🔬 materials science

Interface Energy and Phase Transformations: A Comparative Analysis of Cahn-Hilliard and CALPHAD-based Models in Ternary Substitutional Alloys

This paper presents a thermodynamically consistent finite element scheme derived from the thermodynamic extremal principle that unifies Cahn-Hilliard phase-field models and CALPHAD-based reactive diffusion models into a single framework capable of handling both regularized interfaces and degenerate Stefan-type problems on fixed meshes while ensuring exact mass conservation and discrete energy dissipation.

Original authors: Wolfgang Flachberger, Thomas Antretter, Swaroop Gaddikere-Nagaraja, Silvia Leitner, Manuel Petersmann, Jiri Svoboda

Published 2026-10-08

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Wolfgang Flachberger, Thomas Antretter, Swaroop Gaddikere-Nagaraja, Silvia Leitner, Manuel Petersmann, Jiri Svoboda

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Materials scientists spend much of their time trying to understand how the tiny internal structures of metals and alloys change over time. When a metal is heated, cooled, or stressed, its atoms rearrange themselves, shifting from one arrangement to another in a process called a phase transformation. These shifts are not random; they are driven by the material's desire to reach a state of lower energy, much like a ball rolling down a hill. However, the path the atoms take and the shape they form along the way depend heavily on how the boundary between the old and new structures behaves. Sometimes this boundary is a sharp, distinct line, while at other times it is a blurry, gradual transition. Understanding and predicting these behaviors is crucial because the final microstructure determines whether a bridge beam will hold under a heavy load or whether a jet engine turbine will survive the heat of flight. For decades, researchers have relied on two main ways to simulate these changes, but each approach has had a significant blind spot: one could handle blurry boundaries well but failed when the boundary became sharp, while the other worked for sharp boundaries but struggled to fit into the complex software used to design real-world mechanical parts.

A team of researchers has now developed a new mathematical framework that bridges this gap, allowing them to simulate both types of boundaries with a single, unified method. The work, led by Wolfgang Flachberger and colleagues, introduces a versatile tool capable of modeling how atoms diffuse and rearrange in complex alloys, whether the transition between phases is smooth and gradual or abrupt and jagged. The key to their success lies in a fundamental shift in how they approach the problem. Instead of trying to force the computer to calculate the slope of a chemical driving force—a task that becomes impossible when the boundary is perfectly sharp—they restructured the equations to treat the flow of atoms and the forces pushing them as separate, primary variables. This subtle change allows the simulation to remain stable and accurate even when the interface between phases has no thickness at all, a scenario that causes traditional methods to crash.

The researchers tested their new approach by simulating the behavior of alloys in both one and two dimensions, comparing their results against established models. In one set of experiments, they watched how a mixture of two components separated over time. When they used a model that included a "smoothing" term to represent the energy cost of a blurry boundary, the simulation produced the expected gradual transition. When they removed that smoothing term to simulate a perfectly sharp boundary, the traditional methods would have failed, but their new scheme handled it gracefully. The atoms rearranged themselves until a distinct, sharp line formed between the two phases, exactly as physics predicts for such systems. The simulation showed that even in this difficult, "degenerate" state where the usual mathematical rules break down, the new method still perfectly conserved the total amount of matter and correctly tracked the loss of energy, adhering to the fundamental laws of thermodynamics.

To prove the method's robustness, the team also tackled a more complex scenario involving a ternary alloy, which contains three components, including vacancies. In crystal structures, a vacancy is simply an empty spot where an atom should be. These empty spots are essential because they allow atoms to move; without them, diffusion in solid metals would be nearly impossible. The researchers modeled how these vacancies are created and destroyed as atoms shuffle around, a process that can lead to the Kirkendall effect, where one type of atom moves faster than another, causing the material to shift or even develop voids. Their simulations successfully captured how the vacancies accumulated in certain areas and depleted in others, driving the movement of the other atoms. The results showed that the new method could handle these intricate interactions without losing stability, even when the interface between phases was razor-thin.

