Point defects at the ice (0001) surface

Abstract

Using density functional theory we investigate whether intrinsic defects in ice surface segregate. We predict that hydronium, hydroxide, and the Bjerrum L- and D-defects are all more stable at the surface. However, the energetic cost to create a D-defect at the surface and migrate it into the bulk crystal is smaller than its bulk formation energy. Absolute and relative segregation energies are sensitive to the surface structure of ice, especially the spatial distribution of protons associated with dangling hydrogen bonds. It is found that the basal plane surface of hexagonal ice increases the bulk concentration of Bjerrum defects, strongly favoring D-defects over L-defects. Dangling protons associated with undercoordinated water molecules are preferentially injected into the crystal bulk as Bjerrum D-defects, leading to a surface dipole that attracts hydronium ions. Aside from the disparity in segregation energies for the Bjerrum defects, we find the interactions between defect species to be very finely balanced; surface segregation energies for hydronium and hydroxide species and trapping energies of these ionic species with Bjerrum defects are equal within the accuracy of our calculations. The mobility of the ionic hydronium and hydroxide species is greatly reduced at the surface in comparison to the bulk due to surface sites with high trapping affinities. We suggest that, in pure ice samples, the surface of ice will have an acidic character due to the presence of hydronium ions. This may be important in understanding the reactivity of ice particulates in the upper atmosphere and at the boundary layer.

The surface of ice acts as a catalyst in the upper atmosphere yielding halogen containing species that facilitate the destruction of ozone (1). More generally, probing the structure/activity relationship of ice with trace gases is important, topical, and yet difficult in the laboratory. Here, we use computational approaches to address the fundamental question of whether charged defects in ice show a tendency to locate at the surface in preference to the crystal interior.

The behavior of many properties of ice Ih (the most prevalent ice phase on earth), such as proton conduction, is controlled by the presence of intrinsic charge carrying defects (2). These intrinsic defects violate the “ice rules”—each water molecule donates and receives two hydrogen bonds, and each nearest neighbor oxygen pair has one proton between them (3, 4). Ice is peculiar in that two distinct types of intrinsic charge carrying defects are commonly found: 1) those formed in violation of the first rule from dissociation of the H₂O molecule in ice, H₃O⁺ and OH^- (we will refer to these as ionic defects) and 2) the Bjerrum defects formed by a violation of the second ice rule through molecular rotation(s). Rotation of water molecules on their tetrahedral sites gives zero or two hydrogen atoms between a pair of oxygen atoms; the absence of a proton along a hydrogen bond is referred to as an L-defect (yielding a negative charge), and the defect with two protons along a hydrogen bond is termed a D-defect (bearing a positive charge) (2, 5). Example relaxed structures of these defects are shown in Fig. S1, where movement of water > 1 Å from the ideal lattice site is seen. Charge transfer requires the motion of ionic defects, but due to the topological constraints of the hydrogen bonding network in ice, Bjerrum defects must also be transported in an opposing direction for current to flow (2).

Because of the coupled nature of rotational defects and the ionic defects, it is very difficult to obtain information about the concentration and mobility of each individual defect. However, a series of experiments on various forms of ice have indicated that proton conduction is mediated by the L-defect while the D-defect is inactive and that the hydronium species is much more mobile than the hydroxide (6) (cubic ice), (7) (cubic and amorphous), (8, 9) (polycrystalline ice films), (10) (amorphous ice films).

Most recently, there have been reports of elegant isotopic exchange experiments (11), which show that ice surfaces have a significant role in controlling the concentration of ionic and Bjerrum defects and suggest that hydronium ions are surface segregated (12–14). A further consideration is the disordered proton structure of ice Ih. At the basal surface of ice Ih, the ice rules cannot be fully obeyed by the top layer of water molecules; each water molecule in this top layer is either 1) only a single hydrogen bond acceptor and presents a bare oxygen to the vacuum or 2) only a single hydrogen bond donor with a proton dangling out of the surface into the vacuum. Recent works (15–18) have clearly shown that the energies of water ice surfaces depend upon the ordering of these surface protons. It emerges from our calculations that the spatial arrangement of the dangling surface protons is key to an understanding of defect energetics. Note that the influence of proton disorder on defects within the bulk is a much subtler problem not addressed here, but progress has been made in identifying and tackling the issues arising from it (19, 20).

