| 8 |
|
These models have been used to investigate important physical |
| 9 |
|
phenomena like phase transitions and the hydrophobic |
| 10 |
|
effect.\cite{Yamada02,Marrink94,Gallagher03} With the choice of models |
| 11 |
< |
available, it is only natural to compare the models under interesting |
| 11 |
> |
available, it is only natural to compare them under interesting |
| 12 |
|
thermodynamic conditions in an attempt to clarify the limitations of |
| 13 |
< |
each of the models.\cite{Jorgensen83,Jorgensen98b,Baez94,Mahoney01} |
| 14 |
< |
Two important property to quantify are the Gibbs and Helmholtz free |
| 15 |
< |
energies, particularly for the solid forms of water, as these predict |
| 16 |
< |
the thermodynamic stability of the various phases. Water has a |
| 13 |
> |
each.\cite{Jorgensen83,Jorgensen98b,Baez94,Mahoney01} Two important |
| 14 |
> |
property to quantify are the Gibbs and Helmholtz free energies, |
| 15 |
> |
particularly for the solid forms of water, as these predict the |
| 16 |
> |
thermodynamic stability of the various phases. Water has a |
| 17 |
|
particularly rich phase diagram and takes on a number of different and |
| 18 |
|
stable crystalline structures as the temperature and pressure are |
| 19 |
|
varied. This complexity makes it a challenging task to investigate the |
| 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 |
| 279 |
|
\cmidrule(l){7-8} |
| 280 |
|
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} & \multicolumn{2}{c}{(K)}\\ |
| 281 |
|
\midrule |
| 281 |
– |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(7) & 357(4)\\ |
| 282 |
– |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 262(6) & 354(4)\\ |
| 282 |
|
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 266(7) & 337(4)\\ |
| 283 |
+ |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 262(6) & 354(4)\\ |
| 284 |
+ |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(7) & 357(4)\\ |
| 285 |
|
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 299(6) & 396(4)\\ |
| 286 |
|
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(4) & -\\ |
| 287 |
|
SSD/RF & -11.96(2) & -11.60(2) & -12.53(3) & -12.79(2) & - & 278(7) & 382(4)\\ |
| 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 |
| 403 |
|
|
| 404 |
|
\section{Expanded Results Using Damped Shifted Force Electrostatics} |
| 405 |
|
|
| 406 |
+ |
In chapter \ref{chap:electrostatics}, we discussed in detail a |
| 407 |
+ |
pairwise method for handling electrostatics (shifted force, {\sc sf}) |
| 408 |
+ |
that can be used as a simple and efficient replacement for the Ewald |
| 409 |
+ |
summation. Answering the question of the free energies of these ice |
| 410 |
+ |
polymorphs with varying water models would be an interesting |
| 411 |
+ |
application of this technique. To this end, we set up thermodynamic |
| 412 |
+ |
integrations of all of the previously discussed ice polymorphs using |
| 413 |
+ |
the {\sc sf} technique with a cutoff radius of 12~\AA\ and an $\alpha$ |
| 414 |
+ |
of 0.2125~\AA . These calculations were performed on TIP5P-E and |
| 415 |
+ |
TIP4P-Ew (variants of the root models optimized for the Ewald |
| 416 |
+ |
summation) as well as SPC/E, SSD/RF, and TRED (see section |
| 417 |
+ |
\ref{sec:tredWater}). |
| 418 |
|
|
| 419 |
+ |
\begin{table} |
| 420 |
+ |
\centering |
| 421 |
+ |
\caption{HELMHOLTZ FREE ENERGIES OF ICE POLYMORPHS USING THE DAMPED |
| 422 |
+ |
SHIFTED FORCE CORRECTION} |
| 423 |
+ |
\begin{tabular}{ lccccc } |
| 424 |
+ |
\toprule |
| 425 |
+ |
\toprule |
| 426 |
+ |
Model & I$_\textrm{h}$ & I$_\textrm{c}$ & B & Ice-$i$ & Ice-$i^\prime$ \\ |
