Fickian and non-Fickian diffusion with bimolecular reactions

Mixing zone dynamics of a reaction product C during diffusion of two species (A and B) are examined, using a two-dimensional particle tracking model (PT) for the reaction A+B→C, allowing for both Fickian and non-Fickian transitions. The form of PT we use is equivalent to a continuous time random walk, which is a widely used model for anomalous transport and diffusion. It is shown that the basic patterns of the C dynamics---the temporal evolution of the spatial profile and the temporal C production---are similar for both modes of diffusion. However, the distinctive time scale for the non-Fickian case is very much larger even when the median transition steps are matched with the Fickian case. For immobile C, the spatial profile pattern is a broadening (Gaussian) reaction front evolving to a concentration-fluctuation dominated (Lorentzian) shape. The temporal C production is fit well by a stretched exponential for both diffusion types. In analyzing experiments, the appearance of a Gaussian C profile does not prove that the diffusion process is Fickian.