{"id":"75022490-3436-4b35-b4fd-4b3559eb63de","arxiv_id":"2506.12345","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Diffusion coefficients in a five-component high entropy alloy are extracted over a wide composition range by combining three strategic diffusion couples with a constraint-anchored physics-informed neural network.","lead":"Three diffusion couples plus a physics-informed neural network were used to map how fast each atom moves in the NiCoFeCrMn high-entropy alloy across nearly the whole Ni-Co-Fe composition range. The approach yields tracer, intrinsic and interdiffusion coefficients without radioactive tracers, and could speed up diffusion database building for multicomponent alloys.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Full-range extraction in Sec. 3.2 is underdetermined: 75 polynomial coefficients in 4 composition variables (Eq. 7b) are constrained only by three 1D diffusion paths and a few point constraints, with no held-out validation.","rationale":"The reader's weakest assumption identifies the same load-bearing issue: Eq. 7a/7b as a representation of composition dependence and the 75-parameter inversion are not validated with held-out data. My stress-test sharpens this into an identifiability problem: three 1D profiles and sparse point constraints cannot, by themselves, determine a 4D polynomial over a volume. This is the most load-bearing concern because the experimental part of the paper, the marker-plane estimates of tracer and intrinsic coefficients at three compositions, is internally consistent and agrees with literature values; the questionable step is precisely the extrapolation/interpolation from those point and curve data to the claimed full composition range. The paper's own Figs. 4-6 show that profile-only optimization is non-unique, but the constrained optimization is not shown to be unique, stable, or generalising. The use of self-averaged model outputs as equality constraints in Table 7 is an internal consistency check, not an independent anchor. No code, raw data, or uncertainty quantification is provided, so the inversion cannot be independently audited from the manuscript alone. The reader's CONDITIONAL verdict is appropriate, and I would keep it unchanged: the paper is not fatally flawed, but the central claim should be accepted only after an identifiability or held-out validation check of the type proposed. Credit is due for the experimental design, the marker-plane analysis, and the explicit demonstration that profile-only fits are non-unique; these are valuable regardless of the PINN-based full-range claim.","tokens_in":25765,"tokens_out":6710,"duration_ms":94729,"concrete_test":"Run a synthetic twin experiment: pick a known coefficient set in Eq. 7b, e.g. the authors' optimized coefficients; solve the forward Boltzmann-ODE problem for the exact DC1-DC3 end-member designs to generate noiseless profiles, then add realistic WDS noise; re-run the full PINN inversion with the same constraints. Quantify the recovery error D_recovered/D_true for each element on a grid inside the claimed composition box, excluding points lying on the three training paths. If the error exceeds about a factor of 2 at any composition inside the claimed range that lacks a marker or impurity constraint, the inversion is not identifiable and the central full-range claim is unsupported; if the error stays small, the polynomial ansatz is adequate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract and conclusion claim reliable extraction of tracer, intrinsic, and interdiffusion coefficients over the whole composition range of the couples, i.e. Ni, Co, Fe 0-100 at.%, Cr up to 20 at.%, Mn up to 15 at.%, from only three diffusion couples. The extraction rests on Eq. 7b: for each of the five elements, ln D_i^* is a 15-term polynomial in the four independent fractions Ni, Co, Fe, Cr, giving 75 trainable coefficients. The information supplied to the inversion is (i) three composition profiles, each a one-dimensional curve in the 4D composition space; (ii) tracer D_i^* at three Kirkendall marker planes; (iii) literature impurity/self-diffusivities near the pure Ni, Co, Fe end members; and (iv) the average values at the alloy end, which in Table 7 are themselves averages of the model's own outputs from the three couples. Three 1D curves plus a handful of isolated points do not determine a 4D polynomial over a volume: basis functions that vanish or are linearly dependent along the three profile curves are free, and different coefficient sets can give the same profile fits and the same point constraints. The paper demonstrates profile-only non-uniqueness in Figs. 4-6, but this does not establish uniqueness after constraints are added; it only shows that adding constraints changes the answer. Fig. 8 is an in-sample fit to exactly the profiles used in the data-loss term, so it is not a validation of interpolation to untested compositions. The smooth variation seen on the projected Gibbs triangle (Fig. 7) is partly an artifact of the smooth polynomial ansatz, not independent evidence. The comparison with radiotracer data at the equiatomic composition is suggestive, but it is not a held-out test and is not at the same composition as the alloy end member.