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 |
105 |
|
|
106 |
|
\begin{figure} |
107 |
|
\includegraphics[width=\linewidth]{unitCell.eps} |
108 |
< |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
109 |
< |
elongated variant of Ice-{\it i}. The spheres represent the |
108 |
> |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-{\it i}$^\prime$, |
109 |
> |
the elongated variant of Ice-{\it i}. The spheres represent the |
110 |
|
center-of-mass locations of the water molecules. The $a$ to $c$ |
111 |
|
ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
112 |
|
$a:2.1214c$ and $a:1.7850c$ respectively.} |
138 |
|
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
139 |
|
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
140 |
|
field parametrized single point dipole water model (SSD/RF). It should |
141 |
< |
be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was |
142 |
< |
used in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell |
143 |
< |
of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} |
144 |
< |
unit it is extended in the direction of the (001) face and compressed |
145 |
< |
along the other two faces. |
141 |
> |
be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$) |
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 |
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} |
283 |
– |
\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} |
344 |
– |
\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\\ |
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 |
373 |
< |
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} |
438 |
– |
\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} |