Atypical water lattices and their possible relevance to

Atypical water lattices and their possible relevance to the amorphous ices: A density functional study David J. Anick Citation: AIP Advances 3, 042119 (2013); doi: 10.1063/1.4802877 View online: http://dx.doi.org/10.1063/1.4802877 View Table of Contents: http://aipadvances.aip.org/resource/1/AAIDBI/v3/i4 Published by the AIP Publishing LLC.

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AIP ADVANCES 3, 042119 (2013)

Atypical water lattices and their possible relevance to the amorphous ices: A density functional study David J. Anick Laboratory for Water and Surface Studies, Department of Chemistry, Pearson Lab, Tufts University, Medford, MA 02155, USA (Received 25 August 2012; accepted 10 April 2013; published online 18 April 2013)

Of the fifteen known crystalline forms of ice, eleven consist of a single topologically connected hydrogen bond network with four H-bonds at every O. The other four, Ices VI–VIII and XV, consist of two topologically connected networks, each with four H-bonds at every O. The networks interpenetrate but do not share H-bonds. This article presents two new periodic water lattice families whose topological connectivity is “atypical”: they consist of many two-dimensional layers that share no H-bonds. Layers are held together only by dispersion forces. Within each layer there are still four H-bonds at each O. Called “Hexagonal Bilayer Water” (HBW) and “Pleated Sheet Water” (PSW), they have computed densities of about 1.1 g/mL and 1.3 g/mL respectively, and nearest neighbor O-coordination is 4.5 to 5.5 and 6 to 8 respectively. Using density functional theory (BLYP-D/TZVP), various proton ordered forms of HBW and PSW are optimized and categorized. There are simple pathways connecting Ice-Ih to HBW and HBW to PSW. Their computed properties suggest similarities to the high density and very high density amorphous ices (HDA and VHDA) respectively. It is unknown whether HDA, VHDA, and Low Density Amorphous Ice (LDA) are fully disordered glasses down to the molecular level, or whether there is some short-range local order. Based on estimated radial distribution functions (RDFs), one proton ordered form of HBW matches HDA best. The idea is explored that HDA could contain islands with this underlying structure, and likewise, that VHDA could contain regions of PSW. A “microlattice model version 1” (MLM1) is presented as a device to compare key experimental data on the amorphous ices with these atypical structures and with a microlattice form of Ice-XI for LDA. Resemblances are found with the amorphs’ RDFs, densities, Raman spectra, and transition behaviors. There is not enough information in the static models to assign either a microlattice structure C 2013 Author(s). All or a partial microlattice structure to any amorphous ice phase. article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4802877]

I. INTRODUCTION

Water is one of the most versatile materials known, with fifteen known crystalline phases.1 Water’s versatility comes from its well-known property of being held together by hydrogen bonds (H-bonds), with four H-bonds at each O for ice phases. Water lattices are modeled as directed graphs.2, 3 In all but four of the crystalline phases, the graphs are connected in the topological sense, i.e. they consist of single connected component. For Ices VI, VII, VIII, and XV, the graphs have two connected components interpenetrating in three-dimensional space, but sharing no H-bonds between components.1, 4–9 H-bonding achieves lowest energy at ambient pressure when the four O-H- -O bonds are straight and are arrayed tetrahedrally, but distorted bonds and O-O-O angles far removed from the ideal tetrahedral angle of 109.47◦ occur in many of the higher-pressure phases. Whether a particular arrangement is “tetrahedral” may be a judgment call, so it is clearest to view “tetrahedrality” as a continuous parameter which varies from 1 for a perfect tetrahedral setup to 0 for four randomly distributed nearest neighbors.10 2158-3226/2013/3(4)/042119/27

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TABLE I. Core properties of LDA, HDA and VHDA that one would like to see explained or predicted by a microlattice model. 1. Radial Distribution Functions: a. LDA, HDA, VHDA have 1st shell coordination (defined by 3.5 Å cutoff) of 4, 5, and 6, respectively. b. For HDA the second shell has a broad peak around 3.8 and for VHDA there is a plateau from 3.4 to ∼4.0. c. LDA has a pronounced 2nd shell signal centered at 4.6 A, consistent with high tetrahedrality. Neither HDA nor VHDA has this feature. 2. HDA has two forms, a higher-energy and denser form uHDA and a lower-energy less dense form eHDA. At ambient pressure there is slow but spontaneous unidirectional conversion of uHDA to eHDA. 3. Density of VHDA, projected back to 1 bar, is 1.26 g/cm3 . HDA separates into uHDA with density ∼1.15 g/cm3 and eHDAp with density ∼1.13 g/cm3 . Density of LDA is 0.94 g/cm3 . 4. Raman spectra: the peaks for LDA, HDA, and VHDA occur at 3113, 3170, and 3200 cm-1 respectively. 5. At 77 K when subjected to 1.1 GPa, Ice-Ih converts to HDA. At 1.0 GPa HDA converts to VHDA. Upon heating, at 1 bar, HDA converts to LDA. At 0.6 GPa, LDA converts to HDA. 6. Halo pattern and absence of Bragg lines, suggesting an amorphous structure.

We present here ab initio calculations revealing two new families of periodic water lattices, to be called “Hexagonal Bilayer Water” (HBW) and “Pleated Sheet Water” (PSW). Both are “atypical” in that they are highly disconnected. They consist of two-dimensional layers where each O has four H-bonds within the layer but shares no H-bonds with other layers. The layers are held together by van der Waals forces. Their tetrahedrality measures are 0.84 and 0.66 respectively. (Some other ices’ tetraherality are: Ice-1h – 0.99, Ice-II – 0.83, Ice XIV – 0.78.) This article provides static ab initio calculations of the lattices’ properties. Data are obtained only for periodic structures, which obliges the studied structures to be proton ordered. We feel this is a good place to begin the study of atypical ices, while recognizing that dynamical studies and studies of proton disordered versions will also be necessary for a complete understanding. Because properties can vary considerably depending upon the choice of proton order, we strove to be comprehensive about examining all proton ordered forms that achieve a certain minimal degree of symmetry. There are no known experimental counterparts of HBW and PSW, but they turn out to have some properties in common with the amorphous ices. The idea that our atypical lattices occur as islands or components in the amorphous ices will be developed later and will be called “microlattice model 1” (MLM1). In 1984 Mihsima11, 12 compressed hexagonal ice to 1.0 GPa at 77 K and discovered High Density Amorphous Ice (HDA), a metastable phase of water with density of ∼1.15 g/mL. HDA can persist at 1 atm at temperatures below ∼120 K but spontaneously converts to another amorphous state, Low Density Amorphous Ice (LDA), if warmed. Nearly two decades later Loerting warmed HDA slowly while keeping the pressure at 1.9 GPa and created Very High Density Amorphous Ice (VHDA),13 a phase which distinguished itself by a higher density than HDA and a different Raman spectrum and radial distribution function (RDF). Whether VHDA and HDA were truly distinct was initially controversial, but at this point it is well accepted that they are two distinct phases.14–17 In addition, HDA can be annealed slowly to a variant with slightly different properties. The variant has a subtly different RDF and is about 2% less dense so has been dubbed “expanded HDA” or eHDA.18, 19 To distinguish it, the original HDA is also called “unannealed HDA”, uHDA. There may be two forms of LDA as well, of which we will only consider the form that comes from decompressing HDA. Loerting et al. have written a masterful summary of the history and properties of ice amorphs,20 which is recommended to any reader unfamiliar with these details. Table I summarizes the principal properties of the amorphous ices. The phases LDA, HDA, and VHDA are called “amorphous” because their Xray diffraction yields a halo pattern rather than sharp discrete lines, and the halo pattern is associated with noncrystalline materials such as polymers.21 It is an unresolved controversy, whether the amorphous ices have essentially no long-range order beyond a few Angstroms as in a classical liquid phase or glass, or whether they are microlattices, i.e. there is local long-range order that might apply over domains of hundreds to thousands of molecules.22 The domains would have random orientations and sizes.


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Mixtures of fully disordered and small ordered regions could also be possible. The presence of an endotherm starting at 136 K for LDA suggestive of a glass transition23 has been cited as supporting the former view,24, 25 but this does not rule out the possibility of microdomains for small enough domains.26–30 One argument against microdomains has been that there is no known phase whose local properties match those of HDA or VHDA, especially, their high coordination oxygen numbers (∼5 and 6 respectively). A “mixture of phases” has been suggested but it is not clear how that could generate the experimental RDF. The transition from Ice-Ih to HDA has the hallmarks of a first order phase transition,12, 31, 32 which also weighs against the idea of multiple types of transitions occurring. Density functional theory (DFT), and the Becke–Lee–Yang–Parr functional33, 34 in particular (BLYP), has been used successfully to model a wide variety of water and ice properties.6–8, 35, 36 Grimme37, 38 recognized that DFT could be greatly improved in general through the inclusion of a term that adds in the dispersion force, and further benchmarking specifically supported the use of dispersion-corrected BLYP, called BLYP-D, both for weakly bonded complexes in general39, 40 and specifically for water.41, 42 Yoo and Xantheas43 found that BLYP-D gave good agreement with experiment for RDF, density, and self-diffusion of bulk water, each representing improvement over a prior study using BLYP. However BLYP-D overestimated the melting point of Ice-Ih at 360 K, implying that when it comes to free energy comparisons, BLYP-D overstabilizes ice-Ih compared with other forms of water. BLYP-D was adopted for this study, keeping in mind that ices with high tetrahedrality would probably be overstabilized but other properties would be modeled reasonably well. One of this article’s two purposes is to introduce the HBW and PSW families. Section III contains their descriptions and calculated properties. These forms of water are predicted to be stable at very low temperature if they could be made. Both families’ local geometries share certain features that are worth noting, and there are simple relationships among PSW, HBW, and Ice-Ih. Section III has a technical mathematical flavor. The supplementary files44 contain crystallographic and xyz coordinates that may assist the reader in working through this material. Section III does not need to be understood in detail if the reader is only interested in the possible connection to amorphous ices. Our second purpose is to invite HBW and PSW into the conversation about the amorphous ices. We do this by developing and discussing “microlattice model 1” (MLM1), which states that HBW is microlattice HDA and PSW is microlattice VHDA. There could be islands of short-range order without being everywhere microlattice, as noted above, so we view MLM1 as a useful device for making the comparison with experiment, rather than as a definitive conjecture about HDA and VHDA. We expect MLM1 to undergo refinement or modification over time, but we need some specific version of it, to start a discussion. Section IV uses Table I to drive the comparison. The MLM1 provides a radically different narrative about amorphous ices than has previously been put forth. While it remains speculative and many variants on it are possible, there is value in condensing the highlights of that narrative into a single paragraph. Italicized portions comprise a hypothesized story that is offered for the purposes of consideration. They do not constitute findings, results, or claims: “HBW and PSW are periodic water lattices consisting of layers that are disconnected in the sense of the H-bond network but are held together tightly by dispersion forces, giving them high density. HBW has coordination (around) 5 because in addition to the four H-bonds there can be close approach by one O from the next bilayer. HBW can be proton ordered in a particular way that supports two forms: a denser but slightly higher energy form that corresponds to uHDA, and a less dense and lower energy form that corresponds to eHDA. The forms differ merely by sliding the bilayers relative to one another without altering any H-bonds, and it is proposed this is what is happening during the annealing of uHDA to eHDA. PSW similarly allows an O on a sheet with 4 H-bonds to nestle close to two O’s from the adjacent sheet, giving it coordination 6. Sheets are anisotropic and can remain intact while sliding in one particular direction relative to one another. There is a simple and elegant bond reordering mechanism that converts HBW to PSW, based on preservation of subunits called Z-chains. It is proposed that this mechanism is in play during the interconversion of VHDA and eHDA. Conversion can occur one sheet at a time, and intermediate or mixed eHDA+VHDA states are stable or metastable. A similar mechanism connects HBW with


