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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{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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\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 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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\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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|
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Water has proven to be a challenging substance to depict in |
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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{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04} |
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These models have been used to investigate important physical |
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phenomena like phase transitions, transport properties, and the |
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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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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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|
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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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|
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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-{\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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\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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|
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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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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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|
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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 (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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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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performed using the OOPSE molecular mechanics package.\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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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 was utilized to calculate the free energy of |
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several ice crystals using the TIP3P, TIP4P, TIP5P, SPC/E, and SSD/E |
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water models. Liquid state free energies at 300 and 400 K for all of |
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these water models were also determined using this same technique, in |
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order to determine melting points and generate phase diagrams. |
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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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\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 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$25 |
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simulations ranging from 300 ps (for the unaltered system) to 75 ps |
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(near the reference state) in length. |
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|
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For the thermodynamic integration of molecular crystals, the Einstein |
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Crystal is chosen as the reference state that the system is converted |
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to over the course of the simulation. In an Einstein Crystal, the |
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molecules are harmonically restrained at their ideal lattice locations |
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and orientations. The partition function for a molecular crystal |
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restrained in this fashion has been evaluated, and the Helmholtz Free |
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Energy ({\it A}) is given by |
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crystal was chosen as the reference system. In an Einstein crystal, |
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the molecules are restrained at their ideal lattice locations and |
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orientations. Using harmonic restraints, as applied by B\`{a}ez and |
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Clancy, the total potential for this reference crystal |
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($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
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\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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\end{equation} |
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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 |
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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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A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
