Molecular Dynamics and Self-Diffusion in Supercritical Water by Yuji Kubo B.S. University M.S. University Chemistry of Tokyo, 1985 Chemistry of Tokyo, 1987 SUBMITTED TO THE DEPARTMENT OF CHEMICAL ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN CHEMICAL ENGINEERING AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY ( 2000 Massachusetts Institute of Technology All rights reserved Signature of Author: T} i7 - - - ~)ep~ament of Chemical Engineering September 11, 2000 Certified by: < \ Professor Kenne-t-A. Smith Thesis Supervisor Certified by: ,/'~// Professor Jefferson W. Tester (Thi) s SuDervisor Accepted by: MASSACHUSETTS INST1TUTE OFTECHNOLOGY FEB 0 7 2001 Professor Robert E. Cohen St. Laurent Professor of Chemical Engineering Chairman, Committee for Graduate Students LIBRARIES ARCHIVES Page 2 Molecular Dynamics and Self-Diffusion in Supercritical Water by Yuji Kubo Submitted to the Department of Chemical Engineering in September, 2000 in partial fulfillment of the requirements for the degree of Master of Science in Chemical Engineering ABSTRACT Supercritical water (SCW) which exists beyond the critical point (Tc=647.2K, Pc=221bar) is an innovative solvent to dissolve organic material. Many applications of this new solvent such as oxidation of organic wastes and separation of metals have been researched; however, the properties of SCW have not sufficiently been understood due to the difficulty in experimental measurements at high temperatures and high pressures. Computer simulation is one of the best tools to predict and analyze the properties which are difficult to be experimentally obtained. The goals of this research is to gain a better understanding as to self-diffusivity of SCW, of which experimental data is limited, through molecular dynamic simulation. Extended Simple Point Charge (SPC/E) model reproduced the self-diffusion coefficient of water in the range of the temperature, 673-873K and the density, 0.125- 0.7g/cm3 . In contrast, Simple Potential Charge (SPC) model was not relevant to calculate selfdiffusivity. In order to investigate self-diffusivity in the near critical region, the critical point of SPC/E model was calculated from direct simulation method of two coexisitng phases. Interpolating the simulated orthobaric densities by the scaling law (=0.325) approximation, Tc=616K and pc=0.296g/cm 3 were obtained. This value was explicitly lower than the expected critical point. By comparing reduced pressure/reduced density relationship from simulated data with that of experimental data, the critical point was revised to around 646K and 0.290g/cm3 which are very close to real water's data ( Tc=647K and rc=0.322g/cm3 ). Based on this obtained critical point, the self-diffusion coefficient of water in the near critical region was studied. The small drop of self-diffusion coefficient near critical point was observed from simulation results. The drop point was a little different from the critical point. Thesis supervisor: Title: Thesis supervisor: Title: Kenneth A. Smith Edwin R. Gilliland Professor of Chemical Engineering Jefferson W. Tester He1mann P. Meissner Professor of Chemical Engineering Director, MIT Energy Laboratory Page 3 DEDICATION For my family: Keiko Kubo Miyako Kubo Emniko Kubo Page 4 ACKNOWLEDGMENTS I would like to thank Professor Kenneth A. Smith. He gave me interest in transport properties of SCW through my research and also through 10.52 class. He was very friendly and his comments always encouraged me. I also would like to thank Professor Jefferson W. Tester. Although he is very busy, he always took care of me and my family. He gave us a great experience like ski in Sugerloaf and trip to Canada. Both of professors have interest in Japan and are familiar with Japan, so I enjoyed talking about Japan. Thanks to all Tester research group members. Especially, Mike Kutney taught me NMR experiments and the importance of diffusivity. Matt Reagan greatly helped me to start research on simulation. He often solved the computer trouble I encountered. Joanna DiNaro gave me a first chance to make friends with group members by inviting me to the group trip. I will not forget climbing in White Mountain. Josh Taylor and Mike Timko have a lot of knowledge and experience about SCF. They often answered my trifle questions about SCW. I thank Bonnie Caputo for organizing our meeting schedule and various events and I also appreciate Gillian Killey and Linda L. Mousseau. I have to say great thank to Janet Fischer and Elaine Aufiero advising and helping me through the Chemical Engineering Department. I also thank Nippon Steel Corporation, which gave me a chance to learn and do research at MIT for two years. I would like to thank my parents, Junsuke and Noriko, and my parents in law, Hideo Ohba and Yasuko Ohba. They dealt with many things in Japan instead of me and often sent foods and cloths to us. Finally, I would like to thank greatly my wife, Keiko, my daughters, Miyako and Emiko. They always physically and mentally supported me. I was encouraged to see their smile. I had a wonderful time at MIT and I would like to come back here again. __ _Page _ 5 TABLE OF CONTENTS Chapter 1: Introduction................................................................................ 1.1 Motivation ......................................................................................................... 1.2 Statement of Objectives ....................................................................................... Chapter 2: Background.............. ........................ ....... ........... 2.1 Supercritical Water............................................................................................... 2.2 Diffusion Models ............................................................................................. 2.3 Properties near Critical Point............................................................................... References Chapter 3: Experimental Measurements for Self-Diffusivity ................. 3.1 Introduction 3.2 Experimental measurements for molecular diffusion coefficient................... 3.2.1 Nuclear Magnetic Resonance (NMR)........................................................... 3.2.2 Diaphragm Cell.................................................................................... 3.2.3 Other Techniques ................................................................................ 3.3 Review on Past Research................................ ............................................ 3.4 Sum mary...................................................................... References........................................................................................................... Chapter 4: Molecular Dynamics Simulation for Water...:...................... 4.1 Fundamentals of Molecular Dynamic (MD) simulation...................................... 4.1.1 Equation of Motion.................................................................. 4.1.2 Integration................................................................................................... 4.1.2.1 Verlet Algorithm .......................................................... 4.1.2.2 SHAKE and RATTFE............................. ......... 4.1.3 Simulation Cell and Periodic Boundary Conditions................................ ....... Page 6 4.1.4 Cutoff Distance and Long-Range Correction.................................................. 4.1.5 Temperature calculation and Control 4.1.6 Pressure calculation and Control 4.1.7 Force Fields 4.2 Models for Water........................................................ 4.2.1 Rigid and Unpolarized Models 4.2.2 Flexible Models 4.2.3 Polarizable Models 4.3 Calculation of Properties........................................................................... 4.3.1 Thermal Properties......................................................................... 4.3.2 Dynamic Properties 4.4 Review on Past Research ...................................................................................... References Chapter 5: Properties of supercritical water in SPC/E............... .................. 5.1 Objectives ........................................................ 5.2 Simulation Procedure........................................................................... 5.3. Simulation Results and Discussion 5.3.1 Ambient Water................................. 5.3.2 Supercritical Water at 773K................................. 5.3.3 Supercritical Water at Various Temperatures 5.4 Conclusions ..................................................................................... References Chapter 6: Estimation of the critical point of SPC/E water................................ 6.1 Objectives......................................................... 6.2 Type of Methods..................................................................................... 6.3 Simulation procedure..................... 6.4 Simulation results and Discussion...................................... Page 7 6.4.1 Density profile ....... 6.4.2 Coexistence Curve 6.4.3 Estimation of the Critical Point 6.5 Conclusions References Chapter 7: Self-difiasion of water in near critical region .. 7.1 Objectives................................................................................ 7.2 Simulation procedure............................ 7.3 Simulation Results 7.3.1 Isochoric cases................................... 7.3.2 Isothermal cases.................................. 7.4 Discussion . ... .... ...................................................................... 7.5 Conclusions References Chapter 8: Summary................... Chapter 9: Future work and Recommendations ................................... Appendix ..................................................................................... 1. Code for self-diffusion coefficient Page 8 LIST OF FIGURES Figure 1-1 Typical phase diagram for a pure component. Figure 1-2 The density, viscosity, and diffusivity of pure water at40 °C. Figure 1-3 Density of pure water at 310, 330, and 360 K. Figure 3-1 Basic Hahn Sequence Figure 3-2 Figure 3-3 Figure 3-4 PGSE Sequence Figure 3-5 Figure Figure Figure Figure Figure 3-6 3-7 3-8 3-9 3-10 Figure 4-1 Figure 4-2 Figure 4-3 Figure 5-1 Mean square displacements of SPC and SPC/E Figure 5-2 Relationship between density and self-diffusion coefficient at 773K. Figure 5-3 Figure 5-4 Figure 5-5 Figure 5-6 Figure Figure Figure Figure 5-7 5-8 5-9 6-1 Figure 6-2 GEMC concept (from ) EOS fitting result in EOS method (from Guissani and Guillot, 1993 ) Figure 6-3 Concept of density profile method in direct MD method (from Page 9 Guissani and Guillot, 1993 ) Figure 6-4 A unit cell in simulation Figure 6-5 Density profile at each temperature (timestep= 0.5fs) Figure 6-6 Density profile at each temperature (timestep=5fs) Figure 6-7 Density profile of 512 molecules (Ly=Lz=2.7nm) Figure 6-8 Density profile of 512 molecules Figure 6-9 Density profile of 256 molecules (Lx=Snm) Figure 6-10 Coexistence curve from simulation results Figure 6-11 A unit cell in simulation Figure 6-12 A unit cell in simulation Figure 6-4 A unit cell in simulation Figure 64 A unit cell in simulation Figure 7-1 Viscosity near critical point (Franck Figure 7-2 Binary diffusion coefficient in the vicinity of the critical point Figure 7-3 Isochoric and isothermal approach in near critical region Figure 7-4 Self-diffusion coefficient in isochoric conditions 1 Figure 7-5 Self-diffusion coefficient in isochoric conditions 2 Figure 7-6 Self-diffusion coefficient in isothermal conditions Figure 7-7 The relationship between correlation length and temperature in Tc=646K Figure 7-8 Density in which the length of a unit cell is equal to the correlation length at the given temperature (Ly=Lz=1.9nm) ) Page 10 Figure 7-9 Radial distribution function, goo(r) of water in the near critical region at 0.296g/cm 3 Figure 7-10 Radial distribution function, goo(r) of water in the x;earcritical region at 646K - - LIST OF TABLES Table 1-1 Typical property values for liquid, supercritical fluid, and gases. Table 1-2 Critical points of selected solvents. Table 3-1 Table 3-2 Table 6-1 Table 6-2 Table 6-3 Table 6-4 Table 6-5 Table 6-6 Table 6-7 Table 6-8 Table 7-1 Table 7-2 Table 7-3 Table 7-4 Table 7-5 Table 7-6 Table 8-1 Table 8-2 . 11 A _ ,,,, , .......... u Page 12 LIST OF SIMBOLS T absolute Temperature (K) 27 P 31 P Pressure (bar) Bulk density (g/cm3 ) 44 ¢D(r) Interatomic potential energy of interaction (kJ/mol) 58 Lennard-Jones soft sphere diameter (nm) Eij Lennard-Jones energy well depth parameter (kJ/mol) r Rcut eO Cutt off length qi ds charge a collision diameter d C Cl a b c d e f g h i j k kB Boltzmann constant 1 m n o q S t time Page 13 U V W w x y z A B Introduction ~IIo III I I Page 1 I CHAPTER 1 INTRODUCTION 1.1 Motivation Supercritical water has recently proven to be a novel and clean medium for chemical processes of environmental and industrial importance. In order to control the chemical processes in this unique medium, it is essential to understand the structure and the properties of the solvent water at high temperatures and pressures. There are many promising applications related to supercritical water. Supercritical water oxidation (SCWO) is one of the most famous applications. Since Modell proposed the SCWO process (1981), a lot of research have been done so far and many kinetic mechanism have been elucidated. While kinetic of chemical reaction becomes clear, the research on transport mechanism in supercritical water is recently initiated. Solvation, corrosion, and heterogeneous reaction are strongly dependent on the solute and water mobility.( Hodes, ) Furthermore, the design of the large scale requires the diffusion data. In fact, we notice that the dynamic property data of water over a wide range of temperatures and pressures are limited. For example, the referred data of self-diffusion of water is only Lamb's data in 1981. Surprisingly, the referred data of viscosity is only Dudziak and Franck's data in 1966. When real experiments are extremely difficult and when we cannot see the underlying microscopic factors which control phenomena of our interest, such computer simulations as molecular dynamics (MD) and Monte Carlo (MC) simulation can be helpful to understand microscopically. Both simulations and experiments are complementary and stimulating each other to provide a reliable picture on the molecular level. This work addresses the general research goals of our group which are: 1) to further understand the chemical and physical nature of supercritical water and 2) to characterize the I Introduction --. Page 2 mechanisms and kinetics of reactions of hazardous organic wastes in supercritical water. The specific goals of this thesis are to establish the simulation technique for supercritical water, and to confirm if its simulation technique can reproduce the self-diffusivity of supercritical water. This work is also the first step to study the diffusion properties of aqueous solution at high temperatures and pressures. 1.2 Statement of Objectives This thesis has four main objectives: (1) To develop the simulation code to get the properties including self-diffusivity of water and confirm the molecular dynamic simulation can reproduce the properties in supercritical water. In this thesis, Simple Point Charge (SPC) and Extended Simple Potential Charge (SPC/E) models will be used as water models to be evaluated. (2) To simulate the self-diffusion coefficients of ambient and supercritical water and compare simulated data with the limited experimental data. (3) To estimate the critical point of SPC/E water which is a mainly used simulation model in this thesis. (4) To investigate the self-diffusivity of water in the near critical region. In this context, the drop of diffusivity at the critical point was explored. Based on the critical point given in (3), the self-diffusion coefficients were explored from both the isochoric case and the isothermal case. 1.3 Thesis Outline Chapter 2 reviews the characteristic of supercritical water, diffusion models and the behavior of water near critical point including the scaling law. In Chapter 3, experimental technique to Introduction Page 3 measure self-diffusion and some experimental data used for comparison with the simulation results are introduced. In Chapter 4, a brief overview of the fundamentals and techniques of molecular dynamic simulation used in this thesis. The previous research related to the simulation for supercritical water are also reviewed in Chapter 4. The self-diffusion calculation results in ambient and supercritical condition are described in Chapter 5. Chapter 6 includes the methods to estimate the critical point and the simulation results of the critical point in SPC/E model. In Chapter 7, the simulation results of self diffusivity change in the near critical region. Chapter 8 is summary and Chapter 9 refers to the recommendation for the simulation of the diffusion in supercritical solutions. Background .Back2-_ 7 _a, n I I Page 1 CHAPTER 2 BACKGROUND 2.1. SUPERCRITICAL WATER (SCW) The critical point of a pure fluid marks the end of the vapor-liquid coexistence curve, as shown in Figure 2-1. A pure fluid is considered supercritical when both its temperature and pressure are greater than those at the critical point. In the supercritical region, no matter how much pressure is applied, the fluid will not go through a phase transition as its density increases to a liquid-like state. One attractive feature of supercritical fluids is that their physical properties can vary significantly with just a small change in temperature or pressure relatively close to the critical point. Most traditional liquid solvents' properties are not strong functions of pressure due mainly to the low compressibility of liquids. Their properties also tend to remain fairly constant over the temperature range of operation for chemical synthesis. Aside from obvious thermal activation, it is difficult to vary the physical reaction environment in liquid solvents without changing the solvent itself. However, in the supercritical region, many different solvation environments are permitted. Water is an inexpenxive and nontoxic solvent in its supercritical state. However, its critical temperature (374.1°C) and critical pressure (221bar) are relatively higher than other solvents (e.g. 31.1°C and 73.9 bar for C0 2). As many organic compounds are thermally unstable in this supercritical condition, the most promising application is a medium for the destruction of hazardous organic chemicals by oxidation rather than a medium for synthesis or extraction. Figures 2-2 and 2-3 show how some physical properties (density, permittivity, and ion products) of pure water change as a function of temperature at 250bar which is near the Background _ Page 2 critical pressure (221bar). In the near supercritical region, the reaction environment can be easily manipulated by changing either the temperature or pressure or both in order to enhance the solubilities of reactants and products, to eliminate interface transport limitations on reaction rates, and to integrate the reaction and separation processes. As illustrated in Figures 2-2 and 2-3, these large changes in physical properties with a small change in pressure only occur in the critical region of the fluid (T T, P P). In the supercritical region, especially near the critical point, a fluid's density is a strong function of both temperature and pressure. A small isothermal pressure shift near the critical point causes a large change in the fluid density. This effect is reduced as temperature increases. Changes in other physical properties tend to correlate with changes in density. 2.1.1 Thermodynamic Properties of SCW Density Heat capacity Compressiblity dielectric constant The dielectric constant is a measure of the extent of hydrogen bonding and reflects the concentration of polar molecules in the water ( Marshall, 1975). The high dielectric constant of water at ambient condition is due to the strong hydrogen bonding between water molecules (Josephson, 1982). As the pressure decreases, the extent of hydrogen bonding decreases because it is a short range force (Franck, 1963). In addition, the solvent polarity is reduced due to density drop. The dielectric constant at ambient condition is 78.5 (Uematsu and Franck, 1980), while at the critical point it becomes 5 (Franck, 1976). Background Page 3 2.1.2 Dynamic properties of SCW viscosity The viscosity of water changes from . Near critical point, the viscosity of water is ). The Under supercritical conditions, density and viscosity are directly related ( about viscosity of water as a function of temperature and pressure is illustrated in Figure . As can be seen from Figure, most of the decrease in the viscosity occurs between 25C and 250C. The viscosity exhibits only a weak critical enhancement - dur to critical fluctuations - which can only be observed very near the critical point (Sengers, 1994). Diffusivity Due to te large compressibility of water under supercritical conditions, small changes in pressure can produce substantial changes in density which in turn affect diffusivity, viscosity, dielectric and solvation properties, thus dramatically influence the kinetics of chemical reactions in supercritical water. It is not well understood to what extent diffusion controls the reaction rate . In most cases, there is no data on diffusion of aqueous species under supercritical conditions. Self-diffusion and intra diffusion are denoted as Dii or Di. Both the interdiffusion coefficients and intradifusion coefficients are dependent on concentration. Transport properties in supercritical water are typically divided into (1) asymptotic behavior in the vicinity of the critical point, (2)non-asymptotic behavior further from the critical point. Critical effects (enhancements) occur in the region of asymptotic and nonasymptotic behavior. To describe transport properties throughout the temperature- density(pressure)domain, they are typically divided into a background contribution and critical enhancements. Hence, for molecular diffusion(Luettmer-Strathmann and Sengers, 1996) D=Db+ Dc Background Page 4 Where Db is the background diffusion value and is the critical enhancement. Asymptotically close to the critical point, the transport properties are dominated by critical points, the transport properties are described by their background values, which are typically described by one of the phenomenological expressions presented in Various expressions have been proposed to describe the asymptotic behavior of the molecular diffusion coefficient. The first explanation depends on the assumption that the diffusion flux is proportional to the chemical potential gradient yielding a relation of the form ( Cussier 1984; Debenedetti, 1984): 2.2 Diffusion Models Supercritical fluid can be considered to behave like gas at high pressure and like liquid at low temperature. Therefore, both concepts for diffusion models should be necessary. 2.2.1 The Stokes - Einstein Formula ( liquid-like ) kT D-T6 (2-1) Where a is the radius of the molecule, rl is the viscosity. The origin of this equation is the assumption that the molecule moves like a sphere in a viscous medium under the influence of any driving force. In the case of ambient water, 4/3na3=L3/N where L is the length of a unit cell and N is the number of molecules in a unit cell or 1.38nm ( ) etc. are used as a. 2.2.2 Chapman-Enskog theory (gas-like) 1.86x10T2(M 1 +TM2 (2-1) p3122 2 where D is the diffusion coefficient (cm2 /s), T is the absolute temperature (K). p is the pressure (atm), and the Mi is the molecular weight of i. Background Page 5 Reid et al.