Coupled Surface Diffusion and Mean Curvature Motion: Axisymmetric Steady States with Two Grains and a Hole

Abstract

The evolution of two grains, which lie on a substrate and are in contact with each other, can be roughly described by a simple model in which the exterior surfaces of the grains evolve by surface diffusion and the grain boundary, namely the contact surface between the adjacent grains, evolves by motion by mean curvature. For simplicity we consider an axi-symmetric two grain system, which is contained within an inert bounding semi-infinite cylinder with unit radius and which is bounded below by a planar substrate. The two grain system is assumed to have a hole along the axis of symmetry, where the substrate is exposed. Boundary conditions are imposed reflecting the considerations of W.W. Mullins, 1958. The resultant dynamic problem conserves the total volume of the two grain system and dissipates the total energy, where the total energy is defined as sum of the areas of the various participating surfaces, weighted according to their respective surface free energies.We focus here on the steady states of this system. We demonstrate that at steady state, the exterior surfaces have constant and equal mean curvatures, and the grain boundary has zero mean curvature. Taking into account the geometry and the boundary conditions, it then follows that the exterior surfaces are nodoids and the grain boundary surface is a catenoid. The physical parameters in the model can be expressed via the angles β and θc, which depend on the surface free energies, where, in the meridian cross-section, β ∈ (π/2, π) is the angle between the grain boundary and each of the exterior surfaces, and θc ∈ (0, π], the contact angle, is the angle between the inner grain and the substrate. Typically if a steady state solution exists for given values of (β, θc), then there exists a continuum of such solutions with varying volumes and energies. In particular, we prove that there exists a continuum of solutions with θc = π for any β ∈ (π/2, π). While many open questions remain, with regard to both the steady states and the full dynamic problem, our study already provides insight into the possible steady states and their structure. In particular, the relative volumes and heights of the inner and outer grain can be seen to be roughly in accordance with experimental predictions, for realistic values of the physical parameters.