Surfaces constant mean curvature

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%.

The diffusion behavior implies a rapid, but continuous, change in structure from discrete oil droplets to bicontinuous to discrete water droplets with increasing temperature. The bicontinuous structure appears to be well described by the constant mean curvature surface structures of low mean curvature rather than a tubular structure. The same applies to the bilayer phases, often denoted L3, L4, or sponge phases, also included in Fig. 6. 

Calculations [33] of self-diffusion in ordered bicontinuous structures have, as illustrated in Fig. 21, reproduced the main features of the experimental studies for a large number of microemulsion systems. (For others, a quantitative comparison is difficult because of large influences of effects other than obstruction, such as a high surfactant film concentration or incomplete segregation between domains.) Furthermore, the symmetry of the self-diffusion pattern around the crossover of the oil and water curves implies a symmetry also in structural changes. This symmetry is easy to understand in terms of structures that have a constant mean curvature surface, where changes in spontaneous curvature away from zero in the two directions should be equivalent except for the two solvents changing place. 

Figure 21 The variation of the reduced self-diffusion coefficients, D/Do, of water and oil in model bicontinuous structures of constant mean curvature surfaces according to the results of Ref. 33. The results for two families of constant mean curvature surfaces, the D family (solid lines) and the P family (dashed lines), are given. There is a stronger variation of D/Dq with 4>o for the P family, which has a coordination number of 6, compared to the D family which has a coordination number of 4.

Here, /I % 2 is a numerical coefficient (7) which shows only a weak dependence on the particular family of constant mean curvature surfaces, and hence on the coordination number of the structure. 

Anderson and Wennerstrdm (15) have calculated the diffusion for some different constant mean curvature surface structures, serving as model structures for cubic phases, sponge phases and bicontinuous microemulsions. The D/Dq ratio depends on but there is also a significant dependence on the topology, or the coordination number of the structure, i.e. on the particular family of constant mean curvature surfaces. Near 0Q = 0.5, these authors found D/Dq to vary linearly with (po, and the relationship can be written as follows ... 

As in most ordered two-phase systems forming macrolattices, the interface between the two domains should assume a smooth constant mean-curvature surface, resulting in a cate-noid shape of the perforations. The SAXS of these systems will, of course, show additional peaks depending on the spacing and stacking of the perforations. ... 

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. 

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.

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). 

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