Chemical complexity in high entropy alloys: A pair-interaction perspective

G. Malcolm Stocks; Jiaxin Zhang; Markus Eisenbach; Sirui Bi; Xianglin Liu; Yang Wang

arxiv: 1907.10223 · v1 · pith:I33TBV2Cnew · submitted 2019-07-24 · ❄️ cond-mat.mtrl-sci

Chemical complexity in high entropy alloys: A pair-interaction perspective

Xianglin Liu , Jiaxin Zhang , Sirui Bi , Yang Wang , G. Malcolm Stocks , Markus Eisenbach This is my paper

Pith reviewed 2026-05-24 17:05 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords high entropy alloyspair-interaction modelrefractory HEAsconfiguration energystrengthening mechanismfirst-principles calculationsenergy fluctuationlinear regression

0 comments

The pith

A pair-interaction model fits first-principles energies of refractory high-entropy alloys and yields element-pair parameters that distinguish strengthening effects.

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

The paper tests whether a simple sum of element-element pair interactions can reproduce the configuration energies of NbMoTaW and its vanadium- and titanium-containing variants. Linear regression on first-principles data produces a compact set of pair parameters that both predict energies accurately and separate the strengthening roles of vanadium versus titanium. From the same model an explicit expression for intrinsic energy fluctuation is obtained, offering direct guidance for future calculations and simulations of these alloys.

Core claim

The pair-interaction model demonstrates simplicity, robustness, and high accuracy in predicting the configuration energies of NbMoTaW, NbMoTaWV and NbMoTaWTi. The element-element pair interaction parameters obtained from the linear regression of first-principle data provide a new perspective to understand the strengthening mechanism in HEAs, as revealed by comparing the effects of adding vanadium and titanium. Using the pair-interaction model, an expression for the intrinsic energy fluctuation is derived, which provides guidance on both theoretical modeling and first principles calculation.

What carries the argument

The pair-interaction model, in which total configuration energy is written as a linear sum of independent element-element pair terms whose coefficients are extracted by regression against first-principles energies.

If this is right

The fitted pair parameters allow rapid estimation of energies for any configuration without repeating full first-principles runs.
Differences in vanadium and titanium pair interactions explain their distinct contributions to alloy strengthening.
The derived energy-fluctuation formula supplies an immediate estimate of configurational variance for use in thermodynamic modeling.
The same regression procedure can be repeated for other refractory compositions to map their pair-interaction landscapes.

Where Pith is reading between the lines

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

If the pair model continues to work when the number of elements increases, it would reduce the computational cost of screening new high-entropy compositions.
The element-specific pair values could be used as inputs for faster effective-medium or Monte-Carlo simulations of phase stability.
The approach might generalize to non-refractory high-entropy alloys if their higher-order interactions remain small.
Measuring how the regression residual grows with supercell size would quantify the practical limit of the pair approximation.

Load-bearing premise

The total energy of any atomic arrangement in these alloys is captured by summing independent pair interactions, with negligible contributions from three-body or higher-order terms.

What would settle it

A direct first-principles energy calculation for a large supercell configuration in NbMoTaWTi that deviates from the pair-model prediction by more than the typical regression residual.

Figures

Figures reproduced from arXiv: 1907.10223 by G. Malcolm Stocks, Jiaxin Zhang, Markus Eisenbach, Sirui Bi, Xianglin Liu, Yang Wang.

**Figure 2.** Figure 2: The EPI parameters for the three refractory HEAs. Results from the nearest to the 8-th nearest [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: The nearest-neighbor EPI parameters for the three refractory HEAs. The interactions are grouped [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: The log-log plot of the standard deviation of the averaged energies vs number of atoms in the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

read the original abstract

The recently proposed pair-interaction model is applied to study a series of refractory high entropy alloys. The results demonstrate the simplicity, robustness, and high accuracy of this model in predicting the configuration energies of NbMoTaW, NbMoTaWV and NbMoTaWTi. The element-element pair interaction parameters obtained from the linear regression of first-principle data provide a new perspective to understand the strengthening mechanism in HEAs, as revealed by comparing the effects of adding vanadium and titanium. Using the pair-interaction model, an expression for the intrinsic energy fluctuation is derived, which provides guidance on both theoretical modeling and first principles calculation.

