Kinetics of gelation and universality

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Abstract
The exact solution (size distribution ck(t) and moments Mn(t)) of Smoluchowski's coagulation equation (S-model) and of a modified equation (F-model) with a coagulation rate Kij=ij for i- and j-clusters is obtained for arbitrary ck(0) in the sol (ttc) phases, where tc is the gel point. The behaviour of ck(t) and Mn(t) is given for k to infinity , t to infinity and t to tc. The critical exponents, critical amplitudes and scaling function that characterise the singularities near the non-equilibrium phase transition are calculated. For short-range ck(0) (i.e. all Mn< 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.