Markov models of coarsening in two-dimensional foams with edge rupture

TL;DR

This paper develops Markov models to describe edge rupture in 2D foams, deriving kinetic equations and demonstrating gelation phenomena through simulations, linking network topology with statistical physics.

Contribution

It introduces a Markov process framework for foam edge rupture, connecting network topology with nonlinear kinetic equations and gelation behavior.

Findings

Mean-field model solutions exhibit gelation.

Numerical simulations confirm gelation in the model.

Statistical behaviors are consistent between network and mean-field models.

Abstract

We construct Markov processes for modeling the rupture of edges in a two-dimensional foam. We first describe a network model for tracking topological information of foam networks with a state space of combinatorial embeddings. Through a mean-field rule for randomly selecting neighboring cells of a rupturing edge, we consider a simplified version of the network model in the sequence space $\ell_1(\mathbb N)$ which counts total numbers of cells with $n\ge 3$ sides ($n$-gons). Under a large cell limit, we show that number densities of $n$-gons in the mean field model are solutions of an infinite system of nonlinear kinetic equations. This system is comparable to the Smoluchowski coagulation equation for coalescing particles under a multiplicative collision kernel, suggesting gelation behavior. Numerical simulations reveal gelation in the mean-field model, and also comparable statistical behavior between the network and mean-field models.