(1977) suggested for self-diffusion at high pressure gas: pD = p 0 D0 (2-1) Activation theory Free Volume Theory 2.3 DIFFUSION Self-diffusion in fluids, especially dense fluid Ertl et al., 1974; Tyrell and Harris, 1984; Lee and Thodos, 1988) Enskog theory and its modifications (Chapman and Cowling, 1970; Kincaid et al., 1994) A connection between real and simple model fluids can be made by Chandler's rough hard sphere (RHS) theory ( Chandler 1975; Dymond, 1985). According to the theory, the diffusion coefficient of a nonspherical molecular can be described by the following equation. Marcus (1999) showed that diffusion coefficients of water along the saturation line up to the critical point are shown to depend on the fraction of non- and singly hydrogen-bonded water molecules. The high pressure values of D of supercritical water are a smooth extension of the values for lower temperature water at the same pressure. Background _ Page 6 The important parameter for the calculation of diffusivity is the Coulomic term. Berendsen and coworkers investigated the effect of the reaction field. They found the reaction field does not influence the total energy or radial distribution function but self diffusion coefficient much. This Coulombic effect is also inferred from the difference between SPC and SPC/E. shown to be kinetically controlled (Blankenburg, 1974). The rigorous transition state theory rate constant was derived in Chapter 4 to give: 1 k=K kBT ex, h RT Kr p 2-16) zero. Equations 5-3 and 4-16 may be combined to give: kN(Tp)) Kx(T.p) kx(T, p) Kr (Tp) and r N) 2-5) The Stokes-Einstein Formulae D--~kT 6itaTj 2-5) This equation is based on the assumption that the molecule moves like a sphere in a viscous medium under the influence of any driving force. Corrected Stokes-Einstein Formulae kT DT 6nari, (2-5) Background Bc oU1 Ir -- I Drl _ kT 1 1+3/3pa 6ra 1+2 / a Page 7 I7 (2-5) where J is the coefficient of sliding friction. Li and Chang( ) considered the limiting case 1=0 for small particles to be valid. The result is kT (2-16) Writing 2a as (V/NO)1/3 (V is molar volume and NO is Avogadro's number), we obtain: D=~ kT (No D= m ) 13 (2-) Mechanism of diffusion Supercritical region is considered to be intermediate between gas and liquid. So, both mechanism of diffusion should be discussed. Chapman-Enskog kinetic theory D AB=0.001853T 3/ 2I P(AB )2 An 2D,AB DMB MA 2-4 Properties Near Critical Point Scaling law 1 MB (2-) Background Page 8 Fluids exhibit anomaly behavior near critical point because large fluctuations of the order parameter associated with the critical-point phase transition. For gases near the vaporliquid critical point the order parameter is the density, and for liquid mixtures near the critical mixing point the order parameter is the concentration. The range of the fluctuations can be characterized by a correlation length, 4. When the temperature in one-phase region approaches the critical temperature,Tc, at the critical density, this correlation length diverges as = tolATK I(2-5) where v=0.63 and o=0.216 are generally adopted. The properties near critical region can be described using power laws as follows (Pitzer, 1995): P-_ = DpIC i -P PC Pll I PC (2-6) I = BT -Tc (2-7) TC In many simple fluids, a value of -0.35 is found in intermediate ranges of AT*. Within IAT*<10-3 , the exponent value approaches that of three-dimensional Ising model, 13=0.325. For the dynamic properties, D=AD+D (2-5) (2-5) It appears that the critical part AD of the diffusion coefficient can be represented by a simple generalized Stokes-Einstein diffusion law of the form (Burstyn et al.,1983) is aFor example, Paie 9 Background D-D = AD(q)= kT (q5) (2-5) viscosity of steam in the critical region is shown in figure 2- . Viscosity near critical point is enhanced in the narrow range of temperature. For instance, diffusion coefficient of binary mixtures approaches zero in the critical region and affects the results in the supercritical fluid chromatography (Bartle et al., 1991) TI = ri{ 1 T Jxri{ T r) 5-5) The problem is Guissani et al (1993) obtained the coexistence curve from the EOS. They used 96 simulated states to fit EOS. They assumed that the calculated coexistence curve are invariant as long as correlation length is smaller than the box size L. According to the expansion of the correlation length along the critical isochore, [I=4ot Vll1tl0 +-]-- (5-5) with v=0.63, A=0.5, 0=1.3A and ~1=2.16 (Sengers 1986), the inequality i<L (with L-29A at V=55.2cm3 /mol) is satisfied when T=(1-T/Tc)>O.01.This means that only value of coexisting densities in the immediate vicinity of the critical point( T-Tc<7K) should be modified by increasing the size of the simulated system from N=256 to N--oo. Sengers and Sengers, Ann.Rev.Phys.Chem., 1986, Indirect determination of the critical point from the latent heat data The optimum scaling exponent is 0.3361 in the range 603K-646.73K and 0.325 in the range 643-646.73 Background Page 10 ( In order to obtain values of Pc and .c , fit the data with a scaled equation imposing the Tc value. The optimum parameter values for the scaled potential are b=0.325, Tc=647.07K.(Levelt Sengers, 1983) the critical diffusion coefficient is given by DD=RkT/6pheO(x)( 1+b2x2)zh/2 It appears that the critical part AD of the diffusion coefficient can be represented by a simple generalized Stokes-Einstein diffusion law of the form Which is a function of the scaling variables x=q5 with parameter values Zh--0.06 0.02, R=1.01 0.04, b=0.5 0.2 In many simple fluids, a value of =0.35 is found in intermediate range of AT*. Within the range of AT*I<10-3 , the exponent value approaches that of the three-dimensional Ising model, 3=0.325. This is the region in which the correlation length , indicating the extent of the density fluctuations, is much larger than a typical molecular interaction length. Beyond this range, a large range of crossover is traversed, to a region where mean-field concepts apply. exponent constant, 13,has been calculated by many ways. Most reliable number for (3 is currently 0.325 lead by Sengers. This value is given in the extremely closest region to'the critical point within lmK. Lie et al applied b=0.325 and A 0.33. The problem is that the densities of gas and liquid obtained by simulation are usually ones below 573K. Whether b=0.325 is reliable or not is very skeptical. In fact, warned the treatment of the critical point they derived. et al showed the exponent constant becomes between 0.34 to 0.36 experimentally in the case of In this research, I attempted various exponent constants. Guisanni et al used the Werner-type expansion. This is almost same concept of the scaling law method. x- / Background Page 11 De Pablo also used Wegner expansion as follows. Sengers, J.V.; Levelt Sengers, J.M.H., in Progress in Liquid Physics, edited by Croxton, C.A>(Wiley, Chichester,1978),Chapter 4 Ipt -Pvl = BOAt(1 + AtA At + t2 A +...), T-Tr Ip-v=B toIAt=+ BAtAt2A + B2 At + ...), IPt-Pv- =BoAt3(1+ BAt T-T =C( d-1)= (5-5) (55) -11 Tc P-P =CT-Tj Pc TC T-T <O (6-4) TC (6-5) T = Cl(PL - p )" + Tc The validity of the asymptotic power laws is, however, restricted to a very small region near the critical point. An approach to deal with the nonasymptotic behavior of fluids including the crossover from Ising behavior in the immediate vicinity of the critical point to classical behavior far away from the critical point has been developed by Chen et al.(1990a, b). They used 0.5 as b and also d. They obtained the critical temperature by fitting the simulation data to this expansion, and a rectilinear diameter extrapolation for this critical temperature yielded a critical density. 37,18 Backeround ' I Pae 13l 13 Pa I References: Burtle, K. D.; Baulch, D. L.; Clifford, A. A.; Coleby, S. E., Magnitude of the diffusion coefficient anomaly in the critical region and its effect on supercritical fluid chromatography. J. Chromato., 1991, 557, 69 Hertz, H. G., Self-Diffusion in Liquids. Ber.Bunsenges. Phys. Chem., 1971, 75, 183 Pitzer Sengers, J. V., Transport properties of fluids near critical point. Inter. J. Thermophys., 1985, 6, 203 Hertz, H.G., Self-diffusion in liquids. Ber. Buns.Ges.Phys.Chem., 1971, 75, 183 Mills, R., Isotopic self-difusion in liquids. Ber.Buns.Ges.Phys.Chem., 1971, 75, 195 Background - Lc_ - _ Page 14 r I 1 · I al Pre Ssu re I I Temperature Figure 2-1: Typical phase diagram for a pure component. ii Background --I-· I I ·r I Page 15 I n _ 1.5 180 1.0 60 la I 2 2 40 0.5 20 0 0.0 0 100 200 300 400 500 600 700 Temperature, °C Figure 2-1 Dielectric Constant of Water -10 1.5 -12 -14 1.0 -16 -18 Ko Log -20 0.5 -22 -24 -26 nt~ a.v 0 100 200 300 400 500 600 700 Temperature, C Figure 2-2 Ion Dissociation Constant of Water ExpermentalMeasulrements QfSef-Diffuivit ExeIme l M e o I S lII Page Pg 1 CHAPTER 3 EXPERIMENTAL MEASUREMENTS OF SELF-DIFFUSIVITY 3.1 Introduction In this research, I simulated the self-diffusivity of water in supercritical conditions and compared these simulated data with the experimental values. Although this research does not include the experimental measurements, it is important for the research on diffusion in supercritical water and solutions to deal with the reliable experimental values. In this chapter, I introduce the experimental methods for self-diffusion coefficient with respect to water and refer to actual diffusion coefficients of water. 3.2 Experimental methods for molecular diffusion coefficients There are several experimental methods for measurements of diffusion coefficients. In this chapter, I mainly introduce Nuclear Magnetic Resonance (NMR) and Diaphragm Cell method which have been used for the measurements of self-diffusivity of water. 3.2.1 Nuclear Magnetic Resonance (NMR) The NMR spin-echo technique can measure the relaxation process after magnetic perturbation. Relaxation process itself is related to the dynamic properties of the resonant nuclei (mainly H for H20) and this relaxation process is also influenced due to diffusion of the resonant nuclei in a magnetic field gradient. By measuring the attenuation of a spin echo signal, the diffusion of the nuclei can be calculated as follows. 3.2.1.1 Hahn Sequence ExperimentalMeasurementsof Self-Diffsivity Page 2 Spin echo technique can get rid of Bo inhomogeneity, which manifests itself as an additional precession similar to that of a chemical shift. A 180° pulse applied after an interval Xhas the virtue of refocusing any precession after another interval ; this amounts to canceling any chemical shift effect (Figure 3-1). This pulse sequence, 90°x - 1 - 180'y r-acquisition, is the basic Hahn sequence, which yields in principle the true T 2 because any precession effect is removed, leaving a transverse magnetization attenuated according the transverse relaxation time. By reference to the Bloch equations relating to transverse magnetization, dM, _ dt M M,, (3-1) T2 = ex{- 2TJ (3-2) we can acknowledge that the Fourier transform of the half-echo leads to a signal of amplitude Moexp(-2 r /T2 ). For a set of values, it thus appears possible to extract an accurate value of T2 for each line in the spectrum. This analysis does not take into account translational diffusion phenomena. 3.2.1.2 Diffusion measurement in the presence of a steady field Hahn (1950) first considered the attenuation of a spin echo due to diffusion phenomena. If one assumes a linear field gradient in the z direction, the spins see a different B field depending on their z positions so that after application of a 90' pulse they precess at different rates and therefore become out of phase with each other. The relationship between this loss of coherence and the applied field gradient has been derived by Carr and Purcell (1954). We shall assume that the filed Bo is not perfectly homogeneous; for simplicity and without loss of generality, we shall make the hypothesis that it varies linearly across the whole sample in the X direction of the laboratory frame so that the field sensed by a molecule at abscissa X is of the form B(X) = B + goX (3-3) _ ExperimentalMeasurementsof Self-Dffusivity Da2T a at Page 3 _ (3-4) ax2 (3-5) M,(X,t) = M (X,t)+iMy(X,t) M, (X,t) = TPexp-(2iTv o + )t-6) a2T (3-7) 'T(t) = A(t)exp(- iygoXt) (3-8) aT at iygoX' +DaX( ?X =- + T(t) = M(O)exp(-iygoXt)ex - Dy2g0 2t 3 (3-8) In practice, one needs to consider the dephasing effect which remains at the time of formation of an echo at time 2r, where r is the time delay between the first and second r.f. pulse. The application of a 180° pulse after a time r reverses all the phase shifts which occurred up to that time and the echo attenuation due to diffusion is now: Yup[ (T2) (2 2 M, (2t) = M. exp[ -(( y go (3-8) where Mt is the spin echo amplitude, T 2 the spin-spin relaxation time and y the nuclear gyromagnetic ratio. For protons in pure water and for z >lms and go>lOmT' the first term in the exponent of eqn (3-) may be ignored, and a straight-line plot of ln(Mt) against ' 3 has a slope of -(2/3)y2G2D giving D. The steady gradient method has been used to determine self-diffusion coefficients in liquids to accuracy of the order of 0.5% (Harris et al ., 1978) Experimental Measurements of Self-Diffusivity Page 4 3.2.1.3 Pulsed-gradient spin-echo technique (PGSE) Diffusion is usually slow relative to transverse relaxation, and in this case a large gradient must be applied in order to observe significant echo attenuation before irreversible signal attenuation due to relaxation has occurred. Stejskal and Tanner (1965) developed the Pulsed-Gradient Spin Echo (PGSE) technique in order to overcome this problem. In this pulse sequence, the gradient is applied for a time . either side of 180' pulse and is switched off during r.f. pulse transmission and data acquisition. In the absence of any molecular motion the combination of the 180' pulse and the second gradient would completely refocus the magnetization; the only loss in signal would be due to T 2 effects. However, the existence of self-diffusive motion means that the spins, which have been "phase encoded" by the first gradient pulse, change position during the interval (referred to as the diffusion time) and their contribution to the echo will be reduced as demonstrated in Figure 3-3. The magnitude of the echo attenuation is therefore a measure of the extent of molecular motion and, in this case, the transverse magnetization at the spin echo is given by M,(2T)=M ex p - - Dy2g2(A+6) (3-8) In practice, T and A are fixed and the intensity of spin echo signal, Mt is described as a function of 6. When A>>6, eqn.(3-8) becomes F 2(2T, M,(2t) = M0exp - 2 22A (3-9) - (D]y90 A By varying 6, we can get a straight line from the relationship between 62 and in (M). Its slope corresponds to -D(y 2go 2 6 2A) and yields an accurate value of D. Experimental Measurements n qf of Self-Dfjlfusivity Meas Experimenal Page 5 .r _efs__ 3.2.2 Diaphragm cell The Stokes diaphragm cell is one of the best tools to start research on diffusion in gases or liquids or across membranes (Stokes, 1950) because it is inexpensive to build, rugged enough to use. In this method, diffusion coefficient is calculated based on the steady state diffusion in the diaphragm. Diaphragm cells consist of two compartments separated either by a glass frit or by a porous membrane. The two compartments are most commonly stirred at about 60 rpm with a magnet rotating around the cell. Initially, two compartments are filled with solutions of different concentrations. When the experiment is complete, the two compartments are emptied and the two solution concentrations are measured. The diffusion coefficient D is calculated from the following equation. D = 1 ln 1t (C,. ,, - C,p),ni [(C1 orol-C 1 1,0l"e!J (3- ) in which 13 (in cm2 ) is a diaphragm-cell constant, t is the time, and C is the solute concentration under the various conditions given. It V., V..,, (3-) where A is the area available for diffusion, 1 is the effective thickness of the diaphragm, and Vtopand Vbottmare the volumes of the two cell compartments. If the solute is the isotopic water such as HTO and HDO and solvent is H20, one can get the self diffusivity of HTO and HDO in H2 0. Using mass effect correlation, self-diffusion coefficient can be obtained. 3.2.3 Other techniques Tayler dispersion Experimental Measurements of Self-Diffusivity Page 6 This method is valuable for both gases and liquids. It employs a long tube filled with solvent that slowly moves in laminar flow. A sharp pulse of solute is injected near one end of the tube. Mici owave Spectroscopy technique Yao et al. ( ) showed xDD at various densities as a function of temperature where tD is relaxation time and D is the self-diffusion coefficient. rDDis nearly proportional to l/d2 in the gaseous state and rather insensitive to both temperature and density in the liquid state. 3.2 Review of the Previous research on the self-diffusion of water 3.2.1 Ambient water The large deviation of D values at ambient temperature (298.2K) in the earlier measurements was caused by s.'stematic errors ( Mills, 1976). The best value of D is 2.299 x 10'5cm2/s. Self diffusion in normal water and heavy water was measured by diaphragm cell method ( Mills, 1972). Various binary combination of isotopic water was used. 3.2.2 Water at high temperature and high pressure The first measurement of D at high temperature was carried out by Hausser et al.(1966). They measured D by using NMR spin-echo method along the saturation curve from room temperature up to the critical point. Woolf (1974) measured the tracer diffusion coefficient of THO in H2 0 at temperatures between 277 and 318K for pressure up to about 2.2kbar. He also measured the tracer diffusion coefficient of TDO in D2 0. Krynicki et al (1978) extended the experimental temperature range up to 500K from Woolf s experiment (1971). After that, Krynicki et al (1978) also measured D up to 500K. D obtained by Krynicki et al is slightly higher than that of Hausser at high temperature (>400K). They evaluated D using the modified Stokes-Einstein equation. Experimental Measurements of Self-Diffusivity Hausser et al.( et al. ( Page 7 ) reported Xs=is constant except close to the critical point. Krynicki ) also found the constant, XS=,and calculated from their D data and the literature viscosity data. The result of calculation is (6.9 _+0.3)xlO-5NK 1'. They also applied the corrected Enskog theory. However, all of these measurements were done not in the supercritical region but along the saturation curve. Lamb et al. (1981) measured the self-difusivity of water in the supercritical region of 673K to 973K by the NMR spin-echo method. Their data is only diffusion data related to the supercritical water up to now. They found that diffusion coefficient increases with the increase of the temperature and the decrease of the density. Figure 3-3 shows the region of self-diffusion coefficient experimentally measured. As one can see from figure 3-3 , there are few data around the critical point. Activation analysis They obtained activation energy from 0.88g/cm3 to 1.06g/cm3 and from 278K to 485K. As temperature is increased, activation energy decreases and as the density is increased, activation energy is also decreased. The results are accord with the well known fact that for a non-associated liquid , e.g., benzene. Wilbur et al (1976) also concluded that the dynamic behavior of D2 0 resembles that of a normal molecular liquid at high temperature and high compression. Data given by Mills are shown in Table 3-1. Diffusion coefficient at 25°C is -2.3xl0 cm2/s -5 and the activation energy is 2kJ/mol (1-15°C) and 18.9kJ/mol (15-450C). 3.2.3 Water in the near critical region Strictly speaking, there is no diffusion data near critical point. No one investigated the critical enhancement. Hausser et al. (1966) showed negative activation energy near critical point in the figure of the Arrehnius plot. They did not refer to this data. Self diffusivity at high temperature and high pressure Experimental Measurements of Sel.f-Diffusivity Page 8 The problem for the measurement of self-diffusivity of supercritical water is the equipment which achieve high temperature and pressure. The group of University of Illinois set up the NMR equipment for high pressure (Jonas, 1971; Bull et al.,1971; Parkhurst et al., 1971; Lee et al., 1971; Wilbur et al., 1971). In a separate study, M.Kutney, K.Smith and J.Tester are working with D.Cory of MIT's Francis Bitter Magnet Laboratory to explore the use of NMR to quantitatively capture molecular and bulk fluid motion in supercritical water solutions. This method has the advantage that it is completely non-intrusive. Using a gradient-field pulsed-NMR approach, an electromagnetic signature is assigned to the water molecules in arbitrarily thin cross sections of a control volume at an initial time. These signatures are tracked as time evolves in order to measure the rate of displacement to determine molecular diffusivities or fluid velocity in a 2D cross sections. Exierimental MeasurementsofSelf-Diffusivitv Eemt r t o e .f.D.. i ... Page 9 a e 9 s References Burnett, L. J.; Harmon, J. F., Self-Diffusion in Viscous Liquids: Pulse NMR Measurements. J. Chem. Phys., 1972, 57, 1293 Canet, D., in Nuclear Magnetic Resonance Concepts and Methods (John Wiley & Sons, New York, 1996) DeFries, T. H.; Jonas, J., NMR Probe for High-Pressure and High-Temperature Experiments, J. Magnet. Resonance 1979, 35, 111 Goux, W. J.; Verkruyse, L. A.; Salter, S. J., The Impact of Rayleigh-Benard Convection on NMR Pulsed-Field-Gradient Diffusion Measurements. J. Magnet. Resonance 1990, 88, 609 Gladden, L. F., Nuclear magnetic resonance in chemical engineering: principles and applications. Chem. Eng. Sci., 1994, 49, 3339 Hahn Harris, K. R.; Woolf, L. A., Pressure and temperature dependence of the self-diffusion coefficient of water and oxygen-18 water. J. Chem .Soc. Faraday Trans. 1 , 1980, 76, 377 Hausser, R.; Maier. G.; Noack, F., Kernmagnetische Messungen von SelbstdiffusionsKoeffizzienten in Wasser und Benzol bis zum kritischen Punkt, Z. Naturforschg., 1966, 21a, 1410 Krynicki, K.; Green, C. D.; Sawyer, D. W., Pressure and Temperature Dependence of Self- Diffusion in Water. Faraday Discuss. Chem. Soc.,1978, 66, 199 Lamb, W. J.; Jonas, J.,NMR study of compressed supercritical water. J .Chem. Phys. 1981, 74,913 Lamb, W. J.; Hoffman, G. A.; Jonas,J., Self-diffusion in compressed supercritical water. J .Chem. Phys. 1981, 6875 Matsubayashi, N.; Wakai, C.; Nakahara, M., NMR study of water structure in super- and subcritical conditions. Phys. Rev. Lett., 1997, 78, 2573 Experimental Measurements ofSef-Dffusivity Page 10 Mills, R., Isotopic self-diffusion in liquid. Ber. der Buns. Ges. 1971, 75, 195 Mills, R., Self-Diffusion in Normal and Heavy Water in the Range 145 °. J. Phys. Chem., 1973, 77, 685 Marcus, Y. , On transport properties of hot liquid and supercritical water and their relationship to the hydrogen bonding. Fluid Phase Equilibria, 1999, 164, 131 Price, W. S.; Ide, H.; Arata, Y., Self-Diffusion of Supercooled Water to 238K Using PGSE NMR Diffusion Measurements, J. Phys. Chem. A, 1999, 103, 448 Stokes, R. H., An Improved Diaphragm-cell for Diffusion Studies, and Some Tests of the Method. J. Am. Chem. Soc.,.1950, 72, 763 Wakai ,C.; Nakahara, M., Pressure- and temperature-variable viscosity dependencies of rotational vorrelation times for solitary water molecules in organic solvents. J. Chem. Phys., 1995, 103, 2025 Wakai, C.; Nakahara, M., Attractive potential effect on the self-diffusion coefficients of a solitary water molecule in organic solvents. J. Chem. Phys., 1997, 106, 7512 Wilbur, D. J.; DeFries, T.; Jonas, J. ,Self-diffusion in compressed liquid heavy water. J. Chem. Phys., 1976, 65, 1783 Woolf, L. A., Tracer diffusion of tritiated water (THO) in ordinary water (H20) under pressure. J. Chem. Soc. Faraday Trans. 1 , 1975, 71, 784 Woolf, L. A., Tracer diffusion of tritiated heavy water (DTO) in heavy water (D2 0) under pressure. J. Chem. Soc. Faraday Trans. 1 , 1976, 72, 1267 Yao, M.; Okada, K., Dynamics in supercritical fluid water. J. Phys. Condensed Matter, 1998, 10, 11459 Lamb, W. J.; Hoffman, G. A.; Jonas, J., Self-diffusion in compressed supercritical water. Lamb, W. J.; Jonas, J., NMR study of compressed supercritical water. J. Chem. Phys. 1981, 74, 913 Simulation of Self-Diffucsivitv Smlio o S ef Page 1 -Dffu P - 1 CHAPTER 4 MOLECULAR SIMULATION FOR WATER 4.1 Fundamentals of Molecular Dynamic (MD) Simulation 4.1.1 Equation of Motion Newtonian Equation of Motion Motion is a response to an applied force. The translational motion of a spherical particle and the force, Fi, externally applied to the ith particle are explicitly related through Newton's equation of motion: d2rj Fi = m d 2ri dt2 (4-1) where m is the mass of the particle and ri is a position vector. For N particles, it represents 3N second-order, ordinary differential equations of motion. Hamiltonian Equation of Motion Hamiltonian dynamics does not xpress the applied force explicitly. Instead, motion occurs in such a way to preserve some function of positions and velocities, called the Hamiltonian H, whose value is constant, H(r , p N) = conS tan t (4-2) where pi is the momentum of the ith particle, defined in terms of velocity by p, = mr, (4-3) Simulation of Seif-Diffusivity Page2 .. As the Hamiltonian is constant and has no explicit time dependence, one obtaines, by taking the total time derivative of eqn.