A Commentary on “Diffusion, Mobility and Their Interrelation through Free Energy in Binary Metallic Systems,” L.S. Darken: Trans. AIME, 1948, vol. 175, p. 184ff

As an undergraduate and graduate student of metallurgy, the subject of diffusion at first appeared to me as being rather empirical when compared with thermodynamics. It wasn’t until the end of the course when multicomponent diffusion was discussed and when Darken’s phenomenological equations were described that I appreciated the fundamental nature of diffusion and how the earlier descriptions of flux were really only special cases of Darken’s description in his classic article. Indeed, now that I teach diffusion and transport myself, I often wonder whether the subject should be introduced through Darken’s equations, which related diffusivity to mobility and activity, rather than through Fick’s First Law.

In his classic article[1] titled “Diffusion, Mobility and Their Interrelation Through Free Energy in Binary Metallic Systems,” Dr. Darken presents his phenomenological analysis of diffusion on binary systems. It is divided into two interrelated sections, with the first handling the issue of marker movement and the second handling the effect of nonideality on diffusion.

His treatment is general, but he utilizes the then recent experimental work by Smigelskas and Kirkendall,[2] which showed marker movement in the Cu-brass couple as an example for the first part of his analysis. As a basis for his analysis, he separated the diffusive flux from the flux associated with marker movements (gross material flow) and established a Lagrangian reference frame from which both fluxes were observed. Then, by making a critical assumption that gram atomic volume (density) is constant, he developed two equations: \( v = \left( {D_{2} - D_{1} } \right){\frac{{\partial N_{2} }}{\partial x}} \) and \( D = N_{1} D_{2} + N_{2} D_{1} . \) The first expression identifies individual (intrinsic) diffusion coefficients for the two diffusing species and quantitatively links the marker movement to their difference and the concentration gradient at a given location. The significance of the equation is that, assuming that there are two different diffusivities and markers are found to move, it shows how the problem can be treated. The second equation evolves from the first because the homogenization rate is dependent on the two diffusivities, which are linked through the marker movement. It essentially identifies the chemical diffusion coefficient that is measured through the Bolzmann–Matano analysis.[3]

The work by Johnson[4] shows that the diffusivity in the 0.5Au–0.5Ag system deviates significantly from what is expected from the ideal tracer diffusivity and is used as a basis for the second part of his analysis. This analysis is, in my opinion, possibly even more general in nature because it only assumes that a drift velocity on atoms results from the force arising from the magnitude of a potential gradient. The resulting equation, \( D = \left( {N_{1} D_{2}^{*} + N_{2} D_{1}^{*} } \right)\left( {1 + N_{2} {\frac{{d\ln \gamma_{2} }}{{dN_{2} }}}} \right), \) provides the invaluable link that describes how the chemical diffusion coefficient deviates from the ideal (tracer) coefficient, depending on the nature of the thermodynamic solution. Thus, the equation describes a dynamic phenomenon in terms of a thermodynamic state function in an elegant manner and describes how the interactions between the elements (or the enthalpy of mixing) influences diffusion, and how it could lead to uphill diffusion. In practice, this information provides a way to utilize databases and models on thermodynamic solutions to predict diffusion coefficients in non-ideal systems.

The beauty of Darken’s analysis lies in that it is devoid of any assumptions of mechanisms or structural aspects of the material. Indeed, at the very onset of his introduction, he exemplifies dissimilar ion mobility in nonmetallic systems, such as Ag₂S and FeO, in addition to the classic Cu-Zn system used in the experiments of Smigelskas and Kirkendall.[2]

It comes as no surprise that this article published in 1948 still rates as one of the most cited articles in our community. Beyond the elegance and scientific importance of Darken’s article, it is of practical importance to processing and high-temperature performance of more or less all structural multicomponent alloys that contain substitutional alloying elements (e.g., alloy- and stainless steels and super-alloys). His treatment also has been extended to ceramics,[5] polymers,[6] metallic melts,[7] and has been used in structurally very different systems, such as the molecular diffusion of CH₄ and CF₄ in Zeolite,[8] which in effect confirms the generality of Dr. Darken’s elegant analysis.