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Aqueous foams and a wide range of related systems are believed to coarsen bygas diffusion between neighboring domains into a statistically self-similarscaling state, after the decay of initial transients, such that dimensionlesssize and shape distributions become time independent and the average grows as apower law. Partial integrodifferential equations for the time evolution of thesize distribution for such phase separating systems can be formulated forarbitrary initial conditions, but these are cumbersome for analyzing data onnon-scaling state preparations. Here we show that essential features of theapproach to the scaling state can be captured by an exactly-solvable ordinarydifferential equation for the evolution of the average bubble size. The keyingredient is to characterize the the bubble size distribution approximately,using the average size of all bubbles and the average size of the criticalbubbles, which instantaneously neither grow nor shrink. The difference betweenthese two averages serves as a proxy for the width of the size distribution. Totest our model, we compare with data for quasi-two dimensional dry foamscreated with three different initial amounts of polydispersity. This allows usto readily identify the critical radius from the average area of six-sidedbubbles, whose growth rate is zero by the von~Neumann law. The growth of theaverage and critical radii agree quite well with exact solution, though themost monodisperse sample crosses over to the scaling state faster thanexpected. A simpler approximate solution of our model performs equally well.Our approach is applicable to 3d foams, which we demonstrate by re-analyzingprior data, as well as to froths of dilute droplets and to phase separationkinetics for more general systems such as emulsions, binary mixtures, andalloys.

A simply solvable model capturing the approach to statistical self-similarity for the diffusive coarsening of bubbles, droplets, and grains