Epitaxial crystal growth by depositing atoms from a gas phase onto a substrate is a nonequilibrium process involving both kinetics and thermodynamics. At the early stages of growth, adatoms on the substrate often form small isolated islands. The morphological evolution (e.g. growth or shrink) of these islands is determined by many physical processes such as atom adsorption and desorption, adatom diffusion, adatom attachment to island boundaries or detachment from the boundaries. Mathematical formulation of the problem leads to a moving boundary/interface problem.

In this talk, we present a boundary integral method for computing the quasisteady evolution of an epitaxial island. The problem consists of an adatom diffusion equation (with desorption) on terrace and a kinetic boundary condition at the step (island boundary). The normal velocity for step motion is determined by a two-sided flux. The integral formulation of the problem involves both double and single layer potentials due to the kinetic boundary condition. Numerical tests on a growing/shrinking island are in excellent agreement with the analytical solution and demonstrate that the method is stable, efficient and spectrally accurate in space.
Nonlinear simulations for the growth of perturbed circular islands show that sharp tips and facets will form during growth instead of the usual tip-splitting events for isotropic Laplacian growth. The numerical techniques presented here can be applied generally to a class of free/moving boundary problems in physical and biological science.