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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{Computational free energy studies of a new ice polymorph which |
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exhibits greater stability than Ice $I_h$} |
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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\\ |
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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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\maketitle |
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%\doublespacing |
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\begin{abstract} |
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The absolute free energies of several ice polymorphs were calculated |
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using thermodynamic integration. These polymorphs are predicted by |
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computer simulations using a variety of common water models to be |
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stable at low pressures. A recently discovered ice polymorph that has |
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as yet {\it only} been observed in computer simulations (Ice-{\it i}), |
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was determined to be the stable crystalline state for {\it all} the |
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water models investigated. Phase diagrams were generated, and phase |
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coexistence lines were determined for all of the known low-pressure |
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ice structures. Additionally, potential truncation was shown to play |
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a role in the resulting shape of the free energy landscape. |
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\end{abstract} |
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\maketitle |
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\newpage |
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%\narrowtext |
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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 |
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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 |
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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 |
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the limitations of |
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each.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two important |
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properties to quantify are the Gibbs and Helmholtz free energies, |
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particularly for the solid forms of water as these predict the |
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thermodynamic stability of the various phases. Water has a |
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particularly rich phase diagram and takes on a number of different and |
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stable crystalline structures as the temperature and pressure are |
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varied. It is a challenging task to investigate the entire free |
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energy landscape\cite{Sanz04}; and ideally, research is focused on the |
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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 |
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pressures for the model. |
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|
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The high-pressure phases of water (ice II - ice X as well as ice XII) |
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have been studied extensively both experimentally and |
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computationally. In this paper, standard reference state methods were |
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applied in the {\it low} pressure regime to evaluate the free energies |
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for a few known crystalline water polymorphs that might be stable at |
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these pressures. This work is unique in that one of the crystal |
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lattices was arrived at through crystallization of a computationally |
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efficient water model under constant pressure and temperature |
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conditions. Crystallization events are interesting in and of |
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themselves\cite{Matsumoto02,Yamada02}; however, the crystal structure |
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obtained in this case is different from any previously observed ice |
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polymorphs in experiment or simulation.\cite{Fennell04} We have named |
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this structure Ice-{\it i} to indicate its origin in computational |
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simulation. The unit cell of Ice-{\it i} and an axially-elongated |
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variant named Ice-{\it i}$^\prime$ both consist of eight water |
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molecules that stack in rows of interlocking water tetramers as |
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illustrated in figures \ref{unitcell}A and \ref{unitcell}B. These |
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tetramers form a crystal structure similar in appearance to a recent |
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two-dimensional surface tessellation simulated on silica.\cite{Yang04} |
