Fractal Structures in Carbon Black Reinforced Rubbers

Abstract

Abstract This paper considers the fractal nature of primary and secondary carbon black aggregates in rubber and the implications for the mechanical and electrical properties of rubber goods. In particular, the effects of dispersion and primary aggregate breakup during mixing are investigated. Founded on recent studies of the disordered growth processes of colloids, a model concerning formation, structure and properties of tenuous secondary carbon black aggregates (clusters) in rubber is formulated. We distinguish between two different mechanisms of cluster growth at carbon black concentrations ϕ below and above the gel point ϕ** of the filler network. For ϕ≤ϕ**, the restricted mobility of the dispersed primary aggregates governs the cluster growth and neighboring clusters are seperated by a rubber-specific minimum distance d0. These gaps between neighboring clusters are joined together in a bond percolation model that determines the conductivity in a transition regime ϕ*≤ϕ≤ϕ** above the electrical percolation threshold ϕ*. The mechanical action of carbon black clusters below the gel point ϕ** is estimated by a hydrodynamical amplification factor that is related to a rigidy condition for the clusters. At sufficient large filler concentration for ϕ≥ϕ**, the restricted mobility of primary aggregates in the rubber matrix is insignificant for the cluster growth and a kinetic cluster by cluster aggregation (CCA) process is applied. The resulting fractal carbon black network corresponds to a spacefilling configuration of CCA-clusters. From this network structure, a scaling invariant power law arises for the small-strain modulus as function of carbon black concentration. The conductivity in the networking regime ϕ≥ϕ** shows a typical power law behavior that is implied by an anomalous diffusion of the charge carriers on the fractal clusters. For the frequency dependence of the conductivity a cross-over to a power law regime at large frequencies results. The predicted properties of carbon black filled rubbers are discussed in the framework of experimental results taken from different authors.