Thermodynamics for radiation-induced segregation and phase change needed

Introduction

Radiation-induced segregation (RIS) and phase change phenomena have traditionally been framed as a transport-centric problem, and for good reason: it is fundamentally non-equilibrium diffusion governed by irradiation-generated point defects and their coupling to solute fluxes, which drive the redistribution of alloying elements toward or away from microstructural defects. Yet, this emphasis on transport aspects has created an asymmetry: The thermodynamic counterpart of the problem, namely the local free-energy landscape of the receiving defect, remains under-formalized: In most RIS frameworks, the governing description is closed through coupled transport equations for solutes and irradiation-generated point defects, augmented by source, recombination, and sink terms (see for instance (Nastar and Soisson, 2012; Ardell and Bellon, 2016) and references therein). Accordingly, the microstructural defects typically enters through sink strength, capture-efficiency, geometry, or boundary conditions, whereas their thermodynamic identity is represented in simplified form. Even where segregation energetics are included, they are often projected onto bulk thermodynamic factors or effective interfacial properties (Millet et al., 2011; Piochaud et al., 2016), rather than being based on a sink-specific free-energy landscape. In this Opinion Paper, I argue that a more complete understanding of RIS can emerge when sink-specific thermodynamic landscapes and their interaction with non-equilibrium fluxes are taken explicitly into account.

Recent RIS models are aware of the assumptions of perfect-sink behavior or homogeneous chemical potentials and aim to relax them. For instance, Schuler et al. (Schuler et al., 2022) emphasized the need to move beyond static sink idealizations and to describe the coupled interplay between segregation kinetics and sink evolution itself using atomistic simulations. More generally, multiscale RIS frameworks such as those of Thuinet et al. (Thuinet et al., 2018) and Huang et al. (Huang, 2020) have already combined atomistic transport information with continuum-level segregation modeling, thereby moving beyond purely phenomenological sink treatments. Likewise, phase-field investigations such as by Moladje et al. (Moladje et al., 2022) on RIS near dislocations and low-angle grain boundaries have shown that biased sinks require an explicit treatment of local defect character rather than a neutral-sink approximation. Experimental studies have also reinforced this view point, for example by showing that grain boundary plane and character strongly influence the resulting segregation and precipitation behavior under irradiation (Barres et al., 2022; Wonner and Bellon, 2025). Further recent developments have also broadened the scope of RIS modelling and validation, including two- and three-dimensional treatments of sink geometry and morphology (Nahavandian et al., 2026), nonlinear coupling between neighbouring nanoscale sinks (Xu et al., 2025), machine-learning-assisted parameterization of defect-mediated diffusion kinetics (Huang et al., 2023) and correlative APT/STEM-EDS validation of RIS profiles (Lach et al., 2021). These advances suggest that an important next step is not only to refine transport coefficients, but also to develop thermodynamic representations of sinks as structurally and chemically differentiated states.

A systematic step forward would be to connect RIS frameworks with ongoing advances in the thermodynamic and kinetic theory of defects. Recent advances in developing density-based thermodynamic and phase-field frameworks (DPF) are particularly promising. The DPF theory offers a route to view defects as local thermodynamic states with their own structural, chemical, and energetic degrees of freedom, describing them through a density parameter ρ ≤ 1 that represents their local structural state, while when ρ = 1 the formalism restores the thermodynamic properties of the parent bulk phase (Darvishi Kamachali, 2020). The significance of this idea is far-reaching: defects and bulk phases are no longer treated as conceptually separate objects, but as distinguishable yet continuously connected states within a single free-energy landscape. This provides a natural thermodynamic language for describing defect stability, segregation, and defect-phase transitions in direct continuity with the surrounding bulk material.

Here, DPF can be understood as a strong addition to existing theoretical and multiscale approaches: On the one hand, classical rate-theory and sink-strength models are central for describing defect production, recombination, migration, and biased fluxes under irradiation (Huang et al., 2021; Hu et al., 2021; Huang et al., 2025). On the other hand, atomistic simulations, kinetic Monte Carlo, and cluster-dynamics approaches provide essential information on defect–solute binding, transport pathways, cascade debris, and the evolution of defect clusters (Xiong et al., 2024; Fan, 2025). Within this broad landscape, the DPF framework offers a thermodynamically adaptable and compatible route for representing defect states and sinks. DPF can also be directly integrated with phase-field models, which have become increasingly important for connecting irradiation-modified thermodynamic driving forces to precipitation, phase stability, and mesoscale microstructure evolution (Millet et al., 2011; Li et al., 2017). With these aspects, DPF opens a comprehensive route for integrating defect thermodynamics with existing (bulk) CALPHAD databases, see for instance (Wang and Kamachali, 2023; Marschall et al., 2025).

