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\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER \\ SIMULATIONS} |
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As discussed in the previous chapter, water has proven to be a |
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challenging substance to depict in simulations, and a variety of |
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models have been developed to describe its behavior under varying |
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simulation |
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conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,vanderSpoel98,Urbic00,Mahoney00,Fennell04} |
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These models have been used to investigate important physical |
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phenomena like phase transitions and the hydrophobic |
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effect.\cite{Yamada02,Marrink94,Gallagher03} With the choice of models |
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available, it is only natural to compare them under interesting |
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thermodynamic conditions in an attempt to clarify the limitations of |
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each.\cite{Jorgensen83,Jorgensen98b,Baez94,Mahoney01} Two important |
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property 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. This complexity makes it a challenging task to investigate the |
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entire free energy landscape.\cite{Sanz04} Ideally, research is |
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focused on the phases having the lowest free energy at a given state |
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point, because these phases will dictate the relevant transition |
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temperatures and pressures for the model. |
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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 chapter, standard reference state methods |
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were applied in the {\it low} pressure regime to evaluate the free |
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energies for a few known crystalline water polymorphs that might be |
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stable at these pressures. This work is unique in the fact that one of |
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the crystal lattices was arrived at through crystallization of a |
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computationally efficient water model under constant pressure and |
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temperature conditions. |
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While performing a series of melting simulations on an early iteration |
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of SSD/E, we observed several recrystallization events at a constant |
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pressure of 1 atm. After melting from ice I$_\textrm{h}$ at 235~K, two |
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of five systems recrystallized near 245~K. Crystallization events are |
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interesting in and of themselves;\cite{Matsumoto02,Yamada02} however, |
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the crystal structure extracted from these systems 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-$i$ and an axially elongated variant named Ice-$i^\prime$ 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 figure \ref{fig:iceiCell}A,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 |
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I$_\textrm{h}$; however, the interlocking of these subunits appears to |
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provide significant stabilization to the overall crystal. The |
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arrangement of these tetramers results in open octagonal cavities that |
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are typically greater than 6.3~\AA\ in diameter (see figure |
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\ref{fig:protOrder}). This open structure leads to crystals that are |
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typically 0.07~g/cm$^3$ less dense than ice I$_\textrm{h}$. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{./figures/unitCell.pdf} |
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\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
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elongated variant of Ice-{\it i}. For Ice-{\it i}, the $a$ to $c$ |
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relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a = |
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1.7850c$.} |
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\label{fig:iceiCell} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=3.5in]{./figures/orderedIcei.pdf} |
