54 |
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|
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{Matsumoto02andOthers} 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,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 |
|
|
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,Ichiye96,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 |
411 |
|
in the SSD/E model that the liquid state is preferred under standard |
412 |
|
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
413 |
|
simulations using this model choose interaction truncation radii |
414 |
< |
greater than 9 \AA\. This narrowing trend is much more subtle in the |
414 |
> |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
415 |
|
case of SSD/RF, indicating that the free energies calculated with a |
416 |
|
reaction field present provide a more accurate picture of the free |
417 |
|
energy landscape in the absence of potential truncation. |
423 |
|
correction. This was accomplished by calculation of the potential |
424 |
|
energy of identical crystals with and without PME using TINKER. The |
425 |
|
free energies for the investigated polymorphs using the TIP3P and |
426 |
< |
SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
427 |
< |
are not fully supported in TINKER, so the results for these models |
428 |
< |
could not be estimated. The same trend pointed out through increase of |
429 |
< |
cutoff radius is observed in these PME results. Ice-{\it i} is the |
430 |
< |
preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
431 |
< |
water models; however, there is a narrowing of the free energy |
432 |
< |
differences between the various solid forms. In the case of SPC/E this |
433 |
< |
narrowing is significant enough that it becomes less clear that |
434 |
< |
Ice-{\it i} is the most stable polymorph, and is possibly metastable |
435 |
< |
with respect to ice B and possibly ice $I_c$. However, these results |
436 |
< |
do not significantly alter the finding that the Ice-{\it i} polymorph |
437 |
< |
is a stable crystal structure that should be considered when studying |
438 |
< |
the phase behavior of water models. |
426 |
> |
SPC/E water models are shown in Table \ref{pmeShift}. The same trend |
427 |
> |
pointed out through increase of cutoff radius is observed in these PME |
428 |
> |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
429 |
> |
for both the TIP3P and SPC/E water models; however, the narrowing of |
430 |
> |
the free energy differences between the various solid forms is |
431 |
> |
significant enough that it becomes less clear that it is the most |
432 |
> |
stable polymorph. The free energies of Ice-{\it i} and ice B overlap |
433 |
> |
within error, with ice $I_c$ just outside, indicating that Ice-{\it i} |
434 |
> |
might be metastable with respect to ice B and possibly ice $I_c$ in |
435 |
> |
the SPC/E water model. However, these results do not significantly |
436 |
> |
alter the finding that the Ice-{\it i} polymorph is a stable crystal |
437 |
> |
structure that should be considered when studying the phase behavior |
438 |
> |
of water models. |
439 |
|
|
440 |
|
\begin{table*} |
441 |
|
\begin{minipage}{\linewidth} |
486 |
|
most ideal situation for possible observation. These include the |
487 |
|
negative pressure or stretched solid regime, small clusters in vacuum |
488 |
|
deposition environments, and in clathrate structures involving small |
489 |
< |
non-polar molecules. Fig. \ref{fig:gofr} contains our predictions |
490 |
< |
of both the pair distribution function ($g_{OO}(r)$) and the structure |
491 |
< |
factor ($S(\vec{q})$ for this polymorph at a temperature of 77K. We |
492 |
< |
will leave it to our experimental colleagues to determine whether this |
493 |
< |
ice polymorph should really be called Ice-{\it i} or if it should be |
494 |
< |
promoted to Ice-0. |
489 |
> |
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
490 |
> |
our predictions for both the pair distribution function ($g_{OO}(r)$) |
491 |
> |
and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it |
492 |
> |
i} at a temperature of 77K. In a quick comparison of the predicted |
493 |
> |
S(q) for Ice-{\it i} and experimental studies of amorphous solid |
494 |
> |
water, it is possible that some of the ``spurious'' peaks that could |
495 |
> |
not be assigned in HDA could correspond to peaks labeled in this |
496 |
> |
S(q).\cite{Bizid87} It should be noted that there is typically poor |
497 |
> |
agreement on crystal densities between simulation and experiment, so |
498 |
> |
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. |
501 |
|
|
502 |
|
\begin{figure} |
503 |
|
\includegraphics[width=\linewidth]{iceGofr.eps} |
504 |
< |
\caption{Radial distribution functions of (A) Ice-{\it i} and (B) ice $I_c$ at 77 K from simulations of the SSD/RF water model.} |
504 |
> |
\caption{Radial distribution functions of Ice-{\it i} and ice $I_c$ |
505 |
> |
calculated from from simulations of the SSD/RF water model at 77 K.} |
506 |
|
\label{fig:gofr} |
507 |
|
\end{figure} |
508 |
|
|
509 |
+ |
\begin{figure} |
510 |
+ |
\includegraphics[width=\linewidth]{sofq.eps} |
511 |
+ |
\caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at |
512 |
+ |
77 K. The raw structure factors have been convoluted with a gaussian |
513 |
+ |
instrument function (0.075 \AA$^{-1}$ width) to compensate for the |
514 |
+ |
trunction effects in our finite size simulations. The labeled peaks |
515 |
+ |
compared favorably with ``spurious'' peaks observed in experimental |
516 |
+ |
studies of amorphous solid water.\cite{Bizid87}} |
517 |
+ |
\label{fig:sofq} |
518 |
+ |
\end{figure} |
519 |
+ |
|
520 |
|
\section{Acknowledgments} |
521 |
|
Support for this project was provided by the National Science |
522 |
|
Foundation under grant CHE-0134881. Computation time was provided by |