Smoluchowski recombination

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 central problem in the theory of geminate ion recombination is to describe the relative motion and reaction with each other of two oppositely charged particles initially separated by a distance ro- If we assume that the particles perform an ideal diffusive motion, the time evolution of the probability density, w(r,t), that the two species are separated by r at time t, may be described by the Smoluchowski equation [1,2]... 

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). 

This expression is a fundamental result of the theory of bulk ion recombination and has been extensively used in interpreting experimental results of diffusion controlled reactions. The Debye-Smoluchowski expression can also be written in terms of the mobility,... 

The breakdown of the diffusion theory of bulk ion recombination in high-mobility systems has been clearly demonstrated by the results of the computer simulations by Tachiya [39]. In his method, it was assumed that the electron motion may be described by the Smoluchowski equation only at distances from the cation, which are much larger than the electron mean free path. At shorter distances, individual trajectories of electrons were simulated, and the probability that an electron recombines with the positive ion before separating again to a large distance from the cation was determined. The value of the recombination rate constant was calculated by matching the net inward current of electrons... 

The simulation results of the electron ion recombination rate constant obtained in Ref. 39 are plotted in Fig. 5. The figure shows that the rate constant becomes lower than the Debye-Smoluchowski value when the electron mean free path exceeds —O.Olrc. At higher values of X, the ratio kjk further decreases with increasing mean free path. The simulation results are found to be in good agreement with the experimental data on the electron ion recombination rate constant in liquid methane, which are also plotted in Fig. 5. 

Figure 5 The rate constant of bulk electron-ion recombination, relative to the Debye-Smoluchowski value [Eq. (36)], as a function of the electron mean free path X. The solid line represents the simulation results, and the circles show the experimental data for liquid methane [49]. (From Ref. 39.)...

The remainder of this section considers several experimental studies of reactions to which the Smoluchowski theory of diffusion-controlled chemical reaction rates may be applied. These are fluorescence quenching of aromatic molecules by the heavy atom effect or electron transfer, reactions of the solvated electron with oxidants (where no longe-range transfer is implicated), the recombination of photolytically generated radicals and the reaction of carbon monoxide with microperoxidase. 

The rate of reaction of methyl radicals is in excellent agreement with the predictions of the Smoluchowski theory (see Chap. 2, Sect. 2.6). Consequently, it appears that geminate radicals move towards and away from each other at a diffusion-limited rate. Once an encounter pair is formed, reaction is very rapid (primary recombination). Furthermore, the encounter pair is held together for a considerable time (< 0.1ns in mobile solvents) because the surrounding solvent molecules hinder their separation (solvent caging). There is much evidence which lends some support for this view the most important influences on the recombination probability are listed below. 

In order to solve for the survival and recombination probabilities, p and q in eqn. (126), it is necessary to solve eqn. (122) for p(r, f]r0, f0) and use eqn. (123) to find p or eqn. (125) for q. Again, the boundary and initial conditions are required. Before the pair is formed (f < and ttf is slightly less than f0), the density p is zero, of necessity. The boundary conditions are closely related to the Smoluchowski conditions [eqns. (5), (22), (46) and (47)]. As the radicals approach each other they have a probability of reacting, which can be related to an effective second-order rate coefficient, fcact> f°r the activation-limited process of recombination by... 

The above-described pair problem is treated by the Smoluchowski equation [3, 19] - see Fig. 1.10. It operates with the probability densities (Fig. 1.11) and contains the recombination rate characterizing particle motion. Knowledge of the probability density to find a particle at a given point at time moment t gives us (by means of a trivial integration over reaction volume) the quantity of our primary interest - survival probability of a particle in the system with... 

This means that no recombination occurs unless the particles approach each other to within the critical distance ro which is associated with the unavoidable fall into the recombination sphere. It can be shown that for instant recombination, cto —> oo, w(r < ro, t) = 0, which results in the generally accepted Smoluchowski boundary condition... 

Due to the instant recombination all the dissimilar particles with relative distances r ro disappear, which results in the Smoluchowski boundary condition... 

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,... 

Noolandi and Hong (1978), Hong and Noolandi (1978a, 1978b). and Noolandi (1982) described solutions to the time-dependent Smoluchowski equation. In the long-time limit, Hong and Noolandi s results predict the recombination rate approaches a law as... 

On closer inspection, the combination rate constants are about 1/4 of the estimated diffusion-controlled rate constant. For acetonitrile, for example, fcjj - 2.9 X 10 L mol" s from the von Smoluchowski equation wiA a diffusion coefficient from a modified version of the Stokes-Einstein relation, D - fcT/4jiT r. Owing to the restriction to singlet state recombination, an experimental rate constant 1 /4 of is quite reasonable. On the other hand, for these heavy metals, the spin restriction may not apply, in which case one would argue that the geometrical and orientational requirements of these large species could well give recombination rates somewhat below the theoretical maximum. 

It is possible to reconcile the dilTerence by asserting that a time scale argument applies, namely, the reduced FPE applies on short time scales whereas the reduced Smoluchowski equation applies on time scales that are long compared to the momentum relaxation time scales. The question of the magnitude of these differences is addressed in the next section. For most chemical reactions, the FPE results will be used, except for the diffusive contribution to recombination dynamics considered in the next section. 

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].)...

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