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 (k x -and k y -mers) on a square lattice. These k xand 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 ≥ 6, 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., 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 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.

Calculating entropy by Ma’s method for a system of k-mers on a square lattice

“…The probability of exactly k balls in any box can be calculated as , steady state in the form of diagonal stripes, (g) final "yin-yang" pattern on a torus (the same pattern as in (f)); for clarity, the major radius of the torus is significantly exaggerated. Illustration adopted from [8].…”

“…Random walk of rectangular particles of two mutually perpendicular orientations on a square lattice with periodic boundary conditions demonstrated a pattern formation. The system tends to self-organize, as shown in [8]. This pattern formation is supposed to be entropy-driven.…”

“…(a) t M C = 0, initial jammed state, (b) t M C = 100, end of fluidization, (c) t M C = 10 4 , labyrinth patterns, (d) t M C = 10 5 , labyrinth patterns are transforming into stripes, (e) t M C = 106, final stage of stripe formation, (f) t M C = 10 7 , steady state in the form of diagonal stripes, (g) final "yin-yang" pattern on a torus (the same pattern as in (f)); for clarity, the major radius of the torus is significantly exaggerated. Illustration adopted from[8].…”

Effects of Brownian Motions on Electrical Conductivity and Optical Transparency of Two-Dimensional Films Filled by Needle-Like Particles

“…The analysis included discrete (lattice) [18] and continuous models [19,20], disordered and partially ordered systems, effects of polydispersity of a needle length [21], as well as effects associated with the presence of defects [18]. The anisotropy of the conductivity in monolayers [19,20] and the effects of self-organization/aging that arise in the presence of the Brownian diffusion [22][23][24] or in the presence of a vertical drying have been studied (see, in particular, [25]). The analysis of the behavior of the optical density in systems filled with ordered particles of an anisotropic form (disks or cylinders) [26] was also carried out.…”

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

“…Recently, diffusion-driven pattern formation in a 2D system of k-mers has been studied by means of Monte Carlo (MC) simulation [14,15]. The k-mers were oriented in two mutually perpendicular directions.…”