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\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER \\ SIMULATIONS} |
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As discussed in the previous chapter, water has proven to be a |
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As mentioned in the previous chapter, water has proven to be a |
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challenging substance to depict in simulations, and a variety of |
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models have been developed to describe its behavior under varying |
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simulation |
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available, it is only natural to compare them under interesting |
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thermodynamic conditions in an attempt to clarify the limitations of |
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each.\cite{Jorgensen83,Jorgensen98b,Baez94,Mahoney01} Two important |
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property to quantify are the Gibbs and Helmholtz free energies, |
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properties to quantify are the Gibbs and Helmholtz free energies, |
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particularly for the solid forms of water, as these predict the |
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thermodynamic stability of the various phases. Water has a |
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particularly rich phase diagram and takes on a number of different and |
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particularly rich phase diagram and takes on a number of different |
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stable crystalline structures as the temperature and pressure are |
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varied. This complexity makes it a challenging task to investigate the |
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entire free energy landscape.\cite{Sanz04} Ideally, research is |
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point, because these phases will dictate the relevant transition |
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temperatures and pressures for the model. |
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|
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The high-pressure phases of water (ice II-ice X as well as ice XII) |
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The high-pressure phases of water (ice II-ice X as well as ice XII-ice XIV) |
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have been studied extensively both experimentally and |
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computationally. In this chapter, standard reference state methods |
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were applied in the {\it low} pressure regime to evaluate the free |
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interesting in and of themselves;\cite{Matsumoto02,Yamada02} however, |
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the crystal structure extracted from these systems is different from |
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any previously observed ice polymorphs in experiment or |
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simulation.\cite{Fennell04} We have named this structure Ice-{\it i} |
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to indicate its origin in computational simulation. The unit cell of |
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Ice-$i$ and an axially elongated variant named Ice-$i^\prime$ both |
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consist of eight water molecules that stack in rows of interlocking |
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water tetramers as illustrated in figure \ref{fig:iceiCell}A,B. These |
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tetramers form a crystal structure similar in appearance to a recent |
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two-dimensional surface tessellation simulated on silica.\cite{Yang04} |
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As expected in an ice crystal constructed of water tetramers, the |
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hydrogen bonds are not as linear as those observed in ice |
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I$_\textrm{h}$; however, the interlocking of these subunits appears to |
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provide significant stabilization to the overall crystal. The |
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arrangement of these tetramers results in open octagonal cavities that |
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are typically greater than 6.3~\AA\ in diameter (see figure |
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simulation.\cite{Fennell04} We have named this structure ``imaginary |
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ice'' (Ice-$i$) to indicate its origin in computer simulations. The |
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unit cell of Ice-$i$ and an axially elongated variant named |
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Ice-$i^\prime$ both consist of eight water molecules that stack in |
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rows of interlocking water tetramers as illustrated in figure |
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\ref{fig:iceiCell}. These tetramers form a crystal structure |
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similar in appearance to a recently simulated two-dimensional surface |
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tessellation of water on silica.\cite{Yang04} As expected in an ice |
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crystal constructed of water tetramers, the hydrogen bonds are not as |
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linear as those present in ice I$_\textrm{h}$; however, the |
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interlocking of these subunits appears to provide significant |
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stabilization to the overall crystal. The arrangement of these |
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tetramers results in open octagonal cavities that are typically |
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greater than 6.3~\AA\ in diameter (see figure |
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\ref{fig:protOrder}). This open structure leads to crystals that are |
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typically 0.07~g/cm$^3$ less dense than ice I$_\textrm{h}$. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{./figures/unitCell.pdf} |
