Partially Diffusion Controlled Reaction

The recipe to simulate partially diffusion controlled reactions involves calculating the probability of survival on diffusing from x to y via the encounter radius a in the time step St as 

Similarly, the denominator of Eq. (4.26) can be decomposed into trajectories which do not strike the boundary a and those that do hit a but are reflected. Hence the denominator of Eq. (4.26) can be rewritten as 

As long as St remains sufficiently small, the transition densities for the one dimensional diffusion process with constant drift can be used to calculate the survival probability at every time step St using the equation 

The explicit expression for the numerator and denominator respectively are given as 

To decipher whether reaction has taken place, a uniformly distributed random number is generated between (0,1]. If this random number is greater than S2(x, y, St) then reaction has taken place and the relevant changes are made in the simulation (i.e. update reaction counters and change chemical properties). 

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After the jump, the particle is taken to have reacted with a given probability if its distance from another particle is within the reaction radius. For fully diffusion-controlled reactions, this probability is unity for partially diffusion-controlled reactions, this reaction probability has to be consistent with the specific rate by a defined procedure. The probability that the particle may have reacted while executing the jump is approximated for binary encounters by a Brownian bridge—that is, it is assumed to be given by exp[—(x — a)(y — a)/D St], where a is the reaction radius, x andy are the interparticle separations before and after the jump, and D is the mutual diffusion coefficient of the reactants. After all... 

Green and Pimblott (1989) have extended the IRT model to partially diffusion-controlled reactions between neutrals. They derive an analytical expression that involves an additional parameter, namely the reaction velocity at encounter. For reactions between charged species, W generally cannot be given analytically but must be obtained numerically. Furthermore, numerical inversion to get t then... 

The second boundary condition assures total finite existence probability at any time the first boundary condition implies that the recombination is fully diffusion-controlled, which has been found to be true in various liquid hydrocarbons (Allen and Holroyd, 1974). [The inner boundary condition can be suitably modified for partially diffusion-controlled reactions, which, however, does not seem to have been done.]... 

However, this slowing down may not be observed for reactions with a significant activation energy simulations conducted by Lopez-Quintela and co-workers show that partially diffusion-controlled reactions are more favored in media with a higher degree of compartmentalization because of an increase in the recollision probability. 

In this section the foundations of the theory underlying chemical kinetics are presented. Based on the diffusion equation to describe Brownian motion together with Smoluchowski s theory [ 1, 2], a thorough derivation of the bulk reaction rate constant for neutral species for both diffusion and partially diffusion controlled reactions is presented. This theory is then extended for charged species in subsequent sections. 

Treating partially diffusion controlled reaction involves replacing the inner boundary condition such that pB(a,t) = 0 with a radiation boundary condition [7] of the form... 

In the analysis so far, it is assumed both particles A and B to be uncharged. If however, both the particles are now ions, the diffusion of B reactants about a given A particle has to be modified due to the drift exerted by the electrostatic forces. The steady state solution for both diffusion controlled and partially diffusion controlled reactions is now presented. 

As mentioned previously, for partially diffusion controlled reactions the reactivity of the inner boundary can be controlled using the parameter v which has units of velocity. The required inner boundary condition is of the form... 

Partially Diffusion Controlled Reactions Neutral Species... 

The IRT algorithm has been extended to model partially diffusion controlled reactions by Green and Pimblott [16] and a brief resume is presented in this section. Using the radiation boundary condition such that... 

Unfortunately, the first passage algorithm cannot be used for partially diffusion controlled reactions because of the complex nature of the transition density, making it difficult to use the inversion method to sample from the cumulative distribution function. It is however shown in Chap. 6 (Sect. 6.3.4), that a new model called the centre of diffusion vector approach can accurately model the spatial distribution of reactive products originating from partially diffusion controlled reactions. Like the first passage approach, the implementation is relatively straightforward and does not compromise on the computational resources. 

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... 

Although Eq. (4.122) allows the implementation of partially diffusion controlled reactions, another possibility needs to be considered on reaction at the inner boundary a the radical pair separates to x which is less than B (with B defined to be a+ 8) and survive reaction. The pair then undergo spin relaxation at the point x and re-approach the boundary a without ever hitting the boundary B. The likelihood of this situation can be tested using the recovering boundary formalism. A thorough derivation is now presented. ... 

All terms required to calculate firec(a) (probability on going from a to b, undergoing spin relaxation at an earlier instance, reaching the boundary b without reaction) are now calculated, and the complete expression is shown in Eq. (4.143). Whilst the solution to Brec(fl) assumes diffusion controlled reactivity at the surface a, it can be argued that if Brec(6() is negligible for diffusion controlled reactions, then it must also be negligible for partially diffusion controlled reactions as well. 

The last step to implementing partially diffusion controlled reactions into the Slice... 

For partially diffusion controlled reactions there are two situations possible following an encounter (1) the boundary is unreactive and the particles reflect or (2) the boundary is reactive but the encountering pair is in a repulsive triplet state and consequently unreactive. To handle the above situation, two models have been developed which are described below ... 

Treating spin dependent reactivity poses a special problem in the current model as there are two possibilities which can arise (i) radical pairs encounter and the surface is unreactive or (ii) the radical pairs encounter but are in an unreactive spin-configuration. The two algorithms designed to treat partially diffusion controlled reactions have already been discussed in Sect. 5.4.4. In brief, method 1 collapses the wavefunction (ijr) upon encounter and reaction occurs with a probability / rad (Fig. 5.17) method 2 calculates the probability of reaction (Ps x Prad) and reduces the singlet component of if by -Prad) if no reaction had occurred (Fig. 5.18). To... 

Green [10] has shown that Eq. (6.9) can also be applied for partially diffusion controlled reactions between ions by using the definition of q and b to be... 

Through the other advances made in the IRT algorithm (as described in Chap. 4 of this work), the micelle model can now be extended to model (i) partially diffusion controlled reactions on a spherical surface, (ii) ionic species and (iii) explicit treatment of spin dynamics. Further work in this area would include investigating whether (i) the increased scavenging rate of ions is observed in micelles (i.e. whether the correlation between the scavenging and recombination times is still applicable) and (ii) whether cross-recombination between spin correlated radical pairs in micelles accelerates T) spin-relaxation due to the Einstein-Rosen-Podolsky effect. 

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