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Ice-Ih, but the transition barrier is higher and the intermediate states would be unstable. The end result of this mechanism going from eHDA would be proton ordered microdomain Ice-Ih. The specific proton order would coincide with Ice-XI so it is proposed that LDA has microdomains of Ice-XI. The idea that LDA starts out with the correct proton ordering to become HDA may be part of why LDA more easily (re)converts to HDA compared with starting from (proton disordered) Ice-Ih. Lastly, halo patterns could result from (possibly quite frequent) defects causing misalignment between layers, or from large Debye-Waller factors (DWFs) due to unusually small restorative forces in a dispersion-bonded lattice.” II. METHODS

All reported calculations were done using CRYSTAL0945 running on a PQS Quantum Cube.46 DFT calculations were done using the Becke–Lee–Yang–Parr functional33, 34 with the Grimme dispersion correction37 and the triple-zeta Ahlrichs basis set (BLYP-D/ TZVP).47 A representative input deck is included in the Supplementary Files.44 Crystallographic coordinates for all reported optimized lattices are included in the Supplemental Files.44 For selected lattices, XYZ coordinates are also provided. HBW and PSW lattices are described using axes called North/South, East/West, and Up/Down. We do this in order to allow easy relating of the various proton-ordered forms to one another. Use of these axes is obviously non-standard, since crystallography has specific conventions defining ‘a’, ‘b’, and ‘c’ directions for any lattice. The relationship of our axes to ‘a’, ‘b’, and ‘c’ differs from lattice to lattice and can be inferred from the crystallographic coordinates. The same issue arises with proton ordered forms of Ice-Ih: the ‘c’ direction that is so well known for Ice-Ih is not the crystallographic ‘c’ for some proton ordered variants of Ice-Ih2 . All Ice-Ih variants have a “basal plane” but “basal” may or may not coincide with “a-b plane”. All reported optimizations included optimization of cell parameters and computation of frequencies. CRYSTAL09 only computes “cellular” vibrational modes, i.e. a Hessian is computed numerically for 3N degrees of freedom if there are N atoms in a primitive cell. The harmonic approximation is then used for frequencies. CRYSTAL09’s zero-point energy is based on these 3N modes, so long-range intercellular phonon modes are omitted, and for clarity we will use the notation “cZPE”. How valid is the cZPE as a correction term can be controversial, but the calculation does verify a local minimum or a transition state, and provides approximate vibrational frequencies for O-H stretches, H-O-H bends, and librations. We have included the cZPE correction but our tables also report the uncorrected cellular electronic energies. “Cohesion energy” listed in the tables is the difference between cZPE-corrected molar electronic energy and the ZPE-corrected energy of gas phase monomeric H2 O (−76.422665 a.u.). III. RESULTS AND DISCUSSION: ATYPICAL ICES AND THEIR PROPERTIES

We begin by introducing a construct called a “Z-chain”, which is a building block of HBW and PSW as well as Ice-VIII and Ice-XI. The first eight of Hirsch and Ojamäe’s 16 proton ordered forms of Ice-Ih2 also consist of Z-chains. We go next to PSW, because it is the simpler lattice to describe from a mathematical and crystallographic standpoint, and then to HBW. We glance briefly at a pair of motifs that characterize local geometry for non-H-bonded nearest neighbors. This is followed by a description of HBW’s relationships with both PSW and Ice-Ih/XI. As an aid to the reader, each subsection of Section III is concluded by a short paragraph in italics highlighting its most essential points. A. Z-chains

A “Z-chain” is a homodromic linear chain of H-bonded water molecules in which the O positions lie in a plane and alternate back and forth across an axis. “Z” is for “zig-zag” or for the shape of the letter zee. See Figure 1. Since Z-chains will occur in the context of 3-D structures, each O of a Z-chain will be donor to the next O in its chain as well as to one O external to the chain.


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FIG. 1. Single pleated sheet illustrating the concepts of Z-chains, ridges, and valleys. Z-chains donating ‘E’ have their oxygens colored red (except O*). Z-chains donating ‘W’ have their oxygens colored blue. Sheet notation: an index O* is colored yellow. Directions are chosen so that O* is on a ridge and donates N and E. The O that is directly ‘E’ from O* also donates N [as well as E] and the O that is directly ‘N’ from O* donates W [as well as N]. Notation for this sheet is NE/N/W.

Likewise it serves as the acceptor for one H-bond within the chain as well as (typically) for one H-bond that originates external to the chain. The directions of these bonds can vary but they will run approximately perpendicular to the axis, and the HOH angle for the two donor H’s will normally be 105◦ +/− 3◦ . The O—O—O angles are consistent for a given Z-chain and will typically be between 100◦ and 125◦ . In our examples a Z-chain will always have translation invariance when moved two O positions along the axis, and that includes the directions of the external H-bonds. A Z-chain may or may not have further symmetry that makes all the O’s equivalent. Obviously this is a description of a mathematical abstraction: in the real world Z-chains of H2 O’s can and will wiggle somewhat from this ideal and cannot be infinitely long. “Z-chains” (e.g. the blue or red chains in Figure 1) are an abstraction that will aid in the defining, describing, classification, and naming of our atypical lattices. Z-chains do not necessarily have any inherent chemical significance. B. Pleated sheet water – description

Imagine a Z-chain whose axis runs horizontally across a page, with the O’s alternately above and below the plane of the page. The plane of the Z-chain is perpendicular to the plane of the page. Now imagine making copies of the Z-chain and translating them uniformly up and down the page, so that the above-page O’s and the below-page O’s fall along alternating vertical lines. The resulting lattice is a single “pleated sheet,” with the pleats running vertically (Figure 1). It is two-dimensional in that it is a single layer but its 3-dimensionality is an essential feature of it. Pleats lie alternately above the page (“ridges”) and below the page (“valleys”). Pleated sheets are anisotropic. As a convention, the direction of the pleats is called “North/South” (N/S) and the direction of the component Z-chains’ axes is called “East/West” (E/W). The idea is to be able to refer to these directions as defined relative to the pleated sheet, independently of how the sheet embeds in a larger entity. The topological connectivity of a single sheet is the same as that of “square water”, an abstract model defined by Singer et al. (p. 18 of Ref. 3), but the anisotropy means that their discussion of square water’s symmetries would have to be modified to apply here. Certain “ice rules” severely restrict the possibilities for H-bond directions on a pleated sheet. In “square water” an O would be allowed to donate to both N and S, but for our pleated sheets the result would be an intolerable HOH angle near 180◦ and this is disallowed. Donating E and W from the same O might be OK from the viewpoint of that one O because O—O—O angles in the


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TABLE II. Single pleated sheet geometries and energies. Pleated Sheet

2D sp. group

O-O*-O latticea

Prim. cell #

NS O-O

EW O-O

Z-chain O-O-O

E0 / a.u.

ZPE / a.u.

Ecohb

NE/N/E NE/N/W NE/S/E NE/S/W

P11a P21 ab P21 11 P2/b11

87.86c 94.44 93.07 97.86

2 4 2 4

2.801 2.778 2.799 2.776

2.731 2.742 2.728 2.740

106.66 106.34 107.39 106.82

−152.934474 −305.870027 −152.934374 −305.869982

0.049883 0.100610 0.049817 0.100425

−12.32 −12.35 −12.31 −12.38

a Angle

in degrees, at an index O* between the two O*-O bonds in which O* is donor. energy in kcal/mol, the difference between ZPE-corrected E0 of lattice and gas phase H2 O. c Angle between NS and EW axes for NE/N/E is 87.34◦ ; for the other lattices, it is 90◦ . b Coherence

Z-chains are near 105◦ , but in order to achieve periodicity there would have to be an O somewhere else that accepts from both E and W, forcing that O’s donors to go to N and S, which as we have seen is disallowed. Thus “rules” analogous to the Bernal-Fowler ice rules apply to pleated sheets which force each O to have one E or W donor, one N or S donor, one W or E acceptor (whichever the donor is not), and one S or N acceptor (whichever the donor is not). Notation for an H2 O that donates to N and E is ‘NE’, and likewise for NW, SE, SW. Because “NS” and “EW’ cannot occur, as soon as one H-bond in either a vertical or horizontal line is assigned a direction, the entire line inherits that direction. A list of horizontal directions (e.g. E,W,E,E,W,. . . ) and vertical directions (S,S,S,N,S,. . . ) defines the sheet. For periodic sheets of period 1 or 2 in each direction this reduces the number of possibilities to just four. The notation NE/S/W means that when the directions are assigned so that an index O, call it O*, is ‘above’ (i.e. on a ridge) and ‘NE’, then H-bonding in the valleys just east and west of O* runs “S” and the Z-chains just north and south of O* are directed ‘W’. The four possible sheets where a single H2 O and the space group generate the entire sheet are NE/N/E, NE/N/W, NE/S/E, and NE/S/W. Our focus will be on these four “high symmetry” sheets. Optimization of single sheets as “slabs” finds that the O-O-O angles that contain H-O-H vary with the sheet, from 87.9◦ to 97.9◦ (Table II), making them into lattices of parallelograms. For NE/N/E the angle between the N/S axis and the E/W axis is 87.3◦ but for the other sheets symmetry forces the axis angle to be 90◦ . The unit cell of the lattice consists of either two water molecules (NE/N/E and NE/S/E) or four (NE/N/W and NE/S/W). Figure 1 illustrates NE/N/W. “Pleated Sheet Water” (PSW) is made by stacking sheets so that the pleats line up in the third direction, which will be called Up/Down (U/D). High density is achieved because the pleats “nestle” or “spoon”. Ridges stack over ridges and valleys stack over valleys. See Figure 2 for an example. The question of what determines the equilibrium distance between sheets, and how an O-H bond in one sheet aligns with O-H in an adjacent sheet, is important for a full understanding of PSW. The sheets of PSW are somewhat free to slide N/S relative to one another, but if they try to move E/W then valley crashes with ridge. The “slide PES” (SPES) is the 1-D relaxed potential energy surface in which the scan variable is the intersheet N/S displacement. Because the sheets being considered here are periodic and repeat in the N/S direction every one or two H-bonds, the SPES is a periodic function. Figures 3(a), 3(b) depict two local minima on the same SPES. Note the coordinate frames provided: Figure 2 is viewed from S, whereas Figures 3(a), 3(a) are viewed from W. To render manageable the large number of possible PSW arrangements, we focused on 3-D PSW lattices in which the entire lattice is generated by the space group and one water molecule. In these lattices the sheets are all equivalent and periodicity in the U/D direction is either 1 or 2. The lattice is defined by the bonding of any one sheet (e.g. NE/S/E) and the bonding of the closest O, call it O**, that belongs to the ridge immediately above the index O*. The notation used, e.g., NE/S/E/SW, means that one sheet containing a NE index O* on a ridge is NE/S/E and in the next sheet, the ridge O** that is closest to O*, is SW. Unit cells consist of 2, 4, or 8 H2 O’s. The sixteen NE/(N,S)/(E,W)/(N,S)(E,W) lattices were optimized. Results are reported in Table III. In two cases, symmetry had to be reduced (lattice requires two H2 O generators) to


