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%\documentclass[prb,aps,twocolumn,tabularx]{revtex4} |
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\documentclass[preprint,aps,endfloats]{revtex4} |
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%\documentclass[11pt]{article} |
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\begin{document} |
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\title{A Free Energy Study of Low Temperature and Anomolous Ice} |
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\title{A Free Energy Study of Low Temperature and Anomalous Ice} |
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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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%\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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\end{abstract} |
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\maketitle |
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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{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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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{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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|
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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{nucleationStudies}; 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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|
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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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\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 propogated using the symplectic DLM |
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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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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 |
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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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|
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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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of an Ewald 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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propogate the motion of rigid-bodies, and provides the most direct |
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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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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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advantagious in that it facilitated the spontaneous crystallization of |
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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. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{cutoffChange.eps} |
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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 |
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\AA\. These crystals are unstable at 200 K and rapidly convert into a |
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liquid. The connecting lines are qualitative visual aid.} |
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\label{incCutoff} |
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\end{figure} |
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|
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Increasing the cutoff radius in simulations of the more |
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computationally efficient water models was done in order to evaluate |
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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 |
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free energy of all the ice polymorphs show a substantial dependence on |
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cutoff radius. In general, there is a narrowing of the free energy |
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differences while moving to greater cutoff radius. This trend is much |
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more subtle in the case of SSD/RF, indicating that the free energies |
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calculated with a reaction field present provide a more accurate |
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picture of the free energy landscape in the absence of potential |
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truncation. |
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differences while moving to greater cutoff radius. Interestingly, by |
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increasing the cutoff radius, the free energy gap was narrowed enough |
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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 |
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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 |
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reaction field present provide a more accurate picture of the free |
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energy landscape in the absence of potential truncation. |
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To further study the changes resulting to the inclusion of a |
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long-range interaction correction, the effect of an Ewald summation |
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SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
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are not fully supported in TINKER, so the results for these models |
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could not be estimated. The same trend pointed out through increase of |
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cutoff radius is observed in these results. Ice-{\it i} is the |
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cutoff radius is observed in these PME results. Ice-{\it i} is the |
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preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
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water models; however, there is a narrowing of the free energy |
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differences between the various solid forms. In the case of SPC/E this |
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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 the energy difference attributed to the inclusion of the PME long-range interaction correction. Units are kcal/mol.} |
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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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\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 |
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standard conditions for several common water models via thermodynamic |
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integration. All the water models studied show Ice-{\it i} to be the |
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minimum free energy crystal structure in the with a 9 \AA\ switching |
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function cutoff. Calculated melting and boiling points show |
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surprisingly good agreement with the experimental values; however, the |
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solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
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interaction truncation was investigated through variation of the |
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cutoff radius, use of a reaction field parameterized model, and |
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estimation of the results in the presence of the Ewald summation |
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correction. Interaction truncation has a significant effect on the |
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computed free energy values, and may significantly alter the free |
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energy landscape for the more complex multipoint water models. Despite |
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these effects, these results show Ice-{\it i} to be an important ice |
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polymorph that should be considered in simulation studies. |
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|
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Due to this relative stability of Ice-{\it i} in all manner of |
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investigated simulation examples, the question arises as to possible |
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experimental observation of this polymorph. The rather extensive past |
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and current experimental investigation of water in the low pressure |
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regime leads the authors to be hesitant in ascribing relevance outside |
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of computational models, hence the descriptive name presented. That |
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being said, there are certain experimental conditions that would |
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provide the most ideal situation for possible observation. These |
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include the negative pressure or stretched solid regime, small |
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clusters in vacuum deposition environments, and in clathrate |
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structures involving small non-polar molecules. |
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|
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\section{Acknowledgments} |
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Support for this project was provided by the National Science |
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Foundation under grant CHE-0134881. Computation time was provided by |
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the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
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DMR-0079647. |
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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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\newpage |
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