| 107 |
|
isobaric-isothermal ({\it NPT}) simulations performed at 1 atm and |
| 108 |
|
200~K. Each model (and each crystal structure) was allowed to relax for |
| 109 |
|
300~ps in the {\it NPT} ensemble before averaging the density to obtain |
| 110 |
< |
the volumes for the {\it NVT} simulations.All molecules were treated |
| 110 |
> |
the volumes for the {\it NVT} simulations. All molecules were treated |
| 111 |
|
as rigid bodies, with orientational motion propagated using the |
| 112 |
|
symplectic DLM integration method described in section |
| 113 |
|
\ref{sec:IntroIntegrate}. |
| 119 |
|
been used extensively in the calculation of free energies for |
| 120 |
|
condensed phases of |
| 121 |
|
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
| 122 |
< |
method uses a sequence of simulations over which the system of |
| 122 |
> |
method uses a sequence of simulations during which the system of |
| 123 |
|
interest is converted into a reference system for which the free |
| 124 |
|
energy is known analytically ($A_0$). The difference in potential |
| 125 |
|
energy between the reference system and the system of interest |
| 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 |
| 316 |
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
| 317 |
|
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 318 |
|
the experimental values; however, the solid phases shown are not the |
| 319 |
< |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
| 319 |
> |
experimentally observed forms. Both cubic and hexagonal ice I are |
| 320 |
|
higher in energy and don't appear in the phase diagram.} |
| 321 |
|
\label{fig:tp3PhaseDia} |
| 322 |
|
\end{figure} |
| 359 |
|
B and Ice-{\it i} were omitted, a $T_\textrm{m}$ value around 200~K |
| 360 |
|
would be predicted from this work. However, the $T_\textrm{m}$ from |
| 361 |
|
Ice-{\it i} is calculated to be 262~K, indicating that these |
| 362 |
< |
simulation based structures ought to be included in studies probing |
| 362 |
> |
simulation-based structures ought to be included in studies probing |
| 363 |
|
phase transitions with this model. Also of interest in these results |
| 364 |
|
is that SSD/E does not exhibit a melting point at 1 atm but does |
| 365 |
|
sublime at 355~K. This is due to the significant stability of |
| 447 |
|
\cmidrule(lr){2-6} |
| 448 |
|
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ |
| 449 |
|
\midrule |
| 450 |
< |
TIP5P-E & -11.98(4) & -11.96(4) & & - & -11.95(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) & - \\ |
| 458 |
|
The results of these calculations in table \ref{tab:dampedFreeEnergy} |
| 459 |
|
show similar behavior to the Ewald results in figure |
| 460 |
|
\ref{fig:incCutoff}, at least for SSD/RF and SPC/E which are present |
| 461 |
< |
in both. The ice polymorph Helmholtz free energies for SSD/RF order in |
| 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 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. |
| 461 |
> |
in both. The Helmholtz free energies of the ice polymorphs for SSD/RF |
| 462 |
> |
order in the same fashion; however Ice-$i$ and ice B are quite a bit |
| 463 |
> |
closer in free energy (nearly isoenergetic). The free energy |
| 464 |
> |
differences between ice polymorphs for TRED water parallel SSD/RF, |
| 465 |
> |
with the exception that ice B is destabilized such that it is not very |
| 466 |
> |
close to Ice-$i$. The SPC/E results really show the near isoenergetic |
| 467 |
> |
behavior when using the electrostatic correction. Ice B has the lowest |
| 468 |
> |
Helmholtz free energy; however, all the polymorph results overlap |
| 469 |
> |
within error. |
| 470 |
|
|
| 471 |
|
The most interesting results from these calculations come from the |
| 472 |
|
more expensive TIP4P-Ew and TIP5P-E results. Both of these models were |
| 484 |
|
water model, the experimentally observed polymorph (ice |
| 485 |
|
I$_\textrm{h}$) is the preferred form with ice I$_\textrm{c}$ slightly |
| 486 |
|
higher in energy, though overlapping within error, and the less |
| 487 |
< |
realistic ice B and Ice-$i^\prime$ are destabilized relative to these |
| 488 |
< |
polymorphs. TIP5P-E shows similar behavior to SPC/E, where there is no |
| 489 |
< |
real free energy distinction between the various polymorphs and lend |
| 490 |
< |
credence to other results indicating the preferred form of TIP5P at |
| 491 |
< |
1~atm is a structure similar to ice B.\cite{Yamada02,Vega05,Abascal05} |
| 492 |
< |
These results indicate that TIP4P-Ew is a better mimic of real water |
| 493 |
< |
than these other models when studying crystallization and solid forms |
| 494 |
< |
of water. |
| 487 |
> |
realistic ice B and Ice-$i^\prime$ structures are destabilized |
| 488 |
> |
relative to these polymorphs. TIP5P-E shows similar behavior to SPC/E, |
| 489 |
> |
where there is no real free energy distinction between the various |
| 490 |
> |
polymorphs because many overlap within error. While ice B is close in |
| 491 |
> |
free energy to the other polymorphs, these results fail to support the |
| 492 |
> |
findings of other researchers indicating the preferred form of TIP5P |
| 493 |
> |
at 1~atm is a structure similar to ice |
| 494 |
> |
B.\cite{Yamada02,Vega05,Abascal05} It should be noted that we are |
| 495 |
> |
