Minimal Surfaces With Zero Curvature

Henri Poincare made some key contributions in this area.  

There is a difference between the present situation and the previous experiment where a drop was deposited on a fiber. Here the liquid in the meniscus is in equilibrium with the liquid bath. As a result, we have Ap = 0, which defines a surface with zero curvature. The profile is given by equation (1.15), which reduces to 

This last equation could be obtained directly by noting that the vertical projection of the tension forces must be conserved. At a height x, the tension force is 27T27cos = 27rjb. Since tan0 = i, equation (1.17) is readily 

FIGURE 1.11. Water climbing up a glass fiber of radius b 10 p.m. 

This equation has two solutions for Rm-, corresponding to two surfaces, both with zero curvature. The surface area is minimum for the first and maximum for the second- 

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With more complex frames, one can generate a variety of minimal surfaces with zero curvature. Figure 1.12 shows a cubic structure, a hanging chain, and a spiral. 

However, even without structural studies, Friberg et al. [32], Shinoda [33], and others noted that the broad existence range with respect to the water/oil ratio could not be consistent with a micellar-only picture. Also, the rich polymorphism in general in surfactant systems made such a simplified picture unreasonable. It was natural to try to visualize microemulsions as disordered versions of the ordered liquid crystalline phases occurring under similar conditions, and the rods of hexagonal phases, the layered structure of lamellar phases, and the minimal surface structure of bicontinuous cubic phases formed a starting point. We now know that the minimal surfaces of zero or low mean curvature, as introduced in the field by Scriven [34], offer an excellent description of balanced microemulsions, i.e., microemulsions containing similar volumes of oil and water. 

Bicontinuous cubic phases can be described by periodic minimal surfaces that are well-known in differential geometry [165]. In the case of inverted bicontinuous cubic phases, the periodic minimal surfaces lie along the middle of the bilayer. They are saddle surfaces with mean curvature zero everywhere (i.e., positive and negative curvatures of the bilayers forming the rods balance each other at every point [162] and with negative Gaussian curvature [138,157,159]. 

A special case is the much discussed infinite periodical minimal surface with a mean curvature H=Ho everywhere equal to zero, which separates equal volumes of water and oil. Evidently, interfacial structures of this principal nature can readily account for the water and oil bicontinuity observed for... 

Different types of surface can be categorized in terms of their mean and gaussian curvatures. For example, a sphere of radius R has a mean curvature of / and a gaussian curvature of R, while a cylinder has one principal radius of curvature equal to infinity, so the gaussian curvature is zero. Surfaces with zero mean curvature such that ate known as minimal surfaces and have been proposed as structures for some cubic phases [19]. 

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.

So K is like a chemical potential for topology. Notice that if K is positive, a fluid film system described by Eq.l has a ground state instability leading to the formation of a periodic minimal surface which has many handles, only one component, and zero mean curvature (i.e., with l/i i -f l/i 2 = 0 everywhere). The energy of such a state with unit cell size I diverges as —K/fi, This will eventually be cut off by anharmonic terms, but nonetheless the general expectation is for phase separation to a rather concentrated cubic phase whenever > 0. Similarly if K is too negative (K < —2Kq) there is an instability to the formation of very small spherical vesicles. 

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