Electrical conductivity of randomly placed linear wires: a mean field approach

Abstract

Using the mean-field approximation, a formula for the effective electrical conductivity of a two-dimensional system of randomly arranged conducting sticks with a given orientation distribution was obtained. Both the resistance of the sticks themselves and the resistance of the contacts between them were taken into account. The accuracy in the resulting formula was analyzed. A comparison of the theoretical predictions of mean-field approach with the results of direct electrical conductivity calculations for several model orientation distributions describing systems with crossed sticks demonstrated good agreement. Our study showed that cross-alignment of nanowires should lead to a decreasing in the electrical conductivity compared to electrodes with isotropically arranged nanowires. We suppose that the widely used model with zero-width sticks is quite acceptable for systems of cross-aligned nanowires, but overestimates their connectivity in isotropic systems. Thus, the enhancement of the electrical conductivity of conducting films with cross-aligned nanowires may be due to a significant difference in the network topology.

Figures (10)

• Figure 1: Different ODFs used in our study. Only one part of each distribution is shown. Another part of the same shape is located around the value $\pi/2$. Dash-dotted curve corresponds to \ref{['eq:ODFGauss']}. Solid lines correspond to \ref{['eq:ODF3']}. Dashed curve corresponds to \ref{['eq:ODF4']}.
• Figure 2: The dimensionless electrical conductivity vs the number density of conductive sticks for two different linear sizes of the system under consideration ($L=8$ and $L=32$ while $\Delta = 10^{-6}$) and for two different values of $\Delta$ ($\Delta = 10^{-6}$ and $\Delta = 10^{-3}$ while $L=32$). One half of the sticks is oriented along the $x$ axis, while the other half is oriented along the $y$ axis.
• Figure 3: Examples of backbone fractions obtained using \ref{['eq:BB']} for the two limiting cases, namely, for the isotropic distribution \ref{['eq:ODF1']} and for the perfect cross \ref{['eq:ODF3']}.
• Figure 4: The dimensionless electrical conductivity of a system which obeys the ODF \ref{['eq:ODF2']} ($\omega=0.5$) vs the number density of conductive wires for the three different values of $\Delta$. Filled markers correspond to results published in Grazioli2025, open markers correspond to our direct computations, while curves present the MFA prediction \ref{['eq:MFAsigma-RC0']}.
• Figure 5: The dimensionless electrical conductivity of a system which obeys the ODF \ref{['eq:ODF3']} vs the number density of conductive wires for the three different values of $\Delta$. $\left \langle S^2 \right\rangle = 0.818$ when $\varepsilon = \pi/8$, $\left \langle S^2 \right\rangle = 0.687$ when $\varepsilon = \pi/32$. Open markers correspond to the direct computations, while curves present the MFA prediction \ref{['eq:MFAsigma-RC0']}. For purpose of comparison, results published in Grazioli2025 presented by filled markers.