Molecular Dynamics and Self-Diffusion in Supercritical Water

Molecular Dynamics and Self-Diffusion in Supercritical Water

byYuji Kubo

B.S. ChemistryUniversity of Tokyo, 1985

M.S. ChemistryUniversity of Tokyo, 1987

SUBMITTED TO THE DEPARTMENT OF CHEMICAL ENGINEERING IN PARTIALFULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN CHEMICAL ENGINEERING AT THEMASSACHUSETTS INSTITUTE OF TECHNOLOGY

( 2000 Massachusetts Institute of TechnologyAll rights reserved

Signature of Author:

Certified by:

Certified by:

Accepted by:MASSACHUSETTS INST1TUTE

OF TECHNOLOGY

FEB 0 7 2001

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T} i7 - --

~)ep~ament of Chemical EngineeringSeptember 11, 2000

< \ Professor Kenne-t- A. SmithThesis Supervisor

,/'~// Professor Jefferson W. Tester(Thi) s SuDervisor

Professor Robert E. CohenSt. Laurent Professor of Chemical Engineering

Chairman, Committee for Graduate Students

ARCHIVES

Molecular Dynamics and Self-Diffusionin Supercritical Water

by

Yuji Kubo

Submitted to the Department of Chemical Engineeringin September, 2000

in partial fulfillment of the requirements for thedegree 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 thisnew solvent such as oxidation of organic wastes and separation of metals have beenresearched; however, the properties of SCW have not sufficiently been understood due to thedifficulty in experimental measurements at high temperatures and high pressures. Computersimulation is one of the best tools to predict and analyze the properties which are difficult tobe experimentally obtained. The goals of this research is to gain a better understanding as toself-diffusivity of SCW, of which experimental data is limited, through molecular dynamicsimulation.

Extended Simple Point Charge (SPC/E) model reproduced the self-diffusioncoefficient 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 self-diffusivity. In order to investigate self-diffusivity in the near critical region, the critical pointof 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 theexpected critical point. By comparing reduced pressure/reduced density relationship fromsimulated data with that of experimental data, the critical point was revised to around 646Kand 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 nearcritical region was studied. The small drop of self-diffusion coefficient near critical point wasobserved from simulation results. The drop point was a little different from the critical point.

Thesis supervisor: Kenneth A. SmithTitle: Edwin R. Gilliland Professor of Chemical EngineeringThesis supervisor: Jefferson W. TesterTitle: He1mann P. Meissner Professor of Chemical Engineering

Director, MIT Energy Laboratory

DEDICATION

For my family:

Keiko KuboMiyako KuboEmniko Kubo

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.

_ __ _

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................................

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......................................

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

LIST OF FIGURES

Figure 1-1

Figure 1-2

Figure 1-3

Figure 3-1

Figure 3-2

Figure 3-3

Figure 3-4

Figure 3-5Figure 3-6

Figure 3-7

Figure 3-8

Figure 3-9

Figure 3-10

Figure 4-1Figure 4-2Figure 4-3

Figure 5-1

Figure 5-2

Figure 5-3Figure 5-4Figure 5-5Figure 5-6

Figure 5-7Figure 5-8Figure 5-9Figure 6-1

Figure 6-2

Typical phase diagram for a pure component.

The density, viscosity, and diffusivity of pure water at40 °C.

Density of pure water at 310, 330, and 360 K.

Basic Hahn Sequence

PGSE Sequence

Mean square displacements of SPC and SPC/E

Relationship between density and self-diffusion coefficient at 773K.

GEMC concept (from )

EOS fitting result in EOS method (from Guissani and Guillot, 1993)

Concept of density profile method in direct MD method (fromFigure 6-3

Figure 6-4

Figure 6-5

Figure 6-6

Figure 6-7

Figure 6-8

Figure 6-9

Figure 6-10

Figure 6-11

Figure 6-12

Figure 6-4

Figure 64

Figure 7-1

Figure 7-2

Figure 7-3

Figure 7-4

Figure 7-5

Figure 7-6

Figure 7-7

Figure 7-8

Guissani and Guillot, 1993 )

A unit cell in simulation

Density profile at each temperature (timestep= 0.5fs)

Density profile at each temperature (timestep=5fs)

Density profile of 512 molecules (Ly=Lz=2.7nm)

Density profile of 512 molecules (Ly=Lz=1.9nm)

Density profile of 256 molecules (Lx=Snm)

Coexistence curve from simulation results





Viscosity near critical point (Franck )

Binary diffusion coefficient in the vicinity of the critical point

Isochoric and isothermal approach in near critical region

Self-diffusion coefficient in isochoric conditions 1

Self-diffusion coefficient in isochoric conditions 2

Self-diffusion coefficient in isothermal conditions

The relationship between correlation length and temperature inTc=646K

Density in which the length of a unit cell is equal to the correlationlength at the given temperature

Figure 7-9 Radial distribution function, goo(r) of water in the near critical regionat 0.296g/cm 3

Figure 7-10 Radial distribution function, goo(r) of water in the x;ear critical regionat 646K

- - . A _ ,,,, , .......... u

LIST OF TABLES

Table 1-1 Typical property values for liquid, supercritical fluid, and gases.

Table 1-2 Critical points of selected solvents.

Table 3-1Table 3-2Table 6-1Table 6-2Table 6-3Table 6-4Table 6-5Table 6-6Table 6-7Table 6-8Table 7-1Table 7-2Table 7-3Table 7-4Table 7-5Table 7-6Table 8-1Table 8-2

LIST OF SIMBOLS

absolute Temperature (K)

Pressure (bar)Bulk density (g/cm3)

Interatomic potential energy of interaction (kJ/mol)

Lennard-Jones soft sphere diameter (nm)

Lennard-Jones energy well depth parameter (kJ/mol)

Cutt off length

charge

collision diameter

Boltzmann constant

time

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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

Introduction Page 1

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.

.Back2-_ I I 7 n _a,

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 1

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

about Under supercritical conditions, density and viscosity are directly related ( ). The

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 non-

asymptotic 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 )

kTD-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 = p0D0(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:

kBT , 1k=K exh 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

kTD--~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

kTDT (2-5)6nari,

Bc Ir oU1 -- I I7

Drl _ 1 1+3/3pa

kT 6ra 1+ 2 / a

where J is the coefficient of sliding friction. Li and Chang(

1=0 for small particles to be valid. The result is

kT

) considered the limiting case

(2-16)

Writing 2a as (V/NO)1/3 (V is molar volume and NO is Avogadro's number), we obtain:

D=~ kT (No 1 3D m = )

(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 1

P(AB )2 2D,AB MA MBAn DMB(2-)

2-4 Properties Near Critical Point

Scaling law

(2-5)

Background Page 7

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 vapor-

liquid 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-_ = DpI - Pll (2-6)C I PC I

i -P = BT -Tc (2-7)

PC 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,

k TD-D = AD(q) = (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 Jxri{ 5-5)T T r)

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 Paie 9

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 (3is 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.

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 + At + AtA + t2 A +...),

T-Tr

Ip -v=B toIAt=+ BAtAt2A (5-5)IPt -Pv- =BoAt3(1 + BAt + B2 At + ...), (55)

T-T =C( d-1)=Tc

-11

P-P =CT-Tj T-T <O (6-4)Pc TC 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

' I I Pa 13l

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

Backeround Pae 13

Lc_ -- _ r 1 ii

PreSsu

re

Temperature

Figure 2-1: Typical phase diagram for a pure component.

I · I

al

I I

Background Page 14

la2I

2

1.5

1.0

0.5

0.0

180

60

40

20

00 100 200 300 400 500 600 700

Temperature, °C

Figure 2-1 Dielectric Constant of Water

1.5

1.0

0.5

nt~

-10

-12

-14

-16

-18 Ko

Log-20

-22

-24

-260 100 200 300 400 500 600 700

Temperature, C

Figure 2-2 Ion Dissociation Constant of Water

Background Page 15--I-· I I · r I I

_ n

a.v

ExeIme l M e o S lII I 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

Page Expermental Measulrements Qf Sef-Diffuivit

Experimental Measurements of 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

X has 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, _ M,, (3-1)dt T2

M = 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)

Experimental Measurements of Self-Dffusivity _ _ Page 3

a D a2T (3-4)

at ax2

M,(X,t) = M (X,t)+iMy(X,t) (3-5)

M, (X,t) = TPexp -(2iTv o + )t-6)

aT ?X + a2T (3-7)at =- iygoX' + D aX(

'T(t) = A(t)exp(- iygoXt) (3-8)

T(t) = M(O)exp(- iygoXt)ex - Dy2g02t3 (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 (3-8)M, (2t) = M. exp[ - -(( y go

where Mt is the spin echo amplitude, T2 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 T2 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, 2 22A (3-9)M, (2t) = M0exp - - (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 2go262A) and yields an accurate value of D.