Recent detailed investigations have shown that the density parameter can be derived from atomistic simulations, thus establishing a streamlined bridge between atomistic calculations and continuum thermodynamic and phase-field modelling (Jacobson et al., 2024; Wallis et al., 2025; Wallis and Kamachali, 2025). In these ways, DPF not only extends thermodynamic descriptions to defects in RIS frameworks, but does so in a form that is naturally compatible with both available bulk databases and atomistic input.

There are two profound ways in which DPF can enrich RIS studies. The first is by remapping the phase stability under irradiation, that is, through irradiation-conditioned phase diagrams. The second is by embedding DPF’s defect-sensitive thermodynamics directly into transport and phase-evolution frameworks. These capacities are briefly discussed in the following.

Irradiation-conditioned phase diagrams

The concepts and methods for computing density-based defect phase diagrams have advanced substantially in recent years (Darvishi Kamachali, 2020; Wang and Kamachali, 2023; Marschall et al., 2025; Wang and Kamachali, 2021a; Murugan and Darvishi Kamachali, 2025). At its core, this framework begins with a density-based free-energy functional. For a grain boundary embedded in the microstructure of an N-component alloy, the free-energy functional can be written aswhere the first (volume) integral is over the density-based Gibbs free energy density:with

G = ∫ V G ρ , X , T + e e l ϵ , ρ , X , T d V + ∫ A G ρ G B ρ G B , X , T d A (1)

G = ∑ i N X i G i ρ , T + ρ 2 Δ H mix B X , T − T Δ S mix B X , T + ∑ i N − 1 κ X i 2 ∇ X i 2 (2)

G i ρ , T = ρ 2 E i B T + ρ G i B T − E i B T + κ i ρ 2 ∇ ρ 2 ⋅ (3)

The key parameter here is the atomic density which acts both as a field variable ρ ( r , t ) in the simulations and as a mean-field descriptor ρ ∈ [ ρ G B , 1 ] in the free-energy formulation. Note that density values larger than unity, corresponding for instance to locally compressed regions or interstitial-rich configurations, can be handled by modest modifications of the density-dependent terms in the free-energy formalism. This extension is very naturally allowed, since the bulk atomic density associated with ρ = 1 can be identified with the minimum-energy reference state obtained from a conventional interatomic potential, see also the latest analyses (Wallis et al., 2025; Wallis and Kamachali, 2025). ρ G B ( s , t ) is the in-plane grain boundary density. The elastic energy density e e l = e hom e l + e het e l is readily formalized for multi-component alloys (Kamachali and Wang, 2022; Wang and Kamachali, 2021b) with the homogeneous elastic energy density aswhere λ i h ( ρ ) are the lattice strain coefficients due to solute i (or j) in the host h lattice, q i ( ρ ) and γ i are elastic constants, defined as a function of local atomic density, X i ∈ X and X h + ∑ j N − 1 X j = 1. The heterogeneous elastic energy must be computed using a proper mechanical solver. The second (plane) integral in Equation 1 accounts for the interfacial contribution of the grain boundary, e.g. as:where A is a potential coefficient, κ ρ G B is an in-plane gradient coefficient, and ρ G B is the characteristic grain boundary density, all of which relate to the underlying grain-boundary properties (e.g., atomic planes, misorientation) and can be derived from atomistic simulations (Wallis et al., 2025; Wallis and Kamachali, 2025; Wallis and Kamachali, 2023). The significance of the in-plane treatment lies in the additional degrees of freedom it provides for describing quasi-two-dimensional phases, such as junctions at grain boundaries, and the associated enhancement of kinetics at these defects.

e hom e l = ∑ i N − 1 q i ρ γ h γ h − 1 γ i γ i − 1 λ i h ρ 2 X i − ∑ i j N − 1 q i ρ q j ρ λ i h ρ λ j h ρ X i X j (4)

G ρ G B = A ρ G B − ρ 1 2 ρ G B − ρ 2 2 + κ ρ G B 2 ∇ ρ G B 2 (5)

Equations 1–5 are fed by CALPHAD thermodynamic data of the corresponding alloy system and a minimum set of data concerning the corresponding grain boundary: G i B ( T ) the free energy of the homogeneous bulk substance i, κ i ρ the atomic density gradient coefficient, κ X i the concentration gradient energy coefficients and the mixing enthalpy and entropy Δ H mix B ( X , T ) and Δ S mix B ( X , T ), respectively—the superscript B indicates the bulk properties extractable from a CALPHAD database.

