Bethe lattice model with site and bond correlations for continuum percolation by isotropic systems of monodisperse rods

A model for connectedness percolation in isotropic systems of monodisperse cylinders is developed that employs a generalization of the tree-like Bethe lattice. The traditional Bethe lattice is generalized by incorporating (within a heuristic, mean-field framework) a pair of correlation parameters that describe (i) the states of occupancy of neighboring sites and (ii) the states of directly adjacent bonds, which are also allowed to be in either of two possible states. Averaging over the fluctuating states of neighboring bonds provides an operational means to modulate the dependence upon volume fraction of the average number of next-nearest-neighbor rod-rod contacts without altering the number of such nearest-neighbor interparticle contacts. The percolation threshold is shown to be a sensitive function of the average number of such next-nearest-neighbor contacts, and therefore of the quality of dispersion of the particles.

Figure 1

A schematic illustration of a portion of the treelike lattice employed by our model for the vertex degree $z=4$. Filled and open circles represent occupied and vacant sites, respectively. The solid and broken lines represent bonds corresponding to the correlation parameters $K_{\alpha }$ and $K_{\beta }$, respectively.

Figure 2

The volume fraction at the percolation threshold, ${\varphi }_c$, is shown as a function of $K_{\alpha }$ for monodisperse rods with $L/\phantom{LR}R=200$ and $\lambda /\phantom{\lambda R}R=0.1$. In all cases, $K_{\beta }$ is equal to unity. The solid and broken lines correspond to calculations performed for values of G equal to 0.01 (solid lines) and unity (broken lines), respectively. For each set of curves, solid and broken, ${\psi }_{\alpha }$ is equal to 0.01, 0.1, 0.5, and 0.9, from uppermost to lowermost curve.

Figure 3

The number of nearest-neighbor contacts per particle (or occupied site), $n_c$, and the number of such next-nearest-neighbor contacts, $nn{n}_c$, is shown as a function of the volume fraction. In all cases, $L/\phantom{LR}R=200$, $\lambda /\phantom{\lambda R}R=0.1$, $K_{\beta }=1$, $K_{\alpha }=0.25$, and ${\psi }_{\alpha }=0.1$. The solid lines indicate $nn{n}_c$ calculated from Eq. (9). For the solid lines, from top to bottom, G equals 0.01, 0.1, and unity, respectively. The broken line indicates $n_c$ calculated from Eq. (8). Results for $n_c$ for all values of $G$ superpose onto a single curve indicated by the broken line.

Figure 4

A schematic illustration of the effect of variation in the number of next-nearest-neighbor contacts upon rod clustering and percolation. In each of the panels (a) and (b), there are a total of 15 rods that form 4 distinct clusters. In each case there are a total (summed over all 15 rods) of 22 contacts between nearest-neighbor rods (this is the summation over rods of the numbers of such contacts for each rod). The aggregate number of next-neighbor contacts, similarly calculated as the summation over all rods of the number of such contacts per rod, equals 24 in panel (a) and 16 in panel (b).