Theory of Ostwald ripening invoking Taylor analyticity and a Le Chatelier stability principle

Abstract

As an alternative to a variety of algorithms, and particularly to the much-favoured Lifshitz-Slyozov-Wagner (LSW) mathematical method of regularization of a coarsening precipitate manifold to hydrodynamic continuity controlled locally by quasi-steady state volume diffusion, we have reformulated the asymptotic, terminally discrete time-dependent particle radius distribution function f so as to be consistent with an even Taylor expanded free energy density. This consistency is necessitated by statistical isotropy in a coarse-grained phase space, a condition violated by the LSW continuity condition. Stability is demonstrated through the application of a closely related Le Chatelier Principle which also appeared in the LS formulation. Wagner's equivalent was to assume separability of variables as a scaling postulate. The coefficients in our analytic distribution function and the appropriate order parameter are established through Wagner's quadratic limit as the radius goes to zero and the requirement that the largest particle which terminates the stable distribution must also have the maximal surface velocity. Our smoothed asymptotic particle radius distribution function generates the standard t ^1/3 scaling but is sufficiently different than that of LSW and its generalizations as to be easily distinguishable experimentally. Indeed, the experimental distribution functions are demonstrated to strongly favour the present formulation. Since the explicit volume fraction of particles ϕ does not enter into our treatment and the statistical mean distance between particles goes as (1 − ϕ)^1/3, the onset of functional violations due to diffusion impingement should only occur as ϕ exceeds 0.3, again in accord with many experiments.