| 221 |
|
results in excellent agreement with other established |
| 222 |
|
methods.\cite{Baez95b} |
| 223 |
|
|
| 224 |
+ |
The Helmholtz free energy error was determined in the same manner in |
| 225 |
+ |
both the solid and the liquid free energy calculations . At each point |
| 226 |
+ |
along the integration path, we calculated the standard deviation of |
| 227 |
+ |
the potential energy difference. Addition or subtraction of these |
| 228 |
+ |
values to each of their respective points and integrating the curve |
| 229 |
+ |
again provides the upper and lower bounds of the uncertainty in the |
| 230 |
+ |
Helmholtz free energy. |
| 231 |
+ |
|
| 232 |
|
Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and |
| 233 |
|
Lennard-Jones interactions were gradually reduced by a cubic switching |
| 234 |
|
function. By applying this function, these interactions are smoothly |
| 247 |
|
in the presence and absence of PME was applied to the previous results |
| 248 |
|
in order to predict changes to the free energy landscape. |
| 249 |
|
|
| 250 |
+ |
In addition to the above procedures, we also tested how the inclusion |
| 251 |
+ |
of the Lennard-Jones long-range correction affects the free energy |
| 252 |
+ |
results. The correction for the Lennard-Jones trucation was included |
| 253 |
+ |
by integration of the equation discussed in section |
| 254 |
+ |
\ref{sec:LJCorrections}. Rather than discuss its affect alongside the |
| 255 |
+ |
free energy results, we will just mention that while the correction |
| 256 |
+ |
does lower the free energy of the higher density states more than the |
| 257 |
+ |
lower density states, the effect is so small that it is entirely |
| 258 |
+ |
overwelmed by the error in the free energy calculation. Since its |
| 259 |
+ |
inclusion does not influence the results, the Lennard-Jones correction |
| 260 |
+ |
was omitted from all the calculations below. |
| 261 |
+ |
|
| 262 |
|
\section{Initial Free Energy Results} |
| 263 |
|
|
| 264 |
|
The calculated free energies of proton-ordered variants of three low |
| 447 |
|
\cmidrule(lr){2-6} |
| 448 |
|
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ |
| 449 |
|
\midrule |
| 450 |
< |
TIP5P-E & & & & - & \\ |
| 451 |
< |
TIP4P-Ew & & -13.09(3) & & - & -12.98(3) \\ |
| 452 |
< |
SPC/E & -12.99(3) & -13.00(3) & & - & -12.99(3) \\ |
| 450 |
> |
TIP5P-E & -11.98(4) & -11.96(4) & -11.87(3) & - & -11.95(3) \\ |
| 451 |
> |
TIP4P-Ew & -13.11(3) & -13.09(3) & -12.97(3) & - & -12.98(3) \\ |
| 452 |
> |
SPC/E & -12.99(3) & -13.00(3) & -13.03(3) & - & -12.99(3) \\ |
| 453 |
|
SSD/RF & -11.83(3) & -11.66(4) & -12.32(3) & -12.39(3) & - \\ |
| 454 |
|
TRED & -12.61(3) & -12.43(3) & -12.89(3) & -13.12(3) & - \\ |
| 455 |
|
\end{tabular} |
| 462 |
|
the same fashion; however Ice-$i$ and ice B are quite a bit closer in |
| 463 |
|
free energy (nearly isoenergetic). The free energy differences between |
| 464 |
|
ice polymorphs for TRED water parallel SSD/RF, with the exception that |
| 465 |
< |
ice B is destabilized such that it is no longer nearly isoenergetic |
| 466 |
< |
with Ice-$i$. |
| 465 |
> |
ice B is destabilized such that it is not very close to Ice-$i$. The |
| 466 |
> |
SPC/E results really show the near isoenergetic behavior when using |
| 467 |
> |
the electrostatics correction. Ice B has the lowest Helmholtz free |
| 468 |
> |
energy; however, all the polymorph results overlap within error. |
| 469 |
> |
|
| 470 |
> |
The most interesting results from these calculations come from the |
| 471 |
> |
more expensive TIP4P-Ew and TIP5P-E results. Both of these models were |
| 472 |
> |
optimized for use with an electrostatic correction and are |
| 473 |
> |
geometrically arranged to mimic water following two different |
| 474 |
> |
ideas. In TIP5P-E, the primary location for the negative charge in the |
| 475 |
> |
molecule is assigned to the lone-pairs of the oxygen, while TIP4P-Ew |
| 476 |
> |
places the negative charge near the center-of-mass along the H-O-H |
| 477 |
> |
bisector. There is some debate as to which is the proper choice for |
| 478 |
> |
the negative charge location, and this has in part led to a six-site |
| 479 |
> |
water model that balances both of these options.\cite{Vega05,Nada03} |
| 480 |
> |
The limited results in table \ref{tab:dampedFreeEnergy} support the |
| 481 |
> |
results of Vega {\it et al.}, which indicate the TIP4P charge location |
