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Extracting Composition-Dependent Diffusion Coefficients Over a Very Large Composition Range in NiCoFeCrMn High Entropy Alloy Following Strategic Design of Diffusion Couples and Physics Informed Neural Network Numerical Method

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read From three diffusion couples in NiCoFeCrMn, this paper extracts tracer, intrinsic, and interdiffusion coefficients over nearly the full Ni-Co-Fe composition range using a constraint-enhanced physics-informed neural network.

desk verdict The marker-plane experimental data and the non-uniqueness demonstration are solid, but the full-range PINN extraction rests on an underdetermined fit with no held-out validation. read the letter →

arxiv 2506.12345 v1 pith:XH5BPYSP submitted 2025-06-14 cond-mat.mtrl-sci

Suman Sadhu , Saswata Bhattacharyya , Aloke Paul This is my paper
classification cond-mat.mtrl-sci PACS 66.30.-h
keywords diffusionhighentropyalloyNiCoFeCrMnKirkendallmarkerplanetracercoefficientsphysics-informedneuralnetworkvacancywindeffectmobilitydatabase
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 claims that a strategic set of only three diffusion couples, pure Ni, Co, and Fe each coupled to the same (NiCoFeCr)85Mn15 alloy, can provide composition-dependent tracer, intrinsic, and interdiffusion coefficients across nearly the whole Ni-Co-Fe range, with Cr up to 20 at.% and Mn up to 15 at.%, when combined with a single-profile Kirkendall marker-plane method and a physics-informed neural network inverse solver. The significance is that this replaces the traditional need for many intersecting diffusion paths or radioactive tracers, which have limited multicomponent diffusion studies. The paper further argues that profile-only optimization is ill-posed: matching measured composition profiles does not guarantee physically reliable tracer diffusivities unless experimentally measured tracer and impurity diffusivities enter as equality constraints. If correct, the method gives a scalable, non-radioactive route to mobility databases for high-entropy and other multicomponent alloys, and it sharpens the argument that intrinsic diffusivities, not interdiffusion coefficients, should be used to discuss element interactions in concentrated alloys.

What carries the argument

The load-bearing machinery is the Kirkendall marker-plane flux method, in which inert-marker positions let one compute intrinsic fluxes from a single diffusion profile, combined with the Manning vacancy-wind relation linking tracer and intrinsic coefficients through Onsager cross terms, and the Boltzmann-transformed physics-informed neural network inverse solver. The composition dependence is carried by a quadratic polynomial with pairwise products in the four independent element fractions for $\ln D_i^*$, 15 coefficients per element and 75 total, trained against a four-part loss function: ODE residual, boundary conditions, profile data, and equality constraints such as measured tracer and impurity diffusivities. This machinery turns three end-time composition profiles into a full composition-dependent mobility database.

What would settle it

A decisive check would be to run a fourth diffusion couple outside the fitted set, such as a Ni-Fe alloy coupled to the same (NiCoFeCr)85Mn15 alloy, and compare the predicted interior profile and tracer diffusivities with measurements; disagreement beyond experimental uncertainty would show the polynomial ansatz is too rigid or the constraints too sparse.

Watch

Extended reading notes

Core claim

The central claim, stated on the authors' own terms, is that from three diffusion couples they can estimate and extract diffusion coefficients over the whole composition range covered by the couples: zero to hundred per cent Ni, Co, and Fe, Cr up to 20 at.% and Mn up to 15 at.%, in the FCC NiCoFeCrMn system. At each Kirkendall marker plane, one composition profile supplies intrinsic fluxes, from which tracer coefficients are extracted through Manning's vacancy-wind-corrected Onsager relations, and then intrinsic and interdiffusion coefficients follow directly. The composition dependence is then completed by a physics-informed neural network that solves the Boltzmann-transformed diffusion ODE with a quadratic log-diffusivity polynomial, anchored by measured tracer coefficients at the marker planes, literature impurity and self-diffusivities at the pure ends, and average tracer values at the alloy end. The paper's central negative result is that optimizing only against composition profiles can reproduce the profiles while returning unreliable tracer coefficients; experimental constraints are necessary to avoid this ill-posed inversion. It also shows the vacancy wind effect is large enough on several cross-intrinsic coefficients that it should not be neglected in concentrated alloys, and that interdiffusion coefficients can misrepresent elemental interactions because they average intrinsic contributions.

