| 33 |
|
%\doublespacing |
| 34 |
|
|
| 35 |
|
\begin{abstract} |
| 36 |
< |
The free energies of several ice polymorphs in the low pressure regime |
| 37 |
< |
were calculated using thermodynamic integration. These integrations |
| 38 |
< |
were done for most of the common water models. Ice-{\it i}, a |
| 39 |
< |
structure we recently observed to be stable in one of the single-point |
| 40 |
< |
water models, was determined to be the stable crystalline state (at 1 |
| 41 |
< |
atm) for {\it all} the water models investigated. Phase diagrams were |
| 36 |
> |
The absolute free energies of several ice polymorphs which are stable |
| 37 |
> |
at low pressures were calculated using thermodynamic integration to a |
| 38 |
> |
reference system (the Einstein crystal). These integrations were |
| 39 |
> |
performed for most of the common water models (SPC/E, TIP3P, TIP4P, |
| 40 |
> |
TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed |
| 41 |
> |
crystallizing at room temperature for one of the single-point water |
| 42 |
> |
models, was determined to be the stable crystalline state (at 1 atm) |
| 43 |
> |
for {\it all} the water models investigated. Phase diagrams were |
| 44 |
|
generated, and phase coexistence lines were determined for all of the |
| 45 |
< |
known low-pressure ice structures under all of the common water |
| 46 |
< |
models. Additionally, potential truncation was shown to have an |
| 47 |
< |
effect on the calculated free energies, and can result in altered free |
| 48 |
< |
energy landscapes. |
| 45 |
> |
known low-pressure ice structures under all of these water models. |
| 46 |
> |
Additionally, potential truncation was shown to have an effect on the |
| 47 |
> |
calculated free energies, and can result in altered free energy |
| 48 |
> |
landscapes. Structure factor predictions for the new crystal were |
| 49 |
> |
generated and we await experimental confirmation of the existence of |
| 50 |
> |
this new polymorph. |
| 51 |
|
\end{abstract} |
| 52 |
|
|
| 53 |
|
%\narrowtext |
| 58 |
|
|
| 59 |
|
\section{Introduction} |
| 60 |
|
|
| 57 |
– |
Molecular dynamics is a valuable tool for studying the phase behavior |
| 58 |
– |
of systems ranging from small or simple |
| 59 |
– |
molecules\cite{Matsumoto02andOthers} to complex biological |
| 60 |
– |
species.\cite{bigStuff} Many techniques have been developed to |
| 61 |
– |
investigate the thermodynamic properites of model substances, |
| 62 |
– |
providing both qualitative and quantitative comparisons between |
| 63 |
– |
simulations and experiment.\cite{thermMethods} Investigation of these |
| 64 |
– |
properties leads to the development of new and more accurate models, |
| 65 |
– |
leading to better understanding and depiction of physical processes |
| 66 |
– |
and intricate molecular systems. |
| 67 |
– |
|
| 61 |
|
Water has proven to be a challenging substance to depict in |
| 62 |
|
simulations, and a variety of models have been developed to describe |
| 63 |
|
its behavior under varying simulation |
| 64 |
< |
conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04} |
| 64 |
> |
conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04} |
| 65 |
|
These models have been used to investigate important physical |
| 66 |
< |
phenomena like phase transitions and the hydrophobic |
| 67 |
< |
effect.\cite{Yamada02} With the choice of models available, it |
| 68 |
< |
is only natural to compare the models under interesting thermodynamic |
| 69 |
< |
conditions in an attempt to clarify the limitations of each of the |
| 70 |
< |
models.\cite{modelProps} Two important property to quantify are the |
| 71 |
< |
Gibbs and Helmholtz free energies, particularly for the solid forms of |
| 72 |
< |
water. Difficulty in these types of studies typically arises from the |
| 73 |
< |
assortment of possible crystalline polymorphs that water adopts over a |
| 74 |
< |
wide range of pressures and temperatures. There are currently 13 |
| 75 |
< |
recognized forms of ice, and it is a challenging task to investigate |
| 76 |
< |
the entire free energy landscape.\cite{Sanz04} Ideally, research is |
| 77 |
< |
focused on the phases having the lowest free energy at a given state |
| 78 |
< |
point, because these phases will dictate the true transition |
| 79 |
< |
temperatures and pressures for their respective model. |
| 66 |
> |
phenomena like phase transitions, transport properties, and the |
| 67 |
> |
hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
| 68 |
> |
choice of models available, it is only natural to compare the models |
| 69 |
> |
