1 |
|
%\documentclass[prb,aps,twocolumn,tabularx]{revtex4} |
2 |
< |
\documentclass[11pt]{article} |
3 |
< |
%\documentclass[11pt]{article} |
2 |
> |
\documentclass[12pt]{article} |
3 |
|
\usepackage{endfloat} |
4 |
|
\usepackage{amsmath} |
5 |
|
\usepackage{epsf} |
6 |
< |
\usepackage{berkeley} |
6 |
> |
\usepackage{times} |
7 |
> |
\usepackage{mathptm} |
8 |
|
\usepackage{setspace} |
9 |
|
\usepackage{tabularx} |
10 |
|
\usepackage{graphicx} |
20 |
|
|
21 |
|
\begin{document} |
22 |
|
|
23 |
< |
\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} |
23 |
> |
\title{Computational free energy studies of a new ice polymorph which |
24 |
> |
exhibits greater stability than Ice I$_h$} |
25 |
|
|
26 |
|
\author{Christopher J. Fennell and J. Daniel Gezelter \\ |
27 |
< |
Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
27 |
> |
Department of Chemistry and Biochemistry\\ |
28 |
> |
University of Notre Dame\\ |
29 |
|
Notre Dame, Indiana 46556} |
30 |
|
|
31 |
|
\date{\today} |
34 |
|
%\doublespacing |
35 |
|
|
36 |
|
\begin{abstract} |
37 |
< |
The absolute free energies of several ice polymorphs which are stable |
38 |
< |
at low pressures were calculated using thermodynamic integration to a |
39 |
< |
reference system (the Einstein crystal). These integrations were |
40 |
< |
performed for most of the common water models (SPC/E, TIP3P, TIP4P, |
41 |
< |
TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed |
42 |
< |
crystallizing at room temperature for one of the single-point water |
43 |
< |
models, was determined to be the stable crystalline state (at 1 atm) |
44 |
< |
for {\it all} the water models investigated. Phase diagrams were |
45 |
< |
generated, and phase coexistence lines were determined for all of the |
46 |
< |
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. |
37 |
> |
The absolute free energies of several ice polymorphs were calculated |
38 |
> |
using thermodynamic integration. These polymorphs are predicted by |
39 |
> |
computer simulations using a variety of common water models to be |
40 |
> |
stable at low pressures. A recently discovered ice polymorph that has |
41 |
> |
as yet {\it only} been observed in computer simulations (Ice-{\it i}), |
42 |
> |
was determined to be the stable crystalline state for {\it all} the |
43 |
> |
water models investigated. Phase diagrams were generated, and phase |
44 |
> |
coexistence lines were determined for all of the known low-pressure |
45 |
> |
ice structures. Additionally, potential truncation was shown to play |
46 |
> |
a role in the resulting shape of the free energy landscape. |
47 |
|
\end{abstract} |
48 |
|
|
49 |
|
%\narrowtext |
63 |
|
hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
64 |
|
choice of models available, it is only natural to compare the models |
65 |
|
under interesting thermodynamic conditions in an attempt to clarify |
66 |
< |
the limitations of each of the |
67 |
< |
models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
68 |
< |
important properties to quantify are the Gibbs and Helmholtz free |
69 |
< |
energies, particularly for the solid forms of water. Difficulty in |
70 |
< |
these types of studies typically arises from the assortment of |
71 |
< |
possible crystalline polymorphs that water adopts over a wide range of |
72 |
< |
pressures and temperatures. There are currently 13 recognized forms |
73 |
< |
of ice, and it is a challenging task to investigate the entire free |
74 |
< |
energy landscape.\cite{Sanz04} Ideally, research is focused on the |
66 |
> |
the limitations of |
67 |
> |
each.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two important |
68 |
> |
properties to quantify are the Gibbs and Helmholtz free energies, |
69 |
> |
particularly for the solid forms of water as these predict the |
70 |
> |
thermodynamic stability of the various phases. Water has a |
71 |
> |
particularly rich phase diagram and takes on a number of different and |
72 |
> |
stable crystalline structures as the temperature and pressure are |
73 |
> |
varied. It is a challenging task to investigate the entire free |
74 |
> |
energy landscape\cite{Sanz04}; and ideally, research is focused on the |
75 |
|
phases having the lowest free energy at a given state point, because |
76 |
|
these phases will dictate the relevant transition temperatures and |
77 |
< |
pressures for the model. |
77 |
> |
pressures for the model. |
78 |
|
|
79 |
< |
In this paper, standard reference state methods were applied to known |
80 |
< |
crystalline water polymorphs in the low pressure regime. This work is |
81 |
< |
unique in that one of the crystal lattices was arrived at through |
82 |
< |
crystallization of a computationally efficient water model under |
83 |
< |
constant pressure and temperature conditions. Crystallization events |
84 |
< |
are interesting in and of themselves;\cite{Matsumoto02,Yamada02} |
85 |
< |
however, the crystal structure obtained in this case is different from |
86 |
< |
any previously observed ice polymorphs in experiment or |
87 |
< |
simulation.\cite{Fennell04} We have named this structure Ice-{\it i} |
88 |
< |
to indicate its origin in computational simulation. The unit cell |
89 |
< |
(Fig. \ref{iceiCell}A) consists of eight water molecules that stack in |
90 |
< |
rows of interlocking water tetramers. Proton ordering can be |
91 |
< |
accomplished by orienting two of the molecules so that both of their |
92 |
< |
donated hydrogen bonds are internal to their tetramer |
93 |
< |
(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
94 |
< |
water tetramers, the hydrogen bonds are not as linear as those |
95 |
< |
observed in ice $I_h$, however the interlocking of these subunits |
96 |
< |
appears to provide significant stabilization to the overall |
97 |
< |
crystal. The arrangement of these tetramers results in surrounding |
98 |
< |
open octagonal cavities that are typically greater than 6.3 \AA\ in |
99 |
< |
diameter. This relatively open overall structure leads to crystals |
100 |
< |
that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. |
79 |
> |
