--- trunk/iceiPaper/iceiPaper.tex 2004/09/13 21:28:16 1454 +++ trunk/iceiPaper/iceiPaper.tex 2004/09/17 15:59:25 1477 @@ -1,45 +1,55 @@ %\documentclass[prb,aps,twocolumn,tabularx]{revtex4} -\documentclass[preprint,aps,endfloats]{revtex4} +\documentclass[11pt]{article} %\documentclass[11pt]{article} -%\usepackage{endfloat} +\usepackage{endfloat} \usepackage{amsmath} \usepackage{epsf} \usepackage{berkeley} -%\usepackage{setspace} -%\usepackage{tabularx} +\usepackage{setspace} +\usepackage{tabularx} \usepackage{graphicx} -%\usepackage[ref]{overcite} -%\pagestyle{plain} -%\pagenumbering{arabic} -%\oddsidemargin 0.0cm \evensidemargin 0.0cm -%\topmargin -21pt \headsep 10pt -%\textheight 9.0in \textwidth 6.5in -%\brokenpenalty=10000 +\usepackage[ref]{overcite} +\pagestyle{plain} +\pagenumbering{arabic} +\oddsidemargin 0.0cm \evensidemargin 0.0cm +\topmargin -21pt \headsep 10pt +\textheight 9.0in \textwidth 6.5in +\brokenpenalty=10000 +\renewcommand{\baselinestretch}{1.2} +\renewcommand\citemid{\ } % no comma in optional reference note -%\renewcommand\citemid{\ } % no comma in optional reference note - \begin{document} -\title{A Free Energy Study of Low Temperature and Anomolous Ice} +\title{Ice-{\it i}: a simulation-predicted ice polymorph which is more +stable than Ice $I_h$ for point-charge and point-dipole water models} -\author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote} -\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}} - -\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ +\author{Christopher J. Fennell and J. Daniel Gezelter \\ +Department of Chemistry and Biochemistry\\ University of Notre Dame\\ Notre Dame, Indiana 46556} \date{\today} -%\maketitle +\maketitle %\doublespacing \begin{abstract} +The absolute free energies of several ice polymorphs which are stable +at low pressures were calculated using thermodynamic integration to a +reference system (the Einstein crystal). These integrations were +performed for most of the common water models (SPC/E, TIP3P, TIP4P, +TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed +crystallizing at room temperature for one of the single-point water +models, was determined to be the stable crystalline state (at 1 atm) +for {\it all} the water models investigated. Phase diagrams were +generated, and phase coexistence lines were determined for all of the +known low-pressure ice structures under all of these water models. +Additionally, potential truncation was shown to have an effect on the +calculated free energies, and can result in altered free energy +landscapes. Structure factor for the new crystal were generated and +we await experimental confirmation of the existence of this new +polymorph. \end{abstract} -\maketitle - -\newpage - %\narrowtext %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -48,28 +58,149 @@ Notre Dame, Indiana 46556} \section{Introduction} +Water has proven to be a challenging substance to depict in +simulations, and a variety of models have been developed to describe +its behavior under varying simulation +conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04} +These models have been used to investigate important physical +phenomena like phase transitions, transport properties, and the +hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the +choice of models available, it is only natural to compare the models +under interesting thermodynamic conditions in an attempt to clarify +the limitations of each of the +models.\cite{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two +important properties to quantify are the Gibbs and Helmholtz free +energies, particularly for the solid forms of water. Difficulty in +these types of studies typically arises from the assortment of +possible crystalline polymorphs that water adopts over a wide range of +pressures and temperatures. There are currently 13 recognized forms +of ice, and it is a challenging task to investigate the entire free +energy landscape.\cite{Sanz04} Ideally, research is focused on the +phases having the lowest free energy at a given state point, because +these phases will dictate the relevant transition temperatures and +pressures for the model. + +In this paper, standard reference state methods were applied to known +crystalline water polymorphs in the low pressure regime. This work is +unique in that one of the crystal lattices was arrived at through +crystallization of a computationally efficient water model under +constant pressure and temperature conditions. Crystallization events +are interesting in and of themselves;\cite{Matsumoto02,Yamada02} +however, the crystal structure obtained in this case is different from +any previously observed ice polymorphs in experiment or +simulation.