Experimenal Meas .r n of _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 (C,. ,, - C,p),ni 1

1t [(C1 orol-C 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

Vtop and Vbottm are 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 H20. Using mass effect correlation, self-diffusion coefficient can be obtained.

3.2.3 Other techniques

Tayler dispersion

Experimental Measurements qf Self-Dfjlfusivity Page 5


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. rDD is 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 H20 at temperatures between 277 and 318K for pressure up to about 2.2kbar. He also

measured the tracer diffusion coefficient of TDO in D20. 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.


Hausser et al.( ) reported Xs= is constant except close to the critical point. Krynicki

et al. ( ) 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 - 5

cm2/s 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 2-

D cross sections.

Eemt r t o e .f.D.. i ... s a e 9

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 Selbstdiffusions-

Koeffizzienten 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

Page 9Exierimental Measurements of Self-Diffusivitv

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 (D20) 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

Smlio o S ef -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:

d2rjFi = 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 Self-Diffucsivitv Page 1

Simulation of Seif-Diffusivity .. Page 2

As the Hamiltonian is constant and has no explicit time dependence, one obtaines, by

taking the total time derivative of eqn.(4-2),

Z - - i + . ie =odt i p i ri (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 + U1 2 (4-5)

=- p + U(r")2m

Taking the total time derivative of eqn.(4-4), one obtains for each molecule i,

d P * P1 + E11 =I (4-6)dt m' ' +, =0Comparing 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 ri =0 (4-9)i=l i=1 i=l

Since the work of constraint force caused by displacement is zero, eqn.(4.9) reduces to

N N

XFj1 rj - ja *ari = 0 (4-10)i=l i=l

The old coordinates are transformed into a set of new independent generalized

coordinates qj(j=1, 2, ..... , M), where

ri = ri(q, q2 ,q3 ,', ,qM, t)

i =1,2,3,- , N (4-11)

The generalized force Qj is defined to be

Qj -= Fi - (4-12)i=1 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.

SImliI f ei .- Df - - 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) (4-16)dt 2dt 2 3!dt

r(t - At) = r(t)- d(t At + d2r(t) At2 1 dr(t) At3 + (At4)dt 2 t2 3! dt3 (4-17)

Adding eqn.(4-16) and eqn.(4-17) eliminates all odd-order terms and yields the Verlet

algorithm for positions:

2d2r(t)r(t + At) = 2r(t) -r(t - At) + (At d)2 t + ((At)) (4-18)

Verlet algorithm utilizes the position r(t), acceleration d2r(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-Diffsivitv Page 4


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)r(t + At) = r(t) + v(t)At + 1 At2 dr(t)-20)

2 dt2 (4-20)

and

1 d2r(t) d2 r(t + At)lv(t +At) = v(t)+2[ dt2 + d(t + At) (4-21)

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 position-

oriented 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


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

Simulation of Self-Diffsivity Page 7

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 RCut is 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 mean-

field theory is often used to provide corrections to properties from interactions beyond the cut-

off 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 Rcut is 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:

US4' =4 (J -(C) ) (4-22)

By integrating the force from Rcut to infinity, long range correction term is calculated.

<Rcut

UL J = UL-J + 27Vp r 2UL-J (r)drL-l L-l Rcut (4-23)

<RcuI 87r Ca12 87c C 6

=UL + Ne- p Ne9 R.ut9 3 Ru3

In the case of Coulombic interaction:

1 qIqikUcoulomblc k 4 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., ):

1 qllqJk + If I -( rIkI II R. (4-25)Coulombic I~k 4rr Ik [ 2 ERF + 1J 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 1 qiqk + ERF--1 rk 3RF Uc.o?, = Z 1 / t+ , - (4-26)Coulombic k4 r,,/k 2ERF + R I )ct 2" +1 Rj

RF is infinity in eqn.(4-26) for the interactions between ions:

u =~C~ck4~,1 qIIqk [ 1 ( 4 Jk 3 'IkCoulombc 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/2 associated with it. Hence, one obtains:

Simulation of Self-Diff-usivitv Page 8

N PiZ =i mY2= K)= fkBT

i(£P ) / ) 2 2 (4-28)

and

T ( ) ( 3N )k m (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,,t is 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 mv

f(v)dv m e 2kT4 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

Simulation of Slf-Difusivitv Page 9


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/= sm (4-32)

The extra potential energy associated with s is

Us = (f + 1)kBTIns (4-33)

where f is the number of degrees of freedom and T is the specified temperature.

The kinetic energy associated with s is

I 2 2 .4-_l

K2 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 Ttarget is 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

Simulation of elf-Difusivitv Page 11

(tTJJtT (4-37)

where Turrent is the current kinetic temperature, Ttargct is 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 Pxy Pxz]

P= Pyx Pyy Pyz (4-38)

Pzx Pzy Pzz

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 = . m,v,. v, +~r,. f,P- l IVI a V +r, I, ] (4-39)1

whereiv N N

N L , *v (4-40)I mv, *v, = mv . v, mv * ,Et,v, *vi

LEm-1 Iv ,Im,v v, Em,v, * v,

and

Simulation of Self-Difusivitv Pae 12

Simulation of SeIf-Diffusivity Page 13

N N N Nrrfi (441)ZrL.' fXr, Nr. 'S ~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)

and

ANC (R) = EO(R) (4-45)

where Vi is the electronic wave function and D is the nuclear wave function. v only

parametrically depends on the nuclear positions.

Simulation of Sel-Dififusivity Page 14

(4-46)Hn = + Eo(R)

i1 2i

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. +U (4-47)

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 (4-

49) and the electrostatic terms obey the classical Coulombic point-charge interaction (4-50).

U~,~,.~1 = UL J = 4£L _ C~ 6 ~ (4-49)s1 q - iq (4-50)

U= =Uqqk (4-50)U., Coulombic I,,4mo ro,k 0 111,k

I


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.


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( ), in

which the configuration is based on tetrahedron and . 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 (4-)o i

Eel =YqiVj - iEii l

U = E i + EpoI

EU = jxviqidX = 1qV0 i

Simulation of Self-Diffusivity

(EzPO) = Yi i iiE i i 2

(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)jJj#1

T(r) =(3rr-2,/(47te 0 rij)3

a/Ua, = o

00

Page 18

Smlioo 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 2 1Uintra = -kOH(r-rO) +X-kHoH(O- 0)2

2 2

Table 4- Intramolecular parameters for (1987)

parameter

kOH kJ/(molA 2) 4637

ro 1.0

kHOH kJ/(mol rad2) 383

00 deg 109.47

Simulation o Self-Difusivitv Page 19


TR (Toukan and Rahman)

UOH =2 LIT(r12 +A123 )

2 2UHOH =LrrAr12ArI3 LrrA2(ArA + L Ar3 )Ar2 + 2+LAr 2 3

UDOH = [O(1-exp[- a12 b -- (1 exp[- Arl3 )b2j]

t=cjLrrDOHJ

m-TR

U intra = U OH + UHOH exp[- P3(Ar2 + Ar3 )jThe 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-;- ,-of Sl- If -f-f Pe2 iil

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---Modelsl-

NCCa

PSPCb

PPC

Number of forcecenters

3 charges + 2 dipoles

3 charges + 1 dipole

3 charge.

Timestep

(fs)0.5

0.5

1

Number ofiterations per step

6-9

7-8

3-4

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 Tjk is

the dipole tensor.

Simulation of Self-Diffusivitv Page 21

. .


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

Run-- A B C D EConditionl

POLl 0 3.169 0.155 -0.730 0.465H 0.000 0.000 +0.365 0.135

RPOL 0 3.196 0.160 -0.730 0.528H 0.000 0.000 +0.365 0.170

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.

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 =- with F= -alkBF 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-Diflusivitv Page 23

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

1 dilNrD = lim d (i (t) - ri (0)

6N t---oo dt I P

D = Idt(J(t) J(O)) = 2 Idt( vi(t)) (,vi(O))2[/aN ] i . i

= jaic_ - -1 7dt(JZ(t)-JZ(O)), jZ = ziev i (flux of charge)3kTV i

shear viscosity

rl = - dt(P (t) Pxy (0)) , P = mirnivxviyi + 'rixFa )kBT 0 ' j>i

The thermal conductivity X

x = 1 -JIdt(q(t) q(O)) qV =-I (mivi2)vi +-[ij(v + vj)+Fij (vi+vj)r]3kBTVO 2i 2 j

Smai o SI f Dfs y I 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 vO is 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 cm4P-

mTR 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

Simulation o Se~f-Diffjusivitv Page 25

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 P U Ds t(g/cm 3) (MPa) (kJ/mol) (cm2/s) (D)

270 1 1.3283 1 1.9314 1 4.3391 1 8.4

Check on accuracy Table 6-4: Result of ST2 model

T (K) Density P U Ds J1(g/cm 3) (MPa) (kJ/mol) (cm2/s) (D)

297 0.997 49 -42.7 2.5 2.35575 0.72 89 -25.8 34 2.35667 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) Density P U Ds experimen Ds(g/cm3)[ (MPa) (kJ/mol) (cm2/s) tal (cm2/s)