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FIG. 2. NE/S/E/SW, an example of a PSW Category 2 lattice. See Supplementary Files44 for xyz coordinates. This view highlights nestling of the ridges and valleys.

find local minima. For NE/N/W/NE and NE/N/W/NW, these two local minima lie on the same SPES and are identical structures via a global symmetry operation that converts ridges to valleys. The same applies to NE/S/W/SE and NE/S/W/SW: they lie on the same SPES and are equivalent. Both examples in the prior paragraph invoke a phenomenon that deserves further clarification. If two sheets of NE/N/W/NE are slid North or South relative to each other by one H-bond, then it switches from the O** overlying O* being an NE to the new closest O of that ridge being an NW. According to how the notation is defined, the lattice has become NE/N/W/NW. More generally, whenever a 3-D PSW structure has sheets with both E- and W-oriented Z-chains, i.e. NE/x/W/yz, then NE/x/W/yE and NE/x/W/yW are accessible from each other by sliding the sheets N or S and hence their local minima represent points on the same SPES (cf. Figure 3). Tables III and IV includes the symmetry group used for each optimization; the “beta” angle for lattices that are monoclinic; the density; the electronic energy and ZPE per primitive cell; and relative ZPE-corrected energy per mol H2 O. Also listed are the O—O distances for the N/S and the E/W directions and the O-O-O angle within the Z-chains. Lastly, Table IV contains the distances from any O to its five nearest off-sheet neighbors. An understanding of these neighbors provides insight into PSW geometries but is not essential, so it is relegated to the Appendix. “Pleated sheets” are 2-D layers of 4-coordinated H2 O’s consisting of ridges and valleys in one direction and Z-chains in the other (see Figure 1). “Ice rules” apply that force H-bonding to be homodromic in both directions. Sheets can be stacked to make “Pleated Sheet Water” (PSW) as in Figure 2. High density and high O-coordination occur because the ridges of one sheet abut


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FIG. 3. Illustration of PSW Category 1 monoclinic lattices. (a) and (b) are local minima on the same SPES, using the same monoclinic lattice but with different “beta” angles. Figure 3(a) is the MLM1 microlattice component structure for VHDA. See the Supplementary Files44 for xyz coordinates. For clarity E-going Z-chains’ O’s are red and W-going Z-chains’ O’s are blue. (a) NE/S/W/NE. (b) NE/S/W/NW.

the valleys of the next sheet. A one-dimensional “slide PES” (SPES) characterizes a relatively low-barrier motion that keeps all H-bonds intact but moves sheets relative to each other along the ridges (see Figure 3). C. Pleated sheet water – findings

All sixteen (fourteen distinct) local minima geometries of Tables III and IV have certain features in common. The N- or S-going H-bonds have lengths in the range 2.799 to 2.843 and the E- or Wgoing H-bonds have lengths in the range 2.720 to 2.732. The O—O—O angles of the Z-chains range from 105.4◦ to 109.7◦ . Our PSW geometries fall into two categories. For a PSW structure with notation NE/x/v/yz, Category 1 consists of those where ‘x’ and ‘y’ are different (one N and one S) while Category 2 consists of those where ‘x’ and ‘y’ are the same (both N or both S). Category 1 is characterized by density in the range 1.28 to 1.38 and coordination at each O is 6. Category 2 is characterized


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TABLE III. PSW lattice space groups and energies. PSW lattice

3-D sp. group

beta (if monocl.)

category

Prim. Cell #

E0 / a.u.

ZPE / a.u.

Ecohd

NE/N/E/NE NE/N/E/NW NE/N/E/SE NE/N/E/SW

Cc Pna21 Pca21 P21 /c

94.24a

2 2 1 1

4 4 4 4

−305.878015 −305.877835 −305.876907 −305.877047

0.099907 0.100110 0.099759 0.099821

−13.72 −13.66 −13.57 −13.58

NE/N/W/NE NE/N/W/SE NE/N/W/SW

Pca21 Pbca Pbca

2 1 1

8c 8 8

−611.756498 −611.755590 −611.754597

0.200815 0.201259 0.200922

−13.68 −13.57 −13.52

NE/S/E/NE NE/S/E/NW NE/S/E/SE NE/S/E/SW

P21 P21 /c Pna21 P21 21 21

107.70b 112.64b

1 1 2 2

2 4 4 4

−152.938784 −305.876298 −305.878025 −305.878103

0.050019 0.099725 0.099475 0.099800

−13.63 −13.48 −13.79 −13.75

NE/S/W/NE NE/S/W/NW NE/S/W/SE

P21 /c P21 /c Pca21

73.34b 119.10b

1 1 2

4 4 8c

−305.877977 −305.876767 −611.756871

0.099998 0.099935 0.200839

−13.70 −13.52 −13.70

87.75a

a ‘beta’

is angle between NS and EW axes. is angle between NS and UD axes. c Two H O generators. The geometry for NE/N/W/NW is identical to NE/N/W/NE, and NE/S/W/SW is identical to NE/S/W/SE 2 (see text). d Coherence energy in kcal/mol. b ‘beta’

TABLE IV. PSW lattice densities and geometries. PSW lattice

category

Dens./g/mL

NS O-O

EW O-O

Z-chain O-O-O

OUE1

Distance / Å from O* to OUW1 OUE2 OUW2

NE/N/E/NE NE/N/E/NW NE/N/E/SE NE/N/E/SW

2 2 1 1

1.38 1.38 1.32 1.34

2.837 2.839 2.818 2.821

2.725 2.726 2.726 2.727

107.50 107.86 105.43 108.52

3.14 3.35 3.09 3.25

3.14 3.09 3.09 2.96

3.29 3.35 3.55 3.50

3.29 3.09 3.55 3.61

3.76 3.76 3.68 3.69

NE/N/W/NE NE/N/W/SE NE/N/W/SW

2 1 1

1.38 1.35 1.34

2.820 2.799 2.805

2.725 2.732 2.725

107.58 108.26 108.08

3.16 3.09 3.11

3.26 3.09 3.08

3.19 3.65 3.56

3.27 3.65 3.56

3.76 3.69 3.70

NE/S/E/NE NE/S/E/NW NE/S/E/SE NE/S/E/SW

1 1 2 2

1.36 1.39 1.41 1.40

2.805 2.839 2.836 2.843

2.724 2.724 2.720 2.721

109.46 109.67 109.34 109.30

3.08 3.21 3.20 3.32

3.08 2.98 3.20 3.10

3.55 3.62 3.20 3.32

3.55 3.42 3.20 3.10

3.66 3.68 3.61 3.63

NE/S/W/NE NE/S/W/NW NE/S/W/SE

1 1 2

1.35 1.28 1.40

2.802 2.817 2.821

2.728 2.726 2.725

108.57 105.81 109.58

3.11 3.17 3.17

3.11 3.11 3.21

3.65 3.77 3.20

3.42 3.77 3.27

3.73 3.88 3.62

O**

by density in the range 1.38 to 1.41 and coordination at each O is 8. Category 1 structures have on average higher energies than Category 2 structures. An explanation for the split into two categories is given in the Appendix. We located the transition state on the SPES between NE/S/W/NE and NE/S/W/NW, by scanning on the beta angle and optimizing the other cell parameters and nuclear positions: it occurs around β = 97◦ . The transition state has density 1.29 g/mL and its cZPE-corrected E0 lies 0.39 kcal/mol H2 O higher than NE/S/W/NW. Optimization by BLYP-D identifies two “categories” of PSW that differ in their energy and density. The directions of the H-bonds in the ridges and valleys underlie the differences between Category 1 and Category 2. The lowest-energy Category 1 lattice is the monoclinic lattice shown in Figure 3(a), denoted “NE/S/W/NE.”


042119-10

David J. Anick

AIP Advances 3, 042119 (2013)

TABLE V. HBW lattice space groups and energies. HBW lattice

3-D sp. group

beta (if monocl.)

Dens. / g/mL

Prim. Cell #

Ice-Ihd

E0 / a.u.

ZPE / a.u.

Ecohe

EJ/JEE EJ/JEWa EJ/JEWb EJ/JWEa EJ/JWEb EJ/JWW

Pnc2 P21 /c P21 /c P21 /c P21 /c Pnc2

74.2a 103.4a 107.5a 78.5a

1.02 1.10 1.10 1.10 1.09 1.04

8 8 8 8 8 8

3 6 6 7 7 2

−611.759941 −611.763931 −611.763446 −611.762950 −611.764099 −611.761018

0.202815 0.202849 0.202888 0.202643 0.202827 0.202725

−13.79 −14.10 −14.06 −14.04 −14.11 −13.88

EJ/FEEa EJ/FEEb EJ/FEWa EJ/FEWb EJ/FWEa EJ/FWEb EJ/FWWa EJ/FWWb

C2 C2 P21 /c P21 /c C2/m C2/m P2/c P2/c

73.7b 104.8b 72.8b 109.2b 70.8c 100.1c 74.0b 104.9b

1.03 1.04 1.07 1.11 1.12 1.08 1.06 1.09

4 4 8 8 4 4 8 8

8 8 4 4 1 1 5 5

−305.879857 −305.880518 −611.763005 −611.764466 −305.881981 −305.882432 −611.762177 −611.763626

0.100272 0.100194 0.203226 0.202932 0.101250 0.101461 0.202806 0.202717

−13.95 −14.07 −14.00 −14.14 −14.13 −14.17 −13.97 −14.09

(trigonal) (trigonal)

P3 P31m

1.01 1.03

12 12

−917.641418 −917.641981

0.306016 0.304789

−13.77 −13.87

a ‘beta’

is angle between NS and EW axes. is angle between NS and UD axes. c ‘beta’ is angle between N# S# and U# D# axes. d Corresponding proton ordered Ice-Ih lattice, according to the numbering system of Ref. 48. e Coherence energy in kcal/mol. b ‘beta’

