A technique for the estimation of percolation thresholds in lattice systems: Application to a problem of granular flow through an orifice

Abstract

A simple method to estimate percolation thresholds pc in lattice models is presented. The technique is based on the calculation of the probabilities RLX(p) for finding a percolating cluster of type X [X could be horizontal (H), vertical (V), average (A), intersection (I), or union (U)] on a finite lattice of side length L at concentration p of occupied sites. The functions RLX(p) are obtained by numerical simulations. Then, (1) effective percolation thresholds pcX,r(L) are estimated from the condition RLX(p)=r (where r is a parameter ranging between 0 and 1); and (2) the percolation threshold in the thermodynamic limit is determined by extrapolating the effective percolation thresholds to infinite L. The methodology presented herein contains and extends the classical scheme developed by Yonezawa, Sakamoto and Hori, offering a more complete and versatile theoretical/simulation framework to calculate percolation thresholds. The approach is validated by successfully comparing with well-known results obtained for the problem of random site percolation on square lattices. Taking advantage of the behavior of the curves of pcX,r(L) for different values of the parameter r, more general scaling relations are proposed for the effective percolation thresholds. Finally, the technique is applied to model experimental data of clogging transitions in a two-dimensional silo with a vibrated base.