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+\section{Methods}
-+
+Canonical ensemble (NVT) molecular dynamics calculations were
-+
+performed using the OOPSE (Object-Oriented Parallel Simulation Engine)
-+
+molecular mechanics package. All molecules were treated as rigid
-+
+bodies, with orientational motion propogated using the symplectic DLM
-+
+integration method. Details about the implementation of these
-+
+techniques can be found in a recent publication.\cite{Meineke05}
-+
-+
+Thermodynamic integration was utilized to calculate the free energy of
-+
+several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E,
-+
+SSD/RF, and SSD/E water models. Liquid state free energies at 300 and
-+
+K for all of these water models were also determined using this
-+
+same technique, in order to determine melting points and generate
-+
+phase diagrams. All simulations were carried out at densities
-+
+resulting in a pressure of approximately 1 atm at their respective
-+
+temperatures.
-+
-+
+A single thermodynamic integration involves a sequence of simulations
-+
+over which the system of interest is converted into a reference system
-+
+for which the free energy is known. This transformation path is then
-+
+integrated in order to determine the free energy difference between
-+
+the two states:
-+
+\begin{equation}
-+
+\begin{center}
-+
+\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda
-+
+)}{\partial\lambda}\right\rangle_\lambda d\lambda,
-+
+\end{center}
-+
+\end{equation}
-+
+where $V$ is the interaction potential and $\lambda$ is the
-+
+transformation parameter. Simulations are distributed unevenly along
-+
+this path in order to sufficiently sample the regions of greatest
-+
+change in the potential. Typical integrations in this study consisted
-+
+of $\sim$25 simulations ranging from 300 ps (for the unaltered system)
-+
+to 75 ps (near the reference state) in length.
-+
-+
+For the thermodynamic integration of molecular crystals, the Einstein
-+
+Crystal is chosen as the reference state that the system is converted
-+
+to over the course of the simulation. In an Einstein Crystal, the
-+
+molecules are harmonically restrained at their ideal lattice locations
-+
+and orientations. The partition function for a molecular crystal
-+
+restrained in this fashion has been evaluated, and the Helmholtz Free
-+
+Energy ({\it A}) is given by
-+
+\begin{eqnarray}
-+
+A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left
-+
+[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right
-+
+)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right
-+
+)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right
-+
+)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi
-+
+K_\omega K_\theta)^{\frac{1}{2}}}\exp\left
-+
+(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right
-+
+)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ],
-+
+\label{ecFreeEnergy}
-+
+\end{eqnarray}
-+
+where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation
-+
+\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and
-+
+$K_\mathrm{\omega}$ are the spring constants restraining translational
-+
+motion and deflection of and rotation around the principle axis of the
-+
+molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the
-+
+minimum potential energy of the ideal crystal. In the case of
-+
+molecular liquids, the ideal vapor is chosen as the target reference
-+
+state.
-+
+\begin{figure}
-+
+\includegraphics[scale=1.0]{rotSpring.eps}
-+
+\caption{Possible orientational motions for a restrained molecule.
-+
+$\theta$ angles correspond to displacement from the body-frame {\it
-+
+z}-axis, while $\omega$ angles correspond to rotation about the
-+
+body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring
-+
+constants for the harmonic springs restraining motion in the $\theta$
-+
+and $\omega$ directions.}
-+
+\label{waterSpring}
-+
+\end{figure}
-+
-+
+Charge, dipole, and Lennard-Jones interactions were modified by a
-+
+cubic switching between 100\% and 85\% of the cutoff value (9 \AA ). By
-+
+applying this function, these interactions are smoothly truncated,
-+
+thereby avoiding poor energy conserving dynamics resulting from
-+
+harsher truncation schemes. The effect of a long-range correction was
-+
+also investigated on select model systems in a variety of manners. For
-+
+the SSD/RF model, a reaction field with a fixed dielectric constant of
-+
+was applied in all simulations.\cite{Onsager36} For a series of the
-+
+least computationally expensive models (SSD/E, SSD/RF, and TIP3P),
-+
+simulations were performed with longer cutoffs of 12 and 15 \AA\ to
-+
+compare with the 9 \AA\ cutoff results. Finally, results from the use
-+
+of an Ewald summation were estimated for TIP3P and SPC/E by performing
-+
+calculations with Particle-Mesh Ewald (PME) in the TINKER molecular
-+
+mechanics software package. TINKER was chosen because it can also
-+
+propogate the motion of rigid-bodies, and provides the most direct
-+
+comparison to the results from OOPSE. The calculated energy difference
-+
+in the presence and absence of PME was applied to the previous results
-+
+in order to predict changes in the free energy landscape.
