Post on 02-May-2023
Chemical Physics Letters 376 (2003) 646–652
www.elsevier.com/locate/cplett
A temperature of maximum density in soft sticky dipole water
Ming-Liang Tan a, Justin T. Fischer a, Amalendu Chandra a,Bernard R. Brooks b, Toshiko Ichiye a,b,*
a School of Molecular Biosciences, Washington State University, Pullman, WA 99164-4660, USAb Laboratory of Biophysical Chemistry, National Heart, Lung, and Blood Institute, National Institutes of Health,
Bethesda, MD 20892-8014, USA
Received 30 December 2002; in final form 23 April 2003
Published online: 10 July 2003
Abstract
A temperature of maximum density near 260 K at 1 atm has been found for the soft sticky dipole (SSD) water model
in molecular dynamics simulations. The parameters of SSD have been optimized to reproduce the density of water as
well as other structural, thermodynamic, dielectric, and dynamic properties at room temperature and 1 atm. Re-
markably, this simple model is able to reproduce the anomalous temperature dependence of the density using pa-
rameters optimized at room temperature. Furthermore, these results indicate that the tetrahedral nature of water is
important in determining this anomalous behavior.
� 2003 Elsevier B.V. All rights reserved.
1. Introduction
Due to its crucial role in chemical and biologi-
cal processes as well as its peculiar properties,
water is probably the most extensively studied li-
quid. However, its properties are still not wellunderstood at a molecular level. In the 1980s,
several rigid, pair-wise additive, non-polarizable,
three-site water potentials, such as the SPC [1],
SPC/E [2] and TIP3P [3], were proposed that were
parameterized to reproduce some water properties
near ambient conditions in computer simulations.
These models are advantageous because they are
* Corresponding author. Fax: +1-509-335-9688.
E-mail address: Ichiye@wsu.edu (T. Ichiye).
0009-2614/$ - see front matter � 2003 Elsevier B.V. All rights reserv
doi:10.1016/S0009-2614(03)01044-3
fast in computer simulations. Remarkably, these
simple models are able to reproduce water prop-
erties far from ambient conditions [1–5]. However,
these models have done less well in reproducing
one of water�s most interesting properties, namely,a maximum in the temperature dependence of thedensity. The experimental temperature of maxi-
mum density (TMD) occurs at 277 K while the
TMD of these models, if any, is below 240 K [4,5].
One strategy for improvement is to use models
with more sites. The four-site TIP4P [6] model has
a broad maximum with a TMD of around 248 K
[5]. Moreover, the five-site ST2 [7] and TIP5P [8]
models have TMD around 300 K [7] and 273 K [8],respectively. Another strategy for improvement is
to develop polarizable water models in which
electronic polarization in response to the sur-
ed.
M.-L. Tan et al. / Chemical Physics Letters 376 (2003) 646–652 647
rounding molecules is explicitly accounted for. For
example, the polarizable fluctuating charge TIP4P-
FQ model exhibits TMD around 280 K [9].
Recently, progress has been made in developing
accurate polarizable water models, which was de-
scribed in detail elsewhere and references therein[9–12]. Nevertheless, additional computational
expense is needed to compute the many-body
terms explicitly and unresolved questions remain
about the effects of incomplete iteration and the
optimal method of including polarization. Thus,
simple effective pairwise models, where non-addi-
tive many-body effects are taken into account in an
average way, will continue to be widely used es-pecially in the simulations of large systems.
A single-site non-polarizable water model,
which is referred as the soft sticky dipole (SSD)
model, has been introduced recently [13]. The
SSD model is based on the hard-sphere sticky di-
pole model by Bratko et al. [14,15] and consists of
a Lennard-Jones sphere embedded with a point
dipole and a tetrahedral sticky potential, where theshort-range sticky potential describes the hydro-
gen bonding and regulates the tetrahedral coordi-
nation in the first shell of water. The sticky
potential is an effective description of the energy
terms involving the quadrupole and octopole
moments such as found in the multipole expan-
sions of Kusalik and Patey [16,17], but requires
fewer computations. The center of mass is the onlyinteraction site in the model. Evaluation of the
interaction between two SSD water molecules re-
quires computing only one distance between the
two centers of mass, four spherical angles, and the
angle between the dipole vectors. On the other
hand, three-site models like TIP3P and SPC/E re-
quire computing nine intermolecular distances and
five-site models like ST2 and TIP5P require com-puting 17 intermolecular distances. Simulations
with the SSD potential are about seven and four
times faster than with three-site models in Monte
Carlo [13] and molecular dynamics simulations
[18], respectively.
Previous Monte Carlo and molecular dynamics
simulation results show that SSD water has good
structural, thermodynamic, dielectric, and dy-namic properties at room temperature [13,18,19].
