THE JOURNAL OF CHEMICAL PHYSICS 124, 204715 共2006兲
Molecular dynamics simulation of the density and surface tension
of water by particle-particle particle-mesh method
Bo Shi, Shashank Sinha, and Vijay K. Dhira兲
Department of Mechanical and Aerospace Engineering, University of California,
Los Angeles, California 90095
共Received 9 December 2005; accepted 30 March 2006; published online 31 May 2006兲
In this work, molecular dynamics simulation is performed to study the density and surface tension
of water for a range of temperatures from 300 to 600 K. The extended simple point charge
interaction potential for water is used. The particle-particle particle-mesh method, which
automatically includes untruncated long-range terms, is used for the Lennard-Jones and the
Coulombic terms. The results show that the long-range correction for the Lennard-Jones term is very
important for the calculation of surface tension. It is found that the calculated density and surface
tension of water fit well with experimental data for temperatures less than 500 K. Near the critical
temperature, the simulation results are off from the experimental data. © 2006 American Institute of
Physics. 关DOI: 10.1063/1.2199849兴
I. INTRODUCTION
Due to its ubiquity in our environment, water is certainly
the liquid most investigated through molecular dynamics,
since computer simulation was introduced in the 1960s. After
the Bernal-Fowler 共BF兲 model,1 a variety of interaction potentials for the simulation of water have been proposed. By
using these potentials, many studies have been conducted to
predict the density, surface tension, and other physical properties of water.
Guillot2 presented a review of the density calculations of
water using molecular dynamics. For ambient conditions,
several researchers have successfully calculated the density
by using some water potentials, such as simple point charge
共SPC兲,3 TIP3P, TIP4P,4 TIP5P, extended simple point charge
共SPC/E兲,5,6 and so on. However, all these popular models
failed when the system temperature was raised to near the
critical temperature. Liquid density was found to be lower
and vapor density to be higher than the experimental values.7
The reason for the error is that most of the interaction potentials for water are built empirically for the ambient conditions and are not correct at higher temperatures.
There is a large variation in prediction for surface tension of water. In 1988, Matsumoto and Kataoka8 reported a
value of 30.5 mN/ m at 300 K, which is much smaller than
the experimental value of 72 mN/ m. In 1987, Wilson et al.9
also calculated the surface tension for TIP4P model of water
at 325 K. They reported a surface tension value of
132 mN/ m compared to the experimental value of
68 mN/ m. Later, they claimed that their calculations had errors and reported a corrected value of 59 mN/ m. Another
result reported by Lie et al.,10 also underestimated to onethird of the experimental value at 300 K. Some researchers
have reported higher values. Zhu et al.11 calculated the surface tension of SPC water at 298 K and found a value of
123.6 mN/ m, about 70% higher than the experimental value
a兲
Electronic mail: [email protected]
0021-9606/2006/124共20兲/204715/7/$23.00
of 72.75 mN/ m. They claimed that their calculation had a
large statistical uncertainty and the error may be caused by
the SPC model itself. In 1997, Zakharov et al.12 studied the
surface tension for a water droplet using TIP4P potential
model. They evaluated surface tension as 54 mN/ m at
300 K. This suggested that the error in surface tension did
not only exist in the plane surface, but also in other geometric shapes.
In 1995, while using SPC/E model to simulate water,
Alejandre et al.13 found that the long-range interaction is
very important for surface tension. They introduced the
Ewald sum to correct the long-range interaction of the
Coulombic term and found that their results were closer to
experimental values. However, this method is only applied
for the Coulombic term but not for the Lennard-Jones term
in the SPC/E model. To compensate the discrepancies caused
by the long-range interaction of the Lennard-Jones term, they
introduced an empirical tail correction to their calculation.
