33 |
|
%\doublespacing |
34 |
|
|
35 |
|
\begin{abstract} |
36 |
< |
The free energies of several ice polymorphs in the low pressure regime |
37 |
< |
were calculated using thermodynamic integration. These integrations |
38 |
< |
were done for most of the common water models. Ice-{\it i}, a |
39 |
< |
structure we recently observed to be stable in one of the single-point |
40 |
< |
water models, was determined to be the stable crystalline state (at 1 |
41 |
< |
atm) for {\it all} the water models investigated. Phase diagrams were |
36 |
> |
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 |
|
generated, and phase coexistence lines were determined for all of the |
45 |
< |
known low-pressure ice structures under all of the common water |
46 |
< |
models. Additionally, potential truncation was shown to have an |
47 |
< |
effect on the calculated free energies, and can result in altered free |
48 |
< |
energy landscapes. |
45 |
> |
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 predictions for the new crystal were |
49 |
> |
generated and we await experimental confirmation of the existence of |
50 |
> |
this new polymorph. |
51 |
|
\end{abstract} |
52 |
|
|
53 |
|
%\narrowtext |
58 |
|
|
59 |
|
\section{Introduction} |
60 |
|
|
57 |
– |
Computer simulations are a valuable tool for studying the phase |
58 |
– |
behavior of systems ranging from small or simple molecules to complex |
59 |
– |
biological species.\cite{Matsumoto02,Sanz04,Marrink01} Useful techniques |
60 |
– |
have been developed to investigate the thermodynamic properites of |
61 |
– |
model substances, providing both qualitative and quantitative |
62 |
– |
comparisons between simulations and |
63 |
– |
experiment.\cite{Widom63,Frenkel84} Investigation of these properties |
64 |
– |
leads to the development of new and more accurate models, leading to |
65 |
– |
better understanding and depiction of physical processes and intricate |
66 |
– |
molecular systems. |
67 |
– |
|
61 |
|
Water has proven to be a challenging substance to depict in |
62 |
|
simulations, and a variety of models have been developed to describe |
63 |
|
its behavior under varying simulation |
64 |
< |
conditions.\cite{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04} |
64 |
> |
conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04} |
65 |
|
These models have been used to investigate important physical |
66 |
< |
phenomena like phase transitions, molecule transport, and the |
66 |
> |
phenomena like phase transitions, transport properties, and the |
67 |
|
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 |
< |
important property to quantify are the Gibbs and Helmholtz free |
72 |
> |
important properties to quantify are the Gibbs and Helmholtz free |
73 |
|
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 |
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 |
< |
these phases will dictate the true transition temperatures and |
80 |
> |
these phases will dictate the relevant transition temperatures and |
81 |
|
pressures for the model. |
82 |
|
|
83 |
|
In this paper, standard reference state methods were applied to known |
84 |
|
crystalline water polymorphs in the low pressure regime. This work is |
85 |
< |
unique in the fact that one of the crystal lattices was arrived at |
86 |
< |
through crystallization of a computationally efficient water model |
87 |
< |
under constant pressure and temperature conditions. Crystallization |
88 |
< |
events are interesting in and of |
89 |
< |
themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure |
90 |
< |
obtained in this case is different from any previously observed ice |
91 |
< |
polymorphs in experiment or simulation.\cite{Fennell04} We have named |
92 |
< |
this structure Ice-{\it i} to indicate its origin in computational |
93 |
< |
simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight |
94 |
< |
water molecules that stack in rows of interlocking water |
95 |
< |
tetramers. Proton ordering can be accomplished by orienting two of the |
96 |
< |
molecules so that both of their donated hydrogen bonds are internal to |
97 |
< |
their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal |
98 |
< |
constructed of water tetramers, the hydrogen bonds are not as linear |
99 |
< |
as those observed in ice $I_h$, however the interlocking of these |
100 |
< |
subunits appears to provide significant stabilization to the overall |
101 |
< |
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 |
85 |
> |
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. This crystal structure has a |
95 |
> |
limited resemblence to a recent two-dimensional ice tessellation |
96 |
> |
simulated on a silica surface.\cite{Yang04} Proton ordering can be |
97 |
> |
accomplished by orienting two of the molecules so that both of their |
98 |
> |
donated hydrogen bonds are internal to their tetramer |
99 |
> |
(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
100 |
> |
water tetramers, the hydrogen bonds are not as linear as those |
101 |
> |
observed in ice $I_h$, however the interlocking of these subunits |
102 |
> |
appears to provide significant stabilization to the overall crystal. |
103 |
> |
The arrangement of these tetramers results in surrounding open |
104 |
> |
octagonal cavities that are typically greater than 6.3 \AA\ in |
105 |
> |
diameter. This relatively open overall structure leads to crystals |
