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. Thetechnique is based on the calculation of the probabilities RXL (p) for finding a percolatingcluster 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 occupiedsites. The functions RXL (p) are obtained by numerical simulations. Then, (1) effectivepercolation thresholds pX,r c (L) are estimated from the condition RXL (p) = r (where r isa parameter ranging between 0 and 1); and (2) the percolation threshold in the thermodynamiclimit is determined by extrapolating the effective percolation thresholds toinfinite L. The methodology presented herein contains and extends the classical schemedeveloped by Yonezawa, Sakamoto and Hori, offering a more complete and versatiletheoretical/simulation framework to calculate percolation thresholds. The approach isvalidated by successfully comparing with well-known results obtained for the problemof random site percolation on square lattices. Taking advantage of the behavior of thecurves of pX,r c (L) for different values of the parameter r, more general scaling relationsare proposed for the effective percolation thresholds. Finally, the technique is appliedto model experimental data of clogging transitions in a two-dimensional silo with avibrated base.