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\begin{document} |
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|
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\title{Ice-{\it i}: a simulation-predicted ice polymorph which is more |
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stable than Ice $I_h$ for point-charge and point-dipole water models} |
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\title{Free Energy Analysis of Simulated Ice Polymorphs Using Simple |
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Dipolar and Charge Based Water Models} |
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|
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\author{Christopher J. Fennell and J. Daniel Gezelter \\ |
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Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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crystalline water polymorphs in the low pressure regime. This work is |
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unique in that one of the crystal lattices was arrived at through |
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crystallization of a computationally efficient water model under |
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constant pressure and temperature conditions. Crystallization events |
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constant pressure and temperature conditions. Crystallization events |
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are interesting in and of themselves;\cite{Matsumoto02,Yamada02} |
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however, the crystal structure obtained in this case is different from |
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any previously observed ice polymorphs in experiment or |
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simulation.\cite{Fennell04} We have named this structure Ice-{\it i} |
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to indicate its origin in computational simulation. The unit cell |
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(Fig. \ref{iceiCell}A) consists of eight water molecules that stack in |
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rows of interlocking water tetramers. Proton ordering can be |
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rows of interlocking water tetramers. This crystal structure has a |
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limited resemblence to a recent two-dimensional ice tessellation |
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simulated on a silica surface.\cite{Yang04} Proton ordering can be |
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accomplished by orienting two of the molecules so that both of their |
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donated hydrogen bonds are internal to their tetramer |
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(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
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(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
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water tetramers, the hydrogen bonds are not as linear as those |
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observed in ice $I_h$, however the interlocking of these subunits |
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appears to provide significant stabilization to the overall |
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crystal. The arrangement of these tetramers results in surrounding |
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open octagonal cavities that are typically greater than 6.3 \AA\ in |
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diameter. This relatively open overall structure leads to crystals |
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appears to provide significant stabilization to the overall crystal. |
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The arrangement of these tetramers results in surrounding open |
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octagonal cavities that are typically greater than 6.3 \AA\ in |
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diameter. This relatively open overall structure leads to crystals |
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that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. |
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|
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\begin{figure} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{orderedIcei.eps} |
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\caption{Image of a proton ordered crystal of Ice-{\it i} looking |
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down the (001) crystal face. The rows of water tetramers surrounded by |
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octagonal pores leads to a crystal structure that is significantly |
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down the (001) crystal face. The rows of water tetramers surrounded |
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by octagonal pores leads to a crystal structure that is significantly |
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less dense than ice $I_h$.} |
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\label{protOrder} |
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\end{figure} |
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see our previous work and related |
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articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
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considered energetic stabilization and neglected entropic |
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contributions to the overall free energy. To address this issue, we |
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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 to the free energies of cubic |
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and hexagonal ice $I$ (the experimental low density ice polymorphs) |
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common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
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field parametrized single point dipole water model (SSD/RF). It should |
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be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$) |
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was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit |
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was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit |
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cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it |
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i} unit it is extended in the direction of the (001) face and |
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compressed along the other two faces. There is typically a small |
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Canonical ensemble (NVT) molecular dynamics calculations were |
