Diffusion in multi-component systems with no or dense sources and sinks for vacancies

Abstract

General equations for the multi-component diffusion in crystalline systems are derived in the framework of Onsager's non-equilibrium thermodynamics. The aim is to provide explicit equations needed for computer-modeling of, e.g., diffusion-controlled phase transformations, but avoiding the usual simplifying assumptions, such as the independence of the fluxes of different atomic species. An additional difficulty is introduced into the problem by the fact that the concentration of vacancies, which mediate diffusion of substitutional atoms, may follow different rules, depending on whether there is a sufficient density of sources and sinks to keep the local vacancy concentration in equilibrium. We treat two limiting cases, one where the total vacancy number is conserved (that is, there are no available sources and sinks) and one where the vacancy concentration is kept in equilibrium (the case of dense sources and sinks). First, the diffusive fluxes of all components and of vacancies are expressed by the well known Onsager relation. The kinetic coefficients L ik from the Onsager relation are derived by means of an extremal thermodynamic principle with respect to the atomic mobilities of individual components and taking into account the constraint amongst fluxes resulting from the vacancy diffusion mechanism. In the case where a dense network of sources and sinks of vacancies is active, the number of lattice positions is not necessarily conserved in every region of the specimen. This means, that the second Fick law, derived for the local conservation of lattice positions, is not applicable in this case. Using mass conservation considerations, the second Fick law is modified to account for this effect. The lattice may shrink or expand for two reasons—either due to the generation or annihilation of vacancies or to the change of the molar volume connected with the change of chemical composition. The deformation of the system is expressed quantitatively by strain rates. Finally, the equations for the system evolution are expressed in both the lattice-fixed and in the laboratory-fixed frames of reference.