Autocatalysis Fronts

The simplest possible autocatalytic reaction is the quadratic autocatalysis of Eq. [76]. We now consider the next simplest case, in which a cubic nonlinearity appears in the rate law  

For the distributed system, we can write reaction-diffusion equations identical to Eq. [77], except that the cubic form of the rate law appears in place oi the quadratic form. The reaction-diffusion equations are then rewritten as dimensionless equations to yield 

The front velocity for cubic autocatalysis may take on any value above some minimum velocity just as for the quadratic case. In dimensionless terms, we have 

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Figure C3.6.14 Space-time (y,t) plot of the minima (black) in the cubic autocatalysis front ( )(y,t) in equation C3.6.16 showing the nature of the spatio-temporal chaos.

Figure 17 Phase plane for quadratic autocatalysis front described by Eqs. [83]. The front profile corresponds to a trajectory emanating from the origin (saddle point) along the outset and approadiing the singularity at (1, 0) along the (degenerate) eigenvector (Reprinted from Ref. 43 with permission of the American...

The behavior found for the quadratic system in the one-dimensional case is directly applicable to the two-dimensional configuration planar fronts are exhibited with velocities given by Eq. [88] for the case of equal diffusivities of A and B. When the diffusivities significantly differ, planar fronts are still observed however, now the velocity scales with the diffusion coefficient D = Dr according to Eq. [89]. > The one-dimensional solution for the cubic system is also valid for the two-dimensional configuration however, the cubic front may exhibit lateral instabilities that are not observed in the quadratic system. will now consider the stability of cubic autocatalysis fronts. 

Fig. 11.3. The reaction-diffusion wave for cubic autocatalysis in wave-fixed coordinates. The five fronts correspond to z0 = — 4, — 2, 0, 2, and 4.

Gray, P., Showalter, K., and Scott, S. K. (1987). Propagating reaction-diffusion fronts with cubic autocatalysis the effects of reversibility. J. Chim. Phys., 84, 1329-33. 

Hence for quadratic autocatalysis, constant velocity, constant wave-form propagating fronts are allowed with any velocity c greater than some minimum velocity ... 

Diffusion Fronts with Cubic Autocatalysis The Effeas of Reversibility. 

Synthetic polymer systems can exhibit feedback through several mechanisms. The simplest is thermal autocatalysis, which occurs in any exothermic reaction. The reaction raises the temperature of the system, which increases the rate of reaction through the Arrhenius dependence of the rate constants. In a spatially distributed system, this mechanism allows propagation of thermal fronts. Free-radical polymerizations are highly exothermic. 

Frontal polymerization discovered in 1972 (5) could be realized in free-radical polymerization because of its nonlinear behavior. If the top of a mixture of monomer and initiator in a tube is attached to an external heat source, die initiators are locally decomposed to generate radicals. The polymerization locally initiated is autoaccelerated by the c(xnbinatithermal autocatalysis exclusively at the top of the reaction systmn. An interface between reacted and unreacted regions, called propagating front, is thus formed. Pojman et al. extensively studied the dynamics of frontal polymerization (d-P) and its applicatim in matoials syndiesis (I -I3). 

Direct autocatalysis occurs in biological polymerizations, such as DNA and RNA replication. In normal biological processes, RNA is produced from DNA. The RNA acts as the carrier of genetic information in peptide synthesis. However, RNA has been found to be able to replicate itself, for example, with the assistance of the Q/li replicase enzyme. Bauer and McCaskill have created traveling fronts in populations of short self-replicating RNA variants (Bauer ct ah, 1989 McCaskill and Bauer, 1993). If a solution of monomers of triphosphorylated adenine, gua-... 

The simplest attack [7] anticipates the fact that cubic autocatalysis generates a steady wave-front of quadratic form ... 

SIMPLE AND COMPLEX REACTION-DIFFUSION FRONTS 503 3. Autocatalysis with Decay... 

Next, we consider the wavefronts that may develop for two other geometries a circle and a sphere. These represent the simplest 2-D and 3-D geometries. The natural response to circular or spherical initiations at the center of such reaction zones will be the development of fronts with the same underlying shape. We concentrate here on quadratic and cubic autocatalysis without decay. 

The similarities between the flame equation and that seen earlier, particularly for cubic autocatalysis, provides reason for optimism that a steady flame velocity will emerge in much the same way. There is, however, one distinct operational problem to overcome. This concerns the behavior of the reaction rate function (1 - 9)f 9) ahead of the flame front, i.e., for 0 = 0 close to the origin in Figure 8. With the cubic curve, the rate falls exactly to zero at the origin this allows the reactant ahead of the wave to remain unreacting until the diffusion of autocatalyst initiates the reaction. For the nonisothermal system... 

This diffusive instability mechanism has only recently been examined in reaction-diffusion systems. Continuing with the analogy between isothermal fronts and nonisothermal flames, we pursue the case where > 1 in Equations (4). Thus, the front is destabilized when the diffusivity of the reactant A becomes sufficiently larger than that of the autocatalyst B. We consider systems with pure cubic autocatalysis here in a detailed study [12], pure quadratic and mixed-order fronts are also considered and found to have different sensitivities to the destabilizing effects of diffusion. 

Mcllwaine R, Fenton H, Scott S and Taylor A. 2008. Acid autocatalysis and front propagation in water-in-oQ microemulsions. Journal of Physical Chemistry C 2499-2505. 

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