D. Hexagonal bilayer water — description

A single bilayer of HBW consists of two aligned planar hexagonal lattices with O’s at the vertices and H-bonds along the edges, with H-bonds connecting each O to the corresponding O of the other plane. See Figures 5 and 6(a). The idea of bilayers or parallel planar arrangements of water occurs among finite clusters in the (H2 O)8 cube and stacked cubes, the (H2 O)10 pentagonal prism, the (H2 O)12 hexagonal prism, and the putative global minimum (H2 O)20 consisting of three fused pentagonal prisms.48–53 Directions are assigned as follows. “Up/ Down” and “East/West” lie in the planes of the hexagons, with U/D being parallel to one pair of sides of the hexagons, and E/W running orthogonal to U/D. “North/ South” is the perpendicular direction between planes. A bilayer has a north and a south face. Each face of a bilayer is a 3-coordinated directed graph. The principles of polyhedral water clusters apply.54 To maintain neutrality the number of N/S bonds that are directed north must balance the number directed south. Half of the H2 O’s accept an N/S bond and are “facial” i.e. HOH lies entirely within a face, and half donate in an N/S bond and so are “joining”, i.e. one of their OH’s connects to the other face. These two “types” of H2 O units cannot be equivalent under any symmetry operation on the bilayer so there must be at least two H2 O generators (one facial and one joining) for any 2-D or 3-D bilayer lattice. Notice that the ratio of facial to joining H-bonds is 3:1, while the ratio of facial to joining H2 O’s is 1:1. By stacking repeated copies of a bilayer in the N/S direction, we obtain a 3-D lattice that we call HBW. See Figures 6(a)–6(d). Just as PSW has a 1-D slide PES, there is a 2-D SPES for HBW defined by allowing the offset between adjacent bilayers to slide in the E/W and U/D directions. For this project we focused on lattices where the minimum number of two H2 O units do generate the entire lattice. There are two such trigonal lattices, two that are orthorhombic, and six that are monoclinic. The trigonal lattices are distinguished from each other by the presence or absence of a ¯ ¯ Both trigonal lattices reflection symmetry across an NS-UD plane (symmetry group P 31m vs P 3). have significantly lower density and higher energy than the monoclinic lattices (cf. Table V). The eight high-symmetry orthorhombic or monoclinic HBW lattices are composed of Z-chains whose axes run in the E/W direction. The plane of each Z-chain coincides with either a north or


042119-11

David J. Anick

AIP Advances 3, 042119 (2013)

TABLE VI. Geometries for monoclinic and orthorhombic HBW lattices. HBW lattice

UD O-O

EW O-O

EW O-O

Join O-O

Z-chain O-O-O

Nearest 2 non-HB nbrs for OF and OJ OF -O1 OF -O2 OJ -O1 OJ- O2

EJ/JEE EJ/JEWa EJ/JEWb EJ/JWEa EJ/JWEb EJ/JWW

2.733 2.737 2.725 2.728 2.726 2.733

2.762 2.715 2.715 2.778 2.770 2.703

2.762 2.715 2.715 2.778 2.770 2.703

2.758 2.759 2.764 2.750 2.756 2.755

118.56 122.01 123.92 117.90 120.65 123.81

3.298 3.211 3.189 3.081 3.182 3.184

3.824 3.239 3.297 3.498 3.427 3.810

3.298 3.211 3.189 3.081 3.182 3.184

3.877 3.239 3.297 3.498 3.427 3.864

5.0 6.0 6.0 5.0 5.0 5.0

EJ/FEEa EJ/FEEb EJ/FEWa EJ/FEWb EJ/FWEa EJ/FWEb EJ/FWWa EJ/FWWb

2.831 2.823 2.742 2.736 2.738 2.740 2.738 2.738

2.797 2.785 2.729 2.731 2.739 2.748 2.724 2.730

2.783 2.782 2.735 2.730 2.738 2.740 2.730 2.722

2.763 2.778 2.761 2.778 2.757 2.747 2.756 2.763

114.97 115.75 120.08 120.00 123.48 121.35 123.03 122.82

3.170 3.214 3.138 3.372 3.465 3.069 3.211 3.243

3.407 3.653 3.532 3.591 3.536 3.453 3.536 3.673

3.407 3.214 3.532 3.104 2.964 3.069 3.536 3.243

3.717 3.317 3.855 3.372 3.465 3.798 3.813 3.307

4.5 5.5 4.5 5.5 4.5 5.0 4.5 5.5

a Average

coorda

O coordination using cutoff of 3.4 Å.

a south face, so along a Z-chain the constituent O’s alternate being above (“upper O’s”) or below (“lower O’s”) the Z-chain axis. Two adjacent Z-chains together comprise a row of hexagons. Without loss of generality it is possible to assign directions so that the North face has an Eastward directed Z-chain on which an index O* is of the “Joining” type. With this convention, the three binary parameters which specify a lattice are: (1) whether the O’s next to O* on the index Z-chain are facial (F) or joining (J); (2) whether the corresponding Z-chain on the south face goes E or W; and (3) whether the adjacent Z-chains on the north face go E or W. The notation is EJ/(F or J)(E or W)(E or W), e.g. EJ/FWW. See Figure 5. Of these eight lattices, all but EJ/JEE and EJ/JWW are monoclinic (Table V). For each of the six monoclinic lattices we identified two distinct symmetry-preserving local minima, which are reported in Tables V and VI. Where there are two local minima their notations have ‘a’ or ‘b’ appended to distinguish them (cf. Tables V and VI). For EJ/JEW and EJ/JWE, ‘a’ and ‘b’ differ by having the bilayers slide E or W relative to one another. For EJ/FEE, EJ/FEW, and EJ/FWW, ‘a’ and ‘b’ differ by sliding the bilayers U or D. The remaining monoclinic lattice, EJ/FWE, is unlike the others in that it has a plane of reflection symmetry. Its monoclinic angle “beta” lies on this plane. The plane contains the NS axis and is oriented 30◦ off the EW axis (cf. Fig. 6(a)). It is convenient to define rotated axes U# D# and E# W# for this lattice, so that the U# D# -NS plane is the plane of reflection symmetry. Figures 6(b), 6(c) depict the two local minima for EJ/FWE rotated so that the E# W# axis is horizontal. Axis rotation comes with a caveat that the zig-zag lines running E# /W# in Figure 6(a) are not Z-chains, because their H-bonds alternate direction rather than being homodromic. When it comes to identifying the Z-chains in Sections III G and III H, they run E/W, which is “diagonally” in Fig. 6(a). The O’s of one Z-chain are colored blue in Fig. 6(a). If a bilayers stack directly N or S, as in the example of the orthorhombic EJ/JWW (see xyz coordinates), each O will clearly have 5 nearest neighbors: three that lie in the same face, one on the opposite face to which it is “joined”, and the corresponding O that abuts it from the next bilayer. When bilayers slide relative to each other, the picture is more complex. Depending on the particular geometry and the cutoff used, an O might have 0, 1, or 2 neighbors in the abutting bilayer. Unlike PSW where all H2 O units experience equivalent environments, HBW has two non-equivalent types of H2 O’s, facial (F) and joining (J). For these high-symmetry lattices, coordination at all F H2 O’s will be the same and coordination at all J H2 O’s will be the same, but F and J coordination need not be the same as each other. The lattice RDF is half the sum of the “J-RDF” and the “F-RDF”. The predicted average coordination for a lattice will be the average of the Fs’ and the Js’ coordination numbers,


042119-12

David J. Anick

AIP Advances 3, 042119 (2013)

so it can be a half integer. In an actual lattice, both zero-point motion and thermal motion can blur nuclear positions, so the coordination numbers in Table V should be viewed as approximations only. An “hexagonal bilayer” is illustrated in Figure 5. The H2 O’s of a bilayer are either “facial” and lie entirely in one of the planes, or they donate an H-bond to the other plane and are called “joining”. “Hexagonal Bilayer Water” (HBW) is a 3-D lattice obtained by stacking bilayers (see Figure 6). A two-dimensional SPES describes motions that keep all H-bonds intact while sliding the bilayers relative to each other. E. Hexagonal bilayer water — findings

Sixteen HBW local minimum (no imaginary frequencies) lattices are described in Table V: two trigonal, two orthorhombic, and two minima each for the six monoclinic lattices. Tables V and VI list for each the space group; three O—O facial H-bond lengths and the joining O—O bond length; the density; distances to the next two nearest neighbor O’s for facial and for joining O’s; average coordination based on a cutoff of 3.4 A; the electronic energy and cZPE per primitive cell; and the relative cZPE-corrected electronic energy per mol H2 O. Energy in Table V has a strong inverse correlation with density (r = −0.86). In general, setups that allow for closer inter-bilayer bonding also have lower energy. This applies when comparing ‘a’ and ‘b’ versions as well: the denser lattice typically has lower energy, but EJ/JWE and EJ/FWE are interesting exceptions. No clear-cut pattern emerges from these data as it does for the two categories of PSW. Given that EJ/FWEa is denser but higher energy than EJ/FWEb, there will be a positive pressure at which these two structures attain equal enthalpy. We computed E0 +cZPE+PV at various pressures for both lattices, and looked for where their E0 +cZPE+PV curves cross. This occurs around 190 MPa. We located the transition state between EJ/FWEa and EJ/FWEb, by scanning on beta and optimizing cell parameters and nuclear positions: it occurs around β = 84◦ . The transition state has density 1.015 g/mL and lies 0.25 kcal/mol H2 O above EJ/FWEa. HBW lattices evince a range of densities depending on the geometry. Some HBW lattices attain high density and high O-coordination due to close approach of bilayers to each other. Movement along the SPES can also significantly affect the inter-bilayer distance. Highest density occurs for “EJ/FWEa” (Figure 6(b)), a monoclinic lattice, which lies on the same SPES with the lowest energy lattice, “EJ/FWEb” (Figure 6(c)). F. Local geometry of non-H-bonded interactions

By looking at how non-H-bonded neighbors situate themselves relative to H-bonds in these lattices, a consistent pattern emerges. Let O1 be the nearest non-H-bonded neighbor O to an index oxygen denoted O*. In both the PSW and the HBW lattices, we very often find one of the covalent bonds on O*, call it O*-H*, interacting with O1 so that the O*-H*- -O1 angle is 108◦ ± 10◦ and the O*—O1 distance is ∼3.15 ± 0.2 Å. Commonly, but not always, O*-H* is parallel or antiparallel to one covalent bond on O1, call it O1-H1. Then the four atoms O*,H*,O1,H1 lie (approximately) in a plane. As noted in the Appendix, there are two ways of achieving this local geometry. In the Category 1 PSW lattices and in some HBW lattices, O*-H* and O1-H1 are antiparallel, comprising a small parallelogram in an arrangement we call a Category 1 setup (Figure 4(a)). The O*-H*- -O1 and O1-H1- -O* angles are both ∼105◦ . In the Category 2 lattices and in some HBW lattices, O*-H* aligns parallel to the H1- -O2 of an H-bond to a third O denoted O2, with the O1-H1- -O* and O*-H*- -O2 angles being coplanar and both being ∼115◦ . The atoms O*,H*,O2,H1 form a small trapezoid. We call this arrangement a Category 2 setup (Figure 4(b)). An example of a Category 2 setup in HBW can be seen in Figure 5. If O2N and H2N denote the atoms corresponding to O2 and H2 in the next bilayer to the North, the monoclinic offset for EJ/JEWb positions them so that O*-H*- -O2N -H2N - -O1 comprise a Category 2 setup. A Category 1 setup in HBW occurs in Figure 6(b). Joining O’s from adjacent bilayers have their U#- or D#-going covalent bonds line up antiparallel in the planes of reflection symmetry.