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\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the |
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elongated variant of Ice-{\it i}. For Ice-{\it i}, the $a$ to $c$ |
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\caption{Unit cells for (A) Ice-$i$ and (B) Ice-$i^\prime$, the |
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elongated variant of Ice-$i$. For Ice-$i$, the $a$ to $c$ |
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relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a = |
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1.7850c$.} |
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\label{fig:iceiCell} |
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\begin{figure} |
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\centering |
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\includegraphics[width=3.5in]{./figures/orderedIcei.pdf} |
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\caption{Image of a proton ordered crystal of Ice-{\it i} looking |
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\caption{Image of a proton ordered crystal of Ice-$i$ looking |
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down the (001) crystal face. The rows of water tetramers surrounded by |
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octagonal pores leads to a crystal structure that is significantly |
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less dense than ice I$_\textrm{h}$.} |
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\label{fig:protOrder} |
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\end{figure} |
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Results from our initial studies indicated that Ice-{\it i} is the |
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Results from our initial studies indicated that Ice-$i$ is the |
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minimum energy crystal structure for the single point water models |
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investigated (for discussions on these single point dipole models, see |
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the previous work and related |
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the previous chapter and related |
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articles\cite{Fennell04,Liu96,Bratko85}). These earlier results only |
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considered energetic stabilization and neglected entropic |
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contributions to the overall free energy. To address this issue, we |
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have calculated the absolute free energy of this crystal using |
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thermodynamic integration and compared to the free energies of ice |
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thermodynamic integration and compared it to the free energies of ice |
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I$_\textrm{c}$ and ice I$_\textrm{h}$ (the common low-density ice |
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polymorphs) and ice B (a higher density, but very stable crystal |
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structure observed by B\'{a}ez and Clancy in free energy studies of |
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SPC/E).\cite{Baez95b} This work includes results for the water model |
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from which Ice-{\it i} was crystallized (SSD/E) in addition to several |
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from which Ice-$i$ was crystallized (SSD/E) in addition to several |
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common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
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field parametrized single point dipole water model (SSD/RF). The |
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axially elongated variant, Ice-$i^\prime$, was used in calculations |
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95 degree angles. Under SPC/E, TIP4P, and TIP5P, this geometry is |
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better at forming favorable hydrogen bonds. The degree of rhomboid |
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distortion depends on the water model used but is significant enough |
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to split the peak in the radial distribution function which corresponds |
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to diagonal sites in the tetramers. |
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to split the peak in the radial distribution function which |
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corresponds to diagonal sites in the tetramers. |
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|
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\section{Methods and Thermodynamic Integration} |
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|
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Canonical ensemble ({\it NVT}) molecular dynamics calculations were |
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performed using the OOPSE molecular mechanics package.\cite{Meineke05} |
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The densities chosen for the simulations were taken from |
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isobaric-isothermal ({\it NPT}) simulations performed at 1 atm and |
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200~K. Each model (and each crystal structure) was allowed to relax for |
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300~ps in the {\it NPT} ensemble before averaging the density to obtain |
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the volumes for the {\it NVT} simulations.All molecules were treated |
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as rigid bodies, with orientational motion propagated using the |
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symplectic DLM integration method described in section |
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Canonical ensemble ($NVT$) molecular dynamics calculations were |
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performed using the {\sc oopse} molecular mechanics |
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package.\cite{Meineke05} The densities chosen for the simulations were |
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taken from isobaric-isothermal ($NPT$) simulations performed at 1~atm |
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and 200~K. Each model (and each crystal structure) was allowed to |
