Geminate time dependence

The rate constant for recombination k 3 is time-dependent and can be approximated to t 2 at long times. The geminate recombination explains the residual fluorescence intensity of AH despite the fact that pH pK (see Section 4.5.3). In restricted media, the tail is even longer because of the higher probability of recapturing a photoejected proton by geminate recombination (see Box 4.3). 

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). 

The kinetics data of the geminate ion recombination in irradiated liquid hydrocarbons obtained by the subpicosecond pulse radiolysis was analyzed by Monte Carlo simulation based on the diffusion in an electric field [77,81,82], The simulation data were convoluted by the response function and fitted to the experimental data. By transforming the time-dependent behavior of cation radicals to the distribution function of cation radical-electron distance, the time-dependent distribution was obtained. Subsequently, the relationship between the space resolution and the space distribution of ionic species was discussed. The space distribution of reactive intermediates produced by radiation is very important for advanced science and technology using ionizing radiation such as nanolithography and nanotechnology [77,82]. 

The experimental method and apparatus, and a procedure of the Monte Carlo method that simulates the geminate ion recombination are described, and the time-dependent distribution is elucidated. 

On the other hand, the Monte Carlo method enables us to simultaneously obtain the time-dependent decay curve and the time-dependent distribution function. Therefore we adopted Monte Carlo simulation [18,21,84,85] for the analysis. The geminate ion recombination is also described as follows. 

Bakale et al. [397] pulse irradiated the hydrocarbons cyclopentane, cyclohexane and n-hexane with 0.9 MeV electrons of duration 10 or 100 ns. The transient conductivity decreased approximately exponentially with time for low doses of radiation. The first-order decay of the conductance is probably due to electrons reacting with impurities. With higher doses, the conductance decays approximately as inverse time, characteristic of a second-order recombination of free ions. No evidence for time-dependent geminate ion-pair recombination effects was observed. 

A more tractable theory based on the probability that a reactant pair will react at a time t (pass from reactants to products) is that due to Szabo et al. [282]. If the survival probability of a geminate pair of reactants initially formed with separation r0 is p (r0, t) at time t, the average lifetime of the pair is /dr0 p(r0, t)t and this is longer for larger initial separation distances. It provides a convenient and approximate description of the rate at which a reactant pair can disappear, but it does so without the need of a full time-dependent solution of the appropriate equations. Nevertheless, as a means of comparing time-dependent theory and experiment in order to measure the value of unknown parameters, it cannot be regarded as satisfactory. 

The result is reported in Figure 8-3 where we show the time dependence of the reactant (the geminate complex) and transition state probabilities. 

The geminate-rebinding dynamics measured after photolysis of MbCO and microperoxidase-CO are shown in Fig. 9. The survival fraction denotes the fraction of photolyzed hemes that remain in the deoxy form after CO dissociation. The population was determined by measuring the time dependence of the vibrational absorbance of bound CO. According to Fig. 9, CO rebinds to microperoxidase much more rapidly than to Mb. 

Diffusion models of geminate pair combination connect the time-dependent pair survival probability, P t), with the macroscopic properties of the host solvent. Radicals are treated as spherical particles immersed in a uniformly viscous medium. The pair is assumed to undergo random Brownian movements that ultimately lead to either recombination or escape. The expression of P i) depends on the degree of sophistication of the theory chosen for analyzing the process. In the simplest theory,... 

More rigorous treatments of the geminate combination also take into consideration the probability that the radicals of a pair escape from each other, reencounter in a later event, and finally recombine (Scheme 13.2). This model leads to time-dependent radical pair combination rates and, accordingly, they predict that P t) does not follow a simple exponential decay. For instance, even for the simple case of a contact-start recombination process (ro = o), the survival probabihty is a complex function as shown in Equation 13.2... 

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... 

By Monte Carlo simulations, Ries et al. (1983, 1984) showed that the decay of the geminate recombination rate follows a t x relationship, where x > 3/2. Both the time dependence and the initial recombination rate were dependent upon the thermalization distance. When the final recombination step was impeded by an energy barrier, the recombination kinetics could be described by an exponential time dependence. Results of the simulations show significant deviations from the asymptotic forms of the solutions of Hong and Noolandi. Within the time domain of experimental relevance, the simulated recombination rates were significantly larger. 

The conformational orientation between the excited CNA and CHD should be restricted very much to produce a photocycloadduct in the collision complex indicated in the scheme 1. In the fluid solvents like hexane, the rotational relaxation times of the solute molecules are rather fast compared to the reaction rate, which increases the escape probability of the reactants from the solvent cavity due to the large value of ko. On the other hand, the transit time in the reactive conformation, probably symmetrical face to face, may be longer in the liquid paraffin. This means that the observed kR may be expressed as a function of the mutual rotational relaxation time of reactants and the real reaction rate in the face-to-face conformation. In this sense, it is very important to make precise time-dependent measurements in the course of geminate recombination reaction indicated in Scheme 2, because the initial conformation after photodecomposition of cycloadduct is considered to be close to the face-to-face conformation. The studies on the geminate processes of the system in solution by the time resolved spectroscopy are now progress in our laboratory. 

The diffusive kinetics of geminate pairs have been predicted to show a time-dependent decaying behavior [117-122]. Early experiments showed, in contrast, a decay, with a being dependent on the proton concentration [123]. Experiments on longer time ranges with improved sensitivity are prerequisites for an accurate determination of the asymptotic behavior [124]. In fact, recent measurements on HPTS have demonstrated the validity of the theoretically predicted decay law (see Fig. 14.4) [125]. For 5-(methanesulfonyl)-l-naphthol a kinetic transition from power law to exponential has been reported due to a short photobase lifetime [126]. 

A comparison of Csjitnlions (7A.I4) and (7A.I0I suggests that the time dependence of ft ) is due to the rapid reaction of geminate pairs, though they are not identified as such in the SCK sudden creation approach. That this is correct is clear from an equivalent approach. In a reversible system at equilibrium, let C be made inert at time zero. Geminate pairs are now identifiable, and it is their disappearance that is tracked in the integral of equations (7A. 10) and (7A. 14). If C is sufficient ) dilute, this and the SCK treatments are identical. 

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