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\begin{document} |
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\title{Computational free energy studies of a new ice polymorph which |
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exhibits greater stability than Ice $I_h$} |
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exhibits greater stability than Ice I$_h$} |
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\author{Christopher J. Fennell and J. Daniel Gezelter \\ |
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Department of Chemistry and Biochemistry\\ |
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tetramers form a crystal structure similar in appearance to a recent |
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two-dimensional surface tessellation simulated on silica.\cite{Yang04} |
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As expected in an ice crystal constructed of water tetramers, the |
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hydrogen bonds are not as linear as those observed in ice $I_h$, |
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hydrogen bonds are not as linear as those observed in ice I$_h$, |
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however the interlocking of these subunits appears to provide |
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significant stabilization to the overall crystal. The arrangement of |
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these tetramers results in octagonal cavities that are typically |
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greater than 6.3 \AA\ in diameter (Fig. \ref{iCrystal}). This open |
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structure leads to crystals that are typically 0.07 g/cm$^3$ less |
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dense than ice $I_h$. |
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dense than ice I$_h$. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{orderedIcei.eps} |
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\caption{A rendering of a proton ordered crystal of Ice-{\it i} looking |
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down the (001) crystal face. The presence of large octagonal pores |
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leads to a polymorph that is less dense than ice $I_h$.} |
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leads to a polymorph that is less dense than ice I$_h$.} |
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\label{iCrystal} |
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\end{figure} |
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contributions to the overall free energy. To address this issue, we |
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have calculated the absolute free energy of this crystal using |
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thermodynamic integration and compared it to the free energies of ice |
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$I_c$ and ice $I_h$ (the common low density ice polymorphs) and ice B |
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I$_c$ and ice I$_h$ (the common low density ice polymorphs) and ice B |
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(a higher density, but very stable crystal structure observed by |
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B\`{a}ez and Clancy in free energy studies of SPC/E).\cite{Baez95b} |
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This work includes results for the water model from which Ice-{\it i} |
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of and rotation around the principle axis of the molecule |
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respectively. These spring constants are typically calculated from |
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the mean-square displacements of water molecules in an unrestrained |
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ice crystal at 200 K. For these studies, $K_\mathrm{r} = 4.29$ kcal |
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mol$^{-1}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$, and $K_\omega\ = |
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17.75$ kcal mol$^{-1}$. It is clear from Fig. \ref{waterSpring} that |
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the values of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges |
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from $-\pi$ to $\pi$. The partition function for a molecular crystal |
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restrained in this fashion can be evaluated analytically, and the |
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Helmholtz Free Energy ({\it A}) is given by |
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ice crystal at 200 K. For these studies, $K_\mathrm{v} = 4.29$ kcal |
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mol$^{-1}$ \AA$^{-2}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$ rad$^{-2}$, |
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and $K_\omega\ = 17.75$ kcal mol$^{-1}$ rad$^{-2}$. It is clear from |
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Fig. \ref{waterSpring} that the values of $\theta$ range from $0$ to |
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$\pi$, while $\omega$ ranges from $-\pi$ to $\pi$. The partition |
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function for a molecular crystal restrained in this fashion can be |
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evaluated analytically, and the Helmholtz Free Energy ({\it A}) is |
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given by |
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\begin{eqnarray} |
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A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
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[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
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\section{Results and Discussion} |
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The calculated free energies of proton-ordered variants of three low |
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density polymorphs ($I_h$, $I_c$, and Ice-{\it i} or Ice-{\it |
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density polymorphs (I$_h$, I$_c$, and Ice-{\it i} or Ice-{\it |
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i}$^\prime$) and the stable higher density ice B are listed in Table |
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\ref{freeEnergy}. Ice B was included because it has been |
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shown to be a minimum free energy structure for SPC/E at ambient |
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All other models have similar structure, although the crossing points |
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between the phases move to different temperatures and pressures as |
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indicated from the transition temperatures in Table \ref{freeEnergy}. |
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It is interesting to note that ice $I_h$ (and ice $I_c$ for that |
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It is interesting to note that ice I$_h$ (and ice I$_c$ for that |
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matter) do not appear in any of the phase diagrams for any of the |
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models. For purposes of this study, ice B is representative of the |
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dense ice polymorphs. A recent study by Sanz {\it et al.} provides |
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Kelvin. Calculated error of the final digits is in parentheses.} |
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\begin{tabular}{lccccccc} |
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\hline |
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Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} & Ice-{\it i}$^\prime$ & $T_m$ (*$T_s$) & $T_b$\\ |
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Water Model & I$_h$ & I$_c$ & B & Ice-{\it i} & Ice-{\it i}$^\prime$ & $T_m$ (*$T_s$) & $T_b$\\ |
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\hline |
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TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(4) & 357(2)\\ |
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TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 266(5) & 354(2)\\ |
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TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 271(4) & 337(2)\\ |
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SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 296(3) & 396(2)\\ |
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SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(2) & -\\ |
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SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2) & - & 278(4) & 349(2)\\ |
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SSD/RF & -11.96(2) & -11.60(2) & -12.53(3) & -12.79(2) & - & 287(4) & 382(2)\\ |
