Autocatalytic Reaction Fronts Research Articles

Dispersion curves in the diffusional instability of autocatalytic reaction fronts

A (linear) stability analysis of planar reaction fronts to transverse perturbations is considered for systems based on cubic autocatalysis and a model for the chlorite-tetrathionate reaction. Dispersion curves (plots of the growth rate sigma against a transverse wave-number k) are obtained. In both cases it is seen that there is a nonzero value D0 of D (the ratio of the diffusion coefficients of autocatalyst and substrate) at which sigma(max), the maximum value of sigma for a given value of D, achieves its largest value, with sigma(max) being less for other values of D and becoming small as D decreases to zero. The existence of the optimum value D0 for initiating a diffusional instability is confirmed, in the cubic autocatalysis case, by an asymptotic analysis for small wave numbers.

Coarsening in the buoyancy-driven instability of a reaction-diffusion front

When propagating in a vertical direction an autocatalytic reaction front associated with a change in density can become buoyantly unstable, leading to the formation of a fingerlike pattern. Later on these fingers start to interact. Their temporal evolution is studied experimentally by tracking the horizontal and vertical locations of the extrema of the front pattern. A proceeding development towards larger spatial scales is found. This is reflected in the differences in the vertical speed of neighboring fingers: continually some fingers start to decelerate and vanish finally in the neighboring ones which show a simultaneous acceleration. In addition, weak lateral movements of fingers towards gaps are observed, but there is no evidence for a correlation with the extinction of fingers.

Mixing and reaction fronts in laminar flows.

Autocatalytic reaction fronts between unreacted and reacted mixtures in the absence of fluid flow propagate as solitary waves. In the presence of imposed flow, the interplay between diffusion and advection enhances the mixing, leading to Taylor hydrodynamic dispersion. We present asymptotic theories in the two limits of small and large Thiele modulus (slow and fast reaction kinetics, respectively) that incorporate flow, diffusion, and reaction. For the first case, we show that the problem can be handled to leading order by the introduction of the Taylor dispersion replacing the molecular diffusion coefficient by its Taylor counterpart. In the second case, the leading-order behavior satisfies the eikonal equation. Numerical simulations using a lattice gas model show good agreement with the theory. The Taylor model is relevant to microfluidics applications, whereas the eikonal model applies at larger length scales.

Buoyancy-driven instability of an autocatalytic reaction front in a Hele-Shaw cell.

An autocatalytic reaction-diffusion front between two reacting species may propagate as a solitary wave, namely, at constant velocity and with a stationary concentration profile. Recent experiments on such reactions have been reported to be buoyancy unstable, under certain conditions. We calculate the linear dispersion relation of the resulting instability, by applying our recent analysis of the Rayleigh-Taylor instability of two miscible fluids in a Hele-Shaw cell. The computed dispersion relation as well as our three-dimensional lattice Bhatnagar-Gross-Krook (BGK) simulations fit reasonably well experimental growth rates reported previously.

Gravitational instability of miscible fluids in a Hele-Shaw cell and chemical reaction

Autocatalytic reaction fronts are able to propagate as a solitary wave, that is at a constant velocity and with a stationary concentration profile which result from a balance between diffusion and chemical reaction. Experiments in thin cells have shown a buoyant instability due to the slightly smaller density of the reacted fluid. We provide an extension of our recent analysis of the Rayleigh-Taylor instability of the interface between two miscible fluids by including the chemical process by a reaction-convection-diffusion approach. The computed dispersion relation as well as our lattice BGK simulations compare reasonably well with the growth rates obtained experimentally.

Lateral instabilities in cubic autocatalytic reaction fronts: The effect of autocatalyst decay

The conditions are derived for the onset of lateral instabilities in planar waves propagating in a chemical system based on cubic autocatalysis coupled with autocatalyst decay, represented by the parameter κ. The results from a linear stability analysis of the corresponding traveling wave equations are presented and compared with predictions obtained from a “thin front” analysis. Instabilities arise when δ, the ratio of the diffusion coefficients of the reactant and autocatalyst, is greater than some critical value δcrit, with δcrit being found to depend strongly on κ. Numerical simulations of the full initial-value problem are also determined and confirm the theoretical predictions.

