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

Abstract

Considers possible solutions of Smoluchowski's coagulation equation if the rate constants K(i, j) behave as K(i, j) approximately imu jv 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 tc=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 ck(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.