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A nonhomogeneous boundary value problem for interdiffusion in a solid multicomponent one‐dimensional mixture showing constant concentration is analyzed. Darken's concept of separation of diffusional and drift flows is applied for the general case of diffusional transport in an r‐component compound. The equations of local mass conservation, Darken's flux formula, the postulate of constant molar volume of the mixture, and the initial and boundary conditions form a self‐consistent interdiffusion problem (the quantitative dynamical model). This problem is analyzed in open as well as closed systems and when the component diffusivities vary with composition. The variational form of the interdiffusion problem and the criterion of parabolicity are presented. Results of numerical simulation of interdiffusion in binary and ternary solid solutions of finite dimensions are presented. The development of the “uphill diffusion” concentration profile in the ternary alloy is displayed.

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Over the last two decades, there have been tremendous advances in the computation of diffusion and today many key properties of materials can be accurately predicted by modelling and simulations. In this paper, we present, for the first time, comprehensive studies of interdiffusion in three dimensions, a model, simulations and experiment. The model follows from the local mass conservation with Vegard’s rule and is combined with Darken’s bi-velocity method. The approach is expressed using the nonlinear parabolic–elliptic system of strongly coupled differential equations with initial and nonlinear coupled boundary conditions. Implicit finite difference methods, preserving Vegard’s rule, are generated by some linearization and splitting ideas, in one- and two-dimensional cases. The theorems on the existence and uniqueness of solutions of the implicit difference schemes and the consistency of the difference methods are studied. The numerical results are compared with experimental data for a ternary Fe-Co-Ni system. A good agreement of both sets is revealed, which confirms the strength of the method.

In this work we consider the one and multidimensional diffusional transport in an s-component solid solution. The new model is expressed by the nonlinear parabolic-elliptic system of strongly coupled differential equations with the initial and the nonlinear coupled boundary conditions. It is obtained from the local mass conservation law for fluxes which are a sum of the diffusional and Darken drift terms, together with the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. We construct the implicit finite difference methods (FDM) generated by some linearization idea, in the one and two-dimensional cases. The theorems on existence and uniqueness of solutions of the implicit difference schemes, and the theorems concerned convergence and stability are proved. We present the approximate concentrations, drift and its potential for a ternary mixture of nickel, copper and iron. Such difference methods can be also generalized on the three-dimensional case. The agreement between the theoretical results, numerical simulations and experimental data is shown.