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chrisfen |
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\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER SIMULATIONS} |
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Molecular dynamics is a valuable tool for studying the phase behavior |
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of systems ranging from small or simple |
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molecules\cite{Matsumoto02,andOthers} to complex biological |
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species.\cite{bigStuff} Many techniques have been developed to |
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investigate the thermodynamic properites of model substances, |
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providing both qualitative and quantitative comparisons between |
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simulations and experiment.\cite{thermMethods} Investigation of these |
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properties leads to the development of new and more accurate models, |
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leading to better understanding and depiction of physical processes |
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and intricate molecular systems. |
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Water has proven to be a challenging substance to depict in |
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simulations, and a variety of models have been developed to describe |
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its behavior under varying simulation |
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conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04} |
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These models have been used to investigate important physical |
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phenomena like phase transitions and the hydrophobic |
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effect.\cite{Yamada02} With the choice of models available, it |
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is only natural to compare the models under interesting thermodynamic |
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conditions in an attempt to clarify the limitations of each of the |
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models.\cite{modelProps} Two important property to quantify are the |
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Gibbs and Helmholtz free energies, particularly for the solid forms of |
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water. Difficulty in these types of studies typically arises from the |
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assortment of possible crystalline polymorphs that water adopts over a |
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wide range of pressures and temperatures. There are currently 13 |
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recognized forms of ice, and it is a challenging task to investigate |
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the entire free energy landscape.\cite{Sanz04} Ideally, research is |
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focused on the phases having the lowest free energy at a given state |
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point, because these phases will dictate the true transition |
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temperatures and pressures for their respective model. |
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In this paper, standard reference state methods were applied to known |
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crystalline water polymorphs in the low pressure regime. This work is |
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unique in the fact that one of the crystal lattices was arrived at |
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through crystallization of a computationally efficient water model |
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under constant pressure and temperature conditions. Crystallization |
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events are interesting in and of |
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themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure |
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obtained in this case is different from any previously observed ice |
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polymorphs in experiment or simulation.\cite{Fennell04} We have named |
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this structure Ice-{\it i} to indicate its origin in computational |
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simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight |
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water molecules that stack in rows of interlocking water |
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tetramers. Proton ordering can be accomplished by orienting two of the |
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molecules so that both of their donated hydrogen bonds are internal to |
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their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal |
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constructed of water tetramers, the hydrogen bonds are not as linear |
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as those observed in ice $I_h$, however the interlocking of these |
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subunits 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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that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{./figures/unitCell.pdf} |
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\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
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elongated variant of Ice-{\it i}. For Ice-{\it i}, the $a$ to $c$ |
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relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a = |
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1.7850c$.} |
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\label{iceiCell} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=3.5in]{./figures/orderedIcei.pdf} |
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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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less dense than ice $I_h$.} |
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\label{protOrder} |
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\end{figure} |
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Results from our previous study indicated that Ice-{\it i} is the |
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minimum energy crystal structure for the single point water models we |
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investigated (for discussions on these single point dipole models, see |
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the 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, the |
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absolute free energy of this crystal was calculated 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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and ice B (a higher density, but very stable crystal structure |
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observed by B\`{a}ez and Clancy in free energy studies of |
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SPC/E).\cite{Baez95b} This work includes results for the water model |
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from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
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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-$i^\prime$) was used |
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in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of |
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this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit |
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it is extended in the direction of the (001) face and compressed along |
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the other two faces. |
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\section{Methods} |
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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 described in |
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section \ref{sec:IntroIntegration}. |
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Thermodynamic integration was utilized to calculate the free energy of |
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several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, |
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SSD/RF, and SSD/E water models. Liquid state free energies at 300 and |
