Machine-learning approach for the phase stability and mechanical properties of disordered alloys at finite temperature

Rutchapon Hunkao; Stefano Sanvito; Urvesh Patil

REVIEW 3 major objections 5 minor 81 references

Machine-learning approach for the phase stability and mechanical properties of disordered alloys at finite temperature

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Fcc Ni-Pd is thermodynamically stable across the full composition range only above a critical temperature between 500 K and 700 K, and metastable at room temperature.

desk verdict A useful high-throughput workflow for finite-T alloy thermodynamics whose headline Ni-Pd stability onset is underbounded by omission of magnetic free energy and finite-size entropy. read the letter →

arxiv 2608.10465 v1 pith:36WRAYEI submitted 2026-08-11 cond-mat.mtrl-sci

Rutchapon Hunkao , Urvesh Patil , Stefano Sanvito This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords machinelearningpotentialsdisorderedalloysphasestabilityfinite-temperaturethermodynamicsNi-Pdvibrationalfreeenergyconfigurationalentropyelasticconstants

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a single compact machine-learned potential can settle a question normally reserved for much heavier computation: whether a disordered alloy is thermodynamically stable at a given temperature. Using Ni-Pd as a test case, it claims the fcc solid solution is metastable at room temperature and becomes stable only above roughly 600 K, with Pd-rich compositions stabilizing first. If true, finite-temperature phase stability does not have to be guessed from ideal-mixing entropy; the partition function can be built by explicitly enumerating every symmetry-distinct configuration, each carrying its own harmonic vibrations, at a cost low enough to be practical. The payoff would be a high-throughput route to phase diagrams and mechanical properties of multicomponent alloys, including high-entropy alloys.

What carries the argument

The load-bearing object is the Jacobi-Legendre potential (JLP), a cluster expansion truncated at three-body order in which pair terms use Jacobi polynomials and angular terms use Legendre polynomials; the model is linear in its coefficients, so training is a regression and inference is very fast. The thermodynamic machinery is the coarse-grained partition function $Z_\alpha(\{x_i\},T)=\sum_\sigma \exp[-(E_\alpha(\sigma)+F^{\rm vib}_\alpha(\sigma,T))/k_B T]$, which replaces ideal-mixing entropy with an explicit sum over all symmetry-distinct configurations, each weighted by its own harmonic vibrational free energy. The 24-atom enumeration supplies the exact configurational entropy of that finite system, up to about 10% below the ideal-mixing limit at equiatomic composition; systematic comparisons at fixed supercell size keep composition trends meaningful.

What would settle it

Measure the heat capacity or the order-disorder transition of Ni-Pd alloys across the composition range by calorimetry or in situ diffraction: the predicted 500-700 K stability onset should appear as a composition-dependent feature, with Ni-rich compositions losing stability at higher temperatures than Pd-rich ones. A more direct computational falsifier is to repeat the free-energy sum in 32-atom or larger supercells, or with quasiharmonic volume relaxation and magnetic contributions, and check whether the room-temperature metastability and the roughly 600 K stabilization survive; the 24-atom cell's entropy deficit of about 10% corresponds to an energy scale of tens of meV/atom.

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Extended reading notes

Core claim

The central discovery is that fcc Ni-Pd, while fully miscible when quenched from the melt, is thermodynamically metastable at room temperature and becomes stable only above a critical temperature between 500 K and 700 K, with stability appearing first on the Pd-rich side. This is established by training a Jacobi-Legendre polynomial cluster-expansion potential with 873 parameters on density-functional-theory data from 8-atom cells, then using it to relax all 99,268 symmetry-distinct configurations of a 24-atom supercell and adding each configuration's harmonic vibrational free energy to form the partition function. The resulting Gibbs free energy of formation matches DFT where the comparison is made, and the explicit enumeration gives the exact configurational entropy for the finite supercell instead of the ideal-mixing approximation. The same potential also reproduces relaxed geometries and gives elastic constants within about 10% of DFT, while entropy and heat-capacity curves reveal competition between the solid solution and long-period L1$_0$ ordered forms of NiPd and NiPd$_3$.

Load-bearing premise

The load-bearing bet is that a tiny 24-atom sample of alloy, vibrating harmonically at its zero-temperature size and ignoring magnetism and volume changes, faithfully represents what the real, macroscopic alloy does; if those ignored effects shift the energy balance by more than a few tens of meV per atom, the predicted 500-700 K stability window could move or disappear.

