Asymptotics of reaction–diffusion fronts with one static and one diffusing reactant

Abstract

The long-time behavior of a reaction-diffusion front between one static (e.g. porous solid) reactant A and one initially separated diffusing reactant B is analyzed for the mean-field reaction-rate density R(ρ A , ρ B ) = kρ m A ρ n B . A uniformly valid asymptotic approximation is constructed from matched self-similar solutions in a "reaction front" (of width w ∼ t α where R ∼ t β enters the dominant balance) and a "diffusion layer" (of width W ∼ t 1/2 where R is negligible). The limiting solution exists if and only if m, n ≥ 1, in which case the scaling exponents are uniquely given by α = (m−1)/2(m+1) and β = m/(m + 1). In the diffusion layer, the common ad hoc approximation of neglecting reactions is given mathematical justification, and the exact transient decay of the reaction rate is derived. The physical effects of higher-order kinetics (m, n > 1), such as the broadening of the reaction front and the slowing of transients, are also discussed.

Time Evolution Of The Reaction Front In A Subdiffusive System

“…It was observed [1,2,3,4,5,6] that when the process starts, there appear characteristic regions (see Fig. 1): the depletion zone 'Dep ′ , which is defined as a region where the concentrations are significantly smaller than the initial ones (C A ≪ C 0A and C B ≪ C 0B ), the reaction region where the production of particles P is significant (R(x, t) > 0), and the diffusion region 'Dif ′ , where the reaction term R(x, t) is close to zero and the particle transport appears to be almost subdiffusive (i.e.…”

“…For the normal diffusion the widths of the depletion zone W Dep and the reaction region W R grow as the power functions of time [1,2,3,4,5,6,7], W Dep ∼ t θ , with θ = 1/2, and W R ∼ t µ , where µ < θ. The value of parameter µ depends on the system under study.…”

“…The diffusion-reaction system with two initially separated diffusing particles of spices A and B reacting according to the formula m ′ A + n ′ B → P (inert) has been intensively studied during past years [1,2,3,4,5,6,7,8,9,10,11,12,13]. As the diffusion-reaction equations describing the system are nonlinear, it is difficult to solve them and their general solutions remain unknown (except of very special cases).…”

“…Thus, to simplify the calculations one usually uses various approximations, such as the quasistationary approximation [1,2,3], the scaling method [1,4,5,6,7], or the perturbation one [9,10]. Using these methods, there were derived characteristic functions of the system which include: the time evolution of the reaction front x f (t), the width of the reaction zone W R (t) or the width of the depletion zone W Dep (t) [1,2,4,5,6] which all appear to be the power functions of time f (t) = Kt γ . The results were confirmed by numerical calculations and simulations [3,5,6].…”

Reaction-diffusion analysis for one-step plasma etching and bonding of microfluidic devices

“…Fully resolving the concentrations inside the reaction zone is only possible when the reaction kinetics are known in detail. 12 Here we assume a = b = 1 in Eqs. ͑1͒ and ͑2͒, and find the scaling of ␦ as follows.…”