The paper is organized as follows: We address the formation energetics of Bjerrum defects in bulk and study the migration of L and D defects from bulk to the surface using large scale density functional theory calculations. We then examine the transport of ionic defects from the crystal interior to the external surface. The dynamics of defects are then reported followed by calculations on complexation energetics for the ionic and Bjerrum defects. Finally an interpretation of the predictions and their implications are presented.

Results

Bjerrum Defects.

The formation of rotational defects within the bulk has been studied previously in a series of high quality plane-wave DFT based studies by de Koning and coworkers (21–24) and in earlier works (25), but no concerted study of surface segregation has been reported thus far. Recently, Cwiklik et al. have reported theoretical and joint theoretical/experimental studies of hydroxide surface segregation in ice (26, 27). This study complements and builds on the latter work by considering the role of other defects, their interplay, and using a larger structural model that represents the crystal bulk and the surface environment. We first reproduced the results of de Koning (21) on Bjerrum defect pairs in bulk. A defect pair was formed at neighboring lattice sites, then, by molecular rotations, moved apart. The total supercell energy against the distance separating the defects was fitted to a screened coulombic interaction, giving an estimate of the energy to isolate the defects of 1.2 ± 0.1 eV*, in good agreement with previous calculations (21) and experiment (2).

We then considered the behavior of the Bjerrum defects near the basal surface of ice Ih using a slab consisting of six bilayers of 60 molecules (360 molecules total) giving two ice Ih (0001) surfaces. The schematic (Fig. S2) shows a D-defect formed between molecules in the inner and outer layers of the surface bilayer by rotating a dangling proton 120°. A set of molecular rotations is then carried out (Fig. S2), ending with the D-defect protruding from the opposite surface of the slab. The net effect is the transfer of a dangling proton from one surface to the other, introducing a dipole across the slab. The initial surfaces both contained 15 dangling protons, but the final state contains one surface with only 14 dangling protons and another with 16. The final state of the system fully obeys the ice rules (excepting the top- and bottom-most layers), and it is impossible to identify an individual L-defect on the original surface or the additional D-defect on the opposing face without prior knowledge.

Fig. 1A shows the energy profile for moving a D-defect from one side of the slab to the opposing face, which corresponds to the process shown in Fig. S2. In Fig. 1A layer 0 is the original defect-free surface, and each increment in the layer number is the result of a rotation of a molecule in that layer so as to propagate the D-defect. At layer 12 the defect emerges from the opposing crystal face with a reduction in energy. Fig. 1B shows a similar series of calculations, continuing from the previous state of the system, for the progress of an L-defect through the slab.^†

Fig. 1.

Energies for the formation of D and L-defects in a 6 bilayer slab (see Fig. S2 for further details). The left hand figure shows the energies required to move a D-defect into the bulk and out to the opposite surface. The right hand figure continues the calculation and moves an L-defect across the slab to annihilate the D-defect. Missing points are caused by barrier-free relaxation of the defects to nearby layers. The reference state is the most stable ideal surface calculated.

The energies of the surfaces that do not contain bulk defects depend strongly on the pattern of surface dangling protons, in accord with refs. 15–17. This is clearly seen in Fig. 1A, where the initial state, layer 0, is ca. 0.3 eV higher in energy than the layer 12 states formed after a D-defect has crossed from one side of the slab to the other. The layer 12 states are more stable than the initial state because the D-defect was formed at a surface site with four dangling proton neighbors but emerged to form a dangling surface proton at two sites with only one or two dangling proton neighbors, reducing the total number of neighboring dangling proton interactions by three and two compared to the initial state (layer 0).

