Binary and Multicomponent Diffusion
Authors: Ågren John 
Published: October 2003

Diffusion is intimately related to the irreversible process of mixing. Almost two centuries ago it was observed for the first time that gases tend to mix, no matter how carefully stirring and convection are avoided. A similar tendency was later also observed in liquids.

Diffusion processes play an important role during all processing and usage of materials at elevated temperatures. Most materials are based on a major component and usually several alloy elements that may all be important for the performance of the materials, i.e., the materials are multicomponent systems. In this article the theoretical basis for analyzing diffusion processes in multicomponent solid metals is presented. However, despite their limited practical applicability, for pedagogical reasons are also presented.

The famous Fick's law and mention of its relation to other linear laws such as Fourier's and Ohm's laws are presented. A general linear law is formulated and the concept of mobility is introduced. A special section is devoted to the choice of frame of reference and how this leads to the various diffusion coefficients, e.g., individual diffusion coefficients, chemical diffusion, tracer diffusion, and self‐diffusion.

The general multicomponent formulation, i.e., the Fick‐Onsager law, and the discussion on the frames of reference to the multicomponent systems are given.

In a multicomponent system withncomponents,n− 1 concentration variables may be varied independently, and the concentration of one of the components has to be chosen as a dependent variable. This choice is arbitrary, but it is often convenient to choose one of the major components as the dependent component and sometimes it is even necessary to change the choice of dependent component. Then, a new set of diffusion coefficients have to be calculated from the old set. This is discussed.

The diffusion coefficients, often called diffusivities, express the linear relation between diffusive flux and concentration gradients. However, from a strict thermodynamic point of view, concentration gradients are not forces because real forces are gradients of some potential, e.g., electrical potential or chemical potential, and concentration is not a true potential. The kinetic coefficients relating flux to gradients in chemical potential are called mobilities. In a system withncomponents,n(n− 1) diffusion coefficients are needed to give a full representation of the diffusion behavior whereas there are onlynindependent mobilities, one for each diffusing component. This means that the diffusivities are not independent, and experimental data should thus be stored as mobilities rather than as diffusivities. For binary systems it does not matter whether diffusivities or mobilities are stored, but for ternary systems there are six diffusivities and only three mobilities. For higher‐order systems the difference becomes larger, and in practice it is impossible to construct a database based on experimentally measured diffusivities. A database should thus be based upon mobilities and a method to calculate the diffusivities whenever needed. A method to represent experimental information in terms of mobilities for diffusion in multicomponent interstitial and substitutional metallic systems is presented, and the temperature and concentration dependences of the mobilities are discussed. The influence of magnetic and chemical order are discussed.