Gaussian curvature

The packing parameter of the neighboring surfactant molecules reflects the molecular dimension and is related to the macroscopic curvatures (Gaussian and mean curvature) of the surface imposed by the topology of the coverage relation (127). 

Polyp detection in virtual colonography polyps are characterized by a mushroom-like extrusion of the colon wall, and can be detected by their shape they exhibit higher local 3D curvature" ( Gaussian curvature ) properties. These can be detected with methods from differential geometry (the theory of shapes and how to measure and characterize them), and highlighted as, e.g. colored areas as attention foci for further inspection. 

The connection between molecular shape and interfacial curvature evident from Figure 16.4 is expressed by the following equation that relates the shape parameter to the membrane curvatures (Gaussian curvature, /l, and mean curvature, H) and monolayer thickness, / ... 

FIG. 1 The figure illustrates a piece of surface with non-positive Gaussian curvature. / and / 2 are the principal radii. The Gaussian K) and the mean (H) curvatures are expressed in terms of the principal radii as follows H — j (2/ i) -h 1/(2/ 2), K = l/(/ i/ 2). If R = —Ri at every point, the surface is called minimal. This implies that K is non-positive at every point. 

The surface dividing the components of the mixture formed by a layer of surfactant characterizes the structure of the mixture on a mesoscopic length scale. This interface is described by its global properties such as the surface area, the Euler characteristic or genus, distribution of normal vectors, or in more detail by its local properties such as the mean and Gaussian curvatures. 

The Gaussian (ii (r)) and mean curvatures (//(r)), see Fig. 1, present another characteristic of internal surfaces. By definition we have... 

The Euler characteristic for a closed surface is related to the Gaussian K curvature and genus g of this surface in the following way [33,29]... 

In the standard approaches to the systems in which monolayers or bilayers are formed, one assumes that the width of the film is much smaller than the length characterizing the structure (oil or water domain size, for example). In such a case it is justified to represent the film by a mathematical surface and the structure can be described by the local invariants of the surface, i.e., the mean H and the Gaussian K curvatures and by the global (topological)... 

For a particular field s(r) we can calculate at every point r the mean and the Gaussian curvatures of a corresponding surface passing through r, the normal vector of which is n = s/s. Using the standard geometrical definitions based on h and its derivatives we obtain... 

The line = 0 can be considered as a borderline for applicability of the basic model, in which the Gaussian curvature is always negative. Recall that in the basic model the oil-water interface is saturated by the surfactant molecules by construction of the model. Hence, for equal oil and water volume fractions the Gaussian curvature must be negative, by the definition of the model. 

The surface-averaged Gaussian curvature, K y, introduces the length scale, describing an average radius of curvature of the single,... 

In the microemulsion the role of A is played by the period of damped oscillations of the correlation functions (Eq. (7)). The surface-averaged Gaussian curvature Ky, = 2t x/ S is the topological invariant per unit surface area. Therefore the comparison between Ryyi = Kyy / in the disordered microemulsion and in the ordered periodic phases is justified. We calculate here R= Since K differs for diffused films from cor-... 

FiG. 16 Average Gaussian curvature for 7 = 50 at fixed temperature r = 2.7. is a surfactant volume fraction and a length unit is the size of a surfactant molecule (a bare thickness of the monolayer). 

W. Gozdz, R. Holyst. High genus gyroid surfaces of non-positive gaussian curvature. Phys Rev Lett 76 2726-2729, 1996. 

T. Hofsass, H. Kleinert. Gaussian curvature in an Ising model of microemulsions. J Chem Phys 56 3565-3570, 1987. 

A. Ciach. Four-point correlation functions and average Gaussian curvature in microemuisions. Phys Rev E 55 1954-1964, 1997. 

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. 

The mean, Gaussian, and principal curvatures are well-defined quantities from the standpoint of the differential geometry. If the surface is given by the... 

Here e is a unit vector normal to the surface at the POI defined by e = pH xpv/ pH xpv, and subscripts of p denote the partial derivatives. Thus the mean, H, and Gaussian, K, curvatures are expressed as... 

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