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\begin{document} |
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\title{A Free Energy Study of Low Temperature and Anomalous Ice} |
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|
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\author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote} |
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\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}} |
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|
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\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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%\maketitle |
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\begin{abstract} |
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The free energies of several ice polymorphs in the low pressure regime |
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were calculated using thermodynamic integration of systems consisting |
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of a variety of common water models. Ice-{\it i}, a recent |
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computationally observed solid structure, was determined to be the |
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stable state with the lowest free energy for all the water models |
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investigated. Phase diagrams were generated, and melting and boiling |
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points for all the models were determined and show relatively good |
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agreement with experiment, although the solid phase is different |
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between simulation and experiment. In addition, potential truncation |
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was shown to have an effect on the calculated free energies, and may |
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result in altered free energy landscapes. |
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\end{abstract} |
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\maketitle |
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% BODY OF TEXT |
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|
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\section{Introduction} |
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|
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Molecular dynamics has developed into a valuable tool for studying the |
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phase behavior of systems ranging from small or simple |
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molecules\cite{Matsumoto02andOthers} to complex biological |
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species.\cite{bigStuff} Many techniques have been developed in order |
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to 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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|
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Water has proven to be a challenging substance to depict in |
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simulations, and has resulted in a variety of models that attempt to |
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describe its behavior under a varying simulation |
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conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04} |
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Many of 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{evenMorePapers} With the advent of numerous differing |
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models, it is only natural that attention is placed on the properties |
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of the models themselves in an attempt to clarify their benefits and |
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limitations when applied to a system of interest.\cite{modelProps} One |
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important but challenging property to quantify is the free energy, |
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particularly of the solid forms of water. Difficulty in these types of |
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studies typically arises from the assortment of possible crystalline |
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polymorphs that water that water adopts over a wide range of pressures |
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and temperatures. There are currently 13 recognized forms of ice, and |
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it is a challenging task to investigate the entire free energy |
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landscape.\cite{Sanz04} Ideally, research is focused on the phases |
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having the lowest free energy, because these phases will dictate the |
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true transition temperatures and pressures for their respective model. |
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|
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In this paper, standard reference state methods were applied to the |
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study of crystalline water polymorphs in the low pressure regime. This |
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work is unique in the fact that one of the crystal lattices was |
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arrived at through crystallization of a computationally efficient |
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water model under constant pressure and temperature |
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conditions. Crystallization 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 was different from any previously observed ice |
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polymorphs, in experiment or simulation.\cite{Fennell04} This crystal |
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was termed Ice-{\it i} in homage to 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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waters 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[scale=1.0]{unitCell.eps} |
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\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-2{\it i}, the elongated variant of Ice-{\it i}. For Ice-{\it i}, the $a$ to $c$ relation is given by $a = 2.1214c$, while for Ice-2{\it i}, $a = 1.7850c$.} |
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\label{iceiCell} |
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\end{figure} |
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\begin{figure} |
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\includegraphics[scale=1.0]{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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less dense than ice $I_h$.} |
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\label{protOrder} |
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\end{figure} |
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|
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Results in the previous study indicated that Ice-{\it i} is the |
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minimum energy crystal structure for the single point water models |
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being studied (for discussions on these single point dipole models, |
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see the previous work and related |
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articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only |
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consider energetic stabilization and neglect entropic contributions to |
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the overall free energy. To address this issue, the absolute free |
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energy of this crystal was calculated using thermodynamic integration |
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and compared to the free energies of cubic and hexagonal ice $I$ (the |
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experimental low density ice polymorphs) and ice B (a higher density, |
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but very stable crystal structure observed by B\`{a}ez and Clancy in |
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free energy studies of SPC/E).\cite{Baez95b} This work includes |
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results for the water model from which Ice-{\it i} was crystallized |
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(soft sticky dipole extended, SSD/E) in addition to several common |
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water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field |
