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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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molecules\cite{smallStuff} 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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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 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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conditions.\cite{lotsOfWaterPapers} Many of these models have been |
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used to investigate important physical phenomena like phase |
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transitions and the hydrophobic effect.\cite{evenMorePapers} With the |
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advent of numerous differing models, it is only natural that attention |
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is placed on the properties of the models themselves in an attempt to |
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clarify their benefits and limitations when applied to a system of |
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interest.\cite{modelProps} One important but challenging property to |
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quantify is the free energy, particularly of 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 that water |
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adopts over a wide range of pressures and temperatures. There are |
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currently 13 recognized forms of ice, and it is a challenging task to |
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investigate the entire free energy landscape.\cite{Sanz04} Ideally, |
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research is focused on the phases having the lowest free energy, |
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because these phases will dictate the true transition temperatures and |
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pressures for their respective model. |
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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 the |
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study of crystalline water polymorphs in the low pressure regime. This |
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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{nucleationStudies}; however, the crystal structure |
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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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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]{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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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{orderedIcei.eps} |
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\caption{Image of a proton ordered crystal of Ice-{\it i} looking |
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down the (001) crystal face. The rows of water tetramers surrounded by |
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octagonal pores leads to a crystal structure that is significantly |
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less dense than ice $I_h$.} |
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\label{protOrder} |
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\end{figure} |
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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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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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\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 ). By |
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applying this function, these interactions are smoothly truncated, |
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thereby avoiding poor energy conserving dynamics resulting from |
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harsher truncation schemes. The effect of a long-range correction was |
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also investigated on select model systems in a variety of manners. For |
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the SSD/RF model, a reaction field with a fixed dielectric constant of |
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80 was applied in all simulations.\cite{Onsager36} For a series of the |
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least computationally expensive models (SSD/E, SSD/RF, and TIP3P), |
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simulations were performed with longer cutoffs of 12 and 15 \AA\ to |
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compare with the 9 \AA\ cutoff results. Finally, results from the use |
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of an Ewald summation were estimated for TIP3P and SPC/E by performing |
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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. TINKER was chosen because it can also |
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propagate the motion of rigid-bodies, and provides the most direct |
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comparison to the results from OOPSE. The calculated energy difference |
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in the presence and absence of PME was applied to the previous results |
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in order to predict changes in the free energy landscape. |
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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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\section{Results and discussion} |
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|
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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} |