BF 294 4.3 300 2.30SPC 300 Tble 6-4: Rsuit of RPOL mOl i 300 2.30ST 83 I 1 7R - 283 1 9

TIPS2 T2(1) Density P 3. 3 2130(g/cm3) (MPa) (kJ/mol) (cm/s) (D)

297 0.997 -45 -42.0 2.7 2.62______ 0.66 7?e2 -2U3 -4.7 234 2.36667 0.66 170 -20.8 46 2.31

Simulation of Self-Diffusivitv Pa-e 26

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

It should be roughly an order of magnitude less than the

2p/wE

Einstein period

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

Condtio. Run- A B C D EConditionl

T (K) 673 772 630 680 771Density (g/cm3) 0.1666 0.5282 0.6934 0.9718 1.2840

DMD(Cm 2/s)

196 76 37 23 11

Dexp(cm 2 /s)193 68 44 33 28

Simulation Self-Diff-usivity Page 27

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 self-

diffusivity (Kalinichev,1992)

Mountain ( ) studied TIP4P supercritical water using 108 water molecules and

simulations of 3 .3 2 ps 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


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

SImlIonIII of S Iffusivil y 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

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97, 2659-RPOL

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Halley, J. W.; Rustad, J. R.; Rahman, A., J. Chem. Phys. 1993, 98, 4110

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simulations versus extended RISM computations. Mol.Phys., 1992, 77(4), 769; ibid,

1993, 78(2), 497

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electrolytes. Phys. Condens. Matter, 1994, 6, A141

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Si l .I I II Pe 5

Figure 4-1 2-dimension Periodic boundary condition

Simulation o Self-Diftusivity Page 35

i f S I I - iff J ,, I l

Assu

Acut - oo

Figure 4-2 Concept of long correlation

Simulation of elf-iffusivitv Page 36

-- o Ifl DII P 3

Figure 4-3 Lennard-Jones Potential with and without long

correction

Simulation Sef-Diffu'isivity Page 37

:tI

IJ£j

r (A)

Figure 4-4 Coulombic Potential between O and H with

and without long correction

Simulation of Self-Diffusivitv Page 38__ �I __ LC

Po ri oIf cIial Water ; I SP anI Simlio

CHAPTER 5

PROPERTIES OF SUPERCRITICAL WATER IN SPC AND SPC/ESIMULATIONS

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,.rcs including 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.

Proerties Of SUrerritical Water in SPC and SPCIE Simulations Page 1

Properties of Supercritical Water in SPC and SPC/E Simulations 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 (5-1)site-site

(I)(rLenard-Jones = 4ij( (5-2)

1 qiqj()(r)coulombic = 4 ie r (5-3)

47rEO r

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:

bi qj1 3 r2 (5-4)4Coulombic 4 r 2R cut

2Rcut3


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. Epol is calculated

by using eqn.(4 - ) in Chapter 4:

Properties of Supercritical Water in SPC and SPC/E Simulations Page 4E Y1= ~ ~ A.),(4-Epole -where a is polarizability which is 1.608x10O4°Fm ( Eisenberg et al., 1969 ), 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 , Epol is calculated as 3.74kJ/mol while Epol is 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 5-

1) . 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 self-

diffusion 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.

Properties of Supercritical Water in SPC and SPC/E Simulations _ Page 5

Table 5-1: Simulation result of SPC and SPC/E in ambient condition

Table 5-2: Simulation result of SPC in ambient condition

Althor This work This work This work Berendsen Berendsen1987 1987

Model SPC SPC/E SPC/E i SPC ' SPC/EEnsemble type NVT NVT NPT NVE NVE

Temperature 300 300 300 308 306(K)

Density 0.997 0.997 0.995 0.97 0.998(g/cm3)

Pressure 508±51 67+80 -11.4 -1 6(bar)

Potential energy - 41.2 - 45.8 - 46.5 a - 40.4 - 46.6(kJ/mol)

Corrected - 38.5 - 40.6 -41.3 - 37.7 - 41.4potential energy

(kJ/mol)

N LR rc | Time densi E T P D eference(nm) (psec) (g/cm ) (U/mol (K) (bar) (10 cm'/s)

216 SC 0.85 1.0 12.5 1 -42.2 300 3.6 Berendsen et al. (1981)

216 SC 0.9 1.0 2n 0.97 41.4 300 -0.9 4.3 Berendsen et al. (1987)

216 SC 0.85 1.0 50 0.996 -41.8 301 700 4.4 Telema ct al. (1987)

108 EW MC l -38.6 300 Strauch et al. (1989)

265 EW 0.985 1019 0.965 300 Alper ct al. (1989)

216 EW 700 0.997 -41.8 298 300 3.6 Watanabe et al (1989)

216 EW 200 1 -41.1 300 2000 4.5 Barrat et al. (1990)

216 EW 40 I -39.5 299 4.69 Prevost ct al. (1990)

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)

_ __ _____ I I


Table 5-3: Simulation result of SPC/E in ambient condition

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.

N LR rc Erf Time density E m T P D eference(nm) (psec) (g/cm) (kJ/mol (K) (bar) (10

)= = 0.9 - ~ cm_/s)

216 SC 0.9 1.0 27.5 0.998 -46.3 306 6 2.5 Berendsen et al. (1987)

216 EW 700 0.997 .46.7 298 0 2.4 Watanabe et al (1989)

256 EW 200 300 2.6 Guissani et al. (1993)

216 RF 0.92 150 1 300 2.43 Vaisman et al. (1993)

512 RF 0.9 1.0 10009 1.002 -47.0 300 -5 2.7 Smith et al. (1994a, 1995)

512 RF 0.9 62.3 1000 0.976 300 Smith et al. (1994a, 1995)

512 RF 0.9 o 1000 0.976 -45.9 300 -37 3.2 Barrat ct al. (1990)

360 SC 0.9 1.0 400 1.0013 44.28 307.4 2.51 Bez et al. (1990)

216 EW 6000 0.997 -46.64 298 -80 2.4 Smith et al. (1994b)

256 EW 500 0.998 -46.72 298 2.24 Svishchev et al. (1994)

216 EW 0.95 0.997 -46.3 300.6 4.4 Heyes (1994)

2160 EW 0.85 200 0.999 303.15 69 2.75 Balasubramanian ct al (1996)

512 EW 1.14 200 0.9956 303.15 -7 2.76 Balasubramanian et al (1996)

256 SC 100 0.997 298 2.58 Chandra et al. (1999)

[H F X I F ; g~~~~~~~~~~~~~~~

_ I __

Properties qof Supercnritical Water 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 parameters to the

simulated self-diffusion coefficient were investigated.

Properties of Supercritical Water in SPC and SPC/E Simulations

Table 5-4: Simulation result of SPC and SPC/E at 773K

SPC SPC SPC SPC SPC/E SPC/E SPC/E SPC/E

Ensemble NVT NVT NVT NVT NVT NVT NVT NVTtype

Temperature 773 773 773 773 773 773 773 773

(K)Density 0.125 0.25 0.4 0.6 0.125 0.25 0.4 0.6(g/cm 3)

Pressure 321 555 900 1890 285 469 694 1799

(bar) ±3 ±7 +10 ±23 +4 +6 +40 +

Total energy -7.2 -11.8 -15.8 -20.2 -8.8 -13.9 -18.7 -22.9(kJ/mol) +0.4 ±0.6 +0.5 ±0.6 ±0.5 ±0.6 ±0.6 ±0.5

a: the value is before correction by self-polarize energy

b: Including quantum correction (from Berendsen et al 1987)

Effects of temperature control

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.

Page 8

Table 5-4 The effect of mass stochastic collision

model SPC SPC SPC SPC SPC/E SPC/E SPC/Etimstep(fs) 5 5 0.5 0.5 5 0.5 0.5frequency(collisions /50ps) 2 20 0 20 2 0 20D (cm2/s) 1 105 70 101 78 95 95 78

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, Rcut is 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 Rcut at 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)0.791.87

,,,,

Pressure

(bar)178.8177.2

Total energy(kJ/mol)

-12-12

Diffusion coefficient

(1 05cm2/s)

226.1221.9

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

Page 9Propoerties _qf Suercritical Water in SPCand SPCIE Simuulations

- n I I 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 ERF is 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=80 and infinity, D changed

10%. Since the difference between use eRF=80 and infinity with respect to potential curve is so

small, it is easily predicted that eRF strongly 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

Pressure Total energy Diffusion coefficient(bar) (kJ/mol) (1 0 5cm2/s)

Infinity 178.8 -12 226.180 181.4 -11.8 204.2

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.

Properties of Suercritical Water in SPC and SPCIE Simulations Page 10


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 51cm2 /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


D= kT (5-5)C,lra~,

where kB is the Boltzmann constant, a is the hydrodynamic radius and CSE is 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.

k8T (5-6)a-

- [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


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. In addition,

SPC/E provided the diffusion coefficient similar to the experimental one at 773K, but the

diffusion coefficient by SPC was also higher. As a result, it is elucidated that SPC/E is suitable for

the simulation of supercritical water.