-+
+\section{Results and discussion}
-+
+The free energy of proton ordered Ice-{\it i} was calculated and
-+
+compared with the free energies of proton ordered variants of the
-+
+experimentally observed low-density ice polymorphs, $I_h$ and $I_c$,
-+
+as well as the higher density ice B, observed by B\`{a}ez and Clancy
-+
+and thought to be the minimum free energy structure for the SPC/E
-+
+model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b}
-+
+Ice XI, the experimentally observed proton ordered variant of ice
-+
+$I_h$, was investigated initially, but it was found not to be as
-+
+stable as antiferroelectric variants of proton ordered or even proton
-+
+disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of
-+
+ice $I_h$ used here is a simple antiferroelectric version that has an
-+
+molecule unit cell. The crystals contained 648 or 1728 molecules for
-+
+ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for ice
-+
+$I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes
-+
+were necessary for simulations involving larger cutoff values.
-+
-+
+\begin{table*}
-+
+\begin{minipage}{\linewidth}
-+
+\renewcommand{\thefootnote}{\thempfootnote}
-+
+\begin{center}
-+
+\caption{Calculated free energies for several ice polymorphs with a
-+
+variety of common water models. All calculations used a cutoff radius
-+
+of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are
-+
+kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF.}
-+
+\begin{tabular}{ l  c  c  c  c }
-+
+\hline \\[-7mm]
-+
+\ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \  & \ \quad \ \ \ \ B \ \  & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\
-+
+\hline \\[-3mm]
-+
+\ \quad \ TIP3P  & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\
-+
+\ \quad \ TIP4P  & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\
-+
+\ \quad \ TIP5P  & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\
-+
+\ \quad \ SPC/E  & \ \quad \ -12.67 & \ \quad \ -12.96 & \ \quad \ -13.25 & \quad -13.55\\
-+
+\ \quad \ SSD/E  & \ \quad \ -11.27 & \ \quad \ -11.19 & \ \quad \ -12.09 & \quad -12.54\\
-+
+\ \quad \ SSD/RF & \ \quad \ -11.51 & \ \quad \ NA* & \ \quad \ -12.08 & \quad -12.29\\
-+
+\end{tabular}
-+
+\label{freeEnergy}
-+
+\end{center}
-+
+\end{minipage}
-+
+\end{table*}
-+
-+
+The free energy values computed for the studied polymorphs indicate
-+
+that Ice-{\it i} is the most stable state for all of the common water
-+
+models studied. With the free energy at these state points, the
-+
+temperature and pressure dependence of the free energy was used to
-+
+project to other state points and build phase diagrams. Figures
-+
+\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built
-+
+from the free energy results. All other models have similar structure,
-+
+only the crossing points between these phases exist at different
-+
+temperatures and pressures. It is interesting to note that ice $I$
-+
+does not exist in either cubic or hexagonal form in any of the phase
-+
+diagrams for any of the models. For purposes of this study, ice B is
-+
+representative of the dense ice polymorphs. A recent study by Sanz
-+
+{\it et al.} goes into detail on the phase diagrams for SPC/E and
-+
+TIP4P in the high pressure regime.\cite{Sanz04}
-+
+\begin{figure}
-+
+\includegraphics[width=\linewidth]{tp3PhaseDia.eps}
-+
+\caption{Phase diagram for the TIP3P water model in the low pressure
-+
+regime. The displayed $T_m$ and $T_b$ values are good predictions of
-+
+the experimental values; however, the solid phases shown are not the
-+
+experimentally observed forms. Both cubic and hexagonal ice $I$ are
-+
+higher in energy and don't appear in the phase diagram.}
-+
+\label{tp3phasedia}
-+
+\end{figure}
-+
+\begin{figure}
-+
+\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps}
-+
+\caption{Phase diagram for the SSD/RF water model in the low pressure
