| 34 |
|
The absolute free energies of several ice polymorphs which are stable |
| 35 |
|
at low pressures were calculated using thermodynamic integration with |
| 36 |
|
a variety of common water models. A recently discovered ice polymorph |
| 37 |
< |
that has yet only been observed in computer simulations (Ice-{\it i}), |
| 38 |
< |
was determined to be the stable crystalline state for {\it all} the |
| 39 |
< |
water models investigated. Phase diagrams were generated, and phase |
| 40 |
< |
coexistence lines were determined for all of the known low-pressure |
| 41 |
< |
ice structures. Additionally, potential truncation was show to play a |
| 42 |
< |
role in the resulting shape of the free energy landscape. |
| 37 |
> |
that has as yet only been observed in computer simulations (Ice-{\it |
| 38 |
> |
i}), was determined to be the stable crystalline state for {\it all} |
| 39 |
> |
the water models investigated. Phase diagrams were generated, and |
| 40 |
> |
phase coexistence lines were determined for all of the known |
| 41 |
> |
low-pressure ice structures. Additionally, potential truncation was |
| 42 |
> |
shown to play a role in the resulting shape of the free energy |
| 43 |
> |
landscape. |
| 44 |
|
\end{abstract} |
| 45 |
|
|
| 46 |
|
%\narrowtext |
| 60 |
|
hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
| 61 |
|
choice of models available, it is only natural to compare the models |
| 62 |
|
under interesting thermodynamic conditions in an attempt to clarify |
| 63 |
< |
the limitations of each of the |
| 64 |
< |
models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
| 65 |
< |
important properties to quantify are the Gibbs and Helmholtz free |
| 66 |
< |
energies, particularly for the solid forms of water. Difficulties in |
| 67 |
< |
studies addressing these thermodynamic quantities typically arise from |
| 68 |
< |
the assortment of possible crystalline polymorphs that water adopts |
| 69 |
< |
over a wide range of pressures and temperatures. It is a challenging |
| 70 |
< |
task to investigate the entire free energy landscape\cite{Sanz04}; |
| 71 |
< |
and ideally, research is focused on the phases having the lowest free |
| 63 |
> |
the limitations of |
| 64 |
> |
each.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two important |
| 65 |
> |
properties to quantify are the Gibbs and Helmholtz free energies, |
| 66 |
> |
particularly for the solid forms of water. Difficulties in studies |
| 67 |
> |
addressing these thermodynamic quantities typically arise from the |
| 68 |
> |
assortment of possible crystalline polymorphs that water adopts over a |
| 69 |
> |
wide range of pressures and temperatures. It is a challenging task to |
| 70 |
> |
investigate the entire free energy landscape\cite{Sanz04}; and |
| 71 |
> |
ideally, research is focused on the phases having the lowest free |
| 72 |
|
energy at a given state point, because these phases will dictate the |
| 73 |
|
relevant transition temperatures and pressures for the model. |
| 74 |
|
|
| 118 |
|
considered energetic stabilization and neglected entropic |
| 119 |
|
contributions to the overall free energy. To address this issue, we |
| 120 |
|
have calculated the absolute free energy of this crystal using |
| 121 |
< |
thermodynamic integration and compared to the free energies of cubic |
| 122 |
< |
and hexagonal ice $I$ (the experimental low density ice polymorphs) |
| 123 |
< |
and ice B (a higher density, but very stable crystal structure |
| 124 |
< |
observed by B\`{a}ez and Clancy in free energy studies of |
| 125 |
< |
SPC/E).\cite{Baez95b} This work includes results for the water model |
| 126 |
< |
from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
| 127 |
< |
common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
| 128 |
< |
field parametrized single point dipole water model (SSD/RF). The |
| 129 |
< |
extruded variant, Ice-{\it i}$^\prime$, was used in calculations |
| 130 |
< |
involving SPC/E, TIP4P, and TIP5P. These models exhibit enhanced |
| 131 |
< |
stability with Ice-{\it i}$^\prime$ because of their more |
| 132 |
< |
tetrahedrally arranged internal charge distributions. Additionally, |
| 133 |
< |
there is often a small distortion of proton ordered Ice-{\it |
| 134 |
< |
i}$^\prime$ that converts the normally square tetramer into a rhombus |
| 135 |
< |
with alternating approximately 85 and 95 degree angles. The degree of |
| 136 |
< |
this distortion is model dependent and significant enough to split the |
| 137 |
< |
tetramer diagonal location peak in the radial distribution function. |
| 121 |
> |
thermodynamic integration and compared it to the free energies of ice |
| 122 |
> |
$I_c$ and ice $I_h$ (the experimental low density ice polymorphs) and |
| 123 |
> |
ice B (a higher density, but very stable crystal structure observed by |
| 124 |
> |
B\`{a}ez and Clancy in free energy studies of SPC/E).\cite{Baez95b} |
| 125 |
> |
This work includes results for the water model from which Ice-{\it i} |
| 126 |
> |
was crystallized (SSD/E) in addition to several common water models |
| 127 |
> |
(TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field parametrized |
| 128 |
> |
single point dipole water model (SSD/RF). The extruded variant, |
| 129 |
> |
Ice-{\it i}$^\prime$, was used in calculations involving SPC/E, TIP4P, |
| 130 |
> |
and TIP5P. These models exhibit enhanced stability with Ice-{\it |
| 131 |
> |
i}$^\prime$ because of their more tetrahedrally arranged internal |
| 132 |
> |
charge distributions. Additionally, there is often a small distortion |
| 133 |
> |
of proton ordered Ice-{\it i}$^\prime$ that converts the normally |
| 134 |
> |
square tetramer into a rhombus with alternating approximately 85 and |
| 135 |
> |
95 degree angles. The degree of this distortion is model dependent |
| 136 |
> |
and significant enough to split the tetramer diagonal location peak in |
| 137 |
> |
the radial distribution function. |
| 138 |
|
|
| 139 |
|
Thermodynamic integration was utilized to calculate the free energies |
| 140 |
|
of the listed water models at various state points using a modified |
| 156 |
|
in the calculation of free energies for condensed phases of |
| 157 |
|
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. |
| 158 |
|
|
| 159 |
< |
The calculated free energies of proton-ordered varients of three low |
| 159 |
> |
The calculated free energies of proton-ordered variants of three low |
| 160 |
|
density polymorphs ($I_h$, $I_c$, and Ice-{\it i} or Ice-{\it |
| 161 |
|
i}$^\prime$) and the stable higher density ice B are listed in Table |
| 162 |
< |
\ref{freeEnergy}. The reason for inclusion of ice B was that it was |
| 162 |
> |
\ref{freeEnergy}. Ice B was included because it has been |
| 163 |
|
shown to be a minimum free energy structure for SPC/E at ambient |
| 164 |
|
conditions.\cite{Baez95b} In addition to the free energies, the |
| 165 |
< |
relavent transition temperatures at standard pressure are also |
| 165 |
> |
relevant transition temperatures at standard pressure are also |
| 166 |
|
displayed in Table \ref{freeEnergy}. These free energy values |
| 167 |
|
indicate that Ice-{\it i} is the most stable state for all of the |
| 168 |
|
investigated water models. With the free energy at these state |
| 169 |
|
points, the Gibbs-Helmholtz equation was used to project to other |
| 170 |
< |
state points and to build phase diagrams, and figure \ref{tp3PhaseDia} |
| 171 |
< |
is an example diagram built from the results for the TIP3P water |
| 172 |
< |
model. All other models have similar structure, although the crossing |
| 173 |
< |
points between the phases move to different temperatures and pressures |
| 174 |
< |
as indicated from the transition temperatures in Table |
| 175 |
< |
\ref{freeEnergy}. It is interesting to note that ice $I$ does not |
| 176 |
< |
exist in either cubic or hexagonal form in any of the phase diagrams |
| 177 |
< |
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 at higher pressures than those studied here.\cite{Sanz04} |
| 170 |
> |
state points and to build phase diagrams. Figure \ref{tp3PhaseDia} is |
| 171 |
> |
an example diagram built from the results for the TIP3P water model. |
| 172 |
> |
All other models have similar structure, although the crossing points |
| 173 |
> |
between the phases move to different temperatures and pressures as |
| 174 |
> |
indicated from the transition temperatures in Table \ref{freeEnergy}. |
| 175 |
> |
It is interesting to note that ice $I_h$ (and ice $I_c$ for that |
| 176 |
> |
matter) do not appear in any of the phase diagrams for any of the |
| 177 |
> |
models. For purposes of this study, ice B is representative of the |
| 178 |
> |
dense ice polymorphs. A recent study by Sanz {\it et al.} provides |
| 179 |
> |
details on the phase diagrams for SPC/E and TIP4P at higher pressures |
| 180 |
> |
than those studied here.\cite{Sanz04} |
| 181 |
|
|
| 182 |
|
\begin{table*} |
| 183 |
|
\begin{minipage}{\linewidth} |
| 287 |
|
influence the chosen polymorph upon crystallization. |
| 288 |
|
|
| 289 |
|
So what is the preferred solid polymorph for simulated water? The |
| 290 |
< |
answer appears to be dependent both on conditions and which model is |
| 290 |
> |
answer appears to be dependent both on the conditions and the model |
| 291 |
|
used. In the case of short cutoffs without a long-range interaction |
| 292 |
|
correction, Ice-{\it i} and Ice-{\it i}$^\prime$ have the lowest free |
| 293 |
|
energy of the studied polymorphs with all the models. Ideally, |
| 304 |
|
simulation conditions, the question arises as to possible experimental |
| 305 |
|
observation of this polymorph. The rather extensive past and current |
| 306 |
|
experimental investigation of water in the low pressure regime makes |
| 307 |
< |
us hesitant to ascribe any relevance of this work outside of the |
| 307 |
> |
us hesitant to ascribe any relevance to this work outside of the |
| 308 |
|
simulation community. It is for this reason that we chose a name for |
| 309 |
|
this polymorph which involves an imaginary quantity. That said, there |
| 310 |
|
are certain experimental conditions that would provide the most ideal |
