--- trunk/fennellDissertation/iceChapter.tex 2006/08/29 00:40:05 2979 +++ trunk/fennellDissertation/iceChapter.tex 2006/08/30 22:14:37 2986 @@ -8,12 +8,12 @@ available, it is only natural to compare the models un These models have been used to investigate important physical phenomena like phase transitions and the hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} With the choice of models -available, it is only natural to compare the models under interesting +available, it is only natural to compare them under interesting thermodynamic conditions in an attempt to clarify the limitations of -each of the models.\cite{Jorgensen83,Jorgensen98b,Baez94,Mahoney01} -Two important property to quantify are the Gibbs and Helmholtz free -energies, particularly for the solid forms of water, as these predict -the thermodynamic stability of the various phases. Water has a +each.\cite{Jorgensen83,Jorgensen98b,Baez94,Mahoney01} Two important +property to quantify are the Gibbs and Helmholtz free energies, +particularly for the solid forms of water, as these predict the +thermodynamic stability of the various phases. Water has a particularly rich phase diagram and takes on a number of different and stable crystalline structures as the temperature and pressure are varied. This complexity makes it a challenging task to investigate the @@ -34,8 +34,8 @@ pressure of 1 atm. After melting from ice I$_\textrm{h While performing a series of melting simulations on an early iteration of SSD/E, we observed several recrystallization events at a constant -pressure of 1 atm. After melting from ice I$_\textrm{h}$ at 235K, two -of five systems recrystallized near 245K. Crystallization events are +pressure of 1 atm. After melting from ice I$_\textrm{h}$ at 235~K, two +of five systems recrystallized near 245~K. Crystallization events are interesting in and of themselves;\cite{Matsumoto02,Yamada02} however, the crystal structure extracted from these systems is different from any previously observed ice polymorphs in experiment or @@ -51,9 +51,9 @@ are typically greater than 6.3\AA\ in diameter (see fi I$_\textrm{h}$; however, the interlocking of these subunits appears to provide significant stabilization to the overall crystal. The arrangement of these tetramers results in open octagonal cavities that -are typically greater than 6.3\AA\ in diameter (see figure +are typically greater than 6.3~\AA\ in diameter (see figure \ref{fig:protOrder}). This open structure leads to crystals that are -typically 0.07 g/cm$^3$ less dense than ice I$_\textrm{h}$. +typically 0.07~g/cm$^3$ less dense than ice I$_\textrm{h}$. \begin{figure} \includegraphics[width=\linewidth]{./figures/unitCell.pdf} @@ -85,7 +85,7 @@ structure observed by B\`{a}ez and Clancy in free ener thermodynamic integration and compared to the free energies of ice I$_\textrm{c}$ and ice I$_\textrm{h}$ (the common 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 +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 @@ -105,8 +105,8 @@ isobaric-isothermal ({\it NPT}) simulations performed performed using the OOPSE molecular mechanics package.\cite{Meineke05} The densities chosen for the simulations were taken from isobaric-isothermal ({\it NPT}) simulations performed at 1 atm and -200K. Each model (and each crystal structure) was allowed to relax for -300ps in the {\it NPT} ensemble before averaging the density to obtain +200~K. Each model (and each crystal structure) was allowed to relax for +300~ps in the {\it NPT} ensemble before averaging the density to obtain the volumes for the {\it NVT} simulations.All molecules were treated as rigid bodies, with orientational motion propagated using the symplectic DLM integration method described in section @@ -159,11 +159,12 @@ ice crystal at 200 K. For these studies, $K_\mathrm{v of and rotation around the principle axis of the molecule respectively. These spring constants are typically calculated from the mean-square displacements of water molecules in an unrestrained -ice crystal at 200 K. For these studies, $K_\mathrm{v} = 4.29$ kcal -mol$^{-1}$ \AA$^{-2}$, $K_\theta\ = 13.88$ kcal mol$^{-1}$ rad$^{-2}$, -and $K_\omega\ = 17.75$ kcal mol$^{-1}$ rad$^{-2}$. It is clear from -Fig. \ref{fig:waterSpring} that the values of $\theta$ range from $0$ to -$\pi$, while $\omega$ ranges from $-\pi$ to $\pi$. The partition +ice crystal at 200~K. For these studies, $K_\mathrm{v} = +4.29$~kcal~mol$^{-1}$~\AA$^{-2}$, $K_\theta\ = +13.88$~kcal~mol$^{-1}$~rad$^{-2}$, and $K_\omega\ = +17.75$~kcal~mol$^{-1}$~rad$^{-2}$. It is clear from +Fig. \ref{fig: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 @@ -184,17 +185,17 @@ Einstein crystal reference stat is somewhat arbitrary. \end{equation} where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum potential energy of the ideal crystal.\cite{Baez95a} The choice of an -Einstein crystal reference stat is somewhat arbitrary. Any ideal +Einstein crystal reference state is somewhat arbitrary. Any ideal system for which the partition function is known exactly could be used as a reference point as long as the system does not undergo a phase transition during the integration path between the real and ideal systems. Nada and van der Eerden have shown that the use of different -force constants in the Einstein crystal doesn not affect the total +force constants in the Einstein crystal does not affect the total free energy, and Gao {\it et al.