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\begin{document} |
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\title{Ice-{\it i}: a simulation-predicted ice polymorph which is more |
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stable than Ice $I_h$ for point-charge and point-dipole water models} |
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\title{Computational free energy studies of a new ice polymorph which |
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exhibits greater stability than Ice $I_h$} |
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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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Department of Chemistry and Biochemistry\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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%\doublespacing |
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\begin{abstract} |
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The absolute free energies of several ice polymorphs which are stable |
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at low pressures were calculated using thermodynamic integration to a |
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reference system (the Einstein crystal). These integrations were |
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performed for most of the common water models (SPC/E, TIP3P, TIP4P, |
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TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed |
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crystallizing at room temperature for one of the single-point water |
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models, was determined to be the stable crystalline state (at 1 atm) |
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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 these water models. |
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Additionally, potential truncation was shown to have an effect on the |
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calculated free energies, and can result in altered free energy |
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landscapes. Structure factor predictions for the new crystal were |
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generated and we await experimental confirmation of the existence of |
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this new polymorph. |
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The absolute free energies of several ice polymorphs were calculated |
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using thermodynamic integration. These polymorphs are predicted by |
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computer simulations using a variety of common water models to be |
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stable at low pressures. A recently discovered ice polymorph that has |
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as yet {\it only} been observed in computer simulations (Ice-{\it i}), |
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was determined to be the stable crystalline state for {\it all} the |
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water models investigated. Phase diagrams were generated, and phase |
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coexistence lines were determined for all of the known low-pressure |
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ice structures. Additionally, potential truncation was shown to play |
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a role in the resulting shape of the free energy landscape. |
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\end{abstract} |
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%\narrowtext |
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hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
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choice of models available, it is only natural to compare the models |
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under interesting thermodynamic conditions in an attempt to clarify |
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the limitations of each of the |
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models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
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important properties to quantify are the Gibbs and Helmholtz free |
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energies, particularly for the solid forms of water. Difficulty in |
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these types of studies typically arises from the assortment of |
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possible crystalline polymorphs that water adopts over a wide range of |
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pressures and temperatures. There are currently 13 recognized forms |
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of ice, and it is a challenging task to investigate the entire free |
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energy landscape.\cite{Sanz04} Ideally, research is focused on the |
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the limitations of |
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each.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two important |
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properties to quantify are the Gibbs and Helmholtz free energies, |
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particularly for the solid forms of water as these predict the |
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thermodynamic stability of the various phases. Water has a |
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particularly rich phase diagram and takes on a number of different and |
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stable crystalline structures as the temperature and pressure are |
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varied. It is a challenging task to investigate the entire free |
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energy landscape\cite{Sanz04}; and ideally, research is focused on the |
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phases having the lowest free energy at a given state point, because |
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these phases will dictate the relevant transition temperatures and |
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pressures for the model. |
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pressures for the model. |
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|
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In this paper, standard reference state methods were applied to known |
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crystalline water polymorphs in the low pressure regime. This work is |
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unique in that one of the crystal lattices was arrived at through |
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crystallization of a computationally efficient water model under |
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constant pressure and temperature conditions. Crystallization events |