Poete of II Wt in SP an S/ SImuaIons Pg 14

References

Allen, M. P.; Tildesley, D. J., Computer Simulations of Liquids (Oxford Science, Oxford, 1987)

Alper, H. E.; Levy, R. M., J. Chem. Phys., 1995, 99, 1322

Auffinger, P.; Beveridge, D.L., Chem. Phys. Lett., 1995, 234, 413

Baez, L.A.; Clancy, P., Existence of a density maximum in extended simple point charge water. J.

Chem. Phys., 1994, 101, 9837

Barrat, J.-L.; McDonald, I. R., The role of molecular flexibility in simulations of water. Mol.

Phys., 1990, 70, 535

Belhadj, M.; Alper, H.E.; Levy, R.M., Molecular-dynamics simulations of water with Ewald

summation for the long-range electrostatic interactions. Chem. Phys. Lett., 1991,179, 13

Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Herman, J., in Intermolecular Forces,

edited by B. Pullman (Reidal, Dordrecht, 1981), pp3 3 1 -3 4 2

Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P., The Missing Term in Effective Pair

Potentials. J. Phys. Chem., 1987, 91, 6269

Bernardo, D.N.; Ding, Y.; Krogh-Jespersen, K.; Levy, R.M., An anisotropic polarizable water

model - incorporation of all-atom polarizabilities into molecular mechanics force fields. J.

Phys. Chem., 1994, 98, 4180

Chipot, C.; Milot, C.; Maiggret, B.; Kollman, A.; J. Chem. Phys., 1994, 101, 7953

Coulson, C. A.; Eisenberg, ,Proc. R. Soc. London Ser. A, 1966, 291, 445

Guissani, Y.; Guillot, B.J., J. Chem. Phys., 1993, 98, 8221

Heyes, D.M., Physical properties of liquid water by molecular dynamics simulations. J. Chem.

Soc., Faraday Trans., 1994, 90, 3039

Hummer, G.; Soumpasis, D. M.; Neuman, M., Pair correlations in an NaCI-SPC water model

Simulations versus extended RISM computations. Mol. Phys., 1992, 77, 769

Hummer, G.; Soumpasis, D. M.; Neuman, M., Computer simulatin of aqueous Na-CI electrolytes.

J. Phys. :Condens. Matter, 1994, 6, A141

Properties Of SUercitical Water in SPC and SPCE Simulations Page 14


edlovsky, P.; Brodholt, J. P.; Bruni, F.; Ricci, M. A.; Soper, A. K., Vallauri, R., J. Chem. Phys.,

1998, 108, 8528

Kalinichev, A. G.; Gorbaty, Y.E.; Okhulkov, A.V., Structure and hydrogen bonding of liquid

water at high hydrostatic pressures: Monte Carlo NPT-ensemble simulations up to 10kbar,

J. Mol. Liquids, 1999, 82, 57

Kalinichev, A. G., Ber. Bunsenges. Phys. Chem., 1993, 97, 872

Kataoka, Y., Studies of iiquid water by computer simulations. IV. Transport properties of

Carravetta- Clementi Water. Bull. Chem Soc. Jpn., 1989, 62, 1421

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

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. J. Supercritical

Fluid, 1998, 13, 83

Matsubayashi, N.; Wakai, C.; Nakahara, M., Structural study of supercritical water. II. Computer

simulations. J. Chem. Phys., 1999, 110(16), 8000

Matsubayashi, N.; Wakai, C.; Nakahara, M., Structural study of supercritical water. II. Computer

simulations. J. Chem. Phys., 1999, 110(16), 8000

Mountain, R. D.; J. Chem. Phys., 1989, 90, 1866

Mountain, R., J. Chem. Phys., 1995, 103, 3084

Neumann, M., The dielectric-constant of water - computer-simulations with the MCY potential

Neumann, M.; Steinhauser, O.; Mol. Phys., 1980, 39, 437

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 Properties of

Water and Steam '99


Prevost, M.; van Belle, D.; Lippens, G.; Wodak, S., Computer-simulations of liquid water -

treatment of long-range interactions. Mol. Phys., 1990, 71, 587

Shostak, S. L.; Ebenstein, W. L.; Muenter, J. S., J. Chem. Phys., 1991, 94, 5875

Smith, D.E.; Dang, L.X., Computer-simulations of NaCI association in polarizable water. J.

Chem. Phys., 1994, 100, 3757

Smith, P.E.; van Gunsteren, W.F., Consistent dielectric-properties of the simple point charge and

extended simple point charge water models at 277 and 300K. J. Chem. Phys., 1994, 100,

3169

Smith, P.E.; van Gunsteren, W.F., Reaction field effects on the simulated properties of liquid

water. Mol. 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

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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 T20 - 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


300K, 0.997g/cm3

150

time (psec)

200 250 300

mean square displacements of water molecules by SPC and

SPC/E at ambient conditions

7

6

N2CE(tE

2EUCTI.

{ao0

E~0CLuS

2

5

4

3

2

+ SPCE

3 SPC''' y = 0.0243x + 0.0625 In-Linear (SPCE) R 0 9993 -Lnear (SPC)

'or _ ~y = 0.01 52x + 0.0418

!,.,,.,~~~~~~~~~~R , 0.9988 o

,,~ e .. ........

1

I I050 1000

Figure 5-1

I L __ L �_ ____ _� ___�__I

1


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))

Page 18

2500

2000

Z,~ 1500

JL-

U)0u

o. 1000

500

0

_ _ __ I I

^'^^


* 0.125g/cm3

* 0.4g/cm3-ULinear (0.125g/cm3)

Linear (0.4g/cm3)

0 100 200 300 400 500 600

time (ps)

Figure 5-3 mean square displacements of water molecules at 773K

773K

---vuU

800

C 700E

C 600a)Ea)Oco 500

.ea)X 400

v' 300

0E 200E

100

0

C · L _ _ U L - ��- - -s - �I

Properties of Supercritical Water in SPC and SPC/E Simulations P'age 20

Isothermal (773K)

JOU

300

iZ 250Eu

T

200

10u

0o 1500

u,J2 100

50

0

0 0.1 0.2 0.3 0.4 0.5

Density (g/cm3)

Figure 5-4

0.6 0.7

Diffusion coefficient vs density at 773K

_ _ _ _ _ L _

nrrr


350

300 - * SPC-flex(Mizan et al)

*0~ *0 ~~SPC*SPCE

250 - . O Experimental(Lamb et al)

O200

O

150

100-

O e -50

0

0 200 400 600 800 1000 1200

Pressure (bar)

1400 1600 1800 2000

Diffusion coefficient vs pressure at 773K

E0o.2-

000o

0

.,'6ID

- - - .. .. I ii -- -·I I - - �I - - I

Figure 5-5


1.OE-021.OE-03

timestep (ps)

Diffusion coefficient vs timestep at 773K and 0.25g/cm3

160

140

4* o 773K, .25g/cm3.

-

E

rlcvu0)

._

000._

._

120

100

80

60

40

20

01.OE-04

_ __ __ __ I I

Figure 5-6


0 3 0.35 0.4 0.45 0.5 0.55 0.6, , ,%(.7 0 1

02o 020 I

02 03

020 00

o00o

* O Couipmb(SPC)-HummerCoulpmb(SPC/E)-Hummer infinity

A Coulpmb(SPC/E)-Hummer 80a Coulpmb(SPC/E)-Hummer 1E Coulpmb(SPC/E)-Hummer 0

a~~ O .

r (A)

Figure 5-7

3

Coulombic potential between H and O in SPC/E model

with and without Hummer's Reaction Field

Page 23

-10

-20

-30

0S

0--ar

C-tD

-40

-50

-60

-70

-80

-90

-100

__ I IC __ __ L


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Density (glcm3)

Figure 5-8 Diffusion coefficient vs density at 625K to 873K

4UU

350

300

E.o 250c0@

:: 200o0

u*° 150:3=

100

50

0

_ __ ___ __

AAA


loo0

900

800

700

600

.2 50040

400

300

200

100

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Density (g/cm3)

Figure 5-9 Pressure -density-Temperature data at various conditions

__ __ I ----- II J-Pag 25


500 1000 1500 2000 2500Pressure (bar)

Figure 5-10 Diffusion coefficient vs pressure at 625K to 873K

Page 26

400

350

300

i 250

; 200

c150c 150

100

50-

0

r=0.125g/cm3

-c- 673K(experimentol)-- 773K(experimentol)-o- 873K(experimentol)X 625K* 652K* 673K+ 723KA 773K* 873Kr=0.25g/cm3

r=0.4g/cm3r=0.6g/cm3

0

�· _· · _ II_� _�______ II

I r j r

Properties of SupercriticalWater in SPC and SPC/E Simulations

Ar tJU

400

350

300

E-250

. _

§ 200

-5

150

100

50

0

Page 27

0 2 4 6 8 10 12

1/density (cm3/g)

Figure 5- 11 D

____ L__ L

vs /p


Figuie 5-12 pD vs temperature


1000 1500 2000

Page 29

2500

Pressure (bar)

Figure 5- 13 pD vs pressure

45

40

35 -

30-

25 -

20 -

E

C.

