| 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{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,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,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 |
| 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} |
