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.     

Steady Marangoni flow traveling with chemical fronts

When autocatalytic chemical fronts propagate in thin layers of solution in contact with air, they can induce capillary flows due to surface tension gradients across the front (Marangoni flows). We investigate here such an interplay between autocatalytic reactions, diffusion, and Marangoni effects with a theoretical model coupling the incompressible Navier-Stokes equations to a conservation equation for the autocatalytic product concentration in the absence of gravity and for isothermal conditions. The boundary condition at the open liquid/air interface takes the surface activity of this product into account and introduces the solutal Marangoni number M representing the intensity of the coupling between hydrodynamics and reaction-diffusion processes. Positive and negative Marangoni numbers correspond, respectively, to the cases where the product decreases or increases surface tension behind the front. We show that, in both cases, such coupled systems reach an asymptotic dynamics characterized by a steady fluid vortex traveling at a constant speed with the front and deforming it, with, however, an asymmetry between the results for positive and negative M. A parametric study shows that increased propagation speed, front deformation, and possible transient oscillating dynamics occur when the absolute value of M is increased.     

A + B → C reaction fronts in Hele-Shaw cells under modulated gravitational acceleration

The dynamics of A + B → C reaction fronts is studied under modulated gravitational acceleration by means of a combination of parabolic flight experiments and numerical simulations. During modulated gravity the front position undergoes periodic modulation with an accelerated front propagation under hyper-gravity together with a slowing down under low gravity. The underlying reason for this is an amplification and a decay, respectively, of the buoyancy-driven double vortex associated with the front propagation under standard gravitational acceleration, as explained by reaction-diffusion-convection simulations of convection around an A + B → C front. Deeper insights into the correlation between grey-value changes in the experimental shadowgraph images and characteristic changes in the concentration profiles are obtained by a numerical simulation of the imaging process.     

On the variety of traveling fronts in one-variable multistable reaction-diffusion systems

The still-open problem of the variety of asymptotic solutions to one-variable, one-dimensional infinite multistable reaction-diffusion systems is solved. We show that in such systems, besides monotonic traveling fronts, nonmonotonic traveling fronts can exist for appropriate initial conditions. The dependence of numbers of various types of traveling fronts on the number of stable stationary states also is given. Examples of traveling fronts for the chemical model describing two enzymatic (catalytic) reactions inhibited by an excess of their reactant is presented.     

Rayleigh-Taylor instabilities in reaction-diffusion systems inside Hele-Shaw cell modified by the action of temperature

The influence of temperature in the buoyancy driven Rayleigh-Taylor instability of reaction-diffusion fronts is investigated experimentally in Hele-Shaw cells. The acid autocatalysis of chlorite-tetrathionate reaction coupled to molecular diffusion yields exothermic planar reaction-diffusion fronts separating two miscible reactant and product solutions. The resulting chemical front moves downwards invading the fresh reactants, leaving the products of the reaction behind it. The density of the product solution is higher than the reactant solution; hence, the traveling front is buoyantly unstable and develops density fingers in time (Rayleigh-Taylor instability) when the products are above the reactants. The kinetic constant of a chemical reaction varies due to thermal effects. This may stabilize the exothermic descending front when temperature is increased, so that the mixing zone decreases, modifying the fingering patterns, until it almost disappears. The authors study the influence of the temperature variation on the instability pattern figure observed in the chlorite-tetrathionate reaction for long times, corresponding to the nonlinear regime.     

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.     

Fingering of exothermic reaction-diffusion fronts in Hele-Shaw cells with conducting walls

We consider the influence of heat losses through the walls of a Hele-Shaw cell on the linear stability and nonlinear dynamics of exothermic chemical fronts whose solutal and thermal contributions to density changes have the same signs. Our analysis is based on the reaction-diffusion-convection equations obtained from the Darcy-Boussinesq approximation. The parameters governing the equations are the Damkohler number, a kinetic parameter d, the Lewis number Le, the thermal-expansion coefficient gammaT, and a heat-transfer coefficient alpha which measures heat losses through the walls. We show that for thermally insulating walls, the temperature profile is a front that follows the concentration profile, while in the presence of heat losses, the temperature profile becomes a pulse that leads to a nonmonotonic density profile which in turn may lead to a destabilization of an otherwise stable front.     