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[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
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)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
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\label{ecFreeEnergy} |
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\end{eqnarray} |
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where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
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\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
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$K_\mathrm{\omega}$ are the spring constants restraining translational |
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motion and deflection of and rotation around the principle axis of the |
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molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
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minimum potential energy of the ideal crystal. In the case of |
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molecular liquids, the ideal vapor is chosen as the target reference |
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state. |
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where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
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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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\caption{Possible orientational motions for a restrained molecule. |
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$\theta$ angles correspond to displacement from the body-frame {\it |
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z}-axis, while $\omega$ angles correspond to rotation about the |
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body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
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constants for the harmonic springs restraining motion in the $\theta$ |
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and $\omega$ directions.} |
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\label{waterSpring} |
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\end{figure} |
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|
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In the case of molecular liquids, the ideal vapor is chosen as the |
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target reference state. There are several examples of liquid state |
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free energy calculations of water models present in the |
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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 |
| 239 |
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interaction potential to convert the system to an ideal gas of water |
| 240 |
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molecules. In this study, we applied of one of the most convenient |
| 241 |
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methods and integrated over the $\lambda^4$ path, where all |
| 242 |
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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 |
| 244 |
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results in excellent agreement with other established |
| 245 |
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methods.\cite{Baez95b} |
| 246 |
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|
| 247 |
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Charge, dipole, and Lennard-Jones interactions were modified by a |
| 248 |
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cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
| 249 |
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). By applying this function, these interactions are smoothly |
| 250 |
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truncated, thereby avoiding the poor energy conservation which results |
| 251 |
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from harsher truncation schemes. The effect of a long-range correction |
| 252 |
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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 |
| 254 |
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dielectric constant of 80 was applied in all |
| 255 |
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simulations.\cite{Onsager36} For a series of the least computationally |
| 256 |
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expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
| 257 |
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performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
| 258 |
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\AA\ cutoff results. Finally, the effects of utilizing an Ewald |
| 259 |
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summation were estimated for TIP3P and SPC/E by performing single |
| 260 |
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configuration calculations with Particle-Mesh Ewald (PME) in the |
| 261 |
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TINKER molecular mechanics software package.\cite{Tinker} The |
| 262 |
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calculated energy difference in the presence and absence of PME was |
| 263 |
