Pith. sign in

REVIEW 3 major objections 5 minor 74 references

Computing binary alloy phase diagrams with explicit configurational and vibrational entropy

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read By sampling atomic identity swaps on top of alchemical thermodynamic integration, the authors compute Au-Cu phase diagrams directly from atomistic free energies and show that explicit configurational entropy lowers the order-disorder transi

desk verdict A promising alchemical NE-TI + identity-exchange workflow with honest benchmarking, but the free-energy bookkeeping is under-derived and needs proof before the 97 K shift can be trusted. read the letter →

arxiv 2607.14795 v3 pith:5IW3KJ3D submitted 2026-07-16 cond-mat.mtrl-sci

Sarath Menon , Marvin Poul , Tilmann Hickel , Jörg Neugebauer , Ralf Drautz This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords binaryphasediagramsconfigurationalentropyvibrationalnonequilibriumthermodynamicintegrationalchemicaltransformationidentity-exchangeMonteCarloAu-Cualloyatomicclusterexpansion
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

The paper extends non-equilibrium thermodynamic integration so that a single atomistic free-energy calculation captures both atomic vibrations and the distribution of chemical species over sites. It does this by alchemically interpolating one element into another while Monte Carlo identity-exchange moves reshuffle species identities along the switching path. Applied to Au-Cu with ACE potentials trained on three DFT functionals, the method produces full composition-temperature phase diagrams directly from atomistic free energies. Explicit configurational sampling lowers the predicted AuCu order-disorder transition by about 97 K, from 818 K to 721 K, bringing it close to the experimental 683 K, and markedly widens the solid-solution and solubility fields. The same comparison shows functional choice shifts transition temperatures by hundreds of kelvin, so configurational entropy and functional accuracy must both be handled explicitly.

What carries the argument

The load-bearing mechanism is alchemical non-equilibrium thermodynamic integration: a linear interpolation H(λ)=(1−λ)H_i+λH_f switches a subset of atoms from pure-A to A-B interactions over a finite switching time, and the bidirectional work average gives ΔF. To sample configurational entropy, Metropolis identity-exchange moves swap the species labels of one A and one B atom between MD blocks along the path (Algorithm 1), so that at each λ the simulation equilibrates over both atomic positions and identity assignments in the interacting system. The composition dependence is completed by adding analytic ideal-mixing and de Broglie terms; phase diagrams follow from Legendre-transforming F(T,x)

What would settle it

Recompute the alchemical transformation at xCu=0.5 and 1000 K with the analytic ideal-mixing term removed from the free-energy formula. If the resulting free energy differs from the published value by k_B T ln2 per atom (about 0.060 eV/atom at 1000 K), the term was double-counted; if it agrees, the bookkeeping is consistent.

Watch

Extended reading notes

Core claim

The paper's central claim is that vibrational and non-ideal configurational entropy can be sampled together in one free-energy calculation by combining nonequilibrium thermodynamic integration with an alchemical interpolation that continuously converts one species into another, interspersed with Monte Carlo identity-exchange moves that reshuffle atomic species at fixed positions. Applying this to Au-Cu using ACE potentials trained on LDA, PBE, and r2SCAN, the authors construct full composition-temperature phase diagrams directly from atomistic free energies. The headline result is that explicit configurational sampling lowers the AuCu order-disorder transition from 818 K to 721 K for ACE-LDA

Load-bearing premise

The calculation adds an analytic ideal-mixing entropy term while the simulation at the starting point already randomly rearranges atomic species among sites; if that rearrangement already accounts for all species arrangements, the mixing entropy could be counted twice.

Editorial extensions

If this is right

  • If the bookkeeping is consistent, the 97 K downward shift is a direct measure of non-ideal configurational entropy in the AuCu solid solution, which ideal-mixing treatments miss.
  • The widened stability fields mean ordered compounds like AuCu, AuCu3, and Au4Cu2 acquire finite solubility ranges at temperature, changing predicted phase fractions and tie-lines.
  • The method yields per-point free energies with forward/reverse hysteresis as an uncertainty estimate, so phase boundaries can be assigned uncertainty in composition-temperature space.
  • The framework extends to more than two species and to disorder within stoichiometric phases, so the same path can be reused for higher-component alloys.
  • Functional sensitivity of several hundred kelvin implies that atomistic phase-diagram predictions should be reported across functionals, not only for one exchange-correlation approximation.

