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 1466 by chrisfen, Thu Sep 16 19:28:55 2004 UTC vs.
Revision 1488 by chrisfen, Tue Sep 21 20:17:12 2004 UTC

# Line 33 | Line 33 | The free energies of several ice polymorphs in the low
33   %\doublespacing
34  
35   \begin{abstract}
36 < The free energies of several ice polymorphs in the low pressure regime
37 < were calculated using thermodynamic integration.  These integrations
38 < were done for most of the common water models. Ice-{\it i}, a
39 < structure we recently observed to be stable in one of the single-point
40 < water models, was determined to be the stable crystalline state (at 1
41 < atm) for {\it all} the water models investigated.  Phase diagrams were
36 > The absolute free energies of several ice polymorphs which are stable
37 > at low pressures were calculated using thermodynamic integration to a
38 > reference system (the Einstein crystal).  These integrations were
39 > performed for most of the common water models (SPC/E, TIP3P, TIP4P,
40 > TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed
41 > crystallizing at room temperature for one of the single-point water
42 > models, was determined to be the stable crystalline state (at 1 atm)
43 > for {\it all} the water models investigated.  Phase diagrams were
44   generated, and phase coexistence lines were determined for all of the
45 < known low-pressure ice structures under all of the common water
46 < models.  Additionally, potential truncation was shown to have an
47 < effect on the calculated free energies, and can result in altered free
48 < energy landscapes.
45 > known low-pressure ice structures under all of these water models.
46 > Additionally, potential truncation was shown to have an effect on the
47 > calculated free energies, and can result in altered free energy
48 > landscapes.  Structure factor predictions for the new crystal were
49 > generated and we await experimental confirmation of the existence of
50 > this new polymorph.
51   \end{abstract}
52  
53   %\narrowtext
# Line 54 | Line 58 | Molecular dynamics is a valuable tool for studying the
58  
59   \section{Introduction}
60  
57 – Molecular dynamics is a valuable tool for studying the phase behavior
58 – of systems ranging from small or simple
59 – molecules\cite{Matsumoto02andOthers} to complex biological
60 – species.\cite{bigStuff} Many techniques have been developed to
61 – investigate the thermodynamic properites of model substances,
62 – providing both qualitative and quantitative comparisons between
63 – simulations and experiment.\cite{thermMethods} Investigation of these
64 – properties leads to the development of new and more accurate models,
65 – leading to better understanding and depiction of physical processes
66 – and intricate molecular systems.
67 –
61   Water has proven to be a challenging substance to depict in
62   simulations, and a variety of models have been developed to describe
63   its behavior under varying simulation
64 < conditions.\cite{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04}
64 > conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04}
65   These models have been used to investigate important physical
66 < phenomena like phase transitions and the hydrophobic
67 < effect.\cite{Yamada02} With the choice of models available, it
68 < is only natural to compare the models under interesting thermodynamic
69 < conditions in an attempt to clarify the limitations of each of the
70 < models.\cite{modelProps} Two important property to quantify are the
71 < Gibbs and Helmholtz free energies, particularly for the solid forms of
72 < water.  Difficulty in these types of studies typically arises from the
73 < assortment of possible crystalline polymorphs that water adopts over a
74 < wide range of pressures and temperatures. There are currently 13
75 < recognized forms of ice, and it is a challenging task to investigate
76 < the entire free energy landscape.\cite{Sanz04} Ideally, research is
77 < focused on the phases having the lowest free energy at a given state
78 < point, because these phases will dictate the true transition
79 < temperatures and pressures for their respective model.
