Force vs separation for axisymmetric nearly minimal surfaces

Motivated by the desire to understand the shapes exhibited by colloidal membranes, we study the shape of a surface with bending stiffness and fixed surface tension, stretched between two circular boundaries. The boundary circles lie in parallel planes and have axial symmetry. We work in the limit of large bending stiffness, in which the surface shapes are nearly catenoids, and calculate the force required to hold the rings apart at a fixed separation. We find the the force-separation curve has two branches, with one branch corresponding to the stable branch of shapes observed in catenoidal soap films. As in the case of the soap film, we find a maximal separation of the rings for this branch. We also find a low branch of solutions, as in the case of a soap film, but we do not find an upper limit to the separation.