Autocatalysis with decay

The dependences of wave velocity on the ratio of cubic to quadratic processes, described by eqns (11.43) and (11.44) are shown in Fig. 11.6. The quadratic curve always lies below the cubic result, but the two are tangential at q = 1/2, c = yfl. Numerical computations reveal the following detail concerning the stability of the respective solutions. For q i, where cubic dominates quadratic, it is the velocity determined by eqn (11.43) which is selected by the system. Above q = j, however, the quadratic character takes over, and the minimal velocity described by eqn (11.44) emerges dominant. 

In the various situations we have seen before, allowing a finite decay rate for the catalyst B has had significant results. The concentrations of A and B are then decoupled and this has allowed oscillations, isolas, and mushrooms. In the present case of reaction-diffusion waves, the uncoupling is again an important step upwards in complexity, sufficiently so as to prevent any completely general form of analysis. 

We may still begin by arguing intuitively. In the case of quadratic autocatalysis, the chemistry involves local competition between a production of B which depends on the product of concentrations a/J and a removal of 

We may comment further on the importance of the condition on the dimensionless decay rate k2. This term is the ratio of decay and autocatalytic rate constants, and the condition k2 1 becomes, in terms of the physical rate constants, 

An estimate of the steady pulse velocity in this model can also be obtained. In a similar way to that found in 11.2, the system has a stable minimum velocity cmin which depends on the value of the decay rate constant  

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Fig. 6.15. Flow diagram representation of the origin of isola solutions for cubic autocatalysis with decay. The gradient of the flow line L at first decreases with increasing residence time non-zero intersections appear as R and L become tangential and these move apart as L becomes less steep. After L has attained its minimum slope the non-zero intersections move closer together again, and merge (and disappear) as R and L attain tangency for the second time. The intersection at zero extent of reaction exists for all residence times.

Fig. 8.4. A typical isola for cubic autocatalysis with decay, but for no catalyst inflow fi0 = 0, identifying the notation (1 — aj, (1 — a2), and (1 — a3) in order of increasing extent of reaction. (Exact numerical values correspond to k2 = A-)...

Fig. 8.5. The locus of Hopf bifurcation points t s-k2 described by eqn (8.42) for cubic autocatalysis with decay but no catalyst inflow. Also shown are the loci of the extinction points TreS, marking the ends of the isola, given by eqn (8.27). The three curves meet at the common...

Fig. 8.6. Typical arrangement of local stabilities and development of unstable limit cycle, from a subcritical Hopf bifurcation, appropriate to cubic autocatalysis with decay and no autocatalyst inflow and with 9/256 < k2 < 1/16. The unstable limit cycle grows as t, decreases below t m, and terminates by means of the formation of a homoclinic orbit at rf . Stable stationary states, including the zero conversion branch 1 — a, = 0, are indicated by solid curves, unstable states and limit cycles by broken curves.

Fig. 8.7. Supercritical Hopf bifurcation for cubic autocatalysis with decay and /) = 0, appropriate for small dimensionless decay rate constant k2 < 9/256. A stable limit cycle emerges and grows as the residence time is increased above t s. At higher residence times, this disappears at rj , by merging with an unstable limit cycle born from a homoclinic orbit at t. (With non-zero autocatalyst inflow, (i0 > 0, the stable limit cycle itself may form a homoclinic orbit at long tres.)...

Fig. 8.9. The locus A of double-zero eigenvalue degeneracies of the Hopf bifurcation for cubic autocatalysis with decay. Also shown, as broken curves, are the loci of stationary-state degeneracies, corresponding to the boundaries for isola and mushroom patterns. The curve A lies completely within the parameter regions for multiple stationary states.

FlG. 8.14. The different phase plane portraits identified for cubic autocatalysis with decay (a) unique stable state (b) unique unstable stationary state with stable limit cycle (c) unique stable state with unstable and stable limit cycles (d) two stable stationary states and saddle point (e) stable and unstable states with saddle point (f) stable state, saddle point, and unstable state surrounded by stable limit cycle (g) two unstable states and a saddle point, all surrounded by stable limit cylcle (h) two stable states, one surrounded by an unstable limit cycle, and a saddle point (i) stable state surrounded by unstable limit cycle, unstable state, and saddle point, all surrounded by stable limit cycle (j) stable state, unstable state, and saddle point, all surrounded by stable limit cycle (k) stable state, saddle point, and unstable state, the latter surrounded by concentric stable and unstable limit cycles (1) two stable states, one surrounded by concentric unstable and stable limit cycles, and a saddle point. 

FIG. 9.5. A typical stationary-state solution for the dimensionless concentration profiles a (p) and P (p) for cubic autocatalysis with decay. The reactant concentration shows simply a central minimum, but the autocatalyst profile has three extrema, including two non-central maxima. 

FIG. 9.6. The five qualitative forms for the stationary-state locus ass(0)-D for cubic autocatalysis with decay in a reaction-diffusion cell (a) unique (b) single hysteresis loop (c) mushroom (d) isola (e) isola + hysteresis loop, (f) The division of the Pcx-k2 parameter plane giving the five regions corresponding to the stationary-state forms in (a)-(e) note that the region for response (e), shown inset, is particularly small and has not yet been successfully located. 

