Physical Chemistry Chemical Physics c2cp23438f

Q1Non-hexagonal ice at hexagonal surfaces: the role oflattice mismatch

Stephen J. Cox, Shawn M. Kathmann, John A. Purton,Michael J. Gillan and Angelos Michaelides

Grand canonical Monte Carlo has been employed to studythe role of lattice mismatch in ice nucleation.

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Q1 Non-hexagonal ice at hexagonal surfaces: the role of lattice mismatchw

Stephen J. Cox,a Shawn M. Kathmann,b John A. Purton,c Michael J. Gilland andAngelos Michaelidesa

Received 31st October 2011, Accepted 16th April 2012

DOI: 10.1039/c2cp23438f

It has long been known that ice nucleation usually proceeds

heterogeneously on the surface of a foreign body. However, little

is known at the microscopic level about which properties of a

material determine its effectiveness at nucleating ice. This work

focuses on the long standing, conceptually simple, view on the

role of a good crystallographic match between bulk ice and the

underlying substrate. We use grand canonical Monte Carlo to

generate the first overlayer of water at the surface and find that

the traditional view of heterogeneous nucleation does not ade-

quately account for the array of structures that water may form

at the surface. We find that, in order to describe the structures

formed, a good match between the substrate and the nearest

neighbour oxygen–oxygen distance is a better descriptor than a

good match to the bulk ice lattice constant.

The formation of ice is one of the most common everyday

phenomena, yet the underlying physical processes that deter-

mine its nucleation rate are poorly understood. In the absence

of impurities, water can be held in a supercooled state down to

approximately �40 1C but by far the more prevalent form of

freezing, at temperatures closer to the melting point, is that

facilitated by a surface. This process, termed heterogeneous

nucleation, is a crucial process in atmospheric science where,

for example, it has been found to play an important role in

cirrus and mixed-phase cloud formation,1–3 with consequences

regarding the radiative budget of the Earth’s atmosphere4,5

and the hydrological cycle.6–9 Understanding the microscopic

detail of heterogeneous ice nucleation is also of biological

relevance10–12 and has implications for the airline and food

industries, as well as providing potential insight into clathrate

hydrate formation.13

There are numerous textbook arguments (e.g. insolubility,

size or defects) for a substrate to be an effective ice nucleation

agent (IN) but an important one, which lead to AgI being used

as a cloud seeding agent,14 is a good crystallographic match

with bulk hexagonal ice.15,16 The crystallographic match can

be quantified by the disregistry (or ‘‘mismatch’’), defined in a

simplified manner as

d � a0;S � a0;i

a0;ið1Þ

where a0,S is the lattice constant of the substrate and a0,i is the

bulk lattice constant of ice Ih. The original theory, developed

within the context of classical nucleation theory by Turnball

and Vonnegut,16 envisages a situation where the growing

crystal can be pictured as consisting of regions of good fit

(strained slightly to fit the underlying lattice) bounded by line

dislocations.17 Furthermore, through the interpretation of

early experimental data (in particular from low energy electron

diffraction experiments), the concept of an ice-like bilayer

forming at pristine metal surfaces was developed in which

the water molecules occupy 2/3 of adsorption sites (2/3

monolayer coverage).18

This theory is often used as an explanation for the excellent

ice nucleating abilities for many materials, including kaoli-

nite,15 AgI16 and long chain aliphatic alcohols.19 However, in

certain specific cases, it has been questioned what role, if any,

the lattice mismatch plays in ice nucleation.20–22 Indeed,

density functional theory (DFT) calculations on idealised ice

bilayers at different metal surfaces23 have shown that through

relaxation normal to the surface, the water molecules manage

to maintain near constant hydrogen bonding energy, in con-

tradiction to the traditional theory (Fig. 1B). Also, recent

experimental evidence, especially from scanning tunneling

microscopy, has revealed a number of complex structures of

water on the first overlayer of metal surfaces, inconsistent with

the bilayer model.24–27

Here, through the use of grand canonical Monte Carlo

(GCMC) and simple intermolecular potentials, we shall probe

the effect of the lattice constant of the substrate on the

structure of the first water overlayer formed. It should be

noted that of the four basic modes of heterogeneous ice

nucleation,15 we are most closely simulating the deposition

mode, in which water is adsorbed directly from the vapour

phase onto the surface where it forms the ice phase. Although

there have been previous theoretical works that have used

model surfaces to study the structure of interfacial water (e.g.

ref. 28–31), none has explicitly tackled the effect of changing

the lattice parameter of the substrate alone on the structures

that form in the first water overlayer. What we will see is that,

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a Thomas Young Centre, London Centre for Nanotechnology andDepartment of Chemistry, University College London, LondonWC1E 6BT, U.K.

