Universal Critical Behavior of Percolation in Orientationally Ordered Janus Particles and Other Anisotropic Systems

We combine percolation theory and Monte Carlo simulation to study in two dimensions the connectivity of an equilibrium lattice model of interacting Janus disks which self-assemble into an orientationally ordered stripe phase at low temperature. As the patch size is increased or the temperature is lowered, clusters of patch-connected disks grow, and a percolating cluster emerges at a threshold. In the stripe phase, the critical clusters extend longer in the direction parallel to the stripes than in the perpendicular direction, and percolation is thus anisotropic. It is found that the critical behavior of percolation in the Janus system is consistent with that of standard isotropic percolation, when an appropriate spatial rescaling is made. The rescaling procedure can be applied to understand other anisotropic systems, such as the percolation of aligned rigid rods and of the $q$-state Potts model with anisotropic interactions.

Figure 1

Phase diagram of close-packed Janus disks in 2D. The system is ordered below the phase-transition line with squares, and it is percolated to the right side of the percolation-transition line with triangles. According to whether the system is ordered or percolated, the diagram is divided into four regions: an ordered and (not) percolated region, and a disordered and (not) percolated region. Inset: enlargement near $\theta /\pi =0.503$, as well as a snapshot of an ordered and percolated configuration, with the largest cluster being dark blue. Vertical dashed lines indicate the position of $\theta /\pi =1/2$ and the infinite-temperature percolation threshold ${\theta }_{\mathrm{p}}/\pi \simeq 0.628$ [32]. Lines going through data points are added to guide the eye, and the error bars are smaller than or comparable with the symbols.

Figure 2

Critical wrapping probabilities along the percolation line [i.e., solid curve with triangular points $T(\theta )$ in Fig. 1] of the Janus system. (a) Probability of wrapping in only one direction $R_1$ vs $T$. (b) Wrapping probability $R_2$ vs $T$. (c) Theoretical curve of $R_2$ vs the aspect ratio $\rho$ for standard percolation on parallelogram-shaped periodic lattices. (d) The effective aspect ratio ${\rho }_{\mathrm{e}}$ vs $T$ for the Janus system.

Figure 3

Relations between percolation in the stripe phase of the Janus system and standard site percolation on the triangular lattice. (a) A snapshot of the Janus system at ${\theta }_{\mathrm{p}}/\pi =0.503027(2)$, with $T=0.23$ and $L=256$, which is ordered in the parallel (horizontal) direction. The dark blue region represents the largest wrapping cluster, with its holes being light yellow. (b) Isotropic configuration after rescaling the parallel direction by $\sqrt{3}/(2{\rho }_{\mathrm{e}})$, with ${\rho }_{\mathrm{e}}\simeq 2.85$. (c) Collapse of critical correlations of the Janus system and those of site percolation on the triangular lattice with size $L_{\parallel }\times L=88\times 290$. The light gray line has a slope $-5/24$ from percolation universality. (d) Collapse of critical ratios $Q_1$ and $Q_s$ [78, 79] of the Janus system and those of site percolation with $\rho ={\rho }_{\mathrm{e}}$.

Figure 4

(a) ${\rho }_{\mathrm{e}}$ vs $k$ for percolation of aligned rigid rods. (b) and (c) Sketch of the isoradial mapping for anisotropic bond percolation on triangular and square lattices.