Independent Reaction Times

The IRT model [1,2, 12, 15] is essentially a Monte Carlo algorithm which assumes the independence of reaction times (i.e. each reaction is independent of other such reactions and that the covariance of these reaction times is zero). Unlike the random flights simulation, the diffusive trajectories are not tracked but instead encounter times are generated by sampling from an appropriate probability density function conditioned on the initial separation of the pair. The first encounter takes place at the minimum of the key times generated min(h t2, fs...) and all subsequent reactions occur based on the minimum of surviving reaction times. Unlike random flights 

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The independent reaction time (1RT) model was introduced as a shortcut Monte Carlo simulation of pairwise reaction times without explicit reference to diffusive trajectories (Clifford et al, 1982b). At first, the initial positions of the reactive species (any number and kind) are simulated by convolving from a given (usually gaussian) distribution using random numbers. These are examined for immediate reaction—that is, whether any interparticle separation is within the respective reaction radius. If so, such particles are removed and the reactions are recorded as static reactions. 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

The position approach strives to get the positions of the reactive particles explicitly at the reaction time t obtained in the IRT model. While the nonreac-tive particles are allowed to diffuse freely, the diffusion of the reactive particles is conditioned on having a distance between them equal to the reaction radius at the reaction time. Thus, following a fairly complex procedure, the position of the reactive product can be simulated, and its distance from other radicals or products evaluated, to generate a new sequence of independent reaction times (Clifford et al, 1986). 

FIGURE 7.4 Comparison of Monte Carlo (MC) and independent reaction time (RRT) simulations with respect to the product AB for a spur of two neutral radical-pairs. See text for explanation. From Clifford et al. (1986), by permission of The Royal Society of Chemistry ... 

The methodology of stochastic treatment of e-ion recombination kinetics is basically the same as for neutrals, except that the appropriate electrostatic field term must be included (see Sect. 7.3.1). This means the coulombic field in the dielectric for an isolated pair and, in the multiple ion-pair case, the field due to all unrecombined charges on each electron and ion. All the three methods of stochastic analysis—random flight Monte Carlo (MC), independent reaction time (IRT), and the master equation (ME)—have been used (Pimblott and Green, 1995). 

Analytical treatment of the diffusion-reaction problem in a many-body system composed of Coulombically interacting particles poses a very complex problem. Except for some approximate treatments, most theoretical treatments of the multipair effects have been performed by computer simulations. In the most direct approach, random trajectories and reactions of several ion pairs were followed by a Monte Carlo simulation [18]. In another approach [19], the approximate Independent Reaction Times (IRT) technique was used, in which an actual reaction time in a cluster of ions was assumed to be the smallest one selected from the set of reaction times associated with each independent ion pair. 

In this case it is essential that all the concentrations are measured at the same reaction time, that means simultaneously. If this prerequisite cannot be experimentally achieved the measured values have to be synchronised with respect to equivalent wavelength -independent reaction times (see Fig. 4.6 in Section 4.2.2.3). Usually the concentration at the time tg is taken as a reference ... 

First passage approach The first passage approach calculates the distance of the newly formed product to the remaining reactants by conditioning on the independent reaction time that exists for that pair. An interparticle distance is generated by sampling from the probability density function... 

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