1 |
chrisfen |
1453 |
%\documentclass[prb,aps,twocolumn,tabularx]{revtex4} |
2 |
gezelter |
1463 |
\documentclass[11pt]{article} |
3 |
chrisfen |
1453 |
%\documentclass[11pt]{article} |
4 |
gezelter |
1463 |
\usepackage{endfloat} |
5 |
chrisfen |
1453 |
\usepackage{amsmath} |
6 |
|
|
\usepackage{epsf} |
7 |
|
|
\usepackage{berkeley} |
8 |
gezelter |
1463 |
\usepackage{setspace} |
9 |
|
|
\usepackage{tabularx} |
10 |
chrisfen |
1453 |
\usepackage{graphicx} |
11 |
gezelter |
1463 |
\usepackage[ref]{overcite} |
12 |
|
|
\pagestyle{plain} |
13 |
|
|
\pagenumbering{arabic} |
14 |
|
|
\oddsidemargin 0.0cm \evensidemargin 0.0cm |
15 |
|
|
\topmargin -21pt \headsep 10pt |
16 |
|
|
\textheight 9.0in \textwidth 6.5in |
17 |
|
|
\brokenpenalty=10000 |
18 |
|
|
\renewcommand{\baselinestretch}{1.2} |
19 |
|
|
\renewcommand\citemid{\ } % no comma in optional reference note |
20 |
chrisfen |
1453 |
|
21 |
|
|
\begin{document} |
22 |
|
|
|
23 |
gezelter |
1465 |
\title{Ice-{\it i}: a simulation-predicted ice polymorph which is more |
24 |
|
|
stable than Ice $I_h$ for point-charge and point-dipole water models} |
25 |
chrisfen |
1453 |
|
26 |
gezelter |
1463 |
\author{Christopher J. Fennell and J. Daniel Gezelter \\ |
27 |
|
|
Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
28 |
chrisfen |
1453 |
Notre Dame, Indiana 46556} |
29 |
|
|
|
30 |
|
|
\date{\today} |
31 |
|
|
|
32 |
gezelter |
1463 |
\maketitle |
33 |
chrisfen |
1453 |
%\doublespacing |
34 |
|
|
|
35 |
|
|
\begin{abstract} |
36 |
chrisfen |
1459 |
The free energies of several ice polymorphs in the low pressure regime |
37 |
gezelter |
1463 |
were calculated using thermodynamic integration. These integrations |
38 |
|
|
were done for most of the common water models. Ice-{\it i}, a |
39 |
|
|
structure we recently observed to be stable in one of the single-point |
40 |
|
|
water models, was determined to be the stable crystalline state (at 1 |
41 |
|
|
atm) for {\it all} the water models investigated. Phase diagrams were |
42 |
|
|
generated, and phase coexistence lines were determined for all of the |
43 |
|
|
known low-pressure ice structures under all of the common water |
44 |
|
|
models. Additionally, potential truncation was shown to have an |
45 |
|
|
effect on the calculated free energies, and can result in altered free |
46 |
|
|
energy landscapes. |
47 |
chrisfen |
1453 |
\end{abstract} |
48 |
|
|
|
49 |
|
|
%\narrowtext |
50 |
|
|
|
51 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
52 |
|
|
% BODY OF TEXT |
53 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
54 |
|
|
|
55 |
|
|
\section{Introduction} |
56 |
|
|
|
57 |
chrisfen |
1471 |
Computer simulations are a valuable tool for studying the phase |
58 |
|
|
behavior of systems ranging from small or simple molecules to complex |
59 |
chrisfen |
1472 |
biological species.\cite{Matsumoto02,Li96,Marrink01} Useful techniques |
60 |
chrisfen |
1471 |
have been developed to investigate the thermodynamic properites of |
61 |
|
|
model substances, providing both qualitative and quantitative |
62 |
|
|
comparisons between simulations and |
63 |
|
|
experiment.\cite{Widom63,Frenkel84} Investigation of these properties |
64 |
|
|
leads to the development of new and more accurate models, leading to |
65 |
|
|
better understanding and depiction of physical processes and intricate |
66 |
|
|
molecular systems. |
67 |
chrisfen |
1459 |
|
68 |
|
|
Water has proven to be a challenging substance to depict in |
69 |
gezelter |
1463 |
simulations, and a variety of models have been developed to describe |
70 |
|
|
its behavior under varying simulation |
71 |
chrisfen |
1471 |
conditions.\cite{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04} |
72 |
gezelter |
1463 |
These models have been used to investigate important physical |
73 |
chrisfen |
1471 |
phenomena like phase transitions, molecule transport, and the |
74 |
|
|
hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
75 |
|
|
choice of models available, it is only natural to compare the models |
76 |
|
|
under interesting thermodynamic conditions in an attempt to clarify |
77 |
|
|
