.768 0.823 0.882 0.937 0.999 1.060 1.125 1.195 1.264 1.338 1.341 1.422 1.495 1.574 1.652 1.755 1.843 1.934 2.042 2.145 3.386 5.248 7.835 11.574 16.885 24.773 2.2304 0.274 0.678 0.728 0.779 0.833 0.891 0.947 1.007 1.065 1.136 1.201 1.271 1.268 1.345 1.415 1.495 1.580 1.656 1.742 1.835 1.939 2.025 3.249 4.978 7.476 11.073 16.259 23.932 2.0992 0.258 0.641 0.689 0.737 0.788 0.841 0.892 0.950 1.013 1.071 1.141 1.207 22.897 1.968 0.242 0.603 0.648 0.695 0.742 0.794 0.845 0.895 0.952 1.011 1.072 1.136 1.201 1.272 1.338 1.418 1.490 1.574 1.645 1.736 1.816 1.927 3.044 4.689 7.113 10.632 15.683 22.085 1.8368 0.226 0.566 0.606 0.652 0.694 0.744 0.792 0.841 0.896 0.949 1.008 1.066 1.125 1.193 1.258 1.331 1.396 1.475 1.553 1.630 1.724 1.798 2.863 4.456 6.791 10.083 14.844 21.100 1.7056 0.211 0.527 0.566 0.608 0.648 0.692 0.738 0.785 0.837 0.886 0.941 0.997 1.053 1.108 1.178 1.239 1.305 1.384 1.454 1.529 1.610 1.691 2.712 4.198 6.390 9.573 14.253 T 20.625 1.640 0.203 0.507 0.546 0.582 0.626 0.670 0.712 0.758 0.807 0.855 0.907 0.962 1.015 1.071 1.132 1.191 1.261 1.328 1.395 1.483 1.544 1.628 2.610 4.059 6.196 9.366 13.909 20.349 1.6072 0.198 0.496 0.534 0.572 0.611 0.657 0.697 0.741 0.788 0.835 0.886 0.938 0.992 1.053 1.112 1.168 1.236 1.302 1.371 1.440 1.517 1.601 2.582 4.024 6.060 9.154 13.740 ressure in WCA potential, simulation data. P 20.211 1.5744 0.195 0.488 0.525 0.562 0.602 0.645 0.684 0.731 0.774 0.823 0.879 0.921 0.977 1.035 1.096 1.156 1.221 1.288 1.353 1.413 1.498 1.577 2.539 3.965 6.009 9.076 13.564 19.778 1.53504 0.191 0.476 0.513 0.548 0.588 0.628 0.670 0.713 0.757 0.802 0.855 0.904 0.959 1.017 1.064 1.125 1.185 1.260 1.316 1.387 1.468 1.534 2.503 3.856 5.958 8.952 13.312 Table A.1: 19.499 1.5088 0.187 0.470 0.503 0.543 0.580 0.618 0.659 0.704 0.743 0.787 0.838 0.888 0.941 0.997 1.051 1.116 1.176 1.228 1.308 1.371 1.446 1.518 2.440 3.792 5.897 8.771 13.225 1.386 1.462 2.347 3.720 5.635 8.529 12.815 19.248 0.556 0.594 0.631 0.673 0.715 0.761 0.809 0.854 0.910 0.960 1.016 1.064 1.128 1.182 1.252 1.308 1.4432 0.179 0.450 0.484 0.520 1.320 1.398 2.283 3.565 5.497 8.219 12.615 18.628 0.530 0.567 0.605 0.648 0.686 0.725 0.774 0.815 0.864 0.915 0.969 1.025 1.077 1.143 1.203 1.258 1.3776 0.171 0.431 0.464 0.495 1.280 1.335 2.181 3.394 5.180 7.966 12.125 18.107 0.507 0.542 0.576 0.617 0.656 0.697 0.740 0.785 0.833 0.878 0.932 0.984 1.033 1.092 1.152 1.209 1.312 0.163 0.411 0.441 0.475 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 ρ 0.1 0.2 0.21 0.22 209 1.646 1.651 1.658 1.663 1.670 1.678 1.752 1.847 1.962 2.114 2.262 2.457 2.624 1.529 1.569 1.572 1.577 1.582 1.587 1.590 1.594 1.600 1.606 1.610 1.617 1.621 1.627 1.633 1.638 1.644 1.650 1.657 1.664 1.671 1.677 1.755 1.847 1.967 2.112 2.274 2.498 2.4928 1.530 1.568 1.572 1.577 1.581 1.586 1.591 1.597 1.601 1.606 1.611 1.616 1.622 1.628 1.633 1.640 1.644 1.652 1.659 1.665 1.671 1.679 1.758 1.852 1.975 2.118 2.303 2.498 2.3616 1.529 1.568 1.573 1.578 1.582 1.587 1.592 1.597 1.602 1.606 1.612 1.615 1.624 1.628 1.633 1.639 1.646 1.652 1.660 1.665 1.672 1.677 1.754 1.857 1.979 2.125 2.315 2.540 2.2304 1.529 1.568 1.573 1.577 1.583 1.587 1.592 1.597 1.601 1.606 1.613 1.618 1.624 1.629 1.636 1.640 1.648 1.652 1.660 1.667 1.674 1.680 1.756 1.857 1.981 2.141 2.336 2.577 2.0992 1.530 1.569 1.573 1.579 1.582 1.588 1.592 1.597 1.600 1.607 1.612 1.617 1.623 1.628 1.635 1.641 1.530 1.569 1.574 1.578 1.582 1.587 1.592 1.596 1.602 1.607 1.612 1.618 1.623 1.629 1.633 1.643 1.646 1.655 1.661 1.668 1.675 1.680 1.758 1.862 1.985 2.144 2.340 2.574 1.968 1.530 1.569 1.574 1.578 1.583 1.587 1.592 1.598 1.602 1.607 1.612 1.619 1.625 1.630 1.636 1.641 1.649 1.655 1.661 1.667 1.674 1.681 1.764 1.865 1.995 2.150 2.360 2.589 1.8368 1.530 1.569 1.574 1.578 1.583 1.587 1.593 1.597 1.603 1.608 1.613 1.618 1.626 1.632 1.636 1.643 1.647 1.654 1.662 1.671 1.677 1.685 1.762 1.864 1.999 2.171 2.384 2.641 1.7056 T 1.529 1.569 1.574 1.578 1.583 1.588 1.594 1.597 1.602 1.607 1.615 1.618 1.624 1.630 1.637 1.645 1.649 1.654 1.661 1.669 1.673 1.683 1.764 1.869 2.002 2.178 2.395 2.646 1.640 1.529 1.569 1.573 1.579 1.583 1.588 1.593 1.597 1.602 1.609 1.614 1.619 1.626 1.631 1.636 1.644 1.649 1.657 1.664 1.668 1.677 1.687 1.770 1.871 2.010 2.181 2.380 2.661 1.6072 1.529 1.569 1.573 1.578 1.583 1.587 1.592 1.598 1.602 1.610 1.614 1.619 1.624 1.632 1.638 1.644 1.650 1.656 1.662 1.671 1.675 1.686 1.765 1.872 2.006 2.182 2.388 2.674 1.5744 sochoric heat capacity in WCA potential, simulation data. I 1.530 1.569 1.574 1.578 1.582 1.588 1.592 1.597 1.603 1.608 1.615 1.620 1.625 1.631 1.637 1.643 1.650 1.658 1.664 1.668 1.679 1.684 1.766 1.874 2.004 2.187 2.414 2.684 1.53504 1.529 1.569 1.574 1.579 1.583 1.587 1.593 1.596 1.603 1.608 1.614 1.620 1.623 1.631 1.638 1.644 1.651 1.656 1.664 1.669 1.678 1.685 1.771 1.873 2.013 2.188 2.387 2.669 1.5088 Table A.2: 2.013 2.194 2.405 2.684 1.602 1.608 1.614 1.621 1.627 1.632 1.636 1.643 1.651 1.657 1.662 1.672 1.677 1.682 1.765 1.876 1.4432 1.529 1.568 1.574 1.578 1.583 1.588 1.593 1.597 2.018 2.198 2.413 2.726 1.603 1.608 1.613 1.620 1.625 1.631 1.636 1.644 1.650 1.657 1.665 1.670 1.678 1.683 1.765 1.879 1.3776 1.529 1.569 1.574 1.578 1.582 1.587 1.593 1.597 2.021 2.200 2.421 2.715 1.602 1.609 1.614 1.620 1.623 1.632 1.640 1.646 1.651 1.657 1.663 1.668 1.679 1.685 1.768 1.881 1.312 1.530 1.569 1.573 1.577 1.583 1.588 1.593 1.599 0.7 0.8 0.9 1 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 0.6 ρ 0.1 0.2 0.21 0.22 0.23 0.24 0.25 0.26 210 Simulation data for the WCA potential 2.542 2.558 2.581 2.556 2.603 2.586 2.617 2.601 2.608 2.634 2.701 2.827 2.955 3.166 3.241 3.417 2.624 2.475 2.532 2.496 2.519 2.527 2.539 2.498 2.502 2.521 2.546 2.523 2.558 2.551 2.564 2.567 2.594 2.560 2.562 2.592 2.611 2.623 2.608 2.729 2.815 2.967 3.133 3.255 3.527 2.4928 2.488 2.494 2.508 2.505 2.506 2.521 2.528 2.553 2.521 2.552 2.551 2.535 2.580 2.563 2.563 2.556 2.555 2.603 2.607 2.611 2.607 2.629 2.763 2.842 3.012 3.136 3.341 3.492 2.3616 2.489 2.499 2.502 2.529 2.512 2.536 2.545 2.543 2.545 2.532 2.541 2.513 2.581 2.567 2.598 2.570 2.587 2.580 2.610 2.607 2.613 2.598 2.695 2.879 3.005 3.145 3.358 3.596 2.2304 2.485 2.491 2.512 2.507 2.547 2.531 2.536 2.548 2.521 2.527 2.566 2.560 2.559 2.562 2.586 2.582 2.608 2.567 2.610 2.626 2.638 2.635 2.700 2.846 2.999 3.197 3.414 3.683 2.0992 2.502 2.522 2.516 2.549 2.506 2.540 2.541 2.553 2.506 2.555 2.549 2.546 3.628 1.968 2.498 2.521 2.547 2.523 2.516 2.532 2.546 2.522 2.540 2.547 2.537 2.567 2.553 2.568 2.541 2.625 2.562 2.616 2.628 2.628 2.648 2.633 2.702 2.866 2.997 3.191 3.399 3.637 1.8368 2.499 2.514 2.531 2.531 2.537 2.512 2.525 2.563 2.548 2.527 2.540 2.567 2.582 2.593 2.593 2.580 2.614 2.612 2.612 2.612 2.619 2.628 2.763 2.882 3.044 3.187 3.435 3.761 1.7056 2.499 2.533 2.557 2.525 2.545 2.527 2.565 2.534 2.561 2.562 2.552 2.556 2.608 2.616 2.582 2.614 2.569 2.594 2.619 2.673 2.652 2.682 2.735 2.851 3.037 3.254 3.505 T 3.747 1.640 2.496 2.521 2.528 2.531 2.544 2.542 2.574 2.535 2.537 2.534 2.585 2.544 2.571 2.576 2.597 2.645 2.604 2.596 2.595 2.634 2.595 2.653 2.741 2.882 3.041 3.284 3.528 3.791 1.6072 2.486 2.524 2.508 2.555 2.532 2.549 2.541 2.538 2.526 2.573 2.578 2.567 2.596 2.587 2.578 2.634 2.597 2.637 2.655 2.619 2.653 2.700 2.815 2.908 3.084 3.283 3.454 3.822 1.5744 2.490 2.520 2.517 2.526 2.552 2.522 2.525 2.550 2.543 2.596 2.568 2.571 2.569 2.610 2.600 2.608 2.630 2.622 2.624 2.662 2.620 2.691 2.755 2.899 3.065 3.281 3.480 sobaric heat capacity in WCA potential, simulation data. I 3.836 1.53504 2.497 2.523 2.538 2.527 2.505 2.536 2.526 2.533 2.563 2.551 2.600 2.595 2.580 2.588 2.593 2.612 2.626 2.648 2.657 2.622 2.677 2.663 2.766 2.912 3.045 3.300 3.572 Table A.3: 3.770 1.5088 2.487 2.511 2.535 2.563 2.546 2.534 2.557 2.526 2.570 2.572 2.563 2.596 2.536 2.591 2.610 2.611 2.640 2.623 2.652 2.624 2.673 2.679 2.813 2.900 3.097 3.292 3.451 2.637 2.631 2.735 2.924 3.072 3.312 3.499 3.795 2.550 2.553 2.551 2.531 2.540 2.551 2.577 2.609 2.612 2.621 2.588 2.610 2.638 2.639 2.615 2.674 1.4432 2.489 2.499 2.533 2.533 2.663 2.637 2.733 2.925 3.101 3.304 3.527 3.907 2.526 2.536 2.546 2.537 2.551 2.560 2.566 2.582 2.584 2.601 2.565 2.625 2.612 2.640 2.673 2.642 1.3776 2.485 2.523 2.550 2.525 2.678 2.672 2.767 2.935 3.096 3.300 3.523 3.836 2.549 2.552 2.560 2.578 2.540 2.579 2.559 2.587 2.531 2.596 2.652 2.646 2.627 2.635 2.634 2.603 1.312 2.502 2.520 2.522 2.518 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 ρ 0.1 0.2 0.21 0.22 211 0.294 0.273 0.258 0.241 0.227 0.212 0.119 0.069 0.042 0.026 0.016 0.010 2.624 2.541 0.899 