Catalytic trigger and catalytic oscillator

A simple scheme for the nonlinear kinetic oscillations in heterogeneous catalytic reactions has been proposed by Bykov, Yablonsky, and Kim in 1978.^[9] The authors have started with the catalytic trigger (1976^[10][11]), the simplest catalytic reaction without autocatalysis that allows multiplicity of steady states.

${\displaystyle {\ce {{A2}+2Z <=> 2AZ}}}$

(1)

${\displaystyle {\ce {{B}+Z <=> BZ}}}$

(2)

${\displaystyle {\ce {{AZ}+BZ -> {AB}+2Z}}}$

(3)

Then they have supplemented this classical adsorption mechanism of catalytic oxidation by a "buffer" step

${\displaystyle {\ce {{B}+Z\rightleftharpoons (BZ)}}}$

(4)

Here, A₂, B, and AB are gases (for example, O₂, CO, and CO₂), Z is the "adsorption place" on the surface of the solid catalyst (for example, Pt), AZ and BZ are the intermediates on the surface (adatoms, adsorbed molecules, or radicals), and (BZ) is an intermediate that does not participate in the main reaction.

Let the concentration of the gaseous components be constant. Then the law of mass action gives for this reaction mechanism a system of three ordinary differential equations that describe kinetics on the surface.

${\displaystyle {\dot {x}}=2q_{1}z^{2}-2q_{5}x^{2}-q_{3}xy}$

(5)

${\displaystyle {\dot {y}}=q_{2}z-q_{6}y-q_{3}xy}$

(6)

${\displaystyle {\dot {s}}=q_{4}z-kq_{4}s}$

(7)

where z = 1 − (x + y + s) is the concentration of the free places of adsorption on the surface ("per one adsorption center"), x and y are the concentrations of AZ and BZ, correspondingly (also normalized "per one adsorption center"). and s is the concentration of the buffer component (BZ).

This three-dimensional system includes seven parameters. The detailed analysis shows that there are 23 different phase portraits for this system, including oscillations, multiplicity of steady states, and various types of bifurcations.^[12]

Reactions without the interaction of different components

Let the reaction mechanism consist of reactions.

${\displaystyle \alpha _{r}A_{i_{r}}\to \sum _{j}\beta _{rj}A_{j}\,,}$

where ${\displaystyle A_{i}}$ are symbols of components, r is the number of the elementary reaction and ${\displaystyle \alpha _{r},\beta _{rj}\geq 0}$ are the stoichiometric coefficients (usually they are integer numbers). (The components that are present in excess and the components with almost constant concentrations are not included.)

The Eley–Rideal mechanism of CO oxidation on PT provides a simple example of such a reaction mechanism without interaction of different components on the surface:

${\displaystyle {\ce {2Pt(+O2)<=>2Pt;\;\;{PtO}+CO<=>{Pt}+CO2\!\uparrow }}}$.

Let the reaction mechanism have the conservation law

${\displaystyle \alpha _{r}m_{i_{r}}=\sum _{j}\beta _{rj}m_{j}{\text{ for some }}m_{j}>0{\text{ and all }}r,}$

and let the reaction rate satisfy the mass action law:

${\displaystyle W_{r}=k_{r}c_{i_{r}}^{\alpha _{r}},}$

where ${\displaystyle c_{i}}$ is the concentration of ${\displaystyle A_{i}}$. Then the dynamic of the kinetic system is very simple: the steady states are stable^[13] and all solutions ${\displaystyle \mathbf {c} (t)=(c_{i}(t))}$ with the same value of the conservation law ${\displaystyle m(\mathbf {c} )=\sum m_{i}c_{i}}$ monotonically converge in the weighted ${\displaystyle l_{1}}$ norm: the distance between such solutions ${\displaystyle \mathbf {c} ^{(1)}(t),\mathbf {c} ^{(2)}(t)}$,

${\displaystyle \|\mathbf {c} ^{(1)}-\mathbf {c} ^{(2)}\|=\sum _{i}m_{i}|c_{i}^{(1)}-{c}_{i}^{(2)}|,}$

monotonically decreases in time.^[14]

This quasithermodynamic property of the systems without interaction of different components is important for the analysis of the dynamics of catalytic reactions: nonlinear steps with two (or more) different intermediate reagents are responsible for nontrivial dynamical effects like multiplicity of steady states, oscillations, or bifurcations. Without interaction between different components, the kinetic curves converge into a simple norm, even for open systems.

The extended principle of detailed balance

The detailed mechanism of many real physico-chemical complex systems includes both reversible and irreversible reactions. Such mechanisms are typical in homogeneous combustion, heterogeneous catalytic oxidation, and complex enzyme reactions. The classical thermodynamics of perfect systems is defined for reversible kinetics and has no limit for irreversible reactions.^[8] On the contrary, the mass action law gives the possibility to write the chemical kinetic equations for any combination of reversible and irreversible reactions. Without additional restrictions, this class of equations is extremely wide and can approximate any dynamical system with preservation of positivity of concentrations and the linear conservation laws. (This general approximation theorem was proved in 1986.^[15]) The model of real systems should satisfy some restrictions. Under the standard microscopic reversibility requirement, these restrictions should be formulated as follows: A system with some irreversible reactions should be at the limit of the systems with all reversible reactions and the detailed balance conditions.^[7] Such systems have been completely described in 2011.^[7] The extended principle of detailed balance is the characteristic property of all systems that obey the generalized mass action law and is the limit of systems with detailed balance when some of the reaction rate constants tend to zero (the Gorban-Yablonsky theorem).

The extended principle of detailed balance consists of two parts:

• The algebraic condition: The principle of detailed balance is valid for the reversible part. This means that for the set of all reversible reactions, there exists a positive equilibrium where all the elementary reactions are equilibrated by their reverse reactions.
• The structural condition is that the convex hull of the stoichiometric vectors of the irreversible reactions has an empty intersection with the linear span of the stoichiometric vectors of the reversible reactions. (Physically, this means that the irreversible reactions cannot be included in oriented cyclic pathways.)

The stoichiometric vector of the reaction ${\displaystyle \sum _{i}\alpha _{i}A_{i}\to \sum _{j}\beta _{j}A_{j}}$ is the gain minus loss vector with coordinates ${\displaystyle \gamma _{i}=\beta _{i}-\alpha _{i}}$.

(It may be useful to recall the formal convention: the linear span of an empty set is 0; the convex hull of an empty set is empty.)

The extended principle of detailed balance gives an ultimate and complete answer to the following problem: how to throw away some reverse reactions without violating thermodynamics and microscopic reversibility? The answer is that the convex hull of the stoichiometric vectors of the irreversible reactions should not intersect with the linear span of the stoichiometric vectors of the reversible reactions, and the reaction rate constants of the remaining reversible reactions should satisfy the Wegscheider identities.