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performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
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All molecules were treated as rigid bodies, with orientational motion |
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propagated using the symplectic DLM integration method. Details about |
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propagated using the symplectic DLM integration method. Details about |
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the implementation of this technique can be found in a recent |
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publication.\cite{Dullweber1997} |
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|
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SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
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and 400 K for all of these water models were also determined using |
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this same technique in order to determine melting points and to |
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generate phase diagrams. All simulations were carried out at densities |
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which correspond to a pressure of approximately 1 atm at their |
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respective temperatures. |
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generate phase diagrams. All simulations were carried out at |
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densities which correspond to a pressure of approximately 1 atm at |
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their respective temperatures. |
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|
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Thermodynamic integration involves a sequence of simulations during |
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which the system of interest is converted into a reference system for |
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which the free energy is known analytically. This transformation path |
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which the free energy is known analytically. This transformation path |
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is then integrated in order to determine the free energy difference |
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between the two states: |
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\begin{equation} |
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)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
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\end{equation} |
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where $V$ is the interaction potential and $\lambda$ is the |
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transformation parameter that scales the overall |
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potential. Simulations are distributed strategically along this path |
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in order to sufficiently sample the regions of greatest change in the |
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potential. Typical integrations in this study consisted of $\sim$25 |
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simulations ranging from 300 ps (for the unaltered system) to 75 ps |
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(near the reference state) in length. |
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transformation parameter that scales the overall potential. |
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Simulations are distributed strategically along this path in order to |
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sufficiently sample the regions of greatest change in the potential. |
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Typical integrations in this study consisted of $\sim$25 simulations |
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ranging from 300 ps (for the unaltered system) to 75 ps (near the |
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reference state) in length. |
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|
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For the thermodynamic integration of molecular crystals, the Einstein |
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crystal was chosen as the reference system. In an Einstein crystal, |
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crystal was chosen as the reference system. In an Einstein crystal, |
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the molecules are restrained at their ideal lattice locations and |
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orientations. Using harmonic restraints, as applied by B\`{a}ez and |
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Clancy, the total potential for this reference crystal |
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where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
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the spring constants restraining translational motion and deflection |
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of and rotation around the principle axis of the molecule |
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respectively. It is clear from Fig. \ref{waterSpring} that the values |
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of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
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$-\pi$ to $\pi$. The partition function for a molecular crystal |
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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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\begin{eqnarray} |
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\caption{Possible orientational motions for a restrained molecule. |
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$\theta$ angles correspond to displacement from the body-frame {\it |
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z}-axis, while $\omega$ angles correspond to rotation about the |
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body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
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body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
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constants for the harmonic springs restraining motion in the $\theta$ |
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and $\omega$ directions.} |
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\label{waterSpring} |
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methods.\cite{Baez95b} |
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|
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Charge, dipole, and Lennard-Jones interactions were modified by a |
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cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