0.809 0.745 0.693 0.639 0.574 0.534 0.506 0.472 0.431 0.411 0.377 0.357 0.334 0.308 0.299 0.281 0.269 0.249 0.235 0.218 0.123 0.071 0.043 0.026 0.016 0.010 2.4928 2.721 0.914 0.856 0.777 0.712 0.666 0.618 0.580 0.527 0.495 0.456 0.419 0.396 0.371 0.346 0.327 0.312 0.296 0.279 0.262 0.246 0.233 0.130 0.075 0.045 0.027 0.017 0.011 2.3616 2.872 0.967 0.890 0.827 0.746 0.706 0.652 0.605 0.560 0.512 0.478 0.435 0.420 0.383 0.361 0.332 0.337 0.307 0.293 0.274 0.255 0.236 0.132 0.078 0.046 0.028 0.017 0.011 2.2304 3.027 1.014 0.940 0.860 0.810 0.732 0.682 0.634 0.581 0.533 0.508 0.470 0.444 0.406 0.387 0.355 0.351 0.322 0.309 0.290 0.270 0.254 0.136 0.080 0.048 0.029 0.018 0.011 2.0992 3.232 1.091 0.990 0.929 0.829 0.779 0.727 0.673 0.596 0.575 0.524 0.486 0.460 0.426 0.404 0.373 3.428 1.153 1.072 0.962 0.882 0.816 0.760 0.696 0.649 0.601 0.553 0.522 0.480 0.448 0.411 0.400 0.360 0.345 0.328 0.304 0.289 0.264 0.144 0.085 0.049 0.030 0.018 0.012 1.968 3.674 1.217 1.131 1.026 0.957 0.853 0.795 0.757 0.687 0.629 0.585 0.553 0.520 0.483 0.449 0.412 0.396 0.365 0.340 0.318 0.296 0.281 0.157 0.088 0.052 0.031 0.019 0.012 1.8368 3.926 1.323 1.227 1.092 1.026 0.929 0.875 0.792 0.736 0.686 0.627 0.582 0.560 0.526 0.472 0.451 0.408 0.382 0.362 0.349 0.321 0.305 0.161 0.091 0.054 0.033 0.020 0.012 1.7056 T 4.071 1.359 1.239 1.152 1.052 0.957 0.909 0.818 0.751 0.696 0.664 0.596 0.566 0.529 0.496 0.477 0.430 0.399 0.372 0.349 0.323 0.310 0.167 0.095 0.055 0.033 0.020 0.012 1.640 4.096 1.390 1.249 1.181 1.070 0.976 0.903 0.833 0.763 0.730 0.677 0.622 0.588 0.537 0.496 0.482 0.435 0.415 0.391 0.357 0.339 0.322 0.174 0.097 0.057 0.034 0.020 0.013 1.6072 4.199 1.412 1.278 1.181 1.100 0.982 0.916 0.850 0.787 0.751 0.671 0.637 0.586 0.556 0.509 0.476 0.448 0.414 0.388 0.374 0.335 0.325 0.172 0.097 0.057 0.034 0.020 0.013 1.5744 sothermal compressibility in WCA potential, simulation data. I 4.300 1.448 1.323 1.211 1.089 1.020 0.929 0.860 0.813 0.748 0.707 0.653 0.596 0.553 0.524 0.492 0.461 0.429 0.407 0.370 0.352 0.329 0.174 0.100 0.057 0.035 0.021 0.013 1.53504 4.360 1.445 1.354 1.238 1.130 1.035 0.967 0.864 0.832 0.776 0.706 0.666 0.592 0.566 0.534 0.491 0.466 0.435 0.405 0.373 0.356 0.335 0.183 0.101 0.058 0.035 0.021 0.013 1.5088 Table A.4: 0.060 0.036 0.021 0.013 0.842 0.781 0.732 0.695 0.636 0.595 0.541 0.516 0.483 0.454 0.413 0.403 0.362 0.336 0.182 0.103 1.4432 4.583 1.494 1.393 1.270 1.177 1.081 1.002 0.906 0.061 0.037 0.022 0.014 0.880 0.825 0.757 0.715 0.661 0.619 0.558 0.537 0.498 0.464 0.443 0.408 0.387 0.351 0.185 0.107 1.3776 4.744 1.583 1.465 1.330 1.216 1.125 1.038 0.935 0.064 0.037 0.022 0.014 0.909 0.861 0.784 0.738 0.655 0.637 0.606 0.562 0.520 0.483 0.449 0.412 0.397 0.374 0.196 0.112 1.312 5.035 1.655 1.517 1.368 1.291 1.184 1.101 1.012 0.7 0.8 0.9 1 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 0.6 ρ 0.1 0.2 0.21 0.22 0.23 0.24 0.25 0.26 212 Simulation data for the WCA potential 0.240 0.227 0.212 0.198 0.186 0.174 0.164 0.154 0.145 0.135 0.077 0.045 0.028 0.017 0.011 0.007 2.624 1.570 0.557 0.510 0.466 0.434 0.400 0.365 0.341 0.321 0.298 0.275 0.260 0.252 0.235 0.220 0.207 0.192 0.181 0.172 0.159 0.150 0.140 0.079 0.047 0.028 0.018 0.011 0.007 2.4928 1.672 0.574 0.537 0.489 0.449 0.419 0.389 0.363 0.335 0.312 0.288 0.267 0.265 0.243 0.230 0.213 0.201 0.188 0.177 0.167 0.158 0.149 0.082 0.049 0.029 0.018 0.012 0.008 2.3616 1.764 0.607 0.559 0.516 0.470 0.442 0.408 0.380 0.353 0.325 0.303 0.280 0.280 0.257 0.244 0.227 0.215 0.197 0.186 0.175 0.163 0.152 0.086 0.050 0.030 0.019 0.012 0.008 2.2304 1.863 0.638 0.588 0.541 0.504 0.459 0.428 0.398 0.369 0.339 0.320 0.297 0.292 0.271 0.256 0.237 0.222 0.207 0.196 0.184 0.171 0.162 0.088 0.052 0.031 0.019 0.012 0.008 2.0992 1.976 0.679 0.619 0.575 0.523 0.487 0.455 0.421 0.381 0.362 0.332 0.309 0.008 1.968 2.099 0.717 0.663 0.601 0.554 0.511 0.476 0.441 0.410 0.380 0.351 0.329 0.305 0.284 0.264 0.250 0.231 0.218 0.208 0.193 0.183 0.168 0.094 0.055 0.033 0.020 0.013 0.008 1.8368 2.249 0.760 0.703 0.639 0.597 0.539 0.501 0.472 0.432 0.400 0.371 0.349 0.327 0.304 0.284 0.262 0.250 0.232 0.216 0.203 0.189 0.180 0.100 0.057 0.034 0.021 0.013 0.009 1.7056 2.403 0.820 0.756 0.682 0.638 0.584 0.543 0.499 0.461 0.431 0.396 0.369 0.349 0.328 0.299 0.284 0.262 0.244 0.230 0.218 0.203 0.191 0.104 0.060 0.036 0.022 0.014 T 0.009 1.640 2.495 0.846 0.771 0.718 0.655 0.598 0.563 0.515 0.474 0.441 0.415 0.379 0.357 0.335 0.313 0.297 0.272 0.254 0.238 0.221 0.208 0.197 0.108 0.062 0.036 0.022 0.014 0.009 1.6072 2.520 0.864 0.783 0.730 0.669 0.608 0.566 0.525 0.484 0.456 0.424 0.392 0.368 0.339 0.315 0.301 0.276 0.261 0.245 0.228 0.214 0.201 0.110 0.062 0.037 0.023 0.014 0.009 1.5744 2.579 0.879 0.799 0.738 0.683 0.618 0.577 0.532 0.496 0.465 0.421 0.401 0.370 0.347 0.321 0.300 0.281 0.261 0.246 0.235 0.214 0.204 0.110 0.063 0.037 0.023 0.014 diabatic compressibility in WCA potential, simulation data. A 0.009 1.53504 2.634 0.900 0.820 0.756 0.687 0.639 0.585 0.542 0.509 0.472 0.439 0.408 0.376 0.348 0.331 0.309 0.290 0.268 0.255 0.235 0.221 0.208 0.111 0.065 0.037 0.023 0.014 0.009 1.5088 2.681 0.903 0.840 0.763 0.702 0.648 0.602 0.546 0.519 0.485 0.444 0.416 0.379 0.356 0.335 0.309 0.291 0.275 0.254 0.237 0.224 0.210 0.115 0.065 0.038 0.023 0.014 Table A.5: 0.230 0.215 0.118 0.066 0.039 0.024 0.015 0.009 0.731 0.673 0.626 0.571 0.531 0.492 0.458 0.432 0.396 0.371 0.342 0.325 0.302 0.285 0.262 0.252 1.4432 2.815 0.938 0.865 0.791 0.244 0.224 0.119 0.069 0.040 0.024 0.015 0.009 0.762 0.704 0.649 0.589 0.553 0.518 0.476 0.448 0.416 0.388 0.356 0.336 0.314 0.291 0.276 0.258 1.3776 2.920 0.985 0.905 0.831 0.249 0.236 0.125 0.072 0.042 0.025 0.015 0.010 0.802 0.737 0.685 0.627 0.573 0.537 0.494 0.462 0.420 0.400 0.375 0.349 0.326 0.304 0.283 0.264 1.312 3.079 1.030 0.946 0.857 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 ρ 0.1 0.2 0.21 0.22 213 3.918 3.993 4.059 4.135 4.206 4.292 5.105 6.076 7.202 8.499 10.064 11.780 2.624 2.520 2.994 3.060 3.116 3.166 3.224 3.305 3.360 3.401 3.463 3.540 3.585 3.662 3.716 3.782 3.861 3.857 3.923 3.973 4.061 4.129 4.215 5.014 5.977 7.104 8.410 9.941 11.618 2.4928 2.446 2.946 2.983 3.049 3.109 3.154 3.210 3.258 3.332 3.384 3.455 3.528 3.581 3.643 3.710 3.769 3.775 3.836 3.900 3.968 4.037 4.103 4.909 5.859 6.975 8.288 9.791 11.522 2.3616 2.382 2.870 2.920 2.967 3.038 3.073 3.131 3.187 3.245 3.317 3.376 3.454 3.491 3.576 3.630 3.711 3.661 3.755 3.809 3.879 3.962 4.046 4.829 5.752 6.861 8.160 9.646 11.366 2.2304 2.320 2.800 2.846 2.900 2.939 3.009 3.058 3.111 3.176 3.246 3.286 3.352 3.402 3.483 3.529 3.608 3.584 3.665 3.714 3.782 3.861 3.928 4.744 5.654 6.740 8.017 9.500 11.215 2.0992 2.250 2.713 2.772 2.809 2.882 2.924 2.969 3.025 3.116 3.145 3.219 3.282 3.329 3.397 3.447 3.523 2.184 2.640 2.679 2.746 2.800 2.850 2.898 2.958 3.009 3.068 3.134 3.183 3.252 3.311 3.386 3.421 3.513 3.560 3.613 3.692 3.748 3.844 4.618 5.518 6.621 7.892 9.374 11.072 1.968 2.111 2.563 2.603 2.662 2.702 2.777 2.823 2.857 2.926 2.989 3.047 3.091 3.143 3.205 3.269 3.345 3.386 3.462 3.532 3.599 3.675 3.734 4.476 5.398 6.480 7.747 9.210 10.929 1.8368 2.041 2.470 2.510 2.578 2.613 2.675 2.714 2.779 2.832 2.882 2.949 3.006 3.041 3.093 3.180 3.224 3.308 3.372 3.427 3.470 3.553 3.612 4.383 5.284 6.342 7.589 9.046 10.747 1.7056 T 2.003 2.432 2.483 2.521 2.576 2.634 2.666 2.733 2.792 2.847 2.884 2.963 3.005 3.061 3.116 3.155 3.244 3.310 3.375 3.442 3.515 3.565 4.314 5.202 6.267 7.510 8.966 10.671 1.640 1.990 2.405 2.463 2.494 2.550 2.608 2.656 2.707 2.767 2.800 2.855 2.918 2.964 3.035 3.100 3.131 3.216 3.264 3.324 3.404 3.462 3.521 4.261 5.164 6.206 7.456 8.939 10.609 1.6072 peed of sound in WCA potential, simulation data. 1.969 2.387 2.441 2.483 2.524 2.593 2.635 2.686 2.737 2.773 2.853 2.890 2.955 3.001 3.072 3.130 3.185 3.257 3.317 3.357 3.458 3.499 4.256 5.142 6.183 7.431 8.894 10.575 1.5744 S 1.947 2.359 2.407 2.454 2.515 2.555 2.613 2.663 2.700 2.757 2.802 2.859 2.928 2.989 3.033 3.088 3.146 3.213 3.264 3.347 3.403 3.470 4.224 5.086 6.159 7.384 8.824 10.513 1.53504 Table A.6: 1.931 2.352 2.383 2.435 2.485 2.536 2.578 2.649 2.675 2.722 2.791 2.835 2.920 2.961 3.009 3.080 3.129 3.187 3.257 3.331 3.381 3.445 4.165 5.057 6.117 7.344 8.822 10.485 1.5088 6.036 7.257 8.723 10.411 2.642 2.693 2.740 2.780 2.847 2.899 2.972 3.014 3.073 3.127 3.209 3.241 3.336 3.409 4.128 4.996 1.4432 1.887 2.308 2.344 2.394 2.437 2.486 2.530 2.594 5.958 7.171 8.640 10.290 2.588 2.630 2.691 2.733 2.791 2.841 2.920 2.958 3.020 3.083 3.129 3.200 3.254 3.342 4.080 4.922 1.3776 1.851 2.252 2.292 