On the possible occurrence of instantaneous gelation in Smoluchowski's coagulation equation

Considers possible solutions of Smoluchowski's coagulation equation if the rate constants K(i, j) behave as K(i, j) approximately i mu j v as j to infinity , with an exponent nu satisfying nu >1. The author finds that, for such rate constants. Smoluchowski's equation predicts the instantaneous occurrence of a gelation transition. Thus the gel time t c =0 in such models. This result confirms recent, speculation in the literature. The author also studies the structure of post-gel solutions of Smoluchowski's equation, if they exist. For a given value of nu , the results depend on the value of the exponent mu . If mu >( nu -1), one finds that the cluster size distribution c k (t) approaches a universal form at large times (t to infinity ). No solutions exist if mu <or=( nu -1). Physically this means that the sol phase is depleted instantaneously.