Diffusion of gelation clusters in the Zimm model

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 $n$, ${\hat D}n\sim n^{-{\hat b}}$with${\hat b}=1/d_s-1/2$ 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 $D{\rm eff}\sim \epsilon^a$ of the cluster-averaged effective diffusion constant at the gel point with exponent $a=(3\mathrm{/}2-\tau +1\mathrm{/}{d}_s)\mathrm{/}\sigma$.