Recombination and escape probabilities

Let us suppose that an isolated ion-pair is formed at time to with a separation Tq. The rate of creation of the ion-pair is 5(r—ro)5(t —to) (Chap. 6, Sect. 2.2) and the appropriate diffusion equation must include 

The probability that the ions have a separation r, given that one ion is at r , at time t is the density p(r, f ro, t ), which is also related to the Green s function of the diffusion equation (see Sect. 2.3 and the discussion in Appendix A). As before, the survival probability is the integral of the density over all space [eqn. (123)] and this may be related to the flux, J, crossing the encounter surface [eqn. (124)], which is (following Chap. 3, Sect. 1.1) 

If the potential energy, U, and the boundary and initial conditions are spherically symmetrical, then the survival and recombination probabilities are, respectively 

Because the diffusive flux is enhanced by this drift of a charge under the influence of the coulomb potential [as represented in eqn. (142)], the partially reflecting boundary condition (127) has to be modified to balance the rate of reaction of encounter pairs with the rate of formation of encounter pairs [eqn. (46)]. However, the rate of reaction of ion-pairs at encounter is usually extremely fast and the Smoluchowski condition, eqn. (5), is adequate. The initial and outer boundary conditions are the same as before [eqns. (131) and (128), respectively], representing on ion-pair absent until it is formed at time to and a negligibly small probability of finding the ion-pair with a separation r 

The solution of eqn. (141) with the boundary and initial conditions discussed above and for the coulomb potential between ions of charge 

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Here, Vd = pE is the drift velocity. The recombination and escape probabilities are now given by PR = NR /n+° L0 and Pkc = 1 - Pr. Since Vd = i, but T /r1 these probabilities are independent of mobility. However, the initial separation r0 is expected to depend (increase) with electron mobility, thus making the escape probability indirectly dependent on the mobility. These effects are quite similar to those in the Onsager theory... 

Northrup and Hynes [103] have remarked that the effects of the potential of mean force as well as hydrodynamic repulsion are very much more apparent in their effect on the survival (and escape) probability of a reactant pair of radicals than their effect on the rate coefficient. For instance, considering the escape probability of Fig. 20, suppose that an escape probability of 0.75 had been determined experimentally. Initial distances of separation Tq = 4i or 312 would have been deduced from the diffusion equation analysis alone or from the diffusion equation with the potential of mean force and hydrodynamic repulsion included. Again, the effect of a moderately slow rate of reaction of encounter pairs further reduces the recombination probability. Consequently, as the inherent uncertainty in the magnitudes of U r), D(r) and feact may be as much as a factor of 2, the estimation of an initial separation distance, Tq, of a radical pair from experimental measurements of escape probabilities may be in doubt by a factor of 30% or more. Careful and detailed analysis of the recombination of radical pairs has been made by Northrup and Hynes... 

Figure 11 shows the result of this experiment on a solution of 5 mM N-acetyl tryptophan and 0.2 mM 3-N-carboxy-methyl lumiflavin, hereafter simply called flavin (see Figure 10). Positive enhancements can be observed for the aromatic C-2, C-4 and C-6 protons, while the CH2 group shows emission. This polarization pattern corresponds with a tryptophyl radical in which the electron spin is delocalized over the aromatic ring. It can further be noted that almost no flavin polarization is present in the difference spectrum. Figure 11c (weak lines are present at 2.6 and 4.0 ppm). This is due to cancellation of recombination and escape polarization as will be discussed in Section 5. The mechanism of the photoreaction undoubtedly involves triplet flavin (17). Since 1-N-methyl tryptophan shows similar CIDNP effects, the primary step most probably is electron transfer to the photo-excited flavin. This is also supported by a flash photolysis study by Heelis and Phillips (18). The nature of the primary step in the photoreactions with amino acids is important in view of the interpretation of "accessibility" of an amino acid side chain in a protein as seen by the photo-CIDNP method. This question is therefore the subject of further study.

Williams (1964) derived the relation T = kBTrQV3De2, where T is the recombination time for a geminate e-ion pair at an initial separation of rg, is the dielectric constant of the medium, and the other symbols have their usual meanings. This r-cubed rule is based on the use of the Nernst-Einstein relation in a coulom-bic field with the assumption of instantaneous limiting velocity. Mozumder (1968) criticized the rule, as it connects initial distance and recombination time uniquely without allowance for diffusional broadening and without allowing for an escape probability. Nevertheless, the r-cubed rule was used extensively in earlier studies of geminate ion recombination kinetics. 

