  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 due to other sources, impacts 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, due 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.