the limitations of each of the |
78 |
|
|
models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
79 |
|
|
important property to quantify are the Gibbs and Helmholtz free |
80 |
|
|
energies, particularly for the solid forms of water. Difficulty in |
81 |
|
|
these types of studies typically arises from the assortment of |
82 |
|
|
possible crystalline polymorphs that water adopts over a wide range of |
83 |
|
|
pressures and temperatures. There are currently 13 recognized forms |
84 |
|
|
of ice, and it is a challenging task to investigate the entire free |
85 |
|
|
energy landscape.\cite{Sanz04} Ideally, research is focused on the |
86 |
|
|
phases having the lowest free energy at a given state point, because |
87 |
|
|
these phases will dictate the true transition temperatures and |
88 |
|
|
pressures for the model. |
89 |
chrisfen |
1459 |
|
90 |
gezelter |
1465 |
In this paper, standard reference state methods were applied to known |
91 |
chrisfen |
1471 |
crystalline water polymorphs in the low pressure regime. This work is |
92 |
gezelter |
1465 |
unique in the fact that one of the crystal lattices was arrived at |
93 |
|
|
through crystallization of a computationally efficient water model |
94 |
|
|
under constant pressure and temperature conditions. Crystallization |
95 |
|
|
events are interesting in and of |
96 |
|
|
themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure |
97 |
|
|
obtained in this case is different from any previously observed ice |
98 |
|
|
polymorphs in experiment or simulation.\cite{Fennell04} We have named |
99 |
|
|
this structure Ice-{\it i} to indicate its origin in computational |
100 |
chrisfen |
1459 |
simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight |
101 |
|
|
water molecules that stack in rows of interlocking water |
102 |
|
|
tetramers. Proton ordering can be accomplished by orienting two of the |
103 |
gezelter |
1465 |
molecules so that both of their donated hydrogen bonds are internal to |
104 |
chrisfen |
1459 |
their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal |
105 |
|
|
constructed of water tetramers, the hydrogen bonds are not as linear |
106 |
|
|
as those observed in ice $I_h$, however the interlocking of these |
107 |
|
|
subunits appears to provide significant stabilization to the overall |
108 |
|
|
crystal. The arrangement of these tetramers results in surrounding |
109 |
|
|
open octagonal cavities that are typically greater than 6.3 \AA\ in |
110 |
|
|
diameter. This relatively open overall structure leads to crystals |
111 |
|
|
that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. |
112 |
gezelter |
1463 |
|
113 |
chrisfen |
1460 |
\begin{figure} |
114 |
gezelter |
1463 |
\includegraphics[width=\linewidth]{unitCell.eps} |
115 |
gezelter |
1465 |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
116 |
chrisfen |
1471 |
elongated variant of Ice-{\it i}. The spheres represent the |
117 |
|
|
center-of-mass locations of the water molecules. The $a$ to $c$ |
118 |
|
|
ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
119 |
|
|
$a:2.1214c$ and $a:1.7850c$ respectively.} |
120 |
chrisfen |
1460 |
\label{iceiCell} |
121 |
|
|
\end{figure} |
122 |
gezelter |
1463 |
|
123 |
chrisfen |
1460 |
\begin{figure} |
124 |
gezelter |
1463 |
\includegraphics[width=\linewidth]{orderedIcei.eps} |
125 |
chrisfen |
1460 |
\caption{Image of a proton ordered crystal of Ice-{\it i} looking |
126 |
|
|
down the (001) crystal face. The rows of water tetramers surrounded by |
127 |
|
|
octagonal pores leads to a crystal structure that is significantly |
128 |
|
|
less dense than ice $I_h$.} |
129 |
|
|
\label{protOrder} |
130 |
|
|
\end{figure} |
131 |
chrisfen |
1459 |
|
132 |
gezelter |
1465 |
Results from our previous study indicated that Ice-{\it i} is the |
133 |
|
|
minimum energy crystal structure for the single point water models we |
134 |
|
|
investigated (for discussions on these single point dipole models, see |
135 |
chrisfen |
1471 |
our previous work and related |
136 |
|
|
articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
137 |
gezelter |
1465 |