If a steady irradiation condition is viewed, at a coarse-grained thermodynamic level, as maintaining a supersaturated population of point defects, then it can be connected to density-based phase diagrams after two intermediate linking: (i) the well-established connection between irradiation intensity and the resulting excess concentration of point defects and (ii) the recent demonstration that point defects can be effectively described based on an atomic density field through both a reduction in coordination number and a change in excess free volume (Kamachali and Wallis, 2025). Once these two links are established, CALPHAD-integrated density-based free-energy functionals can be constructed and minimized to compute irradiation-conditioned phase diagrams. Similar phase diagrams have already been constructed for various purposes in the study of defect-dominated phenomena, such as catastrophic failure. The irradiation-conditioned phase diagrams directly serve as concrete tools for predicting phase changes and defect-state-dependent phase stability associated with irradiation. Here the possibility of segregation transitions and defect-state transitions become also important and broaden the spectrum of possible irradiation-induced phase changes.

It should be noted that these irradiation-conditioned phase diagrams differ fundamentally from ‘radiation-modified phase diagrams’ that are constructed essentially in terms of radiation-modified solubility, namely the solute concentration at which the growth rate of an irradiated precipitate vanishes (Turkin et al., 2002). The density-based phase diagrams have a free energy foundation and enable studying the conditions of phase stability, including the nucleation and dissolution of various phases, directly not only in the supersaturated bulk, but also at defects acting as sinks. For instance, it has been shown that the miscibility gap in Fe-Mn-X alloy systems can dramatically change for a reduced atomic density (Wang and Kamachali, 2021a). This means that even modest defect-induced reductions in atomic density can shift the local thermodynamic driving forces and phase equilibria. Such reduction in atomic density can be systematically mapped to specific defects or even super-saturated bulk substance. This opens a wide and still largely unexplored space for studies across defect types, alloy systems, and irradiation conditions, with substantial room for sustained methodological and application-driven developments.

Defect-sensitive transport and phase evolution

In the Onsager framework, the quantities entering the RIS equations are not independent objects, but descend from the underlying thermodynamic structure of the system. Once the free-energy landscape is allowed to depend explicitly on the local defect state, the transport problem can be reformulated on a defect-sensitive thermodynamic basis. The central implication is that both the driving forces and the kinetic couplings become functions of the local defect state.

In this spirit, the generalized diffusional potentials of the alloying species follow from the variational derivatives of the density-based defect free-energy functional Equation 6,with analogous in-plane driving forces on the grain boundary manifold obtained from Equation 7

μ i r , t = δ G δ X i r , t , μ ρ r , t = δ G δ ρ r , t , (6)

μ i G B s , t = δ G δ X i G B s , t , μ ρ G B s , t = δ G δ ρ G B s , t . (7)

The fluxes may then be expressed in Onsager form as linear responses to gradients in these generalized potentials Equations 8, 9 and, correspondingly, within the grain boundary plane Equations 10, 11 where the Onsager coefficients themselves may depend on the local atomic density and hence on the local defect state. Here L ρ j captures the coupling between solute transport and local defect-state evolution, and R ρ represents source/sink terms associated with irradiation-induced defect production and annihilation. Note that the vacancy concentration and the associated vacancy flux, as well as the self-interstitial concentration and flux, can be consistently expressed in terms of the atomic density field. In this sense, both vacancy and self-interstitial concentrations, together with their associated fluxes, may be related to the atomic density field, while still being retained as distinct transport variables when required, allowing interstitial-mediated RIS mechanisms to be incorporated.

J i = − ∑ j N − 1 L i j ρ , X , T ∇ μ j − L i ρ ρ , X , T ∇ μ ρ , (8)

∂ ρ ∂ t = ∇ ⋅ ∑ j N − 1 L ρ j ρ , X , T ∇ μ j + L ρ ρ ρ , X , T ∇ μ ρ + R ρ , (9)

J i G B = − ∑ j N − 1 L i j G B ρ G B , X , T ∇ G B μ j G B − L i ρ G B ρ G B , X , T ∇ G B μ ρ G B , (10)

∂ ρ G B ∂ t = ∇ G B ⋅ ∑ j = 1 N − 1 L ρ j G B ρ G B , X , T ∇ G B μ j G B + L ρ ρ G B ρ G B , X , T ∇ G B μ ρ G B ) + R ρ G B , (11)

This formulation suggests a natural extension of classical RIS theory: segregation, counter-segregation, and coupled solute–defect transport need not be described through fixed sink parameters alone, but can be treated as consequences of a defect-dependent free-energy landscape and its associated mobilities. In such a formulation, the characteristic defect density field ρ, or on the interface ρ G B, acts as a thermodynamic state variable that modifies both the driving forces and the kinetic couplings. From a practical viewpoint, the parameterization of density-dependent Onsager coefficients, mobilities, and defect-state free-energy contributions can be obtained from atomistic simulations, kinetic Monte Carlo calculations, rate-theory estimates, or fitted against spatially resolved irradiation experiments. As a starting point, available knowledge on the diffusion and related phenomena along and across defects can be used to approximate the density dependence of the kinetic coefficients.

