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\begin{document} |
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\title{Free Energy Analysis of Simulated Ice Polymorphs} |
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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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%\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 with |
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a variety of common water models. A recently discovered ice polymorph |
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that has yet 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 show to play a |
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role in the resulting shape of the free energy landscape. |
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\end{abstract} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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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. Difficulties in |
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studies addressing these thermodynamic quantities typically arise from |
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the assortment of possible crystalline polymorphs that water adopts |
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over a wide range of pressures and temperatures. It is a challenging |
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task to investigate the entire free energy landscape\cite{Sanz04}; |
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and ideally, research is focused on the phases having the lowest free |
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energy at a given state point, because these phases will dictate the |
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relevant transition temperatures and pressures for the model. |
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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 of |
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Ice-{\it i} and an extruded variant named Ice-{\it i}$^\prime$ both |
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consist of eight water molecules that stack in rows of interlocking |
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water tetramers as illustrated in figures \ref{iCrystal}A and |
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\ref{iCrystal}B. These tetramers make the crystal structure similar |
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in appearance to a recent two-dimensional ice tessellation simulated |
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on a silica surface.\cite{Yang04} As expected in an ice crystal |
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constructed of water tetramers, the hydrogen bonds are not as linear |
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as those observed in ice $I_h$, however the interlocking of these |
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subunits appears to provide significant stabilization to the overall |
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crystal. The arrangement of these tetramers results in surrounding |
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open octagonal cavities that are typically greater than 6.3 \AA\ in |
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diameter (Fig. \ref{iCrystal}C). This open structure leads to |
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crystals that are typically 0.07 g/cm$^3$ less dense than ice $I_h$. |
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\begin{figure} |
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\includegraphics[width=4in]{iCrystal.eps} |
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\caption{(A) Unit cell for Ice-{\it i}, (B) Ice-{\it i}$^\prime$, |
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and (C) a rendering of a proton ordered crystal of Ice-{\it i} looking |
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down the (001) crystal face. In the unit cells, the spheres represent |
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the 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 $a:1.785c$ respectively. The presence of large |
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octagonal pores in both crystal forms lead to a polymorph that is less |
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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 |
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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} 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). The |
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extruded variant, Ice-{\it i}$^\prime$, was used in calculations |
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involving SPC/E, TIP4P, and TIP5P due to its enhanced stability with |
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these models. There is typically a small distortion of proton ordered |
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Ice-{\it i}$^\prime$ that converts the normally square tetramer into a |
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rhombus with alternating approximately 85 and 95 degree angles. The |
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degree of this distortion is model dependent and significant enough to |
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split the tetramer diagonal location peak in the radial distribution |
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function. |
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Thermodynamic integration was utilized to calculate the free energies |
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of the listed water models at various state points using a modified |
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form of the OOPSE molecular dynamics package.\cite{Meineke05} |
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This calculation method 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 potential. For |
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liquid and solid phases, the ideal gas and harmonically restrained |
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crystal are chosen as the reference states respectively. Thermodynamic |
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integration is an established technique that has been used extensively |
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in the calculation of free energies for condensed phases of |
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materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. |
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The calculated free energies of proton-ordered varients of three low |
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density polymorphs ($I_h$, $I_c$, and Ice-{\it i} or Ice-{\it |
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i}$^\prime$) and the stable higher density ice B are listed in Table |
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\ref{freeEnergy}. The reason for inclusion of ice B was that it was |
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shown to be a minimum free energy structure for SPC/E at ambient |
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conditions.\cite{Baez95b} In addition to the free energies, the |
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relavent transition temperatures at standard pressure are also |
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displayed in Table \ref{freeEnergy}. These free energy values |
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indicate that Ice-{\it i} is the most stable state for all of the |
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investigated water models. With the free energy at these state |
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points, the Gibbs-Helmholtz equation was used to project to other |
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state points and to build phase diagrams, and figure \ref{tp3PhaseDia} |
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is an example diagram built from the results for the TIP3P water |
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model. All other models have similar structure, although the crossing |
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points between the phases move to different temperatures and pressures |
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as indicated from the transition temperatures in Table |
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\ref{freeEnergy}. It is interesting to note that ice $I$ does not |
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exist in either cubic or hexagonal form in any of the phase diagrams |
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for any of the models. For purposes of this study, ice B is |
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representative of the dense ice polymorphs. A recent study by Sanz |
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{\it et al.} goes into detail on the phase diagrams for SPC/E and |
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TIP4P at higher pressures than those studied here.\cite{Sanz04} |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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\caption{Calculated free energies for several ice polymorphs along |
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with the calculated melting (or sublimation) and boiling points for |
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the investigated water models. All free energy calculations used a |
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cutoff radius of 9.0 \AA\ and were performed at 200 K and $\sim$1 atm. |
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Units of free energy are kcal/mol, while transition temperature are in |
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Kelvin. Calculated error of the final digits is in parentheses.} |
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\begin{tabular}{lccccccc} |
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\hline |
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Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} & Ice-{\it i}$^\prime$ & $T_m$ (*$T_s$) & $T_b$\\ |
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\hline |
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TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(4) & 357(2)\\ |
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TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 266(5) & 354(2)\\ |
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TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 271(4) & 337(2)\\ |
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SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 296(3) & 396(2)\\ |
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SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(2) & -\\ |
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SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2) & - & 278(4) & 349(2)\\ |
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\end{tabular} |
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\label{freeEnergy} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
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\caption{Phase diagram for the TIP3P water model in the low pressure |
