Relative shape domains

For the characterization of the shapes of molecular contour surfaces, such as MIDCO s and MEPCO s, one may subdivide the surface into domains fulfilling some local shape criteria. One can distinguish two types of criteria, relative, and absolute, leading to a relative shape domain or an absolute shape domain subdivision of the molecular contour surface. 

For example, the technique of interpenetrating contour surfaces (157] can be applied for a relative shape domain subdivision of a pair of MIDCO and MEPCO surfaces of a molecule. The MIDCO surface can be subdivided into domains using the contour value of the MEPCO as criterion. This procedure is equivalent to generating the interpenetration pattern on the MIDCO surface [157]. 

The numerical value of the reference curvature b can be specified in absolute units or in units scaled relative to the size of the object G(a). If absolute units are used, then a relative convexity characterization of G(a) involves size information if an object G(a) is scaled twofold, then its shape remains the same, but with respect to a fixed, nonzero b value a different relative convexity characterization is obtained. That is, the pattern of relative shape domains Do(b)> D (b), and D2(b) defined with respect to some fixed, nonzero reference curvature value b (b K)) is size-dependent. On the other hand, if the reference curvature b is specified with respect to units proportional to the size of G(a), then a simple. scaling of the object does not alter the pattern of relative shape domains with respect to the scaled reference curvature b. In this case, the shape characterization is size-invariant, that is, a "pure" shape characterization is obtained. 

Several topological methods of shape analysis of molecular contour surfaces have been designed to take advantage of such relative and absolute shape domain subdivisions of the contours, according to. some physical or geometrical conditions [155-158,199]. 

Figure 5.3 The shape domains of relative local convexity of a MIDCO surface G(a) of Figure 5.1, relative to a tangent sphere T of curvature b (radius 1/b) are shown. A geometrical interpretation of the classitication of points r of G(a) into locally concave Dq, locally saddle-type D, and locally convex D2 domains relative to b is given when comparing local neighborhoods of the surface to the tangent sphere T. The classification depends on whether at point r the surface G(a) is curved more in all directions, or more in some and less in some other directions, or less in all directions, than the test sphere T of radius 1/b. In the corresponding three types of domains Do(b). Dm,), and D2(b), or in short Dp, D, and D2, the molecular contour surface G(a) is locally concave, of the saddle-type, and convex, respectively, relative to curvature b.

The relative curvature domains Do(bG)- Dl(bG)> D2(bG). specified in terms of the scaled reference curvature be provide a size-independent shape characterization of the object G(a) for all curvatures. 

Figure 5.4 Two sets of shape domains of oriented relative local convexity of the MIEKDO surface G(a) of Figure 5.1, relative to two orientations of a tangent ellipsoid T are shown.

If the shape domains are defined by relative local convexity, then the notation HP j (a,b), p = 0, 1,2, is u.sed for the shape groups of MIDCO surfaces G(a), where besides the dimension p of the homology group, the truncation type p, the charge density contour parameter a, and the reference curvature parameter b are also specified. For the special case of ordinary local convexity, b=0, the second argument in the parentheses can be omitted and one may simply write HP (a). Usually, we are interested in the Betti numbers of the groups HP (a,b) and HP (a) for these numbers the bp x(a,b) and bpp(a) notations are used, respectively. 

A shape domain partitioning in terms of relative local convexity of parameter b=0.005 leads to a simpler pattern. We obtain... 

The third shape domain partitioning shown has been calculated for the relative local convexity parameter b= - 0.008. We obtain... 

Consider a MIDCO G(a) and a choice for the curvature parameter b, and assume that the shape domains of relative convexity of G(a) have been determined. By using an appropriate neighbor relation to describe the mutual arrangements of the domains along the MIDCO surface G(a), the corresponding shape matrix s(a,b) and the associated shape graph g (a,b) can be defined [109,110,158,193]. 

Figure 5.9 The construction of Shape Globe Invariance Maps (SGIM s), of MIDCO relative convexity shape domain patterns for two reference curvature values, b = 0 and b < 0.

A practical implementation of the above approach is the following a global shape property of the molecule is assigned to each point of the sphere S, followed by the determination of those domains of S where this shape property is invariant. A pair of examples is shown in Figure 5.9, where the shape globe invariance domains of a MIDCO surface for two relative convexity shape domain partitionings (P) with respect to two reference curvatures, b = 0, and b < 0, are given. As... 

The assignment of the index i to the curvature domains D j can be used to encode additional information for example, some indication of relative sizes of the shape domains. In one implementation [109], the index i follows the ordering of all the shape domains according to the decreasing size of their surface areas on... 

Subsequently, these topological methods have been adopted and modified to the significantly simpler, three-dimensional molecular shape problem, where the shape of the molecule is the quantum mechanical shape of the electron density cloud [13-19], This has led to the development of the shape group methods, where the ranks of homology groups describing relative convexity domains of the complete set of all isodensity surfaces of the molecule, the so-called Shape Group Betti numbers, provided a detailed, numerical shape code for the quantum chemical electron density [13-19]. 

In the most common applications of shape groups, the local shape properties are specified in terms of shape domains for example, in terms of the locally convex, concave, or saddle-type regions of MIDCOs, relative to some curvature reference parameter b. 

On the local level, shape complementarity implies matches between locally concave and locally convex domains, as well as matches between properly placed saddle-type domains, where a directional convex-concave match is important. Replacing the simple D (K,a) notation, the more elaborate notation D b),i(K,a) is used sometimes when studying the complementarity of local shape domains, where the notation includes the relative convexity specification fi(b). This quantity takes values... 

T he behavior of liquid crystals in applied electric fields has been the object of several recent studies (1, 3, 4)- The materials primarily used here were those in which the dipole moment of the molecule was not in the same direction as the molecular axis. When an electric field is applied to such a system by transparent electrodes, the characteristic cigar-shaped domains shown in Figure 1 for p-azoxyanisole are readily observed, using relatively low magnification with or without polarized light. The optical behavior of such domains between crossed polarizers indicates that their optic axis is essentially parallel to the electrode surface and essentially perpendicular to the direction of the applied field. 

Rod-coil diblock copolymers with a relatively short rod segment self-assembled into a wavy-lamellar morphology, while increasing the rod fraction resulted in the formation of a zigzag lamellar phase. At rod fractions/phic of 0-96 and 0.98, the PS coils were found to organize into arrowhead-shaped domains embedded in a PHIC matrix. Unlike the self-assembly of the PPP-17-PS and PPQ-b-PS diblock copolymers, the unconventional morphologies discovered for the PHIC-I7-PS rod-coils are related to their macromolecular architecture and are... 

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