Reaction Kinetics under Anomalous Diffusion

Abstract

The present work studies the generalization of reaction–diffusion schemes to subdiffusion. The subdiffusive dynamics was modelled by means of continuous–time random walks on a mesoscopic scale with a heavy–tailed waiting time pdf ψ(t) ∝ t−1−α lacking the first moment. The reaction itself was assumed to take place on a microscopic scale, obeying the classical mass action law. This situation is assumed to apply in a porous medium where the particles are trapped within the catchments, pores and stagnant regions of the flow, but are still able to react during their waiting times. After discussing the subdiffusion equation and different methods of their solution, especially under the aspect of particles being introduced into the system in the course of time, the reaction–subdiffusion equations are addressed. These equations are of integro–differential form and under the assumptions made, the reaction explicitly affects the transport term. The long ranged memory of the subdiffusion kernel is modified by an additional factor accounting for the conversion and survival probabilities due to reaction during the waiting times. In the case of linear reaction kinetics, this factor is governed by the rate coefficients. For nonlinear reaction kinetics the transport kernel depends additionally on the concentrations of the respective reaction partners at all previous times. The simplest linear reaction, the degradation A→ 0 was considered and a general expression for the solution to arbitrary Dirichlet Boundary Value Problems was derived. This solution can be expressed in terms of the solution to the corresponding Dirichlet Problem under mere subdiffusion, i.e. without degradation. The resultant stationary profiles do not differ qualitatively from the stationary profiles in normal reaction diffusion. For stationary solutions to exist in reaction–subdiffusion, the assumption of reactions according to classical rate kinetics is essential. As an example for a nonlinear reaction–subdiffusion system, the irreversible autocatalytic reaction A + B→ 2A under subdiffusion is considered. Under the assumptions of constant overall particle concentration A(x, t) + B(x, t) = const and re–labelling of the converted particles, a subdiffusive analogue of the classical Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) equation was derived and the resultant fronts of A–particles propagating into the B– domain were studied. Two different regimes were detected in numerical simulations. These regimes were discussed using both crossover arguments and analytic calculations. The first regime can be described within the framework of the continuous reaction–subdiffusion equations and is characterized by the front velocity and width going as t α−1 2 at larger times. As the front width decays, the front gets atomically sharp at very large times and a transition to a second regime, the fluctuation dominated one, is expected. The fluctuation dominated regime is not within the scope of the continuous description. In that case, the velocity of the front decays faster in time than in the continuous regime, v f luct ∝ tα−1. Further simulations pertaining the reaction on contact scenario, i.e. the fluctuation dominated regime, revealed additional fluctuation effects that are genuinely due to subdiffusion. Another nonlinear reaction–subdiffusion system where reactants A penetrate a medium initially filled with immobile reactants B and react according to the scheme A+B→ (inert) was considered. Under certain presumptions, this problem can be described in terms of a moving boundary problem, a so–called Stefan–problem, for the concentration of a single species. The main result was that the propagation of the moving boundary between the A– and B–domain goes as R(t) ∝ tα/2. The theoretical predictions concerning the moving boundary were corroborated by numerical simulations.