Diffusion of gelation clusters in the Zimm model

TL;DR

This paper investigates how hydrodynamic interactions influence the diffusion and critical behavior of clusters in a Zimm model near the gelation point, revealing nontrivial scaling laws and decay dynamics.

Contribution

It introduces a detailed analysis of hydrodynamic effects on cluster diffusion in the Zimm model, highlighting shape dependence and deriving new scaling exponents at the gel point.

Findings

Diffusion constant depends on cluster shape, not just size.

Derived a nontrivial scaling law for the Kirkwood diffusion constant.

Identified the critical vanishing of the effective diffusion constant at gelation.

Abstract

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 $a = (3/2 -\tau +1/d_{s})/\sigma$.