Formation and Annihilation of new Phases in Multicomponent Systems

The formation/annihilation of new phases during interdiffusion in multicomponent alloys is the fundamental process in metallurgy and material science. Present mathematical methods combine simple diffusion formalism with highly developed thermodynamic solutions (Thermocalc). They do not produce any quantitative description of the process and leave a lot of unsolved fundamental problems.
We do propose a new original approach to solve the problem of evolution in the multicomponent multiphase systems. We will combine 1) the bi-velocity method of interdiffusion with 2) thermodynamics of the non-reversible processes and 3) advanced mathematical methods.Namely:

1. The bi-velocity method will be transformed to Onsager’s form (the symmetric matrix of diffusivities) and will allow parabolic transforms aimed at the well-posedness in Sobolev spaces.
2. The diffusion fluxes will have the Nernst-Planck form and the Cahn-Hiliard method will allow computing the effective forces. Such formulation leads to symmetrical matrix diffusivities. The entirely new concept is the adding a functional that describes the overall entropy production,and the problem is well posed by minimizing/maximizing certain functionals.
3. Advanced mathematical methods get together functional analysis, variational methods and numerical methods. Darken’s model leads to a system of parabolic equations. Onsager’s formalism provides an elegant framework with many mathematical open questions.

Summing up, the combination of the maximum entropy production principle (Prigogine)with a modern description of interdiffusionwill allow for the fundamental progress in material science and chemistry.The applicationsof the model start from modern metallurgy, kinetics of the reactive diffusion processes and surface treatments. From the mathematical point of view:  well posed nonlocal transport problem and its numerical treatment.