Recently, while simulating an argon system, Sinha
et al.14,15 found that truncating long-range terms of the
Lennard-Jones potential function at 4.5␴ would cause errors
as high as 15% in surface tension, which can be eliminated
by using particle-particle particle-mesh 共PPPM兲 method for
long-range terms. They also suggested that the long-range
interaction of the Lennard-Jones term in SPC/E model of
water is not negligible. In this paper, the PPPM method is
applied to minimize the error in long-range terms for both
Coulombic and Lennard-Jones terms and no tail correction is
needed.
II. THEORY AND APPROACH
A. PPPM method
PPPM or P3M algorithms are a class of hybrid algorithms developed by Hockney and Eastwood.17 The basic
124, 204715-1
© 2006 American Institute of Physics
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204715-2
J. Chem. Phys. 124, 204715 共2006兲
Shi, Sinha, and Dhir
TABLE I. f p and g p in Ref. 16.
p
f p共x兲
g p共x兲
1
2
3
4
5
6
7
8
exp共−x2兲 / 共冑␲x2兲
共冑␲ / x兲erfc共x兲
2⌫关0 , x2兴 / 冑␲
2关exp共−x2兲 − x冑␲ erfc共x兲兴
共4 / 3冑␲兲关exp共−x2兲 − x2⌫关0 , x2兴兴
1
关共1
− 2x2兲exp共−x2兲 + 2x3冑␲ erfc共x兲兴
3
共16x4 / 15冑␲兲关共1 − x2兲exp共−x2兲 + x4⌫关0 , x2兴兴
1
2
2
4
冑 5
45 关exp共−x 兲共3 − 2x + 4x 兲 − 4 ␲x erfc共x兲兴
erfc共x兲
⌫关0 , x2兴
erfc共x兲 + 关2x exp共−x2兲兴 / 冑␲
exp共−x2兲共1 + x2兲
2
exp共−x 兲共6x + 4x2兲 + 3冑␲ erfc共x兲
exp共−x2兲共1 + x2 + x4 / 2兲
erfc共x兲 + 关exp共−x2兲共30x + 20x3 + 8x5兲兴 / 15冑␲
exp共−x2兲共1 + x2 + x4 / 2 + x6 / 6兲
idea of the PPPM method is to split the slow decaying potential function such as the 1 / rn term into two parts by the
following identity:
1 f共r兲 1 − f共␶兲
= n +
.
rn
r
rn
共1兲
There are two requirements for the splitting. 共1兲 The first
part f共r兲 / rn, called the short-range term, should decay exponentially with r increasing and become negligible 共or even
zero兲 beyond some small cutoff radius rc, thus the particleparticle 共PP兲 method with the neighbor-list method could
then be used without introducing large cutoff errors. 共2兲 The
second part 关1 − f共r兲兴 / rn, called the long-range term, should
be a slowly varying function for all r, so that its Fourier
transform can be represented by only a few k vectors with
兩k兩 艋 kmax in Fourier space. This permits an efficient and accurate calculation by using the particle-mesh 共PM兲 method
without introducing large aliasing errors. Since the field
equations are linear, the sum of these two parts gives the
solution for the potential function of the original problem.
However, these two requirements on the function f mentioned above leave a large freedom of choice. One choice
was proposed by Essmann et al.16
Using the definition of the Euler gamma function 共⌫兲
and the Fourier integral expansion of the Gaussian, the following equation is derived:
⌫共p/2兲
=
rp
冕
␤2
t
p/2−1
2
exp共− r t兲dt +
冕
⬁
t
p/2−1
2
exp共− r t兲dt.
␤2
0
共2兲
Here ␤ is an arbitrary positive number and p is a positive
integer. Following the derivation of Essmann et al. we get
1
= ␲3/2␤ p−3
rp
冕
R3
f p共␲兩u兩/␤兲exp共− 2␲iu · r兲d3u +
g p共␤r兲
,
rp
共3兲
where u is the coordinates of points in R3 such that −1 / 2
艋 ai · u 艋 1 / 2 and ai is the lattice vectors of the unit cell.
