50 |
|
|
51 |
|
\section{Methods} |
52 |
|
|
53 |
+ |
Canonical ensemble (NVT) molecular dynamics calculations were |
54 |
+ |
performed using the OOPSE (Object-Oriented Parallel Simulation Engine) |
55 |
+ |
molecular mechanics package. All molecules were treated as rigid |
56 |
+ |
bodies, with orientational motion propogated using the symplectic DLM |
57 |
+ |
integration method. Details about the implementation of these |
58 |
+ |
techniques can be found in a recent publication.\cite{Meineke05} |
59 |
+ |
|
60 |
+ |
Thermodynamic integration was utilized to calculate the free energy of |
61 |
+ |
several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, |
62 |
+ |
SSD/RF, and SSD/E water models. Liquid state free energies at 300 and |
63 |
+ |
400 K for all of these water models were also determined using this |
64 |
+ |
same technique, in order to determine melting points and generate |
65 |
+ |
phase diagrams. All simulations were carried out at densities |
66 |
+ |
resulting in a pressure of approximately 1 atm at their respective |
67 |
+ |
temperatures. |
68 |
+ |
|
69 |
+ |
For the thermodynamic integration of molecular crystals, the Einstein |
70 |
+ |
Crystal is chosen as the reference state that the system is converted |
71 |
+ |
to over the course of the simulation. In an Einstein Crystal, the |
72 |
+ |
molecules are harmonically restrained at their ideal lattice locations |
73 |
+ |
and orientations. The partition function for a molecular crystal |
74 |
+ |
restrained in this fashion has been evaluated, and the Helmholtz Free |
75 |
+ |
Energy ({\it A}) is given by |
76 |
+ |
\begin{eqnarray} |
77 |
+ |
A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
78 |
+ |
[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
79 |
+ |
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right |
80 |
+ |
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right |
81 |
+ |
)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi |
82 |
+ |
K_\omega K_\theta)^{\frac{1}{2}}}\exp\left |
83 |
+ |
(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right |
84 |
+ |
)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
85 |
+ |
\label{ecFreeEnergy} |
86 |
+ |
\end{eqnarray} |
87 |
+ |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
88 |
+ |
\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
89 |
+ |
$K_\mathrm{\omega}$ are the spring constants restraining translational |
90 |
+ |
motion and deflection of and rotation around the principle axis of the |
91 |
+ |
molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
92 |
+ |
minimum potential energy of the ideal crystal. In the case of |
93 |
+ |
molecular liquids, the ideal vapor is chosen as the target reference |
94 |
+ |
state. |
95 |
+ |
\begin{figure} |
96 |
+ |
\includegraphics[scale=1.0]{rotSpring.eps} |
97 |
+ |
\caption{Possible orientational motions for a restrained molecule. |
98 |
+ |
$\theta$ angles correspond to displacement from the body-frame {\it |
99 |
+ |
z}-axis, while $\omega$ angles correspond to rotation about the |
100 |
+ |
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
101 |
+ |
constants for the harmonic springs restraining motion in the $\theta$ |
102 |
+ |
and $\omega$ directions.} |
103 |
+ |
\label{waterSpring} |
104 |
+ |
\end{figure} |
105 |
+ |
|
106 |
+ |
Charge, dipole, and Lennard-Jones interactions were modified by a |
107 |
+ |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA ). By |
108 |
+ |
applying this function, these interactions are smoothly truncated, |
109 |
+ |
thereby avoiding poor energy conserving dynamics resulting from |
110 |
+ |
harsher truncation schemes. The effect of a long-range correction was |
111 |
+ |
also investigated on select model systems in a variety of manners. For |
112 |
+ |
the SSD/RF model, a reaction field with a fixed dielectric constant of |
113 |
+ |
80 was applied in all simulations.\cite{Onsager36} For a series of the |
114 |
+ |
least computationally expensive models (SSD/E, SSD/RF, and TIP3P), |
115 |
+ |
simulations were performed with longer cutoffs of 12 and 15 \AA\ to |
116 |
+ |
compare with the 9 \AA\ cutoff results. Finally, results from the use |
117 |
+ |
of an Ewald summation were estimated for TIP3P and SPC/E by performing |
118 |
+ |
calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
119 |
+ |
mechanics software package. TINKER was chosen because it can also |
120 |
+ |
propogate the motion of rigid-bodies, and provides the most direct |
121 |
+ |
comparison to the results from OOPSE. The calculated energy difference |
122 |
+ |
in the presence and absence of PME was applied to the previous results |
123 |
+ |
in order to predict changes in the free energy landscape. |
124 |
+ |
|
125 |
|
\section{Results and discussion} |
126 |
|
|
127 |
+ |
The free energy of proton ordered Ice-{\it i} was calculated and |
128 |
+ |
compared with the free energies of proton ordered variants of the |
129 |
+ |
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
130 |
+ |
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
