On the shear modulus of two-dimensional liquid foams: A theoretical study of the effect of geometrical disorder

The shear modulus of two-dimensional liquid foams in the dry limit of low liquid content has been studied theoretically. The focus is on the effect of geometrical disorder on the shear modulus (besides the influence of surface tension). Various theoretical predictions are formulated that are all based on the assumptions of isotropic geometrical characteristics, incompressible bubbles, and negligible edge curvature. Three of these predictions are based on a transformation of Princen's theory that is strictly valid only for regular hexagonal bubbles. Another prediction takes into account variations in bubble areas by considering the foam as consisting of approximately regular hexagonal bubbles with varying areas. Two other predictions are solely based on the characteristics of the bubble edges. The first of these is based on the assumption of affine movement of bubble vertices, while the second accounts for nonaffine deformation by considering the interaction with neighboring edges. The theoretical predictions for the shear modulus are compared with the result from a single foam simulation. For the single simulation considered, all predictions, except that based on affine movement of bubble vertices, are close to the value obtained from this simulation.