--- trunk/iceiPaper/iceiPaper.tex 2004/09/17 14:54:49 1473 +++ trunk/iceiPaper/iceiPaper.tex 2004/09/21 20:17:12 1488 @@ -33,17 +33,21 @@ The free energies of several ice polymorphs in the low %\doublespacing \begin{abstract} -The free energies of several ice polymorphs in the low pressure regime -were calculated using thermodynamic integration. These integrations -were done for most of the common water models. Ice-{\it i}, a -structure we recently observed to be stable in one of the single-point -water models, was determined to be the stable crystalline state (at 1 -atm) for {\it all} the water models investigated. Phase diagrams were +The absolute free energies of several ice polymorphs which are stable +at low pressures were calculated using thermodynamic integration to a +reference system (the Einstein crystal). These integrations were +performed for most of the common water models (SPC/E, TIP3P, TIP4P, +TIP5P, SSD/E, SSD/RF). Ice-{\it i}, a structure we recently observed +crystallizing at room temperature for one of the single-point water +models, was determined to be the stable crystalline state (at 1 atm) +for {\it all} the water models investigated. Phase diagrams were generated, and phase coexistence lines were determined for all of the -known low-pressure ice structures under all of the common water -models. Additionally, potential truncation was shown to have an -effect on the calculated free energies, and can result in altered free -energy landscapes. +known low-pressure ice structures under all of these water models. +Additionally, potential truncation was shown to have an effect on the +calculated free energies, and can result in altered free energy +landscapes. Structure factor predictions for the new crystal were +generated and we await experimental confirmation of the existence of +this new polymorph. \end{abstract} %\narrowtext @@ -54,29 +58,18 @@ Computer simulations are a valuable tool for studying \section{Introduction} -Computer simulations are a valuable tool for studying the phase -behavior of systems ranging from small or simple molecules to complex -biological species.\cite{Matsumoto02,Li96,Marrink01} Useful techniques -have been developed to investigate the thermodynamic properites of -model substances, providing both qualitative and quantitative -comparisons between simulations and -experiment.\cite{Widom63,Frenkel84} 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{Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Liu96,Berendsen98,Mahoney00,Fennell04} +conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,Berendsen98,Dill00,Mahoney00,Fennell04} These models have been used to investigate important physical -phenomena like phase transitions, molecule transport, and the +phenomena like phase transitions, transport properties, and the hydrophobic effect.\cite{Yamada02,Marrink94,Gallagher03} 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{Jorgensen83,Jorgensen98b,Clancy94,Mahoney01} Two -important property to quantify are the Gibbs and Helmholtz free +important properties 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 @@ -84,27 +77,27 @@ these phases will dictate the true transition temperat 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 +these phases will dictate the relevant transition temperatures and pressures for the 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 +unique in 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 @@ -112,8 +105,8 @@ that are 0.07 g/cm$^3$ less dense on average than ice \begin{figure} \includegraphics[width=\linewidth]{unitCell.eps} -\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-$i^\prime$, the -elongated variant of Ice-{\it i}. The spheres represent the +\caption{Unit cells for (A) Ice-{\it i} and (B) Ice-{\it i}$^\prime$, +the elongated variant of Ice-{\it i}. The spheres represent the center-of-mass locations of the water molecules. The $a$ to $c$ ratios for Ice-{\it i} and Ice-{\it i}$^\prime$ are given by $a:2.1214c$ and $a:1.7850c$ respectively.} @@ -131,12 +124,12 @@ investigated (for discussions on these single point di 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 -our previous work and related +had investigated (for discussions on these single point dipole models, +see our 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 +contributions to the overall free energy. To address this issue, we +have calculated the absolute free energy of this crystal 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 @@ -145,11 +138,16 @@ be noted that a second version of Ice-{\it i} (Ice-$i^ from which Ice-{\it i} was crystallized (SSD/E) in addition to several common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction field parametrized single point dipole water model (SSD/RF). It should -be noted that a second version of Ice-{\it i} (Ice-$i^\prime$) was used -in calculations involving SPC/E, TIP4P, and TIP5P. The unit cell of -this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it i} unit -it is extended in the direction of the (001) face and compressed along -the other two faces. +be noted that a second version of Ice-{\it i} (Ice-{\it i}$^\prime$) +was used in calculations involving SPC/E, TIP4P, and TIP5P. The unit +cell of this crystal (Fig. \ref{iceiCell}B) is similar to the Ice-{\it +i} unit it is extended in the direction of the (001) face and +compressed along the other two faces. There is typically a small +distortion of proton ordered Ice-{\it i}$^\prime$ that converts the +normally square tetramer into a rhombus with alternating approximately +85 and 95 degree angles. The degree of this distortion is model +dependent and significant enough to split the tetramer diagonal +location peak in the radial distribution function. \section{Methods} @@ -168,16 +166,16 @@ this same technique in order to determine melting poin crystals at 200 K using the TIP3P, TIP4P, TIP5P, SPC/E, SSD/RF, and SSD/E water models.