Networks, Topologies, and Entanglements

Lucia Carlucci, Gianfranco Ciani, and Davide M. Proserpio 

Some advantage can derive from the analysis of related structures reported in the literature, searching for prototypical models that can fit the specific problem. The increasing number of networked species reported in the last years offers a rich variety of new structural types that continuously amplify our knowledge of these organic or metal-organic extended systems. 

A correct analysis of the crystal structures is fundamental in order to avoid misinterpretations, that can easily occur, about the network topology and, when present, the type and extent of the entanglement of distinct motifs. The rationalization of these, often complicated, structures implies a sequence of steps leading to the basic nets of linked nodes. 

The steps, in general, can be summarized as follows (i) simplification process, (ii) identification and separation of the individual motifs, (iii)topological analysis of these motifs, and (iv) topological analysis of the whole entanglement. 

The simplification phase, previously exemplified by some authors [6], consists in the rather obvious operation of removing all the unnecessary elements that have no topological relevance, thus leaving only the essentials, represented by nodes and links (vertices and edges, respectively, in graph theory [7]). For instance, polyatomic nodes (like metal clusters or polyfunctional ligands) can be replaced by their barycenters. 

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These observations can be qualitatively explained in terms of the constrained-junction theory. If a network is cross-linked in solution and the solvent then removed, the chains collapse in such a way that there is reduced overlap in their configurational domains. It is primarily in this regard, namely reduced chain-junction entangling, that solution-cross-linked samples have simpler topologies, and these diminished constraints give correspondingly simpler elastomeric behavior. 

Scheme 2. Pictorial representation of an entanglement 4, an interpenetrated polymer network 5, and a topologically trapped macrocycle in a polymer network 6.

From this rough outline of some examples of current problems in the physics of rubber elasticity, it is clear that it is important to have a well-founded statistical-mechanical theory of equilibrium properties of rubber-elastic networks. Consequently, first junction and entanglement topology are described and discussed. Then a section briefly reviews the theory of the phantom network. In the following two sections, theories of equilibrium properties of networks and a comparison of theoretical results with experimental data are presented. 

The Iwata model probably overestimates the entanglement contribution to the rubber elasticity. This conclusion results from the observation that the topological contribution to the network modulus and the contribution arising from chemical crosslinks are approximately of the same order of magnitude A statistical... 

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