Kinetics of gelation and universality

Abstract

The exact solution (size distribution c_k(t) and moments M_n(t)) of Smoluchowski's coagulation equation (S-model) and of a modified equation (F-model) with a coagulation rate K_ij=ij for i- and j-clusters is obtained for arbitrary c_k(0) in the sol (tt_c) phases, where t_c is the gel point. The behaviour of c_k(t) and M_n(t) is given for k to infinity , t to infinity and t to t_c. The critical exponents, critical amplitudes and scaling function that characterise the singularities near the non-equilibrium phase transition are calculated. For short-range c_k(0) (i.e. all M_n< infinity ) the F-model belongs to the universality class of classical gelation theories and of bond percolation on Cayley trees; the S-model does not.