Mean curvature

Another phase which has attracted recent interest is the gyroid phase, a bicontinuous ordered phase with cubic symmetry (space group Ia3d, cf. Fig. 2 (d) [10]). It consists of two interwoven but unconnected bicontinuous networks. The amphiphile sheets have a mean curvature which is close to constant and intermediate between that of the usually neighboring lamellar and hexagonal phases. The gyroid phase was first identified in lipid/ water mixtures [11], and has been found in many related systems since then, among other, in copolymer blends [12]. 

A = 2k/ /3 for the case of cyhnders. In order to avoid this problem, Gompper and Kroll [241] have recently argued that a more appropriate discretization of the bending free energy should be based directly on the square of the local mean curvature ... 

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

One can discover a special property of the functional (1) by analyzing the formula for the mean curvature (25) expressed in terms of the three dimensional field 4> y). From the form of Eq. (1) one can realize that for some local minima of (1) the average curvature given by... 

Equation (2) is identified as a second-order, nonlinear differential equation once, the curvature is expressed in terms of a shape function of the melt/crystal interface. The mean curvature for the Monge representation y = h(x,t) is... 

Since the right hand side contains the sum of the second derivatives, we will make use of the Laplacian inside the earth and express the second derivative of the potential along the vertical through the mean curvature, density, and angular velocity. Then, 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. 

While the order parameters derived from the self-diffusion data provide quantitative estimates of the distribution of water among the competing chemical equilibria for the various pseudophase microstructures, the onset of electrical percolation, the onset of water self-diffusion increase, and the onset of surfactant self-diffusion increase provide experimental markers of the continuous transitions discussed here. The formation of irregular bicontinuous microstructures of low mean curvature occurs after the onset of conductivity increase and coincides with the onset of increase in surfactant self-diffusion. This onset of surfactant diffusion increase is not observed in the acrylamide-driven percolation. This combination of conductivity and self-diffusion yields the possibility of mapping pseudophase transitions within isotropic microemulsions domains. 

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

Figure 3a. Typical plot from the RS/1 program showing the linear relationship between Mean Curvature and the tested valnes for 40% Compression Deflection of the polynrethane foam samples.

An example of tlie R8/1 table of data from a correlation analysis. Column headings are MC (mean curvature), ES (elongation ratio), MFD (mean free distance between features), MIL (mean intercept length of the features). 

The best-known and simplest class of block copolymers are linear diblock copolymers (AB). Being composed of two immiscible blocks, A and B, they can adopt the following equilibrium microphase morphologies, basically as a function of composition spheres (S), cylinders (C or Hex), double gyroid (G or Gyr), lamellae (L or Lam), cf. Fig. 1 and the inverse structures. With the exception of the double gyroid, all morphologies are ideally characterized by a constant mean curvature of the interface between the different microdomains. 

Fig. 60 Schematic illustration for formation of cylindrical morphology in a blend of slightly asymmetric lower molecular weight PS-b-PI (/3-chain) with large symmetric PS-fc-PI (a-chain). a Molecule of /S-chain with non-zero spontaneous curvature, b Cylindrical morphology formed by neat /3 chains shown in a. Here mean curvature of cylinder (solid line) is larger than spontaneous curvature of /3-chain (dashed lines). c Cylindrical morphology formed by binary blend of /3-chains shown in a and large symmetric copolymers (a-chain). In this case, mean curvature of cylinder closely fits to spontaneous curvature of /3-chain. From [180]. Copyright 2001 American Chemical Society...

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 kinetics of the nonconserved order parameter is determined by local curvature of the phase interface. Lifshitz [137] and Allen and Cahn [138] showed that in the late kinetics, when the order parameter saturates inside the domains, the coarsening is driven by local displacements of the domain walls, which move with the velocity v proportional to the local mean curvature H of the interface. According to the Lifshitz-Cahn-Allen (LCA) theory, typical time t needed to close the domain of size L(t) is t L(t)/v = L(t)/H(t), where H(t) is the characteristic curvature of the system. Thus, under the assumption that H(t) 1 /L(t), the LCA theory predicts the growth law L(t) r1 /2. The late scaling with the growth exponent n = 0.5 has been confirmed for the nonconserved systems in many 2D simulations [139-141]. 

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

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