Jammed systems of oriented needles always percolate on square lattices

Abstract

Random sequential adsorption (RSA) is a standard method of modeling adsorption of large molecules at the liquid-solid interface. Several studies have recently conjectured that in the RSA of rectangular needles, or k-mers, on a square lattice, percolation is impossible if the needles are sufficiently long (k of order of several thousand). We refute these claims and present rigorous proof that in any jammed configuration of nonoverlapping, fixed-length, horizontal, or vertical needles on a square lattice, all clusters are percolating clusters.

Percolation and jamming of random sequential adsorption samples of large lineark-mers on a square lattice

“…We have presented a parallel algorithm which is very efficient in terms of speed and memory usage. Our results are in agreement with the previously obtained results of [11,12,16] and we have obtained the percolation and jamming concentrations for lengths of k-mer up to 2 17 . The ratios of percolation and jamming densities show quite different behavior for k 500 compared to the behavior for k ≤ 512.…”

“…(7) is in agreement with the results of [16]. We have generalized the results of [16] for the case of periodic boundary conditions in Appendix A and have proved that, in thermodynamic limit, percolation always occurs before jamming. 14 and L = 1 638 400 for k > 2 14 .…”

Percolation of aligned rigid rods on two-dimensional triangular lattices

“…This finding reinforces the theoretical analysis in Ref. [15], namely, in the case of linear k-mers on square lattices, percolation always occurs before jamming.…”

“…In a very recent paper, Slutskii et al [16], using simulation techniques, corroborated the result reported by Kondrat et al [15]. Based in a very efficient parallel algorithm, the authors studied the problem of large linear k-mers (up to k = 2 17 ) on a square lattice with periodic boundary conditions.…”

Jamming and percolation of dimers in restricted-valence random sequential adsorption

“…Tarasevich et al [32,33] conjectured, based on simulation, that for k-mers of sufficient length (k 1.2 × 10 4 ) percolation does not occur. More recently, Kondrat et al [34] developed a rigorous proof refuting this conjecture. They showed that for nonoverlapping k-mers, the jammed configuration includes a percolating cluster.…”