Exotic behavior of the reaction front in the A+B-->C reaction-diffusion system

Marangoni- vs. buoyancy-driven flows: competition for spatio-temporal oscillations in A + B → C systems

The emergence of self-organized behaviors such as spatio-temporal oscillations is well-known for complex reactions involving nonlinear chemical or thermal feedback. Recently, it was shown that local oscillations of the chemical species concentration can be induced under isothermal batch conditions for simple bimolecular A + B → C reactions, provided they are actively coupled with hydrodynamics. When two reactants A and B, initially separated in space, react upon diffusive contact, damped spatio-temporal oscillations could develop when the surface tension increases sufficiently in the reaction zone. Additionally, if the density decreases, the coupling of both surface tension- and buoyancy-driven contributions to the flow can further sustain this oscillatory instability. Here, we investigate the opposite case of a reaction inducing a localized decrease in surface tension and an increase in density in the reacting zones. In this case, the competition arising from the two antagonistic flows is needed to create oscillatory dynamics, i.e., no oscillations are observed for pure chemically driven Marangoni flows. We study numerically these scenarios in a 2-dimensional system and show how they are controlled by the following key parameters: (i) ΔM and ΔR governing the surface tension and density variation during the reaction, respectively, (ii) the layer thickness of the system, and (iii) its lateral length. This work is a further step toward inducing and controlling chemical oscillations in simple reactions.

Dynamics of A+B→C reaction fronts under radial advection in a Poiseuille flow

A+B→C reaction fronts describe a wide variety of natural and engineered dynamics, according to the specific nature of reactants and product. Recent works have shown that the properties of such reaction fronts depend on the system geometry, by focusing on one-dimensional plug flow radial injection. Here, we extend the theoretical formulation to radial deformation in two-dimensional systems. Specifically, we study the effect of a Poiseuille advective velocity profile on A+B→C fronts when A is injected radially into B at a constant flow rate in a confined axisymmetric system consisting of two parallel impermeable plates separated by a thin gap. We analyze the front dynamics by computing the temporal evolution of the average over the gap of the front position, the maximum production rate, and the front width. We further quantify the effects of the nonuniform flow on the total amount of product, as well as on its radial concentration profile. Through analytical and numerical analyses, we identify three distinct temporal regimes, namely (i) the early-time regime where the front dynamics is independent of the reaction, (ii) the transient regime where the front properties result from the interplay of reaction, diffusion that smooths the concentration gradients and advection, which stretches the spatial distribution of the chemicals, and (iii) the long-time regime where Taylor dispersion occurs and the system becomes equivalent to the one-dimensional plug flow case.

Dynamics of A+B → C reaction fronts under radial advection in three dimensions

The dynamics of A+B→C reaction fronts is studied both analytically and numerically in three-dimensional systems when A is injected radially into B at a constant flow rate. The front dynamics is characterized in terms of the temporal evolution of the reaction front position, r_{f}, of its width, w, of the maximum local production rate, R^{max}, and of the total amount of product generated by the reaction, n_{C}. We show that r_{f}, w, and R^{max} exhibit the same temporal scalings as observed in rectilinear and two-dimensional radial geometries both in the early-time limit controlled by diffusion, and in the longer time reaction-diffusion-advection regime. However, unlike the two-dimensional cases, the three-dimensional problem admits an asymptotic stationary solution for the reactant concentration profiles where n_{C} grows linearly in time. The timescales at which the transition between the regimes arise, as well as the properties of each regime, are determined in terms of the injection flow rate and reactant initial concentration ratio.