Load-bearing premise

The load-bearing assumption is that a single quadratic polynomial with pairwise cross terms in the four independent element fractions represents $\ln D_i^*$ over the whole composition range, and that three couples plus a handful of point constraints are enough to fix all 75 coefficients without hidden under-determination.

Editorial extensions

If this is right

  • A non-radioactive workflow with three couples can populate a mobility database over a broad composition window, replacing many interdiffusion-couple experiments and avoiding radioactive tracers.
  • Profile-only fits should not be trusted in multicomponent inverse diffusion problems: the same profile can be matched by many diffusivity sets, so experimentally measured tracer and impurity values must enter as constraints.
  • Element interactions in concentrated alloys should be read from intrinsic diffusivities rather than interdiffusion coefficients, because the latter are composition-weighted averages that can flip signs.
  • The vacancy-wind correction is not a small detail at concentrated compositions; it changes several cross-intrinsic coefficients enough to alter mechanistic conclusions.
  • The same strategic couple design, pure end member against one alloy, extends to other high-entropy and multicomponent systems, limited mainly by solubility and the availability of thermodynamic data.

Reading between the lines

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

  • Editorial extension: a held-out fourth diffusion couple or radiotracer measurements at non-equiatomic interior compositions would test whether the quadratic polynomial generalizes, since the present validation is consistency-based rather than predictive.
  • Editorial extension: the closeness of the average alloy-side tracer values to the literature equiatomic radiotracer data is encouraging but not a strong independent check, because the average comes from the same fitted surface and the comparison composition is near the alloy end member.
  • Editorial extension: using the extracted mobility surface to run phase-field or CALPHAD-style simulations would be a natural next test, but the paper stops at showing the data are smooth and consistent.
  • Editorial extension: the method's stated range of Cr up to 20 at.% and Mn up to 15 at.% is dictated by FCC solubility; extending to complete-solubility or non-FCC systems would require re-examining both the polynomial form and the structure factor, 7.15 for FCC, in the vacancy-wind 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.

Referee Report

3 major / 5 minor

Summary. The manuscript combines three marker-containing diffusion couples (pure Ni, Co, and Fe each coupled to (NiCoFeCr)85Mn15) to estimate tracer, intrinsic, and interdiffusion coefficients at the Kirkendall marker planes, and then proposes a physics-informed neural network (PINN) inverse method, built on Boltzmann-transformed ODEs and a polynomial composition dependence of ln D_i^*, to extract composition-dependent tracer diffusivities over the entire composition range of the couples. The paper also argues that intrinsic diffusion coefficients are more appropriate than interdiffusion coefficients for discussing diffusional interactions in concentrated alloys and that Manning's vacancy-wind correction is significant. The claimed outcome is a reliable, non-radioactive route to mobility databases for multicomponent alloys from only three diffusion couples.

Significance. The experimental marker-plane estimates are internally consistent and compare favorably with literature impurity and radiotracer data, and the paper is commendably explicit in showing that profile-only fits are non-unique (Figs. 4-6). If the full-range extraction were sound, the method would substantially reduce the experimental effort needed for mobility database construction in HEAs. The open-source PINN framework and the clear statement of the constraint-enhancement idea are also useful. However, the central claim of reliable extraction over a very large composition range is not yet established: the inversion rests on an unvalidated and underdetermined polynomial ansatz, and one of the key constraints is derived from the model's own earlier outputs.

