| 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 |
| 84 |
|
crystalline water polymorphs in the low pressure regime. This work is |
| 85 |
|
unique in that one of the crystal lattices was arrived at through |
| 86 |
|
crystallization of a computationally efficient water model under |
| 87 |
< |
constant pressure and temperature conditions. Crystallization events |
| 87 |
> |
constant pressure and temperature conditions. Crystallization events |
| 88 |
|
are interesting in and of themselves;\cite{Matsumoto02,Yamada02} |
| 89 |
|
however, the crystal structure obtained in this case is different from |
| 90 |
|
any previously observed ice polymorphs in experiment or |
| 91 |
|
simulation.\cite{Fennell04} We have named this structure Ice-{\it i} |
| 92 |
|
to indicate its origin in computational simulation. The unit cell |
| 93 |
|
(Fig. \ref{iceiCell}A) consists of eight water molecules that stack in |
| 94 |
< |
rows of interlocking water tetramers. Proton ordering can be |
| 94 |
> |
rows of interlocking water tetramers. This crystal structure has a |
| 95 |
> |
limited resemblence to a recent two-dimensional ice tessellation |
| 96 |
> |
simulated on a silica surface.\cite{Yang04} Proton ordering can be |
| 97 |
|
accomplished by orienting two of the molecules so that both of their |
| 98 |
|
donated hydrogen bonds are internal to their tetramer |
| 99 |
< |
(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
| 99 |
> |
(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
| 100 |
|
water tetramers, the hydrogen bonds are not as linear as those |
| 101 |
|
observed in ice $I_h$, however the interlocking of these subunits |
| 102 |
< |
appears to provide significant stabilization to the overall |
| 103 |
< |
crystal. The arrangement of these tetramers results in surrounding |
| 104 |
< |
open octagonal cavities that are typically greater than 6.3 \AA\ in |
| 105 |
< |
diameter. This relatively open overall structure leads to crystals |
| 102 |
> |
appears to provide significant stabilization to the overall crystal. |
| 103 |
> |
The arrangement of these tetramers results in surrounding open |
| 104 |
> |
octagonal cavities that are typically greater than 6.3 \AA\ in |
| 105 |
> |
diameter. This relatively open overall structure leads to crystals |
| 106 |
|
that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. |
| 107 |
|
|
| 108 |
|
\begin{figure} |
| 130 |
|
see our previous work and related |
| 131 |
|
articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
| 132 |
|
considered energetic stabilization and neglected entropic |
| 133 |
< |
contributions to the overall free energy. To address this issue, we |
| 133 |
> |
contributions to the overall free energy. To address this issue, we |
| 134 |
|
have calculated the absolute free energy of this crystal using |
| 135 |
|
thermodynamic integration and compared to the free energies of cubic |
| 136 |
|
and hexagonal ice $I$ (the experimental low density ice polymorphs) |
| 141 |
|
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
| 142 |
|
field parametrized single point dipole water model (SSD/RF). It should |
| 143 |
|
be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$) |
| 144 |
< |
was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit |
| 144 |
> |
was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit |
| 145 |
|
cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it |
| 146 |
|
i} unit it is extended in the direction of the (001) face and |
| 147 |
< |
compressed along the other two faces. There is typically a small unit |
| 148 |
< |
cell distortion of Ice-{\it i}$^\prime$ that converts the normally |
| 149 |
< |
square tetramer into a rhombus with alternating 85 and 95 degree |
| 150 |
< |
angles. The degree of this distortion is model dependent and |
| 151 |
< |
significant enough to split the tetramer diagonal location peak in the |
| 152 |
< |
radial distibution function. |
| 147 |
> |
compressed along the other two faces. There is typically a small |
| 148 |
> |
distortion of proton ordered Ice-{\it i}$^\prime$ that converts the |
| 149 |
> |
normally square tetramer into a rhombus with alternating approximately |
| 150 |
> |
85 and 95 degree angles. The degree of this distortion is model |
| 151 |
> |
dependent and significant enough to split the tetramer diagonal |
| 152 |
> |
location peak in the radial distribution function. |
| 153 |
|
|
| 154 |
|
\section{Methods} |
| 155 |
|
|
| 156 |
|
Canonical ensemble (NVT) molecular dynamics calculations were |
| 157 |
|
performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
| 158 |
|
