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

Transcription

1 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

2 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

3 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 f nancial 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

4 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

5 Notation SPC SPC/E SPC/Fw SPC/Fd TIP3P TIP4P simple point charge extended simple point charge f exible simple point charge f exible simple point charge by Dang et al. transferable intermolecular potential three point transferable intermolecular potential four point TIP4P/2005 transferable intermolecular potential four point 2005 TIP5P MCY MCYL MCYna BNS ST2 PPC GCPM NveD LJ WCA transferable intermolecular potential f ve point Matsuoka-Clementi-Yoshimine water model Matsuoka-Clementi-Yoshimine-Lie f exible water model MCY with non-additive terms water model Ben-Naim and Stillinger model Stillinger and Rahman model polarisable point charge model Gaussian charge polarizable model Nada and van der Eerden six site model Lennard-Jones Weeks-Chandler-Anderson iv

6 v MD MC CFT TMD TM RDF EOS Exp. Ref. fcc IAPWS t t k B molecular dynamics Monte Carlo conventional f uctuation theory temperature of maximum density melting temperature radial distribution function equation of state experiment reference face centered cubic international association for the properties of water and steam simulation step length total simulation time Boltzmann constant ( J/K) R universal gas constant ( J mol 1 K 1 ) N A Avogadro number ( mol 1 ) Å m ns nano second (10 9 s) nm nano meter (10 9 m) ps pico second (10 12 s) Ω ω Ω mn S phase space volume phase space density derivatives ofωwith respect to energy and volume Entropy

7 vi T T C τ ρ ρ C p p C p P G m M V N U U E r ij u i j r v a f i j L temperature critical temperature ratio of temperature to the critical temperature density critical density pressure critical pressure difference in pressure total linear momentum constant related to the initial position of the center of mass mass of a single atom/molecule total mass of the system volume number of molecules/atoms potential energy per particle difference in the potential energy of LJ and WCA potentials internal energy relative position of particles i and j potential energy between particles i and j position of the particle velocity of the particle acceleration of the particle force between particles i and j length of the side of simulation box

8 vii AA q l 1 l 2 θ φ σ ǫ C v C v C p C p β T β T β S β S absolute average charge bond length, between H and O the distance between oxygen and the dummy charge site bond angle, between hydrogen-oxygen-hydrogen angle between hydrogen-oxygen-dummy site Lennard-Jones length constant Lennard-Jones energy constant isochoric heat capacity C v difference between the LJ and WCA potentials isobaric heat capacity C p difference between the LJ and WCA potentials isothermal compressibility β T difference between the LJ and WCA potentials adiabatic compressibility β S difference between the LJ and WCA potentials µ JT Joule-Thomson coefficient µ JT µ JT difference between the LJ and WCA potentials γ v γ v α p α p ω 0 ω 0 isothermal pressure coefficient γ v difference between the LJ and WCA potentials thermal expansion coefficient α p difference between the LJ and WCA potentials speed of sound at zero frequency ω 0 difference between the LJ and WCA potentials

9 Contents Abstract i Acknowledgment ii Declaration iii Notation iv List of Figures xiii List of Tables xv 1 Introduction 1 2 Water Potentials Introduction Why all these models for water? Water models, criteria The interaction between molecules Bond f exibility Values used for parametrization viii

10 CONTENTS ix Polarization The charge distribution Three-site models Four-site models Five-site models Six-site models Review of water models Rigid water models TIP3P and SPC SPC/E TIP4P/ PPC GCPM MCY Flexible water models SPC/Fw SPC/Fd MCYL MCYna water model Improving water models Calculation of Thermodynamic Properties from Molecular Simulation Introduction NVEPG Ensemble Thermodynamic quantities in the NVEPG ensemble

11 x CONTENTS Heat capacities Isothermal pressure coefficient Compressibilities Speed of sound at zero frequency Joule-Thomson coefficient Calculation of thermodynamic quantities in an NVEPG ensemble Thermodynamic quantities from the f uctuation theory Thermodynamics properties near the critical point Simulation Details Introduction Integrators Gear predictor Leap frog Periodic boundaries, cutoff radius and simulation time Ewald summation Simulations of LJ and WCA potentials LJ potential WCA potential MCYna potential Thermodynamic Properties of LJ and WCA f uids and Noble Gases Introduction Thermodynamic properties of LJ and WCA potentials Energy and pressure Isochoric and isobaric heat capacities

12 CONTENTS xi Isothermal pressure coefficient Thermal expansion coefficient Speed of sound Isothermal and adiabatic compressibilities Joule-Thomson coefficient Thermodynamic properties of supercritical argon and krypton Energy and pressure Heat capacities Speed of sound Joule-Thomson coefficient Compressibilities Summary Thermodynamic Properties of MCYna Water Introduction Thermodynamic properties of water Simulation results and discussion Pressure Heat capacities Isothermal pressure coefficient Compressibilities Joule-Thomson coefficient Speed of sound Thermal expansion coefficient Summary

13 xii CONTENTS 7 Conclusions and Recommendations Conclusion Recommendations 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

