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Thermodynamics and phase stability of the AlTiCrMoW high-entropy alloy simulated using machine-learned interatomic potentials

Joseph E. Arnold1, Christopher D. Woodgate2, Roohollah Hafizi3, Matthew J. Harris1, Alessandro Mottura1, Gonzalo García Fuentes4, and Beñat Gurrutxaga-Lerma1,*

  • *Contact author: b.gurrutxagalerma.1@bham.ac.uk
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Phys. Rev. Materials 10, 073605 – Published 21 July, 2026

DOI: https://doi.org/10.1103/5tz1-zg9z

Abstract

High-entropy alloys (HEAs) possess exceptional mechanical properties and thermal stability, yet their vast compositional space and complex atomic arrangements present significant challenges for property prediction and design. To overcome this, we have trained a machine-learned interatomic potential (MLIP) for the AlTiCrMoW HEA using the message-passing atomic cluster expansion (MACE) framework. This new MLIP is trained on a series of high-accuracy density functional theory (DFT) calculations that covers the available configuration space of the HEA. The training set includes key configurations with linear-response-informed short-range ordered configurations, which are generated using a concentration wave analysis within the coherent potential approximation (CPA) and subsequently optimizedd using conventional DFT calculations. Phonon bispersion ands (PDBs) for the HEA are generated using the MACE model, and several physical properties are extracted for the HEA, such as the elastic constants and the bulk modulus. The fine-tuned model reproduces DFT structural and elastic benchmarks with high accuracy, at a significantly reduced computational cost, allowing for larger simulations to be performed, which can more accurately probe the configuration space and composition of the alloy. Here, we perform Monte Carlo based simulations to identify the ordering transition of equiatomic AlTiCrMoW, where we match the experimental value of 1273 K. Vacancy relaxation volumes are investigated for AlTiCrMoW, as an example of a property analysis, which would be computationally intractable via DFT.

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References (111)

  1. High-Entropy Alloys, edited by M. C. Gao, J.-W. Yeh, P. K. Liaw, and Y. Zhang (Springer International Publishing, Cham, 2016).
  2. D. B. Miracle and O. N. Senkov, A critical review of high entropy alloys and related concepts, Acta Mater. 122, 448 (2017).
  3. E. P. George, D. Raabe, and R. O. Ritchie, High-entropy alloys, Nat. Rev. Mater. 4, 515 (2019).
  4. J.-W. Yeh, S.-K. Chen, S.-J. Lin, J.-Y. Gan, T.-S. Chin, T.-T. Shun, C.-H. Tsau, and S.-Y. Chang, Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes, Adv. Eng. Mater. 6, 299 (2004).
  5. Q. F. He, Y. F. Ye, and Y. Yang, The configurational entropy of mixing of metastable random solid solution in complex multicomponent alloys, J. Appl. Phys. 120, 154902 (2016).
  6. P. I. Odetola, B. J. Babalola, A. E. Afolabi, U. S. Anamu, E. Olorundaisi, M. C. Umba, T. Phahlane, O. O. Ayodele, and P. A. Olubambi, Exploring high entropy alloys: A review on thermodynamic design and computational modeling strategies for advanced materials applications, Heliyon 10, e39660 (2024).
  7. S. Praveen and H. S. Kim, High‐entropy alloys: Potential candidates for high‐temperature applications – an overview, Adv. Eng. Mater. 20, 1700645 (2018).
  8. O. El-Atwani, N. Li, M. Li, A. Devaraj, J. K. S. Baldwin, M. M. Schneider, D. Sobieraj, J. S. Wróbel, D. Nguyen-Manh, S. A. Maloy, and E. Martinez, Outstanding radiation resistance of tungsten-based high-entropy alloys, Sci. Adv. 5, eaav2002 (2019).
  9. O. El Atwani, H. T. Vo, M. A. Tunes, C. Lee, A. Alvarado, N. Krienke, J. D. Poplawsky, A. A. Kohnert, J. Gigax, W.-Y. Chen, M. Li, Y. Q. Wang, J. S. Wróbel, D. Nguyen-Manh, J. K. S. Baldwin, O. U. Tukac, E. Aydogan, S. Fensin, and E. Martinez, A quinary WTaCrVHf nanocrystalline refractory high-entropy alloy withholding extreme irradiation environments, Nat. Commun. 14, 2516 (2023).
