Pattern Formation in Reaction−Diffusion Systems: Cellular Acidity Fronts

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.

Fine tuning of pattern selection in the cadmium-hydroxide-system

Controlling self-organization in precipitation reactions has received growing attention in the efforts of engineering highly ordered spatial structures. Experiments have been successful in regulating the band patterns of the Liesegang phenomenon on various scales. Herein, we show that by adjusting the composition of the hydrogel medium, we can switch the final pattern between the classical band structure and the rare precipitate spots with hexagonal symmetry. The accompanying modeling study reveals that besides the modification of gel property, tuning of the time scale of diffusional spreading of hydroxide ions with respect to that of the phase separation drives the mode selection between one-dimensional band and two-dimensional spot patterns.

Directional coupling in spatially distributed nanoreactors

Silica based hollow nanospheres filled with a reactant solution act as nanoreactors. A close packed ensemble of the nanoshells comprise a porous medium through which a chemical front can propagate. The front velocity decreases as the chemical signal, in the shape of a reaction-diffusion front, is transmitted from one sphere to the other due to the high curvature at the contact points. Experiments reveal that front propagation occurs through the cavity of the nanoshells because surface activity of filled nanoparticles itself cannot support chemical front across the medium.

Chemical front propagates through a closed-packed cluster of nanoreactors made of hollow nanoshells filled with reactant solution.

Interaction between amino-functionalized inorganic nanoshells and acid-autocatalytic reactions

Amino-functionalized inorganic silica nanoshells with a diameter of 511 ± 57 nm are efficiently used as hydrogen ion binders with a base dissociation constant of (1.2 ± 0.1) × 10-4. The hydrogen removal has been shown to produce reaction-diffusion fronts of constant propagation velocity in the autocatalytic chlorite-tetrathionate reaction when it is run in thin planar slices of nanoshell-containing agarose gel to exclude all convection related effects. By controlling the exact amount of amino-functionalized hollow nanospheres in the gel matrix it is possible to finely tune the propagation velocity of the chemical front in the 0.1-10 cm h-1 range. Remarkably, this can be achieved with very low amino-functionalized hollow inorganic nanosphere loadings between 0.1-0.01 (m V-1)%. The front width has also been determined experimentally, which increases by a factor of two with one magnitude decrease in the front velocity.

Convection-Induced Fingering Fronts in the Chlorite-Trithionate Reaction

Based upon a former study, the chlorite-trithionate reaction can avoid the side reactions arising from the well-known alkaline decomposition of polythionates, making it a suitable candidate for investigating spatial front instabilities in a reaction-diffusion-convection system. In this work, the chlorite-trithionate reaction was investigated in a Hele-Shaw cell, in which fingering patterns were observed over a wide range of reactant concentrations. A significant density increment crossing the propagating front indicates that the fingering pattern is generated as a consequence of the buoyancy-driven instability due to the density changes of solute when the gap thickness is less than 4 mm. The velocity of the steepest descent in the propagating front depends almost linearly on the gap thickness but displays a saturation-like profile on the trithionate concentration as well as a maximum on the chlorite concentration. Numerical simulation using the Stokes-Brinkman Equation coupled to the reaction-diffusion processes, including hydrogen ion autocatalysis and consumption, reproduces the observed fingering fronts.

Self-Assembly of Charged Nanoparticles by an Autocatalytic Reaction Front

In this work we present that aggregation of charged and pH sensitive nanoparticles can be spatiotemporally controlled by an autonomous way using the chlorite-tetrathionate autocatalytic front, where the front regulates the electrostatic interaction between nanoparticles due to protonation of the capping (carboxylate-terminated) ligand. We found that the aggregation and sedimentation of nanoparticles in liquid phase with the effect of reversible binding of the autocatalyst (H(+)) play important roles in changing the front stability (mixing length) and the velocity of the front in both cases when the fronts propagate upward and downward. Calculation of interparticle interactions (electrostatic and van der Waals) with the measurement of front velocity revealed that the aggregation process occurs fast (within a few seconds) at the front position.

