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Langmuir 2002, 18, 4632-4636
Instability of Ice Films Vlad Sadtchenko and George E. Ewing* Department of Chemistry, Indiana University, Bloomington, Indiana 47405
David R. Nutt and Anthony J. Stone* University Chemical Laboratories, University of Cambridge, Cambridge, England Received January 15, 2002. In Final Form: April 4, 2002 The suggestion that a substrate whose architecture matches that of one of the faces of ice will make an effective nucleating agent goes back over 50 years. Over the years a number of experimental tests of this suggestion have been offered together with a variety of theoretical explanations to rationalize the disappointing nucleation abilities of most substances. In our study we have selected what might appear to be an ideal nucleating substrate, namely, BaF2, whose (111) face lattice constants are a near match to the basal face of Ih ice. Yet thin-film ice on BaF2(111) proves to be unstable. We report a novel infrared spectroscopic probe of thin ice films and present a theoretical analysis to account for the instability of ice on BaF2(111) and similar substrates. We find that the details of the water-surface interaction override the advantage of a close surface-ice lattice match. We summarize the conditions that need to be met for an effective nucleating substrate.
Water is not easy to freeze. This statement might be challenged by anyone who pours water into a tray, places it in the freezer compartment of a refrigerator, and, on returning in several hours, finds ice cubes ready to serve up. But this crystallization has been accomplished with tap water contaminated with traces of salts, dust, and dissolved air and the freezer, rather than being set near the thermodynamic phase transition of 0 °C, has a typical temperature of about -20 °C. Dorsey, in a series of meticulously executed experiments spanning nearly a decade, placed a variety of water samples in carefully cleaned ampules and observed their freezing temperatures.1 In thousands of trials he never observed a sample to freeze higher than -3 °C (e.g., water from a scummedover stagnant pool) and many remained supercooled below -20 °C (e.g., a Washington, DC, tap water sample). Highly purified water fell between these extremes. The possible influence of container walls has been eliminated in other experiments using levitated droplets. Some droplets 10 µm in diameter, a typical size found in clouds, freeze near -40 °C.2 Smaller droplets have an even greater tendency to supercool,2 and a likely record is for water clusters of about 5 nm diameter that do not freeze until -70 °C.3 The presence of foreign substances is known to facilitate the freezing of water. Indeed, Dorsey was convinced that, even in his most highly purified samples, trace impurities served as heterogeneous nucleation sites for the freezing process. Today suspensions of bacteria are mixed in with water sprayed over ski slopes to favor the production of artificial snow at temperatures only slightly below 0 °C.4 The mechanisms of heterogeneous nucleation are still uncertain.5 A logical strategy to promote freezing occurred to cloud physicist Bernard Vonnegut when he reasoned that inorganic substrates whose surface architecture nearly matched that of either the basal or prism faces of * To whom correspondence should be addressed. E-mail:
[email protected],
[email protected]. FAX: (812) 855-8300. (1) Dorsey, N. E. Trans. Am. Philos. Soc. 1948, 38, 246. (2) Fletcher, N. H. The Chemical Physics of Ice; Cambridge University Press: Cambridge, 1970. (3) Huang, J.; Bartell, L. S. J. Phys. Chem. 1995, 99, 3924. (4) Snomax snow inducer. http://www.snomax.com/products/snomax/. (5) Pruppacher, H. R.; Klett, J. D. Microphysics of Clouds and Precipitation, 2nd ed.; Kluwer Academic Publishers: Dordrecht, 1997.