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As expected in an ice crystal constructed of water tetramers, the |
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hydrogen bonds are not as linear as those observed in ice $I_h$, |
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however the interlocking of these subunits appears to provide |
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significant stabilization to the overall crystal. The arrangement of |
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these tetramers results in octagonal cavities that are typically |
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greater than 6.3 \AA\ in diameter (Fig. \ref{iCrystal}). This open |
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structure leads to crystals that are typically 0.07 g/cm$^3$ less |
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dense than ice $I_h$. |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{unitCell.eps} |
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\caption{(A) Unit cells for Ice-{\it i} and (B) Ice-{\it i}$^\prime$. |
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The spheres represent the center-of-mass locations of the water |
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molecules. The $a$ to $c$ ratios for Ice-{\it i} and Ice-{\it |
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i}$^\prime$ are given by 2.1214 and 1.785 respectively.} |
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\label{unitcell} |
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\end{figure} |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{orderedIcei.eps} |
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\caption{A rendering of a proton ordered crystal of Ice-{\it i} looking |
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down the (001) crystal face. The presence of large octagonal pores |
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leads to a polymorph that is less dense than ice $I_h$.} |
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\label{iCrystal} |
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\end{figure} |
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|
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Results from our previous study indicated that Ice-{\it i} is the |
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minimum energy crystal structure for the single point water models |
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investigated (for discussions on these single point dipole models, see |
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our previous work and related |
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articles).\cite{Fennell04,Liu96,Bratko85} Our earlier results |
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considered only energetic stabilization and neglected entropic |
| 131 |
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contributions to the overall free energy. To address this issue, we |
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have calculated the absolute free energy of this crystal using |
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thermodynamic integration and compared it to the free energies of ice |
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$I_c$ and ice $I_h$ (the common low density ice polymorphs) and ice B |
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(a higher density, but very stable crystal structure observed by |
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B\`{a}ez and Clancy in free energy studies of SPC/E).\cite{Baez95b} |
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This work includes results for the water model from which Ice-{\it i} |
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was crystallized (SSD/E) in addition to several common water models |
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(TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field parametrized |
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single point dipole water model (SSD/RF). The axially-elongated |
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variant, Ice-{\it i}$^\prime$, was used in calculations involving |
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SPC/E, TIP4P, and TIP5P. The square tetramers in Ice-{\it i} distort |
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in Ice-{\it i}$^\prime$ to form a rhombus with alternating 85 and 95 |
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degree angles. Under SPC/E, TIP4P, and TIP5P, this geometry is better |
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at forming favorable hydrogen bonds. The degree of rhomboid |
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distortion depends on the water model used, but is significant enough |
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to split a peak in the radial distribution function which corresponds |
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to diagonal sites in the tetramers. |
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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 (Object-Oriented Parallel Simulation Engine) |
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molecular mechanics package. All molecules were treated as rigid |
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bodies, with orientational motion propogated using the symplectic DLM |
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integration method. Details about the implementation of these |
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techniques can be found in a recent publication.\cite{Meineke05} |
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performed using the OOPSE molecular mechanics program.\cite{Meineke05} |
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All molecules were treated as rigid bodies, with orientational motion |
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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 |
