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Two principal differences between the theories of rubber elasticity advanced by James and Guth and by other authors are examined in the light of certain fundamental concepts. First the distribution functions for molecular chain lengths in vulcanized rubber networks are considered from the point of view of symmetry. Assuming a relaxed network to be isotropic and a network subject to uniform stress in one direction to be transversely isotropic, it is possible to formulate very general mathematical forms with respect to which the actual distribution functions must be compatible. It is shown that the functions employed by Wall and by Flory and Rehner are consistent with the required general forms; those of James and Guth are incompatible with any degree of isotropy and suggest an aeolotropic structure for vulcanized rubber. It is also shown that the configurational entropy of vulcanization must be zero and quite independent of the statistical nature of the chains comprising the network, providing the network is not deformed macroscopically. The negative entropy of vulcanization derived by James and Guth arises from their erroneous identification of the configurational probability with the number of configurations which a microscopically specified network structure could assume, rather than with the total number of configurations for all network structures which are consistent with the requirements of the vulcanization process. The assertion of James and Guth that the configurations of the polymer chains are altered in some systematic manner by the introduction of cross-linkages and their concept of ``internal pressure'' originate in this error. The present article attempts to clarify the currently prevalent confusion with respect to rubber elasticity theory.

A new method is described for the generation of excluded volume random walks of contour lengths comparable to those of real polymer molecules. This work was carried out with a high-speed electronic digital computer. An essential feature of the method is an unbiased sample enrichment process used to counteract the attrition resulting from chain intersections. Using the new method, samples were generated for chains of 800 links, a limit imposed by machine storage capacity. In principle the method could be carried even further if more storage were available. Mean-square end-to-end lengths of the chains are expressed by the equation 〈rn2〉=anb,where n equals the number of links and a and b are constants.

The statistical dimensions of macromolecular chains of fixed contour length can be rapidly calculated by Monte Carlo methods applied to a model consisting of dynamic self-avoiding random chains on a lattice. This ’’slithering snake’’ model involves moving the head of a chain one space in a lattice with all other elements of the chain moving forward along the old contour. Possible moves of the head are selected at random, but if such a move is precluded by double occupancy, the old configuration is retained, with head and tail interchanged, and then counted as if a move were made. This technique gives unbiased statistical results except for the effect of double cul-de-sacs. The method can also be applied to interacting chains, either free or confined to a box. Calculations have been made for 10-link chains on a square planar lattice for two different concentrations in infinite space and for two concentrations in a small box.

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