1 |
chrisfen |
861 |
%\documentclass[prb,aps,times,twocolumn,tabularx]{revtex4} |
2 |
chrisfen |
862 |
\documentclass[11pt]{article} |
3 |
|
|
\usepackage{endfloat} |
4 |
chrisfen |
743 |
\usepackage{amsmath} |
5 |
chrisfen |
862 |
\usepackage{epsf} |
6 |
|
|
\usepackage{berkeley} |
7 |
|
|
\usepackage{setspace} |
8 |
|
|
\usepackage{tabularx} |
9 |
chrisfen |
743 |
\usepackage{graphicx} |
10 |
chrisfen |
862 |
\usepackage[ref]{overcite} |
11 |
chrisfen |
743 |
%\usepackage{berkeley} |
12 |
|
|
%\usepackage{curves} |
13 |
chrisfen |
862 |
\pagestyle{plain} |
14 |
|
|
\pagenumbering{arabic} |
15 |
|
|
\oddsidemargin 0.0cm \evensidemargin 0.0cm |
16 |
|
|
\topmargin -21pt \headsep 10pt |
17 |
|
|
\textheight 9.0in \textwidth 6.5in |
18 |
|
|
\brokenpenalty=10000 |
19 |
|
|
\renewcommand{\baselinestretch}{1.2} |
20 |
|
|
\renewcommand\citemid{\ } % no comma in optional reference note |
21 |
chrisfen |
743 |
|
22 |
|
|
\begin{document} |
23 |
|
|
|
24 |
gezelter |
921 |
\title{On the structural and transport properties of the soft sticky |
25 |
gezelter |
1029 |
dipole ({\sc ssd}) and related single point water models} |
26 |
chrisfen |
743 |
|
27 |
chrisfen |
862 |
\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
28 |
|
|
Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
29 |
chrisfen |
743 |
Notre Dame, Indiana 46556} |
30 |
|
|
|
31 |
|
|
\date{\today} |
32 |
|
|
|
33 |
chrisfen |
862 |
\maketitle |
34 |
|
|
|
35 |
chrisfen |
743 |
\begin{abstract} |
36 |
gezelter |
921 |
The density maximum and temperature dependence of the self-diffusion |
37 |
gezelter |
1029 |
constant were investigated for the soft sticky dipole ({\sc ssd}) water |
38 |
gezelter |
921 |
model and two related re-parameterizations of this single-point model. |
39 |
|
|
A combination of microcanonical and isobaric-isothermal molecular |
40 |
|
|
dynamics simulations were used to calculate these properties, both |
41 |
|
|
with and without the use of reaction field to handle long-range |
42 |
|
|
electrostatics. The isobaric-isothermal (NPT) simulations of the |
43 |
|
|
melting of both ice-$I_h$ and ice-$I_c$ showed a density maximum near |
44 |
|
|
260 K. In most cases, the use of the reaction field resulted in |
45 |
|
|
calculated densities which were were significantly lower than |
46 |
|
|
experimental densities. Analysis of self-diffusion constants shows |
47 |
gezelter |
1029 |
that the original {\sc ssd} model captures the transport properties of |
48 |
chrisfen |
861 |
experimental water very well in both the normal and super-cooled |
49 |
gezelter |
1029 |
liquid regimes. We also present our re-parameterized versions of {\sc ssd} |
50 |
gezelter |
921 |
for use both with the reaction field or without any long-range |
51 |
gezelter |
1029 |
electrostatic corrections. These are called the {\sc ssd/rf} and {\sc ssd/e} |
52 |
gezelter |
921 |
models respectively. These modified models were shown to maintain or |
53 |
|
|
improve upon the experimental agreement with the structural and |
54 |
gezelter |
1029 |
transport properties that can be obtained with either the original {\sc ssd} |
55 |
|
|
or the density corrected version of the original model ({\sc ssd1}). |
56 |
gezelter |
921 |
Additionally, a novel low-density ice structure is presented |
57 |
gezelter |
1029 |
which appears to be the most stable ice structure for the entire {\sc ssd} |
58 |
gezelter |
921 |
family. |
59 |
chrisfen |
743 |
\end{abstract} |
60 |
|
|
|
61 |
chrisfen |
862 |
\newpage |
62 |
chrisfen |
743 |
|
63 |
|
|
%\narrowtext |
64 |
|
|
|
65 |
|
|
|
66 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
67 |
|
|
% BODY OF TEXT |
68 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
69 |
|
|
|
70 |
|
|
\section{Introduction} |
71 |
|
|
|
72 |
chrisfen |
862 |
One of the most important tasks in the simulation of biochemical |
73 |
gezelter |
921 |
systems is the proper depiction of the aqueous environment of the |
74 |
|
|
molecules of interest. In some cases (such as in the simulation of |
75 |
|
|
phospholipid bilayers), the majority of the calculations that are |
76 |
|
|
performed involve interactions with or between solvent molecules. |
77 |
|
|
Thus, the properties one may observe in biochemical simulations are |
78 |
|
|
going to be highly dependent on the physical properties of the water |
79 |
|
|
model that is chosen. |
80 |
chrisfen |
743 |
|
81 |
gezelter |
921 |
There is an especially delicate balance between computational |
82 |
|
|
efficiency and the ability of the water model to accurately predict |
83 |
|
|
the properties of bulk |
84 |
|
|
water.\cite{Jorgensen83,Berendsen87,Jorgensen00} For example, the |
85 |
|
|
TIP5P model improves on the structural and transport properties of |
86 |
|
|
water relative to the previous TIP models, yet this comes at a greater |
87 |
|
|
than 50\% increase in computational |
88 |
|
|
cost.\cite{Jorgensen01,Jorgensen00} |
89 |
|
|
|
90 |
|
|
One recently developed model that largely succeeds in retaining the |
91 |
|
|
accuracy of bulk properties while greatly reducing the computational |
92 |
gezelter |
1029 |
cost is the Soft Sticky Dipole ({\sc ssd}) water |
93 |
|
|
model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The {\sc ssd} model was |
94 |
gezelter |
921 |
developed by Ichiye \emph{et al.} as a modified form of the |
95 |
|
|
hard-sphere water model proposed by Bratko, Blum, and |
96 |
gezelter |
1029 |
Luzar.\cite{Bratko85,Bratko95} {\sc ssd} is a {\it single point} model which |
97 |
gezelter |
921 |
has an interaction site that is both a point dipole along with a |
98 |
|
|
Lennard-Jones core. However, since the normal aligned and |
99 |
|
|
anti-aligned geometries favored by point dipoles are poor mimics of |
100 |
|
|
local structure in liquid water, a short ranged ``sticky'' potential |
101 |
|
|
is also added. The sticky potential directs the molecules to assume |
102 |
|
|
the proper hydrogen bond orientation in the first solvation |
103 |
|
|
shell. |
104 |
|
|
|
105 |
gezelter |
1029 |
The interaction between two {\sc ssd} water molecules \emph{i} and \emph{j} |
106 |
gezelter |
921 |
is given by the potential |
107 |
chrisfen |
743 |
\begin{equation} |
108 |
|
|
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
109 |
gezelter |
921 |
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ + |
110 |
chrisfen |
743 |
u_{ij}^{sp} |
111 |
gezelter |
921 |
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j), |
112 |
chrisfen |
743 |
\end{equation} |
113 |
gezelter |
921 |
where the ${\bf r}_{ij}$ is the position vector between molecules |
114 |
|
|
\emph{i} and \emph{j} with magnitude $r_{ij}$, and |
115 |
|
|
${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of |
116 |
|
|
the two molecules. The Lennard-Jones and dipole interactions are given |
117 |
|
|
by the following familiar forms: |
118 |
chrisfen |
743 |
\begin{equation} |
119 |
gezelter |
921 |
u_{ij}^{LJ}(r_{ij}) = 4\epsilon |
120 |
|
|
\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right] |
121 |
|
|
\ , |
122 |
chrisfen |
743 |
\end{equation} |
123 |
gezelter |
921 |
and |
124 |
chrisfen |
743 |
\begin{equation} |
125 |
gezelter |
921 |
u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left( |
126 |
|
|
\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf |
127 |
|
|
r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ , |
128 |
chrisfen |
743 |
\end{equation} |
129 |
gezelter |
921 |
where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along |
130 |
|
|
the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and |
131 |
|
|
$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf |
132 |
|
|
r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule |
133 |
|
|
$i$. |
134 |
|
|
|
135 |
|
|
The sticky potential is somewhat less familiar: |
136 |
chrisfen |
743 |
\begin{equation} |
137 |
|
|
u_{ij}^{sp} |
138 |
gezelter |
921 |
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = |
139 |
|
|
\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) |
140 |
|
|
+ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf |
141 |
|
|
\Omega}_j)]\ . |
142 |
chrisfen |
1017 |
\label{stickyfunction} |
143 |
chrisfen |
743 |
\end{equation} |
144 |
gezelter |
921 |
Here, $\nu_0$ is a strength parameter for the sticky potential, and |
145 |
|
|
$s$ and $s^\prime$ are cubic switching functions which turn off the |
146 |
|
|
sticky interaction beyond the first solvation shell. The $w$ function |
147 |
|
|
can be thought of as an attractive potential with tetrahedral |
148 |
|
|
geometry: |
149 |
chrisfen |
743 |
\begin{equation} |
150 |
gezelter |
921 |
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
151 |
chrisfen |
743 |
\end{equation} |
152 |
gezelter |
921 |
while the $w^\prime$ function counters the normal aligned and |
153 |
|
|
anti-aligned structures favored by point dipoles: |
154 |
chrisfen |
743 |
\begin{equation} |
155 |
chrisfen |
1017 |
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^\circ, |
156 |
chrisfen |
743 |
\end{equation} |
157 |
gezelter |
921 |
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
158 |
|
|
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
159 |
|
|
enhances the tetrahedral geometry for hydrogen bonded structures), |
160 |
|
|
while $w^\prime$ is a purely empirical function. A more detailed |
161 |
|
|