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are interesting in and of themselves;\cite{Matsumoto02,Yamada02} |
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however, the crystal structure obtained in this case is different from |
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any previously observed ice polymorphs in experiment or |
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simulation.\cite{Fennell04} We have named this structure Ice-{\it i} |
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to indicate its origin in computational simulation. The unit cell |
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(Fig. \ref{iceiCell}A) consists of eight water molecules that stack in |
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rows of interlocking water tetramers. Proton ordering can be |
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accomplished by orienting two of the molecules so that both of their |
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donated hydrogen bonds are internal to their tetramer |
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(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
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water tetramers, the hydrogen bonds are not as linear as those |
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observed in ice $I_h$, however the interlocking of these subunits |
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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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The high-pressure phases of water (ice II - ice X as well as ice XII) |
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have been studied extensively both experimentally and |
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computationally. In this paper, standard reference state methods were |
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applied in the {\it low} pressure regime to evaluate the free energies |
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for a few known crystalline water polymorphs that might be stable at |
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these pressures. This work is unique in that one of the crystal |
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lattices was arrived at through crystallization of a computationally |
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efficient 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 is different from any previously observed ice |
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polymorphs in experiment or simulation.\cite{Fennell04} We have named |
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this structure Ice-{\it i} to indicate its origin in computational |
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simulation. The unit cell of Ice-{\it i} and an axially-elongated |
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variant named Ice-{\it i}$^\prime$ both consist of eight water |
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molecules that stack in rows of interlocking water tetramers as |
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illustrated in figures \ref{unitcell}A and \ref{unitcell}B. These |
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tetramers form a crystal structure similar in appearance to a recent |
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two-dimensional surface tessellation simulated on silica.\cite{Yang04} |
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As expected in an ice crystal constructed of water tetramers, the |
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hydrogen bonds are not as linear as those observed in ice $I_h$, |
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however the interlocking of these subunits appears to provide |
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significant stabilization to the overall crystal. The arrangement of |
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these tetramers results in octagonal cavities that are typically |
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greater than 6.3 \AA\ in diameter (Fig. \ref{iCrystal}). This open |
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structure leads to crystals that are typically 0.07 g/cm$^3$ less |
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dense than ice $I_h$. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{unitCell.eps} |
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\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-{\it i}$^\prime$, |
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the elongated variant of Ice-{\it i}. The spheres represent the |
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center-of-mass locations of the water molecules. The $a$ to $c$ |
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ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
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$a:2.1214c$ and $a:1.7850c$ respectively.} |
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\label{iceiCell} |
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\caption{(A) Unit cells for Ice-{\it i} and (B) Ice-{\it i}$^\prime$. |
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The spheres represent the center-of-mass locations of the water |
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molecules. The $a$ to $c$ ratios for Ice-{\it i} and Ice-{\it |
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i}$^\prime$ are given by 2.1214 and 1.785 respectively.} |
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\label{unitcell} |
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\end{figure} |
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\begin{figure} |
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\centering |
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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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\caption{A rendering of a proton ordered crystal of Ice-{\it i} looking |
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down the (001) crystal face. The presence of large octagonal pores |
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leads to a polymorph that is less dense than ice $I_h$.} |
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\label{iCrystal} |
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\end{figure} |
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Results from our previous study indicated that Ice-{\it i} is the |
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minimum energy crystal structure for the single point water models we |
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had investigated (for discussions on these single point dipole models, |
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see our previous work and related |
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articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
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considered energetic stabilization and neglected entropic |
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contributions to the overall free energy. To address this issue, we |
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minimum energy crystal structure for the single point water models |
