Flexoelectricity and the entropic force between fluctuating fluid membranes

Biological membranes undergo noticeable thermal fluctuations at physiological temperatures. When two membranes approach each other, they hinder the out-of-plane fluctuations of the other. This hindrance leads to an entropic repulsive force between membranes which, in an interplay with attractive and repulsive forces owing to other sources, affects a range of biological functions: cell adhesion, membrane fusion, self-assembly, binding-unbinding transition among others. In this work, we take cognizance of the fact that biological membranes are not purely mechanical entities and, owing to the phenomenon of flexoelectricity, exhibit a coupling between deformation and electric polarization. The ensuing coupled mechanics-electrostatics-statistical mechanics problem is analytically intractable. We use a variational perturbation method to analyze, in closed form, the contribution of flexoelectricity to the entropic force between two fluctuating membranes and discuss its possible physical implications. We find that flexoelectricity leads to a correction that switches from an enhanced attraction at close membrane separations and an enhanced repulsion when the membranes are further apart.