Reading between the lines

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

  • A direct test of the double-counting question is to recompute the mixing free energy with the analytic ideal-mixing term omitted; the same alchemical path with identity swaps should then return the ideal term automatically if the bookkeeping is sound.
  • If the method holds, the roughly 100 K shift from configurational entropy should grow in alloys with stronger short-range order or larger size mismatch; AuCu is a mild case, so more dramatic widening of solid-solution fields may appear elsewhere.
  • The method's per-point hysteresis could be exploited for adaptive placement of free-energy calculations, focusing computational effort near the phase boundaries rather than on a uniform grid.
  • The Au3Cu discrepancy being independent of the entropy treatment points to the functional as the limiting factor for this system; a potential trained on hybrid or RPA-level energetics would be a direct follow-up.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents a workflow that extends non-equilibrium thermodynamic integration to composition-dependent alchemical transformations, combining Monte Carlo identity-exchange moves with molecular dynamics so that configurational and vibrational entropy are sampled in a single free-energy calculation. The method is applied to Au–Cu using ACE potentials trained on LDA, PBE, and r2SCAN DFT data. Free energies are computed for the FCC solid solution, ordered intermetallics, and the liquid, and phase diagrams are assembled through a semi-grand-canonical Legendre transformation. For the ACE-LDA potential, explicit configurational sampling lowers the AuCu order–disorder transition temperature from about 818 K to 721 K (closer to the experimental 683 K) and substantially widens the solid-solution field, while the dependence on the exchange-correlation functional is much larger. The authors are careful to caution that the agreement with experiment should not be overinterpreted.

Significance. If the methodology is correct, it offers a direct, general route from machine-learned interatomic potentials to binary phase diagrams that include non-ideal configurational entropy and fully anharmonic vibrational entropy without empirical thermodynamic fitting. The paper has notable strengths: parameter-convergence tests in Fig. 2, validation of the ACE potentials against DFT convex hulls (Fig. 1), a comparison with semi-grand canonical MC/MD, consistency with an independent EAM nested-sampling result, and openly available training data, potentials, and codes. The candid discussion of functional sensitivity is also a strength. However, the central free-energy bookkeeping — specifically the relation between the analytic ideal-mixing term and the identity-exchange sampling — is not derived at the partition-function level, and the benchmark used to support the bookkeeping is reported at inconsistent temperatures. These points must be resolved before the quantitative claims can be accepted.

major comments (3)
  1. [§4.1–4.2, Eqs. (4)–(5), Algorithm 1] The paper does not provide a partition-function-level derivation showing that the analytic ideal-mixing term in Eq. (4) is not already included in the combined MD/MC sampling. At λ=0 the potential is independent of σ, so the identity-exchange moves visit all N!/(NA!NB!) assignments; at λ=1 they sample the equilibrium distribution over σ. The relationship between the sampled partition function and the terms in Eq. (4) should be written out explicitly, including the indistinguishability prefactors and the combinatorial degeneracy, to demonstrate the cancellation that justifies adding kBTN[(1−x)ln(1−x)+x ln x]. Without this, a reader cannot rule out a composition-dependent double counting of the ideal-mixing entropy, which would directly affect the reported 97 K shift and the phase fields in Fig. 5.
  2. [§2.2 and Supplementary Note 2] The benchmark against SGC-MC/MD is presented at inconsistent temperatures: the main text states the comparison is made at 700 K, while Supplementary Note 2 and Fig. 3 state 1000 K, and the convergence figures in the SI are captioned 700 K. The temperature matters because the solid solution is not stable across the full composition range at all temperatures. Moreover, this benchmark is performed at a single temperature, and both methods share the same identity-swap implementation, so a common bias in the energy evaluation or in the treatment of identity labels would not be exposed. The authors should reconcile the reported temperature and, ideally, add a second temperature or provide an explicit argument why one temperature is sufficient to validate the composition-dependent bookkeeping.
  3. [§2.3, Fig. 3; §4.2.2] The 'no-swap' or 'ideal-mixing' baseline uses a fixed random identity assignment rather than an ensemble average over assignments. At λ=1 in the no-swap calculation, the free energy corresponds to one particular configuration of the species plus the analytic ideal-mixing term; it is not the free energy of the ideal-mixing ensemble, which would require averaging the enthalpy over all assignments. It should be clarified how many independent random assignments were used, how the 818 K baseline and the 97 K shift depend on the chosen assignment, and what the associated uncertainty is. As written, the comparison between 'with' and 'without' configurational sampling may conflate the absence of configurational entropy with the absence of configurational enthalpy relaxation.
minor comments (5)
  1. [Abstract vs §2.3] The abstract states the transition is lowered from approximately 810 K to 710 K, while §2.3 reports 818 K to 721 K. The numbers should be made consistent.
  2. [§2.2 vs Supplementary Note 4] The main text says a 1 meV/atom free-energy difference can cause 'more than 50 K' change in transition temperature, whereas Supplementary Note 4 estimates ±29 K for the order–disorder transition. These should be reconciled.
  3. [Supplementary Note 2] The text of the note says the SGC-MC/MD comparison is at 1000 K, while the captions of Supplementary Figs. 1 and 2 say 700 K. This inconsistency must be fixed.
  4. [General] Typographical issues: 'Finaly' in §2.5, 'less then 1.8 Å' in §4.5.1, and a missing article in §4.5.2 ('The exact location of the cross-over point...').
  5. [§5 / Data and code availability] The repositories listed in Refs. [50] and [51] are appropriate, but the version identifiers should be cited consistently so that the exact datasets and potentials used in the paper can be retrieved.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central phase-diagram and transition-temperature results are computed from ACE potentials trained on DFT and are not fitted to experimental phase boundaries; self-citations are to prior, independently published tools and are not load-bearing.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The ACE-LDA/PBE/r2SCAN potentials are fitted to DFT training data generated with ASSYST and validated against DFT convex hulls, elastic constants, equation-of-state curves, and phonon dispersions (Sec. 2.1, Supplementary Notes 6–8); no parameter is tuned to the experimental AuCu transition temperature (683 K) or to CALPHAD boundaries. The central quantitative result, the 97 K lowering of the AuCu order-disorder transition under explicit configurational sampling (Sec. 2.3), is obtained by comparing two computed free-energy curves (with and without identity-exchange moves) and is cross-checked against SGC-MC/MD (Supplementary Note 2) and against EAM/nested-sampling results (Sec. 2.3, Ref. [31]); these are independent checks, not self-referential inputs. The authors explicitly caution that the agreement with experiment should not be overinterpreted because of the large exchange-correlation functional sensitivity (Sec. 2.3, Discussion). Self-citations to calphy [8], landau [21], and ASSYST [21] are to prior, published tools whose underlying methods are restated in Secs. 4.1–4.4, so the argument does not reduce to an unverified self-citation chain. The main substantive concern is a bookkeeping risk, not circularity: Eqs. (4)–(5) add the analytic ideal-mixing term k_B T N[(1−x)ln(1−x)+x ln x] on top of a work integral whose endpoints sample identity assignments via Metropolis exchange moves (§4.2.2), and the paper does not derive the partition function of the combined MD/MC sampler, so a composition-dependent double-counting of ideal-mixing entropy cannot be excluded. However, no equation defines the predicted phase boundaries or transition temperature in terms of the experimental target, nor is any fitted parameter renamed as a prediction, so this is a potential systematic error to be checked rather than a circular derivation.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim depends on the correctness of the alchemical NE-TI free-energy path, the ability of MC identity exchanges to equilibrate chemical order at each λ, and the fidelity of ACE potentials to the chosen DFT references. No new physical entities are introduced.