66 > phenomena like phase transitions, transport properties, and the
67 > hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the
68 > choice of models available, it is only natural to compare the models
69 > under interesting thermodynamic conditions in an attempt to clarify
70 > the limitations of each of the
71 > models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two
72 > important properties to quantify are the Gibbs and Helmholtz free
73 > energies, particularly for the solid forms of water.  Difficulty in
74 > these types of studies typically arises from the assortment of
75 > possible crystalline polymorphs that water adopts over a wide range of
76 > pressures and temperatures.  There are currently 13 recognized forms
77 > of ice, and it is a challenging task to investigate the entire free
78 > energy landscape.\cite{Sanz04} Ideally, research is focused on the
79 > phases having the lowest free energy at a given state point, because
80 > these phases will dictate the relevant transition temperatures and
81 > pressures for the model.
82  
83   In this paper, standard reference state methods were applied to known
84 < crystalline water polymorphs in the low pressure regime. This work is
85 < unique in the fact that one of the crystal lattices was arrived at
86 < through crystallization of a computationally efficient water model
87 < under constant pressure and temperature conditions. Crystallization
88 < events are interesting in and of
89 < themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure
90 < obtained in this case is different from any previously observed ice
91 < polymorphs in experiment or simulation.\cite{Fennell04} We have named
92 < this structure Ice-{\it i} to indicate its origin in computational
93 < simulation. The unit cell (Fig. \ref{iceiCell}A) consists of eight
94 < water molecules that stack in rows of interlocking water
95 < tetramers. Proton ordering can be accomplished by orienting two of the
96 < molecules so that both of their donated hydrogen bonds are internal to
97 < their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal
98 < constructed of water tetramers, the hydrogen bonds are not as linear
99 < as those observed in ice $I_h$, however the interlocking of these
100 < subunits appears to provide significant stabilization to the overall
84 > crystalline water polymorphs in the low pressure regime.  This work is
85 > unique in that one of the crystal lattices was arrived at through
86 > crystallization of a computationally efficient water model under
87 > constant pressure and temperature conditions. Crystallization events
88 > are interesting in and of themselves;\cite{Matsumoto02,Yamada02}
89 > however, the crystal structure obtained in this case is different from
90 > any previously observed ice polymorphs in experiment or
91 > simulation.\cite{Fennell04} We have named this structure Ice-{\it i}
92 > to indicate its origin in computational simulation. The unit cell
93 > (Fig. \ref{iceiCell}A) consists of eight water molecules that stack in
94 > rows of interlocking water tetramers. Proton ordering can be
95 > accomplished by orienting two of the molecules so that both of their
96 > donated hydrogen bonds are internal to their tetramer
97 > (Fig. \ref{protOrder}). As expected in an ice crystal constructed of
98 > water tetramers, the hydrogen bonds are not as linear as those
99 > observed in ice $I_h$, however the interlocking of these subunits
100 > appears to provide significant stabilization to the overall
101   crystal. The arrangement of these tetramers results in surrounding
102   open octagonal cavities that are typically greater than 6.3 \AA\ in
103   diameter. This relatively open overall structure leads to crystals
# Line 110 | Line 105 | that are 0.07 g/cm$^3$ less dense on average than ice
105  
106   \begin{figure}
107   \includegraphics[width=\linewidth]{unitCell.eps}
108 < \caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the
109 < elongated variant of Ice-{\it i}.  For Ice-{\it i}, the $a$ to $c$
110 < relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a =
111 < 1.7850c$.}
108 > \caption{Unit cells for (A) Ice-{\it i} and (B) Ice-{\it i}$^\prime$,
109 > the elongated variant of Ice-{\it i}.  The spheres represent the
110 > center-of-mass locations of the water molecules.  The $a$ to $c$
111 > ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by
112 > $a:2.1214c$ and $a:1.7850c$ respectively.}
113   \label{iceiCell}
114   \end{figure}
115  
# Line 128 | Line 124 | investigated (for discussions on these single point di
124  
125   Results from our previous study indicated that Ice-{\it i} is the
126   minimum energy crystal structure for the single point water models we
127 < investigated (for discussions on these single point dipole models, see
128 < the previous work and related
129 < articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only
127 > had investigated (for discussions on these single point dipole models,
128 > see our previous work and related
129 > articles).\cite{Fennell04,Liu96,Bratko85} Those results only
130   considered energetic stabilization and neglected entropic
131 < contributions to the overall free energy. To address this issue, the
132 < absolute free energy of this crystal was calculated using
131 > contributions to the overall free energy. To address this issue, we
132 > have calculated the absolute free energy of this crystal using
133   thermodynamic integration and compared to the free energies of cubic
134   and hexagonal ice $I$ (the experimental low density ice polymorphs)
135   and ice B (a higher density, but very stable crystal structure
# Line 142 | Line 138 | be noted that a second version of Ice-{\it i} (Ice-$i^
138   from which Ice-{\it i} was crystallized (SSD/E) in addition to several
139   common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction
140   field parametrized single point dipole water model (SSD/RF). It should
141 < be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used
142 < in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of
143 < this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit
144 < it is extended in the direction of the (001) face and compressed along
145 < the other two faces.