A AA

[

X 625K

* 652K

* 673K

+ 723K

A 773K

· 873K

15 -

10-

5

0

0 500

-· -- I -- I

---.

-1


0.2 0.3 0.4 0.5 0.6 0.7 0.8

Density (g/cm3)

CSE vs density

Page 30

7

6

5

4

u

C.)

3

2

1

0

Sticking limit

0

Slipping limit S

A

- -- _ _ O____ _ _ __ _ - - - - - - _ _-_-_ __-- _ -- _-- _--

-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - -. 6 7 3 K--- 773K

* 873K_ - -_ _ _ _ _ _ - - -_ _ _ - --_ - __- - __- - _ _ _ - - _ - -_ - -__ - -__ - - _ _ _

0 0.1

__ ___ L ___ __

Figure 5- 14


qU

350

E300

LI

LO

a' 250

150

0E

0= 200

150

100100 120 140 160 180 200 220 240 260 280 300

diameter from eqn.(5-6)

Figure 5- 15

Page 31_ _ __ __ __ L I __ ___ �_ I

Ann


Activation energy

y = -1097x + 7.01R2 -0 9311y = -1i/8.9x + .6.

R2 = 0.8867 .125g/cm3

... 0..25g/cm3y = -967.46x + 5.753 0 9-._3R2 = 0.9618 -_u/cm

0.6g/cm3y = -1124.9x + 5.5143

R2 = 0.9989

0 0 0 0 0 0 0 0 0

0.0009 0.001 0.0011 0.0012 0.0013 0.0014 0.0015 0.0016 0.0017

1/T(K71 )

Figure 5-15

Page 32

7

6

5

U)

E

0C

4

3

2

1

* InD(0.125)

o InD(0.25)

* InD(0.35)

A InD(0.4)

InD(O.582)

a InD(0.6)

-Linear (InD(0.125))

- Linear (InD(0.25))

-Linear (InD(0.4))

-Linear (InD(0.6))

0

0.0008

- · · · .·I -- - I Y

_

Fuur Work a Ret io n I e dI nI IaI 1

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.( ) proposed simulated in a new

way where they set a different dipole moment based on at . This simulation was on the way

but successful for

Page 1Future Work and Recommendation

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 , The diffusivity of solute are analyzed by

Taylor dispersion ( )

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 (1-1)

6


k T KS(1-1)

k=K K,h c

1 (MRT) 1 2 (1-1)33 3/2 Nd 2

1 (RT) 3/2 (1-1)DAB 37C3/2 NAvd 2 M1 2 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


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


Simuation

Ion

Organic solute

Dilute solutions

Concentrated solutions

Polarizable model was effective for water-C02 binary.


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

Estimation of the critical point in SPC/E water 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.


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)


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 + B2x2 + B3 x3 + B4 x4 + Bl x10 (65)pkT (6-5)

-(-) x(Clx +2C 2x2 +3C 3x3 +4C 4x4 + 5C5x5 )T*

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).


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. LX is 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

Estimation of the critical point in SPC/E water . 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 ) tan d X ) (6-6)where PL is 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)

dp(z )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


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. T Pv CT T (6-8)

T = C (PL - P )"A + TC(6-9)

where f3 is 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.

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 Pc references

(K) (g/cm3) (bar)

SPC 587 0.27 196 de Pablo et al. 1990

SPCG 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 of the critical oinzt i SPCIE water Page 7

Table 6-2: Critical Point of SPC/E Water

__ _ _ _ _ _ _ 1 (1) (2) I (3) jI (4) I (5) I (6) jAuther Errington Guissani Guissani Alejandre Liew Real_ 11 (1998) (1993) (1993) (1995) (1998a) water

Tc (K) 639 651.7 640 636.5 630.4 647.17PC (g/cm ) 0.262 0.326 0.29 0.303 0.308 0.322

Pc(bar) . 189 160 . . 221Estimation

Method GEMC EOS EOS DS DS

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 1000C to 3000C. 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 10x10-5cm2/s

(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

Estimation the ctical point in SPCIE water Page 8


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.

Es n IIf th e I Ia I poi ntI I SIII E Iw aI e I I 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 orthobaric densities 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

Estimzation the citical point in SPCIE water Page 10

Estimation oqf the critical point in SPC/E water 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 of the critical pi i SPC/E wae [ [e 12Table 6-3: Orthobaric densities of SPC/E model

0.5fs

5fs

T(K)373

423

473

523

573

373

473

548

PL(g/cm 3)0.9212

0.869

0.7836

0.7238

0.5699

0.9193

0.7865

0.6307

Pv(g/cm 3)

0.00173

0.00208

0.00742

0.02954

0.06533

-0.00053

0.00959

0.06687

PL- PLe x p

(g/cm3 )

-0.037

-0.048

-0.082

-0.075

-0.198

-0.069

-0.078

-0.128

Pv- pVexP(g/cm3 )

0.0011

0.0004

0.0005

0.0070

0.0233

-0.0011

0.0018

0.0208

d(nm)re

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. =--0.325 is 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=0.325 and =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.

___I _1 _ ___Estimation of the critical point in SPCIE water Page 12

I I I

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. (6-

10). 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 f3

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 Pc " 1 Pc2 32

(K) (g/cm 3) (g/cm3 )

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 0.2959 0.375 0.3151 0.357

641 0.2927 0.390 0.3124 0.368

646 0.2895 0.404 0.3097 0.38

651 0.2863 0.3070

Estimation of the critical point in SPCIE water Page 13


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).


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, 150-

151, 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 liquid-

vapour 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

Estimation qf the critical point in SPC/E water 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 D20. 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. Supercritical Fluid,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


Strauch, H.J.; Cummings, P.T., Comment on molecular simulation of water along the liquid-

vapor 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

Etmto f th I I I S wt Ie II

(1)

0 J 0

0 0_ _ _

- (2)

E,+AENV,

E2 +AE2N2

V2

E2

N2V2

E,N,V,

(3)

E, +fE, E2N, N2V, +AV V2 +AV

E, +AE, E2 +E 2N, +1 N2-1V, V2

Figure 6-1 Concept of GEMC

0 0 o * 0

0 ~0 ,0 0@**e. 6I:' '~al~.--[-Q

__

- |

Estimation of the critical point in SPCIE~ water Page 18

\

EsIm I II cl -- -

Figure 6-2 Example of EOS fitting (from Guissani, 1993)

Estimation o the critical point in SPCIE water Page 19

E i-- i o i - ii i pi i S wt III 0

Liquid

Vacuum Liquid Vacuum

Geometry of a unit cell of direct MD simulation of two coexisting phases

vapor phase I interface I liquidphase I interface Ivaporphase

Den! EITY ~ ~ ~ ~ V·~~IUI~ ,.,,,,. ,.,, Hlas,

isity of gas

Density of profile after simulation

Figure 6-3

- - -

Estimation of the citical point in SPCIE water Page 20

lpnitv nf linllil

Figure -4

Estimation of the critical point in SPC/E water

# of molecules: 256 or 512

Water at 25°C, 0.997g/cm3

equilibrated

L z Vacuum

Ly

Liquid Vacuum

Lx = 7nm(256), 10nm(512)L = 1.97nmL = 1.97nm

density 0.3-0.4gIcm 3

periodic boundary condition

Figure 6-5 Initial configuration of a unit cell in MD

_____ I_ ___ I__ _ ~~~~~Page 21

c-

Ei f e c a oi in SC we Pg 2

.b 0.e.A

o --2 -1.5 -1 -0.5

* rho-100C

#REFI-#REFI

A rho

- 250C* rho

-300C

0

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

1.0

0.9

0.8

0.7

0.6

0.5

0.4

E

u)

aC0

0.3

0.2

0.1

0.0-3.5 -3 -2.5

Estimation of te critical voint in SPCIE water Pape 22

Estimation of t crtia p i

-2.5 -2 -1.5 -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

1.0

0.9

0.8

0.7

1 0.6

0.5

0.5

a1:03

0.3

0.2

0.1

0.0

1-w* rho

-- 100,A rho

-- 200* rho

- 275

* rho

-300

-3.5 -3 0

Page 23

--

,_,,- --

04ir -

1-4

-. M- 1W - -

-1A.-

Bibrl---- ------

I i

Estimation the critical point in SPCIE water

I

IIo f th -Il-- p- in SI w P 4

Figure 6-8 Generation of a bubble inside the liquid phase at 300°C



(a) · .,

I.U

0.9

0.8

_ 0.7CO

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

X (nm)