With r = 28.45 nm, r = 3.0 nm, and r = 8.39 nm, Bartczak and Hummel (1986) compute the escape probability Pesc = 0.0336, 0.0261, and 0.0230 respectively for N = 1, 2, and 3. While the first is comparable to the Onsager value, the latter are new results. The kinetics of recombination for the isolated pair, found by Bartczak and Hummel (1987) using MC, is very similar to that obtained by Abell e al. (1972). For N > 1, these authors found the recombination kinetics to be faster than that for the isolated pair. For two pairs, the calculated escape probability increased with the external field, but not as strongly as for N = 1. 

The latter quantity is called the escape probability, and describes the probability that the two oppositely charged particles of the geminate pair will never recombine with each other and become free ions. 

By solving Eq. (9) subject to boundary conditions (10) and (11b), the escape probability for the totally diffusion-controlled geminate ion recombination is calculated as... 

One of the most important experimental methods of studying the electron-ion recombination processes in irradiated systems are measurements of the external electric field effect on the radiation-induced conductivity. The applied electric field is expected to increase the escape probability of geminate ion pairs and, thus, enhance the number of free ions in the system, which will result in an enhanced conductivity. 

Calculation of the electric field dependence of the escape probability for boundary conditions other than Eq. (11b) with 7 = 0 poses a serious theoretical problem. For the partially reflecting boundary condition imposed at a nonzero R, some analytical treatments were presented by Hong and Noolandi [11]. However, their theory was not developed to the level, where concrete results of (p(ro,F) for the partially diffusion-controlled geminate recombination could be obtained. Also, in the most general case, where the reaction is represented by a sink term, the analytical treatment is very complicated, and the only practical way to calculate the field dependence of the escape probability is to use numerical methods. 

In the preceding part of this section, we have concentrated on the electron escape probability, which is an important quantity in the geminate phase of recombination, and can be experimentally observed. However, modern experimental techniques also give us a possibility to observe the time-resolved kinetics of geminate recombination in some systems. Theoretically, the decay of the geminate ion pairs can be described by the pair survival probability, W t), defined by Eq. (4). One method of calculating W t) is to solve the Smoluchowski equation [Eq. (2)] for w r,t) and, then, to integrate the solution over the space variable. Another method [4] is to directly solve Eq. (7) under relevant conditions. 

An important result of the theoretical studies of the multipair effects is that the recombination kinetics in a cluster of ions, in which the initial separation between neighboring cations is 1 nm, is faster than the corresponding decay kinetics of a single ion pair [18]. Furthermore, the escape probability is lower than the Onsager value [Eq. (15)], and decreases with increasing number of ion pairs in the cluster (a relative decrease of about 30% for two ion pairs, and about 50% for five ion pairs). The average electron escape probability in radiation tracks obviously depends on the distribution of ionization events in the tracks, which is determined by the type of radiations and their energy. 

Mozumder [315] presented an elegant and direct analysis of the longtime or ultimate recombination probability (the complement of which is the escape probability) for an ion-pair. The steady state arises after a time sufficiently long for all transient effects to have decayed away. Only the residual probability that an ion-pair still remains, p(r, t - °°) at some (large) distance of separation is of interest, and dp/dt - 0 as t - °°. Using eqn. (141) with the coulomb potential gives... 

It was pointed out in Chap. 8, Sect. 2.1 that there are primarily two reasons for the failure of the diffusion equation to describe molecular motion on short times. They are connected with each other. A molecule moving in a solvent does not forget entirely the direction it was travelling prior to a collision [271, 502]. The velocity after the collision is, to some degree, correlated with its velocity before the collision. In essence, the Boltzmann assumption of molecular chaos is unsatisfactory in liquids [453, 490, 511—513]. The second consideration relates to the structure of the solvent (discussed in Chap. 8, Sects. 2.5 and 2.6). Because the solvent molecules interact with each other, despite the motion of solvent molecules, some structure develops and persists over several molecular diameters [451,452a]. Furthermore, as two reactants approach each other, the solvent molecules between them have to be squeezed-out of the way before the reactants can collide [70, 456]. These effects have been considered in a rather heuristic fashion earlier. While the potential of mean force has little overall effect on the rate of reaction, its effect on the probability of recombination or escape is rather more significant (Chap. 8, Sect. 2.6). Hydrodynamic repulsion can lead to a reduction in the rate of reaction by as much as 30-40% under the most favourable circumstances (see Chap. 8, Sect. 2.5 and Chap. 9, Sect. 3) [70, 71]. 

There is no general consensus on why the difference in the quantum yield of photosubstitutions is so large for 02-adducts (4> 10 3) and CO-adducts and on which excited states are responsible for this difference. An explanation based on a different efficiency of the recoordination of released 02 or CO molecules (geminate recombination) can be ruled out, as in the systems with the same biocomplex (e.g. Hb02 and HbCO) both molecules (02 and CO) have nearly identical escaping probability from the protein cage due to their similar size, mass and polarity. The reason could, therefore, lie in the different photoreactive excited states involved. 

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