ice should be good nucleating agents.6 A search of the crystallographic literature uncovered AgI, whose lattice constants matched those of ice to within a few percent. And indeed a smoke generated by sublimation of AgI crystals was found to be particularly effective in cloud seeding experiments. But the mechanism by which AgI and many other inorganic substrates7 facilitate ice freezing is not clear. To begin, why does ice grow as hexagonal patches on the substrate rather than as a uniform film?2 Why is testosterone powder, with no obvious morphological similarities to ice, as effective as many inorganic substrates?8 If selected inorganic substrates are such good matches with the ice lattice, why is supercooling required?2 Fletcher presented an elegant analysis of the nucleation of ice films on AgI and similar substrates.9 He argued that the proton alignment in ice, a consequence of the surface bonds with the substrate, lowers its entropy relative to ordinary (proton disordered) ice. The supported ice film is then entropically unfavored over ordinary ice. It is for this reason that certain faces of inorganic substrates may actually inhibit ice nucleation. In a later paper,10 Fletcher considers defects such as steps, corners, and edges, rather than the well-ordered ions on the smooth faces of inorganic substrates, as the favored nucleation sites. In the 50 years since Vonnegut’s original investigation, numerous experiments7,11 and theoretical models12,13 have explored the structures of ice films on inorganic substrates but none has successfully addressed the reasons for their stability or instability. We introduce in this paper a novel apparatus for the preparation of ice films on well-defined substrates, its interrogation by infrared spectroscopy, and an accompanying theoretical analysis based on accurate intermolecular potentials. Because of the very close match between (6) Vonnegut, B. J. Appl. Phys. 1947, 18, 593. (7) Hallett, J.; Mason, B. J. Proc. R. Soc. London, Ser. A 1958, 247, 440. (8) Head, R. B. Nature 1961, 191, 1058. (9) Fletcher, N. H. J. Chem. Phys. 1959, 30, 1476. (10) Fletcher, N. H. Aust. J. Phys. 1959, 13, 408. (11) Bluhm, H.; Inoue, T.; Salmeron, M. Surf. Sci. 2000, 462, L599. (12) Odelius, M.; Bernasconi, M.; Parrinello, M. Phys. Rev. Lett. 1997, 78, 2855. (13) Witek, H.; Buch, V. J. Chem. Phys. 1999, 110, 3168.
10.1021/la0255370 CCC: $22.00 © 2002 American Chemical Society Published on Web 05/16/2002
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Figure 1. Assembly for the preparation and infrared study of ice. A BaF2 prism serves as both a substrate for ice growth and an optical component for spectroscopic interrogation. Water vapor is admitted through the solenoid valve into the assembly chamber. Ice forms at the bottom of the prism, the BaF2(111) face, at -65 °C.
the lattice constant of ice and that of BaF2(111), we have chosen the surface of BaF2(111) as the model substrate. A schematic drawing of the optical cell used for the infrared measurements is shown in Figure 1. The trapezoidal prism is fabricated from a single crystal of BaF2 with its 〈111〉 directions aligned along the long prism axis and the side faces. All faces are polished except the lower one that forms the upper wall of the vapor chamber. Just before cell assembly, this wall is prepared by cleaving with hammer and chisel the lower prism side to expose a new (111) face. This is a (nearly) defect-free (111) face,14 which then becomes the surface for water adsorption. The upper face of the prism is in thermal contact with a coldfinger that can be temperature controlled in the range -70 to 25 °C. At a set temperature, aliquots of water are admitted to the vapor chamber through a solenoid valve. This admission procedure allows a known number of water molecules to be introduced. By design, the coldest part of the vapor chamber is the BaF2(111) face onto which water as liquid or solid condenses. The thermodynamic conditionssof temperature, volume, and number of water molecules introduced into the vapor chambersdefine the thickness of the condensed film and the vapor pressure. The interrogation of the film is by way of the infrared beam, from a Fourier transform infrared spectrometer (Brucker), that enters through the prism face on the left, undergoes multiple reflections on the upper and lower faces, and exits on the right face to the optical detector. We have chosen to examine the -OH stretching region, nominally 2500-4000 cm-1. It is the reflections on the lower (111) face, exposed to vapor, that interrogate the water or ice film. The extinction of the infrared beam is a measure of the light lost to reflection and absorption due to the presence of the film.15 In the experiment to be discussed, the BaF2 prism temperature is set at -65 °C. Over a period of 40 ms, water is admitted through the solenoid valve to the vapor chamber. The initial extinction spectrum, at 1 min, is shown in Figure 2. Its band center near 3200 cm-1 and its profile have a superficial resemblance to the absorbance of ice.16 But the ice film is unstable as is evident from the profile degradations recorded at 4 and 10 min. These spectroscopic changes are consistent with the morphological transformations in the ice film we have represented to the right of Figure 2. Initially a thin film is imagined to be uniformly distributed over the substrate. In a matter (14) Sadtchenko, V.; Conrad, P.; Ewing, G. E. J. Chem. Phys. 2002, 116, 4293. (15) Zhang, Z.; Ewing, G. E. Anal. Chem., in press. (16) Warren, S. G. Appl. Opt. 1984, 8, 1206.
