Transport process of ions in insulating media in the hyperbolic diffusion regime

We extend the microscopic Poisson-Nernst-Planck theory of the effects of mobile charge carriers, completely blocked at the electrodes, on the properties of dielectric materials by incorporating a finite speed of response propagation. The usual microscopic theory is based on the assumption that the diffusion current is given by Fick's law, relating the current density with the gradient of concentration at the same time. On the contrary we assume that the flux of diffusing particles is delayed with respect to the concentration gradient, as suggested by extended thermodynamics formulations. We show that, in the hyperbolic diffusion regime approximation, new trends for the real and imaginary parts of the small-signal electrical impedance of a dielectric containing ions versus the frequency of the applied voltage are expected when the delay time is comparable with the Debye relaxation time. In particular, at sufficiently high frequencies the real part of the conductivity, normally independent of frequency, decreases toward zero because of the finite, rather than the infinite propagation speed present in the hyperbolic diffusion regime model