(b)I.U

0.9

0.8

h 0.7

EE 0.6

,, 0.5

0.40 .3

0.3

0.2

0.1

0.0-5 -4 -3 -2 -1 0 1 2 3 4 5

X (nm)

Figure 6-9 Density Profiles at different timesteps

(a) 5fs timestep at 100°C (b) 0.5fs timestep at 100°C

_ · I _ _ _ __ I __


0.9

0.8

0.7

A

E 0.0

0.4a

0.3

0.2

0.1

0.0

1 .U

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

-5 -4 -3 -2 -1 0X (nm)

2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5x (nm)

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

Page 26

(b)

E

4.a.a-C0(n

I ___ I - - - - _ - . L - _- ___��____. �_

/,\. A


(a)1.0

0.9

0.8

0.7

0.5

c 0.4

a0.3

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

'ag0.42r 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

Page 27_


(a)

E

0,C

-5 -4 -3 -2 -1 0 1 2 3 4 5

X (nm)

(b)

C'E

-)

CaP,a

-5 -4 -3 -2 -1 0 1 2 3 4 5

X (nm)

Figure 6-12 Density Profiles at different cell size

(a) Lx=7nm (b) Lx=5nm

I _ _ _ _ _

- f th crIl I S - I we PaI 2l I

/UU

650

600

550

500

450

400

350

3000.1 0.2 0.3 0.4 0.5 0.6

Density (g/cm3)

0.7 0.8 0.9 1.0

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)

U)I-C2

SH)0.E0

0.0

Estimation th citical point in SPCIE water Page 29

-n-

i th cil poI In S w Pg i

Tc=616.2K, rc=0.308g/cm3, P=0.325

1

0.9

0.8

0.7

) 0.6c)

>, 0.5

c 0.4

0.3

0.2

0.1

0

300 350 400 450 500 550 600 650

Temperature (K)

- - (rhov+rhol)/2__- rL

rV

· rL(O.5fs)* rV(O.5fs)A rL(5fs)A rV(5fs)

---- rL(expl.)---- rV(expl.)

X r+r/2

700

Figure 6-14 Coexistence curve of SPC/E water at 13=0.325

(Tc=616K, pc=0.308g/cm3 )

Page 30Estimation o the critical point in SPCIE water

EImaIonI ofJ the cia oI I S I e P

Figure 6-15 PpT relationship of SPC/E water at 0=0.325

(Tc=616K, pc=0.308g/cm3 )

Estimation of the critical oint in SPCIE water Page 31

- -i m ai -- o SP/ w l 32

Tc=646K, rc=0.2895g/cm3, P=0.404

350 400 450 500 550 600 650

Temperature (K)

Figure 6-16 Coexistence curve of SPC/E water

at Tc=646K and pc=0.2895g/cm3

41

0.9

0.8

0.7

< 0.6

>, 0.5u,- 0.4

0.3

0.2

0.1

0

- - (rhov+rhol)/2rL

rV

* rL(0.5fs)* rV(0.5fs)A rL(5fs)A rV(5fs)

---- rL(expl.)-- -- rV(expl.)

X r+r/2

300 700

---------


Tc=626K, pc=0.3023g/cm3, Pc=140.8bar

1.5 2

- Tr1.3946- Tr=1.2348- Tr=1.1550- Tr=1.0751- Tr=1.0411-Tr--0.9984* 873K(spce)o 773K(spce)A 723K(spce)A 673K(spce)0 651.7K(spce)0 625K(spce)

2.5

Reduced density

Figure 6-17 The relationship between reduced density and reduced pressure



4.5

4

3.5

a~~~~A L

-- ,. " ---i i J

a)

(0

L.

'O

0na)

3

2.5

2

1.5

0.5

0

0 0.5 1

_ ___ __ I I ___ _ _ IL _____ I

Estimation of the 'r I tic I In we

Tc=631 K, rc=0.2991 g/cm3 Pc=1 51.6bar

- Tr=1.3835

-- Tr=1.2250

- Tr=1.1458- Tr=1.0666

- Tr=1.0328

- Tr=0.9905

* 823K(spce)O 773K(spce)

A 723K(spce)

A 673K(spce)* 651.7K(spce)

o 625K(spce)

2.5

Reduced density



4.5

4

3.5

S

0

9/"/ / //A

A

- ~ ~~~~ ~ O- U U H -- -- ^- 7 El

IX/ I I

a)

U)0)L0.0

'O0'

0)

3

2.5

2

1.5

1

0.5

0 0.5 1 1.5 2

Page 3Estimzation o the ctical point in SPCE water


Tc=636K, pc=0.296g/cm3

4.5-0

4 0

3.5 I I Tr=0.9821

_./ -Tr=1.0240

3 3 -Tr=1.0575

(/) . -Tr=1. 136080L2.5 -Tr=1.2146

L '. ' · ^ -Tr=1.371780) 2 823K(spce)O A

0 / °/ A/ 0^ / 0 773K(spce)'O 1.5 //J a 723K(spce)at //A 673K(spce)

1 ] 651.7K(spce)

0 625K(spce)0.5

0 0.5 1 1.5 2 2.5

Reduced density



- I I II-- I- -_- - I __

s- -I -- - - -thII U I

Tc=641 K,0.2927g/cm3

1.5 2

-Tr=-1.3619

- Tr=1.2059

- Tr=1.128

- Tr=-1.0499

-- Tr=1.0167- Tr=0.9750

* 823K(spce)

O 773K(spce)

A 723K(spce)

A 673K(spce)

m 651.7K(spce)

0 625K(spce)

2.5

Reduced density



4.5

4

3.5

U)

CDC)L.

0.'O

:3')

n-C)

a

0Z~~~~~~

/ // //0

-~11/ " /'z ///O - /00 *-//

//3 D -3 / 3 1

1 /// I

3

2.5

2

1.5

0.5

0

0 0.5 1

Estimation of the critical oint in SPCIE water Page 36

-sa f t i p I . . I nIl __In SP/ w P 3

Tc=646K,rc=0.2895g/cm3

2

- Tr=1.3514- Tr=1.1966- Tr=1.1192

Tr=1.0418-Tr=1.0088- Tr=0.9675

* 823K(spce)o 773K(spce)A 723K(spce)A 673K(spce)* 651.7K(spce)o 625K(spce)

2.5

Reduced density



4.5

4

3.5

20(0u)U)

ELla0

'0n-Me

3

2.5

2

1.5

0

/a I -- ,' A-~~~~// oyX // X~~~~// /

0 /3 0 0

1

0.5

O

0 0.5 1 1.5

Estimation o the critical point in SPCIE water Page 37

S IfIon of waIII in nea cIIcIl ri Pe 1I

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

Sef-Difusi'on f water in near critical reion Page 1

Self-Diffusion of 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.


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)


where v=0.630, and A=0.51 are universal exponents, and o=0.13x10' 9m 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 :


rl =1 + (q~)= 1 +l n(q5) (q) (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


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 D20, 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

Properties of Water and 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-Diffusion of water in near critical region

65

X60

55

a.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r~50 ...

Co

45

._40 -]IA 4 Oh 374.2CUr0 Ah 374.5C0A 35 Oh 375C- 35

X h 375.5C+0 *h 376C

30 +h 377CXh 380COh 390C

25

200 0.1 0.2 0.3

Density (g/cm3)

0.4 0.5 0.6

Figure 7-1 Viscosity of water near critical region

Page 8-- ---

Z Add ^t e * - |~~~~~~~~~~~~~~~~~~~~~~~

Figure 7-2 Thermal conductance of water near critical region

- .-Page 9Seldf-Difcusion water in near critical reion

Slf-Di -f-io of wate I

Diffusion coefficient of binary mixture near critical point

__ _ Page 0Self-Diffu~sion of water in near critical reion

Figure 7-3

f I e I

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

0.8

0.7

0.6

0.5

,, 0.i,>, 0.4

C

O 0.3

0.2

0.1

0400

Page 11Self-Diff~sion of water in near critical region

--


105

100

95

90

85

80

75

70

Pane 12

600 610 620 630 640 650 660 670 680

Temperature (K)

Figure 7-5 Diffusion coefficient of water in isochoric conditions 1

E

00-0.

c

A 0.2895g/cm3

0 0.2927g/cm3* 0.2958g/cm3

* [ 0.2991 g/cm30 0 0.3023g/cm3

0.308g/cm3

-- ----- ---

f-Dui i n ii i

Critical region

650 700 750

Temperature (K)

Diffusion coefficient of water in isochoric conditions

130

120

E

a,4.

v-

c

._oC,C~

110

100

90

80

70

60600 800

Page 3Sef-Dif~usion of water in near critical re

( = 0.326 g/cm))Figure 7-6 9

I f Io- - I I In ii i

130673K

120

652K

,-~110E 646K

' 10 0

:D 625K0 o 90C._o

70

60 0.2 0.25 0.3 0.35

Density (g/cm3 )

0.4 0.45

Diffusion coefficient of water in isothermal conditions

- . -Page 14Sellf-Difiusion cf water i nea critical regiono

Figure 7- 7

IffuI .f Iwae I na i e P 1

14

12

10

E

-0, 8

C.

