Coagulation processes with a phase transition

Abstract

Smoluchowski's coagulation equation with a collection kernel K(x, y) ~ (xy)[omega] with describes a gelation transition (formation of an infinite cluster after a finite time tc (gel point)). For general [omega] and t > tc the size distribution is c(x, t) ~ x-[tau] for x --> [infinity] with [tau] = [omega] + 3/2. For [omega] = 1, we determine c(x, t) and the time dependent sol mass M(t) for arbitrary initial distribution in pre- and post-gel stage, where c(x, t) ~ x-5/2 exp(-x/xc) for large x and t tc; for large xt and t > tc. Here xc is a critical cluster size diverging as (t - tc)-2 as t [short up arrow] tc. For initial distributions such that c(x, 0) ~ xp-2 as x --> 0, we find M(t) ~ t-p/(p+1) as t --> [infinity]. New explicit post-gel solutions are obtained for initial gamma distributions, c(x, 0) ~ xp-2e-px (p > 0) in the form of a power series (convergent for all t), and reducing for p = [infinity] to the solution for monodisperse initial conditions. For p = 1, the solution is found in closed form.