Growth rates of the buoyancy-driven instability of an autocatalytic reaction front in a narrow cell

Experimental studies were performed on the buoyancy-driven instability of an autocatalytic reaction front in a quasi-2D cell. The unstable density stratification at an ascending front leads to convection that results in a fingerlike front deformation. The growth rates of the spatial modes of the instability are determined at the initial stage. A stabilization is found at higher wave numbers, while the system is unstable against low wave number perturbations. Whereas comparison with a reported model governed by Hele-Shaw flow fails, a two-dimensional Navier-Stokes model yields more satisfactory results. Still, present deviations suggest the presence of an additional mechanism that suppresses the growth.

Lateral instabilities of cubic autocatalytic reaction fronts in a constant electric field

The region of instability for planar reaction fronts of cubic autocatalysis between ionic species under constant electric field has been determined accurately. The ratio of diffusion coefficients at the onset of instability δcr is substantially varied by the component-dependent drift and directly proportional to the concentration of the autocatalyst behind the front βs as δcr=2.3002βs. This opens the possibility to use electric field as a control parameter for reaction-front instabilities. The dispersion relation calculated from the linear stability analysis of the full system is in good agreement with the initial evolution of the Fourier modes associated with the slightly perturbed planar reaction front obtained by the direct integration of the governing equations in two spatial dimensions.

Electric field induced lateral instability in a simple autocatalytic front

The effect of ionic drift caused by small constant electric field on autocatalytic reaction fronts of ionic species is studied both theoretically and numerically. Besides varying the velocity of propagation, the electric field parallel to the direction of propagation may induce lateral instability in planar fronts resulting in the emergence of cellular structures. The difference in the diffusivities at the onset of instability are lowered when the electric field tends to separate the species spatially. The predictions of the linear stability analysis based on a thin-front approximation are confirmed by the numerical integration of the full two-dimensional system.

Pattern formation and evolution near autocatalytic reaction fronts in a narrow vertical slab.

Linear analysis and nonlinear numerical simulations of autocatalytic reaction fronts ascending in narrow vertically unbounded slabs describe the growth, development, and annihilation of fingers in the front, the dynamics of edge suppression, and a secondary transition to a two-roll state above the onset of convection. The pattern formation and evolution of the reaction fronts are determined by the horizontal aspect ratio \ensuremath{\Gamma}=b/a and the dimensionless driving parameter S=\ensuremath{\delta}${\mathit{ga}}^{3}$/\ensuremath{\nu}${\mathit{D}}_{\mathit{C}}$, which involve the gap thickness a, the slab width b, the fractional density difference \ensuremath{\delta} between the unreacted and reacted solutions, the gravitational acceleration g, the kinematic viscosity \ensuremath{\nu}, and the catalyst molecular diffusivity ${\mathit{D}}_{\mathit{C}}$. The reaction fronts satisfy a chemical reaction-diffusion equation and two-dimensional Navier-Stokes equations describing the average Poiseuille velocity in the vertical plane perpendicular to the gap direction. The wavelength of maximum growth rate reaches a minimum value at a\ensuremath{\approxeq}1 mm. \textcopyright{} 1996 The American Physical Society.

Convective fingering of an autocatalytic reaction front.

We report experimental observations of the convection-driven fingering instability of an iodate-arsenous acid chemical reaction front. The front propagated upward in a vertical slab; the thickness of the slab was varied to control the degree of instability. We observed the onset and subsequent nonlinear evolution of the fingers, which were made visible by a pH indicator. We measured the spacing of the fingers during their initial stages and compared this to the wavelength of the fastest growing linear mode predicted by the stability analysis of Huang et al. [Phys. Rev. E 48, 4378 (1993), and unpublished]. We find agreement with the thickness dependence predicted by the theory. \textcopyright{} 1996 The American Physical Society.