131 |
+ |
and thought to be the minimum free energy structure for the SPC/E |
132 |
+ |
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
133 |
+ |
Ice XI, the experimentally observed proton ordered variant of ice |
134 |
+ |
$I_h$, was investigated initially, but it was found not to be as |
135 |
+ |
stable as antiferroelectric variants of proton ordered or even proton |
136 |
+ |
disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of |
137 |
+ |
ice $I_h$ used here is a simple antiferroelectric version that has an |
138 |
+ |
8 molecule unit cell. The crystals contained 648 or 1728 molecules for |
139 |
+ |
ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for ice |
140 |
+ |
$I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
141 |
+ |
were necessary for simulations involving larger cutoff values. |
142 |
+ |
|
143 |
+ |
\begin{table*} |
144 |
+ |
\begin{minipage}{\linewidth} |
145 |
+ |
\renewcommand{\thefootnote}{\thempfootnote} |
146 |
+ |
\begin{center} |
147 |
+ |
\caption{Calculated free energies for several ice polymorphs with a |
148 |
+ |
variety of common water models. All calculations used a cutoff radius |
149 |
+ |
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
150 |
+ |
kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF.} |
151 |
+ |
\begin{tabular}{ l c c c c } |
152 |
+ |
\hline \\[-7mm] |
153 |
+ |
\ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\ |
154 |
+ |
\hline \\[-3mm] |
155 |
+ |
\ \quad \ TIP3P & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\ |
156 |
+ |
\ \quad \ TIP4P & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\ |
157 |
+ |
\ \quad \ TIP5P & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\ |
158 |
+ |
\ \quad \ SPC/E & \ \quad \ -12.67 & \ \quad \ -12.96 & \ \quad \ -13.25 & \quad -13.55\\ |
159 |
+ |
\ \quad \ SSD/E & \ \quad \ -11.27 & \ \quad \ -11.19 & \ \quad \ -12.09 & \quad -12.54\\ |
160 |
+ |
\ \quad \ SSD/RF & \ \quad \ -11.51 & \ \quad \ NA* & \ \quad \ -12.08 & \quad -12.29\\ |
161 |
+ |
\end{tabular} |
162 |
+ |
\label{freeEnergy} |
163 |
+ |
\end{center} |
164 |
+ |
\end{minipage} |
165 |
+ |
\end{table*} |
166 |
+ |
|
167 |
+ |
The free energy values computed for the studied polymorphs indicate |
168 |
+ |
that Ice-{\it i} is the most stable state for all of the common water |
169 |
+ |
models studied. With the free energy at these state points, the |
170 |
+ |
temperature and pressure dependence of the free energy was used to |
171 |
+ |
project to other state points and build phase diagrams. Figures |
172 |
+ |
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
173 |
+ |
from the free energy results. All other models have similar structure, |
174 |
+ |
only the crossing points between these phases exist at different |
175 |
+ |
temperatures and pressures. It is interesting to note that ice $I$ |
176 |
+ |
does not exist in either cubic or hexagonal form in any of the phase |
177 |
+ |
diagrams for any of the models. For purposes of this study, ice B is |
178 |
+ |
representative of the dense ice polymorphs. A recent study by Sanz |
179 |
+ |
{\it et al.} goes into detail on the phase diagrams for SPC/E and |
180 |
+ |
TIP4P in the high pressure regime.\cite{Sanz04} |
181 |
+ |
\begin{figure} |
182 |
+ |
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
183 |
+ |
\caption{Phase diagram for the TIP3P water model in the low pressure |
184 |
+ |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
185 |
+ |
the experimental values; however, the solid phases shown are not the |
186 |
+ |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
187 |
+ |
higher in energy and don't appear in the phase diagram.} |
188 |
+ |
\label{tp3phasedia} |
189 |
+ |
\end{figure} |
190 |
+ |
\begin{figure} |
191 |
+ |
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
192 |
+ |
\caption{Phase diagram for the SSD/RF water model in the low pressure |
193 |
+ |
regime. Calculations producing these results were done under an |
194 |
+ |
applied reaction field. It is interesting to note that this |
195 |
+ |
computationally efficient model (over 3 times more efficient than |
196 |
+ |
TIP3P) exhibits phase behavior similar to the less computationally |
197 |
+ |
conservative charge based models.} |
198 |
+ |
\label{ssdrfphasedia} |
199 |
+ |
\end{figure} |
200 |
+ |
|
201 |
+ |
\begin{table*} |
202 |
+ |
\begin{minipage}{\linewidth} |
203 |
+ |
\renewcommand{\thefootnote}{\thempfootnote} |
204 |
+ |
\begin{center} |
205 |
+ |
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
206 |
+ |
temperatures of several common water models compared with experiment.} |
207 |
+ |
\begin{tabular}{ l c c c c c c c } |
208 |
+ |
\hline \\[-7mm] |
209 |
+ |
\ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\ |
210 |
+ |
\hline \\[-3mm] |
211 |
+ |
\ \ $T_m$ (K) & \ \ 269 & \ \ 265 & \ \ 271 & 297 & \ \ - & \ \ 278 & \ \ 273\\ |
212 |
+ |
\ \ $T_b$ (K) & \ \ 357 & \ \ 354 & \ \ 337 & 396 & \ \ - & \ \ 349 & \ \ 373\\ |
213 |
+ |
\ \ $T_s$ (K) & \ \ - & \ \ - & \ \ - & - & \ \ 355 & \ \ - & \ \ -\\ |
