Multicomponent diffusion simulation based on finite elements
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I. INTRODUCTION
NUMEROUS models for multicomponent diffusion simulation have been presented in recent decades, e.g., References 1 through 7. The solutions to the partial differential equations that govern the diffusion process were based upon either a finite-difference discretization scheme,[2,3,4] special analytical expressions evaluated for particular boundary conditions,[5,6,7] or numerical formalisms derived with regard to a particular thermodynamic model.[1] In contrast, the present report focuses on a general numerical solution of the diffusion equation based on the finite-element method. The main reasons for favoring this latter technique are (1) an easier handling of the element mesh representing the diffusion problem within a computer program, (2) the relatively simple extension to multiple spatial dimensions, and (3) the possibility of implementing different kinds of boundary conditions directly into the mathematical description. The formalism is particularly suitable for simultaneous multicomponent diffusion and diffusional phase-transformation simulation. The present report gives a brief description of the multicomponent diffusion formalism suggested by Andersson and ˚ Agren.[8] Then, the generalized diffusion equation in its weak form and the finite-element discretization procedure are presented. The finite-element diffusivity matrices and the corresponding load vectors are derived for a linear onedimensional bar element and a linear two-dimensional triangular element. Some general application examples of the model for one-dimensional diffusion simulations, as well as calculations in two dimensions, are presented. The present model has been implemented using the software program MatCalc, recently developed by the author.[9] The thermodynamic information (i.e., the chemical-potential gradients) that is needed to evaluate the diffusion-coefficient matrix is calculated from the SGTE database SSOL.[10] The
kinetic data are taken from the mobility database available in the commercial software package DICTRA.[1] II. MULTICOMPONENT DIFFUSION Atomistic models of diffusion are usually formulated ˜ in the lattice-fixed frame of reference, and the fluxes Jk of atoms k are measured relative to the lattice planes of the crystal. Considering the atom-vacancy exchange mechanism to be predominant, these fluxes can be expressed by[5,11] ˜ Jk 5 2D8k¹ck [1] where ¹ is the gradient operator, D8k is the intrinsic diffusion coefficient of species k, and ck is the corresponding concentration variable. Equation [1] yields a linear relationship between the flux of atoms and a given concentration gradient and is commonly designated as Fick’s first law. However, the lattice-fixed frame of reference is inconvenient with regard to many practical diffusion problems, because the exact treatment of macroscopic diffusion processes requires precise knowledge of position and movement of the particular lattice planes for all times. As pointed out by Andersson ˚ and Agren,[8] a more-convenient (laboratory) frame of reference may be defin
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