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applied to the previous results in order to predict changes to the |
| 264 |
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free energy landscape. |
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|
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\section{Results and discussion} |
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|
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The free energy of proton-ordered Ice-{\it i} was calculated and |
| 269 |
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compared with the free energies of proton ordered variants of the |
| 270 |
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experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
| 271 |
+ |
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
| 272 |
+ |
and thought to be the minimum free energy structure for the SPC/E |
| 273 |
+ |
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
| 274 |
+ |
Ice XI, the experimentally-observed proton-ordered variant of ice |
| 275 |
+ |
$I_h$, was investigated initially, but was found to be not as stable |
| 276 |
+ |
as proton disordered or antiferroelectric variants of ice $I_h$. The |
| 277 |
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proton ordered variant of ice $I_h$ used here is a simple |
| 278 |
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antiferroelectric version that we devised, and it has an 8 molecule |
| 279 |
+ |
unit cell similar to other predicted antiferroelectric $I_h$ |
| 280 |
+ |
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
| 281 |
+ |
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
| 282 |
+ |
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger |
| 283 |
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crystal sizes were necessary for simulations involving larger cutoff |
| 284 |
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values. |
| 285 |
+ |
|
| 286 |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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|
| 290 |
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\caption{Calculated free energies for several ice polymorphs with a |
| 291 |
+ |
variety of common water models. All calculations used a cutoff radius |
| 292 |
+ |
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
| 293 |
+ |
kcal/mol. Calculated error of the final digits is in parentheses.} |
| 294 |
+ |
|
| 295 |
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\begin{tabular}{lcccc} |
| 296 |
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\hline |
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Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
| 298 |
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\hline |
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TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
| 300 |
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TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
| 301 |
+ |
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
| 302 |
+ |
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & -13.55(2)\\ |
| 303 |
+ |
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
| 304 |
+ |
SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2)\\ |
| 305 |
+ |
\end{tabular} |
| 306 |
+ |
\label{freeEnergy} |
| 307 |
+ |
\end{center} |
| 308 |
+ |
\end{minipage} |
| 309 |
+ |
\end{table*} |
| 310 |
+ |
|
| 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 |
+ |
|
| 326 |
+ |
\begin{figure} |
| 327 |
+ |
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
| 328 |
+ |
\caption{Phase diagram for the TIP3P water model in the low pressure |
| 329 |
+ |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 330 |
+ |
the experimental values; however, the solid phases shown are not the |
| 331 |
+ |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
| 332 |
+ |
higher in energy and don't appear in the phase diagram.} |
| 333 |
+ |
\label{tp3phasedia} |
| 334 |
+ |
\end{figure} |
| 335 |
+ |
|
| 336 |
+ |
\begin{figure} |
| 337 |
+ |
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
| 338 |
+ |
\caption{Phase diagram for the SSD/RF water model in the low pressure |
| 339 |
+ |
regime. Calculations producing these results were done under an |
| 340 |
+ |
applied reaction field. It is interesting to note that this |
| 341 |
+ |
computationally efficient model (over 3 times more efficient than |
| 342 |
+ |
TIP3P) exhibits phase behavior similar to the less computationally |
| 343 |
+ |
conservative charge based models.} |
| 344 |
+ |
\label{ssdrfphasedia} |
| 345 |
+ |
\end{figure} |
| 346 |
+ |
|
| 347 |
+ |
\begin{table*} |
| 348 |
+ |
\begin{minipage}{\linewidth} |
| 349 |
+ |
\begin{center} |
| 350 |
+ |
|
| 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 |
+ |
|
| 357 |
+ |
\begin{tabular}{lccccccc} |
| 358 |
+ |
\hline |
| 359 |
+ |
Equilibrium Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
| 360 |
+ |
\hline |
| 361 |
+ |
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
| 362 |
+ |
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
| 363 |
+ |
$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ |
| 364 |
+ |
\end{tabular} |
| 365 |
+ |
\label{meltandboil} |
| 366 |
+ |
\end{center} |
| 367 |
+ |
\end{minipage} |
| 368 |
+ |
\end{table*} |
| 369 |
+ |
|
| 370 |
+ |
Table \ref{meltandboil} lists the melting and boiling temperatures |
| 371 |
+ |
calculated from this work. Surprisingly, most of these models have |
