The multiphase-field method based on internal state variables

The multiphase-field method is widely used to simulate the evolution of complex microstructures in computational materials science and is commonly derived via a variational approach. When coupled with heat conduction, it is desirable to derive both the multiphase-field method and the heat conduction equation from a unified, consistent framework. In the present work, this is accomplished by introducing order parameters as internal state variables and exploiting the entropy production inequality of the diffuse interface region for multiple intersecting phases. The approach represents a generalization of the method of Prahs et al. (Prahs et al. Continuum Mech. Thermodyn. 37 , 55 (2025). https://doi.org/10.1007/s00161-025-01383-y ) via a dual-interaction ansatz and its Lagrange multiplier simplification, while maintaining thermodynamical consistency. This requirement, in turn, restricts the choice of the interpolation functions. Moreover, the coupling effects in the heat conduction equation as well as connections to established evolution equations from the literature are demonstrated through an illustrative derivation.