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regime. The displayed $T_m$ and $T_b$ values are good predictions of |
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the experimental values; however, the solid phases shown are not the |
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experimentally observed forms. Both cubic and hexagonal ice $I$ are |
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higher in energy and don't appear in the phase diagram.} |
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\label{tp3PhaseDia} |
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\end{figure} |
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Most of the water models have melting points that compare quite |
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favorably with the experimental value of 273 K. The unfortunate |
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aspect of this result is that this phase change occurs between |
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Ice-{\it i} and the liquid state rather than ice $I_h$ and the liquid |
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state. Surprisingly, these results are not contrary to other studies. |
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Studies of ice $I_h$ using TIP4P predict a $T_m$ ranging from 214 to |
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238 K (differences being attributed to choice of interaction |
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truncation and different ordered and disordered molecular |
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arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
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Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
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predicted from this work. However, the $T_m$ from Ice-{\it i} is |
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calculated to be 265 K, indicating that these simulation based |
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structures ought to be included in studies probing phase transitions |
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with this model. Also of interest in these results is that SSD/E does |
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not exhibit a melting point at 1 atm, but it rather shows a |
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sublimation point at 355 K. This is due to the significant stability |
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of Ice-{\it i} over all other polymorphs for this particular model |
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under these conditions. While troubling, this behavior resulted in |
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spontaneous crystallization of Ice-{\it i} and led us to investigate |
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this structure. These observations provide a warning that simulations |
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of SSD/E as a ``liquid'' near 300 K are actually metastable and run |
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the risk of spontaneous crystallization. However, when applying a |
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longer cutoff, the liquid state is preferred under standard |
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conditions. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{cutoffChange.eps} |
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\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
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SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
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with an added Ewald correction term. Error for the larger cutoff |
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points is equivalent to that observed at 9.0\AA\ (see Table |
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\ref{freeEnergy}). Data for ice I$_c$ with TIP3P using both 12 and |
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13.5 \AA\ cutoffs were omitted because the crystal was prone to |
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distortion and melting at 200 K. Ice-{\it i}$^\prime$ is the form of |
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Ice-{\it i} used in the SPC/E simulations.} |
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\label{incCutoff} |
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\end{figure} |
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Increasing the cutoff radius in simulations of the more |
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computationally efficient water models was done in order to evaluate |
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the trend in free energy values when moving to systems that do not |
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involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
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free energy of the ice polymorphs with water models lacking a |
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long-range correction show a significant cutoff radius dependence. In |
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general, there is a narrowing of the free energy differences while |
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moving to greater cutoff radii. As the free energies for the |
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polymorphs converge, the stability advantage that Ice-{\it i} exhibits |
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is reduced. Adjacent to each of these model plots is a system with an |
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applied or estimated long-range correction. SSD/RF was parametrized |
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for use with a reaction field, and the benefit provided by this |
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computationally inexpensive correction is apparent. Due to the |
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relative independence of the resultant free energies, calculations |
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performed with a small cutoff radius provide resultant properties |
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similar to what one would expect for the bulk material. In the cases |
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of TIP3P and SPC/E, the effect of an Ewald summation was estimated by |
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applying the potential energy difference do to its inclusion in |
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systems in the presence and absence of the correction. This was |
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accomplished by calculation of the potential energy of identical |
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crystals both with and without particle mesh Ewald (PME). Similar |
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behavior to that observed with reaction field is seen for both of |
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these models. The free energies show less dependence on cutoff radius |
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and span a more narrowed range for the various polymorphs. Like the |
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dipolar water models, TIP3P displays a relatively constant preference |
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for the Ice-{\it i} polymorph. Crystal preference is much more |
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difficult to determine for SPC/E. Without a long-range correction, |
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each of the polymorphs studied assumes the role of the preferred |
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polymorph under different cutoff conditions. The inclusion of the |
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Ewald correction flattens and narrows the sequences of free energies |
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so much that they often overlap within error, indicating that other |
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conditions, such as cell volume in microcanonical simulations, can |
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influence the chosen polymorph upon crystallization. All of these |
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results support the finding that the Ice-{\it i} polymorph is a stable |
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crystal structure that should be considered when studying the phase |
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behavior of water models. |
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Due to this relative stability of Ice-{\it i} in all of the |
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investigated simulation conditions, the question arises as to possible |
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experimental observation of this polymorph. The rather extensive past |
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and current experimental investigation of water in the low pressure |
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regime makes us hesitant to ascribe any relevance of this work outside |
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of the simulation community. It is for this reason that we chose a |
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name for this polymorph which involves an imaginary quantity. That |
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said, there are certain experimental conditions that would provide the |
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most ideal situation for possible observation. These include the |
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negative pressure or stretched solid regime, small clusters in vacuum |
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deposition environments, and in clathrate structures involving small |
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non-polar molecules. Regardless of possible experimental observation, |
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the presence of these stable ice polymorphs has implications in the |
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understanding and depiction of phase changes involving the common |
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water models used in simulations. |
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\section{Acknowledgments} |
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Support for this project was provided by the National Science |
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Foundation under grant CHE-0134881. Computation time was provided by |
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the Notre Dame High Performance Computing Cluster and the Notre Dame |
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Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
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\newpage |
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\bibliographystyle{jcp} |
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\bibliography{iceiPaper} |
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\end{document} |