The high-pressure phases of water (ice II - ice X as well as ice XII) |
80 |
> |
have been studied extensively both experimentally and |
81 |
> |
computationally. In this paper, standard reference state methods were |
82 |
> |
applied in the {\it low} pressure regime to evaluate the free energies |
83 |
> |
for a few known crystalline water polymorphs that might be stable at |
84 |
> |
these pressures. This work is unique in that one of the crystal |
85 |
> |
lattices was arrived at through crystallization of a computationally |
86 |
> |
efficient water model under constant pressure and temperature |
87 |
> |
conditions. Crystallization events are interesting in and of |
88 |
> |
themselves\cite{Matsumoto02,Yamada02}; however, the crystal structure |
89 |
> |
obtained in this case is different from any previously observed ice |
90 |
> |
polymorphs in experiment or simulation.\cite{Fennell04} We have named |
91 |
> |
this structure Ice-{\it i} to indicate its origin in computational |
92 |
> |
simulation. The unit cell of Ice-{\it i} and an axially-elongated |
93 |
> |
variant named Ice-{\it i}$^\prime$ both consist of eight water |
94 |
> |
molecules that stack in rows of interlocking water tetramers as |
95 |
> |
illustrated in figures \ref{unitcell}A and \ref{unitcell}B. These |
96 |
> |
tetramers form a crystal structure similar in appearance to a recent |
97 |
> |
two-dimensional surface tessellation simulated on silica.\cite{Yang04} |
98 |
> |
As expected in an ice crystal constructed of water tetramers, the |
99 |
> |
hydrogen bonds are not as linear as those observed in ice I$_h$, |
100 |
> |
however the interlocking of these subunits appears to provide |
101 |
> |
significant stabilization to the overall crystal. The arrangement of |
102 |
> |
these tetramers results in octagonal cavities that are typically |
103 |
> |
greater than 6.3 \AA\ in diameter (Fig. \ref{iCrystal}). This open |
104 |
> |
structure leads to crystals that are typically 0.07 g/cm$^3$ less |
105 |
> |
dense than ice I$_h$. |
106 |
|
|
107 |
|
\begin{figure} |
108 |
+ |
\centering |
109 |
|
\includegraphics[width=\linewidth]{unitCell.eps} |
110 |
< |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
111 |
< |
elongated variant of Ice-{\it i}. The spheres represent the |
112 |
< |
center-of-mass locations of the water molecules. The $a$ to $c$ |
113 |
< |
ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
114 |
< |
$a:2.1214c$ and $a:1.7850c$ respectively.} |
113 |
< |
\label{iceiCell} |
110 |
> |
\caption{(A) Unit cells for Ice-{\it i} and (B) Ice-{\it i}$^\prime$. |
111 |
> |
The spheres represent the center-of-mass locations of the water |
112 |
> |
molecules. The $a$ to $c$ ratios for Ice-{\it i} and Ice-{\it |
113 |
> |
i}$^\prime$ are given by 2.1214 and 1.785 respectively.} |
114 |
> |
\label{unitcell} |
115 |
|
\end{figure} |
116 |
|
|
117 |
|
\begin{figure} |
118 |
+ |
\centering |
119 |
|
\includegraphics[width=\linewidth]{orderedIcei.eps} |
120 |
< |
\caption{Image of a proton ordered crystal of Ice-{\it i} looking |
121 |
< |
down the (001) crystal face. The rows of water tetramers surrounded by |
122 |
< |
octagonal pores leads to a crystal structure that is significantly |
123 |
< |
less dense than ice $I_h$.} |
122 |
< |
\label{protOrder} |
120 |
> |
\caption{A rendering of a proton ordered crystal of Ice-{\it i} looking |
121 |
> |
down the (001) crystal face. The presence of large octagonal pores |
122 |
> |
leads to a polymorph that is less dense than ice I$_h$.} |
123 |
> |
\label{iCrystal} |
124 |
|
\end{figure} |
125 |
|
|
126 |
|
Results from our previous study indicated that Ice-{\it i} is the |
127 |
< |
minimum energy crystal structure for the single point water models we |
128 |
< |
had investigated (for discussions on these single point dipole models, |
129 |
< |
see our previous work and related |
130 |
< |
articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
131 |
< |
considered energetic stabilization and neglected entropic |
132 |
< |
contributions to the overall free energy. To address this issue, we |
127 |
> |
minimum energy crystal structure for the single point water models |
128 |
> |
investigated (for discussions on these single point dipole models, see |
129 |
> |
our previous work and related |
130 |
> |
articles).\cite{Fennell04,Liu96,Bratko85} Our earlier results |
131 |
> |
considered only energetic stabilization and neglected entropic |
132 |
> |
contributions to the overall free energy. To address this issue, we |
133 |
|
have calculated the absolute free energy of this crystal using |
134 |
< |
thermodynamic integration and compared to the free energies of cubic |
135 |
< |
and hexagonal ice $I$ (the experimental low density ice polymorphs) |
136 |
< |
and ice B (a higher density, but very stable crystal structure |
137 |
< |
observed by B\`{a}ez and Clancy in free energy studies of |
138 |
< |
SPC/E).\cite{Baez95b} This work includes results for the water model |
139 |
< |
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
140 |
< |
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
141 |
< |
field parametrized single point dipole water model (SSD/RF). It should |
142 |
< |
be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was |
143 |
< |
used in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell |
144 |
< |
of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} |
145 |
< |
unit it is extended in the direction of the (001) face and compressed |
146 |
< |
along the other two faces. |
134 |
> |
thermodynamic integration and compared it to the free energies of ice |
135 |
> |
I$_c$ and ice I$_h$ (the common low density ice polymorphs) and ice B |
136 |
> |
(a higher density, but very stable crystal structure observed by |
137 |
> |
B\`{a}ez and Clancy in free energy studies of SPC/E).\cite{Baez95b} |
138 |
> |
This work includes results for the water model from which Ice-{\it i} |
139 |
> |
was crystallized (SSD/E) in addition to several common water models |
140 |
> |
(TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field parametrized |
141 |
> |