Ionic Autocatalytic Reaction Fronts in Electric Fields

The travelling waves of the front type that are initiated in a system in which there is an ionic autocatalytic reaction, with a quadratic rate law, are considered when a constant current I is applied. These waves are seen to depend on both the current and the parameter δB, the ratio of diffusion coefficients of autocatalyst and substrate. For δB > 1 there is a transition from kinetic front waves to Kohlrausch electrophoretic fronts as I is varied. For δB < 1, both kinetic waves and Kohlrausch fronts are seen as well as an additional type of wave where a much enhanced reaction takes place.

Finite thermal diffusivity at onset of convection in autocatalytic systems: Discontinuous fluid density

A linear convective stability analysis for propagating autocatalytic reaction fronts includes density differences due to both thermal and chemical gradients. Critical parameters for the onset of convection are calculated for an unbounded geometry, a vertical slab, and a vertical cylinder. Thermal effects are important at unstable wavelengths well above the critical wavelength for the onset of convection.

Fractal properties of propagating fronts in a strongly stirred fluid

The fractal properties of propagating aqueous autocatalytic chemical reaction fronts are measured in a capillary-wave (CW) flow at values of the ratio of the RMS intensity of the fluid velocity fluctuation (u′) to the laminar propagation rate of the front (SL) up to 220. The images of the fronts are found to exhibit fractal behavior with a fractal dimension (d) of 1.31±0.06, which is very similar to some measurements in gaseous flame fronts, as well as isoscalar contours of passive dyes in CW and other randomly stirred flows. These results suggest that u′/SL, thermal expansion, variations of viscosity and diffusivity across the flame front, and the turbulence spectrum do not significantly affect d in randomly stirred flows.

Derivation of a nonlinear front evolution equation for chemical waves involving convection

The nonlinear stability of exothermic autocatalytic reaction fronts is considered using the viscous thermohydrodynamic fluid equations in the limits of infinite and zero thermal diffusivity. A nonlinear front evolution equation is derived using asymptotic analysis and is solved for an ascending front. The method used is similar to that which has been used in the derivation of a front equation for flame propagation.

Onset of convection for autocatalytic reaction fronts in a vertical slab.

A fully three-dimensional linear stability analysis shows that ascending autocatalytic reaction fronts in vertical slabs are unstable to convection for large-wavelength perturbations at all finite values of the dimensionless driving parameter S=\ensuremath{\delta}${\mathit{ga}}^{3}$/\ensuremath{\nu}${\mathit{D}}_{\mathit{C}}$. This parameter involves a fractional density difference \ensuremath{\delta} between the unreacted and reacted fluids, the acceleration of gravity g, the slab width a, the kinematic viscosity \ensuremath{\nu}, and the catalyst molecular diffusivity ${\mathit{D}}_{\mathit{C}}$. Buoyancy dominates over the competing curvature dependence of the front velocity in a band 0q${\mathit{q}}_{\mathit{c}}$ of unstable dimensionless wave numbers, with ${\mathit{q}}_{\mathit{c}}$\ensuremath{\rightarrow}S/24 as S\ensuremath{\rightarrow}0 and ${\mathit{q}}_{\mathit{c}}$\ensuremath{\rightarrow}(S/4${)}^{1/3}$ as S\ensuremath{\rightarrow}\ensuremath{\infty}. As S\ensuremath{\rightarrow}0, the perturbation with wave number ${\mathit{q}}_{\mathit{m}}$=${\mathit{q}}_{\mathit{c}}$/2 has the maximum dimensionless growth rate ${\mathrm{\ensuremath{\sigma}}}_{\mathit{m}}$=${\mathit{D}}_{\mathit{C}}$${\mathit{S}}^{2}$/48\ensuremath{\nu}. For general S, the cutoff wave number ${\mathit{q}}_{\mathit{c}}$ is calculated using exact analytical solutions for the perturbed fluid velocity. The calculated results should be observable in experiments.

