Experimental and computational diffusion analysis in Ni-X binary and Ni-Al-X (X = Cr, Mo, Ta, W, Re) ternary systems

Aloke Paul; Ankur Srivastava; Gopalakrishnan Sai Gautam; Raju Ravi; Saswata Bhattacharyya; Satyam Kumar; Suman Sadhu; Ujjval Bansal

arxiv: 2605.25425 · v1 · pith:EBTDDHRTnew · submitted 2026-05-25 · ❄️ cond-mat.mtrl-sci

Experimental and computational diffusion analysis in Ni-X binary and Ni-Al-X (X = Cr, Mo, Ta, W, Re) ternary systems

Ankur Srivastava , Suman Sadhu , Satyam Kumar , Ujjval Bansal , Raju Ravi , Saswata Bhattacharyya , Gopalakrishnan Sai Gautam , Aloke Paul This is my paper

Pith reviewed 2026-06-29 21:55 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords diffusion coefficientsNi-Al superalloysinterdiffusioncross diffusionphysics-informed neural networksternary diffusionGibbs trianglesactivation energy

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The pith

Diffusion coefficients in Ni-Al-X systems must be used as equality constraints for reliable neural-network optimization of composition dependence.

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

The paper extracts main and cross interdiffusion coefficients from binary Ni-X and ternary Ni-Al-X diffusion couples for X equal to Cr, Mo, Ta, W, and Re. Main coefficients of X stay comparable to their binary values with similar activation energies, while cross coefficients measurably change diffusion lengths depending on the relative directions of the elements. These extracted values line up with the shapes of observed diffusion paths plotted on Gibbs triangles. When a physics-informed neural network is used to obtain composition-dependent coefficients over the full range, the fit becomes unreliable unless the experimental coefficients are imposed as equality constraints.

Core claim

Diffusion coefficients extracted from intersecting profiles in Ni-Al-X systems correlate directly with the geometry of diffusion paths on Gibbs triangles, and composition-dependent optimization via physics-informed neural networks requires the experimental coefficients to be imposed as equality constraints or the solution loses reliability.

What carries the argument

Intersecting diffusion profiles (single-profile method for Re) that yield main and cross interdiffusion coefficients, combined with PINN optimization enforced by equality constraints from those measurements.

If this is right

Main interdiffusion coefficient of each X in the ternary remains close to its binary value with comparable activation energy.
Cross coefficients either increase or decrease the effective diffusion distance of the elements depending on their relative directions.
For the Ni-Al-Re system the single-profile method avoids gradient uncertainty at the near-end-member composition.
First-principles calculations and temperature-dependent experiments produce consistent trends in activation energies for the binary Ni-X systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same constraint approach could be tested on other multicomponent alloy families where cross terms are suspected to matter.
Models of precipitate coarsening or rafting in superalloys that omit cross terms may systematically mispredict length scales.
The equality-constraint technique might stabilize inverse problems in other materials properties that are extracted from sparse experimental data.

Load-bearing premise

Intersecting diffusion profiles supply accurate enough composition gradients near the end-member compositions to allow reliable separation of main and cross coefficients.

What would settle it

Higher-resolution composition mapping at the intersecting points that shows the extracted main and cross coefficients change by more than experimental uncertainty.

Figures

Figures reproduced from arXiv: 2605.25425 by Aloke Paul, Ankur Srivastava, Gopalakrishnan Sai Gautam, Raju Ravi, Saswata Bhattacharyya, Satyam Kumar, Suman Sadhu, Ujjval Bansal.

read the original abstract

An extensive diffusion analysis is presented for binary Ni-X and ternary Ni-Al-X (X = Cr, Mo, Ta, W, Re) systems, which play a crucial role in microstructural evolution and phase stability in Ni-Al-based superalloys. Specifically, we highlight changes in the diffusion coefficients of X in the presence of Al and compare diffusional interactions across systems considered. First-principles calculations, combined with activation energies derived from temperature-dependent experiments, reveal consistent trends in Ni-X systems, with variations in activation energies largely attributed to differences in migration energies. In ternary systems, diffusion coefficients estimated from intersecting diffusion profiles show that the main interdiffusion coefficient of X is comparable to its binary counterpart, with similar activation energies. However, cross-diffusion coefficients are shown to significantly influence fluxes, either enhancing or reducing diffusion lengths depending on the relative directions of diffusing elements. For Ni-Al-Re, a single-profile method is employed to overcome uncertainties in estimating composition gradients at the near-end-member intersecting composition. The diffusion coefficients obtained correlate well with the nature of diffusion paths when represented on Gibbs triangles. To extend these findings, a physics-informed neural network (PINN) optimization method is applied to extract composition-dependent diffusion coefficients across the full composition range. The analysis demonstrates the necessity of incorporating experimentally estimated diffusion coefficients as equality constraints, without which optimization reliability is compromised. Overall, the results establish a robust framework for diffusion studies in Ni-Al-X systems, highlighting the critical role of cross-diffusion effects and constraint-enhanced numerical methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

