Fingering instability of a retracting solid film edge

Abstract

A thin gold film under annealing on a silica substrate can develop “fingers” at the perimeter of the film. The perimeter retracts to leave behind longer fingers, which eventually pinch off to reduce the surface energy of the system. New fingers then form at the film edge and the process continues until the entire film disintegrates. To maintain the structure integrity of annealed thin films, this fingering instability must be understood. The retraction of a straight film edge via capillarity-driven surface diffusion has been analyzed in two dimensions by Wong et al.[Acta Mater. 48, 1719 (2000)]. They found that a retracting film is thickened at the edge followed by a valley before the film thickness becomes uniform. We study the three-dimensional linear stability of this two-dimensional film profile and find one unstable mode of perturbation. The growth rate of the perturbation is determined as a function of the wavelength of the perturbation and the speed of the receding edge. The results show that a straight film edge becomes wavy when perturbed. The wavelength λm of the fastest growing perturbation agrees with the distance between adjacent fingers observed in a gold-film experiment. Fingers can also form during annealing at the retracting edges of cracks in sapphire, and our predicted λm compares well with the measured finger spacing.