ViewVC Help
View File | Revision Log | Show Annotations | View Changeset | Root Listing
root/group/trunk/iceiPaper/iceiPaper.tex
(Generate patch)

Comparing trunk/iceiPaper/iceiPaper.tex (file contents):
Revision 1458 by chrisfen, Wed Sep 15 06:34:49 2004 UTC vs.
Revision 1464 by chrisfen, Wed Sep 15 21:44:27 2004 UTC

# Line 1 | Line 1
1   %\documentclass[prb,aps,twocolumn,tabularx]{revtex4}
2 < \documentclass[preprint,aps,endfloats]{revtex4}
2 > \documentclass[11pt]{article}
3   %\documentclass[11pt]{article}
4 < %\usepackage{endfloat}
4 > \usepackage{endfloat}
5   \usepackage{amsmath}
6   \usepackage{epsf}
7   \usepackage{berkeley}
8 < %\usepackage{setspace}
9 < %\usepackage{tabularx}
8 > \usepackage{setspace}
9 > \usepackage{tabularx}
10   \usepackage{graphicx}
11 < %\usepackage[ref]{overcite}
12 < %\pagestyle{plain}
13 < %\pagenumbering{arabic}
14 < %\oddsidemargin 0.0cm \evensidemargin 0.0cm
15 < %\topmargin -21pt \headsep 10pt
16 < %\textheight 9.0in \textwidth 6.5in
17 < %\brokenpenalty=10000
11 > \usepackage[ref]{overcite}
12 > \pagestyle{plain}
13 > \pagenumbering{arabic}
14 > \oddsidemargin 0.0cm \evensidemargin 0.0cm
15 > \topmargin -21pt \headsep 10pt
16 > \textheight 9.0in \textwidth 6.5in
17 > \brokenpenalty=10000
18 > \renewcommand{\baselinestretch}{1.2}
19 > \renewcommand\citemid{\ } % no comma in optional reference note
20  
19 – %\renewcommand\citemid{\ } % no comma in optional reference note
20 –
21   \begin{document}
22  
23 < \title{A Free Energy Study of Low Temperature and Anomolous Ice}
23 > \title{Ice-{\it i}: a novel ice polymorph predicted via computer simulation}
24  
25 < \author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote}
26 < \footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}}
27 <
28 < \address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\
25 > \author{Christopher J. Fennell and J. Daniel Gezelter \\
26 > Department of Chemistry and Biochemistry\\ University of Notre Dame\\
27   Notre Dame, Indiana 46556}
28  
29   \date{\today}
30  
31 < %\maketitle
31 > \maketitle
32   %\doublespacing
33  
34   \begin{abstract}
35 + The free energies of several ice polymorphs in the low pressure regime
36 + were calculated using thermodynamic integration.  These integrations
37 + were done for most of the common water models. Ice-{\it i}, a
38 + structure we recently observed to be stable in one of the single-point
39 + water models, was determined to be the stable crystalline state (at 1
40 + atm) for {\it all} the water models investigated.  Phase diagrams were
41 + generated, and phase coexistence lines were determined for all of the
42 + known low-pressure ice structures under all of the common water
43 + models.  Additionally, potential truncation was shown to have an
44 + effect on the calculated free energies, and can result in altered free
45 + energy landscapes.
46   \end{abstract}
47  
39 – \maketitle
40 –
41 – \newpage
42 –
48   %\narrowtext
49  
50   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# Line 47 | Line 52 | Notre Dame, Indiana 46556}
52   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
53  
54   \section{Introduction}
55 +
56 + Molecular dynamics is a valuable tool for studying the phase behavior
57 + of systems ranging from small or simple
58 + molecules\cite{Matsumoto02andOthers} to complex biological
59 + species.\cite{bigStuff} Many techniques have been developed to
60 + investigate the thermodynamic properites of model substances,
61 + providing both qualitative and quantitative comparisons between
62 + simulations and experiment.\cite{thermMethods} Investigation of these
63 + properties leads to the development of new and more accurate models,
64 + leading to better understanding and depiction of physical processes
65 + and intricate molecular systems.
