
The equilibrium shape
of a crystal is the shape it assumes when in complete thermal
equilibrium with its surroundings. For a snow crystal, one can imagine
a small ice crystal floating freely inside a temperaturecontrolled
box, in thermal equilibrium with the water vapor gas inside the box.
The equilibrium shape is generally the shape that minimizes the total
surface energy of the crystal for a constant crystal mass. The
equilibrium shape depends on temperature, and it has not yet been
measured or calculated (with any meaningful accuracy) for ice.
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Basic Considerations
The geometrical shape with the lowest surface area at constant mass is a sphere,
and this would be the equilibrium shape of ice if the surface energy \(\gamma\approx 0.11\) J/m\(^2\) (see Step Energies)
were independent of surface orientation. But we know (or at least we
strongly suspect) that the surface energies of "molecularly smooth"
faceted surfaces are lower than the energies of other "molecularly
rough" surfaces. If the difference in these surface energies were
sufficiently high, then the equilibrium shape of an ice crystal would
be the hexagonal prism, bounded by basal and prism facets (see Snowflake Science).
The real answer is probably somewhere in between  something that
looks like a hexagonal prism with rounded corners, or perhaps something
that looks like a sphere with small faceted regions. We just don't know
for sure, not yet anyway.

2D Square Crystal
It is instructive to look at the particularly simple case of a
2dimensional square crystal, shown on the right. If we cut a corner
off (blue line), then the length of the perimeter of the corner is
reduced by the amount \((2x  {\sqrt 2}x)\), where \(x\) is the length
of one of the sides of the corner. Similarly, the perimeter energy is
reduced by \(2x\gamma_{facet}  {\sqrt 2}x\gamma_0)\). Following this
reasoning, we see that if \(\gamma_{facet} < (\gamma_0/{\sqrt
2}) = 0.71\gamma_0\), then is is energetically favorable to keep the
corner perfectly faceted, with no rounding at all. (This calculation is
exact only in the limit of small \(x\), however. Plus we are not
looking at the detailed atomic structure of the surface or the corner.)

A 3D Cubic Crystal Everything
is more complicated in 3D, but nevertheless we find that if
\(\gamma_{facet}/\gamma_0\) is small enough, then the corners would not
be rounded at all, and the equilibrium shape would be that of a perfectly
faceted prism. This is the case for table salt, aka
sodium chloride, which
has a totally faceted equilibrium shape, as shown on the left (image
from [1]). Note that this picture was taken at 620 C; above 650 C, the
corners do begin to round because \(\gamma_{facet}\) increases
with temperature.  The 2D Hexagonal Case The
sketch on the right shows the corner of a hexagonal shape, and again we
imagine rounding the faceted corner, the same as we did with the square
case above. The math is similar, and this time we find that the corners
of the equilibrium shape remain perfectly faceted if \(\gamma_{facet}
< (\gamma_0 {\sqrt 3}/2) = 0.87\gamma_0\). In other words, if the
surface energy of an ice prism facet is just 13 percent lower than a
molecularly rough surface, then we can expect complete prism facets on
the equilibrium shape of ice (neglecting that the 3D case has some
additional complications). The thing is, however, that 13 percent is a
pretty big number. Surface energies on most simple materials do not
vary more than a few percent with surface orientation, and this is
likely the case for ice also.
3D Nearly Spherical Crystals When
the surface energy is nearly isotropic, the equilibrium shape looks
basically like a sphere, except for small faceted regions. The picture
on the left, for example, shows the equilibrium shape of lead at 300 C
[2].
Although the correct equilibrium shape of ice is not yet
known, the evidence suggests that the nearspherical case is most
likely [3], especially at temperatures not too far from the freezing
point. 
 Equilibration Time For
a small ice crystal of radius \(R\) surrounded by water vapor, the time
required for the crystal to relax to its equilibrium shape is quite
long [3], approximately \[ \tau \approx \frac{R^{2}}{2\alpha v_{kin}\delta } \] where the variables are defined in [3]. This can be written \[\tau
\approx 1.4\Big( \frac{R}{100\textrm{ }\mu \textrm{m}}\Big) ^{2}\Big(
\frac{1}{\alpha }\Big) \Big( \frac{1\textrm{ mm/sec}}{v_{kin}}\Big)
\textrm{ hours} \] Putting in some numbers, we see that it
would take of order a week or more for a 1mmdiameter ice crystal to
relax to its equilibrium shape, and the equilibration time becomes much
longer as the temperature decreases. 

Growth Shape The
shape of a growing ice crystal is typically much different from the
equilibrium shape, even if the growth is very slow (but the growth time
is still shorter than the equilibration time above). The attachment
coefficient \(\alpha\) on rough surfaces is near unity, while
\(\alpha\) on faceted surfaces can be 10,000 times smaller in some
circumstances. As a result, the growth shape of a slowly growing ice
crystal is typically a simple hexagonal prism, as shown at right.
An
important point here is that the growth shape depends mainly on the
attachment kinetics, and hardly at all on surface energies. 

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References
[1] Heyraud and Métois, J.Cryst. Growth 84, 503 (1987)).
[2] (Rottman et al., Phys. Rev. Lett. 52, 1009 (1984).
[3] arXiv:1205.1452 (2012).