free parameters (5)
  • nswap (identity-exchange attempts per trajectory) = 250000
    Chosen from convergence test in Fig. 2(a); free energy within 0.0001 eV/atom of converged value.
  • switching time t_sw = 25 ps
    Chosen from Fig. 2(b); convergence approached around 15 ps.
  • MD steps between swap blocks n_MD = 100
    Chosen from Fig. 2(c); insensitive below ~1000, cost-optimized.
  • ACE basis size / cutoff = 1000 functions/element; 6.2 Å
    Hyperparameters selected for fit quality; not fitted to phase diagram data.
  • grid spacings for phase diagram = 25 K; dx=0.02 (FCC/liquid), 0.01 (compounds)
    Manual resolution choices; could affect coexistence line precision.
assumptions (6)
  • standard math Nonequilibrium work relation gives unbiased free energy differences.
    Eqs. (2)-(3) rely on Jarzynski/Crooks-type bidirectional averaging (Ref. [8]).
  • domain assumption ACE potentials faithfully represent DFT energetics across composition.
    Validated on convex hull, E-V curves, elastic constants, phonons (Sec. 2.1 and SI), but transferability at phase-boundary state points is assumed.
  • domain assumption MC identity-exchange moves sample the canonical distribution of chemical assignments at fixed composition.
    Sec. 4.2.2 Metropolis criterion; convergence only tested at x_Cu=0.5 and one path; stated to be system dependent.
  • domain assumption Eq. (4) decomposition of ideal vs non-ideal configurational entropy does not double count.
    Analytic ideal-mixing term plus identity-swap sampling: no derivation that the sampled path excludes the combinatorial multiplicity; benchmark at one temperature provides indirect support.
  • domain assumption LDA/PBE/r2SCAN with chosen PAW settings are valid electronic-structure references.
    Known limitations: Au3Cu is not stable in any functional (Sec. 2.5); semi-local functionals are challenged for Au-Cu (Refs. 25-28).
  • domain assumption Stoichiometric AuCu can be treated as perfectly ordered without vacancies.
    Stated approximation in Sec. 2.3; may become inaccurate close to the transition.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Computing binary alloy phase diagrams with explicit configurational and vibrational entropy." pith.science (2026). https://pith.science/paper/5IW3KJ3D

@misc{pith2026260714795,
  author       = {Pith},
  title        = {Pith review of: Computing binary alloy phase diagrams with explicit configurational and vibrational entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5IW3KJ3D}},
  note         = {Machine review of arXiv:2607.14795}
}
read the original abstract

Phase stability in multicomponent solid solutions depends on configurational entropy beyond the ideal mixing limit, but capturing it together with vibrational entropy within the same atomistic framework remains challenging. Here, we extend non-equilibrium thermodynamic integration to composition-dependent transformations through an alchemical interpolation of the interactions, combined with Monte Carlo identity exchange moves and molecular dynamics that sample the vibrational and non-ideal configurational entropy along the integration path. We apply the framework to the Au-Cu binary alloy using Atomic Cluster Expansion potentials trained on density functional theory data using the LDA, PBE, and r2SCAN functionals, and construct composition-temperature phase diagrams directly from atomistic free energies. We find that explicit configurational sampling lowers the AuCu order-disorder transition temperature predicted by the ACE potential trained on LDA data from approximately 810 K to 710 K, closer to the experimental value of 683 K, and substantially widens the stability range of the solid solution. At the same time, the much larger sensitivity to the exchange-correlation functional shows that this level of agreement should not be interpreted as general predictive accuracy. Non-ideal configurational entropy must therefore be sampled explicitly, alongside a careful choice of functional, for a reliable atomistic description of binary phase diagrams.

