Reference no: EM133319223

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(γ cos φ -γφ sin φ)

y = TM / (Tm -T)LV (γ sin φ + γφ cos φ)

where φ 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, γ(φ) is the solid-liquid surface energy and T is the equilibrium temperature. Subscripts φ refer to derivatives.

Question 1: Show that the shape obeys the capillary corrected equilibrium Gibbs-Thomson melting temperature condition defining equilibrium between a crystal in its melt.

Question 2: Assume that the anisotropic surface energy has the form:

γ = γo + γk cos(kφ)

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

Question 3: Plot the equilibrium shape for anisotropy values above and below the anisotropy threshold for missing orientations, at the value of k = 6.