Re-encounter

As mentioned earlier, radicals that do not react can go on a random walk and may meet later at time complex constant determined... 

If A is the probability of recombination during a single encounter of a singlet radical pair (close to unity), then the chance of product formation during the second encounter at time t is A Csn( )Pf(0- additional re-encounters, it can be shown that An(<) = A Csn( )P d therefore recombination during the time interval between t and t + dt has a probability Pn(<)d< given by equation (34),... 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

The ambiguity of definition of Re encountered in the concentric annulus case is compounded here because of the fact that no viscosity is definable for non-Newtonian fluids. Thus, in the literature one encounters a bewildering array of definitions of Re-like parameters. We now present friction factor results for the non-Newtonian constitutive relations used above that are common and consistent. Many others are possible. 

As shown in Fig. 3-1, some radicals in separated radical pairs re-encounter their partners within the solvent cages, but others escape from the cages, forming "escape radicals". Since the time scales for the secondary recombination and the S-To conversion rates are 10 " 10 and 10 10 s, [3] respectively, the change in the S-To conversion rate by an external magnetic field and/or the HFC term can influence the yield of cage and escape products. 

Let us consider a reaction where the cage recombination occurs only through the singlet close pair but not through the triplet close pair as shown in Fig. 3-1. In such a reaction, Noyes [4] showed that the probability (f(r)) of the first re-encounter between t and r+dr for a pair, separating from an encounter at /=0, is given from the theory of random flights as... 

The exact behavior at short times is not so important. In Eq. (3-21), p is the total probability at least one re-encounter. 

Fig. 3-3. Probabilities of product formation during first, second, and third re-encounters PN t), Pm(0. and after the formation of radical pairs in solution.

Radicals of a pair that fails to react during the first re-encounter start again their random walk. The total probability Qm t)) for s ch radicals is given as,... 

Fig. 3-3(b) shows the chance of product formation during a second re-encounter (PtdfW) in the interval (r, r + dr) after a first re-encounter at tj. With a similar method used for a first re-encounter, becomes the product of Qmih),... 

Some of the doublet radical and unquenched triplet molecule in the separated pairs continue to diffuse further with each other and become free radical and triplet molecule. This component produces no CIDEP. On the other hand, the other doublet radical and unquenched triplet molecule in the separated pairs re-encounter wnth each other and become the closed pairs again. At the second encounter, the population of D+,2)is increased and... 

The lifetime of encounter complexes between neutral reactants is on the order of 0.1 ns in solvents of low viscosity, that is, k d kd m. Random diffusive displacements of the order of a molecular diameter occur with a frequency of about 1011 s Subsequently, the fragments from a specific dissociation may re-encounter each other and undergo secondary recombination .59 If secondary recombination does not take place within about 1 ns, the fragments will almost certainly have diffused so far apart that the chance of a reencounter becomes negligible. The initial overall electronic multiplicity 2S + 1 of encounter complexes is thus important in determining the fate of the reactants, because their lifetime is usually insufficient to allow for intersystem crossing during an encounter. 

The role of exciplexes in self-sensitized photo-oxidations has been further investigated by Stevens and co-workers.273 - Photoperoxidation of 1,3-diphenyl-isobenzofuran in solution proceeds at a rate which is independent of acceptor concentration when this is very low, and this observation has been interpreted in terms of a re-encounter of x02 and ground-state acceptor molecules generated in the same triplet-triplet annihilation act. This interpretation accounts for the failure of tetramethylethylene to inhibit the reaction completely. Processes 1—3 in Scheme 8 account for the observations if re-encounter effects are included,... 

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