Cross-diffusion and pattern formation in reaction–diffusion systems

Cross-diffusion, the phenomenon in which a gradient in the concentration of one species induces a flux of another chemical species, has generally been neglected in the study of reaction–diffusion systems. We summarize experiments that demonstrate that cross-diffusion coefficients can be quite significant, even exceeding “normal,” diagonal diffusion coefficients in magnitude in systems that involve ions, micelles, complex formation, excluded volume effects (e.g., surface or polymer reactions) and other phenomena commonly encountered in situations of interest to chemists. We then demonstrate with a series of model calculations that cross-diffusion can lead to spatial and spatiotemporal pattern formation, even in relatively simple systems. We also show that, in the absence of cross-diffusion among the reacting species, introduction of a nonreactive species that induces appropriate cross-diffusive fluxes with reactive species can lead to pattern formation.