Other initial distributions and the Greens function

The Smoluchowski theory of diffusion-limited (or controlled) reactions relies heavily on the appropriateness of the inital condition [eqn. (3)]. Though the initial condition does not determine the steady-state rate coefficient [eqn. (20)] because diffusion of B in towards the reactant A is from large separation distances ( 10nm) in the steady-state, it does directly determine the magnitude of the transient component of the rate coefficient because this is due to an excess concentration of B present initially compared with that present in the steady-state. As a first approximation to the initial distribution, the random distribution is intuitively pleasing and there is little experimental evidence available to cast doubt upon its appropriateness. Section 6.6 and Chap. 8 Sect. 2.2 present further comments on this point. 

Other initial distributions have been suggested. Peak and Corbett [29] suggested an initial distribution of the form 

To solve the diffusion equation and obtain the appropriate rate coefficient with these initial distributions is less easy than with the random distribution. As already remarked, the random distribution is a solution of the diffusion equation, while the other distributions are not. The substitution of Z for r(p(r,s) — p(r, 0)/s) is not possible because an inhomogeneous equation results. This requires either the variation of parameters or Green s function methods to be used (they are equivalent). Appendix A discusses these points. The Green s function g(r, t r0) is called the fundamental solution of the diffusion equation and is the solution to the 

Because the density distribution for any given initial condition can be synthesised from it, the Green s function is given here. 

This is valid whether r is greater or less than r0 and generally Green s functions are symmetric to inversion of r and r0. For the same parameters as were used in Fig. 1, namely D = 10 9 m2 s 1 and R = 0.5 nm, Green s function is shown in Fig. 4 with a source point r0 = 2 nm. 

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