under interesting thermodynamic conditions in an attempt to clarify |
| 70 |
> |
the limitations of each of the |
| 71 |
> |
models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
| 72 |
> |
important properties to quantify are the Gibbs and Helmholtz free |
| 73 |
> |
energies, particularly for the solid forms of water. Difficulty in |
| 74 |
> |
these types of studies typically arises from the assortment of |
| 75 |
> |
possible crystalline polymorphs that water adopts over a wide range of |
| 76 |
> |
pressures and temperatures. There are currently 13 recognized forms |
| 77 |
> |
of ice, and it is a challenging task to investigate the entire free |
| 78 |
> |
energy landscape.\cite{Sanz04} Ideally, research is focused on the |
| 79 |
> |
phases having the lowest free energy at a given state point, because |
| 80 |
> |
these phases will dictate the relevant transition temperatures and |
| 81 |
> |
pressures for the model. |
| 82 |
|
|
| 83 |
|
In this paper, standard reference state methods were applied to known |
| 84 |
< |
crystalline water polymorphs in the low pressure regime. This work is |
| 85 |
< |
unique in the fact that one of the crystal lattices was arrived at |
| 86 |
< |
through crystallization of a computationally efficient water model |
| 87 |
< |
under constant pressure and temperature conditions. Crystallization |
| 88 |
< |
events are interesting in and of |
| 89 |
< |
themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure |
| 90 |
< |
obtained in this case is different from any previously observed ice |
| 91 |
< |
polymorphs in experiment or simulation.\cite{Fennell04} We have named |
| 92 |
< |
this structure Ice-{\it i} to indicate its origin in computational |
| 93 |
< |
simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight |
| 94 |
< |
water molecules that stack in rows of interlocking water |
| 95 |
< |
tetramers. Proton ordering can be accomplished by orienting two of the |
| 96 |
< |
molecules so that both of their donated hydrogen bonds are internal to |
| 97 |
< |
their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal |
| 98 |
< |
constructed of water tetramers, the hydrogen bonds are not as linear |
| 99 |
< |
as those observed in ice $I_h$, however the interlocking of these |
| 100 |
< |
subunits appears to provide significant stabilization to the overall |
| 84 |
> |
crystalline water polymorphs in the low pressure regime. This work is |
| 85 |
> |
unique in that one of the crystal lattices was arrived at through |
| 86 |
> |
crystallization of a computationally efficient water model under |
| 87 |
> |
constant pressure and temperature conditions. Crystallization events |
| 88 |
> |
are interesting in and of themselves;\cite{Matsumoto02,Yamada02} |
| 89 |
> |
however, the crystal structure obtained in this case is different from |
| 90 |
> |
any previously observed ice polymorphs in experiment or |
| 91 |
> |
simulation.\cite{Fennell04} We have named this structure Ice-{\it i} |
| 92 |
> |
to indicate its origin in computational simulation. The unit cell |
| 93 |
> |
(Fig. \ref{iceiCell}A) consists of eight water molecules that stack in |
| 94 |
> |
rows of interlocking water tetramers. Proton ordering can be |
| 95 |
> |
accomplished by orienting two of the molecules so that both of their |
| 96 |
> |
donated hydrogen bonds are internal to their tetramer |
| 97 |
> |
(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
| 98 |
> |
water tetramers, the hydrogen bonds are not as linear as those |
| 99 |
> |
observed in ice $I_h$, however the interlocking of these subunits |
| 100 |
> |
appears to provide significant stabilization to the overall |
| 101 |
|
crystal. The arrangement of these tetramers results in surrounding |
| 102 |
|
open octagonal cavities that are typically greater than 6.3 \AA\ in |
| 103 |
|
diameter. This relatively open overall structure leads to crystals |
| 105 |
|
|
| 106 |
|
\begin{figure} |
| 107 |
|
\includegraphics[width=\linewidth]{unitCell.eps} |
| 108 |
< |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
| 109 |
< |
elongated variant of Ice-{\it i}. For Ice-{\it i}, the $a$ to $c$ |
| 110 |
< |
relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a = |
| 111 |
< |
1.7850c$.} |
| 108 |
> |
\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-{\it i}$^\prime$, |
| 109 |
> |
the elongated variant of Ice-{\it i}. The spheres represent the |
| 110 |
> |
center-of-mass locations of the water molecules. The $a$ to $c$ |
| 111 |
> |
ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
| 112 |
> |
$a:2.1214c$ and $a:1.7850c$ respectively.} |
