From the plateau problem to periodic minimal surfaces in lipids,surfactants and diblock copolymers

A novel method is presented for generating periodic surfaces. Such periodic surfaces appear in all systems which are characterized by internal interfaces and which additionally exhibit ordering. One example are systems of AB diblock copolymers, where the internal interfaces are formed by the chemical bonds between the A and B blocks. In these systems at least two bicontinuous phases are formed: the ordered bicontinuous double diamond phase and the gyroid phase. In these phases the ordered domains of A monomers and B monomers are separated by a periodic interface of the same symmetry as the phases themselves. Here we present a novel method for the generation of such periodic surfaces based on the simple Landau-Ginzburg model of microemulsions. We test the method on four known minimal periodic surfaces, find two new surfaces of cubic symmetry, and show how to obtain periodic surfaces of high genus and n-tuply continuous phases (n > 2). So far only bicontinuous (n = 2) phases have been known. We point out that the Landau model used here should be generic for all systems characterized by internal interfaces, including the diblock copolymer systems.