Collins-Kimball model

The more general result extended to the kinetic limit is given by the Collins-Kimball model. The integral over t from the rate constant, Eq. (3.21), is well... 

Only at the fastest diffusion, when Rq < a, the contact theory is applicable to electron transfer. Under this condition the Collins-Kimball expression (3.21) integrated in Eq. (3.4) constitutes a reasonable approximation to R(t) and for both long and short times. However, it was recognized long ago that for slower diffusion the Collins-Kimball model works better if a is considered as a fitting... 

Figure 3.8. (a) The linear viscosity dependence of the inverse ionization rate in the reaction studied in Ref. 98. Bullets—experimental points solid line—fit performed with the generalized Collins—Kimball model, (b) The effective quenching radius for the same reaction in the larger range of the viscosity variation. Bullets—experimental points solid fine—fit performed with the encounter theory for the exponential transfer rate. The diffusion coefficient D given in A2/ns was calculated from the Stokes—Einstein relationship corrected by Spemol and Wirtz [100]. 

We compare and contrast ZB and AB dynamics through treatments based on those of Smoluchowski-Collins-Kimball (SCK) 67.68 and Noyes 69-72. Except where otherwise noted, we treat ideal cases. Molecules A and B arc modeled as hard, spin-free, isotropically-ieactivo spheres surfaces Z are hard, uniform planes media arc structtirelcss. uniform, and isotropic surfaces and media extend to inlinity in all possible directions and Pick s Laws govern diffusion. 

The simplest theory of diffusion-assisted ET assumes that the reaction occurs only when donor and acceptor make contact (Collins-Kimball or the gray sphere model) [334-336]. Some experiments were analyzed on the basis of such theory [16, 337]. However, according to the Marcus expression, the ET rate can exhibit a peculiar dependence on the interparticle distance [17, 329, 338] as a result of the interplay between different dependencies of V, E, and AG, as shown in Figure 9.31. While in the normal region conventional exchange-type exponential dependence is at least qualitatively valid, kjyj(r) is generally nonmonotonic and acquires a bell-shaped form with increasing exothermicity. 

Over molecular length scales, the diffusion distances become very short (< 1 nm) so that only very rapid events can be influenced by these short diffusion times. Necessarily, this limits the number of systems to only relatively few, where the rate at which the reactants can approach one another is slow or comparable with the rate at which the reactants react chemically with each other. Some typical systems which have been studied are discussed in Sect. 2. The Smoluchowski [3] theory of reactions in solution, which occur at a rate limited solely by how fast the reactants can approach each other (sufficiently closely to react chemically almost instantaneously) is discussed in Sect. 3. If the chemical reaction is not so rapid, the observed rate of reaction may be influenced by both the rate of approach and the rate of subsequent chemical reaction. Collins and Kimball [4], and later Noyes [5], have extended the Smoluchowski theory (1917) to consider this situation (Sect. 4). In light of these quantitative theoretical models of diffusion-limited rate processes, some of the more recent and careful experiments on diffusion-controlled reactions in solution are considered briefly in Sect. 5. As the Smoluchowski theory... 

Northrup and Hynes [103] solved these equations for the case of a diffusion model and found the same results as Collins and Kimball [4] of eqn. (25). This case is reasonably easy to solve because the diffusion and reaction of the pair can be separated. When the motion of the pair involves a non-Markovian process, that is the reactants recall which direction they were moving a moment before (i.e. have a memory ) and the process is not diffusional, this elegant separation becomes very difficult or impossible to effect. Under these circumstances, eqn. (368) can only be solved approximately for the pair probability. The initial condition term, l(t), is non-zero if the initial distribution p(0) is other than peq. 

In the contact Smoluchowskii model refined by Collins and Kimball, the reaction takes place only at the closest approach distance a and the non-Markovian rate constant is a product of the contact rate constant kc and the pair density of reactants n r, t) at r = ct [2] ... 

Problems with the radiation boundary Some of the problems with using the radiation boundary condition to model chemical systems have been discussed in the literature [9]. The most important of these are (1) for particles close to the encounter distance, it is not possible to specify a non-zero probability for reaction, since an infinite number of encounters follow an unsuccessful first encounter resulting in reaction (as shown by Collins and Kimball [7]). (2) Schell and Kapral [10] have shown that the probability of reaction on encounter should scale with the ratio of D and a ( D ls, the mutual diffusion coefficient and a the encounter distance) for radiation boundary condition to be applicable. (3) All re-encounters are treated in the same manner. 

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