| 427 |
+ |
\cmidrule(lr){2-6} |
| 428 |
+ |
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ |
| 429 |
+ |
\midrule |
| 430 |
+ |
TIP5P-E & -10.76(4) & -10.72(4) & & - & -10.68(4) \\ |
| 431 |
+ |
TIP4P-Ew & & -11.77(3) & & - & -11.60(3) \\ |
| 432 |
+ |
SPC/E & -12.98(3) & -11.60(3) & & - & -12.93(3) \\ |
| 433 |
+ |
SSD/RF & -11.81(4) & -11.65(3) & & -12.41(4) & - \\ |
| 434 |
+ |
TRED & -12.58(3) & -12.44(3) & & -13.09(4) & - \\ |
| 435 |
+ |
\end{tabular} |
| 436 |
+ |
\label{tab:dampedFreeEnergy} |
| 437 |
+ |
\end{table} |
| 438 |
+ |
|
| 439 |
+ |
|
| 440 |
|
\section{Conclusions} |
| 441 |
|
|
| 442 |
|
In this work, thermodynamic integration was used to determine the |
| 443 |
|
absolute free energies of several ice polymorphs. The new polymorph, |
| 444 |
|
Ice-{\it i} was observed to be the stable crystalline state for {\it |
| 445 |
< |
all} the water models when using a 9.0\AA\ cutoff. However, the free |
| 445 |
> |
all} the water models when using a 9.0~\AA\ cutoff. However, the free |
| 446 |
|
energy partially depends on simulation conditions (particularly on the |
| 447 |
|
choice of long range correction method). Regardless, Ice-{\it i} was |
| 448 |
< |
still observered to be a stable polymorph for all of the studied water |
| 448 |
> |
still observed to be a stable polymorph for all of the studied water |
| 449 |
|
models. |
| 450 |
|
|
| 451 |
|
So what is the preferred solid polymorph for simulated water? As |
| 481 |
|
results, we have calculated the oxygen-oxygen pair correlation |
| 482 |
|
function, $g_\textrm{OO}(r)$, and the structure factor, $S(\vec{q})$ |
| 483 |
|
for the two Ice-{\it i} variants (along with example ice |
| 484 |
< |
I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77K, and they are shown in |
| 484 |
> |
I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77~K, and they are shown in |
| 485 |
|
figures \ref{fig:gofr} and \ref{fig:sofq} respectively. It is |
| 486 |
|
interesting to note that the structure factors for Ice-$i^\prime$ and |
| 487 |
|
Ice-I$_c$ are quite similar. The primary differences are small peaks |
| 488 |
< |
at 1.125, 2.29, and 2.53\AA$^{-1}$, so particular attention to these |
| 488 |
> |
at 1.125, 2.29, and 2.53~\AA$^{-1}$, so particular attention to these |
| 489 |
|
regions would be needed to identify the new $i^\prime$ variant from |
| 490 |
|
the I$_\textrm{c}$ polymorph. |
| 491 |
|
|
| 494 |
|
\includegraphics[width=\linewidth]{./figures/iceGofr.pdf} |
| 495 |
|
\caption{Radial distribution functions of Ice-{\it i} and ice |
| 496 |
|
I$_\textrm{c}$ calculated from from simulations of the SSD/RF water |
| 497 |
< |
model at 77 K.} |
| 497 |
> |
model at 77~K.} |
| 498 |
|
\label{fig:gofr} |
| 499 |
|
\end{figure} |
| 500 |
|
|
| 501 |
|
\begin{figure} |
| 502 |
|
\includegraphics[width=\linewidth]{./figures/sofq.pdf} |
| 503 |
|
\caption{Predicted structure factors for Ice-{\it i} and ice |
| 504 |
< |
I$_\textrm{c}$ at 77 K. The raw structure factors have been |
| 505 |
< |
convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
| 506 |
< |
width) to compensate for the trunction effects in our finite size |
| 504 |
> |
I$_\textrm{c}$ at 77~K. The raw structure factors have been |
| 505 |
> |
convoluted with a gaussian instrument function (0.075~\AA$^{-1}$ |
| 506 |
> |
width) to compensate for the truncation effects in our finite size |
| 507 |
|
simulations. The labeled peaks compared favorably with ``spurious'' |
| 508 |
|
peaks observed in experimental studies of amorphous solid |
| 509 |
|
water.\cite{Bizid87}} |
