Diffusion of gelation clusters in the Zimm model

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Starting from a Zimm model we study selfdiffusion 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 nontrivial scaling of the Kirkwood diffusion constant averaged over all clusters of fixed size n, Dₙ n- b with b=1/dₛ-1/2 given in terms of the spectral dimension dₛ 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 εᵃ of the cluster-averaged effective diffusion constant at the gel point with exponent a = (3/2 -τ+1/dₛ)/σ.