1 |
chrisfen |
2977 |
\chapter{\label{chap:water}SIMPLE MODELS FOR WATER} |
2 |
|
|
|
3 |
|
|
One of the most important tasks in the simulation of biochemical |
4 |
|
|
systems is the proper depiction of the aqueous environment around the |
5 |
|
|
molecules of interest. In some cases (such as in the simulation of |
6 |
|
|
phospholipid bilayers), the majority of the calculations that are |
7 |
|
|
performed involve interactions with or between solvent molecules. |
8 |
|
|
Thus, the motion and behavior of molecules in biochemical simulations |
9 |
|
|
are highly dependent on the properties of the water model that is |
10 |
|
|
chosen as the solvent. |
11 |
|
|
\begin{figure} |
12 |
|
|
\includegraphics[width=\linewidth]{./figures/waterModels.pdf} |
13 |
|
|
\caption{Partial-charge geometries for the TIP3P, TIP4P, TIP5P, and |
14 |
|
|
SPC/E rigid body water models.\cite{Jorgensen83,Mahoney00,Berendsen87} |
15 |
|
|
In the case of the TIP models, the depiction of water improves with |
16 |
|
|
increasing number of point charges; however, computational performance |
17 |
|
|
simultaneously degrades due to the increasing number of distances and |
18 |
|
|
interactions that need to be calculated.} |
19 |
|
|
\label{fig:waterModels} |
20 |
|
|
\end{figure} |
21 |
|
|
|
22 |
|
|
As discussed in the previous chapter, water it typically modeled with |
23 |
|
|
fixed geometries of point charges shielded by the repulsive part of a |
24 |
|
|
Lennard-Jones interaction. Some of the common water models are shown |
25 |
|
|
in figure \ref{fig:waterModels}. The various models all have their |
26 |
|
|
benefits and drawbacks, and these primarily focus on the balance |
27 |
|
|
between computational efficiency and the ability to accurately predict |
28 |
|
|
the properties of bulk water. For example, the TIP5P model improves on |
29 |
|
|
the structural and transport properties of water relative to the TIP3P |
30 |
|
|
and TIP4P models, yet this comes at a greater than 50\% increase in |
31 |
|
|
computational cost.\cite{Mahoney00,Mahoney01} This cost is entirely |
32 |
|
|
due to the additional distance and electrostatic calculations that |
33 |
|
|
come from the extra point charges in the model description. Thus, the |
34 |
|
|
criteria for choosing a water model are capturing the liquid state |
35 |
|
|
properties and having the fewest number of points to insure efficient |
36 |
|
|
performance. As researchers have begun to study larger systems, such |
37 |
|
|
as entire viruses, the choice readily shifts towards efficiency over |
38 |
|
|
accuracy in order to make the calculations |
39 |
|
|
feasible.\cite{Freddolino06} |
40 |
|
|
|
41 |
|
|
\section{Soft Sticky Dipole Model for Water} |
42 |
|
|
|
43 |
|
|
One recently developed model that largely succeeds in retaining the |
44 |
|
|
accuracy of bulk properties while greatly reducing the computational |
45 |
|
|
cost is the Soft Sticky Dipole (SSD) water |
46 |
|
|
model.\cite{Liu96,Liu96b,Chandra99,Tan03} The SSD model was developed |
47 |
|
|
as a modified form of the hard-sphere water model proposed by Bratko, |
48 |
|
|
Blum, and Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} |
49 |
|
|
model which has an interaction site that is both a point dipole and a |
50 |
|
|
Lennard-Jones core. However, since the normal aligned and |
51 |
|
|
anti-aligned geometries favored by point dipoles are poor mimics of |
52 |
|
|
local structure in liquid water, a short ranged ``sticky'' potential |
53 |
|
|
is also added. The sticky potential directs the molecules to assume |
54 |
|
|
the proper hydrogen bond orientation in the first solvation shell. |
55 |
|
|
|
56 |
|
|
The interaction between two SSD water molecules \emph{i} and \emph{j} |
57 |
|
|
is given by the potential |
58 |
|
|
\begin{equation} |
59 |
|
|
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
60 |
|
|
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ + |
61 |
|
|
u_{ij}^{sp} |
62 |
|
|
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j), |
63 |
|
|
\end{equation} |
64 |
|
|
where the ${\bf r}_{ij}$ is the position vector between molecules |
65 |
|
|
\emph{i} and \emph{j} with magnitude $r_{ij}$, and |
66 |
|
|
${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of |
67 |
|
|
the two molecules. The Lennard-Jones and dipole interactions are given |
68 |
|
|
by the following familiar forms: |
69 |
|
|
\begin{equation} |
70 |
|
|
u_{ij}^{LJ}(r_{ij}) = 4\epsilon |
71 |
|
|
\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right] |
72 |
|
|
\ , |
73 |
|
|
\end{equation} |
74 |
|
|
and |
75 |
|
|
\begin{equation} |
76 |
|
|
u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left( |
77 |
|
|
\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf |
78 |
|
|
r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ , |
79 |
|
|
\end{equation} |
80 |
|
|
where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along |
81 |
|
|
the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and |
82 |
|
|
$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf |
83 |
|
|
r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule |
84 |
|
|
$i$. |
85 |
|
|
|
86 |
|
|
The sticky potential is somewhat less familiar: |
87 |
|
|
\begin{equation} |
88 |
|
|
u_{ij}^{sp} |
89 |
|
|
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = |
90 |
|
|
\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) |
91 |
|
|
+ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf |
92 |
|
|
\Omega}_j)]\ . |
93 |
|
|
\label{eq:stickyfunction} |
94 |
|
|
\end{equation} |
95 |
|
|
Here, $\nu_0$ is a strength parameter for the sticky potential, and |
96 |
|
|
$s$ and $s^\prime$ are cubic switching functions which turn off the |
97 |
|
|
sticky interaction beyond the first solvation shell. The $w$ function |
98 |
|
|
can be thought of as an attractive potential with tetrahedral |
99 |
|
|
geometry: |
100 |
|
|
\begin{equation} |
101 |
|
|
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
102 |
|
|
\end{equation} |
103 |
|
|
while the $w^\prime$ function counters the normal aligned and |
104 |
|
|
anti-aligned structures favored by point dipoles: |
105 |
|
|
\begin{equation} |
106 |
|
|
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, |
107 |
|
|
\end{equation} |
108 |
|
|
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
109 |
|
|
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
110 |
|
|
enhances the tetrahedral geometry for hydrogen bonded structures), |
111 |
|
|
while $w^\prime$ is a purely empirical function. A more detailed |
112 |
|
|
description of the functional parts and variables in this potential |
113 |
|
|
can be found in the original SSD |
114 |
|
|
articles.\cite{Liu96,Liu96b,Chandra99,Tan03} |
115 |
|
|
|
116 |
|
|
Since SSD is a single-point {\it dipolar} model, the force |
117 |
|
|
calculations are simplified significantly relative to the standard |
118 |
|
|
{\it charged} multi-point models. In the original Monte Carlo |
119 |
|
|
simulations using this model, Liu and Ichiye reported that using SSD |
120 |
|
|
decreased computer time by a factor of 6-7 compared to other |
121 |
|
|
models.\cite{Liu96b} What is most impressive is that this savings did |
122 |
|
|
not come at the expense of accurate depiction of the liquid state |
123 |
|
|
properties. Indeed, SSD maintains reasonable agreement with the Soper |
124 |
|
|
data for the structural features of liquid water.\cite{Soper86,Liu96b} |
125 |
|
|
Additionally, the dynamical properties exhibited by SSD agree with |
126 |
|
|
experiment better than those of more computationally expensive models |
127 |
|
|
(like TIP3P and SPC/E).\cite{Chandra99} The combination of speed and |
128 |
|
|
accurate depiction of solvent properties makes SSD a very attractive |
129 |
|
|
model for the simulation of large scale biochemical simulations. |
130 |
|
|
|
131 |
|
|
