General random flight models

Though the freely jointed chain is a very simple model, the result R ) xN holds for more general models. Consider for example the model shown in Fig. 2.2, called the freely rotating chain, in wMch the n-th bond is connected to the (n - l)-th bond with a fixed angle 6 and can rotate freely around the (n - l)-th bond. 

Multiplying both ades of tins equation by r and taking the average over 

This recursion equation, with the initial condition (r ) = is solved by 

The constant b is called the effective bond length. The ratio 

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Jump reorientation models may involve activation over barriers to rotation or the migration of lattice defects or holes. Reorientation is in both cases discontinuous and changes in orientation occur-ing in one step are assumed to be large. Both types of jump reorientation models have been discussed by O Reilly [68], In his quasilattice random flight model, for example, O Reilly [69 70] assumes that the liquid structure up to the first coordination shell may be approximated by a lattice. Some of the properties of the solid state such as vacancies and translational diffusion by vacancy migration are considered present. In general difficulties arise when these jump reorientation models are compared with experimental data because several parameters are needed in the analysis. Furthermore, it appears that O Reilly [71] employs results obtained by Huntress [55] which apply only in the limit of small-step reorientation to treat the case of... 

The simplicity of (2.25) is to be contrasted with the complexity of the exact P(R n) for realistic models of flexible chains for all R (and for small n). When dealing with the complicated problems of nonideal polymer solutions, etc., it is therefore customary to replace the real polymer chain by the so-called Kuhn effective random flight chain. An effective chain is one with N (in general different from n) links of size A5 such that N lS.s = L and (R ) is as given by (2.29). This substitution of a real chain by its equivalent chain is often a necessity so that we may separate errors in principle from errors arising from a poor mathematical approximation to the exact P(R n) when dealing with problems which are not exactly soluble. This equivalent chain therefore provides us with reasonable approximations to the properties of real polymer chains, provided the physical properties of interest do not depend heavily upon those chain configurations with i > L or upon chain properties over small distances for which the real chain is stiff. 

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