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\caption{Image of a proton ordered crystal of Ice-{\it i} looking |
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down the (001) crystal face. The rows of water tetramers surrounded by |
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octagonal pores leads to a crystal structure that is significantly |
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less dense than ice I$_\textrm{h}$.} |
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\label{fig:protOrder} |
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\end{figure} |
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Results from our initial studies 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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the previous work and related |
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articles\cite{Fennell04,Liu96,Bratko85}). These earlier 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 ice |
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I$_\textrm{c}$ and ice I$_\textrm{h}$ (the common low-density ice |
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polymorphs) and ice B (a higher density, but very stable crystal |
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structure 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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axially elongated variant, Ice-$i^\prime$, was used in calculations |
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involving SPC/E, TIP4P, and TIP5P. The square tetramer in Ice-$i$ |
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distorts in Ice-$i^\prime$ to form a rhombus with alternating 85 and |
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95 degree angles. Under SPC/E, TIP4P, and TIP5P, this geometry is |
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better 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 the peak in the radial distribution function which corresponds |
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to diagonal sites in the tetramers. |
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\section{Methods and Thermodynamic Integration} |
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Canonical ensemble ({\it NVT}) molecular dynamics calculations were |
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performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
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The densities chosen for the simulations were taken from |
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isobaric-isothermal ({\it NPT}) simulations performed at 1 atm and |
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200~K. Each model (and each crystal structure) was allowed to relax for |
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300~ps in the {\it NPT} ensemble before averaging the density to obtain |
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the volumes for the {\it NVT} simulations.All molecules were treated |
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as rigid bodies, with orientational motion propagated using the |
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symplectic DLM integration method described in section |
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\ref{sec:IntroIntegrate}. |
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We used thermodynamic integration to calculate the Helmholtz free |
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energies ({\it A}) of the listed water models at various state |
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points. Thermodynamic integration is an established technique that has |
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been used extensively in the calculation of free energies for |
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condensed phases of |
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materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
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method uses a sequence of simulations over 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 (Einstein) 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. Figure \ref{fig:integrationPath} |
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shows an example integration path for converting a crystalline system |
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to the Einstein crystal reference state. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{./figures/integrationPath.pdf} |
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\caption{An example integration path to convert an unrestrained |
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crystal ($\lambda = 1$) to the Einstein crystal reference state |
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($\lambda = 0$). Note the increase in samples at either end of the |
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path to improve the smoothness of the curve. For reversible processes, |
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conversion of the Einstein crystal back to the system of interest will |
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give an identical plot, thereby integrating to the same result.} |
