  The behavior of a conductive membrane in a static (DC) electric field is
investigated theoretically. An effective zero-thickness model is constructed
based on a Robin-type boundary condition for the electric potential at the
membrane, originally developed for electrochemical systems. Within such a
framework, corrections to the elastic moduli of the membrane are obtained,
which arise from charge accumulation in the Debye layers due to capacitive
effects and electric currents through the membrane and can lead to an
undulation instability of the membrane. The fluid flow surrounding the membrane
is also calculated, which clarifies issues regarding these flows sharing many
similarities with flows produced by induced charge electro-osmosis (ICEO).
Non-equilibrium steady states of the membrane and of the fluid can be
effectively described by this method. It is both simpler, due to the zero
thickness approximation which is widely used in the literature on fluid
membranes, and more general than previous approaches. The predictions of this
model are compared to recent experiments on supported membranes in an electric
field.
