Irreversible bilayer adsorption of straight semirigid rods on two-dimensional square lattices: Jamming and percolation properties

Inverse percolation by removing straight semirigid rods from bilayer square lattices

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing semirigid rods from a L×L square lattice that contains two layers (and M=L×L×2 sites). 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 removing groups of k consecutive monomers according to a generalized random sequential adsorption mechanism. The study is conducted by following the behavior of two critical concentrations with size k: (1) jamming coverage θ_{j,k}, which represents the concentration of occupied sites at which the jamming state is reached, and (2) inverse percolation threshold θ_{c,k}, which corresponds to the maximum concentration of occupied sites for which connectivity disappears. The obtained results indicate that (1) the jamming coverage exhibits an increasing dependence on the size k-it rapidly increases for small values of k and asymptotically converges towards a definite value for infinitely large k sizes θ_{j,k→∞}≈0.2701-and (2) the inverse percolation threshold is a decreasing function of k in the range 1≤k≤17. For k≥18, all jammed configurations are percolating states (the lattice remains connected even when the highest allowed concentration of removed sites is reached) and, consequently, there is no nonpercolating phase. This finding contrasts with the results obtained in literature for a complementary problem, where straight rigid k-mers are randomly and irreversibly deposited on a square lattice forming two layers. In this case, percolating and nonpercolating phases extend to infinity in the space of the parameter k and the model presents percolation transition for the whole range of k. The results obtained in the present study were also compared with those reported for the case of inverse percolation by removal of rigid linear k-mers from a square monolayer. The differences observed between monolayer and bilayer problems were discussed in terms of vulnerability and network robustness. Finally, the accurate determination of the critical exponents ν, β, and γ reveals that the percolation phase transition involved in the system has the same universality class as the standard percolation problem.

Random adsorption process of linear k-mers on square lattices under the Achlioptas process

We study the explosive percolation with k-mer random sequential adsorption (RSA) process. We consider both the Achlioptas process (AP) and the inverse Achlioptas process (IAP), in which giant cluster formation is prohibited and accelerated, respectively. By employing finite-size scaling analysis, we confirm that the percolation transitions are continuous, and thus we calculate the percolation threshold and critical exponents. This allows us to determine the universality class of the k-mer explosive percolation transition. Interestingly, the numerical simulation suggests that the universality class of the explosive percolation transition with the AP alters when the k-mer size changes. In contrast, the universality class of the transition with the IAP is independent of k, but it differs from that of the RSA without the IAP.

Surface growth during random and irreversible multilayer deposition of straight semirigid rods

Surface growth properties during irreversible multilayer deposition of straight semirigid rods on linear and square lattices have been studied by Monte Carlo simulations and analytical considerations. The filling of the lattice is carried out following a generalized random sequential adsorption mechanism where the depositing objects can be adsorbed on the surface forming multilayers. The results of our simulations show that the roughness evolves in time following two different behaviors: an "homogeneous growth regime" at initial times, where the heights of the columns homogeneously increase, and a "segmented growth regime" at long times, where the adsorbed phase is segmented in actively growing columns and inactive nongrowing sites. Under these conditions, the surface growth generated by the deposition of particles of different sizes is studied. At long times, the roughness of the systems increases linearly with time, with growth exponent β=1, at variance with a random deposition of monomers which presents a sublinear behavior (β=1/2). The linear behavior is due to the segmented growth process, as we show using a simple analytical model.