liquid. The continuous liquid phase is contained in two re-Linear elastic constants are computed for three dry foams that gions: the thin films that separate adjacent bubbles and the have crystal symmetry, bubbles with equal volume V, and films Plateau borders that form an interconnected network of chanwith uniform surface tension T. The Kelvin, Williams, and nels along bubble edges where films join. The thin films are Weaire-Phelan foams contain one, two, and eight bubbles in the stabilized against rupture by soap or other surfactants that unit cell, respectively. All three foams have 14-sided bubbles, but accumulate at the gas-liquid interfaces. Kraynik (1) and these tetrakaidecahedra have different topology; the Weaire-Phe-Weaire and Fortes (2) have reviewed applications and previlan foam also contains pentagonal dodecahedra. In addition to ous studies on foam rheology.
the bulk modulus for volume compression, we calculate two shear
The volume fraction of continuous liquid phase is zero in moduli for the Kelvin and Weaire-Phelan foams, which have cubic symmetry, and four shear moduli for the Williams foam, the hypothetical dry foam limit. Fundamental principles that which has tetragonal symmetry. The Williams foam has five elastic govern the geometry of all dry foams under static conditions, constants, not six, because the stress remains isotropic for uniform whether deformed or undeformed, were recognized in the expansion; this is not guaranteed by symmetry alone. The two 19th century. Observations by Plateau (3) of soap bubble shear moduli for the Weaire-Phelan foam differ by less than 5%. clusters and soap films supported on wires led to the laws The other two foams exhibit much greater elastic anisotropy; their that now bear his name. According to Plateau's laws, each shear moduli differ by a factor of 2. An effective isotropic shear film has uniform mean curvature; three films meet at equal modulus G V , which represents the response averaged over all orien- angles of 120ะ to form cell edges (the curved lines in a dry tations, is evaluated for each foam. Scaled by T/V 1/3 , G V is 0.8070, foam that correspond to Plateau borders in a wet foam); 0.7955, and 0.8684 for the Kelvin, Williams, and Weaire-Phelan and four edges join at each vertex at the tetrahedral angle, foams, respectively. When extrapolated to the dry limit, the shear cos 01 (01/3) ร 109.47ะ. These geometric properties balmodulus data of Princen and Kiss (for concentrated oil-in-water emulsions with polydisperse drop-size distributions) fall within ance forces arising from uniform surface tension in the films the range of our calculations. The Surface Evolver program, develand any pressure differences between the bubbles. Taylor oped by Brakke, was used to compute minimal surfaces for the (4) has shown that Plateau's laws are a mathematical consedry foams. Also reported for each undeformed foam are various quence of minimizing the surface area or surface energy of geometric constants relating to interfacial energy density, cell edge a foam, which is necessary for ''stability'' of the structure length, and bubble pressure.