SELF-SIMILAR COLLAPSE OF STATIONARY BULK FOAMS

In bulk foams, the dependence of bubble size on time can be deduced fr om a hypothesis of statistical self-similarity and the scaling charact eristic of the volume change rate of a foam bubble. If this rate, v, s cales as the mean bubble volume vBAR to the power alpha, the total sur face area of the foam decreases as an in verse of time to the power 1/ [3 (1 - alpha)]. Coarsening of polyhedral foam scales with alpha = 5/6 , when molecular diffusion limits gas transport across lamellae and li quid drainage through Plateau borders limits lamella thinning. Excess liquid is released by disappearing small bubbles and flows into the la mellae and Plateau borders of growing large bubbles. If none of this l iquid accumulates in the foam, coarsening is exponential and alpha = 1 . When resistance to mass transfer at the lamella surfaces is the rate -limiting step, polyhedral foam coarsens with alpha = 113. Coarsening of slowly draining, spherical-bubble foam scales with alpha = 0. The t heory is compared with nine measurements of the total surface area of polyhedral and spherical-bubble foams pregenerated from aqueous soluti ons of sodium lauryl sulfate, hexadecyltrimethylammonium bromide, alph a-olefin sulfonate with alkyl chain lengths from C-14 to C-16, and two shaving creams. The theory also proves that the collapse of a two-dim ensional, polygonal foam is self-similar and scales with alpha = 1. In all cases, our theory agrees well with experiment and numerical calcu lations.