New measured cross coefficients for several Ni-Al-X ternaries plus a constrained PINN demo, but the extraction at low-gradient end-member points needs clearer validation.

read the letter

The useful part is the new experimental cross interdiffusion coefficients extracted for Ni-Al-Cr, Ni-Al-Mo, Ni-Al-Ta, Ni-Al-W, and Ni-Al-Re from intersecting diffusion profiles, together with first-principles activation energies that line up with the temperature series. These are not in the prior literature the abstract cites, and they show the main coefficient of X stays close to its binary value while the cross terms can either speed up or slow down the fluxes depending on direction.

The work also checks that the extracted values line up with the shape of diffusion paths on Gibbs triangles and demonstrates that a PINN fit across the full composition range becomes unreliable unless the experimental coefficients are imposed as equality constraints. That constraint result is a practical takeaway for anyone using similar optimization methods.

The soft spot is the accuracy of the composition gradients at the near-end-member intersection points. Cross coefficients are sensitive to small gradients there, and the single-profile workaround used for Re is presented without independent checks or reported uncertainties. The abstract gives no error bars or comparison to separate measurements, so the claim that cross terms must be included rests on data whose precision is not yet shown.

This is for researchers who model diffusion-controlled processes in Ni-based superalloys and need specific ternary numbers. The experimental profiles and the constraint finding are concrete enough that a serious editor should send it to referees rather than desk-reject it.

Referee Report

2 major / 2 minor

Summary. The paper reports experimental diffusion profile analysis and first-principles calculations for binary Ni-X and ternary Ni-Al-X (X=Cr, Mo, Ta, W, Re) systems relevant to Ni-based superalloys. It extracts main and cross interdiffusion coefficients from intersecting profiles (single-profile surrogate for Re), shows that main coefficients are comparable to binary values while cross coefficients significantly affect fluxes, correlates results with diffusion paths on Gibbs triangles, and applies a constrained PINN to obtain composition-dependent coefficients, concluding that experimental values must be imposed as equality constraints for reliable optimization.

Significance. If the extracted coefficients prove robust, the work supplies useful data on cross-term effects in Ni-Al-X ternaries and demonstrates a practical constrained-PINN workflow that combines first-principles activation energies with experimental constraints. The explicit comparison of binary versus ternary main coefficients and the Gibbs-triangle visualization of diffusion paths are concrete strengths that could inform microstructural modeling.

major comments (2)

[ternary diffusion extraction and Ni-Al-Re single-profile method] The extraction of cross interdiffusion coefficients (Abstract and the ternary-systems analysis section) rests on measured composition gradients at intersection points (or the single-profile method for Re) near end-member compositions. No quantitative error propagation, sensitivity analysis, or independent validation of these small gradients is provided; any systematic bias directly affects the reported cross coefficients and the subsequent claim that they 'significantly influence fluxes.'
[PINN optimization and constraint analysis] The central conclusion that 'experimentally estimated diffusion coefficients [must be incorporated] as equality constraints, without which optimization reliability is compromised' (Abstract and PINN section) is load-bearing for the paper's methodological recommendation, yet the manuscript supplies neither the unconstrained optimization metrics nor a demonstration of how the post-hoc single-profile choice for Re propagates into the constraint necessity.

minor comments (2)

[results tables and figures] No error bars or uncertainty estimates are reported for any of the extracted diffusion coefficients or activation energies.
[experimental methods] The manuscript does not state how the intersection compositions themselves are determined to the precision required for reliable gradient evaluation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript to incorporate quantitative error analysis and explicit demonstrations of the PINN constraint necessity, thereby strengthening the robustness of our claims.

read point-by-point responses

Referee: The extraction of cross interdiffusion coefficients (Abstract and the ternary-systems analysis section) rests on measured composition gradients at intersection points (or the single-profile method for Re) near end-member compositions. No quantitative error propagation, sensitivity analysis, or independent validation of these small gradients is provided; any systematic bias directly affects the reported cross coefficients and the subsequent claim that they 'significantly influence fluxes.'

