Tube formation and spontaneous budding in a fluid charged membrane

We determine the equations governing the equilibrium shape of an axially symmetric charged fluid membrane under the action of an external field. We consider that the charges are free to diffuse along the membrane, and we neglect the effect of screening counterions. By numerically integrating these equations for a membrane spread across a hole by a constant tension, we show that there exists a threshold electric field above which an infinitely long tube can be pulled. The threshold field for pulling a tube decreases as the surface charge density of the membrane increases, reaching zero at a finite critical value. Above this critical density of charge, the membrane spontaneously buds, in the absence of externally applied fields, because of the electrostatic repulsion of its charges.