Surface mean curvature

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

This equation was derived by Bruns, and it establishes a relation between the derivative of the field along the vertical and the mean curvature of the level surface. 

VOLUME DENSITY SURFACE DENSITY SPECIFIC SURFACE MEAN INTERCEPT LENGTH MEAN FREE DISTANCE MEAN DIAMETER MEAN CURVATURE ELONGATION RATIO DEGREE OF ORIENTATION... 

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 integral from the mean curvature H over the surface region S, can be... 

Figure 37. The maximum of the mean curvature distribution scaled with the interface density increases very rapidly (up 2.5 times) within a short time interval, x , after the noise term has been switched off in the simulation. The Euler characteristic and the average domain size, / o, remain constant, and the surfaces area decreases by 3%. This illustrates that the curvature distributions are very sensitive to the thermal undulations of the interface. The times are x = 0.0, 0.032, 0.085, 0.225, 0.896, 2.05 from bottom to top at /// ] 0.

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. 

The expression for weighted mean curvature for any surface in local equilibrium is simplified when the Wulff shape is completely faceted [10, 12], In this case,... 

The Gauss-Bonnet theorem, which relates integrals of Gaussian curvature (1/(/ii) in three dimensions) over a surface to integrals of mean curvature (1/iii + I/R2 in three dimensions) over boundaries of the surface, is particularly simple in two dimensions. In two dimensions, the (N — 6)-rule is equivalent to the Gauss-Bonnet theorem. 

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. 

The convention that a convex interface of a solid body has positive mean curvature and a concave interface has negative mean curvature is adopted throughout this book (see Section 14.1). A table of surface formulae is provided in Table C.l. 

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). 

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