It is important to note that the SSD model was originally developed |
132 |
|
|
for use with the Ewald summation for handling long-range |
133 |
|
|
electrostatics.\cite{Ewald21} In applying this water model in a |
134 |
|
|
variety of molecular systems, it would be useful to know its |
135 |
|
|
properties and behavior under the more computationally efficient |
136 |
|
|
reaction field (RF) technique, the correction techniques discussed in |
137 |
|
|
the previous chapter, or even a simple |
138 |
|
|
cutoff.\cite{Onsager36,Fennell06} This study addresses these issues by |
139 |
|
|
looking at the structural and transport behavior of SSD over a variety |
140 |
|
|
of temperatures with the purpose of utilizing the RF correction |
141 |
|
|
technique. We then suggest modifications to the parameters that |
142 |
|
|
result in more realistic bulk phase behavior. It should be noted that |
143 |
|
|
in a recent publication, some of the original investigators of the SSD |
144 |
|
|
water model have suggested adjustments to the SSD water model to |
145 |
|
|
address abnormal density behavior (also observed here), calling the |
146 |
|
|
corrected model SSD1.\cite{Tan03} In the later sections of this |
147 |
|
|
chapter, we compare our modified variants of SSD with both the |
148 |
|
|
original SSD and SSD1 models and discuss how our changes improve the |
149 |
|
|
depiction of water. |
150 |
|
|
|
151 |
|
|
\section{Simulation Methods} |
152 |
|
|
|
153 |
|
|
Most of the calculations in this particular study were performed using |
154 |
|
|
a internally developed simulation code that was one of the precursors |
155 |
|
|
of the {\sc oopse} molecular dynamics (MD) package.\cite{Meineke05} |
156 |
|
|
All of the capabilities of this code have been efficiently |
157 |
|
|
incorporated into {\sc oopse}, and calculation results are consistent |
158 |
|
|
between the two simulation packages. The later calculations involving |
159 |
|
|
the damped shifted force ({\sc sf}) techniques were performed using |
160 |
|
|
{\sc oopse}. |
161 |
|
|
|
162 |
|
|
In the primary simulations of this study, long-range dipole-dipole |
163 |
|
|
interaction corrections were accounted for by using either the |
164 |
|
|
reaction field technique or a simple cubic switching function at the |
165 |
|
|
cutoff radius. Interestingly, one of the early applications of a |
166 |
|
|
reaction field was in Monte Carlo simulations of liquid |
167 |
|
|
water.\cite{Barker73} In this method, the magnitude of the reaction |
168 |
|
|
field acting on dipole $i$ is |
169 |
|
|
\begin{equation} |
170 |
|
|
\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
171 |
|
|
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} s(r_{ij}), |
172 |
|
|
\label{eq:rfequation} |
173 |
|
|
\end{equation} |
174 |
|
|
where $\mathcal{R}$ is the cavity defined by the cutoff radius |
175 |
|
|
($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the |
176 |
|
|
system, ${\bf\mu}_{j}$ is the dipole moment vector of particle $j$, |
177 |
|
|
and $s(r_{ij})$ is a cubic switching function.\cite{Allen87} The |
178 |
|
|
reaction field contribution to the total energy by particle $i$ is |
179 |
|
|
given by $-\frac{1}{2}{\bf\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque |
180 |
|
|
on dipole $i$ by ${\bf\mu}_{i}\times\mathcal{E}_{i}$.\cite{Allen87} An |
181 |
|
|
applied reaction field will alter the bulk orientational properties of |
182 |
|
|
simulated water, and there is particular sensitivity of these |
183 |
|
|
properties on changes in the length of the cutoff |
184 |
|
|
radius.\cite{vanderSpoel98} This variable behavior makes reaction |
185 |
|
|
field a less attractive method than the Ewald sum; however, for very |
186 |
|
|
large systems, the computational benefit of reaction field is is |
187 |
|
|
significant. |
188 |
|
|
|
189 |
|
|
In contrast to the simulations with a reaction field, we have also |
190 |
|
|
performed a companion set of simulations {\it without} a surrounding |
191 |
|
|
dielectric (i.e. using a simple cubic switching function at the cutoff |
192 |
|
|
radius). As a result, we have developed two reparametrizations of SSD |
193 |
|
|
which can be used either with or without an active reaction field. |
194 |
|
|
|
195 |
|
|
To determine the preferred densities of the models, we performed |
196 |
|
|
simulations in the isobaric-isothermal ({\it NPT}) ensemble. All |
197 |
|
|
dynamical properties for these models were then obtained from |
198 |
|
|
microcanonical ({\it NVE}) simulations done at densities matching the |
199 |
|
|
{\it NPT} density for a particular target temperature. The constant |
200 |
|
|
pressure simulations were implemented using an integral thermostat and |
201 |
|
|
barostat as outlined by Hoover.\cite{Hoover85,Hoover86} All molecules |
202 |
|
|
were treated as non-linear rigid bodies. Vibrational constraints are |
203 |
|
|
not necessary in simulations of SSD, because there are no explicit |
204 |
|
|
hydrogen atoms, and thus no molecular vibrational modes need to be |
205 |
|
|
considered. |
206 |
|
|
|
207 |
|
|
The symplectic splitting method proposed by Dullweber, Leimkuhler, and |
208 |
|
|
McLachlan ({\sc dlm}, see section \ref{sec:IntroIntegrate}) was used |
209 |
|
|
to carry out the integration of the equations of motion in place of |
210 |
|
|
the more prevalent quaternion |
211 |
|
|
method.\cite{Dullweber97,Evans77,Evans77b,Allen87} The reason behind |
212 |
|
|
this decision was that, in {\it NVE} simulations, the energy drift |
213 |
|
|
when using quaternions was substantially greater than when using the |
214 |
|
|
{\sc dlm} method (Fig. \ref{fig:timeStepIntegration}). This steady |
215 |
|
|
drift in the total energy has also been observed in other |
216 |
|
|
studies.\cite{Kol97} |
217 |
|
|
|
218 |
|
|
\begin{figure} |
219 |
|
|
\centering |
220 |
|
|
\includegraphics[width=\linewidth]{./figures/timeStepIntegration.pdf} |
221 |
|
|
\caption{Energy conservation using both quaternion-based integration |
222 |
|
|
and the {\sc dlm} method with increasing time step. The larger time |
223 |
|
|
step plots are shifted from the true energy baseline (that of $\Delta |
224 |
|
|
t$ = 0.1fs) for clarity.} |
225 |
|
|
\label{fig:timeStepIntegration} |
226 |
|
|
\end{figure} |
227 |
|
|
|
228 |
|
|
The {\sc dlm} method allows for Verlet style integration orientational |
229 |
|
|
motion of rigid bodies via a sequence of rotation matrix |
230 |
|
|
operations. Because these matrix operations are more costly than the |
231 |
|
|
simpler arithmetic operations for quaternion propagation, typical SSD |
232 |
|
|
particle simulations using {\sc dlm} are 5-10\% slower than |
233 |
|
|
simulations using the quaternion method and an identical time |
234 |
|
|
step. This additional expense is justified because of the ability to |
235 |
|
|
use time steps that are more that twice as long and still achieve the |
236 |
|
|
same energy conservation. |
237 |
|
|
|
238 |
|
|
Figure \ref{fig:timeStepIntegration} shows the resulting energy drift |
239 |
|
|
at various time steps for both {\sc dlm} and quaternion |
240 |
|
|
integration. All of the 1000 SSD particle simulations started with the |
241 |
|
|
same configuration, and the only difference was the method used to |
242 |
|
|
handle orientational motion. At time steps of 0.1 and 0.5fs, both |
243 |
|
|
methods for propagating the orientational degrees of freedom conserve |
244 |
|
|
energy fairly well, with the quaternion method showing a slight energy |
245 |
|
|
drift over time in the 0.5fs time step simulation. Time steps of 1 and |
246 |
|
|
2fs clearly demonstrate the benefits in energy conservation that come |
247 |
|
|
with the {\sc dlm} method. Thus, while maintaining the same degree of |
248 |
|
|
energy conservation, one can take considerably longer time steps, |
249 |
|
|
leading to an overall reduction in computation time. |