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Figure 2. Instability of thin film ice. On the left is the observed spectroscopic record. On the right is the morphological interpretation of this record. Initially, after 1 min, the spectrum is consistent with a thin film of ice. At later times, 4 and 10 min, the film spontaneously transforms into crystallites of ice.
of minutes, according to this scenario, it spontaneously reassembles into an array of crystallites. The proposed morphological transformations are justified by a quantitative photometric analysis. This analysis takes into account reflection and absorption losses that occur at the interface of the BaF2(111) face and its adsorbed ice film.15 The nominal angle the interrogating beam makes with the normal to the BaF2(111) face is 45°. Since this is close to the critical angle for the BaF2 of 43°,17 the optical properties of the interface become complicated when a film is present. Using the optical constants of ice16 and BaF2,17 we have applied the appropriate Airy equations to this situation15 to allow the calculation of the resulting extinction spectra as a function of film thickness. These equations, described in ref 18, have been programmed to produce theoretical spectra using commercial software.19 The results for uniform films of various thicknesses are shown in the lower panel of Figure 3. The spectra scale linearly with thickness below about 0.1 µm. A profile at 0.01 µm, for example, closely matches that at 0.1 µm; its extinction is simply smaller by a factor of 10. However a nonlinearity sets in somewhat above 0.1 µm. For example, the peak extinction shows a 5-fold increase when the thickness goes from 0.1 to 0.3 µm. In addition the band center is distinctly sharper at 0.3 µm than at 0.1 µm. By contrast, on thickening from 0.3 to 1 µm the peak extinction progressively decreases. This is accompanied by the replacement of the sharp singlet profile for the 0.3 µm film by a diffuse doublet for a thickness of 1 µm. The dramatic spectroscopic profile differences for the various films shown in Figure 3 bring to mind other wellknown shape and film thickness dependent spectra. A well-explored system consists of molecular crystals of CO2 in the form of aerosol clusters20 or supported thin films.21 These morphology dependent spectra have been successfully explained by classical electromagnetism20 or quantum mechanical exciton theory.21 For spherical particles the classical Mie relationships are utilized.18 For our thin films we have employed the Airy equations. These classical approaches both assume bulk homogeneous optical constants for the systems and differ only in the nature of the boundary conditions imposed: a spherical interface for Mie theory and planar interfaces for the Airy treatment. (17) Thomas, M. E.; Tropt, W. J. In Handbook of Optical Constants of Solids; Palik, E. D., Ed.; Academic Press: San Diego, CA, 1998. (18) Born, M.; Wolf, E. Principles of Optics, 7th ed.; Cambridge University Press: New York, 1999. (19) MATHCAD 2000 Mathsoft Inc., Cambridge, MA, 2000. (20) Gough, T. E.; Wang, T. J. Phys. Chem. 1996, 105, 4899. (21) Berg, O.; Disselkamp, R.; Ewing, G. E. Surf. Sci. 1992, 277, 8.
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Figure 3. Calculated spectra of ice on BaF2(111). In keeping with the optical characteristics of the thin film assembly and transfer optics22 the calculated spectra assume 3.5 reflections at the ice/BaF2 interface and TE polarization. Using the optical constants of ice and BaF2, spectra of thin uniform films of several thicknesses are shown in the lower panel. In the upper panel, spectra of ice crystallites on thin films are presented: (a) 99.85% 0.0009 µm + 0.05% 0.3 µm + 0.1% 1.3 µm; (b) 99.85% 0.0006 µm + 0.01% 0.3 µm + 0.1% 1.3 µm; (c) 99.85% 0.0003 µm + 0.01% 0.3 µm + 0.1% 1.3 µm.
