Statistical-mechanical theory of coarsening of spherical droplets

A new statistical-mechanical theory of diffusion-controlled droplet coarsening is presented. With the aid of a scaling expansion method, a spatial graining is carried out in a manner consistent with an expansion in droplet volume fraction φ to obtain kinetic equations for a single distribution function of droplets. It is shown that there two characteristic stages of coarsening, depending on their space-time scales; an intermediate stage and a late stage. In both stages, new kinetic equations are systematically derived to order φ. These equations have two terms at order φ; a collisionless drift term and a collision term. The collision term is shown to be different from the conventional encounter integral discussed by Lifshitz and Slyozov since the former is of order φ and describes distant (soft) collisions, while the latter is of order φ and describes close (hard) collisions. It is shown in both stages that the mean droplet radius increases as the cube root of the time (t13). A scaling behavior of the distribution function is also found in both stages. In particular, in the late stage this scaling behavior is shown to coincide with that obtained by Lifshitz and Slysov in the limit φ→O. It is also pointed out that a naive expansion in powers of φ breaks down due to the long -range nature of interactions among droplets through diffusions. Fluctuations around the kinetic equations are also explicitly explored. They are shown to be nonthermal fluctuations generated by the soft collision process and to be small. Especially, in the late stage they are shown to obey a linear Gaussian Markov process, satisfying a fluctuation-dissipation relation of the second kind.