Explosive Electric Breakdown due to Conducting-Particle Deposition on an Insulating Substrate

We introduce a theoretical model to investigate the electric breakdown of a substrate on which highly conducting particles are adsorbed and desorbed with a probability that depends on the local electric field. We find that, by tuning the relative strength $q$ of this dependence, the breakdown can change from continuous to explosive. Precisely, in the limit in which the adsorption probability is the same for any finite voltage drop, we can map our model exactly onto the $q$-state Potts model and thus the transition to a jump occurs at $q=4$. In another limit, where the adsorption probability becomes independent of the local field strength, the traditional bond percolation model is recovered. Our model is thus an example of a possible experimental realization exhibiting a truly discontinuous percolation transition.

Figure 1

Plot showing a typical state of the system for $q=10$, $L=128$, $\gamma =0.1$, and $p=0.57$. A potential drop $V_0$ is applied from top to bottom and periodic boundary conditions from left to right. The nodes are colored according to their corresponding potential, with the constraint that nodes belonging to the same metallic cluster hold the same potential. Nodes with high potential are shown in red (top), while low potential ones are presented in blue (bottom). The boundaries of several metallic clusters are highlighted for better visualization.

Figure 2

(a) Fraction $P$ of nodes in the largest cluster with the number of iterations $n$ (sampled bonds) for $q=10$, $L=128$, $\gamma =0.1$, and distinct values of $p$ below (black), above (green), and at the transition point $p_c=0.578$ (red). (b) The same as in (a) but for $\gamma =1$ and $p_c=0.5$. The histograms for $P$ at the transition point are shown in (c) and (d) for $\gamma =0.1$ and $\gamma =1$, respectively. For $\gamma =0.1$, the histogram is bimodal and the evolution is characterized by metastability, two typical signs of a discontinuous transition. By contrast, for $\gamma =1$ the transition is continuous since the histogram is unimodal and there is no evidence of metastability. Error bars show the standard error within each bin of the histogram.

Figure 3

Snapshots of steady state configurations for $q=10$, $L=128$, $\gamma =0.1$, with (a) $p=0.57$, (b) $p=0.58$, and (c) $p=0.59$, and for $\gamma =1$ with (d) $p=0.49$, (e) $p=0.5$, and (f) $p=0.51$. Metallic bonds belonging to the largest cluster are colored in black, while the bonds belonging to the other metallic cluster appear in red. For $\gamma =0.1$, the largest metallic cluster is compact and grows abruptly at the threshold, $p_c\approx 0.58$, while for $\gamma =1$ it is fractal and grows rather smoothly with $p$.

Figure 4

Dependence on $p$ the average fraction $⟨P⟩$ of nodes in the largest cluster for $\gamma =0$, 0.1, and 1, for $q=10$ (a) and $q=2$ (b), with $L=128$. The transition point for a given $\gamma$ and $q$ is bounded between the critical point for bond percolation, $p_c(q=1,\gamma =0)=p_c(q,\gamma =\infty )=1/2$ (dotted lines), and the critical point for the $q$-state Potts model, given by Eq. (2), where $p_c(q=2,\gamma =0)\approx 0.5858$ (dashed line), and $p_c(q=10,\gamma =0)\approx 0.7597$ (dotted-dashed line). Error bars show the standard error for a given value of $p$.

Figure 5

(a) Histogram of the fraction $P$ of nodes in the largest metallic cluster for different system sizes $L$ at $p_c$ for $q=10$ and $\gamma =0.1$. In the inset of (a), $\mathrm{\Delta }$ is the difference between the two peaks in the histogram. One clearly sees that $\mathrm{\Delta }$ does not decrease with the system size, as one would expect for a continuous transition. For every $L$, $p_c(L)$ is obtained as the value of $p$ that maximizes the derivative of $P$, $dP/dp$. As shown in (b), $p_c$ can then be extrapolated to the thermodynamic limit through $p_c=0.5671+1.5463L^{-1}$. The error bars shown in (a) are the standard error within each bin of the histogram while in (b) they represent the error of the finite difference approximation to compute the derivative of $p$.