141 > be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$)
142 > was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit
143 > cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it
144 > i} unit it is extended in the direction of the (001) face and
145 > compressed along the other two faces.  There is typically a small
146 > distortion of proton ordered Ice-{\it i}$^\prime$ that converts the
147 > normally square tetramer into a rhombus with alternating approximately
148 > 85 and 95 degree angles.  The degree of this distortion is model
149 > dependent and significant enough to split the tetramer diagonal
150 > location peak in the radial distribution function.
151  
152   \section{Methods}
153  
# Line 154 | Line 155 | the implementation of these techniques can be found in
155   performed using the OOPSE molecular mechanics package.\cite{Meineke05}
156   All molecules were treated as rigid bodies, with orientational motion
157   propagated using the symplectic DLM integration method. Details about
158 < the implementation of these techniques can be found in a recent
159 < publication.\cite{DLM}
158 > the implementation of this technique can be found in a recent
159 > publication.\cite{Dullweber1997}
160  
161 < Thermodynamic integration was utilized to calculate the free energy of
162 < several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E,
163 < SSD/RF, and SSD/E water models. Liquid state free energies at 300 and
164 < 400 K for all of these water models were also determined using this
165 < same technique in order to determine melting points and generate phase
166 < diagrams. All simulations were carried out at densities resulting in a
167 < pressure of approximately 1 atm at their respective temperatures.
161 > Thermodynamic integration is an established technique for
162 > determination of free energies of condensed phases of
163 > materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This
164 > method, implemented in the same manner illustrated by B\`{a}ez and
165 > Clancy, was utilized to calculate the free energy of several ice
166 > crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and
167 > SSD/E water models.\cite{Baez95a} Liquid state free energies at 300
168 > and 400 K for all of these water models were also determined using
169 > this same technique in order to determine melting points and to
170 > generate phase diagrams. All simulations were carried out at densities
171 > which correspond to a pressure of approximately 1 atm at their
172 > respective temperatures.
173  
174 < A single thermodynamic integration involves a sequence of simulations
175 < over which the system of interest is converted into a reference system
176 < for which the free energy is known analytically. This transformation
177 < path is then integrated in order to determine the free energy
178 < difference between the two states:
174 > Thermodynamic integration involves a sequence of simulations during
175 > which the system of interest is converted into a reference system for
176 > which the free energy is known analytically. This transformation path
177 > is then integrated in order to determine the free energy difference
178 > between the two states:
179   \begin{equation}
180   \Delta A = \int_0^1\left\langle\frac{\partial V(\lambda
181   )}{\partial\lambda}\right\rangle_\lambda d\lambda,
182   \end{equation}
183   where $V$ is the interaction potential and $\lambda$ is the
184   transformation parameter that scales the overall
185 < potential. Simulations are distributed unevenly along this path in
186 < order to sufficiently sample the regions of greatest change in the
185 > potential. Simulations are distributed strategically along this path
186 > in order to sufficiently sample the regions of greatest change in the
187   potential. Typical integrations in this study consisted of $\sim$25
188   simulations ranging from 300 ps (for the unaltered system) to 75 ps
189   (near the reference state) in length.