description of the functional parts and variables in this potential |
162 |
gezelter |
1029 |
can be found in the original {\sc ssd} |
163 |
gezelter |
921 |
articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} |
164 |
chrisfen |
743 |
|
165 |
gezelter |
1029 |
Since {\sc ssd} is a single-point {\it dipolar} model, the force |
166 |
gezelter |
921 |
calculations are simplified significantly relative to the standard |
167 |
|
|
{\it charged} multi-point models. In the original Monte Carlo |
168 |
|
|
simulations using this model, Ichiye {\it et al.} reported that using |
169 |
gezelter |
1029 |
{\sc ssd} decreased computer time by a factor of 6-7 compared to other |
170 |
gezelter |
921 |
models.\cite{Ichiye96} What is most impressive is that this savings |
171 |
|
|
did not come at the expense of accurate depiction of the liquid state |
172 |
gezelter |
1029 |
properties. Indeed, {\sc ssd} maintains reasonable agreement with the Soper |
173 |
gezelter |
921 |
data for the structural features of liquid |
174 |
|
|
water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
175 |
gezelter |
1029 |
exhibited by {\sc ssd} agree with experiment better than those of more |
176 |
gezelter |
921 |
computationally expensive models (like TIP3P and |
177 |
|
|
SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction |
178 |
gezelter |
1029 |
of solvent properties makes {\sc ssd} a very attractive model for the |
179 |
gezelter |
921 |
simulation of large scale biochemical simulations. |
180 |
chrisfen |
743 |
|
181 |
gezelter |
1029 |
One feature of the {\sc ssd} model is that it was parameterized for use with |
182 |
gezelter |
921 |
the Ewald sum to handle long-range interactions. This would normally |
183 |
|
|
be the best way of handling long-range interactions in systems that |
184 |
|
|
contain other point charges. However, our group has recently become |
185 |
|
|
interested in systems with point dipoles as mimics for neutral, but |
186 |
|
|
polarized regions on molecules (e.g. the zwitterionic head group |
187 |
|
|
regions of phospholipids). If the system of interest does not contain |
188 |
|
|
point charges, the Ewald sum and even particle-mesh Ewald become |
189 |
|
|
computational bottlenecks. Their respective ideal $N^\frac{3}{2}$ and |
190 |
|
|
$N\log N$ calculation scaling orders for $N$ particles can become |
191 |
|
|
prohibitive when $N$ becomes large.\cite{Darden99} In applying this |
192 |
|
|
water model in these types of systems, it would be useful to know its |
193 |
|
|
properties and behavior under the more computationally efficient |
194 |
|
|
reaction field (RF) technique, or even with a simple cutoff. This |
195 |
|
|
study addresses these issues by looking at the structural and |
196 |
gezelter |
1029 |
transport behavior of {\sc ssd} over a variety of temperatures with the |
197 |
gezelter |
921 |
purpose of utilizing the RF correction technique. We then suggest |
198 |
|
|
modifications to the parameters that result in more realistic bulk |
199 |
|
|
phase behavior. It should be noted that in a recent publication, some |
200 |
gezelter |
1029 |
of the original investigators of the {\sc ssd} water model have suggested |
201 |
|
|
adjustments to the {\sc ssd} water model to address abnormal density |
202 |
gezelter |
921 |
behavior (also observed here), calling the corrected model |
203 |
gezelter |
1029 |
{\sc ssd1}.\cite{Ichiye03} In what follows, we compare our |
204 |
|
|
reparamaterization of {\sc ssd} with both the original {\sc ssd} and {\sc ssd1} models |
205 |
|
|
with the goal of improving the bulk phase behavior of an {\sc ssd}-derived |
206 |
gezelter |
921 |
model in simulations utilizing the Reaction Field. |
207 |
chrisfen |
757 |
|
208 |
chrisfen |
743 |
\section{Methods} |
209 |
|
|
|
210 |
gezelter |
921 |
Long-range dipole-dipole interactions were accounted for in this study |
211 |
|
|
by using either the reaction field method or by resorting to a simple |
212 |
chrisfen |
1019 |
cubic switching function at a cutoff radius. The reaction field |
213 |
|
|
method was actually first used in Monte Carlo simulations of liquid |
214 |
|
|
water.\cite{Barker73} Under this method, the magnitude of the reaction |
215 |
|
|
field acting on dipole $i$ is |
216 |
chrisfen |
743 |
\begin{equation} |
217 |
|
|
\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
218 |
gezelter |
1029 |
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} s(r_{ij}), |
219 |
chrisfen |
743 |
\label{rfequation} |
220 |
|
|
\end{equation} |
221 |
|
|
where $\mathcal{R}$ is the cavity defined by the cutoff radius |
222 |
|
|
($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the |
223 |
gezelter |
921 |
system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole |
224 |
gezelter |
1029 |
moment vector of particle $j$, and $s(r_{ij})$ is a cubic switching |
225 |
chrisfen |
743 |
function.\cite{AllenTildesley} The reaction field contribution to the |
226 |
gezelter |
921 |
total energy by particle $i$ is given by $-\frac{1}{2}{\bf |
227 |
|
|
\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf |
228 |
|
|
\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use of the reaction |
229 |
|
|
field is known to alter the bulk orientational properties, such as the |
230 |
|
|
dielectric relaxation time. There is particular sensitivity of this |
231 |
|
|
property on changes in the length of the cutoff |
232 |
|
|
radius.\cite{Berendsen98} This variable behavior makes reaction field |
233 |
|
|
a less attractive method than the Ewald sum. However, for very large |
234 |
|
|
systems, the computational benefit of reaction field is dramatic. |
235 |
|
|
|
236 |
|
|
We have also performed a companion set of simulations {\it without} a |
237 |
|
|
surrounding dielectric (i.e. using a simple cubic switching function |
238 |
chrisfen |
1022 |
at the cutoff radius), and as a result we have two reparamaterizations |
239 |
gezelter |
1029 |
of {\sc ssd} which could be used either with or without the reaction field |
240 |
gezelter |
921 |
turned on. |
241 |
chrisfen |
777 |
|
242 |
gezelter |
1029 |
Simulations to obtain the preferred densities of the models were |
243 |
|
|
performed in the isobaric-isothermal (NPT) ensemble, while all |
244 |
|
|
dynamical properties were obtained from microcanonical (NVE) |
245 |
|
|
simulations done at densities matching the NPT density for a |
246 |
|
|
particular target temperature. The constant pressure simulations were |
247 |
|
|
implemented using an integral thermostat and barostat as outlined by |
248 |
|
|
Hoover.\cite{Hoover85,Hoover86} All molecules were treated as |
249 |
|
|
non-linear rigid bodies. Vibrational constraints are not necessary in |
250 |
|
|
simulations of {\sc ssd}, because there are no explicit hydrogen atoms, and |
251 |
|
|
thus no molecular vibrational modes need to be considered. |
252 |
chrisfen |
743 |
|
253 |
|
|
Integration of the equations of motion was carried out using the |
254 |
chrisfen |
1027 |
symplectic splitting method proposed by Dullweber, Leimkuhler, and |
255 |
gezelter |
1029 |
McLachlan ({\sc dlm}).\cite{Dullweber1997} Our reason for selecting this |
256 |
chrisfen |
1027 |
integrator centers on poor energy conservation of rigid body dynamics |
257 |
|
|
using traditional quaternion integration.\cite{Evans77,Evans77b} In |
258 |
|
|
typical microcanonical ensemble simulations, the energy drift when |
259 |
gezelter |
1029 |
using quaternions was substantially greater than when using the {\sc dlm} |
260 |
chrisfen |
1027 |
method (fig. \ref{timestep}). This steady drift in the total energy |
261 |
|
|
has also been observed by Kol {\it et al.}\cite{Laird97} |
262 |
chrisfen |
743 |
|
263 |
|
|
The key difference in the integration method proposed by Dullweber |
264 |
|
|
\emph{et al.} is that the entire rotation matrix is propagated from |
265 |
gezelter |
921 |
one time step to the next. The additional memory required by the |
266 |
|
|
algorithm is inconsequential on modern computers, and translating the |
267 |
|
|
rotation matrix into quaternions for storage purposes makes trajectory |
268 |
|
|
data quite compact. |
269 |
chrisfen |
743 |
|
270 |
gezelter |
1029 |
The {\sc dlm} method allows for Verlet style integration of both |
271 |
chrisfen |
1027 |
translational and orientational motion of rigid bodies. In this |
272 |
gezelter |
921 |
integration method, the orientational propagation involves a sequence |
273 |
|
|
of matrix evaluations to update the rotation |
274 |
|
|
matrix.\cite{Dullweber1997} These matrix rotations are more costly |
275 |
|
|
than the simpler arithmetic quaternion propagation. With the same time |
276 |
gezelter |
1029 |
step, a 1000 {\sc ssd} particle simulation shows an average 7\% increase in |
277 |
|
|
computation time using the {\sc dlm} method in place of quaternions. The |
278 |
chrisfen |
1027 |
additional expense per step is justified when one considers the |
279 |
gezelter |
1029 |
ability to use time steps that are nearly twice as large under {\sc dlm} |
280 |
chrisfen |
1027 |
than would be usable under quaternion dynamics. The energy |
281 |
|
|
conservation of the two methods using a number of different time steps |
282 |
|
|
is illustrated in figure |
283 |
gezelter |
921 |
\ref{timestep}. |
284 |
chrisfen |
743 |
|
285 |
|
|
\begin{figure} |
286 |
