Static dielectric function and scaling of the ac conductivity for universal and nonuniversal percolation systems

Abstract

Experimental results and simulations of scaled plots of the normalized conductivity $[\mathrm{log}\mathbf{(}{\ensuremath{\sigma}}_{\mathrm{mr}}(\ensuremath{\phi},\ensuremath{\omega},T)∕{\ensuremath{\sigma}}_{\mathrm{mr}}(\ensuremath{\phi},0,T)\mathbf{)}]$ against the scaled frequency [$\mathrm{log}(\ensuremath{\omega}∕{\ensuremath{\omega}}_{\mathrm{ce}})$ or $\mathrm{log}(\ensuremath{\omega}∕{\ensuremath{\omega}}_{\mathrm{cp}})$], for different conductor volume fractions $\ensuremath{\phi}$, for various percolation systems are examined and analyzed. Here, ${\ensuremath{\omega}}_{\mathrm{ce}}$ is the critical scaling frequency obtained from superimposing experimental results, and ${\ensuremath{\omega}}_{\mathrm{cp}}$ is the $\ensuremath{\omega}$ value at the peak of the imaginary impedance against frequency curve, which is shown to be a valid scaling frequency. The values obtained for the high frequency slopes, as well as ${\ensuremath{\omega}}_{\mathrm{ce}}$ and ${\ensuremath{\omega}}_{\mathrm{cp}}$, of all experimental scaling curves are not in agreement with the widely accepted predictions of percolation theory, which incorporate universality relations. It is proposed that this is due to the behavior of the zero frequency (static) dielectric constant $[{ϵ}_{\mathrm{mr}}(\ensuremath{\phi},0,T)]$ that appears in the equations for the critical scaling frequencies (${\ensuremath{\omega}}_{\mathrm{ce}}$ and ${\ensuremath{\omega}}_{\mathrm{cp}}$) for each $\ensuremath{\phi}$ value. Using an original method, ${ϵ}_{\mathrm{mr}}(\ensuremath{\phi},0,T)$ is calculated for both the simulated and experimental results. The unexpected behavior observed for the nonuniversal experimental $[{ϵ}_{\mathrm{mr}}(\ensuremath{\phi},0,T)]$ results is found to be in qualitative agreement with the simulation results made using the two exponent phenomenological percolation equation when the ratios of the real conductivities of the components are not zero (in practice, ${\ensuremath{\sigma}}_{\mathrm{ir}}∕{\ensuremath{\sigma}}_{\mathrm{cr}}\ensuremath{\leqslant}{10}^{\ensuremath{-}18}$). Experimental results for universal conductivity systems, with different compositions or at different temperatures, can also be scaled onto master curves using the same procedures, as for percolation systems. The similarities and differences between these experimental results are qualitatively discussed.