-+
+regime. Calculations producing these results were done under an
-+
+applied reaction field. It is interesting to note that this
-+
+computationally efficient model (over 3 times more efficient than
-+
+TIP3P) exhibits phase behavior similar to the less computationally
-+
+conservative charge based models.}
-+
+\label{ssdrfphasedia}
-+
+\end{figure}
-+
-+
+\begin{table*}
-+
+\begin{minipage}{\linewidth}
-+
+\renewcommand{\thefootnote}{\thempfootnote}
-+
+\begin{center}
-+
+\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$)
-+
+temperatures of several common water models compared with experiment.}
-+
+\begin{tabular}{ l  c  c  c  c  c  c  c }
-+
+\hline \\[-7mm]
-+
+\ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\
-+
+\hline \\[-3mm]
-+
+\ \ $T_m$ (K)  & \ \ 269 & \ \ 265 & \ \ 271 &  297 & \ \ - & \ \ 278 & \ \ 273\\
-+
+\ \ $T_b$ (K)  & \ \ 357 & \ \ 354 & \ \ 337 &  396 & \ \ - & \ \ 349 & \ \ 373\\
-+
+\ \ $T_s$ (K)  & \ \ - & \ \ - & \ \ - &  - & \ \ 355 & \ \ - & \ \ -\\
-+
+\end{tabular}
-+
+\label{meltandboil}
-+
+\end{center}
-+
+\end{minipage}
-+
+\end{table*}
-+
-+
+Table \ref{meltandboil} lists the melting and boiling temperatures
-+
+calculated from this work. Surprisingly, most of these models have
-+
+melting points that compare quite favorably with experiment. The
-+
+unfortunate aspect of this result is that this phase change occurs
-+
+between Ice-{\it i} and the liquid state rather than ice $I_h$ and the
-+
+liquid state. These results are actually not contrary to previous
-+
+studies in the literature. Earlier free energy studies of ice $I$
-+
+using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences
-+
+being attributed to choice of interaction truncation and different
-+
+ordered and disordered molecular arrangements). If the presence of ice
-+
+B and Ice-{\it i} were omitted, a $T_m$ value around 210 K would be
-+
+predicted from this work. However, the $T_m$ from Ice-{\it i} is
-+
+calculated at 265 K, significantly higher in temperature than the
-+
+previous studies. Also of interest in these results is that SSD/E does
-+
+not exhibit a melting point at 1 atm, but it shows a sublimation point
-+
+at 355 K. This is due to the significant stability of Ice-{\it i} over
-+
+all other polymorphs for this particular model under these
-+
+conditions. While troubling, this behavior turned out to be
-+
+advantagious in that it facilitated the spontaneous crystallization of
-+
+Ice-{\it i}. These observations provide a warning that simulations of
-+
+SSD/E as a ``liquid'' near 300 K are actually metastable and run the
-+
+risk of spontaneous crystallization. However, this risk changes when
-+
+applying a longer cutoff.
-+
-+
+\begin{figure}
-+
+\includegraphics[width=\linewidth]{cutoffChange.eps}
-+
+\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B)
-+
+TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12
-+
+\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9
-+
+\AA\. These crystals are unstable at 200 K and rapidly convert into a
-+
+liquid. The connecting lines are qualitative visual aid.}
-+
+\label{incCutoff}
-+
+\end{figure}
-+
-+
+Increasing the cutoff radius in simulations of the more
-+
+computationally efficient water models was done in order to evaluate
-+
+the trend in free energy values when moving to systems that do not
-+
+involve potential truncation. As seen in Fig. \ref{incCutoff}, the
-+
+free energy of all the ice polymorphs show a substantial dependence on
-+
+cutoff radius. In general, there is a narrowing of the free energy
-+
+differences while moving to greater cutoff radius. This trend is much
-+
+more subtle in the case of SSD/RF, indicating that the free energies
-+
+calculated with a reaction field present provide a more accurate
-+
+picture of the free energy landscape in the absence of potential
-+
+truncation.