However, the original parameters of the SSD
water model give low water density (0.957 g/cm3)
and low heat of vaporization energy at room
temperature and 1 atm, which may be improved by
optimizing the parameters used in the SSD po-
tential. Furthermore, the temperature dependence
of properties of SSD water has not been studied.For example, does SSD water have a temperature
of maximum density, one of the most well known
peculiar properties of real liquid water?
In this Letter, we further optimize the parame-
ters of the SSD water model to reproduce water
properties at room temperature and atmospheric
pressure. Then, molecular dynamics simulations
for SSD liquid water are performed at 1 atmpressure and at different temperatures ranging
from 230 to 330 K to locate the temperature of
maximum density.
2. Methods
Molecular dynamics simulations were carriedout in both the isothermal-isobaric and the mi-
crocanonical ensembles. Periodic boundary con-
ditions with the minimum image convention were
used with a cubic box of 256 water molecules. The
effect of system size was found to be minor in
previous MD simulations [18]. A spherical trun-
cation of the Lennard-Jones interaction was em-
ployed at half of the simulation box, with a longrange correction applied outside of the cutoff ra-
dius [20]. The Ewald method was used to treat the
long range dipole–dipole interactions with the di-
electric constant of the surrounding medium
�0 ¼ 1 and the convergence parameter a ¼ 6:4=Lwhere L is the box length [21,22]. The minimum
image convention was used for the real space
portion and a cut-off at k2max ¼ 54r�2 was used forthe reciprocal space portion in the Ewald sum [18].
The quaternion formulation was employed for the
rotational motion of a water molecule and the
leap-frog algorithm with a time step of 1 fs was
adapted for the integration over time. For the
starting configuration, the water molecules were
located on a face-centered cubic lattice with ran-
dom orientations.For the temperature dependence of the density,
molecular dynamics simulations were carried out
648 M.-L. Tan et al. / Chemical Physics Letters 376 (2003) 646–652
in the NPT ensemble at a pressure of 1 atm and
temperatures between 230 and 330 K at about
every 10 K. The water density was determined
from the average volume of the simulation box by
q ¼ M=ð0:6022� hV i=NÞ;where M is the molecular weight of water, N is the
number of water molecules in the cubic box, hV i isthe averaged volume and q is the density. Every
simulation at a given temperature was performed
long enough so that the average density did not
drift appreciably with simulation time. The equil-ibration and production times were 1.2 and 2.1 ns,
respectively, at T ¼ 230 and 240 K; 1.0 and 1.4 ns,
respectively, at T ¼ 250, 260, and 270 K; 0.8 and
1.2 ns, respectively, at T ¼ 280 K; and 0.6 and 0.9
ns, respectively, at T ¼ 290, 298, 310, 320 and 330
K. All properties reported here for T ¼ 298 K
except the dynamic and dielectric properties were
from the NPT simulation at T ¼ 298 K.For the dynamical and dielectric properties,
molecular dynamics simulations were carried out
in the NVE ensemble at an average temperature of
298 K. The diffusion constant was obtained by
calculating the long-time limit of the mean-square
displacement and the dielectric constant was ob-
tained by calculating the collective dipolar corre-
lation function [18]. For these simulations, thesystem was equilibrated for 800 ps and then the
diffusion and dielectric constants were calculated
over 3.3 ns.
3. Results
3.1. Optimization of parameters
Previous MD simulations in the NVE ensembleat the experimental density of 0.997 g/cm3 show
that SSD water with the original parameters [13]
has a pressure of 660 atm at room temperature,
which is somewhat higher than the experimental
pressure of 1 atm. MD simulations in the NPT
ensemble at room temperature and 1 atm yield a
density of 0.958 g/cm3, which is lower than the
experimental density of 0.997 g/cm3. Thus, theoriginal parameters were further optimized to re-
produce water density at room temperature and
atmospheric pressure, while maintaining other
properties.
A series of MD simulations were performed in
which the parameters in the Lennard-Jones, di-
pole, and sticky potentials were varied. The SSD
water density can be increased by decreasing theLennard-Jones diameter, increasing the dipole
moment, or increasing the sticky potential
strength. However, increasing the original SSD
dipole moment l ¼ 2:35 D, which is the same asthat of a TIP3P or SPC/E monomer, will increase
the already too high first peak in the O–O radial
distribution function and alter the good dielectric
properties. Thus, the Lennard-Jones and stickypotential parameters were varied at fixed l ¼ 2:35D. The original and optimized Lennard-Jones di-
ameters were r ¼ 3:051 �AA and r ¼ 3:016 �AA, re-spectively, and the original and optimized
strengths of the sticky potential parameter were
m0 ¼ 3:7284 kcal/mol and m0 ¼ 3:6613 kcal/mol,
respectively. Henceforth, the original and opti-
mized SSD parameters will be referred to as SSD0and SSD1, respectively.
