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\begin{document} |
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\title{A Free Energy Study of Low Temperature and Anomolous Ice} |
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\author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote} |
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\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}} |
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\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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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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\end{abstract} |
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\maketitle |
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\newpage |
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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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\section{Methods} |
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Canonical ensemble (NVT) molecular dynamics calculations were |
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performed using the OOPSE (Object-Oriented Parallel Simulation Engine) |
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molecular mechanics package. All molecules were treated as rigid |
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bodies, with orientational motion propogated using the symplectic DLM |
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integration method. Details about the implementation of these |
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techniques can be found in a recent publication.\cite{Meineke05} |
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Thermodynamic integration was utilized to calculate the free energy of |
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several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, |
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SSD/RF, and SSD/E water models. Liquid state free energies at 300 and |
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400 K for all of these water models were also determined using this |
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same technique, in order to determine melting points and generate |
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phase diagrams. All simulations were carried out at densities |
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resulting in a pressure of approximately 1 atm at their respective |
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temperatures. |
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A single thermodynamic integration involves a sequence of simulations |
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over which the system of interest is converted into a reference system |
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for which the free energy is known. This transformation path is then |
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integrated in order to determine the free energy difference between |
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the two states: |
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\begin{equation} |
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\begin{center} |
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\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
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)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
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\end{center} |
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\end{equation} |
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where $V$ is the interaction potential and $\lambda$ is the |
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transformation parameter. Simulations are distributed unevenly along |
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this path in order to sufficiently sample the regions of greatest |
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change in the potential. Typical integrations in this study consisted |
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of $\sim$25 simulations ranging from 300 ps (for the unaltered system) |
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to 75 ps (near the reference state) in length. |
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For the thermodynamic integration of molecular crystals, the Einstein |
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Crystal is chosen as the reference state that the system is converted |
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to over the course of the simulation. In an Einstein Crystal, the |
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molecules are harmonically restrained at their ideal lattice locations |
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and orientations. The partition function for a molecular crystal |
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restrained in this fashion has been evaluated, and the Helmholtz Free |
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Energy ({\it A}) is given by |
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\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 $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
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\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
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$K_\mathrm{\omega}$ are the spring constants restraining translational |
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motion and deflection of and rotation around the principle axis of the |
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molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
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minimum potential energy of the ideal crystal. In the case of |
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molecular liquids, the ideal vapor is chosen as the target reference |
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state. |
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\begin{figure} |
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\includegraphics[scale=1.0]{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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Charge, dipole, and Lennard-Jones interactions were modified by a |
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cubic switching between 100\% and 85\% of the cutoff value (9 \AA ). By |
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applying this function, these interactions are smoothly truncated, |
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thereby avoiding poor energy conserving dynamics resulting from |
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harsher truncation schemes. The effect of a long-range correction was |
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also investigated on select model systems in a variety of manners. For |
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the SSD/RF model, a reaction field with a fixed dielectric constant of |
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80 was applied in all simulations.\cite{Onsager36} For a series of the |
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least computationally expensive models (SSD/E, SSD/RF, and TIP3P), |
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simulations were performed with longer cutoffs of 12 and 15 \AA\ to |
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compare with the 9 \AA\ cutoff results. Finally, results from the use |
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of an Ewald summation were estimated for TIP3P and SPC/E by performing |
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calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
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mechanics software package. TINKER was chosen because it can also |
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propogate the motion of rigid-bodies, and provides the most direct |
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comparison to the results from OOPSE. 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 in the free energy landscape. |
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\section{Results and discussion} |
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The free energy of proton ordered Ice-{\it i} was calculated and |
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compared with the free energies of proton ordered variants of the |
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experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
