Generalized Darken's Method; From Diffusional Structures to Nonparabolic Diffusion

Abstract

The generalized Darken method for multicomponent interdiffusion is presented. Its solution enables one to obtain an exact expression for the evolution of component distributions in closed and open systems, for arbitrary initial distributions and time dependent boundary conditions. The model allows to study the evolution of the composition of the solid solution because of its thermal treatment, evaporation and/or oxidation. Introductory commentary on some of the more current experimental and mathematically oriented results (nonparabolic diffusion) as well as the modified Navier-Stokes equation with additional diffusional terms are included. INTRODUCTION The results presented here benefitted immensely from the Darken concepts [1]. The starting point was his fundamental concept in interdiffusion studies which states that as in other media, in solids the flux of a given element equals its diffusional flux (in an internal reference frame) when added to the drift (convection) flux. What is presented here is a sampler of contributions we have made to that field over the last ten years. These contributions were possible due to the tremendous progress made in the modern theory of partial differential equations. The experiments were possible due to collaboration with several Laboratories. In the section that follows, we will put forth the formulation of a generalized Darken method of interdiffusion, GDM, and the method for finding its solution. The metallic alloys and ceramic materials are usually a multicomponent and multiphase. Thus, the understanding of the interdiffusion in solutions containing an unrestricted number of components is stimulating and productive. Such systems often show variable mobilities and exchange mass with the surrounding. Moreover, initial distributions are arbitrary. Although general phenomenological relations for interdiffusion are available, these are hardly simple and their effective solutions are few and narrowed mainly to the closed systems. Investigations of multicomponent systems are usually limited to the determination of interdiffusion coefficients [2], because of experimental difficulties, are few for systems with more than three components. The mathematical model of interdiffusion [3, 4] and the concept of generalized (i.e., weak) solution allow to obtain an exact expression for the evolution of component distributions in a closed system as well as in an open one. Finally, we have included an introductory commentary on some of the more current experimental and mathematically oriented results (nonparabolic diffusion) as well as the modified Navier-Stokes equation with additional diffusional terms. Though this results have not yet been verified on an experimental level, they still offer exciting new frontiers. THEORY We start with the generalized Darken's phenomenological scheme. The particulars of this method for the closed system [5] and the more general description of interdiffusion that incorporates the equation of motion can be found elsewhere [3, 4]. In case of multicomponent solutions the force, being a result of concentration gradient, gives rise to diffusion of a particular element. The velocity of diffusion was defined by Darken in the internal reference frame: . All diffusional fluxes are coupled and their local changes influence the common mixture drift velocity ( ) [1]. Depending of a