106 |
|
that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. |
107 |
|
|
108 |
|
\begin{figure} |
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 |
110 |
> |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-{\it i}$^\prime$, |
111 |
> |
the 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.} |
118 |
|
\begin{figure} |
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 |
121 |
> |
down the (001) crystal face. The rows of water tetramers surrounded |
122 |
> |
by octagonal pores leads to a crystal structure that is significantly |
123 |
|
less dense than ice $I_h$.} |
124 |
|
\label{protOrder} |
125 |
|
\end{figure} |
126 |
|
|
127 |
|
Results from our previous study indicated that Ice-{\it i} is the |
128 |
|
minimum energy crystal structure for the single point water models we |
129 |
< |
investigated (for discussions on these single point dipole models, see |
130 |
< |
our previous work and related |
129 |
> |
had investigated (for discussions on these single point dipole models, |
130 |
> |
see our previous work and related |
131 |
|
articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
132 |
|
considered energetic stabilization and neglected entropic |
133 |
< |
contributions to the overall free energy. To address this issue, the |
134 |
< |
absolute free energy of this crystal was calculated using |
133 |
> |
contributions to the overall free energy. To address this issue, we |
134 |
> |
have calculated the absolute free energy of this crystal using |
135 |
|
thermodynamic integration and compared to the free energies of cubic |
136 |
|
and hexagonal ice $I$ (the experimental low density ice polymorphs) |
137 |
|
and ice B (a higher density, but very stable crystal structure |
140 |
|
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
141 |
|
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
142 |
|
field parametrized single point dipole water model (SSD/RF). It should |
143 |
< |
be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used |
144 |
< |
in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of |
145 |
< |
this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit |
146 |
< |
it is extended in the direction of the (001) face and compressed along |
147 |
< |
the other two faces. |
143 |
> |
be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$) |
144 |
> |
was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit |
145 |
> |
cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it |
146 |
> |
i} unit it is extended in the direction of the (001) face and |
147 |
> |
compressed along the other two faces. There is typically a small |
148 |
> |
distortion of proton ordered Ice-{\it i}$^\prime$ that converts the |
149 |
> |
normally square tetramer into a rhombus with alternating approximately |
150 |
> |
85 and 95 degree angles. The degree of this distortion is model |
151 |
> |
dependent and significant enough to split the tetramer diagonal |
152 |
> |
location peak in the radial distribution function. |
153 |
|
|
154 |
|
\section{Methods} |
155 |
|
|
156 |
|
Canonical ensemble (NVT) molecular dynamics calculations were |
157 |
|
performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
158 |
|
All molecules were treated as rigid bodies, with orientational motion |
159 |
< |
propagated using the symplectic DLM integration method. Details about |
159 |
> |
propagated using the symplectic DLM integration method. Details about |
160 |
|
the implementation of this technique can be found in a recent |
161 |
|
publication.\cite{Dullweber1997} |
162 |
|
|
168 |
|
crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
169 |
|
SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
170 |
|
and 400 K for all of these water models were also determined using |
171 |
< |
this same technique in order to determine melting points and generate |
172 |
< |
phase diagrams. All simulations were carried out at densities |
173 |
< |
resulting in a pressure of approximately 1 atm at their respective |
174 |
< |
temperatures. |
171 |
> |
this same technique in order to determine melting points and to |
172 |
> |
generate phase diagrams. All simulations were carried out at |
173 |
> |
densities which correspond to a pressure of approximately 1 atm at |
174 |
> |
their respective temperatures. |
175 |
|
|
176 |
< |
A single thermodynamic integration involves a sequence of simulations |
177 |
< |
over which the system of interest is converted into a reference system |
178 |
< |
for which the free energy is known analytically. This transformation |
179 |
< |
path is then integrated in order to determine the free energy |
180 |
< |
difference between the two states: |
176 |
> |
Thermodynamic integration involves a sequence of simulations during |
177 |
> |
which the system of interest is converted into a reference system for |
178 |
> |
which the free energy is known analytically. This transformation path |
179 |
> |
is then integrated in order to determine the free energy difference |
180 |
> |
between the two states: |
181 |
|
\begin{equation} |
182 |
|
\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
183 |
|
)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
184 |
|
\end{equation} |
185 |
|
where $V$ is the interaction potential and $\lambda$ is the |
186 |
< |
transformation parameter that scales the overall |
187 |
< |