| 113 |
|
\label{iceiCell} |
| 114 |
|
\end{figure} |
| 115 |
|
|
| 124 |
|
|
| 125 |
|
Results from our previous study indicated that Ice-{\it i} is the |
| 126 |
|
minimum energy crystal structure for the single point water models we |
| 127 |
< |
investigated (for discussions on these single point dipole models, see |
| 128 |
< |
the previous work and related |
| 129 |
< |
articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only |
| 127 |
> |
had investigated (for discussions on these single point dipole models, |
| 128 |
> |
see our previous work and related |
| 129 |
> |
articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
| 130 |
|
considered energetic stabilization and neglected entropic |
| 131 |
< |
contributions to the overall free energy. To address this issue, the |
| 132 |
< |
absolute free energy of this crystal was calculated using |
| 131 |
> |
contributions to the overall free energy. To address this issue, we |
| 132 |
> |
have calculated the absolute free energy of this crystal using |
| 133 |
|
thermodynamic integration and compared to the free energies of cubic |
| 134 |
|
and hexagonal ice $I$ (the experimental low density ice polymorphs) |
| 135 |
|
and ice B (a higher density, but very stable crystal structure |
| 138 |
|
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
| 139 |
|
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
| 140 |
|
field parametrized single point dipole water model (SSD/RF). It should |
| 141 |
< |
be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used |
| 142 |
< |
in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of |
| 143 |
< |
this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit |
| 144 |
< |
it is extended in the direction of the (001) face and compressed along |
| 145 |
< |
the other two faces. |
| 141 |
> |
be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$) |
| 142 |
> |
was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit |
| 143 |
> |
cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it |
| 144 |
> |
i} unit it is extended in the direction of the (001) face and |
| 145 |
> |
compressed along the other two faces. There is typically a small |
| 146 |
> |
distortion of proton ordered Ice-{\it i}$^\prime$ that converts the |
| 147 |
> |
normally square tetramer into a rhombus with alternating approximately |
| 148 |
> |
85 and 95 degree angles. The degree of this distortion is model |
| 149 |
> |
dependent and significant enough to split the tetramer diagonal |
| 150 |
> |
location peak in the radial distribution function. |
| 151 |
|
|
| 152 |
|
\section{Methods} |
| 153 |
|
|
| 155 |
|
performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
| 156 |
|
All molecules were treated as rigid bodies, with orientational motion |
| 157 |
|
propagated using the symplectic DLM integration method. Details about |
| 158 |
< |
the implementation of these techniques can be found in a recent |
| 159 |
< |
publication.\cite{DLM} |
| 158 |
> |
the implementation of this technique can be found in a recent |
| 159 |
> |
publication.\cite{Dullweber1997} |
| 160 |
|
|
| 161 |
< |
Thermodynamic integration was utilized to calculate the free energy of |
| 162 |
< |
several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, |
| 163 |
< |
SSD/RF, and SSD/E water models. Liquid state free energies at 300 and |
| 164 |
< |
400 K for all of these water models were also determined using this |
| 165 |
< |
same technique in order to determine melting points and generate phase |
| 166 |
< |
diagrams. All simulations were carried out at densities resulting in a |
| 167 |
< |
pressure of approximately 1 atm at their respective temperatures. |
| 161 |
> |
Thermodynamic integration is an established technique for |
| 162 |
> |
determination of free energies of condensed phases of |
| 163 |
> |
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
| 164 |
> |
method, implemented in the same manner illustrated by B\`{a}ez and |
| 165 |
> |
Clancy, was utilized to calculate the free energy of several ice |
| 166 |
> |
crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
| 167 |
> |
SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
| 168 |
> |
and 400 K for all of these water models were also determined using |
| 169 |
> |
this same technique in order to determine melting points and to |
| 170 |
> |
generate phase diagrams. All simulations were carried out at densities |
| 171 |
> |
which correspond to a pressure of approximately 1 atm at their |
| 172 |
> |
respective temperatures. |
| 173 |
|
|
| 174 |
< |