Complex dynamics of interacting fronts in a simple A+B→C reaction-diffusion system

Pattern interaction has so far been restricted to systems with relatively complex reaction schemes, such as activator-inhibitor systems, that lead to rich spatio-temporal dynamics. Surprisingly, a simple second-order chemical reaction is capable of generating similar complex phenomena, such as attractive or repulsive interaction modes between the localized reaction zones (or fronts). We illustrate the latter statement both analytically and numerically with two initially separated A+B→C reaction-diffusion fronts when the solution of B is initially confined between two solutions of A. The nature of the front-front interaction changes from an attractive type to a repulsive one above a critical distance separating the two fronts initially. The complexity of the pattern dynamics emerges here due to finite-size effects. A scaling law relating the critical distance d_{c} above which the repulsion occurs and kinetic parameters gives insights into (i) extracting those parameters from experiments for bimolecular reactions and (ii) the control strategy of periodic patterns.

Surface tension- and buoyancy-driven flows across horizontally propagating chemical fronts

Chemical reactions can interplay with hydrodynamic flows to generate various complex phenomena. Because of their relevance in many research areas, chemically-induced hydrodynamic flows have attracted increasing attention in the last decades. In this context, we propose to give a review of the past and recent theoretical and experimental works which have considered the interaction of such flows with chemical fronts, i.e. reactive interfaces, formed between miscible solutions. We focus in particular on the influence of surface tension- (Marangoni) and buoyancy-driven flows on the dynamics of chemical fronts propagating horizontally in the gravity field.

Reaction front formation in contaminant plumes

The formation of successive fronts in contaminated groundwater plumes by subsoil bacterial action is a commonly accepted feature of their propagation, but it is not obviously clear from a mathematical standpoint quite how such fronts are formed or propagate. In this paper we show that these can be explained by combining classical reaction-diffusion theory involving just two reactants (oxidant and reductant), and a secondary reaction in which a reactant on one side of such a front is (re-)formed on the other side of the front via diffusion of its product across the front. We give approximate asymptotic solutions for the reactant profiles, and the propagation rate of the front.

Analytical small-time asymptotic properties of A+B-->C fronts

The small-time asymptotic properties of the reaction front formed by a reaction A+B-->C coupled to diffusion are considered. Reactants A and B are initially separately dissolved in two identical solvents. The solvents are brought into contact and the reactants meet through diffusion. The small-time asymptotic position of the center of mass of the reaction rate is obtained analytically. When one of the reactants diffuses much faster than the other reactant then the position of the local maximum in the reaction rate travels on a length scale related to the diffusion coefficient of the slowest diffusing reactant while the first moment of the reaction rate and the width of the reaction front are on a length scale related to the diffusion coefficient of the fastest diffusing reactant. If the sum of the initial reactant concentrations is fixed, then the fastest reaction rate is obtained when equal concentrations are used. The first-order solutions are analytically obtained, however, each solution involves an integral which requires numerical evaluation. Various small-time asymptotic analytical reaction front properties are obtained. In particular, one finds that the position of the center of mass of the product concentration distribution is initially located at three quarters of the position of the center of mass of the reaction rate.

Death of an A -particle island in the B -particle sea: propagation and evolution of the reaction front A+B<-->C

We present a systematic theory of propagation and evolution of the reaction front A+B<-->C in the reaction-diffusion system where an island of particles A is surrounded by the uniform sea of particles B . In the first part of the work we give a systematic analysis of the crossover from the irreversible to reversible regime of front propagation in terms of the quasistatic approximation (QSA) and derive the key condition for the island death in the quasiequilibrium front regime. We show that the same as in the case of pure annihilation A+B-->0 the QSA enables the description of the quasiequilibrium front propagation only to a critical point t{c} on approaching to which the QSA is violated. In the second part of the work under the assumption of a sufficiently large forward reaction constant k we derive the perturbative expansion in powers of 1k which gives the asymptotically exact description of the quasiequilibrium front evolution up to t-->infinity. We demonstrate that below some critical value of the reduced backward reaction constant g<g{c} there appear two turning points on the front trajectory, the first of which arises at the sharp localized front stage and is due to the finite number of island particles whereas the second is a consequence of radical transformation of the front structure at the passage through the critical point (delocalization of the front). We find a remarkable property of self-similarity of the passage through the critical point, we derive scaling laws for such passage and show that in the limit g-->0 these laws lead to a striking phenomenon of an abrupt delocalization of the front.

