Critical behaviour of the Rouse model for gelling polymers

TL;DR

This paper reveals that the true critical exponents for the gelation transition in the Rouse model differ from traditional values, linking them to spectral properties of fractal clusters and random walk return probabilities.

Contribution

It derives the actual critical behavior of the Rouse model for gelling polymers from spectral properties, challenging previous assumptions and providing new scaling relations.

Findings

Critical exponents differ from traditional values

Spectral properties determine the critical behavior

Scaling relations involve spectral dimension and cluster-size exponents

Abstract

It is shown that the traditionally accepted "Rouse values" for the critical exponents at the gelation transition do not arise from the Rouse model for gelling polymers. The true critical behaviour of the Rouse model for gelling polymers is obtained from spectral properties of the connectivity matrix of the fractal clusters that are formed by the molecules. The required spectral properties are related to the return probability of a "blind ant"-random walk on the critical percolating cluster. The resulting scaling relations express the critical exponents of the shear-stress-relaxation function, and hence those of the shear viscosity and of the first normal stress coefficient, in terms of the spectral dimension $d_{s}$ of the critical percolating cluster and the exponents $\sigma$ and $\tau$ of the cluster-size distribution.