90 |
|
One recently developed model that largely succeeds in retaining the |
91 |
|
accuracy of bulk properties while greatly reducing the computational |
92 |
|
cost is the Soft Sticky Dipole (SSD) water |
93 |
< |
model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model was |
94 |
< |
developed by Ichiye \emph{et al.} as a modified form of the |
93 |
> |
model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model |
94 |
> |
was developed by Ichiye \emph{et al.} as a modified form of the |
95 |
|
hard-sphere water model proposed by Bratko, Blum, and |
96 |
< |
Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} model which |
97 |
< |
has an interaction site that is both a point dipole along with a |
96 |
> |
Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} model |
97 |
> |
which 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. |
102 |
> |
the proper hydrogen bond orientation in the first solvation shell. |
103 |
|
|
104 |
|
The interaction between two SSD water molecules \emph{i} and \emph{j} |
105 |
|
is given by the potential |
168 |
|
SSD decreased computer time by a factor of 6-7 compared to other |
169 |
|
models.\cite{Ichiye96} What is most impressive is that this savings |
170 |
|
did not come at the expense of accurate depiction of the liquid state |
171 |
< |
properties. Indeed, SSD maintains reasonable agreement with the Soper |
172 |
< |
data for the structural features of liquid |
171 |
> |
properties. Indeed, SSD maintains reasonable agreement with the |
172 |
> |
Soper data for the structural features of liquid |
173 |
|
water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
174 |
|
exhibited by SSD agree with experiment better than those of more |
175 |
|
computationally expensive models (like TIP3P and |
177 |
|
of solvent properties makes SSD a very attractive model for the |
178 |
|
simulation of large scale biochemical simulations. |
179 |
|
|
180 |
< |
One feature of the SSD model is that it was parameterized for use with |
181 |
< |
the Ewald sum to handle long-range interactions. This would normally |
182 |
< |
be the best way of handling long-range interactions in systems that |
183 |
< |
contain other point charges. However, our group has recently become |
184 |
< |
interested in systems with point dipoles as mimics for neutral, but |
185 |
< |
polarized regions on molecules (e.g. the zwitterionic head group |
186 |
< |
regions of phospholipids). If the system of interest does not contain |
187 |
< |
point charges, the Ewald sum and even particle-mesh Ewald become |
188 |
< |
computational bottlenecks. Their respective ideal $N^\frac{3}{2}$ and |
189 |
< |
$N\log N$ calculation scaling orders for $N$ particles can become |
190 |
< |
prohibitive when $N$ becomes large.\cite{Darden99} In applying this |
191 |
< |
water model in these types of systems, it would be useful to know its |
192 |
< |
properties and behavior under the more computationally efficient |
193 |
< |
reaction field (RF) technique, or even with a simple cutoff. This |
194 |
< |
study addresses these issues by looking at the structural and |
195 |
< |
transport behavior of SSD over a variety of temperatures with the |
196 |
< |
purpose of utilizing the RF correction technique. We then suggest |
197 |
< |
modifications to the parameters that result in more realistic bulk |
198 |
< |
phase behavior. It should be noted that in a recent publication, some |
199 |
< |
of the original investigators of the SSD water model have suggested |
200 |
< |
adjustments to the SSD water model to address abnormal density |
201 |
< |
behavior (also observed here), calling the corrected model |
202 |
< |
SSD1.\cite{Ichiye03} In what follows, we compare our |
203 |
< |
reparamaterization of SSD with both the original SSD and SSD1 models |
204 |
< |
with the goal of improving the bulk phase behavior of an SSD-derived |
205 |
< |
model in simulations utilizing the Reaction Field. |
180 |
> |
One feature of the SSD model is that it was parameterized for |
181 |
> |
use with the Ewald sum to handle long-range interactions. This would |
182 |
> |
normally be the best way of handling long-range interactions in |
183 |
> |
systems that contain other point charges. However, our group has |
184 |
> |
recently become interested in systems with point dipoles as mimics for |
185 |
> |
neutral, but polarized regions on molecules (e.g. the zwitterionic |
186 |
> |
head group regions of phospholipids). If the system of interest does |
187 |
> |
not contain point charges, the Ewald sum and even particle-mesh Ewald |
188 |
> |
become computational bottlenecks. Their respective ideal |
189 |
> |
$N^\frac{3}{2}$ and $N\log N$ calculation scaling orders for $N$ |
190 |
> |
particles can become prohibitive when $N$ becomes |
191 |
> |
large.\cite{Darden99} In applying this water model in these types of |
192 |
> |
systems, it would be useful to know its properties and behavior under |
193 |
> |
the more computationally efficient reaction field (RF) technique, or |
194 |
> |
even with a simple cutoff. This study addresses these issues by |
195 |
> |
looking at the structural and transport behavior of SSD over a |
196 |
> |
variety of temperatures with the purpose of utilizing the RF |
197 |
> |
correction technique. We then suggest modifications to the parameters |
198 |
> |
that result in more realistic bulk phase behavior. It should be noted |
199 |
> |
that in a recent publication, some of the original investigators of |
200 |
> |
the SSD water model have suggested adjustments to the SSD |
201 |
> |
water model to address abnormal density behavior (also observed here), |
202 |
> |
calling the corrected model SSD1.\cite{Ichiye03} In what |
203 |
> |
follows, we compare our reparamaterization of SSD with both the |
204 |
> |
original SSD and SSD1 models with the goal of improving |
205 |
> |
the bulk phase behavior of an SSD-derived model in simulations |
206 |
> |
utilizing the Reaction Field. |
207 |
|
|
208 |
|
\section{Methods} |
209 |
|
|
215 |
|
field acting on dipole $i$ is |
216 |
|
\begin{equation} |
217 |
|
\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
218 |
< |
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\ , |
218 |
> |
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} s(r_{ij}), |
219 |
|
\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 |
|
system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole |
224 |
< |
moment vector of particle $j$ and $f(r_{ij})$ is a cubic switching |
224 |
> |
moment vector of particle $j$, and $s(r_{ij})$ is a cubic switching |
225 |
|
function.\cite{AllenTildesley} The reaction field contribution to the |
226 |
|
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 |
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 |
|
at the cutoff radius), and as a result we have two reparamaterizations |
239 |
< |
of SSD which could be used either with or without the reaction field |
240 |
< |
turned on. |
239 |
> |
of SSD which could be used either with or without the reaction |
240 |
> |
field turned on. |
241 |
|
|
242 |
< |
Simulations to obtain the preferred density were performed in the |
243 |
< |
isobaric-isothermal (NPT) ensemble, while all dynamical properties |
244 |
< |
were obtained from microcanonical (NVE) simulations done at densities |
245 |
< |
matching the NPT density for a particular target temperature. The |
246 |
< |
constant pressure simulations were implemented using an integral |
247 |
< |
thermostat and barostat as outlined by Hoover.\cite{Hoover85,Hoover86} |
248 |
< |
All molecules were treated as non-linear rigid bodies. Vibrational |
249 |
< |
constraints are not necessary in simulations of SSD, because there are |
250 |
< |
no explicit hydrogen atoms, and thus no molecular vibrational modes |
251 |
< |
need to be considered. |
242 |
> |
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 SSD, because there are no explicit hydrogen |
251 |
> |
atoms, and thus no molecular vibrational modes need to be considered. |
252 |
|
|
253 |
|
Integration of the equations of motion was carried out using the |
254 |
< |
symplectic splitting method proposed by Dullweber {\it et |
