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REVIEW 5 major objections 5 minor 179 references

Thermodynamic modeling of binaries in Cr-Fe-Mo-Nb-Ni supported by first-principles calculations

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that Wyckoff-resolved five-sublattice models backed by DFT endmember enthalpies reproduce measured sigma-phase site occupancies in Cr-Fe and move the Fe-Mo solidus closer to experiment.

desk verdict Real DFT-endmember CALPHAD work underneath a sloppy manuscript: the Cr-Fe and Fe-Mo reassessments deserve review, but the site-fraction 'prediction' is a fit and the sublattice models are stated inconsistently. read the letter →

arxiv 2507.16627 v1 pith:BJ6JQ56J submitted 2025-07-22 cond-mat.mtrl-sci

Hui Sun , Shun-Li Shang , Shuang Lin , Jingjing Li , Allison M. Beese , Zi-Kui Liu This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords CALPHADmodelingtopologicallyclose-packedphasessigmaphasemuLavesC14DFTformationenthalpiesCr-FesystemFe-Mo
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 tries to establish that the brittle topologically close-packed (TCP) phases in the Cr-Fe and Fe-Mo binaries cannot be described accurately by lumping several crystallographic sites together, and that giving each Wyckoff position its own sublattice, backed by DFT-computed endmember formation enthalpies, fixes the problem. If the claim is right, CALPHAD thermodynamic databases for the five-element Cr-Fe-Mo-Nb-Ni system can predict not only which TCP phases appear but also how chromium, iron, and molybdenum distribute among the individual atom sites of those phases. That matters because TCP phases like sigma drain refractory elements from superalloy matrices and embrittle the material, and additive manufacturing of graded Fe-to-Ni parts needs a database that works across the whole composition range. The headline evidence is that sigma-phase Fe occupancies in Cr-Fe match experimental data on all five Wyckoff sites with mean absolute errors between 0.011 and 0.024, and that the Fe-Mo solidus range moves closer to measured values.

What carries the argument

The load-bearing machinery is the complete sublattice model of a TCP phase under the compound energy formalism, a Gibbs-energy expansion in which every sublattice has its own ideal-mixing term: one sublattice for each Wyckoff position, so that each crystallographic site can develop its own occupancy. For $\sigma$, for example, the model is a five-sublattice formula $(\mathrm{Cr}, \mathrm{Fe}, \mathrm{Mo})_1(\mathrm{Cr}, \mathrm{Fe}, \mathrm{Mo})_2(\mathrm{Cr}, \mathrm{Fe}, \mathrm{Mo})_4(\mathrm{Cr}, \mathrm{Fe}, \mathrm{Mo})_4(\mathrm{Cr}, \mathrm{Fe}, \mathrm{Mo})_4$ instead of the simplified $(\mathrm{Cr}, \mathrm{Fe})_{10}(\mathrm{Cr}, \mathrm{Fe})_4(\mathrm{Cr}, \mathrm{Fe})_{16}$ form; mu gets five sublattices and C14 gets three. The Gibbs energy of each endmember in the compound energy sum is supplied by DFT-based phonon and quasiharmonic calculations, giving 243 $\sigma$, 243 mu, and 27 C14 formation enthalpies. The role of this machinery is to make model outputs commensurate with site-resolved experiments: each predicted site fraction is tied to a named Wyckoff position, so agreement or disagreement is per-site rather than averaged.

What would settle it

Measure site occupancies of sigma in Cr-Fe at a composition and temperature not used in the fit, for example a new long anneal near 45 at.% Fe at 1000 K, with atom-probe tomography or neutron diffraction and compare site by site with the model; if the pattern of high- and low-occupancy sites is permuted relative to prediction, the Wyckoff-to-sublattice ordering is wrong. A cheaper check is to recompute the 513 endmember formation enthalpies with a different exchange-correlation functional: if predicted invariant temperatures shift by more than the few kelvin claimed, the PBE energies are doing load-bearing work the paper does not quantify.