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parametrized single point dipole water model (soft sticky dipole |
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reaction field, SSD/RF). In should be noted that a second version of |
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Ice-{\it i} (Ice-2{\it i}) was used in calculations involving SPC/E, |
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TIP4P, and TIP5P. The unit cell of this crystal (Fig. \ref{iceiCell}B) |
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is similar to the Ice-{\it i} unit it is extended in the direction of |
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the (001) face and compressed along the other two faces. |
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|
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\section{Methods} |
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|
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Canonical ensemble (NVT) molecular dynamics calculations were |
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performed using the OOPSE (Object-Oriented Parallel Simulation Engine) |
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molecular mechanics package. All molecules were treated as rigid |
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bodies, with orientational motion propagated using the symplectic DLM |
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integration method. Details about the implementation of these |
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techniques can be found in a recent publication.\cite{Meineke05} |
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|
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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 |
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phase diagrams. All simulations were carried out at densities |
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resulting in a pressure of approximately 1 atm at their respective |
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temperatures. |
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|
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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. This transformation path is then |
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integrated in order to determine the free energy difference between |
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the two states: |
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\begin{equation} |
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\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
177 |
)}{\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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|
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For the thermodynamic integration of molecular crystals, the Einstein |
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Crystal is chosen as the reference state that the system is converted |
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to over the course of the simulation. 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 has been evaluated, and the Helmholtz Free |
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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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\includegraphics[scale=1.0]{rotSpring.eps} |
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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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|
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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 poor energy conserving dynamics resulting |
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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} TINKER was chosen because it |
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can also propagate the motion of rigid-bodies, and provides the most |
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direct comparison to the results from OOPSE. 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 in the free energy |
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landscape. |
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|
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\section{Results and discussion} |
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|
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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 it was found not to be as |
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stable as antiferroelectric variants of proton ordered or even proton |
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disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of |
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ice $I_h$ used here is a simple antiferroelectric version that has an |
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8 molecule unit cell. The crystals contained 648 or 1728 molecules for |
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ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for ice |
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$I_c$, 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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|
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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. *Ice $I_c$ is unstable at 200 K using SSD/RF.} |
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\begin{tabular}{ l c c c c } |
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\hline \\[-7mm] |
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\ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\ |
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\hline \\[-3mm] |
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\ \quad \ TIP3P & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\ |
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\ \quad \ TIP4P & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\ |
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\ \quad \ TIP5P & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\ |
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\ \quad \ SPC/E & \ \quad \ -12.67 & \ \quad \ -12.96 & \ \quad \ -13.25 & \quad -13.55\\ |
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\ \quad \ SSD/E & \ \quad \ -11.27 & \ \quad \ -11.19 & \ \quad \ -12.09 & \quad -12.54\\ |
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\ \quad \ SSD/RF & \ \quad \ -11.51 & \ \quad \ NA* & \ \quad \ -12.08 & \quad -12.29\\ |
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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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|
287 |
The free energy values computed for the studied polymorphs indicate |
288 |
that Ice-{\it i} is the most stable state for all of the common water |
289 |
models studied. With the free energy at these state points, the |
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temperature and pressure dependence of the free energy was used to |
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project to other state points and 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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only the crossing points between these phases exist at different |
295 |
temperatures and pressures. It is interesting to note that ice $I$ |
296 |
does not exist in either cubic or hexagonal form in any of the phase |
297 |
diagrams for any of the models. For purposes of this study, ice B is |
298 |
representative of the dense ice polymorphs. A recent study by Sanz |
299 |
{\it et al.} goes into detail on the phase diagrams for SPC/E and |
300 |
TIP4P in the high pressure regime.\cite{Sanz04} |
301 |
\begin{figure} |
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\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
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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 |
305 |
the experimental values; however, the solid phases shown are not the |
306 |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
307 |
higher in energy and don't appear in the phase diagram.} |
308 |
\label{tp3phasedia} |
309 |
\end{figure} |
310 |
\begin{figure} |
311 |
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
312 |
\caption{Phase diagram for the SSD/RF water model in the low pressure |
313 |
regime. Calculations producing these results were done under an |
314 |
applied reaction field. It is interesting to note that this |
315 |
computationally efficient model (over 3 times more efficient than |
316 |
TIP3P) exhibits phase behavior similar to the less computationally |
317 |
conservative charge based models.} |
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\label{ssdrfphasedia} |
319 |
\end{figure} |
320 |
|
321 |
\begin{table*} |
322 |
\begin{minipage}{\linewidth} |
323 |
\renewcommand{\thefootnote}{\thempfootnote} |
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\begin{center} |
325 |
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
326 |
