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\begin{document} |
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\title{Ice-{\it i}: a novel ice polymorph predicted via computer simulation} |
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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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|
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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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point, because these phases will dictate the true transition |
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temperatures and pressures for their respective model. |
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|
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In this paper, standard reference state methods were applied to the |
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study of crystalline water polymorphs in the low pressure regime. This |
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work is unique in the fact that one of the crystal lattices was |
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arrived at through crystallization of a computationally efficient |
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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 was different from any previously observed ice |
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polymorphs, in experiment or simulation.\cite{Fennell04} This crystal |
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was termed Ice-{\it i} in homage to its origin in computational |
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In this paper, standard reference state methods were applied to known |
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crystalline water polymorphs in the low pressure regime. This work is |
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unique in the fact that one of the crystal lattices was arrived at |
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through crystallization of a computationally efficient water model |
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under constant pressure and temperature conditions. Crystallization |
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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 (Fig. \ref{iceiCell}A) consists of eight |
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water molecules that stack in rows of interlocking water |
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tetramers. Proton ordering can be accomplished by orienting two of the |
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waters so that both of their donated hydrogen bonds are internal to |
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molecules so that both of their donated hydrogen bonds are internal to |
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their tetramer (Fig. \ref{protOrder}). As expected in an ice crystal |
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constructed of water tetramers, the hydrogen bonds are not as linear |
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as those observed in ice $I_h$, however the interlocking of these |
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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-2{\it i}, the elongated variant of Ice-{\it i}. For Ice-{\it i}, the $a$ to $c$ relation is given by $a = 2.1214c$, while for Ice-2{\it i}, $a = 1.7850c$.} |
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\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
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elongated variant of Ice-{\it i}. For Ice-{\it i}, the $a$ to $c$ |
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relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a = |
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1.7850c$.} |
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\label{iceiCell} |
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\end{figure} |
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\label{protOrder} |
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\end{figure} |
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Results in the 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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being studied (for discussions on these single point dipole models, |
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see the previous work and related |
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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 we |
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investigated (for discussions on these single point dipole models, see |
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the previous work and related |
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articles\cite{Fennell04,Ichiye96,Bratko85}). Those results only |
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consider energetic stabilization and neglect entropic contributions to |
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the overall free energy. To address this issue, the absolute free |
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energy of this crystal was calculated using thermodynamic integration |
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and compared to the free energies of cubic and hexagonal ice $I$ (the |
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experimental low density ice polymorphs) and ice B (a higher density, |
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but very stable crystal structure observed by B\`{a}ez and Clancy in |
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free energy studies of SPC/E).\cite{Baez95b} This work includes |
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results for the water model from which Ice-{\it i} was crystallized |
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(soft sticky dipole extended, SSD/E) in addition to several common |
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water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field |
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parametrized single point dipole water model (soft sticky dipole |
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reaction field, SSD/RF). In should be noted that a second version of |
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Ice-{\it i} (Ice-2{\it i}) was used in calculations involving SPC/E, |
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TIP4P, and TIP5P. The unit cell of this crystal (Fig. \ref{iceiCell}B) |
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is similar to the Ice-{\it i} unit it is extended in the direction of |
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the (001) face and compressed along the other two faces. |
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considered energetic stabilization and neglected entropic |
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contributions to the overall free energy. To address this issue, the |
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absolute free energy of this crystal was calculated using |
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thermodynamic integration and compared to the free energies of cubic |
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and hexagonal ice $I$ (the experimental low density ice polymorphs) |
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and ice B (a higher density, but very stable crystal structure |
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observed by B\`{a}ez and Clancy in free energy studies of |
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SPC/E).\cite{Baez95b} This work includes results for the water model |
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from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
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common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
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field parametrized single point dipole water model (SSD/RF). It should |
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be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used |
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in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of |
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this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit |
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it is extended in the direction of the (001) face and compressed along |
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the other two faces. |
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|
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\section{Methods} |
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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 propagated 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 package.\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 these techniques can be found in a recent |
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publication.\cite{DLM} |
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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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same technique in order to determine melting points and generate phase |
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diagrams. All simulations were carried out at densities resulting in a |
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pressure of approximately 1 atm at their respective temperatures. |
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A single thermodynamic integration involves a sequence of simulations |
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over which the system of interest is converted into a reference system |
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for which the free energy is known. This transformation path is then |
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integrated in order to determine the free energy difference between |
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the two states: |
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for which the free energy is known analytically. This transformation |
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path is then integrated in order to determine the free energy |
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difference between the two states: |
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\begin{equation} |
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\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda |
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)}{\partial\lambda}\right\rangle_\lambda d\lambda, |
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(near the reference state) in length. |
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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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crystal was chosen as the reference state. 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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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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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 |
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). By applying this function, these interactions are smoothly |
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truncated, thereby avoiding poor energy conserving dynamics resulting |
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truncated, thereby avoiding the poor energy conservation which results |
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from harsher truncation schemes. The effect of a long-range correction |
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was also investigated on select model systems in a variety of |
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manners. For the SSD/RF model, a reaction field with a fixed |
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\AA\ cutoff results. Finally, results from the use of an Ewald |
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summation were estimated for TIP3P and SPC/E by performing |
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calculations with Particle-Mesh Ewald (PME) in the TINKER molecular |