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

Fig. 6.16. (a) Flow diagram representation of the different stationary-state loci for cubic autocatalysis with inflow and decay of B (b) unique response, minimum slope L, (c) birth of isola as isolated point, minimum slope L , (d) isola, minimum slope L2 (e) transition from isola to mushroom, minimum slope L > (f) mushroom, minimum slope L3. 

We have seen in earlier chapters that kinetic systems with two independent concentrations can show additional complexities of dynamic behaviour beyond those of one-variable systems. Of particular interest are undamped oscillations. The cubic autocatalysis with the additional decay step... 

More challenging is the behaviour of a system with cubic autocatalysis coupled with decay. Here a term af)2 must compete with the removal k2/ . For a 1 and 1 it is not immediately obvious that any wave solutions will exist at all. Such considerations also suggest that wave formation will be favoured by particularly high autocatalyst concentrations at the point of initiation, so / > 1 in a localized region. Thus we may expect some threshold for the initiation. 

Cubic autocatalysis with quadratic decay The scheme is now ... 

FIGURE 1 Different dependence of stationary-state extent of conversion found for cubic autocatalysis with catalyst decay... 

Autocatalysis with an unstable catalyst resembles non-isother-mal reaction under non-adiabatic operation. In each case the species responsible for feedback (B or the heat released) can be removed from the system by a route independent of the reactant A. These extra channels for removal are chemical decay and Newtonian cooling at the walls respectively. These circumstances lead to two independent variables. Only under stationary-state conditions is the concentration of A directly linked to the concentration of the catalyst or the temperature-excess. In our system we find... 

With m = n = 2, the autocatalysis and decay processes compete effectively on equal terms (both depending on at the leading edge where a = 1). The condition for a constant-velocity traveling wave will again be 7 < 1. There is no critical initiation requirement a wave develops for any nonzero /3q if 7 < 1. The wave velocity again tends to zero as 7 1, although now as... 

Fig. 7. Variation of wave speed with dimensionless decay rate constant 7 for cubic autocatalysis with linear decay for 7 < 0.0465 there are two wave speeds, the higher corresponding to the stable wave. No reaction-diffusion waves exist for 7 > 0.0465.

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. 

If we continue to keep (Le) = 1, the appropriate equations for a nonadiabatic flame, for which there is some Newtonian heat loss rate proportional to 0 at each point, will be similar in form to Equations (29) seen earlier for cubic autocatalysis with linear decay, but with the autocatalysis replaced by the Arrhenius function f 9) ... 

In between these tangencies, the curves R and L have three intersections, so the system has multiple stationary states (Fig. 7.3(b)). We see the characteristic S-shaped curve, with a hysteresis loop, similar to that observed with cubic autocatalysis in the absence of catalyst decay ( 4.2). 

The previous two chapters have considered the stationary-state behaviour of reactions in continuous-flow well-stirred reactions. It was seen in chapters 2-5 that stationary states are not always stable. We now address the question of the local stability in a CSTR. For this we return to the isothermal model with cubic autocatalysis. Again we can take the model in two stages (i) systems with no catalyst decay, k2 = 0 and (ii) systems in which the catalyst is not indefinitely stable, so the concentrations of A and B are decoupled. In the former case, it was found from a qualitative analysis of the flow diagram in 6.2.5 that unique states are stable and that when there are multiple solutions they alternate between stable and unstable. In this chapter we become more quantitative and reveal conditions where the simplest exponential decay of perturbations is replaced by more complex time dependences. 

Co-oxidation of indene and thiophenol in benzene solution is a free-radical chain reaction involving a three-step propagation cycle. Autocatalysis is associated with decomposition of the primary hydroperoxide product, but the system exhibits extreme sensitivity to catalysis by impurities, particularly iron. The powerful catalytic activity of N,N -di-sec-butyl-p-phenylenediamine is attributed on ESR evidence to the production of radicals, probably >NO-, and replacement of the three-step propagation by a faster four-step cycle involving R-, RCV, >NO, and RS- radicals. Added iron complexes produce various effects depending on their composition. Some cause a fast initial reaction followed by a strong retardation, then re-acceleration and final decay as reactants are consumed. Kinetic schemes that demonstrate this behavior but are not entirely satisfactory in detail are discussed. 

In the reactor, A and B react and autocatalysis occurs by step (iii) with a rate constant 1, while decay of B to C occurs via step (iv) with the rate constant k2- Further, kf is the flow rate of A and B from the reservoirs. If half-order kinetics for step (iv) is assumed, the kinetic equations (non-linear) are as follows ... 

Consider the simple, irreversible cubic autocatalysis (l) coupled with a first-order decay or poisoning reaction (2). The mass-balance equations can be readily written [2] in the following dimensionless form... 

Figure 2. Flow-diagram for cubic autocatalysis A + 2B -> 3B with catalyst decay B C ...

In this section we consider the passage of a wave through a chemical system with either quadratic or cubic autocatalysis, but in which these processes are in competition with a decay step where the autocatalyst B becomes inactivated by reaction to a stable product species C. The general chemical scheme can be written... 

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