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arxiv: 2607.00203 · v1 · pith:RXV535SBnew · submitted 2026-06-30 · ❄️ cond-mat.mtrl-sci

Dominant-pair free energies predict phase selection in high-entropy alloys

Dennis Boakye , Chuang Deng This is my paper

Pith reviewed 2026-07-02 18:31 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords high-entropy alloysphase selectionB2 orderingthermodynamic modelingdominant-pair mechanismBragg-Williams free energyphase stability mapsmacroscopic atom model
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The pith

A dominant-pair mechanism reduces multicomponent B2 ordering in alloys to an effective pseudo-binary system whose Bragg-Williams free energy predicts the stable phase versus composition and temperature.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a thermodynamic framework that starts from the macroscopic atom model and Dinsdale lattice stabilities. It shows that one interaction family, the Al-transition-metal pair, dominates the ordering enthalpy and thereby converts the full multicomponent problem into a simpler pseudo-binary calculation. The resulting minimum-free-energy classifier produces continuous phase-stability maps rather than single scalar thresholds. On a set of 269 experimentally characterized high-entropy alloy samples the model reaches 77.9 percent accuracy on a three-class task and exceeds the valence-electron-concentration criterion in macro-F1 score. The construction is general and runs quickly across wide ranges of composition and temperature.

Core claim

Phase selection arises from competition between configurational entropy and ordering enthalpy; when the dominant-pair mechanism is invoked, the multicomponent B2-ordering enthalpy collapses to an analytically solvable pseudo-binary Bragg-Williams free energy whose minimum, evaluated at each composition and temperature, directly identifies the lowest-energy phase.

What carries the argument

The dominant-pair mechanism, which isolates the Al-transition-metal interaction family to reduce the multicomponent B2-ordering problem to an effective pseudo-binary system evaluated with an analytic Bragg-Williams free energy.

If this is right

  • Continuous phase-stability maps are generated directly as functions of composition and temperature.
  • The three-class classification accuracy reaches 77.9 percent on 269 experimental samples and exceeds the valence-electron-concentration criterion.
  • The same construction applies without modification to any multicomponent alloy whose ordering is governed by a single dominant pair interaction.
  • Computation remains fast enough to scan broad composition-temperature space without iterative numerical solution of the full multicomponent problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction could be tested on ordering types other than B2 if an analogous dominant pair can be identified.
  • Integration with existing CALPHAD databases would allow the free-energy surfaces to be used as starting points for more detailed kinetic modeling.
  • If the dominant-pair premise holds more generally, similar simplifications may shorten the search for stable phases in other chemically complex materials.

Load-bearing premise

The assumption that the Al-transition-metal interaction family supplies the dominant contribution to the ordering enthalpy across the alloys considered.

What would settle it

An experimental high-entropy alloy composition and temperature at which the phase predicted by the minimum-free-energy classifier is not the observed lowest-energy phase.

Figures

Figures reproduced from arXiv: 2607.00203 by Chuang Deng, Dennis Boakye.

Figure 1
Figure 1. Figure 1: Surface-concentration-corrected macroscopic-atom-model equimolar binary en [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Discrimination of the experimental phase classes by (a) the valence-electron [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The dominant-pair mechanism for AlxCoCrFeNi. (a) The Al–TM pairs con￾tribute more than eighty percent of the chemical mixing enthalpy above approximately ten atomic percent Al, justifying the pseudo-binary reduction. (b) The resulting dominant￾pair B2 ordering free energy falls increasingly below the disordered BCC solid solution as Al is added. The dominance evident in [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 4
Figure 4. Figure 4: Equimolar pairwise mixing enthalpies for the fourteen principal elements. Al– [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Validation of the dominant-pair ordering temperature of Equation (11). (a) The [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Free-energy competition along the AlxCoCrFeNi trajectory at 1273 K. The pre￾dicted phase sequence FCC → duplex → BCC/B2 arises from the crossing of the candidate free-energy curves, the dominant-pair B2 phase descending below the solid solutions as Al increases. For the 3d system AlxCoCrFeNi (panel a), a single-phase FCC field at low Al gives way to a broad duplex band and then to a B2 and intermetallic fi… view at source ↗
Figure 7
Figure 7. Figure 7: Predicted temperature–composition phase maps for (a) the [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Model-predicted phase map of the 3d+Al composition space. Filled points are several thousand sampled compositions colored by predicted phase; open rings are the experimentally characterized alloys at their reported labels. The predicted phases occupy coherent regions, and the experimental alloys fall predominantly within them. 4.6. Four-class benchmark against the parametric criteria The principal comparis… view at source ↗
Figure 9
Figure 9. Figure 9: Four-class phase-prediction performance of the free-energy classifier and the five [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Four-class confusion matrices for the free-energy classifier and two representa [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Per-class (a) precision and (b) recall for the free-energy classifier and two rep [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The phase-label resolution ceiling. A subset of the alloys labeled intermetallic [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Three-class phase prediction (FCC/BCC/FCC [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
read the original abstract

