Stephen K. Scott

Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability

Abstract Autocatalytic reactions are often complicated, and analyses of their behaviour in open systems can seem too particular to permit useful generalisation. We study here the simplest of circumstances (uniform temperatures and concentrations in the isothermal CSTR) and the simplest of reaction schemes: (i) quadratic autocatalysis ( A + B →2 B ); and (ii) cubic autocatalysis ( A + 2 B →3 B ). The catalyst B may be stable or have a finite lifetime (B→ inert products). Allowing for this finite lifetime adds another dimension to the interest. The phenomena encountered include multistability, hysteresis, critical extinctions, critical ignitions, and anomalous relaxation times (though infinite values do not arise). Patterns of stationary states as function of residence time can show isolas and mushrooms. All these aspects yield to simple algebraic analysis. The presence of the catalyst B in the inflow can make qualitative differences of a kind parallelled by an additional, non-catalytic reaction of the same stoichiometry (e.g. A → B ). Invoking the reversibility of the reactions neither increases nor diminishes their variety, and thermodynamic considerations have little to do with the many different patterns of reactivity displayed. The local stability of the various stationary states has also been characterized. Quadratic autocatalysis shows limited variety (stable node, stable focus); cubic autocatalysis generates all the kinds of stationary state possible in a two-variable system. Again all the algebra is straightforward if not always simple. Sustained oscillatory behavior (limit cycles) also occur. All these remarks relate to isothermal systems, but there are the most striking parallels between isothermal autocatalysis and the exothermic, first-order reaction in the CSTR. Behaviour with an autocatalyst of complete stability corresponds to perfect heat insulation (adiabatic operation) in the non-isothermal, exothermic system.

Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C

The prototype, cubic autocatalytic reaction (A + 2B → 3B) forms the basis for the simplest homogeneous system to display “exotic” behaviour. Even under well-stirred, isothermal, open conditions (CSTR) we may find multistability, hysteresis, extinction, ignition and anomalous relaxation times. Allowing for the finite lifetime of the catalyst (B→inert products) adds another dimension. The dependence of the stationary-states on residence-time now yields isolas or mushrooms. Sustained oscillations (stable limit cycles) are also possible. The onset of oscillation corresponds to a change in the character of the stationary-state (from stable focus to unstable focus) and the conditions for this change can be evaluated analytically. The period of the oscillations and their amplitudes increase as the residence time is lengthened. A total of nine different phase-portraits in the a–b plane has been found. The isothermal system is less complex than the exothermic, first-order reaction in a CSTR, but there are strong analogies between the two.

Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system

The excitability properties of a two-variable cubic autocatalysis model for chemical oscillations are examined. The reaction-diffusion behaviour of this model is studied in a one-dimensional configuration with differing relative diffusivities of the species. Wave reflection at no-flux boundaries is examined and described in terms of reactant depletion in the wave front and reactant influx in the wave back. Waves are also reflected upon collision with other waves. Wave splitting, the spontaneous initiation of a wave from the trailing edge of another wave, is found to occur for some relative diffusivities. Successive wave splittings give rise to stationary Turing patterns at long times.

Instabilities in propagating reaction‐diffusion fronts

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 two‐variable 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 v...

Oscillatory chemical reactions in closed vessels

A skeleton kinetic scheme representing the simplest model for oscillatory chemical reactions in a closed vessel can be built around an autocatalytic feedback step precursor decay P → A k0p, uncatalysed step A → B k3a, autocatalysis A + 2B → 3B k1ab2, catalyst decay B → C k2b. The first intermediate A is formed via the slow decay of a reactant or precursor species P, initially in large excess. A is converted to B via two routes: a slow pseudo-first order process and a step in which B acts as its own catalyst. The autocatalyst B is then capable of a simple first order decay to a stable product C. The concentrations of the various species at first change steadily, with that of P decreasing while A, B and C increase. This period is followed by the onset and growth of oscillations in the concentrations of the intermediates A and B. The behaviour at long times, depends upon the uncatalysed conversion of A to B. Provided k3 is not taken as zero, the oscillations finally diminish in amplitude and die out leaving a steady decay of P, A and B until everything has been converted to C. The simplicity of the model allows the first self-consistent test of the ‘pool chemical approximation’, an approach commonly used in the analysis of mechanisms in closed systems in which the precursor concentration is assumed to be constant and set equal to its initial value. The results presented here reveal the range of applicability of the approximation and show clearly how and why it can break down to give unphysical predictions.

On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme

The equations which govern a simple pooled chemical reaction scheme are analysed in detail in terms of a nondimensional parameter

Isolas, mushrooms and oscillations in isothermal, autocatalytic reaction―diffusion equations

\mu

Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis

, which represents the amount of the pooled chemical originally present. It is shown that there is one finite equilibrium point, with a Hopf bifurcation occurring at

Thermal explosions with extensive reactant consumption: a new criterion for criticality

\mu = 1

TRANSIENT CHAOS IN A CLOSED CHEMICAL SYSTEM

. The phase plane at infinity is then examined and it is shown that there are equilibrium points at infinity at the positive ends of both axes, the nature of which are discussed. This, together with a knowledge of the global phase portraits for