2.341 2.390 2.435 2.484 2.549 5.849 7.088 8.540 10.208 2.540 2.578 2.641 2.685 2.767 2.794 2.837 2.898 2.960 3.022 3.086 3.160 3.203 3.261 3.988 4.827 1.312 1.803 2.203 2.246 2.301 2.330 2.380 2.422 2.474 0.7 0.8 0.9 1 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 0.6 ρ 0.1 0.2 0.21 0.22 0.23 0.24 0.25 0.26 214 Simulation data for the WCA potential -0.594 -0.579 -0.559 -0.574 -0.538 -0.547 -0.525 -0.533 -0.526 -0.508 -0.457 -0.397 -0.351 -0.301 -0.277 -0.246 2.624 -0.817 -0.641 -0.687 -0.650 -0.638 -0.621 -0.661 -0.653 -0.629 -0.601 -0.619 -0.585 -0.591 -0.578 -0.572 -0.549 -0.570 -0.565 -0.544 -0.528 -0.520 -0.525 -0.448 -0.402 -0.350 -0.306 -0.277 -0.237 2.4928 -0.781 -0.698 -0.675 -0.673 -0.666 -0.645 -0.635 -0.605 -0.634 -0.599 -0.597 -0.606 -0.572 -0.581 -0.578 -0.580 -0.577 -0.541 -0.537 -0.532 -0.533 -0.518 -0.437 -0.397 -0.343 -0.308 -0.269 -0.241 2.3616 -0.783 -0.695 -0.686 -0.647 -0.664 -0.633 -0.620 -0.620 -0.614 -0.622 -0.611 -0.630 -0.575 -0.582 -0.557 -0.574 -0.560 -0.561 -0.539 -0.538 -0.532 -0.537 -0.467 -0.389 -0.346 -0.308 -0.268 -0.233 2.2304 -0.800 -0.711 -0.678 -0.679 -0.627 -0.642 -0.633 -0.617 -0.640 -0.630 -0.592 -0.594 -0.596 -0.589 -0.569 -0.570 -0.549 -0.573 -0.543 -0.531 -0.521 -0.521 -0.467 -0.400 -0.349 -0.302 -0.263 -0.227 2.0992 -0.753 -0.672 -0.676 -0.632 -0.679 -0.636 -0.632 -0.616 -0.658 -0.609 -0.610 -0.608 -0.233 1.968 -0.771 -0.679 -0.641 -0.669 -0.672 -0.648 -0.631 -0.652 -0.630 -0.621 -0.625 -0.596 -0.604 -0.588 -0.606 -0.543 -0.583 -0.544 -0.535 -0.533 -0.520 -0.524 -0.470 -0.398 -0.352 -0.305 -0.266 -0.234 1.8368 -0.774 -0.693 -0.666 -0.662 -0.653 -0.677 -0.657 -0.616 -0.626 -0.643 -0.627 -0.600 -0.586 -0.573 -0.572 -0.578 -0.553 -0.551 -0.549 -0.546 -0.538 -0.532 -0.451 -0.396 -0.345 -0.307 -0.264 -0.225 1.7056 -0.779 -0.673 -0.638 -0.675 -0.649 -0.666 -0.622 -0.650 -0.620 -0.616 -0.621 -0.614 -0.571 -0.564 -0.584 -0.559 -0.587 -0.566 -0.549 -0.515 -0.525 -0.507 -0.463 -0.407 -0.349 -0.300 -0.258 T -0.227 1.640 -0.793 -0.692 -0.679 -0.672 -0.653 -0.652 -0.616 -0.650 -0.644 -0.643 -0.596 -0.626 -0.600 -0.594 -0.576 -0.542 -0.566 -0.567 -0.565 -0.539 -0.559 -0.525 -0.463 -0.401 -0.349 -0.297 -0.257 cient in WCA potential, simulation data. ffi -0.224 1.6072 -0.828 -0.690 -0.705 -0.643 -0.668 -0.644 -0.651 -0.649 -0.657 -0.611 -0.604 -0.609 -0.584 -0.587 -0.591 -0.550 -0.571 -0.544 -0.531 -0.550 -0.528 -0.501 -0.438 -0.394 -0.342 -0.298 -0.265 -0.222 1.5744 -0.817 -0.697 -0.696 -0.680 -0.646 -0.675 -0.668 -0.639 -0.642 -0.591 -0.612 -0.607 -0.605 -0.572 -0.576 -0.568 -0.551 -0.553 -0.550 -0.527 -0.547 -0.506 -0.460 -0.397 -0.346 -0.298 -0.262 oule-Thomson coe J -0.221 1.53504 -0.798 -0.694 -0.670 -0.681 -0.702 -0.663 -0.669 -0.658 -0.625 -0.633 -0.587 -0.589 -0.598 -0.589 -0.583 -0.567 -0.556 -0.539 -0.533 -0.550 -0.517 -0.523 -0.456 -0.396 -0.350 -0.296 -0.254 -0.227 1.5088 -0.828 -0.713 -0.675 -0.638 -0.655 -0.665 -0.637 -0.664 -0.620 -0.615 -0.620 -0.590 -0.633 -0.588 -0.572 -0.568 -0.547 -0.556 -0.535 -0.550 -0.520 -0.514 -0.441 -0.399 -0.341 -0.298 -0.266 Table A.7: -0.542 -0.542 -0.471 -0.394 -0.347 -0.296 -0.262 -0.225 -0.654 -0.648 -0.646 -0.663 -0.650 -0.636 -0.608 -0.582 -0.577 -0.568 -0.588 -0.572 -0.551 -0.549 -0.560 -0.524 1.4432 -0.826 -0.731 -0.680 -0.677 -0.531 -0.541 -0.473 -0.396 -0.343 -0.299 -0.260 -0.218 -0.684 -0.669 -0.656 -0.659 -0.643 -0.632 -0.622 -0.607 -0.602 -0.586 -0.608 -0.564 -0.570 -0.550 -0.528 -0.545 1.3776 -0.848 -0.701 -0.663 -0.691 -0.524 -0.526 -0.462 -0.395 -0.346 -0.300 -0.262 -0.224 -0.661 -0.656 -0.644 -0.623 -0.656 -0.617 -0.631 -0.605 -0.645 -0.592 -0.550 -0.553 -0.564 -0.556 -0.553 -0.569 1.312 -0.796 -0.710 -0.702 -0.701 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 ρ 0.1 0.2 0.21 0.22 215 0.657 0.686 0.721 0.752 0.787 0.823 1.239 1.799 2.532 3.526 4.601 6.027 2.624 0.119 0.285 0.303 0.324 0.346 0.368 0.387 0.410 0.435 0.461 0.484 0.512 0.537 0.566 0.596 0.622 0.655 0.687 0.723 0.756 0.792 0.824 1.253 1.810 2.560 3.526 4.679 6.284 2.4928 0.119 0.284 0.305 0.325 0.345 0.368 0.391 0.415 0.437 0.463 0.488 0.512 0.541 0.569 0.598 0.630 0.659 0.695 0.728 0.762 0.796 0.835 1.270 1.841 2.612 3.576 4.848 6.304 2.3616 0.119 0.285 0.305 0.327 0.347 0.370 0.393 0.416 0.440 0.464 0.490 0.513 0.546 0.572 0.600 0.628 0.668 0.697 0.733 0.766 0.801 0.833 1.266 1.867 2.641 3.631 4.950 6.563 2.2304 0.119 0.286 0.307 0.327 0.350 0.371 0.394 0.418 0.440 0.465 0.494 0.520 0.549 0.575 0.608 0.634 0.673 0.700 0.739 0.774 0.810 0.846 1.276 1.876 2.669 3.729 5.086 6.793 2.0992 0.120 0.288 0.308 0.330 0.349 0.374 0.396 0.421 0.441 0.470 0.495 0.521 0.550 0.578 0.610 0.640 0.120 0.290 0.311 0.331 0.352 0.374 0.398 0.421 0.446 0.472 0.497 0.526 0.552 0.582 0.608 0.648 0.672 0.710 0.746 0.780 0.819 0.851 1.292 1.911 2.705 3.770 5.132 6.837 1.968 0.120 0.290 0.311 0.332 0.354 0.375 0.399 0.426 0.449 0.473 0.500 0.529 0.559 0.588 0.619 0.647 0.683 0.715 0.749 0.784 0.819 0.859 1.324 1.939 2.768 3.831 5.288 6.956 1.8368 0.120 0.292 0.314 0.333 0.357 0.378 0.404 0.426 0.452 0.479 0.504 0.532 0.565 0.596 0.622 0.657 0.683 0.718 0.756 0.799 0.832 0.875 1.327 1.951 2.810 3.959 5.452 7.300 1.7056 T cient WCA potential simulation data. ffi 0.120 0.293 0.313 0.335 0.357 0.380 0.406 0.427 0.452 0.478 0.509 0.532 0.563 0.594 0.626 0.663 0.691 0.722 0.757 0.796 0.828 0.874 1.339 1.980 2.836 4.011 5.532 7.349 1.640 0.120 0.293 0.313 0.336 0.357 0.381 0.404 0.428 0.452 0.482 0.509 0.536 0.567 0.595 0.625 0.663 0.691 0.729 0.767 0.797 0.839 0.884 1.360 1.992 2.883 4.042 5.464 7.467 1.6072 0.121 0.293 0.314 0.335 0.359 0.379 0.404 0.429 0.454 0.485 0.508 0.538 0.565 0.599 0.629 0.661 0.696 0.728 0.763 0.806 0.835 0.884 1.348 1.996 2.874 4.051 5.525 7.538 1.5744 sothermal pressure coe I 0.121 0.294 0.315 0.336 0.357 0.382 0.404 0.429 0.457 0.482 0.513 0.540 0.567 0.597 0.630 0.663 0.698 0.733 0.771 0.801 0.846 0.884 1.354 2.015 2.866 4.087 5.680 7.639 1.53504 0.121 0.293 0.315 0.338 0.359 0.381 0.407 0.429 0.458 0.485 0.511 0.541 0.564 0.599 0.632 0.663 0.700 0.733 0.770 0.803 0.847 0.886 1.374 2.017 2.907 4.108 5.540 7.578 1.5088 Table A.8: 2.932 4.163 5.672 7.655 0.457 0.484 0.513 0.545 0.573 0.604 0.631 0.667 0.703 0.738 0.769 0.815 0.846 0.883 1.362 2.033 1.4432 0.121 0.294 0.316 0.337 0.361 0.384 0.408 0.431 2.967 4.208 5.735 7.963 0.460 0.487 0.514 0.545 0.574 0.606 0.633 0.672 0.704 0.741 0.781 0.815 0.856 0.889 1.367 2.056 1.3776 0.121 0.295 0.318 0.338 0.360 0.384 0.409 0.432 3.016 4.245 5.821 7.941 0.460 0.490 0.516 0.547 0.570 0.608 0.645 0.677 0.709 0.744 0.779 0.812 0.861 0.901 1.387 2.083 1.312 0.121 0.296 0.317 0.339 0.363 0.387 0.411 0.437 0.7 0.8 0.9 1 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 0.6 ρ 0.1 0.2 0.21 0.22 0.23 0.24 0.25 0.26 216 Simulation data for the WCA potential 0.202 0.202 0.199 0.192 0.193 0.187 0.186 0.181 0.179 0.174 0.147 0.124 0.105 0.092 0.072 0.060 2.624 0.301 0.257 0.245 0.241 0.240 0.235 0.222 0.219 0.220 0.217 0.209 0.210 0.214 0.211 0.207 0.206 0.196 0.193 0.194 0.188 0.187 0.180 0.154 0.129 0.109 0.092 0.075 0.066 2.4928 0.324 0.260 0.261 0.252 0.246 0.245 0.241 0.241 0.230 0.229 0.222 0.214 0.230 0.219 0.217 0.209 0.206 0.205 0.203 0.200 0.196 0.195 0.165 0.137 0.117 0.096 0.081 0.066 2.3616 0.342 0.276 0.272 0.271 0.259 0.262 0.257 0.252 0.247 0.238 0.234 0.223 0.244 0.233 0.235 0.225 0.225 0.214 0.215 0.210 0.204 0.197 0.167 0.145 0.122 0.101 0.086 0.072 2.2304 0.361 0.290 0.289 0.281 0.284 0.272 0.269 0.265 0.256 0.248 0.251 0.244 0.253 0.246 0.247 0.238 0.236 0.226 0.228 0.224 0.219 0.215 0.174 0.150 0.127 0.108 0.092 0.077 2.0992 0.387 0.315 0.305 0.307 0.290 0.291 0.288 0.283 0.263 0.270 0.259 0.253 0.079 1.968 0.411 0.334 0.333 0.318 0.310 0.306 0.303 0.293 0.290 0.284 0.275 0.274 0.265 0.261 0.250 0.259 0.242 0.245 0.245 0.237 0.237 0.225 0.187 0.162 0.133 0.112 0.094 0.082 1.8368 0.441 0.353 0.352 0.341 0.339 0.320 0.317 0.322 0.308 0.297 0.292 0.293 0.291 0.284 0.278 0.267 0.270 0.261 0.255 0.250 0.242 0.241 0.208 0.171 0.143 0.118 0.101 0.090 1.7056 0.473 0.387 0.385 0.364 0.366 0.352 0.353 0.337 0.333 0.328 0.316 0.310 0.317 0.314 0.294 0.296 0.279 0.274 0.274 0.279 0.267 0.267 0.214 0.178 0.152 0.129 0.109 T cient in WCA potential, simulation data. 0.091 1.640 0.491 0.398 0.388 0.386 0.376 0.364 0.369 0.349 0.339 0.332 0.338 0.317 0.319 0.314 0.311 0.317 0.297 0.288 0.282 0.278 0.268 0.271 0.224 0.188 0.157 0.134 0.113 ffi 0.095 1.6072 0.493 0.407 0.390 0.397 0.382 0.371 0.365 0.357 0.345 0.352 0.345 0.333 0.333 0.320 0.310 0.320 0.301 0.303 0.299 0.285 0.285 0.285 0.237 