Constant mean curvature

The best-known and simplest class of block copolymers are linear diblock copolymers (AB). Being composed of two immiscible blocks, A and B, they can adopt the following equilibrium microphase morphologies, basically as a function of composition spheres (S), cylinders (C or Hex), double gyroid (G or Gyr), lamellae (L or Lam), cf. Fig. 1 and the inverse structures. With the exception of the double gyroid, all morphologies are ideally characterized by a constant mean curvature of the interface between the different microdomains. 

In a two-phase composite material of isolated spherical particles embedded in a matrix, there is a driving force to transport material from particles enclosed by isotropic surfaces of larger constant mean curvature to particles of smaller constant mean curvature. This coarsening process and the motion of internal interfaces due to curvature are treated in Chapter 15. 

Capillary forces induce morphological evolution of an interface toward uniform diffusion potential—which is also a condition for constant mean curvature for isotropic free surfaces (Chapter 14). If a microstructure has many internal interfaces, such as one with fine precipitates or a fine grain size, capillary forces drive mass between or across interfaces and cause coarsening (Chapter 15). Capillary-driven processes can occur simultaneously in systems containing both free surfaces and internal interfaces, such as a porous polycrystal. 

Fig. 12. Schematic representation of proposed arrangement of A/B junctions on a triply periodic IMDS of constant mean curvature for SB-4 miktoarm copolymer of Fig. 11 (reproduced with permission from [62])...

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.

Figure 4. Computer line drawings (without hidden line removal) of three representatives of the I-WP family of constant mean curvature surfaces. These surfaces are invoked to describe the polar/apolar dividing surface in the DDAB / water / hydrophobe cubic phases, at water volume fractions of a) 35% b) 47% c) 65%.

In Chapter 4 we shall see how curvature together with global packing constraints conspires to produce and predict the rich diversity of bicontinuous cubic phases and others of constant mean curvature which can be prescribed by variations of solution conditions or temperatures. 

Fig 7.1(a) The diamond (D-) membrane system of the FLB in etiolated leaves. Projection of a section cut approximately normal to the jlOO) plane. The lower mserts show the match between the experimental micrograph and the computer generated constant mean curvature PCS projections for two different distances along the [100] direction. The upper inserts show the Fourier transform (calculated for the regions indicated) for the corresponding experimental and theoretical projections (a and b, and a and b, respectively). 

Fig. 7.1(e) Computer generated projections of the D-PCS for various constant mean curvatures (0-1) for the indicated lattice directions. Shown is also the corresponding 3-D unit cell (right). Notice the change in sub-volume relation between die two spaces. 

The basic equation defining a capillary surface when gravity can be neglected is quite simple - the liquid surface has constant mean curvature However, the application of this equation... 

Many of the remaining twelve examples discussed in Schoen s note must be considered conjectures. In many cases, physical models were built from plastic or with soap films. Although analytic representations for these surfaces have not yet been found, Schoen s contribution to a subject which had seen little progress in over 75 years was substantial. In this regard, it should be recalled that even in the case of rotationally symmetric surfaces, the constant-mean-curvature solutions (Delauney 1841) do not admit closed-form analytic representations (see also Kenmotsu 1980). 

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). 

Figure 3h. A contact angle of 57.3° and a constant mean curvature of H = 1.6 have been prescribed. 

All the structures shown in Fig. 40 have in fact been seen in experimental work applying transmission electron microscopic (TEM) techniques as well as small angle scattering of X rays or neutrons. While most structures have been known for a long time and readily follow from theoretical predictions [41, 42], the OBDD structure has only been found recently (see [303] for a brief review). This structure is interesting since it belongs to a class of geometrical structures with constant mean curvature of the interface [303]. 

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