Statistical Physics Theory of Ostwald Ripening

Abstract

We review recent theoretical developments in our understanding of the late-stage processes of phase separation in binary alloys. When a system is quenched into a metastable state, phase separation occurs by nucleation and growth^1,2 in the late stage, known as Ostwald ripening³, the minority phase takes the form of spherical droplets whose growth and dissolution proceeds by an evaporation-condensation mechanism. There are two theoretical aspects in understanding of the dynamics of such a phase separation, depending on what processes we are interested in. The first is to study the causal motion which is described by a single droplet size distribution function f(R,t) with radius R, and which is experimentally observable by an electron microscope. This was first done in the monumental works by Lifshitz and Slyozov⁴ and independently by Wagner⁵. They found the celebrated scaling law f(R,t)= [n(t)/R(t)]po(R/R(t)), where the average droplet radius grows as R(t)~t^1/3 and the number density of droplets decays as n(t)~t^~l. The relative droplet size distribution function po(p) is a time-independent function of p. Although their works were the origin or later theoretical studies of Ostwald ripening, their results were valid only in the limit of zero volume fraction of the minority phase and did not agree with experimental observations where the volume fraction is not infinitely zero. After their works, many attempts^6–10 to extend their theory to the case of finite volume fraction have been proposed.