A single thermodynamic integration involves a sequence of simulations |
| 175 |
< |
over which the system of interest is converted into a reference system |
| 176 |
< |
for which the free energy is known analytically. This transformation |
| 177 |
< |
path is then integrated in order to determine the free energy |
| 178 |
< |
difference between the two states: |
| 174 |
> |
Thermodynamic integration involves a sequence of simulations during |
| 175 |
> |
which the system of interest is converted into a reference system for |
| 176 |
> |
which the free energy is known analytically. This transformation path |
| 177 |
> |
is then integrated in order to determine the free energy difference |
| 178 |
> |
between the two states: |
| 179 |
|
\begin{equation} |
| 180 |
|
\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
| 181 |
|
)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
| 182 |
|
\end{equation} |
| 183 |
|
where $V$ is the interaction potential and $\lambda$ is the |
| 184 |
|
transformation parameter that scales the overall |
| 185 |
< |
potential. Simulations are distributed unevenly along this path in |
| 186 |
< |
order to sufficiently sample the regions of greatest change in the |
| 185 |
> |
potential. Simulations are distributed strategically along this path |
| 186 |
> |
in order to sufficiently sample the regions of greatest change in the |
| 187 |
|
potential. Typical integrations in this study consisted of $\sim$25 |
| 188 |
|
simulations ranging from 300 ps (for the unaltered system) to 75 ps |
| 189 |
|
(near the reference state) in length. |
| 190 |
|
|
| 191 |
|
For the thermodynamic integration of molecular crystals, the Einstein |
| 192 |
< |
crystal was chosen as the reference state. In an Einstein crystal, the |
| 193 |
< |
molecules are harmonically restrained at their ideal lattice locations |
| 194 |
< |
and orientations. The partition function for a molecular crystal |
| 192 |
> |
crystal was chosen as the reference system. In an Einstein crystal, |
| 193 |
> |
the molecules are restrained at their ideal lattice locations and |
| 194 |
> |
orientations. Using harmonic restraints, as applied by B\`{a}ez and |
| 195 |
> |
Clancy, the total potential for this reference crystal |
| 196 |
> |
($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
| 197 |
> |
\begin{equation} |
| 198 |
> |
V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
| 199 |
> |
\frac{K_\omega\omega^2}{2}, |
| 200 |
> |
\end{equation} |
| 201 |
> |
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
| 202 |
> |
the spring constants restraining translational motion and deflection |
| 203 |
> |
of and rotation around the principle axis of the molecule |
| 204 |
> |
respectively. It is clear from Fig. \ref{waterSpring} that the values |
| 205 |
> |
of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from |
| 206 |
> |
$-\pi$ to $\pi$. The partition function for a molecular crystal |
| 207 |
|
restrained in this fashion can be evaluated analytically, and the |
| 208 |
|
Helmholtz Free Energy ({\it A}) is given by |
| 209 |
|
\begin{eqnarray} |
| 217 |
|
)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
| 218 |
|
\label{ecFreeEnergy} |
| 219 |
|
\end{eqnarray} |
| 220 |
< |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
| 221 |
< |
\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
| 204 |
< |
$K_\mathrm{\omega}$ are the spring constants restraining translational |
| 205 |
< |
motion and deflection of and rotation around the principle axis of the |
| 206 |
< |
molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
| 207 |
< |
minimum potential energy of the ideal crystal. In the case of |
| 208 |
< |
molecular liquids, the ideal vapor is chosen as the target reference |
| 209 |
< |
state. |
| 220 |
> |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
| 221 |
> |
potential energy of the ideal crystal.\cite{Baez95a} |
| 222 |
|
|
| 223 |
|
\begin{figure} |
| 224 |
|
\includegraphics[width=\linewidth]{rotSpring.eps} |
| 231 |
|
\label{waterSpring} |
| 232 |
|
\end{figure} |
| 233 |
|
|
| 234 |
+ |
In the case of molecular liquids, the ideal vapor is chosen as the |
| 235 |
+ |
target reference state. There are several examples of liquid state |
| 236 |
+ |
free energy calculations of water models present in the |
| 237 |
+ |
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
| 238 |
+ |
typically differ in regard to the path taken for switching off the |
| 239 |
+ |
interaction potential to convert the system to an ideal gas of water |
| 240 |
+ |
molecules. In this study, we applied of one of the most convenient |
