Diffusion of gelation clusters in the Zimm model

Starting from a Zimm model, we study self-diffusion in a solution of crosslinked monomers. We focus on the effects of the hydrodynamic interaction on the dynamics and the critical behaviour at the sol-gel point. Hydrodynamic interactions cause the clusters' diffusion constant to depend not only on the cluster's size but also on the cluster's shape --in contrast to the Rouse model. This gives rise to a non-trivial scaling of the Kirkwood diffusion constant averaged over all clusters of fixed size n, Dn approximately equal to n(-b) with b = 1/d(s) - 1/2 given in terms of the spectral dimension d(s) of critical percolation clusters. The long-time decay of the incoherent scattering function is determined by the diffusive motion of the largest clusters. This implies the critical vanishing D(eff) approximately equal to epsilon(a) of the cluster-averaged effective diffusion constant at the gel point with exponent a = (3/2 - tau + 1/d(s))/sigma.