Gel-free experiments of reaction-diffusion front kinetics

Unraveling dispersion and buoyancy dynamics around radial A + B → C reaction fronts: microgravity experiments and numerical simulations

Radial Reaction-Diffusion-Advection (RDA) fronts for A + B → C reactions find wide applications in many natural and technological processes. In liquid solutions, their dynamics can be perturbed by buoyancy-driven convection due to concentration gradients across the front. In this context, we conducted microgravity experiments aboard a sounding rocket, in order to disentangle dispersion and buoyancy effects in such fronts. We studied experimentally the dynamics due to the radial injection of A in B at a constant flow rate, in absence of gravity. We compared the obtained results with numerical simulations using either radial one- (1D) or two-dimensional (2D) models. We showed that gravitational acceleration significantly distorts the RDA dynamics on ground, even if the vertical dimension of the reactor and density gradients are small. We further quantified the importance of such buoyant phenomena. Finally, we showed that 1D numerical models with radial symmetry fail to predict the dynamics of RDA fronts in thicker geometries, while 2D radial models are necessary to accurately describe RDA dynamics where Taylor-Aris dispersion is significant.

Steady state of a two-species annihilation process with separated reactants

We describe the steady state of the annihilation process of a one-dimensional system of two initially separated reactants A and B. The parameters that define the dynamical behavior of the system are the diffusion constant, the reaction rate, and the deposition rate. Depending on the ratio between those parameters, the system exhibits a crossover between a diffusion-limited (DL) regime and a reaction-limited (RL) regime. We found that a key quantity to describe the reaction process in the system is the probability p(x_{A},x_{B}) to find the rightmost A (RMA) particle and the leftmost B (LMB) particle at the positions x_{A} and x_{B}, respectively. The statistical behavior of the system in both regimes is described using the density of particles, the gap length distribution x_{B}-x_{A}, the marginal probabilities p_{A}(x_{A}) and p_{B}(x_{B}), and the reaction kernel. For both regimes, this kernel can be approximated by using p(x_{A},x_{B}). We found an excellent agreement between the numerical and analytical results for all calculated quantities despite the reaction process being quite different in both regimes. In the DL regime, the reaction kernel can be approximated by the probability to find the RMA and LMB particles in adjacent sites. In the RL regime, the kernel depends on the marginal probabilities p_{A}(x_{A}) and p_{B}(x_{B}).

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.

Effects of radial injection and solution thickness on the dynamics of confined A + B → C chemical fronts

The reconstructed amount of product n_C as the volume V of KSCN injected radially into Fe(NO₃)₃ increases and comparison to theory.

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.

Flow Control of A+B→C Fronts by Radial Injection

The dynamics of A+B→C fronts is analyzed theoretically in the presence of passive advection when A is injected radially into B at a constant inlet flow rate Q. We compute the long-time evolution of the front position, r_{f}, of its width, w, and of the local production rate R of the product C at r_{f}. We show that, while advection does not change the well-known scaling exponents of the evolution of corresponding reaction-diffusion fronts, their dynamics is however significantly influenced by the injection. In particular, the total amount of product varies as Q^{-1/2} for a given volume of injected reactant and the front position as Q^{1/2} for a given time, paving the way to a flow control of the amount and spatial distribution of the reaction front product. This control strategy compares well with calcium carbonate precipitation experiments for which the amount of solid product generated in flow conditions at fixed concentrations of reactants and the front position can be tuned by varying the flow rate.

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.

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.

Stationary fronts in an A + B --> 0 reaction under subdiffusion

We consider stationary profiles of reactants' concentrations and of reaction zones of an A + B --> 0 reaction in a flat subdiffusive medium fed by reactants of both types on both sides. The structures formed under such conditions differ strikingly from those in simple diffusion and exhibit accumulation and depletion zones close to the boundaries and nonmonotonic behavior of the reaction intensity with respect to the reactants' concentrations at the boundaries. These findings are connected to an effectively nonlinear character of transport in subdiffusive systems under reactions.

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.

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.