The A B - 0 reaction

A detailed study of individual organs makes us unable to understand a life of a whole organism. 

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The existence of the (quasi) steady-state in the model of particle accumulation (particle creation corresponds to the reaction reversibility) makes its analogy with dense gases or liquids quite convincing. However, it is also useful to treat the possibility of the pattern formation in the A + B —> 0 reaction without particle source. Indeed, the formation of the domain structure here in the diffusion-controlled regime was also clearly demonstrated [17]. Similar patterns of the spatial distributions were observed for the irreversible reactions between immobile particles - Fig. 1.20 [25] and Fig. 1.21 [26] when the long range (tunnelling) recombination takes place (recombination rate a(r) exponentially depends on the relative distance r and could... 

Let us consider a mixture of classical mobile particles A and B participating in the A + B —> 0 reaction occurring in a continuous neutral medium. Assume also that uncorrelated particles are created with rate p and recombine... 

Figures 3.5 and 3.6 present schematic classification of regimes observable for the A + B —> 0 reaction. We will concentrate in further Chapters of the book mainly on diffusion-controlled kinetics and will discuss very shortly an idea of trap-controlled kinetics [47-49]. Any solids contain preradiation defects which are called electron traps and recombination centres -Fig. 3.7. Under irradiation these traps and centres are filled by electrons and holes respectively. The probability of the electron thermal ionization from a trap obeys the usual Arrhenius law 7 = sexp(-E/(kQT)), where s is the so-called frequency factor and E thermal ionization energy. When the temperature is increased, electrons become delocalized, flight over the conduction band and recombine with holes on the recombination centres. Such...

Summing up, note that the direct statistical (computer) simulation does not demonstrate serious errors of the superposition approximation for equal reactant concentrations. Divergence begins first of all for unequal concentrations for not very large reaction depths T 2 it is almost negligible, at T > 2 and especially asymptotically (as t —> oo) it becomes important, but the complete quantitative analysis cannot be done due to unreliable statistics of results. In this Section we have restricted ourselves to the A + B —> 0 reaction without particle generation. Testing of the superposition approximation accuracy for the case of particle creation will be done in Chapter 7. 

The equations derived above, describing the A + B —> B reaction kinetics in terms of the correlation functions g and g2, have the form of the nonlinear generalised multi-dimensional diffusion equation. Ignoring the multidimensionality of the operator terms in (5.2.11), these equations could be formally considered as similar to the basic non-linear equations for the A + B — 0 reaction (Section 5.1). Equations studied in both Sections 5.1 and 5.2 are derived with the help of the Kirkwood superposition approximation, the use of which leads to several equations for the correlation functions of similar and dissimilar reactants. 

The simplest class of bimolecular reactions involves only one type of mobile particles A and could result either in particle coagulation (coalescence, fusion) A + A —> A, or annihilation, A + A — 0 (inert product). Their simplicity in conjunction with the simple topology of d = 1 allows us to solve the problem exactly, which makes it very attractive for testing different approximations and computer simulations. In the standard chemical kinetics (i.e., mean-field theory, Section 2.1.1) we expect in d = 2 and 3 for both reactions mentioned trivial behaviour quite similar to the A+B — 0 reaction, i.e., tia( ) oc t-1, as t — oo. For d = 1 in terms of the Smoluchowski theory the joint density obeys respectively the equation (4.1.56) with V2 = and D = 2Da. 

In this Section following [9], we analyse the A + B —> 0 reaction with immobile reactants on the so-called Sierpinski gasket described below. We will proceed to show that in this case equation (6.1.1) with a = d/2 transforms into... 

Fig. 6.26. Time decay of the particle concentration, tia( ), for cases of equal (curves 1) and unequal (curves 2) concentrations involved into the A+B - 0 reaction for d = 1 [32]. Curves 1 riA(O) = riB(O) =0.1 curves 2 n =0.1, ub(0) = 0.2. Symbols A, B, AB correspond to types of mobile species (one type or both types respectively). Full lines show results of the Monte Carlo simulations whereas dashed lines - solution of a master equation.

An extension of the coupled-cluster approximation to the non-equilibrium classical systems [43-45] has allowed to study asymptotics of bimolecular reactions. It resulted in a rather unexpected conclusion that now the generally-accepted time dependence of the A+B —> 0 reaction for d = 3, n(t) oc f-3/4, is only the pre-asymptotic stage, with the true asymptotics n(t) oc f 1 Similar technique was used also for the study of diffusion-limited aggregation and structure formation processes [47],... 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

Therefore, the approximate treatment of the A+B — 0 reaction for charged particles inavoidably requires a combination of several approximations the Kirkwood superposition approximation for the reaction terms and the Debye-Hvickel approximation for modification of the drift terms with self-consistent potentials. Not discussing here the accuracy of the latter approximation, note... 

The A + B —> 0 reaction with correlated defect creation is characterized by the time-independent quantity 8n = nA(t) — n,Q(t) = const. It is self-evident since the equation for macriscopic density ns(t) or Uv(t) follows from equation (7.1.16) replacing an index A for B (and vice versa). 

These results agree well with what was said above about the A + B 0 reaction (see Fig. 7.5) - the larger reactant diffusities and/or smaller irradiation intensity, the smaller saturation concentrations ns. The Monte Carlo simulations [95] very well confirm these results. These simulations were performed on a lattice of 105 sites, by the direct simulation method. The interparticle probability density was also measured in the simulations, and the results are compared with theory the agreement is excellent. 

As it has been said above, accumulation of radiation-induced (Frenkel) defects takes place in all kinds of solids irrespective which of the two basic recombination mechanisms - annihilation or tunnelling recombination - occurs [9, 11, 13, 17-20, 39, 42, 99-107]. In a good approximation this process could be considered as the A + B —> 0 reaction with the particle input. In many cases strong arguments exist for the clustering of radiation-induced... 

A comparison with the correlation dynamics of the A + B —> 0 reaction, equations (5.1.33) to (5.1.35), shows their similarity, except that now several terms containing functionals J[Z have changed their signs and several singular correlation sources emerged. The accuracy of the superposition approximation in the diffusion-controlled and static reactions was recently confirmed by means of large-scale computer simulations [28]. It was shown to be quite correct up to large reaction depths r = 3 studied. 

The numerical methods for solving equations like (8.2.17), (8.2.22) and (8.2.23) are discussed in Section 5.1. In practice the conservative difference schemes are widely used for solving differential equations with the accuracy of the order 0(At + Ar2) [21, 26, 27] used as well 0(Af2 4- Ar2) [25], Unlike mathematically similar equations for the A + B —> 0 reaction (Section 5.1), where the correlation functions vary monotonously in time, the... 

This statement comes from analytical and topological studies [4], Unlike the Lotka-Volterra model where due to the dependence of the reaction rate K(t) on concentrations NA and NB, the nature of the critical point varied, in the Lotka model the concentration motion is always decaying. Autowave regimes in the Lotka model can arise under quite rigid conditions. It is easy to show that not any time dependence of K(t) emerging due to the correlation motion is able to lead to the principally new results. For example, the reaction rate of the A + B -> 0 reaction considered in Chapter 6 was also time dependent, K(t) oc t1 d/4 but its monotonous change accompanied by a strong decay in the concentration motion has resulted only in a monotonous variation of the quasi-steady solutions of (8.3.20) and (8.3.21) jVa(t) (3/K(t) and N, (t) p/f3 = const. 

Therefore, if the reaction rate K t) is known, the A+B -> 0 reaction kinetics is defined uniquely. 

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