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Revision 1456 by chrisfen, Tue Sep 14 21:55:24 2004 UTC vs.
Revision 1459 by chrisfen, Wed Sep 15 19:06:05 2004 UTC

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1 +
2   %\documentclass[prb,aps,twocolumn,tabularx]{revtex4}
3   \documentclass[preprint,aps,endfloats]{revtex4}
4   %\documentclass[11pt]{article}
# Line 20 | Line 21
21  
22   \begin{document}
23  
24 < \title{A Free Energy Study of Low Temperature and Anomolous Ice}
24 > \title{A Free Energy Study of Low Temperature and Anomalous Ice}
25  
26   \author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote}
27   \footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}}
# Line 34 | Line 35 | Notre Dame, Indiana 46556}
35   %\doublespacing
36  
37   \begin{abstract}
38 + The free energies of several ice polymorphs in the low pressure regime
39 + were calculated using thermodynamic integration of systems consisting
40 + of a variety of common water models. Ice-{\it i}, a recent
41 + computationally observed solid structure, was determined to be the
42 + stable state with the lowest free energy for all the water models
43 + investigated. Phase diagrams were generated, and melting and boiling
44 + points for all the models were determined and show relatively good
45 + agreement with experiment, although the solid phase is different
46 + between simulation and experiment. In addition, potential truncation
47 + was shown to have an effect on the calculated free energies, and may
48 + result in altered free energy landscapes.
49   \end{abstract}
50  
51   \maketitle
# Line 48 | Line 60 | Notre Dame, Indiana 46556}
60  
61   \section{Introduction}
62  
63 + Molecular dynamics has developed into a valuable tool for studying the
64 + phase behavior of systems ranging from small or simple
65 + molecules\cite{smallStuff} to complex biological
66 + species.\cite{bigStuff} Many techniques have been developed in order
67 + to investigate the thermodynamic properites of model substances,
68 + providing both qualitative and quantitative comparisons between
69 + simulations and experiment.\cite{thermMethods} Investigation of these
70 + properties leads to the development of new and more accurate models,
71 + leading to better understanding and depiction of physical processes
72 + and intricate molecular systems.
73 +
74 + Water has proven to be a challenging substance to depict in
75 + simulations, and has resulted in a variety of models that attempt to
76 + describe its behavior under a varying simulation
77 + conditions.\cite{lotsOfWaterPapers} Many of these models have been
78 + used to investigate important physical phenomena like phase
79 + transitions and the hydrophobic effect.\cite{evenMorePapers} With the
80 + advent of numerous differing models, it is only natural that attention
81 + is placed on the properties of the models themselves in an attempt to
82 + clarify their benefits and limitations when applied to a system of
83 + interest.\cite{modelProps} One important but challenging property to
84 + quantify is the free energy, particularly of the solid forms of
85 + water. Difficulty in these types of studies typically arises from the
86 + assortment of possible crystalline polymorphs that water that water
87 + adopts over a wide range of pressures and temperatures. There are
88 + currently 13 recognized forms of ice, and it is a challenging task to
89 + investigate the entire free energy landscape.\cite{Sanz04} Ideally,
90 + research is focused on the phases having the lowest free energy,
91 + because these phases will dictate the true transition temperatures and
92 + pressures for their respective model.
93 +
94 + In this paper, standard reference state methods were applied to the
95 + study of crystalline water polymorphs in the low pressure regime. This
96 + work is unique in the fact that one of the crystal lattices was
97 + arrived at through crystallization of a computationally efficient
98 + water model under constant pressure and temperature
99 + conditions. Crystallization events are interesting in and of
100 + themselves\cite{nucleationStudies}; however, the crystal structure
101 + obtained in this case was different from any previously observed ice
102 + polymorphs, in experiment or simulation.\cite{Fennell04} This crystal
103 + was termed Ice-{\it i} in homage to its origin in computational
104 + simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight
105 + water molecules that stack in rows of interlocking water
106 + tetramers. Proton ordering can be accomplished by orienting two of the
107 + waters so that both of their donated hydrogen bonds are internal to
108 + their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal
109 + constructed of water tetramers, the hydrogen bonds are not as linear
110 + as those observed in ice $I_h$, however the interlocking of these
111 + subunits appears to provide significant stabilization to the overall
112 + crystal. The arrangement of these tetramers results in surrounding
113 + open octagonal cavities that are typically greater than 6.3 \AA\ in
114 + diameter. This relatively open overall structure leads to crystals
115 + that are 0.07 g/cm$^3$ less dense on average than ice $I_h$.
