Explosive percolation in a nanotube-based system

Using the percolation theory, we study the underlying mechanism in the formation of single-walled nanotube bundles with uniform diameter. By applying the cluster repulsion process to stick percolation, we find that the transition becomes explosive. To understand the transition nature, we first investigate the scaling behavior of transition interval $\Delta$. By comparing the results with loopless and loop-allowed bond percolations, we find that the loops crucially affect the scaling behavior of $\Delta$, and $\Delta$ is not universal. Moreover, the scaling behavior of $\Delta$ for the present nanostick systems is the same as that for loopless bond percolation. For more systematic studies on the transition nature, we also measure the changes in order parameter during the stick removal process and show that there exists a hysteresis. The results more clearly show that the transition of the stick system with cluster repulsion is discontinuous.

Figure 1

(Color online) (a) and (b) show the snapshots of the largest cluster for OP. (c) and (d) show those for AP. (a) and (c) are the snapshots at the threshold of percolation transition. (b) and (d) are the snapshots when 200 sticks are added to (a) and (c), respectively. The green (light gray) sticks denote those that form the spanning (largest) cluster. Red (dark gray) sticks are those in the small clusters. (e) Schematic diagram for a loop. Black sticks are connected back to themselves which form a closed loop. (f) Enlarged plot of the dashed square in (a).

Figure 2

(Color online) Plot of $P_{\infty }$ against $N$ when $L=40$. The (black) squares represent $P_{\infty }$ for the OP and the (red) circles denote $P_{\infty }$ under AP.

Figure 3

(Color online) $\Delta /N_0$ for various $L$’s. (a) The (black) squares represent the data obtained from the ordinary stick percolation. The (red) circles display the data for the stick systems with AP. The (black) solid line represents the relation $\Delta /N_0\sim L^{0.40}$ and the (red) dashed line denotes $\Delta /N_0\sim L^{0.65}$. (b) $\Delta /N_0$ for lattice bond percolation, in which the formation of loops is allowed. In the ordinary bond percolation we obtain $\Delta /N_0\sim L^{0.29}$ [solid line in (b)]. For AP with loops in the lattice percolation, we obtain $\Delta /N_0\sim L^{0.43}$ [dashed line in (b)].

Figure 4

(Color online) Hysteresis of $P_{\infty }$ for $L=40$. Black squares represent $P_{\infty }$ when we add sticks and red (gray) circles denote $P_{\infty }$ when we remove sticks. Inset: the same plot for $L=10$.