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relax for 300~ps in the $NPT$ ensemble before averaging the density to |
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obtain the volumes for the $NVT$ simulations. All molecules were |
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treated as rigid bodies, with orientational motion propagated using |
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the symplectic {\sc dlm} integration method described in section |
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\ref{sec:IntroIntegrate}. |
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|
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|
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been used extensively in the calculation of free energies for |
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condensed phases of |
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materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
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method uses a sequence of simulations over which the system of |
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method uses a sequence of simulations during which the system of |
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interest is converted into a reference system for which the free |
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energy is known analytically ($A_0$). The difference in potential |
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energy between the reference system and the system of interest |
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\end{equation} |
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where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
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the spring constants restraining translational motion and deflection |
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of and rotation around the principle axis of the molecule |
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respectively. These spring constants are typically calculated from |
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the mean-square displacements of water molecules in an unrestrained |
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ice crystal at 200~K. For these studies, $K_\mathrm{v} = |
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4.29$~kcal~mol$^{-1}$~\AA$^{-2}$, $K_\theta\ = |
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of and rotation around the principle axis of the molecule respectively |
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(see figure \ref{fig:waterSpring}). These spring constants are |
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typically calculated from the mean-square displacements of water |
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molecules in an unrestrained ice crystal at 200~K. For these studies, |
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$K_\mathrm{v} = 4.29$~kcal~mol$^{-1}$~\AA$^{-2}$, $K_\theta\ = |
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13.88$~kcal~mol$^{-1}$~rad$^{-2}$, and $K_\omega\ = |
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17.75$~kcal~mol$^{-1}$~rad$^{-2}$. It is clear from |
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Fig. \ref{fig:waterSpring} that the values of $\theta$ range from $0$ |
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to $\pi$, while $\omega$ ranges from $-\pi$ to $\pi$. The partition |
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17.75$~kcal~mol$^{-1}$~rad$^{-2}$. It is clear from figure |
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\ref{fig:waterSpring} that the values of $\theta$ range from $0$ to |
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$\pi$, while $\omega$ ranges from $-\pi$ to $\pi$. The partition |
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function for a molecular crystal restrained in this fashion can be |
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evaluated analytically, and the Helmholtz Free Energy ({\it A}) is |
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given by |
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\label{eq:ecFreeEnergy} |
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\end{equation} |
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where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
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potential energy of the ideal crystal.\cite{Baez95a} The choice of an |
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Einstein crystal reference state is somewhat arbitrary. Any ideal |
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system for which the partition function is known exactly could be used |
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as a reference point as long as the system does not undergo a phase |
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transition during the integration path between the real and ideal |
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systems. Nada and van der Eerden have shown that the use of different |
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force constants in the Einstein crystal does not affect the total |
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free energy, and Gao {\it et al.} have shown that free energies |
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computed with the Debye crystal reference state differ from the |
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Einstein crystal by only a few tenths of a |
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potential energy of the ideal crystal.\cite{Baez95a} |
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|
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The choice of an Einstein crystal reference state is somewhat |
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arbitrary. Any ideal system for which the partition function is known |
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exactly could be used as a reference point, as long as the system does |
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not undergo a phase transition during the integration path between the |
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real and ideal systems. Nada and van der Eerden have shown that the |
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use of different force constants in the Einstein crystal does not |
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affect the total free energy, and Gao {\it et al.} have shown that |
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free energies computed with the Debye crystal reference state differ |
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from the Einstein crystal by only a few tenths of a |
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kJ~mol$^{-1}$.\cite{Nada03,Gao00} These free energy differences can |