66 +
67 + Water has proven to be a challenging substance to depict in
68 + simulations, and a variety of models have been developed to describe
69 + its behavior under varying simulation
70 + conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04}
71 + These models have been used to investigate important physical
72 + phenomena like phase transitions and the hydrophobic
73 + effect.\cite{Yamada02} With the choice of models available, it
74 + is only natural to compare the models under interesting thermodynamic
75 + conditions in an attempt to clarify the limitations of each of the
76 + models.\cite{modelProps} Two important property to quantify are the
77 + Gibbs and Helmholtz free energies, particularly for the solid forms of
78 + water.  Difficulty in these types of studies typically arises from the
79 + assortment of possible crystalline polymorphs that water adopts over a
80 + wide range of pressures and temperatures. There are currently 13
81 + recognized forms of ice, and it is a challenging task to investigate
82 + the entire free energy landscape.\cite{Sanz04} Ideally, research is
83 + focused on the phases having the lowest free energy at a given state
84 + point, because these phases will dictate the true transition
85 + temperatures and pressures for their respective model.
86 +
87 + In this paper, standard reference state methods were applied to the
88 + study of crystalline water polymorphs in the low pressure regime. This
89 + work is unique in the fact that one of the crystal lattices was
90 + arrived at through crystallization of a computationally efficient
91 + water model under constant pressure and temperature
92 + conditions. Crystallization events are interesting in and of
93 + themselves\cite{Matsumoto02,Yamada02}; however, the crystal structure
94 + obtained in this case was different from any previously observed ice
95 + polymorphs, in experiment or simulation.\cite{Fennell04} This crystal
96 + was termed Ice-{\it i} in homage to its origin in computational
97 + simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight
98 + water molecules that stack in rows of interlocking water
99 + tetramers. Proton ordering can be accomplished by orienting two of the
100 + waters so that both of their donated hydrogen bonds are internal to
101 + their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal
102 + constructed of water tetramers, the hydrogen bonds are not as linear
103 + as those observed in ice $I_h$, however the interlocking of these
104 + subunits appears to provide significant stabilization to the overall
105 + crystal. The arrangement of these tetramers results in surrounding
106 + open octagonal cavities that are typically greater than 6.3 \AA\ in
107 + diameter. This relatively open overall structure leads to crystals
108 + that are 0.07 g/cm$^3$ less dense on average than ice $I_h$.
109 +
110 + \begin{figure}
111 + \includegraphics[width=\linewidth]{unitCell.eps}
112 + \caption{Unit cells for (A) Ice-{\it i} and (B) Ice-2{\it i}, the elongated variant of Ice-{\it i}.  For Ice-{\it i}, the $a$ to $c$ relation is given by $a = 2.1214c$, while for Ice-2{\it i}, $a = 1.7850c$.}
113 + \label{iceiCell}
114 + \end{figure}
115 +
116 + \begin{figure}
117 + \includegraphics[width=\linewidth]{orderedIcei.eps}
118 + \caption{Image of a proton ordered crystal of Ice-{\it i} looking
119 + down the (001) crystal face. The rows of water tetramers surrounded by
120 + octagonal pores leads to a crystal structure that is significantly
121 + less dense than ice $I_h$.}
122 + \label{protOrder}
123 + \end{figure}
124 +
125 + Results in the previous study indicated that Ice-{\it i} is the
126 + minimum energy crystal structure for the single point water models
127 + being studied (for discussions on these single point dipole models,
128 + see the previous work and related
129 + articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only
130 + consider energetic stabilization and neglect entropic contributions to
131 + the overall free energy. To address this issue, the absolute free
132 + energy of this crystal was calculated using thermodynamic integration
133 + and compared to the free energies of cubic and hexagonal ice $I$ (the
134 + experimental low density ice polymorphs) and ice B (a higher density,
135 + but very stable crystal structure observed by B\`{a}ez and Clancy in
136 + free energy studies of SPC/E).\cite{Baez95b} This work includes
137 + results for the water model from which Ice-{\it i} was crystallized
138 + (soft sticky dipole extended, SSD/E) in addition to several common
139 + water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field
140 + parametrized single point dipole water model (soft sticky dipole
141 + reaction field, SSD/RF). In should be noted that a second version of
142 + Ice-{\it i} (Ice-2{\it i}) was used in calculations involving SPC/E,
143 + TIP4P, and TIP5P. The unit cell of this crystal (Fig. \ref{iceiCell}B)
144 + is similar to the Ice-{\it i} unit it is extended in the direction of
145 + the (001) face and compressed along the other two faces.
