Standard and inverse site percolation of straight rigid rods on triangular lattices: Isotropic and nematic deposition/removal

Abstract

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of straight rigid rods on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, linear k-mers (sets of k linear consecutive sites) are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then, the system is diluted by randomly removing linear k-mers from the lattice. Two schemes are used for the depositing/removing process: isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction; and nematic scheme, where one lattice direction is privileged for depositing (removing) the particles. The study is conducted by following the behavior of four critical concentrations with the size k: (i)[(ii)] standard isotropic[nematic] percolation threshold θ c,k [ϑ c,k ], which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ c,k [ϑ c,k ] is reached by isotropic[nematic] deposition of straight rigid k-mers on an initially empty lattice; and (iii)[(iv)] inverse isotropic[nematic] percolation threshold θ i c,k [ϑ i c,k ], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ i c,k [ϑ i c,k ] is reached after isotropically[nematically] removing straight rigid k-mers from an initially fully occupied lattice. θ c,k , ϑ c,k , θ i c,k and ϑ i c,k are determined for a wide range of k (2 ≤ k ≤ 512). The obtained results indicate that: (1) θ c,k [θ i c,k ] exhibits a non-monotonous dependence on the size k. It decreases[increases] for small particles sizes, goes through a minimum[maximum] around k = 11, and finally increases and asymptotically converges towards a definite value for large segments θ c,k→∞ = 0.500(2)[θ i c,k→∞ = 0.500(1)]; (2) ϑ c,k [ϑ i c,k ] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particles sizes and asymptotically converges towards a definite value for infinitely long k-mers ϑ c,k→∞ = 0.5334(6)[ϑ i c,k→∞ = 0.4666(6)];(3) for both isotropic and nematic models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ = 0.5. Thus, a complementary property is found θ c,k + θ i c,k = 1 (and ϑ c,k + ϑ i c,k = 1), which has not been observed in other regular lattices. This condition is analytically validated by using exact enumeration of configurations for small systems;and (4) in all cases, the critical concentration curves divide the θ-space in a percolating region and a non-percolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for th...