255 |
< |
al.}\cite{Dullweber1997} Our reason for selecting this integrator |
256 |
< |
centers on poor energy conservation of rigid body dynamics using |
257 |
< |
traditional quaternion integration.\cite{Evans77,Evans77b} In typical |
258 |
< |
microcanonical ensemble simulations, the energy drift when using |
259 |
< |
quaternions was substantially greater than when using the symplectic |
260 |
< |
splitting method (fig. \ref{timestep}). This steady drift in the |
261 |
< |
total energy has also been observed by Kol {\it et al.}\cite{Laird97} |
254 |
> |
symplectic splitting method proposed by Dullweber, Leimkuhler, and |
255 |
> |
McLachlan ({\sc dlm}).\cite{Dullweber1997} Our reason for selecting |
256 |
> |
this integrator centers on poor energy conservation of rigid body |
257 |
> |
dynamics using traditional quaternion |
258 |
> |
integration.\cite{Evans77,Evans77b} In typical microcanonical ensemble |
259 |
> |
simulations, the energy drift when using quaternions was substantially |
260 |
> |
greater than when using the {\sc dlm} method (fig. \ref{timestep}). |
261 |
> |
This steady drift in the total energy has also been observed by Kol |
262 |
> |
{\it et al.}\cite{Laird97} |
263 |
|
|
264 |
|
The key difference in the integration method proposed by Dullweber |
265 |
|
\emph{et al.} is that the entire rotation matrix is propagated from |
268 |
|
rotation matrix into quaternions for storage purposes makes trajectory |
269 |
|
data quite compact. |
270 |
|
|
271 |
< |
The symplectic splitting method allows for Verlet style integration of |
272 |
< |
both translational and orientational motion of rigid bodies. In this |
271 |
> |
The {\sc dlm} method allows for Verlet style integration of both |
272 |
> |
translational and orientational motion of rigid bodies. In this |
273 |
|
integration method, the orientational propagation involves a sequence |
274 |
|
of matrix evaluations to update the rotation |
275 |
|
matrix.\cite{Dullweber1997} These matrix rotations are more costly |
276 |
|
than the simpler arithmetic quaternion propagation. With the same time |
277 |
< |
step, a 1000 SSD particle simulation shows an average 7\% increase in |
278 |
< |
computation time using the symplectic step method in place of |
277 |
> |
step, a 1000 SSD particle simulation shows an average 7\% |
278 |
> |
increase in computation time using the {\sc dlm} method in place of |
279 |
|
quaternions. The additional expense per step is justified when one |
280 |
|
considers the ability to use time steps that are nearly twice as large |
281 |
< |
under symplectic splitting than would be usable under quaternion |
282 |
< |
dynamics. The energy conservation of the two methods using a number |
283 |
< |
of different time steps is illustrated in figure |
281 |
> |
under {\sc dlm} than would be usable under quaternion dynamics. The |
282 |
> |
energy conservation of the two methods using a number of different |
283 |
> |
time steps is illustrated in figure |
284 |
|
\ref{timestep}. |
285 |
|
|
286 |
|
\begin{figure} |
287 |
|
\begin{center} |
288 |
|
\epsfxsize=6in |
289 |
|
\epsfbox{timeStep.epsi} |
290 |
< |
\caption{Energy conservation using both quaternion based integration and |
291 |
< |
the symplectic step method proposed by Dullweber \emph{et al.} with |
292 |
< |
increasing time step. The larger time step plots are shifted from the |
293 |
< |
true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
290 |
> |
\caption{Energy conservation using both quaternion-based integration and |
291 |
> |
the {\sc dlm} method with increasing time step. The larger time step plots |
292 |
> |
are shifted from the true energy baseline (that of $\Delta t$ = 0.1 |
293 |
> |
fs) for clarity.} |
294 |
|
\label{timestep} |
295 |
|
\end{center} |
296 |
|
\end{figure} |
297 |
|
|
298 |
|
In figure \ref{timestep}, the resulting energy drift at various time |
299 |
< |
steps for both the symplectic step and quaternion integration schemes |
300 |
< |
is compared. All of the 1000 SSD particle simulations started with |
299 |
> |
steps for both the {\sc dlm} and quaternion integration schemes is |
300 |
> |
compared. All of the 1000 SSD particle simulations started with |
301 |
|
the same configuration, and the only difference was the method used to |
302 |
|
handle orientational motion. At time steps of 0.1 and 0.5 fs, both |
303 |
|
methods for propagating the orientational degrees of freedom conserve |
304 |
|
energy fairly well, with the quaternion method showing a slight energy |
305 |
|
drift over time in the 0.5 fs time step simulation. At time steps of 1 |
306 |
< |
and 2 fs, the energy conservation benefits of the symplectic step |
307 |
< |
method are clearly demonstrated. Thus, while maintaining the same |
308 |
< |
degree of energy conservation, one can take considerably longer time |
309 |
< |
steps, leading to an overall reduction in computation time. |
306 |
> |
and 2 fs, the energy conservation benefits of the {\sc dlm} method are |
307 |
> |
clearly demonstrated. Thus, while maintaining the same degree of |
308 |
> |
energy conservation, one can take considerably longer time steps, |
309 |
> |
leading to an overall reduction in computation time. |
310 |
|
|
311 |
< |
Energy drift in the symplectic step simulations was unnoticeable for |
312 |
< |
time steps up to 3 fs. A slight energy drift on the |
311 |
> |
Energy drift in the simulations using {\sc dlm} integration was |
312 |
> |
unnoticeable for time steps up to 3 fs. A slight energy drift on the |
313 |
|
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
314 |
< |
4 fs, and as expected, this drift increases dramatically |
315 |
< |
with increasing time step. To insure accuracy in our microcanonical |
314 |
> |
4 fs, and as expected, this drift increases dramatically with |
315 |
> |
increasing time step. To insure accuracy in our microcanonical |
316 |
|
simulations, time steps were set at 2 fs and kept at this value for |
317 |
|
constant pressure simulations as well. |
318 |
|
|
319 |
|
Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices |
320 |
|
were generated as starting points for all simulations. The $I_h$ |
321 |
< |
crystals were formed by first arranging the centers of mass of the SSD |
322 |
< |
particles into a ``hexagonal'' ice lattice of 1024 particles. Because |
323 |
< |
of the crystal structure of $I_h$ ice, the simulation box assumed an |
324 |
< |
orthorhombic shape with an edge length ratio of approximately |
325 |
< |
1.00$\times$1.06$\times$1.23. The particles were then allowed to |
326 |
< |
orient freely about fixed positions with angular momenta randomized at |
327 |
< |
400 K for varying times. The rotational temperature was then scaled |
328 |
< |
down in stages to slowly cool the crystals to 25 K. The particles were |
329 |
< |
then allowed to translate with fixed orientations at a constant |
330 |
< |
pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were |
331 |
< |
removed and the ice crystals were allowed to equilibrate for 50 ps at |
332 |
< |
25 K and a constant pressure of 1 atm. This procedure resulted in |
333 |
< |
structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
321 |
> |
crystals were formed by first arranging the centers of mass of the |
322 |
> |
SSD particles into a ``hexagonal'' ice lattice of 1024 |
323 |
> |
particles. Because of the crystal structure of $I_h$ ice, the |
324 |
> |
simulation box assumed an orthorhombic shape with an edge length ratio |
325 |
> |
of approximately 1.00$\times$1.06$\times$1.23. The particles were then |
326 |
> |
allowed to orient freely about fixed positions with angular momenta |
327 |
> |
randomized at 400 K for varying times. The rotational temperature was |
328 |
> |
then scaled down in stages to slowly cool the crystals to 25 K. The |
329 |
> |
particles were then allowed to translate with fixed orientations at a |
330 |
> |
constant pressure of 1 atm for 50 ps at 25 K. Finally, all constraints |
331 |
> |
were removed and the ice crystals were allowed to equilibrate for 50 |
332 |
> |
ps at 25 K and a constant pressure of 1 atm. This procedure resulted |
333 |
> |
in structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
334 |
|
rules.\cite{Bernal33,Rahman72} This method was also utilized in the |
335 |
|
making of diamond lattice $I_c$ ice crystals, with each cubic |
336 |
|
simulation box consisting of either 512 or 1000 particles. Only |
357 |
|
|
358 |
|
\subsection{Density Behavior} |
359 |
|
|
360 |
< |
Our initial simulations focused on the original SSD water model, and |
361 |
< |
an average density versus temperature plot is shown in figure |
360 |
> |
Our initial simulations focused on the original SSD water model, |
361 |
> |
and an average density versus temperature plot is shown in figure |
362 |
|
\ref{dense1}. Note that the density maximum when using a reaction |
363 |
|
field appears between 255 and 265 K. There were smaller fluctuations |
364 |
|
in the density at 260 K than at either 255 or 265, so we report this |
371 |
|
\begin{figure} |
372 |
|
\begin{center} |
373 |
|
\epsfxsize=6in |
374 |
< |
\epsfbox{denseSSD.eps} |
374 |
> |
\epsfbox{denseSSDnew.eps} |
375 |
|
\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}], |
376 |
|
TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD |
377 |
|
without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The |
383 |
|
\end{center} |
384 |
|
\end{figure} |
385 |
|
|
386 |
< |
The density maximum for SSD compares quite favorably to other simple |
387 |
< |
water models. Figure \ref{dense1} also shows calculated densities of |
388 |
< |
several other models and experiment obtained from other |
386 |
> |
The density maximum for SSD compares quite favorably to other |
387 |
> |
simple water models. Figure \ref{dense1} also shows calculated |
388 |
> |
densities of several other models and experiment obtained from other |
389 |
|
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
390 |
< |
models, SSD has a temperature closest to the experimentally observed |
391 |
< |
density maximum. Of the {\it charge-based} models in |
390 |
> |
models, SSD has a temperature closest to the experimentally |
391 |
> |
observed density maximum. Of the {\it charge-based} models in |
392 |
|
Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that |
393 |
< |
seen in SSD. Though not included in this plot, it is useful |
394 |
< |
to note that TIP5P has a density maximum nearly identical to the |
393 |
> |
seen in SSD. Though not included in this plot, it is useful to |
394 |
> |
note that TIP5P has a density maximum nearly identical to the |
395 |
|
experimentally measured temperature. |
396 |
|
|
397 |
|
It has been observed that liquid state densities in water are |
398 |
|
dependent on the cutoff radius used both with and without the use of |
399 |
|
reaction field.\cite{Berendsen98} In order to address the possible |
400 |
|
effect of cutoff radius, simulations were performed with a dipolar |
401 |
< |
cutoff radius of 12.0 \AA\ to complement the previous SSD simulations, |
402 |
< |
all performed with a cutoff of 9.0 \AA. All of the resulting densities |
403 |
< |
overlapped within error and showed no significant trend toward lower |
404 |
< |
or higher densities as a function of cutoff radius, for simulations |
405 |
< |
both with and without reaction field. These results indicate that |
406 |
< |
there is no major benefit in choosing a longer cutoff radius in |
407 |
< |
simulations using SSD. This is advantageous in that the use of a |
408 |
< |
longer cutoff radius results in a significant increase in the time |
409 |
< |
required to obtain a single trajectory. |
401 |
> |
cutoff radius of 12.0 \AA\ to complement the previous SSD |
402 |
> |
simulations, all performed with a cutoff of 9.0 \AA. All of the |
403 |
> |
resulting densities overlapped within error and showed no significant |
404 |
> |
trend toward lower or higher densities as a function of cutoff radius, |
405 |
> |
for simulations both with and without reaction field. These results |
406 |
> |
indicate that there is no major benefit in choosing a longer cutoff |
407 |
> |
radius in simulations using SSD. This is advantageous in that |
408 |
> |
the use of a longer cutoff radius results in a significant increase in |
409 |
> |
the time required to obtain a single trajectory. |
410 |
|
|
411 |
|
The key feature to recognize in figure \ref{dense1} is the density |
412 |
|
scaling of SSD relative to other common models at any given |
413 |
< |
temperature. SSD assumes a lower density than any of the other listed |
414 |
< |
models at the same pressure, behavior which is especially apparent at |
415 |
< |
temperatures greater than 300 K. Lower than expected densities have |
416 |
< |
been observed for other systems using a reaction field for long-range |
417 |
< |
electrostatic interactions, so the most likely reason for the |
418 |
< |
significantly lower densities seen in these simulations is the |
413 |
> |
temperature. SSD assumes a lower density than any of the other |
414 |
> |
listed models at the same pressure, behavior which is especially |
415 |
> |
apparent at temperatures greater than 300 K. Lower than expected |
416 |
> |
densities have been observed for other systems using a reaction field |
417 |
> |
for long-range electrostatic interactions, so the most likely reason |
418 |
> |
for the significantly lower densities seen in these simulations is the |
419 |
|
presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order |
420 |
|
to test the effect of the reaction field on the density of the |
421 |
|
systems, the simulations were repeated without a reaction field |
431 |
|
simulations.\cite{Clancy94,Jorgensen98b} However, even without the |
432 |
|
reaction field, the density around 300 K is still significantly lower |
433 |
|
than experiment and comparable water models. This anomalous behavior |
434 |
< |
was what lead Ichiye {\it et al.} to recently reparameterize |
434 |
> |
was what lead Tan {\it et al.} to recently reparameterize |
435 |
|
SSD.\cite{Ichiye03} Throughout the remainder of the paper our |
436 |
< |
reparamaterizations of SSD will be compared with the newer SSD1 model. |
436 |
> |
reparamaterizations of SSD will be compared with their newer SSD1 |
437 |
> |
model. |
438 |
|
|
439 |
|
\subsection{Transport Behavior} |
440 |
|
|
457 |
|
\epsfxsize=6in |
458 |
|
\epsfbox{betterDiffuse.epsi} |
459 |
|
\caption{Average self-diffusion constant as a function of temperature for |
460 |
< |
SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}], |
461 |
< |
and Experimental data [Refs. \citen{Gillen72} and \citen{Holz00}]. Of |
462 |
< |
the three water models shown, SSD has the least deviation from the |
463 |
< |
experimental values. The rapidly increasing diffusion constants for |
464 |
< |
TIP5P and SSD correspond to significant decrease in density at the |
465 |
< |
higher temperatures.} |
460 |
> |
SSD, SPC/E [Ref. \citen{Clancy94}], and TIP5P |
461 |
> |
[Ref. \citen{Jorgensen01}] compared with experimental data |
462 |
> |
[Refs. \citen{Gillen72} and \citen{Holz00}]. Of the three water models |
463 |
> |
shown, SSD has the least deviation from the experimental values. The |
464 |
> |
rapidly increasing diffusion constants for TIP5P and SSD correspond to |
465 |
> |
significant decreases in density at the higher temperatures.} |
466 |
|
\label{diffuse} |
467 |
|
\end{center} |
468 |
|
\end{figure} |
478 |
|
This behavior at higher temperatures is not particularly surprising |
479 |
|
since the densities of both TIP5P and SSD are lower than experimental |
480 |
|
water densities at higher temperatures. When calculating the |
481 |
< |
diffusion coefficients for SSD at experimental densities (instead of |
482 |
< |
the densities from the NPT simulations), the resulting values fall |
483 |
< |
more in line with experiment at these temperatures. |
481 |
> |
diffusion coefficients for SSD at experimental densities |
482 |
> |
(instead of the densities from the NPT simulations), the resulting |
483 |
> |
values fall more in line with experiment at these temperatures. |
484 |
|
|
485 |
|
\subsection{Structural Changes and Characterization} |
486 |
|
|
499 |
|
\begin{center} |
500 |
|
\epsfxsize=6in |
501 |
|
\epsfbox{corrDiag.eps} |
502 |
< |
