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\begin{document} |
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\title{A Free Energy Study of Low Temperature and Anomolous Ice} |
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\title{Ice-{\it i}: a simulation-predicted ice polymorph which is more |
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stable than Ice $I_h$ for point-charge and point-dipole water models} |
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\author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote} |
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\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}} |
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\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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\author{Christopher J. Fennell and J. Daniel Gezelter \\ |
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Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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\begin{abstract} |
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The absolute free energies of several ice polymorphs which are stable |
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at low pressures were calculated using thermodynamic integration to a |
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reference system (the Einstein crystal). These integrations were |
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performed for most of the common water models (SPC/E, TIP3P, TIP4P, |
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TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed |
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crystallizing at room temperature for one of the single-point water |
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models, was determined to be the stable crystalline state (at 1 atm) |
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for {\it all} the water models investigated. Phase diagrams were |
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generated, and phase coexistence lines were determined for all of the |
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known low-pressure ice structures under all of these water models. |
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Additionally, potential truncation was shown to have an effect on the |
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calculated free energies, and can result in altered free energy |
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landscapes. Structure factor predictions for the new crystal were |
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generated and we await experimental confirmation of the existence of |
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this new polymorph. |
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\end{abstract} |
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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|
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Water has proven to be a challenging substance to depict in |
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simulations, and a variety of models have been developed to describe |
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its behavior under varying simulation |
64 |
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conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04} |
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These models have been used to investigate important physical |
66 |
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phenomena like phase transitions, transport properties, and the |
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hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the |
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choice of models available, it is only natural to compare the models |
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under interesting thermodynamic conditions in an attempt to clarify |
70 |
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the limitations of each of the |
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models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two |
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important properties to quantify are the Gibbs and Helmholtz free |
73 |
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energies, particularly for the solid forms of water. Difficulty in |
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these types of studies typically arises from the assortment of |
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possible crystalline polymorphs that water adopts over a wide range of |
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pressures and temperatures. There are currently 13 recognized forms |
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of ice, and it is a challenging task to investigate the entire free |
78 |
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energy landscape.\cite{Sanz04} Ideally, research is focused on the |
79 |
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phases having the lowest free energy at a given state point, because |
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these phases will dictate the relevant transition temperatures and |
81 |
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pressures for the model. |
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|
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In this paper, standard reference state methods were applied to known |
84 |
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crystalline water polymorphs in the low pressure regime. This work is |
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unique in that one of the crystal lattices was arrived at through |
86 |
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crystallization of a computationally efficient water model under |
87 |
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constant pressure and temperature conditions. Crystallization events |
88 |
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are interesting in and of themselves;\cite{Matsumoto02,Yamada02} |
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however, the crystal structure obtained in this case is different from |
