Relaxation in Aqueous Foams
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of applications. For example, their low density and gas impermeability make aqueous foams an ideal firefighting agent, especially for flammable liquids and in instances when water availability is limited. Their high interfacial area and ability to trap gases, liquids, and solids make aqueous foams useful for physical and chemical separations as well as froth flotation, the isolation and cleanup of toxic spills, and the application of dyes to textiles. And based in part on their ability to resist small static shear like a solid and yet flow like a liquid under high shear, aqueous foams also find use as cosmetics and foods, and in enhancing oil recovery. In all instances, fundamental understanding of relaxation processes in aqueous foams is important for precise control of the material's stability and rheology. Stability Aqueous foams are intrinsically nonequilibrium materials; with time, the gas and liquid phases inexorably separate by some combination of three basic mechanisms: gravitational drainage of liquid from between bubbles, direct coalescence of neighboring bubbles via film rupture, and the diffusion of gas from smaller to larger bubbles, which is known as coarsening. The first two mechanisms can be eliminated in practice, but the last cannot.
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Figure 1. Hierarchy of structure in an aqueous foam: Large gas bubbles are separated by thin liquid films which are stabilized against coalescence by adsorbed surfactants.
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Coarsening fundamentally limits the useful lifetime of even the most stable aqueous foams, and is the focus of our discussion. Issues of foam stability are also important in settings where the inadvertent foaming of a liquid needs to be minimized. The coarsening mechanism is depicted schematically in Figure 3 for extreme examples of spherical and polyhedral foams. In all cases, the diffusion of gas from smaller to larger bubbles is driven by the gas-liquid surface tension and serves to reduce the total interfacial area with time. Another generic feature is that the flux of gas per unit area of interfacial contact is proportional to the local interfacial curvature and, hence, the reciprocal of the bubble size. The rate of change of bubble volume therefore scales as area times curvature, and implies that the average radius grows with time as tV2. The actual coarsening rate depends on material parameters such as the solubility and diffusion coefficient of gas in the liquid, the interfacial tension and surface forces between adjacent bubbles, and geometrical factors such as the volume fraction of liquid and its distribution between neighboring bubbles; however, precise details are not well understood. The problem has long been thought to be mathematically similar to Ostwald ripening3'4 and shares many features with grain growth, including a self-similar domain size distribution. Several techniques are available for characterizing coarsening. The simplest is to o
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