considered energetic stabilization and neglected entropic |
138 |
|
|
contributions to the overall free energy. To address this issue, the |
139 |
|
|
absolute free energy of this crystal was calculated using |
140 |
|
|
thermodynamic integration and compared to the free energies of cubic |
141 |
|
|
and hexagonal ice $I$ (the experimental low density ice polymorphs) |
142 |
|
|
and ice B (a higher density, but very stable crystal structure |
143 |
|
|
observed by B\`{a}ez and Clancy in free energy studies of |
144 |
|
|
SPC/E).\cite{Baez95b} This work includes results for the water model |
145 |
|
|
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
146 |
|
|
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
147 |
|
|
field parametrized single point dipole water model (SSD/RF). It should |
148 |
|
|
be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used |
149 |
|
|
in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of |
150 |
|
|
this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit |
151 |
|
|
it is extended in the direction of the (001) face and compressed along |
152 |
|
|
the other two faces. |
153 |
chrisfen |
1459 |
|
154 |
chrisfen |
1453 |
\section{Methods} |
155 |
|
|
|
156 |
chrisfen |
1454 |
Canonical ensemble (NVT) molecular dynamics calculations were |
157 |
gezelter |
1465 |
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 |
160 |
chrisfen |
1471 |
the implementation of this technique can be found in a recent |
161 |
gezelter |
1468 |
publication.\cite{Dullweber1997} |
162 |
chrisfen |
1454 |
|
163 |
chrisfen |
1471 |
Thermodynamic integration is an established technique for |
164 |
|
|
determination of free energies of condensed phases of |
165 |
|
|
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
166 |
|
|
method, implemented in the same manner illustrated by B\`{a}ez and |
167 |
|
|
Clancy, was utilized to calculate the free energy of several ice |
168 |
|
|
crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
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 generate |
172 |
|
|
phase diagrams. All simulations were carried out at densities |
173 |
|
|
resulting in a pressure of approximately 1 atm at their respective |
174 |
|
|
temperatures. |
175 |
chrisfen |
1454 |
|
176 |
chrisfen |
1458 |
A single thermodynamic integration involves a sequence of simulations |
177 |
|
|
over which the system of interest is converted into a reference system |
178 |
gezelter |
1465 |
for which the free energy is known analytically. This transformation |
179 |
|
|
path is then integrated in order to determine the free energy |
180 |
|
|
difference between the two states: |
181 |
chrisfen |
1458 |
\begin{equation} |
182 |
|
|
\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
183 |
|
|
)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
184 |
|
|
\end{equation} |
185 |
|
|
where $V$ is the interaction potential and $\lambda$ is the |
186 |
chrisfen |
1459 |
transformation parameter that scales the overall |
187 |
chrisfen |
1471 |
potential. Simulations are distributed strategically along this path |
188 |
|
|
in order to sufficiently sample the regions of greatest change in the |
189 |
chrisfen |
1459 |
potential. Typical integrations in this study consisted of $\sim$25 |
190 |
|
|
simulations ranging from 300 ps (for the unaltered system) to 75 ps |
191 |
|
|
(near the reference state) in length. |
192 |
chrisfen |
1458 |
|
193 |
chrisfen |
1454 |
For the thermodynamic integration of molecular crystals, the Einstein |
194 |
chrisfen |
1471 |
crystal was chosen as the reference system. In an Einstein crystal, |
195 |
|
|
the molecules are restrained at their ideal lattice locations and |
196 |
|
|
orientations. Using harmonic restraints, as applied by B\`{a}ez and |
197 |
|
|
Clancy, the total potential for this reference crystal |
198 |
|
|
($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
199 |
|
|
\begin{equation} |
200 |
|
|
V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
201 |
|
|
\frac{K_\omega\omega^2}{2}, |
202 |
|
|
\end{equation} |
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 |
209 |
gezelter |
1465 |