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400 K for all of these water models were also determined using this |
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same technique in order to determine melting points and generate phase |
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diagrams. All simulations were carried out at densities resulting in a |
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pressure of approximately 1 atm at their respective temperatures. |
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A single thermodynamic integration involves a sequence of simulations |
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over which the system of interest is converted into a reference system |
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for which the free energy is known analytically. This transformation |
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path is then integrated in order to determine the free energy |
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difference between the two states: |
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\begin{equation} |
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\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
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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 unevenly along this path in |
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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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For the thermodynamic integration of molecular crystals, the Einstein |
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crystal was chosen as the reference state. In an Einstein crystal, the |
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molecules are harmonically restrained at their ideal lattice locations |
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and orientations. 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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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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)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right |
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)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right |
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)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi |
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K_\omega K_\theta)^{\frac{1}{2}}}\exp\left |
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(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right |
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)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
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\label{ecFreeEnergy} |
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\end{eqnarray} |
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where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
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\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
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$K_\mathrm{\omega}$ are the spring constants restraining translational |
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motion and deflection of and rotation around the principle axis of the |
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molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
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minimum potential energy of the ideal crystal. In the case of |
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molecular liquids, the ideal vapor is chosen as the target reference |
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state. |
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\begin{figure} |
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\centering |
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\includegraphics[width=3.5in]{./figures/rotSpring.pdf} |
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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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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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\end{figure} |
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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, results from the use of an Ewald |
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summation were estimated for TIP3P and SPC/E by performing |
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calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
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mechanics software package.\cite{Tinker} The calculated energy |
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difference in the presence and absence of PME was applied to the |
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previous results in order to predict changes to the free energy |
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landscape. |
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\section{Results and discussion} |
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The free energy of proton ordered Ice-{\it i} was calculated and |
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compared with the free energies of proton ordered variants of the |
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experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
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as well as the higher density ice B, observed by B\`{a}ez and Clancy |
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and thought to be the minimum free energy structure for the SPC/E |
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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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proton ordered variant of ice $I_h$ used here is a simple |
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antiferroelectric version that has an 8 molecule unit |
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cell.\cite{Davidson84} The crystals contained 648 or 1728 molecules |
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for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for |
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ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
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were necessary for simulations involving larger cutoff values. |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\renewcommand{\thefootnote}{\thempfootnote} |
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\begin{center} |
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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$1 atm. Units are |
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kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
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$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
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\begin{tabular}{ l c c c c } |
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\hline |
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Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
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\hline |
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TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\\ |
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TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\ |
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TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\ |
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SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\ |
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SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\ |
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SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\ |
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\end{tabular} |
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\label{freeEnergy} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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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 free energy at these state points, the |
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Gibbs-Helmholtz equation was used to project to other state points and |
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to build phase diagrams. Figures |
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\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
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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 exist at slightly |
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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 |
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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 |