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any previously observed ice polymorphs in experiment or |
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simulation.\cite{Fennell04} We have named this structure Ice-{\it i} |
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to indicate its origin in computational simulation. The unit cell |
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(Fig. \ref{iceiCell}A) consists of eight water molecules that stack in |
94 |
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rows of interlocking water tetramers. This crystal structure has a |
95 |
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limited resemblence to a recent two-dimensional ice tessellation |
96 |
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simulated on a silica surface.\cite{Yang04} Proton ordering can be |
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accomplished by orienting two of the molecules so that both of their |
98 |
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donated hydrogen bonds are internal to their tetramer |
99 |
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(Fig. \ref{protOrder}). As expected in an ice crystal constructed of |
100 |
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water tetramers, the hydrogen bonds are not as linear as those |
101 |
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observed in ice $I_h$, however the interlocking of these subunits |
102 |
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appears to provide significant stabilization to the overall crystal. |
103 |
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The arrangement of these tetramers results in surrounding open |
104 |
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octagonal cavities that are typically greater than 6.3 \AA\ in |
105 |
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diameter. This relatively open overall structure leads to crystals |
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that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{unitCell.eps} |
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\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-{\it i}$^\prime$, |
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the elongated variant of Ice-{\it i}. The spheres represent the |
112 |
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center-of-mass locations of the water molecules. The $a$ to $c$ |
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ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by |
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$a:2.1214c$ and $a:1.7850c$ respectively.} |
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\label{iceiCell} |
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\end{figure} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{orderedIcei.eps} |
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\caption{Image of a proton ordered crystal of Ice-{\it i} looking |
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down the (001) crystal face. The rows of water tetramers surrounded by |
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octagonal pores leads to a crystal structure that is significantly |
123 |
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less dense than ice $I_h$.} |
124 |
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\label{protOrder} |
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\end{figure} |
126 |
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|
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Results from our previous study indicated that Ice-{\it i} is the |
128 |
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minimum energy crystal structure for the single point water models we |
129 |
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had investigated (for discussions on these single point dipole models, |
130 |
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see our previous work and related |
131 |
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articles).\cite{Fennell04,Liu96,Bratko85} Those results only |
132 |
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considered energetic stabilization and neglected entropic |
133 |
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contributions to the overall free energy. To address this issue, we |
134 |
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have calculated the absolute free energy of this crystal using |
135 |
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thermodynamic integration and compared to the free energies of cubic |
136 |
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and hexagonal ice $I$ (the experimental low density ice polymorphs) |
137 |
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and ice B (a higher density, but very stable crystal structure |
138 |
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observed by B\`{a}ez and Clancy in free energy studies of |
139 |
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SPC/E).\cite{Baez95b} This work includes results for the water model |
140 |
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from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
141 |
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common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
142 |
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field parametrized single point dipole water model (SSD/RF). It should |
143 |
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be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$) |
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was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit |
145 |
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cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it |
146 |
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i} unit it is extended in the direction of the (001) face and |
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compressed along the other two faces. There is typically a small |
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distortion of proton ordered Ice-{\it i}$^\prime$ that converts the |
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normally square tetramer into a rhombus with alternating approximately |
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85 and 95 degree angles. The degree of this distortion is model |