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publication.\cite{Dullweber1997} |
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|
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Thermodynamic integration was utilized to calculate the free energy of |
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several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, |
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SSD/RF, and SSD/E water models. Liquid state free energies at 300 and |
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400 K for all of these water models were also determined using this |
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same technique, in order to determine melting points and generate |
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phase diagrams. All simulations were carried out at densities |
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resulting in a pressure of approximately 1 atm at their respective |
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temperatures. |
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Thermodynamic integration was utilized to calculate the Helmholtz free |
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energies ($A$) of the listed water models at various state points |
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using the OOPSE molecular dynamics program.\cite{Meineke05} |
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Thermodynamic integration is an established technique that has been |
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used extensively in the calculation of free energies for condensed |
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phases of |
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materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
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method uses a sequence of simulations during which the system of |
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interest is converted into a reference system for which the free |
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energy is known analytically ($A_0$). The difference in potential |
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energy between the reference system and the system of interest |
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($\Delta V$) is then integrated in order to determine the free energy |
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difference between the two states: |
| 172 |
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\begin{equation} |
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A = A_0 + \int_0^1 \left\langle \Delta V \right\rangle_\lambda d\lambda. |
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\end{equation} |
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Here, $\lambda$ is the parameter that governs the transformation |
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between the reference system and the system of interest. For |
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crystalline phases, an harmonically-restrained (Einsten) crystal is |
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chosen as the reference state, while for liquid phases, the ideal gas |
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is taken as the reference state. |
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|
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For the thermodynamic integration of molecular crystals, the Einstein |
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Crystal is chosen as the reference state that the system is converted |
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to over the course of the simulation. In an Einstein Crystal, the |
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molecules are harmonically restrained at their ideal lattice locations |
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and orientations. The partition function for a molecular crystal |
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restrained in this fashion has been evaluated, and the Helmholtz Free |
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Energy ({\it A}) is given by |
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In an Einstein crystal, the molecules are restrained at their ideal |
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lattice locations and orientations. Using harmonic restraints, as |
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applied by B\`{a}ez and Clancy, the total potential for this reference |
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crystal ($V_\mathrm{EC}$) 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 |
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the spring constants restraining translational motion and deflection |
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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}}\exp(t^2)\mathrm{d}t\right ], |
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\label{ecFreeEnergy} |
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\end{eqnarray} |
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where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation |
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\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and |
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$K_\mathrm{\omega}$ are the spring constants restraining translational |
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motion and deflection of and rotation around the principle axis of the |
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molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the |
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minimum potential energy of the ideal crystal. In the case of |
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molecular liquids, the ideal vapor is chosen as the target reference |