} have shown that free energies computed with the Debye crystal reference state differ from the -Einstein crystal by only a few tenths of a kJ -mol$^{-1}$.\cite{Nada03,Gao00} These free energy differences can lead -to some uncertainty in the computed melting point of the solids. +Einstein crystal by only a few tenths of a +kJ~mol$^{-1}$.\cite{Nada03,Gao00} These free energy differences can +lead to some uncertainty in the computed melting point of the solids. \begin{figure} \centering \includegraphics[width=3.5in]{./figures/rotSpring.pdf} @@ -230,8 +231,8 @@ performed with longer cutoffs of 10.5, 12, 13.5, and 1 dielectric constant of 80 was applied in all simulations.\cite{Onsager36} For a series of the least computationally expensive models (SSD/E, SSD/RF, TIP3P, and SPC/E), simulations were -performed with longer cutoffs of 10.5, 12, 13.5, and 15\AA\ to -compare with the 9\AA\ cutoff results. Finally, the effects of using +performed with longer cutoffs of 10.5, 12, 13.5, and 15~\AA\ to +compare with the 9~\AA\ cutoff results. Finally, the effects of using the Ewald summation were estimated for TIP3P and SPC/E by performing single configuration Particle-Mesh Ewald (PME) calculations for each of the ice polymorphs.\cite{Ponder87} The calculated energy difference @@ -278,9 +279,9 @@ TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) \cmidrule(l){7-8} & \multicolumn{5}{c}{(kcal mol$^{-1}$)} & \multicolumn{2}{c}{(K)}\\ \midrule -TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(7) & 357(4)\\ -TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 262(6) & 354(4)\\ TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 266(7) & 337(4)\\ +TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 262(6) & 354(4)\\ +TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(7) & 357(4)\\ SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 299(6) & 396(4)\\ SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(4) & -\\ SSD/RF & -11.96(2) & -11.60(2) & -12.53(3) & -12.79(2) & - & 278(7) & 382(4)\\ @@ -312,11 +313,11 @@ We note that all of the crystals investigated in this \label{fig:ssdrfPhaseDia} \end{figure} -We note that all of the crystals investigated in this study ar ideal +We note that all of the crystals investigated in this study are ideal proton-ordered antiferroelectric structures. All of the structures obey the Bernal-Fowler rules and should be able to form stable proton-{\it disordered} crystals which have the traditional -$k_\textrm{B}$ln(3/2) residual entropy at 0K.\cite{Bernal33,Pauling35} +$k_\textrm{B}$ln(3/2) residual entropy at 0~K.\cite{Bernal33,Pauling35} Simulations of proton-disordered structures are relatively unstable with all but the most expensive water models.\cite{Nada03} Our simulations have therefore been performed with the ordered @@ -327,47 +328,47 @@ favorably with the experimental value of 273 K. The u of the disordered structures.\cite{Sanz04} Most of the water models have melting points that compare quite -favorably with the experimental value of 273 K. The unfortunate +favorably with the experimental value of 273~K. 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 do not contradict other studies. Studies of ice -I$_h$ using TIP4P predict a $T_m$ ranging from 191 to 238 K +I$_h$ using TIP4P predict a $T_m$ ranging from 191 to 238~K (differences being attributed to choice of interaction truncation and different ordered and disordered molecular arrangements).\cite{Nada03,Vlot99,Gao00,Sanz04} If the presence of ice -B and Ice-{\it i} were omitted, a $T_\textrm{m}$ value around 200 K +B and Ice-{\it i} were omitted, a $T_\textrm{m}$ value around 200~K would be predicted from this work. However, the $T_\textrm{m}$ from -Ice-{\it i} is calculated to be 262 K, indicating that these +Ice-{\it i} is calculated to be 262~K, indicating that these simulation based structures ought to be included in studies probing phase transitions with this model. Also of interest in these results is that SSD/E does not exhibit a melting point at 1 atm but does -sublime at 355 K. This is due to the significant stability of +sublime 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 the spontaneous crystallization of Ice-{\it i} which 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 +of SSD/E as a ``liquid'' near 300~K are actually metastable and run the risk of spontaneous crystallization. However, when a longer cutoff radius is used, SSD/E prefers the liquid state under standard temperature and pressure. -\section{Effects of Potential Trucation} +\section{Effects of Potential Truncation} \begin{figure} \includegraphics[width=\linewidth]{./figures/cutoffChange.pdf} \caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models with an added Ewald correction term. Error for the larger cutoff -points is equivalent to that observed at 9.0\AA\ (see Table +points is equivalent to that observed at 9.0~\AA\ (see Table \ref{tab:freeEnergy}). Data for ice I$_\textrm{c}$ with TIP3P using -both 12 and 13.5\AA\ cutoffs were omitted because the crystal was -prone to distortion and melting at 200K. Ice-$i^\prime$ is the +both 12 and 13.5~\AA\ cutoffs were omitted because the crystal was +prone to distortion and melting at 200~K. Ice-$i^\prime$ is the form of Ice-{\it i} used in the SPC/E simulations.