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dependent and significant enough to split the tetramer diagonal |
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location peak in the radial distribution function. |
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|
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|
\section{Methods} |
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|
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Canonical ensemble (NVT) molecular dynamics calculations were |
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performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
158 |
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All molecules were treated as rigid bodies, with orientational motion |
159 |
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propagated using the symplectic DLM integration method. Details about |
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the implementation of this technique can be found in a recent |
161 |
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publication.\cite{Dullweber1997} |
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|
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Thermodynamic integration is an established technique for |
164 |
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determination of free energies of condensed phases of |
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materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
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method, implemented in the same manner illustrated by B\`{a}ez and |
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Clancy, was utilized to calculate the free energy of several ice |
168 |
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crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and |
169 |
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SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 |
170 |
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and 400 K for all of these water models were also determined using |
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this same technique in order to determine melting points and to |
172 |
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generate phase diagrams. All simulations were carried out at |
173 |
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densities which correspond to a pressure of approximately 1 atm at |
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their respective temperatures. |
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|
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Thermodynamic integration involves a sequence of simulations during |
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which the system of interest is converted into a reference system for |
178 |
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which the free energy is known analytically. This transformation path |
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is then integrated in order to determine the free energy difference |
180 |
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between the two states: |
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\begin{equation} |
182 |
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\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
183 |
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)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
184 |
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\end{equation} |
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where $V$ is the interaction potential and $\lambda$ is the |
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transformation parameter that scales the overall potential. |
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Simulations are distributed strategically along this path in order to |
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sufficiently sample the regions of greatest change in the potential. |
189 |
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Typical integrations in this study consisted of $\sim$25 simulations |
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ranging from 300 ps (for the unaltered system) to 75 ps (near the |
191 |
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reference state) in length. |
192 |
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|
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For the thermodynamic integration of molecular crystals, the Einstein |
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crystal was chosen as the reference system. In an Einstein crystal, |
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the molecules are restrained at their ideal lattice locations and |
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orientations. Using harmonic restraints, as applied by B\`{a}ez and |
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Clancy, the total potential for this reference crystal |
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($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
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\begin{equation} |
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V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
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\frac{K_\omega\omega^2}{2}, |
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\end{equation} |
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where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
204 |
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the spring constants restraining translational motion and deflection |
205 |
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of and rotation around the principle axis of the molecule |
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respectively. These spring constants are typically calculated from |
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the mean-square displacements of water molecules in an unrestrained |
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ice crystal at 200 K. For these studies, $K_\mathrm{r} = 4.29$ kcal |
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mol$^{-1}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$, and $K_\omega\ = |
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17.75$ kcal mol$^{-1}$. It is clear from Fig. \ref{waterSpring} that |