042119-13

David J. Anick

AIP Advances 3, 042119 (2013)

FIG. 4. Illustration of local geometries including weakly bonded O*-H*- -O1 motifs. (a) Category 1 bond setup. Blow-up of a neighborhood of an index O* of NE/S/W/NE (Fig. 3(a)) viewed from S toward N, illustrating the O*-H*-O1-H1 parallelogram with some angles and distances. (b) Category 2 bond setup. Blow-up of a neighborhood of an index O* of NE/S/E/SW (Fig. 2) viewed from E, illustrating the H1-O*-H*-O2 trapezoid.


042119-14

David J. Anick

AIP Advances 3, 042119 (2013)

FIG. 5. Illustration of a single hexagonal bilayer. Each Z-chain has either all red (except O*) or all blue O’s. For this lattice, the red O’s are all joining (J) and the blue O’s are all facial (F). Lattice notation: let the index O* (yellow color) be on an E-going Z-chain, on the N face of the bilayer, and joining (J). This lattice is “EJ/JEW” because same-Z-chain neighbor O1 is J and the adjacent chains’ directions given by donors at O2 and O3 are E and W respectively. See Supplementary Files44 for xyz coordinates of the full HBW lattice EJ/JEWa.

FIG. 6. Illustration of HBW lattice EJ/FWE. (a) Single bilayer. Note planes of reflection symmetry. One E-going Z-chain is colored blue and both coordinate frames are depicted. (b) EJ/FWEa, the MLM1 microlattice component structure for uHDA. (c) EJ/FWEb, the MLM1 microlattice component structure for eHDA. (d) Alternate view of EJ/FEWb from along the W-E axis. See Supplementary Files44 for xyz coordinates.


042119-15

David J. Anick

AIP Advances 3, 042119 (2013)

Empirically, these two motifs appear to have a role in determining the bonding between bilayers of HBW and between sheets of PSW. Imagine two abutting bilayers that are seeking a lowest-energy arrangement by shifting U/D or E/W relative to each other. H-bonds of the North face of one bilayer can align with H-bonds of the South face of the other bilayer, lowering the energy when a Category 1 or Category 2 motif occurs. However, a setup that optimizes one pair of interacting H-bonds will not optimize another pair, so the result is a “frustrated system” in which some motifs occur and others are sacrificed. The specific H-bonding patterns at the abutting faces govern the outcome. In general, higher densities (i.e. closer inter-bilayer spacing) correlate with more motifs and, presumably, with motifs that more closely fit an ideal. As a further test of this idea, we computed a portion of the SPES for EJ/FWE as “beta” is moved from ∼71◦ (for EJ/FWEa) to ∼100◦ (for EJ/FWEb). The inter-bilayer spacing achieves a maximum of 6.11 Å at a point near the transition state (∼85◦ ) where the inter-bilayer H-bond alignments are furthest from the above description, versus 5.52 and 5.68 at the local minima. This illustrates that the inter-bilayer distance can change by 0.4 Å or more due to changing how well aligned are the abutting facial H-bonds. In the optimized geometries for both HBW and PSW, the non-H-bonded interaction is most often characterized by a motif consisting of a particular O*-H* - - O1 angle and a particular O*—O1 distance. Pairs of motifs forming either a small parallelogram (“category 1”) or a small trapezoid (“category 2”) are also commonly seen (see Figure 4). This observation is not essential for either defining the lattices or for relating them to the amorphous ices, but it may help explain the SPES properties and the motif might be worth looking for in other non-H-bonded water interactions.

G. Relationship of HBW and PSW

There is a simple operation that converts HBW lattices to category 1 PSW lattices and vice versa. Specifically, EJ/Jyz interconverts with NE/N/y/Sz and EJ/Fyz interconverts with NE/S/y/Nz (y,z = W or E). To illustrate it, refer to Figure 6(b). Overview: we will rotate each of the facial H2 O’s by 90◦ using one of its OH bonds as the axis of rotation. Providing detail, this expands to a four-step process. Step 1: Identify. Each facial H2 O donates to one bond that goes U or D and another bond that goes E or W and is internal to its Z-chain. Identify the U- and D-going bonds. Step 2: Spin. Treating the internal OH of each facial H2 O as a fixed axis of rotation, rotate the external OH 90◦ away from its bilayer: it will now point N or S toward a joining O of the next bilayer. Step 3: New bond. Allowing that H to form a new H-bond with the adjacent bilayer, notice that Z-chains are now disconnected in the U/D direction because all of the U- and D-going bonds have been redirected N or S. Step 4: Relax. As there are no longer any U or D H-bonds, the result is pleated sheets having the standard NS-EW orientation: allow these sheets to shift or relax to their optimum geometry, with ridges over ridges and valleys over valleys. The four steps can be reversed for the PSW-to-HBW conversion. Referring to Figure 3(a), imagine sliding the sheets N or S so that “blue” O’s line up with “blues” in the U/D direction and “red O’s” line up with “reds”. Identify the blue-to-red bonds that lie on ridges, and rotate them upward so they connect with the nearest “blue” from the next sheet. Likewise rotate valley-based red-to-blue bonds toward the nearest “red” from the next sheet below. After relaxing, the result is EJ/FWEb. The periodic transition state between NE/S/W/NE and EJ/FWEb was computed. It is included in the supplementary material44 as “PSW-ts1”. Its single “imaginary” frequency is –76 cm−1 and the corresponding eigenvector is along the above pathway. The t-state lies 0.31 kcal/mol above NE/S/W/NE. The reader may find this lattice a particularly useful one to view and spin in 3-D, because one can readily see how sheets become hexagons and vice versa. Figure 7 gives a schematic representation of interconversion. In Figure 7, lattices are projected orthographically onto the UD-NS plane. Z-chains are “flattened” and are depicted as rectangles, with external H-bonds depicted as arrows. Z-chains’ directions (i.e. E or W) are suppressed from the drawing for simplicity, allowing the same drawing to represent any of four lattices. Briefly, conversion keeps Z-chains intact, while N- or S-going bonds of PSW flip so that they point U or D respectively. In reverse, all U- or D-going bonds of HBW flip so that they point N or S respectively.


042119-16

David J. Anick

AIP Advances 3, 042119 (2013)

FIG. 7. Schematic representation of the interconversion of PSW (NE/S/y/Nz) and HBW (EJ/Fyz).

FIG. 8. Schematic representation of the interconversion of PSW (NE/N/y/Sz) and HBW (EJ/Jyz).

Figure 7 also shows that flipping need not all occur at once, but can occur one sheet at a time, resulting in a continuum of intermediate or mixed states. Mixed states are not the same as the transition state mentioned above. Mixed states consist of a region of PSW abutting a region of HBW across a phase boundary. In mixed states, notice that each O is still ddaa. The single transitional sheet has intact valleys but the ridges are broken open so they can complete the last hexagons of the HBW region. The fit is not perfect becusee PSW Z-chains prefer a smaller O—O—O angle than HBW Z-chains (Tables IV and VI), but the arrangement provides at least metastability. Any density between that of PSW and HBW can be attained. One can easily picture conversion occurring sequentially via propagation of the phase boundary in the U/D direction, sheet by sheet. Computing E0 +cZPE+PV at various pressures, we find that EJ/FWEb and NE/S/W/NE have the same enthalpy at around 550 MPa. Mixed states are expected to be stable at 550 MPa, although this is hard to verify directly because mixed states have different cell parameters above and below the transitional sheet and so cannot be handled as a unified periodic system. Figure 7 is only for NE/S/y/Nz lattices interconverting with EJ/Fyz. Figure 8 shows the equivalent mechanism for interconversion of NE/N/y/Sz with EJ/Jyz. Category 2 PSW lattices do not have such a relationship with HBW, and we believe they are not directly accessible from HBW. A simple facial bond rotation mechanism converts HBW lattices into Category 1 PSW lattices and vice versa. It is also possible to convert part of a lattice and end up with everywhere-4coordinated “intermediate states” consisting of PSW above and HBW below a transitional plane. For the specific case of interconverting the ground state HBW lattice (EJ/FWEb) and the ground


042119-17

David J. Anick

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state Category 1 PSW lattice (NE/S/W/NE), the pressure at which their enthalpies are equal is about 550 MPa. H. Relationship of HBW and Ice-Ih

