1 |
chrisfen |
1845 |
%\documentclass[prb,aps,twocolumn,tabularx]{revtex4} |
2 |
|
|
\documentclass[11pt]{article} |
3 |
|
|
\usepackage{endfloat} |
4 |
|
|
\usepackage{amsmath} |
5 |
|
|
\usepackage{epsf} |
6 |
|
|
\usepackage{berkeley} |
7 |
|
|
\usepackage{setspace} |
8 |
|
|
\usepackage{tabularx} |
9 |
|
|
\usepackage{graphicx} |
10 |
|
|
\usepackage[ref]{overcite} |
11 |
|
|
\pagestyle{plain} |
12 |
|
|
\pagenumbering{arabic} |
13 |
|
|
\oddsidemargin 0.0cm \evensidemargin 0.0cm |
14 |
|
|
\topmargin -21pt \headsep 10pt |
15 |
|
|
\textheight 9.0in \textwidth 6.5in |
16 |
|
|
\brokenpenalty=10000 |
17 |
|
|
\renewcommand{\baselinestretch}{1.2} |
18 |
|
|
\renewcommand\citemid{\ } % no comma in optional reference note |
19 |
|
|
|
20 |
|
|
\begin{document} |
21 |
|
|
|
22 |
|
|
\title{Free Energy Analysis of Simulated Ice Polymorphs} |
23 |
|
|
|
24 |
|
|
\author{Christopher J. Fennell and J. Daniel Gezelter \\ |
25 |
|
|
Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
26 |
|
|
Notre Dame, Indiana 46556} |
27 |
|
|
|
28 |
|
|
\date{\today} |
29 |
|
|
|
30 |
|
|
\maketitle |
31 |
|
|
%\doublespacing |
32 |
|
|
|
33 |
|
|
\begin{abstract} |
34 |
|
|
The absolute free energies of several ice polymorphs which are stable |
35 |
|
|
at low pressures were calculated using thermodynamic integration with |
36 |
|
|
a variety of common water models. A recently discovered ice polymorph |
37 |
|
|
that has yet only been observed in computer simulations (Ice-{\it i}), |
38 |
|
|
was determined to be the stable crystalline state for {\it all} the |
39 |
|
|
water models investigated. Phase diagrams were generated, and phase |
40 |
|
|
coexistence lines were determined for all of the known low-pressure |
41 |
|
|
ice structures. Additionally, potential truncation was show to play a |
42 |
|
|
role in the resulting shape of the free energy landscape. |
43 |
|
|
\end{abstract} |
44 |
|
|
|
45 |
|
|
%\narrowtext |
46 |
|
|
|
47 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
48 |
|
|
% BODY OF TEXT |
49 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
50 |
|
|
|
51 |
|
|
\section{Introduction} |
52 |
|
|
|
53 |
|
|
Water has proven to be a challenging substance to depict in |
54 |
|
|
simulations, and a variety of models have been developed to describe |
55 |
|
|
its behavior under varying simulation |
56 |
|
|
conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04} |
57 |
|
|
These models have been used to investigate important physical |
58 |
|
|
phenomena like phase transitions, transport properties, and the |
59 |
|
|
hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
60 |
|
|
choice of models available, it is only natural to compare the models |
61 |
|
|
under interesting thermodynamic conditions in an attempt to clarify |
62 |
|
|
the limitations of each of the |
63 |
|
|
models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
64 |
|
|
important properties to quantify are the Gibbs and Helmholtz free |
65 |
|
|
energies, particularly for the solid forms of water. Difficulties in |
66 |
|
|
studies addressing these thermodynamic quantities typically arise from |
67 |
|
|
the assortment of possible crystalline polymorphs that water adopts |
68 |
|
|
over a wide range of pressures and temperatures. It is a challenging |
69 |
|
|
task to investigate the entire free energy landscape\cite{Sanz04}; |
70 |
|
|
and ideally, research is focused on the phases having the lowest free |
71 |
|
|
energy at a given state point, because these phases will dictate the |
72 |
|
|
relevant transition temperatures and pressures for the model. |
73 |
|
|
|
74 |
|
|
In this paper, standard reference state methods were applied to known |
75 |
chrisfen |
1860 |
crystalline water polymorphs to evaluate their free energy in the low |
76 |
|
|
pressure regime. This work is unique in that one of the crystal |
77 |
|
|
lattices was arrived at through crystallization of a computationally |
78 |
|
|
efficient water model under constant pressure and temperature |
79 |
|
|
conditions. Crystallization events are interesting in and of |
80 |
|
|