Watch

Extended reading notes

Core claim

The central discovery is a self-consistent thermodynamic description of the ten binaries in Cr-Fe-Mo-Nb-Ni, with the two most problematic binaries remodeled. In Cr-Fe, sigma is modeled as a five-sublattice compound with formula (Cr,Fe,Mo)1(Cr,Fe,Mo)2(Cr,Fe,Mo)4(Cr,Fe,Mo)4(Cr,Fe,Mo)4, one sublattice per Wyckoff position, so the predicted site fraction of Fe on each position can be compared directly with the measured atom distributions in the paper's two sigma site-occupancy datasets. The reported mean absolute errors are 0.011 to 0.024 at 973 K against one dataset and 0.0288 overall at 923 K against the other. For Fe-Mo, the same treatment is applied to sigma, mu, and Laves_C14, and the resulting model places the solidus at 59.7 to 91.1 at.% Mo between 2191 and 2753 K, closer to the experimental 60.0 to 90.0 at.% Mo range than the previous model. Formation enthalpies of the 243 sigma, 243 mu, and 27 C14 endmembers come from DFT calculations rather than ad hoc assignments.

Load-bearing premise

The claim stands on the assumption that the order of sublattices in each model formula matches the order of the Wyckoff sites as measured, so every predicted site fraction is assigned to the right atom column; it also assumes the DFT endmember enthalpies are accurate enough that the fit does not have to absorb systematic energy errors.

Editorial extensions

If this is right

  • Cr-Fe sigma-phase predictions now resolve all five Wyckoff sites separately, so future site-occupancy experiments can be checked site by site instead of against lumped averages.
  • The Fe-Mo solidus is pulled from 55.6 to 89.8 at.% Mo in the previous model to 59.7 to 91.1 at.% Mo, within about 1 at.% of the measured 60.0 to 90.0 at.% range, improving liquid-solid boundary predictions in Mo-bearing steels and superalloys.
  • Replacing arbitrary endmember enthalpy values with DFT-computed values removes a systematic source of error that would otherwise propagate into ternary and higher-order databases built on these binaries.
  • Using the same Wyckoff-based five-sublattice models for Cr-Fe, Fe-Mo, Fe-Nb, and Nb-Ni makes the quinary database internally consistent for ternary extrapolation.
  • The 973 K and 923 K sigma site-fraction checks give quantitative per-site error metrics that future CALPHAD assessments can use as a baseline.

Reading between the lines

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

  • A stronger test than the paper's would be to hold out one of the two site-occupancy datasets, or a new measurement, during fitting and then compare; the paper's error metrics are computed against data that also helped set the model parameters.
  • If the sublattice ordering in the model formulas is correct, the same five-sublattice machinery should transfer to sigma phases in higher-order Cr-Fe-Mo-Nb-Ni alloys, predicting how Mo and Nb partition among sites, a quantity that governs TCP embrittlement.
  • The paper leaves the R phase, which has eleven Wyckoff positions, in a simplified model; applying the same complete-sublattice philosophy there with DFT endmember enthalpies is the immediate next step the argument points toward.
  • A systematic DFT functional check, such as comparing PBE enthalpies with a meta-GGA, would bound how much of the improved agreement comes from the sublattice structure versus from the DFT energies.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper presents a consolidated CALPHAD thermodynamic database for the ten Cr-Fe-Mo-Nb-Ni binary systems, with a focus on remodeling the Cr-Fe and Fe-Mo systems using complete sublattice models for the TCP phases sigma, mu, and Laves_C14 based on their Wyckoff positions. The TCP endmember enthalpies are computed from DFT (PBE-GGA) and integrated with experimental thermochemical and phase-equilibrium data. The abstract highlights improved accuracy of the models, citing 'excellent agreement' of predicted sigma-site occupancies in Cr-Fe with experiments and a better description of the Fe-Mo solidus. The paper also adopts and slightly modifies earlier models for the other eight binaries. Claims are supported by figures of phase diagrams, thermochemical comparisons, and tables of invariant reactions. The full TDB file and DFT endmember data are promised as supplementary material but are not actually visible in the supplied manuscript.

Significance. If the claims hold, this is a genuinely useful contribution: it provides a self-consistent thermodynamic database for a five-component system of technological importance, upgrades simplified TCP-phase models to Wyckoff-resolved forms, and supplies a large, internally consistent set of DFT endmember enthalpies (243+243+27 endmembers). The use of DFT-derived endmember energies is an independent component that does not reduce to a fit of the same experimental data, giving the central claim partial grounding. The reported invariant reactions and solidus improvements are consistent with established experimental ranges and plausible. However, the headline site-occupancy validation is weakened by two issues: it is a residual of a fit (the cited experimental occupancy data were used in the optimization) and the sublattice-to-Wyckoff mapping is stated inconsistently between the text and the model tables. These issues must be fixed before the paper can be accepted, but they are correctable within the manuscript's scope.