o

4

2

0

645 650 655 660 665

Temperature (K)

670 675 680

The relationship between correlation length and temperature in Tc=646K

Selt-'Diffusion of water in near citical region Page 15

Figure 7-8

Sffs o wate in I III eo Pe 1

1 .U

0.9

0.8

E 0.7

0.6

up

, 0.5

E= 0.4E

' 0.3

0.2

0.1

0.0640 645 650 655 660

Temperature (K)

Figure 7-9 Density in which the length of a unit cell is equal to the correlation length

at the given temperature

Page 16Self-Diffussion of water in near critical region

I -

-- i i n l rI -- I Pg 1

3.5

3.0

2.5

00)cmn

2.0

1.5

1.0

0.5

0.00.0

Figure 7- 10

0.2 0.4 0.6 0.8 1.0

Distance (nm)

Radial distribution function of water in the near critical region at

0.296g/cm3

Page 17Sef-Diffusion of_~! water in near criticazl reg ion


ooCD

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Page 18

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

_ __ Ut

I P 1

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 1

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 calculationprogram newdiftest2c--------------------------------------- .

c 5/17/2000 by Y.Kuboc

cc << set parameters >>c

implicit double precision(a-h,o-z)integer natomx,nmolx,npartx,nmolinfo,natminfo,nspecinx,nconfigmxparameter (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,leng2integer nql,nq2,moltrack,itrack,ntypesinteger nmoltot,npart,nspeciesinteger fstartnum,fendnum,flenlinteger lennam,cmcheckinteger ndigt,nzer,inumbinteger natom,iatom,ipart,sdata,fdata,sstep,fstepinteger dstep,ostep,idump,nnn,nninteger 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,densdouble precision rmdis,ttotmasdouble precision rdsig,sigz,rdeps,epsz,charge,rdcut,zcutdouble precision totmas,tmass,winv,tinit,temper,voldouble precision tstep,ddstep,oostepdouble precision sumt,sumtt,sumtd,sumx,sumy,sumzdouble precision slope,intecept,corr,diffcharacter*80 titlecharacter*40 fcur,finit,frec,filin,fend,faux,flencharacter*40 fpara,ftopo,freccm,frecdif,finpcharacter*40 file8,filer8,file9,filer9,fstringl,fstring2,fstringcharacter*80 names(nspecmx)character*l suffix,inum,enum

cparameter(lennam=5)data suffix/'.'/

200 format(A50,I6)

Appendix Page 2

202 format(A20,I6)210 format(A50,A)220 format(A50,lPG14.5)230 format(' ',A8,I6,A12)9001 format(a)9010 format(x,a,$)9011 format(x,il2)9012 format(x,lpgl4.5)9014 format(x,a)

cwrite(6,*) 'Diffusion coefficient calculation--parameters'write(6,*) '*********************************************$********************,

c write(6,*) 'Please input initial file(ex:spcstartOO00001)'read(5,9001) finitwrite(6,210)'initial file: ',finit

c write(6,*) 'Please input final file(ex:spcstart00100))'read(5,9001) fendwrite(6,210)'final file: ',fend

clengl=index(finit,' ')leng2=index(fend,' ')fstringl=finitfstring2=fend

Cc if (lengl.lt.leng2. or. lengl.gt.leng2) thenc write(6,*) ' file name is wrong'c stopc endifc

j=0jj=0k=Okk=Odo 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)

cc write(6,*) fstringlc write(6,*) fstring2

jj=jj+jkk=kk+knconfig=kk-j j +1l

10 continuec

fstring=fstringlfstartnum=jjfendnum=kkwrite(6,200)'initial number: ',fstartnumwrite(6,200)'final number: ',fendnumwrite(6,200)'# of data files: ',nconfig

Appendi _ Page 3

cif (fstringl.ne.fstring2) thenwrite(6,*) ' file name is wrong'stop

endifcc write(6,*) 'Please input sample data file (aux file)'

read(5,9001) fauxwrite(6,210)'sample data file: ',faux

c write(6,*) 'Please input molecular data file(topol file)'read(5,9001) ftopowrite(6,210)'molecular data file: ',ftopo

c write(6,*) 'Please input atomic data file(params file)'read(5,9001) fparawrite(6,210) 'atomic data file: ',fpara

c write(6,*) 'Please input MD parameter file(inp file)'read(5,9001) finpwrite(6,210)'MD parameter file: ',finp

c write(6,*) 'Please give new diffusion file name'read(5,9001) frec

cc lengl=!ndex(frec,' ')c leng2=lengl-lc freccm=frec (1:leng2)//'cmOOOOO'c f--2'-df=frec(l:leng2)//'dif'c ,½;,_a6,2l0) 'each molecular configuration file: ',freccmc :.'i:5 2l0) 'displacement file: ',frecdif

real(',*' cmcheckwji~ 6, *; '*********************************************

c << oCen initial file - to read npart >>c

open(unit=7,status='old',file=finit)read(7,*) nql,nq2read(7,*) npart

close (7)cc << open .aux file - to read mollist,moltype >>c

open(unit=3,status='old',file=faux)read(3,9001) titlewrite(6,*) titleread(3,*) nspeciesitrack=0moltrack=0nmoltot=0do 20 i=l,nspeciesread(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)

cwrite(6,9002) i,k,itmp(j)

Appendix Page 49002 format(' Species ',i4,' atom#',i4,' is of type ',i2)

if(k.ne.j) thenwrite(6,*) 'Error in deflist reading atom#',j,

$ ' in species#',iwrite(6,*) 'Read in ',k,itmp(j)stop

end if30 continue

do 40 j=l,nnmol(i)moltrack=moltrack+lmoltype (1,moltrack)=imoltype(2,moltrack)=itrackmoltype(3,moltrack)=natoms(i)do 50 k=l,natoms(i)itrack=itrack+1mollist(l,itrack)=kmollist(2,itrack)=moltrackmollist(3,itrack)=itmp(k)lspecies (k, i) =itmp(k)

50 continue40 continue20 continue

close(3)c

if(npart.ne.itrack.or.nmoltot.ne.moltrack) thenwrite(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 ifnmol=nmol tot

c write(6,*) 'Definition of molecule list successful'c write(6,*) 'Defined ',nmol, ' molecules.'

write(6,*) 'You have ',npart,' particles.'

c << open topo file - to read ntypes >>c

open(unit=l0,status='old',file=ftopo)read(10,*) ntypes

c write(6,*) ntypesclose(10)

cc << open params file - to read rmass >>c

open(unit=4,file=fpara,status='old')do 60 i=l,ntypes

read(4,*) j,rdsig,rdeps,rdcut60 continue

do 70 i=l,ntypesread(4,*) j,sigz,epsz,zcut

70 continuedo 80 i=l,ntypesread(4,*) j,rmass(i),charge

c write(6,*) j,rmass(i),charge

Pane 5Apvendix80 continue

close (4)

c << open MD file - to read tstep & idump >>

open(unit=2,file=finp,status='old')read(2,9001) titleread(2,9001) fdebugread(2,9001) filetread(2,9001) filein

c 'tolerance for meeting the constraints: ',tolc 'name for stats file: ',filel

read(2,9001) filelread(2,*) tolread(2,*) istart

c 'random number startup flag: ',istartread(2,9001) fstart

c 'root for file name: ',fstartread(2,*) inumb

c 'sequence number for restart file: ',inumbCINP5 file containing various interaction parmeters-- intermolecularCINP5 interactions, angle, and dihedral parameters.

read(2,9001) flj

CINP5 External temperature used in isothermal, brownian and collisiondynamics.CINP5 pext= external pressure. Note these may be ignored if other inputsareCINP5 set up not to use them.CINP5c 'temperature K/Pressure :

read(2,*) temper,pextc 'New box length in nm(0, to leave unchanged ): '

read(2,*) boxnCINP5 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,*) icmoptCINP5: Brownian friction term: acceleration= normal terms + stochastic termCINP5 - (gamma/mass)*velocityCINP5 stochastic term is also function of gamma and determined to have aCINP5 normal distribution, but so that equilibrium temperature is correct.CINP5 see papers by Gary Grest... Note mistakes in their formulas andCINP5 slightly different definition of gamma (by factor of the mass).CINP5c 'frictional damping coefficient:

read(2,*) gammac 'mean time between brownian collisions:

read(2,*) coltimc 'time step in picoseconds: '

read(2,*) tstepCINP5 REAL*8 WS,WQCINP5 respectively for temperature and pressure.CINP5 for Andersen Haml. these are piston massesCINP5 for Berendsen's bath these are proportional to the relaxation timesCINP5 ws= time(temperature) wq=time(pressure)/compressibility

---- _- I -- I L-.

