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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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\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 using the TIP3P, TIP4P, TIP5P, SPC/E, and SSD/E |
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water models. Liquid state free energies at 300 and 400 K for all of |
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these water models were also determined using this same technique, in |
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order to determine melting points and generate phase diagrams. |
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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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\section{Results and discussion} |
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\section{Conclusions} |
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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 Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
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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} |