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investigated (for discussions on these single point dipole models, see |
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our previous work and related |
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articles).\cite{Fennell04,Liu96,Bratko85} Our earlier results |
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considered only energetic stabilization and neglected entropic |
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contributions to the overall free energy. To address this issue, we |
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have calculated the absolute free energy of this crystal using |
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thermodynamic integration and compared to the free energies of cubic |
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and hexagonal ice $I$ (the experimental low density ice polymorphs) |
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and ice B (a higher density, but very stable crystal structure |
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observed by B\`{a}ez and Clancy in free energy studies of |
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SPC/E).\cite{Baez95b} This work includes results for the water model |
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from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
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common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
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field parametrized single point dipole water model (SSD/RF). It should |
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be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$) |
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was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit |
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cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it |
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i} unit it is extended in the direction of the (001) face and |
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compressed along the other two faces. There is typically a small |
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distortion of proton ordered Ice-{\it i}$^\prime$ that converts the |
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normally square tetramer into a rhombus with alternating approximately |
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85 and 95 degree angles. The degree of this distortion is model |
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dependent and significant enough to split the tetramer diagonal |
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location peak in the radial distribution function. |
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thermodynamic integration and compared it to the free energies of ice |
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$I_c$ and ice $I_h$ (the common low density ice polymorphs) and ice B |
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(a higher density, but very stable crystal structure observed by |
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B\`{a}ez and Clancy in free energy studies of SPC/E).\cite{Baez95b} |
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This work includes results for the water model from which Ice-{\it i} |
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was crystallized (SSD/E) in addition to several common water models |
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(TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field parametrized |
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single point dipole water model (SSD/RF). The axially-elongated |
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variant, Ice-{\it i}$^\prime$, was used in calculations involving |
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SPC/E, TIP4P, and TIP5P. The square tetramers in Ice-{\it i} distort |
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in Ice-{\it i}$^\prime$ to form a rhombus with alternating 85 and 95 |
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degree angles. Under SPC/E, TIP4P, and TIP5P, this geometry is better |
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at forming favorable hydrogen bonds. The degree of rhomboid |
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distortion depends on the water model used, but is significant enough |
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to split a peak in the radial distribution function which corresponds |
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to diagonal sites in the tetramers. |
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\section{Methods} |
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Canonical ensemble (NVT) molecular dynamics calculations were |
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performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
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performed using the OOPSE molecular mechanics program.\cite{Meineke05} |
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All molecules were treated as rigid bodies, with orientational motion |
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propagated using the symplectic DLM integration method. Details about |
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propagated using the symplectic DLM integration method. Details about |
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the implementation of this technique can be found in a recent |
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publication.\cite{Dullweber1997} |
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|
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Thermodynamic integration is an established technique for |
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determination of free energies of condensed phases of |
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materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
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method, implemented in the same manner illustrated by B\`{a}ez and |
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Clancy, was utilized to calculate the free energy of several ice |
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crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
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SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
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and 400 K for all of these water models were also determined using |
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this same technique in order to determine melting points and to |
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generate phase diagrams. All simulations were carried out at densities |
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which correspond to a pressure of approximately 1 atm at their |
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respective temperatures. |
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|
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Thermodynamic integration involves a sequence of simulations during |