To understand in detail the effect of proton ordering at the surface on the energetics of Bjerrum defect formation, we have carried out a further series of calculations, sampling near-surface and bulk Bjerrum defects, at different lattice sites. A series of defects are created from ideal slab geometries by carrying out one, three, and five molecular rotations starting from a particular surface molecule, so as to create the defect of interest one, three, or five layers away from the surface. The formation energy of the defect in a given layer is defined as E_N = E_{(defect in layer N)} - E_{(ideal surface)}. To make these energies directly comparable we always calculate the ideal-surface reference energy using a slab that is exactly identical to the configuration of the defect except for the N-molecular rotations required to construct the defect. To systematize our sampling of the disordered proton configurations we use a local order parameter—the number of dangling protons that neighbor the surface site at which the defect is created. We use this order parameter to sample the possible local neighboring proton environments, i.e., molecules with between zero and six dangling protons at next nearest molecular sites, after Pan et al. (15) An example geometry illustrating these sites for a layer 1 D-defect is shown in Fig. 2A. The results of sampling the possible local environments are shown in Fig. 3 for the Bjerrum defects and show the energetics vary dramatically as a function of the local proton environment. The two types of Bjerrum defect show opposite energetic preferences to unscreened surface protons; the formation energies of subsurface L-defects are increased by dangling protons, while for D-defects the energy cost is reduced by the presence of neighboring protons. This result seems at first counterintuitive but is due to the change in the surface structure when the DL pair is formed. Forming an L-defect inside the crystal leaves a surface D-defect (additional dangling proton), which is destabilized by neighboring protons, while formation of a subsurface D-defect removes a surface dangling proton (surface L-defect) and the subsurface defect is relatively stable compared to the initial state as the order parameter increases. This (de)stabilization is linear with the number of dangling proton neighbors for both species; the slope of the formation energy with respect to the number of proton neighbors reaches ± 0.15 eV per proton for the D- and L-defects in the fifth layer where the Bjerrum defect has moved sufficiently far from the surface that the interaction with surface protons is negligible. The formation energies also describe the energy gain of a Bjerrum defect segregating from the bulk to a surface location; D-defects will tend to migrate to surface locations without dangling protons but L-defects will migrate to dangling proton rich areas.

Fig. 2.

The local environment of layer 1 (A) D and (B,C) L-defects. In this case the D-defect has two neighboring dangling protons, while the L-defect has four. For clarity only water molecules in the upper bilayer are shown in A, B, and C. A is a view across the surface askew to the surface normal. B is a view atop the surface. C is a side view of the surface, and D is a multilayer cross-sectional view of the reconstructed D-defect in the third layer. The top half of the bilayer is red, the lower half blue. The oxygen atom of the defect itself is green.

Fig. 3.

Sampling of D (left) and L (right) formation energies as a function of number of dangling protons neighboring the site. Circles are the energies to form the near-surface defects (i.e., layer 1), triangles layer 3 defects, and red squares the energies to form the bulk-like (layer 5) defects. The different layer 3 D-defect types are detailed in the text. When the D-defect is surrounded by three protons, the formation energy in the surface layer 1 is 0.06 eV and 0.52 eV in bulk layer 5. For the case where the L-defect has three neighboring protons, the formation energy in layer 1 is 0.42 eV and 0.70 eV in layer 5.

