Radical pair modeling diffusion

The photolyses of 1,2-dipheny1-2-methyl-1-propa-none and its 2h and derivatives in micellar solution are now described and further demonstrate the enhanced cage and magnetic isotope effects of mlcelllzatlon. We report also the observation of CIDP during the photolyses of micellar solutions of several ketones, and demonstrate the validity of the radical pair model to these systems. Analyses of the CIDNP spectra in the presence and absence of aqueous free radical scavengers (e.g., Cu2+) allow us to differentiate between radical pairs which react exclusively within the micelle and those that are formed after diffusion into the bulk aqueous phase. In some cases this allows us to estimate a lifetime associated with the exit of free radicals from the micelles. 

Hollander, 1972) using basically the Noyes diffusion model (Noyes, 1954, 1955, 1956, 1961), treats the radicals as if they go on a random walk . It must be remembered, however, that the two radicals must maintain the correlation of their electron spins (Jee f ) or tl distinction between T and S manifolds for a given radical pair would be lost. 

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 ... 

At the initial stage of reactions, the produced intermediate species such as the cation radical and the electron exist in a narrow space, the so-called spur. After the electron thermalization process, a pair of a cation radical and a thermalized electron remain in a spur. The geminate ion recombination of the cation radical and the electron occurs before these ionic species diffuse and spread uniformly in the media. Therefore the geminate ion recombination takes place in the spur. On the condition of a so-called single pair model,... 

The importance of the size of the enclosure (reaction cavity) on a reaction course has been a subject of investigation in several laboratories. On the basis of the proposed mechanistic scheme for DBK fragmentation and on the basis of the effective reaction cavity model presented in Section III, the following predictions can be made (a) a relationship will exist between the cage effect and the reaction cavity size (b) the cage effects observed for singlet and triplet benzyl radical pairs will be different (assuming very similar diffusion rates)... 

Homogeneous kinetics is used instead of diffusion kinetics to express the dependence of intraspur GH, on solute concentration. The rate-determining step for H2 formation is not the combination of reducing species, but first-order disappearance of "excited water." Two physical models of "excited water" are considered. In one model, the HsO + OH radical pair is assumed to undergo geminate recombination in a first-order process with H3O combination to form H2 as a concomitant process. In this model, solute decreases GH, by reaction with HsO. In the other model, "excited water" yields freely diffusing H3O + OH radicals in a first-order process and solute decreases GH, by reaction with "excited water." The dependence of intraspur GH, on solute concentration indicates th,o = 10 9 — 10 10 sec. 

With the establishment of the primary phototriplet reduction mechanism we now turn to the explanation for the effect of flow rates and the formation of polarized phenoxy radicals. Since reaction [2] is a relatively fast secondary process it is readily understood that the observation of the primary ketyl radicals will be dependent upon flow rate. The triplet polarization (E) of the secondary phenacyl radical should not have been affected but the increased contribution of the E/A Radical-Pair polarization altered the overall appearance of the polarization pattern. The diffusion model of the Radical-Pair theory relates the E/A polarization magnitude to the radical concentration within the reaction zone. Since the phenacyl radical is considered to be very chemically reactive, and the product phenol "accumulated" within the reaction zone is also a much better hydrogen donor, the following reactions will proceed within the reaction zone ... 

The diffusion model can be described by a time-dependent Schrodinger equation for a radical pair separated at time t = 0 ... 

A similar representation of a pictorial vector model for CIDEP processes is more difficult to formulate, but recently Monchick and Adrian (100) have succeded in casting the stochastic Liou-ville model of CIDEP into the form of a "Block-type" equation with diffusion which led to a generalized vector model of the radical-pair mechanism to give a clear qualitative picture of both CIDNP and CIDEP. 

With the diffusion model it became apparent that polarization is influenced by the probability of the two radicals being able to reencounter after diffusing apart. In an independent approach to calculate this probability, Adrian (4-7) arrived at the same formulation as developed by Noyes (103) many years ago. The final estimated expectation value of the electron polarization in a geminate radical pair is... 

Recently Kaptein and Adrian have modified the original model by in-trodudng a more adequate function/(/). Noyes ) has treated the diffusive behaviour of radical pairs and has calculated the probability of the first reencounter beween t and t + dt for a pair separating at r = 0 from an encounter to be... 

Diffusion is intricately linked with all aspects of the radical-pair mechanism. The CIDNP kinetics for the reaction of a sensitizer with a large spherical molecule that has only a small reactive spot on its surface were studied theoretically. This situation is t)qjical for protein CIDNP, where only three amino acids are readily polarizable, and where such a polarizable amino acid must be exposed to the bulk solution to be able to react with a photoexcited dye. Goez and Heun carried out Monte Carlo simulations of diffusion for radical ion pairs both in homogeneous phase and in micelles. The advantage of this approach compared to numerical solutions of the diffusion equation is that it can easily accommodate arbitrary boundary conditions, such as non-spherical symmetry, as opposed to the commonly used "model of the microreactor" ° where a diffusive excursion starts at the micelle centre and one radical is kept fixed there. 

In a study of phenacylphenylsulfone photolysis, CIDNP data were taken as evidence that the primary radical pairs cannot recombine to regenerate the starting material because the micelle forces a certain orientation of the radicals [63], From low-field 13C CIDNP and SNP measurements on cleavage of benzylic ketones in sodium dodecyl sulfate micelles, it was inferred [64] that the exchange interaction in these systems is several orders of magnitude smaller ( 10lorads 1 at a reduction distance of 6 A cf. the values in Section IV.B) and the distance dependence is much weaker (a x 0.5 A" cf. the discussion of Eq. 10) than generally assumed for radical pairs. By numerical solutions of the stochastic Liouville equation for a model of the micelle where one of the radicals is kept fixed at the center of the micelle while the other radical is allowed to diffuse, the results of MARY experiments, 13C CIDNP experiments at variable fields, and SNP experiments could be reproduced with the same set of parameters [65],... 

Both the exciton/radical pair equilibrium model and the bipartite model predict formally the same kinetics and thus both give rise also to a biexponential fluorescence decay. However, the two models are fundamentally different. This difference consists in the entirely different meaning of the rate constants involved and thus in the entirely different origin of the two observed lifetimes. In the bipartite model the biexponentiality is due to the equilibration of the excitons between antenna and reaction center. Thus one of the lifetime components reflects an eneigy transfer process. In contrast to the exciton/radical pair equilibrium model the bipartite model basically describes a diffusion-limited kinetics. Despite the fact that it formaUy can describe correctly the observed kinetics, the application of the bipartite model on experimental data leads to physically unreasonable results. First it results in a charge separation time in the reaction centers which is by one to two orders of magnitude too high. 

Here, k is the rate constant for reaction, t is the radical pair lifetime, a is the thickness of the reaction layer, b is the reaction distance, D the mutual diffusion coefficient and T and T2 are the longitudinal and transverse relaxation times of the counter radical. This model neglects any magnetic interactions (i.e. hyperfine and Zeeman) which could possibly influence the observed chemical kinetics. 

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