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). By applying this function, these interactions are smoothly |
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truncated, thereby avoiding the poor energy conservation which results |
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from harsher truncation schemes. The effect of a long-range correction |
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was also investigated on select model systems in a variety of |
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manners. For the SSD/RF model, a reaction field with a fixed |
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dielectric constant of 80 was applied in all |
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simulations.\cite{Onsager36} For a series of the least computationally |
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expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
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performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
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\AA\ cutoff results. Finally, the effects of utilizing an Ewald |
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summation were estimated for TIP3P and SPC/E by performing single |
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configuration calculations with Particle-Mesh Ewald (PME) in the |
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TINKER molecular mechanics software package.\cite{Tinker} The |
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calculated energy difference in the presence and absence of PME was |
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applied to the previous results in order to predict changes to the |
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free energy landscape. |
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cubic switching between 100\% and 85\% of the cutoff value (9 \AA). |
255 |
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By applying this function, these interactions are smoothly truncated, |
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thereby avoiding the poor energy conservation which results from |
257 |
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harsher truncation schemes. The effect of a long-range correction was |
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also investigated on select model systems in a variety of manners. |
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For the SSD/RF model, a reaction field with a fixed dielectric |
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constant of 80 was applied in all simulations.\cite{Onsager36} For a |
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series of the least computationally expensive models (SSD/E, SSD/RF, |
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and TIP3P), simulations were performed with longer cutoffs of 12 and |
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15 \AA\ to compare with the 9 \AA\ cutoff results. Finally, the |
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effects of utilizing an Ewald summation were estimated for TIP3P and |
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SPC/E by performing single configuration calculations with |
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Particle-Mesh Ewald (PME) in the TINKER molecular mechanics software |
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package.\cite{Tinker} The calculated energy difference in the presence |
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and absence of PME was applied to the previous results in order to |
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predict changes to the free energy landscape. |
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|
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\section{Results and discussion} |
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|
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model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
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Ice XI, the experimentally-observed proton-ordered variant of ice |
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$I_h$, was investigated initially, but was found to be not as stable |
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as proton disordered or antiferroelectric variants of ice $I_h$. The |
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as proton disordered or antiferroelectric variants of ice $I_h$. The |
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proton ordered variant of ice $I_h$ used here is a simple |
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antiferroelectric version that we devised, and it has an 8 molecule |
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unit cell similar to other predicted antiferroelectric $I_h$ |
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crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
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molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
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molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger |
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crystal sizes were necessary for simulations involving larger cutoff |
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values. |
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molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The |
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larger crystal sizes were necessary for simulations involving larger |
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cutoff values. |
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|
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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|
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\caption{Calculated free energies for several ice polymorphs with a |
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variety of common water models. All calculations used a cutoff radius |
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of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
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kcal/mol. Calculated error of the final digits is in parentheses.} |
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variety of common water models. All calculations used a cutoff radius |
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of 9.0 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
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kcal/mol. Calculated error of the final digits is in parentheses.} |
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|
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\begin{tabular}{lcccc} |
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\begin{tabular}{lccccc} |
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\hline |
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Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
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Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} & Ice-{\it i}$^\prime$\\ |