The calculated liquid water properties with
SSD1 parameters at 298 K and 1 atm are an im-
provement over those with SSD0 parameters,
compared with experimental data (Table 1). The
water density q has been improved from 0.958 g/
cm3 for SSD0 to 0.990 g/cm3 for SSD1, which is
close to the experimental density of 0.997 g/cm3.The average intermolecular energy, E, has alsobeen improved from )9.23 to )9.61 kcal/mol,
compared to the experimental value of )9.86 kcal/mol [23] obtained by subtracting out the PV work
for expansion to the gas volume from the enthalpy
of vaporization [24].
The overall structure of SSD1 water, as seen in
the radial distribution functions gOOðrÞ, gOHðrÞ,and gHHðrÞ (Fig. 1), is in reasonably good agree-ment with recent neutron diffraction [25] and
X-ray scattering [26,27] measurements. The O–O
radial distribution (Fig. 1a) has two well-defined
peaks: the first is composed of the nearest neigh-
bors and the second is due to long range tetrahe-
dral order. For the first peak, the location rmax 1and the coordination number (found by integrat-ing the O–O radial distribution to 3.36 �AA, thefirst experimental minimum [25]) are close to
Table 1
Computed and experimental properties of water at 298 K
Model P(atm)
q(g/cm3)
�E(kcal/mol)
rmax 1(�AA)
Coordination number � D(10�5 cm2 s�1)
SSD0 660 0.997 9.60 2.78 4.7 80 2.24
SSD0 1 0.958 9.23 2.78 4.6 66 2.37
SSD1 1 0.990 9.61 2.74 4.6 72 2.13
SPC/E 1 1.007a 9.89b 2.75 4.7 68c 2.49d
TIP3P 1 1.002e 9.82e 2.77e 4.9 97d 5.06d
TIP4P 1 1.001e 10.06e 2.76e – 52f 3.29d
TIP5P 1 0.999e 9.87e 2.73e – 82e 2.62d
Expt 1 0.997g 9.86g 2.73h ; i 4.5h 78.3 j 2.30k
a Interpolated from [4].b From [2], T ¼ 306 K.c From [33].d From [34].e From [8].f From [35].g From [23,31].h From [25].i From [27].j From [30].k From [28,29].
M.-L. Tan et al. / Chemical Physics Letters 376 (2003) 646–652 649
experiment (Table 1), even though the peak issomewhat higher than the experimental peak.
More importantly, the presence of the second peak
for SSD1, which is not found in TIP3P, shows that
SSD1 has better tetrahedral structure than TIP3P
water. The second peak still exhibits a small flat-
tened area near 4.0–4.2 �AA and the peak is shifted to
larger r compared to SPC/E water and experi-
mental data, as in previous MC simulations ofSSD0 [13]. This may be partially improved by in-
creasing the strength of dipole and sticky poten-
tial, but stronger dipolar and sticky interactions
would sacrifice good dielectric properties and led
to an even higher first peak of O–O radial distri-
bution function. Nevertheless, the integration of
O–O radial distribution to 5.58 �AA yields coordi-
nation number 23.9, which is very close to thevalue of 23.5 obtained by integrating experimental
O–O radial distribution to second minimum of
5.58 �AA [25]. The overall structure of SSD1 water is
comparable to experiment, and more structured
than TIP3P, although it is not as good as SPC/E.
Dynamical and dielectric properties were also
investigated (Table 1). The translational self-diffu-
sion coefficient D of SSD1 is 2:13� 10�5 cm2/s,which is in good agreement with the experimental
value 2:30� 10�5 cm2/s [28,29]. The calculated di-electric constant � of SSD1 is 72, which is also ingood agreement with the experimental value 78 [30].
The SSD1 parameters improve SSD liquid water
properties at 298 K and 1 atm by reducing the di-
ameter of the Lennard-Jones sphere and decreasing
the strength of the sticky potential. This exemplifies
how in this model, the strength of the sticky po-
tential, which controls the hydrogen bond strength,can be varied independently of the dipole moment,
which controls the dielectric properties. In partic-
ular, the decrease in the diameter of Lennard-Jones
sphere, which is necessary to increase the density,
can be balanced by the increase in the strength of
the sticky potential so that the strength of the hy-
drogen bond at the minimum distance remains
reasonable. This is an advantage over multi-sitemodels, in which the dipole moment and the hy-
drogen bond energy are coupled because they are
both determined by the partial charges.
3.2. Temperature of maximum density
The temperature dependence of SSD1 water
density was studied in the temperature range be-tween 230 and 330 K and was compared with ex-
Fig. 2. Temperature dependence of water density at 1 atm.
Results are shown for SSD1, SPC/E [4], TIP3P [5], TIP4P [5],
TIP5P [8], and experiment [31].
