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

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 (particles occupying k consecutive sites along one of the lattice directions) 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 by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by randomly removing sets of k consecutive monomers (linear k-mers) from the lattice. Two schemes are used for the depositing/removing process: an isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction, and an anisotropic (perfectly oriented) 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 size k: (i) [(ii)] standard isotropic[oriented] percolation threshold theta(c,k) [v(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. theta(c,k) [v(c,k)] is reached by isotropic[oriented] deposition of straight rigid k-mers on an initially empty lattice; and (iii) [(iv)] inverse isotropic[oriented] percolation threshold theta(i)(c,k) [v(c,k)(i)], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. theta(i)(c,k) [v(c,k)(i)] is reached after removing isotropic [completely aligned] straight rigid k-mers from an initially fully occupied lattice. theta(c,k), v(c,k), theta(i)(c,k) and v(c,k)(i) are determined for a wide range of k (2 <= k <= 512). The obtained results indicate that (1) theta(c,k) [theta(i)(c,k)] exhibits a nonmonotonous dependence on the size k. It decreases[increases] for small particle sizes, goes through a minimum[maximum] at around k = 11, and finally increases and asymptotically converges towards a definite value for large segments theta(c,k ->infinity) = 0.500(2) [theta(i)(c,k)(->infinity) = 0.500(1)]; (2) v(c,k)[v(c,k)(i)] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particle sizes and asymptotically converges towards a definite value for infinitely long k-mers v(c,k ->infinity) = 0.5334(6) [v(c,k ->infinity)(i) = 0.4666( 6)]; (3) for both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line theta(v) = 0.5. Thus a complementary property is found theta(c,k) + theta(i)(c,k) = 1 (and v(c,k) + v(c,k)(i) = 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 theta space in a percolating region and a nonpercolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for the whole range of k.

Automated summary

Numerical simulations and finite-size scaling analysis were conducted to study the percolation behaviors of straight rigid rods on triangular lattices, including standard and inverse percolation. The research found that percolation thresholds are dependent on the particle size k, exhibit symmetry, and have different characteristics in various models.