54 |
|
|
55 |
|
\section{Introduction} |
56 |
|
|
57 |
< |
Molecular dynamics is a valuable tool for studying the phase behavior |
58 |
< |
of systems ranging from small or simple |
59 |
< |
molecules\cite{Matsumoto02,andOthers} to complex biological |
60 |
< |
species.\cite{bigStuff} Many techniques have been developed to |
61 |
< |
investigate the thermodynamic properites of model substances, |
62 |
< |
providing both qualitative and quantitative comparisons between |
63 |
< |
simulations and experiment.\cite{thermMethods} Investigation of these |
64 |
< |
properties leads to the development of new and more accurate models, |
65 |
< |
leading to better understanding and depiction of physical processes |
66 |
< |
and intricate molecular systems. |
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,Li96,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 |
|
|
68 |
|
Water has proven to be a challenging substance to depict in |
69 |
|
simulations, and a variety of models have been developed to describe |
70 |
|
its behavior under varying simulation |
71 |
< |
conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04} |
71 |
> |
conditions.\cite{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04} |
72 |
|
These models have been used to investigate important physical |
73 |
< |
phenomena like phase transitions and the hydrophobic |
74 |
< |
effect.\cite{Yamada02} With the choice of models available, it |
75 |
< |
is only natural to compare the models under interesting thermodynamic |
76 |
< |
conditions in an attempt to clarify the limitations of each of the |
77 |
< |
models.\cite{modelProps} Two important property to quantify are the |
78 |
< |
Gibbs and Helmholtz free energies, particularly for the solid forms of |
79 |
< |
water. Difficulty in these types of studies typically arises from the |
80 |
< |
assortment of possible crystalline polymorphs that water adopts over a |
81 |
< |
wide range of pressures and temperatures. There are currently 13 |
82 |
< |
recognized forms of ice, and it is a challenging task to investigate |
83 |
< |
the entire free energy landscape.\cite{Sanz04} Ideally, research is |
84 |
< |
focused on the phases having the lowest free energy at a given state |
85 |
< |
point, because these phases will dictate the true transition |
86 |
< |
temperatures and pressures for their respective model. |
73 |
> |
phenomena like phase transitions, molecule transport, and the |
74 |
> |
hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
75 |
> |
choice of models available, it is only natural to compare the models |
76 |
> |
under interesting thermodynamic conditions in an attempt to clarify |
77 |
> |
the limitations of each of the |
78 |
> |
models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
79 |
> |
important property to quantify are the Gibbs and Helmholtz free |
80 |
> |
energies, particularly for the solid forms of water. Difficulty in |
81 |
> |
these types of studies typically arises from the assortment of |
82 |
> |
possible crystalline polymorphs that water adopts over a wide range of |
83 |
> |
pressures and temperatures. There are currently 13 recognized forms |
84 |
> |
of ice, and it is a challenging task to investigate the entire free |
85 |
> |
energy landscape.\cite{Sanz04} Ideally, research is focused on the |
86 |
> |
phases having the lowest free energy at a given state point, because |
87 |
> |
these phases will dictate the true transition temperatures and |
88 |
> |
pressures for the model. |
89 |
|
|
90 |
|
In this paper, standard reference state methods were applied to known |
91 |
< |
crystalline water polymorphs in the low pressure regime. This work is |
91 |
> |
crystalline water polymorphs in the low pressure regime. This work is |
92 |
|
unique in the fact that one of the crystal lattices was arrived at |
93 |
|
through crystallization of a computationally efficient water model |
94 |
|
under constant pressure and temperature conditions. Crystallization |
113 |
|
\begin{figure} |
114 |
|
\includegraphics[width=\linewidth]{unitCell.eps} |
115 |
|
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
116 |
< |
elongated variant of Ice-{\it i}. For Ice-{\it i}, the $a$ to $c$ |
117 |
< |
relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a = |
118 |
< |
1.7850c$.} |
116 |
> |
elongated variant of Ice-{\it i}. The spheres represent the |
117 |
> |
center-of-mass locations of the water molecules. The $a$ to $c$ |
118 |
> |
ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
119 |
> |
$a:2.1214c$ and $a:1.7850c$ respectively.} |
120 |
|
\label{iceiCell} |
121 |
|
\end{figure} |
122 |
|
|
132 |
|
Results from our previous study indicated that Ice-{\it i} is the |
133 |
|
minimum energy crystal structure for the single point water models we |
134 |
|
investigated (for discussions on these single point dipole models, see |
135 |
< |
the previous work and related |
136 |
< |
articles\cite{Fennell04,Liu96,Bratko85}). Those results only |
135 |
> |
our previous work and related |
136 |
> |
articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
137 |
|
considered energetic stabilization and neglected entropic |
138 |
|
contributions to the overall free energy. To address this issue, the |
139 |
|
absolute free energy of this crystal was calculated using |
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 |
160 |
< |
the implementation of these techniques can be found in a recent |
