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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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\title{Ice-{\it i}: a novel ice polymorph predicted via computer simulation} |
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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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\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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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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Notre Dame, Indiana 46556} |
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\date{\today} |
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%\maketitle |
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\maketitle |
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%\doublespacing |
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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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were calculated using thermodynamic integration. These integrations |
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were done for most of the common water models. Ice-{\it i}, a |
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structure we recently observed to be stable in one of the single-point |
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water models, was determined to be the stable crystalline state (at 1 |
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atm) for {\it all} the water models investigated. Phase diagrams were |
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generated, and phase coexistence lines were determined for all of the |
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known low-pressure ice structures under all of the common water |
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models. Additionally, potential truncation was shown to have an |
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effect on the calculated free energies, and can result in altered free |
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energy landscapes. |
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\end{abstract} |
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\maketitle |
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\newpage |
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%\narrowtext |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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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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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{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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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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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 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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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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Many of these models have been used to investigate important physical |
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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{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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effect.\cite{evenMorePapers} 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 the |
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study of crystalline water polymorphs in the low pressure regime. This |
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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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\includegraphics[width=\linewidth]{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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\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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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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\includegraphics[width=\linewidth]{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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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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\hline |
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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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\hline |
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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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representative of the dense ice polymorphs. A recent study by Sanz |
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{\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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\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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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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\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
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\caption{Phase diagram for the SSD/RF water model in the low pressure |
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\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
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temperatures of several common water models compared with experiment.} |
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\begin{tabular}{ l c c c c c c c } |
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\hline \\[-7mm] |
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\hline |
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\ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\ |
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\hline \\[-3mm] |
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\hline |
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\ \ $T_m$ (K) & \ \ 269 & \ \ 265 & \ \ 271 & 297 & \ \ - & \ \ 278 & \ \ 273\\ |
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\ \ $T_b$ (K) & \ \ 357 & \ \ 354 & \ \ 337 & 396 & \ \ - & \ \ 349 & \ \ 373\\ |
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\ \ $T_s$ (K) & \ \ - & \ \ - & \ \ - & - & \ \ 355 & \ \ - & \ \ -\\ |
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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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\hline |
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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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\hline |
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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} |