Layer 1 and layer 3 D-defects are found to be particularly stable because they preserve the total number of hydrogen bonds within the system but remove unfavorable dangling proton interactions. In Fig. 2 we show representative geometries for these defects while the formation energies are shown in Fig. 3. The layer 1 D-defect involves a surface molecule with a dangling proton flipping over to present the proton to the interior, removing unfavorable interactions between it and neighboring dangling protons and making a very weak hydrogen bond with a layer 3 molecule (see Fig. 2A).^‡

The layer 3 L-defect is found to already be essentially bulk-like, with only rather modest energies to move between layers 3 and 5. Layer 1 L-defects are somewhat different from the other L-defects considered. To stabilize these defects a large motion, > 1 Å, of the oxygen atom in the first layer is required, leading to a configuration that actually resembles a molecular vacancy and a surface admolecule (Fig. 2 B and C). The energies of these surface L-defects are found to vary considerably, 0.27 to 0.54 eV, from site to site. However, this energy cost does not appear to correlate with neighboring protons. Similar to the D-defects, layer 2 and 10 defects “annealed” out during relaxation. It is possible to form a layer 2 L-defect, but like layer 1 L-defects, it required large motion of an oxygen atom out of the surface plane, and the product resembles a vacancy and admolecule.^§

The experimental value for bulk activation energy of Bjerrum defect motion is 0.66–0.79 eV (2). With the contribution to the activation energy coming from the theoretical diffusion barrier of 0.11 eV found by de Koning (21, 23) this means that the surface can be a lower energy source of mobile Bjerrum defects when their formation energy is less than ca. 0.55 eV, which is the case for dangling proton sites with three or more proton neighbors (D-defects) and nondangling proton sites with two or fewer proton neighbors (L-defects).

Because the local environment at the surface has a very significant effect on defect energetics it is necessary to consider the structure of the exposed surface and how this dictates the relevant energies by providing environments for the defects. The observed linear relation between the number of dangling proton neighbors and formation energies is consistent with a short-ranged two-body interaction between neighboring protons, of magnitude 0.15 eV per proton pair. A very similar value was derived by Pan et al. (15) from calculations of surface energies as a function of the average order parameter used here. Using the computed value for the repulsive interaction of neighboring protons, we simulated the ice Ih (0001) surface with an Ising model and estimated the distribution of surface proton environments at 180 K (slightly below the temperature at which premelting takes place), further details of which are given in SI Text. The main findings from the Ising simulation were that a dangling proton can have zero, one, two, or three neighboring protons with a significant frequency but protons surrounded by four neighboring protons are extremely rare, accounting for just 0.003% of sites (1 per 5,625 nm²). Protons surrounded by five or six neighbor protons were not found. It was also found that the distribution of protons around nondangling proton sites is the same as for dangling proton sites but for 6-N neighboring protons. Nondangling protons are likely to have three, four, five, or six dangling proton neighbors. Only 0.003% of sites have two dangling proton neighbors.

The typical number of neighboring dangling protons for the two types of Bjerrum defect is expected to be rather different—sites where a D-defect can be injected into the bulk and therefore must have a dangling proton will have predominantly two proton neighbors (sites with three neighbors have a lower frequency), while the sites lacking a dangling proton, where an L-defect can be created, would most typically have four. However, a significant concentration of dangling and nondangling sites have three neighboring protons, and because these are the most favorable sites for reaction or surface segregation^¶ we consider three neighbors for the segregation process. The energy to move bulk D-defects from their anticipated surface environments is calculated to be slightly smaller than that of the L-defect, 0.52 eV compared to 0.70 eV from the linear fit to our data. This data shows that a significant concentration of dangling proton sites can release D-defects into the bulk at an energy cost lower than through their formation in the bulk. Only the rare nondangling sites with two or fewer proton neighbors could act as sources for L-defects.

Ionic Defects.