\caption{Two dimensional illustration of angles involved in the |
501 |
< |
correlations observed in Fig. \ref{contour}.} |
502 |
> |
\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.} |
503 |
|
\label{corrAngle} |
504 |
|
\end{center} |
505 |
|
\end{figure} |
508 |
|
\begin{center} |
509 |
|
\epsfxsize=6in |
510 |
|
\epsfbox{fullContours.eps} |
511 |
< |
\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at |
512 |
< |
100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for |
513 |
< |
clarity: dark areas signify peaks while light areas signify |
514 |
< |
depressions. White areas have $g(r)$ values below 0.5 and black |
515 |
< |
areas have values above 1.5.} |
511 |
> |
\caption{Contour plots of 2D angular pair correlation functions for |
512 |
> |
512 SSD molecules at 100 K (A \& B) and 300 K (C \& D). Dark areas |
513 |
> |
signify regions of enhanced density while light areas signify |
514 |
> |
depletion relative to the bulk density. White areas have pair |
515 |
> |
correlation values below 0.5 and black areas have values above 1.5.} |
516 |
|
\label{contour} |
517 |
|
\end{center} |
518 |
|
\end{figure} |
551 |
|
|
552 |
|
This complex interplay between dipole and sticky interactions was |
553 |
|
remarked upon as a possible reason for the split second peak in the |
554 |
< |
oxygen-oxygen $g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, |
555 |
< |
the second solvation shell peak appears to have two distinct |
556 |
< |
components that blend together to form one observable peak. At higher |
557 |
< |
temperatures, this split character alters to show the leading 4 \AA\ |
558 |
< |
peak dominated by equatorial anti-parallel dipole orientations. There |
559 |
< |
is also a tightly bunched group of axially arranged dipoles that most |
560 |
< |
likely consist of the smaller fraction of aligned dipole pairs. The |
561 |
< |
trailing component of the split peak at 5 \AA\ is dominated by aligned |
562 |
< |
dipoles that assume hydrogen bond arrangements similar to those seen |
563 |
< |
in the first solvation shell. This evidence indicates that the dipole |
564 |
< |
pair interaction begins to dominate outside of the range of the |
565 |
< |
dipolar repulsion term. The energetically favorable dipole |
566 |
< |
arrangements populate the region immediately outside this repulsion |
567 |
< |
region (around 4 \AA), while arrangements that seek to satisfy both |
568 |
< |
the sticky and dipole forces locate themselves just beyond this |
569 |
< |
initial buildup (around 5 \AA). |
554 |
> |
oxygen-oxygen pair correlation function, |
555 |
> |
$g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, the second |
556 |
> |
solvation shell peak appears to have two distinct components that |
557 |
> |
blend together to form one observable peak. At higher temperatures, |
558 |
> |
this split character alters to show the leading 4 \AA\ peak dominated |
559 |
> |
by equatorial anti-parallel dipole orientations. There is also a |
560 |
> |
tightly bunched group of axially arranged dipoles that most likely |
561 |
> |
consist of the smaller fraction of aligned dipole pairs. The trailing |
562 |
> |
component of the split peak at 5 \AA\ is dominated by aligned dipoles |
563 |
> |
that assume hydrogen bond arrangements similar to those seen in the |
564 |
> |
first solvation shell. This evidence indicates that the dipole pair |
565 |
> |
interaction begins to dominate outside of the range of the dipolar |
566 |
> |
repulsion term. The energetically favorable dipole arrangements |
567 |
> |
populate the region immediately outside this repulsion region (around |
568 |
> |
4 \AA), while arrangements that seek to satisfy both the sticky and |
569 |
> |
dipole forces locate themselves just beyond this initial buildup |
570 |
> |
(around 5 \AA). |
571 |
|
|
572 |
|
From these findings, the split second peak is primarily the product of |
573 |
|
the dipolar repulsion term of the sticky potential. In fact, the inner |
578 |
|
since the second solvation shell would still be shifted too far |
579 |
|
out. In addition, this would have an even more detrimental effect on |
580 |
|
the system densities, leading to a liquid with a more open structure |
581 |
< |
and a density considerably lower than the already low SSD density. A |
582 |
< |
better correction would be to include the quadrupole-quadrupole |
583 |
< |
interactions for the water particles outside of the first solvation |
584 |
< |
shell, but this would remove the simplicity and speed advantage of |
585 |
< |
SSD. |
581 |
> |
and a density considerably lower than the already low SSD |
582 |
> |
density. A better correction would be to include the |
583 |
> |
quadrupole-quadrupole interactions for the water particles outside of |
584 |
> |
the first solvation shell, but this would remove the simplicity and |
585 |
> |
speed advantage of SSD. |
586 |
|
|
587 |
|
\subsection{Adjusted Potentials: SSD/RF and SSD/E} |
588 |
|
|
597 |
|
|
598 |
|
The parameters available for tuning include the $\sigma$ and |
599 |
|
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the |
600 |
< |
strength of the sticky potential ($\nu_0$), and the sticky attractive |
601 |
< |
and dipole repulsive cubic switching function cutoffs ($r_l$, $r_u$ |
602 |
< |
and $r_l^\prime$, $r_u^\prime$ respectively). The results of the |
603 |
< |
reparameterizations are shown in table \ref{params}. We are calling |
604 |
< |
these reparameterizations the Soft Sticky Dipole / Reaction Field |
605 |
< |
(SSD/RF - for use with a reaction field) and Soft Sticky Dipole |
606 |
< |
Extended (SSD/E - an attempt to improve the liquid structure in |
607 |
< |
simulations without a long-range correction). |
600 |
> |
strength of the sticky potential ($\nu_0$), and the cutoff distances |
601 |
> |
for the sticky attractive and dipole repulsive cubic switching |
602 |
> |
function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$ |
603 |
> |
respectively). The results of the reparameterizations are shown in |
604 |
> |
table \ref{params}. We are calling these reparameterizations the Soft |
605 |
> |
Sticky Dipole / Reaction Field (SSD/RF - for use with a reaction |
606 |
> |
field) and Soft Sticky Dipole Extended (SSD/E - an attempt to improve |
607 |
> |
the liquid structure in simulations without a long-range correction). |
608 |
|
|
609 |
|
\begin{table} |
610 |
|
\begin{center} |
632 |
|
\begin{center} |
633 |
|
\epsfxsize=5in |
634 |
|
\epsfbox{GofRCompare.epsi} |
635 |
< |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with SSD/E |
636 |
< |
and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with |
637 |
< |
reaction field turned on (bottom). The insets show the respective |
638 |
< |
first peaks in detail. Note how the changes in parameters have lowered |
639 |
< |
and broadened the first peak of SSD/E and SSD/RF.} |
635 |
> |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with |
636 |
> |
SSD/E and SSD1 without reaction field (top), as well as |
637 |
> |
SSD/RF and SSD1 with reaction field turned on |
638 |
> |
(bottom). The insets show the respective first peaks in detail. Note |
639 |
> |
how the changes in parameters have lowered and broadened the first |
640 |
> |
peak of SSD/E and SSD/RF.} |
641 |
|
\label{grcompare} |
642 |
|
\end{center} |
643 |
|
\end{figure} |
645 |
|
\begin{figure} |
646 |
|
\begin{center} |
647 |
|
\epsfxsize=6in |
648 |
< |
\epsfbox{dualsticky.ps} |
649 |
< |
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& |
650 |
< |
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
651 |
< |
component, and darker areas correspond to the dipolar repulsive |
652 |
< |
component.} |
648 |
> |
\epsfbox{dualsticky_bw.eps} |
649 |
> |
\caption{Positive and negative isosurfaces of the sticky potential for |
650 |
> |
SSD1 (left) and SSD/E \& SSD/RF (right). Light areas |
651 |
> |
correspond to the tetrahedral attractive component, and darker areas |
652 |
> |
correspond to the dipolar repulsive component.} |
653 |
|
\label{isosurface} |
654 |
|
\end{center} |
655 |
|
\end{figure} |
662 |
|
Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering |
663 |
|
data from the Head-Gordon lab indicates a slightly lower and shifted |
