Electrical conductance of two-dimensional random percolating networks based on mixtures of nanowires and nanorings: A mean-field approach along with computer simulation

We have studied the electrical conductance of two-dimensional (2D) random percolating networks of zero-width metallic nanowires (a mixture of rings and sticks). We took into account the nanowire resistance per unit length and the junction (nanowire-nanowire contact) resistance. Using a mean-field approximation (MFA) approach, we derived the total electrical conductance of these nanowire-based networks as a function of their geometrical and physical parameters. The MFA predictions have been confirmed by our Monte Carlo (MC) numerical simulations. The MC simulations were focused on the case when the circumferences of the rings and the lengths of the wires were equal. In this case, the electrical conductance of the network was found to be almost insensitive to the relative proportions of the rings and sticks, provided that the wire resistance and the junction resistance were equal. When the junction resistance dominated over the wire resistance, a linear dependency of the electrical conductance of the network on the proportions of the rings and sticks was observed.

Figure 1

Sample of randomly placed rings and sticks within a domain $L\times L$ with PBCs. All orientations of sticks are equiprobable. The domain size is $L=32$. The ring radius is $r=1$. The stick length is $l=2\pi r$. The number of rings is $N_{\text{r}}=200$. The number of sticks is $N_{\text{s}}=200$. Busbars are shown as the thick lines at top and bottom of the system. The effect of the PBCs is demonstrated using black color.

Figure 2

Dependency of the average number of contacts between a stick and a ring, $\langle k\rangle$, on the relative length of the stick $z$ [Eq. (14)].

Figure 3

Dependencies of the electrical conductance on the number density of fillers, $n=n_{\text{s}}+n_{\text{r}}$, when $n_{\text{r}}=0, n_{\text{s}}=n_{\text{r}}$, and $n_{\text{s}}=0$. The MFA predictions are shown as curves, while markers correspond to the simulations. Here, $l=2\pi r, r=1, R_{\text{j}}=1$, and ${\rho }_{\text{w}}={\left(2\pi r\right)}^{-1}$.

Figure 4

Dependencies of the electrical conductance on the proportions of rings and sticks, $x$, for the three different values of the number density of fillers, $n_{\text{s}}=xn$ and $n_{\text{r}}=(1-x)n$, when $n=2, n=5$, and $n=10$. MFA predictions and the least-squares fitting are shown as lines, while markers correspond to the simulations. Here, $l=2\pi r, r=1, R_{\text{j}}=1$, and ${\rho }_{\text{w}}={\left(2\pi r\right)}^{-1}$.

Figure 5

Dependencies of the electrical conductance on the number density of fillers, $n=n_{\text{s}}+n_{\text{r}}$, when $n_{\text{r}}=0, n_{\text{s}}=n_{\text{r}}, n_{\text{s}}=3n_{\text{r}}, n_{\text{r}}=3n_{\text{s}}$, and $n_{\text{s}}=0$. MFA predictions are shown as curves, while markers correspond to the simulations. Here, $l=2\pi r, r=1, R_{\text{j}}=1$, and ${\rho }_{\text{w}}={\left(2\pi r\right)}^{-1}{10}^{-6}$.

Figure 6

Dependencies of the electrical conductance on the proportions of rings and sticks, $x$, for the three different values of the number density of fillers, $n_{\text{s}}=xn$ and $n_{\text{r}}=(1-x)n$, when $n=2, n=5$, and $n=10$. MFA predictions and the least-squares fitting are shown as lines, while markers correspond to the simulations. For the two larger values of $n$, the difference between the MFA and LSF lines are not visually distinguishable. Here, $l=2\pi r, r=1, R_{\text{j}}=1$, and ${\rho }_{\text{w}}={\left(2\pi r\right)}^{-1}{10}^{-6}$.

Figure 7

Dependencies of the normalized electrical conductance on the proportions of rings and sticks, $x$, for the three different values of the number density of fillers, $n_{\text{s}}=xn$ and $n_{\text{r}}=(1-x)n$, when $n=2, n=5$, and $n=10$. The MFA predictions (34) are shown as a line, while markers correspond to the simulations. Here, $l=2\pi r, r=1, R_{\text{j}}=1$, and ${\rho }_{\text{w}}={\left(2\pi r\right)}^{-1}{10}^{-6}$.

Figure 8

Dependencies of the electrical conductance on the proportions of rings and sticks, $x$, for the fixed values of the number density of fillers and different values of $l$. MFA predictions are shown as curves, while markers correspond to the simulations. Here, $r=1, R_{\text{j}}=1$, and ${\rho }_{\text{w}}={\left(2\pi r\right)}^{-1}{10}^{-6}, n=6$.

Figure 9

Dependencies of the electrical conductance on the relative stick length, $z$, for the fixed values of the number density of fillers and different values of $x$. The MFA predictions are shown as curves, while the markers correspond to the simulations. Here, $r=1, R_{\text{j}}=1$, and ${\rho }_{\text{w}}={\left(2\pi r\right)}^{-1}{10}^{-6}, n=6$.