\cite{Fennell04} We have named this structure Ice-{\it i} +to indicate its origin in computational simulation. The unit cell +(Fig. \ref{iceiCell}A) consists of eight water molecules that stack in +rows of interlocking water tetramers. Proton ordering can be +accomplished by orienting two of the molecules so that both of their +donated hydrogen bonds are internal to their tetramer +(Fig. \ref{protOrder}). As expected in an ice crystal constructed of +water tetramers, the hydrogen bonds are not as linear as those +observed in ice $I_h$, however the interlocking of these subunits +appears to provide significant stabilization to the overall +crystal. The arrangement of these tetramers results in surrounding +open octagonal cavities that are typically greater than 6.3 \AA\ in +diameter. This relatively open overall structure leads to crystals +that are 0.07 g/cm$^3$ less dense on average than ice $I_h$. + +\begin{figure} +\includegraphics[width=\linewidth]{unitCell.eps} +\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the +elongated variant of Ice-{\it i}. The spheres represent the +center-of-mass locations of the water molecules. The $a$ to $c$ +ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by +$a:2.1214c$ and $a:1.7850c$ respectively.} +\label{iceiCell} +\end{figure} + +\begin{figure} +\includegraphics[width=\linewidth]{orderedIcei.eps} +\caption{Image of a proton ordered crystal of Ice-{\it i} looking +down the (001) crystal face. The rows of water tetramers surrounded by +octagonal pores leads to a crystal structure that is significantly +less dense than ice $I_h$.} +\label{protOrder} +\end{figure} + +Results from our previous study indicated that Ice-{\it i} is the +minimum energy crystal structure for the single point water models we +had investigated (for discussions on these single point dipole models, +see our previous work and related +articles).\cite{Fennell04,Liu96,Bratko85} Those results only +considered energetic stabilization and neglected entropic +contributions to the overall free energy. To address this issue, we +have calculated the absolute free energy of this crystal using +thermodynamic integration and compared to the free energies of cubic +and hexagonal ice $I$ (the experimental low density ice polymorphs) +and ice B (a higher density, but very stable crystal structure +observed by B\`{a}ez and Clancy in free energy studies of +SPC/E).\cite{Baez95b} This work includes results for the water model +from which Ice-{\it i} was crystallized (SSD/E) in addition to several +common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction +field parametrized single point dipole water model (SSD/RF). It should +be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was +used in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell +of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} +unit it is extended in the direction of the (001) face and compressed +along the other two faces. + \section{Methods} Canonical ensemble (NVT) molecular dynamics calculations were -performed using the OOPSE (Object-Oriented Parallel Simulation Engine) -molecular mechanics package. All molecules were treated as rigid -bodies, with orientational motion propogated using the symplectic DLM -integration method. Details about the implementation of these -techniques can be found in a recent publication.\cite{Meineke05} +performed using the OOPSE molecular mechanics package.\cite{Meineke05} +All molecules were treated as rigid bodies, with orientational motion +propagated using the symplectic DLM integration method. Details about +the implementation of this technique can be found in a recent +publication.\cite{Dullweber1997} -Thermodynamic integration was utilized to calculate the free energy of -several ice crystals using the TIP3P, TIP4P, TIP5P, SPC/E, and SSD/E -water models. Liquid state free energies at 300 and 400 K for all of -these water models were also determined using this same technique, in -order to determine melting points and generate phase diagrams. +Thermodynamic integration is an established technique for +determination of free energies of condensed phases of +materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This +method, implemented in the same manner illustrated by B\`{a}ez and +Clancy, was utilized to calculate the free energy of several ice +crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and +SSD/E water models.