| 241 |
+ |
methods and integrated over the $\lambda^4$ path, where all |
| 242 |
+ |
interaction parameters are scaled equally by this transformation |
| 243 |
+ |
parameter. This method has been shown to be reversible and provide |
| 244 |
+ |
results in excellent agreement with other established |
| 245 |
+ |
methods.\cite{Baez95b} |
| 246 |
+ |
|
| 247 |
|
Charge, dipole, and Lennard-Jones interactions were modified by a |
| 248 |
|
cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
| 249 |
|
). By applying this function, these interactions are smoothly |
| 255 |
|
simulations.\cite{Onsager36} For a series of the least computationally |
| 256 |
|
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
| 257 |
|
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
| 258 |
< |
\AA\ cutoff results. Finally, results from the use of an Ewald |
| 259 |
< |
summation were estimated for TIP3P and SPC/E by performing |
| 260 |
< |
calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
| 261 |
< |
mechanics software package.\cite{Tinker} The calculated energy |
| 262 |
< |
difference in the presence and absence of PME was applied to the |
| 263 |
< |
previous results in order to predict changes to the free energy |
| 264 |
< |
landscape. |
| 258 |
> |
\AA\ cutoff results. Finally, the effects of utilizing an Ewald |
| 259 |
> |
summation were estimated for TIP3P and SPC/E by performing single |
| 260 |
> |
configuration calculations with Particle-Mesh Ewald (PME) in the |
| 261 |
> |
TINKER molecular mechanics software package.\cite{Tinker} The |
| 262 |
> |
calculated energy difference in the presence and absence of PME was |
| 263 |
> |
applied to the previous results in order to predict changes to the |
| 264 |
> |
free energy landscape. |
| 265 |
|
|
| 266 |
|
\section{Results and discussion} |
| 267 |
|
|
| 268 |
< |
The free energy of proton ordered Ice-{\it i} was calculated and |
| 268 |
> |
The free energy of proton-ordered Ice-{\it i} was calculated and |
| 269 |
|
compared with the free energies of proton ordered variants of the |
| 270 |
|
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
| 271 |
|
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
| 275 |
|
$I_h$, was investigated initially, but was found to be not as stable |
| 276 |
|
as proton disordered or antiferroelectric variants of ice $I_h$. The |
| 277 |
|
proton ordered variant of ice $I_h$ used here is a simple |
| 278 |
< |
antiferroelectric version that has an 8 molecule unit |
| 279 |
< |
cell.\cite{Davidson84} The crystals contained 648 or 1728 molecules |
| 280 |
< |
for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for |
| 281 |
< |
ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
| 282 |
< |
were necessary for simulations involving larger cutoff values. |
| 278 |
> |
antiferroelectric version that we devised, and it has an 8 molecule |
| 279 |
> |
unit cell similar to other predicted antiferroelectric $I_h$ |
| 280 |
> |
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
| 281 |
> |
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
| 282 |
> |
molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger |
| 283 |
> |
crystal sizes were necessary for simulations involving larger cutoff |
| 284 |
> |
values. |
| 285 |
|
|
| 286 |
|
\begin{table*} |
| 287 |
|
\begin{minipage}{\linewidth} |
| 261 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 288 |
|
\begin{center} |
| 289 |
+ |
|
| 290 |
|
\caption{Calculated free energies for several ice polymorphs with a |
| 291 |
|
variety of common water models. All calculations used a cutoff radius |
| 292 |
|
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
| 293 |
< |
kcal/mol. Calculated error of the final digits is in parentheses. *Ice |
| 294 |
< |
$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} |
| 295 |
< |
\begin{tabular}{ l c c c c } |
| 293 |
> |
kcal/mol. Calculated error of the final digits is in |
| 294 |
> |
parentheses. $^{*}$Ice $I_c$ rapidly converts to a liquid at 200 K |
| 295 |
> |
with the SSD/RF model.} |
| 296 |
> |
|
| 297 |
> |
\begin{tabular}{lcccc} |
| 298 |
|
\hline |
| 299 |
|
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
| 300 |
|
\hline |
| 301 |
< |
TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\\ |
| 302 |
< |
TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\ |
| 303 |
< |
TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\ |
| 304 |
< |
SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\ |
| 305 |
< |
SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\ |
| 306 |
< |
SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\ |
| 301 |
> |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
| 302 |
> |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
| 303 |
> |
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
| 304 |
> |
SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\ |
| 305 |
> |
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
| 306 |
> |
SSD/RF & -11.51(2) & NA$^{*}$ & -12.08(3) & -12.29(2)\\ |
| 307 |
|
\end{tabular} |
| 308 |
|
\label{freeEnergy} |
| 309 |
|
\end{center} |
| 312 |
|
|
| 313 |
|
The free energy values computed for the studied polymorphs indicate |
| 314 |
|
that Ice-{\it i} is the most stable state for all of the common water |
| 315 |
< |
models studied. With the free energy at these state points, the |
| 316 |
< |
Gibbs-Helmholtz equation was used to project to other state points and |
| 317 |
< |
to build phase diagrams. Figures |
| 315 |
> |
models studied. With the calculated free energy at these state points, |
| 316 |
> |
the Gibbs-Helmholtz equation was used to project to other state points |
| 317 |
> |
and to build phase diagrams. Figures |
| 318 |
|
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
| 319 |
|
from the free energy results. All other models have similar structure, |
| 320 |
< |
although the crossing points between the phases exist at slightly |
| 320 |
> |
although the crossing points between the phases move to slightly |
| 321 |
|
different temperatures and pressures. It is interesting to note that |
| 322 |
|
ice $I$ does not exist in either cubic or hexagonal form in any of the |
| 323 |
|
phase diagrams for any of the models. For purposes of this study, ice |
| 324 |
|
B is representative of the dense ice polymorphs. A recent study by |
| 325 |
|
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
| 326 |
< |
TIP4P in the high pressure regime.\cite{Sanz04} |
| 326 |
> |
TIP4P at higher pressures than those studied here.\cite{Sanz04} |
| 327 |
|
|
| 328 |
|
\begin{figure} |
| 329 |
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
| 348 |
|
|
| 349 |
|
\begin{table*} |
| 350 |
|
\begin{minipage}{\linewidth} |
| 322 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 351 |
|
\begin{center} |
| 352 |
+ |
|
| 353 |
|
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
| 354 |
|
temperatures at 1 atm for several common water models compared with |
| 355 |
|
experiment. The $T_m$ and $T_s$ values from simulation correspond to a |
| 356 |
|
transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
| 357 |
|
liquid or gas state.} |
| 358 |
< |
\begin{tabular}{ l c c c c c c c } |
| 358 |
> |
|
| 359 |
> |
\begin{tabular}{lccccccc} |
| 360 |
|
\hline |
| 361 |
< |
Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
| 361 |
> |
Equilibrium Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
| 362 |
|
\hline |
| 363 |
< |
$T_m$ (K) & 269(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\ |
| 364 |
< |
$T_b$ (K) & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\ |
| 365 |
< |
$T_s$ (K) & - & - & - & - & 355(3) & - & -\\ |
| 363 |
> |
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
| 364 |
> |
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
| 365 |
> |
$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ |
| 366 |
|
\end{tabular} |
| 367 |
|
\label{meltandboil} |
| 368 |
|
\end{center} |
| 387 |
|
not exhibit a melting point at 1 atm, but it shows a sublimation point |
| 388 |
|
at 355 K. This is due to the significant stability of Ice-{\it i} over |
| 389 |
|
all other polymorphs for this particular model under these |
| 390 |
< |
conditions. While troubling, this behavior turned out to be |
| 391 |
< |
advantageous in that it facilitated the spontaneous crystallization of |
| 392 |
< |
Ice-{\it i}. These observations provide a warning that simulations of |
| 390 |
> |
conditions. While troubling, this behavior resulted in spontaneous |
| 391 |
> |
crystallization of Ice-{\it i} and led us to investigate this |
| 392 |
> |
structure. These observations provide a warning that simulations of |
| 393 |
|
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
| 394 |
< |
risk of spontaneous crystallization. However, this risk changes when |
| 394 |
> |
risk of spontaneous crystallization. However, this risk lessens when |
| 395 |
|
applying a longer cutoff. |
| 396 |
|
|
| 397 |
|
\begin{figure} |
| 415 |
|
in the SSD/E model that the liquid state is preferred under standard |
| 416 |
|
simulation conditions (298 K and 1 atm). Thus, it is recommended that |
| 417 |
|