Functions f p and g p are given by
f p共x兲 =
2x p−3
⌫共p/2兲
冕
⬁
x
s2−p exp共− s2兲ds,
共4兲
g p共x兲 =
2
⌫共p/2兲
冕
⬁
s p−1 exp共− s2兲ds.
共5兲
x
In particular, for the cases p = 1 – 8, we have tabulated
f p共x兲 and g p共x兲 in Table I.
After splitting, the short-range part can be calculated using particle-particle method while employing a neighbor-list
method. The long-range part can be computed by an optimized particle-mesh method following these four steps:
共1兲
共2兲
共3兲
共4兲
Assign charge to the mesh.
Solve the field equation on the mesh.
Calculate the mesh-defined force field.
Interpolate to find forces on the particles.
In the second step, fast Fourier transform 共FFT兲 is used
to solve Poisson’s equation with O共N log N兲. The calculation
is called optimized because an optimized Green function is
applied to make sure that the total error is minimal. The
details can be found in Hockney and Eastwood.17
B. Force Evaluation
Consider the SPC/E potential model,
␾共rij兲 = 4⑀LJ
冋冉 冊 冉 冊 册
␴
rij
12
−
␴
rij
6
+
1 q iq j
.
4␲⑀0⑀r rij
共6兲
Here i and j are atoms in different molecules, rij is the distance between i and j. The Lennard-Jones 共LJ兲 parameters
are ⑀LJ = 1.0797⫻ 10−21 J for oxygen-oxygen, zero for both
hydrogen-oxygen and hydrogen-hydrogen, ␴ = 3.166
⫻ 10−10 m. In Eq. 共6兲, ⑀0 = 8.854187⫻ 10−12 F / m is the permittivity of vacuum, and ⑀r is the relative permittivity which
is 1 for the vacuum case. For oxygen atom, the charge
is qO = −0.8476e and for hydrogen, qH = 0.4238e, where
e = 1.60219⫻ 10−19 C.
This potential function has a Lennard-Jones term and a
Coulombic term. The Lennard-Jones term is only effective
for the oxygen-oxygen interaction and Coulombic term is
effective for all the atoms. Notice that for the Coulombic
term, p = 1. Following Table I, by dropping the constant
1 / 4␲⑀0⑀r, the short-range part of the Coulombic term of the
potential on atom i is
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204715-3
J. Chem. Phys. 124, 204715 共2006兲
Surface tension of water by PPPM method
␾C-SR,i = 兺 qiq j
j
erfc共␤r兲
.
r
= 兺 q iq j
j
冉冑
exp共− ␤2r2兲 +
␲
冊
erfc共␤r兲 r
.
r
r2
共8兲
This term decays exponentially with increasing r so that it
can be calculated by PP method.
Also following Table I, the long-range part of the Coulombic term of the potential can be expressed as
␾C-LR = ␲3/2␤−2
冕
R3
f 1共␲兩u兩/␤兲exp共− 2␲iu · r兲d3u.
共9兲
By introducing a reciprocal k vector from the discrete set
兵2␲n / L : n 苸 Z3其, k = 2␲u, Eq. 共9兲 can be written as
␾C-LR =
1
4␲ 2 2
兺 e−k /4␤ 兩˜␳共k兲兩2 ,
2V k⫽0 k2
共10兲
where the Fourier transformed charge density ˜␳共k兲 is defined
as
N
˜␳共k兲 = 兺 q jeik·r j .
共11兲
j=1
Note that the split for p = 1 gives the well known Ewald
summations.
To calculate the long-range accurately, an optimized
Green function is introduced to minimize the error,
Ĝopt共k兲 =
D̃共k兲 · 兺 m苸Z3 Ũ2共k + 共2␲/h兲m兲R̃共k + 共2␲/h兲m兲
兩D̃共k兲兩2关 兺 m苸Z3 Ũ2共k + 共2␲/h兲m兲兴2
,
共12兲
where D̃共k兲 is the Fourier transform of the employed differentiation operator, Ũ共k兲 = W̃共k兲 / Vc is the Fourier transform
of the charge assignment function per the volume of one
mesh cell, and
W̃共k兲 = h
冉
sin共kh/2兲
kh/2
冊
P
共13兲
is the Pth order assignment scheme which assigns the
charges to the grid and h = L / N M is the mesh spacing. R̃共k兲 is
the Fourier transform of the true reference force, given by
R̃共k兲 = − ikg̃共k兲˜␥共k兲.