\cite{Baez95a} Liquid state free energies at 300 +and 400 K for all of these water models were also determined using +this same technique in order to determine melting points and to +generate phase diagrams. All simulations were carried out at densities +which correspond to a pressure of approximately 1 atm at their +respective temperatures. +Thermodynamic integration involves a sequence of simulations during +which the system of interest is converted into a reference system for +which the free energy is known analytically. This transformation path +is then integrated in order to determine the free energy difference +between the two states: +\begin{equation} +\Delta A = \int_0^1\left\langle\frac{\partial V(\lambda +)}{\partial\lambda}\right\rangle_\lambda d\lambda, +\end{equation} +where $V$ is the interaction potential and $\lambda$ is the +transformation parameter that scales the overall +potential. Simulations are distributed strategically along this path +in order to sufficiently sample the regions of greatest change in the +potential. Typical integrations in this study consisted of $\sim$25 +simulations ranging from 300 ps (for the unaltered system) to 75 ps +(near the reference state) in length. + For the thermodynamic integration of molecular crystals, the Einstein -Crystal is chosen as the reference state that the system is converted -to over the course of the simulation. In an Einstein Crystal, the -molecules are harmonically restrained at their ideal lattice locations -and orientations. The partition function for a molecular crystal -restrained in this fashion has been evaluated, and the Helmholtz Free -Energy ({\it A}) is given by +crystal was chosen as the reference system. In an Einstein crystal, +the molecules are restrained at their ideal lattice locations and +orientations. Using harmonic restraints, as applied by B\`{a}ez and +Clancy, the total potential for this reference crystal +($V_\mathrm{EC}$) is the sum of all the harmonic restraints, +\begin{equation} +V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + +\frac{K_\omega\omega^2}{2}, +\end{equation} +where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are +the spring constants restraining translational motion and deflection +of and rotation around the principle axis of the molecule +respectively. It is clear from Fig. \ref{waterSpring} that the values +of $\theta$ range from $0$ to $\pi$, while $\omega$ ranges from +$-\pi$ to $\pi$. The partition function for a molecular crystal +restrained in this fashion can be evaluated analytically, and the +Helmholtz Free Energy ({\it A}) is given by \begin{eqnarray} A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left [\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right @@ -81,27 +212,314 @@ where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a )^\frac{1}{2}}\exp(t^2)\mathrm{d}t\right ], \label{ecFreeEnergy} \end{eqnarray} -where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$.\cite{Baez95a} In equation -\ref{ecFreeEnergy}, $K_\mathrm{v}$, $K_\mathrm{\theta}$, and -$K_\mathrm{\omega}$ are the spring constants restraining translational -motion and deflection of and rotation around the principle axis of the -molecule respectively (Fig. \ref{waterSpring}), and $E_m$ is the -minimum potential energy of the ideal crystal. In the case of -molecular liquids, the ideal vapor is chosen as the target reference -state. +where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum +potential energy of the ideal crystal.\cite{Baez95a} +\begin{figure} +\includegraphics[width=\linewidth]{rotSpring.eps} +\caption{Possible orientational motions for a restrained molecule. +$\theta$ angles correspond to displacement from the body-frame {\it +z}-axis, while $\omega$ angles correspond to rotation about the +body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring +constants for the harmonic springs restraining motion in the $\theta$ +and $\omega$ directions.} +\label{waterSpring} +\end{figure} +In the case of molecular liquids, the ideal vapor is chosen as the +target reference state. There are several examples of liquid state +free energy calculations of water models present in the +literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods +typically differ in regard to the path taken for switching off the +interaction potential to convert the system to an ideal gas of water +molecules. In this study, we applied of one of the most convenient +methods and integrated over the $\lambda^4$ path, where all +interaction parameters are scaled equally by this transformation +parameter. This method has been shown to be reversible and provide +results in excellent agreement with other established +methods.\cite{Baez95b} +Charge, dipole, and Lennard-Jones interactions were modified by a +cubic switching between 100\% and 85\% of the cutoff value (9 \AA +). By applying this function, these interactions are smoothly +truncated, thereby avoiding the poor energy conservation which results +from harsher truncation schemes. The effect of a long-range correction +was also investigated on select model systems in a variety of +manners. For the SSD/RF model, a reaction field with a fixed +dielectric constant of 80 was applied in all +simulations.