The Cage Effect

There are several criticisms of the diffusion equation approach to radical pair recombination [5]. In particular, the treatment of the radical 

Noyes [269, 270] and, more recently, Northrup and Hynes [103] have endeavoured to incorporate some aspects of the caging process into the Smoluchowski random flight or diffusion equation approach. Both authors develop essentially phenomenological analyses, which introduce further parameters into an expression for escape probabilities for reaction, that are of imprecisely known magnitude and are probably not discrete values but distributed about some mean. Since these theories expose further aspects of diffusion-controlled processes over short distances near encounter, they will be discussed briefly (see also Chap. 8, Sect. 2.6). 

To evaluate j3 and j30, Noyes [269, 270] considered the motion of radicals as random flights (Chandrasekhar [271]). j3 was approximated as 

However, for (r0 — R) a, Noyes [269] developed a different form of j30. Since such motion must be on a molecular scale a R) or even less, the author is doubtful that such approximations are more valid than that of the diffusion equation. To estimate a, it may be noted that Northrup and Hynes [103] have found that 

From studies by Chuang et al. [266] of iodine atom recombination, the probability of recombination is large. The probability would be even greater if the iodine atoms were formed as an encounter pair. Accordingly, a is near to unity. Furthermore, Noyes suggests / /a 10 for iodine atoms and so 0 0.9. Hence, q(r0) R/r0, as might be seen from the long-time limit of eqn. (134). 

Following Noyes [269, 270], recombination of two radicals occurs by diffusion together from a distance Kq to form cm encounter pair this process has a probability Pq. The encounter pair may approach each other a little closer to collide and then react, with probability a, or it may separate before reaction with probability (1 — a). In the latter case, the probability that the separating encounter pair will reform an encounter pair within the cage is p. Further separations and re-encounters may occur. The probability of recombination at the first encounter is o)3o, of the second is o qP 1 — a), of the third oPqP (1 — a), etc. The total recombination probability is the sum of this geometric series 

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The traditional difhision model pemiits estimation of the magnitude of the cage effect in solntion according to [37]... 

The simple difhision model of the cage effect again can be improved by taking effects of the local solvent structure, i.e. hydrodynamic repulsion, into account in the same way as discussed above for bimolecular reactions. The consequence is that the potential of mean force tends to favour escape at larger distances > 1,5R) more than it enliances caging at small distances, leading to larger overall photodissociation quantum yields [H6, 117]. 

Murrell J N, Stace A J and Dammel R 1978 Computer simulation of the cage effect in the photodissociation of iodine J. Chem. Soc. Faraday Trans. II 74 1532... 

However, when MAIs are thermolyzed in solution, the role of the cage effect has to be taken into account. The thermolytically formed macroradicals can, due to their size, diffuse only slowly apart from each other. Therefore, the number of combination events will be much higher for MAIs than for low-molecular weight AIBN derivatives. As was shown by Smith [16], the tendency toward radical combination depends significantly on the rigidity and the bulkiness of the chain. Species such as cyclohexyl or diphenylmethyl incorporated into the MAI s main chain lead to the almost quantitative combination of the radicals formed upon thermolysis. In addition, combination chain transfer reactions may... 

The efficiency of the intitiator is a measure of the extent to which the number of radicals formed reflects the number of polymer chains formed. Typical initiator efficiencies for vinyl polymerisations lie between 0.6 and 1.0. Clearly the efficiency cannot exceed 1.0 but it may fall below this figure for a number of reasons, the most important being the tendency of the newly generated free radicals to recombine before they have time to move apart. This phenomenon is called the cage effect . 

G. W. Hoffman T. J. Chuang, and K. B. Eisenthal, Picosecond smdies of the cage effect and collision induced predissociation of iodine in liquids. Chem. Phys. Lett. 25(2), 201-205 (1974). 

Matheson s widely accepted explanation in terms of the cage effect appears to be based on an unrealistic interpretation of the rates of the processes involved. 

Figure 3. Calculated efficiencies. (1) From the cage effect model and no primary radical termination (Case I) (2) From the assumption of an overall efficiency and no primary radical termination (Case II) (3) From the assumption of an overall efficiency and primary radical termination (Case III) ( l) Calculated from equation (A) with fo - 0.663.

