| 34 |
|
|
| 35 |
|
While performing a series of melting simulations on an early iteration |
| 36 |
|
of SSD/E, we observed several recrystallization events at a constant |
| 37 |
< |
pressure of 1 atm. After melting from ice I$_\textrm{h}$ at 235K, two |
| 38 |
< |
of five systems recrystallized near 245K. Crystallization events are |
| 37 |
> |
pressure of 1 atm. After melting from ice I$_\textrm{h}$ at 235~K, two |
| 38 |
> |
of five systems recrystallized near 245~K. Crystallization events are |
| 39 |
|
interesting in and of themselves;\cite{Matsumoto02,Yamada02} however, |
| 40 |
|
the crystal structure extracted from these systems is different from |
| 41 |
|
any previously observed ice polymorphs in experiment or |
| 51 |
|
I$_\textrm{h}$; however, the interlocking of these subunits appears to |
| 52 |
|
provide significant stabilization to the overall crystal. The |
| 53 |
|
arrangement of these tetramers results in open octagonal cavities that |
| 54 |
< |
are typically greater than 6.3\AA\ in diameter (see figure |
| 54 |
> |
are typically greater than 6.3~\AA\ in diameter (see figure |
| 55 |
|
\ref{fig:protOrder}). This open structure leads to crystals that are |
| 56 |
< |
typically 0.07 g/cm$^3$ less dense than ice I$_\textrm{h}$. |
| 56 |
> |
typically 0.07~g/cm$^3$ less dense than ice I$_\textrm{h}$. |
| 57 |
|
|
| 58 |
|
\begin{figure} |
| 59 |
|
\includegraphics[width=\linewidth]{./figures/unitCell.pdf} |
| 85 |
|
thermodynamic integration and compared to the free energies of ice |
| 86 |
|
I$_\textrm{c}$ and ice I$_\textrm{h}$ (the common low-density ice |
| 87 |
|
polymorphs) and ice B (a higher density, but very stable crystal |
| 88 |
< |
structure observed by B\`{a}ez and Clancy in free energy studies of |
| 88 |
> |
structure observed by B\'{a}ez and Clancy in free energy studies of |
| 89 |
|
SPC/E).\cite{Baez95b} This work includes results for the water model |
| 90 |
|
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
| 91 |
|
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
| 105 |
|
performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
| 106 |
|
The densities chosen for the simulations were taken from |
| 107 |
|
isobaric-isothermal ({\it NPT}) simulations performed at 1 atm and |
| 108 |
< |
200K. Each model (and each crystal structure) was allowed to relax for |
| 109 |
< |
300ps in the {\it NPT} ensemble before averaging the density to obtain |
| 108 |
> |
200~K. Each model (and each crystal structure) was allowed to relax for |
| 109 |
> |
300~ps in the {\it NPT} ensemble before averaging the density to obtain |
| 110 |
|
the volumes for the {\it NVT} simulations.All molecules were treated |
| 111 |
|
as rigid bodies, with orientational motion propagated using the |
| 112 |
|
symplectic DLM integration method described in section |
| 159 |
|
of and rotation around the principle axis of the molecule |
| 160 |
|
respectively. These spring constants are typically calculated from |
| 161 |
|
the mean-square displacements of water molecules in an unrestrained |
| 162 |
< |
ice crystal at 200 K. For these studies, $K_\mathrm{v} = 4.29$ kcal |
| 163 |
< |
mol$^{-1}$ \AA$^{-2}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$ rad$^{-2}$, |
| 164 |
< |
and $K_\omega\ = 17.75$ kcal mol$^{-1}$ rad$^{-2}$. It is clear from |
| 165 |
< |
Fig. \ref{fig:waterSpring} that the values of $\theta$ range from $0$ to |
| 166 |
< |
$\pi$, while $\omega$ ranges from $-\pi$ to $\pi$. The partition |
| 162 |
> |
ice crystal at 200~K. For these studies, $K_\mathrm{v} = |
| 163 |
> |
4.29$~kcal~mol$^{-1}$~\AA$^{-2}$, $K_\theta\ = |
| 164 |
> |
13.88$~kcal~mol$^{-1}$~rad$^{-2}$, and $K_\omega\ = |
