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 |
gezelter |
1475 |
The absolute free energies of several ice polymorphs which are stable |
37 |
|
|
at low pressures were calculated using thermodynamic integration to a |
38 |
|
|
reference system (the Einstein crystal). These integrations were |
39 |
|
|
performed for most of the common water models (SPC/E, TIP3P, TIP4P, |
40 |
|
|
TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed |
41 |
|
|
crystallizing at room temperature for one of the single-point water |
42 |
|
|
models, was determined to be the stable crystalline state (at 1 atm) |
43 |
|
|
for {\it all} the water models investigated. Phase diagrams were |
44 |
gezelter |
1463 |
generated, and phase coexistence lines were determined for all of the |
45 |
gezelter |
1475 |
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. |
51 |
chrisfen |
1453 |
\end{abstract} |
52 |
|
|
|
53 |
|
|
%\narrowtext |
54 |
|
|
|
55 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
56 |
|
|
% BODY OF TEXT |
57 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
58 |
|
|
|
59 |
|
|
\section{Introduction} |
60 |
|
|
|
61 |
chrisfen |
1459 |
Water has proven to be a challenging substance to depict in |
62 |
gezelter |
1463 |
simulations, and a variety of models have been developed to describe |
63 |
|
|
its behavior under varying simulation |
64 |
gezelter |
1477 |
conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04} |
65 |
gezelter |
1463 |
These models have been used to investigate important physical |
66 |
gezelter |
1475 |
phenomena like phase transitions, transport properties, and the |
67 |
chrisfen |
1471 |
hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
68 |
|
|
choice of models available, it is only natural to compare the models |
69 |
|
|
under interesting thermodynamic conditions in an attempt to clarify |
70 |
|
|
the limitations of each of the |
71 |
|
|
models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
72 |
gezelter |
1475 |
important properties to quantify are the Gibbs and Helmholtz free |
73 |
chrisfen |
1471 |
energies, particularly for the solid forms of water. Difficulty in |
74 |
|
|
these types of studies typically arises from the assortment of |
75 |
|
|
possible crystalline polymorphs that water adopts over a wide range of |
76 |
|
|
pressures and temperatures. There are currently 13 recognized forms |
77 |
|
|
of ice, and it is a challenging task to investigate the entire free |
78 |
|
|
energy landscape.\cite{Sanz04} Ideally, research is focused on the |
79 |
|
|
phases having the lowest free energy at a given state point, because |
80 |
gezelter |
1475 |
these phases will dictate the relevant transition temperatures and |
81 |
chrisfen |
1471 |
pressures for the model. |
82 |
chrisfen |
1459 |
|
83 |
gezelter |
1465 |
In this paper, standard reference state methods were applied to known |
84 |
chrisfen |
1471 |
crystalline water polymorphs in the low pressure regime. This work is |
85 |
gezelter |
1475 |
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 |
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 |
95 |
|
|
accomplished by orienting two of the molecules so that both of their |
96 |
|
|
donated hydrogen bonds are internal to their tetramer |
97 |
|
|
(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
98 |
|
|
water tetramers, the hydrogen bonds are not as linear as those |
99 |
|
|
observed in ice $I_h$, however the interlocking of these subunits |
100 |
|
|
appears to provide significant stabilization to the overall |
101 |
chrisfen |
1459 |
crystal. The arrangement of these tetramers results in surrounding |
102 |
|
|
open octagonal cavities that are typically greater than 6.3 \AA\ in |
103 |
|
|
diameter. This relatively open overall structure leads to crystals |
104 |
|
|
that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. |
105 |
gezelter |
1463 |
|
106 |
chrisfen |
1460 |
\begin{figure} |
107 |
gezelter |
1463 |
\includegraphics[width=\linewidth]{unitCell.eps} |
108 |
gezelter |
1465 |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
109 |
chrisfen |
1471 |
elongated variant of Ice-{\it i}. The spheres represent the |
110 |
|
|
center-of-mass locations of the water molecules. The $a$ to $c$ |
111 |
|
|
ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
112 |
|
|
$a:2.1214c$ and $a:1.7850c$ respectively.} |
113 |
chrisfen |
1460 |
\label{iceiCell} |
114 |
|
|
\end{figure} |
115 |