共14兲
Here g̃共k兲 and ˜␥共k兲 depend on the choice of splitting function f. For p = 1 case,
g̃共k兲 =
and
4␲
,
k2
i is
FC-LR,i = − h3
兺
r 苸M
p
2␤
共16兲
Therefore, the final long-range Coulombic force on atom
Therefore, the short-range Coulombic force on atom i can be
obtained as
⳵␾C-SR,i
FC-SR,i = −
⳵r
˜␥共k兲 = exp共− k2/4␤2兲.
共7兲
共15兲
⳵␳M
共r p兲关␳ M 쐓Gopt兴共r p兲,
⳵ri
共17兲
where ␳ M is the mesh based charge density which is obtained
from the charge assignment. The term ␳ M 쐓Gopt can be expressed as
ឈ 关FFT
ជ 关␳ 兴 ⫻ FFT
ជ 关G 兴兴,
␳M 쐓Gopt = FFT
M
opt
共18兲
ជ is the discrete fast Fourier transform
where the operator FFT
ឈ is the inverse discrete fast Fourier transform. As ␳ ,
and FFT
M
Gopt are known, the long-range force can be easily calculated. Notice that, because Gopt can be computed in advance,
the PPPM method is much more efficient than the direct
Ewald summations.
The details of the calculation of Coulombic force by
using PPPM method can be found in Deserno and Holm.18 It
should be mentioned that there is an extra term called the
“dipole” in Ewald summations which cannot be obtained
from our splitting. This term is not large but should also be
included in the calculation,
Fdipole,i = −
4␲qi
兺 q jr j ,
共1 + 2⑀⬘兲V j
共19兲
where ⑀⬘ is the dielectric constant, and for vacuum ⑀⬘ = 1.
The Lennard-Jones term has two parts: 共1兲 the repulsive
term, 1 / r12, to model the electron cloud at small distances
and 共2兲 the attractive dispersive term 1 / r6. The 1 / r12 term
decays fast enough to be evaluated by particle-particle
method with a finite cutoff radius. The 1 / r6 term, however,
decays slowly and has to be evaluated by the PPPM method.
If we include the repulsive term 1 / r12, the total short-range
Lennard-Jones force is
FLJ-SR = −
冉冉冊
冉冊
␴
d
4⑀
dr
r
= 4⑀
␴
r
12
12
− 4⑀␴6
g6共␤r兲
r6
冊
6g6共␤r兲 − rg6⬘共␤r兲
12
− 4⑀␴6
,
r
r7
共20兲
where g6 is obtained from Table I with p = 6, and g6⬘ is the
derivative of g6.
The calculation of the long-range part of the LJ force is
similar to that of the Coulombic force. The only difference is
that because p is six, f 6 should be used, and g̃共k兲 and ˜␥共k兲
can be derived from f 6, respectively.
As the short-range parts and long-range parts of the Coulombic and Lennard-Jones terms are all obtained, the total
force on each atom can be easily calculated by direct summation.
C. Surface tension evaluation
The hydrodynamic definition of surface tension is the
isothermal work of formation per unit area of interface. At
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204715-4
J. Chem. Phys. 124, 204715 共2006兲
Shi, Sinha, and Dhir
the atomic scale, it can be expressed as the integrated imbalance of normal and tangential pressures near the interface.
␥=
冕
phase 2
关PN共z兲 − PT共z兲兴dz.
For a plane interface perpendicular to the z axis, tangential and normal pressure components are defined as
PN共z兲 = Pzz共z兲,
共22兲
PT共z兲 = 共Pxx共z兲 + Pyy共z兲兲/2.