0.193 0.165 0.137 0.110 0.096 1.5744 0.506 0.414 0.401 0.396 0.395 0.373 0.370 0.365 0.357 0.364 0.341 0.342 0.331 0.333 0.320 0.315 0.312 0.301 0.296 0.302 0.280 0.288 0.232 0.194 0.164 0.138 0.113 hermal expansion coe 0.099 1.53504 0.520 0.425 0.417 0.407 0.388 0.389 0.375 0.369 0.371 0.361 0.362 0.353 0.338 0.330 0.330 0.326 0.322 0.314 0.313 0.296 0.298 0.291 0.236 0.202 0.163 0.141 0.120 T 0.098 1.5088 0.526 0.424 0.427 0.418 0.406 0.395 0.393 0.370 0.381 0.376 0.360 0.360 0.334 0.339 0.338 0.326 0.326 0.319 0.312 0.299 0.302 0.297 0.252 0.204 0.170 0.144 0.114 Table A.9: 0.307 0.296 0.248 0.210 0.175 0.149 0.121 0.100 0.425 0.415 0.409 0.390 0.385 0.378 0.375 0.379 0.365 0.360 0.341 0.344 0.339 0.335 0.318 0.329 1.4432 0.554 0.439 0.440 0.429 0.331 0.312 0.253 0.219 0.182 0.154 0.124 0.108 0.438 0.432 0.425 0.404 0.404 0.402 0.389 0.390 0.379 0.375 0.353 0.360 0.350 0.344 0.345 0.333 1.3776 0.574 0.468 0.465 0.450 0.342 0.337 0.272 0.233 0.194 0.159 0.129 0.108 0.468 0.458 0.453 0.442 0.418 0.422 0.404 0.403 0.374 0.387 0.391 0.380 0.368 0.359 0.350 0.335 1.312 0.611 0.490 0.481 0.463 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 ρ 0.1 0.2 0.21 0.22 217 0.387 0.400 0.421 0.435 0.452 0.481 0.690 0.991 1.373 1.894 2.638 3.572 2.624 0.078 0.177 0.187 0.202 0.212 0.226 0.239 0.250 0.260 0.277 0.291 0.305 0.321 0.333 0.352 0.366 0.366 0.379 0.392 0.420 0.436 0.454 0.668 0.944 1.325 1.838 2.525 3.421 2.4928 0.073 0.169 0.177 0.189 0.202 0.212 0.224 0.236 0.247 0.263 0.278 0.292 0.304 0.319 0.333 0.351 0.344 0.367 0.379 0.393 0.406 0.422 0.639 0.892 1.267 1.745 2.407 3.320 2.3616 0.069 0.158 0.168 0.179 0.192 0.200 0.212 0.223 0.234 0.248 0.260 0.273 0.288 0.306 0.315 0.332 0.320 0.342 0.356 0.370 0.390 0.407 0.594 0.863 1.199 1.659 2.276 3.178 2.2304 0.064 0.149 0.159 0.168 0.179 0.191 0.200 0.211 0.220 0.235 0.246 0.259 0.270 0.286 0.297 0.309 0.309 0.319 0.333 0.349 0.368 0.379 0.570 0.809 1.133 1.572 2.174 3.046 2.0992 0.061 0.140 0.150 0.159 0.168 0.178 0.186 0.197 0.210 0.219 0.234 0.245 0.253 0.268 0.278 0.293 0.057 0.131 0.139 0.149 0.157 0.168 0.177 0.184 0.195 0.206 0.217 0.228 0.239 0.252 0.262 0.278 0.288 0.303 0.312 0.328 0.339 0.361 0.524 0.751 1.067 1.499 2.083 2.878 1.968 0.052 0.122 0.129 0.139 0.145 0.156 0.164 0.172 0.183 0.191 0.202 0.212 0.221 0.234 0.244 0.258 0.266 0.280 0.293 0.304 0.322 0.330 0.487 0.708 1.011 1.403 1.941 2.753 1.8368 0.048 0.112 0.119 0.128 0.134 0.142 0.151 0.159 0.169 0.177 0.187 0.197 0.205 0.213 0.227 0.236 0.246 0.261 0.272 0.283 0.296 0.309 0.459 0.660 0.938 1.316 1.848 2.597 1.7056 T 0.046 0.107 0.115 0.120 0.130 0.139 0.145 0.153 0.162 0.170 0.179 0.189 0.197 0.205 0.216 0.224 0.237 0.247 0.257 0.275 0.280 0.294 0.437 0.633 0.903 1.283 1.793 2.523 1.640 0.046 0.105 0.112 0.119 0.126 0.137 0.142 0.149 0.157 0.165 0.174 0.183 0.191 0.203 0.212 0.220 0.232 0.242 0.253 0.263 0.276 0.290 0.434 0.630 0.877 1.245 1.767 2.481 1.6072 0.044 0.102 0.110 0.116 0.124 0.132 0.138 0.147 0.154 0.162 0.174 0.178 0.188 0.198 0.209 0.218 0.229 0.240 0.250 0.256 0.272 0.285 0.425 0.618 0.870 1.233 1.737 2.460 1.5744 otential energy in WCA potential, simulation data. P 0.044 0.100 0.107 0.113 0.120 0.128 0.135 0.143 0.150 0.157 0.167 0.175 0.185 0.196 0.200 0.210 0.220 0.234 0.241 0.252 0.266 0.274 0.420 0.595 0.864 1.214 1.696 2.390 1.53504 Table A.10: 0.042 0.099 0.104 0.113 0.119 0.125 0.133 0.141 0.146 0.153 0.163 0.171 0.180 0.190 0.198 0.211 0.220 0.225 0.241 0.249 0.262 0.272 0.404 0.583 0.854 1.180 1.684 2.345 1.5088 0.804 1.141 1.617 2.312 0.141 0.149 0.157 0.164 0.175 0.183 0.192 0.197 0.209 0.216 0.228 0.234 0.248 0.261 0.386 0.574 1.4432 0.040 0.094 0.100 0.107 0.114 0.120 0.126 0.134 0.783 1.088 1.590 2.213 0.134 0.140 0.149 0.154 0.163 0.171 0.180 0.190 0.197 0.209 0.218 0.224 0.233 0.246 0.376 0.544 1.3776 0.038 0.089 0.096 0.100 0.107 0.113 0.120 0.129 0.722 1.047 1.509 2.134 0.128 0.134 0.142 0.149 0.157 0.164 0.174 0.181 0.188 0.198 0.207 0.215 0.227 0.233 0.355 0.509 1.312 0.036 0.084 0.090 0.097 0.101 0.108 0.113 0.121 0.7 0.8 0.9 1 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 0.6 ρ 0.1 0.2 0.21 0.22 0.23 0.24 0.25 0.26 Appendix B

Simulation data for the LJ f uid

Simulation results of the LJ f uid are given below. The simulation data of system of particles interacting with LJ potential consisting of 2000 particles with a cutoﬀ radius length of 6.5σ are given below. The results are given in reduced unit, where the reduction scheme is given in Chapter Four. This result is also used to calculate the thermodynamic quantities of argon and krypton.

218 219 0.965 0.996 1.037 1.079 1.130 1.161 1.231 1.866 2.907 4.725 7.624 12.029 18.915 2.624 0.253 0.378 0.506 0.534 0.563 0.590 0.619 0.648 0.681 0.710 0.741 0.778 0.805 0.840 0.880 0.920 0.872 0.904 0.948 0.987 1.026 1.075 1.110 1.700 2.708 4.371 7.129 11.545 18.438 2.4928 0.237 0.353 0.471 0.496 0.520 0.543 0.575 0.600 0.628 0.657 0.681 0.713 0.739 0.773 0.812 0.842 0.788 0.832 0.853 0.889 0.925 0.962 1.013 1.510 2.480 4.047 6.758 10.918 17.509 2.3616 0.222 0.326 0.434 0.457 0.481 0.501 0.527 0.546 0.573 0.598 0.623 0.648 0.677 0.705 0.733 0.755 0.630 0.648 0.683 0.701 0.731 0.757 0.787 1.206 1.967 3.280 5.633 9.598 15.912 2.0992 0.190 0.275 0.359 0.375 0.393 0.412 0.427 0.443 0.465 0.483 0.500 0.520 0.539 0.562 0.584 0.603 0.158 0.222 0.279 0.294 0.304 0.317 0.329 0.341 0.355 0.366 0.378 0.391 0.406 0.421 0.424 0.444 0.460 0.481 0.492 0.512 0.527 0.548 0.572 0.867 1.448 2.566 4.622 8.317 14.074 1.8368 0.134 0.183 0.224 0.231 0.238 0.248 0.256 0.263 0.271 0.279 0.283 0.294 0.296 0.309 0.313 0.326 0.330 0.342 0.344 0.359 0.374 0.376 0.392 0.599 1.061 2.038 3.886 7.206 12.811 1.640 0.129 0.176 0.214 0.221 0.228 0.236 0.242 0.246 0.253 0.262 0.267 0.277 0.282 0.290 0.293 0.309 0.308 0.324 0.337 0.333 0.353 0.360 0.364 0.565 1.000 1.871 3.680 7.092 12.251 1.6072 T 0.125 0.170 0.203 0.213 0.218 0.221 0.230 0.234 0.240 0.246 0.257 0.261 0.262 0.268 0.280 0.286 0.294 0.301 0.309 0.323 0.325 0.339 0.344 0.522 0.896 1.823 3.588 6.687 11.996 1.5744 0.121 0.161 0.194 0.199 0.206 0.209 0.215 0.221 0.224 0.229 0.232 0.242 0.246 0.249 0.254 0.258 0.267 0.268 0.277 0.290 0.289 0.305 0.315 0.468 0.804 1.743 3.373 6.538 11.708 1.53504 ressure in the LJ f uid, simulation data. P 0.117 0.155 0.185 0.190 0.194 0.199 0.207 0.208 0.209 0.217 0.223 0.223 0.232 0.235 0.239 0.243 0.248 0.253 0.262 0.268 0.273 0.282 0.286 0.415 0.811 1.663 3.246 6.372 11.605 1.5088 Table B.1: 0.109 0.142 0.165 0.171 0.170 0.174 0.177 0.185 0.184 0.187 0.190 0.198 0.195 0.204 0.203 0.204 0.211 0.208 0.213 0.218 0.218 0.233 0.237 0.354 0.632 1.433 3.026 5.947 11.041 1.4432 0.514 1.265 2.720 5.681 7.636 0.154 0.157 0.155 0.158 0.162 0.163 0.159 0.169 0.169 0.167 0.164 0.172 0.170 0.183 0.176 0.248 1.3776 0.100 0.128 0.144 0.147 0.150 0.149 0.156 0.157 0.385 1.003 2.465 5.260 6.778 0.129 0.123 0.130 0.124 0.128 0.122 0.123 0.126 0.125 0.121 0.126 0.120 0.122 0.120 0.127 0.165 1.312 0.092 0.114 0.125 0.125 0.126 0.126 0.127 0.127 0.339 1.003 2.396 5.096 6.738 0.123 0.123 0.115 0.127 0.125 0.122 0.111 0.123 0.117 0.117 0.119 0.118 0.106 0.113 0.120 0.138 1.300 0.091 0.110 0.119 0.122 0.120 0.123 0.118 0.124 0.6 0.7 0.8 0.9 1 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 ρ 0.1 0.15 0.2 0.21 0.22 0.23 0.24 0.25 220 Simulation data for the LJ f uid 1.663 1.669 1.675 1.679 1.682 1.689 1.694 1.698 1.703 1.712 1.716 1.781 1.881 2.011 2.185 2.420 2.716 2.624 1.564 1.591 1.617 1.622 1.627 1.633 1.636 1.642 1.646 1.651 1.655 1.659 1.674 1.677 1.681 1.685 1.692 1.698 1.701 1.705 1.712 1.716 1.723 1.790 1.886 2.023 2.197 2.431 2.744 2.4928 1.569 1.598 1.624 1.632 1.637 1.642 1.642 1.649 1.652 1.656 1.663 1.667 1.680 1.685 1.689 1.696 1.699 1.699 1.708 1.713 1.717 1.722 1.726 1.796 1.888 2.018 2.178 2.444 2.757 2.3616 1.574 1.610 1.633 1.639 1.642 1.648 1.652 1.659 1.664 1.667 1.669 1.675 1.711 1.713 1.715 1.721 1.724 1.731 1.735 1.741 1.745 1.747 1.752 1.812 1.904 2.043 2.230 2.472 2.793 2.0992 1.596 1.631 1.660 1.668 1.671 1.678 1.686 1.690 1.692 1.698 1.702 1.707 2.494 2.816 1.8368 1.622 1.682 1.730 1.726 1.733 1.739 1.742 1.748 1.749 1.759 1.757 1.757 1.762 1.762 1.777 1.785 1.778 1.776 1.787 1.785 1.793 1.791 1.791 1.835 1.919 2.062 2.244 2.506 2.835 1.640 1.669 1.739 1.815 1.822 1.822 1.812 1.822 1.838 1.835 1.831 1.865 1.856 1.870 1.862 1.867 1.863 1.853 1.858 1.872 1.872 1.861 1.871 1.856 1.868 1.931 2.068 2.270 2.508 2.838 1.6072 1.676 1.741 1.824 1.834 1.836 1.841 1.852 1.863 1.889 1.873 1.875 1.874 1.883 1.881 1.898 1.870 1.894 