restrained in this fashion can be evaluated analytically, and the |
210 |
|
|
Helmholtz Free Energy ({\it A}) is given by |
211 |
chrisfen |
1454 |
\begin{eqnarray} |
212 |
|
|
A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
213 |
|
|
[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
214 |
|
|
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right |
215 |
|
|
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right |
216 |
|
|
)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi |
217 |
|
|
K_\omega K_\theta)^{\frac{1}{2}}}\exp\left |
218 |
|
|
(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right |
219 |
|
|
)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
220 |
|
|
\label{ecFreeEnergy} |
221 |
|
|
\end{eqnarray} |
222 |
chrisfen |
1471 |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
223 |
|
|
potential energy of the ideal crystal.\cite{Baez95a} |
224 |
gezelter |
1463 |
|
225 |
chrisfen |
1456 |
\begin{figure} |
226 |
gezelter |
1463 |
\includegraphics[width=\linewidth]{rotSpring.eps} |
227 |
chrisfen |
1456 |
\caption{Possible orientational motions for a restrained molecule. |
228 |
|
|
$\theta$ angles correspond to displacement from the body-frame {\it |
229 |
|
|
z}-axis, while $\omega$ angles correspond to rotation about the |
230 |
|
|
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
231 |
|
|
constants for the harmonic springs restraining motion in the $\theta$ |
232 |
|
|
and $\omega$ directions.} |
233 |
|
|
\label{waterSpring} |
234 |
|
|
\end{figure} |
235 |
chrisfen |
1454 |
|
236 |
chrisfen |
1471 |
In the case of molecular liquids, the ideal vapor is chosen as the |
237 |
|
|
target reference state. There are several examples of liquid state |
238 |
|
|
free energy calculations of water models present in the |
239 |
|
|
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
240 |
|
|
typically differ in regard to the path taken for switching off the |
241 |
|
|
interaction potential to convert the system to an ideal gas of water |
242 |
|
|
molecules. In this study, we apply of one of the most convenient |
243 |
|
|
methods and integrate over the $\lambda^4$ path, where all interaction |
244 |
|
|
parameters are scaled equally by this transformation parameter. This |
245 |
|
|
method has been shown to be reversible and provide results in |
246 |
|
|
excellent agreement with other established methods.\cite{Baez95b} |
247 |
|
|
|
248 |
chrisfen |
1456 |
Charge, dipole, and Lennard-Jones interactions were modified by a |
249 |
chrisfen |
1462 |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
250 |
|
|
). By applying this function, these interactions are smoothly |
251 |
gezelter |
1465 |
truncated, thereby avoiding the poor energy conservation which results |
252 |
chrisfen |
1462 |
from harsher truncation schemes. The effect of a long-range correction |
253 |
|
|
was also investigated on select model systems in a variety of |
254 |
|
|
manners. For the SSD/RF model, a reaction field with a fixed |
255 |
|
|
dielectric constant of 80 was applied in all |
256 |
|
|
simulations.\cite{Onsager36} For a series of the least computationally |
257 |
|
|
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
258 |
|
|
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
259 |
|
|
\AA\ cutoff results. Finally, results from the use of an Ewald |
260 |
|
|
summation were estimated for TIP3P and SPC/E by performing |
261 |
chrisfen |
1456 |
calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
262 |
gezelter |
1465 |
mechanics software package.\cite{Tinker} The calculated energy |
263 |
chrisfen |
1462 |
difference in the presence and absence of PME was applied to the |
264 |
gezelter |
1465 |
previous results in order to predict changes to the free energy |
265 |
chrisfen |
1462 |
landscape. |
266 |
chrisfen |
1454 |
|
267 |
chrisfen |
1456 |
\section{Results and discussion} |
268 |
chrisfen |
1454 |
|
269 |
chrisfen |
1456 |
The free energy of proton ordered Ice-{\it i} was calculated and |
270 |
|
|
compared with the free energies of proton ordered variants of the |
271 |
|
|
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
