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 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 temperature-controlled 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. Back to Science Contents 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 2-dimensional 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 CrystalEverything 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 CaseThe 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 near-spherical case is most likely [3], especially at temperatures not too far from the freezing point. Equilibration TimeFor 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 1-mm-diameter ice crystal to relax to its equilibrium shape, and the equilibration time becomes much longer as the temperature decreases. Growth ShapeThe 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. Back to Science Contents 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).