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state. |
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where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
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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[scale=1.0]{rotSpring.eps} |
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\centering |
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\includegraphics[width=4in]{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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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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Charge, dipole, and Lennard-Jones interactions were modified by a |
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cubic switching between 100\% and 85\% of the cutoff value (9 \AA ). By |
| 229 |
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applying this function, these interactions are smoothly truncated, |
| 230 |
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thereby avoiding poor energy conserving dynamics resulting from |
| 231 |
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harsher truncation schemes. The effect of a long-range correction was |
| 232 |
< |
also investigated on select model systems in a variety of manners. For |
| 233 |
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the SSD/RF model, a reaction field with a fixed dielectric constant of |
| 234 |
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80 was applied in all simulations.\cite{Onsager36} For a series of the |
| 235 |
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least computationally expensive models (SSD/E, SSD/RF, and TIP3P), |
| 236 |
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simulations were performed with longer cutoffs of 12 and 15 \AA\ to |
| 237 |
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compare with the 9 \AA\ cutoff results. Finally, results from the use |
| 238 |
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of an Ewald summation were estimated for TIP3P and SPC/E by performing |
| 118 |
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calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
| 119 |
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mechanics software package. TINKER was chosen because it can also |
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propogate the motion of rigid-bodies, and provides the most direct |
| 121 |
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comparison to the results from OOPSE. The calculated energy difference |
| 122 |
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in the presence and absence of PME was applied to the previous results |
| 123 |
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in order to predict changes in the free energy landscape. |
| 227 |
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In the case of molecular liquids, the ideal vapor is chosen as the |
| 228 |
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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 |
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literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
| 231 |
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typically differ in regard to the path taken for switching off the |
| 232 |
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interaction potential to convert the system to an ideal gas of water |
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molecules. In this study, we applied one of the most convenient |
| 234 |
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methods and integrated over the $\lambda^4$ path, where all |
| 235 |
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interaction parameters are scaled equally by this transformation |
| 236 |
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parameter. This method has been shown to be reversible and provide |
| 237 |
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results in excellent agreement with other established |
| 238 |
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methods.\cite{Baez95b} |
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|
| 240 |
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\section{Results and discussion} |
| 240 |
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Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and |
| 241 |
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Lennard-Jones interactions were gradually reduced by a cubic switching |
| 242 |
> |
function. By applying this function, these interactions are smoothly |
| 243 |
> |
truncated, thereby avoiding the poor energy conservation which results |
| 244 |
> |
from harsher truncation schemes. The effect of a long-range |
| 245 |
> |
correction was also investigated on select model systems in a variety |
| 246 |
> |
of manners. For the SSD/RF model, a reaction field with a fixed |
| 247 |
> |
dielectric constant of 80 was applied in all |
| 248 |
> |
simulations.\cite{Onsager36} For a series of the least computationally |
| 249 |
> |
expensive models (SSD/E, SSD/RF, TIP3P, and SPC/E), simulations were |
| 250 |
> |
performed with longer cutoffs of 10.5, 12, 13.5, and 15 \AA\ to |
| 251 |
> |
compare with the 9 \AA\ cutoff results. Finally, the effects of using |
| 252 |
> |
the Ewald summation were estimated for TIP3P and SPC/E by performing |
| 253 |
> |
single configuration Particle-Mesh Ewald (PME) |
| 254 |
> |
calculations~\cite{Tinker} for each of the ice polymorphs. The |
| 255 |
> |
calculated energy difference in the presence and absence of PME was |
| 256 |
> |
applied to the previous results in order to predict changes to the |
| 257 |
> |
free energy landscape. |
| 258 |
|
|
| 259 |
< |
The free energy of proton ordered Ice-{\it i} was calculated and |
| 128 |
< |
compared with the free energies of proton ordered variants of the |
| 129 |
< |
experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, |
| 130 |
