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 pres- sure 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 thermody- namic 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, ther- mal 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 com- pressibility 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 encourage- ment. I would like to thank Swinburne University of Technology for fnancial sup- port 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 MC Monte 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