Percolation thresholds of randomly rotating patchy particles on Archimedean lattices

We study the percolation of randomly rotating patchy particles on $11$ Archimedean lattices in two dimensions. Each vertex of the lattice is occupied by a particle, and in each model the patch size and number are monodisperse. When there are more than one patches on the surface of a particle, they are symmetrically decorated. As the proportion $\chi$ of the particle surface covered by the patches increases, the clusters connected by the patches grow and the system percolates at the threshold $\chi_c$. We combine Monte Carlo simulations and the critical polynomial method to give precise estimates of $\chi_c$ for disks with one to six patches and spheres with one to two patches on the $11$ lattices. For one-patch particles, we find that the order of $\chi_c$ values for particles on different lattices is the same as that of threshold values $p_c$ for site percolation on same lattices, which implies that $\chi_c$ for one-patch particles mainly depends on the geometry of lattices. For particles with more patches, symmetry and shapes of particles also become important in determining $\chi_c$. We observe that $\chi_c$ values on different lattices do not follow the same order as those for one-patch particles, and get some rules, e.g., some patchy particle models are equivalent with site percolation on same lattices, and different patchy particles can share the same $\chi_c$ value on a given lattice. By analytically calculating probabilities of different patch-covering structures of a single particle as a function of $\chi$ near $\chi_c$, we provide some understanding of these results, and furthermore, we obtain $\chi_c$ for disks with an arbitrary number of patches on five lattices. For these lattices, when the number of patches increases, $\chi_c$ values for patchy disks repeat in periodic ways.