Hydrodynamic repulsion and the diffusion equation

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. 

As a first approximation to the motion of two spheres in a solvent (which can be regarded as a continuum), the spheres can be presumed to move about the solvent sufficiently slowly that the very much simplified Navier—Stokes equation of fluid flow is applicable. The application of a pressure gradient VP(r) in the fluid develops velocity gradients within the fluid, Vv(r). If another force F(r) is included in the fluid, this can generate a pressure gradient and further affect the velocity gradients. The Navier— Stokes equations [476] becomes 

Hydrodynamic effects were included in the Debye—Smoluchowski equation by Zwanzig [444]. It is a most interesting and readable article. The discussion below follows his analysis quite closely. 

Let the solvent move such that its velocity is v°(r) if there are no particles present, though v° (r) need not be constant but could, for instance, be produced by bulk flow of the solvent in tubes or around obstacles. Now the particle j is located at tj and moves with a velocity ij — v(ry) relative to the solvent velocity, v(tj), when the particle j is absent. In a Newtonian liquid, this velocity difference between particle emd solvent leads to a force between liquid and particle J which is given by 

The velocity at r is changed by Av(r) when a particle is introduced at ry. The velocity in the absence of all the particles is v°(r) and in the presence of all N particles, except for the j particle, the solvent velocity is the sum of all the changes of velocities due to all AT — 1 particles and the solvent velocity in the absence of the particles 

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