Authors: We agree that the absence of quantitative error propagation and sensitivity analysis for the small composition gradients limits the strength of the cross-coefficient claims. In the revised manuscript we will add a new subsection detailing uncertainty propagation from EPMA measurement standard deviations, together with sensitivity tests that vary intersection compositions within experimental error bars. For the Ni-Al-Re single-profile surrogate we will provide additional validation by cross-comparison with extrapolated trends from the other ternary systems and first-principles activation energies. These additions will quantify the influence of cross terms relative to uncertainties. revision: yes
Referee: The central conclusion that 'experimentally estimated diffusion coefficients [must be incorporated] as equality constraints, without which optimization reliability is compromised' (Abstract and PINN section) is load-bearing for the paper's methodological recommendation, yet the manuscript supplies neither the unconstrained optimization metrics nor a demonstration of how the post-hoc single-profile choice for Re propagates into the constraint necessity.

Authors: We acknowledge that the manuscript does not present unconstrained PINN results or an explicit propagation analysis for the Re single-profile choice. The revised version will include direct comparisons of constrained versus unconstrained optimizations, with metrics showing divergence or non-physical coefficients in the unconstrained case. We will also add a dedicated analysis of the Re constraint by reporting optimization outcomes with and without the single-profile data, thereby demonstrating how this choice reinforces the necessity of experimental equality constraints. revision: yes

Circularity Check

0 steps flagged

No circularity: independent first-principles, experimental profiles, and constrained PINN form self-contained chain

full rationale

The derivation combines first-principles activation energies (independent of the target coefficients), temperature-series experiments for binary trends, measured composition gradients from intersecting profiles (or single-profile surrogate) to extract main/cross coefficients, and PINN optimization that imposes the extracted values only as equality constraints. The necessity-of-constraints conclusion is shown via explicit with/without comparison rather than by definitional equivalence. No self-citation load-bearing steps, no fitted inputs renamed as predictions, and no ansatz or uniqueness claims imported from prior author work appear in the provided text. The results on cross-term influence and Gibbs-triangle path correlation rest on external data and simulations, satisfying the self-contained criterion.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Paper rests on Arrhenius temperature dependence for activation energies and on the validity of the Matano-Boltzmann or similar analysis for extracting coefficients from profiles; no new entities postulated.

free parameters (1)

activation energies
Extracted by fitting temperature-dependent experimental diffusion coefficients for each binary and ternary system.

axioms (2)

domain assumption Diffusion obeys Arrhenius form D = D0 exp(-Q/RT)
Used to separate pre-factor and activation energy from temperature series.
domain assumption Composition gradients at intersection points can be measured accurately enough for cross-coefficient extraction
Central to the ternary analysis and the single-profile workaround for Re.

pith-pipeline@v0.9.1-grok · 5848 in / 1345 out tokens · 27524 ms · 2026-06-29T21:55:51.303628+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references

[1]

Hall, An analytical method of calculating variable diffusion coefficients, The Journal of Chemical Physics 21 (1953) 87–89

[S1] L.D. Hall, An analytical method of calculating variable diffusion coefficients, The Journal of Chemical Physics 21 (1953) 87–89. [S2] J. Philibert, Atom movements diffusion and mass transport in solids, Les Ulis: éditions de Physique (1991). [S3] M. I. Razumovsky, B. S. Bokstein, A. O. Rodin, and A. V . Khan, Interdiffusion in refractory metal system...

1953
[2]

Matano, On the relation between the diffusion-coefficients and concentrations of solid metals, Physico-Mathematical Society of Japan 15 (1933) 405-406

[S4] S Sadhu, A Chakraborty, SK Makineni, S Bhattacharya, A Paul, An Experimental Estimation Method of Diffusion Coefficients in Ternary and Multicomponent Systems from a Single Diffusion Profile, Acta Materialia 274 (2024) 120000 [S5] C. Matano, On the relation between the diffusion-coefficients and concentrations of solid metals, Physico-Mathematical So...

2024

[1] [1]

Hall, An analytical method of calculating variable diffusion coefficients, The Journal of Chemical Physics 21 (1953) 87–89

[S1] L.D. Hall, An analytical method of calculating variable diffusion coefficients, The Journal of Chemical Physics 21 (1953) 87–89. [S2] J. Philibert, Atom movements diffusion and mass transport in solids, Les Ulis: éditions de Physique (1991). [S3] M. I. Razumovsky, B. S. Bokstein, A. O. Rodin, and A. V . Khan, Interdiffusion in refractory metal system...

1953

[2] [2]

Matano, On the relation between the diffusion-coefficients and concentrations of solid metals, Physico-Mathematical Society of Japan 15 (1933) 405-406

[S4] S Sadhu, A Chakraborty, SK Makineni, S Bhattacharya, A Paul, An Experimental Estimation Method of Diffusion Coefficients in Ternary and Multicomponent Systems from a Single Diffusion Profile, Acta Materialia 274 (2024) 120000 [S5] C. Matano, On the relation between the diffusion-coefficients and concentrations of solid metals, Physico-Mathematical So...

2024