Coagulation Equations with Gelation

Smoluchowski's equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernel Kij models the bonding mechanism. For different classes of kernels we derive criteria for the occurrence of gelation, and obtain critical exponents in the preand postgelation stage in terms of the model parameters; we calculate bounds on the time of gelation to, and give an exact postgelation solution for the model K,j = (/j)'~ (~0 > 1/2) and Kq = a i+j (a > 1). For the model Kig = i '~ + j ~ (w < 1, without gelation) initial solutions are given. It is argued that the kernel Kij~(/j') '~ with ~0~ 1 l / 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 (o~ = 1). For all ~o the exponents, T = ~0 + 3 /2 and o = ~ 1/2, y = (3/2 o~)/(o~ 1/2) and /3 = 1, characterize the size distribution, at and slightly below the gel point, under the assumption that scaling is valid.