(4-2), dt - - i + . Z i p i ri ie=o (44) For an isolated system, the total energy E is conserved and equals the Hamiltonian. E is the sum of kinetic energy K and potential energy U. H(rN,p") = E = K + U 1 =- p 2m 2 + U(r") (4-5) Taking the total time derivative of eqn.(4-4), one obtains for each molecule i, E11 dt m'dP' *P1++, =I =0 (4-6) Comparing eqns. (4-4) amd (4.6), one obtains that for each molecule i 5H =Pi =. Pi m (4-7) Combining eqn.(4-4) and eqn.(4-7), and satisfying the conditions- that all velocities are independent of each other, one obtains, ri (4-8) Eqn.(4-7) and eqn.(4-8) are the Hamiltonian equations of motion for an isolated system. For a system of N particles, they represent a set of 6N fist-order differential equations and are equivalent to Newton's 3N second-order equations. In the cases where the system can exchange energy with its surrounding, H no longer equals the system's total energy E, but instead contains exta terms to account for the energy exchanges. H is still conserved, but E is not conserved. Lagrangian Equation of Motion Lagrangian dynamics is the most general form of equation of motion and covers all previous versions of equations of motion. In cases where the systems have internal constraints (e.g., rigid bond) which give extra terms in the form of internal forces, Lagrangian Simulation of Self-Diffusivity Page 3 formulation solves dynamics problems in the most economical coordinates by selecting the coordinates that do not violate the physical constraints of the systems. Using Newtonian dynamics, one can include the applied Fi and constraint force Ci and obtain: N N N Fi .r i + Cj .8rj -ma i=l i=1 i ri =0 (4-9) i=l Since the work of constraint force caused by displacement is zero, eqn.(4.9) reduces to N N XFj 1 rj - ja i=l *ari= 0 i=l (4-10) The old coordinates are transformed into a set of new independent generalized coordinates qj(j=1, 2 ,..... , M), where ri = ri(q, q 2 ,q 3 ,', ,qM, t) i =1,2,3,- , N (4-11) The generalized force Qj is defined to be Qj -= Fi i=1 (4-12) 8qj Combining eqns.(4-10),(4-11),(4-12), one obtains Z:Q 8r,, ,(,£mfa, & =0, (4-13) Utilizing the definition of kinetic energy, K, in terms of velocity and the fact that all the generalized coordinates are independent, after much mathematical manipulation, one obtains the Lagrangian equation of motion dKdbLJ AL= 0 (4-14) where the Lagrangian L is defined as L=K-U (4-15) The number of equations of motion are equal to the number of degrees of freedom M of the system. Simulationof Self-Diffsivitv SImliI f ei .- Df - - Page 4 4 4.1,2 Integration 4.1.2.1 Verlet Algorithm Original Verlet Algorithm Verlet algorithm is the most widely used finite difference integration method for molecular dynamics. It is a direct solution of the second-order equations. It results from a combination of two Taylor expansions r(t + At) =rtdr(t) d2r(t) At2+ 1d 3r(t)3 + O(t 4 ) 2dt 2 dt r(t - At) = r(t)- d(t At + d2r(t) At2 2 dt t 2 (4-16) 3!dt 1 dr(t) At3 + (At4) 3! dt3 (4-17) Adding eqn.(4-16) and eqn.(4-17) eliminates all odd-order terms and yields the Verlet algorithm for positions: 2d2 r(t) r(t + At) = 2r(t) -r(t - At)+ (At2 d)t + ((At)) (4-18) Verlet algorithm utilizes the position r(t), acceleration d2 r(t)/dt2 , and the position r(t-At) from the previous step. The local truncation error is on the order of (At)4 . Velocity is not necessary for computing the trajectories but is useful for estimating the kinetic energy of the system. Velocity can be estimated as: r(t +At) - r(t - At) 2At (4-19) The Verlet algorithm has been shown to have excellent stability for relatively large time steps. However, it suffers several deficiencies. First, it is not self-starting. It estimates r(t+At) from the current position r(t) and the previous position r(t-At). To begin a calculation, special technique such as the backward Euler method must be used to get r(-t). Second deficiency of the Verlet algorithm is that, conflicting with the view that phase-space trajectory depends equally on position r(t) and velocity v, it purely rely on positions. Velocities are not explicitly included in the integration and hence this method requires extra computation and storage Simulation of Self-Diffusivity Page 5 effort. A modified version, namely the velocity version of Verlet algorithm, averts these two drawbacks. Velocity Version of the Verlet Algorithm Swope et al.( ) proposed the velocity version of Verlet algorithm which takes the form 1 d 2 r(t) 2 dt (4-20) d2 r(t + At)l + d(t + At) (4-21) r(t + At) = r(t) + v(t)At + 1 At2 dr(t)-20) 2 and 1 d2r(t) 2 v(t +At) = v(t)+2[ dt The velocity version of Verlet is simple, compact and more commomly used compared to the original Verlet algorithm. 4.1.2.2 SHAKE and RATTLE For polyatomic systems where there are internal constraints, the standard Verlet integration method is not sufficient. As mentioned in the earlier section on Lagrangian dynamics, a set of independent generalized coordinates must be constructed to obey the constraint-free equations of motion. SHAKE and RATTICE are two methods that deal with Verlet integration with internal constraints. SHAKE corresponds to the original positionoriented Verlet formulation, and RATTLE corresponds to the velocity version of the Verlet algorithm. SHAKE is a procedure that approaches internal constraints by going through the constraints one by one, cyclically, adjusting the coordinates to satisfy each constraints in turn. The procedure is repeated until all constraints are satisfied to within a specified tolerance level. RATTLE is a modification of SHAKE, based on the velocity version of Verlet algorithm. It calculates the positions and velocities at the next time step from the positions and velocities at the present time step, without requiring information about the earlier history. Like Simulation of Self-Diffusivity Page 6 SHAKE, it retains the simplicity of using Cartesian coordinates for each of the atoms to describe the configuration of molecules with internal constraints. It guarantees that the coordinates and velocities of the atoms within a molecule satisfy the internal constraints at each time step. RATTLE has two advantages over SHAKE: (1) on computers of fixed precision, it is of higher precision than SHAKE ; (2) since RATITE deals directly with the velocities, it is easier to modify RATTIE for use with constant temperature and constant pressure molecular dynamic methods and with the non-equilibrium molecular dynamics methods that make use of rescaling of the atomic velocities. RATTLE is used to conduct integration in this study. 4.1.3 Simulation Cell and Periodic Boundary Conditions In order to perform MD, one must define a simulation cell containing N particles, and specify their interactions.When we simulate the behavior of bulk liquids to overcome the effect of the cell surface, the cell is considered to be surrounded by replicas of itself. In order to conserve the number density in the central cell, periodic boundary conditions are needed. That is, when a particle leaves the central cell and enters one of the surrounding replicas, its image enters the central cell from the opposite surrounding replica. Figure 4-1 illustrates a two-dimensional periodic system. Particles can enter and leave each box across each of the four sides in the 2D-example. In a three-dimensional case, particles would be free to cross any of the six neighboring faces. The limitation of the periodic boundary condition is that it will neglect any density waves with wavelengths longer than the side length of the simulation cell. For liquid system far from the critical point and for which the interactions are short-ranged, periodic boundary conditions are found to be very useful. I usually used 256 molecules and sometimes used 512 molecules. 4.1.4 Cutoff Length and Long Range Correction Page 7 Simulation of Self-Diffsivity In order to calculate the force acting on the target particle, one needs to account for interactions between this particle and all other particles in the simulation cell. In reality, the concept of cut-off length RCutis introduced to decrease the required computational effort, as the potential and force to be calculated are mainly contributed by particles within a close distance. Only interactions within a sphere of radius Rct from each particle are calculated. The radial distribution function of regions beyond the cut-off length is set to unity and meanfield theory is often used to provide corrections to properties from interactions beyond the cutoff distance. In order to avoid checking the distance of all particles from the target particle, the neighboring list is often used. Setting neighboring radius, Rn, which is a little larger than Rut, the particles within the sphere whose radius is Rn are listed in advance. Assuming that all particles within the Rcut sphere in a timestep belong to the Rn sphere, only the distance of the particles in the neighboring list. The list is routinely updated. This procedure saves simulation time. When the interaction beyond Rcutis significant, long range correction should be done. This correction is usually based on the concept in which the field beyond Rct is constant ( mean field). Figure 4-2 shows the concept of the long range correction. In the case of Lennard-Jones interaction: =4 US4' -(C)) (J (4-22) By integrating the force from Rcut to infinity, long range correction term is calculated. <Rcut UL J = UL-J =UL <RcuI + 27Vp r 2 UL-J(r)dr Rcut L-l L-l 2 87r Ca1 87c 9 R.ut9 3 + (4-23) C6 Ru3 Ne Nep In the case of Coulombic interaction: Ucoulomblc k 4 1 qIqik rk (4-24) assuming that the constant dielectric permittivity, ERF,exists in the continuum out of the Rcut sphere ( reaction field method), one obtains long range correction (Cummings et al., ): Simulationof Self-Diff-usivitv Page 8 II If 1 qllqJk 4rr Coulombic I~k Ik [ -( I + 2 ERF R. rIkI + 1J (4-25) c4 This equation is usually discontinuous at Rcut; therefore, switching function which makes potential curve continuous is sometimes used together. Hummer et al (1992, 1994) proposed a different long range correction which is solved by using boundary condition. U~~ibI = Zk4 1 1 Uc.o?, Coulombic + ERF--1 t+ / r,,/k 2ERF+ qiqk rk R 3 , I )ct 2" RF +1 -Rj (4-26) RF is infinity in eqn.(4-26) for the interactions between ions: u =~C~ck4~,1 qIIqk [ Coulombc 1 ( 4 Jk 3 'Ik 2R. (4-27) Figures 4-3 and 4-4 show the potential curve of Lennard-Jones and Coulombic before and after including long range correction. These correlation is based on the mean field theory, but there is another long range correlation method called Ewald summation in which all interactions between the target particle and the replica unit cells surrounding the centered unit cell are integrated. 4.1.5 Temperature Calculation and Control Temperature is a measurable macroscopic property of the system, and can be calculated frcm microscopic details of the system. In addition, in many cases, to mimic experimental conditions, one must be able to maintain the temperature of the system constant during the MD simulation. Temperature Calculation Temperature is related to the average kinetic energy of the system through the equipartition principle which states that every degree of freedom has an average energy of BkT/2associated with it. Hence, one obtains: Simulationof Slf-Difusivitv Page 9 i(£P K)= fk BT mY2= =i N PiZ )2 / ) 2 )k m (4-28) and T ( ) ( 3N (4-29) where f is degree s of freedom. N is the number of particles, NC is the number of constraints on the ensemble, mi, is the mass of the ith particle, Ti,,tis the instantaneous kinetic temperature, T is the temperature, and k is the Boltzmann's constant. Temperature and the distribution of velocities in a system are related through the Maxwell-Boltzmann expression: 312 f(v)dv m mv 2 e kT4 7rv2dv (4-30) which calculates the probability f(v) that a particle of mass m has a velocity of v when it is at temperature T. Maintaining Constant Temperature Even if the initial velocities are generated according to the Maxwell-Boltzmann distribution at the desired temperature, the velocity distribution will not remain constant as the simulation continues. To maintain the correct temperature, the computed velocities needed to be adjusted. Besides getting the temperature to the right target, the temperature-control mechanism should also produce the correct statistical ensembles. Several methods for temperature control have been developed. They are (1) stochastic method, (2) extended system method, (3) direct velocity scaling method, and (4) Berendsen method. (1) Stochastic method A system corresponding to the canonical ensemble (NVT constant) is one that involves interactions between particles of the system and the particles of a heat bath at a specified temperature. Exchange of energy occurs across the system boundaries. At intervals, the Page 10 Simulation of Self-Diffusivity velocity of randomly chosen particle is reassigned with a value selected according to the Maxwell-Boltzmann distribution. This process corresponds to the collision of a system particle with a heat bath particle. When such a collision takes place, the system jumps from one constant energy surface onto a different constant energy surface. If the collisions take place very frequently, it will slow down the speed at which the particles in the system explore the configuration space. If the collisions occur too infrequently, the canonical distribution of energy will be sampled too slowly. For a system to mimic a volume element in real liquid in thermal contact with the heat bath, a collision rate Rcollision=/3N23 (4-31) is suggested by Andersen where XT is the thermal conductivity, N is the number of particles, and p is the liquid density. Instead of changing the velocity of the particles one at a time as described above, massive stochastic collision method assigns the velocities of all the particles at the same time at a much less frequency at equally- spaced time intervals. (2) Extended System Method (Nose Andersen Method) Another way to describe the dynamics of a system in contact with a heat bath is to add an extra degree of freedom to represent the heat bath and carry out a simulation of this extended system. The heat bath has a thermal inertia and energy is allowed to flow between the bath and the system. The extra degree of freedom is denoted s and it has a conjugate momentum Ps. The real particle velocity is v=s/= m s (4-32) The extra potential energy associated with s is Us = (f + 1)kBTIns where f is the number of degrees of freedom and T is the specified temperature. The kinetic energy associated with s is (4-33) Simulationof elf-DifusivitvI 2 K2 Page 11 .4-_l 2 2 (4-34) where Q is the thermal inertia parameter in units of (energy)(time)2 and controls the rate of temperature fluctuations. The extended system Hamiltonian Hs =K+Ks+U+U s (4-35) is conserved and the extended system density function p(rpsPS) = (H, -Es) NVE(r, p s, ps) = drdpdsdps8(Hs -E.) (4-36) Integration over s and Ps leads to a canonical distribution of the variables r and p/s. The parameter Q is often chosen by trial and error. If Q is too high, the energy flow between the system and the heat bath is slow. If Q is too low, there exists long-lived, weakly damped oscillation of the energy, resulting in poor equilibration. (3) Direct Velocity Scaling Method This method involves rescaling the velocities of each particle at each timestep by a factor of (Ttarget/Tcurrent)/2 where Ttargetis the desired thermodynamic temperature and Tcurrnt is the current kinetic temperature. Even though this method transfer energy to/from the system very efficiently, ultimately the speed of this method depends on the potential energy expression, the parameters, the nature of the coupling between the vibrational , rotational, and translational modes, and the system sizes, because the fundamental limitation to achieving equilibrium is how rapidly energy can be transferred to /from/among the various internal degrees of freedoms of the molecule. (4) Berendesen Method of Temperature Coupling Berendesen method is a refined approach to velocity rescaling. Each velocity is multiplied by a factor X at each time step At Simulationof Self-Difusivitv Pae 12 (4-37) (tTJJtT where Turrent is the current kinetic temperature, Ttargctis the desired thermodynamic temperature, and c is a present time constant. This method forces the system towards the desired temperature at a rate determined by 'r, while only slightly perturbing the forces on each molecule. 4.1.6 Pressure calculation Pressure calculation Pressure is a tensor: (Pxx P= Pxy Pxz] Pyx Pyy Pyz Pzx Pzy Pzz (4-38) Each element of the pressure tensor is the force acting on the surface of an infinitesimal cubic volume that has edges parallel to the x, y, and z axes. The first subscript denotes the normal direction to the plane on which the force acts, and the second one denotes that the direction of the force. Pressure is contributed by two components: (1) the momentum carried by the particles as they cross the surface area and (2) the momentum transferred as a result of forces between interacting particles that lie on different sides of the surface. Hence, P can be expressed as P = 1l .a m,v,. PIVI Vv, +~r,. +r, I,f, ] (4-39) where N iv N I mv, *v, = mv LEm-1 and L . , N *v v, mv * ,Et,v, *vi ,Im,v Em,v, v, *v, Iv (4-40) Simulation of SeIf-Diffusivity Page 13 N N N ZrL.' fXr, Nr. 'S Nrrfi (441) ~r ff where .i,vi., and fi. indicate the .( .=x,y, or z) components of the position, velocity, and force vector of the ith particle, respectively. In an isotropic situation, the pressure tensor is diagonal, and the instantaneous hydrostatic pressure is calculated as P=3[Pa+Pi,+P (4-42) 4.1.7 Force fields This section provides an overview of the fundamental of force field, followed by an introduction to various types of force fields related to water. Fundamentals of Force fields The Schrodinger equation, Htp(R,r) = Et(R,r) (4-43) where H is the Hamiltonian operator, v is the wave function, E is the total energy, R is the vector containing the 3N coordinates of the nuclei, and r is the vector of the electrons' coordinate. Since electrons are several thousand times lighter than the nuclei and move much faster than the nuclei, the Born-Oppenheimer approximation can be used to decouple the motion of electrons from that of the nuclei. Two separate equations are derived from: H,Y(R,r) =ET(R,r) (444) ANC (R) = EO(R) (4-45) and where Vi is the electronic wave function and D is the nuclear wave function. v only parametrically depends on the nuclear positions. I Simulation of Sel-Dififusivity Page 14 Hn = (4-46) +Eo(R) i12i where pi and mi are the momentum operator and the mass of the ith nucleus, respectively. Eqn.(4-44) describes the motion of electrons only and eqn.(4-45) describes the motion of nuclei only. Eo(R) is the potential surface and is only a function of the position of the nuclei. In principle, eqn(4-44) and eqn.(4-45) can be solved for E and R. However, this process is often extremely demanding mathematically, therefore further approximations are often made. An empirical fit to the potential energy surface called a force field or potential, is usually used instead of solving eqn.(4-44), and Newton's equation of motion is used instead of eqn.(4-45) based on the fact that non-classical effects are extremely small for the relatively heavy nuclei. Various types of Force field The force fields used for describing water usually consist of the following form: U~j..f,,, = U. (4-47) +U U, famee= Uintr+Un,,,,rW +Utu +Up (448) The total force field includes a combination of intramolecular component, Uint and intermolecular component, Uint. Intramolecular component originates from the vibration energy in a molecule and consists of the potential energies of bond-stretchng and bond-angle bending, etc. Intermolecular component originates from the interactions between molecules and consists of van der Waals potential energy, electrostatic potential energy, and polarizable energy. Van der Waals dispersion is typically represented by Lennard -Jones 12-6 potential (449) and the electrostatic terms obey the classical Coulombic point-charge interaction (4-50). U~,~,.~1 = UL J = C~ 6 4£L s1 q - iq U= U., I,,4mo ,k 0 (4-49) (4-50) =Uqqk Coulombic ~_ ro111,k (4-50) Simulation of Self-Diffusivity Page 15 The most simple force field model for polar fluids consists of van der Waals interactions and electrostatic interactions. By selecting the position and magnitude of charges and Lennard -Jones parameters, and a, various force field model can be produced. When intramolecular component is introduced into the force field model, it is called a flexible model because the bond lengths and angles are not fixed. Otherwise, it is called a rigid model. When polarizable energy term is introduced into the force field model, it is called a polarizable model, otherwise, unpolarizable model. As the number of potential terms is increased, the time required to simulation becomes longer. Therefore, most widely used models are rigid and unpolarizable models; however, flexible and/or polarizable models are currently often used because of the enhancement of the computer ability. 4.2 Models for Water There are many types of force fields (pair potentials) which have been applied for the computer simulation of water. Basically, Tables 4-1 and 4-2 show the list of potentials for water. 4.2.1 Rigid and Unpolarized Models Parameters of models for water are (1) geometry of water, (2) electric charge, and (3) Lennard-Jones parameters. There are two types of models in water. One is the four point charge model which includes 4 point charges considering lone pair of oxygen ( see Table 4-1). The other is the three point charge model ( see Table 4-2). In early days, four point charge model had used, but currently most simulations of water are based on the three point charge models. Among various models, Simple Point Charge (SPC) model ( Berendsen et al., 1981), extended Simple Potential Charge (SPC/E) model ( Berendsen et al.,1987) and Transferable Induced Potential (TIP4P) model are popular due to the simplicity and reliability. 