Curvature local

For model A, the interfaces decouple from the bulk dynamics and their motion is driven entirely by the local curvature, and the surface tension plays only a background, but still an important, role. From this model A... 

The second energy derivatives with respect to the x, y, and z directions of centers a and b (for example, the x, y component for centers a and b is Hax,by = (3 E/dxa3yb)o) form the Hessian matrix H. The elements of H give the local curvatures of the energy surface along the 3N cartesian directions. 

Since the value of depends on the location ofX relative to the surface, the value of r —and therefore the local curvature of the meniscus ( = ljr )—will be similarly dependent. 

The competition between the polar and steric dipoles of molecules may also lead to internal frustration. In this case, the local energetically ideal configuration cannot be extended to the whole space, but tends to be accomodated by the appearance of a periodic array of defects. For example, the presence of the strong steric dipole at the head of a molecule forming bilayers will induce local curvature. As the size of the curved areas increases, an increase in the corresponding elastic energy makes energetically preferable the... 

The classification of critical points in one dimension is based on the curvature or second derivative of the function evaluated at the critical point. The concept of local curvature can be extended to more than one dimension by considering partial second derivatives. d2f/dqidqj, where qt and qj are x or y in two dimensions, or x, y, or z in three dimensions. These partial curvatures are dependent on the choice of the local axis system. There is a mathematical procedure called matrix diagonalization that enables us to extract local intrinsic curvatures independent of the axis system (Popelier 1999). These local intrinsic curvatures are called eigenvalues. In three dimensions we have three eigenvalues, conventionally ranked as A < A2 < A3. Each eigenvalue corresponds to an eigenvector, which yields the direction in which the curvature is measured. 

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]. 

Figure 23. The 2D illustration of the digital pattern analysis for computing the Euler characteristic. The local curvature variables xe —1,0, +1 are assigned to the each lattice site at the boundaries of black pixels. The Euler characteristic is a sum of local curvature variables...

In this method the local curvatures are calculated by using the first and second fundamental forms of differential geometry [7]. The surface is parameterized near the point of interest (POI) as p(u, v) (see Fig. 32). The coordinates (u, v) are set arbitrary on the surface in such a way that POI is located at p(u, v) = (0,0). The first and the second forms of the differential geometry are expressed as... 

Figure 32. Schematic diagram of a surface expressed in a parametric form p (u, v) and a sectioning plane, which is comprised of e and p (0, v). p (0,0) is a point of interest at which the local curvatures are determined [7].

The computation of the curvatures from the bulk field differential geometry has proven to be rather imprecise. The errors produced by the use of the approximate formulas (100)-(104) are especially big if the spatial derivatives of the field sharp peaks at the phase interface. This is a common situation in the late-stage kinetics of the phase separating/ordering process, when the order parameter is saturated and the domains are separated by thin walls. Here, to calculate the curvatures, we propose a much more accurate method. It is based on the observation that the local curvatures are quantities that can be inferred solely from the shape of the interface, without appealing to the properties of the bulk field [Pg.212]

The formulas (120) and (127) define a set of the local curvature variables that can be used for the digital pattern analysis in the case of an arbitrary lattice. Calculation of the curvature distribution is, in principle, impossible within the digital pattern methods. 

The existence of the two lengthscales in the intermediate regime has a simple physical interpretation and can be explained in terms of the LCA theory, which links the velocity of the interface with its local curvature. Namely, one can show [223] that the quantity p(t) = (LH(t)/LK(t))2 estimates the average number of necks piercing the surface of a sphere of radius LH(t) [LH(t) is the characteristic... 

The Euler characteristic, %, of a closed surface is related to the local Gaussian curvature K r) via the Gauss-Bonnet theorem [Eq. (8)]. A number of different schemes have been proposed to calculate the local curvatures and the integral in Eq. (8). 

Eq. (117)] by summing up the local curvatures and dividing the sum over the number of sampling points. Then the Euler characteristic is... 

The local value of Pc depends on the local curvature and has no direct relation with the total surface area. Disregard of this rule invalidates the theories of Sections... 

To pp. 41-43. The criticism of the methods based on Eq. (65) was expanded by J. J. Biker-man in a paper submitted to a magazine. These methods are not justified also because the quantities pj and pj in Eq. (64) refer to two different systems, in each of which a uniform pressure (Pl or pj) acts. In the experiments of Fig. 17 and the analogous tests, two different pressures are supposed to act in one system. A detailed consideration of such systems shows that in them no reversible melting and solidification, fully depending on the local curvature, can take place. Moreover, the actual pressures in the containers used depended on the flexing of the container walls, mentioned in Ref. 

The existence of these defects is fundamental to the covalent modification of the tube, as they can augment the local curvature, acting as catalyzing sites for reaction of neighboring atoms. 

Schematically, the steps in this process are shown in Fig. 14.4. At the beginning, the local radius of the tip end is small. Field emission can be easily established. A high current though the tip end then causes local melting. The local curvature at the end of the tip suddenly decreases. The field emission current is then reduced dramatically. The tip end recrystallizes to have a relatively large radius. Feenstra et al. (1987a) observed that the tips prepared in this way always provide reproducible tunneling spectra, although atomic-resolution topographic images are generally not observed.

There are a number of problems associated with measuring bimolecular rate constants for ET. Only a small set of data can be obtained. In addition, since a wide range of donor anion radicals is used, there are variations in the reorganization energies that influence local curvature (and thus intrinsic barrier equation 53) for each point. In principle, electrochemical measurements such as those described in Section 2 can provide similar information. 

Copper ions have been reduced in colloidal assemblies differing in their structures (55,56). In all cases, copper metal particles are obtained. Figure 9.3.1 shows the freeze-fracture electron microscopy (FFEM) for the various parts of the phase diagram. Their structures have been determined by SAXS, conductivity, FFEM, and by predictions of microstructures that require only notions of local curvature and local and global packing constraints. 

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