bChemical and Materials Sciences Division, Pacific NorthwestNational Laboratory, Richland, Washington 99352, United States

c STFC, Daresbury Laboratory, Warrington WA4 4AD, U.K.d Thomas Young Centre, London Centre for Nanotechnology andDepartment of Physics and Astronomy, University College London,London WC1E 6BT, U.K.

w Electronic supplementary information (ESI) available. See DOI: 10.1039/c2cp23438f

This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, ]]], 1–6 | 1

PCCP Dynamic Article Links

Cite this: DOI: 10.1039/c2cp23438f

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in contrast to the traditional theory, the first water overlayer

consists of a significant fraction of non-hexagonal arrange-

ments, with smaller lattice constants favouring smaller sized

rings and fully hexagonal overlayers only observed on sub-

strates with expanded lattice constants. We will argue that, as

most molecules in the contact layer are of a similar height, the

nearest neighbour oxygen–oxygen distance r0,i becomes more

relevant than the bulk ice lattice constant a0,i in describing the

structures observed (see Fig. 1), consistent with recent experi-

mental observations on close packed metal surfaces.24,32

We wish to obtain a qualitative overview of the influence of

the lattice constant alone on the structures that form in the

first water overlayer. DFT, whilst proven to be a useful

method for investigating interfacial water (e.g.

ref. 24,27,32–34), would be inappropriate for such a study

not only because of the restriction in system size that could be

used (we are using surfaces with areas on the order of 50 � 50

A2 and sampling millions of configurations), but also because

we would be unable to freely vary the lattice parameter of the

surface without changing other properties of the surface. To

this end, we shall use intermolecular potentials to model the

interaction between the water and the surface. Even though

the use of simple potentials cannot describe the ‘chemistry’

present at real surfaces, this is to our advantage in that we can

effectively decouple the lattice parameter from other proper-

ties of the surface. However, to be as careful as we can to

ensure our observed trends are general and not just applicable

to one type of model, we have used two types of water–surface

potential and two standard water models. Details of these tests

can be found in the ESI,w along with tests using Møller–Plesset

perturbation theory on small clusters formed during these

simulations. It must be stressed that this work is not aiming

to model water layers at real surfaces, but to probe an often-

used, long-standing theory that only concerns the geometrical

distribution of adsorption sites, which has so far proved

inadequate in a number of cases.24–27

The first type of surface employed consists of four layers of

points each in a hexagonally closed packed arrangement,

interacting with the oxygen atoms of the water molecules

through a Lennard-Jones interaction. The adsorption sites

are located at the HCP and FCC sites of this surface, giving

rise to a honeycomb arrangement of potential energy minima

for the water monomer – we shall refer to this surface as the

‘‘honeycomb’’ surface. The second surface used is an explicitly

defined external potential acting on the oxygen atoms of the

water molecules. The adsorption sites on this surface are in a

hexagonally close packed arrangement – this surface is re-

ferred to as the ‘‘HCP’’ surface. On both surfaces, the mono-

mer adsorption energy was chosen to be comparable to, but

slightly stronger than, the TIP4P dimer binding energy

(�26.09 kJ mol�1)35 and similar to the optB88-vdW func-

tional36,37 value on Ag(111)38 (�27.112 kJ mol�1).39 Full

details of both these surfaces can be found in the ESI.wThe simulations presented here all used the TIP4P model, a

rigid simple point charge model, to represent the water–water

interaction.40 The TIP4P water model has been shown to

reproduce the phase diagram of water qualitatively well,41

even though it predicts a melting temperature of ice Ih to be

Tm = 232 K. Furthermore, TIP4P has been shown to repro-

duce the intermolecular vibrational density of states of ice Ih

obtained by inelastic neutron scattering reasonably well.42 As

mentioned previously, in order to see how dependent the

results are on the choice of water force field used, the simula-

tions have also been performed with the SPC/E water model,43

with good agreement between the two models (see ESIw).GCMC simulations were performed on six different slab

geometries for the two types of surfaces at 215 K corres-

ponding to a supercooling of 17 K (the SPC/E calculations

were performed at 198 K to achieve the same supercooling),

each with different lattice constant, using the simulation

package DL_MONTE.44 The surface areas varied between

slabs (ranging from 47.51 � 41.14 A2 to 55.41 � 47.99 A2) but

the cell volume was kept constant between simulations, ensur-

ing there was a vacuum of at least 42.65 A between periodic

images in the z-direction. Full periodic boundary conditions

were employed and electrostatic interactions were treated

using the three-dimensional Ewald method45 and short range

interactions were calculated by employing a spherical cut-off

of 17 A. A GCMC step could be either a molecule translation,

rotation, insertion or deletion and these were attempted with a

ratio of 33 : 33 : 17 : 17. The surface was fixed throughout.