  10. B. Gludovatz, A. Hohenwarter, D. Catoor, E. H. Chang, E. P. George, and R. O. Ritchie, A fracture-resistant high-entropy alloy for cryogenic applications, Science 345, 1153 (2014).
  11. D. Liu, Q. Yu, S. Kabra, M. Jiang, P. Forna-Kreutzer, R. Zhang, M. Payne, F. Walsh, B. Gludovatz, M. Asta, A. M. Minor, E. P. George, and R. O. Ritchie, Exceptional fracture toughness of CrCoNi-based medium- and high-entropy alloys at 20 kelvin, Science 378, 978 (2022).
  12. B. Gorr, M. Azim, H.-J. Christ, T. Mueller, D. Schliephake, and M. Heilmaier, Phase equilibria, microstructure, and high temperature oxidation resistance of novel refractory high-entropy alloys, J. Alloys Compd. 624, 270 (2015).
  13. B. Gorr, M. Azim, H.-J. Christ, H. Chen, D. V. Szabo, A. Kauffmann, and M. Heilmaier, Microstructure evolution in a new refractory high-entropy alloy W-Mo-Cr-Ti-Al, Metall. Mater. Trans. A 47, 961 (2016).
  14. C. Zhang, F. Zhang, S. Chen, and W. Cao, Computational thermodynamics aided high-entropy alloy design, JOM 64, 839 (2012).
  15. S. Becker, A. Rahmel, M. Schorr, and M. Schütze, Mechanism of isothermal oxidation of the intel-metallic TiAl and of TiAl alloys, Oxid. Met. 38, 425 (1992).
  16. Y.-F. Kao, T.-J. Chen, S.-K. Chen, and J.-W. Yeh, Microstructure and mechanical property of as-cast, -homogenized, and -deformed AlxCoCrFeNi (0≤x≤2) high-entropy alloys, J. Alloys Compd. 488, 57 (2009).
  17. F. Tian, L. Delczeg, N. Chen, L. K. Varga, J. Shen, and L. Vitos, Structural stability of NiCoFeCrAlx high-entropy alloy from ab initio theory, Phys. Rev. B 88, 085128 (2013).
  18. C. D. Woodgate, G. A. Marchant, L. B. Pártay, and J. B. Staunton, Structure, short-range order, and phase stability of the AlxCrFeCoNi high-entropy alloy: Insights from a perturbative, DFT-based analysis, npj Comput. Mater. 10, 271 (2024).
  19. H. Lukas, S. G. Fries, and B. Sundman, Computational Thermodynamics: The Calphad Method (Cambridge university press, Cambridge, UK, 2007).
  20. Y.-M. Muggianu, M. Gambino, and J.-P. Bros, Enthalpies de formation des alliages liquides bismuth-étain-gallium à 723 k. Choix d'une représentation analytique des grandeurs d'excès intégrales et partielles de mélange, J. Chim. Phys. 72, 83 (1975).
  21. F. Kohler, Zur berechnung der thermodynamischen daten eines ternären systems aus den zugehörigen binären systemen: Kurze mitteilung, Monatsh. Chem. verwandte Teile anderer Wissenschaften 91, 738 (1960).
  22. G. Toop, Predicting ternary activities using binary data, Trans. TMS-AIME 223, 850 (1965).
  23. U. R. Kattner, The thermodynamic modeling of multicomponent phase equilibria, JOM 49, 14 (1997).
  24. M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, Iterative minimization techniques for ab initio total-energy calculations: Molecular dynamics and conjugate gradients, Rev. Mod. Phys. 64, 1045 (1992).
  25. R. O. Jones, Density functional theory: Its origins, rise to prominence, and future, Rev. Mod. Phys. 87, 897 (2015).
  26. J. Neugebauer and T. Hickel, Density functional theory in materials science, WIREs Comput. Mol. Sci. 3, 438 (2013).
  27. M. Finnis and J. Sinclair, A simple empirical n-body potential for transition metals, Philos. Mag. A 50, 45 (1984).
  28. R. A. Johnson, Alloy models with the embedded-atom method, Phys. Rev. B 39, 12554 (1989).
  29. M. I. Baskes, Modified embedded-atom potentials for cubic materials and impurities, Phys. Rev. B 46, 2727 (1992).
  30. C. Chen, W. Ye, Y. Zuo, C. Zheng, and S. P. Ong, Graph networks as a universal machine learning framework for molecules and crystals, Chem. Mater. 31, 3564 (2019).