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the values of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges |
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from $-\pi$ to $\pi$. The partition function for a molecular crystal |
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restrained in this fashion can be evaluated analytically, and the |
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Helmholtz Free Energy ({\it A}) is given by |
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\begin{eqnarray} |
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A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left |
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[\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right |
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)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right |
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)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right |
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)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi |
221 |
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K_\omega K_\theta)^{\frac{1}{2}}}\exp\left |
222 |
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(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right |
223 |
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)^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], |
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\label{ecFreeEnergy} |
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\end{eqnarray} |
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where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
227 |
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potential energy of the ideal crystal.\cite{Baez95a} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{rotSpring.eps} |
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\caption{Possible orientational motions for a restrained molecule. |
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$\theta$ angles correspond to displacement from the body-frame {\it |
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z}-axis, while $\omega$ angles correspond to rotation about the |
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body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
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constants for the harmonic springs restraining motion in the $\theta$ |
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and $\omega$ directions.} |
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\label{waterSpring} |
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\end{figure} |
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|
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In the case of molecular liquids, the ideal vapor is chosen as the |
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target reference state. There are several examples of liquid state |
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free energy calculations of water models present in the |
243 |
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literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
244 |
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typically differ in regard to the path taken for switching off the |
245 |
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interaction potential to convert the system to an ideal gas of water |
246 |
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molecules. In this study, we applied of one of the most convenient |
247 |
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methods and integrated over the $\lambda^4$ path, where all |
248 |
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interaction parameters are scaled equally by this transformation |
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parameter. This method has been shown to be reversible and provide |
250 |
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results in excellent agreement with other established |
251 |
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methods.\cite{Baez95b} |
252 |
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|
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Charge, dipole, and Lennard-Jones interactions were modified by a |
254 |
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cubic switching between 100\% and 85\% of the cutoff value (9 \AA). |
255 |
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By applying this function, these interactions are smoothly truncated, |
256 |
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thereby avoiding the poor energy conservation which results from |
257 |
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harsher truncation schemes. The effect of a long-range correction was |
258 |
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also investigated on select model systems in a variety of manners. |
259 |
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For the SSD/RF model, a reaction field with a fixed dielectric |
260 |
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constant of 80 was applied in all simulations.\cite{Onsager36} For a |
261 |
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series of the least computationally expensive models (SSD/E, SSD/RF, |
262 |
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and TIP3P), simulations were performed with longer cutoffs of 12 and |
263 |
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15 \AA\ to compare with the 9 \AA\ cutoff results. Finally, the |
264 |
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effects of utilizing an Ewald summation were estimated for TIP3P and |
265 |
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SPC/E by performing single configuration calculations with |
266 |
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Particle-Mesh Ewald (PME) in the TINKER molecular mechanics software |
267 |
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package.\cite{Tinker} The calculated energy difference in the presence |
268 |
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and absence of PME was applied to the previous results in order to |
269 |
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predict changes to the free energy landscape. |
270 |
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|
