Hydrodynamic repulsion

R), i.e. there is no effect due to caging of the encounter complex in the common solvation shell. There exist numerous modifications and extensions of this basic theory that not only involve different initial and boundary conditions, but also the inclusion of microscopic structural aspects [31]. Among these are hydrodynamic repulsion at short distances that may be modelled, for example, by a distance-dependent diffiision coefficient... 

The simple difhision model of the cage effect again can be improved by taking effects of the local solvent structure, i.e. hydrodynamic repulsion, into account in the same way as discussed above for bimolecular reactions. The consequence is that the potential of mean force tends to favour escape at larger distances > 1,5R) more than it enliances caging at small distances, leading to larger overall photodissociation quantum yields [H6, 117]. 

Another consequence of the solvent s presence on the rate of reactant diffusion towards (and away from) each other is that solvent has to be squeezed out of ( sucked into ) the intervening space between the reactants. Because this takes time, the approach (or separation) of reactants is slowed. Effectively, the solvent diffusion coefficient is reduced at distances of separation between reactants from one to several solvent diameters. Figure 38 (p. 216) shows the diffusion coefficient as a function of reactant separation distances. This effect is known as hydrodynamic repulsion and it more than cancels the net increase of reaction rate due to the potential of mean force. It is discussed further in Chap. 8 Sect. 2.5 and Chap. 9 Sect. 3. Both the steady-state and transient terms in the rate coefficient depend on these effects. 

In conclusion, the author believes that consideration should be given to the points discussed above and the effects of hydrodynamic repulsion (Chap. 9, Sect. 4) when considering reactions between ions. There are so many factors which may influence such reaction rates, that many experimental studies of ionic reactions may have found agreement with the Debye—Smoluchowski theory (or corrected forms) by cancellation of correction terms. Probable complications due to long-range electron and energy transfer are discussed in Chap. 4. 

Nevertheless, as discussed in Chap. 9 Sect. 3.4, there are several reasons for considering that the Deutch and Felderhof [70] correction of the Debye—Smoluchowski theory [68] to incorporate hydrodynamic repulsive effects overestimates this correction [71]. Furthermore, for reactants... 

U(r) is also shown in these figures. The hydrodynamic repulsion between radicals diffusing together slows their rate of mutual approach because the intervening solvent molecules have to be squeezed out of the way. In Chap. 9, Sect. 3, it is shown that this repulsion can be treated by letting the diffusion coefficient depend upon radical separation, D(r). Northrup and Hynes [103] suggested that... 

With the classifications of spatially correlated reactant recombinations and the cage effect in mind, the effects discussed in Sect. 3.1 are largely due to the spatial correlation. Indeed, changing the solvent viscosity by applying pressure or changing the temperature, or the radical reactivity should have little effect on the potential of mean force and not much on the extent and range of hydrodynamic repulsion. 

Up to now, only hydrodynamic repulsion effects (Chap. 8, Sect. 2.5) have caused the diffusion coefficient to be position-dependent. Of course, the diffusion coefficient is dependent on viscosity and temperature [Stokes—Einstein relationship, eqn. (38)] but viscosity and temperature are constant during the duration of most experiments. There have been several studies which have shown that the drift mobility of solvated electrons in alkanes is not constant. On the contrary, as the electric field increases, the solvated electron drift velocity either increases super-linearly (for cases where the mobility is small, < 10 4 m2 V-1 s-1) or sub-linearly (for cases where the mobility is larger than 10 3 m2 V 1 s 1) as shown in Fig. 28. Consequently, the mobility of the solvated electron either increases or decreases, respectively, as the electric field is increased [341— 348]. 