potential. Simulations are distributed strategically along this path |
188 |
< |
in order to sufficiently sample the regions of greatest change in the |
189 |
< |
potential. Typical integrations in this study consisted of $\sim$25 |
190 |
< |
simulations ranging from 300 ps (for the unaltered system) to 75 ps |
191 |
< |
(near the reference state) in length. |
186 |
> |
transformation parameter that scales the overall potential. |
187 |
> |
Simulations are distributed strategically along this path in order to |
188 |
> |
sufficiently sample the regions of greatest change in the potential. |
189 |
> |
Typical integrations in this study consisted of $\sim$25 simulations |
190 |
> |
ranging from 300 ps (for the unaltered system) to 75 ps (near the |
191 |
> |
reference state) in length. |
192 |
|
|
193 |
|
For the thermodynamic integration of molecular crystals, the Einstein |
194 |
< |
crystal was chosen as the reference system. In an Einstein crystal, |
194 |
> |
crystal was chosen as the reference system. In an Einstein crystal, |
195 |
|
the molecules are restrained at their ideal lattice locations and |
196 |
|
orientations. Using harmonic restraints, as applied by B\`{a}ez and |
197 |
|
Clancy, the total potential for this reference crystal |
203 |
|
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
204 |
|
the spring constants restraining translational motion and deflection |
205 |
|
of and rotation around the principle axis of the molecule |
206 |
< |
respectively. It is clear from Fig. \ref{waterSpring} that the values |
207 |
< |
of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
208 |
< |
$-\pi$ to $\pi$. The partition function for a molecular crystal |
206 |
> |
respectively. These spring constants are typically calculated from |
207 |
> |
the mean-square displacements of water molecules in an unrestrained |
208 |
> |
ice crystal at 200 K. For these studies, $K_\mathrm{r} = 4.29$ kcal |
209 |
> |
mol$^{-1}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$, and $K_\omega\ = |
210 |
> |
17.75$ kcal mol$^{-1}$. It is clear from Fig. \ref{waterSpring} that |
211 |
> |
the values of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges |
212 |
> |
from $-\pi$ to $\pi$. The partition function for a molecular crystal |
213 |
|
restrained in this fashion can be evaluated analytically, and the |
214 |
|
Helmholtz Free Energy ({\it A}) is given by |
215 |
|
\begin{eqnarray} |
231 |
|
\caption{Possible orientational motions for a restrained molecule. |
232 |
|
$\theta$ angles correspond to displacement from the body-frame {\it |
233 |
|
z}-axis, while $\omega$ angles correspond to rotation about the |
234 |
< |
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
234 |
> |
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
235 |
|
constants for the harmonic springs restraining motion in the $\theta$ |
236 |
|
and $\omega$ directions.} |
237 |
|
\label{waterSpring} |
243 |
|
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
244 |
|
typically differ in regard to the path taken for switching off the |
245 |
|
interaction potential to convert the system to an ideal gas of water |
246 |
< |
molecules. In this study, we apply of one of the most convenient |
247 |
< |
methods and integrate over the $\lambda^4$ path, where all interaction |
248 |
< |
parameters are scaled equally by this transformation parameter. This |
249 |
< |
method has been shown to be reversible and provide results in |
250 |
< |
excellent agreement with other established methods.\cite{Baez95b} |
246 |
> |
molecules. In this study, we applied of one of the most convenient |
247 |
> |
methods and integrated over the $\lambda^4$ path, where all |
248 |
> |
interaction parameters are scaled equally by this transformation |
249 |
> |
parameter. This method has been shown to be reversible and provide |
250 |
> |
results in excellent agreement with other established |
251 |
> |
methods.\cite{Baez95b} |
252 |
|
|
253 |
|
Charge, dipole, and Lennard-Jones interactions were modified by a |
254 |
< |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
255 |
< |
). By applying this function, these interactions are smoothly |
256 |
< |
truncated, thereby avoiding the poor energy conservation which results |
257 |
< |
from harsher truncation schemes. The effect of a long-range correction |
258 |
< |
was also investigated on select model systems in a variety of |
259 |
< |
manners. For the SSD/RF model, a reaction field with a fixed |
260 |
< |
dielectric constant of 80 was applied in all |
261 |
< |
simulations.\cite{Onsager36} For a series of the least computationally |
262 |
< |
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
263 |
< |
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
264 |
< |
\AA\ cutoff results. Finally, results from the use of an Ewald |
265 |
< |
summation were estimated for TIP3P and SPC/E by performing |
266 |
< |
calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
267 |
< |
mechanics software package.\cite{Tinker} The calculated energy |
268 |
< |
difference in the presence and absence of PME was applied to the |
269 |
< |
previous results in order to predict changes to the free energy |
265 |
< |
landscape. |
254 |
> |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA). |
255 |
> |
By applying this function, these interactions are smoothly truncated, |
256 |
> |