214 |
+ |
\end{tabular} |
215 |
+ |
\label{meltandboil} |
216 |
+ |
\end{center} |
217 |
+ |
\end{minipage} |
218 |
+ |
\end{table*} |
219 |
+ |
|
220 |
+ |
Table \ref{meltandboil} lists the melting and boiling temperatures |
221 |
+ |
calculated from this work. Surprisingly, most of these models have |
222 |
+ |
melting points that compare quite favorably with experiment. The |
223 |
+ |
unfortunate aspect of this result is that this phase change occurs |
224 |
+ |
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
225 |
+ |
liquid state. These results are actually not contrary to previous |
226 |
+ |
studies in the literature. Earlier free energy studies of ice $I$ |
227 |
+ |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
228 |
+ |
being attributed to choice of interaction truncation and different |
229 |
+ |
ordered and disordered molecular arrangements). If the presence of ice |
230 |
+ |
B and Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
231 |
+ |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
232 |
+ |
calculated at 265 K, significantly higher in temperature than the |
233 |
+ |
previous studies. Also of interest in these results is that SSD/E does |
234 |
+ |
not exhibit a melting point at 1 atm, but it shows a sublimation point |
235 |
+ |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
236 |
+ |
all other polymorphs for this particular model under these |
237 |
+ |
conditions. While troubling, this behavior turned out to be |
238 |
+ |
advantagious in that it facilitated the spontaneous crystallization of |
239 |
+ |
Ice-{\it i}. These observations provide a warning that simulations of |
240 |
+ |
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
241 |
+ |
risk of spontaneous crystallization. However, this risk changes when |
242 |
+ |
applying a longer cutoff. |
243 |
+ |
|
244 |
+ |
Increasing the cutoff radius in simulations of the more |
245 |
+ |
computationally efficient water models was done in order to evaluate |
246 |
+ |
the trend in free energy values when moving to systems that do not |
247 |
+ |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
248 |
+ |
free energy of all the ice polymorphs show a substantial dependence on |
249 |
+ |
cutoff radius. In general, there is a narrowing of the free energy |
250 |
+ |
differences while moving to greater cutoff radius. This trend is much |
251 |
+ |
more subtle in the case of SSD/RF, indicating that the free energies |
252 |
+ |
calculated with a reaction field present provide a more accurate |
253 |
+ |
picture of the free energy landscape in the absence of potential |
254 |
+ |
truncation. |
255 |
+ |
|
256 |
+ |
To further study the changes resulting to the inclusion of a |
257 |
+ |
long-range interaction correction, the effect of an Ewald summation |
258 |
+ |
was estimated by applying the potential energy difference do to its |
259 |
+ |
inclusion in systems in the presence and absence of the |
260 |
+ |
correction. This was accomplished by calculation of the potential |
261 |
+ |
energy of identical crystals with and without PME using TINKER. The |
262 |
+ |
free energies for the investigated polymorphs using the TIP3P and |
263 |
+ |
SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
264 |
+ |
are not fully supported in TINKER, so the results for these models |
265 |
+ |
could not be estimated. The same trend pointed out through increase of |
266 |
+ |
cutoff radius is observed in these results. Ice-{\it i} is the |
267 |
+ |
preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
268 |
+ |
water models; however, there is a narrowing of the free energy |
269 |
+ |
differences between the various solid forms. In the case of SPC/E this |
270 |
+ |
narrowing is significant enough that it becomes less clear cut that |
271 |
+ |
Ice-{\it i} is the most stable polymorph, and is possibly metastable |
272 |
+ |
with respect to ice B and possibly ice $I_c$. However, these results |
273 |
+ |
do not significantly alter the finding that the Ice-{\it i} polymorph |
274 |
+ |
is a stable crystal structure that should be considered when studying |
275 |
+ |
the phase behavior of water models. |
276 |
+ |
|
277 |
+ |
\begin{table*} |
278 |
+ |
\begin{minipage}{\linewidth} |
279 |
+ |
\renewcommand{\thefootnote}{\thempfootnote} |
280 |
+ |
\begin{center} |
281 |
+ |
\caption{The free energy of the studied ice polymorphs after applying the energy difference attributed to the inclusion of the PME long-range interaction correction. Units are kcal/mol.} |
282 |
+ |
\begin{tabular}{ l c c c c } |
283 |
+ |
\hline \\[-7mm] |
284 |
+ |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
285 |
+ |
\hline \\[-3mm] |
286 |
+ |
\ \ TIP3P & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\ |
287 |
+ |
\ \ SPC/E & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\ |
288 |
+ |
\end{tabular} |
289 |
+ |
\label{pmeShift} |
290 |
+ |
\end{center} |
291 |
+ |
\end{minipage} |
292 |
+ |
\end{table*} |
293 |
+ |
|
294 |
|
\section{Conclusions} |
295 |
|
|
296 |
|
\section{Acknowledgments} |