Propagation Behaviors of an Acid Wavefront Through a Microchannel Junction

Waves in reaction-diffusion systems yield a wealth of dynamic self-assembling phenomena in nature. Recent studies have been devoted to utilizing these active waves in conjunction with microscale technology. To provide a compass for controlling reaction-diffusion waves in microspaces, we have investigated the propagation behavior of one specific variety of the reaction-diffusion wave: an acid wave that utilizes an autocatalytic proton-production reaction. Furthermore, the acid wave that we have investigated occurs in a microchannel with a junction connecting circular and straight regions. The obtained results were compared with a neutralization wave that involves only a neutralization reaction. The acid wave was ignited by the addition of the appropriate amount of H2SO4 into the circular region that was filled with a substrate solution, where proton-consuming and proton-producing reactions followed a rapid neutralization reaction. At this stage, the wave penetrated and propagated into the channel region. Comparison between the acid and the neutralization waves clarified that the acid wave required a minimum threshold of H2SO4 concentration in order to be ignited and that the propagation of the acid wave was temporarily delayed because of the presence of intermediate chemical reaction steps. Furthermore, the propagation dynamics was found to be tuned through the configuration of the microchannel. The importance of microchannel configuration, especially for systems with a junction connecting different shapes, is discussed in terms of Fick's law and in terms of the proton flux from the circular to the straight regions.

Diffusion-driven instabilities by immobilizing the autocatalyst in ionic systems

Spatiotemporal coupling of an autocatalytic chemical reaction between ions with diffusion yields various types of reaction-diffusion patterns. The driving force is short range activation and long range inhibition which can be achieved by selective binding of the autocatalyst even for ions with equal mobility. For Turing and lateral instability, we show that identical charge on the autocatalyst and its counterpart has a stabilizing effect on the base state, while opposite charge on them favors the formation of spatial patterns with reversible binding.

Diffusive fingering in a precipitation reaction driven by autocatalysis

The interaction of an autocatalytic reaction with a fast precipitation reaction is shown to produce a permanent precipitate pattern where the major driving force is differential diffusion. The final structure emerges from the leading transient cellular front, the cusps of which evolve into precipitate free zones. The experimental observations are reproduced by a simple model calculation based on the empirical rate-law of the reaction.

Spatiotemporal chaos in the dynamics of buoyantly and diffusively unstable chemical fronts

Nonlinear dynamics resulting from the interplay between diffusive and buoyancy-driven Rayleigh-Taylor (RT) instabilities of autocatalytic traveling fronts are analyzed numerically for various values of the relevant parameters. These are the Rayleigh numbers of the reactant A and autocatalytic product B solutions as well as the ratio D=D(B)/D(A) between the diffusion coefficients of the two key chemical species. The interplay between the coarsening dynamics characteristic of the RT instability and the constant short wavelength modulation of the diffusive instability can lead in some regimes to complex dynamics dominated by irregular succession of birth and death of fingers. By using spectral entropy measurements, we characterize the transition between order and spatial disorder in this system. The analysis of the power spectrum and autocorrelation function, moreover, identifies similarities between the various spatial patterns. The contribution of the diffusive instability to the complex dynamics is discussed.

The effects of a complexing agent on the transverse stability of cubic autocatalytic reaction fronts

The effects of adding a complexing agent on the propagation and transverse stability of reaction fronts in a system based on cubic autocatalysis is considered. Adding the complexing agent is seen to reduce the propagation speed, alter the reaction dynamics and the concentration of the final reaction product of the propagating reaction fronts. A linear stability analysis (LSA) is considered to determine how the complexing agent affects the stability of planar reaction fronts through the numerical calculation of dispersion curves, plots of the growth rate sigma against wavenumber k. These dispersion curves show that adding the complexing agent can make the system unstable when it would otherwise be stable and, when the system is diffusionally unstable without the complexing agent, weaken this instability. An analysis valid for small values of k is undertaken, which confirms the results from the LSA and indicates how the critical value D(c) of the diffusion coefficient ratio D for the onset of an instability is changed by the addition of the complexing agent.

Interaction between buoyancy and diffusion-driven instabilities of propagating autocatalytic reaction fronts. II. Nonlinear simulations

The nonlinear dynamics resulting from the interplay between diffusive and buoyancy-driven Rayleigh-Taylor (RT) instabilities of autocatalytic traveling fronts are analyzed numerically for fronts ascending or descending in the gravity field and for various values of the relevant parameters, the Rayleigh numbers R(a) and R(b) of the reactant A and autocatalytic product B, respectively, and the ratio D=D(B)/D(A) of the diffusion coefficients of the two key chemical species. The interaction between the coarsening dynamics characteristic of the RT instability and the fixed short wavelength dynamics of the diffusive instability leads in some parameter regimes to complex dynamics dominated by the irregular succession of birth and death of fingers. Large single convective fingers with a tip deformed by the short wavelength diffusive instability are also observed. If D is sufficiently small and the RT instability is active, the concentration of the slower diffusing species B can be convected to values above its fully reacted concentration. Experimental conditions that would allow the observation of the dynamics predicted here are described.