Figures

Figures reproduced from arXiv: 2607.14795 by the authors.

Figure 1
Figure 1. Convex hulls of the AuCu system obtained from DFT and reproduced by the corresponding ACE [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Convergence of the simulation parameters used for the alchemical free energy calculations with MC [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Order-disorder transition in AuCu at xCu = 0.5 between the disordered FCC solid solution and the ordered AuCu phase. (a) Free-energy curves obtained with the ACE-LDA potential, shown with and without configurational sampling through atomic identity-exchange moves. The FCC-MD curve (green) includes only the ideal configurational entropy, whereas the FCC-MCMD curve (orange) includes the full configurational contributi… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Melting temperatures of (a) pure Au, (b) pure Cu, and (c) the FCC AuCu solid solution at [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Calculated phase diagrams of AuCu obtained with (a) ACE-LDA, (b) ACE-LDA without explicit [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Schematic of the alchemical transformation path on a fixed set of sites. Sites designated as type [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 1
Figure 1. Figure 1: Convergence of the Au fraction xAu in the SGC-MC/MD simulations of the AuCu FCC solid solution at 700 K, shown as a function of the number of identity-exchange (swap) moves. Each panel corresponds to a different target composition, ranging from xAu = 0.113 to xAu = 0.9…
Figure 2
Figure 2. Figure 2: Convergence of the semi-grand canonical potential [PITH_FULL_IMAGE:figures/full_fig_p017_2.png]
Figure 3
Figure 3. Figure 3: (a) free energy of mixing for the AuCu FCC solid solution at 1000 K by both alchemical transformations [PITH_FULL_IMAGE:figures/full_fig_p018_3.png]
Figure 4
Figure 4. Figure 4: Per-point switching dissipation q for the Cu–Au composition-scaling transformations at 1000 K, computed from the forward and reverse switching runs at each composition xAu. The dissipation stays below 0.5 meV/atom throughout. 72 4. Sensitivity of transition temperature…
Figure 5
Figure 5. Figure 5: Histogram of energy and force errors on the training set for all three functionals, LDA (top), PBE [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Energy-volume curves against DFT for a) ACE-LDA, b) ACE-PBE, c) ACE-r [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Elastic constants for LDA (red), r2SCAN (blue), and PBE (green) for the five stable phases and the experimentally stable Au3Cu. ACE prediction for the independent constants (x-axis) are given as 7 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Phonon spectra for LDA, PBE, and r2SCAN (left to right) descriptions of Cu, Au, AuCu3, Au3Cu, AuCu (top to bottom). Orange dashed lines show the DFT predictions, blue solid lines the ACE predictions. 8 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

74 extracted references · 25 canonical work pages

  1. [1]

    Y., and Reinhardt, A

    Chew, P . Y., and Reinhardt, A. Phase diagrams— why they matter and how to predict them.The Journal of Chemical Physics,158(3), 030902, January

  2. [2]

    The Martensitic Transformation in the Iron-Nickel System.JOM, 8(10), 1393–1401, 1956

    Kaufman, L., and Cohen, M. The Martensitic Transformation in the Iron-Nickel System.JOM, 8(10), 1393–1401, 1956. doi:10.1007/BF03377892

    work page doi:10.1007/bf03377892 1956

  3. [3]

    An insight into using dft data for cal- phad modeling of solid phases in the third generation of calphad databases, a case study for al.Calphad,65, 79–85, 2019

    Bigdeli, S., Zhu, L.-F., Glensk, A., et al. An insight into using dft data for cal- phad modeling of solid phases in the third generation of calphad databases, a case study for al.Calphad,65, 79–85, 2019. doi:https://doi.org/10.1016/j.calphad.2019.02.008

    doi:10.1016/j.calphad.2019.02.008 2019

  4. [4]

    Direct dynami- cal calculation of entropy and free energy by adia- batic switching.Phys

    Watanabe, M., and Reinhardt, W. Direct dynami- cal calculation of entropy and free energy by adia- batic switching.Phys. Rev. Lett.,65(26), 3301, 1990. doi:10.1103/PhysRevLett.65.3301

    work page doi:10.1103/physrevlett.65.3301 1990

  5. [5]

    Optimized Free-Energy Evaluation Using a Single Reversible-Scaling Simu- lation.Phys

    de Koning, M., Antonelli, A., and Yip, S. Optimized Free-Energy Evaluation Using a Single Reversible-Scaling Simu- lation.Phys. Rev. Lett.,83(20), 3973, 1999. doi:10.1103/PhysRevLett.83.3973

    work page doi:10.1103/physrevlett.83.3973 1999

  6. [6]