664 |
|
first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were |
665 |
< |
made while taking into consideration the new experimental |
665 |
> |
made after taking into consideration the new experimental |
666 |
|
findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the |
667 |
|
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing |
668 |
|
the revised SSD model (SSD1), SSD/E, and SSD/RF to the new |
678 |
|
see how altering the cutoffs changes the repulsive and attractive |
679 |
|
character of the particles. With a reduced repulsive surface (darker |
680 |
|
region), the particles can move closer to one another, increasing the |
681 |
< |
density for the overall system. This change in interaction cutoff also |
682 |
< |
results in a more gradual orientational motion by allowing the |
681 |
> |
density for the overall system. This change in interaction cutoff |
682 |
> |
also results in a more gradual orientational motion by allowing the |
683 |
|
particles to maintain preferred dipolar arrangements before they begin |
684 |
|
to feel the pull of the tetrahedral restructuring. As the particles |
685 |
|
move closer together, the dipolar repulsion term becomes active and |
695 |
|
improves the densities, these changes alone are insufficient to bring |
696 |
|
the system densities up to the values observed experimentally. To |
697 |
|
further increase the densities, the dipole moments were increased in |
698 |
< |
both of our adjusted models. Since SSD is a dipole based model, the |
699 |
< |
structure and transport are very sensitive to changes in the dipole |
700 |
< |
moment. The original SSD simply used the dipole moment calculated from |
701 |
< |
the TIP3P water model, which at 2.35 D is significantly greater than |
702 |
< |
the experimental gas phase value of 1.84 D. The larger dipole moment |
703 |
< |
is a more realistic value and improves the dielectric properties of |
704 |
< |
the fluid. Both theoretical and experimental measurements indicate a |
705 |
< |
liquid phase dipole moment ranging from 2.4 D to values as high as |
706 |
< |
3.11 D, providing a substantial range of reasonable values for a |
707 |
< |
dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
708 |
< |
increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF, |
709 |
< |
respectively, leads to significant changes in the density and |
710 |
< |
transport of the water models. |
698 |
> |
both of our adjusted models. Since SSD is a dipole based model, |
699 |
> |
the structure and transport are very sensitive to changes in the |
700 |
> |
dipole moment. The original SSD simply used the dipole moment |
701 |
> |
calculated from the TIP3P water model, which at 2.35 D is |
702 |
> |
significantly greater than the experimental gas phase value of 1.84 |
703 |
> |
D. The larger dipole moment is a more realistic value and improves the |
704 |
> |
dielectric properties of the fluid. Both theoretical and experimental |
705 |
> |
measurements indicate a liquid phase dipole moment ranging from 2.4 D |
706 |
> |
to values as high as 3.11 D, providing a substantial range of |
707 |
> |
reasonable values for a dipole |
708 |
> |
moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
709 |
> |
increasing the dipole moments to 2.42 and 2.48 D for SSD/E and |
710 |
> |
SSD/RF, respectively, leads to significant changes in the |
711 |
> |
density and transport of the water models. |
712 |
|
|
713 |
|
In order to demonstrate the benefits of these reparameterizations, a |
714 |
|
series of NPT and NVE simulations were performed to probe the density |
729 |
|
\begin{center} |
730 |
|
\epsfxsize=6in |
731 |
|
\epsfbox{ssdeDense.epsi} |
732 |
< |
\caption{Comparison of densities calculated with SSD/E to SSD1 without a |
733 |
< |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
734 |
< |
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and |
732 |
> |
\caption{Comparison of densities calculated with SSD/E to |
733 |
> |
SSD1 without a reaction field, TIP3P [Ref. \citen{Jorgensen98b}], |
734 |
> |
TIP5P [Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and |
735 |
|
experiment [Ref. \citen{CRC80}]. The window shows a expansion around |
736 |
|
300 K with error bars included to clarify this region of |
737 |
|
interest. Note that both SSD1 and SSD/E show good agreement with |
740 |
|
\end{center} |
741 |
|
\end{figure} |
742 |
|
|
743 |
< |
Fig. \ref{ssdedense} shows the density profile for the SSD/E model |
744 |
< |
in comparison to SSD1 without a reaction field, other common water |
745 |
< |
models, and experimental results. The calculated densities for both |
746 |
< |
SSD/E and SSD1 have increased significantly over the original SSD |
747 |
< |
model (see fig. \ref{dense1}) and are in better agreement with the |
748 |
< |
experimental values. At 298 K, the densities of SSD/E and SSD1 without |
743 |
> |
Fig. \ref{ssdedense} shows the density profile for the SSD/E |
744 |
> |
model in comparison to SSD1 without a reaction field, other |
745 |
> |
common water models, and experimental results. The calculated |
746 |
> |
densities for both SSD/E and SSD1 have increased |
747 |
> |
significantly over the original SSD model (see |
748 |
> |
fig. \ref{dense1}) and are in better agreement with the experimental |
749 |
> |
values. At 298 K, the densities of SSD/E and SSD1 without |
750 |
|
a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and |
751 |
|
0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with |
752 |
|
the experimental value of 0.997 g/cm$^3$, and they are considerably |
753 |
< |
better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to |
754 |
< |
the dipole moment and sticky switching functions have improved the |
755 |
< |
structuring of the liquid (as seen in figure \ref{grcompare}, but they |
756 |
< |
have shifted the density maximum to much lower temperatures. This |
757 |
< |
comes about via an increase in the liquid disorder through the |
758 |
< |
weakening of the sticky potential and strengthening of the dipolar |
759 |
< |
character. However, this increasing disorder in the SSD/E model has |
760 |
< |
little effect on the melting transition. By monitoring $C_p$ |
761 |
< |
throughout these simulations, the melting transition for SSD/E was |
762 |
< |
shown to occur at 235 K. The same transition temperature observed |
763 |
< |
with SSD and SSD1. |
753 |
> |
better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The |
754 |
> |
changes to the dipole moment and sticky switching functions have |
755 |
> |
improved the structuring of the liquid (as seen in figure |
756 |
> |
\ref{grcompare}, but they have shifted the density maximum to much |
757 |
> |
lower temperatures. This comes about via an increase in the liquid |
758 |
> |
disorder through the weakening of the sticky potential and |
759 |
> |
strengthening of the dipolar character. However, this increasing |
760 |
> |
disorder in the SSD/E model has little effect on the melting |
761 |
> |
transition. By monitoring $C_p$ throughout these simulations, the |
762 |
> |
melting transition for SSD/E was shown to occur at 235 K. The |
763 |
> |
same transition temperature observed with SSD and SSD1. |
764 |
|
|
765 |
|
\begin{figure} |
766 |
|
\begin{center} |
767 |
|
\epsfxsize=6in |
768 |
|
\epsfbox{ssdrfDense.epsi} |
769 |
< |
\caption{Comparison of densities calculated with SSD/RF to SSD1 with a |
770 |
< |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
771 |
< |
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and |
769 |
> |
\caption{Comparison of densities calculated with SSD/RF to |
770 |
> |
SSD1 with a reaction field, TIP3P [Ref. \citen{Jorgensen98b}], |
771 |
> |
TIP5P [Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and |
772 |
|
experiment [Ref. \citen{CRC80}]. The inset shows the necessity of |
773 |
|
reparameterization when utilizing a reaction field long-ranged |
774 |
< |
correction - SSD/RF provides significantly more accurate densities |
775 |
< |
than SSD1 when performing room temperature simulations.} |
774 |
> |
correction - SSD/RF provides significantly more accurate |
775 |
> |
densities than SSD1 when performing room temperature |