250 |
|
|
|
251 |
|
|
Energy drifts in water simulations using {\sc dlm} integration were |
252 |
|
|
unnoticeable for time steps up to 3fs. We observed a slight energy |
253 |
|
|
drift on the order of 0.012~kcal/mol per nanosecond with a time step |
254 |
|
|
of 4fs. As expected, this drift increases dramatically with increasing |
255 |
|
|
time step. To insure accuracy in our {\it NVE} simulations, time steps |
256 |
|
|
were set at 2fs and were also kept at this value for {\it NPT} |
257 |
|
|
simulations. |
258 |
|
|
|
259 |
|
|
Proton-disordered ice crystals in both the I$_\textrm{h}$ and |
260 |
|
|
I$_\textrm{c}$ lattices were generated as starting points for all |
261 |
|
|
simulations. The I$_\textrm{h}$ crystals were formed by first |
262 |
|
|
arranging the centers of mass of the SSD particles into a |
263 |
|
|
``hexagonal'' ice lattice of 1024 particles. Because of the crystal |
264 |
|
|
structure of I$_\textrm{h}$ ice, the simulation boxes were |
265 |
|
|
orthorhombic in shape with an edge length ratio of approximately |
266 |
|
|
1.00$\times$1.06$\times$1.23. We then allowed the particles to orient |
267 |
|
|
freely about their fixed lattice positions with angular momenta values |
268 |
|
|
randomly sampled at 400K. The rotational temperature was then scaled |
269 |
|
|
down in stages to slowly cool the crystals to 25K. The particles were |
270 |
|
|
then allowed to translate with fixed orientations at a constant |
271 |
|
|
pressure of 1atm for 50ps at 25K. Finally, all constraints were |
272 |
|
|
removed and the ice crystals were allowed to equilibrate for 50ps at |
273 |
|
|
25K and a constant pressure of 1atm. This procedure resulted in |
274 |
|
|
structurally stable I$_\textrm{h}$ ice crystals that obey the |
275 |
|
|
Bernal-Fowler rules.\cite{Bernal33,Rahman72} This method was also |
276 |
|
|
utilized in the making of diamond lattice I$_\textrm{c}$ ice crystals, |
277 |
|
|
with each cubic simulation box consisting of either 512 or 1000 |
278 |
|
|
particles. Only isotropic volume fluctuations were performed under |
279 |
|
|
constant pressure, so the ratio of edge lengths remained constant |
280 |
|
|
throughout the simulations. |
281 |
|
|
|
282 |
|
|
\section{SSD Density Behavior} |
283 |
|
|
|
284 |
|
|
Melting studies were performed on the randomized ice crystals using |
285 |
|
|
the {\it NPT} ensemble. During melting simulations, the melting |
286 |
|
|
transition and the density maximum can both be observed, provided that |
287 |
|
|
the density maximum occurs in the liquid and not the supercooled |
288 |
|
|
regime. It should be noted that the calculated melting temperature |
289 |
|
|
($T_\textrm{m}$) will not be the true $T_\textrm{m}$ because of |
290 |
|
|
super-heating due to the relatively short time scales in molecular |
291 |
|
|
simulations. This behavior results in inflated $T_\textrm{m}$ values; |
292 |
|
|
however, these values provide a reasonable initial estimate of |
293 |
|
|
$T_\textrm{m}$. |
294 |
|
|
|
295 |
|
|
An ensemble average from five separate melting simulations was |
296 |
|
|
acquired, each starting from different ice crystals generated as |
297 |
|
|
described previously. All simulations were equilibrated for 100ps |
298 |
|
|
prior to a 200ps data collection run at each temperature setting. The |
299 |
|
|
temperature range of study spanned from 25 to 400K, with a maximum |
300 |
|
|
degree increment of 25K. For regions of interest along this stepwise |
301 |
|
|
progression, the temperature increment was decreased from 25K to 10 |
302 |
|
|
and 5K. The above equilibration and production times were sufficient |
303 |
|
|
in that the fluctuations in the volume autocorrelation function damped |
304 |
|
|
out in all of the simulations in under 20ps. |
305 |
|
|
|
306 |
|
|
Our initial simulations focused on the original SSD water model, and |
307 |
|
|
an average density versus temperature plot is shown in figure |
308 |
|
|
\ref{fig:ssdDense}. Note that the density maximum when using a |
309 |
|
|
reaction field appears between 255 and 265K. There were smaller |
310 |
|
|
fluctuations in the density at 260K than at either 255 or 265K, so we |
311 |
|
|
report this value as the location of the density maximum. Figure |
312 |
|
|
\ref{fig:ssdDense} was constructed using ice I$_\textrm{h}$ crystals |
313 |
|
|
for the initial configuration; though not pictured, the simulations |
314 |
|
|
starting from ice I$_\textrm{c}$ crystal configurations showed similar |
315 |
|
|
results, with a liquid-phase density maximum at the same temperature. |
316 |
|
|
|
317 |
|
|
\begin{figure} |
318 |
|
|
\centering |
319 |
|
|
\includegraphics[width=\linewidth]{./figures/ssdDense.pdf} |
320 |
|
|
\caption{ Density versus temperature for TIP3P, SPC/E, TIP4P, SSD, |
321 |
|
|
SSD with a reaction field, and |
322 |
chrisfen |
2979 |
experiment.\cite{Jorgensen98b,Baez94,CRC80}. Note that using a |
323 |
chrisfen |
2977 |
reaction field lowers the density more than the already lowered SSD |
324 |
|
|
densities. The lower than expected densities for the SSD model |
325 |
|
|
prompted the original reparametrization of SSD to SSD1.\cite{Tan03}} |
326 |
|
|
\label{fig:ssdDense} |
327 |
|
|
\end{figure} |
328 |
|
|
|
329 |
|
|
The density maximum for SSD compares quite favorably to other simple |
330 |
|
|
water models. Figure \ref{fig:ssdDense} also shows calculated |
331 |
|
|
densities of several other models and experiment obtained from other |
332 |
chrisfen |
2979 |
sources.\cite{Jorgensen98b,Baez94,CRC80} Of the listed simple water |
333 |
chrisfen |
2977 |
models, SSD has a temperature closest to the experimentally observed |
334 |
|
|
density maximum. Of the {\it charge-based} models in figure |
335 |
|
|
\ref{fig:ssdDense}, TIP4P has a density maximum behavior most like |
336 |
|
|
that seen in SSD. Though not included in this plot, it is useful to |
337 |
|
|
note that TIP5P has a density maximum nearly identical to the |
338 |
|
|
experimentally measured temperature (see section |
339 |
|
|
\ref{sec:t5peDensity}. |
340 |
|
|
|
341 |
|
|
Liquid state densities in water have been observed to be dependent on |
342 |
|
|
the cutoff radius ($R_\textrm{c}$), both with and without the use of a |
343 |
|
|
reaction field.\cite{vanderSpoel98} In order to address the possible |
344 |
|
|
effect of $R_\textrm{c}$, simulations were performed with a cutoff |
345 |
|
|
radius of 12\AA\, complementing the 9\AA\ $R_\textrm{c}$ used in the |
346 |
|
|
previous SSD simulations. All of the resulting densities overlapped |
347 |
|
|
within error and showed no significant trend toward lower or higher |
348 |
|
|
densities in simulations both with and without reaction field. |
349 |
|
|
|
350 |
|
|
The key feature to recognize in figure \ref{fig:ssdDense} is the |
351 |
|
|
density scaling of SSD relative to other common models at any given |
352 |
|
|
temperature. SSD assumes a lower density than any of the other listed |
353 |
|
|
models at the same pressure, behavior which is especially apparent at |
354 |
|
|
temperatures greater than 300K. Lower than expected densities have |
355 |
|
|
been observed for other systems using a reaction field for long-range |
356 |
|
|
electrostatic interactions, so the most likely reason for the reduced |
357 |
|
|
densities is the presence of the reaction |
358 |
|
|
field.\cite{vanderSpoel98,Nezbeda02} In order to test the effect of |
359 |
|
|
the reaction field on the density of the systems, the simulations were |
360 |
|
|
repeated without a reaction field present. The results of these |
361 |
|
|
simulations are also displayed in figure \ref{fig:ssdDense}. Without |
362 |
|
|
the reaction field, the densities increase to more experimentally |
363 |
|
|
reasonable values, especially around the freezing point of liquid |
364 |
|
|
water. The shape of the curve is similar to the curve produced from |
