# Pattern formation in a two-dimensional two-species diffusion model with anisotropic nonlinear diffusivities: a lattice approach

## TL;DR

This study uses a lattice Monte Carlo approach to analyze anisotropic diffusion of two species of rodlike particles on a 2D lattice, revealing concentration-dependent anisotropic diffusion and boundary-condition-driven pattern formation.

## Contribution

It introduces a lattice simulation of two-species rod diffusion, demonstrating anisotropic, nonlinear diffusion behavior and boundary-condition-dependent steady-state patterns.

## Key findings

- Diffusion coefficients are strongly anisotropic and nonlinear with concentration.
- High concentration and large k lead to organized steady states.
- Boundary conditions determine the pattern formation, such as stripes or yin-yang domains.

## Abstract

Diffusion in a two-species 2D system has been simulated using a lattice approach. Rodlike particles were considered as linear $k$-mers of two mutually perpendicular orientations ($k_x$- and $k_y$-mers) on a square lattice. These $k_x$- and $k_y$-mers were treated as species of two kinds. A random sequential adsorption model was used to produce an initial homogeneous distribution of $k$-mers. The concentration of $k$-mers, $p$, was varied in the range from 0.1 to the jamming concentration, $p_j$. By means of the Monte Carlo technique, translational diffusion of the $k$-mers was simulated as a random walk, while rotational diffusion was ignored. We demonstrated that the diffusion coefficients are strongly anisotropic and nonlinearly concentration-dependent. For sufficiently large concentrations (packing densities) and $k \geq 6$, the system tends toward a well-organized steady state. Boundary conditions (BC) predetermine the final state of the system. When periodic BCs are applied along both directions of the square lattice, the system tends to a steady state in the form of diagonal stripes. The formation of stripe domains takes longer time the larger the lattice size, and is observed only for concentrations above a particular critical value. When insulating (zero flux) BCs are applied along both directions of the square lattice, each kind of $k$-mer tries to completely occupy a half of the lattice divided by a diagonal, e.g., $k_x$-mers locate in the upper left corner, while the $k_y$-mers are situated in the lower right corner ("yin-yang" pattern). From time to time, regions built of $k_x$- and $k_y$-mers exchange their locations through irregular patterns. When mixed BCs are used (periodic BCs are applied along one direction whereas insulating BCs are applied along the other one), the system still tends to form the stripes, but they are unstable and change their spatial orientation.