All molecules were treated as rigid bodies, with orientational motion |
| 159 |
< |
propagated using the symplectic DLM integration method. Details about |
| 159 |
> |
propagated using the symplectic DLM integration method. Details about |
| 160 |
|
the implementation of this technique can be found in a recent |
| 161 |
|
publication.\cite{Dullweber1997} |
| 162 |
|
|
| 169 |
|
SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
| 170 |
|
and 400 K for all of these water models were also determined using |
| 171 |
|
this same technique in order to determine melting points and to |
| 172 |
< |
generate phase diagrams. All simulations were carried out at densities |
| 173 |
< |
which correspond to a pressure of approximately 1 atm at their |
| 174 |
< |
respective temperatures. |
| 172 |
> |
generate phase diagrams. All simulations were carried out at |
| 173 |
> |
densities which correspond to a pressure of approximately 1 atm at |
| 174 |
> |
their respective temperatures. |
| 175 |
|
|
| 176 |
|
Thermodynamic integration involves a sequence of simulations during |
| 177 |
|
which the system of interest is converted into a reference system for |
| 178 |
< |
which the free energy is known analytically. This transformation path |
| 178 |
> |
which the free energy is known analytically. This transformation path |
| 179 |
|
is then integrated in order to determine the free energy difference |
| 180 |
|
between the two states: |
| 181 |
|
\begin{equation} |
| 183 |
|
)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
| 184 |
|
\end{equation} |
| 185 |
|
where $V$ is the interaction potential and $\lambda$ is the |
| 186 |
< |
transformation parameter that scales the overall |
| 187 |
< |
potential. Simulations are distributed strategically along this path |
| 188 |
< |
in order to sufficiently sample the regions of greatest change in the |
| 189 |
< |
potential. Typical integrations in this study consisted of $\sim$25 |
| 190 |
< |
simulations ranging from 300 ps (for the unaltered system) to 75 ps |
| 191 |
< |
(near the reference state) in length. |
| 186 |
> |
transformation parameter that scales the overall potential. |
| 187 |
> |
Simulations are distributed strategically along this path in order to |
| 188 |
> |
sufficiently sample the regions of greatest change in the potential. |
| 189 |
> |
Typical integrations in this study consisted of $\sim$25 simulations |
| 190 |
> |
ranging from 300 ps (for the unaltered system) to 75 ps (near the |
| 191 |
> |
reference state) in length. |
| 192 |
|
|
| 193 |
|
For the thermodynamic integration of molecular crystals, the Einstein |
| 194 |
< |
crystal was chosen as the reference system. In an Einstein crystal, |
| 194 |
> |
crystal was chosen as the reference system. In an Einstein crystal, |
| 195 |
|
the molecules are restrained at their ideal lattice locations and |
| 196 |
|
orientations. Using harmonic restraints, as applied by B\`{a}ez and |
| 197 |
|
Clancy, the total potential for this reference crystal |
| 203 |
|
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
| 204 |
|
the spring constants restraining translational motion and deflection |
| 205 |
|
of and rotation around the principle axis of the molecule |
| 206 |
< |
respectively. It is clear from Fig. \ref{waterSpring} that the values |
| 207 |
< |
of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
| 208 |
< |
$-\pi$ to $\pi$. The partition function for a molecular crystal |
| 206 |
> |
respectively. These spring constants are typically calculated from |
| 207 |
> |
the mean-square displacements of water molecules in an unrestrained |
| 208 |
> |
ice crystal at 200 K. For these studies, $K_\mathrm{r} = 4.29$ kcal |
| 209 |
> |
mol$^{-1}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$, and $K_\omega\ = |
| 210 |
> |
17.75$ kcal mol$^{-1}$. It is clear from Fig. \ref{waterSpring} that |
| 211 |
> |
the values of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges |
| 212 |
> |
from $-\pi$ to $\pi$. The partition function for a molecular crystal |
| 213 |
|
restrained in this fashion can be evaluated analytically, and the |
| 214 |
|
Helmholtz Free Energy ({\it A}) is given by |
| 215 |
|
\begin{eqnarray} |
| 251 |
|
methods.\cite{Baez95b} |
| 252 |
|
|
| 253 |
|
Charge, dipole, and Lennard-Jones interactions were modified by a |
| 254 |
< |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
| 255 |
< |
). By applying this function, these interactions are smoothly |
| 256 |
< |
truncated, thereby avoiding the poor energy conservation which results |
| 257 |
< |
from harsher truncation schemes. The effect of a long-range correction |
