Dielectric Analysis of Carbon Black Networks

For a deeper understanding of structure-property relationships it is useful to consider the effect of carbon black grade and concentration as well as polymer type on the dielectric properties more closely. In Fig. 29 the real part of the a.c.-conductivity o at 20 °C of a series of rubber composites, consisting of the more polar statistical co-polymer NBR and the fine black N220, is depicted for various filler concentrations in the high frequency regime up to 1 GHz. For the lower carbon black concentrations, a power law behavior with exponent around 0.6 is observed, while the highly filled com- 

This behavior becomes more transparent in Fig. 30a,b, where the a.c.-con-ductivity a and relative dielectric constant (permittivity e ), respectively, for a series of less polar S-SBR-samples filled with various amounts of the coarse black N550 are show at 20 °C in a broader frequency range up to 107 Hz. For filler concentrations below the percolation threshold (O 0.15), the conductivity behaves essentially as that of an isolator and increases almost linearly with frequency. Above the percolation threshold (5 0.2), it shows a characteristic conductivity plateau in the small frequency regime. Since at low frequencies the value of the conductivity a agrees fairly well with the d.c.-con-ductivity, the plateau value exhibits the characteristic percolation behavior considered above. In the high frequency regime the conductivity depicted in 

The reduced value of the scaling exponent, observed in Fig. 29 and Fig. 30a for filler concentrations above the percolation threshold, can be related to anomalous diffusion of charge carriers on fractal carbon black clusters. It appears above a characteristic frequency (O when the charge carriers move on parts of the fractal clusters during one period of time. Accordingly, the characteristic frequency for the cross-over of the conductivity from the plateau to the power law regime scales with the correlation length E, of the filler network. 

For a quantitative analysis of the scaling and cross-over behavior of the a.c.-conductivity above the percolation threshold we refer to the predictions of percolation theory [128, 136, 137]  

For the evaluation of the front factor the Einstein equation for the conductivity a0 can be used. It yields 

---