major comments (3)
  1. [3.2 / Eq. (7b)] The central claim that three diffusion couples determine tracer diffusivities over the whole composition range is not established. Eq. (7b) contains 15 trainable coefficients per element (75 total) in four independent composition variables, but the information supplied is only three one-dimensional composition paths plus a small number of point constraints; any coefficient combination that vanishes or is linearly dependent along those paths is unconstrained, so the inverse problem remains underdetermined. The good match in Fig. 8 is an in-sample fit to the very profiles used in the data-loss term, and the smooth variation in Fig. 7 is a property of the polynomial basis, not a validation. Please provide either a held-out validation (e.g., leave-one-couple-out cross-prediction of compositions and diffusivities) or an identifiability analysis showing that all 75 coefficients are bounded by the available data.
  2. [3.2 / Table 7] The use of the averaged PINN-extracted tracer diffusivities as an equality constraint introduces circularity. These averages are computed from the model's own outputs from the three couples and are then inserted into the constraint loss L_c for the final optimization; therefore the final values at the alloy-side composition are forced toward the model's previous estimates, and the subsequent agreement with the radiotracer data of Ref. [35] is not an independent check. The final optimization should be repeated without this self-generated constraint, or the radiotracer data should be reserved as a held-out comparison rather than being used as an input.
  3. [2.3.1 / Eq. (7a)] The quadratic polynomial ansatz for ln D_i^* over the entire composition range is a strong ad hoc assumption and is never tested against independent interior compositions. Literature data on concentration-dependent mobilities, such as Ref. [37], could serve as an external check; without such a test, the extracted diffusivities at untested compositions may be artifacts of the chosen basis. In addition, imposing impurity diffusivities at roughly 0.5 at.% alloying element (Section 3.2) relies on an explicit assumption that the diffusivities are unchanged at that composition; this should be stated as a testable assumption and its sensitivity to the chosen composition examined.
minor comments (5)
  1. [Throughout] The manuscript contains numerous encoding artifacts (e.g., 'di;usion') and inconsistent section numbering: subsections 2.2.3 through 2.2.7 appear after 2.3.1 and 2.3.2, and should be renumbered.
  2. [Figures 4-6] The captions and labels of Figs. 4-6 should be checked carefully; Fig. 5 appears to repeat 'DC1' although the surrounding text describes all three couples, and the reader cannot tell which panels correspond to which couple.
  3. [Table 3] Table 3 cites references [50-59] while the text says [51-59]; the reference range and the in-text citation should be reconciled.
  4. [2.2.7 / 3.2(iv)] The claim that this is the first PINN applied to an actual multicomponent diffusion context should be softened, given the authors' own earlier work in Refs. [8,9] on NiCoFeCr and pseudo-binary couples.
  5. [Supplementary material] The optimized polynomial coefficients are said to be in the supplementary file, but that file was not available for review; for reproducibility, the final coefficient values, loss weights, and network architecture should be included with the manuscript.

Circularity Check

1 steps flagged · score 6.0 of 10

Alloy-side diffusivity values are the PINN's own averaged outputs re-imposed as an equality constraint; the remaining extraction is underdetermined but not circular.

  1. fitted input called prediction [Section 3.2, paragraph after Table 7 and before Fig. 8]
    "To solve this problem, we take an average of the values extended from di;erent di;usion couples, as listed in the table, which can be utilized as another equality constraint in the alloy side of the di;usion couples for generating a consistent mobility database. ... As a next step, the average of optimized tracer di;usion coe;icients is also utilized as equality constraints along with the self and impurity di;usion coe;icients in pure elements, and the data is estimated at the Kirkendall marker plane in this study."

    Table 7's 'average' is computed from the optimizer's own outputs from the three couples, not from a new measurement. The paper itself calls them 'the values extended from di;erent di;usion couples' and 'the average of optimized tracer di;usion coe;icients.' Re-inserting that average as an equality constraint in the final run forces the tracer diffusivities at the (NiCoFeCr)85Mn15 end-member to equal the model's previous values. Thus the final 'extraction' at that composition is a self-consistency condition: the output is equal to its own earlier output by construction, so it cannot validate the extracted database at the alloy side. The marker-plane and literature impurity constraints remain independent, which is why the circularity is partial.

full rationale

The only concrete circular step is the handling of the alloy-side end-member values. The average tracer diffusivities listed in Table 7 are the PINN's own outputs from the three couples; the authors then use that average as an equality constraint in the final optimization, so the displayed alloy-side values are forced to equal the model's previous values. That is a fitted input renamed as an extracted result, and it cannot independently confirm the database at that composition. The broader claim retains independent content: the Kirkendall marker-plane tracer values come from measured marker positions and thermodynamic factors (Eq. 4), the impurity and self-diffusivities are taken from the literature, and the comparison with Gaertner et al. [35] is an external radiotracer benchmark. The 75-parameter polynomial in Eq. 7b and the absence of held-out profiles make the full-range extraction underdetermined, but that is an identifiability and validation risk rather than circularity under the stated rules. The self-citations to the authors' earlier single-profile method are normal applications of prior methodology, not a circular uniqueness argument. Overall score 6: one specific prediction reduces by construction to earlier model outputs, while the central extraction still has independent experimental anchors.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a chain of external inputs (ThermoCalc factors, literature impurity and self-diffusivities) and modeling choices (polynomial mobility, loss weights, averaging constraints). The largest uncharged addition is the 75-coefficient polynomial ansatz plus the self-consistent averaging constraint on the alloy side.

