Reaction-diffusion processes with nonlinear diffusion

We study reaction-diffusion processes with concentration-dependent diffusivity. First, the decay of the concentration in the single-species and two-species diffusion-controlled annihilation processes is determined. We then consider two natural inhomogeneous realizations. The two-species annihilation process is investigated in the situation when the reactants are initially separated, namely each species occupies a half space. In particular, we establish the growth law of the width of the reaction zone. The single-species annihilation process is studied in the situation when the spatially localized source drives the system toward the nonequilibrium steady state. Finally, we investigate a dissolution process with a localized source of diffusing atoms which react with the initially present immobile atoms forming immobile molecules.

Figure 1

Snapshot of the particle positions in two-species annihilation in two dimensions (the particle radii are enlarged for visibility).

Figure 2

Illustration of the three length scales in two-species annihilation in one dimension.

Figure 3

Sketch of the concentration profiles (solid curves) and the reaction rate $K{c}_A{c}_B$ (dashed curve, and considerably magnified) in two-species annihilation with initially separated components. The depletion zone width is $\ell \sim \sqrt{t}$, while the reaction zone width is $w\sim t^{1/(4a+6)}$.

Figure 4

A growing droplet of $B^{*}$ molecules depicted as filled circles. The droplet is still rather small and hence the deviations from the round shape are visible. Each site with a $B^{*}$ molecule can contain $A$ atoms (depicted as squares) diffusing on the lattice and reacting with $B$ atoms (not displayed) which occupy sites outside the droplet.