Effective recombination radius

Therefore, equation (4.2.21) with the substitution of for R cannot describe correctly the process of the steady-state formation if the diffusion process is controlled by the strong tunnelling (x 3> 1). In other words, strong tunnelling could be described in terms of the effective recombination radius i eff analogous to the black sphere in the steady-state reaction stage only. 

The quasi-steady-state hopping recombination rate K(oo) = Kq is related to the coefficient i eff via equation (4.2.14) as in the diffusion-controlled case. As in equation (4.2.15), this Ifu is defined by the asymptotics of the solution, Y(r,oo) = y(r), as r —> oo. It is important, however, that R ff cannot generally be treated as the effective recombination radius. It holds provided that the hop length is much smaller than the distinctive scale ro of tunnelling recombination... 

Joint distribution of BB and AB pairs is shown in Fig. 6.44. The distribution of similar mobile particles B at long times in the asymmetric case practically is the same as in the symmetric case (when X = Xb). The behaviour of Xb (r, t) is determined by the Coulomb repulsion of B s for which the non-equilibrium screening effect does not take place. In its turn, some deviation for the joint dissimilar functions Y(r, t) seen in Fig. 6.44 for the symmetric and asymmetric cases is a direct consequence of different screening effects in the latter case the effective recombination radius increases in time which results in an increase of the Y(r,t) gradient at r = ro at long times this correlation function itself strives for the Heaviside step-like form. 

All local concentrations C of particles entering the non-linear functions F in equation (2.1.40) are taken at the same space points, in other words, the chemical reaction is treated as a local one. Taking into account that for extended systems we shouldn t consider distances greater than the distinctive microscopic scale Ao, the choice of equation (2.1.40) means that inside infinitesimal volumes vo particles are well mixed and their reaction could be described by the phenomenological reaction rates earlier used for systems with complete reactant mixing. This means that Ao value must exceed such distinctive scales of the reaction as contact recombination radius, effective radius of a dynamical interaction and the particle hop length, which imposes quite natural limits on the choice of volumes v0 used for averaging. 

Now the effective recombination (annihilation) radius could be defined similarly to that in the continuous approximation... 

The non-linearity of the equations (5.1.2) to (5.1.4) prevents us from the use of analytical methods for calculating the reaction rate. These equations reveal back-coupling of the correlation and concentration dynamics - Fig. 5.1. Unlike equation (4.1.23), the non-linear terms of equations (5.1.2) to (5.1.4) contain the current particle concentrations n (t), n t) due to which the reaction rate K(t) turns out to be concentration-dependent. (In particular, it depends also on initial reactant concentration.) As it is demonstrated below, in the fluctuation-controlled kinetics (treated in the framework of all joint densities) such fundamental steady-state characteristics of the linear theory as a recombination profile and a reaction rate as well as an effective reaction radius are no longer useful. The purpose of this fluctuation-controlled approach is to study the general trends and kinetics peculiarities rather than to calculate more precisely just mentioned actual parameters. 

In its turn Fig. 6.2 illustrates the effect of the initial concentration on the static tunnelling recombination kinetics. The latter is defined by a competition of three distinctive scales - the tunnelling recombination radius ro, mean distance between particles Iq = n(0) 1/d and lastly, the time-dependent correlation radius . At long time curves corresponding to different initial concentrations could be coincided by their displacements along ordinate axis, which confirms existence of the universal asymptotic decay law. 

The kinetics of the A + B - 0 bimolecular reaction between charged particles (reactants) is treated traditionally in terms of the law of mass action, Section 2.2. In the transient period the reaction rate K(t) depends on the initial particle distribution, but as f -> oo, it reaches the steady-state limit K(oo) = K() = 47rD/ieff, where D — Da + >b is a sum of diffusion coefficients, and /4fr is an effective reaction radius. In terms of the black sphere approximation (when AB pairs approaching to within certain critical distance ro instantly recombine) this radius is [74]... 

The value of the parameter L entering equation (6.4.1) defines whether the Coulomb attraction or recombination is predominant as L effective recombination sphere equals the Onsager radius). 

The cage effect was also analyzed for the model of diffusion of two particles (radical pair) in viscous continuum using the diffusion equation [106], Due to initiator decomposition, two radicals R formed are separated by the distance r( at / = 0. The acceptor of free radicals Q is introduced into the solvent it reacts with radicals with the rate constant k i. Two radicals recombine with the rate constant kc when they come into contact at a distance 2rR, where rR is the radius of the radical R Solvent is treated as continuum with viscosity 17. The distribution of radical pairs (n) as a function of the distance x between them obeys the equation of diffusion ... 

As mentioned earlier, ascorbate and ubihydroquinone regenerate a-tocopherol contained in a LDL particle and by this may enhance its antioxidant activity. Stocker and his coworkers [123] suggest that this role of ubihydroquinone is especially important. However, it is questionable because ubihydroquinone content in LDL is very small and only 50% to 60% of LDL particles contain a molecule of ubihydroquinone. Moreover, there is another apparently much more effective co-antioxidant of a-tocopherol in LDL particles, namely, nitric oxide [125], It has been already mentioned that nitric oxide exhibits both antioxidant and prooxidant effects depending on the 02 /NO ratio [42]. It is important that NO concentrates up to 25-fold in lipid membranes and LDL compartments due to the high lipid partition coefficient, charge neutrality, and small molecular radius [126,127]. Because of this, the value of 02 /N0 ratio should be very small, and the antioxidant effect of NO must exceed the prooxidant effect of peroxynitrite. As the rate constants for the recombination reaction of NO with peroxyl radicals are close to diffusion limit (about 109 1 mol 1 s 1 [125]), NO will inhibit both Reactions (7) and (8) and by that spare a-tocopherol in LDL oxidation. 

The results obtained in Ref 30 for partially diffusion-controlled recombination show that the field dependence of the recombination rate constant is affected by both the reaction radius R and the reactivity parameter p [cf. Eq. (33)]. Depending on their relative values, the rate constant can be increased or decreased by the electric field. The latter effect predominates at low values of p, where the reactants staying at the encounter distance are forced to separate by the electric field. 

Lexp[-L/ro]. For the weak Coulomb interaction, as L —> 0, we naturally obtain from equations (3.2.56) and (3.2.57) that ro the reaction rate again is controlled by the recombination at the black sphere radius. At last, both effective radii - for repulsion and attraction - are trivially related [50, 71] ... 

Strictly speaking, it is correct in the case of complete particle recombination at the black sphere only partial particle reflection is discussed by Doktorov and Kotomin [50]. Incorporation of the back reactions into the kinetics of geminate recombination has been presented quite recently by [74, 75]. The effective radius for an elastic interaction of defects in crystals, (3.1.4), was calculated by Schroder [3], Kotomin and Fabrikant [76],... 

There is a complete analogy here with what has been said about equation (4.1.63). Let us analyze now the effective radius of the diffusion-controlled tunnelling recombination in more detail. From equations (4.2.12) and (4.2.15), one gets... 

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