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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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\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 \ref{unitcell}A and \ref{unitcell}B. These |
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tetramers form a crystal structure similar in appearance to a recent |
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two-dimensional surface tessellation simulated on silica.\cite{Yang04} |
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As expected in an ice crystal constructed of water tetramers, the |
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hydrogen bonds are not as linear as those observed in ice I$_h$, |
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however the interlocking of these subunits appears to provide |
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significant stabilization to the overall crystal. The arrangement of |
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these tetramers results in octagonal cavities that are typically |
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greater than 6.3 \AA\ in diameter (Fig. \ref{iCrystal}). This open |
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structure leads to crystals that are typically 0.07 g/cm$^3$ less |
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dense than ice I$_h$. |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{unitCell.eps} |
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\caption{(A) Unit cells for Ice-{\it i} and (B) Ice-{\it i}$^\prime$. |
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The spheres represent the center-of-mass locations of the water |
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molecules. The $a$ to $c$ ratios for Ice-{\it i} and Ice-{\it |
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i}$^\prime$ are given by 2.1214 and 1.785 respectively.} |
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\label{unitcell} |
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\end{figure} |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{orderedIcei.eps} |
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\caption{A rendering of a proton ordered crystal of Ice-{\it i} looking |
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down the (001) crystal face. The presence of large octagonal pores |
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leads to a polymorph that is less dense than ice I$_h$.} |
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\label{iCrystal} |
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\end{figure} |
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|
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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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\section{Methods} |
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Canonical ensemble (NVT) molecular dynamics calculations were |
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performed using the OOPSE molecular mechanics program.\cite{Meineke05} |
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All molecules were treated as rigid bodies, with orientational motion |
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propagated using the symplectic DLM integration method. Details about |
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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 Helmholtz free |
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energies ($A$) of the listed water models at various state points |
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using the OOPSE molecular dynamics program.\cite{Meineke05} |
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Thermodynamic integration is an established technique that has been |
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used extensively in the calculation of free energies for condensed |
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phases of |
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materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
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method uses a sequence of simulations during which the system of |
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interest is converted into a reference system for which the free |
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energy is known analytically ($A_0$). The difference in potential |
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energy between the reference system and the system of interest |
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($\Delta V$) is then integrated in order to determine the free energy |
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difference between the two states: |
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\begin{equation} |
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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. |
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|
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In an Einstein crystal, the molecules are restrained at their ideal |
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lattice locations and orientations. Using harmonic restraints, as |
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applied by B\`{a}ez and Clancy, the total potential for this reference |
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crystal ($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
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\begin{equation} |
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V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
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\frac{K_\omega\omega^2}{2}, |
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\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. These spring constants are typically calculated from |
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the mean-square displacements of water molecules in an unrestrained |
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ice crystal at 200 K. For these studies, $K_\mathrm{v} = 4.29$ kcal |