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as well as the higher density ice B, observed by B\`{a}ez and Clancy |
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and thought to be the minimum free energy structure for the SPC/E |
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model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
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Ice XI, the experimentally observed proton ordered variant of ice |
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$I_h$, was investigated initially, but it was found not to be as |
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stable as antiferroelectric variants of proton ordered or even proton |
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disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of |
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ice $I_h$ used here is a simple antiferroelectric version that has an |
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8 molecule unit cell. The crystals contained 648 or 1728 molecules for |
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ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for ice |
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$I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
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were necessary for simulations involving larger cutoff values. |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\renewcommand{\thefootnote}{\thempfootnote} |
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\begin{center} |
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\caption{Calculated free energies for several ice polymorphs with a |
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variety of common water models. All calculations used a cutoff radius |
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of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
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kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF.} |
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\begin{tabular}{ l c c c c } |
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\hline \\[-7mm] |
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\ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\ |
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\hline \\[-3mm] |
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\ \quad \ TIP3P & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\ |
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\ \quad \ TIP4P & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\ |
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\ \quad \ TIP5P & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\ |
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\ \quad \ SPC/E & \ \quad \ -12.67 & \ \quad \ -12.96 & \ \quad \ -13.25 & \quad -13.55\\ |
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\ \quad \ SSD/E & \ \quad \ -11.27 & \ \quad \ -11.19 & \ \quad \ -12.09 & \quad -12.54\\ |
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\ \quad \ SSD/RF & \ \quad \ -11.51 & \ \quad \ NA* & \ \quad \ -12.08 & \quad -12.29\\ |
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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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The free energy values computed for the studied polymorphs indicate |
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that Ice-{\it i} is the most stable state for all of the common water |
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models studied. With the free energy at these state points, the |
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temperature and pressure dependence of the free energy was used to |
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project to other state points and build phase diagrams. Figures |
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\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
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from the free energy results. All other models have similar structure, |
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only the crossing points between these phases exist at different |
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temperatures and pressures. It is interesting to note that ice $I$ |
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does not exist in either cubic or hexagonal form in any of the phase |
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diagrams for any of the models. For purposes of this study, ice B is |
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representative of the dense ice polymorphs. A recent study by Sanz |
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{\it et al.} goes into detail on the phase diagrams for SPC/E and |
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TIP4P in the high pressure regime.\cite{Sanz04} |
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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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\begin{figure} |
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\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
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\caption{Phase diagram for the SSD/RF water model in the low pressure |
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regime. Calculations producing these results were done under an |
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applied reaction field. It is interesting to note that this |
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computationally efficient model (over 3 times more efficient than |
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TIP3P) exhibits phase behavior similar to the less computationally |
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conservative charge based models.} |
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\label{ssdrfphasedia} |
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\end{figure} |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\renewcommand{\thefootnote}{\thempfootnote} |
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\begin{center} |
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\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
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temperatures of several common water models compared with experiment.} |
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\begin{tabular}{ l c c c c c c c } |
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\hline \\[-7mm] |
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\ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\ |
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\hline \\[-3mm] |
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\ \ $T_m$ (K) & \ \ 269 & \ \ 265 & \ \ 271 & 297 & \ \ - & \ \ 278 & \ \ 273\\ |
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\ \ $T_b$ (K) & \ \ 357 & \ \ 354 & \ \ 337 & 396 & \ \ - & \ \ 349 & \ \ 373\\ |
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\ \ $T_s$ (K) & \ \ - & \ \ - & \ \ - & - & \ \ 355 & \ \ - & \ \ -\\ |
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\end{tabular} |
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\label{meltandboil} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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Table \ref{meltandboil} lists the melting and boiling temperatures |
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calculated from this work. Surprisingly, most of these models have |
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melting points that compare quite favorably with experiment. The |
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unfortunate aspect of this result is that this phase change occurs |
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between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
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liquid state. These results are actually not contrary to previous |
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studies in the literature. Earlier free energy studies of ice $I$ |
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using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
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being attributed to choice of interaction truncation and different |
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ordered and disordered molecular arrangements). If the presence of ice |
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B and 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 at 265 K, significantly higher in temperature than the |
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previous studies. Also of interest in these results is that SSD/E does |