major comments (5)
  1. [§2.5 and Table 2] The Fe-Mo sigma model is written in Section 2.5 as (Fe,Mo)4(Fe,Mo)4(Fe,Mo)2(Fe,Mo)4(Fe,Mo)1, whereas Table 2 lists the present-work sigma model as (Cr,Fe,Mo)1(Cr,Fe,Mo)2(Cr,Fe,Mo)4(Cr,Fe,Mo)4(Cr,Fe,Mo)4. These orderings are not equivalent: the latter is 1:2:4:4:4 (matching the reduced Wyckoff multiplicities 2b,4f,8i1,8i2,8j from Table 1), while the former is 4:4:2:4:1. Because the site-fraction comparisons in Fig. 1 and the MAEs in Section 5.1 are the abstract's headline evidence, the authors must state explicitly the mapping from each CEF sublattice index to the experimental Wyckoff label and ensure that Section 2.5, Table 2, and the TDB file use the same ordering. If the Section 2.5 ordering is a typo, it must be corrected; otherwise the claimed agreement could be an artifact of permuted sublattice labels.
  2. [§3.1.1 and §5.1] The Cieslak et al. and Yakel et al. site-fraction data are described in Section 3.1.1 as having been incorporated into the modeling. Consequently, the MAE values in Section 5.1 (0.011 to 0.024 for Cieslak et al., and 0.0288 for Yakel et al.) are residuals for fitted data, not independent predictions. The abstract's phrase 'predicted site occupancies ... show excellent agreement' therefore overstates the evidence. The authors should rephrase the claim as reproducing or consistent with the measured site fractions, and ideally include a validation set (for example, hold out one composition or temperature) to demonstrate predictive skill.
  3. [§4.1 and Supplemental Material] Section 4.1 states that all DFT-calculated endmember enthalpies are given in the supplementary TDB file, and the supplemental-material section announces a TDB file, but the supplied supplementary pages contain only figures; the TDB file and the optimized model parameters are not reproduced anywhere in the manuscript. Without those data, the DFT endmember energies (the main independent input) and the actual implemented sublattice models cannot be inspected by the reader. The authors should include the TDB file as supplementary material or, if that is impossible, tabulate the key endmember formation enthalpies and the fitted interaction parameters in an appendix.
  4. [§5.2] The claimed improvement in the Fe-Mo solidus description is based on a narrow margin. The present model is quoted as predicting 59.7 to 91.1 at.% Mo over 2191 to 2753 K, whereas the experimental range is given as 60.0 to 90.0 at.% Mo; this means the model overshoots the experimental range by 0.3 to 1.1 at.% on both ends. The text should quantify the deviations of both the present model and the Rajkumar et al. model from the individual experimental solidus points, and state whether the residual differences are within the reported experimental uncertainty. As written, 'more accurately reflecting experimental trends' leaves this comparison under-specified.
  5. [§4.1] No validation of the PBE-GGA endmember enthalpies against experimental formation enthalpies or alternative calculations is provided. Given that GGA is known to carry systematic errors of the order of 5 to 10 kJ/mol-atom for intermetallic compounds, and that these enthalpies enter Eq. 5 as the endmember energies, the authors should report at least a sanity check for a few well-characterized phases (for example, the formation enthalpy of sigma-CrFe or Fe2Mo against calorimetric data). This would substantiate the claim that the DFT set provides reliable input.
minor comments (5)
  1. [§5.1] In the text after Fig. 4, the invariant reactions are referred to as being summarized in 'Table 3', but Table 3 is titled 'Details of DFT-based first-principles calculations' and the invariant reactions appear in Table 4; the cross-reference is incorrect.
  2. [§5.1 and Table 1] Section 5.1 refers to 'site 2a' when describing the sigma phase, while Table 1 lists the first sigma Wyckoff position as 2b. These labels should be unified across the text, figures, and tables.
  3. [Eq. (4)] Equation (4) appears to be incomplete: the ideal mixing term RT(x_Cr ln x_Cr + x_Fe ln x_Fe) is written without a preceding '+' and without the x_Fe G_Fe term; the expression should be checked for missing algebraic terms.
  4. [Throughout] There are several typographical errors: the abstract uses 'complied' instead of 'compiled', Table 1's caption spells 'strucutre' for 'structure', and Section 2.9 contains a full-width comma after 'MoNi4'.
  5. [Supplemental Material] The numbering of the supplementary figures is inconsistent with the in-text references: for example, the in-text reference to Fig. S1 as the Cr-Mo phase diagram conflicts with the supplemental list in the main text, where Fig. S1 is described as a phonon density-of-states figure. Renumber the supplement so that each figure number is unique and matches the text.