190  
191   For the thermodynamic integration of molecular crystals, the Einstein
192 < crystal was chosen as the reference state. In an Einstein crystal, the
193 < molecules are harmonically restrained at their ideal lattice locations
194 < and orientations. The partition function for a molecular crystal
192 > crystal was chosen as the reference system. In an Einstein crystal,
193 > the molecules are restrained at their ideal lattice locations and
194 > orientations. Using harmonic restraints, as applied by B\`{a}ez and
195 > Clancy, the total potential for this reference crystal
196 > ($V_\mathrm{EC}$) is the sum of all the harmonic restraints,
197 > \begin{equation}
198 > V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} +
199 > \frac{K_\omega\omega^2}{2},
200 > \end{equation}
201 > where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are
202 > the spring constants restraining translational motion and deflection
203 > of and rotation around the principle axis of the molecule
204 > respectively.  It is clear from Fig. \ref{waterSpring} that the values
205 > of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from
206 > $-\pi$ to $\pi$.  The partition function for a molecular crystal
207   restrained in this fashion can be evaluated analytically, and the
208   Helmholtz Free Energy ({\it A}) is given by
209   \begin{eqnarray}
# Line 199 | Line 217 | where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a
217   )^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ],
218   \label{ecFreeEnergy}
219   \end{eqnarray}
220 < where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation
221 < \ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and
204 < $K_\mathrm{\omega}$ are the spring constants restraining translational
205 < motion and deflection of and rotation around the principle axis of the
206 < molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the
207 < minimum potential energy of the ideal crystal. In the case of
208 < molecular liquids, the ideal vapor is chosen as the target reference
209 < state.
220 > where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum
221 > potential energy of the ideal crystal.\cite{Baez95a}
222  
223   \begin{figure}
224   \includegraphics[width=\linewidth]{rotSpring.eps}
# Line 219 | Line 231 | Charge, dipole, and Lennard-Jones interactions were mo
231   \label{waterSpring}
232   \end{figure}
233  
234 + In the case of molecular liquids, the ideal vapor is chosen as the
235 + target reference state.  There are several examples of liquid state
236 + free energy calculations of water models present in the
237 + literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods
238 + typically differ in regard to the path taken for switching off the
239 + interaction potential to convert the system to an ideal gas of water
240 + molecules.  In this study, we applied of one of the most convenient
241 + methods and integrated over the $\lambda^4$ path, where all
242 + interaction parameters are scaled equally by this transformation
243 + parameter.  This method has been shown to be reversible and provide
244 + results in excellent agreement with other established
245 + methods.\cite{Baez95b}
246 +
247   Charge, dipole, and Lennard-Jones interactions were modified by a
248   cubic switching between 100\% and 85\% of the cutoff value (9 \AA
249   ). By applying this function, these interactions are smoothly
# Line 230 | Line 255 | performed with longer cutoffs of 12 and 15 \AA\ to com
255   simulations.\cite{Onsager36} For a series of the least computationally
256   expensive models (SSD/E, SSD/RF, and TIP3P), simulations were
257   performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9
258 < \AA\ cutoff results. Finally, results from the use of an Ewald
259 < summation were estimated for TIP3P and SPC/E by performing
260 < calculations with Particle-Mesh Ewald (PME) in the TINKER molecular
261 < mechanics software package.\cite{Tinker} The calculated energy
262 < difference in the presence and absence of PME was applied to the
263 < previous results in order to predict changes to the free energy
264 < landscape.
258 > \AA\ cutoff results. Finally, the effects of utilizing an Ewald
259 > summation were estimated for TIP3P and SPC/E by performing single
260 > configuration calculations with Particle-Mesh Ewald (PME) in the
261 > TINKER molecular mechanics software package.\cite{Tinker} The
262 > calculated energy difference in the presence and absence of PME was
263 > applied to the previous results in order to predict changes to the
264 > free energy landscape.