} \label{fig:incCutoff} \end{figure} For the more computationally efficient water models, we have also -investigated the effect of potential trunctaion on the computed free +investigated the effect of potential truncation on the computed free energies as a function of the cutoff radius. As seen in Fig. \ref{fig:incCutoff}, the free energies of the ice polymorphs with water models lacking a long-range correction show significant cutoff @@ -382,7 +383,7 @@ Although TIP3P was paramaterized for use without the E field cavity in this model, so small cutoff radii mimic bulk calculations quite well under SSD/RF. -Although TIP3P was paramaterized for use without the Ewald summation, +Although TIP3P was parametrized for use without the Ewald summation, we have estimated the effect of this method for computing long-range electrostatics for both TIP3P and SPC/E. This was accomplished by calculating the potential energy of identical crystals both with and @@ -402,16 +403,49 @@ influence the polymorph expressed upon crystallization \section{Expanded Results Using Damped Shifted Force Electrostatics} +In chapter \ref{chap:electrostatics}, we discussed in detail a +pairwise method for handling electrostatics (shifted force, {\sc sf}) +that can be used as a simple and efficient replacement for the Ewald +summation. Answering the question of the free energies of these ice +polymorphs with varying water models would be an interesting +application of this technique. To this end, we set up thermodynamic +integrations of all of the previously discussed ice polymorphs using +the {\sc sf} technique with a cutoff radius of 12~\AA\ and an $\alpha$ +of 0.2125~\AA . These calculations were performed on TIP5P-E and +TIP4P-Ew (variants of the root models optimized for the Ewald +summation) as well as SPC/E, SSD/RF, and TRED (see section +\ref{sec:tredWater}). +\begin{table} +\centering +\caption{HELMHOLTZ FREE ENERGIES OF ICE POLYMORPHS USING THE DAMPED +SHIFTED FORCE CORRECTION} +\begin{tabular}{ lccccc } +\toprule +\toprule +Model & I$_\textrm{h}$ & I$_\textrm{c}$ & B & Ice-$i$ & Ice-$i^\prime$ \\ +\cmidrule(lr){2-6} +& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ +\midrule +TIP5P-E & -10.76(4) & -10.72(4) & & - & -10.68(4) \\ +TIP4P-Ew & & -11.77(3) & & - & -11.60(3) \\ +SPC/E & -12.98(3) & -11.60(3) & & - & -12.93(3) \\ +SSD/RF & -11.81(4) & -11.65(3) & & -12.41(4) & - \\ +TRED & -12.58(3) & -12.44(3) & & -13.09(4) & - \\ +\end{tabular} +\label{tab:dampedFreeEnergy} +\end{table} + + \section{Conclusions} In this work, thermodynamic integration was used to determine the absolute free energies of several ice polymorphs. The new polymorph, Ice-{\it i} was observed to be the stable crystalline state for {\it -all} the water models when using a 9.0\AA\ cutoff. However, the free +all} the water models when using a 9.0~\AA\ cutoff. However, the free energy partially depends on simulation conditions (particularly on the choice of long range correction method). Regardless, Ice-{\it i} was -still observered to be a stable polymorph for all of the studied water +still observed to be a stable polymorph for all of the studied water models. So what is the preferred solid polymorph for simulated water? As @@ -447,11 +481,11 @@ I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77K, and t results, we have calculated the oxygen-oxygen pair correlation function, $g_\textrm{OO}(r)$, and the structure factor, $S(\vec{q})$ for the two Ice-{\it i} variants (along with example ice -I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77K, and they are shown in +I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77~K, and they are shown in figures \ref{fig:gofr} and \ref{fig:sofq} respectively. It is interesting to note that the structure factors for Ice-$i^\prime$ and Ice-I$_c$ are quite similar. The primary differences are small peaks -at 1.125, 2.29, and 2.53\AA$^{-1}$, so particular attention to these +at 1.125, 2.29, and 2.53~\AA$^{-1}$, so particular attention to these regions would be needed to identify the new $i^\prime$ variant from the I$_\textrm{c}$ polymorph. @@ -460,16 +494,16 @@ model at 77 K.} \includegraphics[width=\linewidth]{./figures/iceGofr.pdf} \caption{Radial distribution functions of Ice-{\it i} and ice I$_\textrm{c}$ calculated from from simulations of the SSD/RF water -model at 77 K.} +model at 77~K.} \label{fig:gofr} \end{figure} \begin{figure} \includegraphics[width=\linewidth]{./figures/sofq.pdf} \caption{Predicted structure factors for Ice-{\it i} and ice -I$_\textrm{c}$ 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 +I$_\textrm{c}$ at 77~K. The raw structure factors have been +convoluted with a gaussian instrument function (0.075~\AA$^{-1}$ +width) to compensate for the truncation effects in our finite size simulations. The labeled peaks compared favorably with ``spurious'' peaks observed in experimental studies of amorphous solid water.\cite{Bizid87}}