The functionalized Cahn-Hilliard (FCH) free energy models interfacial energy in amphiphilic phase separated mixtures. Its minimizers encompass a rich class of morphologies with detailed inner structure, including bilayers, pore networks, pearled pores and micelles. We address the existence and linear stability of α-single curvature bilayer structures in d ≥ 2 spacedimensions for a family of gradient flows associated to the strong functionalization scaling. The existence problem requires the construction of homoclinic solutions in a perturbation of a 4th-order integrable Hamiltonian system while a negative index argument reduces the linear stability analysis to the characterization of the meander and pearling modes of the second variation of the FCH energy on a family of invariant subspaces, independent of the choice of mass-preserving gradient flow.
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, involving nonlocal as well as negative terms. Since the minimizers of the limiting energy have minimal interfaces, the physical interpretation is that, within a sufficiently strong interspecies surface tension and a large enough sample size, raft microdomains are not formed.
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