Buckling and nonlocal elasticity of charged membranes

The elastic behavior of an interacting, and, in specific, of a charged flexible membrane is considered. In the first part of this paper the effective nonlocal elastic energy of a membrane due to a pairwise and arbitrary intra-membrane interaction is derived. Nonlocal elasticity is included to all orders, this description, therefore, corresponds to an infinite resummation of the standard gradient expansion. In the second part, the pair interaction between segments of an (on average) neutral membrane consisting of mobile positive and negative charges is derived both field theoretically in the Gaussian approximation and using a simple ion-pairing approximation. This model might also apply to strongly charged membranes with strongly condensed counter ions. The resulting contribution to the elastic energy is negative and thus favors undulations of the membrane. The bending modulus is extracted from the large-scale or small-momentum behavior of the elastic kernel and found to be comparable to $k_{B}T$ for the case where ion pairing is dominant. The large-momentum elastic response exhibits a markedly different scaling than the small-momentum regime and sensitively depends on the small-distance cutoff and thus on molecular details.