FRACTAL STRUCTURES IN CARBON-BLACK REINFORCED RUBBERS

This paper considers the fractal nature of primary and secondary carbo n black aggregates in rubber and the implications for the mechanical a nd electrical properties of rubber goods. In particular, the effects o f dispersion and primary aggregate breakup during mixing are investiga ted. Founded on recent studies of the disordered growth processes of c olloids, a model concerning formation, structure and properties of ten uous secondary carbon black aggregates (clusters) in rubber is formula ted. We distinguish between two different mechanisms of cluster growth at carbon black concentrations phi below and above the gel point phi of the filler network. For phi less than or equal to phi**, the rest ricted mobility of the dispersed primary aggregates governs the cluste r growth and neighboring clusters are seperated by a rubber-specific m inimum distance do. These gaps between neighboring clusters are joined together in a bond percolation model that determines the conductivity in a transition regime philess than or equal to phi less than or equ al to phi* above the electrical percolation threshold phi*. The mecha nical action of carbon black clusters below the gel point phi* is est imated by a hydrodynamical amplification factor that is related to a r igidy condition for the clusters. At sufficient large filler concentra tion for phi greater than or equal to phi*, the restricted mobility o f primary aggregates in the rubber matrix is insignificant for the clu ster 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 struc ture, a scaling invariant power law arises for the small-strain modulu s as function of carbon black concentration. The conductivity in the n etworking regime phi greater than or equal to phi* shows a typical po wer law behavior that is implied by an anomalous diffusion of the char ge carriers on the fractal clusters. For the frequency dependence of t he conductivity a cross-over to a power law regime at large frequencie s results. The predicted properties of carbon black filled rubbers are discussed in the framework of experimental results taken from differe nt authors.