776 |
> |
simulations.} |
777 |
|
\label{ssdrfdense} |
778 |
|
\end{center} |
779 |
|
\end{figure} |
800 |
|
\begin{center} |
801 |
|
\epsfxsize=6in |
802 |
|
\epsfbox{ssdeDiffuse.epsi} |
803 |
< |
\caption{The diffusion constants calculated from SSD/E and SSD1, |
804 |
< |
both without a reaction field, along with experimental results |
805 |
< |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
806 |
< |
were performed at the average densities observed in the 1 atm NPT |
807 |
< |
simulations for the respective models. SSD/E is slightly more mobile |
808 |
< |
than experiment at all of the temperatures, but it is closer to |
809 |
< |
experiment at biologically relevant temperatures than SSD1 without a |
810 |
< |
long-range correction.} |
803 |
> |
\caption{The diffusion constants calculated from SSD/E and |
804 |
> |
SSD1 (both without a reaction field) along with experimental results |
805 |
> |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations were |
806 |
> |
performed at the average densities observed in the 1 atm NPT |
807 |
> |
simulations for the respective models. SSD/E is slightly more mobile |
808 |
> |
than experiment at all of the temperatures, but it is closer to |
809 |
> |
experiment at biologically relevant temperatures than SSD1 without a |
810 |
> |
long-range correction.} |
811 |
|
\label{ssdediffuse} |
812 |
|
\end{center} |
813 |
|
\end{figure} |
815 |
|
The reparameterization of the SSD water model, both for use with and |
816 |
|
without an applied long-range correction, brought the densities up to |
817 |
|
what is expected for simulating liquid water. In addition to improving |
818 |
< |
the densities, it is important that the excellent diffusive behavior |
819 |
< |
of SSD be maintained or improved. Figure \ref{ssdediffuse} compares |
820 |
< |
the temperature dependence of the diffusion constant of SSD/E to SSD1 |
821 |
< |
without an active reaction field at the densities calculated from the |
822 |
< |
NPT simulations at 1 atm. The diffusion constant for SSD/E is |
823 |
< |
consistently higher than experiment, while SSD1 remains lower than |
824 |
< |
experiment until relatively high temperatures (around 360 K). Both |
825 |
< |
models follow the shape of the experimental curve well below 300 K but |
826 |
< |
tend to diffuse too rapidly at higher temperatures, as seen in SSD1's |
827 |
< |
crossing above 360 K. This increasing diffusion relative to the |
828 |
< |
experimental values is caused by the rapidly decreasing system density |
829 |
< |
with increasing temperature. Both SSD1 and SSD/E show this deviation |
830 |
< |
in diffusive behavior, but this trend has different implications on |
831 |
< |
the diffusive behavior of the models. While SSD1 shows more |
832 |
< |
experimentally accurate diffusive behavior in the high temperature |
833 |
< |
regimes, SSD/E shows more accurate behavior in the supercooled and |
834 |
< |
biologically relevant temperature ranges. Thus, the changes made to |
835 |
< |
improve the liquid structure may have had an adverse affect on the |
836 |
< |
density maximum, but they improve the transport behavior of SSD/E |
837 |
< |
relative to SSD1 under the most commonly simulated conditions. |
818 |
> |
the densities, it is important that the diffusive behavior of SSD be |
819 |
> |
maintained or improved. Figure \ref{ssdediffuse} compares the |
820 |
> |
temperature dependence of the diffusion constant of SSD/E to SSD1 |
821 |
> |
without an active reaction field at the densities calculated from |
822 |
> |
their respective NPT simulations at 1 atm. The diffusion constant for |
823 |
> |
SSD/E is consistently higher than experiment, while SSD1 remains lower |
824 |
> |
than experiment until relatively high temperatures (around 360 |
825 |
> |
K). Both models follow the shape of the experimental curve well below |
826 |
> |
300 K but tend to diffuse too rapidly at higher temperatures, as seen |
827 |
> |
in SSD1's crossing above 360 K. This increasing diffusion relative to |
828 |
> |
the experimental values is caused by the rapidly decreasing system |
829 |
> |
density with increasing temperature. Both SSD1 and SSD/E show this |
830 |
> |
deviation in particle mobility, but this trend has different |
831 |
> |
implications on the diffusive behavior of the models. While SSD1 |
832 |
> |
shows more experimentally accurate diffusive behavior in the high |
833 |
> |
temperature regimes, SSD/E shows more accurate behavior in the |
834 |
> |
supercooled and biologically relevant temperature ranges. Thus, the |
835 |
> |
changes made to improve the liquid structure may have had an adverse |
836 |
> |
affect on the density maximum, but they improve the transport behavior |
837 |
> |
of SSD/E relative to SSD1 under the most commonly simulated |
838 |
> |
conditions. |
839 |
|
|
840 |
|
\begin{figure} |
841 |
|
\begin{center} |
842 |
|
\epsfxsize=6in |
843 |
|
\epsfbox{ssdrfDiffuse.epsi} |
844 |
< |
\caption{The diffusion constants calculated from SSD/RF and SSD1, |
845 |
< |
both with an active reaction field, along with experimental results |
846 |
< |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
847 |
< |
were performed at the average densities observed in the 1 atm NPT |
848 |
< |
simulations for both of the models. Note how accurately SSD/RF |
849 |
< |
simulates the diffusion of water throughout this temperature |
850 |
< |
range. The more rapidly increasing diffusion constants at high |
851 |
< |
temperatures for both models is attributed to lower calculated |
852 |
< |
densities than those observed in experiment.} |
844 |
> |
\caption{The diffusion constants calculated from SSD/RF and |
845 |
> |
SSD1 (both with an active reaction field) along with |
846 |
> |
experimental results [Refs. \citen{Gillen72} and \citen{Holz00}]. The |
847 |
> |
NVE calculations were performed at the average densities observed in |
848 |
> |
the 1 atm NPT simulations for both of the models. SSD/RF |
849 |
> |
simulates the diffusion of water throughout this temperature range |
850 |
> |
very well. The rapidly increasing diffusion constants at high |
851 |
> |
temperatures for both models can be attributed to lower calculated |
852 |
> |
densities than those observed in experiment.} |
853 |
|
\label{ssdrfdiffuse} |
854 |
|
\end{center} |
855 |
|
\end{figure} |
868 |
|
reparameterization when using an altered long-range correction. |
869 |
|
|
870 |
|
\begin{table} |
871 |
+ |
\begin{minipage}{\linewidth} |
872 |
+ |
\renewcommand{\thefootnote}{\thempfootnote} |
873 |
|
\begin{center} |
874 |
< |
\caption{Calculated and experimental properties of the single point waters and liquid water at 298 K and 1 atm. (a) Ref. [\citen{Mills73}]. (b) Calculated by integrating the data in ref. \citen{Head-Gordon00_1}. (c) Calculated by integrating the data in ref. \citen{Soper86}. (d) Ref. [\citen{Eisenberg69}]. (e) Calculated for 298 K from data in ref. \citen{Krynicki66}.} |
874 |
> |
\caption{Properties of the single-point water models compared with |
875 |
> |
experimental data at ambient conditions} |
876 |
|
\begin{tabular}{ l c c c c c } |
877 |
|
\hline \\[-3mm] |
878 |
|
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ |
880 |
|
\hline \\[-3mm] |
881 |
|
\ \ \ $\rho$ (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 \\ |
882 |
|
\ \ \ $C_p$ (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 \\ |
883 |
< |
\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & 2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299$^\text{a}$ \\ |
884 |
< |
\ \ \ Coordination Number & 3.9 & 4.3 & 3.8 & 4.4 & 4.7$^\text{b}$ \\ |
885 |
< |
\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.4$^\text{c}$ \\ |
886 |
< |
\ \ \ $\tau_1^\mu$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & 7.2 $\pm$0.4 & 4.76$^\text{d}$ \\ |
887 |
< |
\ \ \ $\tau_2^\mu$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 $\pm$0.2 & 2.3$^\text{e}$ \\ |
883 |
> |
\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & |
884 |
> |
2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299\cite{Mills73} \\ |
885 |
> |
\ \ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 & |
