A Direct Derivation of Fick’s Law for Multicomponent Diffusion

Abstract

Fick’s law applicable to isothermal solid-solid n-component diffusion couples has been recently derived by the author directly from the continuity equation. The derivation is briefly reviewed in this paper and general expressions applicable to the determination of interdiffusion coefficients, $\tilde{D}_{ij}^{n} (i,j = 1,2, \ldots n - 1)$, are developed for isothermal, diffusion couples with constant molar density. Explicit expressions for ternary and quaternary interdiffusion coefficients, $\tilde{D}_{ij}^{3} \;(i,j = 1,{\kern 1pt} \,2)$ and $\tilde{D}_{ij}^{4} \;(i,j = 1,{\kern 1pt} \,2,3)$, are also presented. These expressions developed for the calculation of both main and cross interdiffusion coefficients at a section x include various partial derivatives of $[(\tilde{J}_{i} ) \cdot (x - x_{o} )]\,(i = 1,2, \ldots n - 1)$ with respect to individual concentrations $C_{j}$, where $\tilde{J}_{i}$ is the interdiffusion flux of component i based on a laboratory-fixed frame and x_o is the Matano plane for the couple. In this paper the analysis is applied to the concentration profiles theoretically calculated for an isothermal, binary diffusion couple characterized by a constant interdiffusion coefficient $\tilde{D}$ to illustrate the validity of the analysis. A parabolic representation of the derivative, ${\text{d}}[(\tilde{J}_{i} ) \cdot (x - x_{o} )]/{\text{d}}C_{i} \,{\kern 1pt} (i = 1,2)$ that is involved in the expression for $\tilde{D}$, is also developed as a function of x, and such representation has been shown to be useful for the calculation of $\tilde{D}$ for the binary diffusion couple. The parabolic representation of the terms employed for the determination of the binary $\tilde{D}$ is presented for the first time in this study.