thereby avoiding the poor energy conservation which results from |
257 |
> |
harsher truncation schemes. The effect of a long-range correction was |
258 |
> |
also investigated on select model systems in a variety of manners. |
259 |
> |
For the SSD/RF model, a reaction field with a fixed dielectric |
260 |
> |
constant of 80 was applied in all simulations.\cite{Onsager36} For a |
261 |
> |
series of the least computationally expensive models (SSD/E, SSD/RF, |
262 |
> |
and TIP3P), simulations were performed with longer cutoffs of 12 and |
263 |
> |
15 \AA\ to compare with the 9 \AA\ cutoff results. Finally, the |
264 |
> |
effects of utilizing an Ewald summation were estimated for TIP3P and |
265 |
> |
SPC/E by performing single configuration calculations with |
266 |
> |
Particle-Mesh Ewald (PME) in the TINKER molecular mechanics software |
267 |
> |
package.\cite{Tinker} The calculated energy difference in the presence |
268 |
> |
and absence of PME was applied to the previous results in order to |
269 |
> |
predict changes to the free energy landscape. |
270 |
|
|
271 |
|
\section{Results and discussion} |
272 |
|
|
273 |
< |
The free energy of proton ordered Ice-{\it i} was calculated and |
273 |
> |
The free energy of proton-ordered Ice-{\it i} was calculated and |
274 |
|
compared with the free energies of proton ordered variants of the |
275 |
|
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
276 |
|
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
278 |
|
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
279 |
|
Ice XI, the experimentally-observed proton-ordered variant of ice |
280 |
|
$I_h$, was investigated initially, but was found to be not as stable |
281 |
< |
as proton disordered or antiferroelectric variants of ice $I_h$. The |
281 |
> |
as proton disordered or antiferroelectric variants of ice $I_h$. The |
282 |
|
proton ordered variant of ice $I_h$ used here is a simple |
283 |
< |
antiferroelectric version that has an 8 molecule unit |
284 |
< |
cell.\cite{Davidson84} The crystals contained 648 or 1728 molecules |
285 |
< |
for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for |
286 |
< |
ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
287 |
< |
were necessary for simulations involving larger cutoff values. |
283 |
> |
antiferroelectric version that we devised, and it has an 8 molecule |
284 |
> |
unit cell similar to other predicted antiferroelectric $I_h$ |
285 |
> |
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
286 |
> |
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
287 |
> |
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The |
288 |
> |
larger crystal sizes were necessary for simulations involving larger |
289 |
> |
cutoff values. |
290 |
|
|
291 |
|
\begin{table*} |
292 |
|
\begin{minipage}{\linewidth} |
287 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
293 |
|
\begin{center} |
294 |
+ |
|
295 |
|
\caption{Calculated free energies for several ice polymorphs with a |
296 |
< |
variety of common water models. All calculations used a cutoff radius |
297 |
< |
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
298 |
< |
kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
299 |
< |
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
300 |
< |
\begin{tabular}{ l c c c c } |
296 |
> |
variety of common water models. All calculations used a cutoff radius |
297 |
> |
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
298 |
> |
kcal/mol. Calculated error of the final digits is in parentheses.} |
299 |
> |
|
300 |
> |
\begin{tabular}{lcccc} |
301 |
|
\hline |
302 |
|
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
303 |
|
\hline |
304 |
< |
TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\\ |
305 |
< |
TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\ |
306 |
< |
TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\ |
307 |
< |
SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\ |
308 |
< |
SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\ |
309 |
< |
SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\ |
304 |
> |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
305 |
> |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
306 |
> |
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
307 |
> |
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & -13.55(2)\\ |
308 |
> |
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
309 |
> |
SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2)\\ |
310 |
|
\end{tabular} |
311 |
|
\label{freeEnergy} |
312 |
|
\end{center} |
315 |
|
|
316 |
|
The free energy values computed for the studied polymorphs indicate |
317 |
|
that Ice-{\it i} is the most stable state for all of the common water |
318 |
< |
models studied. With the free energy at these state points, the |
319 |
< |
Gibbs-Helmholtz equation was used to project to other state points and |
320 |
< |
to build phase diagrams. Figures |
321 |
< |
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
322 |
< |
from the free energy results. All other models have similar structure, |
323 |
< |
although the crossing points between the phases exist at slightly |
324 |
< |
different temperatures and pressures. It is interesting to note that |
325 |
< |
ice $I$ does not exist in either cubic or hexagonal form in any of the |
326 |
< |
phase diagrams for any of the models. For purposes of this study, ice |
327 |
< |