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lead to some uncertainty in the computed melting point of the solids. |
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\begin{figure} |
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molecules. In this study, we applied one of the most convenient |
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methods and integrated over the $\lambda^4$ path, where all |
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interaction parameters are scaled equally by this transformation |
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parameter. This method has been shown to be reversible and provide |
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results in excellent agreement with other established |
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methods.\cite{Baez95b} |
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parameter. This method is reversible and provides results in excellent |
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agreement with other established methods.\cite{Baez95b} |
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|
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The Helmholtz free energy error was determined in the same manner in |
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both the solid and the liquid free energy calculations . At each point |
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both the solid and the liquid free energy calculations. At each point |
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along the integration path, we calculated the standard deviation of |
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the potential energy difference. Addition or subtraction of these |
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values to each of their respective points and integrating the curve |
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again provides the upper and lower bounds of the uncertainty in the |
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Helmholtz free energy. |
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deviations to each of their respective points and integrating the |
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curve again provides the upper and lower bounds of the uncertainty in |
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the Helmholtz free energy. |
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|
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Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and |
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Lennard-Jones interactions were gradually reduced by a cubic switching |
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performed with longer cutoffs of 10.5, 12, 13.5, and 15~\AA\ to |
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compare with the 9~\AA\ cutoff results. Finally, the effects of using |
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the Ewald summation were estimated for TIP3P and SPC/E by performing |
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single configuration Particle-Mesh Ewald (PME) calculations for each |
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of the ice polymorphs.\cite{Ponder87} The calculated energy difference |
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in the presence and absence of PME was applied to the previous results |
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in order to predict changes to the free energy landscape. |
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single configuration smooth particle-mesh Ewald ({\sc spme}) |
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calculations for each of the ice polymorphs.\cite{Ponder87} The |
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calculated energy difference in the presence and absence of {\sc spme} |
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was applied to the previous results in order to predict changes to the |
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free energy landscape. |
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|
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In addition to the above procedures, we also tested how the inclusion |
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of the Lennard-Jones long-range correction affects the free energy |
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results. The correction for the Lennard-Jones truncation was included |
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by integration of the equation discussed in section |
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\ref{sec:LJCorrections}. Rather than discuss its affect alongside the |
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free energy results, we will just mention that while the correction |
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does lower the free energy of the higher density states more than the |
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lower density states, the effect is so small that it is entirely |
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overwhelmed by the error in the free energy calculation. Since its |
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inclusion does not influence the results, the Lennard-Jones correction |
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was omitted from all the calculations below. |
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|
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\section{Initial Free Energy Results} |
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|
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The calculated free energies of proton-ordered variants of three low |
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density polymorphs (I$_\textrm{h}$, I$_\textrm{c}$, and Ice-{\it i} or |
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Ice-$i^\prime$) and the stable higher density ice B are listed in |
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table \ref{tab:freeEnergy}. Ice B was included because it has been |
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shown to be a minimum free energy structure for SPC/E at ambient |
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conditions.\cite{Baez95b} In addition to the free energies, the |
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relevant transition temperatures at standard pressure are also |
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displayed in table \ref{tab:freeEnergy}. These free energy values |
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indicate that Ice-{\it i} is the most stable state for all of the |
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investigated water models. With the free energy at these state |
