At equilibrium (in 2D cross-section), the shape of a crystal immersed in its melt is given in parametric form:

x = TM / (Tm T)LV( cos sin )

y = TM / (Tm T)LV ( sin + cos )

where is the crystallographic orientation of the interface (i.e. the angle that the unit normal vector makes with the cartesian x-axis), TM is the melting point of a planar interface, i.e. the 'bulk' melting point found on a phase diagram, LV is the latent heat per unit volume of crystal, () is the solid-liquid surface energy and T is the equilibrium temperature. Subscripts refer to derivatives.

(a) Show that the shape obeys the capillary corrected equilibrium Gibbs-Thomson melting temperature condition defining equilibrium between a crystal in its melt.

(b) Assume that the anisotropic surface energy has the form:

= o + k cos(k)

where k represents the in-plane rotational symmetry of the 2D crystal. For this form of the surface energy, beyond what threshold ratio k/o does the equilibrium shape exhibit missing crystallographic orientations? Plot the angles range of angles excluded from the equilibrium shape versus k/o. Evaluate these ranges over the ratio k/o from 0 to 1. (This is a form of bifurcation diagram.)

(c) Plot the equilibrium shape for anisotropy values above and below the anisotropy threshold for missing orientations, at the value of k = 6.