272 |
|
|
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
273 |
|
|
and thought to be the minimum free energy structure for the SPC/E |
274 |
|
|
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
275 |
gezelter |
1465 |
Ice XI, the experimentally-observed proton-ordered variant of ice |
276 |
|
|
$I_h$, was investigated initially, but was found to be not as stable |
277 |
|
|
as proton disordered or antiferroelectric variants of ice $I_h$. The |
278 |
|
|
proton ordered variant of ice $I_h$ used here is a simple |
279 |
chrisfen |
1473 |
antiferroelectric version that we divised, and it has an 8 molecule |
280 |
|
|
unit cell similar to other predicted antiferroelectric $I_h$ |
281 |
|
|
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
282 |
|
|
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
283 |
|
|
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger |
284 |
|
|
crystal sizes were necessary for simulations involving larger cutoff |
285 |
|
|
values. |
286 |
chrisfen |
1454 |
|
287 |
chrisfen |
1456 |
\begin{table*} |
288 |
|
|
\begin{minipage}{\linewidth} |
289 |
|
|
\renewcommand{\thefootnote}{\thempfootnote} |
290 |
|
|
\begin{center} |
291 |
|
|
\caption{Calculated free energies for several ice polymorphs with a |
292 |
|
|
variety of common water models. All calculations used a cutoff radius |
293 |
|
|
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
294 |
chrisfen |
1466 |
kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
295 |
|
|
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
296 |
chrisfen |
1456 |
\begin{tabular}{ l c c c c } |
297 |
gezelter |
1463 |
\hline |
298 |
chrisfen |
1466 |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
299 |
gezelter |
1463 |
\hline |
300 |
chrisfen |
1473 |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
301 |
|
|
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
302 |
|
|
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
303 |
|
|
SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\ |
304 |
|
|
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
305 |
|
|
SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\ |
306 |
chrisfen |
1456 |
\end{tabular} |
307 |
|
|
\label{freeEnergy} |
308 |
|
|
\end{center} |
309 |
|
|
\end{minipage} |
310 |
|
|
\end{table*} |
311 |
chrisfen |
1453 |
|
312 |
chrisfen |
1456 |
The free energy values computed for the studied polymorphs indicate |
313 |
|
|
that Ice-{\it i} is the most stable state for all of the common water |
314 |
|
|
models studied. With the free energy at these state points, the |
315 |
gezelter |
1465 |
Gibbs-Helmholtz equation was used to project to other state points and |
316 |
|
|
to build phase diagrams. Figures |
317 |
chrisfen |
1456 |
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
318 |
|
|
from the free energy results. All other models have similar structure, |
319 |
gezelter |
1465 |
although the crossing points between the phases exist at slightly |
320 |
|
|
different temperatures and pressures. It is interesting to note that |
321 |
|
|
ice $I$ does not exist in either cubic or hexagonal form in any of the |
322 |
|
|
phase diagrams for any of the models. For purposes of this study, ice |
323 |
|
|
B is representative of the dense ice polymorphs. A recent study by |
324 |
|
|
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
325 |
|
|
TIP4P in the high pressure regime.\cite{Sanz04} |
326 |
gezelter |
1463 |
|
327 |
chrisfen |
1456 |
\begin{figure} |
328 |
|
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
329 |
|
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
330 |
|
|
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
331 |
|
|
the experimental values; however, the solid phases shown are not the |
332 |
|
|
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
333 |
|
|
higher in energy and don't appear in the phase diagram.} |
334 |
|
|
\label{tp3phasedia} |
335 |
|
|
\end{figure} |
336 |
gezelter |
1463 |
|
337 |
chrisfen |
1456 |
\begin{figure} |
338 |
|
|
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
339 |
|
|