\cite{Onsager36} For a series of the least computationally +expensive models (SSD/E, SSD/RF, and TIP3P), simulations were +performed with longer cutoffs of 12 and 15 \AA\ to compare with the 9 +\AA\ cutoff results. Finally, the effects of utilizing an Ewald +summation were estimated for TIP3P and SPC/E by performing single +configuration calculations with Particle-Mesh Ewald (PME) in the +TINKER molecular mechanics software package.\cite{Tinker} The +calculated energy difference in the presence and absence of PME was +applied to the previous results in order to predict changes to the +free energy landscape. \section{Results and discussion} +The free energy of proton-ordered Ice-{\it i} was calculated and +compared with the free energies of proton ordered variants of the +experimentally observed low-density ice polymorphs, $I_h$ and $I_c$, +as well as the higher density ice B, observed by B\`{a}ez and Clancy +and thought to be the minimum free energy structure for the SPC/E +model at ambient conditions (Table \ref{freeEnergy}).\cite{Baez95b} +Ice XI, the experimentally-observed proton-ordered variant of ice +$I_h$, was investigated initially, but was found to be not as stable +as proton disordered or antiferroelectric variants of ice $I_h$. The +proton ordered variant of ice $I_h$ used here is a simple +antiferroelectric version that we devised, and it has an 8 molecule +unit cell similar to other predicted antiferroelectric $I_h$ +crystals.\cite{Davidson84} The crystals contained 648 or 1728 +molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 +molecules for ice $I_c$, or 1024 molecules for Ice-{\it i}. The larger +crystal sizes were necessary for simulations involving larger cutoff +values. + +\begin{table*} +\begin{minipage}{\linewidth} +\renewcommand{\thefootnote}{\thempfootnote} +\begin{center} +\caption{Calculated free energies for several ice polymorphs with a +variety of common water models. All calculations used a cutoff radius +of 9 \AA\ and were performed at 200 K and $\sim$1 atm. Units are +kcal/mol. Calculated error of the final digits is in parentheses. *Ice +$I_c$ rapidly converts to a liquid at 200 K with the SSD/RF model.} +\begin{tabular}{ l c c c c } +\hline +Water Model & $I_h$ & $I_c$ & B & Ice-{\it i}\\ +\hline +TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3)\\ +TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & -12.33(3)\\ +TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & -12.29(2)\\ +SPC/E & -12.67(2) & -12.96(2) & -13.25(3) & -13.55(2)\\ +SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2)\\ +SSD/RF & -11.51(2) & NA* & -12.08(3) & -12.29(2)\\ +\end{tabular} +\label{freeEnergy} +\end{center} +\end{minipage} +\end{table*} + +The free energy values computed for the studied polymorphs indicate +that Ice-{\it i} is the most stable state for all of the common water +models studied. With the calculated free energy at these state points, +the Gibbs-Helmholtz equation was used to project to other state points +and to build phase diagrams. Figures +\ref{tp3phasedia} and \ref{ssdrfphasedia} are example diagrams built +from the free energy results. All other models have similar structure, +although the crossing points between the phases move to slightly +different temperatures and pressures. It is interesting to note that +ice $I$ does not exist in either cubic or hexagonal form in any of the +phase diagrams for any of the models. For purposes of this study, ice +B is representative of the dense ice polymorphs. A recent study by +Sanz {\it et al.} goes into detail on the phase diagrams for SPC/E and +TIP4P at higher pressures than those studied here.\cite{Sanz04} + +\begin{figure} +\includegraphics[width=\linewidth]{tp3PhaseDia.eps} +\caption{Phase diagram for the TIP3P water model in the low pressure +regime. The displayed $T_m$ and $T_b$ values are good predictions of +the experimental values; however, the solid phases shown are not the +experimentally observed forms. Both cubic and hexagonal ice $I$ are +higher in energy and don't appear in the phase diagram.} +\label{tp3phasedia} +\end{figure} + +\begin{figure} +\includegraphics[width=\linewidth]{ssdrfPhaseDia.eps} +\caption{Phase diagram for the SSD/RF water model in the low pressure +regime. Calculations producing these results were done under an +applied reaction field. It is interesting to note that this +computationally efficient model (over 3 times more efficient than +TIP3P) exhibits phase behavior similar to the less computationally +conservative charge based models.