共23兲
and
For pair potentials with fij = f ijrij / rij, the normal and tangential pressures are
冋
1
V
1
Pyy =
V
册
兺 m jv2xj + 兺 兺 rxij f xij ,
j
i⬎j
冋兺
j
j
册
共25兲
册
共26兲
j
1
V
冋
兺 rzij f zij
兺j m jv2zj + 兺
i⬎j j
.
If we divide the domain into a thin bins of thickness dz
parallel to the x-y plane, V = LxLydz can be obtained where Lx
and Ly are the lengths of the x-y plane, respectively. Equipartition of kinetic energy gives
冓
1
␳共z兲 兺 v2xj
2
i
冔冓
=
1
␳共z兲 兺 v2yj
2
i
冔冓
=
1
␳共z兲 兺 v2zj
2
i
冔
1
= kT具␳共z兲典V.
2
共27兲
Therefore, in terms of atomic position ri for atom i and
potential ␾, the expression for surface tension is
␥=
冓
冔
x2ij + y 2ij − 2z2ij
␾⬘ij .
2Arij
兺
兺
i⬎j j
冉冊
1
1 ⳵2 1
x2
=
−
,
r8 6r6 24 ⳵x2 r4
␥LJ =
冓兺 兺
j
i⬎j
␥C =
冓兺 兺
i⬎j
j
x2ij
y 2ij
+ −
2Arij
2z2ij
␴
rij
1 q iq j
4␲⑀0⑀r rij
12
−
冔
␴
rij
6
⬘
−2
Notice that in Eq. 共30兲 i and j are only for oxygen atoms but
in Eq. 共31兲, i and j are all the atoms in different molecules.
By dropping the subscripts and using the nondimensional form, Eq. 共30兲 is
.
共34兲
2 exp共␤2r2兲 6 6
共␤ r + 3␤4r4 + 6␤2r2 + 6兲
Ar8
⫻共x2 + y 2 − 2z2兲 −
24共x2 + y 2 − 2z2兲
.
Ar14
共35兲
This term decays exponentially and can be calculated by
the PP method with a small cutoff radius at 3.5␴.
Evaluation of the long-range part of the Lennard-Jones
surface tension is similar to the force evaluation except for
p = 4. However, because it has the second partial derivative
of 1 / r4, the three components should be multiplied by −k2x ,
−k2y , and −kz2, respectively, in Fourier space.
For the Coulombic part of the surface tension, Ewald
sums are used to calculate the pressure tensor, which is
VP␣␤ =
1
兺 q jb
兺 兺 qia兺
8␲⑀0 i a
j⫽i b
冋冑
2
␲
⫻exp共− ␬2r2iajb兲 + erfc共␬riajb兲
+
冋 冋
冉
+
␬riajb
册
共rij兲␣共riajb兲␤
r3iajb
1
2␲
兺 Q共h兲S共h兲S共− h兲
4␲⑀0 V h⫽0
⫻ ␦␣␤ −
,
共31兲
⳵2 1
⳵z2 r4
By using the expression given in Table I, for p = 4, the
total short-range Lennard-Jones surface tension is
共30兲
.
冋 冉冊 冉冊
冉 冊册
12共x2 + y 2 − 2z2兲 1 ⳵2 1
⳵2 1
=
+
Ar8
2A ⳵x2 r4
⳵ y 2 r4
共28兲
冋冉 冊 冉 冊 册 冔
共33兲
so that the first part of Eq. 共32兲 is given as
共29兲
x2ij + y 2ij − 2z2ij
4⑀LJ
2Arij
.
In Eq. 共32兲, the 1 / r14 term decays much faster than the 1 / r8
term. So the PPPM method is only applied on the 1 / r8 term.
