Use of stereology to derive a new kinetic equation for mean curvature driven grain growth

Abstract The idea that normal grain growth driven by surface tension may be described theoretically by the hypothesis that the local velocity of an element of grain boundary is proportional to its local mean curvature dates back more than half a century. von Neumann was the first to derive this relation and used it to predict the rate of evolution of a two dimensional cell structure. MacPherson and Srolovitz extended this development to describe growth in three dimensions; however, their result was couched in terms that did not facilitate tests of the theory. In this paper expected value theorems established in stereology are invoked to extend their result to provide a new equation that predicts the rate of change of volume of grains in a microstructure which, while preserving the rigor and generality of the result, expresses it in terms of quantities that can be measured in microstructures. This is illustrated with a set of measurements based upon the theory derived from a grain growth simulation that successfully tests its predictions. It is interesting that this result also exhibits an “ n -6 rule” that is similar to, but not identical with, that contained in von Neumann’s theory.