    Ab ini- tio up to the melting point: Anharmonicity and vacancies in aluminum.Phys

    Grabowski, B., Ismer, L., Hickel, T., et al. Ab ini- tio up to the melting point: Anharmonicity and vacancies in aluminum.Phys. Rev. B,79, 134106,

  7. [7]

    Nonequi- librium free-energy calculation of solids using LAMMPS.Comput

    Freitas, R., Asta, M., and de Koning, M. Nonequi- librium free-energy calculation of solids using LAMMPS.Comput. Mater. Sci,112, 333, 2016. doi:10.1016/j.commatsci.2015.10.050

    doi:10.1016/j.commatsci.2015.10.050 2016

  8. [8]

    Auto- mated free-energy calculation from atomistic sim- ulations.Phys

    Menon, S., Lysogorskiy, Y., Rogal, J., et al. Auto- mated free-energy calculation from atomistic sim- ulations.Phys. Rev. Materials,5(10), 103801, 2021. doi:10.1103/PhysRevMaterials.5.103801

    doi:10.1103/physrevmaterials.5.103801 2021

Show all 74 references
  1. [9]

    Atomic cluster expansion for accurate and transferable interatomic poten- tials.Phys

    Drautz, R. Atomic cluster expansion for accurate and transferable interatomic poten- tials.Phys. Rev. B,99(1), 014104, 2019. doi:10.1103/PhysRevB.99.014104

    2019 doi

  2. [10]

    Performant implementation of the atomic cluster expansion (PACE) and application to cop- per and silicon.Npj Comput

    Lysogorskiy, Y., van der Oord, C., Bochkarev, A., et al. Performant implementation of the atomic cluster expansion (PACE) and application to cop- per and silicon.Npj Comput. Mater.,7(1), 97, 2021. doi:10.1038/s41524-021-00559-9

    2021 doi

  3. [11]

    Menon, S., Lysogorskiy, Y., Knoll, A. L. M., et al. From electrons to phase diagrams with machine learning potentials using pyiron based automated workflows.npj Computational Materials,10(1), 261,

  4. [12]

    Reconsid- ering short-range order in complex concentrated alloys.MRS Bulletin,48(7), 753–761, Jul 2023

    Walsh, F., Abu-Odeh, A., and Asta, M. Reconsid- ering short-range order in complex concentrated alloys.MRS Bulletin,48(7), 753–761, Jul 2023. doi:10.1557/s43577-023-00555-y

    2023 doi

  5. [13]

    A review of calphad modeling of ordered phases.Journal of Phase Equilibria and Diffusion,39(5), 678–693, Oct

    Sundman, B., Chen, Q., and Du, Y. A review of calphad modeling of ordered phases.Journal of Phase Equilibria and Diffusion,39(5), 678–693, Oct

  6. [14]

    Direct obser- vation of chemical short-range order in a medium- entropy alloy.Nature,592(7856), 712–716, Apr 2021

    Chen, X., Wang, Q., Cheng, Z., et al. Direct obser- vation of chemical short-range order in a medium- entropy alloy.Nature,592(7856), 712–716, Apr 2021. doi:10.1038/s41586-021-03428-z

    2021 doi

  7. [15]

    M., and Becker, J

    Sanchez, J. M., and Becker, J. D. The role of the cluster variation method in the first principles calculation of phase diagrams.Progress of Theo- retical Physics Supplement,115, 131–145, 01 1994. doi:10.1143/PTPS.115.131

    1994 doi

  8. [16]

    Z., Cao, Y., et al

    Sheriff, K., Xiao, D. Z., Cao, Y., et al. Machine learning potentials for modeling alloys across com- positions.Science Advances,12(25), eaea9951, June

  9. [17]

    Automat- ing first-principles phase diagram calculations

    van de Walle, A., and Ceder, G. Automat- ing first-principles phase diagram calculations. Journal of Phase Equilibria,23(4), 348, Aug 2002. doi:10.1361/105497102770331596

    2002 doi

  10. [18]

    A., and Glandt, E

    Kofke, D. A., and Glandt, E. D. Monte carlo simula- tion of multicomponent equilibria in a semigrand canonical ensemble.Molecular Physics,64(6), 1105– 1131, 1988. doi:10.1080/00268978800100743

    1988 doi

  11. [19]

    Free energy cal- culations for solid solutions by computer simu- lations.Molecular Physics,72(3), 699–713, 1991

    Kranendonk, W., and Frenkel, D. Free energy cal- culations for solid solutions by computer simu- lations.Molecular Physics,72(3), 699–713, 1991. doi:10.1080/00268979100100511

    1991 doi

  12. [20]

    Scal- able parallel Monte Carlo algorithm for atomistic simulations of precipitation in alloys.Phys

    Sadigh, B., Erhart, P ., Stukowski, A., et al. Scal- able parallel Monte Carlo algorithm for atomistic simulations of precipitation in alloys.Phys. Rev. B, 85(18), 2012. doi:10.1103/physrevb.85.184203

    2012 doi

  13. [21]