| 482 |
> |
geometry is more physically valid.\cite{Vega05} With the TIP4P-Ew |
| 483 |
> |
water model, the experimentally observed polymorph (ice |
| 484 |
> |
I$_\textrm{h}$) is the preferred form with ice I$_\textrm{c}$ slightly |
| 485 |
> |
higher in energy, though overlapping within error, and the less |
| 486 |
> |
realistic ice B and Ice-$i^\prime$ are destabilized relative to these |
| 487 |
> |
polymorphs. TIP5P-E shows similar behavior to SPC/E, where there is no |
| 488 |
> |
real free energy distinction between the various polymorphs because |
| 489 |
> |
many overlap within error. While ice B is close in free energy to the |
| 490 |
> |
other polymorphs, these results fail to support the findings of other |
| 491 |
> |
researchers indicating the preferred form of TIP5P at 1~atm is a |
| 492 |
> |
structure similar to ice B.\cite{Yamada02,Vega05,Abascal05} It should |
| 493 |
> |
be noted that we are looking at TIP5P-E rather than TIP5P, and the |
| 494 |
> |
differences in the Lennard-Jones parameters could be a reason for this |
| 495 |
> |
dissimilarity. Overall, these results indicate that TIP4P-Ew is a |
| 496 |
> |
better mimic of real water than these other models when studying |
| 497 |
> |
crystallization and solid forms of water. |
| 498 |
|
|
| 499 |
|
\section{Conclusions} |
| 500 |
|
|
| 501 |
|
In this work, thermodynamic integration was used to determine the |
| 502 |
|
absolute free energies of several ice polymorphs. The new polymorph, |
| 503 |
< |
Ice-{\it i} was observed to be the stable crystalline state for {\it |
| 503 |
> |
Ice-$i$ was observed to be the stable crystalline state for {\it |
| 504 |
|
all} the water models when using a 9.0~\AA\ cutoff. However, the free |
| 505 |
|
energy partially depends on simulation conditions (particularly on the |
| 506 |
< |
choice of long range correction method). Regardless, Ice-{\it i} was |
| 506 |
> |
choice of long range correction method). Regardless, Ice-$i$ was |
| 507 |
|
still observed to be a stable polymorph for all of the studied water |
| 508 |
|
models. |
| 509 |
|
|
| 510 |
|
So what is the preferred solid polymorph for simulated water? As |
| 511 |
|
indicated above, the answer appears to be dependent both on the |
| 512 |
|
conditions and the model used. In the case of short cutoffs without a |
| 513 |
< |
long-range interaction correction, Ice-{\it i} and Ice-$i^\prime$ have |
| 513 |
> |
long-range interaction correction, Ice-$i$ and Ice-$i^\prime$ have |
| 514 |
|
the lowest free energy of the studied polymorphs with all the models. |
| 515 |
|
Ideally, crystallization of each model under constant pressure |
| 516 |
|
conditions, as was done with SSD/E, would aid in the identification of |
| 519 |
|
insight about important behavior of others. |
| 520 |
|
|
| 521 |
|
We also note that none of the water models used in this study are |
| 522 |
< |
polarizable or flexible models. It is entirely possible that the |
| 523 |
< |
polarizability of real water makes Ice-{\it i} substantially less |
| 524 |
< |
stable than ice I$_h$. However, the calculations presented above seem |
| 525 |
< |
interesting enough to communicate before the role of polarizability |
| 526 |
< |
(or flexibility) has been thoroughly investigated. |
| 522 |
> |
polarizable or flexible models. It is entirely possible that the |
| 523 |
> |
polarizability of real water makes Ice-$i$ substantially less stable |
| 524 |
> |
than ice I$_\textrm{h}$. The dipole moment of the water molecules |
| 525 |
> |
increases as the system becomes more condensed, and the increasing |
| 526 |
> |
dipole moment should destabilize the tetramer structures in |
| 527 |
> |
Ice-$i$. Right now, using TIP4P-Ew with an electrostatic correction |
| 528 |
> |
gives the proper thermodynamically preferred state, and we recommend |
| 529 |
> |
this arrangement for study of crystallization processes if the |
| 530 |
> |
computational cost increase that comes with including polarizability |
| 531 |
> |
is an issue. |
| 532 |
|
|
| 533 |
< |
Finally, due to the stability of Ice-{\it i} in the investigated |
| 533 |
> |
Finally, due to the stability of Ice-$i$ in the investigated |
| 534 |
|
simulation conditions, the question arises as to possible experimental |
| 535 |
|
observation of this polymorph. The rather extensive past and current |
| 536 |
|
experimental investigation of water in the low pressure regime makes |
