I've found this, however, it sound suspicius in multcomponent:

"The diffusion coefficient D corresponds to the solution which means that it is the same for both the electrolyte and the solvent. Certainly the diffusion process involves both substances as it consists in a mutual interdiffusion of solute and solvent."

The condition of equivalent diffusivity in liquid systems is only true when dealing with thermodynamically ideal solutions where the density of the system remains constant. Since electrolytes and solvent will often have large differences in their densities, mutual diffusivity generally only applies in very dilute systems. As mentioned above, this does not apply in multi-component systems.

mutual diffusivity as my understanding also referring to the transportation of A to B, thus we can say that diffusivity DAB. it refers and depends on the mechanism of our process.

For two-component systems the relation between mutual diffusion of solvent and solute is trivial. For multicomponent systems, it is possible to extract the solvent diffusivity from multicomponent diffusion and volumetric data. It is a sort of change of reference frame. In such a way you can evaluate the extent of counterflow.

A general reference is:

Miller, D. G.; Vitagliano, V.; Sartorio, R. J. Phys. Chem. 1986,

90, 1509-1519.

Whereas, a more specific reference on the counterflow is:

Vergara et al. Macromolecules 2002, 35, 1389-1398

The term mutual diffusion was coined by Hartley and Crank (Trans. Faraday. Soc., 1949, 45, 801). Inter-diffusion ( Brown, J.Chem. Soc., Trans., 1918, 113, 559; Albright & Mills, J. Phys. Chem., 1965, 69, 3120) is an equivalent and interchangeable term.

In a two component system, one only needs a single diffusion coefficient to describe mixing due to a composition gradient. Generally experiments yield a value for the volume-fixed frame of reference (flows measured relative to the centre of volume of system, which moves due to the different densities of the solutions that are mixing).

In multicomponent systems, a matrix is required as the diffusion or flow of one component sets up counterflows of the others. For an n-component system, an (n-1)x(n-1) matrix is required. The elements of the matrix are related by the Onsager reciprocal relations, so in a ternary system, only 3 of the 4 matrix elements are independent.

The mutual diffusion coefficient needs to be distinguished from the self or intra-diffusion coefficients of the components in the mixture, which describe their thermal or "Brownian" motion. In an binary electrolyte solution there are three such self-diffusion coefficients, solvent, cation and anion.

In general there is no relation directly linking the self-diffusion and mutual diffusion coefficients, though there are many approximations in the literature.

Appropriate books are Cussler, Diffusion, CUP, 2009; Tyrrell and Harris, Diffusion in Liquids, Butterworths, 1984.