free parameters (2)
  • PINN polynomial coefficients for ln D_i^* = 75 coefficients, 15 per element, values in supplementary file
    Eq. 7a/7b defines each tracer diffusivity as a quadratic polynomial in the independent element fractions; all coefficients are fitted by minimizing the composite loss rather than fixed by physics.
  • Manual loss weights omega_1, omega_2, omega_3, omega_4 = Case 5 values from Table 1
    Section 2.2.5: weights are manually tuned to make the optimizer converge; different weight choices change which constraints dominate and therefore change the extracted diffusivities.
assumptions (7)
  • standard math Fick's second law with constant molar volume holds across the interdiffusion zone.
    Used in Eq. 12 as the governing equation for the composition profiles.
  • domain assumption The Boltzmann transformation collapses the time-dependent PDE to an ODE for the single annealing time.
    Section 2.3.2, Eqs. 13-14; this requires the profiles to be self-similar in the Boltzmann variable.
  • domain assumption Manning's vacancy-wind relations (Eqs. 3 and 9) correctly connect tracer, intrinsic and Onsager transport coefficients in the FCC alloy.
    Used to convert measured intrinsic fluxes at the Kirkendall plane into tracer diffusivities and to build the PINN flux model.
  • domain assumption ThermoCalc database TCHEA7 provides accurate activity profiles and thermodynamic factors at 1200 degrees Celsius.
    Section 3.1 and Fig. 3: thermodynamic factors from these activity profiles enter every tracer diffusion coefficient estimate.
  • ad hoc to paper A quadratic polynomial with pairwise cross terms, Eq. 7a/7b, adequately represents ln D_i^* over the entire composition range.
    This basis is chosen for numerical convenience and is not derived from or validated against an independent physical model.
  • ad hoc to paper Literature impurity diffusion coefficients at pure elements can be imposed at a composition of about 0.5 at.% alloying element because the diffusivities are expected to be similar.
    Section 3.2: the equality constraint cannot be applied at exactly 100 at.% pure element, so the authors shift it to a nearby composition.
  • ad hoc to paper The averaged tracer diffusivities from three diffusion couples at the alloy-side composition can be used as an independent equality constraint.
    Section 3.2 and Table 7: the average is an output of the first round of fitting, so using it as a later constraint is self-referential.

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Pith. "Pith review of Extracting Composition-Dependent Diffusion Coefficients Over a Very Large Composition Range in NiCoFeCrMn High Entropy Alloy Following Strategic Design of Diffusion Couples and Physics Informed Neural Network Numerical Method." pith.science (2026). https://pith.science/paper/XH5BPYSP

@misc{pith2026250612345,
  author       = {Pith},
  title        = {Pith review of: Extracting Composition-Dependent Diffusion Coefficients Over a Very Large Composition Range in NiCoFeCrMn High Entropy Alloy Following Strategic Design of Diffusion Couples and Physics Informed Neural Network Numerical Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XH5BPYSP}},
  note         = {Machine review of arXiv:2506.12345}
}
read the original abstract

Estimating composition dependent diffusion coefficients in multicomponent alloys was a longstanding challenge due to limitations in experimental methods. In this study, we have first demonstrated a strategic design of producing only three diffusion couples to estimate all types, that is tracer, intrinsic, and interdiffusion coefficients at the Kirkendall marker planes. This establishes a systematic variation of diffusion coefficients with composition in a very wide composition range of the NiCoFeCrMn system in comparison to the data available on impurity diffusion coefficients in pure elements and tracer diffusion coefficients at the equiatomic composition estimated by the radiotracer method. Following, a physics-informed Neural Network based numerical inverse method is developed to extract composition-dependent diffusivities over the whole composition range of the diffusion couples.

Figures

Figures reproduced from arXiv: 2506.12345 by the authors.

Figure 7
Figure 7. Impurity diffusion coefficients available [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗

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Reference graph

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Pith tools

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