simulations using this model choose interaction truncation radii |
| 418 |
< |
greater than 9 \AA\. This narrowing trend is much more subtle in the |
| 418 |
> |
greater than 9 \AA\ . This narrowing trend is much more subtle in the |
| 419 |
|
case of SSD/RF, indicating that the free energies calculated with a |
| 420 |
|
reaction field present provide a more accurate picture of the free |
| 421 |
|
energy landscape in the absence of potential truncation. |
| 425 |
|
was estimated by applying the potential energy difference do to its |
| 426 |
|
inclusion in systems in the presence and absence of the |
| 427 |
|
correction. This was accomplished by calculation of the potential |
| 428 |
< |
energy of identical crystals with and without PME using TINKER. The |
| 429 |
< |
free energies for the investigated polymorphs using the TIP3P and |
| 430 |
< |
SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
| 431 |
< |
are not fully supported in TINKER, so the results for these models |
| 432 |
< |
could not be estimated. The same trend pointed out through increase of |
| 433 |
< |
cutoff radius is observed in these PME results. Ice-{\it i} is the |
| 434 |
< |
preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
| 435 |
< |
water models; however, there is a narrowing of the free energy |
| 436 |
< |
differences between the various solid forms. In the case of SPC/E this |
| 437 |
< |
narrowing is significant enough that it becomes less clear that |
| 438 |
< |
Ice-{\it i} is the most stable polymorph, and is possibly metastable |
| 439 |
< |
with respect to ice B and possibly ice $I_c$. However, these results |
| 440 |
< |
do not significantly alter the finding that the Ice-{\it i} polymorph |
| 441 |
< |
is a stable crystal structure that should be considered when studying |
| 442 |
< |
the phase behavior of water models. |
| 428 |
> |
energy of identical crystals both with and without PME. The free |
| 429 |
> |
energies for the investigated polymorphs using the TIP3P and SPC/E |
| 430 |
> |
water models are shown in Table \ref{pmeShift}. The same trend pointed |
| 431 |
> |
out through increase of cutoff radius is observed in these PME |
| 432 |
> |
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
| 433 |
> |
for both the TIP3P and SPC/E water models; however, the narrowing of |
| 434 |
> |
the free energy differences between the various solid forms is |
| 435 |
> |
significant enough that it becomes less clear that it is the most |
| 436 |
> |
stable polymorph with the SPC/E model. The free energies of Ice-{\it |
| 437 |
> |
i} and ice B nearly overlap within error, with ice $I_c$ just outside |
| 438 |
> |
as well, indicating that Ice-{\it i} might be metastable with respect |
| 439 |
> |
to ice B and possibly ice $I_c$ with SPC/E. However, these results do |
| 440 |
> |
not significantly alter the finding that the Ice-{\it i} polymorph is |
| 441 |
> |
a stable crystal structure that should be considered when studying the |
| 442 |
> |
phase behavior of water models. |
| 443 |
|
|
| 444 |
|
\begin{table*} |
| 445 |
|
\begin{minipage}{\linewidth} |
| 416 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 446 |
|
\begin{center} |
| 447 |
+ |
|
| 448 |
|
\caption{The free energy of the studied ice polymorphs after applying |
| 449 |
|
the energy difference attributed to the inclusion of the PME |
| 450 |
|
long-range interaction correction. Units are kcal/mol.} |
| 451 |
< |
\begin{tabular}{ l c c c c } |
| 451 |
> |
|
| 452 |
> |
\begin{tabular}{ccccc} |
| 453 |
|
\hline |
| 454 |
< |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
| 454 |
> |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} \\ |
| 455 |
|
\hline |
| 456 |
< |
TIP3P & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)\\ |
| 457 |
< |
SPC/E & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\ |
| 456 |
> |
TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3) \\ |
| 457 |
> |
SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2) \\ |
| 458 |
|
\end{tabular} |
| 459 |
|
\label{pmeShift} |
| 460 |
|
\end{center} |
| 464 |
|
\section{Conclusions} |
| 465 |
|
|
| 466 |
|
The free energy for proton ordered variants of hexagonal and cubic ice |
| 467 |
< |
$I$, ice B, and recently discovered Ice-{\it i} were calculated under |
| 468 |
< |
standard conditions for several common water models via thermodynamic |
| 469 |
< |
integration. All the water models studied show Ice-{\it i} to be the |
| 470 |
< |
minimum free energy crystal structure in the with a 9 \AA\ switching |
| 471 |
< |
function cutoff. Calculated melting and boiling points show |
| 472 |
