Gradient flow and front propagation with boundary contact energy

Abstract

We consider a gradient flow for the functional F Є (u) = ∫ Ω 1/Є W(u) + Є | with a view towards modelling the interfacial motion associated with domain coarsening in binary alloys in the presence of boundary contact energy. Here If is a non-negative ‘double-well’ free energy density and u is the composition density of the system. The function u represents the contact energy between the alloy and the boundary, 7Q, of the container. The functional was first proposed by Cahn in a different context as a model for the energy of a two-fluid mixture which takes into account boundary contact energy at the boundary of the fluid domain. We derive the first term in an asymptotic expansion, as e->0, for the solution of the associated gradient flow 7 t u = 2eA u — (1/e) W' (u), xeQ, t>0, u(x,0) = g e (x), xeQ, Vu.n = - (1/2e)σ'(u), xe7Q,t >0. Using multiple time scales, we show that fronts rapidly develop and then propagate with normal velocity eK , where K is the mean curvature of the front. At the intersection of a front with 7)Q, the boundary contact energy is shown to imply a contact angle condition for the front. Several examples are presented for this type of propagation in the plane.