365 |
|
|
SSD simulations using reaction field, specifically the rapidly |
366 |
|
|
decreasing densities at higher temperatures; however, a shift in the |
367 |
|
|
density maximum location, down to 245K, is observed. This is a more |
368 |
|
|
accurate comparison to the other listed water models, in that no long |
369 |
|
|
range corrections were applied in those |
370 |
chrisfen |
2979 |
simulations.\cite{Baez94,Jorgensen98b} However, even without the |
371 |
chrisfen |
2977 |
reaction field, the density around 300K is still significantly lower |
372 |
|
|
than experiment and comparable water models. This anomalous behavior |
373 |
|
|
was what lead Tan {\it et al.} to recently reparametrize |
374 |
|
|
SSD.\cite{Tan03} Throughout the remainder of the paper our |
375 |
|
|
reparametrizations of SSD will be compared with their newer SSD1 |
376 |
|
|
model. |
377 |
|
|
|
378 |
|
|
\section{SSD Transport Behavior} |
379 |
|
|
|
380 |
|
|
Accurate dynamical properties of a water model are particularly |
381 |
|
|
important when using the model to study permeation or transport across |
382 |
|
|
biological membranes. In order to probe transport in bulk water, {\it |
383 |
|
|
NVE} simulations were performed at the average densities obtained from |
384 |
|
|
the {\it NPT} simulations at an identical target |
385 |
|
|
temperature. Simulations started with randomized velocities and |
386 |
|
|
underwent 50ps of temperature scaling and 50ps of constant energy |
387 |
|
|
equilibration before a 200ps data collection run. Diffusion constants |
388 |
|
|
were calculated via linear fits to the long-time behavior of the |
389 |
|
|
mean-square displacement as a function of time.\cite{Allen87} The |
390 |
|
|
averaged results from five sets of {\it NVE} simulations are displayed |
391 |
|
|
in figure \ref{fig:ssdDiffuse}, alongside experimental, SPC/E, and TIP5P |
392 |
chrisfen |
2979 |
results.\cite{Gillen72,Holz00,Baez94,Mahoney01} |
393 |
chrisfen |
2977 |
|
394 |
|
|
\begin{figure} |
395 |
|
|
\centering |
396 |
|
|
\includegraphics[width=\linewidth]{./figures/ssdDiffuse.pdf} |
397 |
|
|
\caption{ Average self-diffusion constant as a function of temperature for |
398 |
|
|
SSD, SPC/E, and TIP5P compared with experimental |
399 |
chrisfen |
2979 |
data.\cite{Baez94,Mahoney01,Gillen72,Holz00} Of the three water |
400 |
chrisfen |
2977 |
models shown, SSD has the least deviation from the experimental |
401 |
|
|
values. The rapidly increasing diffusion constants for TIP5P and SSD |
402 |
|
|
correspond to significant decreases in density at the higher |
403 |
|
|
temperatures.} |
404 |
|
|
\label{fig:ssdDiffuse} |
405 |
|
|
\end{figure} |
406 |
|
|
|
407 |
|
|
The observed values for the diffusion constant point out one of the |
408 |
|
|
strengths of the SSD model. Of the three models shown, the SSD model |
409 |
|
|
has the most accurate depiction of self-diffusion in both the |
410 |
|
|
supercooled and liquid regimes. SPC/E does a respectable job by |
411 |
|
|
reproducing values similar to experiment around 290K; however, it |
412 |
|
|
deviates at both higher and lower temperatures, failing to predict the |
413 |
|
|
correct thermal trend. TIP5P and SSD both start off low at colder |
414 |
|
|
temperatures and tend to diffuse too rapidly at higher temperatures. |
415 |
|
|
This behavior at higher temperatures is not particularly surprising |
416 |
|
|
since the densities of both TIP5P and SSD are lower than experimental |
417 |
|
|
water densities at higher temperatures. When calculating the |
418 |
|
|
diffusion coefficients for SSD at experimental densities (instead of |
419 |
|
|
the densities from the {\it NPT} simulations), the resulting values |
420 |
|
|
fall more in line with experiment at these temperatures. |
421 |
|
|
|
422 |
|
|
\section{Structural Changes and Characterization} |
423 |
|
|
|
424 |
|
|
By starting the simulations from the crystalline state, we can get an |
425 |
|
|
estimation of the $T_\textrm{m}$ of the ice structure, and beyond the |
426 |
|
|
melting point, we study the phase behavior of the liquid. The constant |
427 |
|
|
pressure heat capacity ($C_\textrm{p}$) was monitored to locate |
428 |
|
|
$T_\textrm{m}$ in each of the simulations. In the melting simulations |
429 |
|
|
of the 1024 particle ice I$_\textrm{h}$ simulations, a large spike in |
430 |
|
|
$C_\textrm{p}$ occurs at 245K, indicating a first order phase |
431 |
|
|
transition for the melting of these ice crystals (see figure |
432 |
|
|
\ref{fig:ssdCp}. When the reaction field is turned off, the melting |
433 |
|
|
transition occurs at 235K. These melting transitions are considerably |
434 |
|
|
lower than the experimental value of 273K, indicating that the solid |
435 |
|
|
ice I$_\textrm{h}$ is not thermodynamically preferred relative to the |
436 |
|
|
liquid state at these lower temperatures. |
437 |
|
|
\begin{figure} |
438 |
|
|
\centering |
439 |
|
|
\includegraphics[width=\linewidth]{./figures/ssdCp.pdf} |
440 |
|
|
\caption{Heat capacity versus temperature for the SSD model with an |
441 |
|
|
active reaction field. Note the large spike in $C_p$ around 245K, |
442 |
|
|
indicating a phase transition from the ordered crystal to disordered |
443 |
|
|
liquid.} |
444 |
|
|
\label{fig:ssdCp} |
445 |
|
|
\end{figure} |
446 |
|
|
|
447 |
|
|
\begin{figure} |
448 |
|
|
\centering |
449 |
|
|
\includegraphics[width=\linewidth]{./figures/fullContour.pdf} |
450 |
|
|
\caption{ Contour plots of 2D angular pair correlation functions for |
451 |
|
|
512 SSD molecules at 100K (A \& B) and 300K (C \& D). Dark areas |
452 |
|
|
signify regions of enhanced density while light areas signify |
453 |
|
|
depletion relative to the bulk density. White areas have pair |
454 |
|
|
correlation values below 0.5 and black areas have values above 1.5.} |
455 |
|
|
\label{fig:contour} |
456 |
|
|
\end{figure} |
457 |
|
|
|
458 |
|
|
\begin{figure} |
459 |
|
|
\centering |
460 |
|
|
\includegraphics[width=2.5in]{./figures/corrDiag.pdf} |
461 |
|
|
\caption{ An illustration of angles involved in the correlations observed in figure \ref{fig:contour}.} |
462 |
|
|
\label{fig:corrAngle} |
463 |
|
|
\end{figure} |
464 |
|
|
|
465 |
|
|
Additional analysis of the melting process was performed using |
466 |
|
|
two-dimensional structure and dipole angle correlations. Expressions |
467 |
|
|
for these correlations are as follows: |
468 |
|
|
|
469 |
|
|
\begin{equation} |
470 |
|
|
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\ , |
471 |
|
|
\end{equation} |
472 |
|
|
\begin{equation} |
473 |
|
|
g_{\text{AB}}(r,\cos\omega) = |
474 |
|
|
\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\ , |
475 |
|
|
\end{equation} |
476 |
|
|
where $\theta$ and $\omega$ refer to the angles shown in figure |
477 |
|
|
\ref{fig:corrAngle}. By binning over both distance and the cosine of the |
478 |
|
|
desired angle between the two dipoles, the $g(r)$ can be analyzed to |
479 |
|
|
determine the common dipole arrangements that constitute the peaks and |
480 |
|
|
troughs in the standard one-dimensional $g(r)$ plots. Frames A and B |
481 |
|
|
of figure \ref{fig:contour} show results from an ice I$_\textrm{c}$ |
482 |
|
|
simulation. The first peak in the $g(r)$ consists primarily of the |
483 |
|
|
preferred hydrogen bonding arrangements as dictated by the tetrahedral |
484 |
|
|
sticky potential - one peak for the hydrogen bond donor and the other |
485 |
|
|
for the hydrogen bond acceptor. Due to the high degree of |
486 |
|
|
crystallinity of the sample, the second and third solvation shells |
487 |
|
|
show a repeated peak arrangement which decays at distances around the |
488 |
|
|
fourth solvation shell, near the imposed cutoff for the Lennard-Jones |
489 |
|
|
and dipole-dipole interactions. In the higher temperature simulation |
490 |
|
|