1.883 1.858 1.889 1.861 1.878 1.876 1.871 1.940 2.084 2.276 T 2.536 2.829 1.5744 1.694 1.767 1.857 1.831 1.854 1.908 1.873 1.904 1.893 1.915 1.895 1.903 1.922 1.914 1.899 1.903 1.909 1.899 1.898 1.895 1.909 1.887 1.889 1.882 1.947 2.076 2.281 2.539 2.861 1.53504 1.708 1.800 1.848 1.893 1.885 1.892 1.915 1.911 1.911 1.935 1.962 1.922 1.942 1.967 1.942 1.951 1.951 1.957 1.931 1.923 1.938 1.916 1.916 1.890 1.950 2.077 2.279 2.554 2.878 1.5088 1.717 1.817 1.897 1.922 1.943 1.944 1.926 1.980 1.987 1.976 1.958 1.995 1.959 1.960 1.992 1.978 1.967 1.981 1.951 1.941 1.956 1.943 1.944 1.911 1.954 2.087 2.287 I 2.562 2.876 1.4432 1.758 1.891 2.007 1.982 2.061 2.063 2.033 2.037 2.143 2.083 2.073 2.040 2.103 2.052 2.081 2.085 2.048 2.076 2.060 2.062 2.042 2.015 1.992 1.928 1.969 2.096 2.288 Table B.2: sochoric heat capacity in the LJ f uid, simulation data. 2.570 3.412 1.3776 1.797 1.965 2.166 2.177 2.123 2.267 2.198 2.164 2.284 2.310 2.404 2.330 2.287 2.271 2.342 2.285 2.173 2.291 2.267 2.164 2.213 2.106 2.126 1.971 1.978 2.099 2.310 2.532 2.494 2.323 2.032 1.995 2.107 2.325 2.576 3.074 2.569 2.520 2.688 2.764 2.624 3.163 2.491 2.817 2.694 2.934 2.863 2.624 2.747 2.825 2.552 2.688 1.312 1.855 2.120 2.305 2.624 2.804 2.599 2.301 2.054 1.995 2.117 2.319 2.573 3.023 2.687 2.630 3.309 2.614 2.799 2.928 4.207 2.446 2.622 2.727 3.578 2.805 2.725 2.939 2.737 2.538 1.300 1.850 2.201 2.588 2.531 0.38 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 ρ 0.1 0.15 0.2 0.21 221 3.924 3.994 3.921 3.860 3.977 3.929 3.877 3.751 3.671 3.587 3.600 3.708 3.855 2.624 3.051 3.376 3.529 3.624 3.722 3.636 3.736 3.848 3.785 3.873 3.785 3.766 3.792 3.886 3.979 3.945 4.201 4.161 4.028 3.995 4.170 4.036 4.079 3.929 3.739 3.677 3.639 3.743 3.931 2.4928 3.104 3.473 3.730 3.766 3.782 3.870 3.826 4.001 4.084 3.831 4.039 4.014 4.162 4.049 4.108 4.033 4.361 4.181 4.379 4.194 4.286 4.301 4.162 4.087 3.843 3.683 3.604 3.769 3.922 2.3616 3.202 3.625 3.844 3.886 3.886 4.094 3.909 4.025 3.983 4.201 4.057 4.097 4.138 4.299 4.331 4.224 4.660 4.772 4.819 4.876 4.818 4.670 4.780 4.477 4.046 3.901 3.796 3.863 4.004 2.0992 3.373 3.897 4.338 4.492 4.416 4.547 4.659 4.446 4.576 4.838 4.686 4.926 4.672 4.785 4.790 4.766 6.377 5.763 5.964 6.127 5.959 5.679 5.741 5.263 4.325 4.087 3.871 3.911 4.064 1.8368 3.584 4.369 5.393 5.042 5.080 5.206 5.515 5.522 5.303 5.884 5.835 5.489 5.744 5.898 5.957 6.240 3.885 5.121 6.220 6.837 7.001 6.126 7.490 7.445 6.977 8.800 7.856 8.231 7.662 8.256 7.706 7.647 8.561 7.610 8.260 8.006 7.184 8.790 7.089 5.691 4.616 4.158 4.037 3.932 4.118 1.640 4.126 5.378 6.687 6.625 7.569 8.014 7.738 7.635 7.515 8.866 9.349 8.422 9.247 8.858 8.925 9.356 8.747 8.680 8.134 8.267 7.570 8.237 8.070 6.202 4.853 4.395 4.075 3.932 4.110 1.6072 T 4.228 5.689 7.492 7.132 8.144 8.981 8.220 9.108 8.721 9.278 8.826 10.887 8.927 9.903 9.494 9.709 8.471 10.398 9.547 9.316 9.385 8.428 8.050 6.060 5.201 4.261 4.096 4.053 4.072 1.5744 4.365 6.011 7.028 10.313 9.296 8.345 9.886 8.748 8.924 8.684 10.488 10.130 10.689 11.777 11.023 9.916 11.886 11.156 10.015 9.413 9.523 11.129 14.243 6.672 5.039 4.240 4.064 4.053 4.175 1.53504 4.395 6.422 7.780 9.822 9.125 10.165 8.765 14.375 10.129 11.620 10.978 12.200 10.481 10.306 14.640 11.330 10.722 14.368 11.225 9.609 12.384 8.096 10.310 7.583 5.205 4.442 4.108 4.118 4.222 1.5088 I 4.792 7.100 11.320 10.857 13.398 12.419 18.283 11.974 17.817 15.816 14.909 23.699 14.684 13.190 20.564 13.070 12.101 18.351 17.484 16.353 20.287 12.281 11.605 8.830 5.319 4.509 4.114 4.137 4.193 1.4432 Table B.3: sobaric heat capacity in the LJ f uid, simulation data. 5.059 9.083 14.524 40.305 13.531 21.503 16.911 34.881 23.901 33.281 41.073 31.125 52.022 26.243 47.090 22.100 18.222 38.414 171.769 16.906 42.062 24.822 25.179 8.358 5.759 4.591 4.243 4.165 9.646 1.3776 6.459 4.710 4.370 4.152 5.916 63.073 -108.442 318.376 -77.830 40.835 -271.574 -38.177 -32.071 -92.440 -40.660 -21.073 -19.026 1295.720 -29.206 73.926 11.902 1.312 5.600 9.712 198.731 67.887 304.336 36.202 74.647 43041.535 6.488 5.079 4.362 4.173 5.648 -99.680 -231.024 -26.290 269.624 11.750 373.222 -31.781 492.588 -43.678 -25.760 -77.511 3185.617 -25.312 -79.828 -161.044 13.886 1.300 5.966 12.258 32.522 16.677 143.942 47.294 26.849 36.061 0.6 0.7 0.8 0.9 1 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 ρ 0.1 0.15 0.2 0.21 0.22 0.23 0.24 0.25 222 Simulation data for the LJ f uid 0.947 0.917 0.879 0.806 0.740 0.716 0.655 0.599 0.583 0.539 0.487 0.243 0.124 0.063 0.034 0.020 0.012 2.624 3.965 2.663 1.835 1.755 1.670 1.499 1.440 1.397 1.253 1.208 1.084 0.996 1.140 1.018 0.958 0.876 0.871 0.807 0.711 0.661 0.658 0.584 0.559 0.271 0.131 0.067 0.036 0.020 0.012 2.4928 4.286 2.917 2.076 1.922 1.802 1.733 1.544 1.542 1.469 1.243 1.265 1.152 1.176 1.167 1.100 0.999 0.974 0.838 0.855 0.752 0.726 0.678 0.600 0.304 0.141 0.070 0.036 0.021 0.012 2.3616 4.760 3.300 2.286 2.127 1.948 1.980 1.686 1.654 1.498 1.512 1.325 1.257 1.572 1.518 1.415 1.328 1.184 1.165 1.075 1.044 0.938 0.850 0.821 0.380 0.167 0.082 0.041 0.023 0.013 2.0992 5.764 4.057 3.040 2.961 2.668 2.523 2.444 2.137 2.043 2.058 1.829 1.834 2.357 2.280 2.214 2.176 2.075 1.673 1.664 1.605 1.450 1.261 1.185 0.534 0.204 0.095 0.046 0.024 0.014 1.8368 7.122 5.426 4.742 3.835 3.666 3.464 3.503 3.263 2.823 3.021 2.800 2.412 0.026 0.014 1.640 8.863 7.559 6.132 6.410 6.227 4.841 5.725 5.255 4.483 5.584 4.632 4.506 3.871 3.997 3.434 3.168 3.399 2.712 2.895 2.587 2.058 2.557 1.819 0.669 0.247 0.105 0.051 0.026 0.015 1.6072 9.886 8.090 6.966 6.257 6.885 6.789 5.996 5.648 5.156 5.822 5.822 4.710 5.053 4.419 4.206 4.025 3.574 3.269 2.829 2.738 2.271 2.396 2.166 0.753 0.267 0.115 0.052 T 0.027 0.015 1.5744 10.311 8.707 8.053 7.027 7.532 7.936 6.666 7.037 6.210 6.232 5.341 6.481 4.902 5.094 4.529 4.373 3.509 4.182 3.529 3.147 3.068 2.446 2.226 0.760 0.303 0.112 0.053 0.027 0.015 1.53504 11.019 9.477 7.537 11.206 9.110 7.602 8.479 6.905 6.519 5.892 6.985 6.208 6.193 6.414 5.648 4.675 5.374 4.731 3.927 3.332 3.254 3.506 4.324 0.870 0.300 0.112 0.054 0.028 0.015 1.5088 11.368 10.682 8.746 10.818 9.140 9.692 7.334 12.330 7.811 8.616 7.278 8.091 6.065 5.596 8.017 5.618 4.925 6.495 4.462 3.570 4.441 2.502 3.138 1.064 0.307 0.120 0.055 I 0.029 0.015 1.4432 13.182 12.526 13.860 12.178 14.436 12.761 18.374 10.352 15.471 12.350 10.867 17.034 9.667 7.866 12.267 6.957 5.818 8.990 7.901 7.086 8.297 4.302 3.831 1.321 0.336 0.127 0.056 0.029 0.039 1.3776 14.423 17.272 18.579 53.572 15.085 24.045 16.610 34.673 22.190 28.680 36.377 23.718 39.203 17.979 32.446 13.429 10.045 21.384 95.680 7.715 20.037 10.157 9.710 1.351 0.384 0.134 0.059 Table B.4: sothermal compressibility in the LJ f uid, simulation data. 743.240 -16.312 34.032 2.186 0.467 0.146 0.063 0.030 0.027 383.837 41.899 82.753 45883.516 63.397 -107.821 287.906 -71.346 32.922 -220.049 -32.696 -24.481 -63.416 -30.186 -14.541 -12.606 1.312 16.600 18.174 298.149 86.147 -17.828 -42.842 -76.579 2.635 0.484 0.159 0.063 0.030 0.026 173.806 55.670 26.180 36.183 -103.257 -237.978 -27.650 247.513 8.319 325.230 -31.785 372.974 -31.448 -19.597 -51.756 1904.200 1.300 19.040 24.923 46.984 19.320 0.38 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 ρ 0.1 0.15 0.2 0.21 223 0.317 0.303 0.283 0.264 0.250 0.235 0.216 0.116 0.064 0.035 0.021 0.013 0.008 2.624 2.033 1.255 0.840 0.786 0.730 0.673 0.631 0.596 0.545 0.515 0.474 0.439 0.415 0.394 0.370 0.343 0.351 0.329 0.300 0.282 0.270 0.248 0.236 0.124 0.066 0.037 0.022 0.013 0.008 2.4928 2.167 1.342 0.904 0.833 0.780 0.735 0.663 0.636 0.594 0.537 0.521 0.478 0.459 0.422 0.392 0.366 0.379 0.341 0.334 0.307 0.291 0.272 0.249 0.134 0.069 0.038 0.022 0.014 0.009 2.3616 2.339 1.466 0.971 0.897 0.823 0.797 0.712 0.682 0.626 0.600 0.545 0.514 0.477 0.457 0.429 0.401 0.438 0.422 0.387 0.373 0.340 0.318 0.301 0.154 0.078 0.043 0.024 0.014 0.009 2.0992 2.727 1.698 1.164 1.100 1.010 0.931 0.884 0.813 0.755 0.722 0.664 0.636 0.576 0.544 0.507 0.480 3.223 2.088 1.522 1.313 1.251 1.157 1.106 1.033 0.931 0.903 0.843 0.772 0.723 0.681 0.660 0.622 0.579 0.516 0.499 0.468 0.436 0.398 0.370 0.186 0.091 0.048 0.026 0.015 0.010 1.8368 3.807 2.566 1.789 1.708 1.621 1.432 1.392 1.297 1.179 1.162 1.100 1.016 0.945 0.902 0.832 0.772 0.736 0.662 0.656 0.605 0.533 0.544 0.476 0.220 0.103 0.052 0.028 0.016 0.010 1.640 4.016 2.620 1.900 1.732 1.670 1.560 1.435 1.378 1.296 1.230 1.168 1.048 1.029 0.938 0.895 0.804 0.774 0.709 0.646 0.626 0.558 0.546 0.503 0.227 0.107 0.055 0.029 0.016 0.010 1.6072 T 4.133 2.704 1.997 1.804 1.714 1.686 1.519 1.471 1.348 1.286 1.146 1.133 1.055 0.984 0.906 