Percolation of aligned rigid rods on two-dimensional square lattices

P Longone,A J Ramirez-Pastor,P M Centres

doi:10.1103/physreve.85.011108

Abstract

The percolation behavior of aligned rigid rods of length k (kmers) on two-dimensional square lattices has been studied by numerical simulations and finite-size scaling analysis. The kmers, containing k identical units (each one occupying a lattice site), were irreversibly deposited along one of the directions of the lattice. The process was monitored by following the probability R(L,k)(p) that a lattice composed of L×L sites percolates at a concentration p of sites occupied by particles of size k. The results, obtained for k ranging from 1 to 14, show that (i) the percolation threshold exhibits a decreasing function when it is plotted as a function of the kmer size; (ii) for any value of k (k>1), the percolation threshold is higher for aligned rods than for rods isotropically deposited; (iii) the phase transition occurring in the system belongs to the standard random percolation universality class regardless of the value of k considered; and (iv) in the case of aligned kmers, the intersection points of the curves of R(L,k)(p) for different system sizes exhibit nonuniversal critical behavior, varying continuously with changes in the kmer size. This behavior is completely different to that observed for the isotropic case, where the crossing point of the curves of R(L,k)(p) do not modify their numerical value as k is increased.

Highlights

Percolation is a very active field of research and applied to a wide range of fields, such as biology, nanotechnology, device physics, physical chemistry, and materials science [1,2,3,4]
This finding contrasts with the decreasing tendency observed for pc(k) in square lattices, showing that (1) it is of interest and of value to inquire how a specific lattice structure influences the main percolation properties of particles occupying more than one site; and (2) the structure of the lattice plays a fundamental role in determining the statistics of extended objects
The result indicates that for finite systems the anisotropy of the deposited layer favors the percolation along the direction of the nematic phase. This scenario does not occur in isotropic systems, where for a fixed L, the vertical and horizontal percolation probabilities are indistinguishable [57]; and (b) RkP ∗ and RkT ∗ crossing points are located at the same point on the p-axis, indicating that, in the thermodynamic limit, the value of the percolation threshold is the same in both parallel and transversal directions

Summary

INTRODUCTION

Percolation is a very active field of research and applied to a wide range of fields, such as biology, nanotechnology, device physics, physical chemistry, and materials science [1,2,3,4]. The main objective of the present paper is to study the percolation behavior of aligned rigid rods on 2D triangular lattices For this purpose, extensive numerical simulations (with 2 ≤ k ≤ 80 and 75 ≤ L/k ≤ 640) supplemented by analysis using finite-size scaling theory have been carried out. The obtained results revealed that the percolation threshold pc(k) is an increasing function with k This finding contrasts with the decreasing tendency observed for pc(k) in square lattices, showing that (1) it is of interest and of value to inquire how a specific lattice structure influences the main percolation properties of particles occupying more than one site; and (2) the structure of the lattice plays a fundamental role in determining the statistics of extended objects. A complete study of critical exponents and universality is presented in the Supplemental Material [66]

MODEL AND BASIC DEFINITIONS

THE PERCOLATION THRESHOLD

CONCLUSIONS