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