Simulations were run until a single overlayer of water was

obtained. After the GCMC simulations, final structures were

annealed at 15 K and water molecules whose oxygen atom was

greater than 3.5 A above the surface were removed. The

structures were then annealed for a further 5 � 106 MC steps

at 15 K.

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Fig. 1 Schematic of the first water overlayer at a surface. (A) Part of

a single isolated ice bilayer shown from the side (left) and top (right).

The bulk ice lattice constant a0,i and nearest neighbour oxygen–oxygen

distance r0,i are labelled. (B) Ice bilayer at the surface according to the

traditional theory. On the left, the ice bilayer has strained to fit the

lattice of the surface, which has a lattice constant a0,S slightly larger

than bulk ice. On the right, the bilayer is allowed to relax normal to the

surface. (C) Along with relaxation normal to the surface, we can also

envisage a scenario where changes in morphology in the plane of the

surface are favourable.

2 | Phys. Chem. Chem. Phys., 2012, ]]], 1–6 This journal is c the Owner Societies 2012

GCMC is a powerful technique most often used as a method

for calculating adsorption isotherms. However, here we are

using it to efficiently sample configuration space when gener-

ating the single water overlayers.46 The GCMC simulations

generated an array of structures in the first overlayer on the

different surfaces, with coverages on the HCP surfaces (d =

�0.07 to 0.10) increasing approximately linearly from B0.61

monolayer coverage to B0.78 monolayer coverage. This is in

contrast to the traditional theory in which 2/3 monolayer

coverage is predicted. Fig. 2 presents snapshots of water layers

obtained for selected values of d on both the honeycomb and

HCP surfaces. One trend is immediately clear; as the mismatch

is increased, more open ring-like structures are favoured,

whereas as d is decreased, we begin to see more closely packedarrangements. However, perhaps the most striking feature of

these pictures is that on the d = 0.00 surfaces, there is still a

significant proportion of non-hexagonal structures present,

with the only fully hexagonal overlayers being formed on the

honeycomb surface for d Z 0.07.

Presented with these results, it is interesting to ask why we

see such an array of structures in the different overlayers and

why it is only on larger d that we see hexagons starting to

become more favourable. To address these questions, it is

informative to look at the radial distribution functions

(RDFs) and the energies of different overlayers, which are

given in Fig. 3 (the corresponding plots for the honeycomb

surface are included in ESIw). From the RDFs, we can see that

the position of the first peak is quite insensitive with regard to

the underlying substrate, lying at approximately 2.75 A. The

significance of this can be understood by revisiting the defini-

tion of the disregistry, d (eqn (1)) and looking more closely at

the ‘ice bilayer’ model.

In bulk ice Ih, the oxygen atoms lie in layers perpendicular

to the c-axis, each layer being built up of puckered hexagonal

rings (a single ice ‘bilayer’), arranged in ABABAB. . . stacks. A

schematic of a single ice bilayer is shown on the left of Fig. 1.

The lattice constant, a0,i corresponds to the distance between

two nearest coplanar oxygen atoms and not the distance

between two nearest hydrogen bonded water molecules (la-

belled r0,i in Fig. 1). From the RDFs it appears that the water

is able to relinquish long range order in favour of maintaining

a constant nearest neighbour distance. It should be stressed

that this ‘‘constant’’ nearest neighbour distance refers to the

position of the first peak and that this peak exhibits a finite

width (ranging from just under 2.6 A to approximately 3.2 A).

In addition, rather than forming an ice bilayer at the surface, a

flat monolayer forms (see Fig. S5 in ESIw). With these last two

points in mind, simple geometry is enough to explain why we

see such a significant proportion of pentagons at the d = 0.00

surface; the distance between adjacent sites of potential energy

minima r0,S is not large enough to accommodate a fully

hexagonal overlayer with a favourable r0,i. As the surface is

stretched, r0,S becomes larger allowing easier formation of

hexagons at the surface. When the surface is compressed,

however, one way that the water molecules can arrange

themselves such that they keep a favourable r0,i and bind near

the adsorption sites is to adopt structures with smaller O–O–O

angles.

