# Border-Crossing Model for the Diffusive Coarsening of Two-Dimensional and Quasi-Two-Dimensional Wet Foams

## TL;DR

This paper introduces a border-crossing model for wet foam coarsening that accounts for gas flux through borders and vertices, improving understanding of coarsening dynamics in wet and quasi-2D foams.

## Contribution

It develops a new model incorporating border-crossing flux and modifies the von Neumann law for better accuracy in wet foam coarsening analysis.

## Key findings

- Border-crossing flux scales with geometric average of film and border thicknesses.
- Numerical solutions verify the proposed flux scaling.
- Modified von Neumann law accounts for bubble size and shape effects.

## Abstract

For dry foams, the transport of gas from small high-pressure bubbles to large low-pressure bubbles is dominated by diffusion across the thin soap films separating neighboring bubbles. For wetter foams, the film areas become smaller as the Plateau borders and vertices inflate with liquid. So-called "border-blocking" models can explain some features of wet-foam coarsening based on the presumption that the inflated borders totally block the gas flux, however, this approximation dramatically fails in the wet/unjamming limit where the bubbles become close-packed spheres and coarsening proceeds even though there are no films. Here, we account for the ever-present border-crossing flux by a new length scale defined by the average gradient of gas concentration inside the borders. We compute that it is proportional to the geometric average of film and border thicknesses, and we verify this scaling by numerical solution of the diffusion equation. We similarly consider transport across inflated vertices and surface Plateau borders in quasi-2d foams. And we show how the $dA/dt=K_0(n-6)$ von~Neumann law is modified by the appearance of terms that depend on bubble size and shape as well as the concentration gradient length scales. Finally, we use the modified von~Neumann law to compute the growth rate of the average bubble area, which is not constant.