Theory of modulated phases in lipid bilayers and liquid crystal films

We present a general theory for equilibrium-modulated phases of chiral and achiral bilayer membranes and liquid crystal films. Both bulk smectics as well as freely suspended films are considered. For flexoelectric systems, continuous structural phase transitions are predicted among square-lattice, hexagonal, and distorted two-dimensional modulated phases as a function of the applied electric field. It is also shown that only uniform flat phases are predicted for thin films. One-dimensional ripple phases and two-dimensional square-lattice phases can occur with increasing film thickness.