themselves\cite{Matsumoto02,Yamada02}; however, the crystal structure |
81 |
|
|
obtained in this case is different from any previously observed ice |
82 |
|
|
polymorphs in experiment or simulation.\cite{Fennell04} We have named |
83 |
|
|
this structure Ice-{\it i} to indicate its origin in computational |
84 |
|
|
simulation. The unit cell of Ice-{\it i} and an extruded variant named |
85 |
|
|
Ice-{\it i}$^\prime$ both consist of eight water molecules that stack |
86 |
|
|
in rows of interlocking water tetramers as illustrated in figures |
87 |
|
|
\ref{iCrystal}A and |
88 |
chrisfen |
1845 |
\ref{iCrystal}B. These tetramers make the crystal structure similar |
89 |
|
|
in appearance to a recent two-dimensional ice tessellation simulated |
90 |
|
|
on a silica surface.\cite{Yang04} As expected in an ice crystal |
91 |
|
|
constructed of water tetramers, the hydrogen bonds are not as linear |
92 |
|
|
as those observed in ice $I_h$, however the interlocking of these |
93 |
|
|
subunits appears to provide significant stabilization to the overall |
94 |
|
|
crystal. The arrangement of these tetramers results in surrounding |
95 |
|
|
open octagonal cavities that are typically greater than 6.3 \AA\ in |
96 |
|
|
diameter (Fig. \ref{iCrystal}C). This open structure leads to |
97 |
|
|
crystals that are typically 0.07 g/cm$^3$ less dense than ice $I_h$. |
98 |
|
|
|
99 |
|
|
\begin{figure} |
100 |
|
|
\includegraphics[width=4in]{iCrystal.eps} |
101 |
|
|
\caption{(A) Unit cell for Ice-{\it i}, (B) Ice-{\it i}$^\prime$, |
102 |
|
|
and (C) a rendering of a proton ordered crystal of Ice-{\it i} looking |
103 |
|
|
down the (001) crystal face. In the unit cells, the spheres represent |
104 |
|
|
the center-of-mass locations of the water molecules. The $a$ to $c$ |
105 |
|
|
ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
106 |
|
|
$a:2.1214c$ and $a:1.785c$ respectively. The presence of large |
107 |
|
|
octagonal pores in both crystal forms lead to a polymorph that is less |
108 |
|
|
dense than ice $I_h$.} |
109 |
|
|
\label{iCrystal} |
110 |
|
|
\end{figure} |
111 |
|
|
|
112 |
|
|
Results from our previous study indicated that Ice-{\it i} is the |
113 |
|
|
minimum energy crystal structure for the single point water models |
114 |
|
|
investigated (for discussions on these single point dipole models, see |
115 |
|
|
our previous work and related |
116 |
|
|
articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
117 |
|
|
considered energetic stabilization and neglected entropic |
118 |
|
|
contributions to the overall free energy. To address this issue, we |
119 |
|
|
have calculated the absolute free energy of this crystal using |
120 |
|
|
thermodynamic integration and compared to the free energies of cubic |
121 |
|
|
and hexagonal ice $I$ (the experimental low density ice polymorphs) |
122 |
|
|
and ice B (a higher density, but very stable crystal structure |
123 |
|
|
observed by B\`{a}ez and Clancy in free energy studies of |
124 |
|
|
SPC/E).\cite{Baez95b} This work includes results for the water model |
125 |
|
|
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
126 |
|
|
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
127 |
|
|
field parametrized single point dipole water model (SSD/RF). The |
128 |
|
|
extruded variant, Ice-{\it i}$^\prime$, was used in calculations |
129 |
chrisfen |
1872 |
involving SPC/E, TIP4P, and TIP5P. These models exhibit enhanced |
130 |
|
|
stability with Ice-{\it i}$^\prime$ because of their more |
131 |
|
|
tetrahedrally arranged internal charge distributions. Additionally, |
132 |
|
|
there is often a small distortion of proton ordered Ice-{\it |
133 |
|
|
i}$^\prime$ that converts the normally square tetramer into a rhombus |
134 |
|
|
with alternating approximately 85 and 95 degree angles. The degree of |
135 |
|
|
this distortion is model dependent and significant enough to split the |
136 |
|
|
tetramer diagonal location peak in the radial distribution function. |
137 |
chrisfen |
1845 |
|
138 |
|
|
Thermodynamic integration was utilized to calculate the free energies |
139 |
|
|
of the listed water models at various state points using a modified |
