| 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 at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, |
| 62 |
+ |
SSD/RF, and SSD/E water models. Liquid state free energies at 300 and |
| 63 |
+ |
400 K for all of these water models were also determined using this |
| 64 |
+ |
same technique, in order to determine melting points and generate |
| 65 |
+ |
phase diagrams. All simulations were carried out at densities |
| 66 |
+ |
resulting in a pressure of approximately 1 atm at their respective |
| 67 |
+ |
temperatures. |
| 68 |
+ |
|
| 69 |
+ |
For the thermodynamic integration of molecular crystals, the Einstein |
| 70 |
+ |
Crystal is chosen as the reference state that the system is converted |
| 71 |
+ |
to over the course of the simulation. In an Einstein Crystal, the |
| 72 |
+ |
molecules are harmonically restrained at their ideal lattice locations |
| 73 |
+ |
and orientations. The partition function for a molecular crystal |
| 74 |
+ |
restrained in this fashion has been evaluated, and the Helmholtz Free |
| 75 |
+ |
Energy ({\it A}) is given by |
| 76 |
+ |
\begin{eqnarray} |
| 77 |
+ |
A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
| 78 |
+ |
[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
| 79 |
+ |
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right |
| 80 |
+ |
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right |
| 81 |
+ |
)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi |
| 82 |
+ |
K_\omega K_\theta)^{\frac{1}{2}}}\exp\left |
| 83 |
+ |
(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right |
| 84 |
+ |
)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
| 85 |
+ |
\label{ecFreeEnergy} |
| 86 |
+ |
\end{eqnarray} |
| 87 |
+ |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
| 88 |
+ |
\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
| 89 |
+ |
$K_\mathrm{\omega}$ are the spring constants restraining translational |
| 90 |
+ |
motion and deflection of and rotation around the principle axis of the |
| 91 |
+ |
molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
| 92 |
+ |
minimum potential energy of the ideal crystal. In the case of |
| 93 |
+ |
molecular liquids, the ideal vapor is chosen as the target reference |
| 94 |
+ |
state. |
| 95 |
+ |
\begin{figure} |
| 96 |
+ |
\includegraphics[scale=1.0]{rotSpring.eps} |
| 97 |
+ |
\caption{Possible orientational motions for a restrained molecule. |
| 98 |
+ |
$\theta$ angles correspond to displacement from the body-frame {\it |
| 99 |
+ |
z}-axis, while $\omega$ angles correspond to rotation about the |
| 100 |
+ |
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
| 101 |
+ |
constants for the harmonic springs restraining motion in the $\theta$ |
| 102 |
+ |
and $\omega$ directions.} |
| 103 |
+ |
\label{waterSpring} |
| 104 |
+ |
\end{figure} |
| 105 |
+ |
|
| 106 |
+ |
Charge, dipole, and Lennard-Jones interactions were modified by a |
| 107 |
+ |
cubic switching between 100\% and 85\% of the cutoff value (9 \AA ). By |
| 108 |
+ |
applying this function, these interactions are smoothly truncated, |
| 109 |
+ |
thereby avoiding poor energy conserving dynamics resulting from |
| 110 |
+ |
harsher truncation schemes. The effect of a long-range correction was |
| 111 |
+ |
also investigated on select model systems in a variety of manners. For |
| 112 |
+ |
the SSD/RF model, a reaction field with a fixed dielectric constant of |
| 113 |
+ |
80 was applied in all simulations.\cite{Onsager36} For a series of the |
| 114 |
+ |
least computationally expensive models (SSD/E, SSD/RF, and TIP3P), |
| 115 |
+ |
simulations were performed with longer cutoffs of 12 and 15 \AA\ to |
| 116 |
+ |
compare with the 9 \AA\ cutoff results. Finally, results from the use |
| 117 |
+ |
of an Ewald summation were estimated for TIP3P and SPC/E by performing |
| 118 |
+ |
calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
| 119 |
+ |
mechanics software package. TINKER was chosen because it can also |
