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Comparing trunk/iceiPaper/iceiPaper.tex (file contents):
Revision 1454 by chrisfen, Mon Sep 13 21:28:16 2004 UTC vs.
Revision 1456 by chrisfen, Tue Sep 14 21:55:24 2004 UTC

+techniques can be found in a recent publication.\cite{Meineke05}
+Thermodynamic integration was utilized to calculate the free energy of
-<
+several ice crystals using the TIP3P, TIP4P, TIP5P, SPC/E, and SSD/E
-<
+water models. Liquid state free energies at 300 and 400 K for all of
-<
+these water models were also determined using this same technique, in
-<
+order to determine melting points and generate phase diagrams.
->
+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.
+For the thermodynamic integration of molecular crystals, the Einstein
+Crystal is chosen as the reference state that the system is converted
+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.
-<
+\section{Results and discussion}
->
+\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.
-+
-+
-+
+\section{Conclusions}
+\section{Acknowledgments}

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Comparing trunk/iceiPaper/iceiPaper.tex (file contents): Revision 1454 by chrisfen, Mon Sep 13 21:28:16 2004 UTC vs. Revision 1456 by chrisfen, Tue Sep 14 21:55:24 2004 UTC

Diff Legend

Comparing trunk/iceiPaper/iceiPaper.tex (file contents):
Revision 1454 by chrisfen, Mon Sep 13 21:28:16 2004 UTC vs.
Revision 1456 by chrisfen, Tue Sep 14 21:55:24 2004 UTC