Coagulation equations with gelation

Abstract

Smoluchowski's equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernel K ij models the bonding mechanism. For different classes of kernels we derive criteria for the occurrence of gelation, and obtain critical exponents in the pre- and postgelation stage in terms of the model parameters; we calculate bounds on the time of gelation t c , and give an exact postgelation solution for the model K ij =( ij ω ) (ω>1/2) and K ij =a i+j ( a >1). For the model K ij = i ω + j ω ( ω <1, without gelation) initial solutions are given. It is argued that the kernel K ij ∼ ij ω with ω≃1−1/d ( d is dimensionality) effectively models the sol-gel transformation in polymerizing systems and approximately accounts for the effects of cross-linking and steric hindrance neglected in the classical theory of Flory and Stockmayer ( Ω =1). For all Ω the exponents, t=Ω +3/2 and σ=Ω −1/2, γ =(3/2− Ω)/(Ω − 1/2) and Β =1, characterize the size distribution, at and slightly below the gel point, under the assumption that scaling is valid.