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points, the Gibbs-Helmholtz equation was used to project to other |
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state points and to build phase diagrams. Figures |
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\ref{fig:tp3PhaseDia} and \ref{fig:ssdrfPhaseDia} are example diagrams |
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built from the results for the TIP3P and SSD/RF water models. All |
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other models have similar structure, although the crossing points |
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between the phases move to different temperatures and pressures as |
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indicated from the transition temperatures in table |
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\ref{tab:freeEnergy}. It is interesting to note that ice |
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I$_\textrm{h}$ (and ice I$_\textrm{c}$ for that matter) do not appear |
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in any of the phase diagrams for any of the models. For purposes of |
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this study, ice B is representative of the dense ice polymorphs. A |
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recent study by Sanz {\it et al.} provides details on the phase |
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diagrams for SPC/E and TIP4P at higher pressures than those studied |
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here.\cite{Sanz04} |
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\begin{table} |
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\centering |
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\caption{HELMHOLTZ FREE ENERGIES AND TRANSITION TEMPERATURES AT 1 |
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\begin{tabular}{lccccccc} |
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\toprule |
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\toprule |
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Water Model & I$_\textrm{h}$ & I$_\textrm{c}$ & B & Ice-{\it i} & Ice-$i^\prime$ & $T_\textrm{m}$ (*$T_\textrm{s}$) & $T_\textrm{b}$\\ |
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Water Model & I$_\textrm{h}$ & I$_\textrm{c}$ & B & Ice-$i$ & Ice-$i^\prime$ & $T_\textrm{m}$ (*$T_\textrm{s}$) & $T_\textrm{b}$\\ |
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\cmidrule(lr){2-6} |
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\cmidrule(l){7-8} |
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& \multicolumn{5}{c}{(kcal mol$^{-1}$)} & \multicolumn{2}{c}{(K)}\\ |
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\end{tabular} |
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\label{tab:freeEnergy} |
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\end{table} |
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|
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The calculated free energies of proton-ordered variants of three low |
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density polymorphs (I$_\textrm{h}$, I$_\textrm{c}$, and Ice-$i$ or |
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Ice-$i^\prime$) and the stable higher density ice B are listed in |
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table \ref{tab:freeEnergy}. Ice B was included because it has been |
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shown to be a minimum free energy structure for SPC/E at ambient |
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conditions.\cite{Baez95b} In addition to the free energies, the |
| 296 |
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relevant transition temperatures at standard pressure are also |
| 297 |
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displayed in table \ref{tab:freeEnergy}. These free energy values |
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indicate that Ice-$i$ is the thermodynamically preferred state for |
| 299 |
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all of the investigated water models under the chosen simulation |
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conditions. With the free energy at these state points, the |
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Gibbs-Helmholtz equation was used to project to other state points and |
| 302 |
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to build phase diagrams. Figures \ref{fig:tp3PhaseDia} and |
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\ref{fig:ssdrfPhaseDia} are example phase diagrams built from the |
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results for the TIP3P and SSD/RF water models. All other models have |
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similar structure, although the crossing points between the phases |
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move to different temperatures and pressures as indicated from the |
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transition temperatures in table |
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\ref{tab:freeEnergy}. It is interesting to note that ice |
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I$_\textrm{h}$ (and ice I$_\textrm{c}$ for that matter) do not appear |
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in any of the phase diagrams for any of the models. For purposes of |
| 311 |
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this study, ice B is representative of the dense ice polymorphs. A |
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recent study by Sanz {\it et al.} provides details on the phase |
| 313 |
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diagrams for SPC/E and TIP4P at higher pressures than those studied |
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here.\cite{Sanz04} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{./figures/tp3PhaseDia.pdf} |
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\caption{Phase diagram for the TIP3P water model in the low pressure |
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regime. The displayed $T_m$ and $T_b$ values are good predictions of |
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the experimental values; however, the solid phases shown are not the |
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experimentally observed forms. Both cubic and hexagonal ice $I$ are |
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experimentally observed forms. Both cubic and hexagonal ice I are |
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higher in energy and don't appear in the phase diagram.} |
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\label{fig:tp3PhaseDia} |