Editorial extensions

If this is right

Above roughly 600 K, quenching Ni-Pd from the melt should produce a thermodynamically stable fcc solid solution across the whole composition range, while at room temperature the solution is metastable with respect to ordered intermetallic phases.
Because the stability onset is composition-dependent, Pd-rich alloys should remain stable at lower temperatures than Ni-rich ones; experiments probing phase stability as a function of composition can test this asymmetry directly.
Vibrations shift the order-disorder transition down: the heat-capacity peak moves from about 700 K to 500 K at 75% Pd and from about 350 K to 250 K at 50% Pd, so ignoring vibrational free energy would overestimate transition temperatures by up to roughly 30%.
The configurational entropy of a 24-atom cell falls about 10% short of the ideal-mixing value at equiatomic composition, so the predicted 500-700 K stability window is likely to shift slightly when larger supercells are used.
Configuration-averaged elastic constants change sharply near the configurational transition temperatures, coupling mechanical response to the same entropy-driven stability physics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same explicit-enumeration pipeline could resolve whether other 'fully miscible' binaries are in fact metastable at room temperature; the Ni-Pd result suggests that many experimentally known solid solutions may be entropy-stabilized rather than ground-state alloys.
Editorial inference: the claim that stability begins at the Pd-rich end is testable by calorimetry or diffraction across several compositions; one should see the first loss of thermodynamic stability on cooling at the Ni-rich side.
Editorial inference: scaling the model to ternary and quaternary alloys multiplies the number of clusters by about 2.7 and 5.6, but the real bottleneck is the combinatorial growth of distinct supercell configurations, so subcell sampling or active learning will be needed before the method reaches high-entropy alloys.
Editorial inference: the paper's fixed-volume harmonic treatment leaves out magnetic and anharmonic free energies; for a magnetic element like Ni those terms could plausibly move the 600 K onset, and comparing the predicted heat-capacity peaks with measured calorimetry would settle the size of the correction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know. First, this is a solid workflow paper: a compact linear Jacobi-Legendre potential, trained on 8-atom cells, generalizes cleanly to 16-atom cells (~1.9 meV/atom energy RMSE, 1.6 meV/atom after relaxation, 0.64–3.14 meV/atom on vibrational free energies) and is then pushed through 99,268 relaxations and 14 million force evaluations in 24-atom supercells. The throughput claim is real, and the per-configuration phonon treatment is a genuine step beyond SQS-plus-ideal-entropy. Second, the headline quantitative claim—stability onset between 500 and 700 K, room-temperature metastability—is not as robust as the framing suggests. The free energy omits finite-temperature magnetic contributions in a ferromagnetic system whose Ni end has TC ~631 K, inside the claimed window, and the 24-atom cell's configurational entropy is still ~10% below ideal mixing. Both omissions shift ΔG by a few to tens of meV/atom in the relevant temperature range, and neither is bounded by the validation, because the DFT reference also omits the finite-T magnetic term.

The new combination is exact enumeration of all symmetry-distinct configurations at finite T with per-configuration harmonic vibrations from an interpretable, linear potential. The JLP itself was published earlier, so the contribution is the workflow and the Ni-Pd prototype result. The paper is honest about many limitations: it discusses the fixed 0 K volume, tests QHA on a representative configuration, and openly reports the noisy shear elastic constants. The qualitative conclusion, complete miscibility with stability first at the Pd-rich end, is plausible and consistent with the known phase diagram.

The main soft spot is magnetism. Section III E explicitly defers electronic/magnetic terms to "where necessary," but for Ni-Pd it is necessary to bound them. Spin-polarized DFT contains the ground-state magnetic energy, so the omission does not cancel in formation free energies. An approximate F_mag(T,x), even from a mean-field or effective-Hamiltonian estimate, would materially strengthen the central claim. Second, the 24-atom finite-size entropy shortfall is acknowledged but dismissed as minor; at 700 K the shortfall relative to ideal mixing is worth several meV/atom, so the dismissal is too quick. A simple extrapolation of configurational entropy to the thermodynamic limit would help. Third, no code, data, or trained model is released, which limits reproducibility and reuse.

Overall, the paper deserves a serious referee. The workflow is valuable, the validation is honest, and the central claim is plausible but not fully established. I would send it to review and ask for the magnetic estimate, an entropy extrapolation, and a data release.