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\label{fig:integrationPath} |
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\end{figure} |
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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, $K_\mathrm{v} = |
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4.29$~kcal~mol$^{-1}$~\AA$^{-2}$, $K_\theta\ = |
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13.88$~kcal~mol$^{-1}$~rad$^{-2}$, and $K_\omega\ = |
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17.75$~kcal~mol$^{-1}$~rad$^{-2}$. It is clear from |
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Fig. \ref{fig:waterSpring} that the values of $\theta$ range from $0$ |
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to $\pi$, 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{equation} |
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\begin{split} |
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A = E_m &- kT\ln\left(\frac{kT}{h\nu}\right)^3 \\ |
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&- kT\ln\left[\pi^\frac{1}{2}\left( |
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\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right)^\frac{1}{2} |
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\left(\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right)^\frac{1}{2} |
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\left(\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right)^\frac{1}{2} |
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\right] \\ |
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&- kT\ln\left[\frac{kT}{2(\pi K_\omega K_\theta)^{\frac{1}{2}}} |
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\exp\left(-\frac{kT}{2K_\theta}\right) |
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\int_0^{\left(\frac{kT}{2K_\theta}\right)^\frac{1}{2}} |
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\exp(t^2)\mathrm{d}t\right], |
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\end{split} |
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\label{eq:ecFreeEnergy} |
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\end{equation} |
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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} The choice of an |
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Einstein crystal reference state is somewhat arbitrary. Any ideal |
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system for which the partition function is known exactly could be used |
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as a reference point as long as the system does not undergo a phase |
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transition during the integration path between the real and ideal |
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systems. Nada and van der Eerden have shown that the use of different |
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force constants in the Einstein crystal does not affect the total |
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free energy, and Gao {\it et al.} have shown that free energies |
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computed with the Debye crystal reference state differ from the |
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Einstein crystal by only a few tenths of a |
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kJ~mol$^{-1}$.\cite{Nada03,Gao00} These free energy differences can |
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lead to some uncertainty in the computed melting point of the solids. |
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\begin{figure} |
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\centering |
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\includegraphics[width=3.5in]{./figures/rotSpring.pdf} |
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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{fig: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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The Helmholtz free energy error was determined in the same manner in |
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both the solid and the liquid free energy calculations . At each point |
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along the integration path, we calculated the standard deviation of |
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the potential energy difference. Addition or subtraction of these |
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values to each of their respective points and integrating the curve |
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again provides the upper and lower bounds of the uncertainty in the |
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Helmholtz free energy. |
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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) calculations for each |
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of the ice polymorphs.\cite{Ponder87} The calculated energy difference |
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in the presence and absence of PME was applied to the previous results |
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in order to predict changes to the free energy landscape. |