146  
147   \section{Methods}
148  
149   Canonical ensemble (NVT) molecular dynamics calculations were
150   performed using the OOPSE (Object-Oriented Parallel Simulation Engine)
151   molecular mechanics package. All molecules were treated as rigid
152 < bodies, with orientational motion propogated using the symplectic DLM
152 > bodies, with orientational motion propagated using the symplectic DLM
153   integration method. Details about the implementation of these
154   techniques can be found in a recent publication.\cite{Meineke05}
155  
# Line 72 | Line 168 | the two states:
168   integrated in order to determine the free energy difference between
169   the two states:
170   \begin{equation}
75 – \begin{center}
171   \Delta A = \int_0^1\left\langle\frac{\partial V(\lambda
172   )}{\partial\lambda}\right\rangle_\lambda d\lambda,
78 – \end{center}
173   \end{equation}
174   where $V$ is the interaction potential and $\lambda$ is the
175 < transformation parameter. Simulations are distributed unevenly along
176 < this path in order to sufficiently sample the regions of greatest
177 < change in the potential. Typical integrations in this study consisted
178 < of $\sim$25 simulations ranging from 300 ps (for the unaltered system)
179 < to 75 ps (near the reference state) in length.
175 > transformation parameter that scales the overall
176 > potential. Simulations are distributed unevenly along this path in
177 > order to sufficiently sample the regions of greatest change in the
178 > potential. Typical integrations in this study consisted of $\sim$25
179 > simulations ranging from 300 ps (for the unaltered system) to 75 ps
180 > (near the reference state) in length.
181  
182   For the thermodynamic integration of molecular crystals, the Einstein
183   Crystal is chosen as the reference state that the system is converted
# Line 110 | Line 205 | state.
205   minimum potential energy of the ideal crystal. In the case of
206   molecular liquids, the ideal vapor is chosen as the target reference
207   state.
208 +
209   \begin{figure}
210 < \includegraphics[scale=1.0]{rotSpring.eps}
210 > \includegraphics[width=\linewidth]{rotSpring.eps}
211   \caption{Possible orientational motions for a restrained molecule.
212   $\theta$ angles correspond to displacement from the body-frame {\it
213   z}-axis, while $\omega$ angles correspond to rotation about the
# Line 122 | Line 218 | cubic switching between 100\% and 85\% of the cutoff v
218   \end{figure}
219  
220   Charge, dipole, and Lennard-Jones interactions were modified by a
221 < cubic switching between 100\% and 85\% of the cutoff value (9 \AA ). By
222 < applying this function, these interactions are smoothly truncated,
223 < thereby avoiding poor energy conserving dynamics resulting from
224 < harsher truncation schemes. The effect of a long-range correction was
225 < also investigated on select model systems in a variety of manners. For
226 < the SSD/RF model, a reaction field with a fixed dielectric constant of
227 < 80 was applied in all simulations.\cite{Onsager36} For a series of the
228 < least computationally expensive models (SSD/E, SSD/RF, and TIP3P),
229 < simulations were performed with longer cutoffs of 12 and 15 \AA\ to
230 < compare with the 9 \AA\ cutoff results. Finally, results from the use
231 < of an Ewald summation were estimated for TIP3P and SPC/E by performing
221 > cubic switching between 100\% and 85\% of the cutoff value (9 \AA
222 > ). By applying this function, these interactions are smoothly
223 > truncated, thereby avoiding poor energy conserving dynamics resulting
224 > from harsher truncation schemes. The effect of a long-range correction
225 > was also investigated on select model systems in a variety of
226 > manners. For the SSD/RF model, a reaction field with a fixed
227 > dielectric constant of 80 was applied in all
228 > simulations.\cite{Onsager36} For a series of the least computationally
229 > expensive models (SSD/E, SSD/RF, and TIP3P), simulations were
230 > performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9
231 > \AA\ cutoff results. Finally, results from the use of an Ewald
232 > summation were estimated for TIP3P and SPC/E by performing
233   calculations with Particle-Mesh Ewald (PME) in the TINKER molecular
234 < mechanics software package. TINKER was chosen because it can also
235 < propogate the motion of rigid-bodies, and provides the most direct
236 < comparison to the results from OOPSE. The calculated energy difference
237 < in the presence and absence of PME was applied to the previous results
238 < in order to predict changes in the free energy landscape.