< |
surprisingly good agreement with the experimental values; however, the |
| 473 |
< |
solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
| 474 |
< |
interaction truncation was investigated through variation of the |
| 475 |
< |
cutoff radius, use of a reaction field parameterized model, and |
| 467 |
> |
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
| 468 |
> |
calculated under standard conditions for several common water models |
| 469 |
> |
via thermodynamic integration. All the water models studied show |
| 470 |
> |
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
| 471 |
> |
\AA\ switching function cutoff. Calculated melting and boiling points |
| 472 |
> |
show surprisingly good agreement with the experimental values; |
| 473 |
> |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
| 474 |
> |
effect of interaction truncation was investigated through variation of |
| 475 |
> |
the cutoff radius, use of a reaction field parameterized model, and |
| 476 |
|
estimation of the results in the presence of the Ewald |
| 477 |
|
summation. Interaction truncation has a significant effect on the |
| 478 |
|
computed free energy values, and may significantly alter the free |
| 480 |
|
these effects, these results show Ice-{\it i} to be an important ice |
| 481 |
|
polymorph that should be considered in simulation studies. |
| 482 |
|
|
| 483 |
< |
Due to this relative stability of Ice-{\it i} in all manner of |
| 484 |
< |
investigated simulation examples, the question arises as to possible |
| 483 |
> |
Due to this relative stability of Ice-{\it i} in all of the |
| 484 |
> |
investigated simulation conditions, the question arises as to possible |
| 485 |
|
experimental observation of this polymorph. The rather extensive past |
| 486 |
|
and current experimental investigation of water in the low pressure |
| 487 |
|
regime makes us hesitant to ascribe any relevance of this work outside |
| 491 |
|
most ideal situation for possible observation. These include the |
| 492 |
|
negative pressure or stretched solid regime, small clusters in vacuum |
| 493 |
|
deposition environments, and in clathrate structures involving small |
| 494 |
< |
non-polar molecules. Fig. \ref{fig:gofr} contains our predictions |
| 495 |
< |
of both the pair distribution function ($g_{OO}(r)$) and the structure |
| 496 |
< |
factor ($S(\vec{q})$ for this polymorph at a temperature of 77K. We |
| 497 |
< |
will leave it to our experimental colleagues to determine whether this |
| 498 |
< |
ice polymorph should really be called Ice-{\it i} or if it should be |
| 499 |
< |
promoted to Ice-0. |
| 494 |
> |
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
| 495 |
> |
our predictions for both the pair distribution function ($g_{OO}(r)$) |
| 496 |
> |
and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for |
| 497 |
> |
ice-{\it i} at a temperature of 77K. In studies of the high and low |
| 498 |
> |
density forms of amorphous ice, ``spurious'' diffraction peaks have |
| 499 |
> |
been observed experimentally.\cite{Bizid87} It is possible that a |
| 500 |
> |
variant of Ice-{\it i} could explain some of this behavior; however, |
| 501 |
> |
we will leave it to our experimental colleagues to make the final |
| 502 |
> |
determination on whether this ice polymorph is named appropriately |
| 503 |
> |
(i.e. with an imaginary number) or if it can be promoted to Ice-0. |
| 504 |
|
|
| 505 |
|
\begin{figure} |
| 506 |
|
\includegraphics[width=\linewidth]{iceGofr.eps} |
| 507 |
< |
\caption{Radial distribution functions of (A) Ice-{\it i} and (B) ice $I_c$ at 77 K from simulations of the SSD/RF water model.} |
| 507 |
> |
\caption{Radial distribution functions of ice $I_h$, $I_c$, |
| 508 |
> |
Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations |
| 509 |
> |
of the SSD/RF water model at 77 K.} |
| 510 |
|
\label{fig:gofr} |
| 511 |
|
\end{figure} |
| 512 |
|
|
| 513 |
+ |
\begin{figure} |
| 514 |
+ |
\includegraphics[width=\linewidth]{sofq.eps} |
| 515 |
+ |
\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, |
| 516 |
+ |
and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
| 517 |
+ |
been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
| 518 |
+ |
width) to compensate for the trunction effects in our finite size |
| 519 |
+ |
simulations.} |
| 520 |
+ |
\label{fig:sofq} |
| 521 |
+ |
\end{figure} |
| 522 |
+ |
|
| 523 |
|
\section{Acknowledgments} |
| 524 |
|
Support for this project was provided by the National Science |
| 525 |
|
Foundation under grant CHE-0134881. Computation time was provided by |
