Effects of Topological Classification

The simplest example of topological classification is the knotting of individual strands, i.e., self-entanglement. However, the effects attributed to entanglement in non-crosslinked polymers are clearly intermolecular. The simplest such case is that of pair-wise classification without self-entanglement. Consider the following three examples of strand pairs  

The equilibrium positions of the four junction points define the spatial relationship between the strands of each pair. For simplicity, the mean relative positions (internal coordinates) of the junction points of each pair are taken to be the same. The junction points of each strand are separately anchored to the network by at least two of their remaining strands, so each is an elastically effective strand according to Scanlan s criterion. The network itself in effect completes the loop for each strand, making the A, B, and C pairs as structurally distinct as catenane molecules (301). 

Edwards (300) was able to make some progress on the problem by showing that topological classification can serve only to raise the modulus. As a simple example, consider all strand pairs in a network which have, within some small tolerance, a specified set of internal junction coordinates. Suppose there are B such pairs, and that the strands of each pair, labeled 1 and 2, have co1 and co2 distinguishable configurations each as free strands, and fractions (gt)0 and (g2)o respectively which have the end-to-end distances specified by the equilibrium junction coordinates. If the crosslinks were formed in the system at equilibrium, then the total number of configurations for each strand of the pair is cj1(g1)0 and o)2(02h and the number available to the pair is (o1(o2(gl)0(g2)0. 

Suppose that (/ )0 is the fraction of the B pairs in topological class i from a total of Q such classes. The number of configurations available to a pair in class i will be fi i i 2(ffi)o(02)o(/ )o and the total number of configurations in the undeformed state for B pairs is 

The pairs will all have the same new internal coordinates in the deformed state. Suppose that the fractions of configurations for free strands which have the specified end-to-end distances of the deformed state are gx and g2, and that the fractions in the various classes for pairs formed from equilibrium with these coordinates are fh The number of configurations available to a pair in class i in the deformed state is therefore ft i 2 0i02/i- However, the number of strands in each class is always (/j)o B, so the total number of configurations in the deformed state is  

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