The Trouble with Diffusion

Abstract

The phenomenological formalism, which yields Fick's Laws for diffusion in single phase multicomponent systems, is widely accepted as the basis for the mathematical description of diffusion. This paper focuses on problems associated with this formalism. This mode of description of the process is cumbersome, defining as it does matrices of interdiffusion coefficients (the central material properties) that require a large experimental investment for their evaluation in three component systems, and, indeed cannot be evaluated for systems with more than three components. It is also argued that the physical meaning of the numerical values of these properties with respect to the atom motions in the system remains unknown. The attempt to understand the physical content of the diffusion coefficients in the phenomenological formalism has been the central fundamental problem in the theory of diffusion in crystalline alloys. The observation by Kirkendall that the crystal lattice moves during diffusion led Darken to develop the concept of intrinsic diffusion, i.e., atom motion relative to the crystal lattice. Darken and his successors sought to relate the diffusion coefficients computed for intrinsic fluxes to those obtained from the motion of radioactive tracers in chemically homogeneous samples which directly report the jump frequencies of the atoms as a function of composition and temperature. This theoretical connection between tracer, intrinsic and interdiffusion behavior would provide the basis for understanding the physical content of interdiffusion coefficients. Definitive tests of the resulting theoretical connection have been carried out for a number of binary systems for which all three kinds of observations are available. In a number of systems predictions of intrinsic coefficients from tracer data do not agree with measured values although predictions of interdiffusion coefficients appear to give reasonable agreement. Thus, the complete connection has not been made, even for binary systems. The theory has never been tested in multicomponent systems. An alternative path to understanding diffusion behavior in multicomponent systems is presented which is based upon a kinetically derived version of the flux equations. While this approach has problems of its own, it has the potential for providing a new range of insights into the process, and for devising simple models for predicting composition evolution in multicomponent systems

Investigation of the diffusion behavior in Sn-xAg-yCu/Cu solid state diffusion couples

“…However, to obtain the four diffusion coefficients as shown in equation ( 5) for a ternary system, two diffusion couples with intersection of the composition profiles on the interested point are needed [27,28], which is actually quite difficult to be designed in experiments.…”

Diffusion in metals: the flux of history

“…Therefore, one would indeed expect the net flux to depend on how this mathematical product varies with distance. Some researchers maintain this approach despite the antipathy of the general community, 25,26 usually expanded in terms of the jump frequency rather than expressed by a diffusion coefficient strictly defined as being proportional to the jump frequency (which is what most people assume, anyway). The CdD/dx term expected from the expansion of this variant of the flux equation is indeed suggested by the full, (weighted) random walk analysis of an arbitrary assemblage of atoms, but several texts claim that this somehow cancels out, and leaves a drift velocity term CV, 27 where it can easily be shown that V falls short of being dD/dx by a factor c as above.…”

Understanding the Growth of Interfacial Reaction Product Layers between Dissimilar Materials

“…16 It is frequently found out that when dealing with diffusion in ternary or even higher-order systems the mathematics becomes very complicated and the physical picture of the phenomena 74 T. LAURILA AND A. PAUL under investigation may be blurred or disappears totally. 17 It will be shown below that the thermodynamic kinetic method can be used to greatly simplify the analyses and bring back the physics to the phenomena under investigation. Finally, as graphical methods are frequently used in the thermodynamic kinetic method it will be shown that a large amount of quantitative data can be condensed into relatively simple diagrams.…”