Average Evolution and Size-Topology Relations for Coarsening 2d Dry Foams

TL;DR

This study derives theoretical relations for 2D dry foam coarsening based on the von Neumann law and confirms them through high-precision image analysis, revealing consistent area and side-number distribution behaviors.

Contribution

The paper introduces new theoretical relations linking bubble area moments with side-number averages in coarsening foams, validated by experimental data.

Findings

Good agreement between theory and experiment for moments 2 to 20

Derived relations accurately predict area and side-number distribution moments

Experimental data supports the self-similar scaling state of coarsening foams

Abstract

Two-dimensional dry foams coarsen according to the von Neumann law as $dA/dt \propto (n-6)$ where $n$ is the number of sides of a bubble with area $A$. Such foams reach a self-similar scaling state where area and side-number distributions are stationary. Combining self-similarity with the von Neumann law, we derive time derivatives of moments of the bubble area distribution and a relation connecting area moments with averages of the side-number distribution that are weighted by powers of bubble area. To test these predictions, we collect and analyze high precision image data for a large number of bubbles squashed between parallel acrylic plates and allowed to coarsen into the self-similar scaling state. We find good agreement for moments ranging from two to twenty.