Interaction between buoyancy and diffusion-driven instabilities of propagating autocatalytic reaction fronts. I. Linear stability analysis

The interaction between buoyancy-driven and diffusion-driven instabilities that can develop along a propagating reaction front is discussed for a system based on an autocatalytic reaction. Twelve different cases are possible depending on whether the front is ascending or descending in the gravity field, whether the reactant is heavier or lighter than the products, and whether the reactant diffuses faster, slower, or at the same rate as the product. A linear stability analysis (LSA) is undertaken, in which dispersion curves (plots of the growth rate sigma against wave number k) are derived for representative cases as well as an asymptotic analysis for small wave numbers. The results from the LSA indicate that, when the initial reactant is denser than the reaction products, upward propagating fronts remain unstable with the diffusion-driven instability enhancing this instability. Buoyantly stable downward propagating fronts become unstable when the system is also diffusionally unstable. When the initial reactant is lighter than the reaction products, any diffusionally unstable upward propagating front is stabilized by small buoyancy effects. A diffusional instability enhances the buoyant instability of a downward propagating front with there being a very strong interaction between these effects in this case.

Front fingering and complex dynamics driven by the interaction of buoyancy and diffusive instabilities

Traveling fronts can become transversally unstable either because of a diffusive instability arising when the key variables diffuse at sufficiently different rates or because of a buoyancy-driven Rayleigh-Taylor mechanism when the density jump across the front is statically unfavorable. The interaction between such diffusive and buoyancy instabilities of fronts is analyzed theoretically for a simple model system. Linear stability analysis and nonlinear simulations show that their interplay changes considerably the stability properties with regard to the pure Rayleigh-Taylor or diffusive instabilities of fronts. In particular, an instability scenario can arise which triggers convection around statically stable fronts as a result of differential diffusion. Moreover, spatiotemporal chaos can be observed when both buoyancy and diffusive effects cooperate to destabilize the front. Experimental conditions to test our predictions are suggested.

Thermodiffusion-induced instabilities in reactive systems

We examine the influence of a thermal gradient on a classical reaction-diffusion system. The different instability regions in the appropriate parameter space are examined. We show how the imposed temperature gradient destabilizes a chemical front via the Soret effect, giving rise to both absolute and convective instability.

Effects of a constant electric field on the diffusional instability of cubic autocatalytic reaction fronts

An electric field applied in the direction of propagation of a chemical reaction-diffusion front can affect the stability of this front with regard to diffusive instabilities. The influence of an applied constant electric field is investigated by a linear stability analysis and by nonlinear simulations of a simple chemical system based on the cubic autocatalytic reaction A-+2B--->3B-. The diffusional stability of the front is seen to depend on the intensity E and sign of the applied field, and D, the ratio diffusion coefficients of the reactant species. Depending on E, the front can become more or less diffusively unstable for a given value of D. Above a critical value of E, which depends on D, electrophoretic separation of the two fronts is observed.

Periodic heterogeneity-driven resonance amplification in density fingering

Periodic heterogeneity is introduced in experiments with thin solution layers where downward propagating planar autocatalytic fronts are hydrodynamically unstable and cellular patterns develop. The evolution of fingers is greatly affected by the spatial heterogeneity when the wave number associated with it falls in the vicinity of the most unstable mode of the reference system with uniform thickness. The imposed heterogeneity will drive the instability by amplifying the modes with the matching wave numbers as indicated by the experimentally constructed dispersion curves.

Introduction: Self-organization in nonequilibrium chemical systems

The field of self-organization in nonequilibrium chemical systems comprises the study of dynamical phenomena in chemically reacting systems far from equilibrium. Systematic exploration of this area began with investigations of the temporal behavior of the Belousov-Zhabotinsky oscillating reaction, discovered accidentally in the former Soviet Union in the 1950s. The field soon advanced into chemical waves in excitable media and propagating fronts. With the systematic design of oscillating reactions in the 1980s and the discovery of Turing patterns in the 1990s, the scope of these studies expanded dramatically. The articles in this Focus Issue provide an overview of the development and current state of the field.

Periodic spatiotemporal patterns in a two-dimensional two-variable reaction-diffusion model

Periodic spatiotemporal two-dimensional (2D) asymptotic patterns in an excitable two-variable thermochemical (reaction-diffusion) system are shown. In a one-dimensional system the traveling impulse which reflects from impermeable boundaries is a stable asymptotic solution if the diffusion coefficient of the reactant is greater than the thermal diffusivity of the system. Periodic patterns of two symmetries are presented in the 2D system: the impulse of excitation propagating along the diagonal of a square spatial domain and a structure consisting of curved impulses which propagate in the direction perpendicular to one side of a rectangular domain.