14 List of Figures 2.1 Simple representation of water Tetrahedral shape of water Hydrogen bonding in water Schematic representation of water models Schematic representation of MCY water model Schematic representation of three body interaction LJ, WCA and Buckingham potentials U and p in a WCA potential, different system size U and p in WCA potentials, different simulation time U and p comparison in LJ f uid U in WCA and LJ potentials as a function of density U as a function of density and temperature p(lj) and p as a function of density p as a function of temperature C v (LJ) and C v as a function of density C p (LJ) and C p as a function of density γ v in LJ and WCA as a function of temperature xiii

15 xiv LIST OF FIGURES 5.11 α p as a function of density ω 0 (LJ) and ω 0 as a function of density β S and β T as a function of density µ JT (LJ) and µ JT as a function of density Internal energy in supercritical argon and krypton Pressure in argon and krypton Heat capacities in supercritical argon Speed of sound in argon and krypton Joule-Thomson coefficient in argon and krypton Adiabatic and isothermal compressibilities in argon Pressure of water Isochoric heat capacity of water Isobaric heat capacity of water Isothermal pressure coefficient of water Isothermal compressibility of water Adiabatic compressibility of water Joule-Thomson coefficient of water Speed of sound in water Thermal expansion coefficient of water

16 List of Tables 2.1 Summary of properties used for model parametrization Constants of MCY model Parameters for different water models Thermodynamics quantities in terms of phase space function Values for some of multinomial combinations Values for phase space function elements Fluctuation formulas in thermodynamics Summary of reduced thermodynamic quantities Triple and critical points of the LJ f uid Absolute average in C v and C p Locus of the inversion curve in LJ f uid Constants of argon and krypton Locus of the inversion curve in argon and krypton Absolute average inβ S andβ T Experimental triple and critical points of water Thermodynamic properties of water models xv

17 xvi LIST OF TABLES A.1 Pressure in WCA potential A.2 Isochoric heat capacity in WCA potential A.3 Isobaric heat capacity in WCA potential A.4 Isothermal compressibility in WCA potential A.5 Adiabatic compressibility in WCA potential A.6 Speed of sound in WCA potential A.7 Joule-Thomson coefficient in WCA potential A.8 Isothermal pressure coefficient WCA potential A.9 Thermal expansion coefficient in WCA potential A.10 Potential energy in WCA potential B.1 Pressure in LJ f uid B.2 Isochoric heat capacity in LJ f uid B.3 Isobaric heat capacity in LJ f uid B.4 Isothermal compressibility in LJ f uid B.5 Adiabatic compressibility in LJ f uid B.6 Speed of sound in LJ f uid B.7 Joule-Thomson coefficient in LJ f uid B.8 Isothermal pressure coefficient in LJ f uid B.9 Thermal expansion coefficient LJ f uid B.10 Potential energy in LJ f uid B.11 Total energy in LJ f uid C.1 Thermodynamic quantities in MCYna water

18 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

19 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 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-

20 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 f uids. 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

21 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 gm/cm 3 is given in Appendix C.

22 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

23 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]

24 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 Figure 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

25 8 Water Potentials ammonia (NH 3 ) and water have a tetrahedral shape with a bond angle of about (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 positively 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

26 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 f eld 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-

27 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

28 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 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

29 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-

30 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 Laasonen et al. [72] and Matsuoka et al. [23] to mention some 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

31 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 Values used for parametrization The validity of molecular simulation depends only on the availability of reliable data [80]. Generally each model is developed to f t 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.

32 Table 2.1: Summary of properties used for parametrization of different water models. Model Properties used for parametrization Ref. SPC Energy and pressure of liquid water [64] SPC/E Vaporization enthalpy at room temperature, pressure of liquid water, [69] vaporization correction energy and density of water TIP3P Vaporization enthalpy and liquid density of water at ambient condition [87] SPC/Fw Bulk diffusion and dielectric constant, equilibrium bond length and angle [74] SPC/Fd Ground vibrational state frequency [88] PPC Not applicable - ab initio potential [89] GCPM Internal energy and pressure of water [90] TIP4P Vaporization enthalpy and liquid density of water at ambient condition [87] TIP4P/2005 Temperature of maximum density, phase diagram, melting temperature of [57] hexagonal ice and Vaporization correction energy MCY Not applicable - ab initio potential [23] MCYL Not applicable - ab initio potential [79] MCYna Not applicable - ab initio potential [21] BNS Water vapor and second virial coefficient [29] ST2 Water vapor and second virial coefficient [66] TIP5P Vaporization energy, density maximum of liquid water and density of water [54] at ambient conditions NveD Melting point of ice and densities of ice and water near the melting point [30] Water models, criteria 15

33 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 Schrödinger equation or Hartree-Fock approximation. However, the accuracy of force f eld calculated in ab initio is limited by unavoidable approximations 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] 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. Molecular 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].

34 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 working 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, ]. Sprik and Klein [107] considered polarizability as an explicit degree of freedom with an artif cial inertial

35 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 [ ]. Different authors [22, 101, 111, 119] described how the polarization term in a molecular dynamics simulation can be calculated 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.

36 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 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 solvating water 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] Four-site models The basic philosophy of 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

37 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 l 2 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 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] 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 f ve site model. BNS has many simplifying approximations, 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

38 Water models, criteria 21 ±ηe, where the value of η is chosen to be 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 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 There is no charge on

39 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 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 positive charge 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,