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\end{figure} |
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– |
|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{./figures/ssdrfPhaseDia.pdf} |
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Most of the water models have melting points that compare quite |
| 352 |
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favorably with the experimental value of 273~K. The unfortunate |
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aspect of this result is that this phase change occurs between |
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Ice-{\it i} and the liquid state rather than ice I$_h$ and the liquid |
| 354 |
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Ice-$i$ and the liquid state rather than ice I$_\textrm{h}$ and the liquid |
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state. These results do not contradict other studies. Studies of ice |
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I$_h$ using TIP4P predict a $T_m$ ranging from 191 to 238~K |
| 356 |
> |
I$_\textrm{h}$ using TIP4P predict a $T_m$ ranging from 191 to 238~K |
| 357 |
|
(differences being attributed to choice of interaction truncation and |
| 358 |
|
different ordered and disordered molecular |
| 359 |
|
arrangements).\cite{Nada03,Vlot99,Gao00,Sanz04} If the presence of ice |
| 360 |
< |
B and Ice-{\it i} were omitted, a $T_\textrm{m}$ value around 200~K |
| 360 |
> |
B and Ice-$i$ were omitted, a $T_\textrm{m}$ value around 200~K |
| 361 |
|
would be predicted from this work. However, the $T_\textrm{m}$ from |
| 362 |
< |
Ice-{\it i} is calculated to be 262~K, indicating that these |
| 363 |
< |
simulation based structures ought to be included in studies probing |
| 362 |
> |
Ice-$i$ is calculated to be 262~K, indicating that these |
| 363 |
> |
simulation-based structures ought to be included in studies probing |
| 364 |
|
phase transitions with this model. Also of interest in these results |
| 365 |
|
is that SSD/E does not exhibit a melting point at 1 atm but does |
| 366 |
|
sublime at 355~K. This is due to the significant stability of |
| 367 |
< |
Ice-{\it i} over all other polymorphs for this particular model under |
| 367 |
> |
Ice-$i$ over all other polymorphs for this particular model under |
| 368 |
|
these conditions. While troubling, this behavior resulted in the |
| 369 |
< |
spontaneous crystallization of Ice-{\it i} which led us to investigate |
| 369 |
> |
spontaneous crystallization of Ice-$i$ which led us to investigate |
| 370 |
|
this structure. These observations provide a warning that simulations |
| 371 |
|
of SSD/E as a ``liquid'' near 300~K are actually metastable and run |
| 372 |
|
the risk of spontaneous crystallization. However, when a longer |
| 376 |
|
\section{Effects of Potential Truncation} |
| 377 |
|
|
| 378 |
|
\begin{figure} |
| 379 |
+ |
\centering |
| 380 |
|
\includegraphics[width=\linewidth]{./figures/cutoffChange.pdf} |
| 381 |
|
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
| 382 |
|
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
| 385 |
|
\ref{tab:freeEnergy}). Data for ice I$_\textrm{c}$ with TIP3P using |
| 386 |
|
both 12 and 13.5~\AA\ cutoffs were omitted because the crystal was |
| 387 |
|
prone to distortion and melting at 200~K. Ice-$i^\prime$ is the |
| 388 |
< |
form of Ice-{\it i} used in the SPC/E simulations.} |
| 388 |
> |
form of Ice-$i$ used in the SPC/E simulations.} |
| 389 |
|
\label{fig:incCutoff} |
| 390 |
|
\end{figure} |
| 377 |
– |
|
| 391 |
|
For the more computationally efficient water models, we have also |
| 392 |
|
investigated the effect of potential truncation on the computed free |
| 393 |
< |
energies as a function of the cutoff radius. As seen in |
| 394 |
< |
Fig. \ref{fig:incCutoff}, the free energies of the ice polymorphs with |
| 393 |
> |
energies as a function of the cutoff radius. As seen in figure |
| 394 |
> |
\ref{fig:incCutoff}, the free energies of the ice polymorphs with |
| 395 |
|
water models lacking a long-range correction show significant cutoff |
| 396 |
< |
dependence. In general, there is a narrowing of the free energy |
| 397 |
< |
differences while moving to greater cutoff radii. As the free |
| 396 |
> |
radius dependence. In general, there is a narrowing of the free |
| 397 |
> |
energy differences while moving to greater cutoff radii. As the free |
| 398 |
|
energies for the polymorphs converge, the stability advantage that |
| 399 |
< |
Ice-{\it i} exhibits is reduced. Adjacent to each of these plots are |
| 399 |
> |
Ice-$i$ exhibits is reduced. Adjacent to each of these plots are |
| 400 |
|
results for systems with applied or estimated long-range corrections. |
| 401 |
|
SSD/RF was parametrized for use with a reaction field, and the benefit |
| 402 |
|
provided by this computationally inexpensive correction is apparent. |
| 404 |
|
field cavity in this model, so small cutoff radii mimic bulk |
| 405 |
|
calculations quite well under SSD/RF. |
| 406 |
|
|
| 407 |
< |
Although TIP3P was parametrized for use without the Ewald summation, |
| 408 |
< |
we have estimated the effect of this method for computing long-range |
| 409 |
< |
electrostatics for both TIP3P and SPC/E. This was accomplished by |
| 410 |
< |
calculating the potential energy of identical crystals both with and |
| 411 |
< |
without particle mesh Ewald (PME). Similar behavior to that observed |
| 412 |
< |
with reaction field is seen for both of these models. The free |
| 413 |
< |
energies show reduced dependence on cutoff radius and span a narrower |
| 414 |
< |
range for the various polymorphs. Like the dipolar water models, |
| 415 |
< |
TIP3P displays a relatively constant preference for the Ice-{\it i} |
| 416 |
< |
polymorph. Crystal preference is much more difficult to determine for |
| 417 |
< |
SPC/E. Without a long-range correction, each of the polymorphs |
| 418 |
< |
studied assumes the role of the preferred polymorph under different |
| 419 |
< |
cutoff radii. The inclusion of the Ewald correction flattens and |
| 420 |
< |
narrows the gap in free energies such that the polymorphs are |
| 421 |
< |
isoenergetic within statistical uncertainty. This suggests that other |
| 422 |
< |
conditions, such as the density in fixed-volume simulations, can |
| 423 |
< |
influence the polymorph expressed upon crystallization. |
| 407 |
> |
Although the point-charge models investigated here were parametrized |
| 408 |
> |
for use without the Ewald summation, we have estimated the effect of |
| 409 |
> |
this method for computing long-range electrostatics for both TIP3P and |
| 410 |
> |
SPC/E. This was accomplished by calculating the potential energy of |
| 411 |
> |
identical crystals both with and without {\sc spme}. Similar behavior |
| 412 |
> |
to that observed with reaction field is seen for both of these models. |
| 413 |
