| 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 for the new crystal were generated and |
| 49 |
> |
we await experimental confirmation of the existence of this new |
| 50 |
> |
polymorph. |
| 51 |
|
\end{abstract} |
| 52 |
|
|
| 53 |
|
%\narrowtext |
| 58 |
|
|
| 59 |
|
\section{Introduction} |
| 60 |
|
|
| 57 |
– |
Computer simulations are a valuable tool for studying the phase |
| 58 |
– |
behavior of systems ranging from small or simple molecules to complex |
| 59 |
– |
biological species.\cite{Matsumoto02,Li96,Marrink01} Useful techniques |
| 60 |
– |
have been developed to investigate the thermodynamic properites of |
| 61 |
– |
model substances, providing both qualitative and quantitative |
| 62 |
– |
comparisons between simulations and |
| 63 |
– |
experiment.\cite{Widom63,Frenkel84} Investigation of these properties |
| 64 |
– |
leads to the development of new and more accurate models, leading to |
| 65 |
– |
better understanding and depiction of physical processes and intricate |
| 66 |
– |
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{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04} |
| 64 |
> |
conditions.\cite{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04,Amoeba,POL3} |
| 65 |
|
These models have been used to investigate important physical |
| 66 |
< |
phenomena like phase transitions, molecule transport, and the |
| 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 property to quantify are the Gibbs and Helmholtz free |
| 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 |
| 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 true transition temperatures and |
| 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 |
| 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 |
| 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 |
< |
our previous work and related |
| 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-$i^\prime$) was |
| 142 |
> |
used in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell |
| 143 |
> |
of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} |
| 144 |
> |
unit it is extended in the direction of the (001) face and compressed |
| 145 |
> |
along the other two faces. |
| 146 |
|
|
| 147 |
|
\section{Methods} |
| 148 |
|
|
| 161 |
|
crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
| 162 |
|
SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
| 163 |
|
and 400 K for all of these water models were also determined using |
| 164 |
< |
this same technique in order to determine melting points and generate |
| 165 |
< |
phase diagrams. All simulations were carried out at densities |
| 166 |
< |
resulting in a pressure of approximately 1 atm at their respective |
| 167 |
< |
temperatures. |
| 164 |
> |
this same technique in order to determine melting points and to |
| 165 |
> |
generate phase diagrams. All simulations were carried out at densities |
| 166 |
> |
which correspond to a pressure of approximately 1 atm at their |
| 167 |
> |
respective temperatures. |
| 168 |
|
|
| 169 |
< |
A single thermodynamic integration involves a sequence of simulations |
| 170 |
< |
over which the system of interest is converted into a reference system |
| 171 |
< |
for which the free energy is known analytically. This transformation |
| 172 |
< |
path is then integrated in order to determine the free energy |
| 173 |
< |
difference between the two states: |
| 169 |
> |
Thermodynamic integration involves a sequence of simulations during |
| 170 |
> |
which the system of interest is converted into a reference system for |
| 171 |
> |
which the free energy is known analytically. This transformation path |
| 172 |
> |
is then integrated in order to determine the free energy difference |
| 173 |
> |
between the two states: |
| 174 |
|
\begin{equation} |
| 175 |
|
\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
| 176 |
|
)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
| 232 |
|
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
| 233 |
|
typically differ in regard to the path taken for switching off the |
| 234 |
|
interaction potential to convert the system to an ideal gas of water |
| 235 |
< |
molecules. In this study, we apply of one of the most convenient |
| 236 |
< |
methods and integrate over the $\lambda^4$ path, where all interaction |
| 237 |
< |
parameters are scaled equally by this transformation parameter. This |
| 238 |
< |
method has been shown to be reversible and provide results in |
| 239 |
< |
excellent agreement with other established methods.\cite{Baez95b} |
| 235 |
> |
molecules. In this study, we applied of one of the most convenient |
| 236 |
> |
methods and integrated over the $\lambda^4$ path, where all |
| 237 |
> |
interaction parameters are scaled equally by this transformation |
| 238 |
> |
parameter. This method has been shown to be reversible and provide |
| 239 |
> |
results in excellent agreement with other established |
| 240 |
> |
methods.\cite{Baez95b} |
| 241 |
|
|
| 242 |
|
Charge, dipole, and Lennard-Jones interactions were modified by a |
| 243 |