gezelter |
1463 |
|
116 |
chrisfen |
1460 |
\begin{figure} |
117 |
gezelter |
1463 |
\includegraphics[width=\linewidth]{orderedIcei.eps} |
118 |
chrisfen |
1460 |
\caption{Image of a proton ordered crystal of Ice-{\it i} looking |
119 |
|
|
down the (001) crystal face. The rows of water tetramers surrounded by |
120 |
|
|
octagonal pores leads to a crystal structure that is significantly |
121 |
|
|
less dense than ice $I_h$.} |
122 |
|
|
\label{protOrder} |
123 |
|
|
\end{figure} |
124 |
chrisfen |
1459 |
|
125 |
gezelter |
1465 |
Results from our previous study indicated that Ice-{\it i} is the |
126 |
|
|
minimum energy crystal structure for the single point water models we |
127 |
gezelter |
1475 |
had investigated (for discussions on these single point dipole models, |
128 |
|
|
see our previous work and related |
129 |
chrisfen |
1471 |
articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
130 |
gezelter |
1465 |
considered energetic stabilization and neglected entropic |
131 |
gezelter |
1475 |
contributions to the overall free energy. To address this issue, we |
132 |
|
|
have calculated the absolute free energy of this crystal using |
133 |
gezelter |
1465 |
thermodynamic integration and compared to the free energies of cubic |
134 |
|
|
and hexagonal ice $I$ (the experimental low density ice polymorphs) |
135 |
|
|
and ice B (a higher density, but very stable crystal structure |
136 |
|
|
observed by B\`{a}ez and Clancy in free energy studies of |
137 |
|
|
SPC/E).\cite{Baez95b} This work includes results for the water model |
138 |
|
|
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
139 |
|
|
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
140 |
|
|
field parametrized single point dipole water model (SSD/RF). It should |
141 |
gezelter |
1475 |
be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was |
142 |
|
|
used in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell |
143 |
|
|
of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} |
144 |
|
|
unit it is extended in the direction of the (001) face and compressed |
145 |
chrisfen |
1482 |
along the other two faces. There is typically a small unit cell |
146 |
|
|
distortion of Ice-{\it i}$^\prime$ that converts the normally square |
147 |
|
|
tetramer into a rhombus with alternating 85 and 95 degree angles. The |
148 |
|
|
degree of this distortion is model dependent and significant enough to |
149 |
|
|
split the tetramer diagonal location peak in the radial distibution |
150 |
|
|
function. |
151 |
chrisfen |
1459 |
|
152 |
chrisfen |
1453 |
\section{Methods} |
153 |
|
|
|
154 |
chrisfen |
1454 |
Canonical ensemble (NVT) molecular dynamics calculations were |
155 |
gezelter |
1465 |
performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
156 |
|
|
All molecules were treated as rigid bodies, with orientational motion |
157 |
|
|
propagated using the symplectic DLM integration method. Details about |
158 |
chrisfen |
1471 |
the implementation of this technique can be found in a recent |
159 |
gezelter |
1468 |
publication.\cite{Dullweber1997} |
160 |
chrisfen |
1454 |
|
161 |
chrisfen |
1471 |
Thermodynamic integration is an established technique for |
162 |
|
|
determination of free energies of condensed phases of |
163 |
|
|
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
164 |
|
|
method, implemented in the same manner illustrated by B\`{a}ez and |
165 |
|
|
Clancy, was utilized to calculate the free energy of several ice |
166 |
|
|
crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
167 |
|
|
SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
168 |
|
|
and 400 K for all of these water models were also determined using |
169 |
gezelter |
1475 |
this same technique in order to determine melting points and to |
170 |
|
|
generate phase diagrams. All simulations were carried out at densities |
171 |
|
|
which correspond to a pressure of approximately 1 atm at their |
172 |
|
|
respective temperatures. |
173 |
chrisfen |
1454 |
|
174 |
gezelter |
1475 |
Thermodynamic integration involves a sequence of simulations during |
175 |
|
|
which the system of interest is converted into a reference system for |
176 |
|
|
which the free energy is known analytically. This transformation path |
177 |
|
|
is then integrated in order to determine the free energy difference |
178 |
|
|
between the two states: |
179 |
chrisfen |
1458 |
\begin{equation} |
180 |
|
|