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\section{Initial Free Energy Results} |
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The calculated free energies of proton-ordered variants of three low |
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density polymorphs (I$_\textrm{h}$, I$_\textrm{c}$, and Ice-{\it i} or |
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Ice-$i^\prime$) and the stable higher density ice B are listed in |
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table \ref{tab: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{tab: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. Figures |
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\ref{fig:tp3PhaseDia} and \ref{fig:ssdrfPhaseDia} are example diagrams |
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built from the results for the TIP3P and SSD/RF water models. All |
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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 |
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\ref{tab:freeEnergy}. It is interesting to note that ice |
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I$_\textrm{h}$ (and ice I$_\textrm{c}$ for that matter) do not appear |
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in any of the phase diagrams for any of the models. For purposes of |
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this study, ice B is representative of the dense ice polymorphs. A |
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recent study by Sanz {\it et al.} provides details on the phase |
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diagrams for SPC/E and TIP4P at higher pressures than those studied |
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here.\cite{Sanz04} |
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\begin{table} |
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\centering |
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\caption{HELMHOLTZ FREE ENERGIES AND TRANSITION TEMPERATURES AT 1 |
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ATMOSPHERE FOR SEVERAL WATER MODELS} |
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\footnotesize |
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\begin{tabular}{lccccccc} |
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\toprule |
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\toprule |
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Water Model & I$_\textrm{h}$ & I$_\textrm{c}$ & B & Ice-{\it i} & Ice-$i^\prime$ & $T_\textrm{m}$ (*$T_\textrm{s}$) & $T_\textrm{b}$\\ |
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\cmidrule(lr){2-6} |
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\cmidrule(l){7-8} |
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& \multicolumn{5}{c}{(kcal mol$^{-1}$)} & \multicolumn{2}{c}{(K)}\\ |
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\midrule |
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TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 266(7) & 337(4)\\ |
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TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 262(6) & 354(4)\\ |
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TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(7) & 357(4)\\ |
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SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 299(6) & 396(4)\\ |
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SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(4) & -\\ |
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SSD/RF & -11.96(2) & -11.60(2) & -12.53(3) & -12.79(2) & - & 278(7) & 382(4)\\ |
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\bottomrule |
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\end{tabular} |
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\label{tab:freeEnergy} |
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\end{table} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{./figures/tp3PhaseDia.pdf} |
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\caption{Phase diagram for the TIP3P water model in the low pressure |
305 |
|
|
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
306 |
|
|
the experimental values; however, the solid phases shown are not the |
307 |
|
|
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
308 |
|
|
higher in energy and don't appear in the phase diagram.} |
309 |
|
|
\label{fig:tp3PhaseDia} |
310 |
|
|
\end{figure} |
311 |
|
|
|
312 |
|
|
\begin{figure} |
313 |
|
|
\centering |
314 |
|
|
\includegraphics[width=\linewidth]{./figures/ssdrfPhaseDia.pdf} |
315 |
|
|
\caption{Phase diagram for the SSD/RF water model in the low pressure |
316 |
|
|
regime. Calculations producing these results were done under an |
317 |
|
|
applied reaction field. It is interesting to note that this |
318 |
|
|
computationally efficient model (over 3 times more efficient than |
319 |
|
|
TIP3P) exhibits phase behavior similar to the less computationally |
320 |
|
|
conservative charge based models.} |
321 |
|
|
\label{fig:ssdrfPhaseDia} |
322 |
|
|
\end{figure} |
323 |
|
|
|
324 |
|
|
We note that all of the crystals investigated in this study are ideal |
325 |
|
|