  31. C. Chen and S. P. Ong, A universal graph deep learning interatomic potential for the periodic table, Nat. Comput. Sci. 2, 718 (2022).
  32. B. Deng, P. Zhong, K. Jun, J. Riebesell, K. Han, C. J. Bartel, and G. Ceder, Chgnet as a pretrained universal neural network potential for charge-informed atomistic modelling, Nat. Mach. Intell. 5, 1031 (2023).
  33. I. Batatia, P. Benner, Y. Chiang, A. M. Elena, D. P. Kovács, J. Riebesell, X. R. Advincula, M. Asta, M. Avaylon, W. J. Baldwin, F. Berger, N. Bernstein, A. Bhowmik, F. Bigi, S. M. Blau, V. Cărare, M. Ceriotti, S. Chong, J. P. Darby, S. De, et al., A foundation model for atomistic materials chemistry, J. Chem. Phys. 163, 184110 (2025).
  34. J. S. Smith, O. Isayev, and A. E. Roitberg, Ani-1: An extensible neural network potential with DFT accuracy at force field computational cost, Chem. Sci. 8, 3192 (2017).
  35. J. S. Smith, B. T. Nebgen, R. Zubatyuk, N. Lubbers, C. Devereux, K. Barros, S. Tretiak, O. Isayev, and A. E. Roitberg, Approaching coupled cluster accuracy with a general-purpose neural network potential through transfer learning, Nat. Commun. 10, 2903 (2019).
  36. O. Egorova, R. Hafizi, D. C. Woods, and G. M. Day, Multifidelity statistical machine learning for molecular crystal structure prediction, J. Phys. Chem. A 124, 8065 (2020).
  37. P. W. Butler, R. Hafizi, and G. M. Day, Machine-learned potentials by active learning from organic crystal structure prediction landscapes, J. Phys. Chem. A 128, 945 (2024).
  38. C. W. Rosenbrock, K. Gubaev, A. V. Shapeev, L. B. Pártay, N. Bernstein, G. Csányi, and G. L. W. Hart, Machine-learned interatomic potentials for alloys and alloy phase diagrams, npj Comput. Mater. 7, 24 (2021).
  39. F. Körmann, T. Kostiuchenko, A. Shapeev, and J. Neugebauer, B2 ordering in body-centered-cubic AlNbTiV refractory high-entropy alloys, Phys. Rev. Mater. 5, 053803 (2021).
  40. Y. Zhou, P. Srinivasan, F. Körmann, B. Grabowski, R. Smith, P. Goddard, and A. I. Duff, Thermodynamics up to the melting point in a TaVCrW high entropy alloy: Systematic abinitio study aided by machine learning potentials, Phys. Rev. B 105, 214302 (2022).
  41. I. Batatia, D. P. Kovacs, G. Simm, C. Ortner, and G. Csanyi, Mace: Higher order equivariant message passing neural networks for fast and accurate force fields, in Advances in Neural Information Processing Systems, edited by S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Curran Associates, Inc., 2022), Vol. 35, pp. 11423–11436.
  42. H. Kaur, F. Della Pia, I. Batatia, X. R. Advincula, B. X. Shi, J. Lan, G. Csányi, A. Michaelides, and V. Kapil, Data-efficient fine-tuning of foundational models for first-principles quality sublimation enthalpies, Faraday Discuss. 256, 120 (2025).
  43. P. Novelli, L. Bonati, P. J. Buigues, G. Meanti, L. Rosasco, M. Parrinello, and M. Pontil, Fine-tuning foundation models for molecular dynamics: A data-efficient approach with random features, Proc. Adv. Neural Inf. Process. Syst. (NeurIPS) (2024).
  44. J. S. Wróbel, M. R. Zemła, D. Nguyen-Manh, P. Olsson, L. Messina, C. Domain, T. Wejrzanowski, and S. L. Dudarev, Elastic dipole tensors and relaxation volumes of point defects in concentrated random magnetic Fe-Cr alloys, Comput. Mater. Sci. 194, 110435 (2021).