Circularity Check

2 steps flagged · score 6.0 of 10

Headline σ site-occupancy 'prediction' is a fit residual; the sublattice-to-Wyckoff mapping that the comparison depends on is stated inconsistently.

  1. fitted input called prediction [Section 3.1.1 and Section 5.1 (Cr-Fe σ site occupancy)]
    "Two sets of site fraction data for the σ phase are available from Cieslak et al. [34] and Yakel et al. [6], which are in good agreement. Both datasets are incorporated into the present modeling work. // Fig. 1 indicates that the present predictions show excellent agreement with measurements across all five sites. ... the mean absolute error (MAE) values from the present model are 0.022 for site 2a, 0.024 for site 4f, 0.011 for site 8i1, 0.011 for site 8i2, and 0.013 for site 8j, respectively, with respect to the measurements by Cieslak et al. [34]."

    The same Cieslak/Yakel site-fraction data are entered into the CALPHAD optimization as fitting data (Section 3.1.1), and Section 5.1 then reports the optimized model's agreement with those same data as 'predicted site occupancies' with MAEs. In a CEF fit, the adjustable endmember and interaction parameters are chosen to reproduce the assessed experimental site fractions; agreement with those points is a residual of the fit, not an independent test. The abstract's 'predicted site occupancies ... excellent agreement' therefore reduces, for this headline quantity, to a restatement of the fit input. The DFT endmember enthalpies and the Fe-Mo comparisons to other experimental data are independent, but the σ site-fraction claim is forced by construction.

  2. self definitional [Section 2.5, Table 2, Table 1, and Section 5.1 (sublattice-to-Wyckoff mapping)]
    "a five-sublattice model is used for σ ((Fe, Mo)4(Fe, Mo)4(Fe, Mo)2(Fe, Mo)4((Fe, Mo)1) // σ (Cr,Fe,Mo)1(Cr,Fe,Mo)2(Cr,Fe,Mo)4(Cr,Fe,Mo)4(Cr,Fe,Mo)4 // σ (sigma) ... 2b, 4f, 8i (1), 8i (2), 8j"

    The site-fraction validation in Section 5.1 compares experimental occupations labeled by Wyckoff sites (Table 1: 2b, 4f, 8i1, 8i2, 8j) with CEF sublattice site fractions. The mapping between the five CEF sublattices and these Wyckoff positions is not fixed: Section 2.5 lists the σ model as (Fe,Mo)4(Fe,Mo)4(Fe,Mo)2(Fe,Mo)4(Fe,Mo)1, while Table 2 lists it as (Cr,Fe,Mo)1(Cr,Fe,Mo)2(Cr,Fe,Mo)4(Cr,Fe,Mo)4(Cr,Fe,Mo)4, and neither is explicitly keyed to the Table 1 order.

full rationale

The derivation chain is not wholly circular: the DFT endmember enthalpies (Section 4.1) are independent first-principles inputs, and the Fe-Mo comparisons against Rajkumar's model (solidus, phase boundaries, activity) use experimental data that are not identical to the parameters being optimized. However, the paper's most prominent validation—the abstract's 'predicted site occupancies of σ in Cr-Fe show excellent agreement'—is circular with respect to its own fitting procedure. Section 3.1.1 states that the Cieslak and Yakel site-fraction datasets were incorporated into the modeling; Section 5.1 then reports MAEs against those same data as if they were predictions. Those are fit residuals, not independent predictions. Additionally, the sublattice-to-Wyckoff mapping on which the site-fraction comparison depends is not fixed in the manuscript: Section 2.5 gives the σ model with a 4:4:2:4:1 sublattice order, while Table 2 gives a 1:2:4:4:4 order, and neither is explicitly tied to the Table 1 Wyckoff order (2b, 4f, 8i1, 8i2, 8j). Since permuting sublattice labels changes the computed MAE against experimental site labels without changing the Gibbs energy, part of the claimed agreement is an artifact of an unfixed labeling convention. These issues affect the headline site-occupancy claim, while the independent DFT and phase-equilibrium content keeps the paper from being entirely circular; the appropriate overall assessment is partial circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The modeling rests on the SGTE unary database, the compound energy formalism, and PBE-level DFT with no reported error bars for the 513 endmember enthalpies. The dataset selections and the site-fraction data used for validation are the main user-controlled inputs. No new phases, particles, or mediators are postulated: all phases are established TCP phases from prior crystallographic and CALPHAD literature (Tables 1 and 2).

