Medium effects self-diffusion

A. Horner. Self-diffusion in metallic glasses Approximation of the effective medium and molecular simulation. PhD thesis, Stuttgart University, 1993. In German. 

Here, Ds and Dd are the coefficients representing the Soret and Dufour effects, respectively, Du is the self-diffusion coefficient, and Dik is the diffusion coefficient between components / and k. Equations (7.149) and (7.150) may be nonlinear because of, for example, reference frame differences, an anisotropic medium for heat and mass transfer, and temperature- and concentration-dependent thermal conductivity and diffusion coefficients. 

There is no reason why there should be no general medium effect upon the resistivities. The approximate nature of Eqs. (44) and (48a) gives rise to speculations regarding their improvement, which however do not belong to the field of macro-dynamics. The self diffusion of a medium containing solvated macro-molecules was studied by Wang. ... 

Deviations from normal diffusion, i.e. from molecular propagation within a quasi-homogeneous, essentially infinitely extended medium, may be taken into account by introducing an effective diffusivity Dgff. It is defined in the same way as the self-diffusivity, i.e. via Eqs. 6 and 7, however, without the requirement of the validity of Pick s laws 1 and 2. Therefore, Dgff may become a function of the (observation) time. In the considered case of zeolitic diffusion and for intracrystalline diffusion paths being sufficiently small in comparison with the crystallite radii, the effective diffusivity may be shown to be represented by a power series [106-108], leading to... 

Fig. 17 Intracrystalline self-diffusivity of methane ( 2 molecules per supercage, at 25 °C) as a function of the amount of co-adsorbed molecules per window . The solid lines are predictions based on the effective medium approximation of percolation theory with / denoting the ratio of the transition rates through blocked and open windows. From [158] with permission...

Equation (6.12) faces one main difficulty, namely the determination of the explicit functional form of D(s). It is important to stress that D s) does not have exactly the same properties as the classical diffusion coefficient, and we refer to it here as the conductivity. Likewise, we define the resistivity of the medium as R = l/D. We expect the resistivity to be proportional to the number of steps of the particle, and arguments from random walks on fractals should be useful to determine R. Walks on fractals are characterized by the existence of two scales. Divide the medium into small blocks of size, such that the diffusion is normal within the small blocks, f. At scales larger than f, the effect of the heterogeneities becomes important, and motion depends on the fractal parameters. The self-invariance properties of the fractal are not valid at short distances. Similarly, the idealized concept of self-similarity at all scales does not hold for fractals in practice. 

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