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

Applies existing pair model to three refractory HEAs, reports fitted parameters and a derived fluctuation expression, but does not test whether pairs alone suffice.

read the letter

The main takeaway is that this paper takes the pair-interaction model from prior work, fits element-pair parameters to DFT data on NbMoTaW, NbMoTaWV, and NbMoTaWTi, and derives an expression for intrinsic energy fluctuation from that model. The specific parameters and the fluctuation formula are the concrete new pieces here. They also use the fits to compare how vanadium and titanium additions shift the interactions, which they tie to strengthening mechanisms. That part gives a practical angle for people screening refractory compositions. The model comes across as straightforward and they report that it matches the configuration energies for these alloys. The derivation of the fluctuation expression follows directly once the pair form is assumed, so that step is clean on its own terms. The soft spot is the untested assumption that higher-order interactions can be ignored. The abstract and claims rest on the pair sum being adequate, yet there is no comparison to a triplet-inclusive expansion or residual checks that would show whether the fit is missing systematic contributions. Because the same first-principles data serve for both fitting and accuracy assessment, the validation has some built-in dependence on the model itself. No error bars or cross-validation details appear in the summary. This work is aimed at computational materials researchers who already use or want to try simple pair models for high-entropy alloy energies, especially in the refractory space. A reader looking for transferable parameters or a quick way to estimate fluctuations would find the numbers and the formula useful. The results are specific enough and the derivation is explicit enough that the paper deserves referee attention even with the model limitations. I would send it out for peer review.

Referee Report

3 major / 2 minor

Summary. The manuscript applies a pair-interaction model to refractory high-entropy alloys NbMoTaW, NbMoTaWV, and NbMoTaWTi. Linear regression on first-principles DFT data is used to extract element-element pair interaction parameters; the work claims that this yields a simple, robust, and highly accurate description of configuration energies, offers new insight into strengthening mechanisms via comparison of V and Ti additions, and derives an analytic expression for intrinsic energy fluctuation to guide further modeling and calculations.

Significance. If the pair approximation is shown to be sufficient, the approach supplies a low-parameter, transferable framework for configuration energetics in refractory HEAs and a concrete route to connect pair parameters to strengthening trends. The derived fluctuation expression is a useful theoretical byproduct that could inform both analytic models and the design of DFT sampling sets.

major comments (3)

[Abstract] Abstract and results sections: the central claim of 'high accuracy' and 'robustness' for the three alloys rests on linear regression fits, yet no error bars, cross-validation statistics, or performance on held-out configurations are reported, leaving the quantitative support for the accuracy assertion unclear.
[Model derivation] Model and validation sections: the expression for intrinsic energy fluctuation is obtained directly from the same pair-interaction Hamiltonian whose parameters were regressed against the DFT data used to demonstrate accuracy; this creates a degree of circularity that must be addressed by an explicit transferability test on an independent set of configurations or compositions.
[Results] Results on NbMoTaW, NbMoTaWV, NbMoTaWTi: no comparison is presented to a cluster-expansion model that includes triplet or higher-order interactions, nor is a residual analysis versus composition or local environment provided to demonstrate that three-body and higher terms remain negligible across the sampled configurations.

minor comments (2)

[Methods] Notation for the pair-interaction parameters should be defined once in a dedicated subsection and used consistently; occasional redefinition in the text reduces readability.
[Figures] Figure captions should explicitly state the number of configurations used for fitting versus validation and the DFT settings (k-point mesh, cutoff, etc.) employed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each of the major comments below and indicate the revisions we will make to strengthen the work.

read point-by-point responses

Referee: [Abstract] Abstract and results sections: the central claim of 'high accuracy' and 'robustness' for the three alloys rests on linear regression fits, yet no error bars, cross-validation statistics, or performance on held-out configurations are reported, leaving the quantitative support for the accuracy assertion unclear.

Authors: We agree that providing quantitative measures of the fit quality is essential to support the claims of high accuracy and robustness. In the revised manuscript, we will include the root mean square error (RMSE) of the linear regression fits, error bars on the extracted pair interaction parameters, results from cross-validation (e.g., k-fold), and the model's performance on a held-out set of configurations not included in the training data. revision: yes
Referee: [Model derivation] Model and validation sections: the expression for intrinsic energy fluctuation is obtained directly from the same pair-interaction Hamiltonian whose parameters were regressed against the DFT data used to demonstrate accuracy; this creates a degree of circularity that must be addressed by an explicit transferability test on an independent set of configurations or compositions.

Authors: While the fluctuation expression is analytically derived from the pair Hamiltonian and is independent of the specific fitted values, we acknowledge the referee's point regarding potential circularity in validation. To address this, we will add an explicit transferability test by applying the model to an independent set of configurations for at least one alloy and comparing to additional DFT calculations. revision: yes
Referee: [Results] Results on NbMoTaW, NbMoTaWV, NbMoTaWTi: no comparison is presented to a cluster-expansion model that includes triplet or higher-order interactions, nor is a residual analysis versus composition or local environment provided to demonstrate that three-body and higher terms remain negligible across the sampled configurations.