    Auto- mated generation of structure datasets for machine learning potentials and alloys.npj Computational Materials,11(1), 174, Jun 2025

    Poul, M., Huber, L., and Neugebauer, J. Auto- mated generation of structure datasets for machine learning potentials and alloys.npj Computational Materials,11(1), 174, Jun 2025. doi:10.1038/s41524- 025-01669-4

    2025 doi

  14. [22]

    Efficient determina- tion of free energies of non-ideal solid solutions via hybrid Monte Carlo simulations.Computer Physics Communications,304, 109307, November

    Li, Z., and Scandolo, S. Efficient determina- tion of free energies of non-ideal solid solutions via hybrid Monte Carlo simulations.Computer Physics Communications,304, 109307, November

  15. [23]

    G., and Oates, W

    Sundman, B., Fries, S. G., and Oates, W. A thermodynamic assessment of the au-cu system.Calphad,22(3), 335–354,

  16. [24]

    I., Perdew, J

    Csonka, G. I., Perdew, J. P ., Ruzsinszky, A., et al. Assessing the performance of recent density func- tionals for bulk solids.Phys. Rev. B,79, 155107, Apr 2009. doi:10.1103/PhysRevB.79.155107

    2009 doi

  17. [25]

    Nonlocal first-principles calculations in Cu-Au and other intermetallic alloys.Phys

    Zhang, Y., Kresse, G., and Wolverton, C. Nonlocal first-principles calculations in Cu-Au and other intermetallic alloys.Phys. Rev. Lett.,112, 075502, Feb 2014. doi:10.1103/PhysRevLett.112.075502

    2014 doi

  18. [26]

    Exchange-correlation catastrophe in Cu-Au: A challenge for semilocal density functional approx- imations.Phys

    Tian, L.-Y., Levämäki, H., Ropo, M., et al. Exchange-correlation catastrophe in Cu-Au: A challenge for semilocal density functional approx- imations.Phys. Rev. Lett.,117, 066401, Aug 2016. doi:10.1103/PhysRevLett.117.066401

    2016 doi

  19. [27]

    Gradient- level and nonlocal density functional descrip- tions of Cu-Au intermetallic compounds.The 13 European Physical Journal B,91(6), 128, Jun 2018

    Levämäki, H., Tian, L., Kokko, K., et al. Gradient- level and nonlocal density functional descrip- tions of Cu-Au intermetallic compounds.The 13 European Physical Journal B,91(6), 128, Jun 2018. doi:10.1140/epjb/e2018-90166-9

    2018 doi

  20. [28]

    doi:10.1016/j.cpc.2024.109307

    2024

  21. [29]

    Atomistic modeling of the γ and γ′-phases of the Ni–Al system

    Mishin, Y. Atomistic modeling of the γ and γ′-phases of the Ni–Al system. Acta Materialia,52(6), 1451–1467, 2004. doi:https://doi.org/10.1016/j.actamat.2003.11.026

    2004 doi

  22. [30]

    Embedded atom method potential for studying mechanical proper- ties of binary Cu–Au alloys.Modelling Simul

    Gola, A., and Pastewka, L. Embedded atom method potential for studying mechanical proper- ties of binary Cu–Au alloys.Modelling Simul. Mater. Sci. Eng.,26(5), 055006, 2018. doi:10.1088/1361- 651X/aabce4

    2018 doi

  23. [31]

    Baldock, R. J. N., Bernstein, N., Salerno, K. M., et al. Constant-pressure nested sampling with atomistic dynamics.Phys. Rev. E,96(4), 043311,

  24. [32]

    Determination of the melting curve of gold up to 110 gpa.Phys

    Weck, G., Recoules, V ., Queyroux, J.-A., et al. Determination of the melting curve of gold up to 110 gpa.Phys. Rev. B,101, 014106, Jan 2020. doi:10.1103/PhysRevB.101.014106

    2020 doi

  25. [33]

    Efficient approach to compute melting prop- erties fully from ab initio with application to cu.Phys

    Zhu, L.-F., Grabowski, B., and Neugebauer, J. Efficient approach to compute melting prop- erties fully from ab initio with application to cu.Phys. Rev. B,96, 224202, Dec 2017. doi:10.1103/PhysRevB.96.224202

    2017 doi

  26. [34]

    pycalphad: Calphad-based computational thermodynamics in python

    Otis, R., and Liu, Z.-K. pycalphad: Calphad-based computational thermodynamics in python. In Zentropy, pp. 373–392, Jenny Stanford Publishing,

  27. [35]

    K., Adhikari, S., Bates, J

    Nepal, N. K., Adhikari, S., Bates, J. E., et al. Treating different bonding situations: Revisit- ing au-cu alloys using the random phase ap- proximation.Physical Review B,100(4), 2019. doi:10.1103/physrevb.100.045135

    2019 doi

  28. [36]

    pyiron: An integrated development environment for computational materials science

    Janssen, J., Surendralal, S., Lysogorskiy, Y., et al. pyiron: An integrated development environment for computational materials science. Computational Materials Science,163, 24 – 36, 2019. doi:https://doi.org/10.1016/j.commatsci.2018.07.043