The other source of an effective electric field dependence of the diffusion coefficient is due to hydrodynamic repulsion. As the ions approach (or recede from) one another, the intervening solvent has to be squeezed out of (or flow into) the intervening space. The faster the ions move, the more rapidly does the solvent have to move. A Coulomb interaction will markedly increase the rate of approach of ions of opposite charge and so the hydrodynamic repulsion is correspondingly larger. It is necessary to include such an effect in an analysis of escape probabilities. Again, the force is directed parallel to the electric field and so the hydro-dynamic repulsion is also directed parallel to the electric field. Perpendicular to the electric field, there is no hydrodynamic repulsion. Hence, like the complication of the electric field-dependent drift mobility, hydro-dynamic repulsion leads to a tensorial diffusion coefficient, D, which is similarly diagonal, with components... 

However, when the Onsager distance, rc, is much larger than the radii of either ion, hydrodynamic repulsion is unimportant (see Fig. 45, p. 268). The tensorial form of the diffusion equation is [cf. eqn. (141)]... 

The hydrodynamic repulsion between ions is stronger as ions approach each other closely. Recombination is therefore appreciably less likely compared with the case of no hydrodynamic repulsion. Escape becomes slightly less probable compared with the no hydrodynamic repulsion case... 

Consequently, while the effect of an electric field dependence of both drift mobility and diffusion coefficient and also hydrodynamic repulsion decreases, the recombination probability, dielectric saturation and relaxation effects increase the recombination probability. 

In the previous chapters, the diffusion equation has been used extensively to model fast chemical reaction in solution. By addition of various correction factors (such as intermolecular forces, long-range transfer, solvent structure, hydrodynamic repulsion, etc.), the agreement between experiment and theory can be improved as the model becomes more realistic. Nevertheless, the reactants have been presumed to execute Brownian motion. This is only the long-time limit of their actual behaviour. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

It is interesting to note that eqn. (190) is reminiscent of the steady-state Collins and Kimball rate coefficient [4] [eqn. (27)] with kact replaced by kacig R) and 4ttRD by eqn. (189). Equation (190) for the rate coefficient is significantly less than the Smoluchowski rate coefficient on two counts hydrodynamics repulsion and rate of encounter pair reaction. Had experimental studies shown that a measured rate coefficient was within a factor of two of the Smoluchowski rate coefficient, it would be tempting to invoke partial diffusion control of the reaction rate. The reduction of rate due to hydrodynamic repulsion should be included first and then the effect of moderately slow reaction rates between encounter pairs. 

In the previous chapter, several factors which complicate the simple diffusion equation analysis of chemical reactions in solution were discussed rather qualitatively. However, the magnitude of these effects can only be gauged satisfactorily by a detailed physical and mathematical analysis. In particular, the hydrodynamic repulsion and competitive effects have been studied recently by a number of workers. Reactions between ionic species in solutions containing a high concentration of ionic species is a similarly involved subject. These three instances of complications to the diffusion equation all involve aspects of many-body effects. 

The hydrodynamic repulsion of molecules in solution has already been discussed (see Chap. 8, Sect. 2.5) and is analysed in much more detail in Sect. 3 of this chapter. It appears to be a very significant complication, yet it can be analysed relatively straightforwardly. The basic effect arises when one molecule approaches another. Solvent molecules between these two molecules have to be squeezed out in a direction approximately perpendicular to the motion of the approaching molecules. Their approach (and similarly their separation) is impeded and the effective diffusion coefficient is less than that when the molecules are more widely separated. 

When there are two or more reactants diffusing throughout space, the motion of each reactant influences that of all the others due to the solvent being squeezed from between the approaching reactants. The effect of this hydrodynamic repulsion on the rate of a diffusion-limited reaction was discussed in Chap. 8, Sect. 2.5. In this section, this discussion is amplified. First, the nature of the hydrodynamic repulsion is discussed further and then a general diffusion equation for many particles is derived. The two-particle diffusion equation is selected and solved subject to the usual Smoluchowski initial and boundary conditions to obtain the rate coefficient. Finally, this is compared with the rate coefficients in the absence of hydrodynamic repulsion and from experiments. 

INCORPORATION OF THE HYDRODYNAMIC REPULSION INTO THE DIFFUSION EQUATION... 

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