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mol$^{-1}$ \AA$^{-2}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$ rad$^{-2}$, |
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and $K_\omega\ = 17.75$ kcal mol$^{-1}$ rad$^{-2}$. It is clear from |
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Fig. \ref{waterSpring} that the values of $\theta$ range from $0$ to |
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$\pi$, while $\omega$ ranges from $-\pi$ to $\pi$. The partition |
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function for a molecular crystal restrained in this fashion can be |
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evaluated analytically, and the Helmholtz Free Energy ({\it A}) is |
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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}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right |
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)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right |
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)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi |
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K_\omega K_\theta)^{\frac{1}{2}}}\exp\left |
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(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\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 $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, 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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\centering |
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\includegraphics[width=4in]{rotSpring.eps} |
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\caption{Possible orientational motions for a restrained molecule. |
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$\theta$ angles correspond to displacement from the body-frame {\it |
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z}-axis, while $\omega$ angles correspond to rotation about the |
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body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
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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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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 |
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interaction potential to convert the system to an ideal gas of water |
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molecules. In this study, we applied one of the most convenient |
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methods and integrated over the $\lambda^4$ path, where all |
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interaction parameters are scaled equally by this transformation |
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parameter. This method has been shown to be reversible and provide |
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results in excellent agreement with other established |
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methods.\cite{Baez95b} |
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|
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Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and |
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Lennard-Jones interactions were gradually reduced by a cubic switching |
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function. By applying this function, these interactions are smoothly |
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truncated, thereby avoiding the poor energy conservation which results |
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from harsher truncation schemes. The effect of a long-range |
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correction was also investigated on select model systems in a variety |
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of manners. For the SSD/RF model, a reaction field with a fixed |
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dielectric constant of 80 was applied in all |
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simulations.\cite{Onsager36} For a series of the least computationally |
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expensive models (SSD/E, SSD/RF, TIP3P, and SPC/E), simulations were |
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performed with longer cutoffs of 10.5, 12, 13.5, and 15 \AA\ to |
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compare with the 9 \AA\ cutoff results. Finally, the effects of using |
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the Ewald summation were estimated for TIP3P and SPC/E by performing |
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single configuration Particle-Mesh Ewald (PME) |
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calculations~\cite{Tinker} for each of the ice polymorphs. The |
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calculated energy difference in the presence and absence of PME was |
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applied to the previous results in order to predict changes to the |
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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 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 |
268 |
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conditions.\cite{Baez95b} In addition to the free energies, the |
269 |
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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 |
272 |
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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 |
281 |
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models. For purposes of this study, ice B is representative of the |
282 |
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dense ice polymorphs. A recent study by Sanz {\it et al.} provides |
283 |
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details on the phase diagrams for SPC/E and TIP4P at higher pressures |
284 |
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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 |
291 |
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the investigated water models. All free energy calculations used a |