The significance of this work extends beyond just getting the numbers right. The new scheme is built on a foundation that is compatible with the standard software tools engineers use to design mechanical parts, such as those for cars or aircraft. Because the method works on a fixed grid and does not require the computer to constantly redraw the mesh to follow the moving boundary, it can be easily coupled with simulations of mechanical stress and damage. This means that in the future, engineers could potentially run a single simulation that predicts not only how a material's internal structure will evolve during heat treatment but also how that evolution will affect the part's strength and its likelihood of cracking under pressure. The researchers demonstrated that their approach works for both smooth and sharp interfaces, offering a single, reliable tool for a wide range of materials problems.

In their tests, the team observed that the choice of mathematical elements used to represent the data on the computer grid influenced the speed and precision of the results. While some choices made the interface appear slightly jagged during the transition, the final outcome was always consistent with the laws of physics. The simulations confirmed that the method conserves mass exactly and satisfies the second law of thermodynamics, ensuring that energy is dissipated correctly in every step. The researchers noted that while the method operates at the very edge of what is mathematically stable for sharp boundaries, it does so without breaking the physical rules. This allows them to model the "mushy" region where phases coexist without needing to resolve the interface as a separate, moving object, a feat that would otherwise require prohibitively complex calculations.

The study concludes that this new methodology provides a powerful, flexible framework for understanding phase transformations in materials. By unifying the treatment of smooth and sharp interfaces, it removes the need to choose between different models depending on the specific problem. The ability to incorporate thermodynamic data directly, including the complex driving forces derived from the CALPHAD method, gives researchers full control over the simulation inputs. This opens the door to more accurate predictions of how materials will behave in real-world conditions, from the formation of voids in high-temperature alloys to the complex evolution of microstructures in multi-component systems. The work stands as a significant step forward in computational materials science, offering a tool that is both mathematically rigorous and practically applicable to the challenges of modern engineering.

Technical Summary: Interface Energy and Phase Transformations in Ternary Substitutional Alloys

Problem Statement
The paper addresses the challenge of modeling diffusional phase transformations in alloys, specifically focusing on the numerical treatment of interface energy. Existing approaches are divided into two distinct categories with incompatible numerical schemes:

Cahn-Hilliard (CH) Models: These utilize a double-well free energy potential with a gradient term to impose interface energy. While effective for simulating spinodal decomposition with smooth interfaces, standard finite element methods (FEM) for CH models break down when the interface energy term (gradient coefficient $\kappa$ ) is set to zero.
Reactive Diffusion (CALPHAD) Models: These derive driving forces from the convex hull of the molar free energy (common tangent construction), implicitly assuming zero interface energy. These models often rely on finite difference methods or specific discontinuous Galerkin schemes because the resulting equations become degenerate parabolic problems (Stefan-type) where diffusivity vanishes in the two-phase region. Standard FEM schemes requiring continuous gradients of the chemical affinity fail in this limit.

The authors identify a gap in the literature: a unified, thermodynamically consistent finite element scheme capable of handling both limits (smooth interfaces with energy and sharp interfaces without energy) on a fixed mesh, suitable for coupling with continuum mechanics and damage models.

Methodology
The authors derive a single numerical framework based on the Thermodynamic Extremal Principle (TEP). The core of the methodology involves the Rayleighian functional, defined as the sum of the free energy rate and one-half of the dissipation, subject to mass conservation constraints enforced via Lagrange multipliers.

Key methodological steps include:

Variational Formulation: The evolution equations are derived from the stationarity of the Rayleighian with respect to rates, fluxes, and multipliers at a "frozen" state. This formulation treats fluxes and chemical affinities (defined as the negative variation of molar free energy with respect to mole fraction) as primary unknowns.
Avoidance of Affinity Differentiation: By solving for fluxes explicitly, the scheme avoids taking derivatives of the chemical affinity. This is crucial for handling the "degenerate" case (Case ii) where the free energy is a convex hull (linear in the two-phase region), causing the second derivative of the free energy to vanish.
Time Discretization: The time-discrete scheme is formulated as an incremental minimization problem (backward Euler). This approach ensures exact mass conservation and satisfies a discrete energy-dissipation inequality for any convex molar free energy, including the unregularized convex hull.
Finite Element Spaces: The authors propose a mixed finite element formulation. While they utilize equal-order continuous elements (pragmatic for coupling with mechanical solvers like ABAQUS), they also demonstrate the use of Raviart-Thomas spaces for fluxes and discontinuous elements for scalars, which are naturally suited for the degenerate case.
Extension to Ternary Systems: The framework is extended to ternary substitutional alloys, treating vacancies as a non-conserved component with sources and sinks, allowing for the modeling of the Kirkendall effect.