4.2.1.1 Ab initio and Semi-empirical model Ab initio Simulation qf Self-Diffusivity Page 16 Matsuoka-Clementi-Yoshimine(MCY) Matsuoka, O.; Clementi, E.; Yoshimine, M., J.Chem.Phys. 1976, 64, 1351 Clementi-Habitz(CH) Carravetta-Clementi(CC) Yoon-Morokuma-Davidson(YMD) Clementi, E.; Habitz, H., J.Phys.Chem.1983,87,2815 Carravetta, V.; Clementi, E., J.Chem.Phys. 1984, 81, 2646 Yoon, B.J.; Morokuma, K.; Davidson, E.R., J.Chem.Phys. 1985, 83,1223 4.2.1.2 SPC and SPC/E In this section, I introduce two water models, SPC (Simple Point Charge) and SPC/E (Extended Simple Point Charge). The importance for the simulation model is the accuracy and the simplicity. If we make and use a complex simulation model including a lot of adjustable parameters, we could obtain a much more accurate simulation data. However, if it took a long time to simulate, it would not be practical. Indeed, as the calculation speed becomes faster with the development of the computer, Berendsen et al. (1981) developed Simple Potential Charge (SPC) model which consists of a three point charge on each atomic site and Lennard-Jones interaction between oxygen centers. Configuration of SPC model is described in Figure 4- . They fixed A as 0.37122 nm(kJ/mol)"6 which is the experimental value derived from the London expression. Then, they adjusted charge, q, and B by comparing the experimental internal energy and pressure. Finally, they obtained q= 0.41e, B=0.3428nm(kJ/mol)" 2. These values of A and B corresponds to e=0.31656 (kJ/mol) and o=0.65017(nm) for Lennard-Jones equation. According to the researchers, the parameters are slightly changed in SPC. Some examples are described in Table 3-. While both SPC and SPC/E models are simple in configurations and easy to calculate, the results from them are plausible. Page 17 Simulation of Self-Diffusivity Both models are semi-empirical models, and the configuration and parameters are set so that the data of ambient water becomes same. SPC was exploited by Berendesn( which the configuration is based on tetrahedron and ), in . It takes account in the contribution of Lennard-Jones and Coulomb interactions. The parameters in the SPC potential were determined by fitting thermodynamic properties at 298.15K and lg/cm3 to experimental data. ( Berendsen et al 1981)The disadvantage of SPC is that it does not include the effect of polarizability for polar fluid like water. As a result, the self-diffusion coefficient of water at ambient condition is higher than the experimental data. Berendesen et al ( ) calculated the contribution to the total electric energy of polarizability and improved the parameter of SPC code to include They concluded that the change of charge from -0.82 e to -0.8476e on O atom is effective. This revised code is called SPC/E . The most outstanding effect of SPC/E is to introduce dipole-dipole interaction into the model by changing the charges. U = ZjViqidX = ZqjVi o i Eel=YqiVj - iEi i l U = E i + EpoI EU= jxviqidX 0 = 1qV i (4-) Simulationof Self-Diffusivity (EzPO)=i Yii i 2 iiE (p2) = (1i)2 (EP) = (2 a Vi j1 /,,4nrij) U = E +Eoi E, =/2 qiVi EpI=XY&-X Vi /1[S4neorij) + G(rij)Ii] E = I [G(rii)qi+T(rij)jJ j#1 T(r)=(3rr-2, /(47te a/Ua, = o 0 0 0 rij) 3 Page 18 Simulationo Self-Difusivitv Smlioo Page 19 I 19 4.2.2 Flexible models Rigid Models Rigid models mean that the geometry of each molecule is fixed. The bond length and the bond angle do not change; therefore, parameters are only positions and velocities of each atom site. Flexible Models By adding Intra molecLemberg and Stillinger(1975) made a pioneering attempt at incorporating intramolecular degrees of freedom into a potential for water. Their central force (CF) concept does not distinguish between intra- and inter molecular interactions among atoms. The potential represents intra molecular interactions within a certain range of distances and intermolecular interactions elsewhere. Bopp et al. (1983) improved CF model and expressed the vibration in liquid phase. TJE Further simpification of the flexible SPC model was advocated by Teleman et al. (1987) who suggested simple harmonic forms for all vibration terms in the intramolecular potential. This new flexible SPC model is called TJE(Teleman, Jinsson, and Engstr6m). In TJE, he intramolecular part consists of harmonic bond and angle vibration terms. 1 Uintra = Table 4- 2 -kOH(r-rO) 2 1 +X-kHoH(O2 Intramolecular parameters for (1987) parameter kOH kJ/(molA 2 ) 4637 1.0 ro 0) 2 kHOH kJ/(mol rad 2) 383 00 deg 109.47 Simulationof Self-Diffusivity Page 20 TR (Toukan and Rahman) UOH =2LIT(r 21 +A 23 1) 2 UHOH =LrrAr12ArI3 LrrA2(ArA L +Ar 3 )Ar2 ++LAr 2 UDOH= [O(1-exp[- a12 b-- (1 exp[- 2 23 Arl3 )b2j] t=cjLrrDOHJ m-TR U intra = U OH+ UHOHexp[- P3(Ar2 + Ar3 )j The advantage of flexible model In summary, experience with flexible water molecules has indicated that the inclusion of anharmonicity is necessary in order to reproduce the observed gas - liquid shifts in the stretching bands. The effects of anharmonicity on the other observables is probably marginal at best. Moreover, neither the harmonic (TJE) nor the anharmonic (TR) potential reproduces the experimental value of the bending peak position correctly. The former underestimates the position while the latter overestimates it. In any case, 4.2.3 Polarizable models Water is known as a highly polarized liquid. Fixed charge model such as SPC cannot behave exact polarized liquid. Polarized model was developed to reproduce more real water. Simulation Simulation-;,-of of Self-Diffusivitv Sl-If -f-f iil Page Pe2 21 Mountain (1995) compared ST2 fixed-charge model and RPOL model. For the range from 298K to 673K in temperature and from 0.660g/cm3 to 0.997 g/cm3, the explicit inclusion of polarization in the interactions had a relatively small effect on the pair functions. Dang and co-workers(1990) developed a polarizable water potential model (POLl) in which the atomic polarizabilities developed by Applequist et al. This model gave good agreement with the structural and thermodynamic properties of liquid water Dang improved this POLL and developed RPOL. Table 4-: Polarizable water model Run--ModelslNCCa PSPCb Number of force Time centers step (fs) Number of iterations per step 3 charges + 2 dipoles 0.5 6-9 3 charges + 1 dipole 0.5 7-8 1 3-4 PPC 3 charge. . . Induced dipole moments for the RPOL model were obtained by self-consistently solving at each timestep the set of equations for the induced moments A = IE, + a, jTg,k where 1j is the induced moment at site j, Ej° is the electric field at site j due to charges on other molecules, the sum is over all sites not located on the same molecule as site j, and Tjkis the dipole tensor. Page 22 Simulation of Self-Diffusivity Svishchev et al (1996) exploited a new polarizable model. They obtained the values of hydrogen charges by using quantum chemical calculations in the presence of homogeneous static electric field. rangi PPC- Polarizable Point Charge model (PPC) Simple potential charge models are now used widely as condensed phase potentials in computer simulations of water. Two of the more popular and successful models are the extended single point charge (SPC/E) and TIP4P potentials. These rigid nonpolarizable Table 6-4: Result of Polarizable SPC model Conditionl POLl RPOL D E B C 0 H 3.169 0.000 0.155 0.000 -0.730 +0.365 0.465 0.135 0 H 3.196 0.000 0.160 0.000 -0.730 +0.365 0.528 0.170 A Run-- models incorporate three fixed charge sites in a single Lennard-Jones sphere; the effective pair potentials that result have been parameterized for ambient condition. One of their shortcomings is that the effects of electronic polarization which play an important role in physical-chemical processes in water are not explicitly included. Also, their ability to reproduce the behavior of real water over a wide range of state parameters is rather limited. In recent years, considerable effort has been devoted to the development of more refined water models that explicitly incorporate polarizability ( Zhu et al, 1994, Niesar et al, 1990, AhlstrDlm et al, 1989, Sprik et al, 1988, Cieplak et al, 1990, Kozack et al, 1992, Bernardo et al, 1994, Zhu et al, 1993, Rick et al, 1993 and Halley et al, 1993) Mountain compared ST2 model and RPOL model and concluded that the difference due to the inclusion of polarizability has a small effect. Simulationof Self-Diflusivitv Page 23 4.3 Calculation of Properties 4.3.1 Thermodynamic Properties 2N <T2 >-<T> 2 dielectric constant 4.3.2 Dynamic Properties Diffusion coefficient Chemical Diffusion J =-DVC Tracer Diffusivity D,= (a2) 2dt Self-diffusion In an infinite system at equilibrium, the self-diffusion constant D may be obtained from the long time limit of the mean square displacement of a selected molecule j, Ds =- F with F=-alkB alnC In what follows Drj(t) refers specifically to the displacement of the center of mass for molecule j over time interval t, though any other fixed position in the molecule would serve as Simulation of Self-Diffasivity Page 24 well ( such as the oxygen or a hydrogen nucleus position). Because molecular dynamics simulations are limited in both space and time, it is necessary to infer D from the slope of <(Drj)2> vs t. One has the identity where vj is the center -of-mass velocity for molecule j. It is obvious that the slope method for evaluating D will be acceptable only if the molecular dynamics simulation shows that at times for which it is applied, the velocity autocorrelation function has decayed substantially to zero. 2Ndt (s2) 2dt D= dilNr 1 lim d (i 6N t---oodt I I D = dt(J(t) 2[/aN = - (t) - ri (0) P J(O)) = 2 ] 7dt(JZ(t)-JZ(O)), jaic_ -1 3kTV Idt( vi(t)) (,vi(O)) i . i jZ = ziev i (flux of charge) i shear viscosity rl = - dt(P kBT 0 (t) Pxy(0)) , P = ' mirnivxviyi+ 'rixFa ) j>i The thermal conductivity X x = -JIdt(q(t) 1 3kBTVO q(O)) qV =-I 2i (mivi2)vi +-[ij(v 2 j + vj)+Fij (vi+vj)r] Simulation o Se~f-Diffjusivitv Smai o SI f Dfs y I Page 25 2 Self-diffusion In 1980s, self-diffusion of water at room temperature. There are few data about the self-diffusion simulation of the supercritical water. First research was done by Kalinichev(1993). He used ST model along the coexistence curve. SELF=DIFFUSION Self diffusion coefficient of liquid water at ambient condition were obtained by many researchers MCY flex(Lie et al 1986), The linear behavior of <Dr2> for t>0.5psec means that the motions of the atoms in liquid water are beginning to be dominated by random processes after that time. At times much shorter than the characteristic collision time, any tagged particle is expected to move like a free particle and hence its mean square displacement is given by v02t2, where vOis the thermal velocity. Kataoka et al (1989) investigated the effect of temperature and volume (density) by using CC potential in the range of the density 16 - 22cm3/mol and temperature 260-700K. The values of self-diffusion coefficient were different from the experimental data, but the qualitatively trend like the anomaly at low temperature was reproduced. Recently, Yoshii et al.(1998) carried out the simulation by using RPOL polarized model. First, they obtained results along the isochore at lg/cm3(3.4pc) between 280 and 600K. According to their activation energy analysis, room temperature region near 280K indicates 13kJ/mol and high temperature region near 600K shows 6.8kJ/mol. They concluded that the barrier of the diffusion of water molecules becomes small, reflecting the break of the tetrahedral structure. Second, they investigated the isotherm effect at 600K(1.07Tc). They found that the diffusion coefficient is proportional to the inverse of the density and is in good agreement with the experimental data based on reduced temperature. Liew et al utilized cm4PmTR which is a flexible TIP4P model. They indicated the good accord with the experimental data in reduced pressure and volume relationship. They also showed the self diffusion Simulationof Self-Diffusivitv Pa-e 26 coefficient at 673K is in excellent agreement with the experimental data beyond the critical density. However, their data at low density became 30% lower than the experimental data. Baez and Clancy used SPC/E to investigate low temperature water. Table 6-4: Result of ST2 model T (K) Density (g/cm 3) P (MPa) U (kJ/mol) Ds t (cm2/s) (D) 270 283 1 1 1.3 1.9 314 1 4.3 391 1 Check on accuracy 8.4 Table 6-4: Result of ST2 model T (K) Density (g/cm 3 ) P (MPa) 49 U (kJ/mol) -42.7 Ds (cm2/s) 2.5 J1 (D) 297 0.997 2.35 575 0.72 89 -25.8 34 2.35 667 0.66 140 -22.4 45 2.35 Jorgensen et al.(1983) compared the various simple potential functions , ST2, BF, SPC, TIPS2, TIP3P, TIP4P. From the radial distribution function and self-diffusivity. With respect to diffusivity, ST2 is the best potential. 1. conservation laws are properly obeyed, and in particular that the energy should be constant. Table 6-4: Result of RPOL model Type T (K) BF SPC 294 300 ST TIPS2 Density (g/cm3)[ P (MPa) I 1 7R - T2(1) Density 297 667 Ds (cm2/s) 4.3 Tble 6-4: Rsuit of RPOL mOli 83 ______ U (kJ/mol) 0.66 (g/cm3) 0.997 7?e2 0.66 P 3. (MPa) -45 (kJ/mol) -42.0 -2U3 170 -4.7 -20.8 experimen tal Ds (cm2/s) 300 300 2.30 2.30 283 19 3 (cm/s) 2.7 234 46 2130 (D) 2.62 2.36 2.31 Simulation Self-Diff-usivity Page 27 For a simple Lennard -Jones system, the order of 10-4 are generally considered to be acceptable. Energy fluctuations may be reduced by decreasing the time step. If one of the Verlet algorithms is being used, then a suggestion due to Andersen may be useful. several short runs should be undertaken, each starting from the same initial configuration and covering the same total time:each run should employ a different timestep and consist of a different number of steps RMS energy fluctuation should be calculated RMS fluctuations are proportional to timestep2 Good initial estimate of timestep Table 6-4: Result of Polarizable SPC model Run- Condtio. A B C D E Conditionl T (K) Density (g/cm3) 673 0.1666 772 0.5282 630 0.6934 680 0.9718 771 1.2840 DMD(Cm 2 /s) 196 76 37 23 11 193 68 44 33 28 2 Dexp(cm /s) It should be roughly an order of magnitude less than the Einstein period 2p/wE WE2=<fi2>/mi2<vi2> OH-bonding in SCW Mountain (1989) examined the structure of water over a range of densities from 0.1g/cm3 to 1 g/cm3 and for a range of temperatures from the coexistence temperature up to supercritical temperatures. He elucidated that hydrogen bonding exists even in the supercritical water and Simulation of Self-Diffusivity _ Page 28 the number of hydrogen bonds per molecule scales as a single function of the temperature but does not scale for dense vapor densities. Simulation of Seif-Diffusivity Page 29 4.4 Review on Previous Work 4.4.1 Simulation for Ambient Water 4.3.2 Simulation for Supercritical Water O'Shear et al. (1980) reported the simulation of supercritical water. They investigated the thermodynamics of supercritical water by using Monte Carlo simulation based on the MCY potential (Matsuoka et al, 1976). Kalinichev (1986) conducted Monte Carlo simulations of SCW using the TIPS2 model (Jorgensen, 1982). He used only 64 molecules but did obtain reasonable thermodynamic properties of supercritical water. He also obtained good agreement with experimental thermodynamic data by using the TIP4P potential ( Jorgensen et al, 1983) but found that the critical point of TIP4P water was far from that of real water. He reported that the selfdiffusivity (Kalinichev,1992) Mountain ( simulations of 3 .3 2 ps ) studied TIP4P supercritical water using 108 water molecules and duration. The simulations were conducted over a wide range of temperatures and densities from ambient to 1100K and 100kg/m3 . Another supercritical water study using the TIP4P potential was carried out by Gao ( ), who using Monte Carlo technique found that the densities and dielectric constants for TIP4P supercritical water at 673K at various pressures corresponded suggested that the critical temperature of TIP4P water deviated greatly from the experimental value. normal liquid water Kataoka showed the anomality of water simple water like model was used for analysis (Kataoka, 1986) TIP4P was used (Reddy 1987) Yoshii et al used RPOL model and . They evaluated the translation of oxygen atoms because the center of mass is very close to the position of the oxygen. They evaluated Simulation of Self-Diffusivity Page 30 diffusion by Arrhenius plot. The plot was divided into two limiting regions with different slopes. At room temperature region, the activation energy was 13kJ/mol and at high temperature region near 600K, the activation energy was 6.8kJ/mol. They proposed that the diffusion of water molecules becomes small, reflecting the break of the tetrahedral icelike structure. ST2 potential was used (Stillinger 1974) SPCE model was explored to treat ( Berendsen et al 1987) Table 3- X shows the Simulationof Self-Diflsivity SImlIonIII of S Iffusivil y Page 31 Page 31 References Applequist, J.; Carl, J. R.; Fung, K.-K., J. Am. Chem. Soc., 1972, 94, 2952 Baez, L.A.; Clancy P., Existence of a density maximum in extended simple point charge water. 9837 Balbuena, P. B.; Johnston, K. P.; Rossky, P. J.; Hyun, J. K., Aqueous ion transport properties and water reorientation dynamics from ambient to supercritical conditions. J. Phys. Chem. B, 1998, 102, 3806 Ben-Naim, A.; Stillinger, F. H., Aspects of the statisitical mechanical theory of water. in Structure and Transport Processes in Water and Aqueous Solutions, edited by R.A. Home (Wiley-Interscience, New York, 1969) Berendsen, H. J. C.; Potsma, J. P. M.; von Gunsteren, W. F.; Hermans, J., in Intermolecular Forces, edited by B.Pullman ( Reidel, Dordrecht, 1981) 331 Berendsen, H. J.; Grigera, J. R.; Straatsma, T. P., The missing term in effective pair potentials. J. Phys. Chem., 1987, 91, 6269 Bernard, D.N.; Ding, Y.; Krogh-Jespersen, K.; Levy, R.M., J.Phys. Chem. 1994, 98, 4180 Bopp, P.; Jancso, G.; Heinzinger, K., An improved potential for non-rigid water molecules in the liquid phase. Chemical Phys. Lett., 1983, 98, 129 Cardwell, J.; Dang, L. X.; Kollman, P. A., J. Am. Chem. Soc. 1990, 112, 9145-POL1 Coulson, C.A.; Eisenberg, D., Proc. R.Soc.London, Ser.A 1966, 291, 445 Cieplak, P.; Kollman, P.; Lybrand, T., J. Chem. Phys. 1990, 92 ,6755 Dang L. X.; Pettitt, B. M., Simple intramolecular model potentials for water. J. Phys. Chem., 1987, 91, 3349 Dang L. X., The nonadditive intermolecular potential for water revised. J. Chem. Phys., 1992, 97, 2659-RPOL Page 32 Simulation of Self-Diffusivity Errington, J. R.; Kiyohara, K.; Gubbins, K. E.; Panagiotopoulos, A. Z., Monte Carlo simulation of high-pressure phase equilibria in aqueous systems. Fluid Phase Equil., 1998, 150-151, 33 Halley, J. W.; Rustad, J. R.; Rahman, A., J. Chem. Phys. 1993, 98, 4110 Hummer, G.; Soumpasis, D.M.; Neumann, M., Pair corrections in an NaCI-SPC water model simulations versus extended RISM computations. Mol.Phys., 1992, 77(4), 769; ibid, 1993, 78(2), 497 Hummer, G.; Soumpasis, D.M.; Neumann, M., Computer simulation of aqueous Na-Cl electrolytes. Phys. Condens. Matter, 1994, 6, A141 Jorgensen, W. J. Chem. Phys. 1982, 77, 4156 Jorgensen, W.; Chandrasekhar, J.; Madura, J.; Impey, R.; Klein, M., Comparison of simple potential functions for simulating liquid water. J. Chem. Phys. 1983, 79, 926 Kalinichev, A., Z. Naturforsch. 1992, 46a, 433 Kalinichev, A. G., Molecular Dynamics and Self-Diffusion in Supercritical Water, Ber. Bunsenges. Phys. Chem., 1993, 97, 872 Kataoka, Y., Bull. Chem. Soc. Jpn., 1986, 59,1425 Kataoka, Y., Studies of liquid water by computer simulations . V. Equation of state of fluid water with Carraveta-Clementi potential. J. Chem. Phys., 1987, 87,589 Kataoka, Y., Studies of liquid water by computer simulations . VI. Transport Properties of Carraveta-Clementi water. Bull. Chem. Soc. Jpn., 1989, 62,1421 Kozack, R. E.; Jordan, P. C., J. Chem. Phys., 1992, 96, 3120 Lemberg, H. L.; Stillinger, F.H., Central-force model for liquid water. J. Chem. Phys. 1975, 62, 1677 Lemberg, H.; Stillinger, F., J. Chem. Phys. 1989, 90, 1866 Simulation of Self-Diffusivity Page 33 Lie, G. C.; Clementi, E,Molecular-dynamics simulation of liquid water with an ab initio flexible water-water interaction potential. Phys.Rev. A, 1986, 33, 2679 Matsuoka, 0.; Clementi, E.; Yoshimine, M. J., CI study of the water dimer potential surface. J. Chem. Phys. 1976, 64, 1351 Mountain, R. D.; Molecular dynamics investigation of expanded water at elevated temperature. J Chem. Phys., 1988, 80, 1866 Mountain, R.D.; Comparison of a flexed-charge and a polarizable water model. J. Chem. Phys., 1995, 103(8), 3084 Mizan, T.I.; Savage, P.E.; Ziff, R.M., Molecular dynamics of supercritical water using a flexible SPC Model, J. Phys. Chem., 1994, 98, 13067 O'Shea, S.; Tremaine, P., J. Phys .Chem., 1980, 84, 3304 Rami Reddy, M.; Berkowitz., Structure and dynamics of high-pressure TIP4P water, J. Chem. Phys., 1987, 87, 6682 Reddy, M. R.; Berkowitz, M., J. Chem.Phys. 1987,87,6682 Rick, S.W.; Stuart, S. J.; Berne, B. J., J. Chem. Phys., 1994, 101, 614 Roberts, J. E.; Schnither, J., Boundary conditions in simulations of aqueous ionic solutions: Asystematic study. J. Phys. Chem., 1995, 99, 1322 Stillinger, F. H.;Rahman, A., Improved simulation of liquid water by molecular dynamics. J. Chem. Phys., 1974, 60, 1545 Sprik, M.; Klein, M. L., J. Chem. Phys. 1990, 92, 6755 Stillinger, F.; Rahman, A., J. Chem.Phys. 1993, 98, 8892 Svishchev, I.M.; Kusalik, P.G.; Wang, J.; Boyd, R.J., Polarizable point-charge model for water: Results under normal and extreme conditions, J. Chem.Phys. 1996, 105(11), 4742 Toukman, Teleman,O.;Jonsson,B.;Engstrom,S., A molecular dynamics simulation of a water model with intramolecular degrees of freedom.Mol.Phys. 1987,60,193 Simulation of Self-Diffusivity Page 34 Toukan, K.; Rahman, A., Molecular-dynamics study of atomic motions in water. Phys. Rev. B, 1985, 31(5), 2643 Yoshii, N.; Yoshie, H.; Miura, S.; Okazaki, S., A molecular dynamics study of sub- and supercritical water using a polarizable potential model. J. Chem. Phys., 1998, 109(12), 4873 Zhu, S.-B.; Singh, S.; Robinson, G. W., J. Chem. Phys. 1991, 95, 2791 Simulation o Self-Diftusivity Sil I II Figure 4-1 2-dimension Periodic boundary condition Page 35 Pe .I 5 Simulationof elf-iffusivitv i f S I - iff J Page 36 ,, Il Assu Acut- oo Figure 4-2 Concept of long correlation Simulation --o Sef-Diffu'isivity Ifl DII Figure 4-3 correction Lennard-Jones Potential with and without long Page 37 P 3 Simulation of Self-Diffusivitv __ I __ LC I :t I £J j r (A) Figure 4-4 Coulombic Potential between O and H with and without long correction Page 38 Proerties OfSUrerritical Waterin SPC and SPCIESimulations Po ri oIf cIial Water I SP ; anI Simlio Page 1 CHAPTER 5 PROPERTIES OF SUPERCRITICAL WATER IN SPC AND SPC/E SIMULATIONS 5.1 Objectives There are lots of literatures on the simulation of water. One model after another has been developed to describe the properties of water. However, most of the models were established only for the ambient water because both ab initio and semi-empirical models were adjusted to the properties of ambient water. Therefore, to date, the simulation results of water at high temperatures and pressu,.rcsincluding supercritical condition are limited and no one exactly know which models are applicable for the supercritical water and whether a new model is required. My objective i this chapter is to confirm the validity of molecular dynamic simulation for the supercritical water espfcially with respect to self-diffusivity. Our research group at MIT has used SPC model to understand the behavior of water and binary solution in supercritical region and to evaluate Zeno line ( Reagan et al, 1999). Since this SPC model which has been widely used for the simulation of water due to their simplicity and accuracy is qualitatively successful to express the behavior of supercritical water so far, I select SPC model first. Then, I also investigate the usefulness of SPC/E model because this model is as simple as SPC model and is known to reproduce good transport properties compared with SPC model. In this chapter, I firstly simulated ambient water by using SPC and SPC/E models and compared the results with the experimental data (Lamb et al., 1981) and other literature (Mizan et al., 1994). Next, I simulated supercritical water at 773K by using both models and confirmed the reliability of models in supercritical region. Finally, by using the reliable model, the wide range of supercritical water from 673K to 873K was simulated and compared with the experimental data. Propertiesof SupercriticalWaterin SPC and SPC/ESimulations Page 2 5.2 Simulation Procedure Two models, rigid SPC and rigid SPC/E , were used in this chapter. In the rigid SPC model (Berendsen et al, 1981), the distance of the oxygen site and the hydrogen site is invariantly O.lnm and the angle of H-O-H is fixed at 109.47° . The interaction between water molecules includes a Lennard-Jones potential, centered on the oxygen site, with a well depth, E ,of 0.648kJ/mol and a core diameter, a, of 0.3166nm between oxygen centers. In addition, point charges of -0.82e at the oxygen center and +0.41e at each hydrogen site interact through a Coulomb potential where e represents the charge of one electron. As a result, the pair potential is described as eqn.(5-1). In the rigid SPC/E model (Berendsen et al.,1987), only charges are different compared to SPC model. The point charge at the oxygen site is -0.8476e and that at the hydrogen site is +0.4236e in SPC/E model. ((r) water-water = )(r) LenardJones + ((r)Coulombic site-site (I)(rLenard-Jones = 4ij( 1 (5-1) (5-2) qiqj ()(r)coulombic = 4 ie r 47rEO r (5-3) Site-site interactions were neglected beyond a cutoff radius, RIut, of 0.7915nm (2.5aij) to avoid double counting interactions across system periodic boundary condition. To account for long- range interactions, I used the standard long range correction to the Lennard-Jones potential ( Allen et al.,1990 ) and the site-site reaction field method of Hummer et al.