An interesting application of the DPF model for RIS studies is reported by Kadambi et al. (Kadambi et al., 2025), where the atomic density field is explicitly coupled to CALPHAD-informed free-energy descriptions to capture segregation and phase stability in the vicinity of defects, with particular emphasis on maintaining thermodynamic consistency across bulk and defect regions in ternary Fe–Cr–Ni alloys. A more extensive and systematically developed effort in this direction has been carried out by Calbert et al. (Calbert et al., 2026), where the DPF concept, together with a density-dependent Gibbs free-energy functional directly linked to CALPHAD-accessible quantities, is embedded into a classical phase-field framework to describe grain boundaries and radiation-induced segregation in Fe–Cr alloys. The density at the grain boundary, ρ G B, is treated as a governing parameter and systematically varied, demonstrating how the grain-boundary state directly modifies segregation profiles and sink efficiencies.

A key aspect of the elastic description in the work of Calbert et al. lies in making the defect–elastic interaction explicitly dependent on the local atomic density field. Rather than prescribing a constant eigenstrain at the grain boundary, the eigenstrain of each defect species is derived as a continuous function of the local density, leading towhich directly implies a density-dependent relaxation volume for solutes, vacancies, and interstitials. This formulation offers a particularly promising basis for integration with recent density-based elasticity descriptions (Marschall et al., 2025; Wang and Kamachali, 2021b), where elastic energy contributions are consistently formulated within a density-dependent thermodynamic framework.

ε i j 0 , p r = ρ r ε i j 0 , p ρ = 1 + δ i j 3 − δ i j 3 , (12)

In terms of experimental validations, while classical RIS descriptions primarily predict solute enrichment or depletion profiles resulting from defect fluxes toward idealized sinks, a thermodynamic defect-state framework generates additional measurable predictions. These include possible transitions between distinct chemical or structural states of the sink, sink-specific excess compositions, defect-state-dependent segregation tendencies and variations in segregation behavior with grain-boundary character or dislocation structure. Such predictions can be tested using correlative APT and STEM-EDS/EELS measurements, together with TEM-based characterization of sink type, sink density, dislocation structure, cavity morphology, and grain-boundary character. A particularly useful validation strategy would be to compare chemically resolved segregation profiles, or phase formations, at different sinks exposed to the same irradiation condition. If two sinks experience comparable irradiation fluxes but display different segregation levels, excess compositions, or precipitation tendencies due to their structural character, this would, for instance, suggest the necessity for adopting thermodynamic-aware sink-strength parameters.

To summarize, the significant progress of RIS modeling has brought the field to a point where the next missing layer is now coming into view: Classical rate-theory, sink-strength, and transport-centered descriptions have provided a powerful foundation, yet recent developments point to the need for a fully resolved thermodynamic representation of defects as active states of matter. The view of defects as passive sinks characterized by fixed parameters can and should be replaced by a capable and dynamic framework that captures their thermodynamic identity. A consistent formulation requires that defects enter the theory not only through transport coefficients, but as explicit thermodynamic states with their own free-energy landscapes and evolution equations. In this perspective, the DPF concept and, in particular, the atomic density field emerge as natural state variables that bridge bulk and defect thermodynamics and couple directly to transport processes. Embedding such a description into RIS frameworks offers major advantages, not only for analysis but also for design, first through the possibilities enabled by CALPHAD-integrated irradiation-conditioned phase diagrams. Even closer in reach, an Onsager-consistent density-based framework provides a pathway toward a unified description of defect chemistry, transport, and phase stability under irradiation. Developing this connection systematically is, in the author’s view, a necessary step for advancing RIS modeling beyond its current transport-centric paradigm and toward solution-seeking, preventative, and ultimately design-driven strategies for materials under irradiation conditions.

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Footnotes

Edited by: Sourabh Bhagwan Kadambi, Idaho National Laboratory (DOE), United States

Reviewed by: Daniel Vizoso, Georgia Institute of Technology, United States