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which the system of interest is converted into a reference system for |
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which the free energy is known analytically. This transformation path |
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is then integrated in order to determine the free energy difference |
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between the two states: |
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Thermodynamic integration was utilized to calculate the Helmholtz free |
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energies ($A$) of the listed water models at various state points |
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using the OOPSE molecular dynamics program.\cite{Meineke05} |
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Thermodynamic integration is an established technique that has been |
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used extensively in the calculation of free energies for condensed |
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phases of |
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materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
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method uses a sequence of simulations during which the system of |
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interest is converted into a reference system for which the free |
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energy is known analytically ($A_0$). The difference in potential |
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energy between the reference system and the system of interest |
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($\Delta V$) is then integrated in order to determine the free energy |
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difference between the two states: |
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\begin{equation} |
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\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
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)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
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A = A_0 + \int_0^1 \left\langle \Delta V \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 strategically along this path |
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in 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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Here, $\lambda$ is the parameter that governs the transformation |
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between the reference system and the system of interest. For |
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crystalline phases, an harmonically-restrained (Einsten) crystal is |
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chosen as the reference state, while for liquid phases, the ideal gas |
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is taken as the reference state. |
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|
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For the thermodynamic integration of molecular crystals, the Einstein |
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crystal was chosen as the reference system. In an Einstein crystal, |
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the molecules are restrained at their ideal lattice locations and |
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orientations. Using harmonic restraints, as applied by B\`{a}ez and |
| 195 |
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Clancy, the total potential for this reference crystal |
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($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
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In an Einstein crystal, the molecules are restrained at their ideal |
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lattice locations and orientations. Using harmonic restraints, as |
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applied by B\`{a}ez and Clancy, the total potential for this reference |
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crystal ($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
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\begin{equation} |
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V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
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\frac{K_\omega\omega^2}{2}, |
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where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
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the spring constants restraining translational motion and deflection |
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|
of and rotation around the principle axis of the molecule |
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respectively. It is clear from Fig. \ref{waterSpring} that the values |
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of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
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$-\pi$ to $\pi$. The partition function for a molecular crystal |
| 192 |
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respectively. These spring constants are typically calculated from |
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the mean-square displacements of water molecules in an unrestrained |
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ice crystal at 200 K. For these studies, $K_\mathrm{r} = 4.29$ kcal |
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mol$^{-1}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$, and $K_\omega\ = |
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17.75$ kcal mol$^{-1}$. It is clear from Fig. \ref{waterSpring} that |
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the values of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges |
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from $-\pi$ to $\pi$. The partition function for a molecular crystal |
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restrained in this fashion can be evaluated analytically, and the |
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Helmholtz Free Energy ({\it A}) is given by |
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\begin{eqnarray} |
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potential energy of the ideal crystal.\cite{Baez95a} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{rotSpring.eps} |
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\centering |
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\includegraphics[width=4in]{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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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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literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
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typically differ in regard to the path taken for switching off the |
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|
interaction potential to convert the system to an ideal gas of water |
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molecules. In this study, we applied of one of the most convenient |