} +\label{ssdrfphasedia} +\end{figure} + +\begin{table*} +\begin{minipage}{\linewidth} +\renewcommand{\thefootnote}{\thempfootnote} +\begin{center} +\caption{Melting ($T_m$), boiling ($T_b$), and sublimation ($T_s$) +temperatures at 1 atm for several common water models compared with +experiment. The $T_m$ and $T_s$ values from simulation correspond to a +transition between Ice-{\it i} (or Ice-{\it i}$^\prime$) and the +liquid or gas state.} +\begin{tabular}{ l c c c c c c c } +\hline +Equilibria Point & TIP3P & TIP4P & TIP5P & SPC/E & SSD/E & SSD/RF & Exp.\\ +\hline +$T_m$ (K) & 269(4) & 266(5) & 271(4) & 296(3) & - & 278(4) & 273\\ +$T_b$ (K) & 357(2) & 354(2) & 337(2) & 396(2) & - & 348(2) & 373\\ +$T_s$ (K) & - & - & - & - & 355(2) & - & -\\ +\end{tabular} +\label{meltandboil} +\end{center} +\end{minipage} +\end{table*} + +Table \ref{meltandboil} lists the melting and boiling temperatures +calculated from this work. Surprisingly, most of these models have +melting points that compare quite favorably with experiment. The +unfortunate aspect of this result is that this phase change occurs +between Ice-{\it i} and the liquid state rather than ice $I_h$ and the +liquid state. These results are actually not contrary to previous +studies in the literature. Earlier free energy studies of ice $I$ +using TIP4P predict a $T_m$ ranging from 214 to 238 K (differences +being attributed to choice of interaction truncation and different +ordered and disordered molecular +arrangements).\cite{Vlot99,Gao00,Sanz04} If the presence of ice B and +Ice-{\it i} were omitted, a $T_m$ value around 210 K would be +predicted from this work. However, the $T_m$ from Ice-{\it i} is +calculated at 265 K, significantly higher in temperature than the +previous studies. Also of interest in these results is that SSD/E does +not exhibit a melting point at 1 atm, but it shows a sublimation point +at 355 K. This is due to the significant stability of Ice-{\it i} over +all other polymorphs for this particular model under these +conditions. While troubling, this behavior resulted in spontaneous +crystallization of Ice-{\it i} and led us to investigate this +structure. These observations provide a warning that simulations of +SSD/E as a ``liquid'' near 300 K are actually metastable and run the +risk of spontaneous crystallization. However, this risk lessens when +applying a longer cutoff. + +\begin{figure} +\includegraphics[width=\linewidth]{cutoffChange.eps} +\caption{Free energy as a function of cutoff radius for (A) SSD/E, (B) +TIP3P, and (C) SSD/RF. Data points omitted include SSD/E: $I_c$ 12 +\AA\, TIP3P: $I_c$ 12 \AA\ and B 12 \AA\, and SSD/RF: $I_c$ 9 +\AA . These crystals are unstable at 200 K and rapidly convert into +liquids. The connecting lines are qualitative visual aid.} +\label{incCutoff} +\end{figure} + +Increasing the cutoff radius in simulations of the more +computationally efficient water models was done in order to evaluate +the trend in free energy values when moving to systems that do not +involve potential truncation. As seen in Fig. \ref{incCutoff}, the +free energy of all the ice polymorphs show a substantial dependence on +cutoff radius. In general, there is a narrowing of the free energy +differences while moving to greater cutoff radius. Interestingly, by +increasing the cutoff radius, the free energy gap was narrowed enough +in the SSD/E model that the liquid state is preferred under standard +simulation conditions (298 K and 1 atm). Thus, it is recommended that +simulations using this model choose interaction truncation radii +greater than 9 \AA\ . This narrowing trend is much more subtle in the +case of SSD/RF, indicating that the free energies calculated with a +reaction field present provide a more accurate picture of the free +energy landscape in the absence of potential truncation. + +To further study the changes resulting to the inclusion of a +long-range interaction correction, the effect of an Ewald summation +was estimated by applying the potential energy difference do to its +inclusion in systems in the presence and absence of the +correction. This was accomplished by calculation of the potential +energy of identical crystals both with and without PME. The free +energies for the investigated polymorphs using the TIP3P and SPC/E +water models are shown in Table \ref{pmeShift}. The same trend pointed +out through increase of cutoff radius is observed in these PME +results. Ice-{\it i} is the preferred polymorph at ambient conditions +for both the TIP3P and SPC/E water models; however, the narrowing of +the free energy differences between the various solid forms is +significant enough that it becomes less clear that it is the most +stable polymorph with the SPC/E model. The free energies of Ice-{\it +i} and ice B nearly overlap within error, with ice $I_c$ just outside +as well, indicating that Ice-{\it i} might be metastable with respect +to ice B and possibly ice $I_c$ with SPC/E. However, these results do +not significantly alter the finding that the Ice-{\it i} polymorph is +a stable crystal structure that should be considered when studying the +phase behavior of water models. + +\begin{table*} +\begin{minipage}{\linewidth} +\renewcommand{\thefootnote}{\thempfootnote} +\begin{center} +\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.