| 258 |
< |
was also investigated on select model systems in a variety of |
| 259 |
< |
manners. For the SSD/RF model, a reaction field with a fixed |
| 260 |
< |
dielectric constant of 80 was applied in all |
| 261 |
< |
simulations.\cite{Onsager36} For a series of the least computationally |
| 262 |
< |
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
| 263 |
< |
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
| 264 |
< |
\AA\ cutoff results. Finally, the effects of utilizing an Ewald |
| 265 |
< |
summation were estimated for TIP3P and SPC/E by performing single |
| 266 |
< |
configuration calculations with Particle-Mesh Ewald (PME) in the |
| 267 |
< |
TINKER molecular mechanics software package.\cite{Tinker} The |
| 268 |
< |
calculated energy difference in the presence and absence of PME was |
| 269 |
< |
applied to the previous results in order to predict changes to the |
| 264 |
< |
free energy landscape. |
| 254 |
> |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA). |
| 255 |
> |
By applying this function, these interactions are smoothly truncated, |
| 256 |
> |
thereby avoiding the poor energy conservation which results from |
| 257 |
> |
harsher truncation schemes. The effect of a long-range correction was |
| 258 |
> |
also investigated on select model systems in a variety of manners. |
| 259 |
> |
For the SSD/RF model, a reaction field with a fixed dielectric |
| 260 |
> |
constant of 80 was applied in all simulations.\cite{Onsager36} For a |
| 261 |
> |
series of the least computationally expensive models (SSD/E, SSD/RF, |
| 262 |
> |
and TIP3P), simulations were performed with longer cutoffs of 12 and |
| 263 |
> |
15 \AA\ to compare with the 9 \AA\ cutoff results. Finally, the |
| 264 |
> |
effects of utilizing an Ewald summation were estimated for TIP3P and |
| 265 |
> |
SPC/E by performing single configuration calculations with |
| 266 |
> |
Particle-Mesh Ewald (PME) in the TINKER molecular mechanics software |
| 267 |
> |
package.\cite{Tinker} The calculated energy difference in the presence |
| 268 |
> |
and absence of PME was applied to the previous results in order to |
| 269 |
> |
predict changes to the free energy landscape. |
| 270 |
|
|
| 271 |
|
\section{Results and discussion} |
| 272 |
|
|
| 278 |
|
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
| 279 |
|
Ice XI, the experimentally-observed proton-ordered variant of ice |
| 280 |
|
$I_h$, was investigated initially, but was found to be not as stable |
| 281 |
< |
as proton disordered or antiferroelectric variants of ice $I_h$. The |
| 281 |
> |
as proton disordered or antiferroelectric variants of ice $I_h$. The |
| 282 |
|
proton ordered variant of ice $I_h$ used here is a simple |
| 283 |
|
antiferroelectric version that we devised, and it has an 8 molecule |
| 284 |
|
unit cell similar to other predicted antiferroelectric $I_h$ |
| 285 |
|
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
| 286 |
|
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
| 287 |
< |
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger |
| 288 |
< |
crystal sizes were necessary for simulations involving larger cutoff |
| 289 |
< |
values. |
| 287 |
> |
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The |
| 288 |
> |
larger crystal sizes were necessary for simulations involving larger |
| 289 |
> |
cutoff values. |
| 290 |
|
|
| 291 |
|
\begin{table*} |
| 292 |
|
\begin{minipage}{\linewidth} |
| 288 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 293 |
|
\begin{center} |
| 294 |
+ |
|
| 295 |
|
\caption{Calculated free energies for several ice polymorphs with a |
| 296 |
|
variety of common water models. All calculations used a cutoff radius |
| 297 |
|
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
| 298 |
< |
kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
| 299 |
< |
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
| 300 |
< |
\begin{tabular}{ l c c c c } |
| 298 |
> |
kcal/mol. Calculated error of the final digits is in parentheses.} |
| 299 |
> |
|
| 300 |
> |
\begin{tabular}{lcccc} |
| 301 |
|
\hline |
| 302 |
|
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
| 303 |
|
\hline |
| 304 |
|
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
| 305 |
|
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
| 306 |
|
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
| 307 |
< |
SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\ |
| 307 |
> |
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & -13.55(2)\\ |
| 308 |
|
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
| 309 |
< |
SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\ |
| 309 |
> |
SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2)\\ |
| 310 |
|
\end{tabular} |
| 311 |
|
\label{freeEnergy} |
| 312 |
|
\end{center} |
| 315 |
|
|
| 316 |
|
The free energy values computed for the studied polymorphs indicate |
| 317 |
|
that Ice-{\it i} is the most stable state for all of the common water |
| 318 |
< |
models studied. With the calculated free energy at these state points, |
| 319 |
< |
the Gibbs-Helmholtz equation was used to project to other state points |
| 320 |
< |
and to build phase diagrams. Figures |
| 321 |
< |
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
| 322 |
< |
from the free energy results. All other models have similar structure, |
| 323 |
< |
although the crossing points between the phases move to slightly |
| 324 |
< |
different temperatures and pressures. It is interesting to note that |
| 325 |
< |
ice $I$ does not exist in either cubic or hexagonal form in any of the |
| 326 |
< |
phase diagrams for any of the models. For purposes of this study, ice |
| 327 |
< |
B is representative of the dense ice polymorphs. A recent study by |
| 328 |
< |
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
| 318 |
> |
models studied. With the calculated free energy at these state |
| 319 |
> |
points, the Gibbs-Helmholtz equation was used to project to other |
| 320 |
> |
state points and to build phase diagrams. Figures \ref{tp3phasedia} |
| 321 |
> |
and \ref{ssdrfphasedia} are example diagrams built from the free |
| 322 |
> |
energy results. All other models have similar structure, although the |
| 323 |
> |
crossing points between the phases move to slightly different |
| 324 |
> |
temperatures and pressures. It is interesting to note that ice $I$ |
| 325 |
> |
does not exist in either cubic or hexagonal form in any of the phase |
| 326 |
> |
diagrams for any of the models. For purposes of this study, ice B is |
| 327 |
> |
representative of the dense ice polymorphs. A recent study by Sanz |
| 328 |
> |
{\it et al.} goes into detail on the phase diagrams for SPC/E and |
| 329 |
|
TIP4P at higher pressures than those studied here.\cite{Sanz04} |
| 330 |
|
|
| 331 |
|
\begin{figure} |
| 332 |
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
| 333 |
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
| 334 |
< |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 334 |
> |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 335 |
|
the experimental values; however, the solid phases shown are not the |
| 336 |
< |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
| 336 |
> |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
| 337 |
|
higher in energy and don't appear in the phase diagram.} |
| 338 |
|
\label{tp3phasedia} |
| 339 |
|
\end{figure} |
| 341 |
|
\begin{figure} |
| 342 |
|
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
| 343 |
|
\caption{Phase diagram for the SSD/RF water model in the low pressure |
| 344 |
< |
regime. Calculations producing these results were done under an |
| 345 |
< |
applied reaction field. It is interesting to note that this |
| 344 |
> |
regime. Calculations producing these results were done under an |
| 345 |
> |
applied reaction field. It is interesting to note that this |
| 346 |
|
computationally efficient model (over 3 times more efficient than |
| 347 |
|
TIP3P) exhibits phase behavior similar to the less computationally |
| 348 |
|
conservative charge based models.} |
| 351 |
|
|
| 352 |
|
\begin{table*} |
| 353 |
|
\begin{minipage}{\linewidth} |
| 349 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 354 |
|
\begin{center} |
| 355 |
+ |
|
| 356 |
|
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
| 357 |
|
temperatures at 1 atm for several common water models compared with |
| 358 |
< |
experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
| 359 |
< |
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
| 358 |
> |
experiment. The $T_m$ and $T_s$ values from simulation correspond to |
| 359 |
> |
a transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
| 360 |
|
liquid or gas state.} |
| 361 |
< |
\begin{tabular}{ l c c c c c c c } |
| 361 |
> |
|
| 362 |
> |
\begin{tabular}{lccccccc} |
| 363 |
|
\hline |
| 364 |
< |
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
| 364 |
> |
Equilibrium Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
| 365 |
|
\hline |
| 366 |
|
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
| 367 |
|
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
| 373 |
|
\end{table*} |
| 374 |
|
|
| 375 |
|
Table \ref{meltandboil} lists the melting and boiling temperatures |