Lock-exchange experiments with an autocatalytic reaction front

A viscous lock-exchange gravity current corresponds to the reciprocal exchange of two fluids of different densities in a horizontal channel. The resulting front between the two fluids spreads as the square root of time, with a diffusion coefficient reflecting the buoyancy, viscosity, and geometrical configuration of the current. On the other hand, an autocatalytic reaction front between a reactant and a product may propagate as a solitary wave, namely, at a constant velocity and with a stationary concentration profile, resulting from the balance between molecular diffusion and chemical reaction. In most systems, the fluid left behind the front has a different density leading to a lock-exchange configuration. We revisit, with a chemical reaction, the classical situation of lock-exchange. We present an experimental analysis of buoyancy effects on the shape and the velocity of the iodate arsenous acid autocatalytic reaction fronts, propagating in horizontal rectangular channels and for a wide range of aspect ratios (1/3 to 20) and cylindrical tubes. We do observe stationary-shaped fronts, spanning the height of the cell and propagating along the cell axis. Our data support the contention that the front velocity and its extension are linked to each other and that their variations scale with a single variable involving the diffusion coefficient of the lock-exchange in the absence of chemical reaction. This analysis is supported by results obtained with lattice Bathnagar-Gross-Krook (BGK) simulations Jarrige et al. [Phys. Rev. E 81, 06631 (2010)], in other geometries (like in 2D simulations by Rongy et al. [J. Chem. Phys. 127, 114710 (2007)] and experiments in cylindrical tubes by Pojman et al. [J. Phys. Chem. 95, 1299 (1991)]), and for another chemical reaction Schuszter et al. [Phys. Rev. E 79, 016216 (2009)].     

Reaction-diffusion travelling waves in the acidic nitrate-ferroin reaction

A two-variable model proposed for the acidic nitrate-ferroin reaction is considered in the reaction-diffusion context. An initial-value problem in which an amount of nitrate is introduced locally into ferroin at uniform concentration is treated both analytically and numerically. It is shown that the large time structure is a reaction-diffusion travelling wave of permanent form propagating with constant speed. This asymptotic wave speed is shown to be the minimum possible wave speed and the asymptotic approach to this value is estimated. Properties of the permanent-form travelling waves are derived and solutions valid for small and large values of a parameter , involved in the kinetic mechanism, are obtained.     

Differential diffusion effects on buoyancy-driven instabilities of acid-base fronts: the case of a color indicator

Buoyancy-driven hydrodynamic instabilities of acid-base fronts are studied both experimentally and theoretically in the case where an aqueous solution of a strong acid is put above a denser aqueous solution of a color indicator in the gravity field. The neutralization reaction between the acid and the color indicator as well as their differential diffusion modifies the initially stable density profile in the system and can trigger convective motions both above and below the initial contact line. The type of patterns observed as well as their wavelength and the speed of the reaction front are shown to depend on the value of the initial concentrations of the acid and of the color indicator and on their ratio. A reaction-diffusion model based on charge balances and ion pair mobility explains how the instability scenarios change when the concentration of the reactants are varied.     

Systematic front distortion and presence of consecutive fronts in a precipitation system

A new simple reaction-diffusion system is presented focusing on pattern formation phenomena as consecutive precipitation fronts and distortion of the precipitation front. The chemical system investigated here is based on the amphoteric property of aluminum hydroxide and exhibits two unique phenomena. Both the existence of consecutive precipitation fronts and distortion are reported for the first time. The precipitation patterns could be controlled by the pH field, and the distortion of the precipitation front can be practical for microtechnological applications of reaction-diffusion systems.     

Migration-driven instability in the chlorite-tetrathionate reaction

We have studied the lateral stability of planar reaction-diffusion fronts in an autocatalytic reaction between aqueous ions in an externally imposed electric field. In our experiments, migration drives the pattern formation leading to cellular structures where the sufficiently greater migrational flux of the reactant with respect to that of the autocatalyst is the driving force. The difference in electric field strength between the two sides of the thin reaction front results from the significant increase in conductivity during the reaction. The results of the theoretical analysis based on the empirical rate-law model of the reaction reproduce the behavior observed experimentally.     

Turbulent fronts in resonantly forced oscillatory systems

Phase fronts in the forced complex Ginzburg-Landau equation, a model of a resonantly forced oscillatory reaction-diffusion system, are studied in the 3:1 resonance regime. The focus is on the turbulent (Benjamin-Feir-unstable) regime of the corresponding unforced system; in the forced system, phase fronts between spatially uniform phase-locked states exhibit complex dynamics. In one dimension, for strong forcing, phase fronts move with constant velocity. As the forcing intensity is lowered there is a bifurcation to oscillatory motion, followed by a bifurcation to a regime in which fronts multiply via the nucleation of domains of the third homogeneous phase in the front. In two dimensional systems, rough fronts with turbulent, complex internal structure may arise. For a critical value of the forcing intensity there is a nonequilibrium phase transition in which the turbulent interface grows to occupy the entire system. The phenomena we explore can be probed by experiments on periodically forced light sensitive reaction-diffusion systems.     

