Conductivity Of Random Resistor Network Research Articles

Effective electrical conductivity of random resistor networks generated using a Poisson–Voronoi tessellation

We studied the effective electrical conductivity of dense random resistor networks (RRNs) produced using a Voronoi tessellation when its seeds are generated by means of a homogeneous Poisson point process in the two-dimensional Euclidean space. Such RRNs are isotropic and in average homogeneous; however, local fluctuations of the number of edges per unit area are inevitable. These RRNs may mimic, e.g., crack-template-based transparent conductive films. The RRNs were treated within a mean-field approach. We found an analytical dependency of the effective electrical conductivity on the number of conductive edges (resistors) per unit area, nE. The effective electrical conductivity is proportional to nE when nE≫1.

Electrical conductivity of nanorod-based transparent electrodes: Comparison of mean-field approaches.

We mimic nanorod-based transparent electrodes as random resistor networks (RRNs) produced by the homogeneous, isotropic, and random deposition of conductive zero-width sticks onto an insulating substrate. We suppose that the number density (the number of objects per unit area of the surface) of these sticks exceeds the percolation threshold, i.e., the system under consideration is a conductor. We computed the electrical conductivity of random resistor networks versus the number density of conductive fillers for the wire-resistance-dominated case, for the junction-resistance-dominated case, and for an intermediate case. We also offer a consistent continuous variant of the mean-field approach. The results of the RRN computations were compared with this mean-field approach. Our computations suggest that, for a qualitative description of the behavior of the electrical conductivity in relation to the number density of conductive wires, the mean-field approximation can be successfully applied when the number density of the fillers n>2n_{c}, where n_{c} is the percolation threshold. However, note the mean-field approach slightly overestimates the electrical conductivity. We demonstrate that this overestimate is caused by the junction potential distribution.

Transparent electrodes with nanorings: A computational point of view

Four samples of transparent conductive films with different numbers of silver nanorings per unit area were produced. The sheet resistance, transparency, and haze were measured for each sample. Using Monte Carlo simulation, we studied the electrical conductivity of random resistor networks produced by the random deposition of the conducting rings onto the substrate. Both systems of equal-sized rings and systems with rings of different sizes were simulated. Our simulations demonstrated the linear dependence of the electrical conductivity on the number of rings per unit area. Size dispersity decreased the percolation threshold, but without having any other significant effect on the behavior of the electrical conductance. Analytical estimations obtained for dense systems of equal-sized conductive rings were consistent with the simulations.

Conductivity of stick percolation clusters with anisotropic alignments

The conductivity of random resistor networks composed from percolating clusters of two-dimensional (2D) stick systems with anisotropic alignments is analyzed by using a finite-size scaling analysis for comparison to the conductivity of single-walled carbon-nanotube bundle film networks. For the conductivity analysis, we first calculate the critical properties of the percolation transition of 2D stick systems with anisotropic alignments. Even though the percolation transition stick density increases rapidly as the anisotropy is enhanced, the conductivity and the critical properties hardly vary. The resultant conductivity exponent of the stick networks at the percolation threshold is nearly the same as that of the lattice critical percolation clusters regardless of the anisotropy and the resistance ratio r = Rjct/RNT, where Rjct is the stick-to-stick junction resistance and RNT is the resistance of a stick.

Anisotropic generalization of Stinchcombe's solution for the conductivity of random resistor networks on a Bethe lattice

Our study is based on the work of Stinchcombe (1974 J. Phys. C: Solid State Phys. 7 179) and is devoted to the calculations of average conductivity of random resistor networks placed on an anisotropic Bethe lattice. The structure of the Bethe lattice is assumed to represent the normal directions of the regular lattice. We calculate the anisotropic conductivity as an expansion in powers of the inverse coordination number of the Bethe lattice. The expansion terms retained deliver an accurate approximation of the conductivity at resistor concentrations above the percolation threshold. We make a comparison of our analytical results with those of Bernasconi (1974 Phys. Rev. B 9 4575) for the regular lattice.

Elastic properties of randomly cross-linked polymers

We have carried out extensive molecular-dynamics simulations of randomly cross-linked polymers and studied the onset of rigidity as the number of cross-links is increased. We find that for our systems, consisting of chains of length N=10, 20, 30, and 50, the shear modulus E vanishes at a concentration of cross-links that is well above the geometric percolation threshold and that it seems to approach zero as E\ensuremath{\sim}(n-${\mathit{n}}_{\mathit{c}}$${)}^{\mathit{f}}$, where the exponent f is considerably smaller than either the classical value f=3 or the corresponding exponent t\ensuremath{\approxeq}2.0 of the conductivity of random resistor networks. \textcopyright{} 1996 The American Physical Society.

Conductivity of random resistor networks: An investigation of the accuracy of the effective-medium approximation.

We investigate the electrical conductivity of disordered random resistor networks. A systematic perturbative weak-disorder expansion of the conductivity is derived, in terms of the moments of the probability distribution of the random-bond conductances. This diagrammatic technique applies to any distribution of (possibly complex) conductances, and any regular lattice of arbitrary dimensionality d. Explicit quantitative results are given, up to the sixth order of perturbation theory, for (hyper)cubic lattices in all dimensionalities of physical interest, and compared with the predictions of the effective-medium approximation (EMA). The conductivity generically departs from the EMA formula at the fourth order of perturbation theory. On the square lattice, due to the duality symmetry, the EMA starts to be incorrect only at the fifth order. In all the situations considered, the discrepancy between the EMA prediction and the exact conductivity is affected by a very small numerical factor. The limit of a large dimensionality d is also investigated. The conductivity is shown to have a systematic 1/d expansion, three terms of which are given explicitly. The discrepancy with the EMA formula is again very weak, and starts with the 1/${\mathit{d}}^{3}$ terms. This study yields, therefore, a quantitative understanding of the currently observed fact that the EMA prediction is very accurate for a large class of conductance distributions.

Flory theory for conductivity of random resistor networks

Theorie de Flory pour le probleme de la conductivite d'un reseau de resistances aleatoires de dimensions d. On montre que l'exposant t~ de la conductivite est relie a la dimension fractale «df» en accord avec la conjecture d'Alexander et d'Orbach t~=d−2+df/2 ou selon la theorie de Flory df=(d+2)/2 a pour la percolation et df=2(d+2)/5 pour «les animaux» sur reseau. L'accord entre cette theorie et les estimations numeriques de «t~» est excellent pour la percolation et satisfaisant pour les «animaux»

A transfer matrix program to calculate the conductivity of random resistor networks

We describe technical details on a new method of calculating the conductivity of random resistor networks which uses transfer matrix ideas. We give a program which calculates the conductivity of three-dimensional bars, and we provide a few comments on the advantages of this method and its performances.