Zero mean curvature

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). 

By decoration of these various infinite two-dimensional manifolds (just as the sphere has been decorated with closed networks) several related structures have been proposed for graphite nets. These are mostly based on the P, D and G surfaces (the first two due to Schwarz (1890) and the last, the gyroid, discovered by Schoen (1970). However, many other surfaces (perhaps 50) are available for consideration. Some fit naturally with hexagonal sheets and others with sheets of square or lower symmetry. In general, the P, D and G surfaces are the least curved from planarity. Surfaces parallel to the surfaces of zero mean curvature have lower symmetry than those with H = 0. When decorated with graphite nets the symmetry may be further lowered to that of a sub-group of the symmetry group of the surface itself. 

A. L. Mackay. Yes, if you use the same materials to build C60 it is clearly strained. But I should say that the definition of a minimal surface is that it has zero mean curvature i.e. you can say that either the divergence of the normal is zero, or that it has zero splay. Thus if you take the p-orbitals in one direction they are spread out, whereas in another they are compressed or folded in. So the whole evens out. Minimal surfaces have, characteristically, zero splay energy. This is an argument in its favour, whereas spheres are splayed in both directions. 

The curvature of the oil-water interface in microemulsions varies from highly curved towards oil (o/w) or water (w/o) to zero mean curvature in bicontinuous... 

In this chapter, we will focus on the formulation of systems in which the surfactant has equal affinity for both O and W phases. These formulations do form bicontinuous microemulsions of zero mean curvature and have important properties such as minimum interfacial tension and maximum solubilisation. Such condition has been called optimum formulation in the 1970s, because it matches the attainment of an ultralow interfacial tension that guarantees an enhanced oil recovery from petroleum reservoirs, which was the driving force behind the research effort on microemulsions (see Chapter 10, Section 10.3 of this book) [3,4]. High solubilisation performance micro emulsions which are able to cosolubilise approximately equal amount of oil and water with less than 15-20% surfactant are attainable only at an optimum formulation. 

Fig. 19 Left) Schematic representation of the proposed mechanism for topological changes in dioleoyl phosphoethanolamine (DOPE) based liposomal membranes upon ultrasound irradiation. Right) (a) Giant DOPE-based unilamellar vesicle, before sonication, which shows an inhomogeneous membrane DOPE-rich domains of negative curvature are marked in red, embedded in zones rich in dioleoyl phosphocholine (DOPC) of zero mean curvature, (b) Illustration of shape changes upon ultrasound stimuli. Reproduced with permission from [99]. Copyright 2014 The Royal Society of Chemistry...

In principle, we can distinguish (for surfactant self-assemblies in general) between a microstructure in which either oil or water forms discrete domains (droplets, micelles) and one in which both form domains that extend over macroscopic distances (Fig. 7a). It appears that there are few techniques that can distinguish between the two principal cases uni- and bicontinuous. The first technique to prove bicontinuity was self-diffusion studies in which oil and water diffusion were monitored over macroscopic distances [35]. It appears that for most surfactant systems, microemulsions can be found where both oil and water diffusion are uninhibited and are only moderately reduced compared to the neat liquids. Quantitative agreement between experimental self-diffusion behavior and Scriven s suggestion of zero mean curvature surfactant monolayers has been demonstrated [36]. Independent experimental proof of bicontinuity has been obtained by cryo-electron microscopy, and neutron diffraction by contrast variation has demonstrated a low mean curvature surfactant film under balanced conditions. The bicontinuous microemulsion structure (Fig. 7b) has attracted considerable interest and has stimulated theoretical work strongly. 

Alternative structures, termed tubular or interconnected rod models, seem not to explain these features well and are considered less probable for the case of maximal bicontinuity or in the vicinity of maximal bicontinuity. On the other hand, a branched tubular or interconnected rod structure is consistent with observations slightly away from the crossover point, assigned to zero mean curvature. In this region, D/Do of both oil and water are high but differ by a significant factor. The transition between discrete droplet and bicontinuous structure is more easily understood with the interconnected rod model. 

A simultaneously high D/Do value (above 0.6) for both solvents provides evidence for the highest conceivable degree of bicontinuity. This observation indicates a structure of close to zero mean curvature but appears to be inconsistent with a tubular structure. 

Despite the reasonable tolerance of nonionic surfactants, particularly in topical applications, microemulsions prepared from (phospho)lipids seem to be preferred over those prepared by synthetic surfactants from a toxicity point of view. As discussed by Shinoda et al. [13], lecithin in water-oil systems does not spontaneously form the zero mean curvature amphiphile layers required for the formation of balanced microemulsions but rather forms reverse structures. On decreasing the polarity of the aqueous phase by addition of a short-chain alcohol, e.g., propanol, lecithin was found to form microemulsions at low amphiphile concentrations over wide ranges of solvent composition. The structure of the microemulsions formed was investigated by NMR self-diffusion measurements, and it was found that with a decreasing propanol concentration there was a gradual transition from oil droplets in water, over a bicontinuous structure, to water droplets in oil [13]. 

Several theories of surfactant phase are available. Following Scriven (1976), this phase is assumed to be bicontinuous in oil and water, and the interface is assumed to have zero mean curvature, hence the pressure difference between oil and water is zero. Talmon and Prager (1978, 1982) divided up the medium into random polyhedra. The flat walls ensure no pressure difference between oil and water. They placed oil and water randomly into the polyhedra so that both oil and water were continuous when sufficient amounts of both phases were present. As in the earlier models of oil-in-water microemulsions, this randomness gave rise to an increased entropy which overcame the increased surface energy to yield a negative free energy of formation, reached only when the interfadal tension is ultralow. Such structures can form spontaneously. This random structure is characterized by a length scale. This led Jouffrey et al. (1982) to postulate that... 

We also suppose the pressures across the liquid-vapor interface are equal. From Equation (4.1) the Hquid-vapor interface has zero mean curvature. That is ... 

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