Appendix

CINP5 use negative ws(wq) to not use thermostat(pressure-stat)CINP5

read(2,*) ws,wqCINP5 LOGICAL altoptCINP5 .true. = use Berendsen's techniqueCINP5 .false. = use Andersen's HamiltonianCINP5

read(2,*) altoptc 'Number of subgroups:

read(2,*) ngroupread(2,*) nstep

CINP5 collect statistics every "istat'th" timestepCINP5 (note: you must collect statistics from at least two timestepsCINP5 in every group or a divide by zero error will occur)

read(2,*) istatc 'Frequency for making restart dump:'

read(2,*) idump

close(2)

c read(5,*) tstepc read(5,*) idump

cc << get configuration of molecules by center of mass >>

iconfig=0.dOdo 90 i=fstartnum,fendnumiconfig=iconfif;+l

c << get file number >>ndigt=int(log(i*1.0000001)/log(10.d0))+1

c write(6,*) ndigtC lengl=index(freccm, ')

leng2=index(fstring,' ')do 100 j=l,ndigtflenl=mod(int(i/(l0**(j-l))),10)

c freccm(lengl-j:lengl-j)=char(flenl+48)fstring(leng2-j:leng2-j)=char(flenl+48)

100 continuecc filer8=freccm

fcur=fstringcc << make center of mass file of each data >>c open(unit=8,status='unknown',file=filer8)cc << 1. read ratom file >>

open(unit=7,file=fcur,status='old')read(7,*) nql,nq2read(7,*) npartread(7,*) temperread(7,*) tcur(iconfig)

cread(7,*) box(l),box(2),box(3)do 110 j=l,npartread(7,*) ratom(l,j),ratom(2,j),ratom(3,j)

Page 6· · I Y

Appendix Page 7

110 continuec write(6,*) ratom(l,l),ratom(2,1),ratom(3,1)

close(7)c

ttotmas=O.dOc << 2. calculation of center of mass of each atom >>

sumx=O.OdOsumy=O.OdOsumz=O.OdO

do 120 mol=l,nrmolrcm(1)=O.dOrcm(2)=0.dOrcm(3)=O.dOtotmas=O.dOnatom=-noltype (3,mol)

cdo 130 iatom=l,natomipart=moltype(2,mol)+iatom

tmass=rmnass(mollist(3,ipart))totmas=totmas+tmassttotmas=ttotmas+tmass

c 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)

130 continuec

winv=l.dO/totmas

cc < 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)*winvrcm(2)=rcm (2)*winvrcm (3) =rcm(3) *winv

ccm(l,mol,iconfig)=rcm(1)cm(2,mol,iconfig)=rcm(2)cm(3,mol,iconfig)=rcm(3)

c120 continue

if (cmcheck.ne.O) goto 90sumx = sumx/ttotmassumy = sumy/ttotmassumz = sumz/ttotmas

c

Appendix Page 8

do 105 mol=l,nmol

cm(l,mol,iconfig)=cm(l,mol,iconfig)-sumxcm(2,mol,iconfig)=cm(2,mol,iconfig)-sumycm(3,mol,iconfig)=cm(3,mol,iconfig)-sumz

105 continuec

90 continuecc

c

c fstring=fstringlc fstartnum=jjc fendnum=kk

calcnum=fendnum-fstartnum+1write(6,200)'initial number: ',fstartnumwrite(6,200)'final number: ',fendnumwrite(6,200)'# of data files: ',calcnum

c

dispsum=O.dOdisptsum=O.dO

c

cc < calculation of displacement of each molecule(nm) >c

open(unit=9,status='unknown',file=frec)c dstep- displacement time stepc ostep- origin time step

read(5,*) dstepddstep=dstep*tstep*idump

write(6,*)'displacement time step (ps): ',ddstepwrite(6,*)'= dstep*timestep*idump: dstep ',dstepread(5,*) ostepoostep=ostep*ddstep

write(6,*)'origin time step (ps): ',oostepwrite(6,*) '= ostep*dstep*timestep*idump: ostep',ostep

* norigin=Int(nconfig/ostep)

nnn= O. dOc << # of configuratons corresponds tc kinds of displacementsteps.>>

do 1900 nn=l,nconfigif (mod(nn-l,dstep).ne.0) goto 1900

disp(1)=0 .dOdisp(2)=0.dOdisp(3)=0.dOdispt=O.dOdispsum=O.dOendorigin(nnn)=int(Int((nconfig-nn)/dstep)/ostep)+1

Appendix

tdif(nnn)=tcur(nn)-tcur(1)do 2000 iorigin=l,endorigin(nnn)

ori=dstep*ostep*(iorigin-1)+1

do 2100 mol=l,nmoldisp(l)=(cm(l,mol,ori+nn-1)-cm(l,mol,ori))**2disp(2)=(cm(2,mol,ori+nn-1)-cm(2,mol,ori))**2disp(3)=(cm(3,mol,ori+nn-1)-cm(3,mol,ori))**2dispt=disp(l)+disp(2)+disp(3)dispsum=dispsum+dispt

cc write(6,*) disptt

2100 continue2000 continue

cclose(8)

c << average of displacement and temperature with time >>c write(6,*) dispsum

c tdif=tcur-tinitnnconfig=nnnavedis(nnn)=dispsum/(nmol*endorigin(nnn))

c sddis=sqrt(ABS(disptsum/nmol-avedis**2))

c write(6,*) disptsum/nmol,avedis**2c << write data in ...dif file as a line >>

write(9,*) tdif(nnn),avedis(nnn)

nnn=nnn+l

1900 continuewrite(6,*)'total time(ps)',tdif(nnn-1)

cclose(9)

cc << Least Square Approximation - Self Diffusion Coefficient >>c

sstep=0fstep=0calcnum=0stime=0.dOslope=0.dOintercept=0.dOsloped=0.dOresid=0.dOresidsum=0.dOresidave=0.dOdiff=0.dOcorr=0.dO

cc write(6,*) 'Please input diffusion file name:'c read (5,9001), frecdif

Pane 10Avvendix

c read 'Please input diffusion file name:',filecompc open(unit=7,status='old',file=frecdif)c zead(7,9001) frecdifc write(6,200) 'file name: ', frecdifc read(7,9001) titlec write(6,200) 'title: ', title

read(5,*) sstepsdata=sstep*dstep*tstep*idumpwrite(6,200) 'initial time(ps): ', sdataread(5,*) fstepfdata=fstep*dstep*tstep*idumpwrite(6,200) 'final time(ps): ', fdata

c read(7,*) calcnumc write(6,200) '# of data points: ', calcnumc write(6,220)'start point of calc.?(from 1 to ', calcnum

if (sstep.lt.1) thenwrite(6,*)'incorrect'stop

endifif (fstep.gt.int(nconfig/dstep)) then

write(6,*) 'incorrect'stop

endifwrite(6,200) 'start point: ', sstep

c

write(6,220)'end point of calc.?(from start to ', calcnumread(5,*) fstepif (fstep.lt.l.or.fstep.gt.calcnum.or.fstep.le.sstep) then

write(6, *) 'incorrect'stop

endifwrite(6,200) 'end point: ', fstep

sumt=0.dOsumd=O.dOsumdd=O.dOsumtd=O.dOsumtt=O.dO

3010

do 3010 i=l,sstep-1read(7,*) tdif(i),avedis(i)

continuec

do 3020 i=sstep,fstepc read(7,*) tdif(i),avedis(i)c write(6,*) i,tdif(i),avedis(i),sddis(i),calcnum

sumt=sumt+tdif(i)sumd=sumd+avedis(i)sumtt=sumtt+tdif(i)**2sumdd=sumdd+avedis(i)**2sumtd=sumrtd+tdif(i)*avedis(i)

3020 continuec close(7)cc 'calculation of least square approximation'

ccccccc

c

cccc

AnDendix

cstime=fstep-sstep+lslope=(sumt*sumd-stime*sumtd) / (sumt**2-stime*sumtt)intercept=(sumd-slope*sumt)/(stime)

cc 'error'

sloperrl=sumtd-sumt*sumd/stimesloperr2=((sumtt-sumt**2/stime)*(sumdd-sumd**2/stime))**0.5sloperr3=sloperrl/sloperr2sloperr=slope/sloperr3*((1-sloperr3**2)/stime)**0.5

c 'correlation function'c

corr=slope/sqrt(sumdd-sumd**2/stime)c 30 i=sstep,fstepc sloped=tdif(i)*slope+interceptc resid=(sloped-avedis(i)) **2c residsum=residsum+residcc 30 continuecc residave=residsum/(stime)

diff=(slope)*1000/6diffdev=sloperr*1000/6

cwrite(6,*) 'slope by least square approx.(nm^2/psec):',slopewrite(6,*) ' error', sloperrwrite(6,*) 'intercept by least square approx.(nm^2):',interceptwrite(6,*) 'correlation function:',corrwrite(6,*) 'Diffusion coefficient(10^-5cm2/sec):',diffwrite(6,*) 'Coefficient deviation',diffdev

c234567c

end

c

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Molecular Dynamics and Self-Diffusion in Supercritical Water

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