Phase-Field Thermodynamics Overview

• Phase-Field Thermodynamics is a rigorous framework that uses continuous order parameters to simulate diffuse-interface phenomena in nonequilibrium systems.
• It employs both Helmholtz and grand potential formulations to derive variational gradient flows while strictly adhering to the second law and Onsager reciprocity.
• The approach integrates multiphysics effects, such as elasticity, chemical transport, and interface kinetics, to accurately capture complex microstructure evolution.

Phase-field thermodynamics provides the rigorous nonequilibrium thermodynamic and variational foundation for a broad class of diffuse-interface models that describe mesoscale microstructure evolution in materials and physical systems. Phase-field models encode interfacial phenomena using continuous order parameters (phase fields) and, when appropriate, conserved fields such as composition or energy densities, coupled through a free-energy (or appropriate thermodynamic potential) functional and dissipative kinetics. Current formulations explicitly enforce the second law and Onsager reciprocity, correctly capture interface energy and mobility, and provide a dynamic, extensible platform for complex multiphysics phenomena including nonequilibrium solidification, multi-phase coexistence, elasticity, and transport.

1. Thermodynamic Potentials: Helmholtz vs. Grand Potential

The choice of thermodynamic potential is critical for correct phase-field modeling. For isothermal, closed systems with conserved quantities (e.g., mass), the free energy functional should take the Helmholtz form: $$F[\phi,c] = \int_V \left[ \frac{k_\phi}{2} |\nabla\phi|^2 + m\,g(\phi) + f(\phi,c) \right]dV$$ where $\phi(\mathbf x,t)$ is the phase field, $c(\mathbf x,t)$ is composition, $g(\phi)$ is a double-well barrier, and $f(\phi,c)$ interpolates bulk free energies (Zhang et al., 2024). The local equilibrium constraint sets the phase-specific chemical potentials equal.

A grand-potential phase-field model instead performs a Legendre transformation $c\mapsto\mu$ (chemical potential), yielding: $$\Omega[\phi,\mu] = \int_V \left[ \frac{k_\phi}{2}|\nabla\phi|^2 + m\,g(\phi) + w(\phi, \mu) \right] dV$$ with $$w(\phi,\mu)=p(\phi)w^\alpha(\mu)+[1-p(\phi)]w^\beta(\mu)$$ the interpolated Legendre-dual bulk densities.

While both functionals attain minima at equilibrium, only $F[\phi,c]$ is guaranteed to be a true Lyapunov functional under mass conservation: $$\frac{dF}{dt}=-\int_V \left[ M_\phi \left(\frac{\delta F}{\delta\phi}\right)^2 + M |\nabla\mu|^2 \right] dV \leq 0$$ In contrast, $\phi(\mathbf x,t)$0 is strictly decreasing only for systems in which mass is not conserved (iso-$\phi(\mathbf x,t)$1 ensemble), such as open systems with a pervasive reservoir. Therefore, for modeling closed, mass-conserving systems, the Helmholtz formalism is thermodynamically correct (Zhang et al., 2024).

2. Variational Principles and Nonequilibrium Dynamics

Phase-field thermodynamics is built on variational calculus, with equations of motion derived as dissipative gradient flows of the chosen functional, consistent with irreversible thermodynamics (Zhou, 2014, Zhang et al., 2024). In a mass-conserving binary system,

$\phi(\mathbf x,t)$2

The Onsager principle and positive-definite mobilities ensure positive entropy production. Coupling of nonconserved (Allen–Cahn) and conserved (Cahn–Hilliard) kinetics arises naturally from this variational prescription (Zhou, 2014, Ván, 2019).

Generalizations include nonequilibrium kinetic entropy corrections leading to hyperbolic phase-field equations (Zhou, 2014) and models with coupled mechanics (phase–elastic/viscoelastic/chemical), for which the energy balance and entropy inequality are maintained in the presence of additional reversible fields.

3. Non-equilibrium Interface Thermodynamics: Diffuse and Sharp Descriptions

Recent work unifies sharp-interface and diffuse-interface thermodynamics in a fully nonequilibrium framework. In phase-field models for rapid transformations, distinct concentration fields are introduced per phase, with decompositions into nonconserved (“short-range”, interface-tracking) and conserved (“long-range”, bulk-diffusing) fields (Li et al., 2023, Li et al., 2023). This yields kinetic equations that capture both trans-sharp-interface exchange and trans-diffuse-interface diffusion, consistently with Onsager reciprocity: $\phi(\mathbf x,t)$3 Explicit balance equations for driving forces and dissipation decompose the interfacial mechanics into local migration, solute trapping, and drag phenomena. The energy dissipation rate per control volume is quadratic in these fluxes and exactly recovers classical friction and drag effects in the thin-interface limit (Li et al., 2023, Li et al., 2023).