292 |
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cutoff radius of 9.0 \AA\ and were performed at 200 K and $\sim$1 atm. |
293 |
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Units of free energy are kcal/mol, while transition temperature are in |
294 |
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Kelvin. Calculated error of the final digits is in parentheses.} |
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\begin{tabular}{lccccccc} |
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\hline |
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Water Model & I$_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)\\ |
301 |
|
|
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 271(4) & 337(2)\\ |
302 |
|
|
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 296(3) & 396(2)\\ |
303 |
|
|
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(2) & -\\ |
304 |
chrisfen |
2132 |
SSD/RF & -11.96(2) & -11.60(2) & -12.53(3) & -12.79(2) & - & 322(4) & 366(2)\\ |
305 |
chrisfen |
1456 |
\end{tabular} |
306 |
|
|
\label{freeEnergy} |
307 |
|
|
\end{center} |
308 |
|
|
\end{minipage} |
309 |
|
|
\end{table*} |
310 |
chrisfen |
1453 |
|
311 |
chrisfen |
1456 |
\begin{figure} |
312 |
|
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
313 |
|
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
314 |
chrisfen |
1812 |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
315 |
chrisfen |
1456 |
the experimental values; however, the solid phases shown are not the |
316 |
chrisfen |
1812 |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
317 |
chrisfen |
1456 |
higher in energy and don't appear in the phase diagram.} |
318 |
chrisfen |
1905 |
\label{tp3PhaseDia} |
319 |
chrisfen |
1456 |
\end{figure} |
320 |
gezelter |
1463 |
|
321 |
chrisfen |
1905 |
Most of the water models have melting points that compare quite |
322 |
|
|
favorably with the experimental value of 273 K. The unfortunate |
323 |
|
|
aspect of this result is that this phase change occurs between |
324 |
chrisfen |
2132 |
Ice-{\it i} and the liquid state rather than ice I$_h$ and the liquid |
325 |
chrisfen |
1905 |
state. These results do not contradict other studies. Studies of ice |
326 |
chrisfen |
2132 |
I$_h$ using TIP4P predict a $T_m$ ranging from 214 to 238 K |
327 |
chrisfen |
1905 |
(differences being attributed to choice of interaction truncation and |
328 |
|
|
different ordered and disordered molecular |
329 |
chrisfen |
1466 |
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
330 |
|
|
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
331 |
chrisfen |
1812 |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
332 |
chrisfen |
1806 |
calculated to be 265 K, indicating that these simulation based |
333 |
|
|
structures ought to be included in studies probing phase transitions |
334 |
chrisfen |
1812 |
with this model. Also of interest in these results is that SSD/E does |
335 |
chrisfen |
1905 |
not exhibit a melting point at 1 atm but does sublime at 355 K. This |
336 |
|
|
is due to the significant stability of Ice-{\it i} over all other |
337 |
|
|
polymorphs for this particular model under these conditions. While |
338 |
|
|
troubling, this behavior resulted in the spontaneous crystallization |
339 |
|
|
of Ice-{\it i} which led us to investigate this structure. These |
340 |
|
|
observations provide a warning that simulations of SSD/E as a |
341 |
|
|
``liquid'' near 300 K are actually metastable and run the risk of |
342 |
|
|
spontaneous crystallization. However, when a longer cutoff radius is |
343 |
|
|
used, SSD/E prefers the liquid state under standard temperature and |
344 |
|
|
pressure. |
345 |
chrisfen |
1456 |
|
346 |
chrisfen |
1458 |
\begin{figure} |
347 |
|
|
\includegraphics[width=\linewidth]{cutoffChange.eps} |
348 |
chrisfen |
1806 |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
349 |
|
|
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
350 |
chrisfen |
1905 |
with an added Ewald correction term. Error for the larger cutoff |
351 |
|
|
points is equivalent to that observed at 9.0\AA\ (see Table |
352 |
|
|
\ref{freeEnergy}). Data for ice I$_c$ with TIP3P using both 12 and |
353 |
|
|
13.5 \AA\ cutoffs were omitted because the crystal was prone to |
354 |
chrisfen |
1834 |
distortion and melting at 200 K. Ice-{\it i}$^\prime$ is the form of |
355 |
|
|
Ice-{\it i} used in the SPC/E simulations.} |
356 |
chrisfen |
1458 |
\label{incCutoff} |
357 |
|
|
\end{figure} |
358 |
|
|
|
359 |
chrisfen |
1905 |
For the more computationally efficient water models, we have also |
360 |
|
|
investigated the effect of potential trunctaion on the computed free |
361 |
|
|
energies as a function of the cutoff radius. As seen in |
362 |
|
|
Fig. \ref{incCutoff}, the free energies of the ice polymorphs with |
363 |
|
|
water models lacking a long-range correction show significant cutoff |
364 |
|
|
dependence. In general, there is a narrowing of the free energy |
365 |
|
|
differences while moving to greater cutoff radii. As the free |
366 |
|
|
energies for the polymorphs converge, the stability advantage that |
367 |
|
|
Ice-{\it i} exhibits is reduced. Adjacent to each of these plots are |
368 |
|
|
results for systems with applied or estimated long-range corrections. |
369 |
|
|
SSD/RF was parametrized for use with a reaction field, and the benefit |
370 |
|
|
provided by this computationally inexpensive correction is apparent. |
371 |
|
|
The free energies are largely independent of the size of the reaction |
372 |
|
|
field cavity in this model, so small cutoff radii mimic bulk |
373 |
|
|
calculations quite well under SSD/RF. |
374 |
|
|
|
375 |
|
|
Although TIP3P was paramaterized for use without the Ewald summation, |
376 |
|
|
we have estimated the effect of this method for computing long-range |
377 |
|
|
electrostatics for both TIP3P and SPC/E. This was accomplished by |
378 |
|
|
calculating the potential energy of identical crystals both with and |
379 |
|
|
without particle mesh Ewald (PME). Similar behavior to that observed |
380 |
|
|
with reaction field is seen for both of these models. The free |
381 |
|
|
energies show reduced dependence on cutoff radius and span a narrower |
382 |
|
|