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not exhibit a melting point at 1 atm, but it shows a sublimation point |
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at 355 K. This is due to the significant stability of Ice-{\it i} over |
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all other polymorphs for this particular model under these |
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conditions. While troubling, this behavior turned out to be |
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advantagious in that it facilitated the spontaneous crystallization of |
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Ice-{\it i}. These observations provide a warning that simulations of |
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SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
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risk of spontaneous crystallization. However, this risk changes when |
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applying a longer cutoff. |
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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 (A) SSD/E, (B) |
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TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 |
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\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 |
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\AA\. These crystals are unstable at 200 K and rapidly convert into a |
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liquid. The connecting lines are qualitative visual aid.} |
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\label{incCutoff} |
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\end{figure} |
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Increasing the cutoff radius in simulations of the more |
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computationally efficient water models was done in order to evaluate |
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the trend in free energy values when moving to systems that do not |
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involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
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free energy of all the ice polymorphs show a substantial dependence on |
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cutoff radius. In general, there is a narrowing of the free energy |
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differences while moving to greater cutoff radius. This trend is much |
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more subtle in the case of SSD/RF, indicating that the free energies |
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calculated with a reaction field present provide a more accurate |
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picture of the free energy landscape in the absence of potential |
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truncation. |
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To further study the changes resulting to the inclusion of a |
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long-range interaction correction, the effect of an Ewald summation |
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was estimated by applying the potential energy difference do to its |
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inclusion in systems in the presence and absence of the |
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correction. This was accomplished by calculation of the potential |
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energy of identical crystals with and without PME using TINKER. The |
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free energies for the investigated polymorphs using the TIP3P and |
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SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
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are not fully supported in TINKER, so the results for these models |
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could not be estimated. The same trend pointed out through increase of |
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cutoff radius is observed in these results. Ice-{\it i} is the |
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preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
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water models; however, there is a narrowing of the free energy |
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differences between the various solid forms. In the case of SPC/E this |
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narrowing is significant enough that it becomes less clear cut that |
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Ice-{\it i} is the most stable polymorph, and is possibly metastable |
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with respect to ice B and possibly ice $I_c$. However, these results |
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do not significantly alter the finding that the Ice-{\it i} polymorph |
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is a stable crystal structure that should be considered when studying |
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the phase behavior of water models. |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\renewcommand{\thefootnote}{\thempfootnote} |
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\begin{center} |
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\caption{The free energy of the studied ice polymorphs after applying |
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the energy difference attributed to the inclusion of the PME |
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long-range interaction correction. Units are kcal/mol.} |
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\begin{tabular}{ l c c c c } |
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\hline \\[-7mm] |
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\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
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\hline \\[-3mm] |
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\ \ TIP3P & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\ |
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\ \ SPC/E & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\ |
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\end{tabular} |
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\label{pmeShift} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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\section{Conclusions} |
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The free energy for proton ordered variants of hexagonal and cubic ice |
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$I$, ice B, and recently discovered Ice-{\it i} where calculated under |
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standard conditions for several common water models via thermodynamic |
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integration. All the water models studied show Ice-{\it i} to be the |
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minimum free energy crystal structure in the with a 9 \AA\ switching |
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function cutoff. Calculated melting and boiling points show |
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surprisingly good agreement with the experimental values; however, the |
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solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
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interaction truncation was investigated through variation of the |
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cutoff radius, use of a reaction field parameterized model, and |
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estimation of the results in the presence of the Ewald summation |
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correction. Interaction truncation has a significant effect on the |
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computed free energy values, but Ice-{\it i} is still observed to be a |
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relavent ice polymorph in simulation studies. |
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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). |
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\newpage |
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\bibliographystyle{jcp} |
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\bibliography{iceiPaper} |
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\end{document} |