B is representative of the dense ice polymorphs. A recent study by |
328 |
< |
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
329 |
< |
TIP4P in the high pressure regime.\cite{Sanz04} |
318 |
> |
models studied. With the calculated free energy at these state |
319 |
> |
points, the Gibbs-Helmholtz equation was used to project to other |
320 |
> |
state points and to build phase diagrams. Figures \ref{tp3phasedia} |
321 |
> |
and \ref{ssdrfphasedia} are example diagrams built from the free |
322 |
> |
energy results. All other models have similar structure, although the |
323 |
> |
crossing points between the phases move to slightly different |
324 |
> |
temperatures and pressures. It is interesting to note that ice $I$ |
325 |
> |
does not exist in either cubic or hexagonal form in any of the phase |
326 |
> |
diagrams for any of the models. For purposes of this study, ice B is |
327 |
> |
representative of the dense ice polymorphs. A recent study by Sanz |
328 |
> |
{\it et al.} goes into detail on the phase diagrams for SPC/E and |
329 |
> |
TIP4P at higher pressures than those studied here.\cite{Sanz04} |
330 |
|
|
331 |
|
\begin{figure} |
332 |
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
333 |
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
334 |
< |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
334 |
> |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
335 |
|
the experimental values; however, the solid phases shown are not the |
336 |
< |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
336 |
> |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
337 |
|
higher in energy and don't appear in the phase diagram.} |
338 |
|
\label{tp3phasedia} |
339 |
|
\end{figure} |
341 |
|
\begin{figure} |
342 |
|
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
343 |
|
\caption{Phase diagram for the SSD/RF water model in the low pressure |
344 |
< |
regime. Calculations producing these results were done under an |
345 |
< |
applied reaction field. It is interesting to note that this |
344 |
> |
regime. Calculations producing these results were done under an |
345 |
> |
applied reaction field. It is interesting to note that this |
346 |
|
computationally efficient model (over 3 times more efficient than |
347 |
|
TIP3P) exhibits phase behavior similar to the less computationally |
348 |
|
conservative charge based models.} |
351 |
|
|
352 |
|
\begin{table*} |
353 |
|
\begin{minipage}{\linewidth} |
348 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
354 |
|
\begin{center} |
355 |
+ |
|
356 |
|
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
357 |
|
temperatures at 1 atm for several common water models compared with |
358 |
< |
experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
359 |
< |
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
358 |
> |
experiment. The $T_m$ and $T_s$ values from simulation correspond to |
359 |
> |
a transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
360 |
|
liquid or gas state.} |
361 |
< |
\begin{tabular}{ l c c c c c c c } |
361 |
> |
|
362 |
> |
\begin{tabular}{lccccccc} |
363 |
|
\hline |
364 |
< |
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
364 |
> |
Equilibrium Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
365 |
|
\hline |
366 |
< |
$T_m$ (K) & 269(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\ |
367 |
< |
$T_b$ (K) & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\ |
368 |
< |
$T_s$ (K) & - & - & - & - & 355(3) & - & -\\ |
366 |
> |
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
367 |
> |
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
368 |
> |
$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ |
369 |
|
\end{tabular} |
370 |
|
\label{meltandboil} |
371 |
|
\end{center} |
373 |
|
\end{table*} |
374 |
|
|
375 |
|
Table \ref{meltandboil} lists the melting and boiling temperatures |
376 |
< |
calculated from this work. Surprisingly, most of these models have |
377 |
< |
melting points that compare quite favorably with experiment. The |
376 |
> |
calculated from this work. Surprisingly, most of these models have |
377 |
> |
melting points that compare quite favorably with experiment. The |
378 |
|
unfortunate aspect of this result is that this phase change occurs |
379 |
|
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
380 |
< |
liquid state. These results are actually not contrary to previous |
381 |
< |
studies in the literature. Earlier free energy studies of ice $I$ |
382 |
< |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
383 |
< |
being attributed to choice of interaction truncation and different |
377 |
< |
ordered and disordered molecular |
380 |
> |
liquid state. These results are actually not contrary to other |
381 |
> |
studies. Studies of ice $I_h$ using TIP4P predict a $T_m$ ranging |
382 |
> |
from 214 to 238 K (differences being attributed to choice of |
383 |
> |
interaction truncation and different ordered and disordered molecular |
384 |
|
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
385 |
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
386 |
< |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
387 |
< |
calculated at 265 K, significantly higher in temperature than the |
388 |
< |
previous studies. Also of interest in these results is that SSD/E does |
386 |
> |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