| 120 |
+ |
propogate the motion of rigid-bodies, and provides the most direct |
| 121 |
+ |
comparison to the results from OOPSE. The calculated energy difference |
| 122 |
+ |
in the presence and absence of PME was applied to the previous results |
| 123 |
+ |
in order to predict changes in the free energy landscape. |
| 124 |
+ |
|
| 125 |
|
\section{Results and discussion} |
| 126 |
|
|
| 127 |
+ |
The free energy of proton ordered Ice-{\it i} was calculated and |
| 128 |
+ |
compared with the free energies of proton ordered variants of the |
| 129 |
+ |
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
| 130 |
+ |
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
| 131 |
+ |
and thought to be the minimum free energy structure for the SPC/E |
| 132 |
+ |
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
| 133 |
+ |
Ice XI, the experimentally observed proton ordered variant of ice |
| 134 |
+ |
$I_h$, was investigated initially, but it was found not to be as |
| 135 |
+ |
stable as antiferroelectric variants of proton ordered or even proton |
| 136 |
+ |
disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of |
| 137 |
+ |
ice $I_h$ used here is a simple antiferroelectric version that has an |
| 138 |
+ |
8 molecule unit cell. The crystals contained 648 or 1728 molecules for |
| 139 |
+ |
ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for ice |
| 140 |
+ |
$I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
| 141 |
+ |
were necessary for simulations involving larger cutoff values. |
| 142 |
+ |
|
| 143 |
+ |
\begin{table*} |
| 144 |
+ |
\begin{minipage}{\linewidth} |
| 145 |
+ |
\renewcommand{\thefootnote}{\thempfootnote} |
| 146 |
+ |
\begin{center} |
| 147 |
+ |
\caption{Calculated free energies for several ice polymorphs with a |
| 148 |
+ |
variety of common water models. All calculations used a cutoff radius |
| 149 |
+ |
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
| 150 |
+ |
kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF.} |
| 151 |
+ |
\begin{tabular}{ l c c c c } |
| 152 |
+ |
\hline \\[-7mm] |
| 153 |
+ |
\ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\ |
| 154 |
+ |
\hline \\[-3mm] |
| 155 |
+ |
\ \quad \ TIP3P & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\ |
| 156 |
+ |
\ \quad \ TIP4P & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\ |
| 157 |
+ |
\ \quad \ TIP5P & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\ |
| 158 |
+ |
\ \quad \ SPC/E & \ \quad \ -12.67 & \ \quad \ -12.96 & \ \quad \ -13.25 & \quad -13.55\\ |
| 159 |
+ |
\ \quad \ SSD/E & \ \quad \ -11.27 & \ \quad \ -11.19 & \ \quad \ -12.09 & \quad -12.54\\ |
| 160 |
+ |
\ \quad \ SSD/RF & \ \quad \ -11.51 & \ \quad \ NA* & \ \quad \ -12.08 & \quad -12.29\\ |
| 161 |
+ |
\end{tabular} |
| 162 |
+ |
\label{freeEnergy} |
| 163 |
+ |
\end{center} |
| 164 |
+ |
\end{minipage} |
| 165 |
+ |
\end{table*} |
| 166 |
+ |
|
| 167 |
+ |
The free energy values computed for the studied polymorphs indicate |
| 168 |
+ |
that Ice-{\it i} is the most stable state for all of the common water |
| 169 |
+ |
models studied. With the free energy at these state points, the |
| 170 |
+ |
temperature and pressure dependence of the free energy was used to |
| 171 |
+ |
project to other state points and build phase diagrams. Figures |
| 172 |
+ |
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
| 173 |
+ |
from the free energy results. All other models have similar structure, |
| 174 |
+ |
only the crossing points between these phases exist at different |
| 175 |
+ |
temperatures and pressures. It is interesting to note that ice $I$ |
| 176 |
+ |
does not exist in either cubic or hexagonal form in any of the phase |
| 177 |
+ |
diagrams for any of the models. For purposes of this study, ice B is |
| 178 |
+ |
representative of the dense ice polymorphs. A recent study by Sanz |
| 179 |
+ |
{\it et al.} goes into detail on the phase diagrams for SPC/E and |
| 180 |