(1992 ) to correct the Coulombic interaction. The method replaces the site-site Coulombic potential with an effective potential: biqj1 3 4Coulombic 4 r 2R cut r2 2Rcut3 (5-4) Propertiesof SupercriticalWaterin SPC and SPC/ESimulations Page 3 where Rut is the cutoff radius as described before. In addition, each dipole with dipole moment A, contributes a self-energy of -1/2I 2 /Rcut3 to the total potential energy of the system to account for the interaction with the dielectric continuum outside the sphere of radius Rout. System equilibration was checked by tracking total potential energy and total system energy, and by dividing the simulation into 5000 cycle blocks to check the evolution of statistics over time. The equations of motion were integrated with the Verlet algorithm, and bond-length constraints were maintained using the RATTLE algorithm in which tolerance parameter is 2x10-8. A constant -NVT ensemble was maintained with Nose-Andersen thermostat. Partly, I also used massive stochastic collision (rescaling) as a controller of temperature, and a constant -NPT ensemble was also carried out with Nose-Andersen pressure control . The number of water molecules in a unit cell was 256. 5fs, 1 fs and 0.5fs (10-'ls) were used as a timestep. Equilibration time runs of 2.5 - 50ps (10-12s) preceded each 2.5-50ps production runs. The trajectory of the center of mass of each molecule was traced during production time and mean square displacements of all molecules were calculated every 5000cycles. The slope between time and mean square displacement provides the self-diffusion coefficients. 5.3 Simulation Results and Discussion 5.3.1 Study of water in the ambient condition Table 5-1 shows the simulation results of SPC and SPC/E for the ambient water. The total energy of SPC is in good agreement with the experimental data, -41.5kJ/mol which is derived from the heat of vaporization (Watanabe et al., 1989). However, as the matter of fact, the correction of self-polarizable energy, Epol,is required to the total internal energy. Epolis calculated by using eqn.(4 - ) in Chapter 4: Y E ~ A.),(4= 1 Epole - Propertiesof SupercriticalWaterin SPC and SPC/ESimulations where a is polarizability which is 1.608x10O4°Fm( Eisenberg et al., 1969 ), Page4 is the dipole moment of water in the model, and go is the dipole moment of the isolated water molecule which is 1.85D. From Ix=2.27D in SPC model , Epolis calculated as 3.74kJ/mol while Epolis 5.22kJ/mol from gp=2.35D in SPC/E model. By Adding these values to each simulated internal (potential) energy, internal energy, 37.8kJ/mol, is obtained for SPC water and 40.6kJ/mol for SPC/E water (Table 51) . These values were almost same as Berendsen's data (187). The outstanding point in the results is the difference between self-diffusion coefficient of SPC and SPC/E. Although the self-diffusion coefficient obtained by SPC/E is very close to the experimental data, 2.3x10'5cm 2/s (Krynicki, 1978) and 2.4x10 5cm 2/s (Mills, 1973), the selfdiffusion coefficient by SPC is almost twice as large as the experimental one. Figure 5-1 shows the relationship between time and mean square displacements of water molecules. Self-diffusion coefficient of SPC water is explicitly larger than that of SPC/E water. The similar result was also given by Berendsen et al. (1987). The reason is not still clear, but Coulombic interactions including dipole-dipole interactions seem to influence the simulated transport properties much. Tables 5-2 and 5-3 show simulation results of SPC model and SPC/E model, respectively, in ambient conditions by other researchers. Due to the difference of simulation techniques, there is small deviation, but potential energy and self-diffusion coefficients are in good accord. This work also indicates the similar results to others. _Propertiesof Supercritical __ _____ Waterin SPC and SPC/E Simulations _ I I Page 5 Table 5-1: Simulationresultof SPC and SPC/E in ambient condition Althor This work This work This work Berendsen Berendsen 1987 1987 Model Ensemble type SPC NVT SPC/E NVT SPC/E NPT SPC NVE Temperature 300 300 300 308 306 0.997 0.997 0.995 0.97 0.998 508±51 67+80 -11.4 -1 6 - 41.2 - 45.8 - 46.5 a - 40.4 - 46.6 - 38.5 - 40.6 -41.3 - 37.7 - 41.4 i ' SPC/E NVE (K) Density (g/cm3) Pressure (bar) Potential energy (kJ/mol) Corrected potential energy (kJ/mol) Table 5-2: Simulation result of SPC in ambient condition N LR rc (nm) 216 SC 0.85 1.0 216 SC 0.9 1.0 216 SC 0.85 1.0 108 EW 265 EW 216 EW | Time (psec) 0.985 densi (g/cm ) E (U/mol T (K) 1 -42.2 300 2n 0.97 41.4 300 50 0.996 -41.8 301 MC l -38.6 300 Strauch et al. (1989) 1019 0.965 300 Alper ct al. (1989) 700 0.997 -41.8 298 300 3.6 Watanabe et al (1989) 2000 4.5 Barrat et al. (1990) 4.69 Prevost ct al. (1990) 12.5 216 EW 200 1 -41.1 300 216 EW 40 I -39.5 299 P (bar) D (10 cm'/s) eference 3.6 Berendsen et al. (1981) -0.9 4.3 Berendsen et al. (1987) 700 4.4 Telema ct al. (1987) 126 EW 0.775 800 0.992 -40.9 300 Belhadj et al. (1991) 345 EW 1.085 500 0.993 -40.9 300 Bclhadj et al. (1991) 216 SC 0.93 1.0 120 0.996 -41.3 309 4.1 Wallqvist et al (1991) 1000 SC 1.55 1.0 25 0.996 -41.91 300 4.2 Wallqvist et al (1991) 216 EW 144 0.996 -41.1 300 4.2 Wallqvist et al (1991) 216 SF 0.85 1.0 50 -41.8 300 4.6 van Belle t al. (1992) 512 SC 0.9 1.0 1000 0.98 300 28 3.9 Smith ct al. (1995) 512 RF 0.9 ca 1000 0.953 300 2 5.3 Smith t al (1994,1995) 216 SC 0.9 1.0 10 I 298 4.4 Bernardo ct al. (1994) Properties of SupercriticalWater in SPC and SPC/ESimulations _ I __ Page 6 Table 5-3: Simulation result of SPC/E in ambient condition N LR rc (nm) Erf Time (psec) density (g/cm) Em (kJ/mol T (K) )= 216 SC 216 0.9 1.0 = P (bar) 0.9 - ~ D (10 eference cm_/s) 27.5 0.998 -46.3 306 6 2.5 Berendsen et al. (1987) EW 700 0.997 .46.7 298 0 2.4 Watanabe et al (1989) 256 EW 200 300 2.6 Guissani et al. (1993) 300 2.43 Vaisman et al. (1993) 2.7 Smith et al. (1994a, 1995) 216 RF 0.92 150 1 512 RF 0.9 1.0 10009 1.002 512 RF 0.9 62.3 1000 0.976 512 RF 0.9 o 1000 0.976 -45.9 300 360 SC 0.9 1.0 400 1.0013 44.28 307.4 216 EW 6000 0.997 -46.64 298 256 EW 500 0.998 -46.72 216 EW 0.95 0.997 -46.3 2160 EW 0.85 200 0.999 303.15 512 EW 1.14 200 0.9956 303.15 256 SC 100 0.997 298 [H F X -47.0 300 -5 300 I Smith et al. (1994a, 1995) -37 3.2 Barrat ct al. (1990) 2.51 Bez et al. (1990) 2.4 Smith et al. (1994b) 298 2.24 Svishchev et al. (1994) 300.6 4.4 Heyes (1994) 69 2.75 Balasubramanian ct al (1996) -7 2.76 Balasubramanian et al (1996) 2.58 Chandra et al. (1999) -80 F ; g~~~~~~~~~~~~~~~ 5.3.2 Results and Discussion on supercritical water 5.3.2.1 Comparison between SPC and SPC/E at 773K Figure 5-2 shows the relationship between density and pressure at 773K. The pressure of SPC/E seems to be very close to the experimental data of pure water. On the other hand, the pressure of SPC is higher than the experimental values at each density. From these results, SPC/E model is found to be effective with respect to the pressure even in the supercritical region and better to describe the pressure than SPC model. SPC flexible model (Mizan et al, 1994) is also in good agreement with the experimental data. Table 5-3 shows the total internal energy. Properties qofSupercnriticalWater in SPC and SPC/E Simulations Page 7 Assuming that polarization energy is constant at any temperature and density, correlated internal energy were obtained. In the case of subcritical region, heat of vaporization provides the potential energy. In the case of supercritical region, the difference of enthalpy from the reference state can give the internal energy. The calculation of the self-diffusion coefficient was carried out by obtaining the mean square displacements of water molecules. In Figure 5-3, the relationship between time and mean square displacements at 773K was plotted for the SPC model and the SPC/E model at 773K and 0.4g/cm3 . The slope of the SPC model was explicitly steeper than that of the SPC/E model as well as at ambient conditions. Figure 5-4 shows the simulated self-diffusion coefficients of supercritical water with respect to density by SPC and SPC/E. The figure also includes the experimental data of Lamb et al (1974) and simulation results of SPC-flexible model (Mizan et al, 1994) as references. At isothermal (773K) condition, the value derived from SPC/E model is surprisingly in good accord with the experimental data. Self-diffusion coefficient by flexible-SPC model (Mizan et al., 1994) is also close to the experimental data, but the data from SPC model is about 10% higher than the experimental data. Moreover, Figure 5-5 which shows the relationship pressure and D gives the information about the reliability of SPC/E. In this way, SPC/E which is known to be effective in ambient condition is found to be also effective in the supercritical region. Then, the effects of some simulation simulated self-diffusion coefficient were investigated. parameters to the Propertiesof SupercriticalWaterin SPC and SPC/ESimulations Page 8 Table 5-4: Simulation result of SPC and SPC/E at 773K SPC/E SPC/E SPC/E SPC NVT NVT NVT NVT NVT NVT NVT NVT 773 773 773 773 773 773 773 773 0.125 0.25 0.4 0.6 0.125 0.25 0.4 0.6 Pressure (bar) 321 ±3 555 ±7 Total energy -7.2 -11.8 -15.8 -20.2 -8.8 -13.9 -18.7 -22.9 +0.4 ±0.6 +0.5 ±0.6 ±0.5 ±0.6 ±0.6 ±0.5 Ensemble type Temperature (K) Density (g/cm 3) (kJ/mol) SPC SPC SPC 900 +10 1890 ±23 285 +4 469 +6 694 +40 SPC/E 1799 + a: the value is before correction by self-polarize energy b: Including quantum correction (from Berendsen et al 1987) Effectsof temperaturecontrol In order to compare the simulated data to the experimental data, temperature is usually controlled in MD simulation. Mass stochastic collision is one of the effective methods to control temperature. In this method, the velocities of all atoms are reassigned at given frequency so that the distribution of the velocity match Boltzmann distribution at the targeted temperature. This method does not affect thermodynamic properties, but it must affect the transport properties because the velocities of atoms suddenly change. Table 5-4 shows the influence of mass stochastic collision. At 773K and 0.4g/cm 3 , the diffusion coefficient decreases by increasing collision times per time. Surprisingly, high frequent collisions eliminates the difference of the self-diffusion coefficients between SPC and SPC/E. Hence, the condition of temperature control is important for the simulation of diffusivity. Propoerties_qfSuercritical Water in SPCand SPCIE Simuulations Page 9 Table 5-4 The effect of mass stochastic collision model SPC SPC SPC SPC SPC/E SPC/E SPC/E 5 5 0.5 0.5 5 0.5 0.5 2 20 0 20 2 0 20 70 101 78 95 95 78 timstep(fs) frequency(collisions /50ps) D (cm2/s) 1 105 Effects of timstep Since diffusion coefficient is derived from the mean square displacement, timestep seems to be important. Too short timestep spends a lot of waste time, but too long timestep cannot reproduce. Especially diffusion coefficient is calculated from the mean square displacement, so relatively small timestep is required in order to trace proper trajectories. According to other literatures, timestep from 0.1 fs to 5fs are used for the diffusion simulation. Figure 5-6 shows D vs timestep at 773K and 0.25g/cm3 . Maximum 10 % deviation is generated during 0.1 to 5fs; therefore, it is important to use a constant timestep to collect reliable data. 5.3.2.2 Effect of simulation conditions Effects of cutoff length Cut off length, Rcutis considered to influence long range corrections. It is better to be as long RCt as possible, but simulation time becomes lor, and not practical. if it is too short, wrong calculation results will be obtained. In this research, se Rct = 2.50 = 0.7915nm. In order to see if this RCt is long enough, simulation data were compared to long Rcutat low density in Table 5-5 Table 5-5 The effect of cuttcutoff length to D at 673K and 0.125g/cm3 cutoff length (nm) 0.79 ,,,, 1.87 Pressure (bar) 178.8 177.2 Total energy (kJ/mol) -12 -12 Diffusion coefficient (1 05cm2/s) 226.1 221.9 When I used twice as large as Rc,, pressure and total energy were same within 1% error. D becomes slightly(2%) lower, but it is within error. As a result, Rcut=2.50. is long enough to Propertiesof Suercritical Waterin SPC and SPCIESimulations - n I I Page 10 I simulate appropriately. Spoel et al.(1998) found that the density increases on increasing the cutoff and the diffusion constant is reduced by increasing cutoff. However, I did not find significant effect. Effect of lonE range correction Table 5-6 shows the simulation results of different external dielectric constant based on Hummer's site-site reaction field method. When ERFis 80 ( close to the permittivity of ambient water) and infinity, pressure and total energy are in good agreement within 1% error. Diffusion coefficient is, however, 10% different. Figure 5-7 shows the Coulombic potential energy between oxygen atom and hydrogen atom with respect to the distance between them. When we use eRF=80and infinity, D changed 10%. Since the difference between use eRF=80and infinity with respect to potential curve is so small, it is easily predicted that eRFstrongly affects D. Due to the number of eR, it may cause larger difference compared to SPC model as you can see in Figure 5-10. Table 5-6 The effect of external dielectric permittivity Infinity 80 Pressure (bar) 178.8 181.4 Total energy (kJ/mol) -12 -11.8 Diffusion coefficient (10 5 cm2/s) 226.1 204.2 Spoel et al. (1998) investigated the effect of reaction field at ambient water. Diffusion coefficient became higher by more than 10% for SPC, SPC/E, TIP3P, and TIP4P due to the introduction of reaction field. This result is similar to my result. It is necessary to deal deliberately with the methods and parameters of long range correction. 5.3.2.3 SPC/E simulation at various temperatures As the SPC/E model produced reliable properties at 773K, it was also done over wide range of temperatures ( 625K - 873K ). Figure 5-8 shows the relationship between density and diffusion coefficient. In the supercritical region, D increases with decrease of density and D at different temperatures converges at high density. Such trends are reproduced by SPC/E model. Propertiesof SupercriticalWaterin SPC and SPC/ESimulations Page 11 Subcritical water behaves in different ways from the supercritical water. At low density, the self- diffusion coefficient in supercritical water is much larger than that in the steam even if their denisities are same. D of saturated water is also reproduced by SPC/E. For example, D is / experimentally 51cm 2 /s at 623K and 0.582g/cm 3 (Lamb et al. 1981) while D is 48cm2/s at 625K and 0.582g/cm 3 in SPC/E MD simulation. Figure 5-9 shows PpT data at 625K to 873K. Although the pressures at lower temperatures are a little less than experimental pressures, the simulation data are relatively in good agreement with the experimental data. That is, SPC/E can reproduce D in the wide range and may predict D of water at various conditions. The relationship between pressure and diffusivity is displayed in Figure 5-10. As well as the relationship between density and diffusivity, the simulation results are in good agreement with the experimental data even though the pressure has about 10% error. The figure also indicates that the radius of curvature is reduced with the decrease of temperature. In the liquid-like phase, diffusion coefficient is independent on the pressure; however, it changes steeply with the change of the pressure in the gas-like phase. In addition, it is found that the dependency of temperature on D decreases as the pressure is increased. In the supercritical region, D seems to be proportional to the pressure at constant densities in Figure 5-11. That means pD is constant. From Figure 5-1 1 and 5-12, pD is not dependent on pressure but temperature in the supercritical region. pD is proportional to temperature. Diffusion Model Analysis Generally, the self-diffusion coefficient of liquid water is around the order of 1x10'5cm2/s and that of vapor is the order of 1x10' lx10°m2 /s. Simulation results indicate that the self- diffusion coefficient of the supercritical water is the order of 1x103-1x10-4cm2/s. Probably, the diffusion behavior of the supercritical water is expected to behave like a dense gas at low pressure and like a liquid at high pressure. Stokes-Einstcin equation (Eqn.5-5 ) is known to represent the diffusion of the liquid. The hydrodynamic relationship for a particle diffusing in a medium of viscosity r is Propertiesof SupercriticalWaterin SPC and SPC/E Simulations D= kT Page 12 (5-5) C,lra~, where kB is the Boltzmann constant, a is the hydrodynamic radius and CSEis a friction constant. When the diffusing particles are much larger than those of the medium (sticking boundary limit), Cs=6 and eqn.(5-5) becomes familiar Stokes-Einstein relation. When diffusing particles of size approximately equal to those of the medium (slipping boundary limit), Cs=4. Using simulated D, experimental rl and fixed a, we can obtain CSE. Figure 5-14 shows the value of CsE. From this Figure, Stokes-Einstein relationship is apparantly neglected. On the other hand, Marcus (1999) thought effective radius,a should change in the supercritical region. He introduced the equation which expresses the effective hydodynamic diameter of water in terms of temperature and pressure. Truly, self-diffusivity is dependent on the density because it is related to the free volume. As the density increases, D should be decreased. On the other hand, temperature is also considered to be important. When we think hard sphere model, D is related to the effective collision diameter. As the temperature increases and thermal energy is increased, the effective collision diameter decreases; as a result, D should increases. However, we are not sure the degree of contribution from temperature and pressure. Marcus tried to explain the change of D only due to the change of the effective collision diameter. k 8T a- (5-6) [427 - 0.367(T(K))]+ [- 0.125 + 0.0016(T(K))](P(MPa)) Figure 5-15 shows the relationship between 2a from Stokes-Einstein eqn. where CsE=4 and 2a from eqn. (5-5). As a result, Marcus' s equation does not fit my simulation data, but it would be possibe to make optimum equation. Activation analysis 5.5 Conclusions Propertiesof SupercriticalWaterin SPC and SPC/ESimulations Page 13 By using SPC and SPC/E models, self-diffusion coefficients of water were studied. From the simulation for ambient water, SPC showed higher diffusion coefficients than the experimental data while SPC/E presented almost same diffusion coefficient as experimental one. 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Simul., 1995, 15, 233 Smith, P.E.; van Schaik, R.C.; Szyperski, T.; Wuthrich, K.; van Gunsteren, W.F., Internal mobility of the basic pancreatic trypsin-inhibitor in solution-a comparison of NMR spin relaxation measurements and molecular-dynamics simulations. J. Mol. Biol., 1995, 246, 356 Strauch, H. J.; Cummings, P. T., Mol. Simul., 1989, 2, 89 Svishchev, I.M.; Kusalik, P.G., Dynamics in liquid H2 0, D2 0, and T 20 - A comparative simulation study. J. Phys. Chem., 1994, 98, 728 Teleman, O.; Jnsson, B.; Engstrom, , Mol. Phys., 1987, 60, 193 van Belle, D.; Froeyen, M.; Lippens, G.; Wodak, S. J., Molecular-dynamics simulation of polarizable water by an extended lagrangian method. Mol. Phys., 1992, 77, 239 Vaisman, I.I.; Perera, L.; Berkowitz, M.L., Mobility of stretched water. J. Chem. Phys., 1993, 98, 9859 Wallen, S. L.; Palmer, B. J.; Fulton, J. L., J. Chem. Phys., 1998, 108, 4039 Watanabe, K.; Klein, M. L., Chem. Phys., 1989, 131, 157 Walqvist, A.; Teleman, O., Mol. Phys., 1991, 74, 515 Whalley, E., Chem. Phys. Lett., 1978, 53, 449 Propertiesof SupercriticalWaterin SPC and SPC/ESimulations __ L I _ L ____ _____I _ 300K, 0.997g/cm3 7 + SPCE 6 N 2 C 5 E y =0.0243x+ 0.0625 R 0 9993 3 SPC''' -Linear (SPCE) (SPC) -Lnear In (t E 2 4 E U CT I. 3 +0.0418 =0.0152x~y _ 'or E ~0 , !,.,,.,~~~~~~~~~~R {a C o0 0.9988 o Lu 2 S .. e ,,~ 2 ........ 1 1 0 II 0 50 100 150 200 250 300 time (psec) Figure 5-1 mean square displacements of water molecules by SPC and SPC/E at ambient conditions Page 17 Properties of Supercritical Waterin SPC and SPC/ESimulations _ _ __ I I Page 18 ^'^^ 2500 2000 Z,~ 1500 J L- U) 0u o. 1000 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Density (g/cm3) Figure 5-2 The relationship between pressure and density at 773K (solid line: experimental data (NIST)) Properties of Supercritical C · L _ Water _ U in SPC and L SPC/E - -Simulations - -s - Page I 19 773K --- vuU 800 C 700 E C 600 a) E a) O co * 0.125g/cm3 * 0.4g/cm3 500 .e -ULinear (0.125g/cm3) Linear (0.4g/cm3) a) 400 X v' 300 0 E E 200 100 0 0 100 200 300 400 500 600 time (ps) Figure 5-3 mean square displacements of water molecules at 773K Propertiesof SupercriticalWaterin SPC and SPC/ESimulations _ _ _ _ _ L P'age 20 _ Isothermal (773K) nrrr JOU 300 iZ 250 E u T 200 u 10 0 150 0o u, J2 100 50 0 0 0.1 0.2 0.3 0.4 Density (g/cm3) Figure 5-4 Diffusion coefficient vs density at 773K 0.5 0.6 0.7 Properties Simulations -of- SupercriticalWater..in ..SPC I iiand -- SPC/E -·I I - - I 21 - - Page I 350 300 - * SPC-flex(Mizan et al) *0 ~~SPC *0~ - 250 E 0 o .2 - *SPCE O Experimental(Lamb et al) . O 200 O 0 0 0o 150 0 .,'6 100ID O 50 e - 0 0 200 400 600 800 1000 1200 1400 Pressure (bar) Figure 5-5 Diffusion coefficient vs pressure at 773K 1600 1800 2000 of Supercritical Properties _ __ __Water in SPC and SPC/E __ Simulations I I Page 22 160 4* o 773K, .25g/cm3 . 140 - E 120 vrlu 100 c 0) ._ 0 0 80 60 ._0 40 ._ 20 0 1.OE-04 1.OE-03 1.OE-02 timestep (ps) Figure 5-6 Diffusion coefficient vs timestep at 773K and 0.25g/cm3 Properties __of SupercriticalWater in SPC andI SPC/ESimulations IC __ 03 0.35 0.4 0.45 0.5 0.55 0.6, , Page L 23 __ ,%(.7 0 13 -10 -20 02o 02 0 -30 0 S I 02 03 -40 02 -r 0 -a 0 0 -50 0 o0 -60 0o C- tD -70 O Couipmb(SPC)-Hummer Coulpmb(SPC/E)-Hummerinfinity ACoulpmb(SPC/E)-Hummer80 a Coulpmb(SPC/E)-Hummer1 0 E Coulpmb(SPC/E)-Hummer * -80 -90 a~~ O . -100 r (A) Figure 5-7 Coulombicpotential between H and O in SPC/E model with and without Hummer's Reaction Field Properties of __ Supercritical _ ___ Waterin __ SPC and SPC/E Simulations Page 24 AAA 4UU 350 300 E .o 250 c 0@ :: 200 0o u 150 *° :3= 100 50 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Density (glcm3) Figure 5-8 Diffusion coefficient vs density at 625K to 873K 0.8 SPC and SPC/ESimulations Properties of Supercritical Water __ __ I in----II Pag J- 25 loo0 900 800 700 600 . 2 500 40 400 300 200 100 0 0 0.1 Figure 5-9 0.2 0.3 0.4 0.5 Density(g/cm3) 0.6 0.7 0.8 0.9 Pressure -density-Temperature data at various conditions of Waterin SPC and Properties · _· Supercritical ·_ II_SPC/E Simulations _______ -- 400 r=0.125g/cm3 350 -c- 673K(experimentol) 773K(experimentol) -oX * * + A * 300 i 250 r=0.25g/cm3 873K(experimentol) 625K 652K 673K 723K 773K 873K ; 200 c150 c 150 r=0.4g/cm3 r=0.6g/cm3 100 50- 0 0 I 500 r j 1500 1000 r 2000 2500 Pressure (bar) Figure 5-10 Diffusion coefficient vs pressure at 625K to 873K Page 26 II Propertiesof SupercriticalWaterin SPC and SPC/ESimulations ____ L__ L Page 27 Ar tJU 400 350 300 E -250 . _ § 200 -5 150 100 50 0 0 2 4 6 1/density(cm3/g) Figure 5- 11 D vs /p 8 10 12 Propertiesof SupercriticalWaterin SPC and SPC/ESimulations Figuie 5-12 pD vs temperature Page 28 Propertiesof Supercritical Waterin -· -I SPC and SPC/ESimulations --- 45 Page 29 I -- . 40 A A 35 - 30- A [ 25 E C. X 625K 20 - * 652K * 673K 15 - + 723K A 773K 10- · 873K 5 0 -1 0 500 1000 Pressure(bar) Figure 5- 13 pD vs pressure 1500 2000 2500 Properties of Supercritical Waterin SPC and SPC/E Simulations __ ___ L ___ Page __ 30 7 Sticking limit 6 0 5 S Slipping limit 4 A u C.) - 3 -- _ _ O____ - -_ -_ __- -_ - _ _-_-_ -- __-- _ _-- _-- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - -. 673 K 2 --- 773K * 1 _ - -_ _ _ _ _ _ - - -_ _ _ 0.1 0.2 -_ -- - __- - __- - _ _ _ - - _ - -_ - -__ 873K - -__ - - _ _ _ 0 0 0.3 0.4 Density (g/cm3) Figure 5- 14 CSE vs density 0.5 0.6 0.7 0.8 Propertiesof SupercriticalWaterin SPC and SPC/ESimulations _ _ __ __ __ L I __ ___ Ann qU 350 E 300 LI LO a ' 250 E0 0 = 200 150 150 100 100 120 140 160 180 200 220 diameterfrom eqn.(5-6) Figure 5- 15 240 260 280 300 Page 31I _ Propertiesof SupercriticalWater in SPC and SPC/E Simulations - · ·· .·I -- - Page 32 Y I Activation energy 7 6 * InD(0.125) y = -1097x + 7.01 R2 -0 9311 y = -1i/8.9x + .6. R 2 = 0.8867 o InD(0.25) .125g/cm3 * InD(0.35) 5 ... y = -967.46x + 5.753 U) 2 R = 0.9618 0..25g/cm3 9-._3 0 _u/cm A InD(0.4) - E 4 0 InD(O.582) y = -1124.9x + 5.5143 3 0.6g/cm3 a InD(0.6) R2 = 0.9989 C -Linear 2 - (InD(0.125)) Linear (InD(0.25)) -Linear (InD(0.4)) -Linear (InD(0.6)) 1 _ 0 0 0.0008 0.0009 0 0 0 0.001 0.0011 0.0012 0.0013 0.0014 0.0015 0.0016 0.0017 0 1/T(K71 ) Figure 5-15 0 0 0 0 Future Workand a Recommendation Ret e dI ionI n Fuur Work I IaI Page 11 CHAPTER 9 FUTURE WORK AND RECOMMENDATION For the analysis of supercritical water, this research should be extended as follows. 