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\section{Results and discussion} |
272 |
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|
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The free energy of proton-ordered Ice-{\it i} was calculated and |
274 |
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compared with the free energies of proton ordered variants of the |
275 |
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experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
276 |
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as well as the higher density ice B, observed by B\`{a}ez and Clancy |
277 |
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and thought to be the minimum free energy structure for the SPC/E |
278 |
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model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
279 |
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Ice XI, the experimentally-observed proton-ordered variant of ice |
280 |
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$I_h$, was investigated initially, but was found to be not as stable |
281 |
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as proton disordered or antiferroelectric variants of ice $I_h$. The |
282 |
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proton ordered variant of ice $I_h$ used here is a simple |
283 |
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antiferroelectric version that we devised, and it has an 8 molecule |
284 |
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unit cell similar to other predicted antiferroelectric $I_h$ |
285 |
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crystals.\cite{Davidson84} The crystals contained 648 or 1728 |
286 |
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molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 |
287 |
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molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The |
288 |
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larger crystal sizes were necessary for simulations involving larger |
289 |
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cutoff values. |
290 |
|
|
291 |
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\begin{table*} |
292 |
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\begin{minipage}{\linewidth} |
293 |
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\begin{center} |
294 |
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|
295 |
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\caption{Calculated free energies for several ice polymorphs with a |
296 |
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variety of common water models. All calculations used a cutoff radius |
297 |
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of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
298 |
+ |
kcal/mol. Calculated error of the final digits is in parentheses.} |
299 |
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|
300 |
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\begin{tabular}{lcccc} |
301 |
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\hline |
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Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ |
303 |
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\hline |
304 |
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TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ |
305 |
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TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ |
306 |
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TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ |
307 |
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SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & -13.55(2)\\ |
308 |
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SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ |
309 |
+ |
SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2)\\ |
310 |
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\end{tabular} |
311 |
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\label{freeEnergy} |
312 |
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\end{center} |
313 |
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\end{minipage} |
314 |
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\end{table*} |
315 |
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|
316 |
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The free energy values computed for the studied polymorphs indicate |
317 |
+ |
that Ice-{\it i} is the most stable state for all of the common water |
318 |
+ |
models studied. With the calculated free energy at these state |
319 |
+ |
points, the Gibbs-Helmholtz equation was used to project to other |
320 |
+ |
state points and to build phase diagrams. Figures \ref{tp3phasedia} |
321 |
+ |
and \ref{ssdrfphasedia} are example diagrams built from the free |
322 |
+ |
energy results. All other models have similar structure, although the |
323 |
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crossing points between the phases move to slightly different |
324 |
+ |
temperatures and pressures. It is interesting to note that ice $I$ |
325 |
+ |
does not exist in either cubic or hexagonal form in any of the phase |
326 |
+ |
diagrams for any of the models. For purposes of this study, ice B is |
327 |
+ |
representative of the dense ice polymorphs. A recent study by Sanz |
328 |
+ |
{\it et al.} goes into detail on the phase diagrams for SPC/E and |
329 |
+ |
TIP4P at higher pressures than those studied here.\cite{Sanz04} |
330 |
+ |
|
331 |
+ |
\begin{figure} |
332 |
+ |
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
333 |
+ |
\caption{Phase diagram for the TIP3P water model in the low pressure |
334 |
+ |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
335 |
+ |
the experimental values; however, the solid phases shown are not the |
336 |
+ |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
337 |
+ |
higher in energy and don't appear in the phase diagram.} |
338 |
+ |
\label{tp3phasedia} |
339 |
+ |
\end{figure} |
340 |
+ |
|
341 |
+ |
\begin{figure} |
342 |
+ |
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
343 |
+ |
\caption{Phase diagram for the SSD/RF water model in the low pressure |
344 |
+ |
regime. Calculations producing these results were done under an |
345 |
