Reinterpretation of the Mean Field Hypothesis in Analytical Models of Ostwald Ripening and Grain Growth

Abstract

A long right tail is a common feature of experimental quasi-stationary size distributions of particles and grains that is not explained by the classical theories based on the mean field hypothesis. In this work, it is shown that the “pairwise interaction” approach, here presented in a comprehensive exposition involving both Ostwald ripening and grain growth, is a valid alternative to classical mean field theories since it produces more realistic predictions of the distribution shapes. The new analytical models are based on the mean field concept but rely on a detailed physical description of the elementary interactions responsible for the exchange of matter. They are jointly reviewed and compared with the corresponding classical Lifshitz–Slyozov–Wagner and Hillert models. The interactions are treated as a sum of elementary and specific contributions rather than as a generalized exchange with the mean field. The framework is complemented by the introduction of the “interaction volume” in Ostwald ripening and of the “local grain boundary curvature” in grain growth which are both size-dependent and permit to represent more precisely the local physics of the exchanges. The excellent results obtained in reproducing the experiments without any ad hoc parameters suggest that the mean field hypothesis adopted in the classical theories to describe the environment of a growing particle or grain represents a too drastic approximation. Therefore, it is proposed to replace the classical mean field by a “local mean field,” i.e., the ensemble of actual mean environments interacting with any single element of a given size. This alternative assumption induces a higher growth rate for large particles or grains compared with their respective mean field theories, thus producing right-skewed asymptotic distributions. For particles at small volume fraction the stationary distribution resembles a lognormal function, whereas for grains in normal grain growth regime the Rayleigh distribution is found as solution.