Spinodal Decomposition in the Presence of Elastic Interactions

Summary

Spinodal decomposition, i.e., the separation of a homogeneous mixture into different phases, can be modeled by the Cahn-Hilliard equation — a fourth order semilinear parabolic equation. If elastic stresses due to a lattice misfit become important, the Cahn-Hilliard equation has to be coupled to an elasticity system to take this into account.

It is the goal of this paper to understand how elastic effects influence the formation of patterns during spinodal decomposition and to analyze what kind of morphologies one has to expect. It is shown that with a probability close to one, the dynamics of randomly chosen initial data in the neighborhood of a uniform mixture will be dominated by an invariant manifold which is tangential to the most unstable eigenfunctions of the linearized operator. For example in the case of cubic anisotropy it is shown that the most unstable eigenfunctions reflect the cubic anisotropy and the anisotropy will influence the dynamics quite drastically.