single point dipole water model (SSD/RF). The axially-elongated |
142 |
> |
variant, Ice-{\it i}$^\prime$, was used in calculations involving |
143 |
> |
SPC/E, TIP4P, and TIP5P. The square tetramers in Ice-{\it i} distort |
144 |
> |
in Ice-{\it i}$^\prime$ to form a rhombus with alternating 85 and 95 |
145 |
> |
degree angles. Under SPC/E, TIP4P, and TIP5P, this geometry is better |
146 |
> |
at forming favorable hydrogen bonds. The degree of rhomboid |
147 |
> |
distortion depends on the water model used, but is significant enough |
148 |
> |
to split a peak in the radial distribution function which corresponds |
149 |
> |
to diagonal sites in the tetramers. |
150 |
|
|
151 |
|
\section{Methods} |
152 |
|
|
153 |
|
Canonical ensemble (NVT) molecular dynamics calculations were |
154 |
< |
performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
154 |
> |
performed using the OOPSE molecular mechanics program.\cite{Meineke05} |
155 |
|
All molecules were treated as rigid bodies, with orientational motion |
156 |
< |
propagated using the symplectic DLM integration method. Details about |
156 |
> |
propagated using the symplectic DLM integration method. Details about |
157 |
|
the implementation of this technique can be found in a recent |
158 |
|
publication.\cite{Dullweber1997} |
159 |
|
|
160 |
< |
Thermodynamic integration is an established technique for |
161 |
< |
determination of free energies of condensed phases of |
162 |
< |
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
163 |
< |
method, implemented in the same manner illustrated by B\`{a}ez and |
164 |
< |
Clancy, was utilized to calculate the free energy of several ice |
165 |
< |
crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
166 |
< |
SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
167 |
< |
and 400 K for all of these water models were also determined using |
168 |
< |
this same technique in order to determine melting points and to |
169 |
< |
generate phase diagrams. All simulations were carried out at densities |
170 |
< |
which correspond to a pressure of approximately 1 atm at their |
171 |
< |
respective temperatures. |
172 |
< |
|
169 |
< |
Thermodynamic integration involves a sequence of simulations during |
170 |
< |
which the system of interest is converted into a reference system for |
171 |
< |
which the free energy is known analytically. This transformation path |
172 |
< |
is then integrated in order to determine the free energy difference |
173 |
< |
between the two states: |
160 |
> |
Thermodynamic integration was utilized to calculate the Helmholtz free |
161 |
> |
energies ($A$) of the listed water models at various state points |
162 |
> |
using the OOPSE molecular dynamics program.\cite{Meineke05} |
163 |
> |
Thermodynamic integration is an established technique that has been |
164 |
> |
used extensively in the calculation of free energies for condensed |
165 |
> |
phases of |
166 |
> |
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
167 |
> |
method uses a sequence of simulations during which the system of |
168 |
> |
interest is converted into a reference system for which the free |
169 |
> |
energy is known analytically ($A_0$). The difference in potential |
170 |
> |
energy between the reference system and the system of interest |
171 |
> |
($\Delta V$) is then integrated in order to determine the free energy |
172 |
> |
difference between the two states: |
173 |
|
\begin{equation} |
174 |
< |
\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
176 |
< |
)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
174 |
> |
A = A_0 + \int_0^1 \left\langle \Delta V \right\rangle_\lambda d\lambda. |
175 |
|
\end{equation} |
176 |
< |
where $V$ is the interaction potential and $\lambda$ is the |
177 |
< |
transformation parameter that scales the overall |
178 |
< |
potential. Simulations are distributed strategically along this path |
179 |
< |
in order to sufficiently sample the regions of greatest change in the |
180 |
< |
potential. Typical integrations in this study consisted of $\sim$25 |
183 |
< |
simulations ranging from 300 ps (for the unaltered system) to 75 ps |
184 |
< |
(near the reference state) in length. |
176 |
> |
Here, $\lambda$ is the parameter that governs the transformation |
177 |
> |
between the reference system and the system of interest. For |
178 |
> |
crystalline phases, an harmonically-restrained (Einsten) crystal is |
179 |
> |
chosen as the reference state, while for liquid phases, the ideal gas |
180 |
> |
is taken as the reference state. |
181 |
|
|
182 |
< |
For the thermodynamic integration of molecular crystals, the Einstein |
183 |
< |
crystal was chosen as the reference system. In an Einstein crystal, |
184 |
< |
the molecules are restrained at their ideal lattice locations and |
185 |
< |
orientations. Using harmonic restraints, as applied by B\`{a}ez and |
190 |
< |
Clancy, the total potential for this reference crystal |
191 |
< |
($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
182 |
> |
In an Einstein crystal, the molecules are restrained at their ideal |
183 |
> |
lattice locations and orientations. Using harmonic restraints, as |
184 |
> |
applied by B\`{a}ez and Clancy, the total potential for this reference |
185 |
> |
crystal ($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
186 |
|
\begin{equation} |
187 |
|
V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
188 |
|
\frac{K_\omega\omega^2}{2}, |
190 |
|
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
191 |
|
the spring constants restraining translational motion and deflection |
192 |
|
of and rotation around the principle axis of the molecule |
193 |
< |
respectively. It is clear from Fig. \ref{waterSpring} that the values |
194 |
< |
of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
195 |
< |
$-\pi$ to $\pi$. The partition function for a molecular crystal |
196 |
< |
restrained in this fashion can be evaluated analytically, and the |
197 |
< |
Helmholtz Free Energy ({\it A}) is given by |
193 |
> |
respectively. These spring constants are typically calculated from |
194 |
> |