We notice that
Since the SPC/E potential can be described as the sum of two
terms, the surface tension can also be expressed as
␥ = ␥LJ + ␥C ,
册冔
共32兲
␥LJ-SR =
and
Pzz =
12共x2 + y 2 − 2z2兲 24共x2 + y 2 − 2z2兲
+
Ar8
Ar14
共24兲
m jv2yj + 兺 兺 ryij f yij ,
i⬎j
冓兺 冋
共21兲
phase 1
Pxx =
␥LJ =
再
2h␣h␤ h␣h␤
−
h2
2␬2
冊册
1
2␲
兺 兺 共Pia兲␤qiah⫽0
兺 Q共h兲ih␣关S共h兲
4␲⑀0 V i a
冎
⫻exp共− ih · r兲 − S共− h兲exp共ih · r兲兴 .
共36兲
Here
S共h兲 = 兺 兺 qia exp共ih · ria兲,
共37兲
Q共h兲 = exp共− h2/4␬2兲/ប2 ,
共38兲
i
a
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204715-5
J. Chem. Phys. 124, 204715 共2006兲
Surface tension of water by PPPM method
TABLE II. Simulation parameters.
FIG. 1. Simulation domain configuration.
riajb = ria − r jb ,
共39兲
where ria is the position of atom a in the molecule i and r jb
is the position of atom b in the molecule j. Reciprocal lattice
vector h is defined as
h = 2␲共l/Lx,m/Ly,n/Lz兲,
共40兲
where l, m, and n are the values 0, ±1 , ± 2 , . . . , ± ⬁, ␦␣␤ is the
Kronecker delta, ␣ and ␤ are x, y, and z, pia = ria − ri is a
vector between atom a in the molecule i and the center of
mass of molecule i. V = LxLyLz is the volume and ␬ is the
Ewald parameter which turns the relative weight of the real
space and reciprocal space contribution. More detail can be
found in Ref. 13.
Once Pxx, Pyy, and Pzz have been obtained, the Coulombic surface tension can be calculated as
␥C =
1
2
冕
⬁
Parameters
Water
Temperature
Density
Time step
T = 300– 600 K
␳ = 1000 kg/ m3
⌬t = 1.00⫻ 10−15 s
tances are constrained at 1.0 Å. The HOH bond angle is
␪0 = 109.47°. The intermolecular interactions consist of three
point charges located at the oxygen and hydrogen sites, and
the LJ potential used for O–O interactions.
The equations of motion are solved using the Verlet algorithm with a time step of 1 ⫻ 10−15 s. The constraints inside the water molecules are handled using the SHAKE
method.19
The simulations are carried out in an NVT ensemble. The
system was equilibrated at a desired temperature for 100 000
time steps by velocity rescaling and 100 000 time steps of
nonthermostat equilibration. Statistics were sampled 共density
and surface tension兲 for an additional 100 000 time steps.
The density and surface tension are calculated every ten
steps. Simulation parameters are given in Table II.
In using of PPPM method, the arbitrary positive number
␤ is chosen as 0.9 as suggested by Deserno and Holm.18,20
IV. RESULTS AND DISCUSSION
A. Density
关PN共z兲 − PT共z兲兴dz,
共41兲
−⬁
where
PN共z兲 = Pzz ,
共42兲
1
PT共z兲 = 关Pxx + Pyy兴.
2
共43兲
and
III. MODELLING AND SIMULATION
In order to simulate the liquid/vapor interface of water, a
rectangular box is used for the simulation domain. Its dimensions are 32, 32, and 64 Å, respectively. The x, y, and z
boundary conditions are set periodically. The initial condition is a liquid slab in the middle of the z direction in the
simulation domain. 800 water molecules are placed in the
liquid region using a cubic crystal structure as shown in Fig.
1. The initial velocities of molecules are set randomly corresponding to the system temperature. No external force field
is applied.
The cutoff radius for the short-range part is chosen as
0.98 nm. For the evaluation of the long-range part, 32⫻ 32
⫻ 64 mesh points are used for both force and surface tension
evaluation.