Referee Report

3 major / 5 minor

Summary. The paper presents a complete machine-learning workflow for finite-temperature phase-stability and mechanical-property prediction in disordered alloys, using Ni-Pd as the prototype. A Jacobi-Legendre potential (JLP) with 873 linear parameters is fitted to 1,059 DFT calculations on 8-atom cells and strained elemental cells, then validated on 673 held-out 16-atom supercells, achieving energy RMSEs of about 1.9 meV/atom (unrelaxed) and 1.6 meV/atom (relaxed), and vibrational free-energy RMSEs from 0.64 meV/atom at 300 K to 3.14 meV/atom at 1500 K. The authors then enumerate all 99,268 symmetry-inequivalent configurations in 24-atom cells, relax each with the JLP, compute harmonic phonon free energies, and evaluate the canonical partition function with configurational plus vibrational contributions. From this they predict that the fcc solid solution is thermodynamically stable across the full composition range above a critical temperature between 500 K and 700 K, is metastable at room temperature, and becomes stable first at the Pd-rich end. They also compute heat capacities and thermally averaged elastic constants, and propose a scaling route to high-entropy alloys.

Significance. If the quantitative thermodynamic predictions are reliable, this is a significant advance: an 873-parameter linear JLP trained on small cells is shown to reproduce DFT-level energies, relaxed structures, and vibrational free energies on larger held-out cells, and it enables explicit enumeration of the configurational ensemble without invoking ideal-mixing entropy or a single SQS representative. The validation is solid and unusually complete for a machine-learning-potential paper: independent 16-atom test sets, structural parity checks, symmetry comparison after relaxation, and a large DFT phonon reference set. The finite-temperature stability prediction is concrete and falsifiable, and the throughput numbers (99,268 relaxations, 14.3 million force evaluations in 12 hours) make the proposed workflow attractive for multicomponent alloys. However, the central stability claim rests on approximations whose magnitudes are not bounded within the manuscript, most notably the neglect of finite-temperature magnetic free energy in a system with Ni-rich ferromagnetism.

major comments (3)

[Section III E, Eqs. (15)-(17), Fig. 11] The central finite-temperature stability claim neglects the magnetic free energy. Equations (15)-(17) include only configurational and harmonic vibrational contributions, and the text states that electronic and magnetic terms can be added 'where necessary'; for Ni-Pd they are necessary. Pure Ni has a Curie temperature around 631 K, inside the claimed 500-700 K stabilization window, and the magnetism of Ni_xPd_{1-x} is composition-dependent. The finite-temperature magnetic free energy of the alloy relative to the fcc elemental references is therefore T- and x-dependent and can plausibly be of order a few to tens of meV/atom in this window, which is comparable to the free-energy differences controlling the ΔG zero-crossings in Fig. 11. The vibrational free-energy validation in Fig. 10(d) does not bound this term, because the DFT phonon reference also omits the finite-T magnetic contribution; the ground-state magnetism already contained in the spin-polarized DFT energies does not cancel in formation free energies. Without an estimate or bound on F_mag(T,x), the quantitative claims of room-temperature metastability and of stability onset between 500 K and 700 K are not established.
[Section III E and Appendix B] The partition function is evaluated only for 24-atom supercells, and Appendix B shows that the maximum configurational entropy for this cell size is about 0.62 k_B/atom at equiatomic composition, roughly 10% below the ideal-mixing value of 0.693 k_B/atom, with comparable deficits at other compositions. Since the paper attributes the Ni-rich side of the stability onset to configurational entropy, this systematic finite-size entropy deficit can bias the predicted critical temperature; the 16-atom versus 24-atom comparison in Section III E shows a trend but not convergence. A quantitative finite-size correction, an extrapolation to larger cells, or an explicit error estimate on the resulting ΔG is needed to support the specific claim that the onset occurs between 500 K and 700 K rather than, for example, being shifted by the missing entropy.
[Section III E and Section III H] Equations (15)-(17) are evaluated at the 0 K equilibrium volume for all temperatures, with harmonic vibrations only, while the quasi-harmonic treatment in Section III H is applied to a representative configuration rather than to the full ensemble used for the phase-stability analysis. Thermal expansion and anharmonicity typically change vibrational free energies by several meV/atom at 600-700 K, which is again comparable to the ΔG differences in Fig. 11. The manuscript acknowledges these approximations, but it does not quantify how much they could shift the stability onset or the room-temperature metastability boundary; a bound from the QHA calculations, or at least from the volume dependence of the free energy at the relevant compositions, is needed before the quantitative central claim can be considered established.

minor comments (5)

