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The two-dimensional (2D) nucleation of molecular layers on crystalline solids is the an important process governing the formation of snow crystal facets. This section focuses on the fundamental physics underlying 2D nucleation and growth.

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Molecular motions
Our overarching picture begins with the concept of a "smooth" faceted ice surface. The molecular structure of this surface is such that impinging water vapor molecules do not readily bind to the surface. A vapor molecule that strikes the surface may:
1) immediately ricochet off the surface, thus never really leaving the vapor phase,
2) initially bind to the surface, diffuse along the surface for a while, and then pop off again via thermal fluctuations,
3) diffuse along the surface, encounter a terrace (or island) of molecules already attached to the surface, and bind to the edge of the terrace.

In this model, a single admolecule on the surface will not long remain bound to the surface in an isolated state. If it does not encounter an existing terrace to attach to, thermal fluctuations will soon cause it to pop off and return to the vapor phase.

There is, however, a fourth option for the fate of an admolecule, sufficiently rare that it is not listed with the three option above. It is also possible that a group of isolated admolecules will (by chance) come together and form a new terrace, large enough that it is stable against thermal fluctuations. This phenomenon is called the nucleation of a new terrace, and the nucleation rate largely governs the slow addition of new material to faceted surfaces. Once it forms, the terrace accumulates additional admolecules when they diffuse to its edge.

When the supersaturation is extremely low, terraces may form one at a time, each terrace growing to cover the entire faceted surface before the next terrace appears. This is called layer-by-layer growth. Under more typical conditions, however, multiple terraces are present at any given time, resulting in a polynuclear growth model. (Image from [1].)

Theory
The theory describing 2D nucleation and growth on faceted crystalline surfaces was developed in the 1950s, and it has changed little since that time. The derivation is somewhat lengthy, as it describes the statistical mechanics of molecular deposition, surface diffusion, and the nucleation of new terraces. Nevertheless, it is described in detail in most books on crystal growth, for example [2].

Skipping ahead to the answer, the perpendicular growth rate of a faceted surface can be written \[
v_{normal}=\alpha _{nucl}v_{kin}\sigma _{surf} \] where \(v_{kin}\) is the kinetic coefficient and \(\sigma _{surf}\) is the water vapor supersaturation at the ice surface (this equation, and all its factors, all all defined in Attachment Kinetics). All the nucleation physics is contained in the attachment coefficient \(\alpha _{nucl},\) which is convenient to write in the form \[
\alpha _{nucl}\approx A(\sigma _{surf})\exp (-\sigma _{0}/\sigma _{surf})
\] where \(A(\sigma _{surf})\) is a dimensionless factor that depends on things like the surface diffusion length of admolecules [2], and the critical
supersaturation \(\sigma _{0}\) is another dimensionless factor given by \[
\sigma _{0}\approx S\frac{\beta ^{2}a^{2}}{k_{B}^{2}T_{m}^{2}}
\] where \(S\) is a geometrical factor near unity, \(\beta \) is the step energy (see Step Energies), \(a\) is the molecular size (about 3 nm), \(k_{B}\) is the Boltzmann constant, and \(T_{m}\) is the melting point of ice. (This all comes essentially from [2], with additional context described in [3]. It applies specifically to the growth of a faceted ice crystal surface from the deposition of water vapor molecules.)

Breaking it down
Nucleation theory can be broken down into two parts. The first part is the initial nucleation of new terraces, which results in the \(\exp (-\sigma _{0}/\sigma _{surf})\) factor in the expression above for \(\alpha _{nucl}.\) The underlying physics is that small terraces containing just a few molecules are unstable to disintrigation from thermal fluctuations. Especially when \(\sigma _{surf}\) is small, admolcules are likely to break away from the terrace cluster before additional admolecules attach. The net result is that small clusters break up. Above a certain critical size, however, more molecules join the cluster than are lost, so the cluster is stable and grows to form a large terrace. The formation of clusters is governed by the statistical mechanics of admolecules present on the faceted surface.

Although the many-body physics of nucleation is quite complex, the \(\exp (-\sigma _{0}/\sigma _{surf})\) factor is a robust feature in the theory. Changing the underlying assumptions a bit does not change this factor. Moreover, \(\sigma _{0}\) does not depend on the details of the theory, but only on the step energy (along with fundamental parameters) as described above. This is important, as nucleation theory takes a highly dynamical, many-body process (crystal growth) and explains a great deal of it in terms of the terrace step energy, which is a much simpler, equilibrium property of the ice surface.

The second part of nucleation theory involves how quickly a stable terrace grows once it has formed. This is a much messier, and thus less robust, part of the theory, which means that the \(A(\sigma _{surf})\) factor in \(\alpha _{nucl}\) is not so easily calculated. It depends on surface diffusion rates, assumptions about admolecule attachment to terraces, and exactly how one deals with polynucleation processes, among other things..

There is, however, a very useful limiting case for the \(A\) factor. If rapid terrace nucleation makes the surface behave essentially like a rough surface, then we have \(A \approx 1\), independent of all other uncertainties in the terrace growth model. It appears that this limiting case applies over a broad range of conditions in ice crystal growth (see Step Energies), but with a dramatic exception that appears for the case of prism facets growing in air near -5 C (see ***Terrace Erosion at -5C).

Application to ice crystal growth
The good news is that 2D nucleation theory works very well for describing the growth of faceted ice crystals. As we show in Step Energies, measurements of growth rates as a function of \(\sigma _{surf}\) show a clear \(\exp (-\sigma _{0}/\sigma _{surf})\) behavior, indicitive of nucleation-limited growth. We can extract \(\sigma _{0}\) directly from the measurements, and from this determine the step energy from \[
\sigma _{0}\approx S\frac{\beta ^{2}a^{2}}{k_{B}^{2}T_{m}^{2}}
\] This is an excellent step forward, because the step energy \(\beta\) is a fundamental, equilibrium property of ice as a material, in the same way that the bulk energy (latent heat) and surface energy (surface tension) are fundamental, equilibrium properties of ice. Better still, it should be possible to calculate \(\beta\) from ab initio molecular-dynamics models of the ice surface. Such calculations have not been done to date, and it will likely take a considerable amount of work to do so, but it appears to be possible using existing computational methods. Especially exciting is that fact that \(\beta\) depends strongly on temperature, and is quite different on the basal and prism facets. This suggests that we could learn a great deal about the surface structure of ice by calculating step energies using molecular dynamics simulations. Measurements of step energies are in hand (see Step Energies), so the ball is in the theorist's court for now.

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References
[1] Wollschlager, J; Bierkandt, M; Larsson, MI, Kinetic phase diagram for terrace and step nucleation of CaF/Si(111), Applied Surface Science. 219, 107-116 (2003).

[2] Saito Y, Statistical Physics of Crystal Growth. Singapore: World Scientific (1996).

[3] Libbrecht KG, Physical Dynamics of Ice Crystal Growth, Annual Reviews of Materials Research, 47, 271–95 (2017).