\caption{Phase diagram for the SSD/RF water model in the low pressure |
340 |
|
|
regime. Calculations producing these results were done under an |
341 |
|
|
applied reaction field. It is interesting to note that this |
342 |
|
|
computationally efficient model (over 3 times more efficient than |
343 |
|
|
TIP3P) exhibits phase behavior similar to the less computationally |
344 |
|
|
conservative charge based models.} |
345 |
|
|
\label{ssdrfphasedia} |
346 |
|
|
\end{figure} |
347 |
|
|
|
348 |
|
|
\begin{table*} |
349 |
|
|
\begin{minipage}{\linewidth} |
350 |
|
|
\renewcommand{\thefootnote}{\thempfootnote} |
351 |
|
|
\begin{center} |
352 |
|
|
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
353 |
chrisfen |
1466 |
temperatures at 1 atm for several common water models compared with |
354 |
|
|
experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
355 |
|
|
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
356 |
|
|
liquid or gas state.} |
357 |
chrisfen |
1456 |
\begin{tabular}{ l c c c c c c c } |
358 |
gezelter |
1463 |
\hline |
359 |
chrisfen |
1466 |
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
360 |
gezelter |
1463 |
\hline |
361 |
chrisfen |
1473 |
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
362 |
|
|
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
363 |
|
|
$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ |
364 |
chrisfen |
1456 |
\end{tabular} |
365 |
|
|
\label{meltandboil} |
366 |
|
|
\end{center} |
367 |
|
|
\end{minipage} |
368 |
|
|
\end{table*} |
369 |
|
|
|
370 |
|
|
Table \ref{meltandboil} lists the melting and boiling temperatures |
371 |
|
|
calculated from this work. Surprisingly, most of these models have |
372 |
|
|
melting points that compare quite favorably with experiment. The |
373 |
|
|
unfortunate aspect of this result is that this phase change occurs |
374 |
|
|
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
375 |
|
|
liquid state. These results are actually not contrary to previous |
376 |
|
|
studies in the literature. Earlier free energy studies of ice $I$ |
377 |
|
|
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
378 |
|
|
being attributed to choice of interaction truncation and different |
379 |
chrisfen |
1466 |
ordered and disordered molecular |
380 |
|
|
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
381 |
|
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
382 |
chrisfen |
1456 |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
383 |
|
|
calculated at 265 K, significantly higher in temperature than the |
384 |
|
|
previous studies. Also of interest in these results is that SSD/E does |
385 |
|
|
not exhibit a melting point at 1 atm, but it shows a sublimation point |
386 |
|
|
at 355 K. This is due to the significant stability of Ice-{\it i} over |
387 |
|
|
all other polymorphs for this particular model under these |
388 |
|
|
conditions. While troubling, this behavior turned out to be |
389 |
chrisfen |
1459 |
advantageous in that it facilitated the spontaneous crystallization of |
390 |
chrisfen |
1456 |
Ice-{\it i}. These observations provide a warning that simulations of |
391 |
|
|
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
392 |
|
|
risk of spontaneous crystallization. However, this risk changes when |
393 |
|
|
applying a longer cutoff. |
394 |
|
|
|
395 |
chrisfen |
1458 |
\begin{figure} |
396 |
|
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
397 |
|
|
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
398 |
|
|
TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
399 |
|
|
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
400 |
chrisfen |
1466 |
\AA . These crystals are unstable at 200 K and rapidly convert into |
401 |
|
|
liquids. The connecting lines are qualitative visual aid.} |
402 |
chrisfen |
1458 |
\label{incCutoff} |
403 |
|
|
\end{figure} |
404 |
|
|
|
405 |
chrisfen |
1457 |
Increasing the cutoff radius in simulations of the more |
406 |
|
|
computationally efficient water models was done in order to evaluate |
407 |
|
|
the trend in free energy values when moving to systems that do not |
408 |
|
|