387 |
> |
calculated to be 265 K, indicating that these simulation based |
388 |
> |
structures ought to be included in studies probing phase transitions |
389 |
> |
with this model. Also of interest in these results is that SSD/E does |
390 |
|
not exhibit a melting point at 1 atm, but it shows a sublimation point |
391 |
< |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
392 |
< |
all other polymorphs for this particular model under these |
393 |
< |
conditions. While troubling, this behavior turned out to be |
394 |
< |
advantageous in that it facilitated the spontaneous crystallization of |
395 |
< |
Ice-{\it i}. These observations provide a warning that simulations of |
391 |
> |
at 355 K. This is due to the significant stability of Ice-{\it i} |
392 |
> |
over all other polymorphs for this particular model under these |
393 |
> |
conditions. While troubling, this behavior resulted in spontaneous |
394 |
> |
crystallization of Ice-{\it i} and led us to investigate this |
395 |
> |
structure. These observations provide a warning that simulations of |
396 |
|
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
397 |
< |
risk of spontaneous crystallization. However, this risk changes when |
397 |
> |
risk of spontaneous crystallization. However, this risk lessens when |
398 |
|
applying a longer cutoff. |
399 |
|
|
400 |
|
\begin{figure} |
401 |
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
402 |
< |
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
403 |
< |
TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
404 |
< |
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
405 |
< |
\AA . These crystals are unstable at 200 K and rapidly convert into |
406 |
< |
liquids. The connecting lines are qualitative visual aid.} |
402 |
> |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
403 |
> |
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
404 |
> |
with an added Ewald correction term. Calculations performed without a |
405 |
> |
long-range correction show noticable free energy dependence on the |
406 |
> |
cutoff radius and show some degree of converge at large cutoff radii. |
407 |
> |
Inclusion of a long-range correction reduces the cutoff radius |
408 |
> |
dependence of the free energy for all the models. Data for ice I$_c$ |
409 |
> |
with TIP3P using both 12 and 13.5 \AA\ cutoffs were omitted because |
410 |
> |
the crystal was prone to distortion and melting at 200 K. Ice-{\it |
411 |
> |
i}$^\prime$ is the form of Ice-{\it i} used in the SPC/E simulations.} |
412 |
|
\label{incCutoff} |
413 |
|
\end{figure} |
414 |
|
|
415 |
|
Increasing the cutoff radius in simulations of the more |
416 |
|
computationally efficient water models was done in order to evaluate |
417 |
|
the trend in free energy values when moving to systems that do not |
418 |
< |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
419 |
< |
free energy of all the ice polymorphs show a substantial dependence on |
420 |
< |
cutoff radius. In general, there is a narrowing of the free energy |
421 |
< |
differences while moving to greater cutoff radius. Interestingly, by |
422 |
< |
increasing the cutoff radius, the free energy gap was narrowed enough |
423 |
< |
in the SSD/E model that the liquid state is preferred under standard |
424 |
< |
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
418 |
> |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
419 |
> |
free energy of the ice polymorphs with water models lacking a |
420 |
> |
long-range correction show a significant cutoff radius dependence. In |
421 |
> |
general, there is a narrowing of the free energy differences while |
422 |
> |
moving to greater cutoff radii. As the free energies for the |
423 |
> |
polymorphs converge, the stability advantage that Ice-{\it i} exhibits |
424 |
> |
is reduced. Interestingly, increasing the cutoff radius a mere 1.5 |
425 |
> |
\AA\ with the SSD/E model destabilizes the Ice-{\it i} polymorph |
426 |
> |
enough that the liquid state is preferred under standard simulation |
427 |
> |
conditions (298 K and 1 atm). Thus, it is recommended that |
428 |
|
simulations using this model choose interaction truncation radii |
429 |
< |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
430 |
< |
case of SSD/RF, indicating that the free energies calculated with a |
431 |
< |
reaction field present provide a more accurate picture of the free |
432 |
< |
energy landscape in the absence of potential truncation. |
429 |
> |
greater than 9 \AA. Considering the stabilization of Ice-{\it i} with |
430 |
> |
smaller cutoffs, it is not surprising that crystallization was |
431 |
> |
observed with SSD/E. The choice of a 9 \AA\ cutoff in the previous |
432 |
> |
simulations gives the Ice-{\it i} polymorph a greater than 1 kcal/mol |
433 |
> |
lower free energy than the ice $I_\textrm{h}$ starting configurations. |
434 |
> |
Additionally, it should be noted that ice $I_c$ is not stable with |
435 |
> |
cutoff radii of 12 and 13.5 \AA\ using the TIP3P water model. These |
436 |
> |
simulations showed bulk distortions of the simulation cell that |
437 |
> |
rapidly deteriorated crystal integrity. |
438 |
|
|
439 |
< |
To further study the changes resulting to the inclusion of a |
440 |
< |
long-range interaction correction, the effect of an Ewald summation |
441 |
< |
was estimated by applying the potential energy difference do to its |
