Phase transitions and shapes of two component membranes and vesicles I: strong segregation limit

We investigate unilamellar membranes and vesicles composed of an A/B mixture of partially miscible amphiphiles. Assuming a simple bilinear coupling between relative composition and local curvature, and in the strong segregation limit of the A/B mixture, we show for unilamellar open-shape membranes that the competition between surface tension and curvature results in a phase with a selected periodicity (modulated phase) both in the shape and in the A/B composition. The limits of large and small surface tension are discussed separately. These findings extend previous results obtained close to the A/B critical point (shallow quench). For the same limit of strong segregation, we also investigate the coupling between the separation of the system into A and B domains, and the overall shape of closed-shape vesicles. For cylindrical vesicles of fixed overall area (or equivalently vesicles embedded in a two-dimensional space), equilibrium shapes and phase diagrams are obtained. We also consider the effect of an added pressure difference (osmotic pressure) across the vesicle. The results are extended to axial symmetric vesicles embedded in a three-dimensional space