Validation against molecular dynamics simulations (e.g., rapid solidification in Al–Cu) demonstrates that these formulations reproduce solute trapping, partial and complete drag, and interface kinetics accurately over a wide range of undercoolings and velocities (Li et al., 2023).

4. Multiphase, Multicomponent, and Gradient-Enriched Models

For microstructures involving multiple phases and/or components, phase-field thermodynamic models incorporate a set of phase fraction fields $\phi(\mathbf x,t)$4 (summed to unity), composition fields $\phi(\mathbf x,t)$5, and, when needed, higher-order gradients (Pogorelov et al., 2013, Cogswell et al., 2010, Espath et al., 2019).

The general free-energy functional takes the form: $\phi(\mathbf x,t)$6 A Lagrange projection ensures $\phi(\mathbf x,t)$7 (Pogorelov et al., 2013). Multi-obstacle barrier functions prevent unphysical "ghost" phases. Flatness and convexity requirements on the barrier ensure correct interfacial stability and the absence of spurious phases.

When higher-order (second-gradient or microhyperstress) terms are included, as in phase-field gradient or phase-field crystal models, generalized microforce and microtorque balances, as well as advanced boundary and edge conditions, are required for full thermodynamic consistency (Espath et al., 2019, Abdalla et al., 2022).

5. Boundary Conditions and Thermodynamic Consistency at Interfaces

Phase-field thermodynamic models rigorously account for both bulk and interfacial (surface) thermodynamics via dynamic boundary conditions derived from a generalized Onsager principle (Jing et al., 2022, Jing et al., 28 Oct 2025). The total free energy includes both volume and surface terms: $\phi(\mathbf x,t)$8 Varied boundary couplings (purely irreversible, reversible-irreversible, reactive, etc.) can be systematically constructed using a generalized Onsager mobility matrix, yielding dynamic or Robin-type boundary conditions that link bulk and surface chemical potentials via flux balances. The net system (bulk + surface) is closed and energy-dissipative (or, more generally, entropy-producing), with explicit dependence of exchange rates on the system’s geometric and thermodynamic scales (Jing et al., 28 Oct 2025, Jing et al., 2022, Miranville et al., 2013).

6. Extensions: Coupled Thermomechanics, Chemoelasticity, and Stochastic Thermodynamics

Phase-field thermodynamics frameworks have been extended to:

• Non-isothermal, pressurized, and elastic multiphysics systems (Noii et al., 2019, Maraldi et al., 2010), with each field (mechanical, chemical, thermal) derived consistently from the Helmholtz free energy and subjected to the entropy inequality. The models retain variational structure and correct interface thermodynamics under strong coupling.
• Stochastic thermodynamics with internal variables (STIV), providing a nonphenomenological microscopic derivation of phase-field kinetics, capturing nucleation phenomena and entropy production directly from molecular dynamics (Leadbetter et al., 2024).
• Contact thermodynamics frameworks, which embed phase-field thermodynamics in contact geometry and the maximum dissipation principle, fully generalizing the relaxation equations and Onsager symmetry well beyond linear nonequilibrium (Guevel et al., 2019).

7. Summary of Core Principles and Applicability

Phase-field thermodynamics is governed by the following interrelated core principles:

• Construction of free-energy functionals (Helmholtz or grand potential) in the correct thermodynamic ensemble, consistent with the constraints (e.g., mass conservation, chemical potential reservoirs).
• Variational formulation of governing equations as gradient flows, respecting Onsager reciprocity and the second law, with clear distinction between conserved and nonconserved dynamics.
• Explicit decomposition of driving forces and dissipation channels at interfaces, enabling unified treatment of sharp/diffuse transitions, solute trapping, drag, and other nonequilibrium effects, closely matching atomistic simulations.
• Encompassing framework for complex microstructures, multiple phases/components, and higher-gradient or multiphysics couplings, with rigorous treatments of boundary and interface thermodynamics.
• Universally, the thermodynamically consistent phase-field model reduces to the classical sharp-interface results in the appropriate limits and satisfies all requirements of the Gibbsian and continuum nonequilibrium thermodynamic framework (Zhang et al., 2024, Li et al., 2023, Pogorelov et al., 2013, Cogswell et al., 2010, Espath et al., 2019, Zhou, 2014, Jing et al., 2022, Jing et al., 28 Oct 2025).

These functions define the state-of-the-art in mesoscale interface modeling, microstructural evolution, and non-equilibrium thermodynamics in phase-field systems.