Long-range transfer and the diffusion equation

In Chap. 2 and 3, the motion of two reactants was considered and a diffusion equation was derived based upon the equation of continuity and Pick s first law of diffusion (see, for instance. Chap. 2 and Chap. 3, Sect. 

When one reactant (say D) can transfer energy or an electron to the other reactant (say A) over distances greater than the encounter separation, an additional term must be considered in the equation of continuity. The two-body density n x-y,X2,t) decays with a rate coefficient f(r, — rj) due to long-range transfer. Furthermore, if energy is being transferred from an excited donor to an acceptor, the donor molecular excited state will decay, even in the absence of acceptor molecules with a natural lifetime Tq. Hence, the equation of continuity (42) becomes extended to include two such terms and is 

The terms relate the loss of both or either excited D and A molecules to three causes (a) diffusion of both molecules towards each other, leading to reaction (i.e. energy or electron transfer) on contact or away from each other to infinity, (b) reaction between D and A molecules by long-range transfer over distance greater than the encounter separation, and (c) decay of the excited state of D which can donate energy to A (in the case of electron transfer, Tq may be set equal to infinity). Again, eqn. (65) may be simplified because Jj and J2 are linear in n. This follows the discussion of Chap. 3, Sect. 1.1 where the co-ordinates r, and r2 are transferred to the centre of diffusion coefficient and relative co-ordinates, X and r, respectively, n factors into p x,t) exp (— I/tq) as before. 

The exponential exp (— t/T ) describes the natural decay of the excited donor molecules, even in the absence of acceptor molecules. Only the density p (r, t) is of interest since this leads only to reaction by change of separation distances whence the equation 

Comparison of eqns. (44) and (66) indicates that the effect of an interaction energy between donor and acceptor can be included in eqn. 

This has been discussed by Butler and Pilling [128]. A rather more thorough discussion of the derivation of eqn. (66) has been given by Gosele et al. [129, 130] using the method of Waite [30]. 

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