    2019 doi

  29. [37]

    W., Rosenbluth, M

    Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., et al. Equation of state calculations by fast computing machines.The Journal of Chemical Physics,21(6), 1087–1092, 06 1953. doi:10.1063/1.1699114

    1953 doi

  30. [38]

    Efficiency of ab- initio total energy calculations for metals and semiconductors using a plane-wave basis set

    Kresse, G., and Furthmüller, J. Efficiency of ab- initio total energy calculations for metals and semiconductors using a plane-wave basis set. Computational Materials Science,6(1), 15–50, 1996. doi:10.1016/0927-0256(96)00008-0

    1996 doi

  31. [39]

    Efficient itera- tive schemes for<i>ab initio</i>total-energy cal- culations using a plane-wave basis set.Physi- cal Review B,54(16), 11169–11186, October 1996

    Kresse, G., and Furthmüller, J. Efficient itera- tive schemes for<i>ab initio</i>total-energy cal- culations using a plane-wave basis set.Physi- cal Review B,54(16), 11169–11186, October 1996. doi:10.1103/physrevb.54.11169

    1996 doi

  32. [40]

    Blöchl, P . E. Projector augmented-wave method. Physical Review B,50(24), 17953–17979, December

  33. [41]

    From ultrasoft pseu- dopotentials to the projector augmented-wave method.Physical Review B,59(3), 1758–1775, Jan- uary 1999

    Kresse, G., and Joubert, D. From ultrasoft pseu- dopotentials to the projector augmented-wave method.Physical Review B,59(3), 1758–1775, Jan- uary 1999. doi:10.1103/physrevb.59.1758

    1999 doi

  34. [42]

    Methfessel, M., and Paxton, A. T. High-precision sampling for brillouin-zone integration in metals. Physical Review B,40(6), 3616–3621, August 1989. doi:10.1103/physrevb.40.3616

    1989 doi

  35. [43]

    doi:10.1201/9781003514466-18

    doi

  36. [44]

    pycalphad: Calphad- based computational thermodynamics in python.Journal of Open Research Software, 2017

    Otis, R., and Liu, Z.-K. pycalphad: Calphad- based computational thermodynamics in python.Journal of Open Research Software, 2017. doi:10.5334/jors.140

    2017 doi

  37. [45]

    N., Scuseria, G

    Staroverov, V . N., Scuseria, G. E., Tao, J., et al. Tests of a ladder of density functionals for bulk solids and surfaces.Phys. Rev. B,69, 075102, Feb 2004. doi:10.1103/PhysRevB.69.075102

    2004 doi

  38. [46]

    M., Tkatchenko, A., et al

    Zhang, G.-X., Reilly, A. M., Tkatchenko, A., et al. Performance of various density-functional approx- imations for cohesive properties of 64 bulk solids. New Journal of Physics,20(6), 063020, jun 2018. doi:10.1088/1367-2630/aac7f0

    2018 doi

  39. [47]

    Random phase approximation up to the melt- ing point: Impact of anharmonicity and nonlo- cal many-body effects on the thermodynamics of Au.Phys

    Grabowski, B., Wippermann, S., Glensk, A., et al. Random phase approximation up to the melt- ing point: Impact of anharmonicity and nonlo- cal many-body effects on the thermodynamics of Au.Phys. Rev. B,91, 201103(R), May 2015. doi:10.1103/PhysRevB.91.201103

    2015 doi

  40. [48]

    Sun, J., Ruzsinszky, A., and Perdew, J. P . Strongly constrained and appropriately normed semilocal density functional.Phys. Rev. Lett.,115, 036402, Jul

  41. [49]

    W., Kaplan, A

    Furness, J. W., Kaplan, A. D., Ning, J., et al. Ac- curate and numerically efficient r 2SCAN meta- generalized gradient approximation.The Journal of Physical Chemistry Letters,11(19), 8208–8215, 2020. doi:10.1021/acs.jpclett.0c02405

    2020 doi

  42. [50]

    S., Menon, S., Poul, M., et al

    of Thermodynamic Properties, A. S., Menon, S., Poul, M., et al. Cuau (revision 84ff9f0), 2026. doi:10.57967/hf/9602

    2026 doi

  43. [51]

    S., Menon, S., Poul, M., et al

    of Thermodynamic Properties, A. S., Menon, S., Poul, M., et al. Cuau-ace (revision 2ca8ecc), 2026. doi:10.57967/hf/9721. 14 Supplementary Material: Computing binary alloy phase diagrams with explicit configurational entropy Sarath Menon 1,2*, Marvin Poul 2, Tilmann Hickel 3, J...

