Simple reaction-diffusion fronts are examined in one and two dimensions. In one-dimensional configurations, fronts arising from either quadratic or cubic autocatalysis typically choose the minimum allowable velocity from an infinite spectrum of possible wave speeds. These speeds depend on both the diffusion coefficient of the autocatalytic species and the pseudo-first-order rate constant for the autocatalytic reaction. In the mixed-order case, where both quadratic and cubic channels contribute, the wave speed depends on the rate constants for both channels, provided the cubic channel dominates. Wave propagation is completely determined by the quadratic contribution when it is more heavily weighted. In two-dimensional configurations, with unequal diffusion coefficients, the corresponding twovariable planar fronts may become unstable to perturbations. The instability occurs when the ratio of the diffusion coefficient for the reactant to that for the autocatalyst exceeds some critical value. This critical value, in turn, depends on the relative weights of the quadratic and cubic contributions to the overall kinetics. The spatiotemporal form of the nonplanar wave in such systems depends on the width of the reaction zone, and a sequence showing Hopf, symmetry-breaking, and period-doubling bifurcations leading to chaotic behavior is observed as the width is increased.
Density fingering of the chlorite–tetrathionate reaction is studied experimentally in a Hele-Shaw cell. It is shown that the dispersion curve describing the linear regime of the evolution of pattern formation is strongly affected by the orientation of the cell. Both the growth rate and the range of wave numbers associated with the unstable modes decrease on increasing the tilt angle from the vertical. From the dispersion curves, the dependence of the most unstable mode on the tilt angle is determined experimentally and compared with that of existing theories. The ratio of the marginal wave number, separating the stable and unstable modes, to the wave number with the maximum growth rate indicates that the high-frequency disturbances are stabilized by the diffusion of the components in the reaction.
An extensive study on the instabilities of planar fronts leading to the formation of cellular structures has been carried out in the acid-catalyzed chlorite–tetrathionate reaction. A simple two-variable model based on the empirical rate law of the reaction is developed to describe the observed pattern formation. The calculated onset of instability and the size of the patterns in the cellular fronts are in good agreement with experimental observations.
Density fingering of exothermic autocatalytic fronts in vertically oriented porous media and Hele-Shaw cells is studied theoretically for chemical reactions where the solutal and thermal contribution to density changes have opposite signs. The competition between these two effects leads to thermal plumes for ascending fronts. The descending fronts behave strikingly differently as they can feature, for some values of the parameters, fingers of constant amplitude and wavelength. The differences between up and down going fronts are discussed in terms of dispersion curves and nonlinear dynamics. The theoretically predicted dispersion curves are experimentally evidenced with the chlorite-tetrathionate reaction.
We have analyzed the emerging precipitate pattern of calcium-oxalate in a flow system. The circular symmetry is broken because of the hydrodynamic instability at the tip of the underlying gravity current. The presence of a concentration gradient maintained by the flow leads to the enrichment of the thermodynamically unstable calcium oxalate dihydrate form.
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