| 372 |
+ |
melting points that compare quite favorably with experiment. The |
| 373 |
+ |
unfortunate aspect of this result is that this phase change occurs |
| 374 |
+ |
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
| 375 |
+ |
liquid state. These results are actually not contrary to previous |
| 376 |
+ |
studies in the literature. Earlier free energy studies of ice $I$ |
| 377 |
+ |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
| 378 |
+ |
being attributed to choice of interaction truncation and different |
| 379 |
+ |
ordered and disordered molecular |
| 380 |
+ |
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
| 381 |
+ |
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
| 382 |
+ |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 383 |
+ |
calculated at 265 K, significantly higher in temperature than the |
| 384 |
+ |
previous studies. Also of interest in these results is that SSD/E does |
| 385 |
+ |
not exhibit a melting point at 1 atm, but it shows a sublimation point |
| 386 |
+ |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
| 387 |
+ |
all other polymorphs for this particular model under these |
| 388 |
+ |
conditions. While troubling, this behavior resulted in spontaneous |
| 389 |
+ |
crystallization of Ice-{\it i} and led us to investigate this |
| 390 |
+ |
structure. These observations provide a warning that simulations of |
| 391 |
+ |
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
| 392 |
+ |
risk of spontaneous crystallization. However, this risk lessens when |
| 393 |
+ |
applying a longer cutoff. |
| 394 |
+ |
|
| 395 |
+ |
\begin{figure} |
| 396 |
+ |
\includegraphics[width=\linewidth]{cutoffChange.eps} |
| 397 |
+ |
\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) |
| 398 |
+ |
TIP3P, and (C) SSD/RF with a reaction field. Both SSD/E and TIP3P show |
| 399 |
+ |
significant cutoff radius dependence of the free energy and appear to |
| 400 |
+ |
converge when moving to cutoffs greater than 12 \AA. Use of a reaction |
| 401 |
+ |
field with SSD/RF results in free energies that exhibit minimal cutoff |
| 402 |
+ |
radius dependence.} |
| 403 |
+ |
\label{incCutoff} |
| 404 |
+ |
\end{figure} |
| 405 |
+ |
|
| 406 |
+ |
Increasing the cutoff radius in simulations of the more |
| 407 |
+ |
computationally efficient water models was done in order to evaluate |
| 408 |
+ |
the trend in free energy values when moving to systems that do not |
| 409 |
+ |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
| 410 |
+ |
free energy of all the ice polymorphs for the SSD/E and TIP3P models |
| 411 |
+ |
show a substantial dependence on cutoff radius. In general, there is a |
| 412 |
+ |
narrowing of the free energy differences while moving to greater |
| 413 |
+ |
cutoff radii. As the free energies for the polymorphs converge, the |
| 414 |
+ |
stability advantage that Ice-{\it i} exhibits is reduced; however, it |
| 415 |
+ |
remains the most stable polymorph for both of these models over the |
| 416 |
+ |
depicted range for both models. This narrowing trend is not |
| 417 |
+ |
significant in the case of SSD/RF, indicating that the free energies |
| 418 |
+ |
calculated with a reaction field present provide, at minimal |
| 419 |
+ |
computational cost, a more accurate picture of the free energy |
| 420 |
+ |
landscape in the absence of potential truncation. Interestingly, |
| 421 |
+ |
increasing the cutoff radius a mere 1.5 \AA\ with the SSD/E model |
| 422 |
+ |
destabilizes the Ice-{\it i} polymorph enough that the liquid state is |
| 423 |
+ |
preferred under standard simulation conditions (298 K and 1 |
| 424 |
+ |
atm). Thus, it is recommended that simulations using this model choose |
| 425 |
+ |
interaction truncation radii greater than 9 \AA. Considering this |
| 426 |
+ |
stabilization provided by smaller cutoffs, it is not surprising that |
| 427 |
+ |
crystallization into Ice-{\it i} was observed with SSD/E. The choice |
| 428 |
+ |
of a 9 \AA\ cutoff in the previous simulations gives the Ice-{\it i} |
| 429 |
+ |
polymorph a greater than 1 kcal/mol lower free energy than the ice |
| 430 |
+ |
$I_\textrm{h}$ starting configurations. |
| 431 |
+ |
|
| 432 |
+ |
To further study the changes resulting to the inclusion of a |
| 433 |
+ |
long-range interaction correction, the effect of an Ewald summation |
| 434 |
+ |
was estimated by applying the potential energy difference do to its |
| 435 |
+ |
inclusion in systems in the presence and absence of the correction. |
| 436 |
+ |
This was accomplished by calculation of the potential energy of |
| 437 |
+ |
identical crystals both with and without PME. The free energies for |
| 438 |
+ |
the investigated polymorphs using the TIP3P and SPC/E water models are |
| 439 |
+ |
shown in Table \ref{pmeShift}. The same trend pointed out through |
| 440 |
+ |
increase of cutoff radius is observed in these PME results. Ice-{\it |
| 441 |
+ |
i} is the preferred polymorph at ambient conditions for both the TIP3P |
| 442 |
+ |
and SPC/E water models; however, the narrowing of the free energy |
| 443 |
+ |
differences between the various solid forms with the SPC/E model is |
| 444 |
+ |
significant enough that it becomes less clear that it is the most |
| 445 |
+ |
stable polymorph. The free energies of Ice-{\it i} and $I_\textrm{c}$ |
| 446 |
+ |
overlap within error, while ice B and $I_\textrm{h}$ are just outside |
| 447 |
+ |
at t slightly higher free energy. This indicates that with SPC/E, |