< |
as well as the higher density ice B, observed by B\`{a}ez and Clancy |
| 131 |
< |
and thought to be the minimum free energy structure for the SPC/E |
| 132 |
< |
model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
| 133 |
< |
Ice XI, the experimentally observed proton ordered variant of ice |
| 134 |
< |
$I_h$, was investigated initially, but it was found not to be as |
| 135 |
< |
stable as antiferroelectric variants of proton ordered or even proton |
| 136 |
< |
disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of |
| 137 |
< |
ice $I_h$ used here is a simple antiferroelectric version that has an |
| 138 |
< |
8 molecule unit cell. The crystals contained 648 or 1728 molecules for |
| 139 |
< |
ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for ice |
| 140 |
< |
$I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
| 141 |
< |
were necessary for simulations involving larger cutoff values. |
| 259 |
> |
\section{Results and Discussion} |
| 260 |
|
|
| 261 |
+ |
The calculated free energies of proton-ordered variants of three low |
| 262 |
+ |
density polymorphs ($I_h$, $I_c$, and Ice-{\it i} or Ice-{\it |
| 263 |
+ |
i}$^\prime$) and the stable higher density ice B are listed in Table |
| 264 |
+ |
\ref{freeEnergy}. Ice B was included because it has been |
| 265 |
+ |
shown to be a minimum free energy structure for SPC/E at ambient |
| 266 |
+ |
conditions.\cite{Baez95b} In addition to the free energies, the |
| 267 |
+ |
relevant transition temperatures at standard pressure are also |
| 268 |
+ |
displayed in Table \ref{freeEnergy}. These free energy values |
| 269 |
+ |
indicate that Ice-{\it i} is the most stable state for all of the |
| 270 |
+ |
investigated water models. With the free energy at these state |
| 271 |
+ |
points, the Gibbs-Helmholtz equation was used to project to other |
| 272 |
+ |
state points and to build phase diagrams. Figure \ref{tp3PhaseDia} is |
| 273 |
+ |
an example diagram built from the results for the TIP3P water model. |
| 274 |
+ |
All other models have similar structure, although the crossing points |
| 275 |
+ |
between the phases move to different temperatures and pressures as |
| 276 |
+ |
indicated from the transition temperatures in Table \ref{freeEnergy}. |
| 277 |
+ |
It is interesting to note that ice $I_h$ (and ice $I_c$ for that |
| 278 |
+ |
matter) do not appear in any of the phase diagrams for any of the |
| 279 |
+ |
models. For purposes of this study, ice B is representative of the |
| 280 |
+ |
dense ice polymorphs. A recent study by Sanz {\it et al.} provides |
| 281 |
+ |
details on the phase diagrams for SPC/E and TIP4P at higher pressures |
| 282 |
+ |
than those studied here.\cite{Sanz04} |
| 283 |
+ |
|
| 284 |
|
\begin{table*} |
| 285 |
|
\begin{minipage}{\linewidth} |
| 145 |
– |
\renewcommand{\thefootnote}{\thempfootnote} |
| 286 |
|
\begin{center} |
| 287 |
< |
\caption{Calculated free energies for several ice polymorphs with a |
| 288 |
< |
variety of common water models. All calculations used a cutoff radius |
| 289 |
< |
of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are |
| 290 |
< |
kcal/mol. *Ice $I_c$ is unstable at 200 K using SSD/RF.} |
| 291 |
< |
\begin{tabular}{ l c c c c } |
| 292 |
< |
\hline \\[-7mm] |
| 293 |
< |
\ \quad \ Water Model\ \ & \ \quad \ \ \ \ $I_h$ \ \ & \ \quad \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \quad \ \ \ Ice-{\it i} \ \quad \ \\ |
| 294 |
< |
\hline \\[-3mm] |
| 295 |
< |
\ \quad \ TIP3P & \ \quad \ -11.41 & \ \quad \ -11.23 & \ \quad \ -11.82 & \quad -12.30\\ |
| 296 |
< |
\ \quad \ TIP4P & \ \quad \ -11.84 & \ \quad \ -12.04 & \ \quad \ -12.08 & \quad -12.33\\ |
| 297 |
< |
\ \quad \ TIP5P & \ \quad \ -11.85 & \ \quad \ -11.86 & \ \quad \ -11.96 & \quad -12.29\\ |
| 298 |
< |
\ \quad \ SPC/E & \ \quad \ -12.67 & \ \quad \ -12.96 & \ \quad \ -13.25 & \quad -13.55\\ |
| 299 |
< |
\ \quad \ SSD/E & \ \quad \ -11.27 & \ \quad \ -11.19 & \ \quad \ -12.09 & \quad -12.54\\ |
| 300 |
< |
\ \quad \ SSD/RF & \ \quad \ -11.51 & \ \quad \ NA* & \ \quad \ -12.08 & \quad -12.29\\ |
| 287 |
> |
\caption{Calculated free energies for several ice polymorphs along |
| 288 |
> |
with the calculated melting (or sublimation) and boiling points for |
| 289 |
> |
the investigated water models. All free energy calculations used a |
| 290 |
> |
cutoff radius of 9.0 \AA\ and were performed at 200 K and $\sim$1 atm. |
| 291 |
> |
Units of free energy are kcal/mol, while transition temperature are in |
| 292 |
> |
Kelvin. Calculated error of the final digits is in parentheses.} |
| 293 |
> |
\begin{tabular}{lccccccc} |
| 294 |
> |
\hline |
| 295 |
> |
Water Model & $I_h$ & $I_c$ & B & Ice-{\it i} & Ice-{\it i}$^\prime$ & $T_m$ (*$T_s$) & $T_b$\\ |
| 296 |
> |
\hline |
| 297 |
> |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(4) & 357(2)\\ |
| 298 |
> |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 266(5) & 354(2)\\ |
| 299 |
> |
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 271(4) & 337(2)\\ |
| 300 |
> |
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 296(3) & 396(2)\\ |
| 301 |
> |
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(2) & -\\ |
| 302 |
> |
SSD/RF & -11.51(2) & -11.47(2) & -12.08(3) & -12.29(2) & - & 278(4) & 349(2)\\ |
| 303 |
|
\end{tabular} |
| 304 |
|
\label{freeEnergy} |
| 305 |
|
\end{center} |
| 306 |
|
\end{minipage} |
| 307 |
|
\end{table*} |
| 308 |
|
|
| 167 |
– |
The free energy values computed for the studied polymorphs indicate |
| 168 |
– |
that Ice-{\it i} is the most stable state for all of the common water |
| 169 |
– |
models studied. With the free energy at these state points, the |
| 170 |
– |
temperature and pressure dependence of the free energy was used to |
| 171 |
– |
project to other state points and build phase diagrams. Figures |
| 172 |
– |
\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
| 173 |
– |
from the free energy results. All other models have similar structure, |
| 174 |
– |
only the crossing points between these phases exist at different |
| 175 |
– |