The energy profiles in Fig. 3 are useful in explaining these

observations further, where we have separated the different

contributions to the total energy (water–water and water–sur-

face interactions)47 for the different overlayers. The first thing

to note is that there is surprisingly little variation in the total,

water–water and water–surface interactions as the lattice

constant of the substrate is changed (the scale on each panel

is 4 (kJ mol�1)/H2O E 41 meV/H2O). There is a slight

stabilisation in the total energy on the expanded substrates,

due to an increase in binding to the surface. More importantly,

there is very little change in water–water energy, indicating

that water is able to relinquish long range order without

incurring a severe energy penalty. This tells us that on surfaces

where the water–water and water–surface interactions are

comparable, the cost of forming an overlayer that is commen-

surate with the substrate (with r0,S o r0,i) outweighs the gain

in binding to the surface. Therefore, provided the water

molecules can still bind near to the favoured adsorption sites,

the water molecules will alter their O–O–O angles in order to

maintain a favourable nearest–neighbour distance.

With these results, we are now in a position to compare to

the traditional theory of heterogeneous nucleation,16 where it

is assumed that for d sufficiently small, the growing crystal willstrain to fit the substrate lattice. What is unaccounted for in

this formalism (and what is observed in these simulations),

however, is the ability of the first overlayer to change topol-

ogy; firstly, the water molecules bind as a flat layer to increase

their binding to the surface and secondly, they distort their

O–O–O angles so as to maximise their nearest–neighbour

water–water interactions. To make a semi-quantitative com-

parison to the traditional theory, we can try to estimate the

stability of the traditional ice bilayers relative to the structures

found through GCMC. In order to do this, we took a single ice

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Fig. 2 Snapshots for selected values of d. The top row shows the

honeycomb surface and the bottom row shows the HCP surface. The

value of d (disregistry, as defined in eqn (1)) is shown in the top left

corner. We can see that on both d= 0.00 surfaces there is a significant

proportion of non-hexagonal arrangements of water molecules, the

effect being more pronounced on the HCP surface. As d is increased

hexagons become more favoured and conversely, as d is decreased, wesee an increase of more compact structures. For clarity, only a section

of the entire simulation cell is shown and the atoms have been depicted

to be the same size.

This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, ]]], 1–6 | 3

bilayer (consisting of 24 molecules) from a bulk ice Ih struc-

ture and geometry optimised for various sizes of the lattice

constant using the GROMACS simulation package.48 We

performed two types of geometry optimisations; one set where

we constrain the positions of the water molecules such that the

c/a ratio is constant and another set where we allow the

heights of the water molecules to vary, only constraining the

layer to maintain hexagonal symmetry.49 The results from this

analysis are shown in Fig. 4, where we can see that the

structures generated by GCMC are more stable by approxi-

mately 2–3 kJ mol�1 per H2O relative to the most stable

bilayer (the total adsorption energy is shown in the bottom

panel). From inspection of the contribution to the total energy

arising from water–water interactions (middle panel) we can

see that, on the whole, the hydrogen bonding network is

slightly destabilised compared to the bilayer in which the

height of the water molecules is allowed to relax. In fact, for

d o 0.03, the conventional ice bilayer has a more stable

hydrogen bonding network than the GCMC overlayers.

Where the GCMC overlayers gain their overall stabilisation,

however, is by binding in a flat layer to the surface; the

increase in water–surface interaction offsets the very small

energy penalty in forming a flat hydrogen bonded network

relative to a buckled one.

Although we have only used simple potentials to describe

the water–surface interaction, we note here that this conclu-

sion is consistent with a recent DFT study50 in which ‘‘H-

down’’ and ‘‘H-up’’ bilayers were studied at Cu(110) and

Ru(0001): the H-up bilayers tend to have a corrugation

comparable to that of bulk ice and stronger hydrogen bonding

than the H-down bilayers, whereas the latter form flatter

overlayers with a weaker hydrogen bonding network, but have

an increased binding to the surface. We include a more

comprehensive comparison, along with a detailed assessment

of the water-surface interaction, in the ESI.wIn this work we have investigated the role of lattice mis-

match alone on the structure of the first water overlayer

through the use of GCMC and model hexagonal surfaces.

We find that, in direct contrast to the traditional theory, the

first overlayer alters its topology away from that of the

traditional bilayer model in order to maximise its interaction

with the surface and in doing so, suffers little energy cost

regarding its water–water interactions. We explain this ob-

servation by the fact that once the water molecules relax along

the surface normal, there is very little energy associated with

changing the O–O–O angles and that independent of the

underlying substrate lattice constant, the water molecules

maintain a near constant O–O nearest neighbour distance

(although our results do not preclude the existence of short

O–O distances due to other driving forces, for example, charge

transfer to the surface). Our results also suggest that, unless

there is an inherent corrugation of the adsorption sites on the

surface, one should not expect to observe fully hexagonal

overlayers for substrates that have a mismatch close to zero,

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Fig. 3 Left: radial distribution functions for different d (HCP surface). Note that the position of the nearest–neighbour peak is essentially

invariant as the substrate lattice constant is changed, located at approximately 2.75 A. Right: contributions to the total energy as d varies. There isa slight stabilisation in the total energy (bottom) as d is increased, which arises from an increased binding to the surface (top). This trend is seen as

r0,S is closer to r0,i for expanded d, meaning more water molecules can bind close to preferred adsorption sites with a favourable nearest–neighbourdistance. The water–water interaction (middle) is very flat, indicating that varying the O–O–O angle is a facile process.

