Interfacial architecture in surfactant-water mixtures: Beyond spheres, cylinders and planes

(4) Mesh. These surfaces are less well known. They are saddle-shaped everywhere, thus their (integral) Gaussian curvature is negative, and their geometry is hyperbolic. They are confined between two planes, and can be described as porous lamellae. They can adopt constant mean curvature (ref. 1) , although the Gaussian curvature varies over the surface. Mesh surfaces are not minimal surfaces, since their mean curvature cannot be identically zero. The surfaces generated by the equation above exhibit a square planar network of tunnels. The genus of a single "unit cell" of the surface (embedded in the three torus) is two the simplest topology for doubly-periodic surfaces. Presumably, any two-dimensional mesh network can be realised. The other simple two-dimensional net the hexagonal net describes an alternative tunnel morphology. Parallel layers of these square and hexagonal mesh surfaces (with the nodes of one layer positioned over the holes of neighbouring layers) result in three-dimensional arrangements of rhombohedral and body-centred tetragonal symmetry respectively (space groups R3m and I422 resp.) . Their tunnel geometries match those of the rod-structures proposed by Luzzati to describe the so-called T and R phases found in surfactant systems.