shown in frames C and D, these long-range features deteriorate |
491 |
|
|
rapidly. The first solvation shell still shows the strong effect of |
492 |
|
|
the sticky-potential, although it covers a larger area, extending to |
493 |
|
|
include a fraction of aligned dipole peaks within the first solvation |
494 |
|
|
shell. The latter peaks lose due to thermal motion and as the |
495 |
|
|
competing dipole force overcomes the sticky potential's tight |
496 |
|
|
tetrahedral structuring of the crystal. |
497 |
|
|
|
498 |
|
|
This complex interplay between dipole and sticky interactions was |
499 |
|
|
remarked upon as a possible reason for the split second peak in the |
500 |
|
|
oxygen-oxygen pair correlation function, |
501 |
|
|
$g_\textrm{OO}(r)$.\cite{Liu96b} At low temperatures, the second |
502 |
|
|
solvation shell peak appears to have two distinct components that |
503 |
|
|
blend together to form one observable peak. At higher temperatures, |
504 |
|
|
this split character alters to show the leading 4\AA\ peak dominated |
505 |
|
|
by equatorial anti-parallel dipole orientations. There is also a |
506 |
|
|
tightly bunched group of axially arranged dipoles that most likely |
507 |
|
|
consist of the smaller fraction of aligned dipole pairs. The trailing |
508 |
|
|
component of the split peak at 5\AA\ is dominated by aligned dipoles |
509 |
|
|
that assume hydrogen bond arrangements similar to those seen in the |
510 |
|
|
first solvation shell. This evidence indicates that the dipole pair |
511 |
|
|
interaction begins to dominate outside of the range of the dipolar |
512 |
|
|
repulsion term. The energetically favorable dipole arrangements |
513 |
|
|
populate the region immediately outside this repulsion region (around |
514 |
|
|
4\AA), while arrangements that seek to satisfy both the sticky and |
515 |
|
|
dipole forces locate themselves just beyond this initial buildup |
516 |
|
|
(around 5\AA). |
517 |
|
|
|
518 |
|
|
This analysis indicates that the split second peak is primarily the |
519 |
|
|
product of the dipolar repulsion term of the sticky potential. In |
520 |
|
|
fact, the inner peak can be pushed out and merged with the outer split |
521 |
|
|
peak just by extending the switching function ($s^\prime(r_{ij})$) |
522 |
|
|
from its normal 4\AA\ cutoff to values of 4.5 or even 5\AA. This |
523 |
|
|
type of correction is not recommended for improving the liquid |
524 |
|
|
structure, since the second solvation shell would still be shifted too |
525 |
|
|
far out. In addition, this would have an even more detrimental effect |
526 |
|
|
on the system densities, leading to a liquid with a more open |
527 |
|
|
structure and a density considerably lower than the already low SSD |
528 |
|
|
density. A better correction would be to include the |
529 |
|
|
quadrupole-quadrupole interactions for the water particles outside of |
530 |
|
|
the first solvation shell, but this would remove the simplicity and |
531 |
|
|
speed advantage of SSD. |
532 |
|
|
|
533 |
|
|
\section{Adjusted Potentials: SSD/RF and SSD/E} |
534 |
|
|
|
535 |
|
|
The propensity of SSD to adopt lower than expected densities under |
536 |
|
|
varying conditions is troubling, especially at higher temperatures. In |
537 |
|
|
order to correct this model for use with a reaction field, it is |
538 |
|
|
necessary to adjust the force field parameters for the primary |
539 |
|
|
intermolecular interactions. In undergoing a reparametrization, it is |
540 |
|
|
important not to focus on just one property and neglect the others. In |
541 |
|
|
this case, it would be ideal to correct the densities while |
542 |
|
|
maintaining the accurate transport behavior. |
543 |
|
|
|
544 |
|
|
The parameters available for tuning include the $\sigma$ and |
545 |
|
|
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the |
546 |
|
|
strength of the sticky potential ($\nu_0$), and the cutoff distances |
547 |
|
|
for the sticky attractive and dipole repulsive cubic switching |
548 |
|
|
function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$ |
549 |
|
|
respectively). The results of the reparametrizations are shown in |
550 |
|
|
table \ref{tab:ssdParams}. We are calling these reparametrizations the |
551 |
|
|
Soft Sticky Dipole Reaction Field (SSD/RF - for use with a reaction |
552 |
|
|
field) and Soft Sticky Dipole Extended (SSD/E - an attempt to improve |
553 |
|
|
the liquid structure in simulations without a long-range correction). |
554 |
|
|
|
555 |
|
|
\begin{table} |
556 |
|
|
\caption{PARAMETERS FOR THE ORIGINAL AND ADJUSTED SSD MODELS} |
557 |
|
|
|
558 |
|
|
\centering |
559 |
|
|
\begin{tabular}{ lcccc } |
560 |
|
|
\toprule |
561 |
|
|
\toprule |
562 |
|
|
Parameters & SSD & SSD1 & SSD/E & SSD/RF \\ |
563 |
|
|
\midrule |
564 |
|
|
$\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
565 |
|
|
$\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
566 |
|
|
$\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ |
567 |
|
|
$\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
568 |
|
|
$\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\ |
569 |
|
|
$r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ |
570 |
|
|
$r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ |
571 |
|
|
$r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ |
572 |
|
|
$r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ |
573 |
|
|
\bottomrule |
574 |
|
|
\end{tabular} |
575 |
|
|
\label{tab:ssdParams} |
576 |
|
|
\end{table} |
577 |
|
|
|
578 |
|
|
\begin{figure} |
579 |
|
|
\centering |
580 |
|
|
\includegraphics[width=4.5in]{./figures/newGofRCompare.pdf} |
581 |
|
|
\caption{ Plots showing the experimental $g(r)$ (Ref. \cite{Hura00}) |
582 |
|
|
with SSD/E and SSD1 without reaction field (top), as well as SSD/RF |
583 |
|
|
and SSD1 with reaction field turned on (bottom). The changes in |
584 |
|
|
parameters have lowered and broadened the first peak of SSD/E and |
585 |
|
|
SSD/RF, resulting in a better fit to the first solvation shell.} |
586 |
|
|
\label{fig:gofrCompare} |
587 |
|
|
\end{figure} |
588 |
|
|
|
589 |
|
|
\begin{figure} |
590 |
|
|
\centering |
591 |
|
|
\includegraphics[width=\linewidth]{./figures/dualPotentials.pdf} |
592 |
|
|
\caption{ Positive and negative isosurfaces of the sticky potential for |
593 |
|
|
SSD and SSD1 (A) and SSD/E \& SSD/RF (B). Gold areas correspond to the |
594 |
|
|
tetrahedral attractive component, and blue areas correspond to the |
595 |
|
|
dipolar repulsive component.} |
596 |
|
|
\label{fig:isosurface} |
597 |
|
|
\end{figure} |
598 |
|
|
|
599 |
|
|
In the original paper detailing the development of SSD, Liu and Ichiye |
600 |
|
|
placed particular emphasis on an accurate description of the first |
601 |
|
|
solvation shell. This resulted in a somewhat tall and narrow first |
602 |
|
|
peak in $g(r)$ that integrated to give similar coordination numbers to |
603 |
|
|
the experimental data obtained by Soper and |
604 |
|
|
Phillips.\cite{Liu96b,Soper86} New experimental x-ray scattering data |
605 |
|
|
from Hura {\it et al.} indicates a slightly lower and shifted first |
606 |
|
|
peak in the $g_\textrm{OO}(r)$, so our adjustments to SSD were made |
607 |
|
|
after taking into consideration the new experimental |
608 |
|
|
findings.\cite{Hura00} Figure \ref{fig:gofrCompare} shows the |
609 |
|
|
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing |
610 |
|
|
the revised SSD model (SSD1), SSD/E, and SSD/RF to the new |
611 |
|
|
experimental results. Both modified water models have shorter peaks |
612 |
|
|
that match more closely to the experimental peak (as seen in the |
613 |
|
|
insets of figure \ref{fig:gofrCompare}). This structural alteration |
614 |
|
|
was accomplished by the combined reduction in the Lennard-Jones |
615 |
|
|
$\sigma$ variable and adjustment of the sticky potential strength and |
616 |
|
|