886 |
> |
4.7\footnote{Calculated by integrating $g_{\text{OO}}(r)$ in |
887 |
> |
Ref. \citen{Head-Gordon00_1}} \\ |
888 |
> |
\ \ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 & |
889 |
> |
3.5\footnote{Calculated by integrating $g_{\text{OH}}(r)$ in |
890 |
> |
Ref. \citen{Soper86}} \\ |
891 |
> |
\ \ \ $\tau_1$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & |
892 |
> |
7.2 $\pm$0.4 & 5.7\footnote{Calculated for 298 K from data in Ref. \citen{Eisenberg69}} \\ |
893 |
> |
\ \ \ $\tau_2$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 |
894 |
> |
$\pm$0.2 & 2.3\footnote{Calculated for 298 K from data in |
895 |
> |
Ref. \citen{Krynicki66}} |
896 |
|
\end{tabular} |
897 |
|
\label{liquidproperties} |
898 |
|
\end{center} |
899 |
+ |
\end{minipage} |
900 |
|
\end{table} |
901 |
|
|
902 |
|
Table \ref{liquidproperties} gives a synopsis of the liquid state |
903 |
|
properties of the water models compared in this study along with the |
904 |
|
experimental values for liquid water at ambient conditions. The |
905 |
< |
coordination number and hydrogen bonds per particle were calculated by |
906 |
< |
integrating the following relation: |
905 |
> |
coordination number ($n_C$) and number of hydrogen bonds per particle |
906 |
> |
($n_H$) were calculated by integrating the following relations: |
907 |
|
\begin{equation} |
908 |
< |
4\pi\rho\int_{0}^{a}r^2\text{g}(r)dr, |
908 |
> |
n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr, |
909 |
|
\end{equation} |
910 |
< |
where $\rho$ is the number density of pair interactions, $a$ is the |
911 |
< |
radial location of the minima following the first solvation shell |
912 |
< |
peak, and g$(r)$ is either g$_\text{OO}(r)$ or g$_\text{OH}(r)$ for |
913 |
< |
calculation of the coordination number or hydrogen bonds per particle |
914 |
< |
respectively. The number of hydrogen bonds stays relatively constant |
915 |
< |
across all of the models, but the coordination numbers of SSD/E and |
916 |
< |
SSD/RF show an improvement over SSD1. This improvement is primarily |
917 |
< |
due to the widening of the first solvation shell peak, allowing the |
918 |
< |
first minima to push outward. Comparing the coordination number with |
919 |
< |
the number of hydrogen bonds can lead to more insight into the |
920 |
< |
structural character of the liquid. Because of the near identical |
921 |
< |
values for SSD1, it appears to be a little too exclusive, in that all |
922 |
< |
molecules in the first solvation shell are involved in forming ideal |
923 |
< |
hydrogen bonds. The differing numbers for the newly parameterized |
924 |
< |
models indicate the allowance of more fluid configurations in addition |
925 |
< |
to the formation of an acceptable number of ideal hydrogen bonds. |
910 |
> |
\begin{equation} |
911 |
> |
n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr, |
912 |
> |
\end{equation} |
913 |
> |
where $\rho$ is the number density of the specified pair interactions, |
914 |
> |
$a$ and $b$ are the radial locations of the minima following the first |
915 |
> |
peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ respectively. The number |
916 |
> |
of hydrogen bonds stays relatively constant across all of the models, |
917 |
> |
but the coordination numbers of SSD/E and SSD/RF show an |
918 |
> |
improvement over SSD1. This improvement is primarily due to |
919 |
> |
extension of the first solvation shell in the new parameter sets. |
920 |
> |
Because $n_H$ and $n_C$ are nearly identical in SSD1, it appears |
921 |
> |
that all molecules in the first solvation shell are involved in |
922 |
> |
hydrogen bonds. Since $n_H$ and $n_C$ differ in the newly |
923 |
> |
parameterized models, the orientations in the first solvation shell |
924 |
> |
are a bit more ``fluid''. Therefore SSD1 overstructures the |
925 |
> |
first solvation shell and our reparameterizations have returned this |
926 |
> |
shell to more realistic liquid-like behavior. |
927 |
|
|
928 |
< |
The time constants for the self orientational autocorrelation function |
928 |
> |
The time constants for the orientational autocorrelation functions |
929 |
|
are also displayed in Table \ref{liquidproperties}. The dipolar |
930 |
< |
orientational time correlation function ($\Gamma_{l}$) is described |
930 |
> |
orientational time correlation functions ($C_{l}$) are described |
931 |
|
by: |
932 |
|
\begin{equation} |
933 |
< |
\Gamma_{l}(t) = \langle P_l[\mathbf{u}_j(0)\cdot\mathbf{u}_j(t)]\rangle, |
933 |
> |
C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle, |
934 |
|
\end{equation} |
935 |
< |
where $P_l$ is a Legendre polynomial of order $l$ and $\mathbf{u}_j$ |
936 |
< |
is the unit vector of the particle dipole.\cite{Rahman71} From these |
937 |
< |
correlation functions, the orientational relaxation time of the dipole |
938 |
< |
vector can be calculated from an exponential fit in the long-time |
939 |
< |
regime ($t > \tau_l^\mu$).\cite{Rothschild84} Calculation of these |
940 |
< |
time constants were averaged from five detailed NVE simulations |
941 |
< |
performed at the STP density for each of the respective models. It |
942 |
< |
should be noted that the commonly cited value for $\tau_2$ of 1.9 ps |
943 |
< |
was determined from the NMR data in reference \citen{Krynicki66} at a |
944 |
< |
temperature near 34$^\circ$C.\cite{Rahman73} Because of the strong |
945 |
< |
temperature dependence of $\tau_2$, it is necessary to recalculate it |
946 |
< |
at 298 K to make proper comparisons. The value shown in Table |
935 |
> |
where $P_l$ are Legendre polynomials of order $l$ and |
936 |
> |
$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular |
937 |
> |
dipole.\cite{Rahman71} From these correlation functions, the |
938 |
> |
orientational relaxation time of the dipole vector can be calculated |
939 |
> |
from an exponential fit in the long-time regime ($t > |
940 |
> |
\tau_l$).\cite{Rothschild84} Calculation of these time constants were |
941 |
> |
averaged over five detailed NVE simulations performed at the ambient |
942 |
> |
conditions for each of the respective models. It should be noted that |
943 |
> |
the commonly cited value of 1.9 ps for $\tau_2$ was determined from |
944 |
> |
the NMR data in Ref. \citen{Krynicki66} at a temperature near |
945 |
> |
34$^\circ$C.\cite{Rahman71} Because of the strong temperature |
946 |
> |
dependence of $\tau_2$, it is necessary to recalculate it at 298 K to |
947 |
> |
make proper comparisons. The value shown in Table |
948 |
|
\ref{liquidproperties} was calculated from the same NMR data in the |
949 |
< |
fashion described in reference \citen{Krynicki66}. Again, SSD/E and |
950 |
< |
SSD/RF show improved behavior over SSD1, both with and without an |
951 |
< |
active reaction field. Turning on the reaction field leads to much |
952 |
< |
improved time constants for SSD1; however, these results also include |
953 |
< |
a corresponding decrease in system density. Numbers published from the |
954 |
< |
original SSD dynamics studies appear closer to the experimental |
955 |
< |
values, and this difference can be attributed to the use of the Ewald |
956 |
< |
sum technique versus a reaction field.\cite{Ichiye99} |
949 |
> |
fashion described in Ref. \citen{Krynicki66}. Similarly, $\tau_1$ was |
950 |
> |
recomputed for 298 K from the data in Ref. \citen{Eisenberg69}. |
951 |
> |
Again, SSD/E and SSD/RF show improved behavior over SSD1, both with |
952 |
> |
and without an active reaction field. Turning on the reaction field |
953 |
> |
leads to much improved time constants for SSD1; however, these results |
954 |
> |
also include a corresponding decrease in system density. |
955 |
> |
Orientational relaxation times published in the original SSD dynamics |
956 |
> |
papers are smaller than the values observed here, and this difference |
957 |
> |
can be attributed to the use of the Ewald sum.\cite{Ichiye99} |
958 |
|
|
959 |
|
\subsection{Additional Observations} |
960 |
|
|
961 |
|
\begin{figure} |
962 |
|
\begin{center} |
963 |
|
\epsfxsize=6in |
964 |
< |
\epsfbox{povIce.ps} |
965 |
< |
\caption{A water lattice built from the crystal structure assumed by |
