Topological structure cluster relationships

The combined use of fractal analysis and cluster models for the structure of the condensed state of crosslinked polymers allows their quantitative treatment on different structural levels, molecular, topological and suprasegmental, to be obtained for the first time and also the interconnection between the indicated levels to be determined. In turn, elaboration of solid-phase crosslinked polymer structure quantitative models allows structure-properties relationships to be obtained for the first time, which is one of the main goals of polymer physics. 

Fig. 2.3-12. Molecular structures of 59, 60, 60a, and topological relationships of 59 and 60 to the corresponding sections from the solid-state structure of elemental aluminum, and structural similarities of the clusters Ali2R6 and In Rs- In the latter cluster the octahedral sections are highlighted.

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). 

Descriptors based on the 2D structure or simply on the connectivity matrix of a structure have long been used for chemical similarity and for property correlations. Because they often lack any relationship to mechanism, these descriptors are best used within a congeneric series or at least a set of similar structures. They may be empirically useful for cluster analysis and chemical library design, because they are effective at representing structure differences and similarities. A few programs and providers of topological descriptors include the following ... 

Figure 6. Topological relationship between discrete and linked clusters based on triangular building blocks and the electron counts in known structures. Two units build up an octahedron (left). After a distortion, three such units form a tricapped trigonal prism (middle). An infinite number of triangular units builds up a chain of face-sharing octahedra (right).

The same observations are made for clusters 9 and 10, and to a greater extent for the latter [17]. Unlike the clusters described above, clusters 9 and 10 are topologically independent from smaller complexes. They are, however, themselves structurally related and are indeed produced side-by-side from the same reaction. The molecular structures of 9 and 10 are given in Figure 3.38, emphasizing their structural relationship. In Figure 3.39, the based Sj4 or S25 polyhedra are contrasted. 

The fractal dimension D of the chain part between its topological fixation points (entanglements, clusters, crosslinking nodes) is the most important structural parameter, checking molecular mobility and deformability of polymers. Two of the main factors, due to application of dimension D, are its clearly defined variation limits (1 < D < 2) and the dependence on polymer supersegmental (supermolecular) structure. Let us especially note that all fractal relationships contain, at any rate, two variables. 

---