Abstract
Percolation and jamming of k×k square tiles (k^{2}-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 k^{2}-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 θ_{j,k} was determined for a wide range of k values (2≤k≤100 with many intermediate k values to allow a fine scaling analysis). θ_{j,k} exhibits a decreasing behavior with increasing k, being θ_{j,k=∞}=0.5623(3) the limit value for large k^{2}-mer sizes. In addition, a finite-size scaling analysis of the jamming transition was carried out, and the corresponding spatial correlation length critical exponent ν_{j} was measured, being ν_{j}≈1. On the other hand, the obtained results for the percolation threshold θ_{c,k} showed that θ_{c,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 θ_{c,k=2}(∞)=0.60355(8) and θ_{c,k=3}=0.63110(9). Our results significantly improve the previously reported values of θ_{c,k=2}^{Naka}=0.601(7) and θ_{c,k=3}^{Naka}=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.
Highlights
Random sequential adsorption (RSA) is one of the simplest model used for studying of irreversible adsorption processes [1]
If the concentration of the deposited objects on the substrate exceeds a critical value, a cluster extends from one side of the system to the other. This particular value of concentration rate is named critical concentration or percolation threshold θc, and determines a phase transition in the system. This transition is a geometrical phase transition where the critical concentration separates a phase of finite clusters from a phase where an infinite cluster is present
The present paper deals with jamming and percolation aspects of k × k square plaquettes deposited on 3D simple cubic lattices
Summary
Random sequential adsorption (RSA) is one of the simplest model used for studying of irreversible adsorption processes [1]. If the concentration of the deposited objects on the substrate exceeds a critical value, a cluster (a group of occupied sites in such a way that each site has at least one occupied nearest neighbor site) extends from one side of the system to the other This particular value of concentration rate is named critical concentration or percolation threshold θc, and determines a phase transition in the system. In contrast to the statistic for the simple particles, the degeneracy of arrangements of extended objects is strongly influenced by the structure and dimensionality of the lattice In this context, the present paper deals with jamming and percolation aspects of k × k square plaquettes deposited on 3D simple cubic lattices. Using extensive simulations supplemented by finite-size scaling analysis, jamming coverage and percolation thresholds were determined for a wide range of k values.
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