Multicomponent Diffusion (Chapter 5, Materials Kinetics)

Multicomponent diffusion occurs in systems having three or more mobile species. The equations for multicomponent diffusion can be derived from irreversible thermodynamics, accounting for both conjugate driving forces and interaction terms among the various species. If the zero net flux condition is satisfied for an N-component diffusion problem, then only N − 1 of the species need to be considered, leading to an (N − 1) × (N − 1) matrix of interdiffusion coefficients. However, if the zero net flux condition is not satisfied, then the full N × N matrix of diffusivities must be included. As in the case of anisotropic diffusion, the multicomponent diffusion problem can be solved using the methods of linear algebra, viz., by diagonalizing the interdiffusivity matrix and solving the problem in principal component space. Transformation into principal component space removes the off-diagonal terms from the matrix, so that standard solutions of the diffusion equation can be employed. The final solution can then be obtained by transforming the solution from principal component space back to the original coordinate system. Interaction effects in multicomponent diffusion problems result directly from the coupling terms in Onsager's formulation of irreversible thermodynamics. The interaction terms are essential for accurately capturing the complicated diffusion pathways in multicomponent systems. In many cases, these interaction effects can lead to the surprising result of “uphill” diffusion, i.e., where the flux of a diffusing species acts to increase its own concentration gradient. This phenomenon would not be allowed in the original formulation of Fick's first law, where the flux always acts to lower the concentration gradient. Keywords: Diffusion, Multicomponent Diffusion, Uphill Diffusion, Linear Algebra, Matrix Methods, Matrix Diagonalization, Irreversible Thermodynamics