| 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} |
| 289 |
|
|
| 290 |
|
\begin{table*} |
| 291 |
|
\begin{minipage}{\linewidth} |
| 288 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 292 |
|
\begin{center} |
| 293 |
+ |
|
| 294 |
|
\caption{Calculated free energies for several ice polymorphs with a |
| 295 |
|
variety of common water models. All calculations used a cutoff radius |
| 296 |
|
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
| 297 |
< |
kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
| 298 |
< |
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
| 299 |
< |
\begin{tabular}{ l c c c c } |
| 297 |
> |
kcal/mol. Calculated error of the final digits is in parentheses.} |
| 298 |
> |
|
| 299 |
> |
\begin{tabular}{lcccc} |
| 300 |
|
\hline |
| 301 |
|
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
| 302 |
|
\hline |
| 303 |
|
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
| 304 |
|
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
| 305 |
|
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
| 306 |
< |
SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\ |
| 306 |
> |
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & -13.55(2)\\ |
| 307 |
|
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
| 308 |
< |
SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\ |
| 308 |
> |
SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2)\\ |
| 309 |
|
\end{tabular} |
| 310 |
|
\label{freeEnergy} |
| 311 |
|
\end{center} |
| 350 |
|
|
| 351 |
|
\begin{table*} |
| 352 |
|
\begin{minipage}{\linewidth} |
| 349 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 353 |
|
\begin{center} |
| 354 |
+ |
|
| 355 |
|
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
| 356 |
|
temperatures at 1 atm for several common water models compared with |
| 357 |
|
experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
| 358 |
|
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
| 359 |
|
liquid or gas state.} |
| 360 |
< |
\begin{tabular}{ l c c c c c c c } |
| 360 |
> |
|
| 361 |
> |
\begin{tabular}{lccccccc} |
| 362 |
|
\hline |
| 363 |
< |
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
| 363 |
> |
Equilibrium Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
| 364 |
|
\hline |
| 365 |
|
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
| 366 |
|
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
| 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. Data points omitted include SSD/E: $I_c$ 12 |
| 403 |
< |
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
| 404 |
< |
\AA . These crystals are unstable at 200 K and rapidly convert into |
| 405 |
< |
liquids. The connecting lines are qualitative visual aid.} |
| 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.} |
| 407 |
|
\label{incCutoff} |
| 408 |
|
\end{figure} |
| 409 |
|
|
| 411 |
|
computationally efficient water models was done in order to evaluate |
| 412 |
|
the trend in free energy values when moving to systems that do not |
| 413 |
|
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
| 414 |
< |
free energy of all the ice polymorphs show a substantial dependence on |
| 415 |
< |
cutoff radius. In general, there is a narrowing of the free energy |
| 416 |
< |
differences while moving to greater cutoff radius. Interestingly, by |
| 417 |
< |
increasing the cutoff radius, the free energy gap was narrowed enough |
| 418 |
< |
in the SSD/E model that the liquid state is preferred under standard |
| 419 |
< |
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
| 420 |
< |
simulations using this model choose interaction truncation radii |
| 421 |
< |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
| 422 |
< |
case of SSD/RF, indicating that the free energies calculated with a |
| 423 |
< |
reaction field present provide a more accurate picture of the free |
| 424 |
< |
energy landscape in the absence of potential truncation. |
| 414 |
> |
free energy of all the ice polymorphs for the SSD/E and TIP3P models |
| 415 |
> |
show a substantial dependence on cutoff radius. In general, there is a |
| 416 |
> |
narrowing of the free energy differences while moving to greater |
| 417 |
> |
cutoff radii. As the free energies for the polymorphs converge, the |
| 418 |
> |
stability advantage that Ice-{\it i} exhibits is reduced; however, it |
| 419 |
> |
remains the most stable polymorph for both of these models over the |
| 420 |
> |
depicted range for both models. This narrowing trend is not |
| 421 |
> |
significant in the case of SSD/RF, indicating that the free energies |
| 422 |
> |
calculated with a reaction field present provide, at minimal |
| 423 |
> |
computational cost, a more accurate picture of the free energy |