These calculated changes in spectroscopic profiles with film thickness, using the Airy equations, allow us a semiquantitative means to interpret the spectra in Figure 2 as morphological transformations. Not one of the spectroscopic profiles in the lower panel of Figure 3 can match any of the extinction spectra of Figure 2. For example, scaling the theoretical 0.1 µm spectral profile to fit the maximum extinction of the 1 min spectrum fails to account for either its sharpness at the band center or the diffuse shoulders. We do however get a close resemblance to the 1 min spectrum by a simple combination that assumes a nonuniform ice adlayer, with 99.85% of the BaF2(111) surface covered by a 0.9 nm film, 0.05% by patches 300 nm thick, and the remaining 0.1% to be 1.3 µm crystallites. This combination is shown in (a) of the upper panel of Figure 3 and may be compared to the 1 min spectrum in Figure 2. The sharp band center and small diffuse shoulders are well represented. Other combinations of thin and thick films shown in (b) and (c) of the upper panel of Figure 3 can account for the 4 and 10 min spectra, respectively, in Figure 2. For example, the diffuse triplet of features at 10 min are reproduced in (c) of Figure 3. While these assumed combinations are somewhat arbitrary, they point to a consistent pattern in the transformation of an ice film to crystallites. We have taken the bulk ice optical constants16 as appropriate for all thin film thicknesses in these calculations of spectra. This assumption is certainly not true for films 1 nm and below where the influence of the substrate alters the adlayer structure. This has been demonstrated in our recent thin film experiments22 and in the calculations we present later in this paper. However in the calculated spectra of Figure 3 the contributions of films 1 nm or below contribute less than 1% to the overall profile. In the fitting procedure it was necessary to reduce the total number of molecules by 30% at 10 min compared (22) Sadtchenko, V.; Ewing, G. E. J. Chem. Phys. 2002, 116, 4686.
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with the 1 min spectrum and somewhat less at 3 min. (This is the reason that the coverage percentages in spectra (b) and (c) of Figure 3 do not quite add up to 100%.) Thus some molecules deposited on the prism face have migrated to other parts of the vapor chamber. Extrapolating to the initial introduction of water vapor (in 40 ms), we suggest a uniform film about 3 nm thick. Taking the ice bilayer thickness to be 365 pm,23 this would correspond to eight bilayers. After 10 min, this unstable film has been transformed principally into occasional 1.3 µm crystallites and a residual 0.3 nm film covering the BaF2(111) surface. If we assume that the crystallites formed after 10 min have extensions in the plane of BaF2(111) of comparable lengths to their thickness, i.e., ∼1 µm, we arrive at an estimate of their number density of 109 m-2. This figure is exceedingly small compared to the surface ion density of 2 × 1017 m-2 24 and suggests that the crystallites might form on specific but undefined defects on the BaF2(111) surface. These occasional crystallites may well be embryos for the hexagonal patches observed in previous microscopic observations.2 The instabilities that we have described in Figure 2 can be interpreted as dewetting.25 In the more familiar case of a liquid film on a surface (e.g., water on paraffin), an initially prepared uniform film spontaneously forms droplets each with a finite contact angle.25,26 The right side of Figure 2 is the solid film analogy to the liquid case: the initial uniform solid film transforms into a multitude of crystallites rather than droplets. We now consider a molecular interpretation of the instability of the ice film. One possible explanation is based on the analysis of Fletcher,9 which shows that if an Ih bilayer, infinite in extent, is anchored to a hexagonal surface by either lone pairs or hydrogen bonds, the other face of the bilayer will comprise the complementary feature, i.e., hydrogen bonds or lone pairs, respectively. Consequently the bilayer will be ferroelectric, and moreover the ferroelectric property will propagate through all subsequent bilayers. This behavior was observed by Su et al. in their studies of water on Pt(111);27 they report that the ferroelectric structure persisted through some 30 layers. The greater degree of proton ordering in the ferroelectric structure leads to a decrease in entropy of the order of about 2.2 J K-1 mol-1. At the temperature in question, the stable form of bulk ice is the proton disordered Ih structure.28 However, the template effect of the surface may be able to stabilize a ferroelectric thin film as found by Su et al.,27 assuming that the ice rules hold. We therefore initially envisaged that under the conditions of the experiment a ferroelectric film was being formed but that it was metastable with respect to disordered Ih ice and that a rearrangement to Ih crystallites followed. Our calculations do not however support this interpretation. We have calculated the minimum-energy geometry of isolated water molecules and water adlayers on the BaF2 surface, using the methods described in previous papers.29-32 The water-water potential was a (23) Petrenko, V. F.; Whitworth, R. W. Physics of Ice; Oxford University Press: Oxford, 1999. (24) Swanson, H. E.; Tatge, E. Natl. Bur. Stand. Circ. 1953, 539, 70. (25) Dash, J. G. Phys. Rev. B 1977, 15, 3136. (26) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1985. (27) Su, X. C.; Lianos, L.; Shen, Y. R.; Somorjai, G. A. Phys. Rev. Lett. 1998, 80, 1533. (28) Jackson, S. M.; Whitworth, R. W. J. Chem. Phys. 1995, 103, 7647. (29) Engkvist, O.; Stone, A. J. J. Chem. Phys. 1999, 110, 12089. (30) Engkvist, O.; Stone, A. J. Surf. Sci. 1999, 437, 239. (31) Engkvist, O.; Stone, A. J. J. Chem. Phys. 2000, 112, 6827.
