| 45 |
|
known low-pressure ice structures under all of these water models. |
| 46 |
|
Additionally, potential truncation was shown to have an effect on the |
| 47 |
|
calculated free energies, and can result in altered free energy |
| 48 |
< |
landscapes. Structure factor for the new crystal were generated and |
| 49 |
< |
we await experimental confirmation of the existence of this new |
| 50 |
< |
polymorph. |
| 48 |
> |
landscapes. Structure factor predictions for the new crystal were |
| 49 |
> |
generated and we await experimental confirmation of the existence of |
| 50 |
> |
this new polymorph. |
| 51 |
|
\end{abstract} |
| 52 |
|
|
| 53 |
|
%\narrowtext |
| 142 |
|
was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit |
| 143 |
|
cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it |
| 144 |
|
i} unit it is extended in the direction of the (001) face and |
| 145 |
< |
compressed along the other two faces. There is typically a small unit |
| 146 |
< |
cell distortion of Ice-{\it i}$^\prime$ that converts the normally |
| 147 |
< |
square tetramer into a rhombus with alternating 85 and 95 degree |
| 148 |
< |
angles. The degree of this distortion is model dependent and |
| 149 |
< |
significant enough to split the tetramer diagonal location peak in the |
| 150 |
< |
radial distibution function. |
| 145 |
> |
compressed along the other two faces. There is typically a small |
| 146 |
> |
distortion of proton ordered Ice-{\it i}$^\prime$ that converts the |
| 147 |
> |
normally square tetramer into a rhombus with alternating approximately |
| 148 |
> |
85 and 95 degree angles. The degree of this distortion is model |
| 149 |
> |
dependent and significant enough to split the tetramer diagonal |
| 150 |
> |
location peak in the radial distribution function. |
| 151 |
|
|
| 152 |
|
\section{Methods} |
| 153 |
|
|
| 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} |
| 353 |
< |
\renewcommand{\thefootnote}{\thempfootnote} |
| 354 |
< |
\begin{center} |
| 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\\ |
| 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 |
| 378 |
< |
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. 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.} |
| 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 show a substantial dependence on |
| 418 |
< |
cutoff radius. In general, there is a narrowing of the free energy |
| 419 |
< |
differences while moving to greater cutoff radius. Interestingly, by |
| 420 |
< |
increasing the cutoff radius, the free energy gap was narrowed enough |
| 421 |
< |
in the SSD/E model that the liquid state is preferred under standard |
| 422 |
< |
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
| 423 |
< |
simulations using this model choose interaction truncation radii |
| 424 |
< |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
| 425 |
< |
case of SSD/RF, indicating that the free energies calculated with a |
| 426 |
< |
reaction field present provide a more accurate picture of the free |
| 427 |
< |
energy landscape in the absence of potential truncation. |
| 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 |
| 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 |
| 441 |
|
was estimated by applying the potential energy difference do to its |
| 442 |
< |
inclusion in systems in the presence and absence of the |
| 443 |
< |
correction. This was accomplished by calculation of the potential |
| 444 |
< |
energy of identical crystals both with and without PME. The free |
| 445 |
< |
energies for the investigated polymorphs using the TIP3P and SPC/E |
| 446 |
< |
water models are shown in Table \ref{pmeShift}. The same trend pointed |
| 447 |
< |
out through increase of cutoff radius is observed in these PME |
| 448 |
< |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
| 449 |
< |
for both the TIP3P and SPC/E water models; however, the narrowing of |
| 450 |
< |
the free energy differences between the various solid forms is |
| 442 |
> |
inclusion in systems in the presence and absence of the correction. |
| 443 |
> |
This was accomplished by calculation of the potential energy of |
| 444 |
> |
identical crystals both with and without PME. The free energies for |
| 445 |
> |
the investigated polymorphs using the TIP3P and SPC/E water models are |
| 446 |
> |
shown in Table \ref{pmeShift}. The same trend pointed out through |
| 447 |
> |
increase of cutoff radius is observed in these PME results. Ice-{\it |
| 448 |
> |
i} is the preferred polymorph at ambient conditions for both the TIP3P |
| 449 |
> |
and SPC/E water models; however, the narrowing of the free energy |
| 450 |
> |
differences between the various solid forms with the SPC/E model is |
| 451 |
|
significant enough that it becomes less clear that it is the most |
| 452 |
< |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
| 453 |
< |
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
| 454 |
< |
as well, indicating that Ice-{\it i} might be metastable with respect |
| 455 |
< |
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
| 456 |
< |
not significantly alter the finding that the Ice-{\it i} polymorph is |
| 457 |
< |
a stable crystal structure that should be considered when studying the |
| 452 |
> |
stable polymorph. The free energies of Ice-{\it i} and $I_\textrm{c}$ |
| 453 |
> |
overlap within error, while ice B and $I_\textrm{h}$ are just outside |
| 454 |
> |
at t slightly higher free energy. This indicates that with SPC/E, |
| 455 |
> |
Ice-{\it i} might be metastable with all the studied polymorphs, |
| 456 |
> |
particularly ice $I_\textrm{c}$. However, these results do not |
| 457 |
> |
significantly alter the finding that the Ice-{\it i} polymorph is a |
| 458 |
> |
stable crystal structure that should be considered when studying the |
| 459 |
|
phase behavior of water models. |
| 460 |
|
|
| 461 |
|
\begin{table*} |
| 462 |
|
\begin{minipage}{\linewidth} |
| 443 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 463 |
|
\begin{center} |
| 464 |
+ |
|
| 465 |
|
\caption{The free energy of the studied ice polymorphs after applying |
| 466 |
|
the energy difference attributed to the inclusion of the PME |
| 467 |
|
long-range interaction correction. Units are kcal/mol.} |
| 468 |
< |
\begin{tabular}{ l c c c c } |
| 468 |
> |
|
| 469 |
> |
\begin{tabular}{ccccc} |
| 470 |
|
\hline |
| 471 |
< |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
| 471 |
> |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} \\ |
| 472 |
|
\hline |
| 473 |
< |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\ |
| 474 |
< |
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\ |
| 473 |
> |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3) \\ |
| 474 |
> |
SPC/E & -12.97(2) & -13.00(2) & -12.96(3) & -13.02(2) \\ |
| 475 |
|
\end{tabular} |
| 476 |
|
\label{pmeShift} |
| 477 |
|
\end{center} |
