Wilemski

In the Collins-Kimball theory, the rate constant for the reaction was assumed to be distance-independent. Further refinement proposed by Wilemski and Fix-manc) consists of considering that the reaction rate constant has an exponential dependence on distance, which is indeed predicted for electron transfer reactions and energy transfer via electron exchange (see Dexter s formula in Section 4.6.3). The rate constant can thus be written in the following form ... 

H.C. Mam, A. Pigeaud, R. Chamberlin, G. Wilemski, in Proceedings of the Symposium on Electrochemical Modeling of Battery, Fuel Cell, and Photoenergy Conversion Systems, edited by J.R Selman and H.C. Mam, The Electrochemical Society, Inc., Pennington, NJ, Pg. 398, 1986. 

A. Pigeaud, ERC, and G. Wilemski, Physical Sciences, "Effects of Coal-Derived Trace Species on the Performance of Carbonate Fuel Cells," in Proceedings of the Fourth Annual Fuel Cells Contractors Review Meeting, U.S. DOE/METC, Pgs. 42-45, July 1992. 

Baird and Escott [49] suggested that the departures from the Stern— Volmer law could be explained if there was competition between the excited fluorophors and quenchers. However, because the concentration of excited fluorophors is 103—106 times less than the quencher concentration, to an excellent approximation the excited fluorophor molecules are statistically independent. Baird et al. [50] have also developed a more detailed analysis of fluorescence quenching based upon the Wilemski and Fixman [51] approach (see Chap. 9). They wrote many-body equations... 

This discussion highlights the difficulty of deciding at what separation A and B form an encounter pair and then whether this reacts or separates. Noyes [5] and Wilemski and Fixman [51] have taken the encounter distance to be that separation which, if reduced slightly, will lead to reaction. Where these authors disagree is that Noyes [5] only allows reaction to occur in a very narrow range of separation distances about R (which is the usual assumption) and Wilemski and Fixman [51] assume that any separation distance less than the encounter distance, R, can lead to reaction between A and B and that A and B can diffuse through each other till their centres of mass coincide (Chap. 9, Sect. 4). Neither assumption is good, but the differences in predicted rate coefficients are so small that an experimental test of these theories could not be definitive. 

Wilemski and Fixman [51] have suggested that it is intrinsically more satisfactory to treat chemical reaction by means of a sink term included in the diffusion equation than as a boundary condition imposed on the density distribution. They recommended writing the diffusion equation as... 

The first reasonably successful theory of diffusion-limited chemical reactions which specifically endeavoured to develop a model that could described, in principle, the competitive effect was introduced by Wilemski and Fixman [51], They considered the fluorescence of a species A which can be quenched by natural decay (lifetime t) and by a quencher, Q, of concentration [Q]... 

In the previous section, the diffusion equation for motion of a large number of particles was developed [see eqn. (211)]. When the solvent is at rest and both hydrodynamic forces and inter-reactants potential energy terms can be ignored, the equation becomes much simpler. This equation provides the basis for the analysis by Wilemski and Fixman. They chose to consider just one excited A molecule in a volume, V, together with m quencher molecules. The co-ordinates of all these molecules are rA and rQj, rQ2... rQ,n at a time f. Initially, thefluorophorand quencher molecules were positioned at rA° and rqj... r m. As a shorthand notation, these co-ordinates are called (r) and r0, respectively. The fluorophor was excited at time f°. 

Wilemski and Fixman [51] have suggested that the probability that the excited fluorophor and all m quenchers are located at r in volume elements d r = drAdry,... is n((r), f)d(r, where n is the density of the m + 1 particles in the volume V. This is a microscopic quantity. It cannot be observed or probed directly. All that can be observed is the probability, that the fluorophor remains excited. Since all m quenchers remain unaffected by the decay of the fluorophor, the density, n, averaged through space only decays due to de-excitation of the fluorophor. Hence the probability that the fluorophor A remains excited is... 

Equation (219) is a complicated equation and difficult to solve without a considerable simplification. Wilemski and Fixman [51] suggested the equation could be simplified considerably if the rate of the quenching reaction was slow compared with diffusion. The last term on the right-hand side perturbs the density, n, slightly under such circumstances and the equation can be solved by standard Green s function techniques (Appendix A) to give... 

The source term describes the formation of the m + 1 reactants at time t — t0 with these initial positions at rA° and r ls r 2... etc. The integral in eqn. (220) describe the reduction in the density of the quencher and fluorophor distribution if the quenching process is very slow. Unfortunately, within the integral is n( r0, f°), which is the very density that is sought. As a first approximation, ne<1 could be used. Wilemski and Fixman suggested a more satisfactory (closure) approximation... 

Wilemski and Fixman [51] have also discussed the excitation of the fluorophor with steady-state light, at a rate F per second. When the excited fluorophor decays with a natural lifetime, r, the probability that the fluorophor is excited is [Pg.276]

To extend this analysis further to include the cations as well as anions, all that is required is to use the density nNtM and the potential energy UNrM. Rather than represent reaction by a boundary condition, there is much advantage using the Wilemski and Fixman approach of sink terms [ 51 ]. Let the anion and cation p react when they overlap in the region, say, where i r, — p j = R, and specify it by iix — 5 (i — jry — p i). The rate of... 

In the previous chapter, the theory developed by Wilemski and Fixman [51] is discussed in some detail (see Chap. 9, Sect. 4). While there are a number of reservations about this approach to describing diffusion-limited reaction rates (see Chap. 9, Sect. 4.3), it is very useful analysis because it is capable of further refinement. A most interesting case in point is the variation analysis by Doi [485]. This section discusses his analysis in more detail. 

The general approach of Wilemski and Fixman [51] was followed by Doi [485], but the possibility of interactions between reactants was also included. Hence, the spatial diffusion and drift operator becomes... 

By taking Laplace transforms and using the approximate expression for 0, developed by Wilemski and Fixman [see eqn. (220)]... 

Taking a very simple trial function, which is equivalent to the Wilemski and Fixman value... 

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