Instability of a biaxially stressed thin film on a substrate due to material diffusion over its free surface

Abstract

The configuration of an elastically strained thin film bonded to a relatively thick elastic substrate over a plane interface is considered. The free energy of the system is taken to be the surface energy of the free surface, which is initially flat, and the elastic strain energy. It is assumed that the film material can change the shape of its free surface by means of mass diffusion along the surface, and that this mass transport occurs coherently. As a result of this diffusion process, the free energy of the system changes due to a change in surface shape and due to a change in the elastic energy. If the change in free energy as the surface shape departs from planar is positive, then this change will tend to occur spontaneously and the flat shape is unstable. The stability of a biaxially stressed thin film due to material diffusion over its surface is considered here under both two- and three-dimensional conditions. The stability condition is derived in the form of a difference between two positive definite quantities, one associated with surface energy and the other associated with strain energy, and the sign of this difference depends on the relative stiffnesses of the materials, the thickness of the film, the surface energy of the film material, and the initial elastic stress in the film. It is demonstrated that the configuration with a flat free surface is unstable under sinusoidal perturbations in the shape of the surface for any combination of parameters, provided that the wavelength of the perturbation is larger than some critical value. Numerical results are presented for the critical value as a function of film thickness for several values of the ratio of elastic stiffnesses of the materials.