Electrostatic rigidity of charged membranes in systems without added salt

The thermodynamics of surfactants in solution has largely been understood in the framework of the flexible surface model. We calculate the bending rigidity for a case where this model might be put to a strong test: ionic surfactant systems with no added salt to screen the electrostatics. In so doing, we present a perturbative solution to the Poisson-Boltzmann equation for the case of two undulating sheets with intervening counterions. The predictions to which this leads for the bending modulus are shown (for a particular choice of undulation mode) to be identical to that of the solution for the geometry of concentric cylinders. This indicates that the curvature free energy is independent of the global aggregate geometry, even in systems with counterions only.