442 |
< |
inclusion in systems in the presence and absence of the |
443 |
< |
correction. This was accomplished by calculation of the potential |
444 |
< |
energy of identical crystals with and without PME using TINKER. The |
445 |
< |
free energies for the investigated polymorphs using the TIP3P and |
446 |
< |
SPC/E water models are shown in Table \ref{pmeShift}. The same trend |
447 |
< |
pointed out through increase of cutoff radius is observed in these PME |
448 |
< |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
449 |
< |
for both the TIP3P and SPC/E water models; however, the narrowing of |
450 |
< |
the free energy differences between the various solid forms is |
451 |
< |
significant enough that it becomes less clear that it is the most |
452 |
< |
stable polymorph. The free energies of Ice-{\it i} and ice B overlap |
453 |
< |
within error, with ice $I_c$ just outside, indicating that Ice-{\it i} |
454 |
< |
might be metastable with respect to ice B and possibly ice $I_c$ in |
455 |
< |
the SPC/E water model. However, these results do not significantly |
456 |
< |
alter the finding that the Ice-{\it i} polymorph is a stable crystal |
457 |
< |
structure that should be considered when studying the phase behavior |
458 |
< |
of water models. |
439 |
> |
Adjacent to each of these model plots is a system with an applied or |
440 |
> |
estimated long-range correction. SSD/RF was parametrized for use with |
441 |
> |
a reaction field, and the benefit provided by this computationally |
442 |
> |
inexpensive correction is apparent. Due to the relative independence |
443 |
> |
of the resultant free energies, calculations performed with a small |
444 |
> |
cutoff radius provide resultant properties similar to what one would |
445 |
> |
expect for the bulk material. In the cases of TIP3P and SPC/E, the |
446 |
> |
effect of an Ewald summation was estimated by applying the potential |
447 |
> |
energy difference do to its inclusion in systems in the presence and |
448 |
> |
absence of the correction. This was accomplished by calculation of |
449 |
> |
the potential energy of identical crystals both with and without |
450 |
> |
particle mesh Ewald (PME). Similar behavior to that observed with |
451 |
> |
reaction field is seen for both of these models. The free energies |
452 |
> |
show less dependence on cutoff radius and span a more narrowed range |
453 |
> |
for the various polymorphs. Like the dipolar water models, TIP3P |
454 |
> |
displays a relatively constant preference for the Ice-{\it i} |
455 |
> |
polymorph. Crystal preference is much more difficult to determine for |
456 |
> |
SPC/E. Without a long-range correction, each of the polymorphs |
457 |
> |
studied assumes the role of the preferred polymorph under different |
458 |
> |
cutoff conditions. The inclusion of the Ewald correction flattens and |
459 |
> |
narrows the sequences of free energies so much that they often overlap |
460 |
> |
within error, indicating that other conditions, such as cell volume in |
461 |
> |
microcanonical simulations, can influence the chosen polymorph upon |
462 |
> |
crystallization. All of these results support the finding that the |
463 |
> |
Ice-{\it i} polymorph is a stable crystal structure that should be |
464 |
> |
considered when studying the phase behavior of water models. |
465 |
|
|
466 |
|
\begin{table*} |
467 |
|
\begin{minipage}{\linewidth} |
442 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
468 |
|
\begin{center} |
469 |
< |
\caption{The free energy of the studied ice polymorphs after applying |
470 |
< |
the energy difference attributed to the inclusion of the PME |
471 |
< |
long-range interaction correction. Units are kcal/mol.} |
472 |
< |
\begin{tabular}{ l c c c c } |
469 |
> |
|
470 |
> |
\caption{The free energy versus cutoff radius for the studied ice |
471 |
> |
polymorphs using SPC/E after the inclusion of the PME long-range |
472 |
> |
interaction correction. Units are kcal/mol.} |
473 |
> |
|
474 |
> |
\begin{tabular}{ccccc} |
475 |
|
\hline |
476 |
< |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
476 |
> |
Cutoff (\AA) & $I_h$ & $I_c$ & B & Ice-{\it i} \\ |
477 |
|
\hline |
478 |
< |
TIP3P & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)\\ |
479 |
< |
SPC/E & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\ |
478 |
> |
9.0 & -12.98(2) & -13.00(2) & -12.97(3) & -13.02(2) \\ |
479 |
> |
10.5 & -13.13(3) & -13.09(4) & -13.17(3) & -13.11(2) \\ |
480 |
> |
12.0 & -13.06(2) & -13.09(2) & -13.15(4) & -13.12(2) \\ |
481 |
> |
13.5 & -13.02(2) & -13.02(2) & -13.08(2) & -13.07(2) \\ |
482 |
> |
15.0 & -13.11(4) & -12.97(2) & -13.09(2) & -12.95(2) \\ |
483 |
|
\end{tabular} |
484 |
|
\label{pmeShift} |
485 |
|
\end{center} |
489 |
|
\section{Conclusions} |
490 |
|
|
491 |
|
The free energy for proton ordered variants of hexagonal and cubic ice |
492 |
< |
$I$, ice B, and recently discovered Ice-{\it i} were calculated under |
493 |
< |
standard conditions for several common water models via thermodynamic |
494 |
< |
integration. All the water models studied show Ice-{\it i} to be the |
495 |
< |
minimum free energy crystal structure in the with a 9 \AA\ switching |
496 |
< |
function cutoff. Calculated melting and boiling points show |
497 |
< |
surprisingly good agreement with the experimental values; however, the |