cutoffs. As can be seen in table \ref{tab:ssdParams}, the cutoffs for |
617 |
|
|
the tetrahedral attractive and dipolar repulsive terms were nearly |
618 |
|
|
swapped with each other. Isosurfaces of the original and modified |
619 |
|
|
sticky potentials are shown in figure \ref{fig:isosurface}. In these |
620 |
|
|
isosurfaces, it is easy to see how altering the cutoffs changes the |
621 |
|
|
repulsive and attractive character of the particles. With a reduced |
622 |
|
|
repulsive surface, the particles can move closer to one another, |
623 |
|
|
increasing the density for the overall system. This change in |
624 |
|
|
interaction cutoff also results in a more gradual orientational motion |
625 |
|
|
by allowing the particles to maintain preferred dipolar arrangements |
626 |
|
|
before they begin to feel the pull of the tetrahedral |
627 |
|
|
restructuring. As the particles move closer together, the dipolar |
628 |
|
|
repulsion term becomes active and excludes unphysical nearest-neighbor |
629 |
|
|
arrangements. This compares with how SSD and SSD1 exclude preferred |
630 |
|
|
dipole alignments before the particles feel the pull of the ``hydrogen |
631 |
|
|
bonds''. Aside from improving the shape of the first peak in the |
632 |
|
|
$g(r)$, this modification improves the densities considerably by |
633 |
|
|
allowing the persistence of full dipolar character below the previous |
634 |
|
|
4\AA\ cutoff. |
635 |
|
|
|
636 |
|
|
While adjusting the location and shape of the first peak of $g(r)$ |
637 |
|
|
improves the densities, these changes alone are insufficient to bring |
638 |
|
|
the system densities up to the values observed experimentally. To |
639 |
|
|
further increase the densities, the dipole moments were increased in |
640 |
|
|
both of our adjusted models. Since SSD is a dipole based model, the |
641 |
|
|
structure and transport are very sensitive to changes in the dipole |
642 |
|
|
moment. The original SSD simply used the dipole moment calculated from |
643 |
|
|
the TIP3P water model, which at 2.35~D is significantly greater than |
644 |
|
|
the experimental gas phase value of 1.84~D. The larger dipole moment |
645 |
|
|
is a more realistic value and improves the dielectric properties of |
646 |
|
|
the fluid. Both theoretical and experimental measurements indicate a |
647 |
|
|
liquid phase dipole moment ranging from 2.4~D to values as high as |
648 |
|
|
3.11~D, providing a substantial range of reasonable values for a |
649 |
|
|
dipole moment.\cite{Sprik91,Gubskaya02,Badyal00,Barriol64} Moderately |
650 |
|
|
increasing the dipole moments to 2.42 and 2.48~D for SSD/E and SSD/RF, |
651 |
|
|
respectively, leads to significant changes in the density and |
652 |
|
|
transport of the water models. |
653 |
|
|
|
654 |
|
|
\subsection{Density Behavior} |
655 |
|
|
|
656 |
|
|
In order to demonstrate the benefits of these reparametrizations, we |
657 |
|
|
performed a series of {\it NPT} and {\it NVE} simulations to probe the |
658 |
|
|
density and transport properties of the adapted models and compare the |
659 |
|
|
results to the original SSD model. This comparison involved full {\it |
660 |
|
|
NPT} melting sequences for both SSD/E and SSD/RF, as well as {\it NVE} |
661 |
|
|
transport calculations at the calculated self-consistent |
662 |
|
|
densities. Again, the results were obtained from five separate |
663 |
|
|
simulations of 1024 particle systems, and the melting sequences were |
664 |
|
|
started from different ice I$_\textrm{h}$ crystals constructed as |
665 |
|
|
described previously. Each {\it NPT} simulation was equilibrated for |
666 |
|
|
100ps before a 200ps data collection run at each temperature step, |
667 |
|
|
and the final configuration from the previous temperature simulation |
668 |
|
|
was used as a starting point. All {\it NVE} simulations had the same |
669 |
|
|
thermalization, equilibration, and data collection times as stated |
670 |
|
|
previously. |
671 |
|
|
|
672 |
|
|
\begin{figure} |
673 |
|
|
\centering |
674 |
|
|
\includegraphics[width=\linewidth]{./figures/ssdeDense.pdf} |
675 |
|
|
\caption{ Comparison of densities calculated with SSD/E to |
676 |
|
|
SSD1 without a reaction field, TIP3P, SPC/E, TIP5P, and |
677 |
chrisfen |
2979 |
experiment.\cite{Jorgensen98b,Baez94,Mahoney00,CRC80} Both SSD1 and |
678 |
chrisfen |
2977 |
SSD/E show good agreement with experiment when the long-range |
679 |
|
|
correction is neglected.} |
680 |
|
|
\label{fig:ssdeDense} |
681 |
|
|
\end{figure} |
682 |
|
|
|
683 |
|
|
Figure \ref{fig:ssdeDense} shows the density profiles for SSD/E, SSD1, |
684 |
|
|
TIP3P, TIP4P, and SPC/E alongside the experimental results. The |
685 |
|
|
calculated densities for both SSD/E and SSD1 have increased |
686 |
|
|
significantly over the original SSD model (see figure |
687 |
|
|
\ref{fig:ssdDense}) and are in better agreement with the experimental |
688 |
|
|
values. At 298 K, the densities of SSD/E and SSD1 without a long-range |
689 |
|
|
correction are 0.996 g/cm$^3$ and 0.999 g/cm$^3$ respectively. These |
690 |
|
|
both compare well with the experimental value of 0.997 g/cm$^3$, and |
691 |
|
|
they are considerably better than the SSD value of 0.967 g/cm$^3$. The |
692 |
|
|
changes to the dipole moment and sticky switching functions have |
693 |
|
|
improved the structuring of the liquid (as seen in figure |
694 |
|
|
\ref{fig:gofrCompare}), but they have shifted the density maximum to |
695 |
|
|
much lower temperatures. This comes about via an increase in the |
696 |
|
|
liquid disorder through the weakening of the sticky potential and |
697 |
|
|
strengthening of the dipolar character. However, this increasing |
698 |
|
|
disorder in the SSD/E model has little effect on the melting |
699 |
|
|
transition. By monitoring $C_p$ throughout these simulations, we |
700 |
|
|
observed a melting transition for SSD/E at 235K, the same as SSD and |
701 |
|
|
SSD1. |
702 |
|
|
|
703 |
|
|
\begin{figure} |
704 |
|
|
\centering |
705 |
|
|
\includegraphics[width=\linewidth]{./figures/ssdrfDense.pdf} |
706 |
|
|
\caption{ Comparison of densities calculated with SSD/RF to |
707 |
|
|
SSD1 with a reaction field, TIP3P, SPC/E, TIP5P, and |
708 |
chrisfen |
2979 |
experiment.\cite{Jorgensen98b,Baez94,Mahoney00,CRC80} This plot |
709 |
chrisfen |
2977 |
shows the benefit afforded by the reparametrization for use with a |
710 |
|
|
reaction field correction - SSD/RF provides significantly more |
711 |
|
|
accurate densities than SSD1 when performing room temperature |
712 |
|
|
simulations.} |
713 |
|
|
\label{fig:ssdrfDense} |
714 |
|
|
\end{figure} |
715 |
|
|
|
716 |
|
|
Including the reaction field long-range correction results in a more |
717 |
|
|
interesting comparison. A density profile including SSD/RF and SSD1 |
718 |
|
|
with an active reaction field is shown in figure \ref{fig:ssdrfDense}. |
719 |
|
|
As observed in the simulations without a reaction field, the densities |
720 |
|
|
of SSD/RF and SSD1 show a dramatic increase over normal SSD (see |
721 |
|
|
figure \ref{fig:ssdDense}). At 298 K, SSD/RF has a density of 0.997 |
722 |
|
|
g/cm$^3$, directly in line with experiment and considerably better |
723 |
|
|
than the original SSD value of 0.941 g/cm$^3$ and the SSD1 value of |
724 |
|
|
0.972 g/cm$^3$. These results further emphasize the importance of |
725 |
|
|
reparametrization in order to model the density properly under |
726 |
|
|
different simulation conditions. Again, these changes have only a |
727 |
|
|
minor effect on the melting point, which observed at 245K for SSD/RF, |
728 |
|
|
is identical to SSD and only 5K lower than SSD1 with a reaction |
729 |
|
|
field. Additionally, the difference in density maxima is not as |
730 |
|
|
extreme, with SSD/RF showing a density maximum at 255K, fairly close |
731 |
|
|
to the density maxima of 260K and 265K, shown by SSD and SSD1 |