range for the various polymorphs. Like the dipolar water models, |
383 |
|
|
TIP3P displays a relatively constant preference for the Ice-{\it i} |
384 |
chrisfen |
1812 |
polymorph. Crystal preference is much more difficult to determine for |
385 |
|
|
SPC/E. Without a long-range correction, each of the polymorphs |
386 |
|
|
studied assumes the role of the preferred polymorph under different |
387 |
chrisfen |
1905 |
cutoff radii. The inclusion of the Ewald correction flattens and |
388 |
|
|
narrows the gap in free energies such that the polymorphs are |
389 |
|
|
isoenergetic within statistical uncertainty. This suggests that other |
390 |
|
|
conditions, such as the density in fixed-volume simulations, can |
391 |
|
|
influence the polymorph expressed upon crystallization. |
392 |
chrisfen |
1456 |
|
393 |
chrisfen |
1906 |
\section{Conclusions} |
394 |
chrisfen |
1453 |
|
395 |
chrisfen |
1909 |
In this work, thermodynamic integration was used to determine the |
396 |
|
|
absolute free energies of several ice polymorphs. The new polymorph, |
397 |
|
|
Ice-{\it i} was observed to be the stable crystalline state for {\it |
398 |
|
|
all} the water models when using a 9.0 \AA\ cutoff. However, the free |
399 |
|
|
energy partially depends on simulation conditions (particularly on the |
400 |
|
|
choice of long range correction method). Regardless, Ice-{\it i} was |
401 |
|
|
still observered to be a stable polymorph for all of the studied water |
402 |
|
|
models. |
403 |
chrisfen |
1906 |
|
404 |
|
|
So what is the preferred solid polymorph for simulated water? As |
405 |
|
|
indicated above, the answer appears to be dependent both on the |
406 |
|
|
conditions and the model used. In the case of short cutoffs without a |
407 |
|
|
long-range interaction correction, Ice-{\it i} and Ice-{\it |
408 |
|
|
i}$^\prime$ have the lowest free energy of the studied polymorphs with |
409 |
|
|
all the models. Ideally, crystallization of each model under constant |
410 |
|
|
pressure conditions, as was done with SSD/E, would aid in the |
411 |
|
|
identification of their respective preferred structures. This work, |
412 |
|
|
however, helps illustrate how studies involving one specific model can |
413 |
chrisfen |
1909 |
lead to insight about important behavior of others. |
414 |
chrisfen |
1906 |
|
415 |
chrisfen |
1905 |
We also note that none of the water models used in this study are |
416 |
|
|
polarizable or flexible models. It is entirely possible that the |
417 |
|
|
polarizability of real water makes Ice-{\it i} substantially less |
418 |
chrisfen |
2132 |
stable than ice I$_h$. However, the calculations presented above seem |
419 |
chrisfen |
1905 |
interesting enough to communicate before the role of polarizability |
420 |
|
|
(or flexibility) has been thoroughly investigated. |
421 |
chrisfen |
1458 |
|
422 |
chrisfen |
1905 |
Finally, due to the stability of Ice-{\it i} in the investigated |
423 |
|
|
simulation conditions, the question arises as to possible experimental |
424 |
|
|
observation of this polymorph. The rather extensive past and current |
425 |
|
|
experimental investigation of water in the low pressure regime makes |
426 |
|
|
us hesitant to ascribe any relevance to this work outside of the |
427 |
|
|
simulation community. It is for this reason that we chose a name for |
428 |
|
|
this polymorph which involves an imaginary quantity. That said, there |
429 |
|
|
are certain experimental conditions that would provide the most ideal |
430 |
|
|
situation for possible observation. These include the negative |
431 |
|
|
pressure or stretched solid regime, small clusters in vacuum |
432 |
gezelter |
1465 |
deposition environments, and in clathrate structures involving small |
433 |
chrisfen |
1909 |
non-polar molecules. For the purpose of comparison with experimental |
434 |
|
|
results, we have calculated the oxygen-oxygen pair correlation |
435 |
|
|
function, $g_{OO}(r)$, and the structure factor, $S(\vec{q})$ for the |
436 |
chrisfen |
2132 |
two Ice-{\it i} variants (along with example ice I$_h$ and I$_c$ |
437 |
chrisfen |
1909 |
plots) at 77K, and they are shown in figures \ref{fig:gofr} and |
438 |
|
|
\ref{fig:sofq} respectively. It is interesting to note that the |
439 |
|
|
structure factors for Ice-{\it i}$^\prime$ and Ice-I$_c$ are quite similar. |
440 |
|
|
The primary differences are small peaks at 1.125, 2.29, and 2.53 |
441 |
|
|
\AA${-1}$, so particular attention to these regions would be needed |
442 |
|
|
to identify the new {\it i}$^\prime$ variant from the I$_{c}$ variant. |
443 |
chrisfen |
1459 |
|
444 |
chrisfen |
1467 |
\begin{figure} |
445 |
chrisfen |
1905 |
\centering |
446 |
chrisfen |
1467 |
\includegraphics[width=\linewidth]{iceGofr.eps} |
447 |
chrisfen |
2132 |
\caption{Radial distribution functions of ice I$_h$, I$_c$, and |
448 |
chrisfen |
1905 |
Ice-{\it i} calculated from from simulations of the SSD/RF water model |
449 |
|
|
at 77 K. The Ice-{\it i} distribution function was obtained from |
450 |
|
|
simulations composed of TIP4P water.} |
451 |
chrisfen |
1467 |
\label{fig:gofr} |
452 |
|
|
\end{figure} |
453 |
|
|
|
454 |
gezelter |
1469 |
\begin{figure} |
455 |
chrisfen |
1905 |
\centering |
456 |
gezelter |
1469 |
\includegraphics[width=\linewidth]{sofq.eps} |
457 |
chrisfen |
2132 |
\caption{Predicted structure factors for ice I$_h$, I$_c$, Ice-{\it i}, |
458 |
chrisfen |
1479 |
and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
459 |
|
|
been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
460 |
|
|
width) to compensate for the trunction effects in our finite size |
461 |
|
|
simulations.} |
462 |
gezelter |
1469 |
\label{fig:sofq} |
463 |
|
|
\end{figure} |
464 |
|
|
|
465 |
chrisfen |
1453 |
\section{Acknowledgments} |
466 |
|
|
Support for this project was provided by the National Science |
467 |
|
|
Foundation under grant CHE-0134881. Computation time was provided by |
468 |
chrisfen |
1458 |
the Notre Dame High Performance Computing Cluster and the Notre Dame |
469 |
|
|
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
470 |
chrisfen |
1453 |
|
471 |
|
|
\newpage |
472 |
|
|
|
473 |
chrisfen |
1909 |
\bibliographystyle{achemso} |
474 |
chrisfen |
1453 |
\bibliography{iceiPaper} |
475 |
|
|
|
476 |
|
|
|
477 |
|
|
\end{document} |