temperatures of several common water models compared with experiment.} |
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\begin{tabular}{ l c c c c c c c } |
328 |
\hline \\[-7mm] |
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\ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\ |
330 |
\hline \\[-3mm] |
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\ \ $T_m$ (K) & \ \ 269 & \ \ 265 & \ \ 271 & 297 & \ \ - & \ \ 278 & \ \ 273\\ |
332 |
\ \ $T_b$ (K) & \ \ 357 & \ \ 354 & \ \ 337 & 396 & \ \ - & \ \ 349 & \ \ 373\\ |
333 |
\ \ $T_s$ (K) & \ \ - & \ \ - & \ \ - & - & \ \ 355 & \ \ - & \ \ -\\ |
334 |
\end{tabular} |
335 |
\label{meltandboil} |
336 |
\end{center} |
337 |
\end{minipage} |
338 |
\end{table*} |
339 |
|
340 |
Table \ref{meltandboil} lists the melting and boiling temperatures |
341 |
calculated from this work. Surprisingly, most of these models have |
342 |
melting points that compare quite favorably with experiment. The |
343 |
unfortunate aspect of this result is that this phase change occurs |
344 |
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
345 |
liquid state. These results are actually not contrary to previous |
346 |
studies in the literature. Earlier free energy studies of ice $I$ |
347 |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
348 |
being attributed to choice of interaction truncation and different |
349 |
ordered and disordered molecular arrangements). If the presence of ice |
350 |
B and Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
351 |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
352 |
calculated at 265 K, significantly higher in temperature than the |
353 |
previous studies. Also of interest in these results is that SSD/E does |
354 |
not exhibit a melting point at 1 atm, but it shows a sublimation point |
355 |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
356 |
all other polymorphs for this particular model under these |
357 |
conditions. While troubling, this behavior turned out to be |
358 |
advantageous in that it facilitated the spontaneous crystallization of |
359 |
Ice-{\it i}. These observations provide a warning that simulations of |
360 |
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
361 |
risk of spontaneous crystallization. However, this risk changes when |
362 |
applying a longer cutoff. |
363 |
|
364 |
\begin{figure} |
365 |
\includegraphics[width=\linewidth]{cutoffChange.eps} |
366 |
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
367 |
TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
368 |
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
369 |
\AA\. These crystals are unstable at 200 K and rapidly convert into a |
370 |
liquid. The connecting lines are qualitative visual aid.} |
371 |
\label{incCutoff} |
372 |
\end{figure} |
373 |
|
374 |
Increasing the cutoff radius in simulations of the more |
375 |
computationally efficient water models was done in order to evaluate |
376 |
the trend in free energy values when moving to systems that do not |
377 |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
378 |
free energy of all the ice polymorphs show a substantial dependence on |
379 |
cutoff radius. In general, there is a narrowing of the free energy |
380 |
differences while moving to greater cutoff radius. Interestingly, by |
381 |
increasing the cutoff radius, the free energy gap was narrowed enough |
382 |
in the SSD/E model that the liquid state is preferred under standard |
383 |
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
384 |
simulations using this model choose interaction truncation radii |
385 |
greater than 9 \AA\. This narrowing trend is much more subtle in the |
386 |
case of SSD/RF, indicating that the free energies calculated with a |
387 |
reaction field present provide a more accurate picture of the free |
388 |
energy landscape in the absence of potential truncation. |
389 |
|
390 |
To further study the changes resulting to the inclusion of a |
391 |
long-range interaction correction, the effect of an Ewald summation |
392 |
was estimated by applying the potential energy difference do to its |
393 |
inclusion in systems in the presence and absence of the |
394 |
correction. This was accomplished by calculation of the potential |
395 |
energy of identical crystals with and without PME using TINKER. The |
396 |
free energies for the investigated polymorphs using the TIP3P and |
397 |
SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
398 |
are not fully supported in TINKER, so the results for these models |
399 |
could not be estimated. The same trend pointed out through increase of |
400 |
cutoff radius is observed in these PME results. Ice-{\it i} is the |
401 |
preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
402 |
water models; however, there is a narrowing of the free energy |
403 |
differences between the various solid forms. In the case of SPC/E this |
404 |
narrowing is significant enough that it becomes less clear cut that |
405 |
Ice-{\it i} is the most stable polymorph, and is possibly metastable |
406 |
with respect to ice B and possibly ice $I_c$. However, these results |
407 |
do not significantly alter the finding that the Ice-{\it i} polymorph |
408 |
is a stable crystal structure that should be considered when studying |
409 |
the phase behavior of water models. |
410 |
|
411 |
\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{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 \\[-7mm] |
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\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
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\hline \\[-3mm] |
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\ \ TIP3P & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\ |
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\ \ SPC/E & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\ |
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\end{tabular} |
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\label{pmeShift} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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|
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\section{Conclusions} |
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|
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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} where calculated under |
434 |
standard conditions for several common water models via thermodynamic |
435 |
integration. All the water models studied show Ice-{\it i} to be the |
436 |
minimum free energy crystal structure in the with a 9 \AA\ switching |
437 |
function cutoff. Calculated melting and boiling points show |
438 |
surprisingly good agreement with the experimental values; however, the |
439 |
solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
440 |
interaction truncation was investigated through variation of the |
441 |
cutoff radius, use of a reaction field parameterized model, and |
442 |
estimation of the results in the presence of the Ewald summation |
443 |
correction. Interaction truncation has a significant effect on the |
444 |
computed free energy values, and may significantly alter the free |
445 |
energy landscape for the more complex multipoint water models. Despite |
446 |
these effects, these results show Ice-{\it i} to be an important ice |
447 |
polymorph that should be considered in simulation studies. |
448 |
|
449 |
Due to this relative stability of Ice-{\it i} in all manner of |
450 |
investigated simulation examples, the question arises as to possible |
451 |
experimental observation of this polymorph. The rather extensive past |
452 |
and current experimental investigation of water in the low pressure |
453 |
regime leads the authors to be hesitant in ascribing relevance outside |
454 |
of computational models, hence the descriptive name presented. That |
455 |
being said, there are certain experimental conditions that would |
456 |
provide the most ideal situation for possible observation. These |
457 |
include the negative pressure or stretched solid regime, small |
458 |
clusters in vacuum deposition environments, and in clathrate |
459 |
structures involving small non-polar molecules. |
460 |
|
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\section{Acknowledgments} |
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Support for this project was provided by the National Science |
463 |
Foundation under grant CHE-0134881. Computation time was provided by |
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the Notre Dame High Performance Computing Cluster and the Notre Dame |
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Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
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|
467 |
\newpage |
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|
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\bibliographystyle{jcp} |
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\bibliography{iceiPaper} |
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|
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|
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\end{document} |