Periodic surfaces mean curvature

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

However, H. A. Schwarz found before 1865 that patches of varying negative gaussian curvature and constant H = 0 could be smoothly joined to give an infinite triply periodic surface of zero mean curvature. About five different types were found by Schwarz and Neovius, but now about 50 more have been described (Schoen 1970 Fischer Koch, 1989 a e). 

Figure 1. A. Computer graphic portion of a periodic surface of constant mean curvature, having the same space group and topological type as the Schwarz D minimal surfhce. This surbce, together with an identical displaced copy, would represent the polar/apolar dividing surface in a cubic phase with space group 224 (Pn3m). The two graphs shown would thread the two aqueous subspaces. B. Computer graphic of a portion of the Schwarz D minimal sur ce (mean curvature identically zero). In the 224 cubic phase structure, this sur ce would bisect the surfactant bilayer.

Two-periodic surfaces deserve some comment. The most interesting examples of these surfaces can be visualised as confined between two parallel bounding planes, with a regular network of pores joining the two parallel sheets. We call these surfaces "mesh surfaces", due to their characteristic two-dimensional porous network, which resembles a mesh. A square mesh surface is shown in Fig. 1.12. The mean curvature of these surfaces can be... 

The simplest three-periodic hyperbolic surfaces are "Infinite Periodic Minimal Surfaces" (EPMS, named by Alan Schoen [6]). For these surfaces, the mean curvature is constant on the surface, and everywhere identically zero. This is a defining characteristic of minimal surfaces. For these structures, the sub-volumes can be geometrically identical. This occurs if the IPMS contains straight lines. Such surfaces have been called "balance surfaces" by Koch and Fischer [7]. We focus primarily on IPMS in this book. Some further discussion of general properties of minimal surfaces is in order here, since a number of their geometrical and topological properties will be required for later chapters. 

These surfaces of zero potential formed in different salts are very close to periodic minimal surfaces [9], whose mean curvature, defined as the arithmetic mean of the main curvatures, is everywhere zero (see Chapter 1) . On these minimal surfaces the Gaussian curvature is everywhere negative or... 

A rigorous mathematical existence proof for a periodic surface of small, nonzero constant mean curvature can be obtained with the methods of the theory of nonlinear elliptic differential equations. The resulting surface would be a perturbation of a known periodic minimal surface, but the intent of chapter is rather to exhibit numerical solutions that extend over wide ranges in mean curvature. 

In this section, we introduce the computational method in the form used for the surfaces exhibited in Section IV, i.e., where the prescribed mean curvature of the computed surface is everywhere constant and the boundary conditions are determined by two dual periodic graphs. We also give generalizations of the method for the computation for a surface of prescribed—not necessarily constant—mean curvature, with prescribed contact angle against surface. Generalization to the computation of space curves of prescribed curvature or geodesic curvature is available (Anderson 1986). 

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