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 behaviorof square tiles of side k (k-tiles) on two-dimensional square lattices. The k-tiles, containing k × k identicalunits (each one occupying a lattice site), were reversibly adsorbed on the lattice. The process was monitored byfollowing the probability RL,k (θ ) that a lattice composed of L × L sites percolates at a concentration θ of sitesoccupied by particles of side k. The classical percolation problem is recovered for k = 1 giving the well-knownsite percolation threshold θc,k=1 = 0.592746 . . . . A slight decrease is observed at the percolation threshold whenk goes from 1 to 2, with θc,k=2 = 0.58483(9). For k 2, the percolation threshold θc,k monotonically increaseswith k and asymptotically converges toward a definite value for large k-tiles θc,k→∞ ≈ 0.963. Accordingly, themodel presents a percolation transition for the whole range of k. This behavior is completely different fromthat observed for the percolation problem of k × k square tiles irreversibly deposited on square lattices, wherethe percolation threshold is an increasing function of k in the range of 1 k 3. For k 4, the percolationphase transition disappears. This stark contrast between the behaviors of reversible and irreversible adsorptionmodels is a significant outcome of this research. Our findings could guide future studies exploring possibleformation mechanisms in conductivity experiments of composite materials. Finally, the accurate determinationof the critical exponents α, β, and γ , along with the measurement of the fractal dimension of the percolatingcluster and the shortest-path exponent, indicates that although the deposition mechanism drastically affects thebehavior of the percolation threshold with k, it does not alter the nature of the percolation transition occurring inthe system. Accordingly, the universality class of the reversible adsorption model was found to be the same asfor the random percolation model.