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

Abstract

Diffusion in a two-species two-dimensional system has been simulated using a lattice approach. Rodlike particles were considered as linear k-mers of two mutually perpendicular orientations (kx- and ky-mers) on a square lattice. These kx- and ky-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, pj. 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 , the system tends toward a well-organized steady state. Boundary conditions predetermine the final state of the system. When periodic boundary conditions 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) boundary conditions 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. kx-mers locate in the upper left corner, while the ky-mers are situated in the lower right corner (‘yin–yang’ pattern). From time to time, regions built of kx- and ky-mers exchange their locations through irregular patterns. When mixed boundary conditions are used (periodic boundary conditions are applied along one direction whereas insulating boundary conditions are applied along the other one), the system still tends to form the stripes, but they are unstable and change their spatial orientation.