Mean Dimensions of Rubber-Like Polymer Molecules

The mean square end-to-end separations of coiling type polymer molecules have been calculated by two different approximate methods, taking into account the effect of excluded volumes. One of the methods is the equivalent of a first-order perturbation calculation using the unrestricted random walk as a starting approximation. The other method involves the use of certain plausible functional forms valid for large values of the chain lengths. The calculations have been carried out for three, two, and one dimensions and the results are analyzed to provide an insight into factors which might determine convergence or divergence of the quotient 〈rn2〉Av/n (the mean-square length divided by the number of links). It appears that the quotient converges in three (or more) dimensions, but diverges for a lower number of dimensions.