Authors: We note that performing a complete cluster expansion with higher-order terms is outside the primary scope of this study, which aims to demonstrate the effectiveness of the pair-interaction model. However, we will incorporate a residual analysis plotting the model errors against composition and local environment descriptors to confirm that higher-order contributions are negligible within the sampled data. revision: partial

Circularity Check

2 steps flagged

Fitted pair parameters used to 'predict' energies and derive fluctuation expression by construction

specific steps

fitted input called prediction [Abstract]
"The results demonstrate the simplicity, robustness, and high accuracy of this model in predicting the configuration energies of NbMoTaW, NbMoTaWV and NbMoTaWTi. The element-element pair interaction parameters obtained from the linear regression of first-principle data provide a new perspective to understand the strengthening mechanism in HEAs"

Parameters are obtained from linear regression on the first-principles configuration energies; the reported 'prediction' accuracy is therefore the regression fit quality on the same data by construction rather than an out-of-sample test or independent derivation.
self definitional [Abstract]
"Using the pair-interaction model, an expression for the intrinsic energy fluctuation is derived, which provides guidance on both theoretical modeling and first principles calculation."

The fluctuation expression is obtained by direct algebraic manipulation of the pair-interaction energy formula whose parameters were fitted to the target data; it therefore follows tautologically from the model definition rather than constituting an additional empirical result.

full rationale

The paper fits pair-interaction parameters via linear regression to first-principles configuration energies, then presents the model's reproduction of those energies as demonstrating 'high accuracy' and 'robustness' in prediction. It further derives an intrinsic energy fluctuation expression directly from the same fitted pair model. These steps create partial dependence on the fitted inputs rather than independent validation, but the core model application and comparison to adding V/Ti retain independent content outside the fit itself. No self-citation chain or uniqueness theorem is load-bearing.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the pair-interaction model being sufficient for these alloys and on the linear regression accurately extracting parameters from first-principles data; the fluctuation expression follows mathematically once the model is accepted.

free parameters (1)

element-element pair interaction parameters
Obtained from linear regression of first-principles configuration energies for the studied alloys; each pair type (e.g., Nb-Mo) receives its own fitted value.

axioms (1)

domain assumption Configuration energy of the refractory HEAs can be expressed as a sum of independent pair interactions.
Invoked when the recently proposed pair-interaction model is applied to the three alloys.

pith-pipeline@v0.9.0 · 5644 in / 1296 out tokens · 28316 ms · 2026-05-24T17:05:50.450590+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Yeh, S.-K

J.-W. Yeh, S.-K. Chen, S.-J. Lin, J.-Y. Gan, T.-S. Chin, T.-T. Shun, C.-H. Tsau, S.- Y. Chang, Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes, Advanced Engineering Materials 6 (2004) 299–303

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Cantor, I

B. Cantor, I. Chang, P. Knight, A. Vincent, Microstructural development in equiatomic multicomponent alloys, Materials Science and Engineering: A 375-377 (2004) 213 – 218

work page 2004

[3] [3]

Gludovatz, A

B. Gludovatz, A. Hohenwarter, D. Catoor, E. H. Chang, E. P. George, R. O. Ritchie, A fracture-resistant high-entropy alloy for cryogenic applications, Science 345 (2014) 1153–1158. 9

work page 2014

[4] [4]

Gludovatz, A

B. Gludovatz, A. Hohenwarter, K. V. S. Thurston, H. Bei, Z. Wu, E. P. George, R. O. Ritchie, Exceptional damage-tolerance of a medium-entropy alloy CrCoNi at cryogenic temperatures, Nature Communications 7 (2016) 10602

work page 2016

[5] [5]

Senkov, G

O. Senkov, G. Wilks, J. Scott, D. Miracle, Mechanical properties of Nb25Mo25Ta25W25 and V20Nb20Mo20Ta20W20 refractory high entropy alloys, Intermetallics 19 (2011) 698 – 706

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Z. Fu, L. Jiang, J. L. Wardini, B. E. MacDonald, H. Wen, W. Xiong, D. Zhang, Y. Zhou, T. J. Rupert, W. Chen, E. J. Lavernia, A high-entropy alloy with hierarchical nanopre- cipitates and ultrahigh strength, Science Advances 4 (2018)

work page 2018

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O. N. Senkov, D. B. Miracle, K. J. Chaput, J.-P. Couzinie, Development and exploration of refractory high entropy alloys-A review, Journal of Materials Research 33 (2018) 3092–3128

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Yeh, Alloy design strategies and future trends in high-entropy alloys, JOM 65 (2013) 1759–1771

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work page 2016

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S. Wu, G. Wang, Q. Wang, Y. Jia, J. Yi, Q. Zhai, J. Liu, B. Sun, H. Chu, J. Shen, P. Liaw, C. Liu, T. Zhang, Enhancement of strength-ductility trade-oﬀ in a high-entropy alloy through a heterogeneous structure, Acta Materialia 165 (2019) 444 – 458

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Kikuchi, A theory of cooperative phenomena, Phys

