# Impact of packing fraction on diffusion-driven pattern formation in a two-dimensional system of rod-like particles

## TL;DR

This study uses lattice simulations to explore how packing density influences pattern formation in a 2D system of rod-like particles, revealing conditions under which diagonal stripe patterns emerge.

## Contribution

It introduces a lattice-based Monte Carlo simulation approach to analyze diffusion-driven pattern formation in rod-like particles with varying packing fractions.

## Key findings

- Diagonal stripe patterns form for long rods ($k \,\geq\, 6$) at moderate packing densities.
- Pattern formation depends on the rod length and packing fraction.
- The system reaches a non-equilibrium steady state with organized patterns.

## Abstract

Pattern formation in a two-dimensional system of rod-like particles has been simulated using a lattice approach. Rod-like particles were modelled as linear $k$-mers of two mutually perpendicular orientations ($k_x$- and $k_y$-mers) on a square lattice with periodic boundary conditions (torus). Two kinds of random sequential adsorption model were used to produce the initial homogeneous and isotropic distribution of $k$-mers with different values of packing fraction. By means of the Monte Carlo technique, translational diffusion of the $k$-mers was simulated as a random walk, while rotational diffusion was ignored, so, $k_x$- and $k_y$-mers were considered as individual species. The system tends toward a well-organized nonequilibrium steady state in the form of diagonal stripes for the relatively long $k$-mer ($k \geq 6$) and moderate packing densities (in the interval $p_{down} < p < p_{up}$, where both the critical packing fractions $p_{down}$ and $p_{up}$ are depended on $k$).