|
cubic switching between 100\% and 85\% of the cutoff value (9 \AA |
| 250 |
|
simulations.\cite{Onsager36} For a series of the least computationally |
| 251 |
|
expensive models (SSD/E, SSD/RF, and TIP3P), simulations were |
| 252 |
|
performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 |
| 253 |
< |
\AA\ cutoff results. Finally, results from the use of an Ewald |
| 254 |
< |
summation were estimated for TIP3P and SPC/E by performing |
| 255 |
< |
calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
| 256 |
< |
mechanics software package.\cite{Tinker} The calculated energy |
| 257 |
< |
difference in the presence and absence of PME was applied to the |
| 258 |
< |
previous results in order to predict changes to the free energy |
| 259 |
< |
landscape. |
| 253 |
> |
\AA\ cutoff results. Finally, the effects of utilizing an Ewald |
| 254 |
> |
summation were estimated for TIP3P and SPC/E by performing single |
| 255 |
> |
configuration calculations with Particle-Mesh Ewald (PME) in the |
| 256 |
> |
TINKER molecular mechanics software package.\cite{Tinker} The |
| 257 |
> |
calculated energy difference in the presence and absence of PME was |
| 258 |
> |
applied to the previous results in order to predict changes to the |
| 259 |
> |
free energy landscape. |
| 260 |
|
|
| 261 |
|
\section{Results and discussion} |
| 262 |
|
|
| 263 |
< |
The free energy of proton ordered Ice-{\it i} was calculated and |
| 263 |
> |
The free energy of proton-ordered Ice-{\it i} was calculated and |
| 264 |
|
compared with the free energies of proton ordered variants of the |
| 265 |
|
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
| 266 |
|
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
| 270 |
|
$I_h$, was investigated initially, but was found to be not as stable |
| 271 |
|
as proton disordered or antiferroelectric variants of ice $I_h$. The |
| 272 |
|
proton ordered variant of ice $I_h$ used here is a simple |
| 273 |
< |
antiferroelectric version that we divised, and it has an 8 molecule |
| 273 |
> |
antiferroelectric version that we devised, and it has an 8 molecule |
| 274 |
|
unit cell similar to other predicted antiferroelectric $I_h$ |
| 275 |
|
crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
| 276 |
|
molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
| 305 |
|
|
| 306 |
|
The free energy values computed for the studied polymorphs indicate |
| 307 |
|
that Ice-{\it i} is the most stable state for all of the common water |
| 308 |
< |
models studied. With the free energy at these state points, the |
| 309 |
< |
Gibbs-Helmholtz equation was used to project to other state points and |
| 310 |
< |
to build phase diagrams. Figures |
| 308 |
> |
models studied. With the calculated free energy at these state points, |
| 309 |
> |
the Gibbs-Helmholtz equation was used to project to other state points |
| 310 |
> |
and to build phase diagrams. Figures |
| 311 |
|
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
| 312 |
|
from the free energy results. All other models have similar structure, |
| 313 |
< |
although the crossing points between the phases exist at slightly |
| 313 |
> |
although the crossing points between the phases move to slightly |
| 314 |
|
different temperatures and pressures. It is interesting to note that |
| 315 |
|
ice $I$ does not exist in either cubic or hexagonal form in any of the |
| 316 |
|
phase diagrams for any of the models. For purposes of this study, ice |
| 317 |
|
B is representative of the dense ice polymorphs. A recent study by |
| 318 |
|
Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
| 319 |
< |
TIP4P in the high pressure regime.\cite{Sanz04} |
| 319 |
> |
TIP4P at higher pressures than those studied here.\cite{Sanz04} |
| 320 |
|
|
| 321 |
|
\begin{figure} |
| 322 |
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
| 379 |
|
not exhibit a melting point at 1 atm, but it shows a sublimation point |
| 380 |
|
at 355 K. This is due to the significant stability of Ice-{\it i} over |
| 381 |
|
all other polymorphs for this particular model under these |
| 382 |
< |
conditions. While troubling, this behavior turned out to be |
| 383 |
< |
advantageous in that it facilitated the spontaneous crystallization of |
| 384 |
< |
Ice-{\it i}. These observations provide a warning that simulations of |
| 382 |
> |
conditions. While troubling, this behavior resulted in spontaneous |
| 383 |
> |
crystallization of Ice-{\it i} and led us to investigate this |
| 384 |
> |
structure. These observations provide a warning that simulations of |
| 385 |
|
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
| 386 |
< |
risk of spontaneous crystallization. However, this risk changes when |
| 386 |
> |
risk of spontaneous crystallization. However, this risk lessens when |
| 387 |
|
applying a longer cutoff. |
| 388 |
|
|
| 389 |
|
\begin{figure} |
| 417 |
|
was estimated by applying the potential energy difference do to its |
| 418 |
|
inclusion in systems in the presence and absence of the |
| 419 |
|