265  
266   \section{Results and discussion}
267  
268 < The free energy of proton ordered Ice-{\it i} was calculated and
268 > The free energy of proton-ordered Ice-{\it i} was calculated and
269   compared with the free energies of proton ordered variants of the
270   experimentally observed low-density ice polymorphs, $I_h$ and $I_c$,
271   as well as the higher density ice B, observed by B\`{a}ez and Clancy
# Line 250 | Line 275 | antiferroelectric version that has an 8 molecule unit
275   $I_h$, was investigated initially, but was found to be not as stable
276   as proton disordered or antiferroelectric variants of ice $I_h$. The
277   proton ordered variant of ice $I_h$ used here is a simple
278 < antiferroelectric version that has an 8 molecule unit
279 < cell.\cite{Davidson84} The crystals contained 648 or 1728 molecules
280 < for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for
281 < ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes
282 < were necessary for simulations involving larger cutoff values.
278 > antiferroelectric version that we devised, and it has an 8 molecule
279 > unit cell similar to other predicted antiferroelectric $I_h$
280 > crystals.\cite{Davidson84} The crystals contained 648 or 1728
281 > molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000
282 > molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger
283 > crystal sizes were necessary for simulations involving larger cutoff
284 > values.
285  
286   \begin{table*}
287   \begin{minipage}{\linewidth}
# Line 269 | Line 296 | TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\
296   \hline
297   Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\
298   \hline
299 < TIP3P & -11.41(4) & -11.23(6) & -11.82(5) & -12.30(5)\\
300 < TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\
301 < TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\
302 < SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\
303 < SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\
304 < SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\
299 > TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\
300 > TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\
301 > TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\
302 > SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\
303 > SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\
304 > SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\
305   \end{tabular}
306   \label{freeEnergy}
307   \end{center}
# Line 283 | Line 310 | models studied. With the free energy at these state po
310  
311   The free energy values computed for the studied polymorphs indicate
312   that Ice-{\it i} is the most stable state for all of the common water
313 < models studied. With the free energy at these state points, the
314 < Gibbs-Helmholtz equation was used to project to other state points and
315 < to build phase diagrams.  Figures
313 > models studied. With the calculated free energy at these state points,
314 > the Gibbs-Helmholtz equation was used to project to other state points
315 > and to build phase diagrams.  Figures
316   \ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built
317   from the free energy results. All other models have similar structure,
318 < although the crossing points between the phases exist at slightly
318 > although the crossing points between the phases move to slightly
319   different temperatures and pressures. It is interesting to note that
320   ice $I$ does not exist in either cubic or hexagonal form in any of the
321   phase diagrams for any of the models. For purposes of this study, ice
322   B is representative of the dense ice polymorphs. A recent study by
323   Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and
324 < TIP4P in the high pressure regime.\cite{Sanz04}
324 > TIP4P at higher pressures than those studied here.\cite{Sanz04}
325  
326   \begin{figure}
327   \includegraphics[width=\linewidth]{tp3PhaseDia.eps}
# Line 330 | Line 357 | $T_m$ (K)  & 269(8) & 266(10) & 271(7) & 296(5) & - &
357   \hline
358   Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\
359   \hline
360 < $T_m$ (K)  & 269(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\
361 < $T_b$ (K)  & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\
362 < $T_s$ (K)  & - & - & - & - & 355(3) & - & -\\
360 > $T_m$ (K)  & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\
361 > $T_b$ (K)  & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\
362 > $T_s$ (K)  & - & - & - & - & 355(2) & - & -\\
363   \end{tabular}
364   \label{meltandboil}
365   \end{center}
# Line 357 | Line 384 | conditions. While troubling, this behavior turned out
384   not exhibit a melting point at 1 atm, but it shows a sublimation point
385   at 355 K. This is due to the significant stability of Ice-{\it i} over
386   all other polymorphs for this particular model under these
387 < conditions. While troubling, this behavior turned out to be
388 < advantageous in that it facilitated the spontaneous crystallization of
389 < Ice-{\it i}. These observations provide a warning that simulations of
387 > conditions. While troubling, this behavior resulted in spontaneous
388 > crystallization of Ice-{\it i} and led us to investigate this
389 > structure. These observations provide a warning that simulations of
390   SSD/E as a ``liquid'' near 300 K are actually metastable and run the
391 < risk of spontaneous crystallization. However, this risk changes when
391 > risk of spontaneous crystallization. However, this risk lessens when
392   applying a longer cutoff.