+ |
applied reaction field. It is interesting to note that this |
346 |
+ |
computationally efficient model (over 3 times more efficient than |
347 |
+ |
TIP3P) exhibits phase behavior similar to the less computationally |
348 |
+ |
conservative charge based models.} |
349 |
+ |
\label{ssdrfphasedia} |
350 |
+ |
\end{figure} |
351 |
+ |
|
352 |
+ |
\begin{table*} |
353 |
+ |
\begin{minipage}{\linewidth} |
354 |
+ |
\begin{center} |
355 |
+ |
|
356 |
+ |
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
357 |
+ |
temperatures at 1 atm for several common water models compared with |
358 |
+ |
experiment. The $T_m$ and $T_s$ values from simulation correspond to |
359 |
+ |
a transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the |
360 |
+ |
liquid or gas state.} |
361 |
+ |
|
362 |
+ |
\begin{tabular}{lccccccc} |
363 |
+ |
\hline |
364 |
+ |
Equilibrium Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ |
365 |
+ |
\hline |
366 |
+ |
$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ |
367 |
+ |
$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ |
368 |
+ |
$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ |
369 |
+ |
\end{tabular} |
370 |
+ |
\label{meltandboil} |
371 |
+ |
\end{center} |
372 |
+ |
\end{minipage} |
373 |
+ |
\end{table*} |
374 |
+ |
|
375 |
+ |
Table \ref{meltandboil} lists the melting and boiling temperatures |
376 |
+ |
calculated from this work. Surprisingly, most of these models have |
377 |
+ |
melting points that compare quite favorably with experiment. The |
378 |
+ |
unfortunate aspect of this result is that this phase change occurs |
379 |
+ |
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
380 |
+ |
liquid state. These results are actually not contrary to other |
381 |
+ |
studies. Studies of ice $I_h$ using TIP4P predict a $T_m$ ranging |
382 |
+ |
from 214 to 238 K (differences being attributed to choice of |
383 |
+ |
interaction truncation and different ordered and disordered molecular |
384 |
+ |
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
385 |
+ |
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
386 |
+ |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
387 |
+ |
calculated to be 265 K, indicating that these simulation based |
388 |
+ |
structures ought to be included in studies probing phase transitions |
389 |
+ |
with this model. Also of interest in these results is that SSD/E does |
390 |
+ |
not exhibit a melting point at 1 atm, but it shows a sublimation point |
391 |
+ |
at 355 K. This is due to the significant stability of Ice-{\it i} |
392 |
+ |
over all other polymorphs for this particular model under these |
393 |
+ |
conditions. While troubling, this behavior resulted in spontaneous |
394 |
+ |
crystallization of Ice-{\it i} and led us to investigate this |
395 |
+ |
structure. These observations provide a warning that simulations of |
396 |
+ |
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
397 |
+ |
risk of spontaneous crystallization. However, this risk lessens when |
398 |
+ |
applying a longer cutoff. |
399 |
+ |
|
400 |
+ |
\begin{figure} |
401 |
+ |
\includegraphics[width=\linewidth]{cutoffChange.eps} |
402 |
+ |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
403 |
+ |
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
404 |
+ |
with an added Ewald correction term. Calculations performed without a |
405 |
+ |
long-range correction show noticable free energy dependence on the |
406 |
+ |
cutoff radius and show some degree of converge at large cutoff radii. |
407 |
+ |
Inclusion of a long-range correction reduces the cutoff radius |
408 |
+ |
dependence of the free energy for all the models. Data for ice I$_c$ |
409 |
+ |
with TIP3P using 12 and 13.5 \AA\ cutoff radii were omitted being that |
410 |
+ |
the crystal was prone to distortion and melting at 200 K.} |
411 |
+ |
\label{incCutoff} |
412 |
+ |
\end{figure} |
413 |
+ |
|
414 |
+ |
Increasing the cutoff radius in simulations of the more |
415 |
+ |
computationally efficient water models was done in order to evaluate |
416 |
+ |
the trend in free energy values when moving to systems that do not |
417 |
+ |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
418 |
+ |
free energy of the ice polymorphs with water models lacking a |
419 |
+ |
long-range correction show a significant cutoff radius dependence. In |
420 |
+ |
general, there is a narrowing of the free energy differences while |
421 |
+ |
moving to greater cutoff radii. As the free energies for the |
422 |
+ |
polymorphs converge, the stability advantage that Ice-{\it i} exhibits |
423 |
+ |
is reduced. Interestingly, increasing the cutoff radius a mere 1.5 |
424 |
+ |
\AA\ with the SSD/E model destabilizes the Ice-{\it i} polymorph |
425 |
+ |
enough that the liquid state is preferred under standard simulation |
426 |
+ |
conditions (298 K and 1 atm). Thus, it is recommended that |
427 |
+ |
simulations using this model choose interaction truncation radii |
428 |
+ |
greater than 9 \AA. Considering the stabilization of Ice-{\it i} with |
429 |
+ |
smaller cutoffs, it is not surprising that crystallization was |
430 |
+ |
observed with SSD/E. The choice of a 9 \AA\ cutoff in the previous |
431 |
+ |
simulations gives the Ice-{\it i} polymorph a greater than 1 kcal/mol |
432 |
+ |
lower free energy than the ice $I_\textrm{h}$ starting configurations. |
433 |
+ |
Additionally, it should be noted that ice $I_c$ is not stable with |
434 |
+ |
cutoff radii of 12 and 13.5 \AA\ using the TIP3P water model. These |
435 |
+ |
simulations showed bulk distortions of the simulation cell that |
436 |
+ |
rapidly deteriorated crystal integrity. |
437 |
+ |
|
438 |
+ |
Adjacent to each of these model plots is a system with an applied or |
439 |
+ |
estimated long-range correction. SSD/RF was parametrized for use with |
440 |
+ |
a reaction field, and the benefit provided by this computationally |
441 |
+ |
inexpensive correction is apparent. Due to the relative independence |
442 |
+ |
of the resultant free energies, calculations performed with a small |
443 |
+ |
cutoff radius provide resultant properties similar to what one would |
444 |
+ |
expect for the bulk material. In the cases of TIP3P and SPC/E, the |
445 |
+ |
effect of an Ewald summation was estimated by applying the potential |
446 |
+ |
energy difference do to its inclusion in systems in the presence and |
447 |
+ |
absence of the correction. This was accomplished by calculation of |
