Surface topology/curvatures

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. 

In the standard approaches to the systems in which monolayers or bilayers are formed, one assumes that the width of the film is much smaller than the length characterizing the structure (oil or water domain size, for example). In such a case it is justified to represent the film by a mathematical surface and the structure can be described by the local invariants of the surface, i.e., the mean H and the Gaussian K curvatures and by the global (topological)... 

In the microemulsion the role of A is played by the period of damped oscillations of the correlation functions (Eq. (7)). The surface-averaged Gaussian curvature Ky, = 2t x/ S is the topological invariant per unit surface area. Therefore the comparison between Ryyi = Kyy / in the disordered microemulsion and in the ordered periodic phases is justified. We calculate here R= Since K differs for diffused films from cor-... 

Usually, the zeolite inner surface characteristics are rather complex as a consequence of the (3D) character of the porous topologies of most of the zeolite types. The porous framework is a (3D) organization of cavities connected by channels. Inner surfaces are composed of several sorption sites characterized by their local geometry and curvature. Illustrative examples of such inner surface complexity are represented on Figures 1 and 2 they concern the Faujasite and Silicalite-I inner surfaces respectively. 

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). 

To construct a spherical host from two subunits (n = 2), each unit must cover half the surface of the sphere. This can only be achieved if the subunits exhibit curvature and are placed such that their centroids lie at a maximum distance from each other. These criteria place two points along the surface of a sphere separated by a distance equal to the diameter of the shell. As a consequence of this arrangement, there exist two structure types one with two identical subunits attached at the equator and one belonging to the point group Ax/ which is topologically equivalent to a tennis ball. In both cases, the subunits must exhibit curvature. 

Figure 1. A. Computer graphic portion of a periodic surface of constant mean curvature, having the same space group and topological type as the Schwarz D minimal surfhce. This surbce, together with an identical displaced copy, would represent the polar/apolar dividing surface in a cubic phase with space group 224 (Pn3m). The two graphs shown would thread the two aqueous subspaces. B. Computer graphic of a portion of the Schwarz D minimal sur ce (mean curvature identically zero). In the 224 cubic phase structure, this sur ce would bisect the surfactant bilayer.

There is a remarkable relationship between the average Gaussian curvature of a surface and its topology as quantified by the genus, which is the number of holes in a multiply connected surface. The relationship, the Gauss-Bonnet theorem, when applied to a surface of constant Gaussian curvature, is... 

An absolute shape characterization is obtained if a molecular contour surface is compared to some standard surface, such as a plane, or a sphere, or an ellipsoid, or any other clo.sed surface selected as standard. For example, if the contour surface is compared to a plane, then the plane can be moved along the contour as a tangent plane, and the local curvature properties of the molecular surface can be compared to the plane. This leads to a subdivision of the molecular contour surface into locally convex, locally concave, and locally saddle-type shape domains. These shape domains are absolute in the above sense, since they are compared to a selected standard, to the plane. A similar technique can be applied when using a different standard. By a topological analysis and characterization of these absolute shape domains, an absolute shape characterization of the molecular surface is obtained. 

Shape codes [43,109,196,351,408]. The simplest topological shape codes derived from the shape group approach are the (a,b) parameter maps, where a is the isodensity contour value and b is a reference curvature against which the molecular contour surface is compared. Alternative shape codes and local shape codes are derived from shape matrices and the Density Domain Approach to functional groups [262], as well as from Shape Globe Invariance Maps (SGIM). 

The Gaussian curvature has the dimensions of inverse area and the mean curvature has dimensions of inverse length. The topology of the surface... 

Remarkably, the topology is linked to the integral curvature of a surface by the simple equation ... 

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