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Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
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TIP4P in the high pressure regime.\cite{Sanz04} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{./figures/tp3PhaseDia.pdf} |
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\caption{Phase diagram for the TIP3P water model in the low pressure |
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regime. The displayed $T_m$ and $T_b$ values are good predictions of |
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the experimental values; however, the solid phases shown are not the |
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experimentally observed forms. Both cubic and hexagonal ice $I$ are |
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higher in energy and don't appear in the phase diagram.} |
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\label{tp3phasedia} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{./figures/ssdrfPhaseDia.pdf} |
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\caption{Phase diagram for the SSD/RF water model in the low pressure |
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regime. Calculations producing these results were done under an |
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applied reaction field. It is interesting to note that this |
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computationally efficient model (over 3 times more efficient than |
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TIP3P) exhibits phase behavior similar to the less computationally |
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conservative charge based models.} |
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\label{ssdrfphasedia} |
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\end{figure} |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\renewcommand{\thefootnote}{\thempfootnote} |
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\begin{center} |
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\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
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temperatures at 1 atm for several common water models compared with |
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experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
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transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
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liquid or gas state.} |
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\begin{tabular}{ l c c c c c c c } |
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\hline |
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Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
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\hline |
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$T_m$ (K) & 269(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\ |
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$T_b$ (K) & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\ |
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$T_s$ (K) & - & - & - & - & 355(3) & - & -\\ |
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\end{tabular} |
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\label{meltandboil} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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Table \ref{meltandboil} lists the melting and boiling temperatures |
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calculated from this work. Surprisingly, most of these models have |
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melting points that compare quite favorably with experiment. The |
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unfortunate aspect of this result is that this phase change occurs |
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between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
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liquid state. These results are actually not contrary to previous |
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studies in the literature. Earlier free energy studies of ice $I$ |
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using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
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being attributed to choice of interaction truncation and different |
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ordered and disordered molecular |
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arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
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Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
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predicted from this work. However, the $T_m$ from Ice-{\it i} is |
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calculated at 265 K, significantly higher in temperature than the |
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previous studies. Also of interest in these results is that SSD/E does |
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not exhibit a melting point at 1 atm, but it shows a sublimation point |
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at 355 K. This is due to the significant stability of Ice-{\it i} over |
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all other polymorphs for this particular model under these |
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conditions. While troubling, this behavior turned out to be |
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advantageous in that it facilitated the spontaneous crystallization of |
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Ice-{\it i}. These observations provide a warning that simulations of |
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SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
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risk of spontaneous crystallization. However, this risk changes when |
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applying a longer cutoff. |
315 |
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|
316 |
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\begin{figure} |
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\includegraphics[width=\linewidth]{./figures/cutoffChange.pdf} |
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\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
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TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
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\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
321 |
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\AA . These crystals are unstable at 200 K and rapidly convert into |
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liquids. The connecting lines are qualitative visual aid.} |
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\label{incCutoff} |
324 |
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\end{figure} |
325 |
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|
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Increasing the cutoff radius in simulations of the more |
327 |
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computationally efficient water models was done in order to evaluate |
328 |
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the trend in free energy values when moving to systems that do not |
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involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
330 |
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free energy of all the ice polymorphs show a substantial dependence on |
331 |
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cutoff radius. In general, there is a narrowing of the free energy |
332 |
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differences while moving to greater cutoff radius. Interestingly, by |
333 |
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increasing the cutoff radius, the free energy gap was narrowed enough |
334 |
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in the SSD/E model that the liquid state is preferred under standard |
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simulation conditions (298 K and 1 atm). Thus, it is recommended that |
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simulations using this model choose interaction truncation radii |
337 |
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greater than 9 \AA\ . This narrowing trend is much more subtle in the |
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case of SSD/RF, indicating that the free energies calculated with a |
339 |
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reaction field present provide a more accurate picture of the free |
340 |
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energy landscape in the absence of potential truncation. |