| 448 |
+ |
Ice-{\it i} might be metastable with all the studied polymorphs, |
| 449 |
+ |
particularly ice $I_\textrm{c}$. However, these results do not |
| 450 |
+ |
significantly alter the finding that the Ice-{\it i} polymorph is a |
| 451 |
+ |
stable crystal structure that should be considered when studying the |
| 452 |
+ |
phase behavior of water models. |
| 453 |
+ |
|
| 454 |
+ |
\begin{table*} |
| 455 |
+ |
\begin{minipage}{\linewidth} |
| 456 |
+ |
\begin{center} |
| 457 |
+ |
|
| 458 |
+ |
\caption{The free energy of the studied ice polymorphs after applying |
| 459 |
+ |
the energy difference attributed to the inclusion of the PME |
| 460 |
+ |
long-range interaction correction. Units are kcal/mol.} |
| 461 |
+ |
|
| 462 |
+ |
\begin{tabular}{ccccc} |
| 463 |
+ |
\hline |
| 464 |
+ |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} \\ |
| 465 |
+ |
\hline |
| 466 |
+ |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3) \\ |
| 467 |
+ |
SPC/E & -12.97(2) & -13.00(2) & -12.96(3) & -13.02(2) \\ |
| 468 |
+ |
\end{tabular} |
| 469 |
+ |
\label{pmeShift} |
| 470 |
+ |
\end{center} |
| 471 |
+ |
\end{minipage} |
| 472 |
+ |
\end{table*} |
| 473 |
+ |
|
| 474 |
|
\section{Conclusions} |
| 475 |
|
|
| 476 |
+ |
The free energy for proton ordered variants of hexagonal and cubic ice |
| 477 |
+ |
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
| 478 |
+ |
calculated under standard conditions for several common water models |
| 479 |
+ |
via thermodynamic integration. All the water models studied show |
| 480 |
+ |
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
| 481 |
+ |
\AA\ switching function cutoff. Calculated melting and boiling points |
| 482 |
+ |
show surprisingly good agreement with the experimental values; |
| 483 |
+ |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
| 484 |
+ |
effect of interaction truncation was investigated through variation of |
| 485 |
+ |
the cutoff radius, use of a reaction field parameterized model, and |
| 486 |
+ |
estimation of the results in the presence of the Ewald |
| 487 |
+ |
summation. Interaction truncation has a significant effect on the |
| 488 |
+ |
computed free energy values, and may significantly alter the free |
| 489 |
+ |
energy landscape for the more complex multipoint water models. Despite |
| 490 |
+ |
these effects, these results show Ice-{\it i} to be an important ice |
| 491 |
+ |
polymorph that should be considered in simulation studies. |
| 492 |
+ |
|
| 493 |
+ |
Due to this relative stability of Ice-{\it i} in all of the |
| 494 |
+ |
investigated simulation conditions, the question arises as to possible |
| 495 |
+ |
experimental observation of this polymorph. The rather extensive past |
| 496 |
+ |
and current experimental investigation of water in the low pressure |
| 497 |
+ |
regime makes us hesitant to ascribe any relevance of this work outside |
| 498 |
+ |
of the simulation community. It is for this reason that we chose a |
| 499 |
+ |
name for this polymorph which involves an imaginary quantity. That |
| 500 |
+ |
said, there are certain experimental conditions that would provide the |
| 501 |
+ |
most ideal situation for possible observation. These include the |
| 502 |
+ |
negative pressure or stretched solid regime, small clusters in vacuum |
| 503 |
+ |
deposition environments, and in clathrate structures involving small |
| 504 |
+ |
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
| 505 |
+ |
our predictions for both the pair distribution function ($g_{OO}(r)$) |
| 506 |
+ |
and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for |
| 507 |
+ |
ice-{\it i} at a temperature of 77K. In studies of the high and low |
| 508 |
+ |
density forms of amorphous ice, ``spurious'' diffraction peaks have |
| 509 |
+ |
been observed experimentally.\cite{Bizid87} It is possible that a |
| 510 |
+ |
variant of Ice-{\it i} could explain some of this behavior; however, |
| 511 |
+ |
we will leave it to our experimental colleagues to make the final |
| 512 |
+ |
determination on whether this ice polymorph is named appropriately |
| 513 |
+ |
(i.e. with an imaginary number) or if it can be promoted to Ice-0. |
| 514 |
+ |
|
| 515 |
+ |
\begin{figure} |
| 516 |
+ |
\includegraphics[width=\linewidth]{iceGofr.eps} |
| 517 |
+ |
\caption{Radial distribution functions of ice $I_h$, $I_c$, |
| 518 |
+ |
Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations |
| 519 |
+ |
of the SSD/RF water model at 77 K.} |
| 520 |
+ |
\label{fig:gofr} |
| 521 |
+ |
\end{figure} |
| 522 |
+ |
|
| 523 |
+ |
\begin{figure} |
| 524 |
+ |
\includegraphics[width=\linewidth]{sofq.eps} |
| 525 |
+ |
\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, |
| 526 |
+ |
and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
| 527 |
+ |
been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
| 528 |
+ |
width) to compensate for the trunction effects in our finite size |
| 529 |
+ |
simulations.} |
| 530 |
+ |
\label{fig:sofq} |
| 531 |
+ |
\end{figure} |
| 532 |
+ |
|
| 533 |
|
\section{Acknowledgments} |
| 534 |
|
Support for this project was provided by the National Science |
| 535 |
|
Foundation under grant CHE-0134881. Computation time was provided by |
| 536 |
< |
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
| 537 |
< |
DMR-0079647. |
| 536 |
> |
the Notre Dame High Performance Computing Cluster and the Notre Dame |
| 537 |
> |
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
| 538 |
|
|
| 539 |
|
\newpage |
| 540 |
|
|