116 +
117 + Results in the previous study indicated that Ice-{\it i} is the
118 + minimum energy crystal structure for the single point water models
119 + being studied (for discussions on these single point dipole models,
120 + see the previous work and related
121 + articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only
122 + consider energetic stabilization and neglect entropic contributions to
123 + the overall free energy. To address this issue, the absolute free
124 + energy of this crystal was calculated using thermodynamic integration
125 + and compared to the free energies of cubic and hexagonal ice $I$ (the
126 + experimental low density ice polymorphs) and ice B (a higher density,
127 + but very stable crystal structure observed by B\`{a}ez and Clancy in
128 + free energy studies of SPC/E).\cite{Baez95b} This work includes
129 + results for the water model from which Ice-{\it i} was crystallized
130 + (soft sticky dipole extended, SSD/E) in addition to several common
131 + water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field
132 + parametrized single point dipole water model (soft sticky dipole
133 + reaction field, SSD/RF). In should be noted that a second version of
134 + Ice-{\it i} (Ice-2{\it i}) was used in calculations involving SPC/E,
135 + TIP4P, and TIP5P. The unit cell of this crystal (Fig. \ref{iceiCell}B)
136 + is similar to the Ice-{\it i} unit it is extended in the direction of
137 + the (001) face and compressed along the other two faces.
138 +
139   \section{Methods}
140  
141   Canonical ensemble (NVT) molecular dynamics calculations were
142   performed using the OOPSE (Object-Oriented Parallel Simulation Engine)
143   molecular mechanics package. All molecules were treated as rigid
144 < bodies, with orientational motion propogated using the symplectic DLM
144 > bodies, with orientational motion propagated using the symplectic DLM
145   integration method. Details about the implementation of these
146   techniques can be found in a recent publication.\cite{Meineke05}
147  
# Line 65 | Line 153 | temperatures.
153   phase diagrams. All simulations were carried out at densities
154   resulting in a pressure of approximately 1 atm at their respective
155   temperatures.
156 +
157 + A single thermodynamic integration involves a sequence of simulations
158 + over which the system of interest is converted into a reference system
159 + for which the free energy is known. This transformation path is then
160 + integrated in order to determine the free energy difference between
161 + the two states:
162 + \begin{equation}
163 + \Delta A = \int_0^1\left\langle\frac{\partial V(\lambda
164 + )}{\partial\lambda}\right\rangle_\lambda d\lambda,
165 + \end{equation}
166 + where $V$ is the interaction potential and $\lambda$ is the
167 + transformation parameter that scales the overall
168 + potential. Simulations are distributed unevenly along this path in
169 + order to sufficiently sample the regions of greatest change in the
170 + potential. Typical integrations in this study consisted of $\sim$25
171 + simulations ranging from 300 ps (for the unaltered system) to 75 ps
172 + (near the reference state) in length.
173  
174   For the thermodynamic integration of molecular crystals, the Einstein
175   Crystal is chosen as the reference state that the system is converted
# Line 117 | Line 222 | propogate the motion of rigid-bodies, and provides the
222   of an Ewald summation were estimated for TIP3P and SPC/E by performing
223   calculations with Particle-Mesh Ewald (PME) in the TINKER molecular
224   mechanics software package. TINKER was chosen because it can also
225 < propogate the motion of rigid-bodies, and provides the most direct
225 > propagate the motion of rigid-bodies, and provides the most direct
226   comparison to the results from OOPSE. The calculated energy difference
227   in the presence and absence of PME was applied to the previous results
228   in order to predict changes in the free energy landscape.
# Line 235 | Line 340 | advantagious in that it facilitated the spontaneous cr
340   at 355 K. This is due to the significant stability of Ice-{\it i} over
341   all other polymorphs for this particular model under these
342   conditions. While troubling, this behavior turned out to be
343 < advantagious in that it facilitated the spontaneous crystallization of
343 > advantageous in that it facilitated the spontaneous crystallization of
344   Ice-{\it i}. These observations provide a warning that simulations of
345   SSD/E as a ``liquid'' near 300 K are actually metastable and run the
346   risk of spontaneous crystallization. However, this risk changes when
347   applying a longer cutoff.