Higher-order large-time asymptotics for a reaction of the form nA+mB-->C

This study examines the large time asymptotic behavior for the family of reactions of the form nA+mB-->C when the reactants A and B are initially separated. Once the reactants are brought into contact they are assumed to react with a kinetic rate proportional to A;{n}B;{m} . A planar reaction front forms and usually moves away from its initial position to invade one of the reactant solutions. The position of the reaction front relative to the initial position where the reactants were put in contact x_{f} for large times t , is found theoretically to satisfy the expansion x_{f}=2sqrt[t][alpha+alpha_{2}t;{-2sigma}+alpha_{3}t;{-3sigma}+O(t;{-4sigma})] , where sigma=1(n+m+1) and alpha , alpha_{2} , and alpha_{3} are constants. This expansion is valid provided that n and m are positive constants less than or equal to 3. The implication of this is that when n+m not equal3 , x_{f} either tends to zero or infinity, while if n+m=3 then there exists the possibility of x_{f} tending to a finite nonzero constant. For fractional order kinetics, n and m are arbitrary positive constants, however, for simple reactions n and m are positive integers. Hence, the reaction A+2B-->C is the only reaction of the form nA+mB-->C with n and m being positive integers less than 4 in an infinite domain that can lead to a reaction front approaching at a finite but nonzero distance from the position at which the two liquids first met.

Time evolution of the reaction front in a subdiffusive system

Using the quasistatic approximation, we show that in a subdiffusion-reaction system with arbitrary nonzero values of subdiffusion coefficients, the reaction front x_{f}(t) evolves in time as x_{f}(t)=Kt;{alpha2} , with alpha being the subdiffusion parameter and K being controlled by the subdiffusion coefficients. To check the correctness of our analysis, we compare approximate analytical solutions of the subdiffusion-reaction equations with the numerical ones.

Dynamics of A + B --> C reaction fronts in the presence of buoyancy-driven convection

The dynamics of A+B-->C fronts in horizontal solution layers can be influenced by buoyancy-driven convection as soon as the densities of A, B, and C are not all identical. Such convective motions can lead to front propagation even in the case of equal diffusion coefficients and initial concentration of reactants for which reaction-diffusion (RD) scalings predict a nonmoving front. We show theoretically that the dynamics in the presence of convection can in that case be predicted solely on the basis of the knowledge of the one-dimensional RD density profile across the front.

Analytical asymptotic solutions of nA+mB-->C reaction-diffusion equations in two-layer systems: a general study

Large time evolution of concentration profiles is studied analytically for reaction-diffusion systems where the reactants A and B are each initially separately contained in two immiscible solutions and react upon contact and transfer across the interface according to a general nA+mB-->C reaction scheme. This study generalizes to immiscible two-layer systems the large time analytical asymptotic limits of concentrations derived by Koza [J. Stat. Phys. 85, 179 (1996)] for miscible fluids and for reaction rates of the form A;{n}B;{m} with arbitrary diffusion coefficients and homogeneous initial concentrations. In addition to a dependence on the parameters already characterizing the miscible case, the asymptotic concentration profiles in immiscible systems depend now also on the partition coefficients of the chemical species between the two solution layers and on the ratio of diffusion coefficients of a given species in the two fluids. The miscible time scalings are found to remain valid for the immiscible fluids case. However, for immiscible systems, the reaction front speed is enhanced by increasing the stoichiometry of the invading species over that of the species being invaded. The direction of the front propagation is found to depend on the diffusion coefficient of the invading species in its initial fluid but not on its value in the invading fluid. Hence, a reaction front in immiscible fluids can travel in the opposite direction to the reaction front formed in miscible fluids for a range of parameter values. The value of the invading species partition coefficient affects the magnitude of the front speed but it cannot alter the direction of the front. For sufficiently large times, the total amount of product produced in time is independent of the rate of the reaction. The centre of mass of the product can move in the opposite direction to the center of mass of the reaction rate.

