Stationary medium, particle diffusion equations

In the hydrodynamic theory, the diffusion coefficient of a solute molecule A or single particle through a stationary medium B, DAB, is given by the Nemst-Einstein equation ... 

The validity of the generalized Langevin equation (22) is restricted to a stationary medium. In other situations, for instance when the diffusing particle evolves in an aging medium such as a glassy colloidal suspension of Laponite [8,12,55,56], another equation of motion has to be used. 

Let us consider again the particular case of a particle diffusing in a stationary medium, in order to see how the generalized Langevin equation (22) can be deduced from the more general equation (169). When the medium is stationary, the response function x (M0 reduces to a function of t — t (% (t, t ) = X (f — t j). Introducing then the causal function y(f) as defined by... 

This equation was derived above for the movement of a liquid through a stationary solid phase. Its application here to the movement of colloidal particles under experimental conditions that render the liquid medium immobile implies that the solid particle is large compared with the dimensions of the diffuse double layer k 1. It is customary to term this movement of the solid phase electrophoresis. The phenomenon is observed with particles suspended in a liquid (Fig. 6.139). 

Statement of the problem. Following [367, 368], let us consider stationary diffusion to a particle of finite size in a stagnant medium, which corresponds to the case Pe = 0. We assume that the concentration on the surface of the particle and remote from it is constant and equal to Cs and C), respectively. The concentration field outside the particle is described by the Laplace equation... 

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