| 220 |
|
parameter. This method has been shown to be reversible and provide |
| 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 |
| 435 |
|
\cmidrule(lr){2-6} |
| 436 |
|
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ |
| 437 |
|
\midrule |
| 438 |
< |
TIP5P-E & & & & - & \\ |
| 439 |
< |
TIP4P-Ew & & -13.09(3) & & - & -12.98(3) \\ |
| 440 |
< |
SPC/E & -12.99(3) & -13.00(3) & & - & -12.99(3) \\ |
| 438 |
> |
TIP5P-E & -11.98(4) & -11.96(4) & & - & -11.95(3) \\ |
| 439 |
> |
TIP4P-Ew & -13.11(3) & -13.09(3) & -12.97(3) & - & -12.98(3) \\ |
| 440 |
> |
SPC/E & -12.99(3) & -13.00(3) & -13.03(3) & - & -12.99(3) \\ |
| 441 |
|
SSD/RF & -11.83(3) & -11.66(4) & -12.32(3) & -12.39(3) & - \\ |
| 442 |
|
TRED & -12.61(3) & -12.43(3) & -12.89(3) & -13.12(3) & - \\ |
| 443 |
|
\end{tabular} |
| 450 |
|
the same fashion; however Ice-$i$ and ice B are quite a bit closer in |
| 451 |
|
free energy (nearly isoenergetic). The free energy differences between |
| 452 |
|
ice polymorphs for TRED water parallel SSD/RF, with the exception that |
| 453 |
< |
ice B is destabilized such that it is no longer nearly isoenergetic |
| 454 |
< |
with Ice-$i$. |
| 453 |
> |
ice B is destabilized such that it is not very close to Ice-$i$. The |
| 454 |
> |
SPC/E results really show the near isoenergetic behavior when using |
| 455 |
> |
the electrostatics correction. Ice B has the lowest Helmholtz free |
| 456 |
> |
energy; however, all the polymorph results overlap within error. |
| 457 |
> |
|
| 458 |
> |
The most interesting results from these calculations come from the |
| 459 |
> |
more expensive TIP4P-Ew and TIP5P-E results. Both of these models were |
| 460 |
> |
optimized for use with an electrostatic correction and are |
| 461 |
> |
geometrically arranged to mimic water following two different |
| 462 |
> |
ideas. In TIP5P-E, the primary location for the negative charge in the |
| 463 |
> |
molecule is assigned to the lone-pairs of the oxygen, while TIP4P-Ew |
| 464 |
> |
places the negative charge near the center-of-mass along the H-O-H |
| 465 |
> |
bisector. There is some debate as to which is the proper choice for |
| 466 |
> |
the negative charge location, and this has in part led to a six-site |
| 467 |
> |
water model that balances both of these options.\cite{Vega05,Nada03} |
| 468 |
> |
The limited results in table \ref{tab:dampedFreeEnergy} support the |
| 469 |
> |
results of Vega {\it et al.}, which indicate the TIP4P charge location |
| 470 |
> |
geometry is more physically valid.\cite{Vega05} With the TIP4P-Ew |
| 471 |
> |
water model, the experimentally observed polymorph (ice |
| 472 |
> |
I$_\textrm{h}$) is the preferred form with ice I$_\textrm{c}$ slightly |
| 473 |
> |
higher in energy, though overlapping within error, and the less |
| 474 |
> |
realistic ice B and Ice-$i^\prime$ are destabilized relative to these |
| 475 |
> |
polymorphs. TIP5P-E shows similar behavior to SPC/E, where there is no |
| 476 |
> |
real free energy distinction between the various polymorphs and lend |
| 477 |
> |
credence to other results indicating the preferred form of TIP5P at |
| 478 |
> |
1~atm is a structure similar to ice B.\cite{Yamada02,Vega05,Abascal05} |
| 479 |
> |
These results indicate that TIP4P-Ew is a better mimic of real water |
| 480 |
> |
than these other models when studying crystallization and solid forms |
| 481 |
> |
of water. |
| 482 |
|
|
| 483 |
|
\section{Conclusions} |
| 484 |
|
|
| 485 |
|
In this work, thermodynamic integration was used to determine the |
| 486 |
|
absolute free energies of several ice polymorphs. The new polymorph, |
| 487 |
< |
Ice-{\it i} was observed to be the stable crystalline state for {\it |
| 487 |
> |
Ice-$i$ was observed to be the stable crystalline state for {\it |
| 488 |
|
all} the water models when using a 9.0~\AA\ cutoff. However, the free |
| 489 |
|
energy partially depends on simulation conditions (particularly on the |
| 490 |
< |
choice of long range correction method). Regardless, Ice-{\it i} was |
| 490 |
> |
choice of long range correction method). Regardless, Ice-$i$ was |
| 491 |
|
still observed to be a stable polymorph for all of the studied water |
| 492 |
|
models. |
| 493 |
|
|
| 494 |
|
So what is the preferred solid polymorph for simulated water? As |
| 495 |
|
indicated above, the answer appears to be dependent both on the |
| 496 |
|
conditions and the model used. In the case of short cutoffs without a |
| 497 |
< |
long-range interaction correction, Ice-{\it i} and Ice-$i^\prime$ have |
| 497 |
> |
long-range interaction correction, Ice-$i$ and Ice-$i^\prime$ have |
| 498 |
|
the lowest free energy of the studied polymorphs with all the models. |
| 499 |
|
Ideally, crystallization of each model under constant pressure |
| 500 |
|
conditions, as was done with SSD/E, would aid in the identification of |
| 503 |
|
insight about important behavior of others. |
| 504 |
|
|
| 505 |
|
We also note that none of the water models used in this study are |
| 506 |
< |
polarizable or flexible models. It is entirely possible that the |
| 507 |
< |
polarizability of real water makes Ice-{\it i} substantially less |
| 508 |
< |
stable than ice I$_h$. However, the calculations presented above seem |
| 509 |
< |
interesting enough to communicate before the role of polarizability |
| 510 |
< |
(or flexibility) has been thoroughly investigated. |
| 506 |
> |
polarizable or flexible models. It is entirely possible that the |
| 507 |
> |
polarizability of real water makes Ice-$i$ substantially less stable |
| 508 |
> |
than ice I$_\textrm{h}$. The dipole moment of the water molecules |
| 509 |
> |
increases as the system becomes more condensed, and the increasing |
| 510 |
> |
dipole moment should destabilize the tetramer structures in |
| 511 |
> |
Ice-$i$. Right now, using TIP4P-Ew with an electrostatic correction |
| 512 |
> |
gives the proper thermodynamically preferred state, and we recommend |
| 513 |
> |
this arrangement for study of crystallization processes if the |
| 514 |
> |
computational cost increase that comes with including polarizability |
| 515 |
> |
is an issue. |
| 516 |
|
|
| 517 |
< |
Finally, due to the stability of Ice-{\it i} in the investigated |
| 517 |
> |
Finally, due to the stability of Ice-$i$ in the investigated |
| 518 |
|
simulation conditions, the question arises as to possible experimental |
| 519 |
|
observation of this polymorph. The rather extensive past and current |
| 520 |
|
experimental investigation of water in the low pressure regime makes |