[Section III A, Fig. 6] The reported stress units are dimensionally inconsistent: the text gives training and test stress RMSEs in 'eV/atom' and 'meV/atom', while stress is an energy per volume (eV/Å^3 or GPa), and the force units appear truncated as 'meV/'. Please standardize the units in the text and figure legends.
[Section III C] The text states that NiPd_3 has Eform ~ 1.5 eV/atom and is 'just below the stability line'; since the convex-hull boundary is at zero formation energy, this must be ~1.5 meV/atom (as also suggested by Fig. 9).
[Section II B] The relaxation convergence criterion is reported as 'forces on each atom are less than 0.02 eV/atom', and later as 0.005 eV/Å; forces should be in eV/Å, and the two statements should be made consistent.
[Fig. 5 and Section II D] 'trail' should be 'trial' throughout the hyperparameter-optimization section and in the Fig. 5 caption.
[Section III I] 'one has to bare in mind' should read 'one has to bear in mind'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Ni-Pd finite-temperature stability prediction is an extrapolation of a JLP fitted to DFT data and validated on held-out supercells, with no target quantity used to set model parameters.

full rationale

The derivation chain is self-contained. The JLP is trained on DFT energies, forces, and stresses of small 8-atom and 2-atom cells (Sections II A-II D), and its transferability is demonstrated against held-out 16-atom DFT data (Section III A), including relaxed energies with RMSE 1.6 meV/atom and vibrational free energies with RMSE 0.64-3.14 meV/atom (Fig. 10). The thermodynamic stability prediction (Section III E, Eqs. 13-17, Fig. 11) is then obtained by explicit enumeration of 24-atom supercell configurations using this fitted model. No experimental phase-stability datum, no finite-temperature free energy, and no target phase-diagram quantity enters the fit or the hyperparameter selection; the claimed 500-700 K stability onset is therefore an extrapolation rather than a fitted input renamed as a prediction. The JLP formalism is cited to Refs. 39-40, which include coauthors of this paper, but that cited work is an externally published, parameter-free construction of the descriptor basis, and the present conclusions rest on the model's validated accuracy against DFT, not on any uniqueness theorem or unverified self-cited result. Other self-citations (Refs. 15-16, 55, 80) are contextual and not load-bearing for the central Ni-Pd claim. The main scientific caveat is completeness: the calculation neglects electronic and magnetic free energy, uses the 0 K volume, and retains finite-size configurational entropy about 10% below ideal mixing; these are accuracy and validity concerns, not circularity, because they concern omitted physics rather than an equation reducing the output to the input. No equation or construction step equates the predicted Gibbs free energy with the fitting labels or with a self-cited theorem, so the appropriate circularity score is 0.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The central thermodynamic predictions derive from a JLP fitted to DFT data on small cells, combined with several physical approximations (harmonic phonons, adiabatic decoupling, fixed 0 K volume, no magnetic or electronic free energy) and a finite 24-atom enumeration. No new physical entities are introduced; the approximations are standard in the field but directly relevant to the quantitative accuracy of the 600 K stability claim.

free parameters (6)

JLP expansion coefficients (873 features)
Linear-regression weights fitted to 1,059 DFT training energies, forces, and stresses; these determine all subsequent free-energy, phonon, and elastic predictions.
2B hyperparameters rcut, nmax, alpha, beta = rcut=6.392 Å, nmax=7, alpha=beta=1
Selected by tree-structured Parzen estimator optimization with 5-fold cross-validation (Table I); the resulting model errors are sensitive to these choices.
3B hyperparameters rcut, nmax, lmax, c3B = rcut=4.925 Å, nmax=6, lmax=9, c3B=57
Selected by the same TPE optimization (Table I); c3B=57 excludes the Ni-Pd-Ni cluster and defines the 873-feature descriptor set.
Force weight cF in loss function = 0.317
Optimized as a hyperparameter; scales force residuals relative to energy in Eq. (9).
Stress-validation weight ch = 0.013
Set heuristically in Eq. (10) to balance energy and stress errors during hyperparameter selection; the authors describe it as 'heuristically set'.
Training perturbation amplitudes and strain ranges = delta = 2%, 3%, 3.5%; lattice strain 5%
Chosen to sample the potential energy surface around equilibrium; affects model accuracy in strained regions and elastic constants.