Oscillatory and stationary convective patterns in a reaction driven gravity current

Horizontally propagating chemical fronts are studied in a thin solution layer of the acid-catalyzed chlorite-tetrathionate reaction. Unusual cellular patterns develop when significant amount of autocatalyst is bound to polyelectrolyte with low mobility: both oscillatory and stationary patterns evolve as a result of the interaction between the reaction front and the superposed gravity current. The concentration of the polyelectrolyte regulating the velocity of front propagation serves as a bifurcation parameter for switching between the two basic patterns.     

Pattern formation and self-organization in a simple precipitation system

Various types of pattern formation and self-organization phenomena can be observed in biological, chemical, and geochemical systems due to the interaction of reaction with diffusion. The appearance of static precipitation patterns was reported first by Liesegang in 1896. Traveling waves and dynamically changing patterns can also exist in reaction-diffusion systems: the Belousov-Zhabotinsky reaction provides a classical example for these phenomena. Until now, no experimental evidence had been found for the presence of such dynamical patterns in precipitation systems. Pattern formation phenomena, as a result of precipitation front coupling with traveling waves, are investigated in a new simple reaction-diffusion system that is based on the precipitation and complex formation of aluminum hydroxide. A unique kind of self-organization, the spontaneous appearance of traveling waves, and spiral formation inside a precipitation front is reported. The newly designed system is a simple one (we need just two inorganic reactants, and the experimental setup is simple), in which dynamically changing pattern formation can be observed. This work could show a new perspective in precipitation pattern formation and geochemical self-organization.     

Factors affecting propagating fronts of addition polymerization: Velocity,front curvature,temperatue profile,conversion, and molecular weight distribution

Several properties of propagating fronts of addition polymerization were studied. A power function could be fit to the velocity dependence on initiator concentration, but not with the exponents predicted by current models or in agreement with other published work. Bubbles from the volatile by-products of initiator decomposition were found to affect the front velocity and curvature. The front velocity for triethylene glycol dimethacrylate polymerization was found to depend linearly on temperature over a moderate range. The conversion of methacrylic acid in fronts varied greatly with initiator type and concentration. Benzoyl peroxide produced much lower conversion than t-butyl peroxide, but fronts with tBPO propagated slower. A dual initiator system of BPO and tBPO produced rapidly propagating fronts with good conversion but the contribution of each initiator to the velocity was not additive. The possibility of chain branching was considered. The apparent molecular weight distributions were very broad, often trimodal, and found to depend on initiator type and concentration as well as the tube diameter. The temperature profiles were measured and found to be very sharp for BPO and broader for tBPO but both had front temperatures in excess of 200°C, indicating a high ceiling temperature. © 1995 John Wiley & Sons, Inc.     

Propagating fronts in reaction–diffusion systems

Propagating reaction–diffusion fronts constitute one of the paradigms in the realm of the nonlinear chemical phenomena. In this paper different situations are considered. Firstly, we discuss the problem of front propagation in a two‐variable chemical system exhibiting multiple stationary states. Emphasis is put on the question of velocity selection. In section 3 of our contribution, the question of front propagating in spatially modulated and noisy media is addressed. Finally, we also briefly comment on the problem of reaction–diffusion fronts in non‐quiescent media. In this particular scenario we simply aim at introducing the two basic propagation modes, i.e., thin versus distributed reaction fronts, that are identified in our numerical simulations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Influence of an electric field on the buoyancy-driven instabilities

The influence of dc electric fields (EFs) on the development of buoyancy-driven instabilities of reaction fronts is investigated experimentally in a modified Hele-Shaw cell for the arsenous acid-iodate system. Assessment of effects of external EFs is made both visually and through dispersion curves. It is shown that density fingering, observed on ascending fronts, is suppressed by the EF if the front propagates towards the positive electrode and is enhanced when the front propagates towards the negative electrode. The stabilizing (destabilizing) effects include slower (faster) development of fingers and the decrease (increase) in their numbers. The descending front, stable under no EF conditions, remains stable when an EF is applied with the positive electrode facing the approaching front. When the descending front faces the negative electrode, the tiny fingerlike structure develops after quite a long time.