A similar but more subtle mechanism permits conversion of HBW lattices to Ice-Ih and vice versa. This will be described for EJ/FWEb, although it can be done for any of the EJ/xyz lattices. (The sixth column of Table V gives the correspondence between HBW lattices and proton-ordered Ice-Ih isomers, using the classification of Hirsch and Ojamäe2 ). Referring to Fig. 6(a) as an illustration of EJ/FWEb, ignore the Z-chain and focus on the joining H2 O’s of the North faces. Joining H2 O’s of EJ/FWEb lie on planes of mirror symmetry. This mechanism maintains the mirror symmetry throughout. Label a North face joining H2 O as HJ OHF , where HJ is the proton that is initially in the S-going joining H-bond and HF is the proton that is initially in the facial H-bond (see right-hand edge of Figure 6(a)). Imagine rotating HJ OHF by 90◦ in the mirror plane, so that Hj comes to rest where HF started, and HF now points out of the bilayer. The facial bonds remain intact (although there has been a substitution of one H by another H) but all the S-going joining bonds have been lost and there are now H’s pointing North between bilayers. The facial H2 O’s of the south faces have become dda because they lost the “HJ ” that used to donate to them from the North. At the same time, a new “HF ” proton is suddenly pointing toward them from the south. The HF forms a new inter-bilayer H-bond to the dda O, and all O’s again have four H-bonds. The resulting arrangement can easily be recognized as Ice-Ih once it expands in the N/S direction and relaxes. Faces of HBW have become basal layers of Ice-Ih. It is a proton ordered form of Ice Ih: specifically, by starting with EJ/FWEb, one obtains the form that has space group Cmc21 , which has been identified as the proton ordering for Ice-XI.2, 35, 55 The process can be reversed to make EJ/FWEb (resp. any EJ/xyz lattice) from Ice-XI (resp. from any of Hirsch and Ojamäe’s lattices 1–8 2) . To make them from Ice-Ih would require a disorder-order transition somewhere along the way. We computed the periodic transition state between EJ/FWEb and Ice-XI. It is included in the supplementary material44 as “HBW-ts1.” Its single “imaginary” frequency is –185 cm−1 and the corresponding eigenvector is along the above pathway. The t-state lies 0.74 kcal/mol H2 O above EJ/FWb. Again, the reader may find this lattice a particularly useful one to view in 3-D, especially with a viewing program that permits the H-bond distance criterion to be set sequentially at 2.2, 2.33, 2.41, and 2.5 Å. Figure 9 gives the projected schematic representation of this operation. The same conventions apply as in Figures 7 and 8 (cf. Fig. 1 of Ref. 56). In-plane rotation of 25% of the H2 O’s turns any EJ/Fyz (resp. EJ/FWEb) into a proton ordered Ice-Ih (resp. Ice-XI). These four (for y,z = E or W) ices are ferroelectric, because rotating H2 O’s of only the North faces introduces a net dipole moment. A similar mechanism (depiction omitted) converts EJ/Jyz lattices into antiferroelectric proton ordered forms of Ice-Ih. In-plane rotation involves simultaneous breaking and re-formation of both donor H-bonds of a single H2 O. Its periodic transition state HBW-ts1 has 5 H-bonds (two are distorted) at the rotating O and 3 at the facial O, a setup that would normally be very unstable. Its transition energy barrier is more than double that of the PSW-ts1 transition, and the curvature of the transition pathway is six times greater (based on the square of the ratio of −185 to −76). Unlike PSW HBW interconversion, this mechanism is not expected to support mixed or intermediate states. If one performs the operation on a portion of the lattice, at the boundary there will be O’s having 3 and 5 H-bonds, and the buckled basal face of Ice-Ih will not mesh with the flat face of HBW to allow for stabilization by dispersion forces. Sequential bilayer-by-bilayer conversion to Ice-Ih is a likely mechanism, but it would be a rapid process like dominoes flipping that would be difficult to freeze at an intermediate state. A simple mechanism, consisting of rotating half of the joining H2 O’s, converts HBW lattices to Ice-Ih lattices. For the ground state lattice EJ/FWEb, the corresponding Ice-Ih lattice is Ice-XI. Unlike the PSW HBW mechanism of Section III G, interconversion of HBW with Ice-Ih does not support stable-appearing intermediate states, and the transition state barrier is considerably higher.


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FIG. 9. Schematic representation of the interconversion of HBW (EJ/Fyz) and Ice-Ih.

I. Defects

All crystals come with the possibility of defects. In addition to the usual possibility of a vacancy, interstitial, or dislocation defect, HBW and PSW would be prone to other defect types. For the six HBW lattices that have an ‘a’ and a ‘b’ form in Table III, sliding of some bilayers but not others would create a mixed lattice in which some adjacent bilayers are aligned as in Figure 6(b) while others are aligned as in Figure 6(c). An analogy would be a mixed crystal of Ice-Ih and Ice-Ic, in which there are hexagonal basal layers but each layer’s relationship to its neighbor layers may follow either of two patterns. Similarly, we have noted there are two local minima on the SPES of PSW whenever the definition is NE/x/W/yz, hence a PSW structure could be a sheet-by-sheet mix of NE/x/W/yE and NE/x/W/yW. For either HBW or PSW, sliding grossly disrupts the regularity of the O positions in one direction, and a structure with (say) between 25% and 75% of each alignment pattern in a random sequence would have local “partial order” with low configurational entropy but it could not be considered a crystal. It might be considered to be a quasi-crystal. In most forms of ice, cyclic proton transfers are minimally disruptive to the overall positions of the O’s, and the oxygens adhere to a regular lattice despite randomized H-bond directions. This would not be true of HBW and PSW. While it is true that an isolated individual bilayer or sheet might not be greatly affected, Section III F argues that optimal inter-layer alignment and inter-layer distance depend heavily on the specific H-bond positions and directions that abut each other. For example, replacing a few N-going lines of NE/N/W/SE with S-going lines would create some ridgevalley combinations of Category 2 in an otherwise Category 1 arrangement, forcing the structure to “choose” which pattern to satisfy, or trying to switch back and forth between them. In effect, a


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dislocation-like defect would be induced by an H-bond direction change. As noted above for HBW, introducing a patch of local cyclic proton transfers could alter the local inter-bilayer distance by tens of pm, causing a seriously distorted interface. With enough such patches, randomly distributed, the structure might become barely recognizable as crystalline. HBW and PSW could also undergo L and D Bjerrum defects, as occur in other ices. PSW might be particularly subject to these because of the requirement that the N/S and E/W lines be homodromic. One can imagine an N/S line with an N-going segment that collides with an S-going segment, creating a novel type of point defect consisting of a DAAA water with one free H pointing out of the sheet. Despite the mathematically precise descriptions we have given, actual HBW and PSW systems would be very prone to a variety of structural defects, making it potentially difficult to detect their periodicity. The presence of two local minima on the SPES in most cases means that one would be most likely to obtain at best a quasi-crystal. Also, unlike other ices, structural parameters of the atypical ices are highly sensitive to local H-bond direction changes. IV. DISCUSSION: POSSIBLE RELEVANCE TO AMORPHOUS ICES

Can the atypical water lattices be relevant to the amorphous ices? Section IV is an extended discussion of this question. While no definite conclusions will be drawn, in order to engage the question we develop “microlattice model 1” (MLM1) and consider how the model might account for experimental findings on HDA, VHDA, and LDA. In crudest form, the model simply states that HDA could contain regions of HBW and VHDA could contain regions of PSW. Such a vague assertion would be nearly impossible to disprove and would have little predictive value, so to be useful the model should require that a significant fraction of each amorph, say 50% or more, is locally ordered. Then the evidence is better matched by positing particular proton ordered HBW and PSW lattices as the best candidates. Interestingly, proton ordered phases are also consistent with the unexpectedly small configurational entropy of HDA and LDA.57, 58 We begin our tour of MLM1 with the properties of uHDA, moving from there through eHDA, LDA, and VHDA. A. HBW as a model for uHDA

The RDF and the density are our main clues for deciding whether a structure is compatible with uHDA. As Table IV shows there are at least sixteen variants in the HBW family of lattices, evincing a range of densities and much variation in their RDF’s, especially for radial distances between 2.9 and 4.0 Å. Experimentally uHDA has density 1.15 g/mL.20 It has “about 5” nearest neighbors when the RDF is integrated to 3.5 Å. The O—O RDF and the partial O—H and H—H RDF’s found by Finney14, 59 are reproduced in Figure 10. The experimental O—O RDF for HDA has a clear trough suggesting that there should be few O—O pairs at distances between 3.1 and 3.3 Å (at 1 atm). For high-symmetry periodic lattices with unit cells of 4 or 8 H2 O’s, when any O—O distance occurs, it occurs frequently, so the curve essentially rules out any O—O distances between 3.1 and 3.3 Å. Only one of the entries in Table IV comes reasonably close to meeting both the density and the RDF trough criteria. That one is EJ/FWEa, with predicted density 1.12 and nearest non-H-bonded neighbors at 2.96 Å followed by 3.46 Å. On this basis, the MLM1 (for uHDA) is refined to be more specific: microdomains of EJ/FWEa (Figure 6(b)) as components in uHDA. To compare Finney’s curves with the RDFs of the microlattice model, we would ideally need to know how large are the model’s microdomains, and we would need to run a finite temperature simulation of the system and collect the actual internuclear distances. In lieu of this we present in Figure 10 estimated RDF’s. To draw the lattice’s estimated RDF, gest OO (r), first a list {dj } was made, of all distances from an index OF to all other O’s, and from an index OJ to all other O’s. All distances under 9 Å were included, for an RDF up to r = 7 Å. The positions of the {dj } are given by the “spikes” in each figure. [To be precise, the “spikes” curve is a vertically scaled graph of the number of dj ’s which fall into each 2 pm interval.] Then each dj position is replaced by a normalized Gaussian distribution centered at dj , with a spread σ (dj ), σ (x) being an increasing function of x, and


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FIG. 10. Reproductions of Finney’s RDF graphs from Refs. 14 and 59; and internuclear distances and estimated RDFs for lattices. In each plot, the uppermost curve is Finney’s, offset vertically by 2 units; the lowermost curve displays “ideal” internuclear positions; and the middle curve is the estimated RDF. In each column, the top plot is for O—O RDFs; the middle plot is for O—H partial RDFs; and the bottom plot is for H—H partial RDFs. Left column: for LDA and Ice-XI; middle column: for uHDA and EJ/FWEa; right column: for VHDA and NE/S/W/NE. NOTE: The experimental RDFs are reprinted with permission, from J. L. Finney, D. T. Bowron, A. K. Soper, T. Loerting, E. Mayer, and A. Hallbrucker, Phys. Rev. Lett. 89, 205503 (2002), http://link.aps.org/abstract/PRL/v89/e205503, and from J. L. Finney, A. Hallbrucker, I. Kohl, C 2002 A. K. Soper, and D. T. Bowron, Phys. Rev. Lett. 88, 225503 (2002), http://link.aps.org/abstract/PRL/v88/e225503; by the American Physical Society. Readers may view, browse, and/or download Finney et al.’s graphs for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.

divided by 4δπ r2 , δ being the O density. The estimated RDF’s for O—H and H—H were similarly obtained. Given the paucity of theoretical data from which to derive a spread function σ , parameters defining σ (but not the {dj }) were tweaked to improve the visual match with Finney’s curves. (The curves’ overall shape and peak positions are not strongly dependent on σ .) Because of this, and due to our use of Gaussians for distributions that are not exactly Gaussian, these curves are offered as ballpark estimates only. For all three RDF’s, there is decent agreement between the estimated and the experimental, out to about 4.5 Å. Focusing on the O—O comparison, the experimental curve has small broad peaks around 4.6 and 6.4, and a broad trough around 5.3, that are not reproduced by the lattice. These peak and trough positions would be consistent with a high-tetrahedrality component, raising the interesting possibility that HDA consists of some HBW and some Ice-Ih or ASW. We will suggest in Section IV C that thin transition zones or boundaries between microdomains might be enough to account for this. One could also explain the deviation if the BLYP-D functional were off by just a little in its computation of the hexagonal shapes. Slightly thinner, more elongated hexagons would increase the density to 1.15 and would split the spikes between 5 and 6 Å into a group that would shift left and a group that would shift right, in line with Finney’s curve. Clearly one cannot draw any definitive conclusion, but neither can the presence of 50% or more of EJ/FWEa in uHDA be ruled out. Why microdomains? Winkel et al.32 have noted that the transition from Ice-Ih to uHDA starts in many places independently. Suppose the conversion pathway includes momentary conversion to


