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

Abstract

How to predict and better understand the effective properties of disordered material mixtures has been a long-standing problem in different research fields, especially in condensed matter physics. In order to address this subject and achieve a better understanding of the frequency-dependent properties of these systems, a large 2D L × L square structure of resistors and capacitors was used to calculate the immittance response of a network formed by random filling of binary conductor/insulator phases with 1000 Ω resistors and 10 nF capacitors. The effects of percolating clusters on the immittance response were studied statistically through the generation of 10 000 different random network samples at the percolation threshold. The scattering of the imaginary part of the immittance near the dc limit shows a clear separation between the responses of percolating and non-percolating samples, with the gap between their distributions dependent on both network size and applied frequency. These results could be used to monitor connectivity in composite materials. The effects of the content and structure of the percolating path on the nature of the observed dispersion were investigated, with special attention paid to the geometrical fractal concept of the backbone and its influence on the behavior of relaxation-time distributions. For three different resistor-capacitor proportions, the appropriateness of many fitting models was investigated for modeling and analyzing individual resistor-capacitor network dispersed frequency responses using complex-nonlinear-least-squares fitting. Several remarkable new features were identified, including a useful duality relationship and the need for composite fitting models rather than either a simple power law or a single Davidson-Cole one. Good fits of data for fully percolating random networks required two dispersive fitting models in parallel or series, with a cutoff at short times of the distribution of relaxation times of one of them. In addition, such fits surprisingly led to cutoff parameters, including a primitive relaxation or crossover time, with estimated values comparable to those found for real dispersive materials.

Understanding the anomalous frequency responses of composite materials using very large random resistor-capacitor networks

“…An example of this is shown in Figures 3e-3h, where one bond is missing [20]. The final result of this sequence of transformations, as in Figure 4, is to reduce the lattice to a single bond that has the same admittance (Fig.…”

“…An example network with S = 4 and thus N = 32 components is shown in Figure 2a. The total admittance Y (ω) of this network is obtained through the equivalent impedance Z eq (ω) calculated using the Frank and Lobb [20] reduction scheme. The method of Frank and Lobb is based on transformation of star-delta and delta-star, as shown in Figure 2b, and the transformation is defined in both directions.…”

“…This has limited the possible insights on aspects of the influence of RC network size on the low and high frequency distribution of AC conductivity, the statistical variation of conductivity with network size and finite size effects on percolation. As an example, Almond et al [8] examined networks where the number of components (N ) was approaching 10 000 and Hamou et al provided a detailed study on large numbers of networks with up to 500 000 components [20]. The influence of network size is of interest to understand the influence of scale in real materials or composite systems; for example bulk materials can be effectively considered as infinitely large networks while composite thin films at the micrometer scale containing nanometer sized inclusions are effectively smaller finite electrical networks.…”

Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered Double Hydroxide Using Impedance Spectroscopy

“…The different kinetic models including pseudo-first-order, pseudo-secondorder, and intraparticle diffusion are employed to investigate the mechanism of adsorption and potential rate controlling steps such as chemical reaction mass transport and diffusion control processes [28]. The pseudo-first-order and pseudo-second-order are generally expressed as Eqs.…”

The effective conductivity of arrays of squares: Large random unit cells and extreme contrast ratios

“…All these difficulties add up and may manifest themselves as artificial ill-conditioning, slow convergence with mesh refinement, critical slowing down in iterative solvers, and severe loss of precision. Several methods have been suggested to alleviate these problems including variants of the finite element method [1,6], network models [23,12], renormalization schemes [20], mode-matching methods [27], and Brownian motion simulation [21]. See also Section 3 of [26] for state-of-the-art algorithms to combat critical slowing down in network models and [8] for a discussion of future directions in the research field at large.…”