  45. O. Sherby, J. Robbins, and A. Goldberg, Calculation of activation volumes for self-diffusion and creep at high temperature, J. Appl. Phys. 41, 3961 (1970).
  46. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5tz1-zg9z for MACE training results, lattice parameter and lattice volume data, an account of how phonon calculations were performed, a sweep by sweep analysis of the Monte Carlo simulations, ASRO and total energy curves, and a statistical analysis of the vacancy formation.
  47. L. Shenoy, C. D. Woodgate, J. B. Staunton, A. P. Bartók, C. S. Becquart, C. Domain, and J. R. Kermode, Collinear-spin machine learned interatomic potential for Fe7Cr2Ni alloy, Phys. Rev. Mater. 8, 033804 (2024).
  48. J. Korringa, On the calculation of the energy of a Bloch wave in a metal, Physica 13, 392 (1947).
  49. W. Kohn and N. Rostoker, Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium, Phys. Rev. 94, 1111 (1954).
  50. H. Ebert, D. Ködderitzsch, and J. Minár, Calculating condensed matter properties using the KKR-Green's function method—recent developments and applications, Rep. Prog. Phys. 74, 096501 (2011).
  51. P. Soven, Coherent-potential model of substitutional disordered alloys, Phys. Rev. 156, 809 (1967).
  52. B. L. Gyorffy, Coherent-potential approximation for a nonoverlapping-muffin-tin-potential model of random substitutional alloys, Phys. Rev. B 5, 2382 (1972).
  53. G. M. Stocks, W. M. Temmerman, and B. L. Győrffy, Complete solution of the Korringa-Kohn-Rostoker coherent-potential-approximation equations: Cu-Ni alloys, Phys. Rev. Lett. 41, 339 (1978).
  54. J. S. Faulkner, G. M. Stocks, and Y. Wang, Multiple Scattering Theory: Electronic Structure of Solids, 1st ed. (IOP Publishing, Bristol, UK, 2018).
  55. M. Hoffmann, A. Ernst, W. Hergert, V. N. Antonov, W. A. Adeagbo, R. M. Geilhufe, and H. Ben Hamed, Magnetic and electronic properties of complex oxides from first‐principles, Physica Status Solidi (b) 257, 1900671 (2020).
  56. S. N. Khan, J. B. Staunton, and G. M. Stocks, Statistical physics of multicomponent alloys using KKR-CPA, Phys. Rev. B 93, 054206 (2016).
  57. C. D. Woodgate and J. B. Staunton, Compositional phase stability in medium-entropy and high-entropy Cantor-Wu alloys from an ab initio all-electron Landau-type theory and atomistic modeling, Phys. Rev. B 105, 115124 (2022).
  58. C. D. Woodgate and J. B. Staunton, Short-range order and compositional phase stability in refractory high-entropy alloys via first-principles theory and atomistic modeling: NbMoTa, NbMoTaW, and VNbMoTaW, Phys. Rev. Mater. 7, 013801 (2023).
  59. B. L. Gyorffy and G. M. Stocks, Concentration waves and Fermi surfaces in random metallic alloys, Phys. Rev. Lett. 50, 374 (1983).
  60. W. L. Bragg and E. J. Williams, The effect of thermal agitation on atomic arrangement in alloys, Proc. R. Soc. Lond. Ser. A 145, 699 (1934).
  61. W. L. Bragg and E. J. Williams, The effect of thermal agitaion on atomic arrangement in alloys-II, Proc. R. Soc. Lond. Ser. A 151, 540 (1935).
  62. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21, 1087 (1953).
  63. H. J. Naguszewski, L. B. Pártay, D. Quigley, and C. D. Woodgate, BraWl: Simulating the thermodynamics and phase stability of multicomponent alloys using conventional and enhanced sampling techniques, J. Open Source Software 10, 8346 (2025).
  64. C. D. Woodgate, H. J. Naguszewski, D. Redka, M. Ján, Q. David, and J. B. Staunton, Emergent B2 chemical orderings in the AlTiVNb and AlTiCrMo refractory high-entropy superalloys studied via first-principles theory and atomistic modelling, J. Phys.: Mater. 8, 045002 (2025).
