Random walks and drift in chemical diffusion

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Abstract
In self diffusion the average displacement X (t) of an atom after a time t is zero. It is pointed out that in chemical diffusion π is not necessarily zero, for there are several mechanisms tending to make an atom jump preferentially in one direction rather than the other along a chemical concentration gradient, i.e. tending to produce a drift of atoms superimposed upon their otherwise random movement. This leads to a substantial modification of the classical Einstein equation D = X 2(t)/2t connecting the self-diffusion coefficient D with the mean square displacement X 2(t) and the equation for chemical diffusion which replaces it is derived. This is shown to lead, on the basis of a simplo model, to Darken's equation for the chemical diffusion coefficient. Although in both this and a more general model there are usually two independent mechanisms contributing to X, it turns out to be an interesting feature of chemical diffusion that only one of these is manifest in the chemical diffusion coefficient. The total drift X can be measured independently of the diffusion coefficient and values calculated from the theoretical expressions derived agree well with experimental measurements for zinc and for copper in α-brass/copper diffusion couples. The results are generalized for chemical diffusion in multicomponent systems and expressions obtained for the constants in the familiar schemes of equations used to describe multicomponent diffusion. It is concluded that the cross terms Lij in the Onsager scheme are not primarily the result of correlation between the directions of successive jumps of atoms, as has been suggested.