Convective instability of autocatalytic reaction fronts in vertical cylinders

Linear stability analysis predicts that the onset of convection for an ascending autocatalytic reaction front in a vertical cylinder corresponds to a nonaxisymmetric mode. This mode consists of a single convective roll confined to the region near the reaction front, with fluid rising in half of the cylinder and falling in the other half. Experiments show a flat front below the onset of convection and an axisymmetric front well above the onset of convection. New experiments are called for to closely examine the onset of convection in order to test this prediction.

Finite thermal diffusivity at onset of convection in autocatalytic systems: Continuous fluid density.

The linear stability of exothermic autocatalytic reaction fronts is considered using the viscous thermohydrodynamic equations for a fluid with finite thermal diffusivity. For upward front propagation and a thin front, the vertical thermal gradient near the front is reminiscent of the Rayleigh-B\'enard problem of a fluid layer heated from below. The problem is also similar to flame propagation, except that here the front propagation speed is limited by catalyst diffusion rather than by activation kinetics. For small density changes in a laterally unbounded system, the curvature dependence of the front propagation speed stabilizes perturbations with short wavelengths \ensuremath{\lambda}${\ensuremath{\lambda}}_{\mathit{c}}$, whereas long wavelengths are unstable to convection. The critical wavelengths ${\ensuremath{\lambda}}_{\mathit{c}}$ are calculated and compared with experiments and with theoretical results for a similar problem that is driven by a Rayleigh-Taylor instability arising from a discontinuous density difference at the reaction front.

Onset of convection for autocatalytic reaction fronts: Laterally bounded systems.

The linear stability of exothermic autocatalytic reaction fronts that convert unreacted fluid into a lighter reacted fluid is considered using the viscous thermohydrodynamic equations. For upward front propagation and a thin front, the discontinuous jump in density at the front is reminiscent of the Rayleigh-Taylor problem of an interface between two immiscible fluids, whereas the vertical thermal gradient near the front is reminiscent of the Rayleigh-B\'enard problem of a fluid layer heated from below. The problem is also similar to flame propagation, except that here the front propagation speed is limited by catalyst diffusion rather than by activation kinetics. For a thin ascending front and small density changes in a laterally unbounded system, the curvature dependence of the front speed stabilizes perturbations with short wavelengths \ensuremath{\lambda}${\ensuremath{\lambda}}_{\mathit{c}}$, whereas long wavelengths are unstable to convection, indicating that the density discontinuity dominates over thermal gradients. Simple analytical results for the critical wavelength ${\ensuremath{\lambda}}_{\mathit{c}}$ for onset of convection, the growth rate near onset of convection, and the maximum growth rate are found. Agreement with experiments on iodate--arsenous acid solutions in vertical tubes motivates linear and nonlinear calculations in cylindrical geometries.

Onset of convection for autocatalytic reaction fronts: Laterally unbounded system.

The linear stability of exothermic autocatalytic reaction fronts that convert unreacted fluid into a lighter reacted fluid is considered using the viscous thermohydrodynamic equations. For upward front propagation and a thin front, the discontinuous jump in density at the front is reminiscent of the Rayleigh-Taylor problem of an interface between two immiscible fluids, whereas the vertical thermal gradient near the front is reminiscent of the Rayleigh-B\'enard problem of a fluid layer heated from below. The problem is also similar to flame propagation, except that here the front propagation speed is limited by catalyst diffusion rather than by activation kinetics. For a thin ascending front and small density changes in a laterally unbounded system, the curvature dependence of the front speed stabilizes perturbations with short wavelengths \ensuremath{\lambda}${\ensuremath{\lambda}}_{\mathit{c}}$, whereas long wavelengths are unstable to convection, indicating that the density discontinuity dominates over thermal gradients. Simple analytical results for the critical wavelength ${\ensuremath{\lambda}}_{\mathit{c}}$ for onset of convection, the growth rate near onset of convection, and the maximum growth rate are found. Agreement with experiments on iodate--arsenous acid solutions in vertical tubes motivates linear and nonlinear calculations in cylindrical geometries.