+ |
TIP4P in the high pressure regime.\cite{Sanz04} |
| 181 |
+ |
\begin{figure} |
| 182 |
+ |
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
| 183 |
+ |
\caption{Phase diagram for the TIP3P water model in the low pressure |
| 184 |
+ |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 185 |
+ |
the experimental values; however, the solid phases shown are not the |
| 186 |
+ |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
| 187 |
+ |
higher in energy and don't appear in the phase diagram.} |
| 188 |
+ |
\label{tp3phasedia} |
| 189 |
+ |
\end{figure} |
| 190 |
+ |
\begin{figure} |
| 191 |
+ |
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
| 192 |
+ |
\caption{Phase diagram for the SSD/RF water model in the low pressure |
| 193 |
+ |
regime. Calculations producing these results were done under an |
| 194 |
+ |
applied reaction field. It is interesting to note that this |
| 195 |
+ |
computationally efficient model (over 3 times more efficient than |
| 196 |
+ |
TIP3P) exhibits phase behavior similar to the less computationally |
| 197 |
+ |
conservative charge based models.} |
| 198 |
+ |
\label{ssdrfphasedia} |
| 199 |
+ |
\end{figure} |
| 200 |
+ |
|
| 201 |
+ |
\begin{table*} |
| 202 |
+ |
\begin{minipage}{\linewidth} |
| 203 |
+ |
\renewcommand{\thefootnote}{\thempfootnote} |
| 204 |
+ |
\begin{center} |
| 205 |
+ |
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
| 206 |
+ |
temperatures of several common water models compared with experiment.} |
| 207 |
+ |
\begin{tabular}{ l c c c c c c c } |
| 208 |
+ |
\hline \\[-7mm] |
| 209 |
+ |
\ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\ |
| 210 |
+ |
\hline \\[-3mm] |
| 211 |
+ |
\ \ $T_m$ (K) & \ \ 269 & \ \ 265 & \ \ 271 & 297 & \ \ - & \ \ 278 & \ \ 273\\ |
| 212 |
+ |
\ \ $T_b$ (K) & \ \ 357 & \ \ 354 & \ \ 337 & 396 & \ \ - & \ \ 349 & \ \ 373\\ |
| 213 |
+ |
\ \ $T_s$ (K) & \ \ - & \ \ - & \ \ - & - & \ \ 355 & \ \ - & \ \ -\\ |
| 214 |
+ |
\end{tabular} |
| 215 |
+ |
\label{meltandboil} |
| 216 |
+ |
\end{center} |
| 217 |
+ |
\end{minipage} |
| 218 |
+ |
\end{table*} |
| 219 |
+ |
|
| 220 |
+ |
Table \ref{meltandboil} lists the melting and boiling temperatures |
| 221 |
+ |
calculated from this work. Surprisingly, most of these models have |
| 222 |
+ |
melting points that compare quite favorably with experiment. The |
| 223 |
+ |
unfortunate aspect of this result is that this phase change occurs |
| 224 |
+ |
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
| 225 |
+ |
liquid state. These results are actually not contrary to previous |
| 226 |
+ |
studies in the literature. Earlier free energy studies of ice $I$ |
| 227 |
+ |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
| 228 |
+ |
being attributed to choice of interaction truncation and different |
| 229 |
+ |
ordered and disordered molecular arrangements). If the presence of ice |
| 230 |
+ |
B and Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
| 231 |
+ |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 232 |
+ |
calculated at 265 K, significantly higher in temperature than the |
| 233 |
+ |
previous studies. Also of interest in these results is that SSD/E does |
| 234 |
+ |
not exhibit a melting point at 1 atm, but it shows a sublimation point |
| 235 |
+ |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
| 236 |
+ |
all other polymorphs for this particular model under these |
| 237 |
+ |
conditions. While troubling, this behavior turned out to be |
| 238 |
+ |
advantagious in that it facilitated the spontaneous crystallization of |
| 239 |
+ |
Ice-{\it i}. These observations provide a warning that simulations of |
| 240 |
+ |
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
| 241 |
+ |
risk of spontaneous crystallization. However, this risk changes when |
| 242 |
+ |
applying a longer cutoff. |
| 243 |
+ |
|
| 244 |
+ |
|
| 245 |
+ |
|
| 246 |
|
\section{Conclusions} |
| 247 |
|
|
| 248 |
|
\section{Acknowledgments} |