assumptions (8)

domain assumption DFT-PBE provides the reference potential energy surface for training and validation.
All JLP energies, forces, and stresses are fitted to spin-polarized PBE data (Section II E); accuracy of the thermodynamic predictions is bounded by DFT accuracy.
domain assumption The harmonic approximation is used for all vibrational free energies.
F_vib is computed from phonopy finite-displacement harmonic phonons (Eq. 12); anharmonicity is not included in the main free-energy results.
domain assumption Vibrational and configurational degrees of freedom are adiabatically decoupled (partition-function coarse graining).
Eq. 15 writes Z as a sum over configurations weighted by E + F_vib, following Ref. 64; this separation relies on the timescale gap between atomic vibration and diffusion.
domain assumption The cluster expansion truncated at three-body terms is sufficient for Ni-Pd energetics.
The JLP is truncated at 3B (Section II C); the good 16-atom validation suggests sufficiency, but no systematic convergence study versus 4B is shown.
domain assumption The free energy includes only configurational and vibrational contributions; electronic and magnetic contributions are neglected.
Section III E states this explicitly. Ni is magnetic and Pd is a strong paramagnet, so magnetic entropy near the 600 K stability onset may be non-negligible.
domain assumption All thermochemistry is evaluated at the 0 K equilibrium volume for every configuration.
Section III E states 'the volume dependence is neglected'; the QHA is applied only to representative configurations in Section III H, not to the phase-stability calculation.
domain assumption A 24-atom supercell provides a quantitatively accurate configurational entropy for the macroscopic alloy.
All 99,268 unique configurations of 3x2x2 bct cells are enumerated (Section III E); Appendix B shows the maximum entropy is still about 10% below the ideal-mixing value at equiatomic composition, so finite-size bias is present though systematic.
domain assumption The JLP trained on 8-atom supercells transfers reliably to 24-atom supercells.
The model is validated on 16-atom cells (test set 1), but the 24-atom Gibbs free energies and phonon data are not directly checked against DFT; transferability is assumed to hold at the larger size.

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Pith. "Pith review of Machine-learning approach for the phase stability and mechanical properties of disordered alloys at finite temperature." pith.science (2026). https://pith.science/paper/36WRAYEI

@misc{pith2026260810465,
  author       = {Pith},
  title        = {Pith review of: Machine-learning approach for the phase stability and mechanical properties of disordered alloys at finite temperature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/36WRAYEI}},
  note         = {Machine review of arXiv:2608.10465}
}

abstract

The prediction of stable alloys forming solid-state solutions across large portions of the composition space is a serious theoretical challenge, since one has to evaluate the Gibbs free energy, including both configurational and vibrational contributions. This requires an energy theory capable of extremely high throughput. By taking the Ni-Pd system as prototype, we construct an efficient Jacobi-Legendre machine-learning potential based on density-functional-theory data, which provides accurate energies and forces across the entire composition space. Based on a cluster expansion up to three-body terms and only 873 trainable parameters, this allows us to compute the partition function by directly integrating all accessible microstates, differing for composition, atomic configuration and thermal agitation. We confirm that Ni and Pd are fully miscible, forming an $fcc$ solid-state solution. This is only metastable at room temperature, while becomes thermodynamically stable at around 600~K, with the stability achieved first at the Pd-rich end of the composition range. Interestingly, entropy and heat capacity analysis reveal a competition between the solid-state solution and two intermetallic phases with long-period L1$_0$ structure for NiPd and NiPd$_3$. All in all, our approach offers a powerful and high-throughput workflow for the study of disordered alloys, an approach that can be extended to multi-component systems such as high-entropy alloys.

Figures

Figures reproduced from arXiv: 2608.10465 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (18 more)

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

**Figure 13.** Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

**Figure 14.** Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]

**Figure 15.** Figure 15: for Ni–Pd alloys with Pd fractions of 37.5%, 50%, and 75%. We observe pronounced variations in ⟨Cij ⟩ near the configurational transition temperatures identified from the heat capacity (see [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]

**Figure 16.** Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]

**Figure 17.** Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p016_17.png]

**Figure 18.** Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p017_18.png]

**Figure 21.** Figure 21: FIG. 21 [PITH_FULL_IMAGE:figures/full_fig_p018_21.png]

**Figure 19.** Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p018_19.png]

**Figure 20.** Figure 20: FIG. 20 [PITH_FULL_IMAGE:figures/full_fig_p018_20.png]

**Figure 22.** Figure 22: FIG. 22 [PITH_FULL_IMAGE:figures/full_fig_p020_22.png]

**Figure 23.** Figure 23: FIG. 23 [PITH_FULL_IMAGE:figures/full_fig_p020_23.png]

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