348  
349 + \begin{figure}
350 + \includegraphics[width=\linewidth]{cutoffChange.eps}
351 + \caption{Free energy as a function of cutoff radius for (A) SSD/E, (B)
352 + TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12
353 + \AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9
354 + \AA\. These crystals are unstable at 200 K and rapidly convert into a
355 + liquid. The connecting lines are qualitative visual aid.}
356 + \label{incCutoff}
357 + \end{figure}
358  
359 + Increasing the cutoff radius in simulations of the more
360 + computationally efficient water models was done in order to evaluate
361 + the trend in free energy values when moving to systems that do not
362 + involve potential truncation. As seen in Fig. \ref{incCutoff}, the
363 + free energy of all the ice polymorphs show a substantial dependence on
364 + cutoff radius. In general, there is a narrowing of the free energy
365 + differences while moving to greater cutoff radius. Interestingly, by
366 + increasing the cutoff radius, the free energy gap was narrowed enough
367 + in the SSD/E model that the liquid state is preferred under standard
368 + simulation conditions (298 K and 1 atm). Thus, it is recommended that
369 + simulations using this model choose interaction truncation radii
370 + greater than 9 \AA\. This narrowing trend is much more subtle in the
371 + case of SSD/RF, indicating that the free energies calculated with a
372 + reaction field present provide a more accurate picture of the free
373 + energy landscape in the absence of potential truncation.
374  
375 + To further study the changes resulting to the inclusion of a
376 + long-range interaction correction, the effect of an Ewald summation
377 + was estimated by applying the potential energy difference do to its
378 + inclusion in systems in the presence and absence of the
379 + correction. This was accomplished by calculation of the potential
380 + energy of identical crystals with and without PME using TINKER. The
381 + free energies for the investigated polymorphs using the TIP3P and
382 + SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P
383 + are not fully supported in TINKER, so the results for these models
384 + could not be estimated. The same trend pointed out through increase of
385 + cutoff radius is observed in these PME results. Ice-{\it i} is the
386 + preferred polymorph at ambient conditions for both the TIP3P and SPC/E
387 + water models; however, there is a narrowing of the free energy
388 + differences between the various solid forms. In the case of SPC/E this
389 + narrowing is significant enough that it becomes less clear cut that
390 + Ice-{\it i} is the most stable polymorph, and is possibly metastable
391 + with respect to ice B and possibly ice $I_c$. However, these results
392 + do not significantly alter the finding that the Ice-{\it i} polymorph
393 + is a stable crystal structure that should be considered when studying
394 + the phase behavior of water models.
395 +
396 + \begin{table*}
397 + \begin{minipage}{\linewidth}
398 + \renewcommand{\thefootnote}{\thempfootnote}
399 + \begin{center}
400 + \caption{The free energy of the studied ice polymorphs after applying
401 + the energy difference attributed to the inclusion of the PME
402 + long-range interaction correction. Units are kcal/mol.}
403 + \begin{tabular}{ l  c  c  c  c }
404 + \hline \\[-7mm]
405 + \ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
406 + \hline \\[-3mm]
407 + \ \ TIP3P  & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\
408 + \ \ SPC/E  & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\
409 + \end{tabular}
410 + \label{pmeShift}
411 + \end{center}
412 + \end{minipage}
413 + \end{table*}
414 +
415   \section{Conclusions}
416  
417 + The free energy for proton ordered variants of hexagonal and cubic ice
418 + $I$, ice B, and recently discovered Ice-{\it i} where calculated under
419 + standard conditions for several common water models via thermodynamic
420 + integration. All the water models studied show Ice-{\it i} to be the
421 + minimum free energy crystal structure in the with a 9 \AA\ switching
422 + function cutoff. Calculated melting and boiling points show
423 + surprisingly good agreement with the experimental values; however, the
424 + solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of
425 + interaction truncation was investigated through variation of the
426 + cutoff radius, use of a reaction field parameterized model, and
427 + estimation of the results in the presence of the Ewald summation
428 + correction. Interaction truncation has a significant effect on the
429 + computed free energy values, and may significantly alter the free
430 + energy landscape for the more complex multipoint water models. Despite
431 + these effects, these results show Ice-{\it i} to be an important ice
432 + polymorph that should be considered in simulation studies.
433 +
434 + Due to this relative stability of Ice-{\it i} in all manner of
435 + investigated simulation examples, the question arises as to possible
436 + experimental observation of this polymorph. The rather extensive past
437 + and current experimental investigation of water in the low pressure
438 + regime leads the authors to be hesitant in ascribing relevance outside
439 + of computational models, hence the descriptive name presented. That
440 + being said, there are certain experimental conditions that would
441 + provide the most ideal situation for possible observation. These
442 + include the negative pressure or stretched solid regime, small
443 + clusters in vacuum deposition environments, and in clathrate
444 + structures involving small non-polar molecules.
445 +
446   \section{Acknowledgments}
447   Support for this project was provided by the National Science
448   Foundation under grant CHE-0134881. Computation time was provided by
449 < the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant
450 < DMR-0079647.
449 > the Notre Dame High Performance Computing Cluster and the Notre Dame
450 > Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647).
451  
452   \newpage
453  

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