proton-ordered antiferroelectric structures. All of the structures |
326 |
|
|
obey the Bernal-Fowler rules and should be able to form stable |
327 |
|
|
proton-{\it disordered} crystals which have the traditional |
328 |
|
|
$k_\textrm{B}$ln(3/2) residual entropy at 0~K.\cite{Bernal33,Pauling35} |
329 |
|
|
Simulations of proton-disordered structures are relatively unstable |
330 |
|
|
with all but the most expensive water models.\cite{Nada03} Our |
331 |
|
|
simulations have therefore been performed with the ordered |
332 |
|
|
antiferroelectric structures which do not require the residual entropy |
333 |
|
|
term to be accounted for in the free energies. This may result in some |
334 |
|
|
discrepancies when comparing our melting temperatures to the melting |
335 |
|
|
temperatures that have been calculated via thermodynamic integrations |
336 |
|
|
of the disordered structures.\cite{Sanz04} |
337 |
|
|
|
338 |
|
|
Most of the water models have melting points that compare quite |
339 |
|
|
favorably with the experimental value of 273~K. The unfortunate |
340 |
|
|
aspect of this result is that this phase change occurs between |
341 |
|
|
Ice-{\it i} and the liquid state rather than ice I$_h$ and the liquid |
342 |
|
|
state. These results do not contradict other studies. Studies of ice |
343 |
|
|
I$_h$ using TIP4P predict a $T_m$ ranging from 191 to 238~K |
344 |
|
|
(differences being attributed to choice of interaction truncation and |
345 |
|
|
different ordered and disordered molecular |
346 |
|
|
arrangements).\cite{Nada03,Vlot99,Gao00,Sanz04} If the presence of ice |
347 |
|
|
B and Ice-{\it i} were omitted, a $T_\textrm{m}$ value around 200~K |
348 |
|
|
would be predicted from this work. However, the $T_\textrm{m}$ from |
349 |
|
|
Ice-{\it i} is calculated to be 262~K, indicating that these |
350 |
|
|
simulation based structures ought to be included in studies probing |
351 |
|
|
phase transitions with this model. Also of interest in these results |
352 |
|
|
is that SSD/E does not exhibit a melting point at 1 atm but does |
353 |
|
|
sublime at 355~K. This is due to the significant stability of |
354 |
|
|
Ice-{\it i} over all other polymorphs for this particular model under |
355 |
|
|
these conditions. While troubling, this behavior resulted in the |
356 |
|
|
spontaneous crystallization of Ice-{\it i} which led us to investigate |
357 |
|
|
this structure. These observations provide a warning that simulations |
358 |
|
|
of SSD/E as a ``liquid'' near 300~K are actually metastable and run |
359 |
|
|
the risk of spontaneous crystallization. However, when a longer |
360 |
|
|
cutoff radius is used, SSD/E prefers the liquid state under standard |
361 |
|
|
temperature and pressure. |
362 |
|
|
|
363 |
|
|
\section{Effects of Potential Truncation} |
364 |
|
|
|
365 |
|
|
\begin{figure} |
366 |
|
|
\includegraphics[width=\linewidth]{./figures/cutoffChange.pdf} |
367 |
|
|
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
368 |
|
|
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
369 |
|
|
with an added Ewald correction term. Error for the larger cutoff |
370 |
|
|
points is equivalent to that observed at 9.0~\AA\ (see Table |
371 |
|
|
\ref{tab:freeEnergy}). Data for ice I$_\textrm{c}$ with TIP3P using |
372 |
|
|
both 12 and 13.5~\AA\ cutoffs were omitted because the crystal was |
373 |
|
|
prone to distortion and melting at 200~K. Ice-$i^\prime$ is the |
374 |
|
|
form of Ice-{\it i} used in the SPC/E simulations.} |
375 |
|
|
\label{fig:incCutoff} |
376 |
|
|
\end{figure} |
377 |
|
|
|
378 |
|
|
For the more computationally efficient water models, we have also |
379 |
|
|
investigated the effect of potential truncation on the computed free |
380 |
|
|
energies as a function of the cutoff radius. As seen in |
381 |
|
|
Fig. \ref{fig:incCutoff}, the free energies of the ice polymorphs with |
382 |
|
|
water models lacking a long-range correction show significant cutoff |
383 |
|
|
dependence. In general, there is a narrowing of the free energy |
384 |
|
|
differences while moving to greater cutoff radii. As the free |
385 |
|
|
energies for the polymorphs converge, the stability advantage that |
386 |
|
|
Ice-{\it i} exhibits is reduced. Adjacent to each of these plots are |
387 |
|
|
results for systems with applied or estimated long-range corrections. |
388 |
|
|
SSD/RF was parametrized for use with a reaction field, and the benefit |
389 |
|
|
provided by this computationally inexpensive correction is apparent. |
390 |
|
|
The free energies are largely independent of the size of the reaction |
391 |
|
|
field cavity in this model, so small cutoff radii mimic bulk |
392 |
|
|
calculations quite well under SSD/RF. |
393 |
|
|
|
394 |
|
|
Although TIP3P was parametrized for use without the Ewald summation, |
395 |
|
|
we have estimated the effect of this method for computing long-range |
396 |
|
|
electrostatics for both TIP3P and SPC/E. This was accomplished by |
397 |
|
|
calculating the potential energy of identical crystals both with and |
398 |
|
|
without particle mesh Ewald (PME). Similar behavior to that observed |
399 |
|
|
with reaction field is seen for both of these models. The free |
400 |
|
|
energies show reduced dependence on cutoff radius and span a narrower |
401 |
|
|
