The evolution of a foam is determined by drainage flow of the continuous (liquid) phase and coarsening (aging) of the dispersed phase (gas bubbles). Free-drainage experiments with slow-and fast-coarsening gases show markedly different dynamics and elucidate the importance of the coupling of the two effects. Strong coarsening leads to drainage times that are shorter (accelerated drainage) and independent of the initial liquid content (self-limiting drainage). A model incorporating the physics of

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... both drainage and diffusive coarsening shows quantitative agreement with experiment. Foams [1,2] are a prime example of a multiphase "soft condensed matter" system. They have important applications in the food and chemical industries, firefighting, mineral processing, and structural material science [2, 3] . Recent research in foams and emulsions has centered on three topics which are often treated separately, but are in fact interdependent: drainage, coarsening, and rheology; see Fig. 1 . We focus here on a quantitative description of the coupling of drainage and coarsening. Foam drainage is the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity [5] [6] [7] . The foam drainage equation models the dynamics of the liquid volume fraction e in the foam on length scales larger than the bubble size. The exact form of this equation depends on the mobility of liquid-gas interface and thus on the choice of surfactant [8] . Rigid interfaces result in the channel-dominated model [6, 9] , where viscous dissipation of the flow occurs in all of the liquid volume, most of which is in the channels. For mobile interfaces, dissipation in the nodes can dominate [10] . Both models can be treated as limiting cases of a generalized theory [11] . Foams evolve towards thermodynamic equilibrium by reducing their total surface area as the average size of the bubbles grows over time, or coarsens, by either rupture of the liquid films between bubbles or growth through diffusive exchange of gas. The gas exchange is only appreciable through the thin, almost flat film areas of the polyhedral bubbles in a dry foam. We concentrate on diffusive coarsening, as rupture can be minimized using a surfactant that generates stable films. Previously [10, 11] we have minimized coarsening in order to study drainage alone. In general, however, the time scales of diffusive coarsening and drainage are not well separated, e.g., foams with small bubbles tend to coarsen quickly and drain slowly. Also, foams with gases of high solubility and diffusivity coarsen rapidly. We choose C 2 F 6 , an almost insoluble gas, and highly soluble CO 2 , to create initially identical aqueous foams in order to study the effects of coarsening on drainage. We revisit the classic free-drainage experiment that has been in use for over 40 years [12] . A vertical foam column of height H with uniform volume fraction e 0 at time t 0 drains liquid that accumulates at the bottom with height h͑t͒ He 0 2 R 0 2H e͑z, t͒ dz (see the inset in Fig. 2a ). The liquid height increases until all liquid has drained out of the foam, so that h͑t !'͒ ϵ h' e 0 H. To generate the foam, gas (either CO 2 or C 2 F 6 ) and a soap solution containing 0.5% SDS (sodium dodecyl sulfate) by weight in distilled water are pumped separately into a single line through a filter. The resultant coarse froth is then forced through a porous brass plug, which extrudes the final foam. Imaging bubbles at the tube wall, we measure initial bubble diameters of 0.5 mm with ഠ10% polydispersity. We assume the initial bubble volume to be equal to that of polyhedral bubbles in the bulk, for which V 0 d V L 3 0 with d V ഠ 11.3 as a typical value. The average initial edge (channel) length of bulk bubbles is then L 0 ഠ 0.014 cm. The liquid volume fraction can be adjusted by the pumping rates for gas and liquid to yield foams as FIG. 1. Schematic of the interdependence of drainage, coarsening, and rheology of foams. For example, drainage results in a drier foam with increased shear modulus and accelerated coarsening. Coarsening in turn enhances drainage, but also decreases the shear modulus [4]. 4704 0031-9007͞01͞86(20)͞4704(4)$15.00