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":26146,"tokens_out":4127,"duration_ms":51787,"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":[{"comment":"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.","section":"3.2 / Eq. (7b)"},{"comment":"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.","section":"3.2 / Table 7"},{"comment":"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.","section":"2.3.1 / Eq. (7a)"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Figures 4-6"},{"comment":"Table 3 cites references [50-59] while the text says [51-59]; the reference range and the in-text citation should be reconciled.","section":"Table 3"},{"comment":"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.","section":"2.2.7 / 3.2(iv)"},{"comment":"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.","section":"Supplementary material"}],"recommendation":"major_revision","confidential_remarks":"The experimental marker-plane dataset and the honest demonstration of profile-only non-uniqueness are valuable. The main risk is the overclaim of a validated full-range extraction: the 75-coefficient polynomial is underdetermined by three 1D paths and the alloy-side constraint is self-referential. The authors can address this by performing a held-out validation or by reformulating the claim as an interpolation tool with explicit uncertainty, but the current wording asserts reliability that the evidence does not support."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I want you to know two things about this paper: the experimental marker-plane data are worth taking seriously, and the claim that the PINN extracts reliable diffusivities over the whole composition range is not established. The three couples are well designed—Ni, Co, Fe each coupled to the same alloy—and the tracer diffusivities at the three Kirkendall planes are internally consistent, fall in the expected ordering, and line up with literature impurity and radiotracer values. The paper also shows, cleanly, that fitting only composition profiles gives non-unique tracer coefficients. That is a real caution for the field.\n\nThe soft spot is the whole-range extraction. The model has 75 polynomial coefficients for the five ln D* functions in four independent composition variables. The information is three one-dimensional diffusion paths, a few point constraints, and the boundary impurity values. That does not determine a 4D polynomial over a volume. The paper demonstrates profile-only non-uniqueness but never shows that adding the constraints makes the solution unique; it only shows that adding constraints changes the answer. The 'smooth' Gibbs-triangle trends are partly a consequence of the smooth polynomial ansatz. And one of the key constraints—the average tracer value at the alloy end—is an average of the model's own outputs from the three couples, then fed back as an equality constraint, which is circular. There is no held-out validation, no uncertainty quantification, and the final Fig. 8 is an in-sample fit. The comparison with the equiatomic radiotracer data is suggestive, but those compositions are close, not identical, and it is not a blind test.\n\nThese are not nitpicks. They bear directly on the headline claim. The experimental contribution and the method architecture are still valuable, and the paper is honest about the ill-posedness of profile-only fits. But the full-range mobility database should not be treated as reliable without an independent validation couple or a synthetic test with known diffusivities.\n\nWho is this for: anyone working on multicomponent diffusion, CALPHAD mobility databases, or PINN inverse methods. It deserves a serious referee: the experimental part should survive review, and the numerical claim can be fixed with a held-out couple and de-circularized constraints. I would recommend sending it to review with a clear request for those revisions.","headline":"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.","tokens_in":26685,"tokens_out":2335,"would_cite":false,"duration_ms":27559,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["66.30.-h"],"model":"deepseek-v4-flash","headline":"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.","keywords":["diffusion","high entropy alloy","NiCoFeCrMn","Kirkendall marker plane","tracer diffusion coefficients","physics-informed neural network","vacancy wind effect","mobility database"],"falsifier":"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.","tokens_in":25537,"feed_emoji":"🧪","tokens_out":11352,"duration_ms":130111,"temperature":0.7,"pith_summary":"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.","feed_headline":"Three diffusion couples map five-metal alloy diffusivity","feed_subtitle":"A physics-informed neural network recovers tracer, intrinsic, and interdiffusion coefficients without radioactive tracers.