0.857 0.791 0.764 0.702 0.640 0.624 0.548 0.522 0.236 0.113 0.055 0.029 0.017 0.010 1.5744 4.312 2.838 1.982 2.057 1.848 1.723 1.643 1.508 1.396 1.313 1.307 1.178 1.125 1.071 0.995 0.920 0.882 0.830 0.757 0.681 0.662 0.604 0.582 0.246 0.116 0.055 0.030 0.017 0.010 1.53504 diabatic compressibility in LJ f uid, simulation data. 4.440 3.022 2.133 2.117 1.946 1.854 1.612 1.699 1.533 1.465 1.298 1.323 1.134 1.065 1.091 0.981 0.904 0.896 0.776 0.721 0.701 0.600 0.592 0.268 0.115 0.056 0.030 0.017 0.010 1.5088 A 4.835 3.336 2.457 2.223 2.221 2.120 2.043 1.761 1.861 1.626 1.511 1.466 1.384 1.224 1.241 1.110 0.985 1.017 0.931 0.893 0.835 0.706 0.658 0.289 0.125 0.059 0.031 0.018 0.011 1.4432 Table B.5: 5.124 3.736 2.771 2.893 2.367 2.534 2.158 2.151 2.121 1.991 2.129 1.775 1.723 1.556 1.614 1.389 1.198 1.275 1.263 0.987 1.054 0.862 0.820 0.319 0.132 0.061 0.032 0.018 0.014 1.3776 0.144 0.065 0.033 0.018 0.014 2.637 3.145 2.252 2.582 2.172 2.377 2.452 2.003 1.885 2.098 1.761 1.781 1.452 1.393 1.069 0.373 1.312 5.497 3.967 3.458 3.329 3.240 2.916 2.980 2.946 0.149 0.066 0.034 0.019 0.014 2.900 3.016 4.425 2.246 1.856 2.376 3.579 2.124 1.962 2.236 1.828 1.517 1.975 1.395 1.094 0.390 1.300 5.903 4.475 3.739 2.933 3.245 3.095 3.226 2.622 0.6 0.7 0.8 0.9 1 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 ρ 0.1 0.15 0.2 0.21 0.22 0.23 0.24 0.25 224 Simulation data for the LJ f uid 2.838 2.868 2.906 2.969 3.035 3.072 3.134 3.205 3.243 3.316 3.399 4.160 5.127 6.337 7.734 9.308 11.053 2.624 2.217 2.304 2.445 2.466 2.496 2.546 2.573 2.593 2.654 2.684 2.746 2.798 2.700 2.767 2.817 2.875 2.899 2.952 3.041 3.096 3.125 3.210 3.258 4.025 5.020 6.207 7.606 9.182 10.932 2.4928 2.147 2.226 2.351 2.389 2.416 2.439 2.500 2.506 2.540 2.619 2.622 2.684 2.639 2.652 2.697 2.757 2.790 2.886 2.891 2.971 3.012 3.077 3.164 3.885 4.893 6.092 7.505 9.053 10.811 2.3616 2.066 2.137 2.268 2.300 2.343 2.335 2.412 2.425 2.476 2.483 2.556 2.590 2.406 2.432 2.480 2.515 2.587 2.603 2.671 2.695 2.781 2.841 2.885 3.605 4.601 5.766 7.182 8.768 10.544 2.0992 1.915 1.982 2.071 2.080 2.120 2.153 2.169 2.221 2.251 2.261 2.317 2.328 8.485 10.250 1.8368 1.759 1.786 1.824 1.901 1.909 1.938 1.939 1.967 2.026 2.021 2.056 2.110 2.142 2.168 2.185 2.209 2.254 2.348 2.361 2.402 2.458 2.538 2.595 3.274 4.283 5.459 6.876 8.238 10.012 1.640 1.621 1.611 1.673 1.671 1.678 1.740 1.725 1.753 1.802 1.780 1.805 1.838 1.884 1.889 1.944 1.980 2.005 2.080 2.071 2.121 2.224 2.185 2.301 3.021 4.010 5.220 6.615 8.202 9.957 1.6072 1.579 1.594 1.623 1.657 1.651 1.667 1.703 1.711 1.728 1.734 1.752 1.810 1.802 1.852 1.876 1.934 1.957 2.004 2.064 2.088 2.168 2.168 2.240 2.961 3.946 5.128 6.559 T 8.124 9.922 1.5744 1.556 1.566 1.587 1.617 1.623 1.614 1.654 1.652 1.692 1.700 1.757 1.741 1.783 1.817 1.854 1.878 1.926 1.931 1.987 2.045 2.054 2.158 2.189 2.906 3.845 5.113 6.523 8.078 9.858 1.53504 1.523 1.531 1.582 1.518 1.561 1.590 1.591 1.625 1.660 1.680 1.658 1.705 1.719 1.736 1.773 1.820 1.825 1.864 1.920 1.989 2.002 2.058 2.070 2.844 3.809 5.080 6.468 peed of sound in the LJ f uid, simulation data. S 8.033 9.826 1.5088 1.504 1.490 1.529 1.500 1.530 1.534 1.599 1.535 1.594 1.592 1.655 1.622 1.713 1.742 1.696 1.761 1.808 1.790 1.892 1.935 1.940 2.065 2.060 2.741 3.788 5.014 6.428 7.934 9.739 1.4432 1.442 1.415 1.424 1.454 1.439 1.438 1.434 1.498 1.440 1.512 1.538 1.523 1.554 1.614 1.585 1.654 1.724 1.683 1.731 1.742 1.784 1.901 1.946 2.622 3.669 4.912 6.337 Table B.6: 1.586 1.716 1.752 2.511 3.558 4.810 6.230 7.844 8.515 1.383 1.317 1.382 1.359 1.352 1.367 1.304 1.398 1.391 1.439 1.402 1.472 1.563 1.500 1.493 1.655 1.3776 1.399 1.337 1.347 1.284 1.351 1.366 1.528 2.316 3.397 4.679 6.118 7.751 8.507 1.186 1.224 1.185 1.167 1.206 1.097 1.255 1.164 1.238 1.174 1.136 1.230 1.251 1.176 1.255 1.241 1.312 1.346 1.297 1.199 1.197 1.171 1.362 1.507 2.277 3.359 4.625 6.100 7.726 8.505 1.192 1.183 1.146 1.231 1.152 1.107 0.913 1.228 1.334 1.162 0.950 1.189 1.227 1.133 1.230 1.333 1.300 1.302 1.227 1.162 1.273 0.38 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 ρ 0.1 0.15 0.2 0.21 225 0.167 0.165 0.123 0.084 0.100 0.064 0.035 -0.110 -0.186 -0.227 -0.236 -0.227 -0.211 2.624 0.804 0.727 0.498 0.509 0.514 0.409 0.421 0.436 0.361 0.361 0.284 0.245 0.225 0.234 0.239 0.200 0.251 0.214 0.151 0.123 0.149 0.095 0.088 -0.077 -0.177 -0.217 -0.233 -0.224 -0.206 2.4928 0.909 0.832 0.641 0.591 0.546 0.540 0.466 0.506 0.495 0.343 0.394 0.343 0.362 0.291 0.280 0.227 0.295 0.218 0.247 0.176 0.179 0.159 0.108 -0.049 -0.161 -0.217 -0.237 -0.223 -0.208 2.3616 1.107 0.973 0.709 0.666 0.602 0.661 0.507 0.514 0.448 0.498 0.396 0.374 0.350 0.374 0.349 0.284 0.365 0.366 0.340 0.329 0.283 0.233 0.232 0.014 -0.135 -0.193 -0.221 -0.218 -0.204 2.0992 1.382 1.197 0.982 0.977 0.867 0.841 0.820 0.679 0.673 0.703 0.604 0.628 0.508 0.503 0.465 0.428 1.693 1.534 1.394 1.153 1.093 1.047 1.071 0.999 0.870 0.940 0.874 0.750 0.744 0.728 0.691 0.690 0.665 0.535 0.532 0.520 0.465 0.395 0.374 0.114 -0.101 -0.176 -0.216 -0.216 -0.202 1.8368 2.046 1.967 1.587 1.619 1.560 1.287 1.439 1.335 1.183 1.333 1.165 1.134 1.004 1.020 0.905 0.853 0.877 0.746 0.765 0.705 0.589 0.681 0.523 0.157 -0.069 -0.170 -0.203 -0.217 -0.200 1.640 2.354 2.066 1.719 1.570 1.640 1.598 1.450 1.362 1.262 1.340 1.305 1.147 1.164 1.065 1.009 0.977 0.888 0.837 0.760 0.720 0.633 0.650 0.594 0.201 -0.045 -0.148 -0.201 -0.217 -0.201 1.6072 T cient in the LJ f uid, simulation data. 2.429 2.170 1.845 1.690 1.712 1.684 1.523 1.519 1.392 1.361 1.240 1.320 1.124 1.120 1.044 1.004 0.874 0.943 0.849 0.786 0.758 0.652 0.601 0.193 -0.013 -0.161 -0.199 -0.209 -0.204 1.5744 ffi 2.563 2.257 1.764 2.091 1.860 1.645 1.685 1.495 1.410 1.304 1.376 1.287 1.248 1.224 1.138 1.017 1.059 0.976 0.881 0.796 0.767 0.785 0.831 0.237 -0.028 -0.164 -0.203 -0.209 -0.198 1.53504 2.582 2.412 1.894 2.037 1.824 1.816 1.565 1.834 1.502 1.533 1.393 1.407 1.222 1.149 1.277 1.097 1.013 1.094 0.926 0.819 0.880 0.637 0.720 0.296 -0.014 -0.145 -0.199 -0.204 -0.195 1.5088 oule-Thomson coe J 2.905 2.585 2.259 2.091 2.089 1.958 2.080 1.715 1.839 1.665 1.558 1.676 1.418 1.298 1.405 1.165 1.063 1.177 1.102 1.048 1.046 0.835 0.780 0.358 -0.005 -0.140 -0.200 -0.204 -0.198 1.4432 Table B.7: 3.043 2.961 2.385 2.752 2.096 2.231 1.982 2.161 1.939 1.907 1.923 1.713 1.754 1.545 1.599 1.377 1.245 1.358 1.425 1.057 1.212 1.053 0.999 0.342 0.027 -0.133 -0.190 -0.203 -0.044 1.3776 0.073 -0.124 -0.181 -0.205 -0.116 2.147 2.241 2.046 2.103 1.734 1.859 2.013 1.906 1.681 1.784 1.811 1.769 1.399 1.507 1.177 0.464 1.312 3.328 2.887 3.102 2.688 2.713 2.352 2.356 2.361 0.076 -0.096 -0.182 -0.203 -0.125 2.346 2.260 2.482 2.025 1.269 1.901 2.192 1.724 1.774 1.881 1.619 1.449 1.720 1.385 1.271 0.500 1.300 3.673 3.233 2.770 2.183 2.595 2.419 1.975 2.097 0.6 0.7 0.8 0.9 1 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 ρ 0.1 0.15 0.2 0.21 0.22 0.23 0.24 0.25 226 Simulation data for the LJ f uid 0.509 0.536 0.565 0.594 0.623 0.655 0.684 0.714 0.750 0.787 0.820 1.242 1.821 2.569 3.534 4.766 6.166 2.624 0.119 0.196 0.283 0.303 0.324 0.343 0.366 0.389 0.411 0.436 0.458 0.482 0.514 0.539 0.568 0.595 0.628 0.657 0.687 0.717 0.756 0.787 0.824 1.258 1.841 2.622 3.595 4.834 6.300 2.4928 0.120 0.196 0.285 0.305 0.325 0.346 0.367 0.390 0.414 0.433 0.460 0.487 0.514 0.541 0.570 0.598 0.630 0.659 0.692 0.721 0.756 0.794 0.827 1.272 1.866 2.645 3.605 4.915 6.382 2.3616 0.120 0.198 0.286 0.306 0.326 0.347 0.367 0.390 0.412 0.437 0.461 0.486 0.519 0.545 0.574 0.601 0.632 0.661 0.698 0.728 0.769 0.800 0.839 1.291 1.910 2.751 3.809 5.141 6.683 2.0992 0.121 0.200 0.289 0.309 0.328 0.351 0.372 0.393 0.416 0.442 0.466 0.492 5.340 7.034 1.8368 0.122 0.201 0.293 0.314 0.331 0.354 0.375 0.396 0.420 0.447 0.470 0.493 0.524 0.550 0.577 0.607 0.640 0.671 0.702 0.737 0.772 0.809 0.850 1.319 1.958 2.850 3.946 5.509 7.312 1.640 0.123 0.202 0.296 0.317 0.335 0.353 0.379 0.402 0.425 0.452 0.471 0.499 0.525 0.549 0.578 0.606 0.642 0.674 0.702 0.735 0.775 0.809 0.843 1.322 1.990 2.902 4.112 5.523 7.403 1.6072 0.124 0.205 0.295 0.316 0.338 0.360 0.383 0.401 0.422 0.449 0.474 0.500 0.522 0.551 0.579 0.615 0.640 0.672 0.701 0.737 0.769 0.803 0.849 1.334 2.012 2.971 4.158 T cient in the LJ f uid, simulation data. ffi 5.701 7.393 1.5744 0.125 0.206 0.299 0.315 0.340 0.363 0.380 0.404 0.427 0.451 0.478 0.504 0.524 0.558 0.583 0.611 0.634 0.671 0.703 0.740 0.767 0.811 0.839 1.318 2.033 2.940 4.181 5.722 7.595 1.53504 0.125 0.208 0.298 0.320 0.339 0.357 0.383 0.400 0.427 0.449 0.474 0.498 0.525 0.556 0.579 0.608 0.640 0.669 0.697 0.734 0.764 0.816 0.860 1.335 