160 |
> |
the implementation of this technique can be found in a recent |
161 |
|
publication.\cite{Dullweber1997} |
162 |
|
|
163 |
< |
Thermodynamic integration was utilized to calculate the free energy of |
164 |
< |
several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, |
165 |
< |
SSD/RF, and SSD/E water models. Liquid state free energies at 300 and |
166 |
< |
400 K for all of these water models were also determined using this |
167 |
< |
same technique in order to determine melting points and generate phase |
168 |
< |
diagrams. All simulations were carried out at densities resulting in a |
169 |
< |
pressure of approximately 1 atm at their respective temperatures. |
163 |
> |
Thermodynamic integration is an established technique for |
164 |
> |
determination of free energies of condensed phases of |
165 |
> |
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
166 |
> |
method, implemented in the same manner illustrated by B\`{a}ez and |
167 |
> |
Clancy, was utilized to calculate the free energy of several ice |
168 |
> |
crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
169 |
> |
SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
170 |
> |
and 400 K for all of these water models were also determined using |
171 |
> |
this same technique in order to determine melting points and generate |
172 |
> |
phase diagrams. All simulations were carried out at densities |
173 |
> |
resulting in a pressure of approximately 1 atm at their respective |
174 |
> |
temperatures. |
175 |
|
|
176 |
|
A single thermodynamic integration involves a sequence of simulations |
177 |
|
over which the system of interest is converted into a reference system |
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 unevenly along this path in |
188 |
< |
order to sufficiently sample the regions of greatest change in the |
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$25 |
190 |
|
simulations ranging from 300 ps (for the unaltered system) to 75 ps |
191 |
|
(near the reference state) in length. |
192 |
|
|
193 |
|
For the thermodynamic integration of molecular crystals, the Einstein |
194 |
< |
crystal was chosen as the reference state. In an Einstein crystal, the |
195 |
< |
molecules are harmonically restrained at their ideal lattice locations |
196 |
< |
and orientations. The partition function for a molecular crystal |
194 |
> |
crystal was chosen as the reference system. In an Einstein crystal, |
195 |
> |
the molecules are restrained at their ideal lattice locations and |
196 |
> |
orientations. Using harmonic restraints, as applied by B\`{a}ez and |
197 |
> |
Clancy, the total potential for this reference crystal |
198 |
> |
($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
199 |
> |
\begin{equation} |
200 |
> |
V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
201 |
> |
\frac{K_\omega\omega^2}{2}, |
202 |
> |
\end{equation} |
203 |
> |
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
204 |
> |
the spring constants restraining translational motion and deflection |
205 |
> |
of and rotation around the principle axis of the molecule |
206 |
> |
respectively. It is clear from Fig. \ref{waterSpring} that the values |
207 |
> |
of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
208 |
> |
$-\pi$ to $\pi$. The partition function for a molecular crystal |
209 |
|
restrained in this fashion can be evaluated analytically, and the |
210 |
|
Helmholtz Free Energy ({\it A}) is given by |
211 |
|
\begin{eqnarray} |
219 |
|
)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
220 |
|
\label{ecFreeEnergy} |
221 |
|
\end{eqnarray} |
222 |
< |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
223 |
< |
\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
204 |
< |
$K_\mathrm{\omega}$ are the spring constants restraining translational |
205 |
< |
motion and deflection of and rotation around the principle axis of the |
206 |
< |
molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
207 |
< |
minimum potential energy of the ideal crystal. In the case of |
208 |
< |
molecular liquids, the ideal vapor is chosen as the target reference |
209 |
< |
state. |
222 |
> |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
223 |
> |
potential energy of the ideal crystal.\cite{Baez95a} |
224 |
|
|
225 |
|
\begin{figure} |
226 |
|
\includegraphics[width=\linewidth]{rotSpring.eps} |
233 |
|
\label{waterSpring} |
234 |
|
\end{figure} |
235 |
|
|
236 |
+ |
In the case of molecular liquids, the ideal vapor is chosen as the |
237 |
+ |
target reference state. There are several examples of liquid state |
238 |
+ |
free energy calculations of water models present in the |
239 |
+ |
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
240 |
+ |
typically differ in regard to the path taken for switching off the |
241 |
+ |
interaction potential to convert the system to an ideal gas of water |
242 |
+ |
molecules. In this study, we apply of one of the most convenient |
243 |
+ |
methods and integrate over the $\lambda^4$ path, where all interaction |
244 |
+ |
parameters are scaled equally by this transformation parameter. This |
245 |
+ |
method has been shown to be reversible and provide results in |
246 |
+ |
excellent agreement with other established methods.\cite{Baez95b} |
247 |
+ |
|
248 |
|
Charge, dipole, and Lennard-Jones interactions were modified by a |
249 |
|
cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