498 |
< |
solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
499 |
< |
interaction truncation was investigated through variation of the |
500 |
< |
cutoff radius, use of a reaction field parameterized model, and |
501 |
< |
estimation of the results in the presence of the Ewald |
502 |
< |
summation. Interaction truncation has a significant effect on the |
503 |
< |
computed free energy values, and may significantly alter the free |
504 |
< |
energy landscape for the more complex multipoint water models. Despite |
505 |
< |
these effects, these results show Ice-{\it i} to be an important ice |
506 |
< |
polymorph that should be considered in simulation studies. |
492 |
> |
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
493 |
> |
calculated under standard conditions for several common water models |
494 |
> |
via thermodynamic integration. All the water models studied show |
495 |
> |
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
496 |
> |
\AA\ switching function cutoff. Calculated melting and boiling points |
497 |
> |
show surprisingly good agreement with the experimental values; |
498 |
> |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
499 |
> |
effect of interaction truncation was investigated through variation of |
500 |
> |
the cutoff radius, use of a reaction field parameterized model, and |
501 |
> |
estimation of the results in the presence of the Ewald summation. |
502 |
> |
Interaction truncation has a significant effect on the computed free |
503 |
> |
energy values, and may significantly alter the free energy landscape |
504 |
> |
for the more complex multipoint water models. Despite these effects, |
505 |
> |
these results show Ice-{\it i} to be an important ice polymorph that |
506 |
> |
should be considered in simulation studies. |
507 |
|
|
508 |
< |
Due to this relative stability of Ice-{\it i} in all manner of |
509 |
< |
investigated simulation examples, the question arises as to possible |
508 |
> |
Due to this relative stability of Ice-{\it i} in all of the |
509 |
> |
investigated simulation conditions, the question arises as to possible |
510 |
|
experimental observation of this polymorph. The rather extensive past |
511 |
|
and current experimental investigation of water in the low pressure |
512 |
|
regime makes us hesitant to ascribe any relevance of this work outside |
518 |
|
deposition environments, and in clathrate structures involving small |
519 |
|
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
520 |
|
our predictions for both the pair distribution function ($g_{OO}(r)$) |
521 |
< |
and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it |
522 |
< |
i} at a temperature of 77K. In a quick comparison of the predicted |
523 |
< |
S(q) for Ice-{\it i} and experimental studies of amorphous solid |
524 |
< |
water, it is possible that some of the ``spurious'' peaks that could |
525 |
< |
not be assigned in HDA could correspond to peaks labeled in this |
526 |
< |
S(q).\cite{Bizid87} It should be noted that there is typically poor |
527 |
< |
agreement on crystal densities between simulation and experiment, so |
528 |
< |
such peak comparisons should be made with caution. We will leave it |
499 |
< |
to our experimental colleagues to determine whether this ice polymorph |
500 |
< |
is named appropriately or if it should be promoted to Ice-0. |
521 |
> |
and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for |
522 |
> |
ice-{\it i} at a temperature of 77K. In studies of the high and low |
523 |
> |
density forms of amorphous ice, ``spurious'' diffraction peaks have |
524 |
> |
been observed experimentally.\cite{Bizid87} It is possible that a |
525 |
> |
variant of Ice-{\it i} could explain some of this behavior; however, |
526 |
> |
we will leave it to our experimental colleagues to make the final |
527 |
> |
determination on whether this ice polymorph is named appropriately |
528 |
> |
(i.e. with an imaginary number) or if it can be promoted to Ice-0. |
529 |
|
|
530 |
|
\begin{figure} |
531 |
|
\includegraphics[width=\linewidth]{iceGofr.eps} |
532 |
< |
\caption{Radial distribution functions of Ice-{\it i} and ice $I_c$ |
533 |
< |
calculated from from simulations of the SSD/RF water model at 77 K.} |
532 |
> |
\caption{Radial distribution functions of ice $I_h$, $I_c$, |
533 |
> |
Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations |
534 |
> |
of the SSD/RF water model at 77 K.} |
535 |
|
\label{fig:gofr} |
536 |
|
\end{figure} |
537 |
|
|
538 |
|
\begin{figure} |
539 |
|
\includegraphics[width=\linewidth]{sofq.eps} |
540 |
< |
\caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at |
541 |
< |
77 K. The raw structure factors have been convoluted with a gaussian |
542 |
< |
instrument function (0.075 \AA$^{-1}$ width) to compensate for the |
543 |
< |
trunction effects in our finite size simulations. The labeled peaks |
544 |
< |
compared favorably with ``spurious'' peaks observed in experimental |
516 |
< |
studies of amorphous solid water.\cite{Bizid87}} |
540 |
> |
\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, |
541 |
> |
and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
542 |
> |
been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
543 |
> |
width) to compensate for the trunction effects in our finite size |
544 |
> |
simulations.} |
545 |
|
\label{fig:sofq} |
546 |
|
\end{figure} |
547 |
|
|