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molecules. In this study, we applied one of the most convenient |
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methods and integrated over the $\lambda^4$ path, where all |
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interaction parameters are scaled equally by this transformation |
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parameter. This method has been shown to be reversible and provide |
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results in excellent agreement with other established |
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methods.\cite{Baez95b} |
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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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Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and |
| 241 |
> |
Lennard-Jones interactions were gradually reduced by a cubic switching |
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function. By applying this function, these interactions are smoothly |
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truncated, thereby avoiding the poor energy conservation which results |
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from harsher truncation schemes. The effect of a long-range correction |
| 245 |
< |
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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from harsher truncation schemes. The effect of a long-range |
| 245 |
> |
correction was also investigated on select model systems in a variety |
| 246 |
> |
of manners. For the SSD/RF model, a reaction field with a fixed |
| 247 |
|
dielectric constant of 80 was applied in all |
| 248 |
|
simulations.\cite{Onsager36} For a series of the least computationally |
| 249 |
< |
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
| 250 |
< |
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
| 251 |
< |
\AA\ cutoff results. Finally, the effects of utilizing an Ewald |
| 252 |
< |
summation were estimated for TIP3P and SPC/E by performing single |
| 253 |
< |
configuration calculations with Particle-Mesh Ewald (PME) in the |
| 254 |
< |
TINKER molecular mechanics software package.\cite{Tinker} The |
| 249 |
> |
expensive models (SSD/E, SSD/RF, TIP3P, and SPC/E), simulations were |
| 250 |
> |
performed with longer cutoffs of 10.5, 12, 13.5, and 15 \AA\ to |
| 251 |
> |
compare with the 9 \AA\ cutoff results. Finally, the effects of using |
| 252 |
> |
the Ewald summation were estimated for TIP3P and SPC/E by performing |
| 253 |
> |
single configuration Particle-Mesh Ewald (PME) |
| 254 |
> |
calculations~\cite{Tinker} for each of the ice polymorphs. The |
| 255 |
|
calculated energy difference in the presence and absence of PME was |
| 256 |
|
applied to the previous results in order to predict changes to the |
| 257 |
|
free energy landscape. |
| 258 |
|
|
| 259 |
< |
\section{Results and discussion} |
| 259 |
> |
\section{Results and Discussion} |
| 260 |
|
|
| 261 |
< |
The free energy of proton-ordered Ice-{\it i} was calculated and |
| 262 |
< |
compared with the free energies of proton ordered variants of the |
| 263 |
< |
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
| 264 |
< |
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
| 265 |
< |
and thought to be the minimum free energy structure for the SPC/E |
| 266 |
< |
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
| 267 |
< |
Ice XI, the experimentally-observed proton-ordered variant of ice |
| 268 |
< |
$I_h$, was investigated initially, but was found to be not as stable |
| 269 |
< |
as proton disordered or antiferroelectric variants of ice $I_h$. The |
| 270 |
< |
proton ordered variant of ice $I_h$ used here is a simple |
| 271 |
< |
antiferroelectric version that we devised, and it has an 8 molecule |
| 272 |
< |
unit cell similar to other predicted antiferroelectric $I_h$ |
| 273 |
< |
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
| 274 |
< |
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
| 275 |
< |
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger |
| 276 |
< |
crystal sizes were necessary for simulations involving larger cutoff |
| 277 |
< |
values. |
| 261 |
> |
The calculated free energies of proton-ordered variants of three low |
| 262 |
> |
density polymorphs ($I_h$, $I_c$, and Ice-{\it i} or Ice-{\it |
| 263 |
> |
i}$^\prime$) and the stable higher density ice B are listed in Table |
| 264 |
> |
\ref{freeEnergy}. Ice B was included because it has been |
| 265 |
> |
shown to be a minimum free energy structure for SPC/E at ambient |
| 266 |
> |
conditions.\cite{Baez95b} In addition to the free energies, the |
| 267 |
> |
relevant transition temperatures at standard pressure are also |
| 268 |
> |
displayed in Table \ref{freeEnergy}. These free energy values |
| 269 |
> |
indicate that Ice-{\it i} is the most stable state for all of the |
| 270 |
> |
investigated water models. With the free energy at these state |
| 271 |
> |
points, the Gibbs-Helmholtz equation was used to project to other |
| 272 |
> |
state points and to build phase diagrams. Figure \ref{tp3PhaseDia} is |
| 273 |
> |
an example diagram built from the results for the TIP3P water model. |
| 274 |
> |
All other models have similar structure, although the crossing points |
| 275 |
> |
between the phases move to different temperatures and pressures as |
| 276 |
> |
indicated from the transition temperatures in Table \ref{freeEnergy}. |
| 277 |
> |
It is interesting to note that ice $I_h$ (and ice $I_c$ for that |
| 278 |
> |
matter) do not appear in any of the phase diagrams for any of the |
| 279 |
> |
models. For purposes of this study, ice B is representative of the |
| 280 |
> |
dense ice polymorphs. A recent study by Sanz {\it et al.} provides |
| 281 |
> |
details on the phase diagrams for SPC/E and TIP4P at higher pressures |
| 282 |
> |
than those studied here.\cite{Sanz04} |
| 283 |
|
|
| 284 |
|
\begin{table*} |
| 285 |
|
\begin{minipage}{\linewidth} |
| 288 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 286 |
|
\begin{center} |
| 287 |
< |
\caption{Calculated free energies for several ice polymorphs with a |
| 288 |
< |
variety of common water models. All calculations used a cutoff radius |
| 289 |
< |
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
| 290 |
< |
kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
| 291 |
< |
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
| 292 |
< |
\begin{tabular}{ l c c c c } |
| 293 |
< |
\hline |
| 297 |
< |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
| 287 |
> |
\caption{Calculated free energies for several ice polymorphs along |
| 288 |
> |
with the calculated melting (or sublimation) and boiling points for |
| 289 |
> |
the investigated water models. All free energy calculations used a |
| 290 |
> |
cutoff radius of 9.0 \AA\ and were performed at 200 K and $\sim$1 atm. |
| 291 |
> |
Units of free energy are kcal/mol, while transition temperature are in |
| 292 |
> |
Kelvin. Calculated error of the final digits is in parentheses.} |
| 293 |
> |
\begin{tabular}{lccccccc} |
| 294 |
|
\hline |
| 295 |
< |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
| 296 |
< |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
| 297 |
< |
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
| 298 |
< |
SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\ |
| 299 |
< |
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
| 300 |
< |
SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\ |
| 295 |
> |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} & Ice-{\it i}$^\prime$ & $T_m$ (*$T_s$) & $T_b$\\ |
| 296 |
> |
\hline |
| 297 |
> |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(4) & 357(2)\\ |
| 298 |
> |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 266(5) & 354(2)\\ |
| 299 |
> |
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 271(4) & 337(2)\\ |
| 300 |
> |
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 296(3) & 396(2)\\ |
| 301 |
> |
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(2) & -\\ |
| 302 |
> |
SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2) & - & 278(4) & 349(2)\\ |
| 303 |
|
\end{tabular} |
| 304 |
|
\label{freeEnergy} |
| 305 |
|
\end{center} |
| 306 |
|
\end{minipage} |
| 307 |
|
\end{table*} |
| 308 |
|
|
| 311 |
– |
The free energy values computed for the studied polymorphs indicate |
| 312 |
– |
that Ice-{\it i} is the most stable state for all of the common water |
| 313 |
– |
models studied. With the calculated free energy at these state points, |
| 314 |
– |
the Gibbs-Helmholtz equation was used to project to other state points |
| 315 |
– |
and to build phase diagrams. Figures |
| 316 |
– |
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
| 317 |
– |
from the free energy results. All other models have similar structure, |
| 318 |
– |
although the crossing points between the phases move to slightly |
| 319 |
– |
different temperatures and pressures. It is interesting to note that |
| 320 |
– |
ice $I$ does not exist in either cubic or hexagonal form in any of the |
| 321 |
– |
phase diagrams for any of the models. For purposes of this study, ice |
| 322 |
– |
B is representative of the dense ice polymorphs. A recent study by |
| 323 |
– |
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
| 324 |
– |
TIP4P at higher pressures than those studied here.\cite{Sanz04} |
| 325 |
– |
|
| 309 |
|
\begin{figure} |
| 310 |
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
| 311 |
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
| 312 |
< |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 312 |
> |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 313 |
|
the experimental values; however, the solid phases shown are not the |
| 314 |
< |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
| 314 |
> |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
| 315 |
|
higher in energy and don't appear in the phase diagram.} |
| 316 |
< |
\label{tp3phasedia} |
| 316 |
> |
\label{tp3PhaseDia} |
| 317 |
|
\end{figure} |
| 318 |
|
|
| 319 |
< |
\begin{figure} |
| 320 |
< |
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
| 321 |
< |
\caption{Phase diagram for the SSD/RF water model in the low pressure |
| 322 |
< |
regime. Calculations producing these results were done under an |
| 323 |
< |
applied reaction field. It is interesting to note that this |
| 324 |
< |
computationally efficient model (over 3 times more efficient than |
| 325 |
< |
TIP3P) exhibits phase behavior similar to the less computationally |
| 326 |
< |
conservative charge based models.} |
| 344 |
< |
\label{ssdrfphasedia} |
| 345 |
< |
\end{figure} |
| 346 |
< |
|
| 347 |
< |
\begin{table*} |
| 348 |
< |
\begin{minipage}{\linewidth} |
| 349 |
< |
\renewcommand{\thefootnote}{\thempfootnote} |
| 350 |
< |
\begin{center} |
| 351 |
< |
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
| 352 |
< |
temperatures at 1 atm for several common water models compared with |
| 353 |
< |
experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
| 354 |
< |
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
| 355 |
< |
liquid or gas state.} |
| 356 |
< |
\begin{tabular}{ l c c c c c c c } |
| 357 |
< |
\hline |
| 358 |
< |
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
| 359 |
< |
\hline |
| 360 |
< |
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
| 361 |
< |
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
| 362 |
< |
$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ |
| 363 |
< |
\end{tabular} |
| 364 |
< |
\label{meltandboil} |
| 365 |
< |
\end{center} |
| 366 |
< |
\end{minipage} |
| 367 |
< |
\end{table*} |
| 368 |
< |
|
| 369 |
< |
Table \ref{meltandboil} lists the melting and boiling temperatures |
| 370 |
< |
calculated from this work. Surprisingly, most of these models have |
| 371 |
< |
melting points that compare quite favorably with experiment. The |
| 372 |
< |
unfortunate aspect of this result is that this phase change occurs |
| 373 |
< |
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
| 374 |
< |
liquid state. These results are actually not contrary to previous |
| 375 |
< |
studies in the literature. Earlier free energy studies of ice $I$ |
| 376 |
< |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
| 377 |
< |
being attributed to choice of interaction truncation and different |
| 378 |
< |
ordered and disordered molecular |
| 319 |
> |
Most of the water models have melting points that compare quite |
| 320 |
> |
favorably with the experimental value of 273 K. The unfortunate |
| 321 |
> |
aspect of this result is that this phase change occurs between |
| 322 |
> |
Ice-{\it i} and the liquid state rather than ice $I_h$ and the liquid |
| 323 |
> |
state. These results do not contradict other studies. Studies of ice |
| 324 |
> |
$I_h$ using TIP4P predict a $T_m$ ranging from 214 to 238 K |
| 325 |
> |
(differences being attributed to choice of interaction truncation and |
| 326 |
> |
different ordered and disordered molecular |
| 327 |
|
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
| 328 |
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
| 329 |
< |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 330 |
< |
calculated at 265 K, significantly higher in temperature than the |
| 331 |
< |
previous studies. Also of interest in these results is that SSD/E does |
| 332 |
< |
not exhibit a melting point at 1 atm, but it shows a sublimation point |
| 333 |
< |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
| 334 |
< |
all other polymorphs for this particular model under these |
| 335 |
< |
conditions. While troubling, this behavior resulted in spontaneous |
| 336 |
< |
crystallization of Ice-{\it i} and led us to investigate this |
| 337 |
< |
structure. These observations provide a warning that simulations of |
| 338 |
< |
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
| 339 |
< |
risk of spontaneous crystallization. However, this risk lessens when |
| 340 |
< |
applying a longer cutoff. |
| 329 |
> |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 330 |
> |
calculated to be 265 K, indicating that these simulation based |
| 331 |
> |
structures ought to be included in studies probing phase transitions |
| 332 |
> |
with this model. Also of interest in these results is that SSD/E does |
| 333 |
> |
not exhibit a melting point at 1 atm but does sublime at 355 K. This |
| 334 |
> |
is due to the significant stability of Ice-{\it i} over all other |
| 335 |
> |
polymorphs for this particular model under these conditions. While |
| 336 |
> |
troubling, this behavior resulted in the spontaneous crystallization |
| 337 |
> |
of Ice-{\it i} which led us to investigate this structure. These |
