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\begin{document} |
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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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|
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\author{Christopher J. Fennell and J. Daniel Gezelter \\ |
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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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\maketitle |
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%\doublespacing |
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\begin{abstract} |
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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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 |
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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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|
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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 |
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\ref{iCrystal}A and |
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\ref{iCrystal}B. These tetramers form a crystal structure similar |
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in appearance to a recent two-dimensional surface tessellation |
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simulated on silica.\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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|
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\begin{figure} |
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\centering |
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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 2.1214 |
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and 1.785 respectively. The presence of large octagonal pores in |
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both crystal forms lead to a polymorph that is less dense than ice |
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$I_h$.} |
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\label{iCrystal} |
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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 |
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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 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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|
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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} 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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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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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. There is some subtlety in the path |
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taken as $\lambda$ goes from $0$ to $1$. However, 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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|
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The calculated free energies of proton-ordered variants 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}. Ice B was included because it has been |
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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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relevant 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. Figure \ref{tp3PhaseDia} is |
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an example diagram built from the results for the TIP3P water model. |
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All other models have similar structure, although the crossing points |
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between the phases move to different temperatures and pressures as |
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indicated from the transition temperatures in Table \ref{freeEnergy}. |
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It is interesting to note that ice $I_h$ (and ice $I_c$ for that |
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matter) do not appear in any of the phase diagrams for any of the |
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models. For purposes of this study, ice B is representative of the |
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dense ice polymorphs. A recent study by Sanz {\it et al.} provides |
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details on the phase diagrams for SPC/E and TIP4P at higher pressures |
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than those studied here.\cite{Sanz04} |
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|
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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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|
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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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|
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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. These results do not contradict other studies. Studies of ice |
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$I_h$ using TIP4P predict a $T_m$ ranging from 214 to 238 K |
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(differences being attributed to choice of interaction truncation and |
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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 does sublime at 355 K. This |
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is due to the significant stability of Ice-{\it i} over all other |
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polymorphs for this particular model under these conditions. While |
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troubling, this behavior resulted in the spontaneous crystallization |
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of Ice-{\it i} which led us to investigate this structure. These |
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observations provide a warning that simulations of SSD/E as a |
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``liquid'' near 300 K are actually metastable and run the risk of |
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spontaneous crystallization. However, when a longer cutoff radius is |
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used, SSD/E prefers the liquid state under standard temperature and |
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pressure. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{cutoffChange.eps} |
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\caption{Free energy as a function of cutoff radius for 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 |
| 259 |
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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|
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For the more computationally efficient water models, we have also |
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investigated the effect of potential trunctaion on the computed free |
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energies as a function of the cutoff radius. As seen in |
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Fig. \ref{incCutoff}, the free energies of the ice polymorphs with |
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water models lacking a long-range correction show significant cutoff |
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dependence. In general, there is a narrowing of the free energy |
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differences while moving to greater cutoff radii. As the free |
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energies for the polymorphs converge, the stability advantage that |
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Ice-{\it i} exhibits is reduced. Adjacent to each of these plots are |
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results for systems with applied or estimated long-range corrections. |
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SSD/RF was parametrized for use with a reaction field, and the benefit |
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provided by this computationally inexpensive correction is apparent. |
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The free energies are largely independent of the size of the reaction |
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field cavity in this model, so small cutoff radii mimic bulk |
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calculations quite well under SSD/RF. |
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|
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Although TIP3P was paramaterized for use without the Ewald summation, |
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we have estimated the effect of this method for computing long-range |
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electrostatics for both TIP3P and SPC/E. This was accomplished by |
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calculating the potential energy of identical crystals both with and |
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without particle mesh Ewald (PME). Similar behavior to that observed |
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with reaction field is seen for both of these models. The free |
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energies show reduced dependence on cutoff radius and span a narrower |
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range for the various polymorphs. Like the dipolar water models, |
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TIP3P displays a relatively constant preference for the Ice-{\it i} |
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polymorph. Crystal preference is much more difficult to determine for |
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SPC/E. Without a long-range correction, each of the polymorphs |
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studied assumes the role of the preferred polymorph under different |
| 292 |
cutoff radii. The inclusion of the Ewald correction flattens and |
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narrows the gap in free energies such that the polymorphs are |
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isoenergetic within statistical uncertainty. This suggests that other |
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conditions, such as the density in fixed-volume simulations, can |
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influence the polymorph expressed upon crystallization. |
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|
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So what is the preferred solid polymorph for simulated water? The |
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answer appears to be dependent both on the conditions and the model |
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used. In the case of short cutoffs without a long-range interaction |
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correction, Ice-{\it i} and Ice-{\it i}$^\prime$ have the lowest free |
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energy of the studied polymorphs with all the models. Ideally, |
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crystallization of each model under constant pressure conditions, as |
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was done with SSD/E, would aid in the identification of their |
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respective preferred structures. This work, however, helps illustrate |
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how studies involving one specific model can lead to insight about |
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important behavior of others. In general, the above results support |
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the finding that the Ice-{\it i} polymorph is a stable crystal |
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structure that should be considered when studying the phase behavior |
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of water models. |
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|
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We also note that none of the water models used in this study are |
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polarizable or flexible models. It is entirely possible that the |
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polarizability of real water makes Ice-{\it i} substantially less |
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stable than ice $I_h$. However, the calculations presented above seem |
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interesting enough to communicate before the role of polarizability |
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(or flexibility) has been thoroughly investigated. |
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|
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Finally, due to the stability of Ice-{\it i} in the investigated |
| 320 |
simulation conditions, the question arises as to possible experimental |
| 321 |
observation of this polymorph. The rather extensive past and current |
| 322 |
experimental investigation of water in the low pressure regime makes |
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us hesitant to ascribe any relevance to this work outside of the |
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simulation community. It is for this reason that we chose a name for |
| 325 |
this polymorph which involves an imaginary quantity. That said, there |
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are certain experimental conditions that would provide the most ideal |
| 327 |
situation for possible observation. These include the negative |
| 328 |
pressure or stretched solid regime, small clusters in vacuum |
| 329 |
deposition environments, and in clathrate structures involving small |
| 330 |
non-polar molecules. |
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|
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\section{Acknowledgments} |
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Support for this project was provided by the National Science |
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Foundation under grant CHE-0134881. Computation time was provided by |
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the Notre Dame High Performance Computing Cluster and the Notre Dame |
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Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
| 337 |
|
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\newpage |
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|
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\bibliographystyle{jcp} |
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\bibliography{iceiPaper} |
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|
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|
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\end{document} |