-+
-+
+To further study the changes resulting to the inclusion of a
-+
+long-range interaction correction, the effect of an Ewald summation
-+
+was estimated by applying the potential energy difference do to its
-+
+inclusion in systems in the presence and absence of the
-+
+correction. This was accomplished by calculation of the potential
-+
+energy of identical crystals with and without PME using TINKER. The
-+
+free energies for the investigated polymorphs using the TIP3P and
-+
+SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P
-+
+are not fully supported in TINKER, so the results for these models
-+
+could not be estimated. The same trend pointed out through increase of
-+
+cutoff radius is observed in these results. Ice-{\it i} is the
-+
+preferred polymorph at ambient conditions for both the TIP3P and SPC/E
-+
+water models; however, there is a narrowing of the free energy
-+
+differences between the various solid forms. In the case of SPC/E this
-+
+narrowing is significant enough that it becomes less clear cut that
-+
+Ice-{\it i} is the most stable polymorph, and is possibly metastable
-+
+with respect to ice B and possibly ice $I_c$. However, these results
-+
+do not significantly alter the finding that the Ice-{\it i} polymorph
-+
+is a stable crystal structure that should be considered when studying
-+
+the phase behavior of water models.
-+
-+
+\begin{table*}
-+
+\begin{minipage}{\linewidth}
-+
+\renewcommand{\thefootnote}{\thempfootnote}
-+
+\begin{center}
-+
+\caption{The free energy of the studied ice polymorphs after applying
-+
+the energy difference attributed to the inclusion of the PME
-+
+long-range interaction correction. Units are kcal/mol.}
-+
+\begin{tabular}{ l  c  c  c  c }
-+
+\hline \\[-7mm]
-+
+\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
-+
+\hline \\[-3mm]
-+
+\ \ TIP3P  & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\
-+
+\ \ SPC/E  & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\
-+
+\end{tabular}
-+
+\label{pmeShift}
-+
+\end{center}
-+
+\end{minipage}
-+
+\end{table*}
-+
+\section{Conclusions}
-+
+The free energy for proton ordered variants of hexagonal and cubic ice
-+
+$I$, ice B, and recently discovered Ice-{\it i} where calculated under
-+
+standard conditions for several common water models via thermodynamic
-+
+integration. All the water models studied show Ice-{\it i} to be the
-+
+minimum free energy crystal structure in the with a 9 \AA\ switching
-+
+function cutoff. Calculated melting and boiling points show
-+
+surprisingly good agreement with the experimental values; however, the
-+
+solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of
-+
+interaction truncation was investigated through variation of the
-+
+cutoff radius, use of a reaction field parameterized model, and
-+
+estimation of the results in the presence of the Ewald summation
-+
+correction. Interaction truncation has a significant effect on the
-+
+computed free energy values, but Ice-{\it i} is still observed to be a
-+
+relavent ice polymorph in simulation studies.
-+
+\section{Acknowledgments}
+Support for this project was provided by the National Science
+Foundation under grant CHE-0134881. Computation time was provided by
-<
+the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
-<
+DMR-0079647.
->
+the Notre Dame High Performance Computing Cluster and the Notre Dame
->
+Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647).
+\newpage

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Comparing trunk/iceiPaper/iceiPaper.tex (file contents): Revision 1453 by chrisfen, Mon Sep 13 18:39:53 2004 UTC vs. Revision 1458 by chrisfen, Wed Sep 15 06:34:49 2004 UTC

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Comparing trunk/iceiPaper/iceiPaper.tex (file contents):
Revision 1453 by chrisfen, Mon Sep 13 18:39:53 2004 UTC vs.
Revision 1458 by chrisfen, Wed Sep 15 06:34:49 2004 UTC