Nonlinear fingering dynamics of reaction-diffusion acidity fronts: Self-similar scaling and influence of differential diffusion

Nonlinear interactions between chemical reactions and buoyancy-driven Rayleigh-Taylor instability of reaction-diffusion acidity fronts of the chlorite-tetrathionate (CT) reaction are studied theoretically in a vertical Hele-Shaw cell or a porous medium. To do so, we perform a numerical integration of a two-variable reaction-diffusion model of the CT system coupled through an advection term to Darcy's law ruling the evolution of the velocity field of the fluid. The fingering dynamics of these chemical fronts is characterized by the appearance of several fingers at onset. These fingers then undergo coarsening and eventually merge to form one single symmetric finger. We study this asymptotic dynamics as a function of the three dimensionless parameters of the problem, i.e., the Damkohler number Da, the diffusivity ratio delta of the two chemical species, and the Rayleigh number Ra constructed here on the basis of the width L(y) of the system. For moderate values of Ra, the asymptotic single finger is shown to have self-similar scaling properties while above a given value of Ra, which depends on the other values of the parameters, tip splitting comes into play. Increasing the difference of diffusivities of the two chemical species (i.e., increasing delta) leads to more efficient coarsening and smaller asymptotic fingers. Experimental procedures to verify our predictions are proposed.

A Three-Variable Model for the Explanation of the “Supercatalytic” Effect of Hydrogen Ion in the Chlorite−Tetrathionate Reaction

It has been shown that not only the slow direct but also the indirect (HOCl-catalyzed) reaction between chlorite and tetrathionate ions is second order with respect to hydrogen ion. Since the direct reaction was found to be orders of magnitude slower than the parallel HOCl-catalyzed pathway, a three-variable model is derived from the previously published five-step model taking into account the experimentally determined H+ concentration dependence of its rate coefficients by neglecting the direct reaction. The new three-variable model indicates that the "supercatalytic" effect of the hydrogen ion in the HOCl-catalyzed pathway arises from the pH dependence of the individual reactions of the five-step model. The new three-variable model also accounts for the continuous change of the stoichiometric ratio of the reactants and provides a simple kinetic law for involving it in the partial differential equation systems widely used in the study of spatiotemporal behavior of the chlorite-tetrathionate reaction.

Spatial bistability and excitability in the chlorite-tetrathionate reaction in cylindrical and conical geometries

Spatial bistability and excitability in the chlorite-tetrathionate reaction, performed in gels fed by diffusion from one boundary, have been extensively studied, both experimentally and numerically, in a flat annular striplike geometry. We first complement these numerical results. Afterwards, we extend the calculations to the cylindrical and conical geometries. In the cylinder, we compute the limits of bistability and of excitability which are important for experiments in chemomechanics but cannot be directly measured. The results of the simulations in the conical geometry agree with previous experiments on the corresponding setup. We show that the characteristics of the traveling waves which spontaneously arise in the latter geometry provide a simple and direct experimental access to these limits.

Autocatalysis and Self-Inhibition: Coupled Kinetic Phenomena in the Chlorite−Tetrathionate Reaction

The initial rate of formation of chlorine dioxide in the chlorite-tetrathionate reaction changes in an unusual fashion. The formal kinetic order of both reactants varies over a very wide range. Moreover, chlorite ion behaves not just as a simple reactant, but also as a self-inhibitor. A five-step scheme, derived from an eight-step mechanism, is proposed in which the autocatalytic formation of HOCl plays a central role in accounting for this kinetic behavior.

Simple chemomechanical process for self-generation of rhythms and forms

We show that cross-coupling between the geometrical changes and the chemical reaction-diffusion regimes within a piece of gel embedded in a stationary reactive medium kept far from equilibrium can destabilize the trivial steady state and lead to spatiotemporal dissipative structures. The involved physical processes are the spatial bistability and the swelling properties of the gel. As an illustration of this morphogenetic process, we show that a sphere of gel immersed within such a medium with autocatalytic properties can exhibit periodic radius pulsations, the amplitude of which are controlled by chemistry.

Dynamical effects induced by long range activation in a nonequilibrium reaction-diffusion system

We show both experimentally and numerically that the time scales separation introduced by long range activation can induce oscillations and excitability in nonequilibrium reaction-diffusion systems that would otherwise only exhibit bistability. Namely, we show that in the chlorite-tetrathionate reaction, where the autocatalytic species H+ diffuses faster than the substrates, the spatial bistability domain in the nonequilibrium phase diagram is extended with oscillatory and excitability domains. A simple model and a more realistic model qualitatively account for the observed dynamical behavior. The latter model provides quantitative agreement with the experiments.