
The twodimensional
(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 layerbylayer 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 manybody 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, manybody 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 nucleationlimited 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
moleculardynamics 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, 107116 (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). 