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
409 |
|
|
free energy of all the ice polymorphs show a substantial dependence on |
410 |
|
|
cutoff radius. In general, there is a narrowing of the free energy |
411 |
chrisfen |
1459 |
differences while moving to greater cutoff radius. Interestingly, by |
412 |
|
|
increasing the cutoff radius, the free energy gap was narrowed enough |
413 |
|
|
in the SSD/E model that the liquid state is preferred under standard |
414 |
|
|
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
415 |
|
|
simulations using this model choose interaction truncation radii |
416 |
gezelter |
1469 |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
417 |
chrisfen |
1459 |
case of SSD/RF, indicating that the free energies calculated with a |
418 |
|
|
reaction field present provide a more accurate picture of the free |
419 |
|
|
energy landscape in the absence of potential truncation. |
420 |
chrisfen |
1456 |
|
421 |
chrisfen |
1457 |
To further study the changes resulting to the inclusion of a |
422 |
|
|
long-range interaction correction, the effect of an Ewald summation |
423 |
|
|
was estimated by applying the potential energy difference do to its |
424 |
|
|
inclusion in systems in the presence and absence of the |
425 |
|
|
correction. This was accomplished by calculation of the potential |
426 |
|
|
energy of identical crystals with and without PME using TINKER. The |
427 |
|
|
free energies for the investigated polymorphs using the TIP3P and |
428 |
chrisfen |
1471 |
SPC/E water models are shown in Table \ref{pmeShift}. The same trend |
429 |
|
|
pointed out through increase of cutoff radius is observed in these PME |
430 |
|
|
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
431 |
|
|
for both the TIP3P and SPC/E water models; however, the narrowing of |
432 |
|
|
the free energy differences between the various solid forms is |
433 |
|
|
significant enough that it becomes less clear that it is the most |
434 |
chrisfen |
1474 |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
435 |
|
|
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
436 |
|
|
as well, indicating that Ice-{\it i} might be metastable with respect |
437 |
|
|
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
438 |
|
|
not significantly alter the finding that the Ice-{\it i} polymorph is |
439 |
|
|
a stable crystal structure that should be considered when studying the |
440 |
|
|
phase behavior of water models. |
441 |
chrisfen |
1456 |
|
442 |
chrisfen |
1457 |
\begin{table*} |
443 |
|
|
\begin{minipage}{\linewidth} |
444 |
|
|
\renewcommand{\thefootnote}{\thempfootnote} |
445 |
|
|
\begin{center} |
446 |
chrisfen |
1458 |
\caption{The free energy of the studied ice polymorphs after applying |
447 |
|
|
the energy difference attributed to the inclusion of the PME |
448 |
|
|
long-range interaction correction. Units are kcal/mol.} |
449 |
chrisfen |
1457 |
\begin{tabular}{ l c c c c } |
450 |
gezelter |
1463 |
\hline |
451 |
chrisfen |
1457 |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
452 |
gezelter |
1463 |
\hline |
453 |
chrisfen |
1473 |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\ |
454 |
|
|
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\ |
455 |
chrisfen |
1457 |
\end{tabular} |
456 |
|
|
\label{pmeShift} |
457 |
|
|
\end{center} |
458 |
|
|
\end{minipage} |
459 |
|
|
\end{table*} |
460 |
|
|
|
461 |
chrisfen |
1453 |
\section{Conclusions} |
462 |
|
|
|
463 |
chrisfen |
1458 |
The free energy for proton ordered variants of hexagonal and cubic ice |
464 |
gezelter |
1465 |
$I$, ice B, and recently discovered Ice-{\it i} were calculated under |
465 |
chrisfen |
1458 |
standard conditions for several common water models via thermodynamic |
466 |
|
|
integration. All the water models studied show Ice-{\it i} to be the |
467 |
|
|
minimum free energy crystal structure in the with a 9 \AA\ switching |
468 |
|
|
function cutoff. Calculated melting and boiling points show |
469 |
|
|
surprisingly good agreement with the experimental values; however, the |
470 |
|
|
solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