9.1. Analysis of the vicinity of the critical point When we consider the correlation length near critical region and the behavior of phase splitting, the large size of a unit cell is essential. For example, spinodal decomposition was observed in large size of a cell ( ) and Reagan et al.(1999) found the solid- supercritical phase splitting in the solution by increasing the number of water molecules from 256 to 838. From Figure 9 - 1, The increase of the number of molecules 9.2. Improvement of Simulation Model for Water As I discussed in Chapter 4, there exist lots of models for simulation of water. Therefore, constructing completely new effective potential model is ineffective. Introducing flexibility and polarizability is one method to improve the model. With the viewpoint of application to the supercitical region, the introduction of polarizability seems to be important. In fact, the Coulomb potential is a dominant term in total internal energy and SPC/E which took the polariability into account drastically improved the transport properties compared to SPC. Because of difficulty in making a universal polarizable model, we have no satisfactory polarizable model.yet; however, Matsubayashi et al.( way where they set a different dipole moment based on at but successful for ) proposed simulated in a new . This simulation was on the way Future Work and Recommendation Page 2 When we think of the supercritical water, The newly water model. Slightly changeThe final goal of this study is to understand the behavior of Diffusion of gases 9.3. The difffusion in binary mixture In addition, this simulation method can be applied to the binary mixture. Experimental data of transport properties in binary mixtures in supercritical region are also limited like that of supercritical water. In the case of dilute solution, NMR and Taylor dispersion can be used. While NMR requires Taylor dispersion ( , The diffusivity of solute are analyzed by ) In concentated solution, 9.4. Additional Experiment The diffusion data in supercritical water is limited. Therefore, In fact, .only the data of Lamb et al. (1981) NMR imaging Simulation If we have intermolecular potential Model can be used. Critical point Binary Diffusivities for Liquids at Infinite Dilution ( Deen, 199X) ul 6 (1-1) Future Work and Recommendation Page 3 k T KS(1-1) k=K h K,c 1 (MRT) 33 3/2 1 DAB 1 2 (1-1) Nd 2 (RT) 3 /2 37C3/2 NAvd 2 M1 2 (1-1) P Diffusivities for liquids Stokes-Einstein model DAB= kBT 6nrlr A The design and operation of high pressure reactors are extremely important for obtaining accurate rate and selectivity information from reactions performed in supercritical fluids. Reactions in supercritical fluid systems are often highly nonideal and phase behavior can have dramatic effects on the course of reactions. Unknown or unverified phase behavior should be avoided. Visual monitoring of phase behavior is an easy and effective method of confirming phase behavior. It is difficult to properly interpret rate and selectivity data obtained from environment in which the phase behavior is unknown. Interestingly, mixing effects arc often neglected in these systems. It might be difficult to design a mixing system for high pressure operation, but mixing can have dramatic effects on reactions, especially on diffusion controlled and heterogeneous reactions. All reactions performed in these studies were agitated. The rate of mixing was high enough so that an increase in mixing would not alter Future Work and Recommendation Page 4 the results obtained. In addition, sampling from high pressure systems must be done with care. It is important to design a system which can accurately remove a high pressure sample and then depressurize it without the loss of material. Of course, in situ techniques would eliminate many of the problems associated with sampling from supercritical fluids. Recently Balbuena et al.(1998) began to study aqueous ion transport properties by molecular dynamic simulation Experimental work Nakahara et al investigated the diffusion of water in organic solvents. Nakahara, M.; Wakai, C., Monomeric and Cluster States of Water Molecules in Organic Solvent. Chem.Lett. 1992, 80 Roberts et al. ( ) studied the ambient properties of aqueous solutions by using SPC, TIP4P and MCY water. They used 200 water molecules and 1 ion which corresponds to 0.5mol%. In the case of solution, we have to define the additional intermolecular potentials, which are pair potential of solute-solute and solute-water. This should be Roberts, J.E.; Schnitker, J., Boundary Conditions in Simulations of Aqueous Ionic Solutions: A Systematic Study. J.Phys.Chem, 1995, 99, 1322 Taylor dispersion Future Work and Recommendation Simuation Ion Organic solute Dilute solutions Concentrated solutions Polarizable model was effective for water-C02 binary. Page 5 Future Work and Recommendation Page 6 References Balbuena, P.B.; Johnston, K.P.; Rossky, P.J.; Hyun, J-K, Aqueous Ion Transport Properties and Water Reorientation Dynamics from Ambient to Supercritical Conditions, J.Phys.Chem. B 1998,102,3806 Clifford, A. A.; Coley, S. E., Diffusion of a solute in dilute solution in a supercritical fluid. Proc. R. Soc. Lond. A, 1991, 433, 63 Jost, S.; Bar, N-K.; Fritzsche, S.; Haberlandt, R.; Karger, J., Diffusion of a mixture of methane and xenon in silicalite: A Molecular Dynamics study and pulse field gradient nuvlear magnetic resonance experiments. J. Phys. Chem. B, 1998, 102, 6375 Koneshan, S.; Rasaiah, J. C.; Lynden-Bell, R. M.; Lee, S. H., Solvent structure, dynamics, and ion mobility in aqueous solutions at 25C. J. Phys. Chem. B, 1998, 102, 4193 Lee, S. H.; Cummings, P. T., Molecular dynamics simulation of limiting conductances for LiCI, NaBr, and CsBr in supercritical water. J. Chem. Phys., 2000, 112, 864 Estimationof the criticalpoint in SPC/Ewater Page 1 CHAPTER 6 ESTIMATION OF THE CRITICAL POINT OF SPC/E WATER 6.1 Objectives Main objective of the work presented in his chapter is to estimate the critical point of SPC/E water. When one accurately evaluates the properties from the simulation results, one should use the reduced properties. For example, in a simple Lennard-Jones fluid, the properties reduced by Lennard-Jones parameters, E and ca, are often used as follows. T*=T/ke (6-1) p*=pa 3 (6-2) Generally speaking, it is convenient for the critical point to be used for reduced parameters as below. T*=T/Tc (6-3) P*=P/Pc (6-4) In addition, the critical point of a simulation model is essential when one analyzes the behavior in the vicinity of the critical point because the small change of temperature and pressure from the critical point makes the properties of water change drastically like in Figures 1-1 and 1-2. As there is no perfect simulation model in water currently, the critical point of the model is considered to be different form that of real water. In the complex water model like SPC/E water which includes both Lennard-Jones interaction and Coulomb interaction, however, researchers seldom use the reduced properties even if the critical point of models are found to be far from the real water's because they have to obtain the critical point directly from the simulation results. As far as the ambient water is analyzed, the influence of the critical point seems to be small, but it will be large for the analysis of near and supercritical water. Estimationof the criticalpoint in SPC/E water Page 2 The critical points for some models have been already calculated, but there are too few data to discuss the influence of simulation technique and the methods of calculation. In fact, the different critical points are obtained even in the same water model by different researchers (Guissani et al., 1993; Alejandre et al.,1995 ). Therefore, the further information is necessary. In this chapter, I introduce some methods to estimate the critical point and calculate the critical point of SPC/E water based on my simulation technique to help to understand the results in Chapter 5 and to explore the self-diffusivity near critical point in Chapter 7. 6.2 Types of Methods to Estimate the Critical Point The critical point is calculated through two steps. First, the saturated densities of gas and liquid at a given temperature are calculated. Then, the critical point is estimated via interpolation of the coexistence curve which is plotted along the orthobaric densities of gas and liquid. I briefly introduce three general methods to calculate densities of gas and liquid below. 6.2.1 Gibbs Ensemble Monte Carlo Simulation (GEM1C) One method is the Gibbs Ensemble Monte Carlo Simulation (GEMC). In this method, two simulation boxes are prepared. These two boxes have different densities and compositions and are at thermodynamic equilibrium both internally and with each other. The Monte Carlo technique uses three types of move. One is ()independent particle displacements in each box which are made using the normal Metropolis algorithm. Another is (2) a combined attempted volume-move in which the volume of one box changes by AV while the volume of the other box changes by -AV. The other is (3) a combined attempted creation/destruction-move in which a randomly chosen particle is extracted from one box and placed at random in the other box. The chemical potential in the two boxes is equal but its precise value is not required. ( see Figure 6-1) Estimationof the criticalpoint in SPC/Ewater Page 3 The advantage of this method is the speed with which the coexisting phases are established from an initial configuration in the two-phase region. In addition, the properties of saturated gas and liquid can be obtained respectively from each simulation box. The main disadvantage of this method is the difficulty in the molecular insertions at the high density close to triple point. Nor can this method present information about the interface. 6.2.2 Calculation of Equation of States The second method is the direct method. The simulations at many states are done and the equation of state is directly obtained from the results. For example, Ree(1980) used 99 states and Kataoka(1989) used 367 states to obtain the equation of state of their water models. In Ree's method, the following equation was used as an equation of state: p=1 + Blx + B2x 2 + B3 x3 + B4 x 4 + Bl10 x pkT -(-) T* (65) (6-5) x(Clx +2C 2x2 +3C 3x3 +4C 4x 4 + 5C5 x5 ) where x is p*/(T*)" 4 and where reduced temperature, T*=T/T, and reduced density, p*= P/Pc are expressed. All parameters are determined by least square approximation. From the fitted EOS, the orthobaric densities of gas and liquid at given temperatures and the critical point is estimated. The advantage of this method is that the results are accurate; however, many states have to be simulated in order to fit EOS. Guissani and Guillot(1993) proposed that 60 states are sufficient to obtain EOS. In this method, the simulation results at the subcritical water is important. Van der Waals type EOS can be fitted if the phase is homogeneous, but it cannot be fitted when the phase splitting is observed in the large unit system. Figure 6-2 shows the example in which EOS was fitted by Guissani and Guillot(1993). Estimationof the criticalpoint in SPC/Ewater Page 4 6.2.3 Direct MD Simulation of Two Coexisting Phases The third method is the direct simulation of two coexisting phases. In this method, the box of a liquid phase is first made. The two vacuum boxes are set next to both sides of the liquid cell in one direction (Figure 6-3). MD or MC simulation is carried out at a given temperature, and particles in a liquid phase diffuse in the vacuum phase, then finally gas/liquid/gqs coexisting phases are produced at equilibrium. After ensemble averaged density profile is measured, minimal flat density is assigned to the density of gas and maximal flat density is assigned to the density of liquid (Figure 6-4). These detail procedures are described in next section. 6.3 Simulation Procedure 6.3.1 Measurement of the density profile In this research, the approach of direct simulation of two coexisting phases by MD simulation was employed to determine the saturation densities of water because diffusion coefficient was simulated by MD. The MD parameters were almost same as that in Chapter 5 except initial configuration. The simulation system was an NVT ensemble in a rectangular box of dimension L, Ly, and LI, as illustrated in Figure 6-5, with periodic boundary conditions in all directions. L and LI are fixed at 1.97nm which is the length of a unit cubic cell at 0.997g/cm3 for 256 water molecules. 1.97nm is a reasonable value for the periodic boundary condition as explained in Chapter 5. LXis fixed at 5nm which corresponds to the total density, 0.39g/cm3 in a unit cell, a little higher than the critical density of the real water. In order to make two phase equilibrium, total density should be slightly higher than the critical density which is in the range of 0.3 - 0.4g/cm3 . At the beginning of runs, I made a cubic unit cell which was equilibrated at 300K and 0.997g/cm3 and this configuration of water molecules was set in the center of the rectangular box. I attempted to use 256 molecules because simulations for self-difffusivity measurement are carried out using 256 molecules. 512 molecules were also used to check the effect of the Estimationof the criticalpoint in SPC/Ewater . Page 5 number of molecules. Two types of timestep were used. One is 0.5fs which was the same timestep used for the simulation to measure self-diffusivity. The other is 5fs which can achieve longer production time at the same production cycles. 100,000 timesteps corresponding to 50 ps for 0.5fs timestep and 500ps for 5fs timestep were spent for equilibration and same timesteps were used for production runs. The simulated configurations were collected every 25 timesteps. The density profile of water was obtained by averaging 4000 configurations. The density profile is assumed to show the hyperbolic function as follows. P P+2 - 2 )tanXd ) (6-6) where PLis the density of liquid, Pv is the density of gas, xo is position of the Gibbs' dividing surface and d is parameter for the thickness of the liquid-vapor interface. This d is a measure of the thickness of the interface defined by the following equation: (Toxvaerd et al., 1975) )z dp(z d=- (PL - PV,[, d--jz =z0 (6-7) The origin of the density profile in each configuration was adjusted by calculating the center of mass and all collected density profiles were summed and averaged by the number of configurations. The simulated density profile was fitted by eqn.(6-2) using non-linear least square method. Fitting results gave the orthobaric densities of gas and liquid at given temperatures. By plotting the relationship between orthobaric densities and temperaturse, the coexistence curve ( i.e., binodal curve) of pure water was obtained. 6.3.2 Estimation Methods of a Critical Point There are some methods to estimate of the critical point from the coexistence curve. One of the most popular method is the application of the scaling law. A lot of research has Estimationof the critical point in SPC/Ewater Page 6 been theoretically and experimentally done on the scaling law near the critical point ( for example, Levelt Sengers, 1985; Sengers, 1985; Levelt Sengers, 1993). In the scaling law, temperature, T, pressure, P ,and density of saturated vapor, Pv, and density of saturated liquid, PL, are described as follows. P2. Pv (6-8) CT TT T = C (PL - P )"A + TC (6-9) where f3is the scaling exponent, Tc is the critical temperature and Pc is the critical density. C and C1 are constants. The scaling exponent, 3, is given theoretically and experimentally. Classical theory shows that =0.5, but Ising model suggests 13=0.325. T, PL, and Pv are known from simulation results and 3 from the literature, so T is easily provided from eqn.(6-9) by non-linear least square method. Then, by using the following rectilinear diameter method: T-TC2 PL+PV P) (6-10) Pc is easily calculated. Finally, we obtain the critical temperature and the critical density. 6.3.3 Review on the critical points of simulation models Although many models for water exist, the critical point of each model has not been enough researched. In Table 6-1, reported data on the critical points are described for each water model. Some models such SPC and RPOL models show very different critical points compared to the real water. The critical point of SPC water which has been widely used was reported by de Pablo et al. (1990). Only is his data available up to now and no one has validated it. Furthermore, the critical points of many models have not been explored yet. Due to the lack of reports on the critical point, the influence of simulation technique including long range correction methods to the critical points has not been elucidated, either. Estimationof the critical oinzti SPCIEwater Page 7 For SPC/E model, Guissani and Guillot (1993), Alejandre et al (1995) and Errington et al.(1998) reported the critical points by using three different methods. Table 6-2 shows the reported values of critical points of SPC/E water. The critical point of SPC/E are relatively close to that of real water; however, all simulated critical points are slightly different. Even the same data make different critical points due to the difference of estimation methods. For example, (2) and (3) in Table 6-2 originated from same data but only method to estimate the critical point was different. (4) and (5) in Table 6-2 originated from same data and same direct simulation method was used to estimate the critical point. Liew et al.(1998) used Alejandre's data and calculated the critical point using scaling exponent, 3=0.325 in (4) while Alejandre et al. used 3=0.33 in (5). Table 6-1 Critical point of various water models model Tc pc SPC (K) 587 (g/cm3) (bar) 0.27 196 SPCG Pc references de Pablo et al. 1990 606 0.27 - Strauch et al. 1992 SPC-ZW 710.5 0.29 - Liew et al 1998a SPC-mTR 643.3 0.32 - Liew et al 1998a TIP3P-mTR 593 0.288 - Liew et al 1998a cm4P-mTR 641.4 0.307 180 Liew et al. 1998b RPOL 561 0.3 331 CC 603 0.29032 280 Kataoka 1987 Estimation the ctical point in SPCIEwater Page 8 Table 6-2: Critical Point of SPC/E Water ___ _ _ _ Auther _ _ _ Tc (K) PC (g/cm ) Pc(bar) Estimation Method (3) jI I (1) (2) Errington 11(1998) Guissani Guissani Alejandre Liew Real (1993) 651.7 (1993) 640 (1995) 636.5 (1998a) 630.4 water 647.17 0.262 . 0.326 189 0.29 160 0.303 . 0.308 . 0.322 221 GEMC EOS EOS DS DS 1 639 I (4) I (5) (6) j 6.4 Simulation Results and DIscussion 6.4.1 Density profile The effect of temperature Figure 6-6 shows the density profile at 0.5fs timestep from 1000 C to 300 0 C. In this figure, density profiles are folded at the center of a unit cell and averaged. As the temperature increases, the density of liquid phase decreases and that of gas phase increases. In addition, the width of interface increases with increasing temperature. At low temperature, the deviation of the density in the liquid phase is larger than the deviation at high temperature. This deviation depends on the degree of the displacement of the water molecule. As the diffusion coefficient of saturated water at 100°C is about 5 cm 2/s 10x10- (Krinicki et al, 1978), during total production time, 50ps, each molecule moves only 0.1 nm. Therefore, even averaged profile has deviations. When temperature is increased, diffusivity is also enhanced, then profile is a little smoother. Figure 6-7 shows the density profile at 5fs timestep from 100°C to 3000 C. The relationship between profiles and temperatures is almost same as that of Figure 6-6. In order to get coexistence curve, we should obtain as high temperature data as possible. However, at high temperature, it is difficult to distinguish between gas phase and liquid phase and Estimationof the critical point in SPC/E water Page 9 sometimes initial liquid phase generates a bubble inside the phase like in Figure 6-8. Therefore, 300°C seems to be a maximum limit in this direct simulation method. The effect of timesteps Figure 6-9 shows the density profile of different timesteps, 0.5fs and 5fs at 1000C. Although the number of configurations utilized for averaging at 5fs is equal to that at 0.5fs timestep, the deviation becomes smaller due to 10 times longer total production time. Obtained isobaric densities and the widths of interface are close to the case at 0.5fs timestep. So, the results of orthobaric densities are independent of the length of timestep. The effect of number of molecules In Figure 6-10, the density profiles at 250C were depicted. In (a), a unit cell was filled with 256 molecules and in (b) it was filled with 512 molecule. The unit cell of (b) consists of Lx=8nm, Ly=2.7nm and Lz=2.7nm. Initially, liquid water was equilibrated in a 2.7nmx2.7nmx2.7nm cube corresponding to 0.997g/cm3 . In (b), both of the lengths of gas phase and liquid phase become larger and flatter than that in (a). Although the values theselves of orthobaric densities were same in (a) and (b), a large number of molecules is easy to grasp the orthobaric densities. Chapela et al. ( ) used the density profile method for the Lennard-Jones fluid by using MD and MC in order to analyze the interface between gas and liquid. They used 255, 1020, 4080 molecules, respectively. They also found that the simulation results do not depend on the number of molecules as well as my result. The importance of the number of molecules is related to the reliability of the liquid phase. When the bulk of liquid phase are constructed, the minimum thickness is necessary to maintain the liquid phase. If the second nearest neighbor molecules are required for the liquid phase, the thickness of the liquid phase should be larger than lnm because the second nearest neighbor molecules in liquid water are usually located around 0.5nm from the center of a target particle. In order to keep the thicker liquid phase, the number of molecules should be large as far as the calculation time is reasonable. Estimzation Es n the theI citical point IIf pointIinIaSPCIE SIII E water aIe I I I I Iw Page 10 10 The effect of cell size (the length of x direction) Since the periodical boundary condition were carried out, the minimum length of a cell has to be larger than 2Rcut to avoid calculation error. In this research, I have used Rcut= 2.5 a = 0.7915nm; therefore, the minimum length is required to be > 1.6nm. Figure 6-10 also shows the density profile of 512 molecules. However, the unit cell consists of Lx=lOnm, Ly=1.97nm, Lz=1.97nm and liquid water was initially equilibrated at Lx=3.94nm, Lx=1.97nm and Lz=1.97nm. In order to keep thick liquid layer, this configuration was applied. The orthobaricdensities at high temperature At high temperature, the amount of vapor layer and interface layer increases while the liquid layer decreases. As a result, the density of liquid might be underestimated because molecules are supplied for the gas and interface and the number of molecules in liquid phase does not reach the required number. Or the thickness of layer might not achieve the thickness required to keep a liquid phase. According to the radial distribution function of water, at least lnm thickness is necessary to include the second nearest neighbor molecules. One method to overcome this shortcoming is to enlarge the initial thickness of the liquid phase in x direction. Due to the minimum length of a unit cell which has to be longer than 2.5c x 2=1.6nm, it is difficult to change the initial configuration in the case of 256 molecules keeping a total density. Hence, the combination of the increase of the number of molecules and the increase of Lx should be a good solution. Figure 6-11 shows the density profiles of different x lengths. In both (a) and (b), Ly =Lz=1.97nm. In (a), Lx=7nm and 256 molecules, and In (b), Lx=lOnm and 512 molecules. By extending only Lx with the increase of the number of molecules, the thickness of liquid was increased by the ratio 512/256=2. The other method to enlarge the thickness of a liquid phase is to decrease the layer of a gas phase. By shortening the length of x-axis of a unit cell, final thickness of a liquid phase is expected to be large. Figure 6-12 shows the density profile at Lx=Snm corresponding to the Estimationoqfthe criticalpoint in SPC/Ewater Page 11 density (0.55g/cm3 ). As a result, the density of a liquid phase slightly increased, but a gas phase was not be able to be generated. In this situation, the cell includes only a liquid phase and the interface, so one does not call the situation the coexistence. Hence, the density of liquid which slightly increased cannot be defined as an orthobaric density. In summary, it is difficult to optimize the size of a unit cell to get reasonable two phases' densities keeping small number of molecules. 