Quadratic autocatalysis

In the hydrogen-oxygen and B-Z reactions considered above, the autocatalytic cycles correspond to a value for n of unity. The resulting rate law, rate = klab, involves the product of two concentrations and is known as quadratic autocatalysis . In the reaction between iodate and iodide ions, I is produced through an autocatalytic cycle which, at its simplest, corresponds... 

Autocatalysis will play a central role in driving the oscillations and other non-linear phenomena of interest in this book. Usually, an autocatalytic process will be combined in a larger mechanism with other steps. Before considering such systems, however, we investigate the properties and behaviour of autocatalysis on its own—in particular how the concentrations and rate vary with time and with respect to each other. We start with quadratic autocatalysis, and then look at the cubic form. 

Fig. 1.4. The variation of reactant concentration, a, and reaction rate, - da/dt, with time for a system obeying quadratic autocatalysis with non-zero initial catalyst concentration a0 = 0.1 mol dm"3 b0 = 10 6 mol dm"3 kq - 3 dm3 mol 1 s. ...

As well as deceleratory reactions, kineticists often find that some chemical systems show a rate which increases as the extent of reaction increases (at least over some ranges of composition). Such acceleratory, or autocatalytic, behaviour may arise from a complex coupling of more than one elementary kinetic step, and may be manifest as an empirically determined rate law. Typical dependences of R on y for such systems are shown in Figs 6.6(a) and (b). In the former, the curve has a basic parabolic character which can be approximated at its simplest by a quadratic autocatalysis, rate oc y(l - y). 

Fig. 6.6. Dependences of reaction rate R on extent of reaction y typical of self-accelerating (autocatalytic) systems (a) prototype quadratic autocatalysis (b) prototype cubic autocatalysis.

Thus, with the simple cubic autocatalytic rate law, we have been able to find an analytical expression for the time and space dependence of a steady reaction-diffusion wave and make various quantitative and qualitative comments about the behaviour of the wave in terms of the kinetic and diffusion parameters. We now turn to the apparently simpler kinetics of a quadratic autocatalysis, hoping for similar rewards. 

In the previous section, full analytical solutions for constant velocity travelling waves in reaction with cubic autocatalysis were obtained. We might hope that the equivalent system with quadratic autocatalysis... 

Fig. 11.5. Phase plane representation of a travelling wavefront for quadratic autocatalysis. The trajectory emerges from the initial singularity at p = 1, g = 0 and tends to the final state fi = 0,...

The reaction between iodate and arsenite ions appears to have contributions from both cubic and quadratic autocatalysis (the autocatalyst is the product, iodide ion). In the previous sections we have treated these two rate laws separately and by different methods. Both methods can be applied to the system in which these processes occur simultaneously, yielding results which, despite not being consistent at first sight, can be resolved by the idea of stability. 

In the limit of pure cubic autocatalysis (q — 0), we regain the previous result, c - /sf2. When cubic and quadratic processes combine, the speed of the wave increases. 

If we consider the opposite extreme, where quadratic autocatalysis dominates, we should also change our characteristic timescale. So far we have based the wave velocity c on the cubic chemical time fch = /klal ... 

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... 

Beyond the linear autocatalyses, nonlinear effects such as a quadratic autocatalysis have been considered [6,24,31] ... 

In order to understand the molecular mechanisms of how the quadratic autocatalysis is brought about, the concept of dimer catalyst introduced by Kagan and coworkers, may be relevant [ 19,30,32,33]. Assume that monomers R and S react to form homodimers R2 and S2 with a rate Vhom and a heterodimer RS with a rate VheP... 

When A is positive, as in the case of Fig. 6c, the coefficient of the cubic termB is also positive, and the velocity 4>i vanishes at three values of 0i in the range of - 1 < 0i < 1. This is possible if a strong quadratic autocatalysis > k 2 exists together with a linear recycling X > 0, or if a linear autocatalysis k and a nonlinear recycling x > 0 coexist. By following the direction indicated by the arrows for positive order parameter ends up at a definite value ... 

As for the second example, we consider the case with a quadratic autocatalysis k2 > 0 and a linear recycling A > 0. Because the linear recycling takes place whenever there is nonzero concentrations of enantiomers, a few achiral substrate always remain, a t = oo) 0. Thus, the diagonal line q = r + s = c is no longer a fixed line. Instead, there appear fixed points in general. In the present case with a finite A. > 0, a total of seven fixed points appear three stable (0, Si,2) and four unstable ones (1/1-4), as shown in Fig. 7b. As the reaction proceeds, the system approaches a state associated with one of the stable fixed points. The origin 0 is not interesting, since all the chiral products are recycled back to the achiral substrate. Also its influence extends only in a small... 

Fig. 7 Flow diagrams with a linear autocatalysis and nonlinear recycling (ki = /z > 0, k(1 = k2 =X = 0) similar to the Frank model, and b quadratic autocatalysis and linear recycling (k2c2 = /. > 0, fc, = ki = [ = 0). O origin, S stable fixed points, U instable fixed points...

In order to give these idealized feedback curves a chemical face , they are frequently represented in terms of autocatalysis [5,6]. A purely quadratic autocatalytic curve would arise if the A + B reaction does not give rise to products C and D, but instead, produces two molecules of species B... 

The higher order, with respect to the autocatalyst, skews the rate curve so that the maximum lies at higher extents of conversion and there is a longer induction phase during which the reaction rate is close to zero at low extents of conversion. Cubic autocatalysis is apparently less significant than quadratic which is relatively common as chemical feedback in combustion systems, although cubic-type curves have been reported and exploited in the oxidation of H2 for which a rate expression of the form d[H20]/dr = / [H2][H20] was observed [7] and also the oxidation of CS2 in heavily-diluted air mixtures [8]. 

The unrealistic "cubic autocatalysis" form 14.4 was chosen here for simplicity s sake (a "quadratic" form X+Y— 2Y does not produce instability). Gray and Scott [34] discuss this case in much greater detail. They include reverse steps as well as an uncatalyzed, parallel step X— Y the latter keeps the mathematics from... 

To make this quadratic-autocatalysis network capable of periodic behavior, the "step B— R is given a nonlinear rate equation —rB = kBRCB l(CB+K) as might result from saturation kinetics (see Section 8.3.1). To allow chaos, another step producing an otherwise inert species C is added ... 

We begin by considering the simplest form of autocatalysis, which is characterized by a rate equation with a quadratic nonlinearity ... 

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...

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

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 ... 

---