| 165 |
> |
17.75$~kcal~mol$^{-1}$~rad$^{-2}$. It is clear from |
| 166 |
> |
Fig. \ref{fig:waterSpring} that the values of $\theta$ range from $0$ |
| 167 |
> |
to $\pi$, while $\omega$ ranges from $-\pi$ to $\pi$. The partition |
| 168 |
|
function for a molecular crystal restrained in this fashion can be |
| 169 |
|
evaluated analytically, and the Helmholtz Free Energy ({\it A}) is |
| 170 |
|
given by |
| 185 |
|
\end{equation} |
| 186 |
|
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
| 187 |
|
potential energy of the ideal crystal.\cite{Baez95a} The choice of an |
| 188 |
< |
Einstein crystal reference stat is somewhat arbitrary. Any ideal |
| 188 |
> |
Einstein crystal reference state is somewhat arbitrary. Any ideal |
| 189 |
|
system for which the partition function is known exactly could be used |
| 190 |
|
as a reference point as long as the system does not undergo a phase |
| 191 |
|
transition during the integration path between the real and ideal |
| 192 |
|
systems. Nada and van der Eerden have shown that the use of different |
| 193 |
< |
force constants in the Einstein crystal doesn not affect the total |
| 193 |
> |
force constants in the Einstein crystal does not affect the total |
| 194 |
|
free energy, and Gao {\it et al.} have shown that free energies |
| 195 |
|
computed with the Debye crystal reference state differ from the |
| 196 |
< |
Einstein crystal by only a few tenths of a kJ |
| 197 |
< |
mol$^{-1}$.\cite{Nada03,Gao00} These free energy differences can lead |
| 198 |
< |
to some uncertainty in the computed melting point of the solids. |
| 196 |
> |
Einstein crystal by only a few tenths of a |
| 197 |
> |
kJ~mol$^{-1}$.\cite{Nada03,Gao00} These free energy differences can |
| 198 |
> |
lead to some uncertainty in the computed melting point of the solids. |
| 199 |
|
\begin{figure} |
| 200 |
|
\centering |
| 201 |
|
\includegraphics[width=3.5in]{./figures/rotSpring.pdf} |
| 231 |
|
dielectric constant of 80 was applied in all |
| 232 |
|
simulations.\cite{Onsager36} For a series of the least computationally |
| 233 |
|
expensive models (SSD/E, SSD/RF, TIP3P, and SPC/E), simulations were |
| 234 |
< |
performed with longer cutoffs of 10.5, 12, 13.5, and 15\AA\ to |
| 235 |
< |
compare with the 9\AA\ cutoff results. Finally, the effects of using |
| 234 |
> |
performed with longer cutoffs of 10.5, 12, 13.5, and 15~\AA\ to |
| 235 |
> |
compare with the 9~\AA\ cutoff results. Finally, the effects of using |
| 236 |
|
the Ewald summation were estimated for TIP3P and SPC/E by performing |
| 237 |
|
single configuration Particle-Mesh Ewald (PME) calculations for each |
| 238 |
|
of the ice polymorphs.\cite{Ponder87} The calculated energy difference |
| 313 |
|
\label{fig:ssdrfPhaseDia} |
| 314 |
|
\end{figure} |
| 315 |
|
|
| 316 |
< |
We note that all of the crystals investigated in this study ar ideal |
| 316 |
> |
We note that all of the crystals investigated in this study are ideal |
| 317 |
|
proton-ordered antiferroelectric structures. All of the structures |
| 318 |
|
obey the Bernal-Fowler rules and should be able to form stable |
| 319 |
|
proton-{\it disordered} crystals which have the traditional |
| 320 |
< |
$k_\textrm{B}$ln(3/2) residual entropy at 0K.\cite{Bernal33,Pauling35} |
| 320 |
> |
$k_\textrm{B}$ln(3/2) residual entropy at 0~K.\cite{Bernal33,Pauling35} |
| 321 |
|
Simulations of proton-disordered structures are relatively unstable |
| 322 |
|
with all but the most expensive water models.\cite{Nada03} Our |
| 323 |
|
simulations have therefore been performed with the ordered |
| 328 |
|
of the disordered structures.\cite{Sanz04} |
| 329 |
|
|
| 330 |
|