the mean-square displacements of water molecules in an unrestrained |
195 |
> |
ice crystal at 200 K. For these studies, $K_\mathrm{v} = 4.29$ kcal |
196 |
> |
mol$^{-1}$ \AA$^{-2}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$ rad$^{-2}$, |
197 |
> |
and $K_\omega\ = 17.75$ kcal mol$^{-1}$ rad$^{-2}$. It is clear from |
198 |
> |
Fig. \ref{waterSpring} that the values of $\theta$ range from $0$ to |
199 |
> |
$\pi$, while $\omega$ ranges from $-\pi$ to $\pi$. The partition |
200 |
> |
function for a molecular crystal restrained in this fashion can be |
201 |
> |
evaluated analytically, and the Helmholtz Free Energy ({\it A}) is |
202 |
> |
given by |
203 |
|
\begin{eqnarray} |
204 |
|
A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
205 |
|
[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
215 |
|
potential energy of the ideal crystal.\cite{Baez95a} |
216 |
|
|
217 |
|
\begin{figure} |
218 |
< |
\includegraphics[width=\linewidth]{rotSpring.eps} |
218 |
> |
\centering |
219 |
> |
\includegraphics[width=4in]{rotSpring.eps} |
220 |
|
\caption{Possible orientational motions for a restrained molecule. |
221 |
|
$\theta$ angles correspond to displacement from the body-frame {\it |
222 |
|
z}-axis, while $\omega$ angles correspond to rotation about the |
223 |
< |
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
223 |
> |
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
224 |
|
constants for the harmonic springs restraining motion in the $\theta$ |
225 |
|
and $\omega$ directions.} |
226 |
|
\label{waterSpring} |
232 |
|
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
233 |
|
typically differ in regard to the path taken for switching off the |
234 |
|
interaction potential to convert the system to an ideal gas of water |
235 |
< |
molecules. In this study, we applied of one of the most convenient |
235 |
> |
molecules. In this study, we applied one of the most convenient |
236 |
|
methods and integrated over the $\lambda^4$ path, where all |
237 |
|
interaction parameters are scaled equally by this transformation |
238 |
|
parameter. This method has been shown to be reversible and provide |
239 |
|
results in excellent agreement with other established |
240 |
|
methods.\cite{Baez95b} |
241 |
|
|
242 |
< |
Charge, dipole, and Lennard-Jones interactions were modified by a |
243 |
< |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
244 |
< |
). By applying this function, these interactions are smoothly |
242 |
> |
Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and |
243 |
> |
Lennard-Jones interactions were gradually reduced by a cubic switching |
244 |
> |
function. By applying this function, these interactions are smoothly |
245 |
|
truncated, thereby avoiding the poor energy conservation which results |
246 |
< |
from harsher truncation schemes. The effect of a long-range correction |
247 |
< |
was also investigated on select model systems in a variety of |
248 |
< |
manners. For the SSD/RF model, a reaction field with a fixed |
246 |
> |
from harsher truncation schemes. The effect of a long-range |
247 |
> |
correction was also investigated on select model systems in a variety |
248 |
> |
of manners. For the SSD/RF model, a reaction field with a fixed |
249 |
|
dielectric constant of 80 was applied in all |
250 |
|
simulations.\cite{Onsager36} For a series of the least computationally |
251 |
< |
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
252 |
< |
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
253 |
< |
\AA\ cutoff results. Finally, the effects of utilizing an Ewald |
254 |
< |
summation were estimated for TIP3P and SPC/E by performing single |
255 |
< |
configuration calculations with Particle-Mesh Ewald (PME) in the |
256 |
< |
TINKER molecular mechanics software package.\cite{Tinker} The |
251 |
> |
expensive models (SSD/E, SSD/RF, TIP3P, and SPC/E), simulations were |
252 |
> |
performed with longer cutoffs of 10.5, 12, 13.5, and 15 \AA\ to |
253 |
> |
compare with the 9 \AA\ cutoff results. Finally, the effects of using |
254 |
> |
the Ewald summation were estimated for TIP3P and SPC/E by performing |
255 |
> |
single configuration Particle-Mesh Ewald (PME) |
256 |
> |
calculations~\cite{Tinker} for each of the ice polymorphs. The |
257 |
|
calculated energy difference in the presence and absence of PME was |
258 |
|
applied to the previous results in order to predict changes to the |
259 |
|
free energy landscape. |
260 |
|
|
261 |
< |
\section{Results and discussion} |
261 |
> |
\section{Results and Discussion} |
262 |
|
|
263 |
< |
The free energy of proton-ordered Ice-{\it i} was calculated and |
264 |
< |
compared with the free energies of proton ordered variants of the |
265 |
< |
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
266 |
< |
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
267 |
< |
and thought to be the minimum free energy structure for the SPC/E |
268 |
< |
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
269 |
< |
Ice XI, the experimentally-observed proton-ordered variant of ice |
270 |
< |
$I_h$, was investigated initially, but was found to be not as stable |
271 |
< |
as proton disordered or antiferroelectric variants of ice $I_h$. The |
272 |
< |
proton ordered variant of ice $I_h$ used here is a simple |
273 |
< |
antiferroelectric version that we devised, and it has an 8 molecule |
274 |
< |
unit cell similar to other predicted antiferroelectric $I_h$ |
275 |
< |
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
276 |
< |
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
277 |
< |
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger |
278 |
< |
crystal sizes were necessary for simulations involving larger cutoff |
279 |
< |
values. |
263 |
> |
The calculated free energies of proton-ordered variants of three low |
264 |
> |
density polymorphs (I$_h$, I$_c$, and Ice-{\it i} or Ice-{\it |
265 |
> |
i}$^\prime$) and the stable higher density ice B are listed in Table |
266 |
> |
\ref{freeEnergy}. Ice B was included because it has been |
267 |
> |
shown to be a minimum free energy structure for SPC/E at ambient |
