Black reaction sphere

It is convenient to analyze the general equations (3.2.7) and (3.2.11) in terms of the simplest model of the recombination event called clear-cut radius or black reaction sphere - equation (3.1.1) ... 

It is convenient to consider a model of an anisotropic recombination region the reflecting recombination sphere (white sphere) with black reaction spots on its surface [77, 78], The measure of the reaction anisotropy here is the geometrical steric factor Q which is a ratio of a black spot square to a total surface square. Such a model could be actual for reactions of complex biologically active molecules and tunnelling recombination when the donor electron has an asymmetric (e.g., p-like) wavefunction. Note the non-trivial result that at small Q, due to the partial averaging of the reaction anisotropy by rotational motion arising due to numerous repeated contacts of reactants before the reaction, the reaction rate is K() oc J 1/2 rather than the intuitive estimate Kq oc Q. 

Equation (8.3.14) is not an asymptotically exact result for the black sphere model due to the superposition approximation used. When deriving (8.3.14), we neglected in (8.3.11) small terms containing functionals I[Z], i.e., those terms which came due to Kirkwood s approximation. However, the study of the immobile particle accumulation under permanent source (Chapter 7) has demonstrated that direct use of the superposition approximation does not reproduce the exact expression for the volume fraction covered by the reaction spheres around B s. The error arises due to the incorrect estimate of the order of three-point density p2,i for a large parameter op at some relative distances ( f — f[ < tq, [r 2 - r[ > ro) the superposition approximation is correct, p2,i oc ct 1, however, it gives a wrong order of magnitude fn, oc Oq2 instead of the exact p2,i oc <7q 1 (if n — r[ < ro, fi — f[ < ro). It was... 

If Rq > black sphere radius. This is the radius of a reaction sphere outside which neither excitation is yet quenched while inside the sphere all of them are already deactivated. In reality, of course, the border between outskirts and interior of the sphere is not as sharp, but Rq fixes this boundary and specifies the stationary (Markovian) rate of quenching (3.41). 

This phenoxyl radical recombines with the rate constant k = 1.5x 107 L mol-1 s-1 in toluene where its kD 2-4 x 109 L mol 1 s. This phenomenon was explained within the scope of conception of bimolecular reaction as the interaction of two spheres with black spots [79-85], Two different situations are possible for reactions that occur without an activation energy ... 

Figure 3.1. Schematic representation of dimensional reduction for a framework of corner-sharing MX6 octahedra. The M and X atoms are represented by black and white spheres, respectively. In a) though d), reaction with AbX incorporates additional X atoms into the M—X framework, progressively reducing the connectedness and effective dimensionality of the M—X framework. In d), after incorporating n units of AbX (n > 2), the structure is reduced to isolated oligomeric or monomeric components. For clarity, the A atoms are not shown in the figure. [Adapted with permission from [Ref. 16]. Copyright 2001 American Chemical Society.]...

Figure 13.24 Structures of sMMO components and proposed reaction cycle, (a) MMOH (b) the MMOR FAD and ferredoxin (Fd) domains (c) MMOB. In MMOH the ot, P and y subunits are coloured blue, green and purple, respectively. Iron, sulfur and FAD are coloured orange, yellow and red, respectively and are depicted as spheres. The MMO reaction cycle is shown on the right, with atoms coloured [Fe (black), C (grey), O (red) and N (blue)]. (Reprinted with permission from Sazinsky and Lippard, 2006. Copyright (2006) American Chemical Society.)...

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],... 

The black sphere approximation permits us to obtain the most simple and physically transparent results for the kinetics of diffusion-controlled reactions. We should remind that this approximation involves a strong negative correlation of dissimilar particles at r ro, where Y(r, t) = 0, described by the Smoluchowski boundary condition... 

Similarly to the black sphere model, equation (4.1.63), the effective reaction radius i eff could be defined through Kq — 47rDi eff. Comparing the i eff obtained in such a way, with the results of Chapter 3, the conclusion suggests that they coincide, i.e., both definitions of the effective radii turn... 

Considering the reaction kinetics in the preceding Sections of Chapter 4, we have restricted ourselves to the simplest case of the recombination rate er(r) corresponding to the black sphere approximation, equation (3.2.16). However, if recombination is long-range, like that described by equations (4.1.44) or (3.1.2), one has to use equations (4.1.23) and (4.1.24), which yield essentially more complicated kinetics, especially for the transient period. Let us discuss briefly the main features of the diffusion-controlled kinetics controlled by tunnelling recombination, equation (3.1.2) (see also [32-34]). 

The conclusion could be drawn from Fig. 4.2 that the steady-state profile depends essentially on the defect mobility or temperature - unlike the black sphere model, equation (4.1.70). The steady-state solution y(r) defines the stationary reaction rate K(00) through the effective radius of reaction R n-... 

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. 

Diffusion-controlled reactions. Black sphere model... 

Direct establishment of the asymptotic reaction law (2.1.78) requires performance of computer simulations up to certain reaction depths r, equation (5.1.60). In general, it depends on the initial concentrations of reactants. Since both computer simulations and real experiments are limited in time, it is important to clarify which values of the intermediate asymptotic exponents a(t), equation (4.1.68), could indeed be observed for, say, r 3. The relevant results for the black sphere model (3.2.16) obtained in [25, 26] are plotted in Figs 6.21 to 6.23. The illustrative results for the linear approximation are also presented there. 

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]... 

This equation differs from (5.1.4) which served us as an example for calculating the reaction rate in the black sphere model. Introducing the function h(r,t) = a(r)Y(r,t), and taking into account (3.2.16), we arrive at... 

Sodium was stored under mineral oil and washed with pentane before use. For convenience the checkers used 1/6 to 1/4 inch sodium spheres (Matheson Coleman and Bell) that were weighed in mineral oil, then wiped free of oil, rinsed in hexane, cut in half, rinsed in hexane again, and immediately added to the reaction over a 2-hr period, during which time the dark-black mixture became extremely viscous. 

The R obtained from this condition is approximately equal to Rq and follows the same logarithmic dependence on diffusion (3.62). All excitations crossing the reaction layer of width 1/2 adjacent to the black sphere are quenched during their residence time there, l2/AD. However, most of those that were initially inside this sphere are quenched where they were at the moment of excitation. These excitations disappear first, during static quenching, which is often considered as instantaneous compared to subsequent stages limited by diffusion, which delivers the excitations from outside into the black sphere. 

First, the diffusional radical reactions in solutions whose rates are proportional to I) sometimes have rate constants that are much smaller than their contact estimate for the isotropic black sphere, ki> = 4naD. It was proved that this is the result of chemical anisotropy of the reactants. Partially averaged by translational and rotational diffusion of reactants, this anisotropy manifests itself via the encounter efficiency w < 1, which enters the rate constant kn = w4nak) [249]. Even the model of white spheres with black spots is more appropriate for such reactions than the conventional Smoluchowski model. 

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