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\hline |
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TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
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TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
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TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
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SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\ |
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SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
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SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2)\\ |
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TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & -\\ |
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TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3)\\ |
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TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2)\\ |
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SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2)\\ |
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SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & -\\ |
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SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2) & -\\ |
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\end{tabular} |
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\label{freeEnergy} |
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\end{center} |
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|
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The free energy values computed for the studied polymorphs indicate |
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that Ice-{\it i} is the most stable state for all of the common water |
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models studied. With the calculated free energy at these state points, |
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the Gibbs-Helmholtz equation was used to project to other state points |
320 |
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and to build phase diagrams. Figures |
321 |
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\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
322 |
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from the free energy results. All other models have similar structure, |
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although the crossing points between the phases move to slightly |
324 |
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different temperatures and pressures. It is interesting to note that |
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ice $I$ does not exist in either cubic or hexagonal form in any of the |
326 |
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phase diagrams for any of the models. For purposes of this study, ice |
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B is representative of the dense ice polymorphs. A recent study by |
328 |
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Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
318 |
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models studied. With the calculated free energy at these state |
319 |
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points, the Gibbs-Helmholtz equation was used to project to other |
320 |
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state points and to build phase diagrams. Figures \ref{tp3phasedia} |
321 |
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and \ref{ssdrfphasedia} are example diagrams built from the free |
322 |
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energy results. All other models have similar structure, although the |
323 |
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crossing points between the phases move to slightly different |
324 |
> |
temperatures and pressures. It is interesting to note that ice $I$ |
325 |
> |
does not exist in either cubic or hexagonal form in any of the phase |
326 |
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diagrams for any of the models. For purposes of this study, ice B is |
327 |
> |
representative of the dense ice polymorphs. A recent study by Sanz |
328 |
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{\it et al.} goes into detail on the phase diagrams for SPC/E and |
329 |
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TIP4P at higher pressures than those studied here.\cite{Sanz04} |
330 |
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|
331 |
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\begin{figure} |
332 |
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
333 |
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\caption{Phase diagram for the TIP3P water model in the low pressure |
334 |
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regime. The displayed $T_m$ and $T_b$ values are good predictions of |
334 |
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regime. The displayed $T_m$ and $T_b$ values are good predictions of |
335 |
|
the experimental values; however, the solid phases shown are not the |
336 |
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experimentally observed forms. Both cubic and hexagonal ice $I$ are |
336 |
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experimentally observed forms. Both cubic and hexagonal ice $I$ are |
337 |
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higher in energy and don't appear in the phase diagram.} |
338 |
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\label{tp3phasedia} |
339 |
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\end{figure} |
341 |
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\begin{figure} |
342 |
|
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
343 |
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\caption{Phase diagram for the SSD/RF water model in the low pressure |
344 |
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regime. Calculations producing these results were done under an |
345 |
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applied reaction field. It is interesting to note that this |
344 |
> |
regime. Calculations producing these results were done under an |
345 |
> |
applied reaction field. It is interesting to note that this |
346 |
|
computationally efficient model (over 3 times more efficient than |
347 |
|
TIP3P) exhibits phase behavior similar to the less computationally |
348 |
|
conservative charge based models.} |
355 |
|
|
356 |
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\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
357 |
|
temperatures at 1 atm for several common water models compared with |
358 |
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experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
359 |
< |
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
358 |
> |
experiment. The $T_m$ and $T_s$ values from simulation correspond to |
359 |
> |
a transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
360 |
|
liquid or gas state.} |
361 |
|
|
362 |
|
\begin{tabular}{lccccccc} |
373 |
|
\end{table*} |
374 |
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|
375 |
|
Table \ref{meltandboil} lists the melting and boiling temperatures |
376 |
< |
calculated from this work. Surprisingly, most of these models have |
377 |
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melting points that compare quite favorably with experiment. The |
376 |
> |
calculated from this work. Surprisingly, most of these models have |
377 |
> |
melting points that compare quite favorably with experiment. The |
378 |
|
unfortunate aspect of this result is that this phase change occurs |
379 |
|
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
380 |
< |
liquid state. These results are actually not contrary to previous |
381 |
< |
studies in the literature. Earlier free energy studies of ice $I$ |
382 |
< |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
383 |
< |
being attributed to choice of interaction truncation and different |
379 |
< |
ordered and disordered molecular |
380 |
> |
liquid state. These results are actually not contrary to other |
381 |
> |
studies. Studies of ice $I_h$ using TIP4P predict a $T_m$ ranging |
382 |
> |
from 214 to 238 K (differences being attributed to choice of |
383 |
> |
interaction truncation and different ordered and disordered molecular |
384 |
|
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
385 |
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
386 |
< |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
387 |
< |
calculated at 265 K, significantly higher in temperature than the |
388 |
< |
previous studies. Also of interest in these results is that SSD/E does |
386 |
> |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
387 |
> |
calculated to be 265 K, indicating that these simulation based |
388 |
> |
structures ought to be included in studies probing phase transitions |
389 |
> |
with this model. Also of interest in these results is that SSD/E does |
390 |
|
not exhibit a melting point at 1 atm, but it shows a sublimation point |
391 |
< |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
392 |
< |
all other polymorphs for this particular model under these |
393 |
< |
conditions. While troubling, this behavior resulted in spontaneous |
391 |
> |
at 355 K. This is due to the significant stability of Ice-{\it i} |
392 |
> |
over all other polymorphs for this particular model under these |
393 |
> |
conditions. While troubling, this behavior resulted in spontaneous |
394 |
|
crystallization of Ice-{\it i} and led us to investigate this |
395 |
< |
structure. These observations provide a warning that simulations of |
395 |
> |
structure. These observations provide a warning that simulations of |
396 |
|
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
397 |
< |
risk of spontaneous crystallization. However, this risk lessens when |
397 |
> |
risk of spontaneous crystallization. However, this risk lessens when |
398 |
|
applying a longer cutoff. |
399 |
|
|
400 |
|
\begin{figure} |
401 |
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
402 |
< |
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
403 |
< |
TIP3P, and (C) SSD/RF with a reaction field. Both SSD/E and TIP3P show |
404 |
< |
significant cutoff radius dependence of the free energy and appear to |
405 |
< |
converge when moving to cutoffs greater than 12 \AA. Use of a reaction |
406 |
< |
field with SSD/RF results in free energies that exhibit minimal cutoff |
407 |
< |
radius dependence.} |
402 |
> |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
403 |
> |
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
404 |
> |
with an added Ewald correction term. Calculations performed without a |
405 |
> |
long-range correction show noticable free energy dependence on the |
406 |
> |
cutoff radius and show some degree of converge at large cutoff radii. |
407 |
> |
Inclusion of a long-range correction reduces the cutoff radius |
408 |
> |
dependence of the free energy for all the models. Error for the |
409 |
> |
larger cutoff points is equivalent to that observed at 9.0 \AA\ (see |
410 |
> |
Table \ref{freeEnergy}). Data for ice I$_c$ with TIP3P using both 12 |
411 |
> |
and 13.5 \AA\ cutoffs were omitted because the crystal was prone to |
412 |
> |
distortion and melting at 200 K. Ice-{\it i}$^\prime$ is the form of |
413 |
> |
Ice-{\it i} used in the SPC/E simulations.} |
414 |
|
\label{incCutoff} |
415 |
|
\end{figure} |
416 |
|
|
417 |
|
Increasing the cutoff radius in simulations of the more |
418 |
|
computationally efficient water models was done in order to evaluate |
419 |
|
the trend in free energy values when moving to systems that do not |
420 |
< |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
421 |
< |
free energy of all the ice polymorphs for the SSD/E and TIP3P models |
422 |
< |
show a substantial dependence on cutoff radius. In general, there is a |
423 |
< |
narrowing of the free energy differences while moving to greater |
424 |
< |
cutoff radii. As the free energies for the polymorphs converge, the |
425 |
< |
stability advantage that Ice-{\it i} exhibits is reduced; however, it |
426 |
< |
remains the most stable polymorph for both of these models over the |
427 |
< |
depicted range for both models. This narrowing trend is not |
428 |
< |
significant in the case of SSD/RF, indicating that the free energies |
429 |
< |
calculated with a reaction field present provide, at minimal |
430 |
< |
computational cost, a more accurate picture of the free energy |
431 |
< |
landscape in the absence of potential truncation. Interestingly, |
432 |
< |
increasing the cutoff radius a mere 1.5 \AA\ with the SSD/E model |
433 |
< |
destabilizes the Ice-{\it i} polymorph enough that the liquid state is |
434 |
< |
preferred under standard simulation conditions (298 K and 1 |
435 |
< |
atm). Thus, it is recommended that simulations using this model choose |
436 |
< |
interaction truncation radii greater than 9 \AA. Considering this |
437 |
< |
stabilization provided by smaller cutoffs, it is not surprising that |
438 |
< |
crystallization into Ice-{\it i} was observed with SSD/E. The choice |
439 |
< |
of a 9 \AA\ cutoff in the previous simulations gives the Ice-{\it i} |
429 |
< |
polymorph a greater than 1 kcal/mol lower free energy than the ice |
430 |
< |
$I_\textrm{h}$ starting configurations. |
420 |
> |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
421 |
> |
free energy of the ice polymorphs with water models lacking a |
422 |
> |
long-range correction show a significant cutoff radius dependence. In |
423 |
> |
general, there is a narrowing of the free energy differences while |
424 |
> |
moving to greater cutoff radii. As the free energies for the |
425 |
> |
polymorphs converge, the stability advantage that Ice-{\it i} exhibits |
426 |
> |
is reduced. Interestingly, increasing the cutoff radius a mere 1.5 |
427 |
> |
\AA\ with the SSD/E model destabilizes the Ice-{\it i} polymorph |
428 |
> |
enough that the liquid state is preferred under standard simulation |
429 |
> |
