Radiation boundary conditions, probability

Monchick [36, 273] has used the diffusion equation and radiation boundary conditions [eqns. (122) and (127)] to discuss photodissociative recombination probabilities. His results are similar to those of Collins and Kimball [4] and Noyes [269]. However, Monchick extended the analysis to probe the effect of a time delay in the dissociation of the encounter pair. It was hoped that such an effect would mimic the caging of an encounter pair. Since the cage oscillations have periods < 1 ps, and the diffusion equation is hardly adequate over such times (see Chap. 11, Sect. 2), this is a doubtful improvement. Nor does using the telegraphers equation (Chap. 11, Sect. 3.3) help significantly as it is only valid for times longer than a few picoseconds. 

Consider the stochastic translocation process given by Equation 10.39. Let mo be the number of monomers, which have been nucleated in the receiver compartment, to begin with. Starting from this initial condition, the probability distribution function of the first passage time r is given by Equation 6.86. Let us now consider the key results for the boundary conditions BCl and BC2 (Table 6.1). The results for the radiation boundary condition BCi can be obtained similarly by looking up the results in Section 6.7.4. 

Problems with the radiation boundary Some of the problems with using the radiation boundary condition to model chemical systems have been discussed in the literature [9]. The most important of these are (1) for particles close to the encounter distance, it is not possible to specify a non-zero probability for reaction, since an infinite number of encounters follow an unsuccessful first encounter resulting in reaction (as shown by Collins and Kimball [7]). (2) Schell and Kapral [10] have shown that the probability of reaction on encounter should scale with the ratio of D and a ( D ls, the mutual diffusion coefficient and a the encounter distance) for radiation boundary condition to be applicable. (3) All re-encounters are treated in the same manner. 

Sometimes for a spin controlled reaction, the probability of reaction of first encounter has a physical origin, and if this first encounter is unreactive then the spin state is also unreactive, and therefore all subsequent rapid re-encounters will not react either [due to condition (3)]. The radiation boundary condition is clearly not appropriate to use for such reactions, where an appropriate model for spin dynamics is not incorporated. 

Unlike diffusion controlled reactions, where reaction takes place as soon as the interparticle separation hits the boundary a partially diffusion controlled reactions involve an extra complexity, such that the probability of reaction must be calculated based on the surface reactivity. This probability can be calculated by solving the backward diffusion equation to And the survival probability (a, B ) on going from boundary a to B (defined as a + 5) subject to a radiation boundary condition at surface a (situation 2 as shown in Fig. 4.7). Using the boundary condition (a) = v/D )p a) and Q. B ) = 1, the survival probability Q(a, B ) is found to be... 

As the outer surface is now reactive, it is necessary to modify the random flights algorithm to take this reactivity into account using the radiation boundary condition. As discussed in Sect. 4.3.5, the simulation proceeds by assuming the outer boundary is reflective. If this outer boundary is hit during the diffusive motion of the particle, then the probabiUty of escape is calculated based on the parameter v which controls the surface reactivity. This section presents the algorithm to (i) handle the reflection of a particle subject to an upper reflective boundary and (ii) calculating the probability of reaction. 

This section extends the outer radiation boundary condition in the previous model to include geminate reaction within the micelle. In this model a single particle diffuses with a sink at the centre and an outer radiation boundary. The required initial condition is S2(( = 0) = 1, inner boundary S2(r = a) = 0 and outer boundary condition — wS2(r = R). Solving the backward diffusion equation with these boundary conditions gives the analytical expression for the survival probability to be... 

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