2.023 2.944 4.189 5.808 7.677 1.5088 0.125 0.208 0.298 0.319 0.339 0.360 0.382 0.408 0.427 0.448 0.478 0.495 0.528 0.554 0.580 0.605 0.634 0.667 0.704 0.725 0.770 0.797 0.843 1.336 2.040 2.996 4.230 I 5.893 7.741 1.4432 0.127 0.208 0.305 0.323 0.348 0.361 0.385 0.405 0.428 0.457 0.479 0.502 0.521 0.548 0.577 0.602 0.636 0.665 0.700 0.720 0.765 0.801 0.832 1.338 2.044 3.028 4.254 5.959 10.767 1.3776 0.128 0.212 0.312 0.330 0.347 0.368 0.390 0.412 0.431 0.461 0.468 0.507 0.526 0.548 0.571 0.592 0.626 0.657 0.685 0.717 0.744 0.791 0.833 1.314 2.073 3.052 4.363 Table B.8: sothermal pressure coe 0.713 0.765 0.800 1.313 2.089 3.088 4.456 6.021 9.068 0.363 0.376 0.400 0.424 0.434 0.466 0.482 0.503 0.514 0.547 0.557 0.597 0.625 0.624 0.668 0.702 1.312 0.131 0.219 0.316 0.348 0.688 0.763 0.808 1.322 2.080 3.150 4.463 6.081 8.866 0.374 0.376 0.412 0.420 0.446 0.451 0.494 0.486 0.500 0.520 0.531 0.575 0.623 0.629 0.654 0.689 1.300 0.129 0.217 0.315 0.344 0.38 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 ρ 0.1 0.15 0.2 0.21 227 0.461 0.469 0.448 0.428 0.437 0.424 0.399 0.302 0.226 0.163 0.121 0.094 0.072 2.624 0.473 0.521 0.519 0.532 0.542 0.515 0.527 0.543 0.514 0.527 0.497 0.480 0.482 0.492 0.497 0.478 0.547 0.530 0.489 0.474 0.497 0.459 0.461 0.341 0.241 0.176 0.129 0.098 0.075 2.4928 0.513 0.572 0.592 0.587 0.585 0.600 0.567 0.602 0.609 0.539 0.582 0.561 0.586 0.548 0.544 0.521 0.613 0.552 0.592 0.542 0.549 0.539 0.496 0.387 0.263 0.185 0.131 0.103 0.078 2.3616 0.572 0.652 0.654 0.650 0.635 0.686 0.619 0.645 0.617 0.661 0.611 0.611 0.604 0.631 0.627 0.597 0.749 0.770 0.750 0.760 0.722 0.680 0.689 0.491 0.318 0.226 0.158 0.116 0.086 2.0992 0.699 0.811 0.879 0.914 0.875 0.885 0.910 0.839 0.851 0.909 0.853 0.903 0.816 0.828 0.812 0.798 1.328 1.123 1.168 1.183 1.119 1.020 1.007 0.704 0.400 0.271 0.180 0.129 0.097 1.8368 0.870 1.091 1.388 1.203 1.215 1.226 1.312 1.293 1.186 1.349 1.317 1.189 1.234 1.255 1.278 1.321 1.094 1.528 1.817 2.031 2.085 1.708 2.171 2.114 1.905 2.521 2.181 2.247 2.034 2.194 1.986 1.918 2.182 1.827 2.033 1.901 1.596 2.069 1.533 0.884 0.492 0.305 0.208 0.141 0.105 1.640 1.229 1.655 2.054 1.977 2.328 2.443 2.294 2.264 2.176 2.612 2.759 2.353 2.639 2.436 2.437 2.475 2.287 2.195 1.982 2.017 1.747 1.925 1.839 1.005 0.537 0.342 0.217 0.142 0.109 1.6072 T cient the LJ f uid simulation data. 1.289 1.796 2.411 2.216 2.563 2.883 2.535 2.843 2.652 2.810 2.551 3.267 2.568 2.843 2.638 2.671 2.226 2.806 2.481 2.328 2.353 1.984 1.868 1.001 0.616 0.329 0.220 0.154 0.109 1.5744 ffi 1.382 1.972 2.244 3.584 3.092 2.714 3.245 2.764 2.783 2.646 3.308 3.090 3.249 3.567 3.272 2.840 3.437 3.167 2.737 2.447 2.485 2.860 3.719 1.162 0.606 0.330 0.224 0.156 0.115 1.53504 1.426 2.220 2.608 3.450 3.098 3.493 2.804 5.034 3.336 3.861 3.481 4.006 3.202 3.100 4.647 3.398 3.125 4.332 3.141 2.590 3.422 1.994 2.645 1.421 0.627 0.360 0.231 0.162 0.117 1.5088 hermal expansion coe T 1.672 2.608 4.222 3.935 5.030 4.611 7.077 4.192 6.624 5.647 5.206 8.547 5.035 4.307 7.080 4.186 3.701 5.983 5.527 5.103 6.349 3.444 3.188 1.768 0.688 0.385 0.238 0.169 0.120 1.4432 1.852 3.663 5.792 17.655 5.230 8.841 6.485 14.292 9.556 13.226 17.036 12.035 20.616 9.844 18.511 7.953 6.289 14.041 65.534 5.529 14.898 8.038 8.092 1.774 0.795 0.409 0.259 0.174 0.419 1.3776 Table B.9: 0.975 0.451 0.280 0.179 0.243 27.518 -50.276 138.844 -35.911 16.936 -120.403 -18.198 -14.616 -39.605 -18.843 -9.706 -8.844 529.561 -12.472 27.231 2.870 1.312 2.171 3.975 94.222 30.022 139.500 15.769 33.079 19432.294 1.006 0.502 0.282 0.185 0.230 -46.015 -107.413 -13.670 120.351 4.162 169.078 -16.888 214.300 -19.580 -12.334 -33.846 1312.266 -12.258 -32.701 -61.848 3.483 1.300 2.456 5.414 14.791 6.638 64.934 20.942 10.774 15.200 0.6 0.7 0.8 0.9 1 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 ρ 0.1 0.15 0.2 0.21 0.22 0.23 0.24 0.25 228 Simulation data for the LJ f uid -1.812 -1.871 -1.927 -1.984 -2.038 -2.102 -2.162 -2.221 -2.276 -2.342 -2.390 -2.966 -3.513 -3.971 -4.321 -4.517 -4.420 2.624 -0.617 -0.920 -1.222 -1.280 -1.338 -1.399 -1.457 -1.516 -1.573 -1.634 -1.693 -1.748 -1.834 -1.891 -1.946 -2.007 -2.070 -2.132 -2.187 -2.245 -2.306 -2.361 -2.425 -3.004 -3.550 -4.041 -4.413 -4.597 -4.490 2.4928 -0.626 -0.931 -1.234 -1.294 -1.356 -1.418 -1.470 -1.533 -1.591 -1.650 -1.714 -1.771 -1.851 -1.911 -1.972 -2.038 -2.096 -2.147 -2.216 -2.274 -2.334 -2.393 -2.447 -3.050 -3.594 -4.100 -4.473 -4.707 -4.650 2.3616 -0.635 -0.948 -1.251 -1.310 -1.368 -1.432 -1.490 -1.555 -1.612 -1.673 -1.732 -1.794 -1.906 -1.965 -2.025 -2.090 -2.147 -2.212 -2.266 -2.332 -2.391 -2.454 -2.515 -3.117 -3.703 -4.254 -4.690 -4.941 -4.918 2.0992 -0.658 -0.978 -1.290 -1.352 -1.414 -1.472 -1.537 -1.601 -1.658 -1.720 -1.783 -1.844 -5.164 -5.232 1.8368 -0.686 -1.017 -1.347 -1.402 -1.470 -1.531 -1.593 -1.656 -1.715 -1.780 -1.842 -1.904 -1.965 -2.024 -2.099 -2.155 -2.214 -2.273 -2.340 -2.398 -2.462 -2.522 -2.582 -3.200 -3.813 -4.391 -4.876 -5.362 -5.443 1.640 -0.715 -1.058 -1.392 -1.459 -1.524 -1.583 -1.645 -1.711 -1.774 -1.837 -1.906 -1.963 -2.033 -2.089 -2.158 -2.214 -2.281 -2.340 -2.411 -2.468 -2.526 -2.596 -2.655 -3.273 -3.895 -4.490 -5.009 -5.379 -5.546 1.6072 -0.721 -1.064 -1.402 -1.465 -1.533 -1.594 -1.660 -1.732 -1.794 -1.852 -1.918 -1.974 -2.042 -2.102 -2.172 -2.221 -2.296 -2.349 -2.406 -2.482 -2.534 -2.599 -2.668 -3.280 -3.908 -4.527 -5.049 T -5.458 -5.590 1.5744 -0.727 -1.073 -1.417 -1.472 -1.540 -1.617 -1.672 -1.740 -1.804 -1.867 -1.921 -1.989 -2.058 -2.121 -2.175 -2.239 -2.300 -2.361 -2.424 -2.480 -2.549 -2.607 -2.674 -3.292 -3.934 -4.533 -5.065 -5.482 -5.639 1.53504 -0.734 -1.085 -1.420 -1.490 -1.551 -1.625 -1.687 -1.750 -1.818 -1.881 -1.950 -2.002 -2.069 -2.134 -2.196 -2.262 -2.318 -2.387 -2.446 -2.500 -2.572 -2.624 -2.686 -3.308 -3.956 -4.547 -5.106 otential energy in the LJ f uid, simulation data. P -5.512 -5.654 1.5088 -0.743 -1.099 -1.435 -1.504 -1.571 -1.638 -1.691 -1.766 -1.842 -1.894 -1.953 -2.029 -2.080 -2.146 -2.212 -2.272 -2.333 -2.396 -2.453 -2.514 -2.580 -2.637 -2.702 -3.327 -3.949 -4.563 -5.130 -5.588 -5.751 1.4432 -0.759 -1.120 -1.464 -1.523 -1.612 -1.675 -1.740 -1.788 -1.867 -1.930 -1.994 -2.043 -2.119 -2.168 -2.240 -2.306 -2.359 -2.433 -2.490 -2.549 -2.617 -2.664 -2.728 -3.341 -3.993 -4.609 -5.168 Table B.10: -5.631 -6.394 1.3776 -0.775 -1.147 -1.508 -1.573 -1.636 -1.720 -1.764 -1.832 -1.917 -1.978 -2.052 -2.109 -2.162 -2.223 -2.304 -2.340 -2.400 -2.472 -2.542 -2.587 -2.653 -2.697 -2.775 -3.382 -4.018 -4.640 -5.225 -2.708 -2.771 -2.811 -3.413 -4.048 -4.695 -5.270 -5.706 -6.546 -1.695 -1.770 -1.842 -1.909 -1.968 -2.074 -2.085 -2.186 -2.225 -2.313 -2.369 -2.403 -2.472 -2.543 -2.583 -2.661 1.312 -0.791 -1.181 -1.545 -1.627 -2.747 -2.777 -2.819 -3.426 -4.061 -4.693 -5.284 -5.738 -6.552 -1.721 -1.770 -1.886 -1.902 -1.982 -2.045 -2.172 -2.150 -2.223 -2.297 -2.421 -2.403 -2.486 -2.544 -2.591 -2.648 1.300 -0.798 -1.199 -1.581 -1.635 0.38 0.39 0.4 0.5 0.6 0.7 0.8 0.9 1 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 ρ 0.1 0.15 0.2 0.21 229 1.93924 1.83166 1.76417 1.70079 1.67092 1.5402 1.56846 0.96595 0.39759 -0.01251 -0.36508 2.624 3.32766 3.01638 2.68009 2.63026 2.5927 2.51142 2.46253 2.40221 2.37647 2.28688 2.23349 2.21098 2.09491 2.03473 2.00612 1.96394 1.65342 1.57507 1.54985 1.48939 1.41874 1.38966 1.29191 0.72912 0.21379 -0.28803 -0.66989 2.4928 3.1205 2.82389 2.51149 2.4542 2.3708 2.28104 2.30613 2.21718 2.16676 2.11851 2.00672 1.97148 1.88244 1.84257 1.82553 1.74253 1.41937 1.432 1.30187 1.24769 1.19262 1.12941 1.11201 0.4371 -0.0106 -0.53084 -0.86396 2.3616 2.91069 2.56802 2.28966 2.24812 2.20763 2.11014 2.08206 1.96906 1.9444 1.87657 1.82235 1.74585 1.70527 1.64765 1.57388 1.46553 1.01536 0.92526 0.91188 0.8099 0.76284 0.68616 0.62722 0.03698 -0.53569 -1.12118 -1.55918 2.0992 2.48796 2.16421 1.86806 1.80025 1.74376 1.71096 1.61969 1.5347 1.51506 1.44301 1.36894 1.30655 1.24168 1.1972 1.13728 1.05257 2.06176 1.73784 1.36723 1.34686 1.26061 1.19977 1.2234 1.08929 0.98618 0.97631 0.91065 0.8724 0.76525 0.70031 0.69598 0.55216 0.45995 0.4821 0.35472 0.32987 0.27382 0.22812 0.19501 -0.49504 -1.0656 -1.71264 -2.18152 1.8368 1.74315 1.40683 1.06136 0.99513 0.92375 0.88639 0.83235 0.76006 0.70202 0.64172 0.54202 0.51047 0.40543 0.37877 0.28054 0.2491 0.15946 0.11098 0.00511 -0.03302 -0.07634 -0.17935 -0.22654 -0.81988 -1.4251 -2.00875 -2.53608 1.64 T 1.68406 1.35221 1.00615 0.94931 0.86948 0.8264 0.75292 0.65243 0.59447 0.56378 0.48257 0.44671 0.36068 0.31409 0.21537 0.21258 0.09088 0.06974 0.03293 -0.10016 -0.1146 -0.19448 -0.289 -0.85819 -1.48161 -2.14244 -2.65846 1.6072 