High-Dilution Carbon-Black/Polymer Composites: Hierarchical Percolating Network Derived from Hz to THz ac Conductivity

We have studied conducting carbon-black (CB) composites with a percolation threshold $\left(p_c\right)$ of $3\times {10}^{-4\mathrm{}}$ in volume fraction. The low $p_c$ is explained in terms of a fractal distribution of CB aggregates. To prove this model we measured the frequency dependence of the complex conductivity (up to 0.6 THz) for various CB concentrations. Two successive scaling regimes are found, the first dominated by the infinite percolating cluster and the second by finite clusters. We extend the anomalous diffusion model to explain this observation and show the parameters found to be consistent with the proposed structure.