Jamming and percolation for deposition of k2-mers on square lattices: A Monte Carlo simulation study

Abstract

Percolation and jamming of k×k square tiles (k2-mers) deposited on square lattices have been studied by numerical simulations complemented with finite-size scaling theory and exact enumeration of configurations for small systems. The k2-mers were irreversibly deposited into square lattices of sizes L×L with L/k ranging between 128 and 448 (64 and 224) for jamming (percolation) calculations. Jamming coverage θj,k was determined for a wide range of k values (2≤k≤100 with many intermediate k values to allow a fine scaling analysis). θj,k exhibits a decreasing behavior with increasing k, being θj,k= 0.5623(3) the limit value for large k2-mer sizes. In addition, a finite-size scaling analysis of the jamming transition was carried out, and the corresponding spatial correlation length critical exponent νj was measured, being νj≈1. On the other hand, the obtained results for the percolation threshold θc,k showed that θc,k is an increasing function of k in the range 1≤k≤3. For k≥4, all jammed configurations are nonpercolating states and, consequently, the percolation phase transition disappears. An explanation for this phenomenon is offered in terms of the rapid increase with k of the number of surrounding occupied sites needed to reach percolation, which gets larger than the sufficient number of occupied sites to define jamming. In the case of k=2 and 3, the percolation thresholds are θc,k=2(-)=0.60355(8) and θc,k=3=0.63110(9). Our results significantly improve the previously reported values of θc,k=2Naka=0.601(7) and θc,k=3Naka=0.621(6) [Nakamura, Phys. Rev. A 36, 2384 (1987)0556-279110.1103/PhysRevA.36.2384]. In parallel, a comparison with previous results for jamming on these systems is also done. Finally, a complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system has the same universality class as the ordinary random percolation, regardless of the size k considered.