Partially Diffusion Controlled Reactions Neutral Species

3 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 

In order to apply the IRT method, Eq. (4.79) needs to be inverted for t, in order to extract a reaction time, which is not straightforward. The authors have devised two methods to make sampling from this distribution function possible. The first method involves fitting this probability distribution to an incomplete y-function for a range of r. A random uniform number is then generated from the appropriate y-distribution [16]. The second method generates a random number f/(0,l] uniform in the interval (0,1] and solves the equation 

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. 

---

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... 

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. 

---