732 |
|
|
respectively. |
733 |
|
|
|
734 |
|
|
\subsection{Transport Behavior} |
735 |
|
|
|
736 |
|
|
\begin{figure} |
737 |
|
|
\centering |
738 |
|
|
\includegraphics[width=\linewidth]{./figures/ssdeDiffuse.pdf} |
739 |
|
|
\caption{ The diffusion constants calculated from SSD/E and |
740 |
|
|
SSD1 (both without a reaction field) along with experimental |
741 |
|
|
results.\cite{Gillen72,Holz00} The {\it NVE} calculations were |
742 |
|
|
performed at the average densities from the {\it NPT} simulations for |
743 |
|
|
the respective models. SSD/E is slightly more mobile than experiment |
744 |
|
|
at all of the temperatures, but it is closer to experiment at |
745 |
|
|
biologically relevant temperatures than SSD1 without a long-range |
746 |
|
|
correction.} |
747 |
|
|
\label{fig:ssdeDiffuse} |
748 |
|
|
\end{figure} |
749 |
|
|
|
750 |
|
|
The reparametrization of the SSD water model, both for use with and |
751 |
|
|
without an applied long-range correction, brought the densities up to |
752 |
|
|
what is expected for proper simulation of liquid water. In addition to |
753 |
|
|
improving the densities, it is important that the diffusive behavior |
754 |
|
|
of SSD be maintained or improved. Figure \ref{fig:ssdeDiffuse} |
755 |
|
|
compares the temperature dependence of the diffusion constant of SSD/E |
756 |
|
|
to SSD1 without an active reaction field at the densities calculated |
757 |
|
|
from their respective {\it NPT} simulations at 1 atm. The diffusion |
758 |
|
|
constant for SSD/E is consistently higher than experiment, while SSD1 |
759 |
|
|
remains lower than experiment until relatively high temperatures |
760 |
|
|
(around 360K). Both models follow the shape of the experimental curve |
761 |
|
|
below 300K but tend to diffuse too rapidly at higher temperatures, as |
762 |
|
|
seen in SSD1 crossing above 360K. This increasing diffusion relative |
763 |
|
|
to the experimental values is caused by the rapidly decreasing system |
764 |
|
|
density with increasing temperature. Both SSD1 and SSD/E show this |
765 |
|
|
deviation in particle mobility, but this trend has different |
766 |
|
|
implications on the diffusive behavior of the models. While SSD1 |
767 |
|
|
shows more experimentally accurate diffusive behavior in the high |
768 |
|
|
temperature regimes, SSD/E shows more accurate behavior in the |
769 |
|
|
supercooled and biologically relevant temperature ranges. Thus, the |
770 |
|
|
changes made to improve the liquid structure may have had an adverse |
771 |
|
|
affect on the density maximum, but they improve the transport behavior |
772 |
|
|
of SSD/E relative to SSD1 under the most commonly simulated |
773 |
|
|
conditions. |
774 |
|
|
|
775 |
|
|
\begin{figure} |
776 |
|
|
\centering |
777 |
|
|
\includegraphics[width=\linewidth]{./figures/ssdrfDiffuse.pdf} |
778 |
|
|
\caption{ The diffusion constants calculated from SSD/RF and |
779 |
|
|
SSD1 (both with an active reaction field) along with experimental |
780 |
|
|
results.\cite{Gillen72,Holz00} The {\it NVE} calculations were |
781 |
|
|
performed at the average densities from the {\it NPT} simulations for |
782 |
|
|
both of the models. SSD/RF captures the self-diffusion of water |
783 |
|
|
throughout most of this temperature range. The increasing diffusion |
784 |
|
|
constants at high temperatures for both models can be attributed to |
785 |
|
|
lower calculated densities than those observed in experiment.} |
786 |
|
|
\label{fig:ssdrfDiffuse} |
787 |
|
|
\end{figure} |
788 |
|
|
|
789 |
|
|
In figure \ref{fig:ssdrfDiffuse}, the diffusion constants for SSD/RF are |
790 |
|
|
compared to SSD1 with an active reaction field. Note that SSD/RF |
791 |
|
|
tracks the experimental results quantitatively, identical within error |
792 |
|
|
throughout most of the temperature range shown and exhibiting only a |
793 |
|
|
slight increasing trend at higher temperatures. SSD1 tends to diffuse |
794 |
|
|
more slowly at low temperatures and deviates to diffuse too rapidly at |
795 |
|
|
temperatures greater than 330K. As stated above, this deviation away |
796 |
|
|
from the ideal trend is due to a rapid decrease in density at higher |
797 |
|
|
temperatures. SSD/RF does not suffer from this problem as much as SSD1 |
798 |
|
|
because the calculated densities are closer to the experimental |
799 |
|
|
values. These results again emphasize the importance of careful |
800 |
|
|
reparametrization when using an altered long-range correction. |
801 |
|
|
|
802 |
|
|
\subsection{Summary of Liquid State Properties} |
803 |
|
|
|
804 |
|
|
\begin{table} |
805 |
|
|
\caption{PROPERTIES OF THE SINGLE-POINT WATER MODELS COMPARED WITH |
806 |
|
|
EXPERIMENTAL DATA AT AMBIENT CONDITIONS} |
807 |
|
|
\footnotesize |
808 |
|
|
\centering |
809 |
chrisfen |
2978 |
\begin{tabular}{ llccccc } |
810 |
chrisfen |
2977 |
\toprule |
811 |
|
|
\toprule |
812 |
chrisfen |
2978 |
& & SSD1 & SSD/E & SSD1 (RF) & SSD/RF & Experiment [Ref.] \\ |
813 |
chrisfen |
2977 |
\midrule |
814 |
chrisfen |
2978 |
$\rho$ & (g cm$^{-3}$) & 0.999(1) & 0.996(1) & 0.972(2) & 0.997(1) & 0.997 \cite{CRC80}\\ |
815 |
|
|
$C_p$ & (cal mol$^{-1}$ K$^{-1}$) & 28.80(11) & 25.45(9) & 28.28(6) & 23.83(16) & 18.005 \cite{Wagner02}\\ |
816 |
|
|
$D$ & ($10^{-5}$ cm$^2$ s$^{-1}$) & 1.78(7) & 2.51(18) & 2.00(17) & 2.32(6) & 2.299 \cite{Mills73}\\ |
817 |
|
|
$n_C$ & & 3.9 & 4.3 & 3.8 & 4.4 & 4.7 \cite{Hura00}\\ |
818 |
|
|
$n_H$ & & 3.7 & 3.6 & 3.7 & 3.7 & 3.5 \cite{Soper86}\\ |
819 |
|
|
$\tau_1$ & (ps) & 10.9(6) & 7.3(4) & 7.5(7) & 7.2(4) & 5.7 \cite{Eisenberg69}\\ |
820 |
|
|
$\tau_2$ & (ps) & 4.7(4) & 3.1(2) & 3.5(3) & 3.2(2) & 2.3 \cite{Krynicki66}\\ |
821 |
chrisfen |
2977 |
\bottomrule |
822 |
|
|
\end{tabular} |
823 |
|
|
\label{tab:liquidProperties} |
824 |
|
|
\end{table} |
825 |
|
|
|
826 |
|
|
Table \ref{tab:liquidProperties} gives a synopsis of the liquid state |
827 |
|
|
properties of the water models compared in this study along with the |
828 |
|
|
experimental values for liquid water at ambient conditions. The |
829 |
|
|
coordination number ($n_C$) and number of hydrogen bonds per particle |
830 |
|
|
($n_H$) were calculated by integrating the following relations: |
831 |
|
|
\begin{equation} |
832 |
|
|
n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2g_{\textrm{OO}}(r)dr, |
833 |
|
|
\end{equation} |
834 |
|
|
\begin{equation} |
835 |
|
|
n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2g_{\textrm{OH}}(r)dr, |
836 |
|
|
\end{equation} |
837 |
|
|
where $\rho$ is the number density of the specified pair interactions, |
838 |
|
|
$a$ and $b$ are the radial locations of the minima following the first |
839 |
|
|
peak in $g_\textrm{OO}(r)$ or $g_\textrm{OH}(r)$ respectively. The |
840 |
|
|
number of hydrogen bonds stays relatively constant across all of the |
841 |
|
|
models, but the coordination numbers of SSD/E and SSD/RF show an |
842 |
|
|
improvement over SSD1. This improvement is primarily due to extension |
843 |
|
|
of the first solvation shell in the new parameter sets. Because $n_H$ |
844 |
|
|
and $n_C$ are nearly identical in SSD1, it appears that all molecules |
845 |
|
|
in the first solvation shell are involved in hydrogen bonds. Since |
846 |
|
|
$n_H$ and $n_C$ differ in the newly parameterized models, the |
847 |
|
|
orientations in the first solvation shell are a bit more ``fluid''. |
848 |
|
|
Therefore SSD1 over-structures the first solvation shell and our |
849 |
|
|
reparametrizations have returned this shell to more realistic |
850 |
|
|
liquid-like behavior. |
851 |
|
|
|
852 |
|
|
The time constants for the orientational autocorrelation functions |
853 |
|
|
are also displayed in Table \ref{tab:liquidProperties}. The dipolar |
854 |
|
|
orientational time correlation functions ($C_{l}$) are described |
855 |
|
|
by: |
856 |
|
|
\begin{equation} |
857 |
|
|
