The Gaussian (ii (r)) and mean curvatures (//(r)), see Fig. 1, present another characteristic of internal surfaces. By definition we have [Pg.698]

VOLUME DENSITY SURFACE DENSITY SPECIFIC SURFACE MEAN INTERCEPT LENGTH MEAN FREE DISTANCE MEAN DIAMETER MEAN CURVATURE ELONGATION RATIO DEGREE OF ORIENTATION [Pg.162]

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). [Pg.166]

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 [Pg.303]

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. [Pg.80]

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 [Pg.80]

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]. [Pg.176]

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 [Pg.670]

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. [Pg.262]

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