Topology transformation

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 topological transformations in an incompatible blend can be described by the dynamic phase diagram that is usually determined experimentally at a constant shear rate. For equal viscosities, a bicontinuous morphology is observed within a broad interval of the volume fractions. When the viscosity ratio increases, the bicontinuous region of the phase diagram shrinks. At large viscosity ratios, the droplets of a more viscous component in a continuous matrix of a less viscous component are observed practically for all allowed geometrically volume fractions. 

A few considerations about possible schemes of relationships between inorganic crystal structures based on a systematic construction of complex structural types by means of a few operations (symmetry operations, topological transformations) applied to some building units (point systems, clusters, rods, sheets), have been previously reported in 3.9.1, following criteria suggested, for instance, by Hyde and Andersson (1989) and by Zvyagin (1993). 

The ever-increasing interest in the catenanes and knots of DNA stems not only from their widespread occurrence or from their topological novelty determination of their structures provides precious information about the biological processes which generate them. A whole class of enzymes - topoisomerases - effects these topological transformations perfectly [30, 31]. Their possible role in a large vari-... 

Table 7. Proposed reaction scheme for topological transformation during micelle formation in nonpolar solvents. AOT in C6H12 [Ber. Bunsenges. Physikal. Chem. 79, 667 (1975)]... 

Figure 1 The topological transformation of ice XI (at 50 kbar) to ice II with 02, easy directions displayed as arrows for hydrogen and O—O BCPs, (a) ice XI at 50 kbar, (b) supercell of ice II (most of O—O BCPs removedfor clarity).

Figure 3 The topological transformation of ice II to ice IX. - twelve molecule unit cell of ice IX...

Figure 5 The topological transformation of ice VIII to ice X. (a) Sixteen molecule unit cell of ice VIII with hydrogen and O—O Ci directions and BCPs. (b) and (c) show the ice X O-H and O—O BCP easy directions respectively.

In the following chapters of this book, when a pseudorotation mechanism is involved in the overall sequence of reactions, it will be denoted in the reaction schemes using the formalism shown below, according to the convention introduced by Oae. This formalism does not bear any significance on the nature of the pathway followed by the topological transformation. 

Scheme 2.11 Topological transformations during the thermolysis of triarylbis(phenylethynyl)antimony compounds...

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