448 |
+ |
the potential energy of identical crystals both with and without |
449 |
+ |
particle mesh Ewald (PME). Similar behavior to that observed with |
450 |
+ |
reaction field is seen for both of these models. The free energies |
451 |
+ |
show less dependence on cutoff radius and span a more narrowed range |
452 |
+ |
for the various polymorphs. Like the dipolar water models, TIP3P |
453 |
+ |
displays a relatively constant preference for the Ice-{\it i} |
454 |
+ |
polymorph. Crystal preference is much more difficult to determine for |
455 |
+ |
SPC/E. Without a long-range correction, each of the polymorphs |
456 |
+ |
studied assumes the role of the preferred polymorph under different |
457 |
+ |
cutoff conditions. The inclusion of the Ewald correction flattens and |
458 |
+ |
narrows the sequences of free energies so much that they often overlap |
459 |
+ |
within error, indicating that other conditions, such as cell volume in |
460 |
+ |
microcanonical simulations, can influence the chosen polymorph upon |
461 |
+ |
crystallization. All of these results support the finding that the |
462 |
+ |
Ice-{\it i} polymorph is a stable crystal structure that should be |
463 |
+ |
considered when studying the phase behavior of water models. |
464 |
+ |
|
465 |
+ |
\begin{table*} |
466 |
+ |
\begin{minipage}{\linewidth} |
467 |
+ |
\begin{center} |
468 |
+ |
|
469 |
+ |
\caption{The free energy versus cutoff radius for the studied ice |
470 |
+ |
polymorphs using SPC/E after the inclusion of the PME long-range |
471 |
+ |
interaction correction. Units are kcal/mol.} |
472 |
+ |
|
473 |
+ |
\begin{tabular}{ccccc} |
474 |
+ |
\hline |
475 |
+ |
Cutoff (\AA) & $I_h$ & $I_c$ & B & Ice-{\it i} \\ |
476 |
+ |
\hline |
477 |
+ |
9.0 & -12.98(2) & -13.00(2) & -12.97(3) & -13.02(2) \\ |
478 |
+ |
10.5 & -13.13(3) & -13.09(4) & -13.17(3) & -13.11(2) \\ |
479 |
+ |
12.0 & -13.06(2) & -13.09(2) & -13.15(4) & -13.12(2) \\ |
480 |
+ |
13.5 & -13.02(2) & -13.02(2) & -13.08(2) & -13.07(2) \\ |
481 |
+ |
15.0 & -13.11(4) & -12.97(2) & -13.09(2) & -12.95(2) \\ |
482 |
+ |
\end{tabular} |
483 |
+ |
\label{pmeShift} |
484 |
+ |
\end{center} |
485 |
+ |
\end{minipage} |
486 |
+ |
\end{table*} |
487 |
+ |
|
488 |
|
\section{Conclusions} |
489 |
|
|
490 |
+ |
The free energy for proton ordered variants of hexagonal and cubic ice |
491 |
+ |
$I$, ice B, and our recently discovered Ice-{\it i} structure were |
492 |
+ |
calculated under standard conditions for several common water models |
493 |
+ |
via thermodynamic integration. All the water models studied show |
494 |
+ |
Ice-{\it i} to be the minimum free energy crystal structure with a 9 |
495 |
+ |
\AA\ switching function cutoff. Calculated melting and boiling points |
496 |
+ |
show surprisingly good agreement with the experimental values; |
497 |
+ |
however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The |
498 |
+ |
effect of interaction truncation was investigated through variation of |
499 |
+ |
the cutoff radius, use of a reaction field parameterized model, and |
500 |
+ |
estimation of the results in the presence of the Ewald summation. |
501 |
+ |
Interaction truncation has a significant effect on the computed free |
502 |
+ |
energy values, and may significantly alter the free energy landscape |
503 |
+ |
for the more complex multipoint water models. Despite these effects, |
504 |
+ |
these results show Ice-{\it i} to be an important ice polymorph that |
505 |
+ |
should be considered in simulation studies. |
506 |
+ |
|
507 |
+ |
Due to this relative stability of Ice-{\it i} in all of the |
508 |
+ |
investigated simulation conditions, the question arises as to possible |
509 |
+ |
experimental observation of this polymorph. The rather extensive past |
510 |
+ |
and current experimental investigation of water in the low pressure |
511 |
+ |
regime makes us hesitant to ascribe any relevance of this work outside |
512 |
+ |
of the simulation community. It is for this reason that we chose a |
513 |
+ |
name for this polymorph which involves an imaginary quantity. That |
514 |
+ |
said, there are certain experimental conditions that would provide the |
515 |
+ |
most ideal situation for possible observation. These include the |
516 |
+ |
negative pressure or stretched solid regime, small clusters in vacuum |
517 |
+ |
deposition environments, and in clathrate structures involving small |
518 |
+ |
non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain |
519 |
+ |
our predictions for both the pair distribution function ($g_{OO}(r)$) |
520 |
+ |
and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for |
521 |
+ |
ice-{\it i} at a temperature of 77K. In studies of the high and low |
522 |
+ |
density forms of amorphous ice, ``spurious'' diffraction peaks have |
523 |
+ |
been observed experimentally.\cite{Bizid87} It is possible that a |
524 |
+ |
variant of Ice-{\it i} could explain some of this behavior; however, |
525 |
+ |
we will leave it to our experimental colleagues to make the final |
526 |
+ |
determination on whether this ice polymorph is named appropriately |
527 |
+ |
(i.e. with an imaginary number) or if it can be promoted to Ice-0. |
528 |
+ |
|
529 |
+ |
\begin{figure} |
530 |
+ |
\includegraphics[width=\linewidth]{iceGofr.eps} |
531 |
+ |
\caption{Radial distribution functions of ice $I_h$, $I_c$, |
532 |
+ |
Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations |
533 |
+ |
of the SSD/RF water model at 77 K.} |
534 |
+ |
\label{fig:gofr} |
535 |
+ |
\end{figure} |
536 |
+ |
|
537 |
+ |
\begin{figure} |
538 |
+ |
\includegraphics[width=\linewidth]{sofq.eps} |
539 |
+ |
\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, |
540 |
+ |
and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
541 |
+ |
been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
542 |
+ |
width) to compensate for the trunction effects in our finite size |
543 |
+ |
simulations.} |
544 |
+ |
\label{fig:sofq} |
545 |
+ |
\end{figure} |
546 |
+ |
|
547 |
|
\section{Acknowledgments} |
548 |
|
Support for this project was provided by the National Science |
549 |
|
Foundation under grant CHE-0134881. Computation time was provided by |
550 |
< |
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
551 |
< |
DMR-0079647. |
550 |
> |
the Notre Dame High Performance Computing Cluster and the Notre Dame |
551 |
> |
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
552 |
|
|
553 |
|
\newpage |
554 |
|
|