Maximal Density, Kinetics of Deposition and Percolation Threshold of Loose Packed Lattices

TL;DR

This study uses Monte Carlo simulations to analyze how particles fill lattices with limited adjacency, revealing maximum densities and percolation thresholds depending on rules and dimensions.

Contribution

It introduces a model for particle deposition on lattices with controlled adjacency rules, quantifying maximum densities and percolation thresholds in 2D and 3D.

Findings

Maximum density in 2D lattice is approximately 0.364 with strict rules.

Allowing edge sharing increases maximum density and enables percolation.

Maximum density in 3D lattice is approximately 0.326.

Abstract

In many areas of research it is interesting how lattices can be filled with particles that have no nearest neighbors, or they are in limited quantities. Examples may be found in statistical physics, chemistry, materials science, discrete mathematics, etc. Using Monte Carlo (MC) simulation we study the kinetics of filling of square lattice (2D). Two complementary rules are used to fill the lattice. We study their influence on the kinetics of the process as well as on the properties of the obtained systems. According to the first rule the occupied sites may not share edges (nearest neighbors occupations are not permitted). Under this condition, the maximum possible concentration is 0.5, forming a checkerboard type structure. However, we found that if the deposition is done by random selection of sites the concentration of 0.5 is inaccessible and the maximum concentration is Cmax(2D)=0.3638 (0.0003) for 2D lattice. If the lattice is 3D we find that the maximal concentration is even lower Cmax(3D)=0.326 (0.001). The second rule establishes permission to break the first one with certain probability 0<=p<=1, thus the occupied sites can start to share edges when p>0. In this case higher then 0.3638 concentrations are accessible, even up to C=1. Therefore the percolation threshold Pc can be reached. Its value depends on the value of the probability p. Our model describes the kinetics of formation of thin films of particles attracted by the substrate but repulsing each other.