chrisfen |
862 |
\begin{center} |
287 |
|
|
\epsfxsize=6in |
288 |
|
|
\epsfbox{timeStep.epsi} |
289 |
gezelter |
1029 |
\caption{Energy conservation using both quaternion-based integration and |
290 |
|
|
the {\sc dlm} method with increasing time step. The larger time step plots |
291 |
|
|
are shifted from the true energy baseline (that of $\Delta t$ = 0.1 |
292 |
|
|
fs) for clarity.} |
293 |
chrisfen |
743 |
\label{timestep} |
294 |
chrisfen |
862 |
\end{center} |
295 |
chrisfen |
743 |
\end{figure} |
296 |
|
|
|
297 |
|
|
In figure \ref{timestep}, the resulting energy drift at various time |
298 |
gezelter |
1029 |
steps for both the {\sc dlm} and quaternion integration schemes is compared. |
299 |
|
|
All of the 1000 {\sc ssd} particle simulations started with the same |
300 |
chrisfen |
1027 |
configuration, and the only difference was the method used to handle |
301 |
|
|
orientational motion. At time steps of 0.1 and 0.5 fs, both methods |
302 |
|
|
for propagating the orientational degrees of freedom conserve energy |
303 |
|
|
fairly well, with the quaternion method showing a slight energy drift |
304 |
|
|
over time in the 0.5 fs time step simulation. At time steps of 1 and 2 |
305 |
gezelter |
1029 |
fs, the energy conservation benefits of the {\sc dlm} method are clearly |
306 |
chrisfen |
1027 |
demonstrated. Thus, while maintaining the same degree of energy |
307 |
|
|
conservation, one can take considerably longer time steps, leading to |
308 |
|
|
an overall reduction in computation time. |
309 |
chrisfen |
743 |
|
310 |
gezelter |
1029 |
Energy drift in the simulations using {\sc dlm} integration was unnoticeable |
311 |
chrisfen |
1027 |
for time steps up to 3 fs. A slight energy drift on the order of 0.012 |
312 |
|
|
kcal/mol per nanosecond was observed at a time step of 4 fs, and as |
313 |
|
|
expected, this drift increases dramatically with increasing time |
314 |
|
|
step. To insure accuracy in our microcanonical simulations, time steps |
315 |
|
|
were set at 2 fs and kept at this value for constant pressure |
316 |
|
|
simulations as well. |
317 |
chrisfen |
743 |
|
318 |
gezelter |
921 |
Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices |
319 |
|
|
were generated as starting points for all simulations. The $I_h$ |
320 |
gezelter |
1029 |
crystals were formed by first arranging the centers of mass of the {\sc ssd} |
321 |
gezelter |
921 |
particles into a ``hexagonal'' ice lattice of 1024 particles. Because |
322 |
|
|
of the crystal structure of $I_h$ ice, the simulation box assumed an |
323 |
|
|
orthorhombic shape with an edge length ratio of approximately |
324 |
chrisfen |
743 |
1.00$\times$1.06$\times$1.23. The particles were then allowed to |
325 |
|
|
orient freely about fixed positions with angular momenta randomized at |
326 |
|
|
400 K for varying times. The rotational temperature was then scaled |
327 |
chrisfen |
862 |
down in stages to slowly cool the crystals to 25 K. The particles were |
328 |
|
|
then allowed to translate with fixed orientations at a constant |
329 |
chrisfen |
743 |
pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were |
330 |
|
|
removed and the ice crystals were allowed to equilibrate for 50 ps at |
331 |
|
|
25 K and a constant pressure of 1 atm. This procedure resulted in |
332 |
|
|
structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
333 |
chrisfen |
862 |
rules.\cite{Bernal33,Rahman72} This method was also utilized in the |
334 |
chrisfen |
743 |
making of diamond lattice $I_c$ ice crystals, with each cubic |
335 |
|
|
simulation box consisting of either 512 or 1000 particles. Only |
336 |
|
|
isotropic volume fluctuations were performed under constant pressure, |
337 |
|
|
so the ratio of edge lengths remained constant throughout the |
338 |
|
|
simulations. |
339 |
|
|
|
340 |
|
|
\section{Results and discussion} |
341 |
|
|
|
342 |
|
|
Melting studies were performed on the randomized ice crystals using |
343 |
gezelter |
921 |
isobaric-isothermal (NPT) dynamics. During melting simulations, the |
344 |
|
|
melting transition and the density maximum can both be observed, |
345 |
|
|
provided that the density maximum occurs in the liquid and not the |
346 |
|
|
supercooled regime. An ensemble average from five separate melting |
347 |
|
|
simulations was acquired, each starting from different ice crystals |
348 |
|
|
generated as described previously. All simulations were equilibrated |
349 |
|
|
for 100 ps prior to a 200 ps data collection run at each temperature |
350 |
|
|
setting. The temperature range of study spanned from 25 to 400 K, with |
351 |
|
|
a maximum degree increment of 25 K. For regions of interest along this |
352 |
|
|
stepwise progression, the temperature increment was decreased from 25 |
353 |
|
|
K to 10 and 5 K. The above equilibration and production times were |
354 |
|
|
sufficient in that fluctuations in the volume autocorrelation function |
355 |
|
|
were damped out in all simulations in under 20 ps. |
356 |
chrisfen |
743 |
|
357 |
|
|
\subsection{Density Behavior} |
358 |
|
|
|
359 |
gezelter |
1029 |
Our initial simulations focused on the original {\sc ssd} water model, and |
360 |
gezelter |
921 |
an average density versus temperature plot is shown in figure |
361 |
|
|
\ref{dense1}. Note that the density maximum when using a reaction |
362 |
|
|
field appears between 255 and 265 K. There were smaller fluctuations |
363 |
|
|
in the density at 260 K than at either 255 or 265, so we report this |
364 |
|
|
value as the location of the density maximum. Figure \ref{dense1} was |
365 |
|
|
constructed using ice $I_h$ crystals for the initial configuration; |
366 |
|
|
though not pictured, the simulations starting from ice $I_c$ crystal |
367 |
|
|
configurations showed similar results, with a liquid-phase density |
368 |
|
|
maximum in this same region (between 255 and 260 K). |
369 |
|
|
|
370 |
chrisfen |
743 |
\begin{figure} |
371 |
chrisfen |
862 |
\begin{center} |
372 |
|
|
\epsfxsize=6in |
373 |
|
|
\epsfbox{denseSSD.eps} |
374 |
gezelter |
921 |
\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}], |
375 |
gezelter |
1029 |
TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], {\sc ssd} |
376 |
|
|
without Reaction Field, {\sc ssd}, and experiment [Ref. \citen{CRC80}]. The |
377 |
gezelter |
921 |
arrows indicate the change in densities observed when turning off the |
378 |
gezelter |
1029 |
reaction field. The the lower than expected densities for the {\sc ssd} |
379 |
|
|
model were what prompted the original reparameterization of {\sc ssd1} |
380 |
gezelter |
921 |
[Ref. \citen{Ichiye03}].} |
381 |
chrisfen |
861 |
\label{dense1} |
382 |
chrisfen |
862 |
\end{center} |
383 |
chrisfen |
743 |
\end{figure} |
384 |
|
|
|
385 |
gezelter |
1029 |
The density maximum for {\sc ssd} compares quite favorably to other simple |
386 |
gezelter |
921 |
water models. Figure \ref{dense1} also shows calculated densities of |
387 |
|
|
several other models and experiment obtained from other |
388 |
chrisfen |
743 |
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
389 |
gezelter |
1029 |
models, {\sc ssd} has a temperature closest to the experimentally observed |
390 |
gezelter |
921 |
density maximum. Of the {\it charge-based} models in |
391 |
|
|
Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that |
392 |
gezelter |
1029 |
seen in {\sc ssd}. Though not included in this plot, it is useful |
393 |
gezelter |
921 |
to note that TIP5P has a density maximum nearly identical to the |
394 |
|
|
experimentally measured temperature. |
395 |
chrisfen |
743 |
|
396 |
gezelter |
921 |
It has been observed that liquid state densities in water are |
397 |
|
|
dependent on the cutoff radius used both with and without the use of |
398 |
|
|
reaction field.\cite{Berendsen98} In order to address the possible |
399 |
|
|
effect of cutoff radius, simulations were performed with a dipolar |
400 |
gezelter |
1029 |
cutoff radius of 12.0 \AA\ to complement the previous {\sc ssd} simulations, |
401 |
gezelter |
921 |
all performed with a cutoff of 9.0 \AA. All of the resulting densities |
402 |
|
|
overlapped within error and showed no significant trend toward lower |
403 |
|
|
or higher densities as a function of cutoff radius, for simulations |
404 |
|
|
both with and without reaction field. These results indicate that |
405 |
|
|
there is no major benefit in choosing a longer cutoff radius in |
406 |
gezelter |
1029 |
simulations using {\sc ssd}. This is advantageous in that the use of a |
407 |
gezelter |
921 |
longer cutoff radius results in a significant increase in the time |
408 |
|
|
required to obtain a single trajectory. |
409 |
chrisfen |
743 |
|
410 |
chrisfen |
862 |
The key feature to recognize in figure \ref{dense1} is the density |
411 |
gezelter |
1029 |
scaling of {\sc ssd} relative to other common models at any given |
412 |
|
|
temperature. {\sc ssd} assumes a lower density than any of the other listed |
413 |
gezelter |
921 |
models at the same pressure, behavior which is especially apparent at |
414 |
|
|
temperatures greater than 300 K. Lower than expected densities have |
415 |
|
|
been observed for other systems using a reaction field for long-range |
416 |
|
|
electrostatic interactions, so the most likely reason for the |
417 |
|
|
