Percolation of binary disk systems: Modeling and theory

The dispersion and connectivity of particles with a high degree of polydispersity is relevant to problems involving composite material properties and reaction decomposition prediction and has been the subject of much study in the literature. This work utilizes Monte Carlo models to predict percolation thresholds for a two-dimensional systems containing disks of two different radii. Monte Carlo simulations and spanning probability are used to extend prior models into regions of higher polydispersity than those previously considered. A correlation to predict the percolation threshold for binary disk systems is proposed based on the extended dataset presented in this work and compared to previously published correlations. A set of boundary conditions necessary for a good fit is presented, and a condition for maximizing percolation threshold for binary disk systems is suggested.

Figure 1

Convergence of the estimated critical filling factor for monodisperse disks. Each data point represents a different relative system size $\left(R/L\right)$. The fitted line to the data gives an intercept—or predicted percolation threshold for an infinite system—of 1.128, which demonstrates reasonably good agreement with the published percolation threshold value for monodisperse disks.

Figure 2

Comparison of results generated here with those published by Quintanilla. The lines correspond to the points generated in this work while the x markers denote the corresponding results published in [24]. Very good agreement is demonstrated.

Figure 3

Percolation threshold for binary dispersions of disks, presented as a contour plot. Each line represents a constant value of percolation thresholds. The dashed line closely tracks the location of the maximum percolation threshold.

Figure 4

Number density of large disks $(1-f)$ at the maximum percolation threshold, as a function of $\lambda$. Location of maximum percolation threshold as predicted by Monte Carlo simulation, the $v$-for-$f$ substitution and average excluded area predictions, the empirical fit proposed by Quintanilla [24], and the correlation proposed in this work.

Figure 5

Percolation threshold predicted by size-density correlation given by Eqs. (21, 22, 23). Contour plot for all considered number fractions and radii ratios. This plot demonstrates a high degree of similarity to the simulation data presented in Fig. 3

Figure 6

The maximum percolation threshold predicted for binary disk dispersions for both the simulation data and the proposed correlation given by Eqs. (21, 22, 23). Very good agreement is demonstrated, and the values predicted by the proposed correlation match the simulation data to within 2%.