  65. A. G. Khachaturyan, Theory of Structural Transformations in Solids (Dover, 2008).
  66. C. Chatfield, The Analysis of Time Series: Theory and Practice (Springer, 2013).
  67. Z. Gurskii and Y. Khokhlov, Atomic static displacements and their effect on the short range order in alloys, Condens. Matter Phys. 1, 389 (1998).
  68. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  69. G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996).
  70. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  71. R. Elmer, M. Berg, L. Carlen, B. Jakobsson, B. Noren, A. Oskarsson, G. Ericsson, J. Julien, T. F. Thorsteinsen, M. Guttormsen, G. Lovhoiden, V. Bellini, E. Grosse, C. Muntz, P. Senger, and L. Westerberg, K+ emission in symmetric heavy ion reactions at subthreshold energies [Phys. Rev. Lett. 77, 4884 (1996)], Phys. Rev. Lett. 78, 1396 (1997).
  72. G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  73. H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  74. G. Dusson, M. Bachmayr, G. Csányi, R. Drautz, S. Etter, C. van der Oord, and C. Ortner, Atomic cluster expansion: Completeness, efficiency and stability, J. Comput. Phys. 454, 110946 (2022).
  75. P. Brommer and F. Gähler, Potfit: Effective potentials from ab initio data, Modell. Simul. Mater. Sci. Eng. 15, 295 (2007).
  76. P. Brommer, A. Kiselev, D. Schopf, P. Beck, J. Roth, and H.-R. Trebin, Classical interaction potentials for diverse materials from ab initio data: A review of potfit, Modell. Simul. Mater. Sci. Eng. 23, 074002 (2015).
  77. P. Brommer and F. Gähler, Effective potentials for quasicrystals from ab-initio data, Philos. Mag. 86, 753 (2006).
  78. R. Jacobs, D. Morgan, S. Attarian, J. Meng, C. Shen, Z. Wu, C. Y. Xie, J. H. Yang, N. Artrith, B. Blaiszik, G. Ceder, K. Choudhary, G. Csanyi, E. D. Cubuk, B. Deng, R. Drautz, X. Fu, J. Godwin, V. Honavar, O. Isayev, et al., A practical guide to machine learning interatomic potentials – status and future, Curr. Opin. Solid State Mater. Sci. 35, 101214 (2025).
  79. R. Ryltsev and N. Chtchelkatchev, Deep machine learning potentials for multicomponent metallic melts: Development, predictability and compositional transferability, J. Mol. Liq. 349, 118181 (2022).
  80. A. Togo, L. Chaput, T. Tadano, and I. Tanaka, Implementation strategies in phonopy and phono3py, J. Phys.: Condens. Matter 35, 353001 (2023).
  81. A. Togo, First-principles phonon calculations with phonopy and Phono3py, J. Phys. Soc. Jpn. 92, 012001 (2023).
  82. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2021).
  83. P. Hirel, Atomsk: A tool for manipulating and converting atomic data files, Comput. Phys. Commun. 197, 212 (2015).
  84. J. M. Cowley, An approximate theory of order in alloys, Phys. Rev. 77, 669 (1950).
  85. J. M. Cowley, Short-range order and long-range order parameters, Phys. Rev. 138, A1384 (1965).
  86. F. Birch, Finite elastic strain of cubic crystals, Phys. Rev. 71, 809 (1947).
  87. A. Stukowski, Visualization and analysis of atomistic simulation data with ovito–the open visualization tool, Modell. Simul. Mater. Sci. Eng. 18, 015012 (2010).
  88. Z. Tang, M. C. Gao, H. Diao, T. Yang, J. Liu, T. Zuo, Y. Zhang, Z. Lu, Y. Cheng, Y. Zhang, et al., Aluminum alloying effects on lattice types, microstructures, and mechanical behavior of high-entropy alloys systems, JOM 65, 1848 (2013).
  89. M. C. Gao, C. Zhang, P. Gao, F. Zhang, L. Ouyang, M. Widom, and J. A. Hawk, Thermodynamics of concentrated solid solution alloys, Curr. Opin. Solid State Mater. Sci. 21, 238 (2017).