250 |
|
). By applying this function, these interactions are smoothly |
276 |
|
$I_h$, was investigated initially, but was found to be not as stable |
277 |
|
as proton disordered or antiferroelectric variants of ice $I_h$. The |
278 |
|
proton ordered variant of ice $I_h$ used here is a simple |
279 |
< |
antiferroelectric version that has an 8 molecule unit |
280 |
< |
cell.\cite{Davidson84} The crystals contained 648 or 1728 molecules |
281 |
< |
for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for |
282 |
< |
ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
283 |
< |
were necessary for simulations involving larger cutoff values. |
279 |
> |
antiferroelectric version that we divised, and it has an 8 molecule |
280 |
> |
unit cell similar to other predicted antiferroelectric $I_h$ |
281 |
> |
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
282 |
> |
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
283 |
> |
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger |
284 |
> |
crystal sizes were necessary for simulations involving larger cutoff |
285 |
> |
values. |
286 |
|
|
287 |
|
\begin{table*} |
288 |
|
\begin{minipage}{\linewidth} |
297 |
|
\hline |
298 |
|
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
299 |
|
\hline |
300 |
< |
TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\\ |
301 |
< |
TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\ |
302 |
< |
TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\ |
303 |
< |
SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\ |
304 |
< |
SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\ |
305 |
< |
SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\ |
300 |
> |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
301 |
> |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
302 |
> |
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
303 |
> |
SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\ |
304 |
> |
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
305 |
> |
SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\ |
306 |
|
\end{tabular} |
307 |
|
\label{freeEnergy} |
308 |
|
\end{center} |
358 |
|
\hline |
359 |
|
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
360 |
|
\hline |
361 |
< |
$T_m$ (K) & 269(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\ |
362 |
< |
$T_b$ (K) & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\ |
363 |
< |
$T_s$ (K) & - & - & - & - & 355(3) & - & -\\ |
361 |
> |
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
362 |
> |
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
363 |
> |
$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ |
364 |
|
\end{tabular} |
365 |
|
\label{meltandboil} |
366 |
|
\end{center} |
425 |
|
correction. This was accomplished by calculation of the potential |
426 |
|
energy of identical crystals with and without PME using TINKER. The |
427 |
|
free energies for the investigated polymorphs using the TIP3P and |
428 |
< |
SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
429 |
< |
are not fully supported in TINKER, so the results for these models |
430 |
< |
could not be estimated. The same trend pointed out through increase of |
431 |
< |
cutoff radius is observed in these PME results. Ice-{\it i} is the |
432 |
< |
preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
433 |
< |
water models; however, there is a narrowing of the free energy |
434 |
< |
differences between the various solid forms. In the case of SPC/E this |
435 |
< |
narrowing is significant enough that it becomes less clear that |
436 |
< |
Ice-{\it i} is the most stable polymorph, and is possibly metastable |
437 |
< |
with respect to ice B and possibly ice $I_c$. However, these results |
438 |
< |
do not significantly alter the finding that the Ice-{\it i} polymorph |
439 |
< |
is a stable crystal structure that should be considered when studying |
440 |
< |
the phase behavior of water models. |
428 |
> |
SPC/E water models are shown in Table \ref{pmeShift}. The same trend |
429 |
> |
pointed out through increase of cutoff radius is observed in these PME |
430 |
> |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
431 |
> |
for both the TIP3P and SPC/E water models; however, the narrowing of |
432 |
> |
the free energy differences between the various solid forms is |
433 |
> |
significant enough that it becomes less clear that it is the most |
434 |
> |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
435 |
> |
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
436 |
> |
as well, indicating that Ice-{\it i} might be metastable with respect |
437 |
> |
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
438 |
> |
not significantly alter the finding that the Ice-{\it i} polymorph is |
439 |
> |
a stable crystal structure that should be considered when studying the |
440 |
> |
phase behavior of water models. |
441 |
|
|
442 |
|
\begin{table*} |
443 |
|
\begin{minipage}{\linewidth} |
450 |
|
\hline |
451 |
|
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
452 |
|
\hline |
453 |
< |
TIP3P & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)\\ |
454 |
< |
SPC/E & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\ |
453 |
> |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\ |
454 |
> |
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\ |
455 |
|
\end{tabular} |
456 |
|
\label{pmeShift} |
457 |
|
\end{center} |