The multiphase-field method based on internal state variables

Continuum Mech. Thermodyn. (2026) 38:47 https://doi.org/10.1007/s00161-026-01470-8 O R I G I NA L A RT I C L E Andreas Prahs · Ahmed Elmoghazy · Daniel Schneider · Britta Nestler The multiphase-field method based on internal state variables Received: 16 October 2025 / Accepted: 26 March 2026 / Published online: 12 May 2026 © The Author(s) 2026 Abstract 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 dualinteraction 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. Keywords Multiphase-field method · Continuum thermodynamics · Internal state variable · Heat conduction · Dual-interaction ansatz · Lagrange multiplier 1 Introduction From two to multiple phases The phase-field method (PFM) was initially developed to describe nanoscale and microscale phenomena (see the seminal works [1–6]). Today, it is widely used for modeling phase transitions and, more generally, the evolution of singular surfaces [7–10]. The PFM employs order parameters, which are continuous indicator functions that satisfy a nonlocal (kinetic) evolution equation [6,11]. This equation is A. Prahs (B) · B. Nestler Institute of Nanotechnology (INT), Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, EggensteinLeopoldshafen 76344, Germany E-mail: B. Nestler E-mail: A. Elmoghazy (B) · D. Schneider · B. Nestler Institute for Applied Materials - Microstructure Modelling and Simulation, Karlsruhe Institute of Technology (KIT), Straße am Forum 7, Karlsruhe 76131, Germany E-mail: D. Schneider E-mail: D. Schneider · B. Nestler Institute of Digital Materials Science (IDM), Karlsruhe University of Applied Sciences, Moltkestraße 30, Karlsruhe 76133, Germany 47 Page 2 of 25 A. Prahs et al. usually derived using a variational approach: the variational derivative of a functional depending on the order parameters is balanced by their rate and a positive mobility factor. Notably, the strong form of the evolution equation remains a postulate. An alternative formulation by Penrose and Fife uses an entropy functional rather than a free-energy functional, which allows for spatiotemporal variations in the temperature field [11]. While the PFM was originally formulated for two phases, e.g., [4,6,11], it was extended to multiple phases by [7,12– 14], referred to as the multiphase-field method (MPFM). Since the order parameters are formulated as scalar fields, the numerical tracking of diffuse interfaces can be performed efficiently, cf., e.g., [15–17]. Hence, the MPFM is a well-established approach for simulating the evolution of complex microstructures and is applied in contexts such as solidification, solid–solid phase transitions, precipitation growth and coarsening, and grain growth, cf. [8–10]. Moreover, the MPFM was further extended to account for mechanical contributions [18–21], with non-linear constitutive behavior, e.g., [22–24], crack propagation [25,26], and many more. Approaches to derive the (M)PFM The evolution equation of the order parameter is typically derived via a variational approach. However, in the case of coupling with heat conduction, it is imperative to adopt a consistent framework in which the derivation of the MPFM and the derivation of the modified heat conduction equation are obtained in a unified manner. A generalization of the PFM was proposed by Gurtin [27] via an extended principle of virtual power (POVP), rooted in continuum thermodynamics. Unlike variational approaches, this framework separates balance equations, reflecting universal physical laws, from constitutive relations, describing material behavior. In continuum mechanics, the evolution of the order parameter is commonly derived either from the extended POVP [27,28] or from the invariance of an extended energy balance under Euclidean transformations [29]. In both cases, the order parameter is introduced as an additional generic degree of freedom (DOF), implying that the PFM is formulated within an extended continuum. Another approach is given by [30], [31], considering the order parameter as an internal state variable (ISV) instead of as an additional DOF, since it represents an observable rather than a controllable quantity, e.g., [32,33]. In this regard, the order parameter is considered to be of dissipative nature [33] and its evolution equation, consequently, follows from the exploitation of the dissipation inequality. In deriving the evolution equation of the order parameter, [31] employs the dissipation inequality associated with the singular surface, in contrast to [30], which bases the derivation on the bulk dissipation inequality. The distinction originates from the modeling framework: [30] assumes a Cauchy continuum with a diffuse interfacial region, from the outset, while [31] employs a Cauchy continuum with a material singular surface for the derivation of the evolution equation. The latter approach interprets the PFM as an approximation of the sharp interface theory for the quasi-static special case of a slowly evolving non-material singular surface without body forces, within the framework of continuum thermodynamics, [34]. For this special case, the balances of mass, linear momentum, and internal energy of a non-material singular surface are identical to those of a material singular surface. Objectives of the current work This work derives the MPFM by introducing the order parameters as ISVs and systematically exploiting the entropy production inequality of the diffuse interface region for multiple intersecting phases within the framework of continuum thermodynamics. Thermodynamically consistent evolution equations for the order parameters are presented, and the corresponding heat conduction equation is derived. A comparison with the classical variational approach is also provided. Originality This work represents a direct generalization of Prahs et al. [31] to multiple intersecting phases. Following this approach, this work: • demonstrates the thermodynamic consistency of the dual-interacti (...truncated)