268 |
> |
conditions.\cite{Baez95b} In addition to the free energies, the |
269 |
> |
relevant transition temperatures at standard pressure are also |
270 |
> |
displayed in Table \ref{freeEnergy}. These free energy values |
271 |
> |
indicate that Ice-{\it i} is the most stable state for all of the |
272 |
> |
investigated water models. With the free energy at these state |
273 |
> |
points, the Gibbs-Helmholtz equation was used to project to other |
274 |
> |
state points and to build phase diagrams. Figure \ref{tp3PhaseDia} is |
275 |
> |
an example diagram built from the results for the TIP3P water model. |
276 |
> |
All other models have similar structure, although the crossing points |
277 |
> |
between the phases move to different temperatures and pressures as |
278 |
> |
indicated from the transition temperatures in Table \ref{freeEnergy}. |
279 |
> |
It is interesting to note that ice I$_h$ (and ice I$_c$ for that |
280 |
> |
matter) do not appear in any of the phase diagrams for any of the |
281 |
> |
models. For purposes of this study, ice B is representative of the |
282 |
> |
dense ice polymorphs. A recent study by Sanz {\it et al.} provides |
283 |
> |
details on the phase diagrams for SPC/E and TIP4P at higher pressures |
284 |
> |
than those studied here.\cite{Sanz04} |
285 |
|
|
286 |
|
\begin{table*} |
287 |
|
\begin{minipage}{\linewidth} |
283 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
288 |
|
\begin{center} |
289 |
< |
\caption{Calculated free energies for several ice polymorphs with a |
290 |
< |
variety of common water models. All calculations used a cutoff radius |
291 |
< |
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
292 |
< |
kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
293 |
< |
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
294 |
< |
\begin{tabular}{ l c c c c } |
289 |
> |
\caption{Calculated free energies for several ice polymorphs along |
290 |
> |
with the calculated melting (or sublimation) and boiling points for |
291 |
> |
the investigated water models. All free energy calculations used a |
292 |
> |
cutoff radius of 9.0 \AA\ and were performed at 200 K and $\sim$1 atm. |
293 |
> |
Units of free energy are kcal/mol, while transition temperature are in |
294 |
> |
Kelvin. Calculated error of the final digits is in parentheses.} |
295 |
> |
\begin{tabular}{lccccccc} |
296 |
|
\hline |
297 |
< |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
297 |
> |
Water Model & I$_h$ & I$_c$ & B & Ice-{\it i} & Ice-{\it i}$^\prime$ & $T_m$ (*$T_s$) & $T_b$\\ |
298 |
|
\hline |
299 |
< |
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)\\ |
299 |
> |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(4) & 357(2)\\ |
300 |
> |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 266(5) & 354(2)\\ |
301 |
> |
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 271(4) & 337(2)\\ |
302 |
> |
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 296(3) & 396(2)\\ |
303 |
> |
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(2) & -\\ |
304 |
> |
SSD/RF & -11.96(2) & -11.60(2) & -12.53(3) & -12.79(2) & - & 322(4) & 366(2)\\ |
305 |
|
\end{tabular} |
306 |
|
\label{freeEnergy} |
307 |
|
\end{center} |
308 |
|
\end{minipage} |
309 |
|
\end{table*} |
310 |
|
|
306 |
– |
The free energy values computed for the studied polymorphs indicate |
307 |
– |
that Ice-{\it i} is the most stable state for all of the common water |
308 |
– |
models studied. With the calculated free energy at these state points, |
309 |
– |
the Gibbs-Helmholtz equation was used to project to other state points |
310 |
– |
and to build phase diagrams. Figures |
311 |
– |
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
312 |
– |
from the free energy results. All other models have similar structure, |
313 |
– |
although the crossing points between the phases move to slightly |
314 |
– |
different temperatures and pressures. It is interesting to note that |
315 |
– |
ice $I$ does not exist in either cubic or hexagonal form in any of the |
316 |
– |
phase diagrams for any of the models. For purposes of this study, ice |
317 |
– |
B is representative of the dense ice polymorphs. A recent study by |
318 |
– |
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
319 |
– |
TIP4P at higher pressures than those studied here.\cite{Sanz04} |
320 |
– |
|
311 |
|
\begin{figure} |
312 |
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
313 |
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
314 |
< |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
314 |
> |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
315 |
|
the experimental values; however, the solid phases shown are not the |
316 |
< |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
316 |
> |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
317 |
|
higher in energy and don't appear in the phase diagram.} |
318 |
< |
\label{tp3phasedia} |
318 |
> |
\label{tp3PhaseDia} |
319 |
|
\end{figure} |
320 |
|
|
321 |
< |
\begin{figure} |
322 |
< |
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
323 |
< |
\caption{Phase diagram for the SSD/RF water model in the low pressure |
324 |
< |
regime. Calculations producing these results were done under an |
325 |
< |
applied reaction field. It is interesting to note that this |
326 |
< |
computationally efficient model (over 3 times more efficient than |
327 |
< |
TIP3P) exhibits phase behavior similar to the less computationally |
328 |
< |
conservative charge based models.} |
339 |
< |
\label{ssdrfphasedia} |
340 |
< |
\end{figure} |
341 |
< |
|
342 |
< |
\begin{table*} |
343 |
< |
\begin{minipage}{\linewidth} |
344 |
< |
\renewcommand{\thefootnote}{\thempfootnote} |
345 |
< |
\begin{center} |
346 |
< |
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
347 |
< |
temperatures at 1 atm for several common water models compared with |
348 |
< |
experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
349 |
< |
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
350 |
< |
liquid or gas state.} |
351 |
< |
\begin{tabular}{ l c c c c c c c } |
352 |
< |
\hline |