} +\begin{tabular}{ l c c c c } +\hline +\ \ Water Model \ \ & \ \ \ \ \ $I_h$ \ \ & \ \ \ \ \ $I_c$ \ \ & \ \quad \ \ \ \ B \ \ & \ \ \ \ \ Ice-{\it i} \ \ \\ +\hline +TIP3P & -11.53(2) & -11.24(3) & -11.51(3) & -11.67(3)\\ +SPC/E & -12.77(2) & -12.92(2) & -12.96(3) & -13.02(2)\\ +\end{tabular} +\label{pmeShift} +\end{center} +\end{minipage} +\end{table*} + \section{Conclusions} +The free energy for proton ordered variants of hexagonal and cubic ice +$I$, ice B, and our recently discovered Ice-{\it i} structure were +calculated under standard conditions for several common water models +via thermodynamic integration. All the water models studied show +Ice-{\it i} to be the minimum free energy crystal structure with a 9 +\AA\ switching function cutoff. Calculated melting and boiling points +show surprisingly good agreement with the experimental values; +however, the solid phase at 1 atm is Ice-{\it i}, not ice $I_h$. The +effect of interaction truncation was investigated through variation of +the cutoff radius, use of a reaction field parameterized model, and +estimation of the results in the presence of the Ewald +summation. Interaction truncation has a significant effect on the +computed free energy values, and may significantly alter the free +energy landscape for the more complex multipoint water models. Despite +these effects, these results show Ice-{\it i} to be an important ice +polymorph that should be considered in simulation studies. + +Due to this relative stability of Ice-{\it i} in all of the +investigated simulation conditions, the question arises as to possible +experimental observation of this polymorph. The rather extensive past +and current experimental investigation of water in the low pressure +regime makes us hesitant to ascribe any relevance of this work outside +of the simulation community. It is for this reason that we chose a +name for this polymorph which involves an imaginary quantity. That +said, there are certain experimental conditions that would provide the +most ideal situation for possible observation. These include the +negative pressure or stretched solid regime, small clusters in vacuum +deposition environments, and in clathrate structures involving small +non-polar molecules. Figs. \ref{fig:gofr} and \ref{fig:sofq} contain +our predictions for both the pair distribution function ($g_{OO}(r)$) +and the structure factor ($S(\vec{q})$ for ice $I_c$ and for ice-{\it +i} at a temperature of 77K. In our initial comparison of the +predicted S(q) for Ice-{\it i} and experimental studies of amorphous +solid water, it is possible that some of the ``spurious'' peaks that +could not be assigned in an early report on high-density amorphous +(HDA) ice could correspond to peaks labeled in this +S(q).\cite{Bizid87} It should be noted that there is typically poor +agreement on crystal densities between simulation and experiment, so +such peak comparisons should be made with caution. We will leave it +to our experimental colleagues to make the final determination on +whether this ice polymorph is named appropriately (i.e. with an +imaginary number) or if it can be promoted to Ice-0. + +\begin{figure} +\includegraphics[width=\linewidth]{iceGofr.eps} +\caption{Radial distribution functions of Ice-{\it i} and ice $I_c$ +calculated from from simulations of the SSD/RF water model at 77 K.} +\label{fig:gofr} +\end{figure} + +\begin{figure} +\includegraphics[width=\linewidth]{sofq.eps} +\caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at +77 K. The raw structure factors have been convoluted with a gaussian +instrument function (0.075 \AA$^{-1}$ width) to compensate for the +trunction effects in our finite size simulations. The labeled peaks +compared favorably with ``spurious'' peaks observed in experimental +studies of amorphous solid water.\cite{Bizid87}} +\label{fig:sofq} +\end{figure} + \section{Acknowledgments} Support for this project was provided by the National Science Foundation under grant CHE-0134881. Computation time was provided by -the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant -DMR-0079647. +the Notre Dame High Performance Computing Cluster and the Notre Dame +Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647). \newpage