free parameters (4)
  • Redlich-Kister interaction parameters for liquid, BCC_A2, and FCC_A1 in Cr-Fe and Fe-Mo = Encoded in the supplementary TDB file; not tabulated in the text
    The a+bT coefficients of Eq. 4 are fitted to selected experimental phase equilibrium and thermochemical data in Sections 3.1 and 3.2.
  • Excess interaction parameters within sigma, mu, and C14 sublattices in Cr-Fe and Fe-Mo = Encoded in the supplementary TDB file
    The L parameters of Eq. 6 are fitted to phase boundaries and to the Cieslak and Yakel site-fraction datasets that are also used as the main validation of the sigma model (Sections 3.1.1 and 5.1).
  • Endmember Gibbs energies of the R phase in Fe-Mo = Inherited or fitted; not documented
    The R phase is remodeled with the sublattice model (Cr,Fe)17(Cr,Mo)14(Cr,Fe,Mo)12 in Table 2, but Section 4.1 reports DFT calculations only for sigma, mu, and C14 endmembers.
  • Dataset selection and weighting in the Cr-Fe and Fe-Mo optimizations = Qualitative choices, e.g., only Thiedemann and Pavars for Cr-Fe liquid; only Iguchi for Fe-Mo liquid; only Dubiel and…
    Exclusions are reasoned but not formalized, and they directly shape the fitted parameters (Section 3.1).
assumptions (5)
  • domain assumption SGTE unary lattice stabilities for Cr, Fe, Mo, Nb, and Ni are accurate and mutually consistent
    Adopted as the fixed reference frame for all binaries (Sections 2.1 and 4.2); this choice excludes assessments based on revised unary data, such as Xiong et al. [25] and Hao et al. [57].
  • domain assumption The compound energy formalism with Wyckoff-based sublattice sizes adequately represents TCP phase thermodynamics
    Central modeling premise behind Eqs. 5 and 6; the paper itself notes that cross-sublattice short-range ordering terms are neglected (Section 4.2).
  • domain assumption PBE-GGA DFT with the stated k-point meshes and convergence criteria yields endmember formation enthalpies accurate enough for CALPHAD
    DFT inputs enter every TCP endmember Gibbs energy, while no error bars, convergence tests, or comparisons to measured compound enthalpies are reported (Section 4.1).
  • standard math The quasiharmonic approximation with phonon DOS and Mermin electronic entropy captures finite-temperature Gibbs energies of TCP phases
    Eqs. 1 to 3 are standard, but their accuracy for sigma, mu, and C14 is not validated against measured heat capacities of these phases.
  • ad hoc to paper Binary descriptions assembled here extrapolate reliably to ternary and higher-order alloys without additional interaction terms
    The Conclusions claim a 'robust thermodynamic foundation' for the quinary system, but no ternary or higher-order validation is presented anywhere in the paper.

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Pith. "Pith review of Thermodynamic modeling of binaries in Cr-Fe-Mo-Nb-Ni supported by first-principles calculations." pith.science (2026). https://pith.science/paper/BJ6JQ56J

@misc{pith2026250716627,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic modeling of binaries in Cr-Fe-Mo-Nb-Ni supported by first-principles calculations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJ6JQ56J}},
  note         = {Machine review of arXiv:2507.16627}
}
read the original abstract

Thermodynamic descriptions of all binaries within the Cr-Fe-Mo-Nb-Ni system have been complied and, where necessary, remodeled. Notably, the Cr-Fe and Fe-Mo systems have been remodeled using comprehensive sublattice models for the topologically close-packed (TCP) phases of Laves_C14, sigma, and mu according to their Wyckoff positions. These refinements are supported by first-principles calculations based on density functional theory (DFT), in conjunction with available experimental data in the literature. The resulting models offer improved accuracy in describing the TCP phases. For instance, the predicted site occupancies of sigma in Cr-Fe show excellent agreement with experimental observations. The present work provides a robust foundation for CALPHAD modeling and the design of complex, multi-component materials, particularly those based on Fe-based and Ni-based alloys.

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Pith tools

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