The behavior of the hydroxide/hydronium ions is considered in a similar manner to that of the Bjerrum defects. We add (or remove) a dangling proton from the ice surface forming the ionic species and, also, an incipient D or L-defect, which is equivalent to a surface charge on the same surface as the ionic defect. This additional Bjerrum defect is then moved to the opposite face of the slab, in exactly the same manner as described in the last section, to form a supercell fully obeying the ice rules (Fig. S3). As was found for the Bjerrum defects, the energy of the ionic defects is bulk-like from the fourth layer, and defects in the near-surface layers tend to relax to surface sites during geometry optimization with significant energy gains. The surface segregation energy of an ionic defect can be estimated by moving the defect from a bulk-like position (sixth layer of the slab) to a location at the surface layer. The segregation energy is defined as E_{(defect in 1st layer)} - E_{(defect in 6th layer)}, hence a negative value indicates a preference for the surface. We always calculate the energies of the first layer and sixth layer defects with exactly the same surface dangling proton patternation.

As can be seen in Fig. 4, there is a clear correlation between the number of neighboring dangling protons and the segregation energy of the ionic defects. Dangling protons, due to their location above the surface, are poorly screened and can interact strongly with the ionic defect’s charge. Fig. 4 shows that the presence of nearby dangling protons stabilizes the hydroxide defect but destabilizes the hydronium. For both species the interaction is linear with the number of proton neighbors, with a very similar energy cost (gain) per proton of 0.14 eV and, again, consistent with a short-ranged proton-proton interaction. In their respective optimal environments, by which we mean the hydroxide surrounded by six protons and the hydronium with no proton neighbors, the hydronium ions have larger segregation energies than the hydroxide species, though only by a modest margin (-0.95 vs. -0.87 eV). However, in their expected environments at low temperatures with three proton neighbors the respective segregation energies (-0.46 eV for hydronium and -0.50 eV for hydroxide) are within the variation in energy expected of a bulk-like species due to proton disorder.

Fig. 4.

Surface segregation energies of ionic defects at the ice surface as function of neighboring dangling protons. H₃O⁺ squares and OH^- as circles.

Having established that both ionic defects show a preference for the ice surface we carried out Born–Oppenheimer molecular dynamics (BOMD) simulations to investigate how their dynamics are influenced by segregation to the surface. In the bulk, both ionic defects can diffuse rather rapidly, on a timescale of 10–100 femtoseconds, consistent with previous reports in the literature (29). However, on segregating to the surface, we find that the mobility of both species is reduced by orders of magnitude. For OH^- and H₃O⁺ at the surface we observe only one and two diffusion events respectively in simulations of over 10 ps at 270 K, suggesting diffusion barriers of the order 0.1 eV. Starting the simulations from configurations with the respective ionic defects in the bulk, we observed segregation to the surface within 2–5 ps (the direct distance to the surface is ≈1 nm, though the defects must take a longer path due to the nature of the hydrogen-bonded network). The barrier estimate of > 0.1 eV is consistent with our calculation that sites with differing numbers of proton neighbors have energy differences of ca. 0.15 eV per change of proton neighbors—ionic defects cannot diffuse freely on the ice surface but experience a barrier in moving between sites with differing local proton environments.

We finally turn to consider the experimental evidence (9) that the trapping of D-defects at certain sites in polycrystalline ice samples is more significant than for the L-defects. Previously, molecular defects—vacancies and interstitials—have been investigated theoretically as possible trapping sites for Bjerrum defects (23, 24), but no clear evidence of a trapping site that differentiated between the two Bjerrum defects emerged from these studies. Another possibility is the complexation of the Bjerrum defects to the opposing charged ionic species; to explain the experimental finding that L-defect mobility is higher than that of the D-defect, the OH^- species should be more efficacious at trapping the D-defect than the L-defect trapped at H₃O⁺ species (6, 9). Recent theoretical work of Cwiklik et al. (26, 27) has identified a stable hydroxide species in an “offsite” position, or interstitial trap, that corresponds to a D-defect complexing to the hydroxide ion. Here we consider and compare the trapping of L- and D-defects at hydronium or hydroxide defects respectively within the bulk.