\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
181 |
|
|
)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
182 |
|
|
\end{equation} |
183 |
|
|
where $V$ is the interaction potential and $\lambda$ is the |
184 |
chrisfen |
1459 |
transformation parameter that scales the overall |
185 |
chrisfen |
1471 |
potential. Simulations are distributed strategically along this path |
186 |
|
|
in order to sufficiently sample the regions of greatest change in the |
187 |
chrisfen |
1459 |
potential. Typical integrations in this study consisted of $\sim$25 |
188 |
|
|
simulations ranging from 300 ps (for the unaltered system) to 75 ps |
189 |
|
|
(near the reference state) in length. |
190 |
chrisfen |
1458 |
|
191 |
chrisfen |
1454 |
For the thermodynamic integration of molecular crystals, the Einstein |
192 |
chrisfen |
1471 |
crystal was chosen as the reference system. In an Einstein crystal, |
193 |
|
|
the molecules are restrained at their ideal lattice locations and |
194 |
|
|
orientations. Using harmonic restraints, as applied by B\`{a}ez and |
195 |
|
|
Clancy, the total potential for this reference crystal |
196 |
|
|
($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
197 |
|
|
\begin{equation} |
198 |
|
|
V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
199 |
|
|
\frac{K_\omega\omega^2}{2}, |
200 |
|
|
\end{equation} |
201 |
|
|
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
202 |
|
|
the spring constants restraining translational motion and deflection |
203 |
|
|
of and rotation around the principle axis of the molecule |
204 |
|
|
respectively. It is clear from Fig. \ref{waterSpring} that the values |
205 |
|
|
of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
206 |
|
|
$-\pi$ to $\pi$. The partition function for a molecular crystal |
207 |
gezelter |
1465 |
restrained in this fashion can be evaluated analytically, and the |
208 |
|
|
Helmholtz Free Energy ({\it A}) is given by |
209 |
chrisfen |
1454 |
\begin{eqnarray} |
210 |
|
|
A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
211 |
|
|
[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
212 |
|
|
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right |
213 |
|
|
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right |
214 |
|
|
)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi |
215 |
|
|
K_\omega K_\theta)^{\frac{1}{2}}}\exp\left |
216 |
|
|
(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right |
217 |
|
|
)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
218 |
|
|
\label{ecFreeEnergy} |
219 |
|
|
\end{eqnarray} |
220 |
chrisfen |
1471 |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
221 |
|
|
potential energy of the ideal crystal.\cite{Baez95a} |
222 |
gezelter |
1463 |
|
223 |
chrisfen |
1456 |
\begin{figure} |
224 |
gezelter |
1463 |
\includegraphics[width=\linewidth]{rotSpring.eps} |
225 |
chrisfen |
1456 |
\caption{Possible orientational motions for a restrained molecule. |
226 |
|
|
$\theta$ angles correspond to displacement from the body-frame {\it |
227 |
|
|
z}-axis, while $\omega$ angles correspond to rotation about the |
228 |
|
|
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
229 |
|
|
constants for the harmonic springs restraining motion in the $\theta$ |
230 |
|
|
and $\omega$ directions.} |
231 |
|
|
\label{waterSpring} |
232 |
|
|
\end{figure} |
233 |
chrisfen |
1454 |
|
234 |
chrisfen |
1471 |
In the case of molecular liquids, the ideal vapor is chosen as the |
235 |
|
|
target reference state. There are several examples of liquid state |
236 |
|
|
free energy calculations of water models present in the |
237 |
|
|
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
238 |
|
|
typically differ in regard to the path taken for switching off the |
239 |
|
|
interaction potential to convert the system to an ideal gas of water |
240 |
gezelter |
1475 |
molecules. In this study, we applied of one of the most convenient |
241 |
|
|
methods and integrated over the $\lambda^4$ path, where all |
242 |
|
|
interaction parameters are scaled equally by this transformation |
243 |
|
|
parameter. This method has been shown to be reversible and provide |
244 |
|
|
results in excellent agreement with other established |
245 |
|
|
methods.\cite{Baez95b} |
246 |
chrisfen |
1471 |
|
247 |
chrisfen |
1456 |
Charge, dipole, and Lennard-Jones interactions were modified by a |
248 |
chrisfen |
1462 |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
249 |
|
|
