Stiffening of fluid membranes and entropy loss of membrane closure: Two effects of thermal undulations

The problem of membrane softening by thermal undulations is revisited. In contrast to general belief, fluid membranes are predicted to be stiffened, not softened, by their undulations. Equal values of the effective bending rigidity are calculated from the interplay of local mean curvature modes (hats) on the basically flat membrane and from the coupling of spherical harmonic modes with spherical curvature. In addition, a conjecture is made on the entropy of membrane closure. It relies on a similarity of membrane closure to periodic boundary conditions.