Exponential temporal asymptotics of the A+B-->0 reaction-diffusion process with initially separated reactants

We study theoretically and numerically the irreversible A+B-->0 reaction-diffusion process of initially separated reactants occupying the regions of lengths LA, LB comparable with the diffusion length (LA,LB approximately sqrt[Dt], here D is the diffusion coefficient of the reactants). It is shown that the process can be divided into two stages in time. For t<<L2/D the front characteristics are described by the well-known power-law dependencies on time, whereas for t>L2/D these are well-approximated by exponential laws. The reaction-diffusion process of about 0.5 of initial quantities of reactants is described by the obtained exponential laws. Our theoretical predictions show good agreement with numerical simulations.

Dynamical localization-delocalization transition of the reaction-diffusion front at a semipermeable cellulose membrane

We study the kinetics of the reaction front in the A+B-->C reaction-diffusion system with reactants initially separated by a semipermeable membrane. The semipermeable membrane allows only one reactant species to go through ("penetrating species") while the other reactant species is sterically prohibited from penetration. Theoretically, the ratio of the diffusive fluxes of the two species has been defined before as a control parameter and it was predicted [Chopard, Phys. Rev. E 56, 5343 (1997)] to give rise to a localization-delocalization transition of the reaction front. In this paper we show the experimental realization of a dynamical localization-delocalization transition, in a system consisting of the reactants Ca2+ and calcium green-1 dextran, separated by a finite-sized cellulose membrane. The dynamical transition results from the continuous change in time of the flux of the penetrating species at the reaction boundary. Here this time-dependent flux is attributed to the free diffusion of the penetrating species through a membrane with a finite thickness. The dynamical transition is exemplified by the kinetic behavior of the front characteristics which exhibits several time regimes--an early time, an intermediate time, and an asymptotic time regime. The crossover times between these regimes are found to depend on the membrane thickness, a parameter not considered before to our knowledge. Monte Carlo simulations show good agreement with the finite-time experiments.

Reaction-diffusion fronts in systems with concentration-dependent diffusivities

We examine properties of a reaction front that forms in irreversible reaction-diffusion systems with concentration-dependent diffusivities. We study two different models of such systems and find that in the limit of a vanishingly small diffusivity of the reaction product, the reaction front dynamics enters a separate universality class, with the front width asymptotically tending to a constant value, and the reaction rate at the reaction front center diminishing with time t as t(-1/2). This behavior can be also observed in systems with nonvanishing (but small) diffusivity of the reaction product at intermediate times.

Perturbation analysis for competing reactions with initially separated components

We study a competitive reaction-diffusion system with initially separated components. In this system, two similar species on one side of the system compete to react with the species on the other side. The competition is due to significant differences in the microscopic reaction constants and the initial densities of the two competing species. In the short-time limit, each of the competitive reactions is considered as perturbation with respect to the diffusion, the latter is essential for the effective mixing of the reactants. We identify the small parameters required for the perturbation analysis of the competitive scheme. The resulting perturbative expressions provide the rich spatiotemporal reaction front patterns, which were experimentally observed for Cr3+ + Xylenol Orange (XO) --> products, where the aggregated and nonaggregated forms of Cr3+ in aqueous solution compete to react with the XO.

Reaction-diffusion front width anomalies in disordered media

We study the front characteristics of the A + B --> C reaction-diffusion system with initially separated reactants in disordered media, exemplified by two-dimensional (2D) percolation. We investigate the front characteristics as a function of the disorder degree in this system, in particular close to criticality. We show that the front width exponent is larger than the mean-field (MF) exponent of 1/6, and at criticality it approaches 1/4, which is the one-dimensional (1D) exponent. We show that previous predictions in the literature for the 2D percolation cluster at criticality are wrong. The results are discussed in the context of other systems with attenuated transport where the front width exponent is smaller than the MF exponent. We also study the short-time behavior of the front width exponent, and discuss the validity of the scaling relations between the relevant exponents.

