Fluctuating lipid bilayer in an arbitrary potential: Theory and experimental determination of bending rigidity

We consider the position fluctuations of a membrane close to a substrate in an external potential and at finite temperature. We derive self-consistent equations for the average membrane position and its root mean square fluctuation amplitude in the range of moderate fluctuations much below the unbinding transition. These self-consistent equations can be solved for arbitrary potentials. We first check them for special known cases, then we apply our approach to lipid bilayers where van der Waals attractions are balanced by hydration repulsion associated to a hard wall. Finally, we compare these results with recent experiments on lipid bilayer.(1).