significantly lower densities seen in these simulations is the |
418 |
|
|
presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order |
419 |
|
|
to test the effect of the reaction field on the density of the |
420 |
|
|
systems, the simulations were repeated without a reaction field |
421 |
|
|
present. The results of these simulations are also displayed in figure |
422 |
|
|
\ref{dense1}. Without the reaction field, the densities increase |
423 |
|
|
to more experimentally reasonable values, especially around the |
424 |
|
|
freezing point of liquid water. The shape of the curve is similar to |
425 |
gezelter |
1029 |
the curve produced from {\sc ssd} simulations using reaction field, |
426 |
gezelter |
921 |
specifically the rapidly decreasing densities at higher temperatures; |
427 |
|
|
however, a shift in the density maximum location, down to 245 K, is |
428 |
|
|
observed. This is a more accurate comparison to the other listed water |
429 |
|
|
models, in that no long range corrections were applied in those |
430 |
|
|
simulations.\cite{Clancy94,Jorgensen98b} However, even without the |
431 |
chrisfen |
861 |
reaction field, the density around 300 K is still significantly lower |
432 |
|
|
than experiment and comparable water models. This anomalous behavior |
433 |
chrisfen |
1027 |
was what lead Tan {\it et al.} to recently reparameterize |
434 |
gezelter |
1029 |
{\sc ssd}.\cite{Ichiye03} Throughout the remainder of the paper our |
435 |
|
|
reparamaterizations of {\sc ssd} will be compared with their newer {\sc ssd1} |
436 |
|
|
model. |
437 |
chrisfen |
861 |
|
438 |
chrisfen |
743 |
\subsection{Transport Behavior} |
439 |
|
|
|
440 |
gezelter |
921 |
Accurate dynamical properties of a water model are particularly |
441 |
|
|
important when using the model to study permeation or transport across |
442 |
|
|
biological membranes. In order to probe transport in bulk water, |
443 |
|
|
constant energy (NVE) simulations were performed at the average |
444 |
|
|
density obtained by the NPT simulations at an identical target |
445 |
|
|
temperature. Simulations started with randomized velocities and |
446 |
|
|
underwent 50 ps of temperature scaling and 50 ps of constant energy |
447 |
|
|
equilibration before a 200 ps data collection run. Diffusion constants |
448 |
|
|
were calculated via linear fits to the long-time behavior of the |
449 |
|
|
mean-square displacement as a function of time. The averaged results |
450 |
|
|
from five sets of NVE simulations are displayed in figure |
451 |
|
|
\ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
452 |
chrisfen |
1022 |
results.\cite{Gillen72,Holz00,Clancy94,Jorgensen01} |
453 |
gezelter |
921 |
|
454 |
chrisfen |
743 |
\begin{figure} |
455 |
chrisfen |
862 |
\begin{center} |
456 |
|
|
\epsfxsize=6in |
457 |
|
|
\epsfbox{betterDiffuse.epsi} |
458 |
gezelter |
921 |
\caption{Average self-diffusion constant as a function of temperature for |
459 |
gezelter |
1029 |
{\sc ssd}, SPC/E [Ref. \citen{Clancy94}], and TIP5P |
460 |
|
|
[Ref. \citen{Jorgensen01}] compared with experimental data |
461 |
|
|
[Refs. \citen{Gillen72} and \citen{Holz00}]. Of the three water models |
462 |
|
|
shown, {\sc ssd} has the least deviation from the experimental values. The |
463 |
|
|
rapidly increasing diffusion constants for TIP5P and {\sc ssd} correspond to |
464 |
|
|
significant decreases in density at the higher temperatures.} |
465 |
chrisfen |
743 |
\label{diffuse} |
466 |
chrisfen |
862 |
\end{center} |
467 |
chrisfen |
743 |
\end{figure} |
468 |
|
|
|
469 |
|
|
The observed values for the diffusion constant point out one of the |
470 |
gezelter |
1029 |
strengths of the {\sc ssd} model. Of the three models shown, the {\sc ssd} model |
471 |
gezelter |
921 |
has the most accurate depiction of self-diffusion in both the |
472 |
|
|
supercooled and liquid regimes. SPC/E does a respectable job by |
473 |
|
|
reproducing values similar to experiment around 290 K; however, it |
474 |
|
|
deviates at both higher and lower temperatures, failing to predict the |
475 |
gezelter |
1029 |
correct thermal trend. TIP5P and {\sc ssd} both start off low at colder |
476 |
gezelter |
921 |
temperatures and tend to diffuse too rapidly at higher temperatures. |
477 |
|
|
This behavior at higher temperatures is not particularly surprising |
478 |
gezelter |
1029 |
since the densities of both TIP5P and {\sc ssd} are lower than experimental |
479 |
gezelter |
921 |
water densities at higher temperatures. When calculating the |
480 |
gezelter |
1029 |
diffusion coefficients for {\sc ssd} at experimental densities (instead of |
481 |
gezelter |
921 |
the densities from the NPT simulations), the resulting values fall |
482 |
|
|
more in line with experiment at these temperatures. |
483 |
chrisfen |
743 |
|
484 |
|
|
\subsection{Structural Changes and Characterization} |
485 |
gezelter |
921 |
|
486 |
chrisfen |
743 |
By starting the simulations from the crystalline state, the melting |
487 |
gezelter |
921 |
transition and the ice structure can be obtained along with the liquid |
488 |
chrisfen |
862 |
phase behavior beyond the melting point. The constant pressure heat |
489 |
|
|
capacity (C$_\text{p}$) was monitored to locate the melting transition |
490 |
|
|
in each of the simulations. In the melting simulations of the 1024 |
491 |
|
|
particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs |
492 |
|
|
at 245 K, indicating a first order phase transition for the melting of |
493 |
|
|
these ice crystals. When the reaction field is turned off, the melting |
494 |
|
|
transition occurs at 235 K. These melting transitions are |
495 |
gezelter |
921 |
considerably lower than the experimental value. |
496 |
chrisfen |
743 |
|
497 |
chrisfen |
862 |
\begin{figure} |
498 |
|
|
\begin{center} |
499 |
|
|
\epsfxsize=6in |
500 |
|
|
\epsfbox{corrDiag.eps} |
501 |
gezelter |
1029 |
\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.} |
502 |
chrisfen |
862 |
\label{corrAngle} |
503 |
|
|
\end{center} |
504 |
|
|
\end{figure} |
505 |
|
|
|
506 |
|
|
\begin{figure} |
507 |
|
|
\begin{center} |
508 |
|
|
\epsfxsize=6in |
509 |
|
|
\epsfbox{fullContours.eps} |
510 |
gezelter |
1029 |
\caption{Contour plots of 2D angular pair correlation functions for |
511 |
|
|
512 {\sc ssd} molecules at 100 K (A \& B) and 300 K (C \& D). Dark areas |
512 |
|
|
signify regions of enhanced density while light areas signify |
513 |
|
|
depletion relative to the bulk density. White areas have pair |
514 |
|
|
correlation values below 0.5 and black areas have values above 1.5.} |
515 |
chrisfen |
743 |
\label{contour} |
516 |
chrisfen |
862 |
\end{center} |
517 |
chrisfen |
743 |
\end{figure} |
518 |
|
|
|
519 |
gezelter |
921 |
Additional analysis of the melting process was performed using |
520 |
|
|
two-dimensional structure and dipole angle correlations. Expressions |
521 |
|
|
for these correlations are as follows: |
522 |
chrisfen |
861 |
|
523 |
chrisfen |
862 |
\begin{equation} |
524 |
gezelter |
921 |
g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ , |
525 |
chrisfen |
862 |
\end{equation} |
526 |
|
|
\begin{equation} |
527 |
|
|
g_{\text{AB}}(r,\cos\omega) = |
528 |
gezelter |
921 |
\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ , |
529 |
chrisfen |
862 |
\end{equation} |
530 |
chrisfen |
861 |
where $\theta$ and $\omega$ refer to the angles shown in figure |
531 |
|
|
\ref{corrAngle}. By binning over both distance and the cosine of the |
532 |
gezelter |
921 |
desired angle between the two dipoles, the $g(r)$ can be analyzed to |
533 |
|
|
determine the common dipole arrangements that constitute the peaks and |
534 |
|
|
troughs in the standard one-dimensional $g(r)$ plots. Frames A and B |
535 |
|
|
of figure \ref{contour} show results from an ice $I_c$ simulation. The |
536 |
|
|
first peak in the $g(r)$ consists primarily of the preferred hydrogen |
537 |
chrisfen |
861 |
bonding arrangements as dictated by the tetrahedral sticky potential - |
538 |
gezelter |
921 |
one peak for the hydrogen bond donor and the other for the hydrogen |
539 |
|
|
bond acceptor. Due to the high degree of crystallinity of the sample, |
540 |
|
|
the second and third solvation shells show a repeated peak arrangement |
541 |
chrisfen |
743 |
which decays at distances around the fourth solvation shell, near the |
542 |
|
|
imposed cutoff for the Lennard-Jones and dipole-dipole interactions. |
543 |
chrisfen |
861 |
In the higher temperature simulation shown in frames C and D, these |
544 |
gezelter |
921 |
long-range features deteriorate rapidly. The first solvation shell |
545 |
|
|
still shows the strong effect of the sticky-potential, although it |
546 |
|
|
covers a larger area, extending to include a fraction of aligned |
547 |
|
|
dipole peaks within the first solvation shell. The latter peaks lose |
548 |
|
|
due to thermal motion and as the competing dipole force overcomes the |
549 |
|
|
sticky potential's tight tetrahedral structuring of the crystal. |
550 |
chrisfen |
743 |
|
551 |
|
|
This complex interplay between dipole and sticky interactions was |
552 |
|
|
remarked upon as a possible reason for the split second peak in the |
553 |
gezelter |
1029 |
oxygen-oxygen pair correlation function, |
554 |
|
|
$g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, the second |
555 |
|
|