> |
The free energies show reduced dependence on cutoff radius and span a |
| 414 |
> |
narrower range for the various polymorphs. Like the dipolar water |
| 415 |
> |
models, TIP3P displays a relatively constant preference for the |
| 416 |
> |
Ice-$i$ polymorph. The crystal preference is much more difficult to |
| 417 |
> |
determine for SPC/E. Without a long-range correction, each of the |
| 418 |
> |
polymorphs studied assumes the role of the thermodynamically preferred |
| 419 |
> |
state under different cutoff radii. The inclusion of the Ewald |
| 420 |
> |
correction flattens and narrows the gap in free energies such that the |
| 421 |
> |
polymorphs are isoenergetic within statistical uncertainty. This |
| 422 |
> |
suggests that other conditions, such as the density in fixed-volume |
| 423 |
> |
simulations, can influence the polymorph expressed upon |
| 424 |
> |
crystallization. |
| 425 |
|
|
| 426 |
|
\section{Expanded Results Using Damped Shifted Force Electrostatics} |
| 427 |
|
|
| 449 |
|
\cmidrule(lr){2-6} |
| 450 |
|
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ |
| 451 |
|
\midrule |
| 452 |
< |
TIP5P-E & -11.98(4) & -11.96(4) & & - & -11.95(3) \\ |
| 452 |
> |
TIP5P-E & -11.98(4) & -11.96(4) & -11.87(3) & - & -11.95(3) \\ |
| 453 |
|
TIP4P-Ew & -13.11(3) & -13.09(3) & -12.97(3) & - & -12.98(3) \\ |
| 454 |
|
SPC/E & -12.99(3) & -13.00(3) & -13.03(3) & - & -12.99(3) \\ |
| 455 |
|
SSD/RF & -11.83(3) & -11.66(4) & -12.32(3) & -12.39(3) & - \\ |
| 456 |
|
TRED & -12.61(3) & -12.43(3) & -12.89(3) & -13.12(3) & - \\ |
| 457 |
+ |
\bottomrule |
| 458 |
|
\end{tabular} |
| 459 |
|
\label{tab:dampedFreeEnergy} |
| 460 |
|
\end{table} |
| 461 |
|
The results of these calculations in table \ref{tab:dampedFreeEnergy} |
| 462 |
|
show similar behavior to the Ewald results in figure |
| 463 |
|
\ref{fig:incCutoff}, at least for SSD/RF and SPC/E which are present |
| 464 |
< |
in both. The ice polymorph Helmholtz free energies for SSD/RF order in |
| 465 |
< |
the same fashion; however Ice-$i$ and ice B are quite a bit closer in |
| 466 |
< |
free energy (nearly isoenergetic). The free energy differences between |
| 467 |
< |
ice polymorphs for TRED water parallel SSD/RF, with the exception that |
| 468 |
< |
ice B is destabilized such that it is not very close to Ice-$i$. The |
| 469 |
< |
SPC/E results really show the near isoenergetic behavior when using |
| 470 |
< |
the electrostatics correction. Ice B has the lowest Helmholtz free |
| 471 |
< |
energy; however, all the polymorph results overlap within error. |
| 464 |
> |
in both. The Helmholtz free energies of the ice polymorphs for SSD/RF |
| 465 |
> |
order in the same fashion; however, Ice-$i$ and ice B are quite a bit |
| 466 |
> |
closer in free energy (nearly isoenergetic). The free energy |
| 467 |
> |
differences between ice polymorphs for TRED water parallel SSD/RF, |
| 468 |
> |
with the exception that ice B is destabilized such that the free |
| 469 |
> |
energy is not very close to Ice-$i$. The SPC/E results really show the |
| 470 |
> |
near isoenergetic behavior when using the electrostatic |
| 471 |
> |
correction. Ice B has the lowest Helmholtz free energy; however, all |
| 472 |
> |
the polymorph results overlap within error. |
| 473 |
|
|
| 474 |
|
The most interesting results from these calculations come from the |
| 475 |
|
more expensive TIP4P-Ew and TIP5P-E results. Both of these models were |
| 486 |
|
geometry is more physically valid.\cite{Vega05} With the TIP4P-Ew |
| 487 |
|
water model, the experimentally observed polymorph (ice |
| 488 |
|
I$_\textrm{h}$) is the preferred form with ice I$_\textrm{c}$ slightly |
| 489 |
< |
higher in energy, though overlapping within error, and the less |
| 490 |
< |
realistic ice B and Ice-$i^\prime$ are destabilized relative to these |
| 491 |
< |
polymorphs. TIP5P-E shows similar behavior to SPC/E, where there is no |
| 492 |
< |
real free energy distinction between the various polymorphs and lend |
| 493 |
< |
credence to other results indicating the preferred form of TIP5P at |
| 494 |
< |
1~atm is a structure similar to ice B.\cite{Yamada02,Vega05,Abascal05} |
| 495 |
< |
These results indicate that TIP4P-Ew is a better mimic of real water |
| 496 |
< |
than these other models when studying crystallization and solid forms |
| 497 |
< |
of water. |
| 489 |
> |
higher in energy, though overlapping within error. Additionally, the |
| 490 |
> |
less realistic ice B and Ice-$i^\prime$ structures are destabilized |
| 491 |
> |
relative to these polymorphs. TIP5P-E shows similar behavior to SPC/E, |
| 492 |
> |
where there is no real free energy distinction between the various |
| 493 |
> |
polymorphs, because many overlap within error. While ice B is close in |
| 494 |
> |
free energy to the other polymorphs, these results fail to support the |
| 495 |
> |
findings of other researchers indicating the preferred form of TIP5P |
| 496 |
> |
at 1~atm is a structure similar to ice |
| 497 |
> |
B.\cite{Yamada02,Vega05,Abascal05} It should be noted that we are |
| 498 |
> |
looking at TIP5P-E rather than TIP5P, and the differences in the |
| 499 |
> |
Lennard-Jones parameters could be a reason for this dissimilarity. |
| 500 |
> |
Overall, these results indicate that TIP4P-Ew is a better mimic of |
| 501 |
> |
real water than these other models when studying crystallization and |
| 502 |
> |
solid forms of water. |
| 503 |
|
|
| 504 |
|
\section{Conclusions} |
| 505 |
|
|
| 506 |
|
In this work, thermodynamic integration was used to determine the |
| 507 |
|
absolute free energies of several ice polymorphs. The new polymorph, |
| 508 |
< |
Ice-{\it i} was observed to be the stable crystalline state for {\it |
| 509 |
< |
all} the water models when using a 9.0~\AA\ cutoff. However, the free |
| 510 |
< |
energy partially depends on simulation conditions (particularly on the |
| 511 |
< |
choice of long range correction method). Regardless, Ice-{\it i} was |
| 512 |
< |
still observed to be a stable polymorph for all of the studied water |
| 513 |
< |
models. |
| 508 |
> |
Ice-$i$ was observed to be the stable crystalline state for {\it all} |
| 509 |
> |
the water models when using a 9.0~\AA\ cutoff. Additional work showed |
| 510 |
> |
that the free energy depends in part on simulation conditions - in |
| 511 |
> |
particular, the choice of long-range correction method. Regardless, |
| 512 |
> |
Ice-$i$ was still observed to be a stable polymorph for all of the |
| 513 |
> |
studied water models. |
| 514 |
|
|
| 515 |
< |
So what is the preferred solid polymorph for simulated water? As |
| 516 |
< |
indicated above, the answer appears to be dependent both on the |
| 517 |
< |
conditions and the model used. In the case of short cutoffs without a |
| 518 |
< |
long-range interaction correction, Ice-{\it i} and Ice-$i^\prime$ have |
| 519 |
< |