966 |
< |
SSD/E when undergoing an extremely restricted temperature NPT |
967 |
< |
simulation. This form of ice is referred to as ice-{\it i} to |
968 |
< |
emphasize its simulation origins. This image was taken of the (001) |
947 |
< |
face of the crystal.} |
964 |
> |
\epsfbox{icei_bw.eps} |
965 |
> |
\caption{The most stable crystal structure assumed by the SSD family |
966 |
> |
of water models. We refer to this structure as Ice-{\it i} to |
967 |
> |
indicate its origins in computer simulation. This image was taken of |
968 |
> |
the (001) face of the crystal.} |
969 |
|
\label{weirdice} |
970 |
|
\end{center} |
971 |
|
\end{figure} |
972 |
|
|
973 |
|
While performing a series of melting simulations on an early iteration |
974 |
< |
of SSD/E not discussed in this paper, we observed recrystallization |
975 |
< |
into a novel structure not previously known for water. After melting |
976 |
< |
at 235 K, two of five systems underwent crystallization events near |
977 |
< |
245 K. The two systems remained crystalline up to 320 and 330 K, |
978 |
< |
respectively. The crystal exhibits an expanded zeolite-like structure |
979 |
< |
that does not correspond to any known form of ice. This appears to be |
980 |
< |
an artifact of the point dipolar models, so to distinguish it from the |
981 |
< |
experimentally observed forms of ice, we have denoted the structure |
982 |
< |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (ice-{\it i}). A large enough |
974 |
> |
of SSD/E not discussed in this paper, we observed |
975 |
> |
recrystallization into a novel structure not previously known for |
976 |
> |
water. After melting at 235 K, two of five systems underwent |
977 |
> |
crystallization events near 245 K. The two systems remained |
978 |
> |
crystalline up to 320 and 330 K, respectively. The crystal exhibits |
979 |
> |
an expanded zeolite-like structure that does not correspond to any |
980 |
> |
known form of ice. This appears to be an artifact of the point |
981 |
> |
dipolar models, so to distinguish it from the experimentally observed |
982 |
> |
forms of ice, we have denoted the structure |
983 |
> |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (Ice-{\it i}). A large enough |
984 |
|
portion of the sample crystallized that we have been able to obtain a |
985 |
< |
near ideal crystal structure of ice-{\it i}. Figure \ref{weirdice} |
985 |
> |
near ideal crystal structure of Ice-{\it i}. Figure \ref{weirdice} |
986 |
|
shows the repeating crystal structure of a typical crystal at 5 |
987 |
|
K. Each water molecule is hydrogen bonded to four others; however, the |
988 |
|
hydrogen bonds are bent rather than perfectly straight. This results |
993 |
|
configuration. Though not ideal, these flexed hydrogen bonds are |
994 |
|
favorable enough to stabilize an entire crystal generated around them. |
995 |
|
|
996 |
< |
Initial simulations indicated that ice-{\it i} is the preferred ice |
996 |
> |
Initial simulations indicated that Ice-{\it i} is the preferred ice |
997 |
|
structure for at least the SSD/E model. To verify this, a comparison |
998 |
|
was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and |
999 |
< |
ice-{\it i} at constant pressure with SSD/E, SSD/RF, and |
999 |
> |
Ice-{\it i} at constant pressure with SSD/E, SSD/RF, and |
1000 |
|
SSD1. Near-ideal versions of the three types of crystals were cooled |
1001 |
< |
to 1 K, and the enthalpies of each were compared using all three water |
1002 |
< |
models. With every model in the SSD family, ice-{\it i} had the lowest |
1003 |
< |
calculated enthalpy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with |
1004 |
< |
SSD/E, and 7.5\% lower with SSD/RF. The enthalpy data is summarized |
1005 |
< |
in Table \ref{iceenthalpy}. |
1001 |
> |
to 1 K, and enthalpies of formation of each were compared using all |
1002 |
> |
three water models. Enthalpies were estimated from the |
1003 |
> |
isobaric-isothermal simulations using $H=U+P_{\text ext}V$ where |
1004 |
> |
$P_{\text ext}$ is the applied pressure. A constant value of -60.158 |
1005 |
> |
kcal / mol has been added to place our zero for the enthalpies of |
1006 |
> |
formation for these systems at the traditional state (elemental forms |
1007 |
> |
at standard temperature and pressure). With every model in the SSD |
1008 |
> |
family, Ice-{\it i} had the lowest calculated enthalpy of formation. |
1009 |
|
|
1010 |
|
\begin{table} |
1011 |
|
\begin{center} |
1012 |
< |
\caption{Enthalpies (in kcal / mol) of the three crystal structures (at 1 |
1013 |
< |
K) exhibited by the SSD family of water models} |
1012 |
> |
\caption{Enthalpies of Formation (in kcal / mol) of the three crystal |
1013 |
> |
structures (at 1 K) exhibited by the SSD family of water models} |
1014 |
|
\begin{tabular}{ l c c c } |
1015 |
|
\hline \\[-3mm] |
1016 |
|
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \ & \ |
1017 |
|
Ice-{\it i} \\ |
1018 |
|
\hline \\[-3mm] |
1019 |
< |
\ \ \ SSD/E & -12.286 & -12.292 & -13.590 \\ |
1020 |
< |
\ \ \ SSD/RF & -12.935 & -12.917 & -14.022 \\ |
1021 |
< |
\ \ \ SSD1 & -12.496 & -12.411 & -13.417 \\ |
1022 |
< |
\ \ \ SSD1 (RF) & -12.504 & -12.411 & -13.134 \\ |
1019 |
> |
\ \ \ SSD/E & -72.444 & -72.450 & -73.748 \\ |
1020 |
> |
\ \ \ SSD/RF & -73.093 & -73.075 & -74.180 \\ |
1021 |
> |
\ \ \ SSD1 & -72.654 & -72.569 & -73.575 \\ |
1022 |
> |
\ \ \ SSD1 (RF) & -72.662 & -72.569 & -73.292 \\ |
1023 |
|
\end{tabular} |
1024 |
|
\label{iceenthalpy} |
1025 |
|
\end{center} |
1037 |
|
\section{Conclusions} |
1038 |
|
|
1039 |
|
The density maximum and temperature dependence of the self-diffusion |
1040 |
< |
constant were studied for the SSD water model, both with and without |
1041 |
< |
the use of reaction field, via a series of NPT and NVE |
1040 |
> |
constant were studied for the SSD water model, both with and |
1041 |
> |
without the use of reaction field, via a series of NPT and NVE |
1042 |
|
simulations. The constant pressure simulations showed a density |
1043 |
|
maximum near 260 K. In most cases, the calculated densities were |
1044 |
|
significantly lower than the densities obtained from other water |
1045 |
< |
models (and experiment). Analysis of self-diffusion showed SSD to |
1046 |
< |
capture the transport properties of water well in both the liquid and |
1047 |
< |
super-cooled liquid regimes. |
1045 |
> |
models (and experiment). Analysis of self-diffusion showed SSD |
1046 |
> |
to capture the transport properties of water well in both the liquid |
1047 |
> |
and supercooled liquid regimes. |
1048 |
|
|
1049 |
|
In order to correct the density behavior, the original SSD model was |
1050 |
|
reparameterized for use both with and without a reaction field (SSD/RF |
1058 |
|
simulations of biochemical systems. |
1059 |
|
|
1060 |
|
The existence of a novel low-density ice structure that is preferred |
1061 |
< |
by the SSD family of water models is somewhat troubling, since liquid |
1062 |
< |
simulations on this family of water models at room temperature are |
1063 |
< |
effectively simulations of super-cooled or metastable liquids. One |
1064 |
< |
way to de-stabilize this unphysical ice structure would be to make the |
1061 |
> |
by the SSD family of water models is somewhat troubling, since |
1062 |
> |
liquid simulations on this family of water models at room temperature |
1063 |
> |
are effectively simulations of supercooled or metastable liquids. One |
1064 |
> |
way to destabilize this unphysical ice structure would be to make the |
1065 |
|
range of angles preferred by the attractive part of the sticky |
1066 |
|
potential much narrower. This would require extensive |
1067 |
|
reparameterization to maintain the same level of agreement with the |
1068 |
|
experiments. |
1069 |
|
|
1070 |
< |
Additionally, our initial calculations show that the ice-{\it i} |
1070 |
> |
Additionally, our initial calculations show that the Ice-{\it i} |
1071 |
|
structure may also be a preferred crystal structure for at least one |
1072 |
|
other popular multi-point water model (TIP3P), and that much of the |
1073 |
|
simulation work being done using this popular model could also be at |