Asymptotic behaviour of initially separated A + B(static) → 0 reaction-diffusion systems

Abstract

We examine the long-time behaviour of A + B(static) → 0 reaction-diffusion systems with initially separated species A and B. All of our analysis is carried out for arbitrary (positive) values of the diffusion constant D A of particles A and initial concentrations a 0 and b 0 of A's and B's. We derive general formulae for the location of the reaction zone centre, the total reaction rate, and the concentration profile of species A outside the reaction zone. The general properties of the reaction zone are studied with a help of the scaling ansatz. Using the mean-field approximation we find the functional forms of 'tails' of the reaction rate R and the dependence of the width of the reaction zone on the external parameters of the system. We also study the change in the kinetics of the system with D B > 0 in the limit D B → 0. Our results are supported by numerical solutions of the mean-field reaction-diffusion equation.

Time Evolution Of The Reaction Front In A Subdiffusive System

“…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]. However, as the methods of extracting the power functions are not based on analytical solutions of subdiffusion-reaction equations (not even on their approximately forms) the proportionality coefficient K is unknown.…”

“…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).…”

“…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 .…”

“…how the diffusion coefficient influences the process. As far as we know, there were only a few attempts to determine of K by means of the quasistationary approximation [2,3].…”

“…An alternative method is the quasistationary one. In the case of normal diffusion-reaction system it is based on the assumption that process proceeds so slowly that changes of concentration of transported substance are small in some regions [2,3]. Since the subdiffusion process is significantly slower than the normal diffusion one, we expect that the quasistationary approximation is also applicable to the subdiffusive case.…”