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ordered Ice-XI, as suggested in Section III H, and suppose there are many seeds where ordering begins. Even if one starts with a perfect crystal of Ice-Ih, the planes of symmetry for Ice-XI can adopt any of three orientations around the c axis at each seed and there are two choices for “North” (cf. Ref. 55, p.6413). Domains of order will not cohere, the result being microdomains. Ferroelectricity of Ice-XI may also be a reason why it takes less energy to create two adjacent microdomains with opposite orientations, than a single larger coherent domain. Lastly, there is the macroscopic problem of anisotropic compression. In “perfect” conversion of Ice-Ih to HBW (Figure 9) the unit cell ‘c’ dimension would shrink from 7.00 Å to 5.85 Å (based on Ice-XI and EJ/FWEa) while the ‘a’ and ‘b’ dimensions would expand from 4.45 to 4.55 and from 8.12 to 8.51 respectively as the basal hexagons become flat. Whether one starts with a crystal or with powder, some cracking of the sample and shuffling of the pieces is probably inevitable. B. HBW as a model for eHDA

As noted in Section III E, EJ/FWEa is one of just two monoclinic HBW lattices that connect along their SPES to a related lattice that has lower energy as well as lower density. When decompressed to 1 atm, EJ/FWEa has a natural exothermic pathway to EJ/FWEb that consists merely of sliding the bilayers relative to one another in the U# or D# direction. The MLM1 postulates that microdomains of EJ/FWEb (Figure 6(c)) are a major component of the structure of eHDA, and the annealing process is this sliding mechanism. It might seem that the large alteration in “beta” during sliding would “tilt” a finite domain into a very different shape, and shape change would be inhibited by crashing into nearby domains, but this is not so. Because of SPES periodicity one layer might go U# while the next goes D# , so the lattice as a whole need not “tilt” as it anneals. It can occupy about the same region of 3-D space after sliding as before. The annealing of uHDA to eHDA has been described as an “ultraviscous liquid”,18 which could be taken as consistent with a description of bilayers staying intact while sliding past each other. Per Table IV, EJ/FWEb has density 1.08 (compared with experimental value of 1.13) so this is a larger difference than for uDHA, but still plausibly consistent with the accuracy of BLYP-D. There is essentially no change in the H-bonding portion of the RDF during annealing of EJ/FWEa to EJ/FWEb, but the region between 2.9 and 4.0 changes. The nearest neighbor moves outward, from 2.96 to 3.06, and this nearest non-H-bonded distance also switches from applying only to the J H2 O’s, to applying to all H2 O’s. Average coordination increases from ∼4.5 to ∼5.0 (for cutoff of 3.2 Å). This is consistent with experiment in that Nelmes et al.18 say of eHDA that it is “characterized by an increase or expansion of the first-peak,” compared to uHDA. The next-nearest neighbor goes from being at 3.46 to 3.45, which helps explain why the eHDA and uHDA RDF curves appear similar beyond that first peak.19 During cycling of HDA and either LDA or VHDA it is noted that uHDA occurs at the upstrokes i.e. when HDA is formed by compression, whereas eHDA occurs at the downstrokes i.e. when HDA is formed by decompression. This is consistent with the idea that under higher pressures, EJ/FWE will adapt to the pressure by moving along the SPES to the denser EJ/FWEa, whereas at lower pressures EJ/FWEb is preferred. C. Implication of MLM1 for LDA

When decompressed and warmed to ∼120 K, HDA converts spontaneously to LDA. LDA has the hallmarks of a high-tetrahedrality ice such as ice-Ih or ice-Ic, with the exception of the XR halo pattern. Upon further warming there is the small endotherm starting at 136 K that has been attributed to a glass transition, followed by crystallization to Ice-Ic starting at ∼150 K.60 If having an amorphous Xray diffractogram can be attributed to a microlattice structure, i.e. to being a mosaic of small randomly oriented microdomains, then that evidence and the RDFs are consistent with these microdomains consisting of either actual Ice-Ih or actual Ice-Ic. As noted in Section III G, the simple pathway connecting Ice-Ih to HBW can also go spontaneously in reverse. Starting from EJ/FWE (either a or b), the result is Ice-XI. The MLM1 should therefore propose that the structure of LDA contains local order in the form of embedded microdomains of Ice-Ih. Moreover,


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these microdomains will inherit proton order from EJ/FWE and would therefore be microdomains of Ice-XI. Estimated RDFs for BLYP-D-optimized Ice-XI are shown in Figure 10 and compared with Finney’s LDA data.59 There is reasonable agreement between estimate and experiment, with the understanding that the estimate, obtained as above, is not definitive. Salzmann et al.60 have questioned whether the endotherm at 136 K represents a glass transition. Arguing by analogy to similar endotherms seen for Ices V, XII, and IV, where the endotherm represents an order-disorder transition, they wondered whether the endotherm for LDA might instead indicate “kinetic unfreezing of the reorientation dynamics of the water molecules.” The MLM1 could take this a step further, attributing the endotherm to an order-disorder transition of (microdomain) Ice-XI. While this transition normally occurs at 72 K, that is for KOH-doped samples in which the purpose of the KOH is to facilitate proton reordering. The MLM1 suggests that undoped microdomain Ice-XI could undergo the order-disorder transition at ∼136 K, which is close to the TO for three other proton-ordered ices (cf. Fig. 7 of Ref. 60). Those ices are all antiferroelectric. Ice-Ih does not have an order-disorder transition near 136 K but being composed of randomly oriented microdomains might alter TO because the random orientations would cancel out Ice-XI’s net ferroelectricity, allowing it to behave more like the antiferroelectric proton ordered ices. Why is HDA reconstituted more easily from LDA (i.e. why is a lower pressure of only 0.6 GPa needed)16 than from Ice-Ih? Johari28 has observed that smaller crystal grain sizes permit transition at lower pressure, so being microdomains may be enough to account for this. In view of MLM1, one can also wonder if part of the answer lies in proton order. We have already noted that making EJ/FWE from Ice-Ih requires transition to Ice-XI at some point before the rotational mechanism of Section III H can occur. If LDA is microlattice Ice-XI, then the ordering step is already done. In other words, LDA’s proton order makes it “poised” to become EJ/FWE, and no work needs to go into proton ordering. If the reduced transition pressure of LDA is partly due to being Ice-XI rather than Ice-Ih, the same pressure reduction should occur if one starts with Ice-XI made with low-concentration KOH doping. Interestingly, Yoshimura and coworkers61–64 have done compression tests of high-concentration LiCl, KCl, KOH, and LiOH solutions. They did find reduction of the pressure necessary to create a high density phase, in support of the MLM1. However their findings have been questioned65, 66 and there are definite differences between HDA and their high-solute phases. A potential problem with LDA being Ice-XI is that Ice-Ih/XI is almost isoenergetic with IceIc,67 but crystallization of LDA to Ice-Ic releases 341 cal/mol (second exotherm of Ref. 68). The MLM1would therefore suggest that (nearly) all of that heat comes from new H-bond formation or straightening of distorted H-bonds, as poor alignment at domain boundaries and any other regions with distorted bonds give way to a coherent crystal. If one makes some approximations this gives a clue to the mean molecule count per domain, denoted D. A reasonable guess for the average enthalpy change from bond formation or straightening on a domain boundary might be 2 kcal/mol. The fraction of H2 O’s that sit on boundaries between domains will be inversely proportional to D1/3 ; e.g. for regular cubic domains with one-molecule thick boundaries it is 3/(D1/3 −1). Setting (3/(D1/3 −1))(2) = (0.341) gives D ≈ 6430 molecules. At a density of 0.94, this occupies 205 nm3 . While the dependence on crude approximations makes this a very rough estimate, it is in the ballpark with Koza and Schober’s estimate of 8 to 9 nm for LDA domain diameters based on the inelastic phonon response.29, 30 Taking this coarse calculation further, 15% of H2 O’s are found to fall on the boundaries and 41% lie either on or immediately adjacent to the boundaries. Proportions like these suggest that even a high-accuracy RDF based on the infinite lattice can be expected to go only so far toward explaining the experimental RDFs. In MLM1, the inter-domain transition zones would also make a significant contribution to the predicted RDFs. D. Implication of MLM1 for VHDA and eHDA-VHDA mixes

The mechanism of Section III G converts EJ/FWEb, the postulated structure for microdomain components of eHDA, to NE/S/W/NE (Figure 3(a)). Conversion of HDA to VHDA can occur at temperatures below ∼120 K, where proton hopping is kinetically suppressed, so the MLM1


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prediction should be that proton order is preserved and NE/S/W/NE is the structure of the converted microdomains embedded in VHDA. By Table III this also happens to be the ground state among Category 1 PSW lattices. Estimated RDFs for NE/S/W/NE are shown in Figure 10 and are compared with Finney’s VHDA data.14 There is reasonable agreement between estimates and experiment, again with the proviso that the estimated RDFs should not be taken too literally. One possibly discordant datum is the experimental density of 1.26 when VHDA is extrapolated back to 1 atm, whereas BLYP-D/TZVP gives 1.35 for NE/S/W/NE. One possible explanation for why BLYP-D gives an anomalously high density (about 7% higher than experiment) for this lattice is that 3-body dispersion terms should not be discounted, given the mutual proximity of O* and its nearest next-sheet neighbors. Three- and four-atom dispersion terms are increasingly being recognized as important corrections that the basic Grimme model omits.69–71 Typically the 3-atom term is a net repulsive force, meaning that we would obtain a lower density prediction if it were included. A benchmark BLYP-D/TZVP optimization of Ice-XIV yielded a density of 1.382, also higher by 7% than the experimental value of 1.294,72 so BLYP-D may have a general bias toward overestimating density for very high density ice lattices. Loerting and coworkers17 found that in the pressure range 0.3 to 1.0 GPa, structural states intermediate between HDA and VHDA can be prepared. MLM1 can account for this by recognizing that interconversion of PSW with HBW can be done sheet by sheet, and can be stopped at any intermediate point along the way (Section III G). VHDA and HDA are distinct phases but they can coexist across a movable phase boundary. At low pressures the energetics will tilt the entire system toward HDA and at high pressures the system will convert spontaneously to all VHDA, but there is a middle regime where the intermediate states have enough kinetic stability to hold their structure when quenched to very low temperatures. By contrast, mixed states of eHDA and LDA have been observed as snapshots during the eHDA-LDA transition73 but they have never been quench recovered. This seems to agree with the analysis in Section III H, which explains why no metastable intermediate is expected for the HBW–Ice-1h transition. E. Spectra

