Flux-independent theory of nonlinear diffusion for Vegard's law solutions

Abstract

The phenomenological Boltzmann–Matano (B–M) analysis argues that a concentration-dependent diffusion coefficient D( X) relates to infinite binary diffusion couple experiments by the solute conserving, flux-formulated non-linear parabolic equation in terms of the non-conserved mole fraction X: − λ 2 d X d λ = d d λ D ( X ) d X d λ In contradiction, this contribution proves that for stable substitutional Vegard's law solutions, in which D( X) is empirically defined by a series of incremental couples, this D( X) must consistently lie outside the second derivative on the right. The base equation has accordingly to be replaced by a Ginzburg–Landau variational equation in which a solute-conserving flux is denied entry to the theoretical structure. In the ternary generalization solute fluxes are likewise not defined so the concept of a “zero-flux plane” must be re-examined. Furthermore, in this ternary formulation volume and molar mass remain conserved, and as in the binary case their corresponding neutral planes are not coincident.