Dispersive dielectric and conductive effects in 2D resistor–capacitor networks

Abstract

How to predict and better understand the effective properties of disordered materialmixtures has been a long-standing problem in different research fields, especially incondensed matter physics. In order to address this subject and achieve a betterunderstanding of the frequency-dependent properties of these systems, a large 2DL × L square structure of resistors and capacitors was used to calculate the immittance responseof a network formed by random filling of binary conductor/insulator phases with1000 Ω resistors and 10 nF capacitors. The effects of percolating clusters on the immittanceresponse were studied statistically through the generation of 10 000 different randomnetwork samples at the percolation threshold. The scattering of the imaginary part of theimmittance near the dc limit shows a clear separation between the responses of percolatingand non-percolating samples, with the gap between their distributions dependent onboth network size and applied frequency. These results could be used to monitorconnectivity in composite materials. The effects of the content and structure of thepercolating path on the nature of the observed dispersion were investigated, withspecial attention paid to the geometrical fractal concept of the backbone and itsinfluence on the behavior of relaxation-time distributions. For three differentresistor–capacitor proportions, the appropriateness of many fitting models wasinvestigated for modeling and analyzing individual resistor–capacitor networkdispersed frequency responses using complex-nonlinear-least-squares fitting. Severalremarkable new features were identified, including a useful duality relationship and theneed for composite fitting models rather than either a simple power law or asingle Davidson–Cole one. Good fits of data for fully percolating random networksrequired two dispersive fitting models in parallel or series, with a cutoff at shorttimes of the distribution of relaxation times of one of them. In addition, such fitssurprisingly led to cutoff parameters, including a primitive relaxation or crossover time,with estimated values comparable to those found for real dispersive materials.