Theory of Filler Reinforcement

TLDR

The reinforcement of rubber by colloidal carbon black at low loadings behaves like a suspension, analogous to solvation, and its elastic, viscous, and dielectric properties are theoretically described by extensions of Einstein, Maxwell, and Rayleigh models. This study generalizes the colloidal filler model to ellipsoidal particles, aiming to compute suspension properties based on matrix and filler characteristics. The authors extend the model to ellipsoidal fillers and derive analytical expressions for viscosity, modulus, stress–strain behavior, and dielectric constant as functions of filler concentration. The derived linear (low loading) and quadratic (high loading) relationships predict that viscosity, modulus, and dielectric constant increase with concentration, stress–strain curves remain similar, tensile strength initially drops due to stress concentrations, then rises with chain/network formation before declining at very high loadings,.

Abstract

For small loadings (up to about 10 percent volume parts) the colloidal carbon black spheres may be considered as suspended in a continuous rubber matrix. In the present paper this model is generalized for ellipsoidal (including plate- and rod-like) filler particles and it is extended to the computation of various properties of the suspension in terms of the properties of the matrix and of the fillers. Viscosity, Young's modulus, stress-strain curve below crystallization, and dielectric constant of the suspension are derived as linear functions of the volume concentration for small, and as quadratic functions for higher loadings. The stress-strain curves for varying amounts of fillers are similar. For small loadings the tensile strength first decreases because of the stress concentrations occurring around the carbon black spheres when the samples are stretched. The increase of the tensile strength observed for greater loadings is caused by the tendency of the carbon black spheres to form chains and finally, a type of network. The stiffness increases with loading, up to the point where the suspension becomes a dilution of carbon black by rubber. There the tensile strength decreases too. Binding of rubber by carbon black is similar to solvation. The theoretical conclusions were checked experimentally, in particular, the dependence of Young's modulus on concentration, the similarity of stress-strain curves, and the decrease of the tensile strength for small loadings. The theory of the elastic properties is very similar to the theory of Einstein on the viscosity of colloidal solutions and to Maxwell's and Rayleigh's theory of dielectric properties.