C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle, |
858 |
|
|
\end{equation} |
859 |
|
|
where $P_l$ are Legendre polynomials of order $l$ and |
860 |
|
|
$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular |
861 |
|
|
dipole.\cite{Rahman71} Note that this is identical to equation |
862 |
|
|
(\ref{eq:OrientCorr}) were $\alpha$ is equal to $z$. From these |
863 |
|
|
correlation functions, the orientational relaxation time of the dipole |
864 |
|
|
vector can be calculated from an exponential fit in the long-time |
865 |
chrisfen |
2978 |
regime ($t > \tau_l$).\cite{Rothschild84} Calculation of these time |
866 |
|
|
constants were averaged over five detailed {\it NVE} simulations |
867 |
|
|
performed at the ambient conditions for each of the respective |
868 |
|
|
models. It should be noted that the commonly cited value of 1.9 ps for |
869 |
|
|
$\tau_2$ was determined from the NMR data in Ref. \cite{Krynicki66} at |
870 |
|
|
a temperature near 34$^\circ$C.\cite{Rahman71} Because of the strong |
871 |
|
|
temperature dependence of $\tau_2$, it is necessary to recalculate it |
872 |
|
|
at 298K to make proper comparisons. The value shown in Table |
873 |
chrisfen |
2977 |
\ref{tab:liquidProperties} was calculated from the same NMR data in the |
874 |
|
|
fashion described in Ref. \cite{Krynicki66}. Similarly, $\tau_1$ was |
875 |
|
|
recomputed for 298K from the data in Ref. \cite{Eisenberg69}. |
876 |
|
|
Again, SSD/E and SSD/RF show improved behavior over SSD1, both with |
877 |
|
|
and without an active reaction field. Turning on the reaction field |
878 |
|
|
leads to much improved time constants for SSD1; however, these results |
879 |
|
|
also include a corresponding decrease in system density. |
880 |
|
|
Orientational relaxation times published in the original SSD dynamics |
881 |
|
|
paper are smaller than the values observed here, and this difference |
882 |
|
|
can be attributed to the use of the Ewald sum.\cite{Chandra99} |
883 |
|
|
|
884 |
|
|
\subsection{SSD/RF and Damped Electrostatics} |
885 |
|
|
|
886 |
chrisfen |
2978 |
In section \ref{sec:dampingMultipoles}, a method was described for |
887 |
|
|
applying the damped {\sc sf} or {\sc sp} techniques to for systems |
888 |
|
|
containing point multipoles. The SSD family of water models is the |
889 |
|
|
perfect test case because of the dipole-dipole (and |
890 |
|
|
charge-dipole/quadrupole) interactions that are present. The {\sc sf} |
891 |
|
|
and {\sc sp} techniques were presented as a pairwise replacement for |
892 |
|
|
the Ewald summation. It has been suggested that models parametrized |
893 |
|
|
for the Ewald summation (like TIP5P-E) would be appropriate for use |
894 |
|
|
with a reaction field and vice versa.\cite{Rick04} Therefore, we |
895 |
|
|
decided to test the SSD/RF water model with this damped electrostatic |
896 |
|
|
technique in place of the reaction field to see how the calculated |
897 |
|
|
properties change. |
898 |
|
|
|
899 |
chrisfen |
2977 |
\begin{table} |
900 |
chrisfen |
2978 |
\caption{PROPERTIES OF SSD/RF WHEN USING DIFFERENT ELECTROSTATIC CORRECTION METHODS} |
901 |
chrisfen |
2977 |
\footnotesize |
902 |
|
|
\centering |
903 |
chrisfen |
2978 |
\begin{tabular}{ llccc } |
904 |
chrisfen |
2977 |
\toprule |
905 |
|
|
\toprule |
906 |
chrisfen |
2978 |
& & Reaction Field & Damped Electrostatics & Experiment [Ref.] \\ |
907 |
|
|
& & $\epsilon = 80$ & $\alpha = 0.2125$\AA$^{-1}$ & \\ |
908 |
chrisfen |
2977 |
\midrule |
909 |
chrisfen |
2978 |
$\rho$ & (g cm$^{-3}$) & 0.997(1) & 1.004(4) & 0.997 \cite{CRC80}\\ |
910 |
|
|
$C_p$ & (cal mol$^{-1}$ K$^{-1}$) & 23.8(2) & 27(1) & 18.005 \cite{Wagner02} \\ |
911 |
|
|
$D$ & ($10^{-5}$ cm$^2$ s$^{-1}$) & 2.32(6) & 2.33(2) & 2.299 \cite{Mills73}\\ |
912 |
|
|
$n_C$ & & 4.4 & 4.4 & 4.7 \cite{Hura00}\\ |
913 |
|
|
$n_H$ & & 3.7 & 3.7 & 3.5 \cite{Soper86}\\ |
914 |
|
|
$\tau_1$ & (ps) & 7.2(4) & 5.86(8) & 5.7 \cite{Eisenberg69}\\ |
915 |
|
|
$\tau_2$ & (ps) & 3.2(2) & 2.45(7) & 2.3 \cite{Krynicki66}\\ |
916 |
chrisfen |
2977 |
\bottomrule |
917 |
|
|
\end{tabular} |
918 |
|
|
\label{tab:dampedSSDRF} |
919 |
|
|
\end{table} |
920 |
|
|
|
921 |
|
|
In addition to the properties tabulated in table |
922 |
|
|
\ref{tab:dampedSSDRF}, we calculated the static dielectric constant |
923 |
|
|
from a 5ns simulation of SSD/RF using the damped electrostatics. The |
924 |
|
|
resulting value of 82.6(6) compares very favorably with the |
925 |
|
|
experimental value of 78.3.\cite{Malmberg56} This value is closer to |
926 |
|
|
the experimental value than what was expected according to figure |
927 |
|
|
\ref{fig:dielectricMap}, raising some questions as to the accuracy of |
928 |
|
|
the visual contours in the figure. This simply enforces the |
929 |
|
|
qualitative nature of contour plotting. |
930 |
|
|
|
931 |
|
|
\section{Tetrahedrally Restructured Elongated Dipole (TRED) Water Model} |
932 |
|
|
|
933 |
chrisfen |
2978 |
\begin{table} |
934 |
|
|
\caption{PROPERTIES OF TRED COMPARED WITH SSD/RF AND EXPERIMENT} |
935 |
|
|
\footnotesize |
936 |
|
|
\centering |
937 |
|
|
\begin{tabular}{ llccc } |
938 |
|
|
\toprule |
939 |
|
|
\toprule |
940 |
|
|
& & SSD/RF & TRED & Experiment [Ref.]\\ |
941 |
|
|
& & $\alpha = 0.2125$\AA$^{-1}$ & $\alpha = 0.2125$\AA$^{-1}$ & \\ |
942 |
|
|
\midrule |
943 |
|
|
$\rho$ & (g cm$^{-3}$) & 1.004(4) & 0.996(4) & 0.997 \cite{CRC80}\\ |
944 |
|
|
$C_p$ & (cal mol$^{-1}$ K$^{-1}$) & 27(1) & & 18.005 \cite{Wagner02} \\ |
945 |
|
|
$D$ & ($10^{-5}$ cm$^2$ s$^{-1}$) & 2.33(2) & 2.30(5) & 2.299 \cite{Mills73}\\ |
946 |
|
|
$n_C$ & & 4.4 & 5.3 & 4.7 \cite{Hura00}\\ |
947 |
|
|
$n_H$ & & 3.7 & 4.1 & 3.5 \cite{Soper86}\\ |
948 |
|
|
$\tau_1$ & (ps) & 5.86(8) & 6.0(1) & 5.7 \cite{Eisenberg69}\\ |
949 |
|
|
$\tau_2$ & (ps) & 2.45(7) & 2.49(5) & 2.3 \cite{Krynicki66}\\ |
950 |
|
|
$\epsilon_0$ & & 82.6(6) & & 78.3 \cite{Malmberg56}\\ |
951 |
|
|
$\tau_D$ & (ps) & & & 8.2(4) \cite{Kindt96}\\ |
952 |
|
|
\bottomrule |
953 |
|
|
\end{tabular} |
954 |
|
|
\label{tab:tredProps} |
955 |
|
|
\end{table} |
956 |
|
|
|
957 |
chrisfen |
2977 |
\section{Conclusions} |
958 |
|
|
|
959 |
|
|
In the above sections, the density maximum and temperature dependence |
960 |
|
|
of the self-diffusion constant were studied for the SSD water model, |
961 |
|
|
both with and without the use of reaction field, via a series of {\it |
962 |
|
|
NPT} and {\it NVE} simulations. The constant pressure simulations |
963 |
|
|
showed a density maximum near 260K. In most cases, the calculated |
964 |
|
|
densities were significantly lower than the densities obtained from |
965 |
|
|
other water models (and experiment). Analysis of self-diffusion showed |
966 |
|
|
SSD to capture the transport properties of water well in both the |
967 |
|
|
liquid and supercooled liquid regimes. |
968 |
|
|
|
969 |
|
|
In order to correct the density behavior, we reparametrized the |
970 |
|
|
original SSD model for use both with and without a reaction field |
971 |
|
|
(SSD/RF and SSD/E), and made comparisons with SSD1, an alternate |
972 |
|
|
density corrected version of SSD. Both models improve the liquid |
973 |
|
|
structure, densities, and diffusive properties under their respective |
974 |
|
|
simulation conditions, indicating the necessity of reparametrization |
975 |
|
|
when changing the method of calculating long-range electrostatic |
976 |
|
|
interactions. |
977 |
|
|
|
978 |
|
|
These simple water models are excellent choices for representing |
979 |
|
|
explicit water in large scale simulations of biochemical systems. They |
980 |
|
|
are more computationally efficient than the common charge based water |
981 |
|
|
models, and, in many cases, exhibit more realistic bulk phase fluid |
982 |
|
|
properties. These models are one of the few cases in which maximizing |
983 |
|
|
efficiency does not result in a loss in realistic representation. |