353 |
< |
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
354 |
< |
\hline |
355 |
< |
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
356 |
< |
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
357 |
< |
$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ |
358 |
< |
\end{tabular} |
359 |
< |
\label{meltandboil} |
360 |
< |
\end{center} |
361 |
< |
\end{minipage} |
362 |
< |
\end{table*} |
363 |
< |
|
364 |
< |
Table \ref{meltandboil} lists the melting and boiling temperatures |
365 |
< |
calculated from this work. Surprisingly, most of these models have |
366 |
< |
melting points that compare quite favorably with experiment. The |
367 |
< |
unfortunate aspect of this result is that this phase change occurs |
368 |
< |
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
369 |
< |
liquid state. These results are actually not contrary to previous |
370 |
< |
studies in the literature. Earlier free energy studies of ice $I$ |
371 |
< |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
372 |
< |
being attributed to choice of interaction truncation and different |
373 |
< |
ordered and disordered molecular |
321 |
> |
Most of the water models have melting points that compare quite |
322 |
> |
favorably with the experimental value of 273 K. The unfortunate |
323 |
> |
aspect of this result is that this phase change occurs between |
324 |
> |
Ice-{\it i} and the liquid state rather than ice I$_h$ and the liquid |
325 |
> |
state. These results do not contradict other studies. Studies of ice |
326 |
> |
I$_h$ using TIP4P predict a $T_m$ ranging from 214 to 238 K |
327 |
> |
(differences being attributed to choice of interaction truncation and |
328 |
> |
different ordered and disordered molecular |
329 |
|
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
330 |
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
331 |
< |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
332 |
< |
calculated at 265 K, significantly higher in temperature than the |
333 |
< |
previous studies. Also of interest in these results is that SSD/E does |
334 |
< |
not exhibit a melting point at 1 atm, but it shows a sublimation point |
335 |
< |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
336 |
< |
all other polymorphs for this particular model under these |
337 |
< |
conditions. While troubling, this behavior resulted in spontaneous |
338 |
< |
crystallization of Ice-{\it i} and led us to investigate this |
339 |
< |
structure. These observations provide a warning that simulations of |
340 |
< |
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
341 |
< |
risk of spontaneous crystallization. However, this risk lessens when |
342 |
< |
applying a longer cutoff. |
331 |
> |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
332 |
> |
calculated to be 265 K, indicating that these simulation based |
333 |
> |
structures ought to be included in studies probing phase transitions |
334 |
> |
with this model. Also of interest in these results is that SSD/E does |
335 |
> |
not exhibit a melting point at 1 atm but does sublime at 355 K. This |
336 |
> |
is due to the significant stability of Ice-{\it i} over all other |
337 |
> |
polymorphs for this particular model under these conditions. While |
338 |
> |
troubling, this behavior resulted in the spontaneous crystallization |
339 |
> |
of Ice-{\it i} which led us to investigate this structure. These |
340 |
> |
observations provide a warning that simulations of SSD/E as a |
341 |
> |
``liquid'' near 300 K are actually metastable and run the risk of |
342 |
> |
spontaneous crystallization. However, when a longer cutoff radius is |
343 |
> |
used, SSD/E prefers the liquid state under standard temperature and |
344 |
> |
pressure. |
345 |
|
|
346 |
|
\begin{figure} |
347 |
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
348 |
< |
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
349 |
< |
TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
350 |
< |
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
351 |
< |
\AA . These crystals are unstable at 200 K and rapidly convert into |
352 |
< |
liquids. The connecting lines are qualitative visual aid.} |
348 |
> |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
349 |
> |
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
350 |
> |
with an added Ewald correction term. Error for the larger cutoff |
351 |
> |
points is equivalent to that observed at 9.0\AA\ (see Table |
352 |
> |
\ref{freeEnergy}). Data for ice I$_c$ with TIP3P using both 12 and |
353 |
> |
13.5 \AA\ cutoffs were omitted because the crystal was prone to |
354 |
> |
distortion and melting at 200 K. Ice-{\it i}$^\prime$ is the form of |
355 |
> |
Ice-{\it i} used in the SPC/E simulations.} |
356 |
|
\label{incCutoff} |
357 |
|
\end{figure} |
358 |
|
|
359 |
< |
Increasing the cutoff radius in simulations of the more |
360 |
< |
computationally efficient water models was done in order to evaluate |
361 |
< |
the trend in free energy values when moving to systems that do not |
362 |
< |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
363 |
< |
free energy of all the ice polymorphs show a substantial dependence on |
364 |
< |
cutoff radius. In general, there is a narrowing of the free energy |
365 |
< |
differences while moving to greater cutoff radius. Interestingly, by |
366 |
< |
increasing the cutoff radius, the free energy gap was narrowed enough |
367 |
< |
in the SSD/E model that the liquid state is preferred under standard |
368 |
< |
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
369 |
< |
simulations using this model choose interaction truncation radii |
370 |
< |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
371 |
< |
case of SSD/RF, indicating that the free energies calculated with a |
372 |
< |
reaction field present provide a more accurate picture of the free |
373 |
< |
energy landscape in the absence of potential truncation. |
359 |
> |
For the more computationally efficient water models, we have also |
360 |
> |
investigated the effect of potential trunctaion on the computed free |
361 |
> |