| 376 |
< |
calculated from this work. Surprisingly, most of these models have |
| 377 |
< |
melting points that compare quite favorably with experiment. The |
| 376 |
> |
calculated from this work. Surprisingly, most of these models have |
| 377 |
> |
melting points that compare quite favorably with experiment. The |
| 378 |
|
unfortunate aspect of this result is that this phase change occurs |
| 379 |
|
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
| 380 |
< |
liquid state. These results are actually not contrary to previous |
| 381 |
< |
studies in the literature. Earlier free energy studies of ice $I$ |
| 382 |
< |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
| 383 |
< |
being attributed to choice of interaction truncation and different |
| 378 |
< |
ordered and disordered molecular |
| 380 |
> |
liquid state. These results are actually not contrary to other |
| 381 |
> |
studies. Studies of ice $I_h$ using TIP4P predict a $T_m$ ranging |
| 382 |
> |
from 214 to 238 K (differences being attributed to choice of |
| 383 |
> |
interaction truncation and different ordered and disordered molecular |
| 384 |
|
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
| 385 |
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
| 386 |
< |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 387 |
< |
calculated at 265 K, significantly higher in temperature than the |
| 388 |
< |
previous studies. Also of interest in these results is that SSD/E does |
| 386 |
> |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 387 |
> |
calculated to be 265 K, indicating that these simulation based |
| 388 |
> |
structures ought to be included in studies probing phase transitions |
| 389 |
> |
with this model. Also of interest in these results is that SSD/E does |
| 390 |
|
not exhibit a melting point at 1 atm, but it shows a sublimation point |
| 391 |
< |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
| 392 |
< |
all other polymorphs for this particular model under these |
| 393 |
< |
conditions. While troubling, this behavior resulted in spontaneous |
| 391 |
> |
at 355 K. This is due to the significant stability of Ice-{\it i} |
| 392 |
> |
over all other polymorphs for this particular model under these |
| 393 |
> |
conditions. While troubling, this behavior resulted in spontaneous |
| 394 |
|
crystallization of Ice-{\it i} and led us to investigate this |
| 395 |
< |
structure. These observations provide a warning that simulations of |
| 395 |
> |
structure. These observations provide a warning that simulations of |
| 396 |
|
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
| 397 |
< |
risk of spontaneous crystallization. However, this risk lessens when |
| 397 |
> |
risk of spontaneous crystallization. However, this risk lessens when |
| 398 |
|
applying a longer cutoff. |
| 399 |
|
|
| 400 |
|
\begin{figure} |
| 401 |
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
| 402 |
< |
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
| 403 |
< |
TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
| 404 |
< |
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
| 405 |
< |
\AA . These crystals are unstable at 200 K and rapidly convert into |
| 406 |
< |
liquids. The connecting lines are qualitative visual aid.} |
| 402 |
> |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
| 403 |
> |
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
| 404 |
> |
with an added Ewald correction term. Calculations performed without a |
| 405 |
> |
long-range correction show noticable free energy dependence on the |
| 406 |
> |
cutoff radius and show some degree of converge at large cutoff radii. |
| 407 |
> |
Inclusion of a long-range correction reduces the cutoff radius |
| 408 |
> |
dependence of the free energy for all the models. Data for ice I$_c$ |
| 409 |
> |
with TIP3P using 12 and 13.5 \AA\ cutoff radii were omitted being that |
| 410 |
> |
the crystal was prone to distortion and melting at 200 K.} |
| 411 |
|
\label{incCutoff} |
| 412 |
|
\end{figure} |
| 413 |
|
|
| 414 |
|
Increasing the cutoff radius in simulations of the more |
| 415 |
|
computationally efficient water models was done in order to evaluate |
| 416 |
|
the trend in free energy values when moving to systems that do not |
| 417 |
< |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
| 418 |
< |
free energy of all the ice polymorphs show a substantial dependence on |
| 419 |
< |
cutoff radius. In general, there is a narrowing of the free energy |
| 420 |
< |
differences while moving to greater cutoff radius. Interestingly, by |
| 421 |
< |