341 |
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|
342 |
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To further study the changes resulting to the inclusion of a |
343 |
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long-range interaction correction, the effect of an Ewald summation |
344 |
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was estimated by applying the potential energy difference do to its |
345 |
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inclusion in systems in the presence and absence of the |
346 |
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correction. This was accomplished by calculation of the potential |
347 |
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energy of identical crystals with and without PME using TINKER. The |
348 |
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free energies for the investigated polymorphs using the TIP3P and |
349 |
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SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
350 |
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|
are not fully supported in TINKER, so the results for these models |
351 |
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|
could not be estimated. The same trend pointed out through increase of |
352 |
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cutoff radius is observed in these PME results. Ice-{\it i} is the |
353 |
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preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
354 |
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|
water models; however, there is a narrowing of the free energy |
355 |
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|
differences between the various solid forms. In the case of SPC/E this |
356 |
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narrowing is significant enough that it becomes less clear that |
357 |
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Ice-{\it i} is the most stable polymorph, and is possibly metastable |
358 |
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|
with respect to ice B and possibly ice $I_c$. However, these results |
359 |
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|
do not significantly alter the finding that the Ice-{\it i} polymorph |
360 |
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is a stable crystal structure that should be considered when studying |
361 |
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the phase behavior of water models. |
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|
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\begin{table*} |
364 |
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\begin{minipage}{\linewidth} |
365 |
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|
\renewcommand{\thefootnote}{\thempfootnote} |
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\begin{center} |
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\caption{The free energy of the studied ice polymorphs after applying |
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|
the energy difference attributed to the inclusion of the PME |
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long-range interaction correction. Units are kcal/mol.} |
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|
\begin{tabular}{ l c c c c } |
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\hline |
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\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
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\hline |
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TIP3P & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)\\ |
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SPC/E & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\ |
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\end{tabular} |
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\label{pmeShift} |
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\end{center} |
379 |
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\end{minipage} |
380 |
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\end{table*} |
381 |
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|
382 |
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\section{Conclusions} |
383 |
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|
384 |
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The free energy for proton ordered variants of hexagonal and cubic ice |
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$I$, ice B, and recently discovered Ice-{\it i} were calculated under |
386 |
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|
standard conditions for several common water models via thermodynamic |
387 |
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|
integration. All the water models studied show Ice-{\it i} to be the |
388 |
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|
minimum free energy crystal structure in the with a 9 \AA\ switching |
389 |
|
|
function cutoff. Calculated melting and boiling points show |
390 |
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|
surprisingly good agreement with the experimental values; however, the |
391 |
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|
solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
392 |
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|
interaction truncation was investigated through variation of the |
393 |
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|
cutoff radius, use of a reaction field parameterized model, and |
394 |
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|
estimation of the results in the presence of the Ewald |
395 |
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|
summation. Interaction truncation has a significant effect on the |
396 |
|
|
computed free energy values, and may significantly alter the free |
397 |
|
|
energy landscape for the more complex multipoint water models. Despite |
398 |
|
|
these effects, these results show Ice-{\it i} to be an important ice |
399 |
|
|
polymorph that should be considered in simulation studies. |
400 |
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|
|
401 |
|
|
Due to this relative stability of Ice-{\it i} in all manner of |
402 |
|
|
investigated simulation examples, the question arises as to possible |
403 |
|
|
experimental observation of this polymorph. The rather extensive past |
404 |
|
|
and current experimental investigation of water in the low pressure |
405 |
|
|
regime makes us hesitant to ascribe any relevance of this work outside |
406 |
|
|
of the simulation community. It is for this reason that we chose a |
407 |
|
|
name for this polymorph which involves an imaginary quantity. That |
408 |
|
|
said, there are certain experimental conditions that would provide the |
409 |
|
|
most ideal situation for possible observation. These include the |
410 |
|
|
negative pressure or stretched solid regime, small clusters in vacuum |
411 |
|
|
deposition environments, and in clathrate structures involving small |
412 |
|
|
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
413 |
|
|
our predictions for both the pair distribution function ($g_{OO}(r)$) |
414 |
|
|
and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it |
415 |
|
|
i} at a temperature of 77K. In a quick comparison of the predicted |
416 |
|
|
S(q) for Ice-{\it i} and experimental studies of amorphous solid |
417 |
|
|
water, it is possible that some of the ``spurious'' peaks that could |
418 |
|
|
not be assigned in HDA could correspond to peaks labeled in this |
419 |
|
|
S(q).\cite{Bizid87} It should be noted that there is typically poor |
420 |
|
|
agreement on crystal densities between simulation and experiment, so |
421 |
|
|
such peak comparisons should be made with caution. We will leave it |
422 |
|
|
to our experimental colleagues to determine whether this ice polymorph |
423 |
|
|
is named appropriately or if it should be promoted to Ice-0. |
424 |
|
|
|
425 |
|
|
\begin{figure} |
426 |
|
|
\includegraphics[width=\linewidth]{./figures/iceGofr.pdf} |
427 |
|
|
\caption{Radial distribution functions of Ice-{\it i} and ice $I_c$ |
428 |
|
|
calculated from from simulations of the SSD/RF water model at 77 K.} |
429 |
|
|
\label{fig:gofr} |
430 |
|
|
\end{figure} |
431 |
|
|
|
432 |
|
|
\begin{figure} |
433 |
|
|
\includegraphics[width=\linewidth]{./figures/sofq.pdf} |
434 |
|
|
\caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at |
435 |
|
|
77 K. The raw structure factors have been convoluted with a gaussian |
436 |
|
|
instrument function (0.075 \AA$^{-1}$ width) to compensate for the |
437 |
|
|
trunction effects in our finite size simulations. The labeled peaks |
438 |
|
|
compared favorably with ``spurious'' peaks observed in experimental |
439 |
|
|
studies of amorphous solid water.\cite{Bizid87}} |
440 |
|
|
\label{fig:sofq} |
441 |
|
|
\end{figure} |
442 |
|
|
|