Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions

We present a method for the generation of periodic embedded surfaces of nonpositive Gaussian curvature and multiply continuous phases. The structures are related to the local minima of the scalar order parameter Landau-Ginzburg Hamiltonian for microemulsions. In the bicontinuous structure the single surface separates the volume into two disjoint subvolumes. In some of our phases (multiply continuous) there is more than one periodic surface that disconnects the volume into three or more disjoint subvolumes. We show that some of these surfaces are triply periodic minimal surfaces. We have generated known minimal surfaces (e.g., Schwarz primitive P, diamond D, and Schoen-Luzatti gyroid G and many surfaces of high genus. We speculate that the structure of microemulsion can be related to the high-genus gyroid structures, since the high-genus surfaces were most easily generated in the phase diagram close to the microemulsion stability region. We study in detail the geometrical characteristics of these phases, such as genus per unit cell, surface area per unit volume, and volume fraction occupied by oil or water in such a structure. Our discovery calls for new experimental techniques, which could be used to discern between bicontinuous and multiply continuous structures. We observe that multiply continuous structures are most easily generated close to the water-oil coexistence region.