Thread analogy

Analytic solutions are also possible based on the idealized Gaussian thread model since the molecular closures simplify dramatically. Because the hard-core diameter is shrunk to zero, Eq. (6.4) applies for all r, thereby allowing cancellation of the convolution integrals and all factors of w. Hence, the thread analogs of Eqs. (6.5) and (6.6) become" ... 

Fig. 7.151. Screw-thread spiral analogy to a screw dislocation.

A totally new situation arises from the presence of defined topological bonds in polymer systems. The last documented example is given by polyrotaxanes 7 in which defined topological bonds occur between the macrocycles and the polymer chain (considered as infinite). The polyrotaxanes are composed of a polymer chain on to which a certain number of macrocycles is threaded. For short polymer chains, the end-capping by stoppers prevents the macrocycles unthreading from the chain [27, 28] (Scheme 3). Multicatenanes 8 are structurally related to polyrotaxanes 7 and can be viewed as cyclic analogs of polyrotaxanes [29]. 

Figure 10.91 The simplest kind of ravel based on a 3-connected vertex. While none of the three threads are knotted or catenated, pulling on all three of them simultaneously results in an entangled unit distinct from the analogous unravelled 3-connected node.110...

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