increasing the cutoff radius, the free energy gap was narrowed enough |
| 422 |
< |
in the SSD/E model that the liquid state is preferred under standard |
| 423 |
< |
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
| 417 |
> |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
| 418 |
> |
free energy of the ice polymorphs with water models lacking a |
| 419 |
> |
long-range correction show a significant cutoff radius dependence. In |
| 420 |
> |
general, there is a narrowing of the free energy differences while |
| 421 |
> |
moving to greater cutoff radii. As the free energies for the |
| 422 |
> |
polymorphs converge, the stability advantage that Ice-{\it i} exhibits |
| 423 |
> |
is reduced. Interestingly, increasing the cutoff radius a mere 1.5 |
| 424 |
> |
\AA\ with the SSD/E model destabilizes the Ice-{\it i} polymorph |
| 425 |
> |
enough that the liquid state is preferred under standard simulation |
| 426 |
> |
conditions (298 K and 1 atm). Thus, it is recommended that |
| 427 |
|
simulations using this model choose interaction truncation radii |
| 428 |
< |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
| 429 |
< |
case of SSD/RF, indicating that the free energies calculated with a |
| 430 |
< |
reaction field present provide a more accurate picture of the free |
| 431 |
< |
energy landscape in the absence of potential truncation. |
| 428 |
> |
greater than 9 \AA. Considering the stabilization of Ice-{\it i} with |
| 429 |
> |
smaller cutoffs, it is not surprising that crystallization was |
| 430 |
> |
observed with SSD/E. The choice of a 9 \AA\ cutoff in the previous |
| 431 |
> |
simulations gives the Ice-{\it i} polymorph a greater than 1 kcal/mol |
| 432 |
> |
lower free energy than the ice $I_\textrm{h}$ starting configurations. |
| 433 |
> |
Additionally, it should be noted that ice $I_c$ is not stable with |
| 434 |
> |
cutoff radii of 12 and 13.5 \AA\ using the TIP3P water model. These |
| 435 |
> |
simulations showed bulk distortions of the simulation cell that |
| 436 |
> |
rapidly deteriorated crystal integrity. |
| 437 |
|
|
| 438 |
< |
To further study the changes resulting to the inclusion of a |
| 439 |
< |
long-range interaction correction, the effect of an Ewald summation |
| 440 |
< |
was estimated by applying the potential energy difference do to its |
| 441 |
< |
inclusion in systems in the presence and absence of the |
| 442 |
< |
correction. This was accomplished by calculation of the potential |
| 443 |
< |
energy of identical crystals both with and without PME. The free |
| 444 |
< |
energies for the investigated polymorphs using the TIP3P and SPC/E |
| 445 |
< |
water models are shown in Table \ref{pmeShift}. The same trend pointed |
| 446 |
< |
out through increase of cutoff radius is observed in these PME |
| 447 |
< |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
| 448 |
< |
for both the TIP3P and SPC/E water models; however, the narrowing of |
| 449 |
< |
the free energy differences between the various solid forms is |
| 450 |
< |
significant enough that it becomes less clear that it is the most |
| 451 |
< |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
| 452 |
< |
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
| 453 |
< |
as well, indicating that Ice-{\it i} might be metastable with respect |
| 454 |
< |
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
| 455 |
< |
not significantly alter the finding that the Ice-{\it i} polymorph is |
| 456 |
< |
a stable crystal structure that should be considered when studying the |
| 457 |
< |
phase behavior of water models. |
| 438 |
> |
Adjacent to each of these model plots is a system with an applied or |
| 439 |
> |
estimated long-range correction. SSD/RF was parametrized for use with |
| 440 |
> |
a reaction field, and the benefit provided by this computationally |
| 441 |
> |
inexpensive correction is apparent. Due to the relative independence |
| 442 |
> |
of the resultant free energies, calculations performed with a small |
| 443 |
> |
cutoff radius provide resultant properties similar to what one would |
| 444 |
> |
expect for the bulk material. In the cases of TIP3P and SPC/E, the |
| 445 |
> |
effect of an Ewald summation was estimated by applying the potential |
| 446 |
> |
energy difference do to its inclusion in systems in the presence and |
| 447 |
> |
absence of the correction. This was accomplished by calculation of |
| 448 |
> |
the potential energy of identical crystals both with and without |
| 449 |
> |
particle mesh Ewald (PME). Similar behavior to that observed with |
| 450 |
> |