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\end{tabular} |
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\label{freeEnergy} |
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\end{center} |
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Most of the water models have melting points that compare quite |
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favorably with the experimental value of 273 K. The unfortunate |
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aspect of this result is that this phase change occurs between |
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Ice-{\it i} and the liquid state rather than ice $I_h$ and the liquid |
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Ice-{\it i} and the liquid state rather than ice I$_h$ and the liquid |
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state. These results do not contradict other studies. Studies of ice |
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$I_h$ using TIP4P predict a $T_m$ ranging from 214 to 238 K |
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I$_h$ using TIP4P predict a $T_m$ ranging from 214 to 238 K |
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(differences being attributed to choice of interaction truncation and |
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different ordered and disordered molecular |
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arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
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\section{Conclusions} |
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In this report, thermodynamic integration was used to determine the |
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absolute free energies of several ice polymorphs. Of the studied |
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crystal forms, Ice-{\it i} was observed to be the stable crystalline |
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state for {\it all} the water models when using a 9.0 \AA\ |
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intermolecular interaction cutoff. Through investigation of possible |
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interaction truncation methods, the free energy was shown to be |
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partially dependent on simulation conditions; however, Ice-{\it i} was |
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still observered to be a stable polymorph of the studied water models. |
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In this work, thermodynamic integration was used to determine the |
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absolute free energies of several ice polymorphs. The new polymorph, |
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Ice-{\it i} was observed to be the stable crystalline state for {\it |
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all} the water models when using a 9.0 \AA\ cutoff. However, the free |
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energy partially depends on simulation conditions (particularly on the |
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choice of long range correction method). Regardless, Ice-{\it i} was |
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still observered to be a stable polymorph for all of the studied water |
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models. |
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So what is the preferred solid polymorph for simulated water? As |
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indicated above, the answer appears to be dependent both on the |
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pressure conditions, as was done with SSD/E, would aid in the |
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identification of their respective preferred structures. This work, |
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however, helps illustrate how studies involving one specific model can |
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lead to insight about important behavior of others. In general, the |
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above results support the finding that the Ice-{\it i} polymorph is a |
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stable crystal structure that should be considered when studying the |
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phase behavior of water models. |
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lead to insight about important behavior of others. |
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We also note that none of the water models used in this study are |
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polarizable or flexible models. It is entirely possible that the |
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polarizability of real water makes Ice-{\it i} substantially less |
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stable than ice $I_h$. However, the calculations presented above seem |
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stable than ice I$_h$. However, the calculations presented above seem |
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interesting enough to communicate before the role of polarizability |
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(or flexibility) has been thoroughly investigated. |
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situation for possible observation. These include the negative |
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pressure or stretched solid regime, small clusters in vacuum |
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deposition environments, and in clathrate structures involving small |
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non-polar molecules. For experimental comparison purposes, example |
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$g_{OO}(r)$ and $S(\vec{q})$ plots were generated for the two Ice-{\it |
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i} variants (along with example ice $I_h$ and $I_c$ plots) at 77K, and |
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they are shown in figures \ref{fig:gofr} and \ref{fig:sofq} |
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respectively. |
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non-polar molecules. For the purpose of comparison with experimental |
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results, we have calculated the oxygen-oxygen pair correlation |
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function, $g_{OO}(r)$, and the structure factor, $S(\vec{q})$ for the |
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two Ice-{\it i} variants (along with example ice I$_h$ and I$_c$ |
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plots) at 77K, and they are shown in figures \ref{fig:gofr} and |
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\ref{fig:sofq} respectively. It is interesting to note that the |
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structure factors for Ice-{\it i}$^\prime$ and Ice-I$_c$ are quite similar. |
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The primary differences are small peaks at 1.125, 2.29, and 2.53 |
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\AA${-1}$, so particular attention to these regions would be needed |
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to identify the new {\it i}$^\prime$ variant from the I$_{c}$ variant. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{iceGofr.eps} |
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\caption{Radial distribution functions of ice $I_h$, $I_c$, and |
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\caption{Radial distribution functions of ice I$_h$, I$_c$, and |
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Ice-{\it i} calculated from from simulations of the SSD/RF water model |
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at 77 K. The Ice-{\it i} distribution function was obtained from |
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simulations composed of TIP4P water.} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{sofq.eps} |
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\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, |
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\caption{Predicted structure factors for ice I$_h$, I$_c$, Ice-{\it i}, |
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and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
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been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
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width) to compensate for the trunction effects in our finite size |
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\newpage |
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\bibliographystyle{jcp} |
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\bibliographystyle{achemso} |
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\bibliography{iceiPaper} |
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