range for the various polymorphs. Like the dipolar water models, |
402 |
|
|
TIP3P displays a relatively constant preference for the Ice-{\it i} |
403 |
|
|
polymorph. Crystal preference is much more difficult to determine for |
404 |
|
|
SPC/E. Without a long-range correction, each of the polymorphs |
405 |
|
|
studied assumes the role of the preferred polymorph under different |
406 |
|
|
cutoff radii. The inclusion of the Ewald correction flattens and |
407 |
|
|
narrows the gap in free energies such that the polymorphs are |
408 |
|
|
isoenergetic within statistical uncertainty. This suggests that other |
409 |
|
|
conditions, such as the density in fixed-volume simulations, can |
410 |
|
|
influence the polymorph expressed upon crystallization. |
411 |
|
|
|
412 |
|
|
\section{Expanded Results Using Damped Shifted Force Electrostatics} |
413 |
|
|
|
414 |
|
|
In chapter \ref{chap:electrostatics}, we discussed in detail a |
415 |
|
|
pairwise method for handling electrostatics (shifted force, {\sc sf}) |
416 |
|
|
that can be used as a simple and efficient replacement for the Ewald |
417 |
|
|
summation. Answering the question of the free energies of these ice |
418 |
|
|
polymorphs with varying water models would be an interesting |
419 |
|
|
application of this technique. To this end, we set up thermodynamic |
420 |
|
|
integrations of all of the previously discussed ice polymorphs using |
421 |
|
|
the {\sc sf} technique with a cutoff radius of 12~\AA\ and an $\alpha$ |
422 |
|
|
of 0.2125~\AA . These calculations were performed on TIP5P-E and |
423 |
|
|
TIP4P-Ew (variants of the root models optimized for the Ewald |
424 |
|
|
summation) as well as SPC/E, SSD/RF, and TRED (see section |
425 |
|
|
\ref{sec:tredWater}). |
426 |
|
|
|
427 |
|
|
\begin{table} |
428 |
|
|
\centering |
429 |
|
|
\caption{HELMHOLTZ FREE ENERGIES OF ICE POLYMORPHS USING THE DAMPED |
430 |
|
|
SHIFTED FORCE CORRECTION} |
431 |
|
|
\begin{tabular}{ lccccc } |
432 |
|
|
\toprule |
433 |
|
|
\toprule |
434 |
|
|
Model & I$_\textrm{h}$ & I$_\textrm{c}$ & B & Ice-$i$ & Ice-$i^\prime$ \\ |
435 |
|
|
\cmidrule(lr){2-6} |
436 |
|
|
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ |
437 |
|
|
\midrule |
438 |
chrisfen |
3004 |
TIP5P-E & -11.98(4) & -11.96(4) & & - & -11.95(3) \\ |
439 |
|
|
TIP4P-Ew & -13.11(3) & -13.09(3) & -12.97(3) & - & -12.98(3) \\ |
440 |
|
|
SPC/E & -12.99(3) & -13.00(3) & -13.03(3) & - & -12.99(3) \\ |
441 |
chrisfen |
3001 |
SSD/RF & -11.83(3) & -11.66(4) & -12.32(3) & -12.39(3) & - \\ |
442 |
|
|
TRED & -12.61(3) & -12.43(3) & -12.89(3) & -13.12(3) & - \\ |
443 |
chrisfen |
2987 |
\end{tabular} |
444 |
|
|
\label{tab:dampedFreeEnergy} |
445 |
|
|
\end{table} |
446 |
chrisfen |
3001 |
The results of these calculations in table \ref{tab:dampedFreeEnergy} |
447 |
|
|
show similar behavior to the Ewald results in figure |
448 |
|
|
\ref{fig:incCutoff}, at least for SSD/RF and SPC/E which are present |
449 |
|
|
in both. The ice polymorph Helmholtz free energies for SSD/RF order in |
450 |
|
|
the same fashion; however Ice-$i$ and ice B are quite a bit closer in |
451 |
|
|
free energy (nearly isoenergetic). The free energy differences between |
452 |
|
|
ice polymorphs for TRED water parallel SSD/RF, with the exception that |
453 |
chrisfen |
3004 |
ice B is destabilized such that it is not very close to Ice-$i$. The |
454 |
|
|
SPC/E results really show the near isoenergetic behavior when using |
455 |
|
|
the electrostatics correction. Ice B has the lowest Helmholtz free |
456 |
|
|
energy; however, all the polymorph results overlap within error. |
457 |
chrisfen |
2987 |
|
458 |
chrisfen |
3004 |
The most interesting results from these calculations come from the |
459 |
|
|
more expensive TIP4P-Ew and TIP5P-E results. Both of these models were |
460 |
|
|
optimized for use with an electrostatic correction and are |
461 |
|
|
geometrically arranged to mimic water following two different |
462 |
|
|
ideas. In TIP5P-E, the primary location for the negative charge in the |
463 |
|
|
molecule is assigned to the lone-pairs of the oxygen, while TIP4P-Ew |
464 |
|
|
places the negative charge near the center-of-mass along the H-O-H |
465 |
|
|
bisector. There is some debate as to which is the proper choice for |
466 |
|
|
the negative charge location, and this has in part led to a six-site |
467 |
|
|
water model that balances both of these options.\cite{Vega05,Nada03} |
468 |
|
|
The limited results in table \ref{tab:dampedFreeEnergy} support the |
469 |
|
|
results of Vega {\it et al.}, which indicate the TIP4P charge location |
470 |
|
|
geometry is more physically valid.\cite{Vega05} With the TIP4P-Ew |
471 |
|
|
water model, the experimentally observed polymorph (ice |
472 |
|
|
I$_\textrm{h}$) is the preferred form with ice I$_\textrm{c}$ slightly |
473 |
|
|
higher in energy, though overlapping within error, and the less |
474 |
|
|
realistic ice B and Ice-$i^\prime$ are destabilized relative to these |
475 |
|
|
polymorphs. TIP5P-E shows similar behavior to SPC/E, where there is no |
476 |
|
|
real free energy distinction between the various polymorphs and lend |
477 |
|
|
credence to other results indicating the preferred form of TIP5P at |
478 |
|
|
1~atm is a structure similar to ice B.\cite{Yamada02,Vega05,Abascal05} |