solvation shell peak appears to have two distinct components that |
556 |
|
|
blend together to form one observable peak. At higher temperatures, |
557 |
|
|
this split character alters to show the leading 4 \AA\ peak dominated |
558 |
|
|
by equatorial anti-parallel dipole orientations. There is also a |
559 |
|
|
tightly bunched group of axially arranged dipoles that most likely |
560 |
|
|
consist of the smaller fraction of aligned dipole pairs. The trailing |
561 |
|
|
component of the split peak at 5 \AA\ is dominated by aligned dipoles |
562 |
|
|
that assume hydrogen bond arrangements similar to those seen in the |
563 |
|
|
first solvation shell. This evidence indicates that the dipole pair |
564 |
|
|
interaction begins to dominate outside of the range of the dipolar |
565 |
|
|
repulsion term. The energetically favorable dipole arrangements |
566 |
|
|
populate the region immediately outside this repulsion region (around |
567 |
|
|
4 \AA), while arrangements that seek to satisfy both the sticky and |
568 |
|
|
dipole forces locate themselves just beyond this initial buildup |
569 |
|
|
(around 5 \AA). |
570 |
chrisfen |
743 |
|
571 |
|
|
From these findings, the split second peak is primarily the product of |
572 |
chrisfen |
861 |
the dipolar repulsion term of the sticky potential. In fact, the inner |
573 |
|
|
peak can be pushed out and merged with the outer split peak just by |
574 |
gezelter |
921 |
extending the switching function ($s^\prime(r_{ij})$) from its normal |
575 |
|
|
4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of |
576 |
chrisfen |
861 |
correction is not recommended for improving the liquid structure, |
577 |
chrisfen |
862 |
since the second solvation shell would still be shifted too far |
578 |
chrisfen |
861 |
out. In addition, this would have an even more detrimental effect on |
579 |
|
|
the system densities, leading to a liquid with a more open structure |
580 |
gezelter |
1029 |
and a density considerably lower than the already low {\sc ssd} density. A |
581 |
gezelter |
921 |
better correction would be to include the quadrupole-quadrupole |
582 |
|
|
interactions for the water particles outside of the first solvation |
583 |
|
|
shell, but this would remove the simplicity and speed advantage of |
584 |
gezelter |
1029 |
{\sc ssd}. |
585 |
chrisfen |
743 |
|
586 |
gezelter |
1029 |
\subsection{Adjusted Potentials: {\sc ssd/rf} and {\sc ssd/e}} |
587 |
gezelter |
921 |
|
588 |
gezelter |
1029 |
The propensity of {\sc ssd} to adopt lower than expected densities under |
589 |
chrisfen |
743 |
varying conditions is troubling, especially at higher temperatures. In |
590 |
chrisfen |
861 |
order to correct this model for use with a reaction field, it is |
591 |
|
|
necessary to adjust the force field parameters for the primary |
592 |
|
|
intermolecular interactions. In undergoing a reparameterization, it is |
593 |
|
|
important not to focus on just one property and neglect the other |
594 |
|
|
important properties. In this case, it would be ideal to correct the |
595 |
gezelter |
921 |
densities while maintaining the accurate transport behavior. |
596 |
chrisfen |
743 |
|
597 |
chrisfen |
1017 |
The parameters available for tuning include the $\sigma$ and |
598 |
|
|
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the |
599 |
gezelter |
1029 |
strength of the sticky potential ($\nu_0$), and the cutoff distances |
600 |
|
|
for the sticky attractive and dipole repulsive cubic switching |
601 |
|
|
function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$ |
602 |
|
|
respectively). The results of the reparameterizations are shown in |
603 |
|
|
table \ref{params}. We are calling these reparameterizations the Soft |
604 |
|
|
Sticky Dipole / Reaction Field ({\sc ssd/rf} - for use with a reaction |
605 |
|
|
field) and Soft Sticky Dipole Extended ({\sc ssd/e} - an attempt to improve |
606 |
|
|
the liquid structure in simulations without a long-range correction). |
607 |
chrisfen |
743 |
|
608 |
|
|
\begin{table} |
609 |
chrisfen |
862 |
\begin{center} |
610 |
chrisfen |
743 |
\caption{Parameters for the original and adjusted models} |
611 |
chrisfen |
856 |
\begin{tabular}{ l c c c c } |
612 |
chrisfen |
743 |
\hline \\[-3mm] |
613 |
gezelter |
1029 |
\ \ \ Parameters\ \ \ & \ \ \ {\sc ssd} [Ref. \citen{Ichiye96}] \ \ \ |
614 |
|
|
& \ {\sc ssd1} [Ref. \citen{Ichiye03}]\ \ & \ {\sc ssd/e}\ \ & \ {\sc ssd/rf} \\ |
615 |
chrisfen |
743 |
\hline \\[-3mm] |
616 |
chrisfen |
856 |
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
617 |
|
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
618 |
|
|
\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ |
619 |
|
|
\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
620 |
chrisfen |
1017 |
\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\ |
621 |
chrisfen |
856 |
\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ |
622 |
|
|
\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ |
623 |
|
|
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ |
624 |
|
|
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ |
625 |
chrisfen |
743 |
\end{tabular} |
626 |
|
|
\label{params} |
627 |
chrisfen |
862 |
\end{center} |
628 |
chrisfen |
743 |
\end{table} |
629 |
|
|
|
630 |
chrisfen |
862 |
\begin{figure} |
631 |
|
|
\begin{center} |
632 |
|
|
\epsfxsize=5in |
633 |
|
|
\epsfbox{GofRCompare.epsi} |
634 |
gezelter |
1029 |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with {\sc ssd/e} |
635 |
|
|
and {\sc ssd1} without reaction field (top), as well as {\sc ssd/rf} and {\sc ssd1} with |
636 |
chrisfen |
743 |
reaction field turned on (bottom). The insets show the respective |
637 |
chrisfen |
862 |
first peaks in detail. Note how the changes in parameters have lowered |
638 |
gezelter |
1029 |
and broadened the first peak of {\sc ssd/e} and {\sc ssd/rf}.} |
639 |
chrisfen |
743 |
\label{grcompare} |
640 |
chrisfen |
862 |
\end{center} |
641 |
chrisfen |
743 |
\end{figure} |
642 |
|
|
|
643 |
chrisfen |
862 |
\begin{figure} |
644 |
|
|
\begin{center} |
645 |
|
|
\epsfxsize=6in |
646 |
chrisfen |
1027 |
\epsfbox{dualsticky_bw.eps} |
647 |
gezelter |
1029 |
\caption{Positive and negative isosurfaces of the sticky potential for |
648 |
|
|
{\sc ssd1} (left) and {\sc ssd/e} \& {\sc ssd/rf} (right). Light areas correspond to the |
649 |
|
|
tetrahedral attractive component, and darker areas correspond to the |
650 |
|
|
dipolar repulsive component.} |
651 |
chrisfen |
743 |
\label{isosurface} |
652 |
chrisfen |
862 |
\end{center} |
653 |
chrisfen |
743 |
\end{figure} |
654 |
|
|
|
655 |
gezelter |
1029 |
In the original paper detailing the development of {\sc ssd}, Liu and Ichiye |
656 |
gezelter |
921 |
placed particular emphasis on an accurate description of the first |
657 |
|
|
solvation shell. This resulted in a somewhat tall and narrow first |
658 |
|
|
peak in $g(r)$ that integrated to give similar coordination numbers to |
659 |
chrisfen |
862 |
the experimental data obtained by Soper and |
660 |
|
|
Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering |
661 |
|
|
data from the Head-Gordon lab indicates a slightly lower and shifted |
662 |
gezelter |
1029 |
first peak in the g$_\mathrm{OO}(r)$, so our adjustments to {\sc ssd} were |
663 |
|
|
made after taking into consideration the new experimental |
664 |
chrisfen |
862 |
findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the |
665 |
gezelter |
921 |
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing |
666 |
gezelter |
1029 |
the revised {\sc ssd} model ({\sc ssd1}), {\sc ssd/e}, and {\sc ssd/rf} to the new |
667 |
chrisfen |
862 |
experimental results. Both modified water models have shorter peaks |
668 |
gezelter |
921 |
that match more closely to the experimental peak (as seen in the |
669 |
|
|
insets of figure \ref{grcompare}). This structural alteration was |
670 |
chrisfen |
862 |
accomplished by the combined reduction in the Lennard-Jones $\sigma$ |
671 |
gezelter |
921 |
variable and adjustment of the sticky potential strength and cutoffs. |
672 |
|
|
As can be seen in table \ref{params}, the cutoffs for the tetrahedral |
673 |
|
|
attractive and dipolar repulsive terms were nearly swapped with each |
674 |
|
|
other. Isosurfaces of the original and modified sticky potentials are |
675 |
|
|
shown in figure \ref{isosurface}. In these isosurfaces, it is easy to |
676 |
|
|
see how altering the cutoffs changes the repulsive and attractive |
677 |
|
|
character of the particles. With a reduced repulsive surface (darker |
678 |
|
|
region), the particles can move closer to one another, increasing the |
679 |
|
|
density for the overall system. This change in interaction cutoff also |
680 |
|
|
results in a more gradual orientational motion by allowing the |
681 |
|
|
particles to maintain preferred dipolar arrangements before they begin |
682 |
|
|
to feel the pull of the tetrahedral restructuring. As the particles |
683 |
|
|
move closer together, the dipolar repulsion term becomes active and |
684 |
|
|
excludes unphysical nearest-neighbor arrangements. This compares with |
685 |
gezelter |
1029 |
how {\sc ssd} and {\sc ssd1} exclude preferred dipole alignments before the |
686 |
gezelter |
921 |
particles feel the pull of the ``hydrogen bonds''. Aside from |
687 |
|
|
improving the shape of the first peak in the g(\emph{r}), this |