reaction field is seen for both of these models. The free energies |
| 451 |
> |
show less dependence on cutoff radius and span a more narrowed range |
| 452 |
> |
for the various polymorphs. Like the dipolar water models, TIP3P |
| 453 |
> |
displays a relatively constant preference for the Ice-{\it i} |
| 454 |
> |
polymorph. Crystal preference is much more difficult to determine for |
| 455 |
> |
SPC/E. Without a long-range correction, each of the polymorphs |
| 456 |
> |
studied assumes the role of the preferred polymorph under different |
| 457 |
> |
cutoff conditions. The inclusion of the Ewald correction flattens and |
| 458 |
> |
narrows the sequences of free energies so much that they often overlap |
| 459 |
> |
within error, indicating that other conditions, such as cell volume in |
| 460 |
> |
microcanonical simulations, can influence the chosen polymorph upon |
| 461 |
> |
crystallization. All of these results support the finding that the |
| 462 |
> |
Ice-{\it i} polymorph is a stable crystal structure that should be |
| 463 |
> |
considered when studying the phase behavior of water models. |
| 464 |
|
|
| 465 |
|
\begin{table*} |
| 466 |
|
\begin{minipage}{\linewidth} |
| 443 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 467 |
|
\begin{center} |
| 468 |
< |
\caption{The free energy of the studied ice polymorphs after applying |
| 469 |
< |
the energy difference attributed to the inclusion of the PME |
| 470 |
< |
long-range interaction correction. Units are kcal/mol.} |
| 471 |
< |
\begin{tabular}{ l c c c c } |
| 468 |
> |
|
| 469 |
> |
\caption{The free energy versus cutoff radius for the studied ice |
| 470 |
> |
polymorphs using SPC/E after the inclusion of the PME long-range |
| 471 |
> |
interaction correction. Units are kcal/mol.} |
| 472 |
> |
|
| 473 |
> |
\begin{tabular}{ccccc} |
| 474 |
|
\hline |
| 475 |
< |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
| 475 |
> |
Cutoff (\AA) & $I_h$ & $I_c$ & B & Ice-{\it i} \\ |
| 476 |
|
\hline |
| 477 |
< |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\ |
| 478 |
< |
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\ |
| 477 |
> |
9.0 & -12.98(2) & -13.00(2) & -12.97(3) & -13.02(2) \\ |
| 478 |
> |
10.5 & -13.13(3) & -13.09(4) & -13.17(3) & -13.11(2) \\ |
| 479 |
> |
12.0 & -13.06(2) & -13.09(2) & -13.15(4) & -13.12(2) \\ |
| 480 |
> |
13.5 & -13.02(2) & -13.02(2) & -13.08(2) & -13.07(2) \\ |
| 481 |
> |
15.0 & -13.11(4) & -12.97(2) & -13.09(2) & -12.95(2) \\ |
| 482 |
|
\end{tabular} |
| 483 |
|
\label{pmeShift} |
| 484 |
|
\end{center} |
| 490 |
|
The free energy for proton ordered variants of hexagonal and cubic ice |
| 491 |
|
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
| 492 |
|
calculated under standard conditions for several common water models |
| 493 |
< |
via thermodynamic integration. All the water models studied show |
| 493 |
> |
via thermodynamic integration. All the water models studied show |
| 494 |
|
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
| 495 |
< |
\AA\ switching function cutoff. Calculated melting and boiling points |
| 495 |
> |
\AA\ switching function cutoff. Calculated melting and boiling points |
| 496 |
|
show surprisingly good agreement with the experimental values; |
| 497 |
< |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
| 497 |
> |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
| 498 |
|
effect of interaction truncation was investigated through variation of |
| 499 |
|
the cutoff radius, use of a reaction field parameterized model, and |
| 500 |
< |
estimation of the results in the presence of the Ewald |
| 501 |
< |
summation. Interaction truncation has a significant effect on the |
| 502 |
< |
computed free energy values, and may significantly alter the free |
| 503 |
< |
energy landscape for the more complex multipoint water models. Despite |
| 504 |
< |
these effects, these results show Ice-{\it i} to be an important ice |
| 505 |
< |
polymorph that should be considered in simulation studies. |
| 500 |
> |
estimation of the results in the presence of the Ewald summation. |
| 501 |
> |
Interaction truncation has a significant effect on the computed free |
| 502 |
> |
energy values, and may significantly alter the free energy landscape |
| 503 |
> |
for the more complex multipoint water models. Despite these effects, |
| 504 |
> |
these results show Ice-{\it i} to be an important ice polymorph that |
| 505 |
> |
should be considered in simulation studies. |
| 506 |
|
|
| 507 |
|
Due to this relative stability of Ice-{\it i} in all of the |
| 508 |
|
investigated simulation conditions, the question arises as to possible |