6.4.2 Coexistence curve Table 6-3 shows the result of the orthobaric densities of the gas and liquid at different temperatures. The data are plotted in Figure 6-13 where some simulation data from other literatures are included. In this research, the orthobaric density of gas was a little higher than that of real steam and the orthobaric density of liquid was a little smaller than that of real water. In addition, the results indicate that this SPC/E model is better fitted to real water than SPC model, but is poorer compared to Alejandre et al.'s SPC/E and SPC/mTR. In comparison with Alejandre et al.'s SPC/E model, the simulation method is almost same except long correction method. They used Ewald summation as a long range Coulombic interaction and I used Hummer's site-site reaction field method. Therefore, this difference caused the difference of coexistence curve. In fact, Alejandre et al. found that the critical point changed by changing the parameters of Ewald summation. Estimation wae [ [e the critical critical point pi in i SPCIE SPC/Ewater Estimationof of the Table 6-3: Orthobaric densities of SPC/E model ___I T 0.5fs 5fs Pv PL PL e x p (g/cm 3 ) PL- Pv- pVexP (g/cm3 ) (K) 373 (g/cm 3) (g/cm 3) 0.9212 0.00173 -0.037 0.0011 423 0.869 0.00208 -0.048 0.0004 473 0.7836 0.00742 -0.082 0.0005 523 0.7238 0.02954 -0.075 0.0070 573 0.5699 0.06533 -0.198 0.0233 373 0.9193 -0.00053 -0.069 -0.0011 473 0.7865 0.00959 -0.078 0.0018 548 0.6307 0.06687 -0.128 0.0208 I Page 12 12 _ ___ _1 d re (nm) II 6.4.3 Estimation of the Critical Point By using eqns (6-9) and (6-10) at =0.325, the critical point was obtained where Tc=616K and pc=0.308g/cm3 (Figure 6-14). Using these values of the critical point, reduced pressure-density-tern:ierature relationship was described in Figure 6-15. Here, I used the critical pressure, 177bar, which was directly obtained from NVT ensemble simulation at T=616K, p=0.308g/cm3. The reduced simulated data were extremely different from those of experimental data. This result means that the calculated critical point is supposed to be much far from the nominal critical point of this model. This difference was considered to be caused due to the selection of the scaling exponent. =--0.325is actually effe,- ive at the temperatures very close to the critical temperature, usually in the range of T-Tc<lmK ( Sengers, 1985). In the direct simulation method of two coexisting phases, Liew et al. (1998) and Alejandre et al. (1995) used 13=0.325and =0.33, respectively, but these selections have a problem because the temperature where the orthobatic densities are employed in this method is at most 573K ( T-Tc-70K). In order to interpolate the coexistence curve, another effective scaling exponent has to be utilized. Estimationof the criticalpoint in SPCIEwater Page 13 Therefore, I attempted a different method to estimate the critical point. Assuming that the rectilinear diameter approximation is correct, I calculated the critical density, Pc, from the simulated orthobaric densities and the specified critical temperature, Tc, using eqn.(6-11). The results in which the specified critical temperature was increased by every 5K from 616K were described in Table 6-4. j3 was calculated via nonlinear least square approximation in eqn. (610). In Table 6-4, B1 shows the result of fitting data at all densities at 0.5fs timestep and 32 shows the result of fitting data except the data at 573K. Figure 6-16 shows the fitting coexistence curve at 646K. At each point, the critical pressure was directly simulated. Comparing reduced properties based on the obtained various critical points with the reduced properties of real water, the most reliable critical point is specified. Reduced pressure-density-temperature relationship from Tc=626K to Tc=646K were depicted in Figure 6-17 to Figure 6-21. Figure 6-21 shows the coexistent curve of pure water at Tc=646K and pc=0.2895g/cm3 . The reduced parameters in this Figure are in good agreement with the reduced real water. Therefore, I set these values as the critical point of this SPC/E model. In this case, the critical component becomes 0.4 which is between 0.325 and 0.5. Table 6-4: Critical densities of SPC/E model via rectilinear diameter method Tc 1 Pc " 3) Pc2 (g/cm3 32 ) (K) (g/cm 616 0.3088 0.319 0.3258 0.311 621 0.3056 0.333 0.3231 0.322 626 0.3023 0.347 0.3205 0.334 631 0.2991 0.361 0.3178 0.345 636 641 0.2959 0.2927 0.375 0.390 0.3151 0.3124 0.357 0.368 646 0.2895 0.404 0.3097 0.38 651 0.2863 0.3070 f3 Estimationof the critical point in SPC/Ewater Page 14 The validity of the asymptotic power laws is, however, restricted to a very small region near the critical point. An approach to deal with the nonasymptotic behavior of fluids including the crossover from Ising behavior in the immediate vicinity of the critical point to classical behavior far away from the critical point has been developed by Chen et al.(1990a, b). They used 0.5 as . They obtained the critical temperature by fitting the simulation data to this expansion, and a rectilinear diameter extrapolation for this critical temperature yielded a critical density. 6.5 Conclusions In conclusion, the critical point of water in SPC/E was explored by using direct simulation method of two coexisting phases. Although the orthobaric densities of SPC/E model were better fitted to the experimental data than those of SPC model; however, a slightly poorer than the other data of the literature (Alejandre et al., 1995). Since the main difference in simulation technique is only the method of long range correction of Coulomb interaction, that should be the origin of the difference. Although the critical point was calculated by using the scaling component like other literatures, the obtained value (Tc=616K, pc=0.308g/cm3 ) was evaluated far from the nominal critical point of this model compared with the reduced properties of real water. Combining the rectilinear diameter method and the evaluation of reduced pressure-reduced density-reduced temperature relationship, another critical point (Tc=646K, pc=0.290g/cm3 ) was obtained. This value produces good agreement with the reduced properties of real water and is very close to the critical point of SPC/E model by Guissani and Guillot (1993). Estimationof the critical point in SPC/Ewater Page 15 References Abraham, F. F., On the thermodynamics, structure and phase stability of the nonuniform fluid state. Phys.Rep., 1979, 53, 93 Alejandre, J.; Tildesley, D. J., Molecular dynamics simulation of the orthobaric densities and surface tension of water. J. Chem. Phys., 1995, 102(11), 4574 Bejan, A., in Advanced Engineering Thermodynamics ( John Wiley & Sons) p.299-338 Burstyn, H. C.; Sengers, J. V.; Bhattacharjee, J. K.; Ferrel, R. A., Dynamic scaling function for critical fluctuations in classical fluids. Phys. Rev .A , 1983, 28(3), 1567 Chapela, G. A.; Saville, G.; Thompson, S. M.; Rowlinson, J. S., Computer Simulation of a Gas-Liquid Surface Partl. Faraday Disc .Chem. Soc., 1977, 1133 Chapela, G. A.; Martinez-Casas, S. E.; Varea, Square well orthobaric densities via spinodal decomposition. J. .Chem. Phys., 1987, 86, 5683 De Pablo, J.J.; Prausnitz, J.M.; Strauch, H.J.; Cummings, P.T., Molecular simulation of water along the liquid-vapor coexistence curve from 25C to the critical point. J. Chem. Phys. 1990, 93, 7355 Errington, J.R.; Kiyohara, K.; Gubbins, K.E.; Panagiotopoulos, A.Z., Monte Carlo simulation of high-pressure phase equilibria in aqueous systems. Fluid Phase Equil., 1998, 150151, 33 Guissani, Y.; Guillot, B., A computer simulation study of the liquid-vapor coexistence curve of water., J. Chem.Phys. , 1993, 98 (10), 8221 Holcomb, C. D.; Clancy, P.; Zollweg, J. A., A critical study of the simulation of the liquidvapour interface of a Lennard-Jones fluid. Mol. Phys., 1993, 78, 437 Lee, C. Y.; Scott, H.L., The surface tension of water: A Monte Carlo calculation using an umbrella sampling algorithm. J. Chem. Phys., 1980, 73, 4591 Estimationqf the criticalpoint in SPC/Ewater Page 16 Levelt Sengers, J.M.H.; Kamgar-Parsi, B.; Balfour, F.W.; Sengers, J.V., J. Phys. Chem. Ref. Data, 1983, 12, 1 Levelt Sengers, J.M.H.; Straub, J.; Watanabe, K.; Hill, P.G., Assessment of critical parameter values for H2 0 and D2 0. J. Phys. Chem. Ref Data, 1985, 14(1), 193 Levelt Sengers, J. M. H.; Given, J. A., Critical behavior of ionic fluids. Mol. Phys., 1993, 80(4), 899 Liew, C.C.; Inomata, H.; Arai, K., Flexible molecular models for molecular dynamics study of near and supercritical water. Fluid Phase Equili., 1998a, 144, 287 Liew, C.C.; Inomata, H.; Arai, K.; Saito, S.,, Three-dimensional structure and hydrogen bonding of water in sub- and supercritical regions: a molecular simulation study. flexible molecular models for molecular dynamics study of near and supercritical water. J. SupercriticalFluid,1998b,13, 83 Mountain, R. D., Comparison of a tlexed-charge and a polarizable water model. J. Chem. Phys., 1995, 103, 3084 Panagiotopoulos, A.Z., Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble. Mol.Phys., 1987, 61, 813 Panagiotopoulos, A.Z.,; Quirke, N.; Stapleton, M. ; Tildesley, Phase equilibria by simulation in the Gibbs ensemble Alternative deviation, generalization and application to mixture and membrane equilibria. Mol.Phys., 1988, 63, 527 Pitzer, K.S., Ionic fluids: Near-critical and related properties. J. Phys. Chem., 1995, 99, 13070 Reagan, M.T.; Teater,J.W., Molecular modeling of dense sodium chloride-water solutions near the critical point. Proc. Int. Conf. Prop. Water and Steam, 1999, 99 Ree, F.H., J.Chem.Phys. 1980, 73, 5401 Estimationof the criticalpoint in SPC/Ewater Page 17 Strauch, H.J.; Cummings, P.T., Comment on molecular simulation of water along the liquidvapor coexistence curve from 25C to the critical point. J.Chem.Phys., 1992, 96 (1), 864 Toxvaerd, S., Statistical Mechanics, edited by K.Singer ( Specialist Periodical Reports, Chem. Soc.,1975) 2, chap4, 256 Watson, J.T.R.; Basu, R.S.; Sengers, J.V., An improved representative equation for the dynamic viscosity of water substance. J.Phys.Chem.Ref.Data, 1980, 9(4), 1255 Wyczalkowska, A. K.; Sengers, J. V., Thermodynamic properties of sulfurhexafluoride in the critical region. J. Chem. Phys., 1999, 111, 1551 Yoshii, N.; Yoshie, H.; Miura, S.; Okazaki, S., Rev. High Pressure Sci. Technol., 1998, 7, 1115 Estimationof the criticalpoint in SPCIE~water Etmto f th I S I wt I Ie __ 0 0 o * (1) 0 ,0*0@* . 6 I:' '~al ~.--[-Q 0 ~0 e 0 0J - _ 0 _ 0 _ E, N, V, E,+AE N V, E2 +AE2 E, +fE, N, V, +AV E2 N2 V2 +AV E, +AE, N, +1 V, E2 +E N2-1 V2 N2 V2 (2) \ E2 N2 V2 (3) - Figure 6-1 Concept of GEMC | 2 Page 18 II Estimationo the critical point in SPCIEwater EsIm I II cl Figure 6-2 Example of EOS fitting (from Guissani, 1993) -- - Page 19 Estimationof the citical point in SPCIE water E i-- i i o - ii pi i i S Page 20 III wt 0 Liquid Liquid Vacuum Figure 6-3 Vacuum Geometry of a unit cell of direct MD simulation of two coexisting phases vapor phase Den! I interface I liquidphase EITY ~ ~ Iinterface Ivaporphase lpnitv nf linllil ~ ~ ,.,,,,. V·~~IUI~ ,.,, Hlas, isity of gas - Figure -4 - - Density of profile after simulation of the Estimation ___ water I_ criticalpoint in SPC/E _____ _ I_ 21 _ ~~~~~Page # of molecules: 256 or 512 Water at 25°C, 0.997g/cm3 equilibrated Vacuum Lz Liquid Vacuum Lx = 7nm(256), 10nm(512) L = 1.97nm L = 1.97nm Ly c- density 0.3-0.4gIcm 3 periodic boundary condition Figure 6-5 Initial configuration of a unit cell in MD Estimation off tee critical water Ei c a voint oi in in SPCIE SC we Pape Pg 22 2 1.0 .e. .b 0.9 0.8 0 0.7 A * rho -100C #REFI E 0.6 -#REFI 0.5 A rho - 250C u) aC 0 * 0.4 o- 0.3 0.2 0.1 0.0 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 X (nm) from center Figure 6-6 Density Profiles at 0.5fs timestep closed symbols show the simulation results and solid lines show fittin2 curves rho -300C 0 Estimation the critical point in SPCIEwater Estimation of t crtia p Page 23 i -- 1.0 ,_,,- 0.9 I 04ir - 1-w 0.8 0.7 1 -- -. M- 1W0.6 - * 0.5 A rho -- 200 0.5 * - a 1:03 rho 275 * rho -300 0.3 0.2 0.1 -1 A.- Bib 1 rl---- 0.0 rho -- 100 ------ -4 -3.5 -3 -2.5 -2 -1.5 I i -1 -0.5 X (nm) from center Figure 6-7 Density Prof.!es at 5fs timestep closed symbols show the simulation results and solid lines show fitting curves 0 Estimationof the citical point in SPCIEwater IIo f th p- in SI w -Il-- Page 24 P Figure 6-8 Generation of a bubble inside the liquid phase at 300°C 4 _ · I __ _ __ I __ Estimation of the critical point in SPC/E water (a) · Page 25 ., I.U 0.9 0.8 _ 0.7 CO o 0.6 >, 0.5 .)2 0.4 " 0.3 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 3 4 5 X (nm) (b) I.U 0.9 0.8 h 0.7 E E 0.6 ,, 0.5 0.4 0 .3 0.3 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 1 X (nm) Figure 6-9 Density Profiles at different timesteps (a) 5fs timestep at 100°C (b) 0.5fs timestep at 100°C IEstimation ___ of the I critical - - point-in SPC/Ewater - _ - . L Page 26_ - _- _______. A /,\. 0.9 0.8 0.7 A E 0.0 0.4 a 0.3 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 X (nm) 2 3 4 5 (b) 1 .U 0.9 0.8 .a0.7 E 0.6 0.5 a 4. 0.4 C 0 (n 0.3 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 x (nm) 1 2 3 4 Figure 6-10 Density Profiles at different number of molecules (a) 256 molecules, Lx=7nm, Ly=Lz=1.97nm (b) 512 molecules, Lx=lOnm. Lv=Lz=2.7nm 5 Estimationof the_ critical point in SPC/Ewater Page 27 (a) 1.0 0.9 0.8 0.7 0.5 c 0.4 a 0.3 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 x (nm) (b) 4 *'' I.U 0.9 0.8 0.7 E 0.6 'a g0.42 r 0.4 0.3 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 X (nm) Figure 6-11 Density Profiles at different number of molecules (a) 256 molecules, Lx=7nm (b) 512 molecules, Lx=lOnm I _ _ _ _ _ Estimation of the critical point in SPC/E water Page 28 (a) E 0, C -5 -4 -3 -2 -1 0 1 2 3 4 5 X (nm) (b) C' E -) a C aP, -5 -4 -3 -2 -1 0 1 2 3 X (nm) Figure 6-12 Density Profiles at different cell size (a) Lx=7nm (b) Lx=5nm 4 5 Estimation th citical point in SPCIEwater - f th crIl I - S I Page 29 we I PaI -n- /UU 650 600 550 U) IC2 500 S H) 0. 450 0E 400 350 300 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Density (g/cm3) Figure 6-13 Relationship between reduced pressure and reduced density of SPC/E at the critical point (Tc=616K, pc=0.308g/cm3 , Pc=177bar) 1.0 2l Estimationo the critical point in SPCIE water i th cil poI In S Page 30 w Pg i Tc=616.2K, rc=0.308g/cm3, P=0.325 1 0.9 0.8 - 0.7 ) 0.6 c) >, 0.5 c 0.4 0.2 0.1 0 350 400 450 500 550 600 650 Temperature (K) Figure 6-14 Coexistencecurve of SPC/E water at 13=0.325 (Tc=616K, pc=0.308g/cm3 ) 700 rL · * A A rV rL(O.5fs) rV(O.5fs) rL(5fs) rV(5fs) ------- rL(expl.) rV(expl.) X 0.3 300 - (rhov+rhol)/2 __- r+r/2 Estimation oint SPCIEwater EImaIonI of ofJthe the critical cia oI Iin S e Figure 6-15 I 31 Page P PpT relationship of SPC/E water at 0=0.325 (Tc=616K, pc=0.308g/cm3 ) Estimationof the citical point in SPCIE water - ai -i o m SP/ w -- Page 32 l 32 Tc=646K, rc=0.2895g/cm3, P=0.404 41 0.9 0.8 - 0.7 < 0.6 - (rhov+rhol)/2 -- * * A A >, 0.5 u, - 0.4 ----- 0.3 X --------- 0.2 0.1 0 300 350 400 450 500 550 600 Temperature (K) Figure 6-16 Coexistence curve of SPC/E water at Tc=646K and pc=0.2895g/cm3 650 700 rL rV rL(0.5fs) rV(0.5fs) rL(5fs) rV(5fs) rL(expl.) rV(expl.) r+r/2 Estimationof the critical point in SPC/Ewater _ ___ __ I I ___ _ _ IL Page 33 I _____ Tc=626K, pc=0.3023g/cm3, Pc=140.8bar 4.5 4 3.5 a) Tr1.3946 Tr=1.2348 Tr=1.1550 Tr=1.0751 Tr=1.0411 -Tr--0.9984 * 873K(spce) 3 (0 L. 'O 0n a) 2.5 2 AL o 773K(spce) a~~~~ A 723K(spce) A 673K(spce) 0 651.7K(spce) 1.5 -- ,. i i " --- 0 625K(spce) J 0.5 0 0 0.5 1 1.5 2 2.5 Reduced density Figure 6-17 The relationship between reduced density and reduced pressure at Tc=626K and pc=0.3023g/cm3 Estimzation oofthe Estimation the ctical 'r Itic point in InSPCE water we Page 3 I Tc=631 K, rc=0.2991 g/cm3 Pc=1 51.6bar 4.5 S /"/ // 4 0 3.5 a) 3 9 U) 0) L 2.5 0. 0 0' 2 0) 1.5 'O -* O A A * / A A - ~~ ~~~~H -- O- -- U U ^- 7 El o 1 IX/ I I 1 1.5 0.5 0 0.5 2 2.5 Reduced density Figure 6-18 The relationship between reduced density and reduced pressure at Tc=631K and pc=0.299g/cm3 Tr=1.3835 Tr=1.2250 Tr=1.1458 Tr=1.0666 Tr=1.0328 Tr=0.9905 823K(spce) 773K(spce) 723K(spce) 673K(spce) 651.7K(spce) 625K(spce) Estimation of the criticalpoint in SPC/E water I I II-- I- -_- - Page 35 __ I Tc=636K, pc=0.296g/cm3 4.50 4 0 '.'· 3.5 _./ 3 3 (/) 0 O ^ -T I Tr=0.9821 -Tr=1.0240 -Tr=1.0575 . L2.5 0) I L 2 0 'O 1.5 at 1 / °/ -Tr=1. -Tr=1.2146 823K(spce) A A/ 13608 / 0^ 0 773K(spce) //J a 723K(spce) //A 673K(spce) ] 651.7K(spce) 0 625K(spce) 0.5 0 0.5 1 1.5 2 2.5 Reduced density Figure 6-19 The relationship between reduced density and reduced pressure at Tc=636K and pc=0.296g/cm3 Estimationof the critical oint in SPCIEwater s- -I U -- - -thII - Page 36 I Tc=641 K,0.2927g/cm3 4.5 4 a 3.5 0 -Tr=-1.3619 Z~~~~~~ U) CD C) L. 0. 'O -~11/ "/'z 2.5 2 //0 // / 3 /// - Tr=1.2059 - Tr=1.128 - Tr=-1.0499 -- Tr=1.0167 - Tr=0.9750 * 823K(spce) O 773K(spce) :3 ') O 1.5 nC) - /00 *-// A 723K(spce) A 673K(spce) m 651.7K(spce) //3 0.5 1 D -3 /3 1 0 625K(spce) I /// 0 0 0.5 1 1.5 2 2.5 Reduced density Figure 6-20 The relationship between reduced density and reduced pressure 3 at Tc=641K and pc=0.293g/cm Estimationo the criticalpoint in SPCIEwater -sa . . f t i p I InIl__In SP/ Page 37 w P 3 Tc=646K,rc=0.2895g/cm3 4.5 4 0 3.5 02 3 - Tr=1.1192 (0 u) U) Tr=1.0418 -Tr=1.0088 2.5 - EL la 0 '0 nMe Tr=1.3514 Tr=1.1966 - /aI-,' 2 * o 773K(spce) // oyX A-~~~~ // X~~~~/// A 723K(spce) A 673K(spce) * 651.7K(spce) o 625K(spce) 1.5 1 0 Tr=0.9675 823K(spce) 0 /3 0 0.5 O 0 0.5 1 1.5 2 2.5 Reduced density Figure 6-21 The relationship between reduced density and reduced pressure at Tc=646K and pc=0.290g/cm3 Sef-Difusi'on S IfIon off water waIII in innear nea critical cIIcIl reion ri Pe 11I Page CHAPTER 7 SELF-DIFFUSION OF WATER IN NEAR CRITICAL REGION 7.1 Objectives Anomalous behavior of properties of fluid at near critical region are well known. For example, the thermal conduction and viscosity increases only in the vicinity of the critical point due to the large density fluctuation ( Figure 7-1 and 7-2). When we consider the Stokes-Einstein model, the self-diffusivity is proportional to the inverse of viscosity. Therefore, the self-diffusivity is also expected to show the anomaly behavior as the drop near critical region. In fact, it is often reported that the diffusion coefficient of a binary mixture approaches to zero in the vicinity of the critical point ( Figure 7-3 ). However, there is no evidence of the drop of self-diffusion coefficient near critical point so far since the experimental measurement of self-diffusivity is very difficult at high temperature and pressure. The objective in this chapter is to grasp the behavior of self diffusivity of water near critical point which was obtained in Chapter 6 by using SPC/E model which can reproduce the properties of water in supercritical region. 7.2 Simulation procedure The same technique as the method in Chapter 5 was used. All simulation parameters are fixed through all calculations as follows. NVT ensemble Number of molecules: 256 Timestep: 0.5fs Self-Diffusionof water in near critical region Page 2 Equilibration time: 50ps Production time: 50ps The position vector of each atom are collected every 25timesteps. 7.3 Simulation results Since NVT ensemble was used, either density or temperature can be fixed. As is shown in Figure 7-3, both isochoric and isothermal cases near critical point were studied with respect to the self-diffusion coefficient of water. 7.3.1 Isochoric approach Figure 7-4 shows the relationship between temperature and self-diffusion coefficient at constant densities. At 0.296g/cm 3 which is a little higher than the obtained critical density(0.290g/cm 3 ), about 5% drop of self-diffusion coefficient was observed around critical temperature (646K). Figure 7-5 shows the case at 0.326g/cm 3 which is a much higher than the obtained critical density and corresponds to the critical density by Guissani and Gillot (1993). This figure also indicates the drop of self-diffusion coefficient occurs around 650K. Guissani's critical temperature is 652K. Surprisingly, the drop at 0.326g/cm 3 is larger than that at 0.296g/cm3 . After the drop, self-diffusion coefficients fluctuated much at low temperature. The problem of isochoric approach is that water enters in thermodynamically unstable region below critical temperature. In practice, supercritical water splits into two phases, gas and liquid. However, in the simulation which includes only 256 molecule, system is too small to split and probably make homogeneous phase which does not actually exist. In fact, from the averaged density profile, there was no evidence of phase splitting. 7.3.2 Isothermal approach Figure 7-6 shows the self-diffusivity near critical region in the isothermal conditions. Self-Diffusion of water in near critical region Page 3 In the range from 0.3 to 0.35 g/cm3 , the drop was observed. Also in this case, the drop at 652K is larger than that at 646K (observed critical temperature). In the case of viscosity, the critical enhancement is observed with the wide range of densities. If the diffusion completely followed Stokes-Einstein equation, the diffusivity would not change much at different temperatures because the change of the viscosity is small. In practice, water behaves like a gas below the critical density, so D changes depending on the temperature as one can see in Figure 7-2. 