Modeling the approach to statistical self-similarity for systems that coarsen by the diffusion of material between neighboring bubbles, droplets, or grains

Aqueous foams are commonly believed to coarsen by gas diffusion between neighboring bub- bles into a statistically self-similar scaling state, such that the shape of size distributions is time- independent. This allows prediction of average growth rates as well as size-topology relations. Integro-differential PDE models for phase separating systems, as first shown by Lifshitz and Sly- ozov, can be written down for the evolution of the size distribution but are generally intractable. Here we show that essential features of the approach to the scaling state can be captured by an exactly-solvable pair of coupled differential equations for the evolution of the average bubble size and of the critical bubble size, which instantaneously neither grows nor shrinks. To test our sim- plified model, we compare with data for two-dimensional dry foams created with different initial polydispersities. This allows us to readily identify the critical radius from the average area of six- sided bubbles, whose growth rate is zero by the von Neumann law. Preliminary results show good agreement. Our approach is applicable to 3d foams, as well to dilute phase-separating systems like very wet froths. We hope it will aid in analysis of data recently-collected aboard the International Space Station for the coarsening of 3d foams with different liquid content.