Fig. 1. Radial distribution functions for liquid water at
T ¼ 298 K and P ¼ 1 atm for (a) O–O, (b) O–H, and (c) H–H.
Results are shown for SSD1, SPC/E, TIP3P, and experiment.
650 M.-L. Tan et al. / Chemical Physics Letters 376 (2003) 646–652
perimental data [31] and simulations of TIPnP
[5,8] and SPC/E [4] models (Fig. 2). The SSD1water model reproduces the experimental water
density near room temperature and at lower tem-
peratures, although the density decreases toorapidly as temperature increases above the TMD.
This is a common feature of non-polarizable water
models [4,5,8], which may be improved by allow-
ing electronic polarization of the models or
perhaps by making a softer repulsive wall. Nev-
ertheless, the temperature of maximum density for
SSD water was found to be around 260 K, which is
shifted somewhat lower than the experimentalTMD at 277 K. This is better than the three-site
TIP3P model, in which the densities monotonically
increase with decreasing temperature over the
temperature range discussed here [5], the three-site
SPC/E, in which the TMD is 235 K [4], and the
five-site ST2, in which the TMD is around 300 K
[7]. The SSD1 TMD is better than that of the four-
site TIP4P, which is around 248 K (Fig. 2) and theSSD1 density is generally closer to experiment
than TIP4P below about 300 K. Only the TMD of
the computationally intensive five-site TIP5P
model at 273 K [8] is significantly better.
The existence and value of the TMD is most
likely determined by the degree of tetrahedrality
[8]. Amongst the multi-site models, there is a
general trend in which increased tetrahedral or-dering as seen in the radial distribution functions
correlates with higher values of the TMD. In
particular, the TIP3P model does not have the
characteristic tetrahedral second peak in the O–O
distribution and also does not appear to have a
M.-L. Tan et al. / Chemical Physics Letters 376 (2003) 646–652 651
TMD while at the other extreme, the ST2 model is
generally thought to be too structured and has a
TMD above the experimental value. Since the
SSD1 potential has an explicitly tetrahedral po-
tential energy term, the results here further sup-
port the idea that tetrahedrality is important forthe TMD.
In terms of the interaction potential, the exis-
tence of TMD and other good properties of SSD1
water are most likely due to a good description of
the hydrogen bond interaction, which leads to the
overall tetrahedral structure of water. The angular
dependence of the energy of the SSD1 hydrogen
bonded dimer at the optimized oxygen–oxygendistance rOO was compared with TIPnP and SPC/Emodels (Fig. 3). The minimum in the hydrogen
bond energy for the SSD model is at a distance
rOO ¼ 2:74 �AA and a tilt angle h ¼ 47�, which is
more bent than the three-site models TIP3P with
h ¼ 27� and SPC/E with h ¼ 26�, and comparableto four-site model TIP4P with h ¼ 46� and five-sitemodel TIP5P with h ¼ 51�. The experimental va-lue for the gas phase D2O dimer [32] is h ¼ 57�.This good SSD dimer structure, governed mostly
by the angular sticky potential, gives a more tet-
rahedral network and is closer to real water than
the three-site models. This is further manifested in
the appearance of the second peak of the O–O
radial distribution function (Fig. 1a).
Fig. 3. Energy of linear water dimer as a function of dimer
angle h. Results are shown at the optimized O–O distances for
SSD1 at rOO ¼ 2:74 �AA, SPC/E at rOO ¼ 2:74 �AA, TIP3P at
rOO ¼ 2:74 �AA, TIP4P at rOO ¼ 2:76 �AA, and TIP5P at rOO ¼ 2:68�AA.
Overall, the sticky potential of the SSD1 model
describes the short range hydrogen bonding in-
teraction quite well by mimicking higher order
terms in a multipole expansion of electrostatic in-
teractions. A true multipole expansion requires
terms up to at least octupoles [16,17] and thus iscomputationally much more expensive than the
SSD model.
4. Conclusions
Classical molecular dynamics simulations of
SSD water with optimized parameters give goodstructural, thermodynamic, dielectric and dynamic
properties at 298 K and 1 atm. The temperature
dependence of the density of SSD water has also
been investigated and a temperature of maximum
density has been found at around 260 K at 1 atm.
The results are in good agreement with the ex-
perimental results. The success of this model may
be attributed to the short range sticky potentialintroduced in SSD model, which gives a good de-
scription of the tetrahedral hydrogen bonding of
liquid water.
As a one-site effective pairwise additive model,
SSD-type potentials significantly increase the
computational speed in computer simulations over
multi-site models while modeling many properties
quite well. Clearly, these features of SSD poten-tials will benefit simulations of biological macro-
molecules in solution.
Acknowledgements
We are grateful to the National Science Foun-
dation for the support of this work through Grantnumber MCB-0131780.
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