688 |
|
|
modification improves the densities considerably by allowing the |
689 |
|
|
persistence of full dipolar character below the previous 4.0 \AA\ |
690 |
|
|
cutoff. |
691 |
chrisfen |
743 |
|
692 |
gezelter |
921 |
While adjusting the location and shape of the first peak of $g(r)$ |
693 |
|
|
improves the densities, these changes alone are insufficient to bring |
694 |
|
|
the system densities up to the values observed experimentally. To |
695 |
|
|
further increase the densities, the dipole moments were increased in |
696 |
gezelter |
1029 |
both of our adjusted models. Since {\sc ssd} is a dipole based model, the |
697 |
gezelter |
921 |
structure and transport are very sensitive to changes in the dipole |
698 |
gezelter |
1029 |
moment. The original {\sc ssd} simply used the dipole moment calculated from |
699 |
gezelter |
921 |
the TIP3P water model, which at 2.35 D is significantly greater than |
700 |
|
|
the experimental gas phase value of 1.84 D. The larger dipole moment |
701 |
|
|
is a more realistic value and improves the dielectric properties of |
702 |
|
|
the fluid. Both theoretical and experimental measurements indicate a |
703 |
|
|
liquid phase dipole moment ranging from 2.4 D to values as high as |
704 |
|
|
3.11 D, providing a substantial range of reasonable values for a |
705 |
|
|
dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
706 |
gezelter |
1029 |
increasing the dipole moments to 2.42 and 2.48 D for {\sc ssd/e} and {\sc ssd/rf}, |
707 |
chrisfen |
862 |
respectively, leads to significant changes in the density and |
708 |
|
|
transport of the water models. |
709 |
chrisfen |
743 |
|
710 |
chrisfen |
861 |
In order to demonstrate the benefits of these reparameterizations, a |
711 |
chrisfen |
743 |
series of NPT and NVE simulations were performed to probe the density |
712 |
|
|
and transport properties of the adapted models and compare the results |
713 |
gezelter |
1029 |
to the original {\sc ssd} model. This comparison involved full NPT melting |
714 |
|
|
sequences for both {\sc ssd/e} and {\sc ssd/rf}, as well as NVE transport |
715 |
chrisfen |
861 |
calculations at the calculated self-consistent densities. Again, the |
716 |
chrisfen |
862 |
results are obtained from five separate simulations of 1024 particle |
717 |
|
|
systems, and the melting sequences were started from different ice |
718 |
|
|
$I_h$ crystals constructed as described previously. Each NPT |
719 |
chrisfen |
861 |
simulation was equilibrated for 100 ps before a 200 ps data collection |
720 |
chrisfen |
862 |
run at each temperature step, and the final configuration from the |
721 |
|
|
previous temperature simulation was used as a starting point. All NVE |
722 |
|
|
simulations had the same thermalization, equilibration, and data |
723 |
gezelter |
921 |
collection times as stated previously. |
724 |
chrisfen |
743 |
|
725 |
chrisfen |
862 |
\begin{figure} |
726 |
|
|
\begin{center} |
727 |
|
|
\epsfxsize=6in |
728 |
|
|
\epsfbox{ssdeDense.epsi} |
729 |
gezelter |
1029 |
\caption{Comparison of densities calculated with {\sc ssd/e} to {\sc ssd1} without a |
730 |
gezelter |
921 |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
731 |
|
|
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and |
732 |
|
|
experiment [Ref. \citen{CRC80}]. The window shows a expansion around |
733 |
|
|
300 K with error bars included to clarify this region of |
734 |
gezelter |
1029 |
interest. Note that both {\sc ssd1} and {\sc ssd/e} show good agreement with |
735 |
chrisfen |
856 |
experiment when the long-range correction is neglected.} |
736 |
chrisfen |
743 |
\label{ssdedense} |
737 |
chrisfen |
862 |
\end{center} |
738 |
chrisfen |
743 |
\end{figure} |
739 |
|
|
|
740 |
gezelter |
1029 |
Fig. \ref{ssdedense} shows the density profile for the {\sc ssd/e} model |
741 |
|
|
in comparison to {\sc ssd1} without a reaction field, other common water |
742 |
chrisfen |
862 |
models, and experimental results. The calculated densities for both |
743 |
gezelter |
1029 |
{\sc ssd/e} and {\sc ssd1} have increased significantly over the original {\sc ssd} |
744 |
gezelter |
921 |
model (see fig. \ref{dense1}) and are in better agreement with the |
745 |
gezelter |
1029 |
experimental values. At 298 K, the densities of {\sc ssd/e} and {\sc ssd1} without |
746 |
chrisfen |
862 |
a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and |
747 |
|
|
0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with |
748 |
|
|
the experimental value of 0.997 g/cm$^3$, and they are considerably |
749 |
gezelter |
1029 |
better than the {\sc ssd} value of 0.967$\pm$0.003 g/cm$^3$. The changes to |
750 |
chrisfen |
862 |
the dipole moment and sticky switching functions have improved the |
751 |
|
|
structuring of the liquid (as seen in figure \ref{grcompare}, but they |
752 |
|
|
have shifted the density maximum to much lower temperatures. This |
753 |
|
|
comes about via an increase in the liquid disorder through the |
754 |
|
|
weakening of the sticky potential and strengthening of the dipolar |
755 |
gezelter |
1029 |
character. However, this increasing disorder in the {\sc ssd/e} model has |
756 |
gezelter |
921 |
little effect on the melting transition. By monitoring $C_p$ |
757 |
gezelter |
1029 |
throughout these simulations, the melting transition for {\sc ssd/e} was |
758 |
gezelter |
921 |
shown to occur at 235 K. The same transition temperature observed |
759 |
gezelter |
1029 |
with {\sc ssd} and {\sc ssd1}. |
760 |
chrisfen |
743 |
|
761 |
chrisfen |
862 |
\begin{figure} |
762 |
|
|
\begin{center} |
763 |
|
|
\epsfxsize=6in |
764 |
|
|
\epsfbox{ssdrfDense.epsi} |
765 |
gezelter |
1029 |
\caption{Comparison of densities calculated with {\sc ssd/rf} to {\sc ssd1} with a |
766 |
gezelter |
921 |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
767 |
|
|
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and |
768 |
|
|
experiment [Ref. \citen{CRC80}]. The inset shows the necessity of |
769 |
|
|
reparameterization when utilizing a reaction field long-ranged |
770 |
gezelter |
1029 |
correction - {\sc ssd/rf} provides significantly more accurate densities |
771 |
|
|
than {\sc ssd1} when performing room temperature simulations.} |
772 |
chrisfen |
743 |
\label{ssdrfdense} |
773 |
chrisfen |
862 |
\end{center} |
774 |
chrisfen |
743 |
\end{figure} |
775 |
|
|
|
776 |
chrisfen |
862 |
Including the reaction field long-range correction in the simulations |
777 |
gezelter |
921 |
results in a more interesting comparison. A density profile including |
778 |
gezelter |
1029 |
{\sc ssd/rf} and {\sc ssd1} with an active reaction field is shown in figure |
779 |
chrisfen |
862 |
\ref{ssdrfdense}. As observed in the simulations without a reaction |
780 |
gezelter |
1029 |
field, the densities of {\sc ssd/rf} and {\sc ssd1} show a dramatic increase over |
781 |
|
|
normal {\sc ssd} (see figure \ref{dense1}). At 298 K, {\sc ssd/rf} has a density |
782 |
chrisfen |
862 |
of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and |
783 |
gezelter |
1029 |
considerably better than the original {\sc ssd} value of 0.941$\pm$0.001 |
784 |
|
|
g/cm$^3$ and the {\sc ssd1} value of 0.972$\pm$0.002 g/cm$^3$. These results |
785 |
gezelter |
921 |
further emphasize the importance of reparameterization in order to |
786 |
|
|
model the density properly under different simulation conditions. |
787 |
|
|
Again, these changes have only a minor effect on the melting point, |
788 |
gezelter |
1029 |
which observed at 245 K for {\sc ssd/rf}, is identical to {\sc ssd} and only 5 K |
789 |
|
|
lower than {\sc ssd1} with a reaction field. Additionally, the difference in |
790 |
|
|
density maxima is not as extreme, with {\sc ssd/rf} showing a density |
791 |
gezelter |
921 |
maximum at 255 K, fairly close to the density maxima of 260 K and 265 |
792 |
gezelter |
1029 |
K, shown by {\sc ssd} and {\sc ssd1} respectively. |
793 |
chrisfen |
743 |
|
794 |
chrisfen |
862 |
\begin{figure} |
795 |
|
|
\begin{center} |
796 |
|
|
\epsfxsize=6in |
797 |
|
|
\epsfbox{ssdeDiffuse.epsi} |
798 |
gezelter |
1029 |
\caption{The diffusion constants calculated from {\sc ssd/e} and {\sc ssd1} (both |
799 |
|
|
without a reaction field) along with experimental results |
800 |
|
|
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations were |
801 |
|
|
performed at the average densities observed in the 1 atm NPT |
802 |
|
|
simulations for the respective models. {\sc ssd/e} is slightly more mobile |
803 |
|
|
than experiment at all of the temperatures, but it is closer to |
804 |
|
|
experiment at biologically relevant temperatures than {\sc ssd1} without a |
805 |
|
|
long-range correction.} |
806 |
chrisfen |
861 |
\label{ssdediffuse} |
807 |
chrisfen |
862 |
\end{center} |
808 |
chrisfen |
861 |
\end{figure} |
809 |
|
|
|
810 |
gezelter |
1029 |
The reparameterization of the {\sc ssd} water model, both for use with and |
811 |
chrisfen |
743 |
without an applied long-range correction, brought the densities up to |
812 |
|
|
what is expected for simulating liquid water. In addition to improving |
813 |
gezelter |
1029 |
the densities, it is important that the diffusive behavior of {\sc ssd} be |
814 |
|
|
maintained or improved. Figure \ref{ssdediffuse} compares the |
815 |
|
|
