Static and dynamic correlations in a charged membrane.
We study static and dynamic correlations of two fluctuations, the charge density fluctuation and height fluctuation (undulation), on a fluid membrane with a finite excess charge in a viscous fluid. For a planar and symmetrical membrane, we consider a model Hamiltonian inclusive of the fluctuations at the Gaussian level, and construct their equations of motion. Within the model, there exists no coupling, either static or dynamic, between the two fluctuations. The correlation function of the charge density has a short-range damped oscillation over the size of lipid heads due to Coulomb attraction between unlike-charged lipids. Its dynamic correlation function is shown to decay much faster in time than that in simple diffusion. The correlation function of height undulation, on the other hand, has a long-range damped oscillation (bud) over the membrane size, due to Coulomb repulsion among the excess charges. As the excess charge density increases to a critical value, a bending instability sets in, where a minute perturbation on the membrane can cause a large bud to form. Due to the excess charge, the dynamic correlation of the undulation decays slowly in time; at the critical density of the instability, the decay becomes infinitely slow.