| 338 |
> |
observations provide a warning that simulations of SSD/E as a |
| 339 |
> |
``liquid'' near 300 K are actually metastable and run the risk of |
| 340 |
> |
spontaneous crystallization. However, when a longer cutoff radius is |
| 341 |
> |
used, SSD/E prefers the liquid state under standard temperature and |
| 342 |
> |
pressure. |
| 343 |
|
|
| 344 |
|
\begin{figure} |
| 345 |
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
| 346 |
< |
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
| 347 |
< |
TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
| 348 |
< |
\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
| 349 |
< |
\AA . These crystals are unstable at 200 K and rapidly convert into |
| 350 |
< |
liquids. The connecting lines are qualitative visual aid.} |
| 346 |
> |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
| 347 |
> |
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
| 348 |
> |
with an added Ewald correction term. Error for the larger cutoff |
| 349 |
> |
points is equivalent to that observed at 9.0\AA\ (see Table |
| 350 |
> |
\ref{freeEnergy}). Data for ice I$_c$ with TIP3P using both 12 and |
| 351 |
> |
13.5 \AA\ cutoffs were omitted because the crystal was prone to |
| 352 |
> |
distortion and melting at 200 K. Ice-{\it i}$^\prime$ is the form of |
| 353 |
> |
Ice-{\it i} used in the SPC/E simulations.} |
| 354 |
|
\label{incCutoff} |
| 355 |
|
\end{figure} |
| 356 |
|
|
| 357 |
< |
Increasing the cutoff radius in simulations of the more |
| 358 |
< |
computationally efficient water models was done in order to evaluate |
| 359 |
< |
the trend in free energy values when moving to systems that do not |
| 360 |
< |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
| 361 |
< |
free energy of all the ice polymorphs show a substantial dependence on |
| 362 |
< |
cutoff radius. In general, there is a narrowing of the free energy |
| 363 |
< |
differences while moving to greater cutoff radius. Interestingly, by |
| 364 |
< |
increasing the cutoff radius, the free energy gap was narrowed enough |
| 365 |
< |
in the SSD/E model that the liquid state is preferred under standard |
| 366 |
< |
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
| 367 |
< |
simulations using this model choose interaction truncation radii |
| 368 |
< |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
| 369 |
< |
case of SSD/RF, indicating that the free energies calculated with a |
| 370 |
< |
reaction field present provide a more accurate picture of the free |
| 371 |
< |
energy landscape in the absence of potential truncation. |
| 357 |
> |
For the more computationally efficient water models, we have also |
| 358 |
> |
investigated the effect of potential trunctaion on the computed free |
| 359 |
> |
energies as a function of the cutoff radius. As seen in |
| 360 |
> |
Fig. \ref{incCutoff}, the free energies of the ice polymorphs with |
| 361 |
> |
water models lacking a long-range correction show significant cutoff |
| 362 |
> |
dependence. In general, there is a narrowing of the free energy |
| 363 |
> |
differences while moving to greater cutoff radii. As the free |
| 364 |
> |
energies for the polymorphs converge, the stability advantage that |
| 365 |
> |
Ice-{\it i} exhibits is reduced. Adjacent to each of these plots are |
| 366 |
> |
results for systems with applied or estimated long-range corrections. |
| 367 |
> |
SSD/RF was parametrized for use with a reaction field, and the benefit |
| 368 |
> |
provided by this computationally inexpensive correction is apparent. |
| 369 |
> |
The free energies are largely independent of the size of the reaction |
| 370 |
> |
field cavity in this model, so small cutoff radii mimic bulk |
| 371 |
> |
calculations quite well under SSD/RF. |
| 372 |
> |
|
| 373 |
> |
Although TIP3P was paramaterized for use without the Ewald summation, |
| 374 |
> |
we have estimated the effect of this method for computing long-range |
| 375 |
> |
electrostatics for both TIP3P and SPC/E. This was accomplished by |
| 376 |
> |
calculating the potential energy of identical crystals both with and |
| 377 |
> |
without particle mesh Ewald (PME). Similar behavior to that observed |
| 378 |
> |
with reaction field is seen for both of these models. The free |
| 379 |
> |
energies show reduced dependence on cutoff radius and span a narrower |
| 380 |
> |
range for the various polymorphs. Like the dipolar water models, |
| 381 |
> |
TIP3P displays a relatively constant preference for the Ice-{\it i} |
| 382 |
> |
polymorph. Crystal preference is much more difficult to determine for |
| 383 |
> |
SPC/E. Without a long-range correction, each of the polymorphs |
| 384 |
> |
studied assumes the role of the preferred polymorph under different |
| 385 |
> |
cutoff radii. The inclusion of the Ewald correction flattens and |
| 386 |
> |
narrows the gap in free energies such that the polymorphs are |
| 387 |
> |
isoenergetic within statistical uncertainty. This suggests that other |
| 388 |
> |
conditions, such as the density in fixed-volume simulations, can |
| 389 |
> |
influence the polymorph expressed upon crystallization. |
| 390 |
|
|
| 391 |
< |
To further study the changes resulting to the inclusion of a |
| 421 |
< |
long-range interaction correction, the effect of an Ewald summation |
| 422 |
< |
was estimated by applying the potential energy difference do to its |
| 423 |
< |
inclusion in systems in the presence and absence of the |
| 424 |
< |
correction. This was accomplished by calculation of the potential |
| 425 |
< |
energy of identical crystals both with and without PME. The free |
| 426 |
< |
energies for the investigated polymorphs using the TIP3P and SPC/E |
| 427 |
< |
water models are shown in Table \ref{pmeShift}. The same trend pointed |
| 428 |
< |
out through increase of cutoff radius is observed in these PME |
| 429 |
< |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
| 430 |
< |
for both the TIP3P and SPC/E water models; however, the narrowing of |
| 431 |
< |
the free energy differences between the various solid forms is |
| 432 |
< |
significant enough that it becomes less clear that it is the most |
| 433 |
< |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
| 434 |
< |
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
| 435 |
< |
as well, indicating that Ice-{\it i} might be metastable with respect |
| 436 |
< |
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
| 437 |
< |
not significantly alter the finding that the Ice-{\it i} polymorph is |
| 438 |
< |
a stable crystal structure that should be considered when studying the |
| 439 |
< |
phase behavior of water models. |
| 391 |
> |
\section{Conclusions} |
| 392 |
|
|
| 393 |
< |
\begin{table*} |
| 394 |
< |
\begin{minipage}{\linewidth} |
| 395 |
< |
\renewcommand{\thefootnote}{\thempfootnote} |
| 396 |
< |
\begin{center} |
| 397 |
< |
\caption{The free energy of the studied ice polymorphs after applying |
| 398 |
< |
the energy difference attributed to the inclusion of the PME |
| 399 |
< |
long-range interaction correction. Units are kcal/mol.} |
| 400 |
< |
\begin{tabular}{ l c c c c } |
| 449 |
< |
\hline |
| 450 |
< |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
| 451 |
< |
\hline |
| 452 |
< |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\ |
| 453 |
< |
