trunk/fennellDissertation/iceChapter.tex

\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER SIMULATIONS}

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]{./figures/unitCell.pdf}
\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}
\centering
\includegraphics[width=3.5in]{./figures/orderedIcei.pdf}
\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 molecular mechanics package.\cite{Meineke05}
All molecules were treated as rigid bodies, with orientational motion
propagated using the symplectic DLM integration method described in
section \ref{sec:IntroIntegration}.

Thermodynamic integration 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. 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 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 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
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right
)^\frac{1}{2}\left (\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right
)^\frac{1}{2}\right ] \nonumber \\ &-& kT\ln \left [\frac{kT}{2(\pi
K_\omega K_\theta)^{\frac{1}{2}}}\exp\left
(-\frac{kT}{2K_\theta}\right)\int_0^{\left (\frac{kT}{2K_\theta}\right
)^\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.

\begin{figure}
\centering
\includegraphics[width=3.5in]{./figures/rotSpring.pdf}
\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}
\centering
\includegraphics[width=\linewidth]{./figures/tp3PhaseDia.pdf}
\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}
\centering
\includegraphics[width=\linewidth]{./figures/ssdrfPhaseDia.pdf}
\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]{./figures/cutoffChange.pdf}
\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.  In a quick 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 HDA 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 determine whether this ice polymorph
is named appropriately or if it should be promoted to Ice-0.

\begin{figure}
\includegraphics[width=\linewidth]{./figures/iceGofr.pdf}
\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]{./figures/sofq.pdf}
\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}

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