energies as a function of the cutoff radius. As seen in |
362 |
> |
Fig. \ref{incCutoff}, the free energies of the ice polymorphs with |
363 |
> |
water models lacking a long-range correction show significant cutoff |
364 |
> |
dependence. In general, there is a narrowing of the free energy |
365 |
> |
differences while moving to greater cutoff radii. As the free |
366 |
> |
energies for the polymorphs converge, the stability advantage that |
367 |
> |
Ice-{\it i} exhibits is reduced. Adjacent to each of these plots are |
368 |
> |
results for systems with applied or estimated long-range corrections. |
369 |
> |
SSD/RF was parametrized for use with a reaction field, and the benefit |
370 |
> |
provided by this computationally inexpensive correction is apparent. |
371 |
> |
The free energies are largely independent of the size of the reaction |
372 |
> |
field cavity in this model, so small cutoff radii mimic bulk |
373 |
> |
calculations quite well under SSD/RF. |
374 |
> |
|
375 |
> |
Although TIP3P was paramaterized for use without the Ewald summation, |
376 |
> |
we have estimated the effect of this method for computing long-range |
377 |
> |
electrostatics for both TIP3P and SPC/E. This was accomplished by |
378 |
> |
calculating the potential energy of identical crystals both with and |
379 |
> |
without particle mesh Ewald (PME). Similar behavior to that observed |
380 |
> |
with reaction field is seen for both of these models. The free |
381 |
> |
energies show reduced dependence on cutoff radius and span a narrower |
382 |
> |
range for the various polymorphs. Like the dipolar water models, |
383 |
> |
TIP3P displays a relatively constant preference for the Ice-{\it i} |
384 |
> |
polymorph. Crystal preference is much more difficult to determine for |
385 |
> |
SPC/E. Without a long-range correction, each of the polymorphs |
386 |
> |
studied assumes the role of the preferred polymorph under different |
387 |
> |
cutoff radii. The inclusion of the Ewald correction flattens and |
388 |
> |
narrows the gap in free energies such that the polymorphs are |
389 |
> |
isoenergetic within statistical uncertainty. This suggests that other |
390 |
> |
conditions, such as the density in fixed-volume simulations, can |
391 |
> |
influence the polymorph expressed upon crystallization. |
392 |
|
|
393 |
< |
To further study the changes resulting to the inclusion of a |
416 |
< |
long-range interaction correction, the effect of an Ewald summation |
417 |
< |
was estimated by applying the potential energy difference do to its |
418 |
< |
inclusion in systems in the presence and absence of the |
419 |
< |
correction. This was accomplished by calculation of the potential |
420 |
< |
energy of identical crystals both with and without PME. The free |
421 |
< |
energies for the investigated polymorphs using the TIP3P and SPC/E |
422 |
< |
water models are shown in Table \ref{pmeShift}. The same trend pointed |
423 |
< |
out through increase of cutoff radius is observed in these PME |
424 |
< |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
425 |
< |
for both the TIP3P and SPC/E water models; however, the narrowing of |
426 |
< |
the free energy differences between the various solid forms is |
427 |
< |
significant enough that it becomes less clear that it is the most |
428 |
< |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
429 |
< |
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
430 |
< |
as well, indicating that Ice-{\it i} might be metastable with respect |
431 |
< |
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
432 |
< |
not significantly alter the finding that the Ice-{\it i} polymorph is |
433 |
< |
a stable crystal structure that should be considered when studying the |
434 |
< |
phase behavior of water models. |
393 |
> |
\section{Conclusions} |
394 |
|
|
395 |
< |
\begin{table*} |
396 |
< |
\begin{minipage}{\linewidth} |
397 |
< |
\renewcommand{\thefootnote}{\thempfootnote} |
398 |
< |
\begin{center} |
399 |
< |
\caption{The free energy of the studied ice polymorphs after applying |
400 |
< |
the energy difference attributed to the inclusion of the PME |
401 |
< |
long-range interaction correction. Units are kcal/mol.} |
402 |
< |
\begin{tabular}{ l c c c c } |
444 |
< |
\hline |
445 |
< |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
446 |
< |
\hline |
447 |
< |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\ |
448 |
< |
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\ |
449 |
< |
\end{tabular} |
450 |
< |
\label{pmeShift} |
451 |
< |
\end{center} |
452 |
< |
\end{minipage} |
453 |
< |
\end{table*} |
395 |
> |
In this work, thermodynamic integration was used to determine the |
396 |
> |
absolute free energies of several ice polymorphs. The new polymorph, |
397 |
> |
Ice-{\it i} was observed to be the stable crystalline state for {\it |
398 |
> |
all} the water models when using a 9.0 \AA\ cutoff. However, the free |
399 |
> |
energy partially depends on simulation conditions (particularly on the |
400 |
> |
choice of long range correction method). Regardless, Ice-{\it i} was |
401 |
> |
still observered to be a stable polymorph for all of the studied water |
402 |
> |
models. |
403 |
|
|
404 |
< |
\section{Conclusions} |
404 |
> |
So what is the preferred solid polymorph for simulated water? As |
405 |
> |
indicated above, the answer appears to be dependent both on the |
406 |
> |
conditions and the model used. In the case of short cutoffs without a |
407 |
> |
long-range interaction correction, Ice-{\it i} and Ice-{\it |
408 |
> |
i}$^\prime$ have the lowest free energy of the studied polymorphs with |
409 |
> |
all the models. Ideally, crystallization of each model under constant |
410 |
> |
pressure conditions, as was done with SSD/E, would aid in the |
411 |
> |
identification of their respective preferred structures. This work, |
412 |
> |
however, helps illustrate how studies involving one specific model can |
413 |
> |
lead to insight about important behavior of others. |
414 |
|
|
415 |
< |
The free energy for proton ordered variants of hexagonal and cubic ice |
