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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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isobaric-isothermal ({\it NPT}) simulations performed at 1 atm and |
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200~K. Each model (and each crystal structure) was allowed to relax for |
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300~ps in the {\it NPT} ensemble before averaging the density to obtain |
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the volumes for the {\it NVT} simulations.All molecules were treated |
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the volumes for the {\it NVT} simulations. All molecules were treated |
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as rigid bodies, with orientational motion propagated using the |
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symplectic DLM integration method described in section |
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\ref{sec:IntroIntegrate}. |
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been used extensively in the calculation of free energies for |
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condensed phases of |
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materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
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method uses a sequence of simulations over which the system of |
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method uses a sequence of simulations during which the system of |
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interest is converted into a reference system for which the free |
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energy is known analytically ($A_0$). The difference in potential |
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energy between the reference system and the system of interest |
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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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\caption{Phase diagram for the TIP3P water model in the low pressure |
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regime. The displayed $T_m$ and $T_b$ values are good predictions of |
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the experimental values; however, the solid phases shown are not the |
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experimentally observed forms. Both cubic and hexagonal ice $I$ are |
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experimentally observed forms. Both cubic and hexagonal ice I are |
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higher in energy and don't appear in the phase diagram.} |
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\label{fig:tp3PhaseDia} |
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\end{figure} |
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B and Ice-{\it i} were omitted, a $T_\textrm{m}$ value around 200~K |
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would be predicted from this work. However, the $T_\textrm{m}$ from |
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Ice-{\it i} is calculated to be 262~K, indicating that these |
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simulation based structures ought to be included in studies probing |
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simulation-based structures ought to be included in studies probing |
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phase transitions with this model. Also of interest in these results |
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is that SSD/E does not exhibit a melting point at 1 atm but does |
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sublime at 355~K. This is due to the significant stability of |
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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 Helmholtz free energies of the ice polymorphs for SSD/RF |
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order in the same fashion; however Ice-$i$ and ice B are quite a bit |
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closer in free energy (nearly isoenergetic). The free energy |
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differences between ice polymorphs for TRED water parallel SSD/RF, |
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with the exception that ice B is destabilized such that it is not very |
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close to Ice-$i$. The SPC/E results really show the near isoenergetic |
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behavior when using the electrostatic correction. Ice B has the lowest |
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Helmholtz free energy; however, all the polymorph results overlap |
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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-$i^\prime$ structures are destabilized |
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relative to these polymorphs. TIP5P-E shows similar behavior to SPC/E, |
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where there is no real free energy distinction between the various |
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polymorphs because many overlap within error. While ice B is close in |
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free energy to the other polymorphs, these results fail to support the |
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findings of other researchers indicating the preferred form of TIP5P |
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at 1~atm is a structure similar to ice |
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B.\cite{Yamada02,Vega05,Abascal05} It should be noted that we are |
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looking at TIP5P-E rather than TIP5P, and the differences in the |
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Lennard-Jones parameters could be a reason for this dissimilarity. |
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Overall, these results indicate that TIP4P-Ew is a better mimic of |
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real water than these other models when studying crystallization and |
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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 the Ice-$i$ structure substantially |
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less stable than ice I$_\textrm{h}$. The dipole moment of the water |
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molecules increases as the system becomes more condensed, and the |
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increasing 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 |
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results, we have calculated the oxygen-oxygen pair correlation |
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function, $g_\textrm{OO}(r)$, and the structure factor, $S(\vec{q})$ |
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for the two Ice-{\it i} variants (along with example ice |
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I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77~K, and they are shown in |
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figures \ref{fig:gofr} and \ref{fig:sofq} respectively. It is |
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I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77~K, and they are shown |
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in figures \ref{fig:gofr} and \ref{fig:sofq} respectively. It is |
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interesting to note that the structure factors for Ice-$i^\prime$ and |
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Ice-I$_c$ are quite similar. The primary differences are small peaks |
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at 1.125, 2.29, and 2.53~\AA$^{-1}$, so particular attention to these |
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regions would be needed to identify the new $i^\prime$ variant from |
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the I$_\textrm{c}$ polymorph. |
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regions would be needed to distinguish Ice-$i^\prime$ from ice |
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I$_\textrm{c}$. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{./figures/sofq.pdf} |
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\caption{Predicted structure factors for Ice-{\it i} and ice |
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I$_\textrm{c}$ at 77~K. The raw structure factors have been |
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convoluted with a gaussian instrument function (0.075~\AA$^{-1}$ |
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convoluted with a Gaussian instrument function (0.075~\AA$^{-1}$ |
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width) to compensate for the truncation effects in our finite size |
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simulations. The labeled peaks compared favorably with ``spurious'' |
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peaks observed in experimental studies of amorphous solid |
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water.\cite{Bizid87}} |
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simulations.} |
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\label{fig:sofq} |
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\end{figure} |
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