479 |
|
|
These results indicate that TIP4P-Ew is a better mimic of real water |
480 |
|
|
than these other models when studying crystallization and solid forms |
481 |
|
|
of water. |
482 |
|
|
|
483 |
chrisfen |
2987 |
\section{Conclusions} |
484 |
|
|
|
485 |
|
|
In this work, thermodynamic integration was used to determine the |
486 |
|
|
absolute free energies of several ice polymorphs. The new polymorph, |
487 |
chrisfen |
3016 |
Ice-$i$ was observed to be the stable crystalline state for {\it |
488 |
chrisfen |
2987 |
all} the water models when using a 9.0~\AA\ cutoff. However, the free |
489 |
|
|
energy partially depends on simulation conditions (particularly on the |
490 |
chrisfen |
3016 |
choice of long range correction method). Regardless, Ice-$i$ was |
491 |
chrisfen |
2987 |
still observed to be a stable polymorph for all of the studied water |
492 |
|
|
models. |
493 |
|
|
|
494 |
|
|
So what is the preferred solid polymorph for simulated water? As |
495 |
|
|
indicated above, the answer appears to be dependent both on the |
496 |
|
|
conditions and the model used. In the case of short cutoffs without a |
497 |
chrisfen |
3016 |
long-range interaction correction, Ice-$i$ and Ice-$i^\prime$ have |
498 |
chrisfen |
2987 |
the lowest free energy of the studied polymorphs with all the models. |
499 |
|
|
Ideally, crystallization of each model under constant pressure |
500 |
|
|
conditions, as was done with SSD/E, would aid in the identification of |
501 |
|
|
their respective preferred structures. This work, however, helps |
502 |
|
|
illustrate how studies involving one specific model can lead to |
503 |
|
|
insight about important behavior of others. |
504 |
|
|
|
505 |
|
|
We also note that none of the water models used in this study are |
506 |
chrisfen |
3016 |
polarizable or flexible models. It is entirely possible that the |
507 |
|
|
polarizability of real water makes Ice-$i$ substantially less stable |
508 |
|
|
than ice I$_\textrm{h}$. The dipole moment of the water molecules |
509 |
|
|
increases as the system becomes more condensed, and the increasing |
510 |
|
|
dipole moment should destabilize the tetramer structures in |
511 |
|
|
Ice-$i$. Right now, using TIP4P-Ew with an electrostatic correction |
512 |
|
|
gives the proper thermodynamically preferred state, and we recommend |
513 |
|
|
this arrangement for study of crystallization processes if the |
514 |
|
|
computational cost increase that comes with including polarizability |
515 |
|
|
is an issue. |
516 |
chrisfen |
2987 |
|
517 |
chrisfen |
3016 |
Finally, due to the stability of Ice-$i$ in the investigated |
518 |
chrisfen |
2987 |
simulation conditions, the question arises as to possible experimental |
519 |
|
|
observation of this polymorph. The rather extensive past and current |
520 |
|
|
experimental investigation of water in the low pressure regime makes |
521 |
|
|
us hesitant to ascribe any relevance to this work outside of the |
522 |
|
|
simulation community. It is for this reason that we chose a name for |
523 |
|
|
this polymorph which involves an imaginary quantity. That said, there |
524 |
|
|
are certain experimental conditions that would provide the most ideal |
525 |
|
|
situation for possible observation. These include the negative |
526 |
|
|
pressure or stretched solid regime, small clusters in vacuum |
527 |
|
|
deposition environments, and in clathrate structures involving small |
528 |
|
|
non-polar molecules. For the purpose of comparison with experimental |
529 |
|
|
results, we have calculated the oxygen-oxygen pair correlation |
530 |
|
|
function, $g_\textrm{OO}(r)$, and the structure factor, $S(\vec{q})$ |
531 |
|
|
for the two Ice-{\it i} variants (along with example ice |
532 |
|
|
I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77~K, and they are shown in |
533 |
|
|
figures \ref{fig:gofr} and \ref{fig:sofq} respectively. It is |
534 |
|
|
interesting to note that the structure factors for Ice-$i^\prime$ and |
535 |
|
|
Ice-I$_c$ are quite similar. The primary differences are small peaks |
536 |
|
|
at 1.125, 2.29, and 2.53~\AA$^{-1}$, so particular attention to these |
537 |
|
|
regions would be needed to identify the new $i^\prime$ variant from |
538 |
|
|
the I$_\textrm{c}$ polymorph. |
539 |
|
|
|
540 |
|
|
|
541 |
|
|
\begin{figure} |
542 |
|
|
\includegraphics[width=\linewidth]{./figures/iceGofr.pdf} |
543 |
|
|
\caption{Radial distribution functions of Ice-{\it i} and ice |
544 |
|
|
I$_\textrm{c}$ calculated from from simulations of the SSD/RF water |
545 |
|
|
model at 77~K.} |
546 |
|
|
\label{fig:gofr} |
547 |
|
|
\end{figure} |
548 |
|
|
|
549 |
|
|
\begin{figure} |
550 |
|
|
\includegraphics[width=\linewidth]{./figures/sofq.pdf} |
551 |
|
|
\caption{Predicted structure factors for Ice-{\it i} and ice |
552 |
|
|
I$_\textrm{c}$ at 77~K. The raw structure factors have been |
553 |
|
|
convoluted with a gaussian instrument function (0.075~\AA$^{-1}$ |
554 |
|
|
width) to compensate for the truncation effects in our finite size |
555 |
|
|
simulations. The labeled peaks compared favorably with ``spurious'' |
556 |
|
|
peaks observed in experimental studies of amorphous solid |
557 |
|
|
water.\cite{Bizid87}} |
558 |
|
|
\label{fig:sofq} |
559 |
|
|
\end{figure} |
560 |
|
|
|