). By applying this function, these interactions are smoothly |
250 |
gezelter |
1465 |
truncated, thereby avoiding the poor energy conservation which results |
251 |
chrisfen |
1462 |
from harsher truncation schemes. The effect of a long-range correction |
252 |
|
|
was also investigated on select model systems in a variety of |
253 |
|
|
manners. For the SSD/RF model, a reaction field with a fixed |
254 |
|
|
dielectric constant of 80 was applied in all |
255 |
|
|
simulations.\cite{Onsager36} For a series of the least computationally |
256 |
|
|
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
257 |
|
|
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
258 |
gezelter |
1475 |
\AA\ cutoff results. Finally, the effects of utilizing an Ewald |
259 |
|
|
summation were estimated for TIP3P and SPC/E by performing single |
260 |
|
|
configuration calculations with Particle-Mesh Ewald (PME) in the |
261 |
|
|
TINKER molecular mechanics software package.\cite{Tinker} The |
262 |
|
|
calculated energy difference in the presence and absence of PME was |
263 |
|
|
applied to the previous results in order to predict changes to the |
264 |
|
|
free energy landscape. |
265 |
chrisfen |
1454 |
|
266 |
chrisfen |
1456 |
\section{Results and discussion} |
267 |
chrisfen |
1454 |
|
268 |
gezelter |
1475 |
The free energy of proton-ordered Ice-{\it i} was calculated and |
269 |
chrisfen |
1456 |
compared with the free energies of proton ordered variants of the |
270 |
|
|
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
271 |
|
|
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
272 |
|
|
and thought to be the minimum free energy structure for the SPC/E |
273 |
|
|
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
274 |
gezelter |
1465 |
Ice XI, the experimentally-observed proton-ordered variant of ice |
275 |
|
|
$I_h$, was investigated initially, but was found to be not as stable |
276 |
|
|
as proton disordered or antiferroelectric variants of ice $I_h$. The |
277 |
|
|
proton ordered variant of ice $I_h$ used here is a simple |
278 |
gezelter |
1475 |
antiferroelectric version that we devised, and it has an 8 molecule |
279 |
chrisfen |
1473 |
unit cell similar to other predicted antiferroelectric $I_h$ |
280 |
|
|
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
281 |
|
|
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
282 |
|
|
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger |
283 |
|
|
crystal sizes were necessary for simulations involving larger cutoff |
284 |
|
|
values. |
285 |
chrisfen |
1454 |
|
286 |
chrisfen |
1456 |
\begin{table*} |
287 |
|
|
\begin{minipage}{\linewidth} |
288 |
|
|
\renewcommand{\thefootnote}{\thempfootnote} |
289 |
|
|
\begin{center} |
290 |
|
|
\caption{Calculated free energies for several ice polymorphs with a |
291 |
|
|
variety of common water models. All calculations used a cutoff radius |
292 |
|
|
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
293 |
chrisfen |
1466 |
kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
294 |
|
|
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
295 |
chrisfen |
1456 |
\begin{tabular}{ l c c c c } |
296 |
gezelter |
1463 |
\hline |
297 |
chrisfen |
1466 |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
298 |
gezelter |
1463 |
\hline |
299 |
chrisfen |
1473 |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
300 |
|
|
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
301 |
|
|
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
302 |
|
|
SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\ |
303 |
|
|
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
304 |
|
|
SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\ |
305 |
chrisfen |
1456 |
\end{tabular} |
306 |
|
|
\label{freeEnergy} |
307 |
|
|
\end{center} |
308 |
|
|
\end{minipage} |
309 |
|
|
\end{table*} |
310 |
chrisfen |
1453 |
|
311 |
chrisfen |
1456 |
The free energy values computed for the studied polymorphs indicate |
312 |
|
|
that Ice-{\it i} is the most stable state for all of the common water |
313 |
gezelter |
1475 |
models studied. With the calculated free energy at these state points, |
314 |
|
|
the Gibbs-Helmholtz equation was used to project to other state points |
315 |
|
|
and to build phase diagrams. Figures |
316 |
chrisfen |
1456 |
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
317 |
|
|
from the free energy results. All other models have similar structure, |
318 |