140 |
|
|
form of the OOPSE molecular dynamics package.\cite{Meineke05} |
141 |
|
|
This calculation method involves a sequence of simulations during |
142 |
|
|
which the system of interest is converted into a reference system for |
143 |
|
|
which the free energy is known analytically. This transformation path |
144 |
chrisfen |
1872 |
is then integrated, in order to determine the free energy difference |
145 |
chrisfen |
1845 |
between the two states: |
146 |
|
|
\begin{equation} |
147 |
|
|
\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
148 |
|
|
)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
149 |
|
|
\end{equation} |
150 |
|
|
where $V$ is the interaction potential and $\lambda$ is the |
151 |
|
|
transformation parameter that scales the overall potential. For |
152 |
|
|
liquid and solid phases, the ideal gas and harmonically restrained |
153 |
|
|
crystal are chosen as the reference states respectively. Thermodynamic |
154 |
|
|
integration is an established technique that has been used extensively |
155 |
|
|
in the calculation of free energies for condensed phases of |
156 |
|
|
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. |
157 |
|
|
|
158 |
|
|
The calculated free energies of proton-ordered varients of three low |
159 |
|
|
density polymorphs ($I_h$, $I_c$, and Ice-{\it i} or Ice-{\it |
160 |
|
|
i}$^\prime$) and the stable higher density ice B are listed in Table |
161 |
|
|
\ref{freeEnergy}. The reason for inclusion of ice B was that it was |
162 |
|
|
shown to be a minimum free energy structure for SPC/E at ambient |
163 |
|
|
conditions.\cite{Baez95b} In addition to the free energies, the |
164 |
|
|
relavent transition temperatures at standard pressure are also |
165 |
|
|
displayed in Table \ref{freeEnergy}. These free energy values |
166 |
|
|
indicate that Ice-{\it i} is the most stable state for all of the |
167 |
|
|
investigated water models. With the free energy at these state |
168 |
|
|
points, the Gibbs-Helmholtz equation was used to project to other |
169 |
|
|
state points and to build phase diagrams, and figure \ref{tp3PhaseDia} |
170 |
|
|
is an example diagram built from the results for the TIP3P water |
171 |
|
|
model. All other models have similar structure, although the crossing |
172 |
|
|
points between the phases move to different temperatures and pressures |
173 |
|
|
as indicated from the transition temperatures in Table |
174 |
|
|
\ref{freeEnergy}. It is interesting to note that ice $I$ does not |
175 |
|
|
exist in either cubic or hexagonal form in any of the phase diagrams |
176 |
|
|
for any of the models. For purposes of this study, ice B is |
177 |
|
|
representative of the dense ice polymorphs. A recent study by Sanz |
178 |
|
|
{\it et al.} goes into detail on the phase diagrams for SPC/E and |
179 |
|
|
TIP4P at higher pressures than those studied here.\cite{Sanz04} |
180 |
|
|
|
181 |
|
|
\begin{table*} |
182 |
|
|
\begin{minipage}{\linewidth} |
183 |
|
|
\begin{center} |
184 |
|
|
\caption{Calculated free energies for several ice polymorphs along |
185 |
|
|
with the calculated melting (or sublimation) and boiling points for |
186 |
|
|
the investigated water models. All free energy calculations used a |
187 |
|
|
cutoff radius of 9.0 \AA\ and were performed at 200 K and $\sim$1 atm. |
188 |
|
|
Units of free energy are kcal/mol, while transition temperature are in |
189 |
|
|
Kelvin. Calculated error of the final digits is in parentheses.} |
190 |
|
|
\begin{tabular}{lccccccc} |
191 |
|
|
\hline |
192 |
|
|
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} & Ice-{\it i}$^\prime$ & $T_m$ (*$T_s$) & $T_b$\\ |
193 |
|
|
\hline |
194 |
|
|
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(4) & 357(2)\\ |
195 |
|
|
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 266(5) & 354(2)\\ |
196 |
|
|
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 271(4) & 337(2)\\ |
197 |
|
|
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 296(3) & 396(2)\\ |
198 |
|
|
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(2) & -\\ |
199 |
|
|
SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2) & - & 278(4) & 349(2)\\ |
200 |
|
|
\end{tabular} |
201 |
|
|
\label{freeEnergy} |
202 |
|
|
\end{center} |
203 |
|
|
\end{minipage} |
204 |
|
|
\end{table*} |
205 |
|
|
|
206 |
|
|
\begin{figure} |
207 |
|
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
208 |
|
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
209 |
|
|
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
210 |
|
|
the experimental values; however, the solid phases shown are not the |
211 |
|
|
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
212 |
|
|
higher in energy and don't appear in the phase diagram.} |
213 |
|
|
\label{tp3PhaseDia} |
214 |
|
|
\end{figure} |
215 |
|
|
|
216 |
|
|
Most of the water models have melting points that compare quite |
217 |
|
|
favorably with the experimental value of 273 K. The unfortunate |
218 |
|
|
aspect of this result is that this phase change occurs between |
219 |
|
|
Ice-{\it i} and the liquid state rather than ice $I_h$ and the liquid |
220 |
|
|
state. Surprisingly, these results are not contrary to other studies. |
221 |
|
|
Studies of ice $I_h$ using TIP4P predict a $T_m$ ranging from 214 to |
222 |
|
|
238 K (differences being attributed to choice of interaction |
223 |
|
|
truncation and different ordered and disordered molecular |
224 |
|
|
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
225 |
|
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
226 |
|
|
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
227 |
|
|
calculated to be 265 K, indicating that these simulation based |
228 |
|
|
structures ought to be included in studies probing phase transitions |
229 |
|
|
with this model. Also of interest in these results is that SSD/E does |
230 |
|
|
not exhibit a melting point at 1 atm, but it rather shows a |
231 |
|
|
sublimation point at 355 K. This is due to the significant stability |
232 |
|
|
of Ice-{\it i} over all other polymorphs for this particular model |
233 |
|
|
under these conditions. While troubling, this behavior resulted in |
234 |
|
|
spontaneous crystallization of Ice-{\it i} and led us to investigate |
235 |
|
|
this structure. These observations provide a warning that simulations |
236 |
|
|
of SSD/E as a ``liquid'' near 300 K are actually metastable and run |
237 |
|
|
the risk of spontaneous crystallization. However, when applying a |
238 |
|
|
longer cutoff, the liquid state is preferred under standard |
239 |
|
|
conditions. |
240 |
|
|
|
241 |
|
|
\begin{figure} |
242 |
|
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
243 |
|
|
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
244 |
|
|
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
245 |
|
|
with an added Ewald correction term. Error for the larger cutoff |
246 |
|
|
points is equivalent to that observed at 9.0\AA\ (see Table |
247 |
|
|
\ref{freeEnergy}). Data for ice I$_c$ with TIP3P using both 12 and |
248 |
|
|
13.5 \AA\ cutoffs were omitted because the crystal was prone to |
249 |
|
|
distortion and melting at 200 K. Ice-{\it i}$^\prime$ is the form of |
250 |
|
|
Ice-{\it i} used in the SPC/E simulations.} |
251 |
|
|
\label{incCutoff} |
252 |
|
|
\end{figure} |
253 |
|
|
|
254 |
|
|
Increasing the cutoff radius in simulations of the more |
255 |
|
|
computationally efficient water models was done in order to evaluate |
256 |
|
|
the trend in free energy values when moving to systems that do not |
257 |
|
|
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
258 |
|
|
free energy of the ice polymorphs with water models lacking a |
259 |
|
|
long-range correction show a significant cutoff radius dependence. In |
260 |
|
|
general, there is a narrowing of the free energy differences while |
261 |
|
|
moving to greater cutoff radii. As the free energies for the |
262 |
|
|
polymorphs converge, the stability advantage that Ice-{\it i} exhibits |
263 |
|
|
is reduced. Adjacent to each of these model plots is a system with an |
264 |
|
|
applied or estimated long-range correction. SSD/RF was parametrized |
265 |
|
|
for use with a reaction field, and the benefit provided by this |
266 |
|
|