Key Contributions

Unified Numerical Scheme: The paper presents a single finite element scheme that seamlessly transitions between Cahn-Hilliard models (with interface energy) and reactive diffusion models (without interface energy) by simply altering the definition of the molar free energy function ( $f_m$ ).
Handling Degeneracy: The authors explicitly characterize the sharp-interface limit (Case ii) as a degenerate parabolic Stefan problem. They demonstrate that while the equation loses parabolicity (diffusivity vanishes) in the two-phase region, the proposed scheme remains valid on a fixed mesh. The scheme operates "on the boundary of degeneracy," preserving mass conservation, energy balance, and the second law of thermodynamics without requiring a moving mesh or explicit interface tracking.
Thermodynamic Consistency: The paper provides a rigorous proof of thermodynamic consistency for multi-component systems, clarifying the relationship between Onsager coefficients and diffusion coefficients. It corrects a common misconception regarding the "chemical affinity" versus "chemical potential" in constrained systems.
Vacancy Modeling: A specific implementation for ternary systems including vacancies is derived, showing how vacancy generation/annihilation couples with diffusion and affects site fractions.

Results
The methodology was implemented using the open-source solver FEniCSx and validated through 1D and 2D simulations:

1D Comparisons: The scheme successfully reproduced the smooth phase separation of the classical Cahn-Hilliard model, the regularized reactive diffusion (Case i), and the sharp-interface reactive diffusion (Case ii). In the sharp-interface case, the mole fraction developed a discontinuity while the affinity remained continuous. The scheme correctly handled the "degenerate region" where the driving force vanishes, consuming the two-phase region from the bulk phases until a sharp interface remained.
2D Morphology: In 2D simulations, the classical Cahn-Hilliard model minimized interface area (forming circular shapes), whereas the sharp-interface models (Cases i and ii) did not prioritize area minimization, resulting in distinct morphological patterns.
Vacancy Effects: In the ternary vacancy model, the simulation successfully captured the Kirkendall effect, showing vacancy accumulation in inclusion regions and depletion in bulk regions due to differing diffusion rates of the alloy components.
Numerical Stability: The scheme satisfied the discrete energy-dissipation inequality (Proposition 1) to machine precision. Mass conservation was exact. The use of equal-order elements was shown to be stable for the time-dependent problem due to the coupling provided by the time-derivative term, though Raviart-Thomas elements provided sharper interface resolution with lower computational cost.

Significance and Claims
The paper claims to offer a versatile and thermodynamically consistent framework for modeling phase transformations that bridges the gap between phase-field methods and reactive diffusion models.

Operational Scope: The authors emphasize that their approach is designed for coupling with continuum mechanics and damage on a fixed mesh. They explicitly state that while moving-mesh methods could remove the degeneracy of the sharp-interface limit, such methods are incompatible with their goal of coupling to complex mechanical problems involving damage and multiple inclusions.
Physical Fidelity: The significance lies in the ability to operate at the "boundary of degeneracy" while strictly enforcing the fundamental laws of thermodynamics (mass, energy, entropy). The authors argue that for continuum-scale modeling, resolving the fine structure of the interface is less critical than ensuring the global conservation laws hold, a condition their scheme satisfies even when the equation is not parabolic.
Clarification: The work aims to clarify the mathematical nature of the sharp-interface limit and the role of the Rayleighian in deriving consistent evolution equations, correcting misconceptions about the necessity of regularization for numerical stability in convex free energy systems.

The authors conclude that this methodology provides a robust foundation for future studies involving complex, multi-physics phase transformation problems in materials science.

More like this