Statistical Thermodynamics of Irreversible Aggregation and Gelation

TL;DR

This paper develops a rigorous thermodynamic framework for irreversible binary aggregation, demonstrating that gelation is a true phase transition with characteristic thermodynamic signatures.

Contribution

It introduces a statistical thermodynamics approach to binary aggregation, providing exact solutions and connecting gelation to phase transition phenomena.

Findings

Gelation exhibits thermodynamic phase transition signatures.

Partition function derived for arbitrary kernels.

Complete solutions for product kernel aggregation.

Abstract

Binary aggregation is known to lead, under certain kinetic rules, to the coexistence of two populations, one consisting of finite-size clusters (sol), and one that contains a single cluster that carries a finite fraction of the total mass (giant component or gel). The sol-gel transition is commonly discussed as a phase transition by qualitative analogy to vapor condensation. Here we show that the connection to thermodynamic phase transition is rigorous. We develop the statistical thermodynamics of irreversible binary aggregation in discrete finite systems, obtain the partition function for arbitrary kernel, and show that the emergence of the gel cluster has all the hallmarks of a phase transition, including an unstable van der Waals loop. We demonstrate the theory by presenting the complete pre- and post-gel solution for aggregation with the product kernel, $k_{ij}=i j$.