Towards finite-dimensional gelation

TL;DR

This paper develops a finite-dimensional model of gelation using replica theory and Mayer-cluster expansion, revealing how connectivity influences localization lengths and the sol-gel transition.

Contribution

It introduces a systematic approach to include corrections beyond mean-field in finite-dimensional gelation models, linking connectivity to localization properties.

Findings

Mean-field theory captures basic gelation behavior.

Higher-order corrections shift the critical point.

Connectivity correlates with localization length.

Abstract

We consider the gelation of particles which are permanently connected by random crosslinks, drawn from an ensemble of finite-dimensional continuum percolation. To average over the randomness, we apply the replica trick, and interpret the replicated and crosslink-averaged model as an effective molecular fluid. A Mayer-cluster expansion for moments of the local static density fluctuations is set up. The simplest non-trivial contribution to this series leads back to mean-field theory. The central quantity of mean-field theory is the distribution of localization lengths, which we compute for all connectivities. The highly crosslinked gel is characterized by a one-to-one correspondence of connectivity and localization length. Taking into account higher contributions in the Mayer-cluster expansion, systematic corrections to mean-field can be included. The sol-gel transition shifts to a higher number of crosslinks per particle, as more compact structures are favored. The critical behavior of the model remains unchanged as long as finite truncations of the cluster expansion are considered. To complete the picture, we also discuss various geometrical properties of the crosslink network, e.g. connectivity correlations, and relate the studied crosslink ensemble to a wider class of ensembles, including the Deam-Edwards distribution.