| 50 |
|
|
| 51 |
|
\section{Methods} |
| 52 |
|
|
| 53 |
+ |
Canonical ensemble (NVT) molecular dynamics calculations were |
| 54 |
+ |
performed using the OOPSE (Object-Oriented Parallel Simulation Engine) |
| 55 |
+ |
molecular mechanics package. All molecules were treated as rigid |
| 56 |
+ |
bodies, with orientational motion propogated using the symplectic DLM |
| 57 |
+ |
integration method. Details about the implementation of these |
| 58 |
+ |
techniques can be found in a recent publication.\cite{Meineke05} |
| 59 |
+ |
|
| 60 |
+ |
Thermodynamic integration was utilized to calculate the free energy of |
| 61 |
+ |
several ice crystals using the TIP3P, TIP4P, TIP5P, SPC/E, and SSD/E |
| 62 |
+ |
water models. Liquid state free energies at 300 and 400 K for all of |
| 63 |
+ |
these water models were also determined using this same technique, in |
| 64 |
+ |
order to determine melting points and generate phase diagrams. |
| 65 |
+ |
|
| 66 |
+ |
For the thermodynamic integration of molecular crystals, the Einstein |
| 67 |
+ |
Crystal is chosen as the reference state that the system is converted |
| 68 |
+ |
to over the course of the simulation. In an Einstein Crystal, the |
| 69 |
+ |
molecules are harmonically restrained at their ideal lattice locations |
| 70 |
+ |
and orientations. The partition function for a molecular crystal |
| 71 |
+ |
restrained in this fashion has been evaluated, and the Helmholtz Free |
| 72 |
+ |
Energy ({\it A}) is given by |
| 73 |
+ |
\begin{eqnarray} |
| 74 |
+ |
A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
| 75 |
+ |
[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
| 76 |
+ |
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right |
| 77 |
+ |
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right |
| 78 |
+ |
)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi |
| 79 |
+ |
K_\omega K_\theta)^{\frac{1}{2}}}\exp\left |
| 80 |
+ |
(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right |
| 81 |
+ |
)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
| 82 |
+ |
\label{ecFreeEnergy} |
| 83 |
+ |
\end{eqnarray} |
| 84 |
+ |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
| 85 |
+ |
\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
| 86 |
+ |
$K_\mathrm{\omega}$ are the spring constants restraining translational |
| 87 |
+ |
motion and deflection of and rotation around the principle axis of the |
| 88 |
+ |
molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
| 89 |
+ |
minimum potential energy of the ideal crystal. In the case of |
| 90 |
+ |
molecular liquids, the ideal vapor is chosen as the target reference |
| 91 |
+ |
state. |
| 92 |
+ |
|
| 93 |
+ |
|
| 94 |
+ |
|
| 95 |
+ |
|
| 96 |
|
\section{Results and discussion} |
| 97 |
|
|
| 98 |
|
\section{Conclusions} |
