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Bi-velocity (Darken) method, which includes material drift and diffusion fluxes, is for the first time applied to describe mass transport in multiphase materials. The respective model is formulated in details and then applied for a ternary diffusion couple in which a two-phase zone can grow during isothermal diffusion. Thanks to the use of a phase-field order parameter, identified here with volume fraction of a chosen phase, the mass transport throughout both phases within a two-phase zone can be considered. The model allows smooth crossing of the type 1 boundary, which makes the mass transport equations valid in both single- and two-phase regions. The solution obtained for 1D geometry provides: (1) a diffusion path in the concentration triangle, (2) element-concentration profiles, (3) volume fractions of the phases in the two-phase zone and (4) drift-velocity distribution along x axis parallel to the mass transport. As an example, the interdiffusion in thediffusion couple of Type 0 boundary is modelled. The zigzag diffusion path is predicted and the profiles of the element concentrations are simulated. For the first time, the drift velocity for the diffusion in two-phase system is determined and correlated with the changes in volume fractions of the phases. [ABSTRACT FROM AUTHOR]