Geminate recombination equation

In an early attempt, Mozumder (1968) used a prescribed diffusion approach to obtain the e-ion geminate recombination kinetics in the pure solvent. At any time t, the electron distribution function was assumed to be a gaussian corresponding to free diffusion, weighted by another function of t only. The latter function was found by substituting the entire distribution function in the Smoluchowski equation, for which an analytical solution was possible. The result may be expressed by... 

The probability of the radical pair to escape from geminate recombination (e) depends on the rate constants according to the following equation ... 

Analysis of pump-probe absorption data of Walther et al. (2005 not shown here) indicated two stretched exponentials characterizing the geminate recombination. In agreement with pump-probe results, their transient grating data (reproduced in Figure 1.12) have been fitted by an equation,... 

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. 

The analytical solution of the Smoluchowski equation for a Coulomb potential has been found by Hong and Noolandi [13]. Their results of the pair survival probability, obtained for the boundary condition (11a) with R = 0, are presented in Fig. 2. The solid lines show W t) calculated for two different values of Yq. The horizontal axis has a unit of r /D, which characterizes the timescale of the kinetics of geminate recombination in a particular system For example, in nonpolar liquids at room temperature r /Z) 10 sec. Unfortunately, the analytical treatment presented by Hong and Noolandi [13] is rather complicated and inconvenient for practical use. Tabulated values of W t) can be found in Ref. 14. The pair survival probability of geminate ion pairs can also be calculated numerically [15]. In some cases, numerical methods may be a more convenient approach to calculate W f), especially when the reaction cannot be assumed as totally diffusion-controlled. 

The geminate recombination in the presence of a scavenger can be described by the Smoluchowski equation [Eq. (2)] with an additional term representing the loss of the geminate pairs by scavenging reactions... 

When the motion of electrons and positive ions in a particular system may be described as ideal diffusion, the process of bulk recombination of these particles is described by the diffusion equation. The mathematical formalism of the bulk recombination theory is very similar to that used in the theory of geminate electron-ion recombination, which was described in Sec. 10.1.2 ( Diffusion-Controlled Geminate Ion Recombination ). Geminate recombination is described by the Smoluchowski equation for the probability density w(r,i) [cf. Eq. (2)], while the bulk recombination is described by the diffusion equation for the space and time-dependent concentration of electrons around a cation (or vice versa), c(r,i). 

When the decomposition in question involves more than one bond in a concerted homolysis, as in an azoalkane (Equation 9.39), the disappearance of substrate is unaffected by recombination, but the number of R radicals available in the bulk solution to initiate other processes is less than two for each molecule of initiator consumed. Most experimental efforts to determine amount of cage recombination in these instances are of either the crossover or the scavenger type. In a crossover experiment, one decomposes a mixture of R—N=N—R and R —N=N—R geminate recombination must yield only R—R and R —R, whereas the separated radicals will recombine randomly to a statistical mixture of R—R, R —R, and R—R. 101... 

Although MMDCR was addressed to the bimolecular reactions in the bulk, the same ideas were successfully applied to the reversible geminate recombination of an ion pair into the exciplex [29-31], The survival probability of the instantaneous created exciplex, Pe(t), and the pair distribution at contact, g([Pg.195]

In the first case, pairs are isolated from each other and the recombination is monomolecular, with a rate which is independent of illumination intensity. The non-geminate electrons and holes act as independent particles and the recombination rate is proportional to the product of the two densities (here assumed equal). The observable difference is a recombination lifetime which is independent of the excitation intensity for geminate recombination, but which decreases with increasing illumination intensity for non-geminate recombination. The simple rate equations also predict a different form of the time dependence, but a more realistic model must also include a distribution of recombination rates due to the tunneling recombination. 

The tunneling mechanism only applies at low temperatures when the electrons and hole are immobile. The luminescence and LESR decay more rapidly above about 50 K. Hong, Noolandi and Street (1981) solved the complicated time-dependent diflfusion equation for geminate recombination when there is a distribution of thermalization distances and temperature-dependent multiple hopping of carriers. The asymptotic solution for the liuninescence intensity is... 

The models that have been most widely used to describe a geminate recombination are based on theories due to Onsager (1934. 1938). The theories are derived from the Smoluchowski (1916) equations,... 

Because the geminate recombination occurs from the singlet radical pair, it will therefore be more convenient to consider the dynamic equation for = p s > Pt = Proro >... 

Equation (12.14) was found to be exact by Gopich and Agmon when the nonreac-tive lifetime of the photoacid equals that of the photobase. Reviews of the extensive kinetic analysis done over the past 20 years in order to refine the basic geminate-recombination model have been recently published by Pines and Pines [25] and by Agmon [48]. 

In Eqs. (13.4) and (13.5) k is rec dissociation and recombination rate constants within a contact ion pair of a radius a, Hj, is the Debye radius. For the systems with pronounced geminate recombination it is possible to fit nonlinear R OH decay to a numerical solution of a system of DSE equations [18]. Equation... 

Figure 14.4 Semi-logarithmic plot of normalized fluorescence decay of excited HPTS. Points are experimental data = 375 nm, = 420 nm) in water acidified by HCIO, after lifetime correction. The geminate recombination data (pH = 6) is fitted by a numerical solution ofthe Debye-von Smoluchowski equation convoluted with the instrument response function after lifetime correction. (Adapted from Ref [125].)...

---