The use of an overall initiator efficiency, appears to be more effective than the cage effect concept in describing both the effect of the initiator concentration on the efficiency, and the initiator loadings. 

The cage effect described above is also referred to as the Franck-Rabinowitch effect (5). It has one other major influence on reaction rates that is particularly noteworthy. In many photochemical reactions there is often an initiatioh step in which the absorption of a photon leads to homolytic cleavage of a reactant molecule with concomitant production of two free radicals. In gas phase systems these radicals are readily able to diffuse away from one another. In liquid solutions, however, the pair of radicals formed initially are caged in by surrounding solvent molecules and often will recombine before they can diffuse away from one another. This phenomenon is referred to as primary recombination, as opposed to secondary recombination, which occurs when free radicals combine after having previously been separated from one another. The net effect of primary recombination processes is to reduce the photochemical yield of radicals formed in the initiation step for the reaction. 

Quantum yield (d>) of molecular photodissociation in the gas phase is equal to unity according to the Einstein law. Frank and Rabinowitch [72] predicted the reduction of the quantum yield in a solution due to the cage effect. The quantum yield

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The influence of pressure on the cage effect was studied by Neuman and colleagues [95-98]. They measured the influence of pressure on the cage effect for competition between recombination and diffusion for the 1,1-dimethylethoxy radical pairs generated from bis(l,l-dimethylethyl)hyponitrite. The empirical activation volume difference (AF(f for the... 

The cage effect can be interpreted within the scope of a simple kinetic scheme [15]. For example, azo-compound decomposes according to the kinetic scheme given below ... 

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 ... 

The very low yield of radicals by the reaction of ozone with cumene was found to be the result of the intensive ozone reaction with the benzene ring of cumene with molozonide formation. The values of the parameter e in other reactions are typical of the cage effect of radical pairs in solutions. The rate constants of ozone reactions with various compounds are presented in Table 3.7 and Table 3.8. 

The cage effect is a component of this scheme. It takes place when the RO radical rapidly (within the time of the cage existence) reacts with the metal ion in the oxidized state. 

The mechanism of antioxidant action on the oxidation of carbon-chain polymers is practically the same as that of hydrocarbon oxidation (see Chapters 14 and 15 and monographs [29 10]). The peculiarities lie in the specificity of diffusion and the cage effect in polymers. As described earlier, the reaction of peroxyl radicals with phenol occurs more slowly in the polymer matrix than in the liquid phase. This is due to the influence of the polymeric rigid cage on a bimolecular reaction (see earlier). The values of rate constants of macromolecular peroxyl radicals with phenols are collected in Table 19.7. 

Azobisisobutyronitrile, 182, reacts thermally or photochemically to give the intermediate 183, which leads, in inert solvents, to combination products 184 and 185, and disproportionation products 186 and 187. The parent compound is dimorphic, and both crystal forms behave similarly on photolysis, yielding 95% disproportionation and 5% 184. In contrast, in both fluid and rigid solution the disproportionation products form only 5% of the total. The cage effect in the solid is almost quantitative. 

Lowering of / by reactions analogous to Eq. 3-68 is referred to as the cage effect [Bamford, 1988 Koenig and Fischer, 1973 Martin, 1973]. It is a general phenomenon observed in almost all initiation systems. About the only exceptions are the initiation systems such as... 

The cage effect has also been well documented as the cause of lowered values in photoinitiation [Berner et al., 1978 Braun and Studenroth, 1979]. Benzoin photolysis yields benzaldehyde ... 

Most of the energy associated with an incident x-ray or y-ray is absorbed by ejected electrons. These secondary electrons are ejected with sufficient energy to cause further ionizations or excitations. The consequences of excitations may not represent permanent change, as the molecule may just return to the ground state by emission or may dissipate the excess energy by radiationless decay. In the gas phase, excitations often lead to molecular dissociations. In condensed matter, new relaxation pathways combined with the cage effect greatly curtail permanent dissociation. Specifically in DNA, it is known that the quantum yields for fluorescence are very small and relaxation is very fast [6]. For these reasons, the present emphasis will be on the effects of ionizations. 

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