Theory for the Single Phase Simulation
The flux of a component 
in the z-direction in an isobarothermal system is in general given by
where 
is a matrix of kinetic coefficients and 
is the chemical potential of component 
. The correlation effects, i.e. the coupling of the flux of 
component to the chemical potential gradients of the other elements, are normally neglected
and thus
where 
is the concentration and 
the mobility of component 
.
The equation for the flux is combined with the equation of continuity, which takes the following form in a planar domain,
which relates the local evolution of the concentration of 
to the divergence of the flux.
The expression for the flux can be expanded in terms of concentration gradients
where the diffusion coefficient of component 
with respect to the concentration gradient of component 
has been introduced.
The flux expressions above are given in the so-called lattice-fixed frame of reference. In practical calculations it is more common to use a volume-fixed frame of reference. For a discussion of these concepts, see Andersson and Ågren [1992And].
Reference
[1992And] J.-O. Andersson, J. Ågren, Models for numerical treatment of multicomponent diffusion in simple phases. J. Appl. Phys. 72, 1350–1355 (1992).