234 > mechanics software package.\cite{Tinker} TINKER was chosen because it
235 > can also propagate the motion of rigid-bodies, and provides the most
236 > direct comparison to the results from OOPSE. The calculated energy
237 > difference in the presence and absence of PME was applied to the
238 > previous results in order to predict changes in the free energy
239 > landscape.
240  
241   \section{Results and discussion}
242  
# Line 167 | Line 265 | kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF
265   of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are
266   kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF.}
267   \begin{tabular}{ l  c  c  c  c }
268 < \hline \\[-7mm]
268 > \hline
269   \ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \  & \ \quad \ \ \ \ B \ \  & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\
270 < \hline \\[-3mm]
270 > \hline
271   \ \quad \ TIP3P  & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\
272   \ \quad \ TIP4P  & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\
273   \ \quad \ TIP5P  & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\
# Line 196 | Line 294 | TIP4P in the high pressure regime.\cite{Sanz04}
294   representative of the dense ice polymorphs. A recent study by Sanz
295   {\it et al.} goes into detail on the phase diagrams for SPC/E and
296   TIP4P in the high pressure regime.\cite{Sanz04}
297 +
298   \begin{figure}
299   \includegraphics[width=\linewidth]{tp3PhaseDia.eps}
300   \caption{Phase diagram for the TIP3P water model in the low pressure
# Line 205 | Line 304 | higher in energy and don't appear in the phase diagram
304   higher in energy and don't appear in the phase diagram.}
305   \label{tp3phasedia}
306   \end{figure}
307 +
308   \begin{figure}
309   \includegraphics[width=\linewidth]{ssdrfPhaseDia.eps}
310   \caption{Phase diagram for the SSD/RF water model in the low pressure
# Line 223 | Line 323 | temperatures of several common water models compared w
323   \caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$)
324   temperatures of several common water models compared with experiment.}
325   \begin{tabular}{ l  c  c  c  c  c  c  c }
326 < \hline \\[-7mm]
326 > \hline
327   \ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\
328 < \hline \\[-3mm]
328 > \hline
329   \ \ $T_m$ (K)  & \ \ 269 & \ \ 265 & \ \ 271 &  297 & \ \ - & \ \ 278 & \ \ 273\\
330   \ \ $T_b$ (K)  & \ \ 357 & \ \ 354 & \ \ 337 &  396 & \ \ - & \ \ 349 & \ \ 373\\
331   \ \ $T_s$ (K)  & \ \ - & \ \ - & \ \ - &  - & \ \ 355 & \ \ - & \ \ -\\
# Line 253 | Line 353 | advantagious in that it facilitated the spontaneous cr
353   at 355 K. This is due to the significant stability of Ice-{\it i} over
354   all other polymorphs for this particular model under these
355   conditions. While troubling, this behavior turned out to be
356 < advantagious in that it facilitated the spontaneous crystallization of
356 > advantageous in that it facilitated the spontaneous crystallization of
357   Ice-{\it i}. These observations provide a warning that simulations of
358   SSD/E as a ``liquid'' near 300 K are actually metastable and run the
359   risk of spontaneous crystallization. However, this risk changes when
# Line 275 | Line 375 | differences while moving to greater cutoff radius. Thi
375   involve potential truncation. As seen in Fig. \ref{incCutoff}, the
376   free energy of all the ice polymorphs show a substantial dependence on
377   cutoff radius. In general, there is a narrowing of the free energy
378 < differences while moving to greater cutoff radius. This trend is much
379 < more subtle in the case of SSD/RF, indicating that the free energies
380 < calculated with a reaction field present provide a more accurate
381 < picture of the free energy landscape in the absence of potential
382 < truncation.
