--- trunk/iceiPaper/iceiPaper.tex 2004/09/13 21:28:16 1454 +++ trunk/iceiPaper/iceiPaper.tex 2004/09/16 21:38:10 1469 @@ -1,45 +1,51 @@ %\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 free energies of several ice polymorphs in the low pressure regime +were calculated using thermodynamic integration. These integrations +were done for most of the common water models. Ice-{\it i}, a +structure we recently observed to be stable in 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 the common water +models. Additionally, potential truncation was shown to have an +effect on the calculated free energies, and can result in altered free +energy landscapes. \end{abstract} -\maketitle - -\newpage - %\narrowtext %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -48,28 +54,140 @@ Notre Dame, Indiana 46556} \section{Introduction} +Molecular dynamics is a valuable tool for studying the phase behavior +of systems ranging from small or simple +molecules\cite{Matsumoto02,andOthers} to complex biological +species.\cite{bigStuff} Many techniques have been developed to +investigate the thermodynamic properites of model substances, +providing both qualitative and quantitative comparisons between +simulations and experiment.\cite{thermMethods} Investigation of these +properties leads to the development of new and more accurate models, +leading to better understanding and depiction of physical processes +and intricate molecular systems. + +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{Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Mahoney00,Fennell04} +These models have been used to investigate important physical +phenomena like phase transitions and the hydrophobic +effect.\cite{Yamada02} 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{modelProps} Two important property 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 true transition +temperatures and pressures for their respective 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 the fact 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}. For Ice-{\it i}, the $a$ to $c$ +relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a = +1.7850c$.} +\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 +investigated (for discussions on these single point dipole models, see +the 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, the +absolute free energy of this crystal was calculated 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 these techniques 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. +several ice crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, +SSD/RF, 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. All simulations were carried out at densities resulting in a +pressure of approximately 1 atm at their respective temperatures. +A single thermodynamic integration involves a sequence of simulations +over 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 unevenly along this path in +order to sufficiently sample the regions of greatest change in the +potential. Typical integrations in this study consisted of $\sim$25 +simulations ranging from 300 ps (for the unaltered system) to 75 ps +(near the reference state) in length. + For the thermodynamic integration of molecular crystals, the Einstein -Crystal is chosen as the reference state that the system is converted -to over the course of the simulation. In an Einstein Crystal, the +crystal was chosen as the reference state. In an Einstein crystal, the molecules are harmonically restrained at their ideal lattice locations and orientations. The partition function for a molecular crystal -restrained in this fashion has been evaluated, and the Helmholtz Free -Energy ({\it A}) is given by +restrained in this fashion can be evaluated analytically, and the +Helmholtz Free Energy ({\it A}) is given by \begin{eqnarray} A = E_m\ -\ kT\ln \left (\frac{kT}{h\nu}\right )^3&-&kT\ln \left [\pi^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right @@ -90,18 +208,285 @@ state. molecular liquids, the ideal vapor is chosen as the target reference state. +\begin{figure} +\includegraphics[width=\linewidth]{rotSpring.eps} +\caption{Possible orientational motions for a restrained molecule. +$\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} +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, results from the use of an Ewald +summation were estimated for TIP3P and SPC/E by performing +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 has an 8 molecule unit +cell.\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(4) & -11.23(6) & -11.82(5) & -12.30(5)\\ +TIP4P & -11.84(5) & -12.04(4) & -12.08(6) & -12.33(6)\\ +TIP5P & -11.85(5) & -11.86(4) & -11.96(4) & -12.29(4)\\ +SPC/E & -12.67(3) & -12.96(3) & -13.25(5) & -13.55(3)\\ +SSD/E & -11.27(3) & -11.19(8) & -12.09(4) & -12.54(4)\\ +SSD/RF & -11.51(4) & NA* & -12.08(5) & -12.29(4)\\ +\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 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 exist at 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 in the high pressure regime.\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(8) & 266(10) & 271(7) & 296(5) & - & 278(7) & 273\\ +$T_b$ (K) & 357(2) & 354(3) & 337(3) & 396(4) & - & 348(3) & 373\\ +$T_s$ (K) & - & - & - & - & 355(3) & - & -\\ +\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 turned out to be +advantageous in that it facilitated the spontaneous crystallization of +Ice-{\it i}. 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 changes 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 with and without PME using TINKER. The +free energies for the investigated polymorphs using the TIP3P and +SPC/E water models are shown in Table \ref{pmeShift}. TIP4P and TIP5P +are not fully supported in TINKER, so the results for these models +could not be estimated. 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, there is a narrowing of the free energy +differences between the various solid forms. In the case of SPC/E this +narrowing is significant enough that it becomes less clear that +Ice-{\it i} is the most stable polymorph, and is possibly metastable +with respect to ice B and possibly ice $I_c$. 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(4) & -11.24(6) & -11.51(5) & -11.67(5)\\ +SPC/E & -12.77(3) & -12.92(3) & -12.96(5) & -13.02(3)\\ +\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 recently discovered Ice-{\it i} 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 in the 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 manner of +investigated simulation examples, 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. We will leave it to our experimental +colleagues to determine whether this ice polymorph is named +appropriately or if it should be promoted to Ice-0. + +\begin{figure} +\includegraphics[width=\linewidth]{iceGofr.eps} +\caption{Radial distribution functions of (A) Ice-{\it i} and (B) ice $I_c$ at 77 K from simulations of the SSD/RF water model.} +\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.} +\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