Diffusion and reaction in a lamellar system: Self-similarity with finite rates of reaction

The evolution of an imperfectly mixed system—mimicked in terms of a distribution of lamellae—is studied. Two reactants A and B, initially placed in alternate striations, diffuse and undergo a reaction A+B→2P with intrinsic rate r=${\mathit{k}}_{\mathit{r}}$(${\mathit{c}}_{\mathit{A}}$ ${\mathit{c}}_{\mathit{B}}$ ${)}^{\mathrm{\alpha }}$. Simulations, scaling analysis, and space-averaged (fractal) kinetics are used to study the evolution of the system for different values of α and ${\mathit{k}}_{\mathit{r}}$. For α=1 and short times, a model based on the dynamics of reaction for a single lamella with infinite neighbors predicts the overall rate of reaction. For α<2.5, diffusion takes control of the dynamics for moderate to large times, and the kinetic parameters become irrelevant. Under these conditions, critical self-organization determines the behavior of the system, and the spatial structure evolves into a self-similar form that is independent of both ${\mathit{k}}_{\mathit{r}}$ and initial conditions. En route to scaling, the system undergoes two independent transitions: (i) from intrinsic chemical kinetics control to diffusion control, and (ii) from a system with several characteristic lengths to a system with only one characteristic length; these transitions might occur in any order, depending on controlling parameters. A combination of both short- and long-time regimes gives an efficient prediction for the average concentration of reactants for all times.