temperature dependence of the diffusion constant of {\sc ssd/e} to {\sc ssd1} |
816 |
chrisfen |
1027 |
without an active reaction field at the densities calculated from |
817 |
|
|
their respective NPT simulations at 1 atm. The diffusion constant for |
818 |
gezelter |
1029 |
{\sc ssd/e} is consistently higher than experiment, while {\sc ssd1} remains lower |
819 |
chrisfen |
1027 |
than experiment until relatively high temperatures (around 360 |
820 |
|
|
K). Both models follow the shape of the experimental curve well below |
821 |
|
|
300 K but tend to diffuse too rapidly at higher temperatures, as seen |
822 |
gezelter |
1029 |
in {\sc ssd1}'s crossing above 360 K. This increasing diffusion relative to |
823 |
chrisfen |
1027 |
the experimental values is caused by the rapidly decreasing system |
824 |
gezelter |
1029 |
density with increasing temperature. Both {\sc ssd1} and {\sc ssd/e} show this |
825 |
chrisfen |
1027 |
deviation in particle mobility, but this trend has different |
826 |
gezelter |
1029 |
implications on the diffusive behavior of the models. While {\sc ssd1} |
827 |
chrisfen |
1027 |
shows more experimentally accurate diffusive behavior in the high |
828 |
gezelter |
1029 |
temperature regimes, {\sc ssd/e} shows more accurate behavior in the |
829 |
chrisfen |
1027 |
supercooled and biologically relevant temperature ranges. Thus, the |
830 |
|
|
changes made to improve the liquid structure may have had an adverse |
831 |
|
|
affect on the density maximum, but they improve the transport behavior |
832 |
gezelter |
1029 |
of {\sc ssd/e} relative to {\sc ssd1} under the most commonly simulated |
833 |
chrisfen |
1027 |
conditions. |
834 |
chrisfen |
743 |
|
835 |
chrisfen |
862 |
\begin{figure} |
836 |
|
|
\begin{center} |
837 |
|
|
\epsfxsize=6in |
838 |
|
|
\epsfbox{ssdrfDiffuse.epsi} |
839 |
gezelter |
1029 |
\caption{The diffusion constants calculated from {\sc ssd/rf} and {\sc ssd1} (both |
840 |
|
|
with an active reaction field) along with experimental results |
841 |
|
|
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations were |
842 |
|
|
performed at the average densities observed in the 1 atm NPT |
843 |
|
|
simulations for both of the models. {\sc ssd/rf} simulates the diffusion of |
844 |
|
|
water throughout this temperature range very well. The rapidly |
845 |
|
|
increasing diffusion constants at high temperatures for both models |
846 |
|
|
can be attributed to lower calculated densities than those observed in |
847 |
|
|
experiment.} |
848 |
chrisfen |
856 |
\label{ssdrfdiffuse} |
849 |
chrisfen |
862 |
\end{center} |
850 |
chrisfen |
743 |
\end{figure} |
851 |
|
|
|
852 |
gezelter |
1029 |
In figure \ref{ssdrfdiffuse}, the diffusion constants for {\sc ssd/rf} are |
853 |
|
|
compared to {\sc ssd1} with an active reaction field. Note that {\sc ssd/rf} |
854 |
gezelter |
921 |
tracks the experimental results quantitatively, identical within error |
855 |
chrisfen |
1017 |
throughout most of the temperature range shown and exhibiting only a |
856 |
gezelter |
1029 |
slight increasing trend at higher temperatures. {\sc ssd1} tends to diffuse |
857 |
chrisfen |
1017 |
more slowly at low temperatures and deviates to diffuse too rapidly at |
858 |
gezelter |
921 |
temperatures greater than 330 K. As stated above, this deviation away |
859 |
|
|
from the ideal trend is due to a rapid decrease in density at higher |
860 |
gezelter |
1029 |
temperatures. {\sc ssd/rf} does not suffer from this problem as much as {\sc ssd1} |
861 |
gezelter |
921 |
because the calculated densities are closer to the experimental |
862 |
|
|
values. These results again emphasize the importance of careful |
863 |
|
|
reparameterization when using an altered long-range correction. |
864 |
chrisfen |
743 |
|
865 |
chrisfen |
1017 |
\begin{table} |
866 |
gezelter |
1029 |
\begin{minipage}{\linewidth} |
867 |
|
|
\renewcommand{\thefootnote}{\thempfootnote} |
868 |
chrisfen |
1017 |
\begin{center} |
869 |
gezelter |
1029 |
\caption{Properties of the single-point water models compared with |
870 |
|
|
experimental data at ambient conditions} |
871 |
chrisfen |
1017 |
\begin{tabular}{ l c c c c c } |
872 |
|
|
\hline \\[-3mm] |
873 |
gezelter |
1029 |
\ \ \ \ \ \ & \ \ \ {\sc ssd1} \ \ \ & \ {\sc ssd/e} \ \ \ & \ {\sc ssd1} (RF) \ \ |
874 |
|
|
\ & \ {\sc ssd/rf} \ \ \ & \ Expt. \\ |
875 |
chrisfen |
1017 |
\hline \\[-3mm] |
876 |
|
|
\ \ \ $\rho$ (g/cm$^3$) & 0.999 $\pm$0.001 & 0.996 $\pm$0.001 & 0.972 $\pm$0.002 & 0.997 $\pm$0.001 & 0.997 \\ |
877 |
|
|
\ \ \ $C_p$ (cal/mol K) & 28.80 $\pm$0.11 & 25.45 $\pm$0.09 & 28.28 $\pm$0.06 & 23.83 $\pm$0.16 & 17.98 \\ |
878 |
gezelter |
1029 |
\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & |
879 |
|
|
2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299\cite{Mills73} \\ |
880 |
|
|
\ \ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 & |
881 |
|
|
4.7\footnote{Calculated by integrating $g_{\text{OO}}(r)$ in |
882 |
|
|
Ref. \citen{Head-Gordon00_1}} \\ |
883 |
|
|
\ \ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 & |
884 |
|
|
3.5\footnote{Calculated by integrating $g_{\text{OH}}(r)$ in |
885 |
|
|
Ref. \citen{Soper86}} \\ |
886 |
|
|
\ \ \ $\tau_1$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & |
887 |
|
|
7.2 $\pm$0.4 & 5.7\footnote{Calculated for 298 K from data in Ref. \citen{Eisenberg69}} \\ |
888 |
|
|
\ \ \ $\tau_2$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 |
889 |
|
|
$\pm$0.2 & 2.3\footnote{Calculated for 298 K from data in |
890 |
|
|
Ref. \citen{Krynicki66}} |
891 |
chrisfen |
1017 |
\end{tabular} |
892 |
|
|
\label{liquidproperties} |
893 |
|
|
\end{center} |
894 |
gezelter |
1029 |
\end{minipage} |
895 |
chrisfen |
1017 |
\end{table} |
896 |
|
|
|
897 |
|
|
Table \ref{liquidproperties} gives a synopsis of the liquid state |
898 |
|
|
properties of the water models compared in this study along with the |
899 |
|
|
experimental values for liquid water at ambient conditions. The |
900 |
gezelter |
1029 |
coordination number ($n_C$) and number of hydrogen bonds per particle |
901 |
|
|
($n_H$) were calculated by integrating the following relations: |
902 |
chrisfen |
1017 |
\begin{equation} |
903 |
gezelter |
1029 |
n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr, |
904 |
chrisfen |
1017 |
\end{equation} |
905 |
chrisfen |
1027 |
\begin{equation} |
906 |
gezelter |
1029 |
n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr, |
907 |
chrisfen |
1027 |
\end{equation} |
908 |
|
|
where $\rho$ is the number density of the specified pair interactions, |
909 |
|
|
$a$ and $b$ are the radial locations of the minima following the first |
910 |
gezelter |
1029 |
peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ respectively. The number |
911 |
|
|
of hydrogen bonds stays relatively constant across all of the models, |
912 |
|
|
but the coordination numbers of {\sc ssd/e} and {\sc ssd/rf} show an improvement |
913 |
|
|
over {\sc ssd1}. This improvement is primarily due to extension of the |
914 |
|
|
first solvation shell in the new parameter sets. Because $n_H$ and |
915 |
|
|
$n_C$ are nearly identical in {\sc ssd1}, it appears that all molecules in |
916 |
|
|
the first solvation shell are involved in hydrogen bonds. Since $n_H$ |
917 |
|
|
and $n_C$ differ in the newly parameterized models, the orientations |
918 |
|
|
in the first solvation shell are a bit more ``fluid''. Therefore {\sc ssd1} |
919 |
|
|
overstructures the first solvation shell and our reparameterizations |
920 |
|
|
have returned this shell to more realistic liquid-like behavior. |
921 |
chrisfen |
1017 |
|
922 |
gezelter |
1029 |
The time constants for the orientational autocorrelation functions |
923 |
chrisfen |
1017 |
are also displayed in Table \ref{liquidproperties}. The dipolar |
924 |
gezelter |
1029 |
orientational time correlation functions ($C_{l}$) are described |
925 |
chrisfen |
1017 |
by: |
926 |
|
|
\begin{equation} |
927 |
gezelter |
1029 |
C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle, |
928 |
chrisfen |
1017 |
\end{equation} |
929 |
gezelter |
1029 |
where $P_l$ are Legendre polynomials of order $l$ and |
930 |
|
|
$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular |
931 |
|
|
dipole.\cite{Rahman71} From these correlation functions, the |
932 |
|
|
orientational relaxation time of the dipole vector can be calculated |
933 |
|
|
from an exponential fit in the long-time regime ($t > |
934 |
|
|
\tau_l$).\cite{Rothschild84} Calculation of these time constants were |
935 |
|
|
averaged over five detailed NVE simulations performed at the ambient |
936 |
|
|
conditions for each of the respective models. It should be noted that |
937 |
|
|
the commonly cited value of 1.9 ps for $\tau_2$ was determined from |
938 |
|
|
the NMR data in Ref. \citen{Krynicki66} at a temperature near |
939 |
|
|
34$^\circ$C.\cite{Rahman71} Because of the strong temperature |
940 |
|
|
dependence of $\tau_2$, it is necessary to recalculate it at 298 K to |
941 |
|
|
make proper comparisons. The value shown in Table |
942 |
chrisfen |
1022 |
\ref{liquidproperties} was calculated from the same NMR data in the |
943 |
gezelter |
1029 |
fashion described in Ref. \citen{Krynicki66}. Similarly, $\tau_1$ was |
944 |
|
|
recomputed for 298 K from the data in Ref. \citen{Eisenberg69}. |
945 |
|
|
