Connection with Lattice Topology

We can relate the dynamics, which, as is clear from the discussion of the previous section, depends solely on the set of elementary divisors i(a ), to the topology C 

We arrive at equation (5.106) by applying this result to all principal minors (each of which is the determinant of a subgraph of of size i ).B 

Using the notation N i) = of subgraphs of type , and N ii. in) of n-disjoint subgraphs the first few coefficients for OT rules are given 

In the more general case of a directed graph, wo have tliat 

Although the actual cycle decomposition (as well as the tree structure) of a particular graph is determined exactly by the set of elementary divisors i(a ), much of the general form of the possible dynamics may be extracted from Pl x) itself. All graphs whf)se characteristic polynomials Pii=P Yi=i Pi AY (mod q), for. some fixed P ( / ), for example, mu.st share the following properties  

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The word W consisting of a sequence of letters corresponding to different entanglements (introduced in Sect. 2.21) plays a role of full topological invariant for the PCAO-model. It is closely connected with the concept of the primitive path obtained by means of roughing of the microscopic chain trajectory up to the scale of the lattice cell and by exclusion of all loop fragments not entangled with the obstacles (Fig. 5). 

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

There exist several ways to treat the zeolite framework structure in modeling. One can take into account either the periodicity of the full lattice or only a small part of the lattice, the latter sometimes being called the cluster approach. The cluster approach is often used in quantum chemistry studies because it requires less computer time. As long as specific properties connected with the framework topology (e.g., the dimensions of the channels) are not dominating the outcome of the calculations, this approach can provide valuable informa-... 

Figure 6.9 pictures the example of the triangular-square network taken from [6.38]. Some 4-coordinated sites are seen comprising inner boundaries of this LRC. It is easy to notice that there are two 5-coordinated atoms in the first coordination sphere of each 5-coordinated atom in LRC. This circumstance follows from the fact that, in P-polyhedra, we have an even number of squares leading to the formation of the MRO, which manifests itself in the formation of chains of 5-coordinated atoms. Collins and Kawamura studied the thermodynamic properties of triangular-square lattices. Kawamura established the existence of the first-order phase transition connected with the transformation of the crystalline structure into a topologically disordered one. 

Correspondingly, the values of nF characterize the connectivity of the lattice of particles or skeleton of a PS and the values of Zc characterize the interconnectivity of the lattice of pores of the same PS. Connectivity of PS is the major topological attribute, which in the general case does not depend on the shape and size of the PS s individual supramolecular elements, although the latter characterize the major geometrical properties of PS [8], Appropriately, the classification of PSs by the degree of interconnectivity with allocation of various types of integrity is possible. 

Coarse-grained polymer models neglect the chemical detail of a specific polymer chain and include only excluded volume and topology (chain connectivity) as the properties determining universal behavior of polymers. They can be formulated for the continuum (off-lattice) as well as for a lattice. For all coarse-grained models, the repeat unit or monomer unit represents a section of a chemically realistic chain. MD techniques are employed to study dynamics with off-lattice models, whereas MC techniques are used for the lattice models and for efficient equilibration of the continuum models.36 2 A tutorial on coarse-grained modeling can be found in this book series.43... 

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