    2026 doi

  44. [53]

    P ., and Zunger, A

    Perdew, J. P ., and Zunger, A. Self-interaction cor- rection to density-functional approximations for many-electron systems.Phys. Rev. B,23, 5048–5079, May 1981. doi:10.1103/PhysRevB.23.5048

    1981 doi

  45. [54]

    P ., Burke, K., and Ernzerhof, M

    Perdew, J. P ., Burke, K., and Ernzerhof, M. Generalized gradient approximation made sim- ple.Phys. Rev. Lett.,77, 3865–3868, Oct 1996. doi:10.1103/PhysRevLett.77.3865

    1996 doi

  46. [63]

    Here we describe how this cost is reduced7 from a scaling linear in the system size to one indepen-8 dent of it

    Local-energy evaluation of identity-1 exchange moves2 The combined MD/MC switching protocol attempts a3 large number of identity-exchange moves along each4 transformation path, and the evaluation of the energy5 change associated with every trial swap dominates the6 overall cos...

  47. [64]

    The 57 convergence of the SGC-MC/MD sampling is shown 58 in Figs

    Comparison with SGC-MC/MD53 To benchmark the alchemical transformation approach, 54 we compare it against semi-grand canonical Monte 55 Carlo / molecular dynamics (SGC-MC/MD) simula- 56 tions of the AuCu FCC solid solution at 1000 K. The 57 convergence of the SGC-MC/MD samplin...

  48. [65]

    Dissipation in alchemical transformations 65 Each alchemical free energy difference is obtained from a forward (increase Au composition in Cu) and a re- verse (decrease Au composition in Cu) switching run. Following the nonequilibrium thermodynamic integra- tion formalism [2],...

  49. [66]

    At the phase76 transition, the two competing phases satisfy Gα(T0) =77 Gβ(T0), where T0 is the transition temperature

    Sensitivity of transition temperatures to72 free energy uncertainty73 To assess how uncertainties in the computed free ener-74 gies propagate to the transition temperatures, we per-75 formed a first order sensitivity analysis. At the phase76 transition, the two competing phase...

  50. [67]

    99 Force MAE at ∼ 30 m eV also indicate a good fit

    Fitting Errors 96 All three potentials fit the training set very well, with 97 mean absolute errors (MAE) of 5 m eV and root mean 98 square errors (RMSE) of 9 m eV, across all references. 99 Force MAE at ∼ 30 m eV also indicate a good fit. Fig. 5 100 shows histograms of the en...

  51. [68]

    6 shows the energy volume curves for all three107 potentials presented in the main text

    Energy Volume Curves106 Fig. 6 shows the energy volume curves for all three107 potentials presented in the main text. Over the full108 range of strains tested, the agreement is excellent, with109 the energy MAE over all strains of 3 meV/atom, rising110 up to 20 meV/atom in the...

  52. [69]

    Elastic Constants117 We calculated elastic tensors using elastic [3]. Fig. 7118 shows the independent components by phase as bar119 plots. For all three potentials, the agreement is good, al-120 though LDA is slightly worse described at 7 GPa MAE121 compared to 3 GPa MAE for P...

  53. [70]

    Phonon Density of States126 Finally, we compare the density of states (DOS) of127 phonons for all three models in fig. 8. Qualitative agree-128 ment is observed throughout, with some frequency129 shifts particularly noticable in Au and AuCu 3. The130 calculations were performe...

  54. [71]

    P ., Aktulga, H

    Thompson, A. P ., Aktulga, H. M., Berger, R.,133 et al. LAMMPS - a flexible simulation tool134 for particle-based materials modeling at the135 atomic, meso, and continuum scales.Com-136 puter Physics Communications,271, 108171, 2022.137 doi:10.1016/j.cpc.2021.108171.138

    2022

  55. [72]

    Auto-139 mated free-energy calculation from atomistic sim-140 ulations.Phys

    Menon, S., Lysogorskiy, Y., Rogal, J., et al. Auto-139 mated free-energy calculation from atomistic sim-140 ulations.Phys. Rev. Materials,5(10), 103801, 2021.141 doi:10.1103/PhysRevMaterials.5.103801.142

    2021 doi

  56. [73]

    T., and Badger, C

    Jochym, P . T., and Badger, C. jochym/elastic:143 Maintenance release, May 2018.144 doi:10.5281/zenodo.1254570.145

    2018 doi

  57. [74]

    First-principles phonon calculations with146 phonopy and phono3py.J

    Togo, A. First-principles phonon calculations with146 phonopy and phono3py.J. Phys. Soc. Jpn.,92(1),147 012001, 2023. doi:10.7566/JPSJ.92.012001.148 6 Figure 7: Elastic constants for LDA (red), r 2SCAN (blue), and PBE (green) for the five stable phases and the experimentally s...

    2023 doi

  58. [1994]

    doi:10.1103/physrevb.50.17953

    doi

  59. [1998]

    doi:https://doi.org/10.1016/S0364- 5916(98)00034-0

    doi

  60. [2009]

    doi:10.1103/PhysRevB.79.134106

    doi

  61. [2015]

    doi:10.1103/PhysRevLett.115.036402

    doi

  62. [2017]

    doi:10.1103/PhysRevE.96.043311

    doi

  63. [2018]

    doi:10.1007/s11669-018-0671-y

    doi

  64. [2023]

    doi:10.1063/5.0131028. 12

    doi

  65. [2024]

    doi:10.1038/s41524-024-01441-0

    doi

  66. [2026]

    doi:10.1126/sciadv.aea9951

    doi

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.