computationally inexpensive correction is apparent. Due to the |
267 |
|
|
relative independence of the resultant free energies, calculations |
268 |
|
|
performed with a small cutoff radius provide resultant properties |
269 |
|
|
similar to what one would expect for the bulk material. In the cases |
270 |
|
|
of TIP3P and SPC/E, the effect of an Ewald summation was estimated by |
271 |
|
|
applying the potential energy difference do to its inclusion in |
272 |
|
|
systems in the presence and absence of the correction. This was |
273 |
|
|
accomplished by calculation of the potential energy of identical |
274 |
|
|
crystals both with and without particle mesh Ewald (PME). Similar |
275 |
|
|
behavior to that observed with reaction field is seen for both of |
276 |
|
|
these models. The free energies show less dependence on cutoff radius |
277 |
|
|
and span a more narrowed range for the various polymorphs. Like the |
278 |
|
|
dipolar water models, TIP3P displays a relatively constant preference |
279 |
|
|
for the Ice-{\it i} polymorph. Crystal preference is much more |
280 |
|
|
difficult to determine for SPC/E. Without a long-range correction, |
281 |
|
|
each of the polymorphs studied assumes the role of the preferred |
282 |
|
|
polymorph under different cutoff conditions. The inclusion of the |
283 |
|
|
Ewald correction flattens and narrows the sequences of free energies |
284 |
chrisfen |
1872 |
such that they often overlap within error, indicating that other |
285 |
|
|
conditions, such as the density in fixed volume simulations, can |
286 |
chrisfen |
1860 |
influence the chosen polymorph upon crystallization. |
287 |
chrisfen |
1845 |
|
288 |
chrisfen |
1860 |
So what is the preferred solid polymorph for simulated water? The |
289 |
|
|
answer appears to be dependent both on conditions and which model is |
290 |
|
|
used. In the case of short cutoffs without a long-range interaction |
291 |
|
|
correction, Ice-{\it i} and Ice-{\it i}$^\prime$ have the lowest free |
292 |
|
|
energy of the studied polymorphs with all the models. Ideally, |
293 |
|
|
crystallization of each model under constant pressure conditions, as |
294 |
|
|
was done with SSD/E, would aid in the identification of their |
295 |
|
|
respective preferred structures. This work, however, helps illustrate |
296 |
|
|
how studies involving one specific model can lead to insight about |
297 |
|
|
important behavior of others. In general, the above results support |
298 |
|
|
the finding that the Ice-{\it i} polymorph is a stable crystal |
299 |
|
|
structure that should be considered when studying the phase behavior |
300 |
|
|
of water models. |
301 |
|
|
|
302 |
|
|
Finally, due to the stability of Ice-{\it i} in the investigated |
303 |
|
|
simulation conditions, the question arises as to possible experimental |
304 |
|
|
observation of this polymorph. The rather extensive past and current |
305 |
|
|
experimental investigation of water in the low pressure regime makes |
306 |
|
|
us hesitant to ascribe any relevance of this work outside of the |
307 |
|
|
simulation community. It is for this reason that we chose a name for |
308 |
|
|
this polymorph which involves an imaginary quantity. That said, there |
309 |
|
|
are certain experimental conditions that would provide the most ideal |
310 |
|
|
situation for possible observation. These include the negative |
311 |
|
|
pressure or stretched solid regime, small clusters in vacuum |
312 |
chrisfen |
1845 |
deposition environments, and in clathrate structures involving small |
313 |
chrisfen |
1860 |
non-polar molecules. |
314 |
chrisfen |
1845 |
|
315 |
|
|
\section{Acknowledgments} |
316 |
|
|
Support for this project was provided by the National Science |
317 |
|
|
Foundation under grant CHE-0134881. Computation time was provided by |
318 |
|
|
the Notre Dame High Performance Computing Cluster and the Notre Dame |
319 |
|
|
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
320 |
|
|
|
321 |
|
|
\newpage |
322 |
|
|
|
323 |
|
|
\bibliographystyle{jcp} |
324 |
|
|
\bibliography{iceiPaper} |
325 |
|
|
|
326 |
|
|
|
327 |
|
|
\end{document} |