Domain Formation in Membranes Near the Onset of Instability

The formation of microdomains, also called rafts, in biomembranes can be attributed to the surface tension of the membrane. In order to model this phenomenon, a model involving a coupling between the local composition and the local curvature was proposed by Seul and Andelman in 1995. In addition to the familiar Cahn-Hilliard/Modica-Mortola energy, there are additional ‘forces’ that prevent large domains of homogeneous concentration. This is taken into account by the bending energy of the membrane, which is coupled to the value of the order parameter, and reflects the notion that surface tension associated with a slightly curved membrane influences the localization of phases as the geometry of the lipids has an effect on the preferred placement on the membrane. The main result of the paper is the study of the Γ-convergence of this family of energy functionals depending on the size of the sample, involving nonlocal as well as negative terms. Since the limiting energy is minimized by a phase function with minimal interfaces, the physical interpretation is that, within a certain parameter range, raft microdomains are not formed.