a commentary on ‘‘diffusion, mobility and their interrelation ......a commentary on...

A Commentary on ‘‘Diffusion, Mobility and Their Interrelationthrough Free Energy in Binary Metallic Systems,’’ L.S. Darken:Trans. AIME, 1948, vol. 175, p. 184ff

SEETHARAMAN SRIDHAR

DOI: 10.1007/s11661-010-0177-7� The Minerals, Metals & Materials Society and ASM International 2010

AS an undergraduate and graduate student ofmetallurgy, the subject of diffusion at first appeared tome as being rather empirical when compared withthermodynamics. It wasn’t until the end of the coursewhen multicomponent diffusion was discussed and whenDarken’s phenomenological equations were describedthat I appreciated the fundamental nature of diffusionand how the earlier descriptions of flux were really onlyspecial cases of Darken’s description in his classicarticle. Indeed, now that I teach diffusion and transportmyself, I often wonder whether the subject should beintroduced through Darken’s equations, which relateddiffusivity to mobility and activity, rather than throughFick’s First Law.

In his classic article[1] titled ‘‘Diffusion, Mobility andTheir Interrelation Through Free Energy in BinaryMetallic Systems,’’ Dr. Darken presents his phenome-nological analysis of diffusion on binary systems. It isdivided into two interrelated sections, with the firsthandling the issue of marker movement and the secondhandling the effect of nonideality on diffusion.

His treatment is general, but he utilizes the then recentexperimental work by Smigelskas and Kirkendall,[2]

which showed marker movement in the Cu-brass coupleas an example for the first part of his analysis. As a basisfor his analysis, he separated the diffusive flux from theflux associated with marker movements (gross materialflow) and established a Lagrangian reference frame fromwhich both fluxes were observed. Then, by making acritical assumption that gram atomic volume (density) isconstant, he developed two equations: v ¼ D2 �D1ð Þ@N2

@xand D ¼ N1D2 þN2D1: The first expression identifiesindividual (intrinsic) diffusion coefficients for the twodiffusing species and quantitatively links the markermovement to their difference and the concentrationgradient at a given location. The significance of theequation is that, assuming that there are two differentdiffusivities and markers are found to move, it showshow the problem can be treated. The second equation

evolves from the first because the homogenization rate isdependent on the two diffusivities, which are linkedthrough the marker movement. It essentially identifiesthe chemical diffusion coefficient that is measuredthrough the Bolzmann–Matano analysis.[3]

The work by Johnson[4] shows that the diffusivity inthe 0.5Au–0.5Ag system deviates significantly fromwhat is expected from the ideal tracer diffusivity and isused as a basis for the second part of his analysis. Thisanalysis is, in my opinion, possibly even more general innature because it only assumes that a drift velocity onatoms results from the force arising from the magnitudeof a potential gradient. The resulting equation,

D ¼ N1D�2 þN2D

�1

� �1þN2

d ln c2dN2

� �; provides the invalu-

able link that describes how the chemical diffusioncoefficient deviates from the ideal (tracer) coefficient,depending on the nature of the thermodynamic solution.Thus, the equation describes a dynamic phenomenon interms of a thermodynamic state function in an elegantmanner and describes how the interactions between theelements (or the enthalpy of mixing) influences diffusion,and how it could lead to uphill diffusion. In practice, thisinformation provides a way to utilize databases andmodels on thermodynamic solutions to predict diffusioncoefficients in non-ideal systems.The beauty of Darken’s analysis lies in that it is

devoid of any assumptions of mechanisms or structuralaspects of the material. Indeed, at the very onset of hisintroduction, he exemplifies dissimilar ion mobility innonmetallic systems, such as Ag2S and FeO, in additionto the classic Cu-Zn system used in the experiments ofSmigelskas and Kirkendall.[2]

It comes as no surprise that this article published in1948 still rates as one of the most cited articles in ourcommunity. Beyond the elegance and scientific impor-tance of Darken’s article, it is of practical importance toprocessing and high-temperature performance of moreor less all structural multicomponent alloys that containsubstitutional alloying elements (e.g., alloy- and stain-less steels and super-alloys). His treatment also has beenextended 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 ofDr. Darken’s elegant analysis.

SEETHARAMAN SRIDHAR, POSCO Professor of MaterialsScience and Engineering, is with Carnegie Mellon University, 5000Forbes Avenue, Wean Hall 4317, Pittsburgh, PA 15213. Contacte-mail: sridhars@andrew.cmu.edu

Article published online February 6, 2010

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 41A, MARCH 2010—543

REFERENCES1. L.S. Darken: Trans. AIME, 1948, vol. 175, p. 184.2. A.C. Smigelskas and E.O. Kirkendall: Trans. AIME, 1947, vol. 171,

p. 130.3. L. Bolzmann: Ann. Phys., 1894, vol. 53, p. 959.4. W.A. Johnson: Trans. AIME, 1942, vol. 147, p. 331.

5. A.R. Cooper, Jr. and J.H. Heasley: J. Am. Ceram. Soc., 2006,vol. 49, p. 280.

6. E.J.Kramer,P.Green,andC.J.Palmstrom:Polymer, 1984,vol.25,p.473.7. C.H. Buell and F.O. Shuck: Metall. Mater. Trans. B, 2007, vol. 1,

p. 1073.8. A.I. Skoulides and D.S. Sholl: J. Phys. Chem. B, 2001, vol. 105,

p. 3151.

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