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Figure 5. Hexagonal chair building block of Ih ice shown in relation to the BaF2 (111) surface.
Figure 4. Equilibrium configuration of a single water molecule adsorbed on BaF2.
slightly simplified version of the ASP-W2 potential.33 The repulsion potential between the water molecule and the Ba2+ and F- ions was determined using ab initio intermolecular perturbation theory (IMPT)34 and fitted to an orientation-dependent exponential function of the form that we have used previously. The electrostatic interaction was described using a distributed multipole analysis of the water molecule charge distribution, with charges of +1.6 and -0.8 on the Ba2+ and F- ions. These ion charges were obtained from self-consistent field and density functional theory Crystal9835 calculations on a BaF2(111) slab. The water-ion dispersion coefficients come from empirical combining rules.36,37 The induction energy of the water molecule was included. Our calculations show that one lone pair of an isolated water molecule is directed toward the Ba2+ ion within the BaF2(111) surface, with the hydrogen atoms pointing toward adjacent surface fluoride ions in an asymmetric manner (Figure 4). One O-H bond is oriented along the Ba-F direction and points slightly toward the surface. The other hydrogen points very slightly away from the surface. The adsorption energy is -39.8 kJ mol-1. This is (32) Stone, A. J.; Dullweber, A.; Engkvist, O.; Fraschini, E.; Hodges, M. P.; Meredith, A. W.; Popelier, P. L. A.; Wales, D. J. Orient: a program for studying interactions between molecules, version 4.4; University of Cambridge: Cambridge, 2001. Enquiries to A. J. Stone,
[email protected]. (33) Millot, C.; Soetens, J.-C.; Martins Costa, M. T. C.; Hodges, M. P.; Stone, A. J. J. Phys. Chem. A 1998, 102, 754. (34) Hayes, I. C.; Stone, A. J. Mol. Phys. 1984, 53, 83. (35) Saunders, V. R.; Dovesi, R.; Roetti, C.; Causa`, M.; Harrison, N. M.; Orlando, R.; Zicovich-Wilson, C. M. Crystal98 Technol. rep. 1998. (36) Zeiss, G. D.; Meath, W. J. Mol. Phys. 1977, 33, 1155. (37) Fowler, P. W.; Pyper, N. C. Proc. R. Soc. London, Ser. A 1985, 398, 377.
quite different from the structure of an adsorbed water molecule on a metal surface such as Ru(001), where the molecule is adsorbed symmetrically with both hydrogen atoms pointing away from the surface.38 Because of the nature of this structure, adsorption of further water molecules does not lead to the formation of a stable ice bilayer. The binding energy of a water molecule in the orientation needed for the ice structure is only 34.3 kJ mol-1, 5.5 kJ mol-1 less than in the equilibrium geometry. The binding energy between the water molecules in an ice lattice appears to be insufficient to overcome this energy difference. We have attempted to optimize the structure of an icelike bilayer on the surface, based on the hexagonal chair building block of Ih ice shown in Figure 5, but there appears to be no stable structure of this form. If the orientations of the water molecules are constrained as in the ice bilayer, and their positions are optimized, the resulting binding energy is about 39 kJ mol-1 per water molecule. When the orientations are allowed to relax, the structure collapses into an irregular form with a much higher binding energy of about 54 kJ mol-1 per water molecule, in satisfactory agreement with the experimental adsorption enthalpy of 58 kJ mol-1.14 An example of these structures is shown in Figure 6. They are very similar to those observed by Wassermann and co-workers,39 which involve chains of water molecules running across the surface, but we also noted the presence of cross-links between chains. They occur when adjacent chains are close enough to distort the geometry of the surface adsorbed species, enabling them to form additional hydrogen bonds. The chain structure cannot act as a template for the formation of ice, and adsorption of subsequent bilayers leads instead to a structure with further positional and orientational disorder that can no longer be compared to that of ice. Similar behavior has been observed in recent calculations by Witek and Buch.13 In molecular dynamics simulations they observed a rapid decrease in the orientational ordering of the water molecules upon the addition of successive bilayers to a perfect fixed ice bilayer. Some positional disorder was also observed. In their case the disorder was attributed to reorganization of the surface structure to reduce the number of dangling hydrogen bonds. In our case the disorder of the underlying structure also has an effect. Our calculations suggest that a second bilayer, i.e., two more water molecules per surface Ba2+ ion, is bound by 48 kJ mol-1 per water molecule, while the experimental measurements lead to a binding energy of 44 kJ mol-1 for multilayer adsorption.14 A further bilayer added to the Ih ice structure, on the other hand, is bound (38) Doering, D. L.; Madey, T. E. Surf. Sci. 1982, 123, 305. (39) Wassermann, B.; Reif, J.; Matthias, E. Phys. Rev. B 1994, 50, 2593.