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\ |
| 454 |
< |
\end{tabular} |
| 455 |
< |
\label{pmeShift} |
| 456 |
< |
\end{center} |
| 457 |
< |
\end{minipage} |
| 458 |
< |
\end{table*} |
| 393 |
> |
In this report, thermodynamic integration was used to determine the |
| 394 |
> |
absolute free energies of several ice polymorphs. Of the studied |
| 395 |
> |
crystal forms, Ice-{\it i} was observed to be the stable crystalline |
| 396 |
> |
state for {\it all} the water models when using a 9.0 \AA\ |
| 397 |
> |
intermolecular interaction cutoff. Through investigation of possible |
| 398 |
> |
interaction truncation methods, the free energy was shown to be |
| 399 |
> |
partially dependent on simulation conditions; however, Ice-{\it i} was |
| 400 |
> |
still observered to be a stable polymorph of the studied water models. |
| 401 |
|
|
| 402 |
< |
\section{Conclusions} |
| 402 |
> |
So what is the preferred solid polymorph for simulated water? As |
| 403 |
> |
indicated above, the answer appears to be dependent both on the |
| 404 |
> |
conditions and the model used. In the case of short cutoffs without a |
| 405 |
> |
long-range interaction correction, Ice-{\it i} and Ice-{\it |
| 406 |
> |
i}$^\prime$ have the lowest free energy of the studied polymorphs with |
| 407 |
> |
all the models. Ideally, crystallization of each model under constant |
| 408 |
> |
pressure conditions, as was done with SSD/E, would aid in the |
| 409 |
> |
identification of their respective preferred structures. This work, |
| 410 |
> |
however, helps illustrate how studies involving one specific model can |
| 411 |
> |
lead to insight about important behavior of others. In general, the |
| 412 |
> |
above results support the finding that the Ice-{\it i} polymorph is a |
| 413 |
> |
stable crystal structure that should be considered when studying the |
| 414 |
> |
phase behavior of water models. |
| 415 |
|
|
| 416 |
< |
The free energy for proton ordered variants of hexagonal and cubic ice |
| 417 |
< |
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
| 418 |
< |
calculated under standard conditions for several common water models |
| 419 |
< |
via thermodynamic integration. All the water models studied show |
| 420 |
< |
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
| 421 |
< |
\AA\ switching function cutoff. Calculated melting and boiling points |
| 468 |
< |
show surprisingly good agreement with the experimental values; |
| 469 |
< |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
| 470 |
< |
effect of interaction truncation was investigated through variation of |
| 471 |
< |
the cutoff radius, use of a reaction field parameterized model, and |
| 472 |
< |
estimation of the results in the presence of the Ewald |
| 473 |
< |
summation. Interaction truncation has a significant effect on the |
| 474 |
< |
computed free energy values, and may significantly alter the free |
| 475 |
< |
energy landscape for the more complex multipoint water models. Despite |
| 476 |
< |
these effects, these results show Ice-{\it i} to be an important ice |
| 477 |
< |
polymorph that should be considered in simulation studies. |
| 416 |
> |
We also note that none of the water models used in this study are |
| 417 |
> |
polarizable or flexible models. It is entirely possible that the |
| 418 |
> |
polarizability of real water makes Ice-{\it i} substantially less |
| 419 |
> |
stable than ice $I_h$. However, the calculations presented above seem |
| 420 |
> |
interesting enough to communicate before the role of polarizability |
| 421 |
> |
(or flexibility) has been thoroughly investigated. |
| 422 |
|
|
| 423 |
< |
Due to this relative stability of Ice-{\it i} in all of the |
| 424 |
< |
investigated simulation conditions, the question arises as to possible |
| 425 |
< |
experimental observation of this polymorph. The rather extensive past |
| 426 |
< |
and current experimental investigation of water in the low pressure |
| 427 |
< |
regime makes us hesitant to ascribe any relevance of this work outside |
| 428 |
< |
of the simulation community. It is for this reason that we chose a |
| 429 |
< |
name for this polymorph which involves an imaginary quantity. That |
| 430 |
< |
said, there are certain experimental conditions that would provide the |
| 431 |
< |
most ideal situation for possible observation. These include the |
| 432 |
< |
negative pressure or stretched solid regime, small clusters in vacuum |
| 423 |
> |
Finally, due to the stability of Ice-{\it i} in the investigated |
| 424 |
> |
simulation conditions, the question arises as to possible experimental |
| 425 |
> |
observation of this polymorph. The rather extensive past and current |
| 426 |
> |
experimental investigation of water in the low pressure regime makes |
| 427 |
> |
us hesitant to ascribe any relevance to this work outside of the |
| 428 |
> |
simulation community. It is for this reason that we chose a name for |
| 429 |
> |
this polymorph which involves an imaginary quantity. That said, there |
| 430 |
> |
are certain experimental conditions that would provide the most ideal |
| 431 |
> |
situation for possible observation. These include the negative |
| 432 |
> |
pressure or stretched solid regime, small clusters in vacuum |
| 433 |
|
deposition environments, and in clathrate structures involving small |
| 434 |
< |
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
| 435 |
< |
our predictions for both the pair distribution function ($g_{OO}(r)$) |
| 436 |
< |
and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for |
| 437 |
< |
ice-{\it i} at a temperature of 77K. In studies of the high and low |
| 438 |
< |
density forms of amorphous ice, ``spurious'' diffraction peaks have |
| 495 |
< |
been observed experimentally.\cite{Bizid87} It is possible that a |
| 496 |
< |
variant of Ice-{\it i} could explain some of this behavior; however, |
| 497 |
< |
we will leave it to our experimental colleagues to make the final |
| 498 |
< |
determination on whether this ice polymorph is named appropriately |
| 499 |
< |
(i.e. with an imaginary number) or if it can be promoted to Ice-0. |
| 434 |
> |
non-polar molecules. For experimental comparison purposes, example |
| 435 |
> |
$g_{OO}(r)$ and $S(\vec{q})$ plots were generated for the two Ice-{\it |
| 436 |
> |
i} variants (along with example ice $I_h$ and $I_c$ plots) at 77K, and |
| 437 |
> |
they are shown in figures \ref{fig:gofr} and \ref{fig:sofq} |
| 438 |
> |
respectively. |
| 439 |
|
|
| 440 |
|
\begin{figure} |
| 441 |
+ |
\centering |
| 442 |
|
\includegraphics[width=\linewidth]{iceGofr.eps} |
| 443 |
< |
\caption{Radial distribution functions of ice $I_h$, $I_c$, |
| 444 |
< |
Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations |
| 445 |
< |
of the SSD/RF water model at 77 K.} |
| 443 |
> |
\caption{Radial distribution functions of ice $I_h$, $I_c$, and |
| 444 |
> |
Ice-{\it i} calculated from from simulations of the SSD/RF water model |
| 445 |
> |
at 77 K. The Ice-{\it i} distribution function was obtained from |
| 446 |
> |
simulations composed of TIP4P water.} |
| 447 |
|
\label{fig:gofr} |
| 448 |
|
\end{figure} |
| 449 |
|
|
| 450 |
|
\begin{figure} |
| 451 |
+ |
\centering |
| 452 |
|
\includegraphics[width=\linewidth]{sofq.eps} |
| 453 |
|
\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, |
| 454 |
|
and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