gezelter |
1475 |
although the crossing points between the phases move to slightly |
319 |
gezelter |
1465 |
different temperatures and pressures. It is interesting to note that |
320 |
|
|
ice $I$ does not exist in either cubic or hexagonal form in any of the |
321 |
|
|
phase diagrams for any of the models. For purposes of this study, ice |
322 |
|
|
B is representative of the dense ice polymorphs. A recent study by |
323 |
|
|
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
324 |
gezelter |
1475 |
TIP4P at higher pressures than those studied here.\cite{Sanz04} |
325 |
gezelter |
1463 |
|
326 |
chrisfen |
1456 |
\begin{figure} |
327 |
|
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
328 |
|
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
329 |
|
|
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
330 |
|
|
the experimental values; however, the solid phases shown are not the |
331 |
|
|
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
332 |
|
|
higher in energy and don't appear in the phase diagram.} |
333 |
|
|
\label{tp3phasedia} |
334 |
|
|
\end{figure} |
335 |
gezelter |
1463 |
|
336 |
chrisfen |
1456 |
\begin{figure} |
337 |
|
|
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
338 |
|
|
\caption{Phase diagram for the SSD/RF water model in the low pressure |
339 |
|
|
regime. Calculations producing these results were done under an |
340 |
|
|
applied reaction field. It is interesting to note that this |
341 |
|
|
computationally efficient model (over 3 times more efficient than |
342 |
|
|
TIP3P) exhibits phase behavior similar to the less computationally |
343 |
|
|
conservative charge based models.} |
344 |
|
|
\label{ssdrfphasedia} |
345 |
|
|
\end{figure} |
346 |
|
|
|
347 |
|
|
\begin{table*} |
348 |
|
|
\begin{minipage}{\linewidth} |
349 |
|
|
\renewcommand{\thefootnote}{\thempfootnote} |
350 |
|
|
\begin{center} |
351 |
|
|
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
352 |
chrisfen |
1466 |
temperatures at 1 atm for several common water models compared with |
353 |
|
|
experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
354 |
|
|
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
355 |
|
|
liquid or gas state.} |
356 |
chrisfen |
1456 |
\begin{tabular}{ l c c c c c c c } |
357 |
gezelter |
1463 |
\hline |
358 |
chrisfen |
1466 |
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
359 |
gezelter |
1463 |
\hline |
360 |
chrisfen |
1473 |
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
361 |
|
|
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
362 |
|
|
$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ |
363 |
chrisfen |
1456 |
\end{tabular} |
364 |
|
|
\label{meltandboil} |
365 |
|
|
\end{center} |
366 |
|
|
\end{minipage} |
367 |
|
|
\end{table*} |
368 |
|
|
|
369 |
|
|
Table \ref{meltandboil} lists the melting and boiling temperatures |
370 |
|
|
calculated from this work. Surprisingly, most of these models have |
371 |
|
|
melting points that compare quite favorably with experiment. The |
372 |
|
|
unfortunate aspect of this result is that this phase change occurs |
373 |
|
|
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
374 |
|
|
liquid state. These results are actually not contrary to previous |
375 |
|
|
studies in the literature. Earlier free energy studies of ice $I$ |
376 |
|
|
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
377 |
|
|
being attributed to choice of interaction truncation and different |
378 |
chrisfen |
1466 |
ordered and disordered molecular |
379 |
|
|
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
380 |
|
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
381 |
chrisfen |
1456 |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
382 |
|
|
calculated at 265 K, significantly higher in temperature than the |
383 |
|
|
previous studies. Also of interest in these results is that SSD/E does |
384 |
|
|
not exhibit a melting point at 1 atm, but it shows a sublimation point |
385 |
|
|
at 355 K. This is due to the significant stability of Ice-{\it i} over |
386 |
|
|
all other polymorphs for this particular model under these |
387 |
gezelter |
1475 |
conditions. While troubling, this behavior resulted in spontaneous |
388 |
|
|
crystallization of Ice-{\it i} and led us to investigate this |
389 |
|
|
structure. These observations provide a warning that simulations of |
390 |
chrisfen |
1456 |
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
391 |
