Statistical Thermodynamics of Rubber Elasticity

Abstract

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.

Modeling semiflexible polymer networks

“…The stress tensor is found by adding contributions from polymer strands over all orientations. These aspects form the essence of affine models of both the semiflexible networks reviewed here, as well as much earlier approaches to rubber elasticity (Doi and Edwards, 1988;James and Guth, 1943;Rubinstein and Colby, 2003;Wall and Flory, 1951) that have been the inspiration for much of what follows in this section.…”

Random Networks, Graphs, and Matrices

“…The advantage of this model over the Wall-Flory [7,8] treatment is that it explicitly acknowledges the many-body aspects, alias connectedness, of the elastomer network. The disadvantage is that it requires more sophisticated methods to extract an elastic equation of state than does the Wall-Flory independent chain model.…”

Finite extensibility and orientational effects in rubbers. A physical interpretation of the ‘van der Waals equation’ for the elastic force

“…We propose a physical origin for these two parameters. The first attempt of basic theory for non interacting chains was the Kuhn argument [2], later improved by Flory [3], that the position of the crosslinks -or the distribution of the position for Flory -was deformed affinely to the macroscopical deformation. This gives a free energy where Nrnes h is the number of monomers in a strand between two crosslinks, and ~t i the principle value of the deformation tensor.…”