393  
394   \begin{figure}
# Line 385 | Line 412 | greater than 9 \AA\. This narrowing trend is much more
412   in the SSD/E model that the liquid state is preferred under standard
413   simulation conditions (298 K and 1 atm). Thus, it is recommended that
414   simulations using this model choose interaction truncation radii
415 < greater than 9 \AA\. This narrowing trend is much more subtle in the
415 > greater than 9 \AA\ . This narrowing trend is much more subtle in the
416   case of SSD/RF, indicating that the free energies calculated with a
417   reaction field present provide a more accurate picture of the free
418   energy landscape in the absence of potential truncation.
# Line 395 | Line 422 | energy of identical crystals with and without PME usin
422   was estimated by applying the potential energy difference do to its
423   inclusion in systems in the presence and absence of the
424   correction. This was accomplished by calculation of the potential
425 < energy of identical crystals with and without PME using TINKER. The
426 < free energies for the investigated polymorphs using the TIP3P and
427 < SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P
428 < are not fully supported in TINKER, so the results for these models
429 < could not be estimated. The same trend pointed out through increase of
430 < cutoff radius is observed in these PME results. Ice-{\it i} is the
431 < preferred polymorph at ambient conditions for both the TIP3P and SPC/E
432 < water models; however, there is a narrowing of the free energy
433 < differences between the various solid forms. In the case of SPC/E this
434 < narrowing is significant enough that it becomes less clear that
435 < Ice-{\it i} is the most stable polymorph, and is possibly metastable
436 < with respect to ice B and possibly ice $I_c$. However, these results
437 < do not significantly alter the finding that the Ice-{\it i} polymorph
438 < is a stable crystal structure that should be considered when studying
439 < the phase behavior of water models.
425 > energy of identical crystals both with and without PME. The free
426 > energies for the investigated polymorphs using the TIP3P and SPC/E
427 > water models are shown in Table \ref{pmeShift}. The same trend pointed
428 > out through increase of cutoff radius is observed in these PME
429 > results. Ice-{\it i} is the preferred polymorph at ambient conditions
430 > for both the TIP3P and SPC/E water models; however, the narrowing of
431 > the free energy differences between the various solid forms is
432 > significant enough that it becomes less clear that it is the most
433 > stable polymorph with the SPC/E model.  The free energies of Ice-{\it
434 > i} and ice B nearly overlap within error, with ice $I_c$ just outside
435 > as well, indicating that Ice-{\it i} might be metastable with respect
436 > to ice B and possibly ice $I_c$ with SPC/E. However, these results do
437 > not significantly alter the finding that the Ice-{\it i} polymorph is
438 > a stable crystal structure that should be considered when studying the
439 > phase behavior of water models.
