Bending Plateau's laws: Equilibrium shape of an elastic ribbon in a 2D bubble column

A foam is a set of gas bubbles trapped in a continuous liquid or solid material. In the low-density limit, liquid foams usually exhibit tetrahedral structures guided by capillarity and Plateau’s laws. Despite recent progress in “liquid foam templating” to produce solid foams in a controlled and self-assembled manner from liquid precursors, we still lack methods to tune explicitly the geometry and topology of foams (shape, size and connection of the bubbles). To reach a better customization of such structures, we integrate deformable solids to modify the structure of foams in the liquid state, using an elastocapillarity approach. The competition between elasticity and capillarity has indeed already proven to be an efficient way to assemble, orient or spontaneously bend slender elastic structures.

We consider a model experiment, consisting of elastomeric ribbons which deform under the action of capillary forces in a quasi-2D foam column. Confining bubbles into square section tubes leads to periodically ordered structures. We focus on the so-called “staircase” structure where bubbles of equal volume are rearranged in a staggered pattern. This structure has central soap films connected with 120° angles, in which we introduce elastic ribbons of different bending rigidities. We establish the key parameters leading to the deformation of the ribbon (capillarity dominated) or to the deformation of the foam (rigidity dominated) in the extreme cases. Using X-ray micro-tomography, we quantify the equilibrium shapes of the quasi-2D foam/ribbon systems: we present a detailed analysis of the ribbon profile, that compares well with theoretical predictions in the whole range of bending rigidities. We also provide a proof-of-concept that such setup can be used as a method to mold materials with characteristic shapes and curves imprinted by the foam structure. Finally, we discuss the extension of this approach to 3D complex materials.