temperatures and pressures. It is interesting to note that ice $I$ |
| 176 |
– |
does not exist in either cubic or hexagonal form in any of the phase |
| 177 |
– |
diagrams for any of the models. For purposes of this study, ice B is |
| 178 |
– |
representative of the dense ice polymorphs. A recent study by Sanz |
| 179 |
– |
{\it et al.} goes into detail on the phase diagrams for SPC/E and |
| 180 |
– |
TIP4P in the high pressure regime.\cite{Sanz04} |
| 309 |
|
\begin{figure} |
| 310 |
|
\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
| 311 |
|
\caption{Phase diagram for the TIP3P water model in the low pressure |
| 312 |
< |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 312 |
> |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 313 |
|
the experimental values; however, the solid phases shown are not the |
| 314 |
< |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
| 314 |
> |
experimentally observed forms. Both cubic and hexagonal ice $I$ are |
| 315 |
|
higher in energy and don't appear in the phase diagram.} |
| 316 |
< |
\label{tp3phasedia} |
| 316 |
> |
\label{tp3PhaseDia} |
| 317 |
|
\end{figure} |
| 318 |
+ |
|
| 319 |
+ |
Most of the water models have melting points that compare quite |
| 320 |
+ |
favorably with the experimental value of 273 K. The unfortunate |
| 321 |
+ |
aspect of this result is that this phase change occurs between |
| 322 |
+ |
Ice-{\it i} and the liquid state rather than ice $I_h$ and the liquid |
| 323 |
+ |
state. These results do not contradict other studies. Studies of ice |
| 324 |
+ |
$I_h$ using TIP4P predict a $T_m$ ranging from 214 to 238 K |
| 325 |
+ |
(differences being attributed to choice of interaction truncation and |
| 326 |
+ |
different ordered and disordered molecular |
| 327 |
+ |
arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and |
| 328 |
+ |
Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
| 329 |
+ |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 330 |
+ |
calculated to be 265 K, indicating that these simulation based |
| 331 |
+ |
structures ought to be included in studies probing phase transitions |
| 332 |
+ |
with this model. Also of interest in these results is that SSD/E does |
| 333 |
+ |
not exhibit a melting point at 1 atm but does sublime at 355 K. This |
| 334 |
+ |
is due to the significant stability of Ice-{\it i} over all other |
| 335 |
+ |
polymorphs for this particular model under these conditions. While |
| 336 |
+ |
troubling, this behavior resulted in the spontaneous crystallization |
| 337 |
+ |
of Ice-{\it i} which led us to investigate this structure. These |
| 338 |
+ |
observations provide a warning that simulations of SSD/E as a |
| 339 |
+ |
``liquid'' near 300 K are actually metastable and run the risk of |
| 340 |
+ |
spontaneous crystallization. However, when a longer cutoff radius is |
| 341 |
+ |
used, SSD/E prefers the liquid state under standard temperature and |
| 342 |
+ |
pressure. |
| 343 |
+ |
|
| 344 |
|
\begin{figure} |
| 345 |
< |
\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} |
| 346 |
< |
\caption{Phase diagram for the SSD/RF water model in the low pressure |
| 347 |
< |
regime. Calculations producing these results were done under an |
| 348 |
< |
applied reaction field. It is interesting to note that this |
| 349 |
< |
computationally efficient model (over 3 times more efficient than |
| 350 |
< |
TIP3P) exhibits phase behavior similar to the less computationally |
| 351 |
< |
conservative charge based models.} |
| 352 |
< |
\label{ssdrfphasedia} |
| 345 |
> |
\includegraphics[width=\linewidth]{cutoffChange.eps} |
| 346 |
> |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
| 347 |
> |
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
| 348 |
> |
with an added Ewald correction term. Error for the larger cutoff |
| 349 |
> |
points is equivalent to that observed at 9.0\AA\ (see Table |
| 350 |
> |
\ref{freeEnergy}). Data for ice I$_c$ with TIP3P using both 12 and |
| 351 |
> |
13.5 \AA\ cutoffs were omitted because the crystal was prone to |
| 352 |
> |
distortion and melting at 200 K. Ice-{\it i}$^\prime$ is the form of |
| 353 |
> |
Ice-{\it i} used in the SPC/E simulations.} |
| 354 |
> |
\label{incCutoff} |
| 355 |
|
\end{figure} |
| 356 |
|
|
| 357 |
< |
\begin{table*} |
| 358 |
< |
\begin{minipage}{\linewidth} |
| 359 |
< |
\renewcommand{\thefootnote}{\thempfootnote} |
| 360 |
< |
\begin{center} |
| 361 |
< |
\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) |
| 362 |
< |
temperatures of several common water models compared with experiment.} |
| 363 |
< |
\begin{tabular}{ l c c c c c c c } |
| 364 |
< |
\hline \\[-7mm] |
| 365 |
< |
\ \ Equilibria Point\ \ & \ \ \ \ \ TIP3P \ \ & \ \ \ \ \ TIP4P \ \ & \ \quad \ \ \ \ TIP5P \ \ & \ \ \ \ \ SPC/E \ \ & \ \ \ \ \ SSD/E \ \ & \ \ \ \ \ SSD/RF \ \ & \ \ \ \ \ Exp. \ \ \\ |
| 366 |
< |
\hline \\[-3mm] |
| 367 |
< |
\ \ $T_m$ (K) & \ \ 269 & \ \ 265 & \ \ 271 & 297 & \ \ - & \ \ 278 & \ \ 273\\ |
| 368 |
< |
\ \ $T_b$ (K) & \ \ 357 & \ \ 354 & \ \ 337 & 396 & \ \ - & \ \ 349 & \ \ 373\\ |
| 369 |
< |
\ \ $T_s$ (K) & \ \ - & \ \ - & \ \ - & - & \ \ 355 & \ \ - & \ \ -\\ |
| 370 |
< |
\end{tabular} |
| 371 |
< |
\label{meltandboil} |
| 372 |
< |
\end{center} |
| 373 |
< |
\end{minipage} |
| 374 |
< |
\end{table*} |
| 357 |
> |
For the more computationally efficient water models, we have also |
| 358 |
> |
investigated the effect of potential trunctaion on the computed free |
| 359 |
> |
energies as a function of the cutoff radius. As seen in |
| 360 |
> |
Fig. \ref{incCutoff}, the free energies of the ice polymorphs with |
| 361 |
> |
water models lacking a long-range correction show significant cutoff |
| 362 |
> |
dependence. In general, there is a narrowing of the free energy |
| 363 |
> |
differences while moving to greater cutoff radii. As the free |
| 364 |
> |
energies for the polymorphs converge, the stability advantage that |
| 365 |
> |
Ice-{\it i} exhibits is reduced. Adjacent to each of these plots are |
| 366 |
> |
results for systems with applied or estimated long-range corrections. |
