Dynamic thresholds and relaxation effects in AC percolation of random resistor-capacitor networks

Abstract

Abstract

Abstract Most random resistor–capacitor (RC) network models of AC percolation represent the conducting phase by pure resistors and the insulating phase by ideal capacitors, neglecting the intrinsic relaxation times of the constituent phases. We study a two-dimensional conductor–dielectric random bond network in which both phases are parallel RC elements, with relaxation times τc and τd . The frequency-dependent impedance of each realisation is computed exactly with the Frank–Lobb reduction, validated against an analytical identity and a direct Kirchhoff solution. The sample-to-sample fluctuations of the maximum imaginary impedance, evaluated separately for spanning and non-spanning realisations, display two maxima located symmetrically about the bond percolation threshold pc = 1/2, at pcd = 0.4885(17) and pcc = 0.5115(27) for a 100 × 100 network. For τc = τd these fluctuations are shown to be identical to those of the DC resistance: the double-peak structure is therefore purely geometric, its symmetry follows from the duality of the square lattice, and the separation of the two maxima decreases with system size. A systematic scan over 10−12 ≤ τc /τd ≤ 107 shows that the relaxation-time contrast leaves the dielectric-side maximum and the positions of both maxima unchanged. It enhances the conductor-side maximum by up to 19% and makes the characteristic frequency fluctuate, but only when τc /τd ≲ Rb /Rd , where Rb is the resistance of the conducting backbone and Rd that of a dielectric bond. The model thus separates the geometric and the dynamical contributions to impedance fluctuations near the percolation threshold of conductor–dielectric composites.