conditions (298 K and 1 atm). Thus, it is recommended that |
430 |
> |
simulations using this model choose interaction truncation radii |
431 |
> |
greater than 9 \AA. Considering the stabilization of Ice-{\it i} with |
432 |
> |
smaller cutoffs, it is not surprising that crystallization was |
433 |
> |
observed with SSD/E. The choice of a 9 \AA\ cutoff in the previous |
434 |
> |
simulations gives the Ice-{\it i} polymorph a greater than 1 kcal/mol |
435 |
> |
lower free energy than the ice $I_\textrm{h}$ starting configurations. |
436 |
> |
Additionally, it should be noted that ice $I_c$ is not stable with |
437 |
> |
cutoff radii of 12 and 13.5 \AA\ using the TIP3P water model. These |
438 |
> |
simulations showed bulk distortions of the simulation cell that |
439 |
> |
rapidly deteriorated crystal integrity. |
440 |
|
|
441 |
< |
To further study the changes resulting to the inclusion of a |
442 |
< |
long-range interaction correction, the effect of an Ewald summation |
443 |
< |
was estimated by applying the potential energy difference do to its |
444 |
< |
inclusion in systems in the presence and absence of the |
445 |
< |
correction. This was accomplished by calculation of the potential |
446 |
< |
energy of identical crystals both with and without PME. The free |
447 |
< |
energies for the investigated polymorphs using the TIP3P and SPC/E |
448 |
< |
water models are shown in Table \ref{pmeShift}. The same trend pointed |
449 |
< |
out through increase of cutoff radius is observed in these PME |
450 |
< |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
451 |
< |
for both the TIP3P and SPC/E water models; however, the narrowing of |
452 |
< |
the free energy differences between the various solid forms is |
453 |
< |
significant enough that it becomes less clear that it is the most |
454 |
< |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
455 |
< |
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
456 |
< |
as well, indicating that Ice-{\it i} might be metastable with respect |
457 |
< |
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
458 |
< |
not significantly alter the finding that the Ice-{\it i} polymorph is |
459 |
< |
a stable crystal structure that should be considered when studying the |
460 |
< |
phase behavior of water models. |
441 |
> |
Adjacent to each of these model plots is a system with an applied or |
442 |
> |
estimated long-range correction. SSD/RF was parametrized for use with |
443 |
> |
a reaction field, and the benefit provided by this computationally |
444 |
> |
inexpensive correction is apparent. Due to the relative independence |
445 |
> |
of the resultant free energies, calculations performed with a small |
446 |
> |
cutoff radius provide resultant properties similar to what one would |
447 |
> |
expect for the bulk material. In the cases of TIP3P and SPC/E, the |
448 |
> |
effect of an Ewald summation was estimated by applying the potential |
449 |
> |
energy difference do to its inclusion in systems in the presence and |
450 |
> |
absence of the correction. This was accomplished by calculation of |
451 |
> |
the potential energy of identical crystals both with and without |
452 |
> |
particle mesh Ewald (PME). Similar behavior to that observed with |
453 |
> |
reaction field is seen for both of these models. The free energies |
454 |
> |
show less dependence on cutoff radius and span a more narrowed range |
455 |
> |
for the various polymorphs. Like the dipolar water models, TIP3P |
456 |
> |
displays a relatively constant preference for the Ice-{\it i} |
457 |
> |
polymorph. Crystal preference is much more difficult to determine for |
458 |
> |
SPC/E. Without a long-range correction, each of the polymorphs |
459 |
> |
studied assumes the role of the preferred polymorph under different |
460 |
> |
cutoff conditions. The inclusion of the Ewald correction flattens and |
461 |
> |
narrows the sequences of free energies so much that they often overlap |
462 |
> |
within error, indicating that other conditions, such as cell volume in |
463 |
> |
microcanonical simulations, can influence the chosen polymorph upon |
464 |
> |
crystallization. All of these results support the finding that the |
465 |
> |
Ice-{\it i} polymorph is a stable crystal structure that should be |
466 |
> |
considered when studying the phase behavior of water models. |
467 |
|
|
453 |
– |
\begin{table*} |
454 |
– |
\begin{minipage}{\linewidth} |
455 |
– |
\begin{center} |
456 |
– |
|
457 |
– |
\caption{The free energy of the studied ice polymorphs after applying |
458 |
– |
the energy difference attributed to the inclusion of the PME |
459 |
– |
long-range interaction correction. Units are kcal/mol.} |
460 |
– |
|
461 |
– |
\begin{tabular}{ccccc} |
462 |
– |
\hline |
463 |
– |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} \\ |
464 |
– |
\hline |
465 |
– |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3) \\ |
466 |
– |
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2) \\ |
467 |
– |
\end{tabular} |
468 |
– |
\label{pmeShift} |
469 |
– |
\end{center} |
470 |
– |
\end{minipage} |
471 |
– |
\end{table*} |
472 |
– |
|
468 |
|
\section{Conclusions} |
469 |
|
|
470 |
|
The free energy for proton ordered variants of hexagonal and cubic ice |
471 |
|
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
472 |
|
calculated under standard conditions for several common water models |
473 |
< |
via thermodynamic integration. All the water models studied show |
473 |
> |
via thermodynamic integration. All the water models studied show |
474 |
|
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
475 |
< |
\AA\ switching function cutoff. Calculated melting and boiling points |
475 |
> |
\AA\ switching function cutoff. Calculated melting and boiling points |
476 |
|
show surprisingly good agreement with the experimental values; |
477 |
< |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
477 |
> |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
478 |
|
effect of interaction truncation was investigated through variation of |
479 |
|
the cutoff radius, use of a reaction field parameterized model, and |
480 |
< |
estimation of the results in the presence of the Ewald |
481 |
< |
summation. Interaction truncation has a significant effect on the |
482 |
< |
computed free energy values, and may significantly alter the free |
483 |
< |
energy landscape for the more complex multipoint water models. Despite |
484 |
< |
these effects, these results show Ice-{\it i} to be an important ice |
485 |
< |
polymorph that should be considered in simulation studies. |
480 |
> |
estimation of the results in the presence of the Ewald summation. |
481 |
> |
Interaction truncation has a significant effect on the computed free |
482 |
> |
energy values, and may significantly alter the free energy landscape |
483 |
> |
for the more complex multipoint water models. Despite these effects, |
484 |
> |
these results show Ice-{\it i} to be an important ice polymorph that |
485 |
> |
should be considered in simulation studies. |
486 |
|
|
487 |
|
Due to this relative stability of Ice-{\it i} in all of the |
488 |
|
investigated simulation conditions, the question arises as to possible |