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Figure 6. Side and top views of the disordered bilayer structure of water on the BaF2(111) surface.
by 59 kJ mol-1 per water molecule,40 so the upper layers of the disordered structure are metastable relative to Ih ice. We therefore suggest the following scenario for the instability of the ice film at -65 °C. The film is prepared under nonequilibrium conditions by admitting aliquots of water vapor over a period of 40 ms. The vapor, initially at room temperature, is cooled by striking the walls of the vapor chamber and sticking to the coldest part, the BaF2(111) face, where it initially forms long chains and then becomes increasingly amorphous. The layer nearest the surface is quite strongly bound, but the upper layers are metastable with respect to ordinary Ih ice. The transformation to the Ih form is allowed, despite the constraints imposed by the BaF2(111) surface, by the growth of crystallites of ice. Only where the crystallites are in contact with the surface is their structure affected (40) Whalley, E. J. Glaciol. 1978, 21, 13.
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by the surface; elsewhere the normal Ih structure can be adopted. However an amorphous adlayer remains on the surface, being as tightly bound as it would be in an Ih crystallite. We shall now comment on the temporal scale that characterizes the transformation of an initial amorphous thin film to crystallites supported on the BaF2(111) surface. The vapor pressure of ice at -65 °C of 5 × 10-3 mbar41 provides for a site collision frequency of the order of 104 s-1.42 The film is then in rapid dynamic equilibrium with the surrounding vapor, allowing it to advance to the stable proton disordered configuration of the crystallites. Thus the initial instability of the amorphous film can be relieved under the conditions of our experiment. This is in contrast to the experiments of Su et al.27 whose ferroelectric film on Pt(111) cannot be warmed above -136 °C before it is lost to sublimation, the vapor being pumped away in the ultrahigh vacuum chamber. This suggests that the ferroelectric ice film on Pt(111) may not actually be stables perhaps it just does not have an opportunity to rearrange in the nonequilibrium environment of their experiment. On the other hand, Gavish et al.43 found that films of long-chain aliphatic alcohols were able to nucleate ice growth successfully near 0 °C. The spacing and orientation of the alcohol OH groups are such that water molecules in a nascent ice layer can form hydrogen bonds to them without distortion. We conclude that in order to grow Ih ice on a substrate, the ice rules need to be strictly obeyed. In the building block of ice, viewed as a six-membered chair ring, not only do the O atom positions need to be accurately spaced but all the water bond angles in the chair need to be close to tetrahedral. An effective nucleating substrate needs to provide sites to which the water molecules can bind without distorting this structure. The implications of our results on the understanding of ice nucleation on surfaces are profound. Similarity of the lattice constants of the surface and ice structures is not enough to allow the epitaxial formation of ice, in either a proton ordered or disordered form. The specific nature of the molecule-surface interaction also plays a critical role. Acknowledgment. Zhenfeng Zhang has guided our understanding of thin film spectroscopy and contributed to many of our discussions. G.E. benefited from discussions with Lane H. Seeley in the early stages of this study. The research was funded in part by the National Science Foundation through Grant CHE-9816299 and by an EPSRC research studentship to D.N. LA0255370 (41) Smithsonian meteorological tables Technol. rep. Washington, DC, 1951. (42) Atkins, P. W. Physical Chemistry, 6th ed.; W. A. Freeman: New York, 1998. (43) Gavish, M.; Popovitz-Biro, R.; Lahav, M.; Leiserowitz, L. Science 1990, 250, 973.
