Linear Elastic Behavior of Dry Soap Foams

Abstract

Linear elastic constants are computed for three dry foams that have crystal symmetry, bubbles with equal volume V, and films with uniform surface tension T. The Kelvin, Williams, and Weaire–Phelan foams contain one, two, and eight bubbles in the unit cell, respectively. All three foams have 14-sided bubbles, but these tetrakaidecahedra have different topology; the Weaire–Phelan foam also contains pentagonal dodecahedra. In addition to the bulk modulus for volume compression, we calculate two shear moduli for the Kelvin and Weaire–Phelan foams, which have cubic symmetry, and four shear moduli for the Williams foam, which has tetragonal symmetry. The Williams foam has five elastic constants, not six, because the stress remains isotropic for uniform expansion; this is not guaranteed by symmetry alone. The two shear moduli for the Weaire–Phelan foam differ by less than 5%. The other two foams exhibit much greater elastic anisotropy; their shear moduli differ by a factor of 2. An effective isotropic shear modulus G, which represents the response averaged over all orientations, is evaluated for each foam. Scaled by T/ V 1/3, Gis 0.8070, 0.7955, and 0.8684 for the Kelvin, Williams, and Weaire–Phelan foams, respectively. When extrapolated to the dry limit, the shear modulus data of Princen and Kiss (for concentrated oil-in-water emulsions with polydisperse drop-size distributions) fall within the range of our calculations. The Surface Evolver program, developed by Brakke, was used to compute minimal surfaces for the dry foams. Also reported for each undeformed foam are various geometric constants relating to interfacial energy density, cell edge length, and bubble pressure.