looking at TIP5P-E rather than TIP5P, and the differences in the |
| 496 |
> |
Lennard-Jones parameters could be a reason for this dissimilarity. |
| 497 |
> |
Overall, these results indicate that TIP4P-Ew is a better mimic of |
| 498 |
> |
real water than these other models when studying crystallization and |
| 499 |
> |
solid forms of water. |
| 500 |
|
|
| 501 |
|
\section{Conclusions} |
| 502 |
|
|
| 503 |
|
In this work, thermodynamic integration was used to determine the |
| 504 |
|
absolute free energies of several ice polymorphs. The new polymorph, |
| 505 |
< |
Ice-{\it i} was observed to be the stable crystalline state for {\it |
| 505 |
> |
Ice-$i$ was observed to be the stable crystalline state for {\it |
| 506 |
|
all} the water models when using a 9.0~\AA\ cutoff. However, the free |
| 507 |
|
energy partially depends on simulation conditions (particularly on the |
| 508 |
< |
choice of long range correction method). Regardless, Ice-{\it i} was |
| 508 |
> |
choice of long range correction method). Regardless, Ice-$i$ was |
| 509 |
|
still observed to be a stable polymorph for all of the studied water |
| 510 |
|
models. |
| 511 |
|
|
| 512 |
|
So what is the preferred solid polymorph for simulated water? As |
| 513 |
|
indicated above, the answer appears to be dependent both on the |
| 514 |
|
conditions and the model used. In the case of short cutoffs without a |
| 515 |
< |
long-range interaction correction, Ice-{\it i} and Ice-$i^\prime$ have |
| 515 |
> |
long-range interaction correction, Ice-$i$ and Ice-$i^\prime$ have |
| 516 |
|
the lowest free energy of the studied polymorphs with all the models. |
| 517 |
|
Ideally, crystallization of each model under constant pressure |
| 518 |
|
conditions, as was done with SSD/E, would aid in the identification of |
| 521 |
|
insight about important behavior of others. |
| 522 |
|
|
| 523 |
|
We also note that none of the water models used in this study are |
| 524 |
< |
polarizable or flexible models. It is entirely possible that the |
| 525 |
< |
polarizability of real water makes Ice-{\it i} substantially less |
| 526 |
< |
stable than ice I$_h$. However, the calculations presented above seem |
| 527 |
< |
interesting enough to communicate before the role of polarizability |
| 528 |
< |
(or flexibility) has been thoroughly investigated. |
| 524 |
> |
polarizable or flexible models. It is entirely possible that the |
| 525 |
> |
polarizability of real water makes the Ice-$i$ structure substantially |
| 526 |
> |
less stable than ice I$_\textrm{h}$. The dipole moment of the water |
| 527 |
> |
molecules increases as the system becomes more condensed, and the |
| 528 |
> |
increasing dipole moment should destabilize the tetramer structures in |
| 529 |
> |
Ice-$i$. Right now, using TIP4P-Ew with an electrostatic correction |
| 530 |
> |
gives the proper thermodynamically preferred state, and we recommend |
| 531 |
> |
this arrangement for study of crystallization processes if the |
| 532 |
> |
computational cost increase that comes with including polarizability |
| 533 |
> |
is an issue. |
| 534 |
|
|
| 535 |
< |
Finally, due to the stability of Ice-{\it i} in the investigated |
| 535 |
> |
Finally, due to the stability of Ice-$i$ in the investigated |
| 536 |
|
simulation conditions, the question arises as to possible experimental |
| 537 |
|
observation of this polymorph. The rather extensive past and current |
| 538 |
|
experimental investigation of water in the low pressure regime makes |
| 547 |
|
results, we have calculated the oxygen-oxygen pair correlation |
| 548 |
|
function, $g_\textrm{OO}(r)$, and the structure factor, $S(\vec{q})$ |
| 549 |
|
for the two Ice-{\it i} variants (along with example ice |
| 550 |
< |
I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77~K, and they are shown in |
| 551 |
< |
figures \ref{fig:gofr} and \ref{fig:sofq} respectively. It is |
| 550 |
> |
I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77~K, and they are shown |
| 551 |
> |
in figures \ref{fig:gofr} and \ref{fig:sofq} respectively. It is |
| 552 |
|
interesting to note that the structure factors for Ice-$i^\prime$ and |
| 553 |
|
Ice-I$_c$ are quite similar. The primary differences are small peaks |
| 554 |
|
at 1.125, 2.29, and 2.53~\AA$^{-1}$, so particular attention to these |
| 555 |
< |
regions would be needed to identify the new $i^\prime$ variant from |
| 556 |
< |
the I$_\textrm{c}$ polymorph. |
| 555 |
> |
regions would be needed to distinguish Ice-$i^\prime$ from ice |
| 556 |
> |
I$_\textrm{c}$. |
| 557 |
|
|
| 558 |
|
|
| 559 |
|
\begin{figure} |
| 568 |
|
\includegraphics[width=\linewidth]{./figures/sofq.pdf} |
| 569 |
|
\caption{Predicted structure factors for Ice-{\it i} and ice |
| 570 |
|
I$_\textrm{c}$ at 77~K. The raw structure factors have been |
| 571 |
< |
convoluted with a gaussian instrument function (0.075~\AA$^{-1}$ |
| 571 |
> |
convoluted with a Gaussian instrument function (0.075~\AA$^{-1}$ |
| 572 |
|
width) to compensate for the truncation effects in our finite size |
| 573 |
< |
simulations. The labeled peaks compared favorably with ``spurious'' |
| 551 |
< |
peaks observed in experimental studies of amorphous solid |
| 552 |
< |
water.\cite{Bizid87}} |
| 573 |
> |
simulations.} |
| 574 |
|
\label{fig:sofq} |
| 575 |
|
\end{figure} |
| 576 |
|
|