| 424 |
> |
landscape in the absence of potential truncation. Interestingly, |
| 425 |
> |
increasing the cutoff radius a mere 1.5 \AA\ with the SSD/E model |
| 426 |
> |
destabilizes the Ice-{\it i} polymorph enough that the liquid state is |
| 427 |
> |
preferred under standard simulation conditions (298 K and 1 |
| 428 |
> |
atm). Thus, it is recommended that simulations using this model choose |
| 429 |
> |
interaction truncation radii greater than 9 \AA. Considering this |
| 430 |
> |
stabilization provided by smaller cutoffs, it is not surprising that |
| 431 |
> |
crystallization into Ice-{\it i} was observed with SSD/E. The choice |
| 432 |
> |
of a 9 \AA\ cutoff in the previous simulations gives the Ice-{\it i} |
| 433 |
> |
polymorph a greater than 1 kcal/mol lower free energy than the ice |
| 434 |
> |
$I_\textrm{h}$ starting configurations. |
| 435 |
|
|
| 436 |
|
To further study the changes resulting to the inclusion of a |
| 437 |
|
long-range interaction correction, the effect of an Ewald summation |
| 438 |
|
was estimated by applying the potential energy difference do to its |
| 439 |
< |
inclusion in systems in the presence and absence of the |
| 440 |
< |
correction. This was accomplished by calculation of the potential |
| 441 |
< |
energy of identical crystals both with and without PME. The free |
| 442 |
< |
energies for the investigated polymorphs using the TIP3P and SPC/E |
| 443 |
< |
water models are shown in Table \ref{pmeShift}. The same trend pointed |
| 444 |
< |
out through increase of cutoff radius is observed in these PME |
| 445 |
< |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
| 446 |
< |
for both the TIP3P and SPC/E water models; however, the narrowing of |
| 447 |
< |
the free energy differences between the various solid forms is |
| 439 |
> |
inclusion in systems in the presence and absence of the correction. |
| 440 |
> |
This was accomplished by calculation of the potential energy of |
| 441 |
> |
identical crystals both with and without PME. The free energies for |
| 442 |
> |
the investigated polymorphs using the TIP3P and SPC/E water models are |
| 443 |
> |
shown in Table \ref{pmeShift}. The same trend pointed out through |
| 444 |
> |
increase of cutoff radius is observed in these PME results. Ice-{\it |
| 445 |
> |
i} is the preferred polymorph at ambient conditions for both the TIP3P |
| 446 |
> |
and SPC/E water models; however, the narrowing of the free energy |
| 447 |
> |
differences between the various solid forms with the SPC/E model is |
| 448 |
|
significant enough that it becomes less clear that it is the most |
| 449 |
< |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
| 450 |
< |
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
| 451 |
< |
as well, indicating that Ice-{\it i} might be metastable with respect |
| 452 |
< |
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
| 453 |
< |
not significantly alter the finding that the Ice-{\it i} polymorph is |
| 454 |
< |
a stable crystal structure that should be considered when studying the |
| 449 |
> |
stable polymorph. The free energies of Ice-{\it i} and $I_\textrm{c}$ |
| 450 |
> |
overlap within error, while ice B and $I_\textrm{h}$ are just outside |
| 451 |
> |
at t slightly higher free energy. This indicates that with SPC/E, |
| 452 |
> |
Ice-{\it i} might be metastable with all the studied polymorphs, |
| 453 |
> |
particularly ice $I_\textrm{c}$. However, these results do not |
| 454 |
> |
significantly alter the finding that the Ice-{\it i} polymorph is a |
| 455 |
> |
stable crystal structure that should be considered when studying the |
| 456 |
|
phase behavior of water models. |
| 457 |
|
|
| 458 |
|
\begin{table*} |
| 459 |
|
\begin{minipage}{\linewidth} |
| 443 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 460 |
|
\begin{center} |
| 461 |
+ |
|
| 462 |
|
\caption{The free energy of the studied ice polymorphs after applying |
| 463 |
|
the energy difference attributed to the inclusion of the PME |
| 464 |
|
long-range interaction correction. Units are kcal/mol.} |
| 465 |
< |
\begin{tabular}{ l c c c c } |
| 465 |
> |
|
| 466 |
> |
\begin{tabular}{ccccc} |
| 467 |
|
\hline |
| 468 |
< |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
| 468 |
> |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} \\ |
| 469 |
|
\hline |
| 470 |
< |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\ |
| 471 |
< |
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\ |
| 470 |
> |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3) \\ |
| 471 |
> |
SPC/E & -12.97(2) & -13.00(2) & -12.96(3) & -13.02(2) \\ |
| 472 |
|
\end{tabular} |
| 473 |
|
\label{pmeShift} |
| 474 |
|
\end{center} |
