Deformed foam structure and transitions in a tube

The "staircase" (2,1.1) foam structure in an ordered foam under expansion in a cylindrical tube and the structural transition to the "bamboo" (1,1,0) structure are studied experimentally, analytically, and numerically using the Surface Evolver minimization program. An analytical expression of the oblique interface ill the (2,1,1) structure, a minimal surface, but curved by stretching of the foam, is given as a classic example of singular perturbation (boundary layer) theory. The structural transition from staircase structure to bamboo structure takes place at a considerable dilation. The transition is not due to an edge length tending to zero continuously with dilation, but results from the equilibrium (2,1,1) solution losing its stability due to a saddle node bifurcation.