correction. This was accomplished by calculation of the potential |
| 420 |
< |
energy of identical crystals with and without PME using TINKER. The |
| 421 |
< |
free energies for the investigated polymorphs using the TIP3P and |
| 422 |
< |
SPC/E water models are shown in Table \ref{pmeShift}. The same trend |
| 423 |
< |
pointed out through increase of cutoff radius is observed in these PME |
| 420 |
> |
energy of identical crystals both with and without PME. The free |
| 421 |
> |
energies for the investigated polymorphs using the TIP3P and SPC/E |
| 422 |
> |
water models are shown in Table \ref{pmeShift}. The same trend pointed |
| 423 |
> |
out through increase of cutoff radius is observed in these PME |
| 424 |
|
results. Ice-{\it i} is the preferred polymorph at ambient conditions |
| 425 |
|
for both the TIP3P and SPC/E water models; however, the narrowing of |
| 426 |
|
the free energy differences between the various solid forms is |
| 455 |
|
\section{Conclusions} |
| 456 |
|
|
| 457 |
|
The free energy for proton ordered variants of hexagonal and cubic ice |
| 458 |
< |
$I$, ice B, and recently discovered Ice-{\it i} were calculated under |
| 459 |
< |
standard conditions for several common water models via thermodynamic |
| 460 |
< |
integration. All the water models studied show Ice-{\it i} to be the |
| 461 |
< |
minimum free energy crystal structure in the with a 9 \AA\ switching |
| 462 |
< |
function cutoff. Calculated melting and boiling points show |
| 463 |
< |
surprisingly good agreement with the experimental values; however, the |
| 464 |
< |
solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
| 465 |
< |
interaction truncation was investigated through variation of the |
| 466 |
< |
cutoff radius, use of a reaction field parameterized model, and |
| 458 |
> |
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
| 459 |
> |
calculated under standard conditions for several common water models |
| 460 |
> |
via thermodynamic integration. All the water models studied show |
| 461 |
> |
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
| 462 |
> |
\AA\ switching function cutoff. Calculated melting and boiling points |
| 463 |
> |
show surprisingly good agreement with the experimental values; |
| 464 |
> |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
| 465 |
> |
effect of interaction truncation was investigated through variation of |
| 466 |
> |
the cutoff radius, use of a reaction field parameterized model, and |
| 467 |
|
estimation of the results in the presence of the Ewald |
| 468 |
|
summation. Interaction truncation has a significant effect on the |
| 469 |
|
computed free energy values, and may significantly alter the free |
| 471 |
|
these effects, these results show Ice-{\it i} to be an important ice |
| 472 |
|
polymorph that should be considered in simulation studies. |
| 473 |
|
|
| 474 |
< |
Due to this relative stability of Ice-{\it i} in all manner of |
| 475 |
< |
investigated simulation examples, the question arises as to possible |
| 474 |
> |
Due to this relative stability of Ice-{\it i} in all of the |
| 475 |
> |
investigated simulation conditions, the question arises as to possible |
| 476 |
|
experimental observation of this polymorph. The rather extensive past |
| 477 |
|
and current experimental investigation of water in the low pressure |
| 478 |
|
regime makes us hesitant to ascribe any relevance of this work outside |
| 485 |
|
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
| 486 |
|
our predictions for both the pair distribution function ($g_{OO}(r)$) |
| 487 |
|
and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it |
| 488 |
< |
i} at a temperature of 77K. In a quick comparison of the predicted |
| 489 |
< |
S(q) for Ice-{\it i} and experimental studies of amorphous solid |
| 490 |
< |
water, it is possible that some of the ``spurious'' peaks that could |
| 491 |
< |
not be assigned in HDA could correspond to peaks labeled in this |
| 488 |
> |
i} at a temperature of 77K. In our initial comparison of the |
| 489 |
> |
predicted S(q) for Ice-{\it i} and experimental studies of amorphous |
| 490 |
> |
solid water, it is possible that some of the ``spurious'' peaks that |
| 491 |
> |
could not be assigned in an early report on high-density amorphous |
| 492 |
> |
(HDA) ice could correspond to peaks labeled in this |
| 493 |
|
S(q).\cite{Bizid87} It should be noted that there is typically poor |
| 494 |
|
agreement on crystal densities between simulation and experiment, so |
| 495 |
|
such peak comparisons should be made with caution. We will leave it |
| 496 |
< |
to our experimental colleagues to determine whether this ice polymorph |
| 497 |
< |
is named appropriately or if it should be promoted to Ice-0. |
| 496 |
> |
to our experimental colleagues to make the final determination on |
| 497 |
> |
whether this ice polymorph is named appropriately (i.e. with an |
| 498 |
> |
imaginary number) or if it can be promoted to Ice-0. |
| 499 |
|
|
| 500 |
|
\begin{figure} |
| 501 |
|
\includegraphics[width=\linewidth]{iceGofr.eps} |
