Self-similar collapse of stationary bulk foams



In bulk foams, the dependence of bubble size on time can be deduced from a hypothesis of statistical self-similarity and the scaling characteristic of the volume change rate of a foam bubble. If this rate, equation image, scales as the mean bubble volume v to the power α, the total surface area of the foam decreases as an inverse of time to the power 1/[3 (1 − α)]. Coarsening of polyhedral foam scales with α = 5/6, when molecular diffusion limits gas transport across lamellae and liquid drainage through Plateau borders limits lamella thinning. Excess liquid is released by disappearing small bubbles and flows into the lamellae and Plateau borders of growing large bubbles. If none of this liquid accumulates in the foam, coarsening is exponential and α = 1. When resistance to mass transfer at the lamella surfaces is the rate-limiting step, polyhedral foam coarsens with α = 1/3. Coarsening of slowly draining, spherical-bubble foam scales with α = 0. The theory is compared with nine measurements of the total surface area of polyhedral and spherical-bubble foams pregenerated from aqueous solutions of sodium lauryl sulfate, hexadecyltrimethylammonium bromide, alpha-olefin sulfonate with alkyl chain lengths from C14 to C16, and two shaving creams. The theory also proves that the collapse of a two-dimensional, polygonal foam is self-similar and scales with α = l. In all cases, our theory agrees well with experiment and numerical calculations.