378 > differences while moving to greater cutoff radius. Interestingly, by
379 > increasing the cutoff radius, the free energy gap was narrowed enough
380 > in the SSD/E model that the liquid state is preferred under standard
381 > simulation conditions (298 K and 1 atm). Thus, it is recommended that
382 > simulations using this model choose interaction truncation radii
383 > greater than 9 \AA\. This narrowing trend is much more subtle in the
384 > case of SSD/RF, indicating that the free energies calculated with a
385 > reaction field present provide a more accurate picture of the free
386 > energy landscape in the absence of potential truncation.
387  
388   To further study the changes resulting to the inclusion of a
389   long-range interaction correction, the effect of an Ewald summation
# Line 291 | Line 395 | cutoff radius is observed in these results. Ice-{\it i
395   SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P
396   are not fully supported in TINKER, so the results for these models
397   could not be estimated. The same trend pointed out through increase of
398 < cutoff radius is observed in these results. Ice-{\it i} is the
398 > cutoff radius is observed in these PME results. Ice-{\it i} is the
399   preferred polymorph at ambient conditions for both the TIP3P and SPC/E
400   water models; however, there is a narrowing of the free energy
401   differences between the various solid forms. In the case of SPC/E this
# Line 310 | Line 414 | long-range interaction correction. Units are kcal/mol.
414   the energy difference attributed to the inclusion of the PME
415   long-range interaction correction. Units are kcal/mol.}
416   \begin{tabular}{ l  c  c  c  c }
417 < \hline \\[-7mm]
417 > \hline
418   \ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
419 < \hline \\[-3mm]
419 > \hline
420   \ \ TIP3P  & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\
421   \ \ SPC/E  & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\
422   \end{tabular}
# Line 335 | Line 439 | computed free energy values, but Ice-{\it i} is still
439   cutoff radius, use of a reaction field parameterized model, and
440   estimation of the results in the presence of the Ewald summation
441   correction. Interaction truncation has a significant effect on the
442 < computed free energy values, but Ice-{\it i} is still observed to be a
443 < relavent ice polymorph in simulation studies.
442 > computed free energy values, and may significantly alter the free
443 > energy landscape for the more complex multipoint water models. Despite
444 > these effects, these results show Ice-{\it i} to be an important ice
445 > polymorph that should be considered in simulation studies.
446  
447 + Due to this relative stability of Ice-{\it i} in all manner of
448 + investigated simulation examples, the question arises as to possible
449 + experimental observation of this polymorph. The rather extensive past
450 + and current experimental investigation of water in the low pressure
451 + regime leads the authors to be hesitant in ascribing relevance outside
452 + of computational models, hence the descriptive name presented. That
453 + being said, there are certain experimental conditions that would
454 + provide the most ideal situation for possible observation. These
455 + include the negative pressure or stretched solid regime, small
456 + clusters in vacuum deposition environments, and in clathrate
457 + structures involving small non-polar molecules.
458 +
459   \section{Acknowledgments}
460   Support for this project was provided by the National Science
461   Foundation under grant CHE-0134881. Computation time was provided by

Diff Legend

– Removed lines
+ Added lines
< Changed lines
> Changed lines