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the single-profile Kirkendall marker-plane method for estimating tracer and intrinsic diffusivities in ternary and multicomponent systems from one diffusion couple.","marker":"[11]"},{"why":"Provides the textbook relations for intrinsic flux, interdiffusion coefficients, and Kirkendall marker-plane analysis used throughout the method.","marker":"[2]"},{"why":"Gives the vacancy-wind factor and the cross-term connection between intrinsic and tracer diffusion coefficients used in the flux equations.","marker":"[47]"},{"why":"Provides the correlation between Onsager phenomenological coefficients and tracer diffusivities underlying the intrinsic diffusion equations.","marker":"[48]"},{"why":"Supplies the Sauer-Freise normalized composition variable used in the intrinsic flux calculation.","marker":"[44]"},{"why":"Is the open-source scientific machine-learning library used to implement the physics-informed neural network inverse solver.","marker":"[50]"},{"why":"Prior physics-informed neural network inverse method for diffusion coefficients in a multicomponent alloy, which this work extends to five components with experimental constraints.","marker":"[8]"},{"why":"Radiotracer tracer diffusivities at the equiatomic NiCoFeCrMn composition used to compare and anchor the extracted data.","marker":"[35]"},{"why":"Bulk radiotracer diffusion data in CoCrFeNi and CoCrFeMnNi used as a consistency check for the estimated tracer coefficients.","marker":"[33]"}],"fun_headline_variants":["Three couples unlock full diffusivity map","No radiotracers: three couples give full diffusivity","AI maps diffusion in five-metal alloy from 3 couples","Three couples cover 0–100% diffusivity in NiCoFeCrMn","Three couples, one network: full diffusivity map"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Three couples unlock full diffusivity map","No radiotracers: three couples give full diffusivity","AI maps diffusion in five-metal alloy from 3 couples","Three couples cover 0–100% diffusivity in NiCoFeCrMn","Three couples, one network: full diffusivity map"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000719,"raw_usage":{"total_tokens":3225,"prompt_tokens":940,"completion_tokens":2285,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":2203}},"tokens_in":556,"tokens_out":2285,"duration_ms":51348,"temperature":1.0,"reasoning_tokens":2203,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:52:46.566291+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Morral, Body-diagonal di;usion couples for hig h entropy alloys, Journal of Phase Equilibria and Di;usion, 39 (2018) 51–56","cited_arxiv_id":null,"evidence_quote":"Supplies the single-profile Kirkendall marker-plane method for estimating tracer and intrinsic diffusivities in ternary and multicomponent systems from one diffusion couple."},{"cited_title":"Melting was repeated 5-6 times for better remixing of the elements","cited_arxiv_id":null,"evidence_quote":"Provides the textbook relations for intrinsic flux, interdiffusion coefficients, and Kirkendall marker-plane analysis used throughout the method."},{"cited_title":"Mohan Muralikrishna, N","cited_arxiv_id":null,"evidence_quote":"Gives the vacancy-wind factor and the cross-term connection between intrinsic and tracer diffusion coefficients used in the flux equations."},{"cited_title":"van Loo, Multiphase di;usion in binary and t ernary solid state systems, Progress in solid state chemistry, 20 (1990) 47-99 45","cited_arxiv_id":null,"evidence_quote":"Provides the correlation between Onsager phenomenological coefficients and tracer diffusivities underlying the intrinsic diffusion equations."},{"cited_title":"Gaertner, K","cited_arxiv_id":null,"evidence_quote":"Supplies the Sauer-Freise normalized composition variable used in the intrinsic flux calculation."},{"cited_title":"Baheti and A","cited_arxiv_id":null,"evidence_quote":"Is the open-source scientific machine-learning library used to implement the physics-informed neural network inverse solver."},{"cited_title":"Kumar, A","cited_arxiv_id":null,"evidence_quote":"Prior physics-informed neural network inverse method for diffusion coefficients in a multicomponent alloy, which this work extends to five components with experimental constraints."},{"cited_title":"We have also shown such reliable outcomes following the augmented Kirkaldy Lane method by producing di; erent types of di;usion paths in 38 the same system [13]","cited_arxiv_id":null,"evidence_quote":"Radiotracer tracer diffusivities at the equiatomic NiCoFeCrMn composition used to compare and anchor the extracted data."},{"cited_title":"Esakkiraja, A","cited_arxiv_id":null,"evidence_quote":"Bulk radiotracer diffusion data in CoCrFeNi and CoCrFeMnNi used as a consistency check for the estimated tracer coefficients."}],"review_version":1}