\cite{Baez95a} 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. +this same technique in order to determine melting points and to +generate phase diagrams. All simulations were carried out at densities +which correspond to 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: +Thermodynamic integration involves a sequence of simulations during +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, @@ -239,11 +237,12 @@ molecules. In this study, we apply of one of the most literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods typically differ in regard to the path taken for switching off the interaction potential to convert the system to an ideal gas of water -molecules. In this study, we apply of one of the most convenient -methods and integrate over the $\lambda^4$ path, where all interaction -parameters are scaled equally by this transformation parameter. This -method has been shown to be reversible and provide results in -excellent agreement with other established methods.\cite{Baez95b} +molecules. In this study, we applied of one of the most convenient +methods and integrated over the $\lambda^4$ path, where all +interaction parameters are scaled equally by this transformation +parameter. This method has been shown to be reversible and provide +results in excellent agreement with other established +methods.\cite{Baez95b} Charge, dipole, and Lennard-Jones interactions were modified by a cubic switching between 100\% and 85\% of the cutoff value (9 \AA @@ -256,17 +255,17 @@ performed with longer cutoffs of 12 and 15 \AA\ to com 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. +\AA\ cutoff results. Finally, the effects of utilizing an Ewald +summation were estimated for TIP3P and SPC/E by performing single +configuration 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 +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 @@ -276,7 +275,7 @@ antiferroelectric version that we divised, and it has $I_h$, 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 we divised, and it has an 8 molecule +antiferroelectric version that we devised, and it has an 8 molecule unit cell similar to other predicted antiferroelectric $I_h$ crystals.\cite{Davidson84} The crystals contained 648 or 1728 molecules for ice B, 1024 or 1280 molecules for ice $I_h$, 1000 @@ -311,18 +310,18 @@ models studied. With the free energy at these state po 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 +models studied. With the calculated 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 +although the crossing points between the phases move to 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} +TIP4P at higher pressures than those studied here.\cite{Sanz04} \begin{figure} \includegraphics[width=\linewidth]{tp3PhaseDia.eps} @@ -385,11 +384,11 @@ conditions. While troubling, this behavior turned out 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 +conditions. While troubling, this behavior resulted in spontaneous +crystallization of Ice-{\it i} and 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 the -risk of spontaneous crystallization. However, this risk changes when +risk of spontaneous crystallization. However, this risk lessens when applying a longer cutoff. \begin{figure} @@ -423,21 +422,21 @@ energy of identical crystals with and without PME usin 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}. The same trend -pointed out through increase of cutoff radius is observed in these PME +energy of identical crystals both with and without PME. The free +energies for the investigated polymorphs using the TIP3P and SPC/E +water models are shown in Table \ref{pmeShift}. 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, the narrowing of the free energy differences between the various solid forms is significant enough that it becomes less clear that it is the most -stable polymorph in the SPC/E. The free energies of Ice-{\it i} and -ice B nearly overlap within error, with ice $I_c$ just outside as -well, indicating that Ice-{\it i} might be metastable with respect to -ice B and possibly ice $I_c$ in the SPC/E water model. 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. +stable polymorph with the SPC/E model. The free energies of Ice-{\it +i} and ice B nearly overlap within error, with ice $I_c$ just outside +as well, indicating that Ice-{\it i} might be metastable with respect +to ice B and possibly ice $I_c$ with SPC/E. 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} @@ -461,15 +460,15 @@ $I$, ice B, and recently discovered Ice-{\it i} were c \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 +$I$, ice B, and our recently discovered Ice-{\it i} structure 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 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 @@ -477,8 +476,8 @@ Due to this relative stability of Ice-{\it i} in all m 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 +Due to this relative stability of Ice-{\it i} in all of the +investigated simulation conditions, 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 @@ -490,32 +489,30 @@ and the structure factor ($S(\vec{q})$ for ice $I_c$ a 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. +and the structure factor ($S(\vec{q})$ for ice $I_h$, $I_c$, and for +ice-{\it i} at a temperature of 77K. In studies of the high and low +density forms of amorphous ice, ``spurious'' diffraction peaks have +been observed experimentally.\cite{Bizid87} It is possible that a +variant of Ice-{\it i} could explain some of this behavior; however, +we will leave it to our experimental colleagues to make the final +determination on whether this ice polymorph is named appropriately +(i.e. with an imaginary number) or if it can be promoted to Ice-0. \begin{figure} \includegraphics[width=\linewidth]{iceGofr.eps} -\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.} +\caption{Radial distribution functions of ice $I_h$, $I_c$, +Ice-{\it i}, and Ice-{\it i}$^\prime$ calculated from from simulations +of the SSD/RF water model at 77 K.} \label{fig:gofr} \end{figure} \begin{figure} \includegraphics[width=\linewidth]{sofq.eps} -\caption{Predicted structure factors for Ice-{\it i} and ice $I_c$ at -77 K. The raw structure factors have been convoluted with a gaussian -instrument function (0.075 \AA$^{-1}$ width) to compensate for the -trunction effects in our finite size simulations. The labeled peaks -compared favorably with ``spurious'' peaks observed in experimental -studies of amorphous solid water.\cite{Bizid87}} +\caption{Predicted structure factors for ice $I_h$, $I_c$, Ice-{\it i}, + and Ice-{\it i}$^\prime$ at 77 K. The raw structure factors have + been convoluted with a gaussian instrument function (0.075 \AA$^{-1}$ + width) to compensate for the trunction effects in our finite size + simulations.} \label{fig:sofq} \end{figure}