Fig. 4 Comparison to the traditional theory of heterogeneous ice

nucleation. The bottom panel shows the total energy of the different

types of overlayer at the HCP surface, the middle panel shows the

water–water energy and the top panel shows the interaction energy of

the water with the surface. ‘‘Fully constrained’’ is a bilayer whose c/a

ratio was kept constant, whereas ‘‘xy-constrained’’ is a bilayer in

which the relative heights of the water molecules was allowed to

change whilst keeping the lateral coordinates of the water molecules

fixed. We can see that by all water molecules binding closely to the

surface, the structures generated by GCMC are stabilised relative to

the bilayer models.

4 | Phys. Chem. Chem. Phys., 2012, ]]], 1–6 This journal is c the Owner Societies 2012

consistent with recent experimental findings.32 Not only does

this work confirm the findings that the lattice match alone is

insufficient to make a material an efficient IN,20,21,28,51,52 but is

also suggests that a good lattice match in the conventional

sense (eqn (1)) may possibly be detrimental to ice nucleation,

as large molecular rearrangement in the first water overlayer

may be required to form an ice nucleus.53 Indeed, if the first

water overlayer bears little resemblance to an ice bilayer, it is

not entirely obvious what role it plays in heterogeneous ice

nucleation. Future work to try and understand this phenom-

enon will focus on details of the water surface interaction (e.g.

binding energy and corrugation), as well as on defects at the

surface, as it is likely that they either give rise to electric fields

(computed from either classical point charges or ab initio via

the nuclear plus the electronic charge density) that enhance the

ordering of interfacial water molecules,54 or allow the facile

reconstruction of the first water overlayer.

Acknowledgements

The authors are grateful to Dr Gregory Schenter for many

useful discussions and Dr Peter Feibelman for his comments

on the manuscript. This research used resources of the Na-

tional Energy Research Scientific Computing Center, which is

supported by the Office of Science of the U.S. Department of

Energy under Contract No. DE-AC02-05CH11231. We are

grateful to the London Centre for Nanotechnology and UCL

Research Computing for computational resources. via our

membership of the UK’s HPC Materials Chemistry Consor-

tium, which is funded by EPSRC (EP/F067496), this work

made use of the facilities of HECToR, the UK’s national high-

performance computing service, which is provided by UoE

HPCx Ltd. at the University of Edinburgh, Cray Inc., and

NAG Ltd., and funded by the Office of Science and Technol-

ogy through EPSRC’s High End Computing Programme.

A.M. is supported by the ERC.

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39 J. Carrasco, J. Klimes and A. Michaelides, unpublished Q2.40 W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey

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configuration space, there is no guarantee that it will generate thelowest energy structures. In fact, it should also be noted that theperiodicity imposed by the boundary conditions may play a role in

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preventing the lowest energy structures being found. However,byusing large simulation cells (between 288 and 400 adsorption sites),we reduce this finite size effect as much as possible within ourcomputational resources.

47 In general, such energy decompositions are arbitrary since it isassumed that the hydrogen bonding within the overlayer does notchange upon approaching the surface. When using forcefields, thisassumption is correct but in general this is not the case. For a moredetailed discussion on this type of energy decomposition see, forexample, ref. 23.

48 B. Hess, C. Kutzner, D. van der Spoel and E. Lindahl, J. Chem.Theory Comput., 2008, 4, 435.

49 The total adsorption energy of the bilayers at the HCP surface byassuming that all water molecules lie directly above the adsorptionsites and that the low-lying water molecules are at the optimalheight. As we know the functional form of the water surfaceinteraction, calculating the energy of the high-lying water mole-cules is trivial.

50 J. Carrasco, B. Santra, J. Klimes and A. Michaelides, Phys. Rev.Lett., 2011, 106.

51 X. L. Hu and A. Michaelides, Surf. Sci., 2008, 602, 960.52 V. Sadtchenko, G. Ewing, D. Nutt and A. Stone, Langmuir, 2002,

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