The Payne effect revisited

Measurements of the storage modulus, mu', vs. strain amplitude, u, for highly filled rubbers exhibit a pronounced decrease of mu' with increasing u. Unfilled rubbers do not show this so called Payne effect. Even though the effect is known since the 1940s, it continues to play a significant role in the research community focusing on rubber materials in general and automobile tires in particular. The key problem is the elucidation of the dependence of the Payne effect on the chemical composition of the rubber material. Based on a scaling approach we derive the functional form mu' (u, T) - mu'(infinity, T) proportional to (1 -(D/d) u)(-(sigma-df+1))g(T), where the parameters D/d and sigma - d(f) + 1 are directly related to the filler network structure. In addition, we explain the temperature dependence of the Payne effect, g(T), in terms of a distribution of activation energies corresponding to different types of filler-filler interactions. Finally, the model is extended to describe the attendant amplitude dependence of the loss modulus.