| 367 |
> |
SSD/RF was parametrized for use with a reaction field, and the benefit |
| 368 |
> |
provided by this computationally inexpensive correction is apparent. |
| 369 |
> |
The free energies are largely independent of the size of the reaction |
| 370 |
> |
field cavity in this model, so small cutoff radii mimic bulk |
| 371 |
> |
calculations quite well under SSD/RF. |
| 372 |
> |
|
| 373 |
> |
Although TIP3P was paramaterized for use without the Ewald summation, |
| 374 |
> |
we have estimated the effect of this method for computing long-range |
| 375 |
> |
electrostatics for both TIP3P and SPC/E. This was accomplished by |
| 376 |
> |
calculating the potential energy of identical crystals both with and |
| 377 |
> |
without particle mesh Ewald (PME). Similar behavior to that observed |
| 378 |
> |
with reaction field is seen for both of these models. The free |
| 379 |
> |
energies show reduced dependence on cutoff radius and span a narrower |
| 380 |
> |
range for the various polymorphs. Like the dipolar water models, |
| 381 |
> |
TIP3P displays a relatively constant preference for the Ice-{\it i} |
| 382 |
> |
polymorph. Crystal preference is much more difficult to determine for |
| 383 |
> |
SPC/E. Without a long-range correction, each of the polymorphs |
| 384 |
> |
studied assumes the role of the preferred polymorph under different |
| 385 |
> |
cutoff radii. The inclusion of the Ewald correction flattens and |
| 386 |
> |
narrows the gap in free energies such that the polymorphs are |
| 387 |
> |
isoenergetic within statistical uncertainty. This suggests that other |
| 388 |
> |
conditions, such as the density in fixed-volume simulations, can |
| 389 |
> |
influence the polymorph expressed upon crystallization. |
| 390 |
|
|
| 391 |
< |
Table \ref{meltandboil} lists the melting and boiling temperatures |
| 221 |
< |
calculated from this work. Surprisingly, most of these models have |
| 222 |
< |
melting points that compare quite favorably with experiment. The |
| 223 |
< |
unfortunate aspect of this result is that this phase change occurs |
| 224 |
< |
between Ice-{\it i} and the liquid state rather than ice $I_h$ and the |
| 225 |
< |
liquid state. These results are actually not contrary to previous |
| 226 |
< |
studies in the literature. Earlier free energy studies of ice $I$ |
| 227 |
< |
using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences |
| 228 |
< |
being attributed to choice of interaction truncation and different |
| 229 |
< |
ordered and disordered molecular arrangements). If the presence of ice |
| 230 |
< |
B and Ice-{\it i} were omitted, a $T_m$ value around 210 K would be |
| 231 |
< |
predicted from this work. However, the $T_m$ from Ice-{\it i} is |
| 232 |
< |
calculated at 265 K, significantly higher in temperature than the |
| 233 |
< |
previous studies. Also of interest in these results is that SSD/E does |
| 234 |
< |
not exhibit a melting point at 1 atm, but it shows a sublimation point |
| 235 |
< |
at 355 K. This is due to the significant stability of Ice-{\it i} over |
| 236 |
< |
all other polymorphs for this particular model under these |
| 237 |
< |
conditions. While troubling, this behavior turned out to be |
| 238 |
< |
advantagious in that it facilitated the spontaneous crystallization of |
| 239 |
< |
Ice-{\it i}. These observations provide a warning that simulations of |
| 240 |
< |
SSD/E as a ``liquid'' near 300 K are actually metastable and run the |
| 241 |
< |
risk of spontaneous crystallization. However, this risk changes when |
| 242 |
< |
applying a longer cutoff. |
| 391 |
> |
\section{Conclusions} |
| 392 |
|
|
| 393 |
< |
Increasing the cutoff radius in simulations of the more |
| 394 |
< |
computationally efficient water models was done in order to evaluate |
| 395 |
< |
the trend in free energy values when moving to systems that do not |
| 396 |
< |
involve potential truncation. As seen in Fig. \ref{incCutoff}, the |
| 397 |
< |
free energy of all the ice polymorphs show a substantial dependence on |
| 398 |
< |
cutoff radius. In general, there is a narrowing of the free energy |
| 399 |
< |
differences while moving to greater cutoff radius. This trend is much |
| 400 |
< |
more subtle in the case of SSD/RF, indicating that the free energies |
| 252 |
< |
calculated with a reaction field present provide a more accurate |
| 253 |
< |
picture of the free energy landscape in the absence of potential |
| 254 |
< |
truncation. |
| 393 |
> |
In this report, thermodynamic integration was used to determine the |
| 394 |
> |
absolute free energies of several ice polymorphs. Of the studied |
| 395 |
> |
crystal forms, Ice-{\it i} was observed to be the stable crystalline |
| 396 |
> |
state for {\it all} the water models when using a 9.0 \AA\ |
| 397 |
> |
intermolecular interaction cutoff. Through investigation of possible |
| 398 |
> |
interaction truncation methods, the free energy was shown to be |
| 399 |
> |
partially dependent on simulation conditions; however, Ice-{\it i} was |
| 400 |
> |
still observered to be a stable polymorph of the studied water models. |
| 401 |
|
|
| 402 |
< |
To further study the changes resulting to the inclusion of a |
| 403 |
< |
long-range interaction correction, the effect of an Ewald summation |
| 404 |
< |
was estimated by applying the potential energy difference do to its |
| 405 |
< |
inclusion in systems in the presence and absence of the |
| 406 |
< |
correction. This was accomplished by calculation of the potential |
| 407 |
< |
energy of identical crystals with and without PME using TINKER. The |
| 408 |
< |
free energies for the investigated polymorphs using the TIP3P and |
| 409 |
< |
SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P |
| 410 |
< |
are not fully supported in TINKER, so the results for these models |
| 411 |
< |
could not be estimated. The same trend pointed out through increase of |
| 412 |
< |
cutoff radius is observed in these results. Ice-{\it i} is the |
| 413 |
< |
preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