Phase selection in multicomponent alloys is governed by the competition between entropic stabilization of disordered solutions and enthalpic driving forces for chemical ordering. However, widely used parametric criteria reduce it to a single scalar, carrying no explicit free energy for any competing ordered phase. Herein, we develop a thermodynamic framework based on the semi-empirical macroscopic atom model and the Dinsdale lattice stability database to fill this gap. We show that a dominant-pair mechanism, in which the Al-transition-metal interaction family dominates the ordering enthalpy, enables the complex multicomponent B2-ordering problem to be reduced to an effective pseudo-binary system with an analytically evaluated Bragg-Williams free energy. Combined with a minimum-free-energy classifier, the framework predicts the lowest-energy phase as a function of composition and temperature. This provides continuous phase stability maps rather than the single-value predictions of conventional descriptors. Demonstrated on high-entropy alloys using a dataset of 269 experimentally characterized samples, the model outperforms widely used phase-selection criteria in the class-balanced macro-F1 metric and achieves 77.9% on the well-posed three-class task, outperforming the valence electron concentration criterion. The model is general by construction and computationally efficient for predicting phase stability in multicomponent alloys over a broad range of compositions and temperatures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a thermodynamic framework using the semi-empirical macroscopic atom model and Dinsdale database to predict phase selection in high-entropy alloys. It introduces a dominant-pair mechanism in which the Al-transition-metal interaction family is assumed to dominate ordering enthalpy, reducing the multicomponent B2-ordering problem to an effective pseudo-binary system whose free energy is evaluated analytically via the Bragg-Williams approximation. Combined with a minimum-free-energy classifier, the model generates continuous composition-temperature phase stability maps and is reported to achieve 77.9% accuracy on a three-class task using 269 experimentally characterized samples, outperforming the valence electron concentration criterion in class-balanced macro-F1.

Significance. If the dominant-pair reduction is shown to hold across the dataset, the framework supplies an explicit, temperature-dependent free-energy surface rather than a single scalar descriptor, enabling continuous stability maps that are computationally efficient and general by construction. The use of an established semi-empirical model with analytic evaluation is a strength when the central assumption is validated.

major comments (2)
  1. [Abstract] Abstract and the description of the dominant-pair mechanism: the claim that the Al-transition-metal interaction family dominates the ordering enthalpy (enabling the pseudo-binary reduction) is load-bearing for the minimum-free-energy classifier and continuous maps, yet the manuscript supplies no explicit validation such as a ranking of all pair-interaction enthalpies from the macroscopic atom model for the experimentally labeled B2 samples to confirm the chosen pair is maximal in magnitude.
  2. [Abstract] Abstract: the reported 77.9% accuracy on the 269-sample set is presented without information on data splits, cross-validation procedure, error bars, or whether any samples overlap with those used to fit the underlying pair-interaction enthalpies; this information is required to assess whether the performance claim is independent of the semi-empirical parameters.
minor comments (1)
  1. The manuscript would benefit from a table or figure explicitly comparing the magnitudes of the dominant Al-TM pair enthalpy against other TM-TM and Al-Al pairs for representative compositions in the dataset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our thermodynamic framework. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the description of the dominant-pair mechanism: the claim that the Al-transition-metal interaction family dominates the ordering enthalpy (enabling the pseudo-binary reduction) is load-bearing for the minimum-free-energy classifier and continuous maps, yet the manuscript supplies no explicit validation such as a ranking of all pair-interaction enthalpies from the macroscopic atom model for the experimentally labeled B2 samples to confirm the chosen pair is maximal in magnitude.

    Authors: We agree that an explicit ranking of pair-interaction enthalpies for the B2 samples would provide direct support for the dominant-pair assumption. The assumption is motivated by the well-documented strength of Al-TM interactions in the macroscopic atom model, but we acknowledge the manuscript does not include a dataset-specific validation. In the revised manuscript we will add a supplementary table (or figure) that ranks the relevant pair enthalpies computed from the model for the experimentally labeled B2 samples, confirming that the Al-TM family is maximal. This addition will be referenced in the abstract and methods. revision: yes

  2. Referee: [Abstract] Abstract: the reported 77.9% accuracy on the 269-sample set is presented without information on data splits, cross-validation procedure, error bars, or whether any samples overlap with those used to fit the underlying pair-interaction enthalpies; this information is required to assess whether the performance claim is independent of the semi-empirical parameters.

    Authors: The 77.9% figure is the direct accuracy of the analytic thermodynamic model evaluated on the full set of 269 experimental samples; no machine-learning training or parameter fitting to this dataset occurs. The pair-interaction enthalpies are taken unchanged from the established macroscopic atom model and Dinsdale database. We will revise the abstract, methods, and results sections to state explicitly that (i) there is no overlap with any fitting procedure, (ii) data splits and cross-validation are not applicable because the model is deterministic and physics-based rather than statistical, and (iii) the reported class-balanced macro-F1 already provides a robustness metric. Error bars in the statistical sense are not generated by the model, but we can note the consistency across the three-class task. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation uses an external semi-empirical macroscopic atom model plus Dinsdale database to obtain pair enthalpies, applies the stated dominant-pair reduction to form a pseudo-binary, evaluates an analytic Bragg-Williams free energy, and feeds the result into a minimum-free-energy classifier. Phase predictions are then tested against an independent experimental dataset of 269 samples (77.9 % accuracy on the three-class task). No equation or step in the provided text reduces the output to the input quantities by construction, and no self-citation chain is invoked to justify the central premise. The framework therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; ledger entries are inferred from stated modeling choices. The macroscopic atom model and Dinsdale database supply the interaction parameters and lattice stabilities; the dominant-pair assumption is an additional modeling choice.

free parameters (1)
  • pair-interaction enthalpies
    Taken from the semi-empirical macroscopic atom model; these are pre-fitted quantities used directly in the free-energy expression.
axioms (2)
  • domain assumption Bragg-Williams mean-field approximation is adequate for the effective pseudo-binary free energy
    Invoked to obtain an analytically evaluated free energy for the ordered phase.
  • ad hoc to paper Al-transition-metal pair dominates ordering enthalpy in the alloys considered
    Central modeling choice that permits reduction to pseudo-binary system.

pith-pipeline@v0.9.1-grok · 5755 in / 1525 out tokens · 30342 ms · 2026-07-02T18:31:21.403860+00:00 · methodology

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