The fact that VHDA and HDA have different Raman spectra was one of the pieces of evidence making the case that they are distinct phases. The peak of the Raman signal for VHDA lies at 3200 cm−1 , about 30 cm-1 higher than for HDA (Fig. 2 of Ref. 13). CRYSTAL09 does not compute Raman intensities, but if one averages the Raman-active O-H stretch modes for EJ/FWEb and for NE/S/W/NE the results are 3240 cm−1 and 3293 cm−1 respectively, i.e. the proposed microlattice component for VDHA has a predicted peak around 50 cm−1 higher than the proposed microlattice component for eHDA. Given the uncertainty caused by cell-only modes that may omit much of the coupling, and averaging of modes whose relative intensities are unknown, our point is only that VHDA would be predicted to have a peak that is blue-shifted relative to eHDA, and that the longer O—O distances for the N/S bonds in PSW versus shorter bond lengths for HBW are what is behind this. The predicted average of Raman-active modes for Ice-XI is 3135 cm−1 , in good agreement with the measured value of 3113 cm−1 for LDA. Recommended frequency scaling factors for BLYP are very close to 1.00.74 A region of the IR phonon spectrum where LDA and Ice-Ih are quite similar, but HDA has a marked difference from them, is the high end of the translational band and the low end of the librational band. The translational and librational bands of the spectrum of ice are well separated: a density of states graph (DoS) for Ice-Ih or Ice-Ic shows the translational band ending at 10 THz and the tail of the librational band beginning around 15 THz (Fig. 2 of Ref. 75. In LDA this band also begins its tail at 15 THz, but HDA’s DoS graph has a tail that extends almost down to 12 THz and has close to linear growth between 13 and 15 THz. We focus on the fact that HDA has its libration band begin about 3 THz lower than Ice-Ih and LDA. Librational modes in a periodic proton-ordered ice will consist of cellular modes and couplings of cellular modes across cells. The result is broadening of the cellular signals. We can provide soft evidence only for MLM1, since we have not computed the expected broadening. The computed spectrum for Ice-XI has its first librations at 18.6 THz, whereas the computed spectrum for EJ/FWEb


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has them at 16.0 THz, a red shift of ∼2.6 THz. The lowest non-translational modes in HBW are hindered in-plane rotations of the joining H2 O’s. These are the motions which, if extended, would begin the conversion of HBW to Ice-Ih. If intercellular coupling and measurement broadening were to replace all cellular modes by signals with a half-height width of about 4 THz, these data show that MLM1 is at least coarsely consistent with Klug’s results for this region of DoS spectrum.

F. Transition enthalpies

The transition from eHDA to VDHA requires an external pressure that is somewhat temperature dependent.76 The transition clearly proceeds at or above 0.8 GPa and proceeds in reverse below 0.4 GPa.76, 77 The predicted value of ∼0.55 GPa for the eHDA-VHDA equilibrium pressure (Section III G) falls nicely in this interval. Handa et al.68 reported enthalpy changes as measured by DSC during stepwise conversion of uHDA to eHDA to LDA to Ice-Ic to Ice-Ih. All steps were induced by progressive slow warming. The H for annealing of uHDA to eHDA was found to be 0.05 kcal/mol, in decent agreement with our prediction of 0.04 kcal/mol for the H for EJ/FWEa → EJ/FWEb. As noted in Section III E we also predicted the equilibrium pressure to be ∼0.19 GPa, i.e. the pressure would have to be kept below this cutoff for the system to anneal toward eHDA. Nelmes et al.18 obtained eHDA at pressures up to 0.3 GPa, so the predicted and measured equilibrium pressures are somewhat similar. Handa’s measured H for eHDA conversion to LDA was as 544 J/mol,68 or 0.13 kcal/mol. This is substantially smaller than the BLYP-D prediction of 1.48 kcal/mol for the cZPE-corrected electronic energy difference between EJ/FWEb and Ice-XI. We have already noted that BLYP-D is biased in favor of Ice-Ih to a significant extent. The same calculation would predict that H0 (H2 O)(g) − H0 (Ice-XI) is 15.65 kcal/mol, “H0 ” meaning enthalpy at 0 K. The experimental value for H0 (H2 O)(g) − H0 (Ice-Ih) is 14.09 kcal/mol,56, 78 so if BLYP-D is truer to experiment for HBW and PSW then the discrepancy could be explained. The best we can say is that there is a large but uncertain error bar comparing BLYP-D enthalpy of Ice-Ih against that of HBW and PSW. A meaningful enthalpy comparison will require a less biased model. The DFT methods vdW-DF, vdW-DF2, and optPBE-vdW, benchmarked for bulk water by Fernández-Serra et al.79 and by Nillson, Møgelhøj et al.,80 show promise for this purpose. The large error bar does not apply to the uHDA-eHDA and VHDA-eHDA transitions because typically DFT methods do best when comparing H-bonding topologies that are similar35 and do worst when comparing very different H-bonding topologies.

G. MLM1 and Halo Pattern

Can the MLM1 be compatible with the absence of sharp peaks or Bragg lines in the neutron scattering and XR diffractograms? Line spread may be due to factors on several levels. First, for a perfect crystal, line spread is normally modeled via ellipsoids defined by Debye-Waller factors (DWFs).81, 82 Although quantitative data will require a dynamical simulation, which is beyond this article, it is expected that dispersion-only interaction between bilayers or sheets will result in exceptionally large DWFs due to the weak restoring forces. For another dispersion-bonded material, graphite, the DWF in the Z direction is 3 to 5 times greater than in the basal plane.83–86 Second, as explained in Section III I, HBW and PSW support many kinds of defects. To recap just two, incomplete annealing would leave an HBW or PSW lattice with a stochastic mix of two inter-layer alignments, undermining periodicity in one direction. In atypical ices, defects that are normally “benign” such as patches of deviation from the ideal H-bond directions can have a large effect on local inter-layer O—O distances and local inter-layer O-O-O angles. Finally, given their ease of interconversion, the presence of small amounts of PSW mixed into HDA or small amounts of HBW in VHDA cannot be ruled out; and obviously there will be complete loss of periodicity at microdomain boundaries. As noted above, H2 O’s in transitional zones may account for 15 to 40% of what is seen in RDFs. Indeed, it could be surprising if these systems, with so many opportunities for departure from idealized crystalline positions, did not appear amorphous.


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V. CONCLUSION

Hexagonal bilayer water and pleated sheet water are BLYP-D-computed “atypical” water lattices in which weak van der Waals forces bind together two-dimensional layers that do not share H-bonds. Their predicted properties include high density, high O coordination numbers despite having just 4 H-bonds at each O, and strong dependency of the symmetry group and cell parameters on the specific proton order. A motif consisting of O-H- -O being ∼108◦ with O—O about 3.15 Å apart for the non-H-bonded interaction, and pairs of these motifs forming either a parallelogram or trapezoid, appear to be common in the optimized geometries. The motifs may be a factor in determining and stabilizing the local minima. There are simple rearrangement pathways that connect HBW lattices with proton ordered forms of Ice-Ih and with PSW. While it would be interesting to try to make HBW and PSW, any such attempts would have to allow for the likelihood of frequent occurrence of a wide variety of defects. The Microlattice Model Version 1 allows one to ponder whether they may already have been made, since certain proton-ordered forms of HBW and PSW have predicted properties that resemble those of HDA and VHDA respectively. MLM1 is presented “for the sake of argument” only. There is not enough information to assign a microlattice structure to any amorphous phase. Still, neither can MLM1 be ruled out. The model offers a novel (hypothesized) explanatory framework for the core properties of the amorphous ices, including their densities, O-coordination, RDFs, transition behaviors, and (apparent) amorphousness. In addition, the model hints at alternate explanations for the amorphs’ low configurational entropy, Raman spectra, librational red shift, and the 136 K endotherm. All of these explanations are inevitably incomplete but hopefully they will inspire further work on partially locally ordered models for the amorphous ices. A “version 2” or “version 1.1” of the MLM might include any of the following: DWFs and more accurate RDFs via dynamical calculations; further exploration of the nature, occurrence frequency, and consequences of defects; more precise modeling and better comparisons to Ice-Ih, via BLYP-D3 or other emerging functionals; better estimates of domain size and shape and description of the transitional zones; in-depth studies of the energy-topology relationships for abutting hexagonal bilayer faces; and examination of lower symmetry forms that were not covered herein. ACKNOWLEDGMENT

The author is grateful to Roberto Orlando of the CRYSTAL09 support team for generous provision of advice and technical assistance. APPENDIX 1. Definition of terminology for nearest off-sheet neighbors

Starting from an index O* lying on a sheet denoted S*, we assign directions (N,E,U) so that O* is NE and lies on a ridge. Let us denote the sheet above S* as S**. As in Section III B, the closest O on a ridge of S** is called O**. The oxygens located in sheet S** lying one H-bond E or W from O** are denoted OUE1 and OUW1 . These lie in valleys of S**. Traveling one H-bond north and one H-bond south from OUE1 , whichever O is closer to O* is denoted OUE2 , and likewise for OUW2 . The distances listed in Table IV are the distances from O* to OUE1 , OUW1 , OUE2 , OUW2 , and O**. 2. Explanation of Category 1 (“small parallelogram”) vs Category 2 (“small trapezoid”)

The greatest inter-sheet dispersion forces on an index oxygen O* are from the neighbors OUE1 and OUW1 . Optimization will include positioning along the SPES to maximize the interactions between the O* ridge and the OUE1 and OUW1 valleys. Empirically, this occurs when the O*-H*- -O angle is between 98◦ and 118◦ , H* being the H attached to O* going N. For Category 1, the H-bonds in the valleys containing OUE1 and OUW1 are directed S, i.e. these valleys run antiparallel to the ridge containing O* (cf. Figure 4(a)). For Category 2, H-bonding in these valleys runs parallel to the ridge containing O*. In the antiparallel setup of Category 1, OUE1 aligns between O* and H*, i.e. only a


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little off from being due east of O* (and up). This maximizes interaction of H* with OUE1 as well as interaction of HUE1 with O*. As a result OUE1 (and OUW1 ) comes close to O* while OUE2 (and OUW2 ) are further away. In the parallel setup of Category 2, interaction is maximized by having the O-H bonds of the overlying valleys align with the H - - O bonds of the ridge containing O* (cf. Figure 4(b)). The positions of OUE1 and OUW1 are staggered relative to O* i.e. about half an H-bond N or S as well as one position E and U. This puts them further from O* (compared with the antiparallel setup) but also means that OUE2 and OUW2 are at about the same distances from O* as OUE1 and OUW1 . In a Category 2 PSW lattice, four next-sheet neighbors (OUE1 , OUW1 , OUE2 , OUW2 ) are about equidistant from O*. Distances from O* to the 4 next-sheet neighbors fall in the range 3.09 to 3.35 Å (Table IV). Coordination at O* is 8, due to 4 H-bonded neighbors and 4 next-sheet neighbors. In a category 1 PSW lattice, only OUE1 and OUW1 are close enough to count toward coordination at O*, and the total coordination at O* is 6. In Category 1, OUE1 and OUW1 are at 2.96 to 3.25 Å while OUE2 and OUW2 sit at least 3.42 Å away. Notice that O** is always at least 3.61 Å from O*. 1 C.

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