Most of the water models have melting points that compare quite |
| 331 |
< |
favorably with the experimental value of 273 K. The unfortunate |
| 331 |
> |
favorably with the experimental value of 273~K. The unfortunate |
| 332 |
|
aspect of this result is that this phase change occurs between |
| 333 |
|
Ice-{\it i} and the liquid state rather than ice I$_h$ and the liquid |
| 334 |
|
state. These results do not contradict other studies. Studies of ice |
| 335 |
< |
I$_h$ using TIP4P predict a $T_m$ ranging from 191 to 238 K |
| 335 |
> |
I$_h$ using TIP4P predict a $T_m$ ranging from 191 to 238~K |
| 336 |
|
(differences being attributed to choice of interaction truncation and |
| 337 |
|
different ordered and disordered molecular |
| 338 |
|
arrangements).\cite{Nada03,Vlot99,Gao00,Sanz04} If the presence of ice |
| 339 |
< |
B and Ice-{\it i} were omitted, a $T_\textrm{m}$ value around 200 K |
| 339 |
> |
B and Ice-{\it i} were omitted, a $T_\textrm{m}$ value around 200~K |
| 340 |
|
would be predicted from this work. However, the $T_\textrm{m}$ from |
| 341 |
< |
Ice-{\it i} is calculated to be 262 K, indicating that these |
| 341 |
> |
Ice-{\it i} is calculated to be 262~K, indicating that these |
| 342 |
|
simulation based structures ought to be included in studies probing |
| 343 |
|
phase transitions with this model. Also of interest in these results |
| 344 |
|
is that SSD/E does not exhibit a melting point at 1 atm but does |
| 345 |
< |
sublime at 355 K. This is due to the significant stability of |
| 345 |
> |
sublime at 355~K. This is due to the significant stability of |
| 346 |
|
Ice-{\it i} over all other polymorphs for this particular model under |
| 347 |
|
these conditions. While troubling, this behavior resulted in the |
| 348 |
|
spontaneous crystallization of Ice-{\it i} which led us to investigate |
| 349 |
|
this structure. These observations provide a warning that simulations |
| 350 |
< |
of SSD/E as a ``liquid'' near 300 K are actually metastable and run |
| 350 |
> |
of SSD/E as a ``liquid'' near 300~K are actually metastable and run |
| 351 |
|
the risk of spontaneous crystallization. However, when a longer |
| 352 |
|
cutoff radius is used, SSD/E prefers the liquid state under standard |
| 353 |
|
temperature and pressure. |
| 354 |
|
|
| 355 |
< |
\section{Effects of Potential Trucation} |
| 355 |
> |
\section{Effects of Potential Truncation} |
| 356 |
|
|
| 357 |
|
\begin{figure} |
| 358 |
|
\includegraphics[width=\linewidth]{./figures/cutoffChange.pdf} |
| 359 |
|
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
| 360 |
|
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
| 361 |
|
with an added Ewald correction term. Error for the larger cutoff |
| 362 |
< |
points is equivalent to that observed at 9.0\AA\ (see Table |
| 362 |
> |
points is equivalent to that observed at 9.0~\AA\ (see Table |
| 363 |
|
\ref{tab:freeEnergy}). Data for ice I$_\textrm{c}$ with TIP3P using |
| 364 |
< |
both 12 and 13.5\AA\ cutoffs were omitted because the crystal was |
| 365 |
< |
prone to distortion and melting at 200K. Ice-$i^\prime$ is the |
| 364 |
> |
both 12 and 13.5~\AA\ cutoffs were omitted because the crystal was |
| 365 |
> |
prone to distortion and melting at 200~K. Ice-$i^\prime$ is the |
| 366 |
|
form of Ice-{\it i} used in the SPC/E simulations.} |
| 367 |
|
\label{fig:incCutoff} |
| 368 |
|
\end{figure} |
| 369 |
|
|
| 370 |
|
For the more computationally efficient water models, we have also |
| 371 |
< |
investigated the effect of potential trunctaion on the computed free |
| 371 |
> |
investigated the effect of potential truncation on the computed free |
| 372 |
|
energies as a function of the cutoff radius. As seen in |