416 |
< |
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
417 |
< |
calculated under standard conditions for several common water models |
418 |
< |
via thermodynamic integration. All the water models studied show |
419 |
< |
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
420 |
< |
\AA\ switching function cutoff. Calculated melting and boiling points |
463 |
< |
show surprisingly good agreement with the experimental values; |
464 |
< |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
465 |
< |
effect of interaction truncation was investigated through variation of |
466 |
< |
the cutoff radius, use of a reaction field parameterized model, and |
467 |
< |
estimation of the results in the presence of the Ewald |
468 |
< |
summation. Interaction truncation has a significant effect on the |
469 |
< |
computed free energy values, and may significantly alter the free |
470 |
< |
energy landscape for the more complex multipoint water models. Despite |
471 |
< |
these effects, these results show Ice-{\it i} to be an important ice |
472 |
< |
polymorph that should be considered in simulation studies. |
415 |
> |
We also note that none of the water models used in this study are |
416 |
> |
polarizable or flexible models. It is entirely possible that the |
417 |
> |
polarizability of real water makes Ice-{\it i} substantially less |
418 |
> |
stable than ice I$_h$. However, the calculations presented above seem |
419 |
> |
interesting enough to communicate before the role of polarizability |
420 |
> |
(or flexibility) has been thoroughly investigated. |
421 |
|
|
422 |
< |
Due to this relative stability of Ice-{\it i} in all of the |
423 |
< |
investigated simulation conditions, the question arises as to possible |
424 |
< |
experimental observation of this polymorph. The rather extensive past |
425 |
< |
and current experimental investigation of water in the low pressure |
426 |
< |
regime makes us hesitant to ascribe any relevance of this work outside |
427 |
< |
of the simulation community. It is for this reason that we chose a |
428 |
< |
name for this polymorph which involves an imaginary quantity. That |
429 |
< |
said, there are certain experimental conditions that would provide the |
430 |
< |
most ideal situation for possible observation. These include the |
431 |
< |
negative pressure or stretched solid regime, small clusters in vacuum |
422 |
> |
Finally, due to the stability of Ice-{\it i} in the investigated |
423 |
> |
simulation conditions, the question arises as to possible experimental |
424 |
> |
observation of this polymorph. The rather extensive past and current |
425 |
> |
experimental investigation of water in the low pressure regime makes |
426 |
> |
us hesitant to ascribe any relevance to this work outside of the |
427 |
> |
simulation community. It is for this reason that we chose a name for |
428 |
> |
this polymorph which involves an imaginary quantity. That said, there |
429 |
> |
are certain experimental conditions that would provide the most ideal |
430 |
> |
situation for possible observation. These include the negative |
431 |
> |
pressure or stretched solid regime, small clusters in vacuum |
432 |
|
deposition environments, and in clathrate structures involving small |
433 |
< |
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
434 |
< |
our predictions for both the pair distribution function ($g_{OO}(r)$) |
435 |
< |
and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for |
436 |
< |
ice-{\it i} at a temperature of 77K. In studies of the high and low |
437 |
< |
density forms of amorphous ice, ``spurious'' diffraction peaks have |
438 |
< |
been observed experimentally.\cite{Bizid87} It is possible that a |
439 |
< |
variant of Ice-{\it i} could explain some of this behavior; however, |
440 |
< |
we will leave it to our experimental colleagues to make the final |
441 |
< |
determination on whether this ice polymorph is named appropriately |
442 |
< |
(i.e. with an imaginary number) or if it can be promoted to Ice-0. |
433 |
> |
non-polar molecules. For the purpose of comparison with experimental |
434 |
> |
results, we have calculated the oxygen-oxygen pair correlation |
435 |
> |
function, $g_{OO}(r)$, and the structure factor, $S(\vec{q})$ for the |
436 |
> |
two Ice-{\it i} variants (along with example ice I$_h$ and I$_c$ |
437 |
> |
plots) at 77K, and they are shown in figures \ref{fig:gofr} and |
438 |
> |
\ref{fig:sofq} respectively. It is interesting to note that the |
439 |
> |
structure factors for Ice-{\it i}$^\prime$ and Ice-I$_c$ are quite similar. |
440 |
> |
The primary differences are small peaks at 1.125, 2.29, and 2.53 |
441 |
> |
\AA${-1}$, so particular attention to these regions would be needed |
442 |
> |
to identify the new {\it i}$^\prime$ variant from the I$_{c}$ variant. |
443 |
|
|
444 |
|
\begin{figure} |
445 |
+ |
\centering |
446 |
|
\includegraphics[width=\linewidth]{iceGofr.eps} |
447 |
< |
\caption{Radial distribution functions of ice $I_h$, $I_c$, |
448 |
< |
Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations |
449 |
< |
of the SSD/RF water model at 77 K.} |
447 |
> |
\caption{Radial distribution functions of ice I$_h$, I$_c$, and |
448 |
> |
Ice-{\it i} calculated from from simulations of the SSD/RF water model |
449 |
> |
at 77 K. The Ice-{\it i} distribution function was obtained from |
450 |
> |
simulations composed of TIP4P water.} |
451 |
|
\label{fig:gofr} |
452 |
|
\end{figure} |
453 |
|
|
454 |
|
\begin{figure} |
455 |
+ |
\centering |
456 |
|
\includegraphics[width=\linewidth]{sofq.eps} |
457 |
< |
\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, |
457 |
> |
\caption{Predicted structure factors for ice I$_h$, I$_c$, Ice-{\it i}, |
458 |
|
and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
459 |
|
been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
460 |
|
width) to compensate for the trunction effects in our finite size |
470 |
|
|
471 |
|
\newpage |
472 |
|
|
473 |
< |
\bibliographystyle{jcp} |
473 |
> |
\bibliographystyle{achemso} |
474 |
|
\bibliography{iceiPaper} |
475 |
|
|
476 |
|
|