--- trunk/iceiPaper/iceiPaper.tex	2004/09/13 21:30:53	1455
+++ trunk/iceiPaper/iceiPaper.tex	2004/09/14 21:55:24	1456
@@ -58,10 +58,13 @@ several ice crystals using the TIP3P, TIP4P, TIP5P, SP
 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
+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. 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
@@ -89,12 +92,157 @@ state.
 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
+80 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
+8 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}
