The statistical self-similarity hypothesis in grain growth and particle coarsening

The time dependence of particle size (e.g., mean volume) in normal grain growth, bubble growth, and late-stage coarsening is deduced from a statistical self-similarity (SSS) hypothesis, according to which consecutive configurations of the system in the self-similar mode are geometrically similar in a statistical sense, and from the scaling characteristic of v̇, the rate of change of the volume of a given particle. It is shown that if v̇ scales as v̄α under uniform magnification, where v̄ is the mean particle volume and α a constant depending on the controlling kinetics and geometry of the system, then v̄1−α is a linear function of the time. Values of α are obtained for a number of cases. The treatment is free of many of the approximations and geometrical simplifications used in most theories. Evidence for the SSS hypothesis is discussed. Special new results are obtained for two-dimensional bubble and idealized grain growth that should be testable by computer simulation or by observation of a well-behaved soap-bubble array.