| 414 |
< |
water models; however, there is a narrowing of the free energy |
| 269 |
< |
differences between the various solid forms. In the case of SPC/E this |
| 270 |
< |
narrowing is significant enough that it becomes less clear cut that |
| 271 |
< |
Ice-{\it i} is the most stable polymorph, and is possibly metastable |
| 272 |
< |
with respect to ice B and possibly ice $I_c$. However, these results |
| 273 |
< |
do not significantly alter the finding that the Ice-{\it i} polymorph |
| 274 |
< |
is a stable crystal structure that should be considered when studying |
| 275 |
< |
the phase behavior of water models. |
| 402 |
> |
So what is the preferred solid polymorph for simulated water? As |
| 403 |
> |
indicated above, the answer appears to be dependent both on the |
| 404 |
> |
conditions and the model used. In the case of short cutoffs without a |
| 405 |
> |
long-range interaction correction, Ice-{\it i} and Ice-{\it |
| 406 |
> |
i}$^\prime$ have the lowest free energy of the studied polymorphs with |
| 407 |
> |
all the models. Ideally, crystallization of each model under constant |
| 408 |
> |
pressure conditions, as was done with SSD/E, would aid in the |
| 409 |
> |
identification of their respective preferred structures. This work, |
| 410 |
> |
however, helps illustrate how studies involving one specific model can |
| 411 |
> |
lead to insight about important behavior of others. In general, the |
| 412 |
> |
above results support the finding that the Ice-{\it i} polymorph is a |
| 413 |
> |
stable crystal structure that should be considered when studying the |
| 414 |
> |
phase behavior of water models. |
| 415 |
|
|
| 416 |
< |
\begin{table*} |
| 417 |
< |
\begin{minipage}{\linewidth} |
| 418 |
< |
\renewcommand{\thefootnote}{\thempfootnote} |
| 419 |
< |
\begin{center} |
| 420 |
< |
\caption{The free energy of the studied ice polymorphs after applying the energy difference attributed to the inclusion of the PME long-range interaction correction. Units are kcal/mol.} |
| 421 |
< |
\begin{tabular}{ l c c c c } |
| 283 |
< |
\hline \\[-7mm] |
| 284 |
< |
\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ |
| 285 |
< |
\hline \\[-3mm] |
| 286 |
< |
\ \ TIP3P & \ \ -11.53 & \ \ -11.24 & \ \ -11.51 & \ \ -11.67\\ |
| 287 |
< |
\ \ SPC/E & \ \ -12.77 & \ \ -12.92 & \ \ -12.96 & \ \ -13.02\\ |
| 288 |
< |
\end{tabular} |
| 289 |
< |
\label{pmeShift} |
| 290 |
< |
\end{center} |
| 291 |
< |
\end{minipage} |
| 292 |
< |
\end{table*} |
| 416 |
> |
We also note that none of the water models used in this study are |
| 417 |
> |
polarizable or flexible models. It is entirely possible that the |
| 418 |
> |
polarizability of real water makes Ice-{\it i} substantially less |
| 419 |
> |
stable than ice $I_h$. However, the calculations presented above seem |
| 420 |
> |
interesting enough to communicate before the role of polarizability |
| 421 |
> |
(or flexibility) has been thoroughly investigated. |
| 422 |
|
|
| 423 |
< |
\section{Conclusions} |
| 423 |
> |
Finally, due to the stability of Ice-{\it i} in the investigated |
| 424 |
> |
simulation conditions, the question arises as to possible experimental |
| 425 |
> |
observation of this polymorph. The rather extensive past and current |
| 426 |
> |
experimental investigation of water in the low pressure regime makes |
| 427 |
> |
us hesitant to ascribe any relevance to this work outside of the |
| 428 |
> |
simulation community. It is for this reason that we chose a name for |
| 429 |
> |
this polymorph which involves an imaginary quantity. That said, there |
| 430 |
> |
are certain experimental conditions that would provide the most ideal |
| 431 |
> |
situation for possible observation. These include the negative |
| 432 |
> |
pressure or stretched solid regime, small clusters in vacuum |
| 433 |
> |
deposition environments, and in clathrate structures involving small |
| 434 |
> |
non-polar molecules. For experimental comparison purposes, example |
| 435 |
> |
$g_{OO}(r)$ and $S(\vec{q})$ plots were generated for the two Ice-{\it |
| 436 |
> |
i} variants (along with example ice $I_h$ and $I_c$ plots) at 77K, and |
| 437 |
> |
they are shown in figures \ref{fig:gofr} and \ref{fig:sofq} |
| 438 |
> |
respectively. |
| 439 |
|
|
| 440 |
+ |
\begin{figure} |
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+ |
\centering |
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+ |
\includegraphics[width=\linewidth]{iceGofr.eps} |
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+ |
\caption{Radial distribution functions of ice $I_h$, $I_c$, and |
| 444 |
+ |
Ice-{\it i} calculated from from simulations of the SSD/RF water model |
| 445 |
+ |
at 77 K. The Ice-{\it i} distribution function was obtained from |
| 446 |
+ |
simulations composed of TIP4P water.} |
| 447 |
+ |
\label{fig:gofr} |
| 448 |
+ |
\end{figure} |
| 449 |
+ |
|
| 450 |
+ |
\begin{figure} |
| 451 |
+ |
\centering |
| 452 |
+ |
\includegraphics[width=\linewidth]{sofq.eps} |
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+ |
\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, |
| 454 |
+ |
and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have |
| 455 |
+ |
been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ |
| 456 |
+ |
width) to compensate for the trunction effects in our finite size |
| 457 |
+ |
simulations.} |
| 458 |
+ |
\label{fig:sofq} |
| 459 |
+ |
\end{figure} |
| 460 |
+ |
|
| 461 |
|
\section{Acknowledgments} |
| 462 |
|
Support for this project was provided by the National Science |
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|
Foundation under grant CHE-0134881. Computation time was provided by |
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< |
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
| 465 |
< |
DMR-0079647. |
| 464 |
> |
the Notre Dame High Performance Computing Cluster and the Notre Dame |
| 465 |
> |
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). |
| 466 |
|
|
| 467 |
|
\newpage |
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|
|