Again, {\sc ssd/e} and {\sc ssd/rf} show improved behavior over {\sc ssd1}, both with |
946 |
chrisfen |
1027 |
and without an active reaction field. Turning on the reaction field |
947 |
gezelter |
1029 |
leads to much improved time constants for {\sc ssd1}; however, these results |
948 |
|
|
also include a corresponding decrease in system density. |
949 |
|
|
Orientational relaxation times published in the original {\sc ssd} dynamics |
950 |
|
|
papers are smaller than the values observed here, and this difference |
951 |
|
|
can be attributed to the use of the Ewald sum.\cite{Ichiye99} |
952 |
chrisfen |
1017 |
|
953 |
chrisfen |
743 |
\subsection{Additional Observations} |
954 |
|
|
|
955 |
|
|
\begin{figure} |
956 |
chrisfen |
862 |
\begin{center} |
957 |
|
|
\epsfxsize=6in |
958 |
chrisfen |
1027 |
\epsfbox{icei_bw.eps} |
959 |
gezelter |
1029 |
\caption{The most stable crystal structure assumed by the {\sc ssd} family |
960 |
|
|
of water models. We refer to this structure as Ice-{\it i} to |
961 |
|
|
indicate its origins in computer simulation. This image was taken of |
962 |
|
|
the (001) face of the crystal.} |
963 |
chrisfen |
743 |
\label{weirdice} |
964 |
chrisfen |
862 |
\end{center} |
965 |
chrisfen |
743 |
\end{figure} |
966 |
|
|
|
967 |
gezelter |
921 |
While performing a series of melting simulations on an early iteration |
968 |
gezelter |
1029 |
of {\sc ssd/e} not discussed in this paper, we observed recrystallization |
969 |
gezelter |
921 |
into a novel structure not previously known for water. After melting |
970 |
|
|
at 235 K, two of five systems underwent crystallization events near |
971 |
|
|
245 K. The two systems remained crystalline up to 320 and 330 K, |
972 |
|
|
respectively. The crystal exhibits an expanded zeolite-like structure |
973 |
|
|
that does not correspond to any known form of ice. This appears to be |
974 |
|
|
an artifact of the point dipolar models, so to distinguish it from the |
975 |
|
|
experimentally observed forms of ice, we have denoted the structure |
976 |
gezelter |
1029 |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (Ice-{\it i}). A large enough |
977 |
gezelter |
921 |
portion of the sample crystallized that we have been able to obtain a |
978 |
gezelter |
1029 |
near ideal crystal structure of Ice-{\it i}. Figure \ref{weirdice} |
979 |
gezelter |
921 |
shows the repeating crystal structure of a typical crystal at 5 |
980 |
|
|
K. Each water molecule is hydrogen bonded to four others; however, the |
981 |
|
|
hydrogen bonds are bent rather than perfectly straight. This results |
982 |
|
|
in a skewed tetrahedral geometry about the central molecule. In |
983 |
|
|
figure \ref{isosurface}, it is apparent that these flexed hydrogen |
984 |
|
|
bonds are allowed due to the conical shape of the attractive regions, |
985 |
|
|
with the greatest attraction along the direct hydrogen bond |
986 |
chrisfen |
863 |
configuration. Though not ideal, these flexed hydrogen bonds are |
987 |
gezelter |
921 |
favorable enough to stabilize an entire crystal generated around them. |
988 |
chrisfen |
743 |
|
989 |
gezelter |
1029 |
Initial simulations indicated that Ice-{\it i} is the preferred ice |
990 |
|
|
structure for at least the {\sc ssd/e} model. To verify this, a |
991 |
|
|
comparison was made between near ideal crystals of ice~$I_h$, |
992 |
|
|
ice~$I_c$, and Ice-{\it i} at constant pressure with {\sc ssd/e}, {\sc |
993 |
|
|
ssd/rf}, and {\sc ssd1}. Near-ideal versions of the three types of |
994 |
|
|
crystals were cooled to 1 K, and enthalpies of formation of each were |
995 |
|
|
compared using all three water models. Enthalpies were estimated from |
996 |
|
|
the isobaric-isothermal simulations using $H=U+P_{\text ext}V$ where |
997 |
|
|
$P_{\text ext}$ is the applied pressure. A constant value of |
998 |
|
|
-60.158 kcal / mol has been added to place our zero for the |
999 |
|
|
enthalpies of formation for these systems at the traditional state |
1000 |
|
|
(elemental forms at standard temperature and pressure). With every |
1001 |
|
|
model in the {\sc ssd} family, Ice-{\it i} had the lowest calculated |
1002 |
|
|
enthalpy of formation. |
1003 |
chrisfen |
743 |
|
1004 |
gezelter |
921 |
\begin{table} |
1005 |
|
|
\begin{center} |
1006 |
gezelter |
1029 |
\caption{Enthalpies of Formation (in kcal / mol) of the three crystal |
1007 |
|
|
structures (at 1 K) exhibited by the {\sc ssd} family of water models} |
1008 |
gezelter |
921 |
\begin{tabular}{ l c c c } |
1009 |
|
|
\hline \\[-3mm] |
1010 |
|
|
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \ & \ |
1011 |
|
|
Ice-{\it i} \\ |
1012 |
|
|
\hline \\[-3mm] |
1013 |
gezelter |
1029 |
\ \ \ {\sc ssd/e} & -12.286 & -12.292 & -13.590 \\ |
1014 |
|
|
\ \ \ {\sc ssd/rf} & -12.935 & -12.917 & -14.022 \\ |
1015 |
|
|
\ \ \ {\sc ssd1} & -12.496 & -12.411 & -13.417 \\ |
1016 |
|
|
\ \ \ {\sc ssd1} (RF) & -12.504 & -12.411 & -13.134 \\ |
1017 |
gezelter |
921 |
\end{tabular} |
1018 |
|
|
\label{iceenthalpy} |
1019 |
|
|
\end{center} |
1020 |
|
|
\end{table} |
1021 |
chrisfen |
743 |
|
1022 |
gezelter |
921 |
In addition to these energetic comparisons, melting simulations were |
1023 |
gezelter |
1029 |
performed with ice-{\it i} as the initial configuration using {\sc ssd/e}, |
1024 |
|
|
{\sc ssd/rf}, and {\sc ssd1} both with and without a reaction field. The melting |
1025 |
|
|
transitions for both {\sc ssd/e} and {\sc ssd1} without reaction field occurred at |
1026 |
|
|
temperature in excess of 375~K. {\sc ssd/rf} and {\sc ssd1} with a reaction field |
1027 |
gezelter |
921 |
showed more reasonable melting transitions near 325~K. These melting |
1028 |
gezelter |
1029 |
point observations clearly show that all of the {\sc ssd}-derived models |
1029 |
gezelter |
921 |
prefer the ice-{\it i} structure. |
1030 |
chrisfen |
743 |
|
1031 |
|
|
\section{Conclusions} |
1032 |
|
|
|
1033 |
gezelter |
921 |
The density maximum and temperature dependence of the self-diffusion |
1034 |
gezelter |
1029 |
constant were studied for the {\sc ssd} water model, both with and without |
1035 |
gezelter |
921 |
the use of reaction field, via a series of NPT and NVE |
1036 |
|
|
simulations. The constant pressure simulations showed a density |
1037 |
|
|
maximum near 260 K. In most cases, the calculated densities were |
1038 |
|
|
significantly lower than the densities obtained from other water |
1039 |
gezelter |
1029 |
models (and experiment). Analysis of self-diffusion showed {\sc ssd} to |
1040 |
gezelter |
921 |
capture the transport properties of water well in both the liquid and |
1041 |
chrisfen |
1027 |
supercooled liquid regimes. |
1042 |
gezelter |
921 |
|
1043 |
gezelter |
1029 |
In order to correct the density behavior, the original {\sc ssd} model was |
1044 |
|
|
reparameterized for use both with and without a reaction field ({\sc ssd/rf} |
1045 |
|
|
and {\sc ssd/e}), and comparisons were made with {\sc ssd1}, Ichiye's density |
1046 |
|
|
corrected version of {\sc ssd}. Both models improve the liquid structure, |
1047 |
gezelter |
921 |
densities, and diffusive properties under their respective simulation |
1048 |
|
|
conditions, indicating the necessity of reparameterization when |
1049 |
|
|
changing the method of calculating long-range electrostatic |
1050 |
|
|
interactions. In general, however, these simple water models are |
1051 |
|
|
excellent choices for representing explicit water in large scale |
1052 |
|
|
simulations of biochemical systems. |
1053 |
|
|
|
1054 |
|
|
The existence of a novel low-density ice structure that is preferred |
1055 |
gezelter |
1029 |
by the {\sc ssd} family of water models is somewhat troubling, since liquid |
1056 |
gezelter |
921 |
simulations on this family of water models at room temperature are |
1057 |
chrisfen |
1027 |
effectively simulations of supercooled or metastable liquids. One |
1058 |
|
|
way to destabilize this unphysical ice structure would be to make the |
1059 |
gezelter |
921 |
range of angles preferred by the attractive part of the sticky |
1060 |
|
|
potential much narrower. This would require extensive |
1061 |
|
|
reparameterization to maintain the same level of agreement with the |
1062 |
|
|
experiments. |
1063 |
|
|
|
1064 |
gezelter |
1029 |
Additionally, our initial calculations show that the Ice-{\it i} |
1065 |
gezelter |
921 |
structure may also be a preferred crystal structure for at least one |
1066 |
|
|
other popular multi-point water model (TIP3P), and that much of the |
1067 |
|
|
simulation work being done using this popular model could also be at |
1068 |
|
|
risk for crystallization into this unphysical structure. A future |
1069 |
|
|
publication will detail the relative stability of the known ice |
1070 |
|
|
structures for a wide range of popular water models. |
1071 |
|
|
|
1072 |
chrisfen |
743 |
\section{Acknowledgments} |
1073 |
chrisfen |
777 |
Support for this project was provided by the National Science |
1074 |
|
|
Foundation under grant CHE-0134881. Computation time was provided by |
1075 |
|
|
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
1076 |
gezelter |
921 |
DMR-0079647. |
1077 |
chrisfen |
743 |
|
1078 |
chrisfen |
862 |
\newpage |
1079 |
|
|
|
1080 |
chrisfen |
743 |
\bibliographystyle{jcp} |
1081 |
|
|
\bibliography{nptSSD} |
1082 |
|
|
|
1083 |
|
|
%\pagebreak |
1084 |
|
|
|
1085 |
|
|
\end{document} |