The freely jointed single chain

Rotation around a single bond in a molecular chain generally gives a cone of positions of one part of the chain with respect to the other. If the rotation is not totally free, as for example when tram or gauche bonds have lower energy than other conformations for an all-carbon backbone, certain positions on this cone are favoured. Nevertheless, in order to develop the simplest form of the statistical theory of polymer-chain con- 

An important quantity in the theory is the root-mean-square (RMS) length of an unperturbed randomly coiled chain, which can be calculated as follows (see fig. 3.6)  

If rl is averaged over a large number of chains, the last term averages to 

The fully extended length of the chain is equal to nl so that the maximum extensibility of a random coil with r = Zrms is from n l to nl, i.e. a factor of rfi. The value of Znns also gives a measure of the spatial extent of a chain. A second useful measure of this is the radius of gyration, r, which is the RMS distance of the atoms of the chain from the centre of gravity of the chain. Debye showed that, provided that n is very large, Zg = r /s/6. 

In order to simplify the theory further it is usual to assume that, for all chains that need to be considered, the actual end-to-end distance is very much less than the fully extended length, i.e. r nl, which becomes true for all chains when n is sufficiently large. A chain for which the assumption is valid is called a Gaussian chain. Consider such a chain with one end fixed at the origin and let the other end, P, be free to move (see fig. 3.7). With OP nl it can then be shown that the probability p x, y, z) that P lies in the small element of volume dx dy dz at (x, y, z) is 

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