Using a bulk cell containing 360 water molecules we created a pair of ionic defects and separated them as far as possible (1.9 nm) within the supercell, at a cost of 1.11 eV, shown in Fig. S4. From this configuration an additional DL-defect pair was created with the D-defect bound to the hydroxide. The L-defect was then migrated through the crystal to bind with the hydronium. The Bjerrum defects bind to the ionic defects creating a [hydroxide-D] complex, which accepts four hydrogen bonds and contains an offsite proton (26) and a [hydronium-L] complex, which donates three hydrogen bonds and leaves an empty space between the hydronium oxygen and one neighboring water molecule (Fig. S4). This configuration with a pair of defect complexes was found to be 0.33 eV higher in energy than the system with a pair of ionic defects only. By moving the D- or L-defects away from their ionic defect partners, the trapping energies of the respective defect pairs in bulk ice, [H₃O⁺-L] or [OH^--D], were found to be 0.40 eV and 0.48 eV, respectively. These results are consistent with measurements in cubic ice by Wooldridge and Devlin, who estimated the trapping energy of hydronium ions to be 0.43 eV and suggested that the trapping sites were L-defects (8). The trapping energies for the two complexes are equal within the expected energy range caused by proton disorder in the bulk. For both bound complexes, the mobility of the complexed pair is reduced by orders of magnitude from that of the isolated OH^- or H₃O⁺ species. This observation comes from a two picosecond BOMD simulation where no migration of either complex is observed. By way of contrast, if we start the simulation from a configuration where an L-defect is five lattice sites away from an H₃O⁺, the hydronium rapidly (within 100 fs) migrates to, and then combines with, the L-defect.

For the trapping of an individual ionic defect by a Bjerrum defect released by the surface to be energetically favored, the surface DL pair formation energy should be less than the trapping energy. Comparison of the data in Fig. 3 and our estimates of trapping energies, 0.40 eV and 0.48 eV, shows that trapping is expected to be unfavorable for all [L-hydronium] pairs beyond the immediate surface. For the D-defects, the energetics suggest that some of those dangling surface proton sites with more than two neighboring protons would be energetically capable of trapping hydroxide ions in the bulk of the crystal. It is notable that the difference in trapping energies (0.40 vs. 0.48 eV) is significantly smaller than the difference in the energy required to release the appropriate Bjerrum defect from the surface (0.70 vs. 0.52 eV). In all cases trapped complex species will be rather shallow and metastable compared to the segregation of their component species to the surface.^∥

Discussion

We now summarize the main findings:

1. The surface segregation energies of the isolated, charged ionic defects cannot be distinguished with the model and level of theory used here, and thus there is no preference for an excess of hydronium or hydroxide at the surface in the absence of Bjerrum defects.

2. The formation energies of the D- and L-defects at the surface are considerably different. Surface D-defects form with a very small energy penalty of 0.06 eV while the formation of L-defects is 0.36 eV less favorable with an overall energy of 0.42 eV. The difference in formation energies arises because creating an L-defect adds one dangling proton to surface, increasing Coloumb repulsion; D-defect formation removes a surface dangling proton, reducing Coulomb repulsion.

3. All the defects considered in this paper show a significant preference for segregation to the surface, especially the D Bjerrum defect (0.46 eV), the hydronium (0.46 eV) and hydroxide defects (0.52 eV). The L Bjerrum defect appears to segregate least strongly of the four defects considered with an energy of 0.28 eV. However, the overall formation energy of the D-defect in bulk is more favorable than the L-defect by 0.18 eV (0.52 eV for D and 0.70 eV for L).

4. Many surface sites can inject individual Bjerrum defects into the bulk at energies greatly reduced from the bulk DL pair formation energy of ca. 1.2 eV. However, when we consider the structure of the surface, we find a significant preference for the movement of D-defects into the bulk compared to L-defects, where relatively common dangling proton sites with three or more proton neighbors can act as low energy sources of D-defects. However, nondangling proton sites with two or fewer proton neighbors, which could act as sources of L-defects, are very rare, estimated to occur once every 5,625 nm². D-defects are peculiar in having low energy states in very near-surface layers, which could provide additional trapping sites for hydroxide ions.

