Interdiffusion under the Chemical Potential Gradient; Comparison of Onsager and Darken Models

Abstract

The generalized Darken method of interdiffusion in multicomponent systems (GDM) enables obtaining an exact expression for the evolution of component distributions for arbitrary initial distributions and time dependent boundary conditions. In this work we studied the consequences of the more general formulae for the diffusional flux (Planck's equation) which permits to take into account thermodynamical driving forces. This paper is based on our studies of diffusion couples in the Cu-Fe-Ni system at 1273 K. The Cu-Fe-Ni system was chosen because it is a single phase in a wide range of compositions and because its thermodynamic properties are fairly well known. However, the solid solutions in this system are not ideal and consequently the diffusivities depend on composition. The driving force for the diffusion in such ternary system is the gradient of the chemical potential which can be calculated from the concentration profiles and using the known thermodynamical data of the system. Consequently the diffusional flux can be expressed as a function of the concentration gradients of all elements in the system, of the thermodynamical terms and of the mobilities. The diffusion paths are discussed in the light of the ternary interdiffusion coefficients and the self diffusivities with the use of generalized Darken method. A comparison of the Onsager and Darken models is presented. The results show the prospect for the future application of the GDM in the modelling of stress affected diffusion in ternary and higher solid solutions. The generalized Darken’s method A key problem in multicomponent diffusion is the prediction of the diffusion path between the two terminal alloys. For the predictive calculations, the data for the intrinsic diffusivities and/or self diffusion coefficients and their concentration dependence have to be known. Moreover the thermodynamic data for the system have to be known for the calculation of the chemical potentials for each component. The generalised Darken method (GDM) allows a complete quantitative description of the complex diffusional transport process and for the unlimited number of elements. It allows calculation of the diffusion paths when the interdiffusion coefficients are concentration dependent. The details of this model for the closed system [1] and the more general description of interdiffusion that incorporates the equation of motion can be found elsewhere [2-3]. In this paper the formulation of the initial-boundaryvalue problem for the interdiffusion in a closed system is presented. It differs from that presented recently [4,5] with the expression for the mass flux. Physical laws: 1) the law of the mass conservation of an i-th element: