| 201 |
|
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
| 202 |
|
the spring constants restraining translational motion and deflection |
| 203 |
|
of and rotation around the principle axis of the molecule |
| 204 |
< |
respectively. It is clear from Fig. \ref{waterSpring} that the values |
| 205 |
< |
of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
| 206 |
< |
$-\pi$ to $\pi$. The partition function for a molecular crystal |
| 204 |
> |
respectively. These spring constants are typically calculated from |
| 205 |
> |
the mean-square displacements of water molecules in an unrestrained |
| 206 |
> |
ice crystal at 200 K. For these studies, $K_\mathrm{r} = 4.29$ kcal |
| 207 |
> |
mol$^{-1}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$, and $K_\omega\ = |
| 208 |
> |
17.75$ kcal mol$^{-1}$. It is clear from Fig. \ref{waterSpring} that |
| 209 |
> |
the values of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges |
| 210 |
> |
from $-\pi$ to $\pi$. The partition function for a molecular crystal |
| 211 |
|
restrained in this fashion can be evaluated analytically, and the |
| 212 |
|
Helmholtz Free Energy ({\it A}) is given by |
| 213 |
|
\begin{eqnarray} |
| 376 |
|
melting points that compare quite favorably with experiment. The |
| 377 |
|
unfortunate aspect of this result is that this phase change occurs |
| 378 |
|
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
| 379 |
< |
liquid state. These results are actually not contrary to previous |
| 380 |
< |
studies in the literature. Earlier free energy studies of ice $I$ |
| 381 |
< |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
| 382 |
< |
being attributed to choice of interaction truncation and different |
| 379 |
< |
ordered and disordered molecular |
| 379 |
> |
liquid state. These results are actually not contrary to other |
| 380 |
> |
studies. Studies of ice $I_h$ using TIP4P predict a $T_m$ ranging from |
| 381 |
> |
214 to 238 K (differences being attributed to choice of interaction |
| 382 |
> |
truncation and different ordered and disordered molecular |
| 383 |
|
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
| 384 |
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
| 385 |
|
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 386 |
< |
calculated at 265 K, significantly higher in temperature than the |
| 387 |
< |
previous studies. Also of interest in these results is that SSD/E does |
| 386 |
> |
calculated to be 265 K, indicating that these simulation based |
| 387 |
> |
structures ought to be included in studies probing phase transitions |
| 388 |
> |
with this model. Also of interest in these results is that SSD/E does |
| 389 |
|
not exhibit a melting point at 1 atm, but it shows a sublimation point |
| 390 |
|
at 355 K. This is due to the significant stability of Ice-{\it i} over |
| 391 |
|
all other polymorphs for this particular model under these |
| 398 |
|
|
| 399 |
|
\begin{figure} |
| 400 |
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
| 401 |
< |
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
| 402 |
< |
TIP3P, and (C) SSD/RF with a reaction field. Both SSD/E and TIP3P show |
| 403 |
< |
significant cutoff radius dependence of the free energy and appear to |
| 404 |
< |
converge when moving to cutoffs greater than 12 \AA. Use of a reaction |
| 405 |
< |
field with SSD/RF results in free energies that exhibit minimal cutoff |
| 406 |
< |
radius dependence.} |
| 401 |
> |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
| 402 |
> |
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
| 403 |
> |
with an added Ewald correction term. Calculations performed without a |
| 404 |
> |
long-range correction show noticable free energy dependence on the |
| 405 |
> |
cutoff radius and show some degree of converge at large cutoff |
| 406 |
> |
radii. Inclusion of a long-range correction reduces the cutoff radius |
| 407 |
> |
