Basic Polymer Topologies

In your mind you might envision a polymer as an indefinitely long, linear chain. In reality, this is rarely the case. Whether by design or by accident, most polymers deviate from perfect linearity, and these deviations profoundly affect a polymer s properties. Here we review some of the basic issues in polymer topology (recall from Chapter 6 that for most structures the topology of a molecule is established solely by its connectivity). 

A special case arises for cross-linked polymers. Consider a polymer with an average DP of —10,000, and with one cross-link for every 100 monomers (termed 1% cross-linking). In such a sample, every chain has many cross-links, and it is inevitable that a covalent path exists between every chain. That is, for all practical purposes, every chain is ultimately connected to every other chain—the polymer is a single molecule Rubber is a highly cross-linked polymer of isoprene (Table 13.2), so a rubber band may very well be a single molecule. 

Protein Folding Modeled by a Two-State Polymer Phase Transition 

Proteins are heteropolymers, and therefore application of the simple analysis given above for mixing homopolymers with diblock copolymers would seem to be a stretch for proteins. Yet, one of the simplest models for proteins divides the twenty natural amino acids into twocate-gories hydrophobic (H) and ionic/polar (P). The attraction between H-typ e monomers models the tendency of hydrophobic polymers in water to collapse in order to minimize their exposed surface area, but also to minimize their interactions with P-type monomers. The minimiza- 

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The basic theories of physics - classical mechanics and electromagnetism, relativity theory, quantum mechanics, statistical mechanics, quantum electrodynamics - support the theoretical apparatus which is used in molecular sciences. Quantum mechanics plays a particular role in theoretical chemistry, providing the basis for the valence theories which allow to interpret the structure of molecules and for the spectroscopic models employed in the determination of structural information from spectral patterns. Indeed, Quantum Chemistry often appears synonymous with Theoretical Chemistry it will, therefore, constitute a major part of this book series. However, the scope of the series will also include other areas of theoretical chemistry, such as mathematical chemistry (which involves the use of algebra and topology in the analysis of molecular structures and reactions) molecular mechanics, molecular dynamics and chemical thermodynamics, which play an important role in rationalizing the geometric and electronic structures of molecular assemblies and polymers, clusters and crystals surface, interface, solvent and solid-state effects excited-state dynamics, reactive collisions, and chemical reactions. 

The basic assumption of our treatment is that polymer structures are topologically characterized by rod-like segments that are packed together to form representative domains. The concept allows us to generate structures of very different shapes with the aid of a unique formalism. The utility of this approach will be discussed with representative examples. 

After these short explanation of basic definitions, it is quite easy to understand that because topological problems are very general they can be formulated easily using polymer language. 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

We have presented only a few basic physical results obtained within the framework of the PCAO-model the mathematical methods for investigating equilibrium and dynamic properties of a test polymer chain of different topology placed in the lattice of obstacles will be reviewed below. 

A fundamental contribution is represented by a famous series of articles and books on crystal chemistry published many years ago by A. F. Wells [4], who analysed and classified a great number of nets. Fie emphasized the importance of describing a crystal structure in terms of its basic topology such a description not only provides a simple and elegant way of representing the structures but also evidences relations between structures that are not always apparent from the conventional descriptions. Wells introduced a method for the systematic generation of 3D arrays from 2D nets and also described many hypothetical motifs that were successively discovered within the realm of coordination polymers or of other extended systems. Flis results included a list of many simple nets described with only one kind of node (uninodal) or with two nodes of different connectivity (mainly binodal 3,4-connected). 

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