Continuum percolation of congruent overlapping spherocylinders

Continuum percolation of randomly orientated congruent overlapping spherocylinders (composed of cylinder of height $H$ with semispheres of diameter $D$ at the ends) with aspect ratio $\alpha =H/D$ in $[0,\infty )$ is studied. The percolation threshold ${\varphi }_c$, percolation transition width Δ, and correlation-length critical exponent $\nu$ for spherocylinders with $\alpha$ in [0, 200] are determined with a high degree of accuracy via extensive finite-size scaling analysis. A generalized excluded-volume approximation for percolation threshold with an exponent explicitly depending on both aspect ratio and excluded volume for arbitrary $\alpha$ values in $[0,\infty )$ is proposed and shown to yield accurate predictions of ${\varphi }_c$ for an extremely wide range of $\alpha$ in [0, 2000] based on available numerical and experimental data. We find ${\varphi }_c$ is a universal monotonic decreasing function of $\alpha$ and is independent of the effective particle size. Our study has implications in percolation theory for nonspherical particles and composite material design.

Figure 1

(a) Realization of a congruent overlapping spherocylinder system with the aspect ratio of $\alpha =8$; (b) identification of a percolating cluster in the system.

Figure 2

Percolation probability $P$ for different sizes $L$ of cubic domain as a function of $\varphi$ for congruent overlapping spherocylinders. The parameters of the particles are $R_{\mathrm{eq}}=1.0$ and $\alpha =100.0$. The cubic symbols are MC simulation results and the lines are fitting results. The percolation probability curves for various $\alpha$ are displayed in Fig. 3.

Figure 3

Percolation probability $P(\varphi ,L)$ versus $\varphi$ for different $\alpha$ and system sizes of $L$. The cubic symbols are MC simulation results and the lines are fitting results. The parameters of the particles are $R_{\mathrm{eq}}=1.0$.

Figure 4

(a) The percolation transition width $\mathrm{\Delta }$ for different $\alpha$ as a function of system size $L$. (b) The effective percolation threshold ${\varphi }_c\left(L\right)$ for different $\alpha$ as a function of $L^{-1/\nu }$, with linear fits to the data.

Figure 5

The percolation threshold ${\varphi }_c$ for different $\alpha$. The cubic symbols are the present MC simulation results, the solid lines are theoretical bounds for the percolation threshold of congruent overlapping spherocylinders derived by Torquato and Jiao [16], and the dashed red line is the approximate solution for the percolation threshold reported by Chatterjee [11].

Figure 6

The effect of aspect ratios $\alpha$ on the exponent $C$ in the proposed theoretical model.

Figure 7

Comparisons of the proposed empirical approximation with (a) the numerical and experimental results from Mutiso and coworkers [36, 37], and with (b) the numerical results from Schilling et al. [19].

Figure 8

Percolation probability $P(\varphi ,L)$ versus $\varphi$ for different $R_{\mathrm{eq}}$ and system sizes of $L$. The parameters of the particles are $\alpha =8.0$.

Figure 9

The effective percolation threshold ${\varphi }_c\left(L\right)$ as a function of $L^{-1/\nu }$ for different $R_{\mathrm{eq}}$, with accompanying linear fits to the data.