gezelter |
1475 |
risk of spontaneous crystallization. However, this risk lessens when |
392 |
chrisfen |
1456 |
applying a longer cutoff. |
393 |
|
|
|
394 |
chrisfen |
1458 |
\begin{figure} |
395 |
|
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
396 |
|
|
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
397 |
|
|
TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
398 |
|
|
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
399 |
chrisfen |
1466 |
\AA . These crystals are unstable at 200 K and rapidly convert into |
400 |
|
|
liquids. The connecting lines are qualitative visual aid.} |
401 |
chrisfen |
1458 |
\label{incCutoff} |
402 |
|
|
\end{figure} |
403 |
|
|
|
404 |
chrisfen |
1457 |
Increasing the cutoff radius in simulations of the more |
405 |
|
|
computationally efficient water models was done in order to evaluate |
406 |
|
|
the trend in free energy values when moving to systems that do not |
407 |
|
|
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
408 |
|
|
free energy of all the ice polymorphs show a substantial dependence on |
409 |
|
|
cutoff radius. In general, there is a narrowing of the free energy |
410 |
chrisfen |
1459 |
differences while moving to greater cutoff radius. Interestingly, by |
411 |
|
|
increasing the cutoff radius, the free energy gap was narrowed enough |
412 |
|
|
in the SSD/E model that the liquid state is preferred under standard |
413 |
|
|
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
414 |
|
|
simulations using this model choose interaction truncation radii |
415 |
gezelter |
1469 |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
416 |
chrisfen |
1459 |
case of SSD/RF, indicating that the free energies calculated with a |
417 |
|
|
reaction field present provide a more accurate picture of the free |
418 |
|
|
energy landscape in the absence of potential truncation. |
419 |
chrisfen |
1456 |
|
420 |
chrisfen |
1457 |
To further study the changes resulting to the inclusion of a |
421 |
|
|
long-range interaction correction, the effect of an Ewald summation |
422 |
|
|
was estimated by applying the potential energy difference do to its |
423 |
|
|
inclusion in systems in the presence and absence of the |
424 |
|
|
correction. This was accomplished by calculation of the potential |
425 |
gezelter |
1475 |
energy of identical crystals both with and without PME. The free |
426 |
|
|
energies for the investigated polymorphs using the TIP3P and SPC/E |
427 |
|
|
water models are shown in Table \ref{pmeShift}. The same trend pointed |
428 |
|
|
out through increase of cutoff radius is observed in these PME |
429 |
chrisfen |
1471 |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
430 |
|
|
for both the TIP3P and SPC/E water models; however, the narrowing of |
431 |
|
|
the free energy differences between the various solid forms is |
432 |
|
|
significant enough that it becomes less clear that it is the most |
433 |
chrisfen |
1474 |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
434 |
|
|
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
435 |
|
|
as well, indicating that Ice-{\it i} might be metastable with respect |
436 |
|
|
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
437 |
|
|
not significantly alter the finding that the Ice-{\it i} polymorph is |
438 |
|
|
a stable crystal structure that should be considered when studying the |
439 |
|
|
phase behavior of water models. |
440 |
chrisfen |
1456 |
|
441 |
chrisfen |
1457 |
\begin{table*} |
442 |
|
|
\begin{minipage}{\linewidth} |
443 |
|
|
\renewcommand{\thefootnote}{\thempfootnote} |
444 |
|
|
\begin{center} |
445 |
chrisfen |
1458 |
\caption{The free energy of the studied ice polymorphs after applying |
446 |
|
|
the energy difference attributed to the inclusion of the PME |
447 |
|
|
long-range interaction correction. Units are kcal/mol.} |
448 |
chrisfen |
1457 |
\begin{tabular}{ l c c c c } |
449 |
gezelter |
1463 |
\hline |
450 |
chrisfen |
1457 |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
451 |
gezelter |
1463 |
\hline |
452 |
chrisfen |
1473 |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\ |
453 |
|
|
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\ |
454 |
chrisfen |
1457 |
\end{tabular} |
455 |
|
|
\label{pmeShift} |
456 |
|
|
\end{center} |
457 |
|
|
\end{minipage} |
458 |
|
|
\end{table*} |
459 |
|
|
|
460 |
chrisfen |
1453 |
\section{Conclusions} |
461 |
|
|
|
462 |
chrisfen |
1458 |