An Approach To Extract Rate Constants from Reaction−Diffusion Dynamics in a Microchannel

A theoretical model is proposed to extract rate constants of second-order chemical reactions down to the millisecond time scale from the observation of reaction-diffusion processes in a microchannel. We validate this theoretical approach by examining an appropriate model reaction. The measured rate constant is in excellent agreement with this obtained from nuclear magnetic resonance experiments.

Reaction front in an A+B-->C reaction-subdiffusion process

We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.

Competing reactions with initially separated components in the asymptotic time region

Two competing irreversible reactions with initially separated components and with essentially different reaction constants are theoretically studied in the asymptotic time region. The description of the two simultaneous reactions is reduced to the consideration of two reactions separated in space. It is shown that the reaction rate profile can have two maxima and their ratio is independent of time. The location and relative value of the maxima are functions of the reaction constants and initial concentrations.

Reaction-diffusion dynamics: confrontation between theory and experiment in a microfluidic reactor

We confront, quantitatively, the theoretical description of the reaction-diffusion process of a second-order reaction to experiment. The reaction at work is Ca(2+)/CaGreen, a fluorescent tracer for calcium. The reactor is a T-shaped microchannel, 10 microm deep, 200 microm wide, and 2 cm long. The experimental measurements are compared with the two-dimensional numerical simulation of the reaction-diffusion equations. We find good agreement between theory and experiment. From this study, one may propose a method of measurement of various quantities, such as the kinetic rate of the reaction, in conditions yet inaccessible to conventional methods.

Asymptotic expansion for reversible A+B<-->C reaction-diffusion process

We study long-time properties of reversible reaction-diffusion systems of type A+B<-->C by means of the perturbation expansion in powers of 1/t (inverse of time). For the case of equal diffusion coefficients we present exact formulas for the asymptotic forms of reactant concentrations and a complete, recursive expression for an arbitrary term of the expansions. Taking an appropriate limit we show that by studying reversible reactions one can obtain "singular" solutions typical of irreversible reactions.

Theory for competing reactions with initially separated components

The asymptotic long-time properties of a system with initially separated components and two competing irreversible reactions A1+B-->C1 and A2+B-->C2 are studied. It is shown that the system is characterized by a single reaction zone, with width growing like t(1/6), in which both reactions occur. Numerical computations of the mean-field kinetic equations confirm these asymptotic results.

Gel-free experiments of reaction-diffusion front kinetics

We present a gel-free experimental system to study the kinetics of the reaction front in the A+B-->C reaction-diffusion system with initially-separated reactants. The experimental setup consists of a CCD camera monitoring the kinetics of the front formed in the reaction-diffusion process Cu(2+) + tetra [disodium ethyl bis(5-tetrazolylazo) acetate trihydrate] -->1:1 complex, in aqueous, gel-free solution, taking place inside a 150 microm gap between two flat microscope slides. The experimental results agree with the theoretical predictions for the time dependence of the front's width, height, and location, as well as the global reaction rate.

Reaction front structure in the diffusion-limited A+B model with initially randomized reactants

Subtle features of the reaction front formation in the A+B-->0 reaction are reported for the initially random and equal A+B reactant distribution. Three nonclassical parameters (initial linewidth, minimum, and maximum), for each interparticle gap and nearest neighbor distance distributions, are derived, as a function of time, using Monte Carlo simulations. These empirical front measures and their temporal scaling exponents are compared with the previously studied ones for the reactant interparticle distributions.

Behavior of the reaction front between initially segregated species in a two-stage reaction

The large-time asymptotic behavior of a two-stage reaction (A+B-->R, B+R-->S) with initially segregated reactants is described. The concentration of the reactants is found to be significantly less than the initial concentrations in a depletion zone of width proportional to t(1/2), where t is time; the reaction takes place in a thinner zone of width proportional to t(1/6). Similarity solutions for the chemical concentration profiles in the reaction zone are calculated, and are compared with numerical simulations of the full partial differential reaction-diffusion equations. The large-time asymptotic scalings reported here are the same as in the absence of the secondary reaction, but we find that the location of the reaction zone is significantly shifted due to the secondary reaction. The reaction zone may behave in an exotic fashion at large time, moving first one way, then reversing its direction.