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mechanics software package.\cite{Tinker} TINKER was chosen because it |
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can also propagate the motion of rigid-bodies, and provides the most |
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direct comparison to the results from OOPSE. The calculated energy |
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mechanics software package.\cite{Tinker} The calculated energy |
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difference in the presence and absence of PME was applied to the |
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previous results in order to predict changes in the free energy |
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previous results in order to predict changes to the free energy |
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landscape. |
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\section{Results and discussion} |
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as well as the higher density ice B, observed by B\`{a}ez and Clancy |
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and thought to be the minimum free energy structure for the SPC/E |
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model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} |
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Ice XI, the experimentally observed proton ordered variant of ice |
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$I_h$, was investigated initially, but it was found not to be as |
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stable as antiferroelectric variants of proton ordered or even proton |
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disordered ice$I_h$.\cite{Davidson84} The proton ordered variant of |
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ice $I_h$ used here is a simple antiferroelectric version that has an |
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8 molecule unit cell. The crystals contained 648 or 1728 molecules for |
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ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for ice |
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$I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
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Ice XI, the experimentally-observed proton-ordered variant of ice |
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$I_h$, was investigated initially, but was found to be not as stable |
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as proton disordered or antiferroelectric variants of ice $I_h$. The |
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proton ordered variant of ice $I_h$ used here is a simple |
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antiferroelectric version that has an 8 molecule unit |
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cell.\cite{Davidson84} The crystals contained 648 or 1728 molecules |
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for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 molecules for |
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ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger crystal sizes |
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were necessary for simulations involving larger cutoff values. |
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|
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\begin{table*} |
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The free energy values computed for the studied polymorphs indicate |
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that Ice-{\it i} is the most stable state for all of the common water |
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models studied. With the free energy at these state points, the |
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temperature and pressure dependence of the free energy was used to |
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project to other state points and build phase diagrams. Figures |
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Gibbs-Helmholtz equation was used to project to other state points and |
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to build phase diagrams. Figures |
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\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built |
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from the free energy results. All other models have similar structure, |
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only the crossing points between these phases exist at different |
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temperatures and pressures. It is interesting to note that ice $I$ |
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does not exist in either cubic or hexagonal form in any of the phase |
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diagrams for any of the models. For purposes of this study, ice B is |
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representative of the dense ice polymorphs. A recent study by Sanz |
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{\it et al.} goes into detail on the phase diagrams for SPC/E and |
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TIP4P in the high pressure regime.\cite{Sanz04} |
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although the crossing points between the phases exist at slightly |
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different temperatures and pressures. It is interesting to note that |
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ice $I$ does not exist in either cubic or hexagonal form in any of the |
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phase diagrams for any of the models. For purposes of this study, ice |
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B is representative of the dense ice polymorphs. A recent study by |
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Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and |
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TIP4P in the high pressure regime.\cite{Sanz04} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{tp3PhaseDia.eps} |
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preferred polymorph at ambient conditions for both the TIP3P and SPC/E |
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water models; however, there is a narrowing of the free energy |
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differences between the various solid forms. In the case of SPC/E this |
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narrowing is significant enough that it becomes less clear cut that |
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narrowing is significant enough that it becomes less clear that |
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Ice-{\it i} is the most stable polymorph, and is possibly metastable |
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with respect to ice B and possibly ice $I_c$. However, these results |
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do not significantly alter the finding that the Ice-{\it i} polymorph |
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\section{Conclusions} |
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|
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The free energy for proton ordered variants of hexagonal and cubic ice |
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$I$, ice B, and recently discovered Ice-{\it i} where calculated under |
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$I$, ice B, and recently discovered Ice-{\it i} were calculated under |
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standard conditions for several common water models via thermodynamic |
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integration. All the water models studied show Ice-{\it i} to be the |
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minimum free energy crystal structure in the with a 9 \AA\ switching |
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solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The effect of |
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interaction truncation was investigated through variation of the |
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cutoff radius, use of a reaction field parameterized model, and |
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estimation of the results in the presence of the Ewald summation |
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correction. Interaction truncation has a significant effect on the |
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estimation of the results in the presence of the Ewald |
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summation. Interaction truncation has a significant effect on the |
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computed free energy values, and may significantly alter the free |
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energy landscape for the more complex multipoint water models. Despite |
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these effects, these results show Ice-{\it i} to be an important ice |
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Due to this relative stability of Ice-{\it i} in all manner of |
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investigated simulation examples, the question arises as to possible |
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experimental observation of this polymorph. The rather extensive past |
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experimental observation of this polymorph. The rather extensive past |
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and current experimental investigation of water in the low pressure |
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regime leads the authors to be hesitant in ascribing relevance outside |
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of computational models, hence the descriptive name presented. That |
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being said, there are certain experimental conditions that would |
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provide the most ideal situation for possible observation. These |
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include the negative pressure or stretched solid regime, small |
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clusters in vacuum deposition environments, and in clathrate |
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structures involving small non-polar molecules. |
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regime makes us hesitant to ascribe any relevance of this work outside |
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of the simulation community. It is for this reason that we chose a |
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name for this polymorph which involves an imaginary quantity. That |
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said, there are certain experimental conditions that would provide the |
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most ideal situation for possible observation. These include the |
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negative pressure or stretched solid regime, small clusters in vacuum |
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deposition environments, and in clathrate structures involving small |
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non-polar molecules. Fig. \ref{fig:sofkgofr} contains our predictions |
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of both the pair distribution function ($g_{OO}(r)$) and the structure |
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factor ($S(\vec{q})$ for this polymorph at a temperature of 77K. We |
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will leave it to our experimental colleagues to determine whether this |
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ice polymorph should really be called Ice-{\it i} or if it should be |
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promoted to Ice-0. |
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|
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\section{Acknowledgments} |
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Support for this project was provided by the National Science |
