Surfactants system Gaussian 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. 

Furthermore, Oda et al. pointed out that there are two topologically distinct types of chiral bilayers, as shown in Figure 5.46.165 Helical ribbons (helix A) have cylindrical curvature with an inner face and an outer face and are the precursors of tubules. These are, for example, the same structures that are observed in the diacetylenic lipid systems discussed in Section 4.1. By contrast, twisted ribbons (helix B) have Gaussian saddlelike curvature, with two equally curved faces and a C2 symmetry axis. They are similar to the aldonamide and peptide ribbons discussed in Sections 2 and 3, respectively. The twisted ribbons in the tartrate-gemini surfactant system were found to be stable in water for alkyl chains with 14-16 carbons. Only micelles form... 

As in binary surfactant-water systems considered previously, two constraints on the geometry of the surfactant interface are active a local constraint, which is due to the surfactant molecular architecture, and a global constraint, set by the composition. These constraints alone are sufficient to determine the microstructure of the microemulsion. They imply that the expected microstructure must vary continuously as a function of the composition of tile microemulsion. Calculations show - and small-angle X-ray and neutron scattering studies confirm - that the DDAB/water/alkane microemulsions consist of a complex network of water tubes within the hydrocarbon matrix. As water is added to the mixture, the Gaussian curvature - and topology -decreases [41]. Thus the connectivity of the water networks drops (Fig. 4.20). 

In the case of binary systems, the values of V and t are linked to micellar size. In water-oil-surfactant ternary solutions, more degrees of freedom for the shape come into play. Thus, two more quantities are usually considered, i.e. the average and Gaussian curvatures of the interface, averaged over the whole sample, (//) (A ) and K) (A ), respectively. The condition... 

Abstract We introduce a new technique using small-angle neutron scattering (SANS) to measure the average Gaussian curvature and the average square-mean curvature of the oil-water interface in a three-component, nearly isometric (equal volume fractions of water and oil) ionic microemulsion system. The microemulsion is composed of AOT/brine/decane. SANS measurements are made as a function of both the volume fraction of surfactant and salinity at a constant temperature,... 

YatciUa et al. investigated the conversion of a mixtime of cationic and anionic surfactants to form vesicles. In this case, the reaction was veiy slow and a kinetic phase was associated with the evolution and growth of vesicles over a period of weeks (see Figure 6.12) to a final vesicular system, which, as already mentioned, was thought by the authors to be thermodynamically stable. A subsequent study explored the system CTAB mixed with sodium perfiuorooctanoate. Cylinders, disks, and spherical uni-lamellar vesicles were found to coexist at equilibrium by cryo-TEM. This observation confirms the importance of structural confirmation by cryo-TEM when this technique can be applied. Erom their analysis of the data, the mean curvature modulus, the Gaussian curvature modulus, and the spontaneous curvature could all be evaluated. 

In this chapter, we have surveyed a wide range of chiral molecules that self-assemble into helical structures. The molecules include aldonamides, cere-brosides, amino acid amphiphiles, peptides, phospholipids, gemini surfactants, and biological and synthetic biles. In all of these systems, researchers observe helical ribbons and tubules, often with helical markings. In certain cases, researchers also observe twisted ribbons, which are variations on helical ribbons with Gaussian rather than cylindrical curvature. These structures have a large-scale helicity which manifests the chirality of the constituent molecules. 

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