A review. This chapter provides a semiformal, though largely qual., look at the mathematics of triply periodic minimal surfaces (TPMS) in relation to bicontinuous and polycontinuous liq. crystals. It is motivated largely by research of Stephen Hyde and others at the Applied Mathematics Department, Australian National University, Canberra, Australia and by the discovery of these TPMS liq. crystal partitions by Kare Larsson and the late Krister Fontell at Lund University. The article is meant to complement the existing, more rigorous papers and provide an introduction to the more fundamental topics of differential geometry and topol. of TPMS. Bicontinuous liq. crystals contain 2 mutually interpenetrating labyrinths sepd. by a hyperbolic partition that is often best described as a triply periodic minimal surface (TPMS). Properties of these surfaces can be broken down into local and global types. Local properties pertain to small, isolated surface patches and include intrinsic measures such as the metric, surface areas, and angles and the Gaussian curvature. The global properties require a look at the surface as a whole, embedded in 3D space. The global measures include parameters such as the Cartesian coordinates, topol., curvature distributions, and overall symmetries. These local and global properties are connected through the Gauss-Bonnet theorem that relates the topol. to the Gaussian curvature. For the simplest TPMS, such as the P, D, and G surfaces, the local Gaussian curvature detd. from small patches of the surface can give information regarding the stability of bicontinuous liq. crystals. For some simpler TPMS, the small surface patches are polygonal (e.g., triangles), according to the various symmetries of the surface. These small 2D surface patches can be built up into the entire 3D surface using various 3D construction algorithms based on symmetries, the same way 3D crystals can be built up from unique atoms. Alternatively these patches can be usefully represented in 2 dimensions by various mappings to the sphere, hyperbolic plane, and complex plane; then built up in those spaces using analogous symmetry operations; and then embedded in 3D space by folding and "gluing" the edges of the 2D representations. The 2D forms of these surfaces offer simple ways of constructing known and novel surfaces, algorithmically, and also explicitly through the Weierstrass parametrization, which gives the exact and explicit (x,y,z) coordinates of the actual minimal surface (not approximates) via the complex plane representation. Armed with this explicit information, modeling the stabilities, phase changes, and other phys. processes involving these minimal surfaces is rigorously quantifiable from a geometric perspective. A novel, topol.-preserving phase transformation between the P, D, and G cubic phases arises directly from the anal. of rhombohedral and tetrahedral distortions of the P, D, and G surfaces using this method. Networks decorating TPMS are of interest for generating and analyzing crystal networks. The simplest networks overlay the polygonally decorated surfaces built up of the fundamental patches. Families of networks can be generated for individual TPMS using supergroup-subgroup symmetry relationships between the decorating tesselation and the underlying symmetry elements of the TPMS itself. Of direct relevance to liq. cryst. mesophases is the set of polycontinuous networks that can be generated on various TPMS. These are generated by commensurately decorating 2D surface representations of TPMS with tree networks. With some well-chosen constraints, folding and gluing operations can result in surfaces decorated by entangled thickets of unconnected networks. In turn, these embedded networks can be used to form tri-, quadra-, octa- and other polycontinuous morphologies, with their resp. mutually interpenetrating labyrinths sepd. by a single triply periodic minimal surface.
2005. Vol. 127, no Bicontinuous Liquid Crystals, 99-127 p.
CAPLUS AN 2005:562473(Journal; General Review). NQC