The gaussian chain in the theory of rubber elasticity

Abstract

AbstractThe importance of the conception of the kinetic theory of rubber elasticity is comparable to that of the kinetic theory of gases. In the theory of rubber elasticity the concept of the Gaussian chain is comparable to that of the Maxwell distribution in a gas. However, whereas the theory of gases has been refined in successive approximations to a coherent set of equations relating meaurable macroscopic properties to the field of force between the molecules and thus allowing the collection of detailed information on molecular forces, no similar progress has been made in the theory of rubber elasticity since its conception 40 years ago. The molecular forces appear in the theory of the Gaussian chain in the quantity r02, the mean square end‐to‐end distance of the “free” chain. In a bulk rubber and also in a swollen rubber this quantity depends on intra‐ and intermolecular forces. If the definition of a Gaussian chain is a chain in which magnified image is independent of r (F(r) being the free energy of the chain with fixed end‐to‐end distance r), then the observation of the Rivlin‐Mooney term in measurements of rubber elasticity appears to be incompatible with the assumption of Gaussian chains. Starting from James's theory of networks of Gaussian chains, it is shown that the Rivlin‐Mooney term can be explained by assuming that the chains are indeed Gaussian, but that r02 depends on the internal pressure pi. However, in order to explain what was observed, the effect of pi on r02 should be much larger than one expected, and hence additional effects are needed to explain the behavior. The additional effects can consist of higher terms in the development of r02 with respect to pi and also in anisotropic terms giving rise to differences between the 3 components of r02 in a nonisotropic stress field. In order to answer the question whether Gaussian chains exist at all in rubber much more experimental information on deformation under different pressures and stress tensors should be available. By ignoring the effect of pressure on r02 one can derive equations between measurable quantities from which r02 is eliminated. These equations should be considered unfounded since already a small effect of pressure in r02 will invalidate them. In the introduction it is shown, in accordance with an earlier conclusion of Lodge, that James's theory of networks of Gaussian chains predicts isotropic elastic behavior of the network, irrespective of any anisotropy of the topology of the network. It is also shown that the expression for the elastic free energy is insensitive to the choice of the junctions and of the chains between junctions.