In this simulation, the SPC/E interaction potential for
water is used. Water 共H2O兲 molecule has a tetrahedral shape
and the two hydrogen atoms are connected to the oxygen
atom through covalent bonds. In this model, OH bond dis-
Once the system has reached equilibrium, the properties
are calculated. The statistics start at 200 001 time step. Figure 2 is a snapshot of the equilibrium state of the simulation
system at a temperature of 572 K.
To obtain the density profile, the system was divided
along the z axis into 64 cells, and the number of water molecules falling within each cell is counted. The density profile
at different temperatures are shown in Fig. 3. It is clear from
the profiles that a well defined interface is established and
pure liquid and vapor regions exist on different sides.
The calculated density profile is fitted to a hyperbolic
tangent function of the form,
冉 冊
1
1
z − z0
␳共z兲 = 共␳l + ␳v兲 + 共␳l − ␳v兲tanh
,
2
2
d
共44兲
where the fitting parameter ␳l is the bulk liquid density, ␳v
the bulk vapor density, and z0 the middle of the interface
between liquid and vapor. The average thickness d is found
to be 1.68 Å at a temperature of 302 K. The distribution is
FIG. 2. Snapshot of the simulation domain at a temperature of 572 K.
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204715-6
J. Chem. Phys. 124, 204715 共2006兲
Shi, Sinha, and Dhir
FIG. 3. Density profiles at various temperatures.
shown in Fig. 4. For the hyperbolic tangent function the
“10–90 thickness,” t, is related to d by t = 2.1972d. The thickness can be obtained from above to be 3.91 Å. This is
smaller than the values of 4.1 and 8.3 Å Matsumoto8 obtained from ellipsometric and x-ray reflectivity experiments.
From Fig. 3, we can see that the thickness of the liquid
region is more than 20 Å, which is wide enough to make an
accurate estimate of the liquid and vapor density. In Fig. 5,
the calculated results are compared with the data from
experiments.21 Here we can see that at low temperatures,
density values agree well with experimental results. In the
region near the critical temperature, the density of the liquid
from simulation is lower than that of the experimental data
while vapor density is higher. This shows that the SPC/E
model itself is not very accurate for the high temperature
simulations. Currently, no potential model accurately simulates that region.
FIG. 5. Comparison with data from literature of the predicted liquid density
and vapor density as a function of temperature.
Simulation results are compared with the thermodynamic correlations based on the equation of the corresponding states for the surface tension of water in Fig. 6.
Filled symbols are the simulation results and the line is
emprical fit to data from experiments.22 From the above figure it is clear that the simulation closely matches the thermodynamic correlation. This agreement shows that the surface
tension should be calculated without truncating the long-rang
part in the LJ potential interaction. By using the PPPM
method, this error can by minimized and no artificial tail
correction is needed at low temperatures. At the high temperature region, because of the low density of liquid water
and high density of vapor, calculated surface tension differs
from the experimental data. Another reason for the difference
is, because of the long-range interactions, the two interfaces
cannot be independent. To obtain more accurate results, a
thicker and more stable film needs to be modeled.
B. Surface Tension
V. CONCLUSION
Surface tension of the liquid-vapor interface is calculated using Eq. 共28兲. Since there are two liquid-vapor interfaces in this geometry, surface tension of the interface is
one-half of the total summation value.
SPC/E water is simulated by using the PPPM method
and the Ewald sum for the LJ part and the Coulomb part,
respectively. The results show that in the low temperature
FIG. 4. Density profile with the tangent hyperbolic correlating function at
302 K.
FIG. 6. Comparison with data from literature of the surface tension as a
function of temperature.
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204715-7
region, density and surface tension agree with experimental
values. In the high temperature region, the model for potential is not very accurate, as a result the density and surface
tension differ from experimental data. The proposed method
does not require any tail corrections.
ACKNOWLEDGMENT
The authors gratefully acknowledge the support this research received from NASA under the Microgravity Fluid
Physics Program.
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