Percolation of k×k square tiles deposited under equilibrium conditions on square lattices.

Abstract

Numerical simulations and finite-size scaling analysis have been carried out to study the percolation behavior of square tiles of side k (k-tiles) on two-dimensional square lattices. The k-tiles, containing k×k identical units (each one occupying a lattice site), were reversibly adsorbed on the lattice. The process was monitored by following the probability R_{L,k}(θ) that a lattice composed of L×L sites percolates at a concentration θ of sites occupied by particles of side k. The classical percolation problem is recovered for k=1 giving the well-known site percolation threshold θ_{c,k=1}=0.592746⋯. A slight decrease is observed at the percolation threshold when k goes from 1 to 2, with θ_{c,k=2}=0.58483(9). For k≥2, the percolation threshold θ_{c,k} monotonically increases with k and asymptotically converges toward a definite value for large k-tiles θ_{c,k→∞}≈0.963. Accordingly, the model presents a percolation transition for the whole range of k. This behavior is completely different from that observed for the percolation problem of k×k square tiles irreversibly deposited on square lattices, where the percolation threshold is an increasing function of k in the range of 1≤k≤3. For k≥4, the percolation phase transition disappears. This stark contrast between the behaviors of reversible and irreversible adsorption models is a significant outcome of this research. Our findings could guide future studies exploring possible formation mechanisms in conductivity experiments of composite materials. Finally, the accurate determination of the critical exponents α, β, and γ, along with the measurement of the fractal dimension of the percolating cluster and the shortest-path exponent, indicates that although the deposition mechanism drastically affects the behavior of the percolation threshold with k, it does not alter the nature of the percolation transition occurring in the system. Accordingly, the universality class of the reversible adsorption model was found to be the same as for the random percolation model.