7.4 Discussion Simulation results indicate that self-diffusion coefficient decreases around the critical point. Generally speaking,, critical enhancement is observed in the narrow region. However, in this simulation, MD simulation results showed the critical behavior relatively in the wide range. Therefore, we have to confirm whether this drop is caused by anomaly critical behavior and if not, what is the origin of this drop. First, I check the availability of MD to the detect of the critical behavior. Currently, the critical behavior is believed to be caused by large density fluctuations near critical point. As the periodic boundary condition is used in MD simulation, when the wave length of density fluctuation is larger than the unit cell length, MD cannot reproduce the effect of fluctuation. That means it is hard to detect the critical behavior at the right critical point in MD. When the unit cell is extremely large, it i's possible to detect the critical behavior in the vicinity of the critical point. On the contrary, MD simulation is usually done in the condition where the correlation length is less than half of the unit cell length to keep reliability without the effect of fluctuations. Correlation length Levelt Sengers and Sengers (1986) give the following equation for the correlation length of water along the critical isochore: = o (AT*)- [1+ 51(AT*)-A + ] (7-1) (7-1) Self-Diffusion of water in near critical region where v=0.630, and A=0.51 are universal exponents, and Page 4 o=0.13x10' 9 m and ,1=2.16 are specific for water. Figure 7-5 shows the relationship between temperature and correlation length when the critical temperature is 646K. As the temperature approaches the critical temperature, the correlation length becomes rapidly large. Figure 7-6 indicates the relationship between temperature and the density where the unit cell length is equal to the correlation length at the temperature. In the range of density I simulated, density fluctuations may be detected above 650K. This value is close to the temperature at which the drop of diffusion coefficient was detected. Figure 7-7 and 7-8 show the radial distribution functions between oxygen atoms in near critical region. The shape of peaks continuously changes with the change of temperature or density. As the temperature increases and the density decreases, the first peak height decreases. This behavior corresponds general tendency. Self-diffusivity is said to be related to the number of hydrogen bonding, but there is no evidence which explains the drop. This trend continues up to subcritical conditions, so in this work, water molecules are considered to exist homogeneously in a unit cell. Prediction from critical viscosity As there is no experimental data of self-diffusivity in the vicinity of the critical point, the behavior of the. viscosity of water which should be highly related to self-diffusion as predicted from Stokes-Einstein relation is helpful to discuss the behavior of diffusion coefficient. Watson et al. (1980) investigated the behavior of viscosity in the vicinity of critical region. They considered viscosity rl(p,T) as the sum of a normal viscosity (p,T) and a critical or anomalous viscosity Ar(p,T). nl=n +Arn (7-1) _l=(q4) (7-1) Several investigators have predicted an equation for the critical viscosity enhancement of the form : Self-Diffusion of water in near critical region rl =1 + (q~)= 1+l n(q5) (q) Page 5 (7-1) with = 8 = 0.054 (7-1) As theoretical value of ~, 0.065 or 0.07 are proposed, but they are reasonable only when the critical point is approached sufficiently. From experimental data, c1 = 0.05 is often used for a number of fluids including water. According to data from Rivkin et al. ( ), the critical enhancement occurs at a little apart from the critical density. Critical behavior of self-diffusivity at low temperature There are two possibilities. (1) the water properties retain the same trend they have just below the freezing point Prielmeier et al. (1987) have shown that in the low and moderate pressure, the power law is superior to Vogel-Fulker law in the interpretation of the strong non- Arrhenius behavior of the water self-diffusioin coefficients. D(T) = ;D,(T)f(T) °0~~~~~~~~~~~~ ~~~(7-1) where 7.5 Conclusions Self-diffusivity in the vicinity of the critical point was first simulated. Due to the periodic boundary condition, the critical enhancement effect is not supposed to be observed in the simulation results. However, the drop of diffusion coefficient in the vicinity of the critical point was observed. In isochoric approach, self-diffusivity plummeted near critical temperature and it fluctuated much below critical temperature. In practice, supercritical water generates phase splitting into gas and liquid below critical temperature. It might cause the fluctuation. However, the system of simulation includes only 256 molecules and the averaged Self-Diffusion of water in near critical region Page 6 density profile is flat, so we cannot find explicit evidence of the phase splitting. In isothermal approach, the drop of self-diffusion coefficient was observed, too. The density where the diffusion coefficient is minimum is a little higher than the critical density in Chapter 6. Possible reasons is (1) due to the difference of the net critical point, (2) due to the effective correlation length by periodic boundary condition and (3) due to the small density fluctuations like local phase splitting. In order to elucidate the origin of the drop of self-diffusion coefficient, more research including a large unit cell is required. Self-Diffision of water in near critical region Page 7 References: Bartle, K.D.; Baulch, D.L.; Clifford, A.A.; Coleby, S.E., Magnitude of the diffusion coefficient anomaly in the critical region and its effect on supercritical fluid chromatography. J. Chromat., 1991, 557, 69 Dzugutov, M, Auniversal scaling law for atomic diffusion in condenced matter. Nature, 1996, 381, 137. Fannjiang, A. C., Phase diagram for turbulent transport: sampling drift, eddy diffusivity and variational priciples. Physica D, 2000, 136, 145 Lamanna, R.; Delmelle, M.; Cannistraro, S., Role of hydrogen-bond cooperatively and free volume fluctuations in the non-Arrhenius behavior of water self-diffusion: A continuity-of-states model. Phys. Rev. E, 1994, 49, 2841 Levelt Sengers, J.M.H.; Sraub,J.; Watanabe,K.; Hill, P.G., Assessment of Critical Parameter Values for H20 and D2 0, J. Phys. Chem. Ref: Data, 1985, 14, 193 Rosenfeld, Y., A quasi-universal scaling law for atomic transport in simple fluids. J. Phys.:Condens.Matter, 1999, 11, 5415. Reagan, M.T.; Tester, J.W., Molecular modeling of dense sodium chloride-water solutions near the critical point. the Proceedings of the International Conference on the Propertiesof Waterand Steam '99 Sengers, J. V.; Levelt Sengers, J. M. H., Thermodynamic behavior of fluids near the critical point. Ann. Rev. Phys. Chem., 1986, 37, 189 Sengers, J.V., Transport properties of fluids near critical points. Inter. J. Thermophys., 1985, 6, 203 Smith, P.E.; van Gunsteren, W.F., The viscosity of SPC and SPC/E water at 277 and 300 K. Chem Phys. Lett., 1933, 215, 315. Yokoyama, I., On Dzugutov's scaling law for atomic diffusion in condensed matter. Physica B, 1999, 269, 244. Watson, J. T. R., Basu, R. S.; Sengers, J.V., An improveed representative equation for the dynamic viscosity of water substance. J. Phys. Reft Data, 1980, 9, 1255 Self-Diffusionof water in near critical region -- Page 8 --- 65 X 60 55 ~50 ... a.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r Co 45 ._40 IA Ur 0 4 -] Oh 374.2C Ah 374.5C 0 A- 35 35 Oh 375C Xh 375.5C *h 376C +h 377C Xh 380C Oh 390C +0 30 25 20 0 0.1 0.2 0.3 0.4 0.5 Density (g/cm3) Figure 7-1 Viscosity of water near critical region 0.6 Seldf-Difcusionwater in near criticalreion Z Figure 7-2 Add ^t e * - |~~~~~~~~~~~~~~~~~~~~~~~ Thermal conductance of water near critical region - .-Page 9 Self-Diffu~sion of water in near critical reion Slf-Di-f-io ofwate I Figure 7-3 __ Diffusion coefficient of binary mixture near critical point _ Page 0 Self-Diff~sion of water in near critical f I region e I Page 11 -- 0.8 0.7 0.6 0.5 ,, 0. i, >, 0.4 C O 0.3 0.2 0.1 0 400 450 500 550 600 650 700 Temperature (K) Figure 7-4 Isochoric approaches and isothermal approaches near critical pointDiffusion coefficient of water in isochoric conditions Self-Diffusion -- ----- of water in near---criticalregion Pane 12 105 100 E 95 90 00 .-0 85 A 0.2895g/cm3 0 0.2927g/cm3 * 0.2958g/cm3 80 c [ * 0 75 0.2991 g/cm3 0 0.3023g/cm3 0.308g/cm3 70 600 610 620 630 640 650 660 670 680 Temperature (K) Figure 7-5 Diffusion coefficient of water in isochoric conditions 1 Sef-Dif~usionof water in near critical re f-Dui i n ii Page i 3 Critical region 130 120 E 110 a, 4. 100 v- c 90 o ._ C, C~ 80 70 60 600 650 700 750 800 Temperature (K) Figure 7-6 Diffusion coefficient of water in isochoric conditions ( = 0.326 g/cm)) 9 Sellf-Difiusioncf water i nea critical regiono I f Io- - In ii I i - . -Page 14 I 130 673K 120 652K ,-~110 E ' 646K 10 0 :D 0 o 90 625K C ._o 70 60 0.2 0.25 0.3 0.35 0.4 Density (g/cm3 ) Figure 7- 7 Diffusion coefficient of water in isothermal conditions 0.45 Selt-'Diffusionof water in near citical region IffuI .f I na i Page 15 Iwae e P 1 14 12 10 E 0, - 8 C. o 4 2 0 645 650 655 660 665 670 675 680 Temperature (K) Figure 7-8 The relationship between correlation length and temperature in Tc=646K Self-Diffussion of water in near critical region Sffs o wate in I eo Page 16 Pe III 1 I1 .U- 0.9 0.8 E 0.7 0.6 up , 0.5 E = 0.4 E ' 0.3 0.2 0.1 0.0 640 645 650 Temperature 655 660 (K) Figure 7-9 Density in which the length of a unit cell is equal to the correlation length at the given temperature Sef-Diffusion of_~! water in -i i near n criticazl l reg rI ion -- Page 1 17 Pg I 3.5 3.0 2.5 2.0 0 0) cmn 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Distance (nm) Figure 7- 10 Radial distribution function of water in the near critical region at 0.296g/cm3 Self-Diffusionof water in near criticalregion _ Ut Page 18 __ 3.5 3.0 2.5 2.0 o o CD 1.5 1.0 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r (nm) Figure 7- 11 Radial distribution function of water in the near critical region at 646K Summary I PPage 11 CHAPTER 8 SUMMARY In this research, the self-diffusion coefficients of water in subcritical and supercritical conditions were calculated by using molecular dynamic simulation. The following results were elucidated: 1) Extended Simple Point Charge (SPC/E) model can reproduce the self-diffusion coefficient which is good agreement with the experimental data in both ambient condition and high temperature and high pressure conditions. 2) Orthobaric densities of liquid and gas of pure water were obtained by simulating the density profile of coexisting phases. The orthobaric densities are not dependent on the number of molecules in the simulation system, but timestep seems to affect orthobaric densities a little. 3) The coexistence curve obtained from orthobaric densities is better fitted to that of the real water than SPC model's coexistence curve in the literature, but is slightly worse than that of the other SPC/E model. It is probably due to the difference of simulation techniques such as long range corrections of electrostatic interactions. 4) The critical point of SPC/E water was obtained by using the scaling law and the rectilinear diameter method. The value of the critical point based on the theoretical critical component, f=0.325, was Tc=616K and pc=0.308g/cm 3 . Summary Page 2 5) From the Pressure- Temperature-Density data, the above critical point seemed to be lower than the nominal critical point of this SPC/E code. Using a given critical temperature and the calculated critical density based on the rectilinear diameter method, the relationship between the reduced pressure and the reduced density was described. By comparing this relationship with the data of real water, the critical point was estimated to be around Tc=646K, pc=0.296g/cm3 . This critical temperature was so close to that of real water. 6) Using the obtained critical point, the self-diffusion coefficients were calculated in the vicinity of the critical point. The drop of the self-diffusion coefficient was observed in both isochoric and isotherm cases. The point where the drop occurred was shifted from the critical point. This point corresponded to the condition where the correlation length is equivalent to the unit cell length. Appendix Page 1 Appendix 1. Code for diffusion coefficient calculation program newdiftest2 c--------------------------------------- c c . 5/17/2000 by Y.Kubo c c << set parameters >> c implicit double precision(a-h,o-z) integer natomx,nmolx,npartx,nmolinfo,natminfo,nspecinx,nconfigmx parameter (natomx=20,nmolx=1000) parameter (npartx=natomx*nmolx) parameter (nmolinfo=3,natminfo=3,nspecmx=4) parameter(nconfigmx=10000) integer i,j,jj,k,kk,mol,nmol,calcnum,ll,lengl,leng2 integer nql,nq2,moltrack,itrack,ntypes integer nmoltot,npart,nspecies integer fstartnum,fendnum,flenl integer lennam,cmcheck integer ndigt,nzer,inumb integer natom,iatom,ipart,sdata,fdata,sstep,fstep integer dstep,ostep,idump,nnn,nn integer mollist(nmolinfo,npartx),moltype(natminfo,nmolx) Integer nnmol(nspecmx),natoms(nspecmx),itmp(natomx) integer lspecies(natomx,nmolx),endorigin(nconfigmx) double precision ratom(3,npartx),box(3),rmass(natomx) double precision cm(3,rmolx,nconfigmx),rcm(3) double precision tdif(nconfigmx),tcur(nconfigmx),avedis(nconfigmx) double precision disp(3),dispsum,dispt,disptt,disptsum,dens double precision rmdis,ttotmas double precision rdsig,sigz,rdeps,epsz,charge,rdcut,zcut double precision totmas,tmass,winv,tinit,temper,vol double precision tstep,ddstep,oostep double precision sumt,sumtt,sumtd,sumx,sumy,sumz double precision slope,intecept,corr,diff character*80 title character*40 fcur,finit,frec,filin,fend,faux,flen character*40 fpara,ftopo,freccm,frecdif,finp character*40 file8,filer8,file9,filer9,fstringl,fstring2,fstring character*80 names(nspecmx) character*l suffix,inum,enum c parameter(lennam=5) data suffix/'.'/ 200 format(A50,I6) Page 2 Appendix 202 210 220 230 9001 9010 9011 9012 9014 format(A20,I6) format(A50,A) format(A50,lPG14.5) format(' ',A8,I6,A12) format(a) format(x,a,$) format(x,il2) format(x,lpgl4.5) format(x,a) c write(6,*) 'Diffusion coefficient calculation--parameters' write(6,*) '********************************************* $********************, c write(6,*) 'Please input initial file(ex:spcstartOO00001)' read(5,9001) finit write(6,210)'initial c file: ',finit write(6,*) 'Please input final file(ex:spcstart00100))' read(5,9001) fend write(6,210)'final file: ',fend c lengl=index(finit,' ') leng2=index(fend,' ') fstringl=finit fstring2=fend C c if c c (lengl.lt.leng2. or. lengl.gt.leng2) write(6,*) ' file name is wrong' c stop endif c j=0 jj=0 k=O kk=O do 10 i=l,lennam inum=finit(lengl-i:lengl-i) fstringl(lengl-i:lengl-i)=char(48) enum=fend(lengl-i:lengl-i) fstrlng2(leng2-i:leng2-i)=char(48) j=(ichar(inum)-48)*10**(i-1) k=(ichar(enum)-48)*10**(i-l) c c c write(6,*) fstringl write(6,*) fstring2 jj=jj+j kk=kk+k nconfig=kk-j j +1l 10 continue c fstring=fstringl fstartnum=jj fendnum=kk write(6,200)'initial number: ',fstartnum write(6,200)'final number: ',fendnum write(6,200)'# of data files: ',nconfig then _ Appendi c if (fstringl.ne.fstring2) then write(6,*) ' file name is wrong' stop endif c c write(6,*) 'Please input sample data file (aux file)' read(5,9001) faux write(6,210)'sample data file: ',faux write(6,*) 'Please input molecular data file(topol file)' read(5,9001) ftopo write(6,210)'molecular data file: ',ftopo write(6,*) 'Please input atomic data file(params file)' read(5,9001) fpara write(6,210) 'atomic data file: ',fpara write(6,*) 'Please input MD parameter file(inp file)' read(5,9001) finp write(6,210)'MD parameter file: ',finp write(6,*) 'Please give new diffusion file name' read(5,9001) frec lengl=!ndex(frec,' ') c c c c cc c leng2=lengl-l c freccm=frec c f--2'-df=frec(l:leng2)//'dif' c c (1:leng2)//'cmOOOOO' ,½;,_a6,2l0)'each molecular configuration file: ',freccm :.'i:5 2l0) 'displacement file: ',frecdif real(',*' cmcheck 6,*; '********************************************* wji~ c << oCen initial file - to read npart >> c open(unit=7,status='old',file=finit) read(7,*) nql,nq2 read(7,*) npart close (7) c c << open .aux file - to read mollist,moltype >> c open(unit=3,status='old',file=faux) read(3,9001) title write(6,*) title read(3,*) nspecies itrack=0 moltrack=0 nmoltot=0 do 20 i=l,nspecies read(3,9001) names(i) write(6,*) names(i) read(3,*) nnmol(i),natoms(i) nmoltot=nmoltot+nnmol $ (i) write(6,*) 'Number of molecules=',nnmol(i), ' number of atoms',natoms(i) do 30 j=l,natoms(i) read(3,*) k,itmp(j) c write(6,9002) i,k,itmp(j) Page 3 Appendix 9002 Page 4 format(' Species ',i4,' atom#',i4,' is of type ',i2) if(k.ne.j) then write(6,*) 'Error in deflist reading atom#',j, $ ' in species#',i 30 write(6,*) 'Read in ',k,itmp(j) stop end if continue do 40 j=l,nnmol(i) moltrack=moltrack+l moltype (1,moltrack)=i moltype(2,moltrack)=itrack moltype(3,moltrack)=natoms(i) do 50 k=l,natoms(i) itrack=itrack+1 mollist(l,itrack)=k mollist(2,itrack)=moltrack mollist(3,itrack)=itmp(k) lspecies (k, i) =itmp(k) 50 40 20 continue continue continue close(3) c if(npart.ne.itrack.or.nmoltot.ne.moltrack) then write(6,*) 'Error: number of molecules/particles discrepancy' write(6,*) 'You have ',npart,' particles.' write(6,*) 'You have defined only ',itrack, ' ofthem' write(6,*) 'You should have ',nmoltot,' molecules.' write(6,*) 'You defined ',moltrack, ' of them' stop end if nmol=nmol tot write(6,*) 'Definition of molecule list successful' write(6,*) 'Defined ',nmol, ' molecules.' write(6,*) 'You have ',npart,' particles.' c c c << open topo file - to read ntypes >> c open(unit=l0,status='old',file=ftopo) read(10,*) ntypes write(6,*) ntypes close(10) c c c << open params file - to read rmass >> c open(unit=4,file=fpara,status='old') do 60 i=l,ntypes read(4,*) j,rdsig,rdeps,rdcut 60 70 c continue do 70 i=l,ntypes read(4,*) j,sigz,epsz,zcut continue do 80 i=l,ntypes read(4,*) j,rmass(i),charge write(6,*) j,rmass(i),charge Apvendix ---_- I 80 -- I L-. Pane 5 continue close (4) c << open MD file - to read tstep & idump >> c open(unit=2,file=finp,status='old') read(2,9001) title read(2,9001) fdebug read(2,9001) filet read(2,9001) filein 'tolerance for meeting the constraints: ',tol c 'name for stats file: ',filel read(2,9001) filel read(2,*) tol read(2,*) istart c 'random number startup flag: ',istart read(2,9001) fstart c 'root for file name: ',fstart read(2,*) inumb c 'sequence number for restart file: ',inumb CINP5 file containing various interaction parmeters-- intermolecular CINP5 interactions, angle, and dihedral parameters. read(2,9001) flj CINP5 External temperature used in isothermal, brownian and collision dynamics. CINP5 pext= external pressure. Note these may be ignored if other inputs are CINP5 set up not to use them. CINP5 c 'temperature K/Pressure : read(2,*) temper,pext c 'New box length in nm(0, to leave unchanged ): ' read(2,*) boxn CINP5 option to make center of mass=(0,0): icmopt=0 means "do nothing" CINP5 icmopt=l means move particles so that center of mass=(0,0) CINP5 read(2,*) icmopt CINP5: Brownian friction term: acceleration= normal terms + stochastic term CINP5 - (gamma/mass)*velocity CINP5 stochastic term is also function of gamma and determined to have a CINP5 normal distribution, but so that equilibrium temperature is correct. CINP5 see papers by Gary Grest... Note mistakes in their formulas and CINP5 slightly different definition of gamma (by factor of the mass). CINP5 c 'frictional damping coefficient: read(2,*) gamma c 'mean time between brownian collisions: read(2,*) coltim c 'time step in picoseconds: ' read(2,*) tstep CINP5 REAL*8 WS,WQ CINP5 respectively for temperature and pressure. CINP5 for Andersen Haml. these are piston masses CINP5 for Berendsen's bath these are proportional to the relaxation times CINP5 ws= time(temperature) wq=time(pressure)/compressibility Appendix ·· CINP5 CINP5 Page Y 6 I use negative ws(wq) to not use thermostat(pressure-stat) read(2,*) ws,wq LOGICAL altopt CINP5 CINP5 .true. = use Berendsen's technique CINP5 .false. = use Andersen's Hamiltonian CINP5 read(2,*) altopt c 'Number of subgroups: read(2,*) ngroup read(2,*) nstep CINP5 collect statistics every "istat'th" timestep CINP5 (note: you must collect statistics from at least two timesteps CINP5 in every group or a divide by zero error will occur) read(2,*) istat c 'Frequency for making restart dump:' read(2,*) idump close(2) read(5,*) tstep read(5,*) idump c c c c << get configuration of molecules by center of mass >> iconfig=0.dO do 90 i=fstartnum,fendnum iconfig=iconfif;+l c << get file number >> ndigt=int(log(i*1.0000001)/log(10.d0))+1 write(6,*) ndigt lengl=index(freccm, ') c C leng2=index(fstring,'') do 100 j=l,ndigt flenl=mod(int(i/(l0**(j-l))),10) c freccm(lengl-j:lengl-j)=char(flenl+48) 100 fstring(leng2-j:leng2-j)=char(flenl+48) continue c c filer8=freccm fcur=fstring c c c << make center of mass file of each data >> open(unit=8,status='unknown',file=filer8) c c << 1. read ratom file >> open(unit=7,file=fcur,status='old') read(7,*) nql,nq2 read(7,*) npart read(7,*) temper read(7,*) tcur(iconfig) c read(7,*) box(l),box(2),box(3) do 110 j=l,npart read(7,*) ratom(l,j),ratom(2,j),ratom(3,j) Appendix 110 c Page 7 continue write(6,*) ratom(l,l),ratom(2,1),ratom(3,1) close(7) c ttotmas=O.dO << 2. calculation of center of mass of each atom >> sumx=O.OdO sumy=O.OdO sumz=O.OdO c do 120 mol=l,nrmol rcm(1)=O.dO rcm(2)=0.dO rcm(3)=O.dO totmas=O.dO natom=-noltype(3,mol) c do 130 iatom=l,natom ipart=moltype(2,mol)+iatom tmass=rmnass(mollist(3,ipart)) totmas=totmas+tmass ttotmas=ttotmas+tmass write(6,*) ipart,tmass,rcm(l) rcm(l)=rcm(l)+tmass*latom(l,ipart) rcm(2)=rcm(2)+tmass*ratom(2,ipart) rcm(3)=rcm(3)+tmass*ratom(3,ipart) c 130 continue c winv=l.dO/totmas c c < center of mass(x,y,z) of each molecule sumx=sumx+rcm(1) sumy=sumy+rcm(2) sumz=sumz+rcm(3) rcm(1)=rcm(l)*winv rcm(2)=rcm (2)*winv rcm (3)=rcm(3)*winv c cm(l,mol,iconfig)=rcm(1) cm(2,mol,iconfig)=rcm(2) cm(3,mol,iconfig)=rcm(3) c 120 continue if (cmcheck.ne.O) goto sumx = sumx/ttotmas sumy = sumy/ttotmas sumz = sumz/ttotmas c 90 > Appendix Page 8 do 105 mol=l,nmol cm(l,mol,iconfig)=cm(l,mol,iconfig)-sumx cm(2,mol,iconfig)=cm(2,mol,iconfig)-sumy cm(3,mol,iconfig)=cm(3,mol,iconfig)-sumz 105 continue c 90 continue c c c c c c fstring=fstringl fstartnum=jj fendnum=kk calcnum=fendnum-fstartnum+1 write(6,200)'initial number: ',fstartnum write(6,200)'final number: ',fendnum write(6,200)'# of data files: ',calcnum c dispsum=O.dO disptsum=O.dO c c c < calculation of displacement of each molecule(nm) > c c c * open(unit=9,status='unknown',file=frec) dstep- displacement time step ostep- origin time step read(5,*) dstep ddstep=dstep*tstep*idump write(6,*)'displacement time step (ps): ',ddstep write(6,*)'= dstep*timestep*idump: dstep ',dstep read(5,*) ostep oostep=ostep*ddstep write(6,*)'origin time step (ps): ',oostep write(6,*) '= ostep*dstep*timestep*idump: ostep',ostep norigin=Int(nconfig/ostep) nnn= O. dO c << # of configuratons corresponds tc kinds of displacement steps.>> do 1900 nn=l,nconfig if (mod(nn-l,dstep).ne.0) goto 1900 disp(1)=0 .dO disp(2)=0.dO disp(3)=0.dO dispt=O.dO dispsum=O.dO endorigin(nnn)=int(Int((nconfig-nn)/dstep)/ostep)+1 Appendix Page 9 tdif(nnn)=tcur(nn)-tcur(1) do 2000 iorigin=l,endorigin(nnn) ori=dstep*ostep*(iorigin-1)+1 do 2100 mol=l,nmol disp(l)=(cm(l,mol,ori+nn-1)-cm(l,mol,ori))**2 disp(2)=(cm(2,mol,ori+nn-1)-cm(2,mol,ori))**2 disp(3)=(cm(3,mol,ori+nn-1)-cm(3,mol,ori))**2 dispt=disp(l)+disp(2)+disp(3) dispsum=dispsum+dispt c c write(6,*) disptt 2100 2000 continue continue c close(8) c c c << average of displacement and temperature with time >> write(6,*) dispsum tdif=tcur-tinit nnconfig=nnn avedis(nnn)=dispsum/(nmol*endorigin(nnn)) c c sddis=sqrt(ABS(disptsum/nmol-avedis**2)) write(6,*) disptsum/nmol,avedis**2 c << write data in ...dif file as a line >> write(9,*) tdif(nnn),avedis(nnn) nnn=nnn+l 1900 continue write(6,*)'total time(ps)',tdif(nnn-1) c close(9) c c << Least Square Approximation - Self Diffusion Coefficient >> c sstep=0 fstep=0 calcnum=0 stime=0.dO slope=0.dO intercept=0.dO sloped=0.dO resid=0.dO residsum=0.dO residave=0.dO diff=0.dO corr=0.dO c c c write(6,*) 'Please input diffusion file name:' read (5,9001), frecdif Avvendix Pane 10 c read 'Please input diffusion file name:',filecomp c open(unit=7,status='old',file=frecdif) c c c c zead(7,9001) frecdif write(6,200) 'file name: ', frecdif read(7,9001) title write(6,200) 'title: ', title read(5,*) sstep sdata=sstep*dstep*tstep*idump write(6,200) 'initial time(ps): ', sdata read(5,*) fstep fdata=fstep*dstep*tstep*idump write(6,200) 'final time(ps): ', fdata read(7,*) calcnum write(6,200) '# of data points: ', calcnum write(6,220)'start point of calc.?(from 1 to ', calcnum c c c if (sstep.lt.1) then write(6,*)'incorrect' stop endif if (fstep.gt.int(nconfig/dstep)) then write(6,*) 'incorrect' stop endif write(6,200) 'start point: ', sstep c c c c c c write(6,220)'end point of calc.?(from start to ', calcnum read(5,*) fstep if (fstep.lt.l.or.fstep.gt.calcnum.or.fstep.le.sstep) then write(6, *) 'incorrect' stop endif write(6,200) 'end point: ', fstep c c c sumt=0.dO sumd=O.dO sumdd=O.dO sumtd=O.dO sumtt=O.dO c c c c 3010 do 3010 i=l,sstep-1 read(7,*) tdif(i),avedis(i) continue c c c 3020 c c c do 3020 i=sstep,fstep read(7,*) tdif(i),avedis(i) write(6,*) i,tdif(i),avedis(i),sddis(i),calcnum sumt=sumt+tdif(i) sumd=sumd+avedis(i) sumtt=sumtt+tdif(i)**2 sumdd=sumdd+avedis(i)**2 sumtd=sumrtd+tdif(i)*avedis(i) continue close(7) 'calculation of least square approximation' AnDendix r ---- r -- r c stime=fstep-sstep+l slope=(sumt*sumd-stime*sumtd) / (sumt**2-stime*sumtt) intercept=(sumd-slope*sumt)/(stime) c c 'error' c sloperrl=sumtd-sumt*sumd/stime sloperr2=((sumtt-sumt**2/stime)*(sumdd-sumd**2/stime))**0.5 sloperr3=sloperrl/sloperr2 sloperr=slope/sloperr3*((1-sloperr3**2)/stime)**0.5 'correlation function' c corr=slope/sqrt(sumdd-sumd**2/stime) c c c c 30 i=sstep,fstep sloped=tdif(i)*slope+intercept **2 resid=(sloped-avedis(i)) residsum=residsum+resid c c continue 30 c c residave=residsum/(stime) diff=(slope)*1000/6 diffdev=sloperr*1000/6 c write(6,*) write(6,*) write(6,*) write(6,*) write(6,*) write(6,*) c234567 c end c 'slope by least square approx.(nm^2/psec):',slope ' error', sloperr 'intercept by least square approx.(nm^2):',intercept 'correlation function:',corr 'Diffusion coefficient(10^-5cm2/sec):',diff 'Coefficient deviation',diffdev Page 11 Y THESIS PROCESSING SLIP FIXED FIELD: , COPIES: ill. name index biblio Archives, Aero Dewey Barker Hum Lindgren Music Rotch Sciencee Sche-Plough n TITLE VARIES: · NAME VARIE .0_-IMPRINT: (COPYRIGHT) · COLLATION: - - -`- ADD: DEGREE: ADD: DEGREE: DEPT.: ·- CEPT.: Q IP:FRVI"RJRn. Il -~ ""l -_ -- ! IU, ,--------- NOTES: cat'r date page i6 · DEPT: YEAR: bNAME: DEGREE: _y_ - _ J