Mullins' Self-Similar Grooving Solution Revisited

Abstract

In W.W. Mullins' classical 1957 paper on thermal grooving, motion by surface diffusion was proposed to describe the development of a thermal groove separating two grains in a simple semi-infinite planar geometry. After making a small slope approximation which is often realistic, Mullins' sought self-similar solutions, and obtained an explicit time series solution for the groove depth. In the years since, Mullins' grooving solution has become a standard tool; however it has yet to be rigorously demonstrated that self-similar solutions exist when the small slope approximation is not applicable. Here we demonstrate that reformulation of Mullins' nonlinear problem in arc-length variables yields a particularly simple fully nonlinear formulation, which is useful for verifying large slope grooving properties and which should aid in proving existence.