| 373 |
|
Fig. \ref{fig:incCutoff}, the free energies of the ice polymorphs with |
| 374 |
|
water models lacking a long-range correction show significant cutoff |
| 383 |
|
field cavity in this model, so small cutoff radii mimic bulk |
| 384 |
|
calculations quite well under SSD/RF. |
| 385 |
|
|
| 386 |
< |
Although TIP3P was paramaterized for use without the Ewald summation, |
| 386 |
> |
Although TIP3P was parametrized for use without the Ewald summation, |
| 387 |
|
we have estimated the effect of this method for computing long-range |
| 388 |
|
electrostatics for both TIP3P and SPC/E. This was accomplished by |
| 389 |
|
calculating the potential energy of identical crystals both with and |
| 409 |
|
In this work, thermodynamic integration was used to determine the |
| 410 |
|
absolute free energies of several ice polymorphs. The new polymorph, |
| 411 |
|
Ice-{\it i} was observed to be the stable crystalline state for {\it |
| 412 |
< |
all} the water models when using a 9.0\AA\ cutoff. However, the free |
| 412 |
> |
all} the water models when using a 9.0~\AA\ cutoff. However, the free |
| 413 |
|
energy partially depends on simulation conditions (particularly on the |
| 414 |
|
choice of long range correction method). Regardless, Ice-{\it i} was |
| 415 |
< |
still observered to be a stable polymorph for all of the studied water |
| 415 |
> |
still observed to be a stable polymorph for all of the studied water |
| 416 |
|
models. |
| 417 |
|
|
| 418 |
|
So what is the preferred solid polymorph for simulated water? As |
| 448 |
|
results, we have calculated the oxygen-oxygen pair correlation |
| 449 |
|
function, $g_\textrm{OO}(r)$, and the structure factor, $S(\vec{q})$ |
| 450 |
|
for the two Ice-{\it i} variants (along with example ice |
| 451 |
< |
I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77K, and they are shown in |
| 451 |
> |
I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77~K, and they are shown in |
| 452 |
|
figures \ref{fig:gofr} and \ref{fig:sofq} respectively. It is |
| 453 |
|
interesting to note that the structure factors for Ice-$i^\prime$ and |
| 454 |
|
Ice-I$_c$ are quite similar. The primary differences are small peaks |
| 455 |
< |
at 1.125, 2.29, and 2.53\AA$^{-1}$, so particular attention to these |
| 455 |
> |
at 1.125, 2.29, and 2.53~\AA$^{-1}$, so particular attention to these |
| 456 |
|
regions would be needed to identify the new $i^\prime$ variant from |
| 457 |
|
the I$_\textrm{c}$ polymorph. |
| 458 |
|
|
| 461 |
|
\includegraphics[width=\linewidth]{./figures/iceGofr.pdf} |
| 462 |
|
\caption{Radial distribution functions of Ice-{\it i} and ice |
| 463 |
|
I$_\textrm{c}$ calculated from from simulations of the SSD/RF water |
| 464 |
< |
model at 77 K.} |
| 464 |
> |
model at 77~K.} |
| 465 |
|
\label{fig:gofr} |
| 466 |
|
\end{figure} |
| 467 |
|
|
| 468 |
|
\begin{figure} |
| 469 |
|
\includegraphics[width=\linewidth]{./figures/sofq.pdf} |
| 470 |
|
\caption{Predicted structure factors for Ice-{\it i} and ice |
| 471 |
< |
I$_\textrm{c}$ at 77 K. The raw structure factors have been |
| 472 |
< |
convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
| 473 |
< |
width) to compensate for the trunction effects in our finite size |
| 471 |
> |
I$_\textrm{c}$ at 77~K. The raw structure factors have been |
| 472 |
> |
convoluted with a gaussian instrument function (0.075~\AA$^{-1}$ |
| 473 |
> |
width) to compensate for the truncation effects in our finite size |
| 474 |
|
simulations. The labeled peaks compared favorably with ``spurious'' |
| 475 |
|
peaks observed in experimental studies of amorphous solid |
| 476 |
|
water.\cite{Bizid87}} |
