Undulation Instability of Lipid Membranes under an Electric Field

The influence of an external electric field on a poorly conductive membrane such as a lipid bilayer is studied theoretically. The unbalanced electric stress created by an ionic current across a nonperfectly flat membrane gives rise to a destabilizing surface energy enhancing undulations. The deformation of a membrane attached to a frame is derived and the electrohydrodynamic instability of a free floating membrane is studied. We find a most unstable mode of undulation, of wavelength in the $\mu \mathrm{m}$ range, connected to the crossover between membrane and solvent dominated dissipations.