471 |
|
|
interaction truncation was investigated through variation of the |
472 |
|
|
cutoff radius, use of a reaction field parameterized model, and |
473 |
gezelter |
1465 |
estimation of the results in the presence of the Ewald |
474 |
|
|
summation. Interaction truncation has a significant effect on the |
475 |
chrisfen |
1459 |
computed free energy values, and may significantly alter the free |
476 |
|
|
energy landscape for the more complex multipoint water models. Despite |
477 |
|
|
these effects, these results show Ice-{\it i} to be an important ice |
478 |
|
|
polymorph that should be considered in simulation studies. |
479 |
chrisfen |
1458 |
|
480 |
chrisfen |
1459 |
Due to this relative stability of Ice-{\it i} in all manner of |
481 |
|
|
investigated simulation examples, the question arises as to possible |
482 |
gezelter |
1465 |
experimental observation of this polymorph. The rather extensive past |
483 |
chrisfen |
1459 |
and current experimental investigation of water in the low pressure |
484 |
gezelter |
1465 |
regime makes us hesitant to ascribe any relevance of this work outside |
485 |
|
|
of the simulation community. It is for this reason that we chose a |
486 |
|
|
name for this polymorph which involves an imaginary quantity. That |
487 |
|
|
said, there are certain experimental conditions that would provide the |
488 |
|
|
most ideal situation for possible observation. These include the |
489 |
|
|
negative pressure or stretched solid regime, small clusters in vacuum |
490 |
|
|
deposition environments, and in clathrate structures involving small |
491 |
gezelter |
1469 |
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
492 |
|
|
our predictions for both the pair distribution function ($g_{OO}(r)$) |
493 |
|
|
and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it |
494 |
chrisfen |
1470 |
i} at a temperature of 77K. In a quick comparison of the predicted |
495 |
|
|
S(q) for Ice-{\it i} and experimental studies of amorphous solid |
496 |
|
|
water, it is possible that some of the ``spurious'' peaks that could |
497 |
|
|
not be assigned in HDA could correspond to peaks labeled in this |
498 |
|
|
S(q).\cite{Bizid87} It should be noted that there is typically poor |
499 |
|
|
agreement on crystal densities between simulation and experiment, so |
500 |
|
|
such peak comparisons should be made with caution. We will leave it |
501 |
|
|
to our experimental colleagues to determine whether this ice polymorph |
502 |
|
|
is named appropriately or if it should be promoted to Ice-0. |
503 |
chrisfen |
1459 |
|
504 |
chrisfen |
1467 |
\begin{figure} |
505 |
|
|
\includegraphics[width=\linewidth]{iceGofr.eps} |
506 |
chrisfen |
1470 |
\caption{Radial distribution functions of Ice-{\it i} and ice $I_c$ |
507 |
|
|
calculated from from simulations of the SSD/RF water model at 77 K.} |
508 |
chrisfen |
1467 |
\label{fig:gofr} |
509 |
|
|
\end{figure} |
510 |
|
|
|
511 |
gezelter |
1469 |
\begin{figure} |
512 |
|
|
\includegraphics[width=\linewidth]{sofq.eps} |
513 |
|
|
\caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at |
514 |
|
|
77 K. The raw structure factors have been convoluted with a gaussian |
515 |
chrisfen |
1470 |
instrument function (0.075 \AA$^{-1}$ width) to compensate for the |
516 |
|
|
trunction effects in our finite size simulations. The labeled peaks |
517 |
|
|
compared favorably with ``spurious'' peaks observed in experimental |
518 |
|
|
studies of amorphous solid water.\cite{Bizid87}} |
519 |
gezelter |
1469 |
\label{fig:sofq} |
520 |
|
|
\end{figure} |
521 |
|
|
|
522 |
chrisfen |
1453 |
\section{Acknowledgments} |
523 |
|
|
Support for this project was provided by the National Science |
524 |
|
|
Foundation under grant CHE-0134881. Computation time was provided by |
525 |
chrisfen |
1458 |
the Notre Dame High Performance Computing Cluster and the Notre Dame |
526 |
|
|
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
527 |
chrisfen |
1453 |
|
528 |
|
|
\newpage |
529 |
|
|
|
530 |
|
|
\bibliographystyle{jcp} |
531 |
|
|
\bibliography{iceiPaper} |
532 |
|
|
|
533 |
|
|
|
534 |
|
|
\end{document} |