Asymptotic properties of a reversible A+B<-->C (static) reaction-diffusion process with initially separated reactants

The asymptotic properties of the reaction front formed in a reversible reaction-diffusion process A+B<-->C (static) with initially separated reactants are investigated. The case of arbitrary nonzero values of the diffusion constants D(A) and D(B) and initial concentrations a(0) and b(0) of the reactants A and B is considered. The system is studied in the limit of t-->infinity and g-->0, where t and g are the time and the backward reaction rate constant, respectively. The dynamics of the reaction front is described as a crossover between the "irreversible" regime at times t<<g(-1) and the "reversible" regime at times t>>g(-1). The general properties of the crossover are studied with the help of an extended scaling approach formulated in this work. On the basis of the mean-field equations the analytical solutions in the reversible regime t>>g(-1) inside the reaction zone are discussed. It is shown that in the immobile reaction zone the reaction rate profile has two distinct maxima. This profile differs drastically from the usual single-maximum reaction rate profile inherent in the mobile reaction zone. The two-hump reaction zone profile is the result of the influence of C on the reaction rate in the reversible regime. Numerical computation of the mean-field kinetics equations supports the results of the asymptotic consideration.

Crossover from nonclassical to classical chemical kinetics in an initially separated A + B<-->C reaction-diffusion system with arbitrary diffusion constants

The asymptotic long-time properties of the reaction front formed in a reversible reaction-diffusion process A + B<-->C with initially separated reactants are investigated. The case of arbitrary nonzero values of the diffusion constants DA, DB, DC of the components A, B, C and the initial concentrations a0 and b0 of A and B is considered. The system is studied in the limit of g-->0, where g is the backward reaction rate constant. In accordance with previous work, the dynamics of the reaction front is described as a crossover between the "irreversible" regime at times t << g-1 and the "reversible" regime at times t >> g-1. It is shown that through this crossover the macroscopic properties of the reaction front, such as the global rate of C production, the motion of the reaction zone center, and the concentration profiles of the components outside the reaction front, are unchanged. The concentration profiles of the components inside the reaction zone are described by quasistatic equations. The results of the theoretical consideration are confirmed by computing the mean-field kinetics equations.

Properties of the crossover from nonclassical to classical chemical kinetics in a reversible A+B<-->C reaction diffusion process

We study the properties of the reaction front formed in a reversible reaction diffusion process A+B<-->C, with initially separated reactants. The case of the mobile C component is considered. In accordance with Chopard et al. [Phys. Rev. E 47, R40 (1993)] the dynamics of the front is described as a crossover between the "irreversible" regime at short times and the "reversible" regime at long times. A refined definition for the rate of C production is suggested, taking into account both the forward and the backward reaction rates. By this definition within the framework of the mean-field equations it is shown that the reversible regime is characterized by scaling of the local rate of C production as R(local) approximately t(-1) and by scaling of the global rate of C production as R(global) approximately t(-1/2). It is also established that in the considered special case of equal diffusion coefficients and equal initial concentrations, the macroscopic properties of the reaction front, such as the global rate of the C production R(global) and the concentration profiles of the components outside the front reaction, are unchanged through this crossover.

Reaction kinetics of diffusing particles injected into a reactive substrate

We analyze the kinetics of trapping (A+B-->B) and annihilation (A+B-->0) processes on a one-dimensional substrate with homogeneous distribution of immobile B particles while the A particles are supplied by a localized source. For the imperfect reaction case, we analyze both problems by means of a stochastic model and compare the results with numerical simulations. In addition, we present the exact analytical results of the stochastic model for the case of perfect trapping.