1.64428 1.30767 0.98417 0.825 0.8298 0.79522 0.67581 0.6596 0.54319 0.56799 0.42906 0.36292 0.3014 0.20718 0.20469 0.13138 0.05592 0.0154 -0.06779 -0.10717 -0.18762 -0.2807 -0.3404 1.5744 otal energy in the LJ f uid, simulation data. T 1.57742 1.22161 0.88457 0.83169 0.77702 0.71782 0.59518 0.55293 0.50555 0.44821 0.36362 0.30653 0.22543 0.21105 0.10829 0.05764 -7.25E-04 -0.07204 -0.15718 -0.1982 -0.24946 -0.35466 -0.39871 -1.00762 -1.59974 -2.26427 -2.85261 1.53504 Table B.11: 1.50168 1.14536 0.83405 0.7565 0.685 0.6143 0.60304 0.4943 0.38623 0.36219 0.3208 0.20826 0.18774 0.11327 0.04131 -0.02043 -0.08198 -0.1469 -0.18896 -0.24978 -0.32591 -0.37192 -0.45065 -1.08639 -1.66036 -2.26989 -2.89646 1.5088 -1.84228 -2.43209 -2.97831 0.28585 0.2813 0.17961 0.11372 0.07911 -0.042 -0.0431 -0.10649 -0.24437 -0.30601 -0.36252 -0.38573 -0.47954 -0.52048 -0.56159 -1.21231 1.4432 1.42472 1.03716 0.71878 0.59736 0.55315 0.49377 0.41728 0.29352 -1.95505 -2.54557 -3.17119 0.12991 0.07792 -0.01721 -0.05828 -0.09814 -0.15555 -0.27125 -0.25924 -0.32184 -0.41623 -0.50335 -0.52332 -0.60337 -0.60928 -0.72459 -1.32755 1.3776 1.28046 0.9136 0.54369 0.49033 0.43994 0.32046 0.32718 0.24924 -2.0768 -2.73604 -3.30177 0.00449 -0.14711 -0.10486 -0.24575 -0.25635 -0.37596 -0.4249 -0.43573 -0.51069 -0.60303 -0.61467 -0.71979 -0.75578 -0.82732 -1.44864 1.312 1.18633 0.7819 0.43296 0.3374 0.26816 0.18703 0.11596 0.05103 0.6 0.7 0.8 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.5 ρ 0.1 0.15 0.2 0.21 0.22 0.23 0.24 0.25 Appendix C

Simulation data for the MCYna water potential

Simulation results of the MCYna water are given below. The MD simulation was for a system consisting of 500 water molecules and density 0.998 gm/cm3.

230 231 K) / 4 − ( p 6.18375 6.26625 6.33745 6.3985 6.4504 6.49415 6.5306 6.5604 6.58425 6.60265 6.616 6.62465 6.62895 6.62915 6.6254 6.618 6.6071 6.5929 6.5757 6.5557 α 1.71865 10 1.9524 2.4912 2.9707 3.3981 3.7801 4.1224 4.42975 4.7061 4.9545 5.17755 5.3774 5.5559 5.7148 5.85555 5.9797 6.08865 . 3 MPa) cm / / ( v 1.7358528171 1.7769790293 1.8152328959 1.8508715599 1.8841041129 1.9151643659 1.9442327387 1.9714576778 1.9969943101 2.0209513024 2.0434195774 2.0644677755 2.084175677 2.1025897204 2.1197345773 2.1356792028 2.1504476912 2.1640757323 2.1766489464 2.188201994 γK 0.3834252492 0.4394753487 0.5725817202 0.6959636591 0.8102598096 0.9162523845 1.0146846709 1.1062261823 1.1914689999 1.2708636332 1.3448111958 1.4136581237 1.4776997896 1.5372489946 1.5925495806 1.6439037801 1.6915829948 K) / (MPa T J -0.184426 -0.184126 -0.18387 -0.183658 -0.227855 -0.224165 -0.221043 -0.218282 -0.215751 -0.213373 -0.21111 -0.208948 -0.206886 -0.20493 -0.20309 -0.201372 -0.199782 -0.198319 -0.196981 -0.19576 -0.194647 -0.193631 -0.1927 -0.191841 -0.191043 -0.190296 -0.189591 -0.188924 -0.188289 -0.187685 -0.187112 -0.18657 -0.186062 -0.185591 -0.18516 -0.184771 µ -0.229556 s) / ( 0 1950.98482 1962.202814 1973.293976 1984.259088 1505.960829 1524.429907 1542.239974 1559.480015 1576.22428 1592.53467 1608.462746 1624.051402 1639.336285 1654.346991 1669.108063 1683.639848 1697.959202 1712.08009 1726.014089 1739.770804 1753.358218 1766.782977 1780.050631 1793.165831 1806.132485 1818.953897 1831.632869 1844.17179 1856.572712 1868.837402 1880.967398 1892.964046 1904.828535 1916.56193 1928.165193 1939.639212 ωm 1498.365465 GPa) / ( s 0.263376 0.260373 0.257454 0.254617 0.442034 0.431388 0.421483 0.412215 0.403504 0.395281 0.387491 0.380088 0.373033 0.366294 0.359844 0.353659 0.347719 0.342007 0.336508 0.331207 0.326094 0.321157 0.316387 0.311776 0.307315 0.302998 0.298818 0.294768 0.290844 0.287039 0.283349 0.279768 0.276294 0.272922 0.269647 0.266466 β1 0.446527 continued ... GPa) / ( T 0.307243 0.304652 0.302102 0.299593 0.444257 0.435082 0.426847 0.419384 0.412561 0.406274 0.400438 0.394983 0.389853 0.385002 0.380389 0.375983 0.371755 0.367684 0.36375 0.359938 0.356237 0.352635 0.349126 0.345702 0.342359 0.339091 0.335896 0.332769 0.329708 0.32671 0.323771 0.320889 0.318061 0.315285 0.312558 0.309878 β1 0.448236 molK) / ( p 68.103742 67.954832 67.804481 67.652727 74.609209 74.431967 74.16423 73.863021 73.561867 73.277827 73.017102 72.779238 72.560165 72.354307 72.155979 71.960252 71.76341 71.563122 71.358397 71.149409 70.937235 70.723562 70.510402 70.299818 70.093712 69.893654 69.700771 69.515689 69.338525 69.16892 69.006109 68.849014 68.696351 68.546754 68.398882 68.251535 CJ 74.636182 hermodynamic quantities in MCYna water of density 0.998 gm T molK) / ( v 61.1651 60.7700 60.3907 60.0263 59.6753 59.3364 59.0083 58.6898 58.3800 58.0781 57.7836 57.4963 69.4295 68.8385 68.2587 67.6877 67.1235 66.5656 66.0141 65.4700 64.9348 64.4103 63.8983 63.4007 62.9189 62.4541 62.0068 61.5772 CJ 74.3516 74.2359 73.8000 73.2322 72.6004 71.9468 71.2949 70.6563 70.0346 Table C.1: 180.6819 190.6030 200.6213 210.7300 220.9229 231.1942 241.5384 251.9503 262.4251 272.9582 283.5453 294.1823 43.0399 49.9440 57.1569 64.6565 72.4220 80.4346 88.6766 97.1319 105.7854 114.6232 123.6326 132.8015 142.1191 151.5751 161.1600 170.8650 p (MPa) 1.0738 2.2596 5.6438 9.5926 14.0650 19.0229 24.4309 30.2563 36.4686 420 425 430 435 440 445 450 455 460 465 470 475 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 T (K) 298 300 305 310 315 320 325 330 335 232 Simulation data for the MCYna water potential K) / 4 − ( p 6.1555 6.1198 6.08345 6.0464 6.00855 5.9698 5.9302 5.8898 5.8488 5.80745 5.76605 5.72505 5.68485 5.64585 5.60825 5.572 5.53665 5.50085 5.4623 5.41735 5.36045 5.28385 α 6.5331 10 6.50825 6.48145 6.4529 6.42295 6.3918 6.35975 6.32695 6.29355 6.2597 6.2254 6.19065 MPa) / ( v 2.2749867873 2.2784065525 2.2813593391 2.2838494255 2.285827871 2.28727093 2.2881947794 2.2885985841 2.288558382 2.2881542917 2.2874705936 2.2866723117 2.2858906362 2.2852603671 2.2848569787 2.2846833549 2.2845960544 2.2841405484 2.282530118 2.2784951211 2.2700785568 2.2544715239 γK 2.1987715608 2.2084622798 2.2173434095 2.2254602391 2.2329123341 2.2397583564 2.2460868521 2.2519202158 2.2573222958 2.2623268388 2.266939046 2.271148075 K) / (MPa T J -0.184482 -0.184465 -0.184477 -0.184639 -0.185148 -0.186307 -0.188563 µ -0.183485 -0.183348 -0.183242 -0.18316 -0.183097 -0.183046 -0.183003 -0.182963 -0.182922 -0.182881 -0.182838 -0.182797 -0.182762 -0.182739 -0.182735 -0.182756 -0.182808 -0.182898 -0.183027 -0.183193 -0.183392 -0.183614 -0.183844 -0.184063 -0.184251 -0.184389 -0.184466 s) / ( 0 2244.89886 2252.765759 2260.549471 2268.252713 2275.878894 2283.432334 2290.918532 ωm 1995.09895 2005.814393 2016.406292 2026.875581 2037.223259 2047.4504 2057.558157 2067.547768 2077.420553 2087.177914 2096.821332 2106.352352 2115.772584 2125.083681 2134.28733 2143.385235 2152.3791 2161.270614 2170.061436 2178.753182 2187.347414 2195.845638 2204.249306 2212.559828 2220.778595 2228.907015 2236.946564 GPa) / ( s 0.198925 0.197538 0.19618 0.19485 0.193546 0.192268 0.191014 β1 0.251857 0.249174 0.246563 0.244022 0.24155 0.239142 0.236799 0.234516 0.232292 0.230125 0.228014 0.225955 0.223947 0.221989 0.220079 0.218214 0.216394 0.214618 0.212882 0.211187 0.209531 0.207912 0.20633 0.204783 0.20327 0.20179 0.200342 continued ... GPa) / ( T 0.243885 0.242347 0.240828 0.239309 0.23776 0.236135 0.234372 β1 0.297125 0.294696 0.292307 0.289958 0.287649 0.285379 0.283148 0.280958 0.278806 0.276693 0.274617 0.272578 0.270573 0.2686 0.266659 0.264746 0.262861 0.261001 0.259165 0.257354 0.255567 0.253805 0.252071 0.250366 0.248693 0.247055 0.245453 molK) / ( p 64.281692 64.229649 64.173188 64.085144 63.921081 63.61406 63.070072 CJ 67.49996 67.346885 67.194455 67.043786 66.896064 66.752431 66.613882 66.481162 66.35468 66.234441 66.120014 66.010526 65.904701 65.800941 65.69745 65.592396 65.484116 65.371337 65.253407 65.130506 65.003813 64.875591 64.749138 64.628576 64.518391 64.422678 64.343993 molK) / ( v 53.2944 53.1447 52.9997 52.8618 52.7341 52.6192 52.5185 52.4315 52.3540 52.2760 52.1793 52.0343 51.7963 51.4021 56.1751 55.9374 55.7095 55.4919 55.2845 55.0871 54.8991 54.7197 54.5478 54.3821 54.2213 54.0637 53.9083 53.7539 53.6001 53.4467 CJ 57.2162 56.9435 56.6788 56.4225 523.8070 534.8557 545.9027 556.9468 567.9869 579.0220 590.0510 601.0731 612.0874 623.0930 634.0895 645.0762 656.0529 667.0193 347.9891 358.8520 369.7420 380.6564 391.5928 402.5488 413.5223 424.5112 435.5134 446.5272 457.5506 468.5822 479.6201 490.6630 501.7094 512.7578 p (MPa) 304.8654 315.5910 326.3556 337.1560 580 585 590 595 600 605 610 615 620 625 630 635 640 645 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 T (K) 480 485 490 495