Interface mean curvature

There are several geometrical aspects of interface mean curvature that are particularly important when the interfacial energy is isotropic and the curvature becomes a driving force for mass transport. We present several equivalent cursory statements regarding mean curvature that have rigorous counterparts in differential geometry [6]. 

In the case of monodisperse foam, another approximation formula for the interface mean curvature was obtained in [156] by using the deformation theory [430] and the rounded dodecahedron model namely,... 

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... 

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. 

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]. 

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.

Equation (6.42) introduces a new independent variable of the system the mean curvature c = (c1 +C2). This variable must be taken into account in the Gibbs phase rule, which now reads F + Ph = C + 2 + 1. The number of degrees of freedom (F) of a two-phase system (Ph = 2) with a curved interface is given by... 

Analysis is simplified if 7 is isotropic—i.e., independent of geometrical attributes such as interfacial inclination n and, for internal interfaces in crystalline materials, the crystallographic misorientation across the interface. All interfacial energy reduction then results from a reduction of interfacial area through interface motion. The rate of interfacial area reduction per volume transferred across the interface is the local geometric mean curvature. Thus, local driving forces derived from variations in mean curvature allow tractable models for the capillarity-induced morphological evolution of isotropic interfaces. 

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. 

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. 

There are two different varieties of the curvature of an interface which are convenient to use in capillarity studies mean curvature, denoted by k, and the weighted mean curvature, denoted by k7. 

Mean Curvature of an Interface. The mean curvature is simply the sum of the curvatures of two curves on the interface that intersect at right angles. Any two such curves, C and c2, can be obtained by the intersection of orthogonal planes with the interface, as illustrated in Fig. C.2. The planes are chosen so that the interface normal h lies completely in each plane the line of intersection between the planes is parallel to h (a crystallographer might think of h as a zone axis). There are an infinite number of choices for these planes, but all are related by rotation around the axis h. 

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. 

Weighted Mean Curvature of an Interface. The weighted mean curvature, k7, has exactly the same geometrical properties as the mean curvature except that it is weighted by the possibly orientation-dependent magnitude of the interfacial tension. It is particularly useful for addressing capillarity problems when the interfacial energy is anisotropic, that is, dependent upon the interface orientation (Section C.3). 

The mean curvature is the local rate of interface area change with a local addition of volume.4 This is perhaps the most important aspect of curvature, especially when combined with 7, which is the work required to create an interface per unit area, A. Imagine that in a pure material the addition of a small volume makes an interface develop a localized small bump. The statement above implies that k = AA/AV in the limit of small volumes therefore, the work to create the bump is 7 A A = yuAV, where 7 is the interfacial energy per unit area (see Eq. 3.73). Equating this work to the work done by the system P AV, where P is the net pressure on the interface... 

The quantity 7/c may be regarded as the local potential due to the interface curvature to add a chemical species per unit volume of the species. On interfaces where the mean curvature is constant everywhere (such as on a sphere where k = 2/Rc, on a cylinder where k = 1/RC. and on a plane and a catenoid where k = 0), this potential is uniform and thus these are equilibrium interfaces. There is an infinite number of equilibrium interfaces a three-parameter family of minimal interfaces has been described [9]. 

The weighted mean curvature is the local rate of interfacial energy change with a local addition of volume. This establishes the connection to the work, 8W, to pass a small volume of material, 8V, through an interface. 8W/8V = /c7(f), in the limit of small volumes. 

Of all local motions, v(r), of an interface that pass the same amount of volume from one side to the other, the motion that is normal to the interface with magnitude proportional to the weighted mean curvature, v f) oc /c7n, increases the interfacial energy the fastest. However, fastest depends on how distance is measured. How this distance metric alters the variational principles that generate the kinetic equations is discussed elsewhere [14]. 

The weighted mean curvature is the interface divergence of the evaluated on the unit sphere. The interface divergence is defined within the interface, and if the interface is not differentiable, subgradients must be used. The convex portion of is equivalent to the the Wulff shape, so the interface divergence is operating from one interface onto another. This form can get very complicated. 

Therefore, in a system with a small (3 particle that is rich in component B, the value of 2 characteristic of equilibrium is raised compared with a system in which the mean curvature of the a/(3 interface is zero. 

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