dependence of the free energy for all the models. Data for ice I$_c$ |
| 408 |
> |
with TIP3P using 12 and 13.5 \AA\ cutoff radii were omitted being that |
| 409 |
> |
the crystal was prone to distortion and melting at 200 K.} |
| 410 |
|
\label{incCutoff} |
| 411 |
|
\end{figure} |
| 412 |
|
|
| 414 |
|
computationally efficient water models was done in order to evaluate |
| 415 |
|
the trend in free energy values when moving to systems that do not |
| 416 |
|
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
| 417 |
< |
free energy of all the ice polymorphs for the SSD/E and TIP3P models |
| 418 |
< |
show a substantial dependence on cutoff radius. In general, there is a |
| 419 |
< |
narrowing of the free energy differences while moving to greater |
| 420 |
< |
cutoff radii. As the free energies for the polymorphs converge, the |
| 421 |
< |
stability advantage that Ice-{\it i} exhibits is reduced; however, it |
| 422 |
< |
remains the most stable polymorph for both of these models over the |
| 423 |
< |
depicted range for both models. This narrowing trend is not |
| 424 |
< |
significant in the case of SSD/RF, indicating that the free energies |
| 425 |
< |
calculated with a reaction field present provide, at minimal |
| 426 |
< |
computational cost, a more accurate picture of the free energy |
| 417 |
> |
free energy of the ice polymorphs with water models lacking a |
| 418 |
> |
long-range correction show a cutoff radius dependence. In general, |
| 419 |
> |
there is a narrowing of the free energy differences while moving to |
| 420 |
> |
greater cutoff radii. As the free energies for the polymorphs |
| 421 |
> |
converge, the stability advantage that Ice-{\it i} exhibits is |
| 422 |
> |
reduced; however, it remains the most stable polymorph for both of |
| 423 |
> |
these models over the depicted range for both models. This narrowing |
| 424 |
> |
trend is not significant in the case of SSD/RF, indicating that the |
| 425 |
> |
free energies calculated with a reaction field present provide, at |
| 426 |
> |
minimal computational cost, a more accurate picture of the free energy |
| 427 |
|
landscape in the absence of potential truncation. Interestingly, |
| 428 |
< |
increasing the cutoff radius a mere 1.5 \AA\ with the SSD/E model |
| 429 |
< |
destabilizes the Ice-{\it i} polymorph enough that the liquid state is |
| 430 |
< |
preferred under standard simulation conditions (298 K and 1 |
| 431 |
< |
atm). Thus, it is recommended that simulations using this model choose |
| 432 |
< |
interaction truncation radii greater than 9 \AA. Considering this |
| 433 |
< |
stabilization provided by smaller cutoffs, it is not surprising that |
| 434 |
< |
crystallization into Ice-{\it i} was observed with SSD/E. The choice |
| 435 |
< |
of a 9 \AA\ cutoff in the previous simulations gives the Ice-{\it i} |
| 436 |
< |
polymorph a greater than 1 kcal/mol lower free energy than the ice |
| 437 |
< |
$I_\textrm{h}$ starting configurations. |
| 428 |
> |
increasing the cutoff radius a mere 1.5 |
| 429 |
> |
\AA\ with the SSD/E model destabilizes the Ice-{\it i} polymorph |
| 430 |
> |
enough that the liquid state is preferred under standard simulation |
| 431 |
> |
conditions (298 K and 1 atm). Thus, it is recommended that simulations |
| 432 |
> |
using this model choose interaction truncation radii greater than 9 |
| 433 |
> |
\AA. Considering this stabilization provided by smaller cutoffs, it is |
| 434 |
> |
not surprising that crystallization into Ice-{\it i} was observed with |
| 435 |
> |
SSD/E. The choice of a 9 \AA\ cutoff in the previous simulations |
| 436 |
> |
gives the Ice-{\it i} polymorph a greater than 1 kcal/mol lower free |
| 437 |
> |
energy than the ice $I_\textrm{h}$ starting configurations. |
| 438 |
|
|
| 439 |
|
To further study the changes resulting to the inclusion of a |
| 440 |
|
long-range interaction correction, the effect of an Ewald summation |