440  
441   \begin{table*}
442   \begin{minipage}{\linewidth}
# Line 422 | Line 449 | TIP3P  & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)
449   \hline
450   \ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\
451   \hline
452 < TIP3P  & -11.53(4) & -11.24(6) & -11.51(5) & -11.67(5)\\
453 < SPC/E  & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\
452 > TIP3P  & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\
453 > SPC/E  & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\
454   \end{tabular}
455   \label{pmeShift}
456   \end{center}
# Line 433 | Line 460 | $I$, ice B, and recently discovered Ice-{\it i} were c
460   \section{Conclusions}
461  
462   The free energy for proton ordered variants of hexagonal and cubic ice
463 < $I$, ice B, and recently discovered Ice-{\it i} were calculated under
464 < standard conditions for several common water models via thermodynamic
465 < integration. All the water models studied show Ice-{\it i} to be the
466 < minimum free energy crystal structure in the with a 9 \AA\ switching
467 < function cutoff. Calculated melting and boiling points show
468 < surprisingly good agreement with the experimental values; however, the
469 < solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of
470 < interaction truncation was investigated through variation of the
471 < cutoff radius, use of a reaction field parameterized model, and
463 > $I$, ice B, and our recently discovered Ice-{\it i} structure were
464 > calculated under standard conditions for several common water models
465 > via thermodynamic integration. All the water models studied show
466 > Ice-{\it i} to be the minimum free energy crystal structure with a 9
467 > \AA\ switching function cutoff. Calculated melting and boiling points
468 > show surprisingly good agreement with the experimental values;
469 > however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The
470 > effect of interaction truncation was investigated through variation of
471 > the cutoff radius, use of a reaction field parameterized model, and
472   estimation of the results in the presence of the Ewald
473   summation. Interaction truncation has a significant effect on the
474   computed free energy values, and may significantly alter the free
# Line 449 | Line 476 | Due to this relative stability of Ice-{\it i} in all m
476   these effects, these results show Ice-{\it i} to be an important ice
477   polymorph that should be considered in simulation studies.
478  
479 < Due to this relative stability of Ice-{\it i} in all manner of
480 < investigated simulation examples, the question arises as to possible
479 > Due to this relative stability of Ice-{\it i} in all of the
480 > investigated simulation conditions, the question arises as to possible
481   experimental observation of this polymorph.  The rather extensive past
482   and current experimental investigation of water in the low pressure
483   regime makes us hesitant to ascribe any relevance of this work outside
# Line 460 | Line 487 | non-polar molecules.  Fig. \ref{fig:sofkgofr} contains
487   most ideal situation for possible observation. These include the
488   negative pressure or stretched solid regime, small clusters in vacuum
489   deposition environments, and in clathrate structures involving small
490 < non-polar molecules.  Fig. \ref{fig:sofkgofr} contains our predictions
491 < of both the pair distribution function ($g_{OO}(r)$) and the structure
492 < factor ($S(\vec{q})$ for this polymorph at a temperature of 77K.  We
493 < will leave it to our experimental colleagues to determine whether this
494 < ice polymorph should really be called Ice-{\it i} or if it should be
495 < promoted to Ice-0.
490 > non-polar molecules.  Figs. \ref{fig:gofr} and \ref{fig:sofq} contain
491 > our predictions for both the pair distribution function ($g_{OO}(r)$)
492 > and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for
493 > ice-{\it i} at a temperature of 77K.  In studies of the high and low
494 > density forms of amorphous ice, ``spurious'' diffraction peaks have
495 > been observed experimentally.\cite{Bizid87} It is possible that a
496 > variant of Ice-{\it i} could explain some of this behavior; however,
497 > we will leave it to our experimental colleagues to make the final
498 > determination on whether this ice polymorph is named appropriately
499 > (i.e. with an imaginary number) or if it can be promoted to Ice-0.
500  
501 + \begin{figure}
502 + \includegraphics[width=\linewidth]{iceGofr.eps}
503 + \caption{Radial distribution functions of ice $I_h$, $I_c$,
504 + Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations
505 + of the SSD/RF water model at 77 K.}
506 + \label{fig:gofr}
507 + \end{figure}
508 +
509 + \begin{figure}
510 + \includegraphics[width=\linewidth]{sofq.eps}
511 + \caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i},
512 + and Ice-{\it i}$^\prime$ at 77 K.  The raw structure factors have
513 + been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$
514 + width) to compensate for the trunction effects in our finite size
515 + simulations.}
516 + \label{fig:sofq}
517 + \end{figure}
518 +
519   \section{Acknowledgments}
520   Support for this project was provided by the National Science
521   Foundation under grant CHE-0134881. Computation time was provided by

Diff Legend

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