The free energy for proton ordered variants of hexagonal and cubic ice |
463 |
gezelter |
1475 |
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
464 |
|
|
calculated under standard conditions for several common water models |
465 |
|
|
via thermodynamic integration. All the water models studied show |
466 |
|
|
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
467 |
|
|
\AA\ switching function cutoff. Calculated melting and boiling points |
468 |
|
|
show surprisingly good agreement with the experimental values; |
469 |
|
|
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
470 |
|
|
effect of interaction truncation was investigated through variation of |
471 |
|
|
the cutoff radius, use of a reaction field parameterized model, and |
472 |
gezelter |
1465 |
estimation of the results in the presence of the Ewald |
473 |
|
|
summation. Interaction truncation has a significant effect on the |
474 |
chrisfen |
1459 |
computed free energy values, and may significantly alter the free |
475 |
|
|
energy landscape for the more complex multipoint water models. Despite |
476 |
|
|
these effects, these results show Ice-{\it i} to be an important ice |
477 |
|
|
polymorph that should be considered in simulation studies. |
478 |
chrisfen |
1458 |
|
479 |
gezelter |
1475 |
Due to this relative stability of Ice-{\it i} in all of the |
480 |
|
|
investigated simulation conditions, the question arises as to possible |
481 |
gezelter |
1465 |
experimental observation of this polymorph. The rather extensive past |
482 |
chrisfen |
1459 |
and current experimental investigation of water in the low pressure |
483 |
gezelter |
1465 |
regime makes us hesitant to ascribe any relevance of this work outside |
484 |
|
|
of the simulation community. It is for this reason that we chose a |
485 |
|
|
name for this polymorph which involves an imaginary quantity. That |
486 |
|
|
said, there are certain experimental conditions that would provide the |
487 |
|
|
most ideal situation for possible observation. These include the |
488 |
|
|
negative pressure or stretched solid regime, small clusters in vacuum |
489 |
|
|
deposition environments, and in clathrate structures involving small |
490 |
gezelter |
1469 |
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
491 |
|
|
our predictions for both the pair distribution function ($g_{OO}(r)$) |
492 |
chrisfen |
1479 |
and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for |
493 |
|
|
ice-{\it i} at a temperature of 77K. In studies of the high and low |
494 |
|
|
density forms of amorphous ice, ``spurious'' diffraction peaks have |
495 |
|
|
been observed experimentally.\cite{Bizid87} It is possible that a |
496 |
|
|
variant of Ice-{\it i} could explain some of this behavior; however, |
497 |
|
|
we will leave it to our experimental colleagues to make the final |
498 |
|
|
determination on whether this ice polymorph is named appropriately |
499 |
|
|
(i.e. with an imaginary number) or if it can be promoted to Ice-0. |
500 |
chrisfen |
1459 |
|
501 |
chrisfen |
1467 |
\begin{figure} |
502 |
|
|
\includegraphics[width=\linewidth]{iceGofr.eps} |
503 |
chrisfen |
1479 |
\caption{Radial distribution functions of ice $I_h$, $I_c$, |
504 |
|
|
Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations |
505 |
|
|
of the SSD/RF water model at 77 K.} |
506 |
chrisfen |
1467 |
\label{fig:gofr} |
507 |
|
|
\end{figure} |
508 |
|
|
|
509 |
gezelter |
1469 |
\begin{figure} |
510 |
|
|
\includegraphics[width=\linewidth]{sofq.eps} |
511 |
chrisfen |
1479 |
\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, |
512 |
|
|
and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
513 |
|
|
been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
514 |
|
|
width) to compensate for the trunction effects in our finite size |
515 |
|
|
simulations.} |
516 |
gezelter |
1469 |
\label{fig:sofq} |
517 |
|
|
\end{figure} |
518 |
|
|
|
519 |
chrisfen |
1453 |
\section{Acknowledgments} |
520 |
|
|
Support for this project was provided by the National Science |
521 |
|
|
Foundation under grant CHE-0134881. Computation time was provided by |
522 |
chrisfen |
1458 |
the Notre Dame High Performance Computing Cluster and the Notre Dame |
523 |
|
|
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
524 |
chrisfen |
1453 |
|
525 |
|
|
\newpage |
526 |
|
|
|
527 |
|
|
\bibliographystyle{jcp} |
528 |
|
|
\bibliography{iceiPaper} |
529 |
|
|
|
530 |
|
|
|
531 |
|
|
\end{document} |