Thermodynamically consistent multi-phase-field model incorporating passive external domains

Phase-field (PF) modeling of multiphase and multicomponent material processes where material phases and their external domains coexist, such as sintering, lacks a thermodynamically consistent formulation. To address this issue, this paper introduces a novel multi-phase-field (MPF) model with passive external domain (PED), which is defined as an external region that remains inert and does not contribute to microstructural evolution in the material system. An additional variable associated with concentration fields is incorporated to distinguish between the material and PED without thermodynamic inconsistencies. The model is validated through 2D single-phase simulations, which highlight its ability to capture surface triple junction behavior, consistent with Young’s equation. Its capability is further demonstrated through 3D simulations of a three-phase, ternary system coexisting with a PED. These simulations correctly predict the evolution of the system toward thermodynamic equilibrium, including phase transformations and solute partitioning. Overall, the MPF-PED model provides a robust and physically consistent framework, overcoming the limitations of conventional models in handling external domain to facilitate the realistic simulation of complex material processes.