R. Kikuchi, A theory of cooperative phenomena, Phys. Rev. 81 (1951) 988–1003

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J. Sanchez, F. Ducastelle, D. Gratias, Generalized cluster description of multicomponent systems, Physica A: Statistical Mechanics and its Applications 128 (1984) 334 – 350. 10

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van de Walle, G

A. van de Walle, G. Ceder, Automating ﬁrst-principles phase diagram calculations, Journal of Phase Equilibria 23 (2002) 348

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Widom, Modeling the structure and thermodynamics of high-entropy alloys, Journal of Materials Research 33 (2018) 2881–2898

M. Widom, Modeling the structure and thermodynamics of high-entropy alloys, Journal of Materials Research 33 (2018) 2881–2898

work page 2018

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Smith, M

T. Smith, M. Hooshmand, B. Esser, F. Otto, D. McComb, E. George, M. Ghazisaeidi, M. Mills, Atomic-scale characterization and modeling of 60 ◦ dislocations in a high- entropy alloy, Acta Materialia 110 (2016) 352 – 363

work page 2016

[17] [17]

X. Liu, Z. Pei, M. Eisenbach, Dislocation core structures and peierls stresses of the high- entropy alloy nicofecrmn and its subsystems, Materials & Design 180 (2019) 107955

work page 2019

[18] [18]

Ikeda, F

Y. Ikeda, F. K¨ ormann, I. Tanaka, J. Neugebauer, Impact of chemical ﬂuctuations on stacking fault energies of CrCoNi and CrMnFeCoNi high entropy alloys from ﬁrst principles, Entropy 20 (2018)

work page 2018

[19] [19]

J. Ding, Q. Yu, M. Asta, R. O. Ritchie, Tunable stacking fault energies by tailoring local chemical order in CrCoNi medium-entropy alloys, Proceedings of the National Academy of Sciences 115 (2018) 8919–8924

work page 2018

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S. Zhao, Y. Osetsky, G. M. Stocks, Y. Zhang, Local-environment dependence of stacking fault energies in concentrated solid-solution alloys, npj Computational Materials 5 (2019) 13

work page 2019

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Varvenne, A

C. Varvenne, A. Luque, W. A. Curtin, Theory of strengthening in fcc high entropy alloys, Acta Materialia 118 (2016) 164 – 176

work page 2016

[22] [22]

C. R. LaRosa, M. Shih, C. Varvenne, M. Ghazisaeidi, Solid solution strengthening theories of high-entropy alloys, Materials Characterization 151 (2019) 310 – 317

work page 2019

[23] [23]

Yoshida, T

S. Yoshida, T. Ikeuchi, T. Bhattacharjee, Y. Bai, A. Shibata, N. Tsuji, Eﬀect of ele- mental combination on friction stress and Hall-Petch relationship in face-centered cubic high / medium entropy alloys, Acta Materialia 171 (2019) 201 – 215. 11

work page 2019

[24] [24]

Zhang, Y

L. Zhang, Y. Xiang, J. Han, D. J. Srolovitz, The eﬀect of randomness on the strength of high-entropy alloys, Acta Materialia 166 (2019) 424 – 434

work page 2019

[25] [25]

X. Liu, J. Zhang, M. Eisenbach, Y. Wang, Machine learning modeling of high entropy alloy: the role of short-range order, arXiv e-prints (2019) arXiv:1906.02889

work page internal anchor Pith review Pith/arXiv arXiv 2019

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Pei, Theory of the energy ﬂuctuation of multicomponent alloys, Scripta Materialia 162 (2019) 503 – 506

Z. Pei, Theory of the energy ﬂuctuation of multicomponent alloys, Scripta Materialia 162 (2019) 503 – 506

work page 2019

[27] [27]

Zhang, H

Z. Zhang, H. Sheng, Z. Wang, B. Gludovatz, Z. Zhang, E. P. George, Q. Yu, S. X. Mao, R. O. Ritchie, Dislocation mechanisms and 3d twin architectures generate excep- tional strength-ductility-toughness combination in crconi medium-entropy alloy, Nature Communications 8 (2017) 14390

work page 2017

[28] [28]

A high-bias, low-variance introduction to Machine Learning for physicists

P. Mehta, M. Bukov, C.-H. Wang, A. r. G. R. Day, C. Richardson, C. K. Fisher, D. J. Schwab, A high-bias, low-variance introduction to Machine Learning for physicists, arXiv e-prints (2018) arXiv:1803.08823

work page internal anchor Pith review Pith/arXiv arXiv 2018

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S. Mu, S. Wimmer, S. Mankovsky, H. Ebert, G. Stocks, Inﬂuence of local lattice distortions on electrical transport of refractory high entropy alloys, Scripta Materialia 170 (2019) 189 – 194. 12

work page 2019