--- 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}}