5. The migration of D-defects from the surface to the crystal interior produces a dipole with negative charge at the surface. The presence of the dipole will preferentially attract hydronium species to the surface to counter the dipole leading to a build up of positively charged sites at the surface leading to surface acidity.

6. The trapping energies of [hydronium-L-defect] and [hydroxide-D-defect] complexes are 0.40 and 0.48 eV, respectively, within the crystal bulk. Taking into account the formation energies of the DL pairs at the surface, some surface dangling bond sites with three or more neighboring protons could trap bulk-like hydroxide ions with modest energy gain. However, these states will be metastable compared to the hydroxide segregated to the surface. The greater cost of forming subsurface L-defects means that only extremely rare surface sites would favorably trap hydronium species.

7. Ionic defect diffusion is greatly reduced at the surface due to the inequivalence of surface sites with distinct neighboring proton environments. A crude estimate of the diffusion barrier is ≈0.15 eV, from our calculation of the dependence of ionic defect binding energy on local proton environment. The patterning of protons at the surface leads to less facile diffusion pathways than within the crystal bulk.

Like previous predictions of ionic defects at water surfaces (30), the most prevalent water ice surface also appears to present excess proton sites at the crystal surface, but the mechanism by which this space charge effect takes place is manifestly different to that in water. While the onset of premelting will undoubtedly influence the distribution of defects, our calculations provide insight into the complex surface structure of real crystalline hexagonal ice, in a temperature regime (< 180 K) relevant to upper atmosphere heterogenous chemistry. In particular, the existence of charged sites at the surface of ice may help to explain the uptake behavior and reaction chemistry of the ice surface.

Materials and Methods

We used the Quickstep (31) module of the CP2K program suite to perform density functional theory calculations using the PBE functional (32). GTH pseudopotentials (33) were used with TZV2P quality gaussian basis sets and a 300 Ry energy cutoff for the auxiliary plane-wave basis. A large slab containing 360 water molecules with cell dimensions of 2.215 × 2.302 nm² was used along with a 2D Poisson solver (34) for surface calculations. Calculations within the bulk are carried out in a conventional 3D supercell with dimensions 2.215 × 2.302 × 2.155nm³. In static calculations all ionic coordinates were optimized using a limited memory BFGS algorithm (35) until the maximum force on any ion was less than 0.0045 AU/bohr. BOMD simulations were carried out within NVE or NVT ensembles at nominal temperatures of 270–350 K, and the equations of motion were integrated using a time step of 0.4 fs with the mass of hydrogen atoms set to 2 AU for computational expediency. Our supercells of ice were taken from the work of Hayward and Reimers (36) for a system containing 360 water molecules.The surfaces of our slab model are reasonably stable with an order parameter averaged over all dangling proton sites of ca. 2.3 order parameter (15), and thus the model is likely to be a realistic representation of the surface at moderate temperatures before substantial premelting occurs. In order to sample less likely arrangements of surface protons we have been forced to modify our surfaces to artificially produce sites with extreme values of the order parameter. To create sites with 0–1 and 5–6 neighbors it has been necessary to modify the surface; the reported segregation energies always use the appropriate reference state with consistent dangling proton distributions. We observe that the energies of the defects in the interior of the cell rapidly converge to a bulk value and only show a variation of ca. 50 meV from site to site, which we attribute to slight changes of environment due to proton disorder (20, 22). Additional details are in SI Text.

Fig. 5.

Geometry of hydronium and hydroxide surface defects. Only the upper bilayer or water molecules is shown for clarity. The bond length of the defect to its neighboring second layer water molecules are shown (in Å). For comparison bulk hydrogen bond distances are ≈1.70 Å. Top of the bilayer is coloured red while the lower layer is blue.