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\usepackage{amsmath} |
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\usepackage{berkeley} |
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\usepackage{times} |
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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 surrounding open octagonal cavities that |
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are typically greater than 6.3 \AA\ in diameter |
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(Fig. \ref{iCrystal}). This open structure leads to crystals that |
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are typically 0.07 g/cm$^3$ less dense than ice $I_h$. |
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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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\begin{figure} |
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\centering |
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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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conditions, such as the density in fixed-volume simulations, can |
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influence the polymorph expressed upon crystallization. |
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|
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So what is the preferred solid polymorph for simulated water? The |
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answer appears to be dependent both on the conditions and the model |
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used. In the case of short cutoffs without a long-range interaction |
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correction, Ice-{\it i} and Ice-{\it i}$^\prime$ have the lowest free |
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energy of the studied polymorphs with all the models. Ideally, |
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crystallization of each model under constant pressure conditions, as |
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was done with SSD/E, would aid in the identification of their |
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respective preferred structures. This work, however, helps illustrate |
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how studies involving one specific model can lead to insight about |
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important behavior of others. In general, the above results support |
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the finding that the Ice-{\it i} polymorph is a stable crystal |
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structure that should be considered when studying the phase behavior |
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of water models. |
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\section{Conclusions} |
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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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|
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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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conditions and the model used. In the case of short cutoffs without a |
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long-range interaction correction, Ice-{\it i} and Ice-{\it |
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i}$^\prime$ have the lowest free energy of the studied polymorphs with |
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all the models. Ideally, crystallization of each model under constant |
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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. |
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|
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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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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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|
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\begin{figure} |
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\centering |
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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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