A Finite Element Method for Simulating Interface Motion

This paper describes our recent progress in developing a finite element method for simulating interface motion. Attention is focused on two mass transport mechanisms: interface migration and surface diffusion. A classical theory states that, for interface migration, the local normal velocity of an interface is proportional to the free energy reduction associated with a unit volume of atoms detach from one side of the interface and attach to the other side. We express this theory into a weak statement, in which the normal velocity and any arbitrary virtual motion of the interface relate to the free energy change associated with the virtual motion. An example with two degrees of freedom shows how the weak statement works. For a general case, we divide the interface into many elements, and use the positions of the nodes as the generalized coordinates. The variations of the free energy associated with the variations of the nodal positions define the generalized forces. The weak statement connects the velocity components at all the nodes to the generalized forces. A symmetric, positive-definite matrix appears, which we call the viscosity matrix. A set of nonlinear ordinary differential equations evolve the nodal positions. We then treat combined surface diffusion and evaporation-condensation in a similar method with generalized coordinates including both nodal positions and mass fluxes. Three numerical examples are included. The first example shows the capability of the method in dealing with anisotropic surface energy. The second example is pore-grain boundary separation in the final stage of ceramic sintering. The third example relates to the process of mass reflow in VLSI fabrication.