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\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER SIMULATIONS} |
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\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER \\ SIMULATIONS} |
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As discussed in the previous chapter, water has proven to be a |
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challenging substance to depict in simulations, and a variety of |
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results in excellent agreement with other established |
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methods.\cite{Baez95b} |
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The Helmholtz free energy error was determined in the same manner in |
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both the solid and the liquid free energy calculations . At each point |
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along the integration path, we calculated the standard deviation of |
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the potential energy difference. Addition or subtraction of these |
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values to each of their respective points and integrating the curve |
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again provides the upper and lower bounds of the uncertainty in the |
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Helmholtz free energy. |
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Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and |
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Lennard-Jones interactions were gradually reduced by a cubic switching |
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function. By applying this function, these interactions are smoothly |
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of the ice polymorphs.\cite{Ponder87} The calculated energy difference |
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in the presence and absence of PME was applied to the previous results |
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in order to predict changes to the free energy landscape. |
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In addition to the above procedures, we also tested how the inclusion |
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of the Lennard-Jones long-range correction affects the free energy |
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results. The correction for the Lennard-Jones trucation was included |
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by integration of the equation discussed in section |
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\ref{sec:LJCorrections}. Rather than discuss its affect alongside the |
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free energy results, we will just mention that while the correction |
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does lower the free energy of the higher density states more than the |
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lower density states, the effect is so small that it is entirely |
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overwelmed by the error in the free energy calculation. Since its |
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inclusion does not influence the results, the Lennard-Jones correction |
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was omitted from all the calculations below. |
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\section{Initial Free Energy Results} |
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\cmidrule(lr){2-6} |
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& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ |
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\midrule |
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TIP5P-E & -10.76(4) & -10.72(4) & & - & -10.68(4) \\ |
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TIP4P-Ew & & -11.77(3) & & - & -11.60(3) \\ |
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SPC/E & -12.98(3) & -11.60(3) & & - & -12.93(3) \\ |
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SSD/RF & -11.81(4) & -11.65(3) & & -12.41(4) & - \\ |
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TRED & -12.58(3) & -12.44(3) & & -13.09(4) & - \\ |
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TIP5P-E & -11.98(4) & -11.96(4) & -11.87(3) & - & -11.95(3) \\ |
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TIP4P-Ew & -13.11(3) & -13.09(3) & -12.97(3) & - & -12.98(3) \\ |
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SPC/E & -12.99(3) & -13.00(3) & -13.03(3) & - & -12.99(3) \\ |
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SSD/RF & -11.83(3) & -11.66(4) & -12.32(3) & -12.39(3) & - \\ |
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TRED & -12.61(3) & -12.43(3) & -12.89(3) & -13.12(3) & - \\ |
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\end{tabular} |
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\label{tab:dampedFreeEnergy} |
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\end{table} |
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The results of these calculations in table \ref{tab:dampedFreeEnergy} |
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show similar behavior to the Ewald results in figure |
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\ref{fig:incCutoff}, at least for SSD/RF and SPC/E which are present |
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in both. The ice polymorph Helmholtz free energies for SSD/RF order in |
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the same fashion; however Ice-$i$ and ice B are quite a bit closer in |
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free energy (nearly isoenergetic). The free energy differences between |
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ice polymorphs for TRED water parallel SSD/RF, with the exception that |
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ice B is destabilized such that it is not very close to Ice-$i$. The |
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SPC/E results really show the near isoenergetic behavior when using |
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the electrostatics correction. Ice B has the lowest Helmholtz free |
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energy; however, all the polymorph results overlap within error. |
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The most interesting results from these calculations come from the |
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more expensive TIP4P-Ew and TIP5P-E results. Both of these models were |
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optimized for use with an electrostatic correction and are |
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geometrically arranged to mimic water following two different |
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ideas. In TIP5P-E, the primary location for the negative charge in the |
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molecule is assigned to the lone-pairs of the oxygen, while TIP4P-Ew |
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places the negative charge near the center-of-mass along the H-O-H |
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bisector. There is some debate as to which is the proper choice for |
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the negative charge location, and this has in part led to a six-site |
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water model that balances both of these options.\cite{Vega05,Nada03} |
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The limited results in table \ref{tab:dampedFreeEnergy} support the |
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results of Vega {\it et al.}, which indicate the TIP4P charge location |
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geometry is more physically valid.\cite{Vega05} With the TIP4P-Ew |
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water model, the experimentally observed polymorph (ice |
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I$_\textrm{h}$) is the preferred form with ice I$_\textrm{c}$ slightly |
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higher in energy, though overlapping within error, and the less |
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realistic ice B and Ice-$i^\prime$ are destabilized relative to these |
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polymorphs. TIP5P-E shows similar behavior to SPC/E, where there is no |
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real free energy distinction between the various polymorphs because |
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many overlap within error. While ice B is close in free energy to the |
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other polymorphs, these results fail to support the findings of other |
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researchers indicating the preferred form of TIP5P at 1~atm is a |
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structure similar to ice B.\cite{Yamada02,Vega05,Abascal05} It should |
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be noted that we are looking at TIP5P-E rather than TIP5P, and the |
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differences in the Lennard-Jones parameters could be a reason for this |
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dissimilarity. Overall, these results indicate that TIP4P-Ew is a |
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better mimic of real water than these other models when studying |
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crystallization and solid forms of water. |
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\section{Conclusions} |
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In this work, thermodynamic integration was used to determine the |
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absolute free energies of several ice polymorphs. The new polymorph, |
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Ice-{\it i} was observed to be the stable crystalline state for {\it |
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Ice-$i$ was observed to be the stable crystalline state for {\it |
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all} the water models when using a 9.0~\AA\ cutoff. However, the free |
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energy partially depends on simulation conditions (particularly on the |
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choice of long range correction method). Regardless, Ice-{\it i} was |
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choice of long range correction method). Regardless, Ice-$i$ was |
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still observed to be a stable polymorph for all of the studied water |
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models. |
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So what is the preferred solid polymorph for simulated water? As |
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indicated above, the answer appears to be dependent both on the |
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conditions and the model used. In the case of short cutoffs without a |
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long-range interaction correction, Ice-{\it i} and Ice-$i^\prime$ have |
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long-range interaction correction, Ice-$i$ and Ice-$i^\prime$ have |
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the lowest free energy of the studied polymorphs with all the models. |
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Ideally, crystallization of each model under constant pressure |
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conditions, as was done with SSD/E, would aid in the identification of |
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insight about important behavior of others. |
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We also note that none of the water models used in this study are |
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polarizable or flexible models. It is entirely possible that the |
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polarizability of real water makes Ice-{\it i} substantially less |
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stable than ice I$_h$. However, the calculations presented above seem |
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interesting enough to communicate before the role of polarizability |
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(or flexibility) has been thoroughly investigated. |
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polarizable or flexible models. It is entirely possible that the |
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polarizability of real water makes Ice-$i$ substantially less stable |
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than ice I$_\textrm{h}$. The dipole moment of the water molecules |
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increases as the system becomes more condensed, and the increasing |
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dipole moment should destabilize the tetramer structures in |
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Ice-$i$. Right now, using TIP4P-Ew with an electrostatic correction |
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gives the proper thermodynamically preferred state, and we recommend |
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this arrangement for study of crystallization processes if the |
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computational cost increase that comes with including polarizability |
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is an issue. |
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Finally, due to the stability of Ice-{\it i} in the investigated |
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Finally, due to the stability of Ice-$i$ in the investigated |
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simulation conditions, the question arises as to possible experimental |
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observation of this polymorph. The rather extensive past and current |
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experimental investigation of water in the low pressure regime makes |