the lowest free energy of the studied polymorphs with all the models. |
| 520 |
< |
Ideally, crystallization of each model under constant pressure |
| 521 |
< |
conditions, as was done with SSD/E, would aid in the identification of |
| 522 |
< |
their respective preferred structures. This work, however, helps |
| 523 |
< |
illustrate how studies involving one specific model can lead to |
| 503 |
< |
insight about important behavior of others. |
| 515 |
> |
So what is the preferred solid polymorph for simulated water? The |
| 516 |
> |
answer appears to be dependent both on the conditions and the model |
| 517 |
> |
used. In the case of short cutoffs without a long-range interaction |
| 518 |
> |
correction, Ice-$i$ and Ice-$i^\prime$ have the lowest free energy of |
| 519 |
> |
the studied polymorphs with all the models. Ideally, crystallization |
| 520 |
> |
of each model under constant pressure conditions, as was done with |
| 521 |
> |
SSD/E, would aid in identifying their respective preferred structures. |
| 522 |
> |
This work, however, helps illustrate how studies involving one |
| 523 |
> |
specific model can lead to insight about important behavior of others. |
| 524 |
|
|
| 525 |
|
We also note that none of the water models used in this study are |
| 526 |
< |
polarizable or flexible models. It is entirely possible that the |
| 527 |
< |
polarizability of real water makes Ice-{\it i} substantially less |
| 528 |
< |
stable than ice I$_h$. However, the calculations presented above seem |
| 529 |
< |
interesting enough to communicate before the role of polarizability |
| 530 |
< |
(or flexibility) has been thoroughly investigated. |
| 526 |
> |
polarizable or flexible models. It is entirely possible that the |
| 527 |
> |
polarizability of real water makes the Ice-$i$ structure substantially |
| 528 |
> |
less stable than ice I$_\textrm{h}$. The dipole moment of the water |
| 529 |
> |
molecules increases as the system becomes more condensed, and the |
| 530 |
> |
increasing dipole moment should destabilize the tetramer structures in |
| 531 |
> |
Ice-$i$. Right now, using TIP4P-Ew with an electrostatic correction |
| 532 |
> |
gives the proper thermodynamically preferred state, and we recommend |
| 533 |
> |
this arrangement for study of crystallization processes if the |
| 534 |
> |
computational cost increase that comes with including polarizability |
| 535 |
> |
is an issue. |
| 536 |
|
|
| 537 |
< |
Finally, due to the stability of Ice-{\it i} in the investigated |
| 538 |
< |
simulation conditions, the question arises as to possible experimental |
| 539 |
< |
observation of this polymorph. The rather extensive past and current |
| 540 |
< |
experimental investigation of water in the low pressure regime makes |
| 541 |
< |
us hesitant to ascribe any relevance to this work outside of the |
| 542 |
< |
simulation community. It is for this reason that we chose a name for |
| 543 |
< |
this polymorph which involves an imaginary quantity. That said, there |
| 544 |
< |
are certain experimental conditions that would provide the most ideal |
| 545 |
< |
situation for possible observation. These include the negative |
| 546 |
< |
pressure or stretched solid regime, small clusters in vacuum |
| 547 |
< |
deposition environments, and in clathrate structures involving small |
| 548 |
< |
non-polar molecules. For the purpose of comparison with experimental |
| 549 |
< |
results, we have calculated the oxygen-oxygen pair correlation |
| 550 |
< |
function, $g_\textrm{OO}(r)$, and the structure factor, $S(\vec{q})$ |
| 551 |
< |
for the two Ice-{\it i} variants (along with example ice |
| 552 |
< |
I$_\textrm{h}$ and I$_\textrm{c}$ plots) at 77~K, and they are shown in |
| 553 |
< |
figures \ref{fig:gofr} and \ref{fig:sofq} respectively. It is |
| 554 |
< |
interesting to note that the structure factors for Ice-$i^\prime$ and |
| 555 |
< |
Ice-I$_c$ are quite similar. The primary differences are small peaks |
| 531 |
< |
at 1.125, 2.29, and 2.53~\AA$^{-1}$, so particular attention to these |
| 532 |
< |
regions would be needed to identify the new $i^\prime$ variant from |
| 533 |
< |
the I$_\textrm{c}$ polymorph. |
| 537 |
> |
Finally, the stability of Ice-$i$ in these simulations raises the |
| 538 |
> |
possibility of experimental observation. The extensive body of |
| 539 |
> |
research on water in the low pressure regime makes us hesitant to |
| 540 |
> |
ascribe any relevance to this work outside the simulation community. |
| 541 |
> |
It is for this reason that we chose a name for this polymorph |
| 542 |
> |
involving an imaginary quantity. That said, there are certain |
| 543 |
> |
conditions that would be ideal for experimental observation of |
| 544 |
> |
Ice-$i$. These include the negative pressure or stretched solid |
| 545 |
> |
regime, clusters deposited in vacuum environments, and clathrate |
| 546 |
> |
structures involving small non-polar molecules. For the purpose of |
| 547 |
> |
comparison with future experimental results, we calculated the |
| 548 |
> |
oxygen-oxygen pair correlation function, $g_\textrm{OO}(r)$, and the |
| 549 |
> |
structure factor, $S(\vec{q})$ for the two Ice-$i$ variants (along |
| 550 |
> |
with ice I$_\textrm{h}$ and I$_\textrm{c}$) at 77~K (figures |
| 551 |
> |
\ref{fig:gofr} and \ref{fig:sofq}). It is interesting to note that |
| 552 |
> |
the structure factors for Ice-$i^\prime$ and ice I$_\textrm{c}$ are |
| 553 |
> |
quite similar. The primary differences are small peaks at 1.125, |
| 554 |
> |
2.29, and 2.53~\AA$^{-1}$, so particular attention to these regions |
| 555 |
> |
would be needed to distinguish Ice-$i^\prime$ from ice I$_\textrm{c}$. |
| 556 |
|
|
| 535 |
– |
|
| 557 |
|
\begin{figure} |
| 558 |
|
\includegraphics[width=\linewidth]{./figures/iceGofr.pdf} |
| 559 |
< |
\caption{Radial distribution functions of Ice-{\it i} and ice |
| 559 |
> |
\caption{Radial distribution functions of Ice-$i$ and ice |
| 560 |
|
I$_\textrm{c}$ calculated from from simulations of the SSD/RF water |
| 561 |
|
model at 77~K.} |
| 562 |
|
\label{fig:gofr} |
| 564 |
|
|
| 565 |
|
\begin{figure} |
| 566 |
|
\includegraphics[width=\linewidth]{./figures/sofq.pdf} |
| 567 |
< |
\caption{Predicted structure factors for Ice-{\it i} and ice |
| 567 |
> |
\caption{Predicted structure factors for Ice-$i$ and ice |
| 568 |
|
I$_\textrm{c}$ at 77~K. The raw structure factors have been |
| 569 |
< |
convoluted with a gaussian instrument function (0.075~\AA$^{-1}$ |
| 569 |
> |
convoluted with a Gaussian instrument function (0.075~\AA$^{-1}$ |
| 570 |
|
width) to compensate for the truncation effects in our finite size |
| 571 |
< |
simulations. The labeled peaks compared favorably with ``spurious'' |
| 551 |
< |
peaks observed in experimental studies of amorphous solid |
| 552 |
< |
water.\cite{Bizid87}} |
| 571 |
> |
simulations.} |
| 572 |
|
\label{fig:sofq} |
| 573 |
|
\end{figure} |
| 574 |
|
|
