Non-Gaussian Single Chain Statistics

The Gaussian distribution (Eq. III-l) arises if the number of statistical chain elements per chain is large and if the extension of the chain is not too extreme. Treloar 171) has shown that the exact distribution for a chain of N links of length A for any value of N is given by 

The exact result shown in Eq. (IV-4) indicates that appreciable errors are introduced if the Gaussian approximation is used below N = 6. The use of Eq. (IV-4) in a complete network theory (compare e.g. Eq. III-2) has not been undertaken because of mathematical difficulties. Instead, Treloar has shown that in a tetrahedral four chain arrangement around a central crosslink, the fluctuations in the crosslink 

If one proceeds in this way, it does not cause problems in treating constant-volume deformations (i.e. dry network deformations), but it meets with obstacles in swelling type strains because the logarithmic term HvlnXxXyXz in Eq. (III-9) is absent in the simplified treatment of Eq. (IV-5). A full derivation according to the HFW approach but employing the series distribution Eq. (IV-4), is, however, not available. 

The qualitative, overall effect of small N is a stress in the deformed network which is larger than the corresponding Gaussian stress. This effect is, however, very small in normally crosslinked rubbers, whereas these rubbers exhibit large deviations from Gaussian behaviour. The tentative conclusion must therefore be that the deviations from Gaussian behaviour at moderate extensions cannot be caused by short chains only. 

Another reason for deviations from Gaussian behaviour, even at large N, lies in the finite extensibility of polymer chains. To account for this, one utilizes the complete expression for the partition function qt of a chain of N chain elements, rather than just the Gaussian approximation to it. A very clear exposition of the statistical mechanics of a chain with length rt under tension, can be found in Hill s book (85) yielding 

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