  90. P. Singh, A. Marshal, A. V. Smirnov, A. Sharma, G. Balasubramanian, K. G. Pradeep, and D. D. Johnson, Tuning phase stability and short-range order through Al doping in (CoCrFeMn)100−xAlx high-entropy alloys, Phys. Rev. Mater. 3, 075002 (2019).
  91. C. Li, J. Li, M. Zhao, and Q. Jiang, Effect of aluminum contents on microstructure and properties of AlxCoCrFeNi alloys, J. Alloys Compd. 504, S515 (2010).
  92. C. Freysoldt, B. Grabowski, T. Hickel, J. Neugebauer, G. Kresse, A. Janotti, and C. G. Van de Walle, First-principles calculations for point defects in solids, Rev. Mod. Phys. 86, 253 (2014).
  93. A. Goyal, S. R. Phillpot, G. Subramanian, D. A. Andersson, C. R. Stanek, and B. P. Uberuaga, Impact of homogeneous strain on uranium vacancy diffusion in uranium dioxide, Phys. Rev. B 91, 094103 (2015).
  94. W. Chen, X. Ding, Y. Feng, X. Liu, K. Liu, Z. Lu, D. Li, Y. Li, C. Liu, and X.-Q. Chen, Vacancy formation enthalpies of high-entropy FeCoCrNi alloy via first-principles calculations and possible implications to its superior radiation tolerance, J. Mater. Sci. Technol. 34, 355 (2018).
  95. S. E. Daigle and D. W. Brenner, Statistical approach to obtaining vacancy formation energies in high-entropy crystals from first principles calculations: Application to a high-entropy diboride, Phys. Rev. Mater. 4, 123602 (2020).
  96. S. Kang and A. Tamm, Density functional study of atomic arrangements in CrMnFeCoNi high-entropy alloy and their impact on vacancy formation energy and segregation, Comput. Mater. Sci. 230, 112456 (2023).
  97. C. Gao, S. Wang, X. Liu, and C. V. Singh, Defect energetics in an high-entropy alloy FCC CoCrFeMnNi, Mater. Adv. 5, 4231 (2024).
  98. A. Roy, P. Singh, G. Balasubramanian, and D. D. Johnson, Vacancy formation energies and migration barriers in multi-principal element alloys, Acta Mater. 226, 117611 (2022).
  99. X. Zhou, S. He, and J. Marian, Vacancy energetics and diffusivities in the equiatomic multielement Nb-Mo-Ta-W alloy, Materials 15, 5468 (2022).
  100. I. A. Zhuravlev, J. M. An, and K. D. Belashchenko, Microscopic first-principles model of strain-induced interaction in concentrated size-mismatched alloys, Phys. Rev. B 90, 214108 (2014).
  101. H. Song, F. Tian, Q.-M. Hu, L. Vitos, Y. Wang, J. Shen, and N. Chen, Local lattice distortion in high-entropy alloys, Phys. Rev. Mater. 1, 023404 (2017).
  102. H. Kanzaki, Point defects in face-centred cubic lattice—i distortion around defects, J. Phys. Chem. Solids 2, 24 (1957).
  103. Y. Ikeda, K. Gubaev, J. Neugebauer, B. Grabowski, and F. Körmann, Chemically induced local lattice distortions versus structural phase transformations in compositionally complex alloys, npj Comput. Mater. 7, 34 (2021).
  104. R. Nazarov, J. S. Majevadia, M. Patel, M. R. Wenman, D. S. Balint, J. Neugebauer, and A. P. Sutton, First-principles calculation of the elastic dipole tensor of a point defect: Application to hydrogen in α-zirconium, Phys. Rev. B 94, 241112(R) (2016).
  105. B. Gurrutxaga-Lerma and J. Verschueren, Elastic models of dislocations based on atomistic Kanzaki forces, Phys. Rev. B 98, 134104 (2018).
  106. R. W. Balluffi, Introduction to Elasticity Theory for Crystal Defects (World Scientific Publishing Company, 2016).
  107. B. Gurrutxaga-Lerma, Static and dynamic multipolar field expansions of dislocations and cracks in solids, Int. J. Eng. Sci. 128, 165 (2018).
  108. S. Dudarev and A. Sutton, Elastic interactions between nano-scale defects in irradiated materials, Acta Mater. 125, 425 (2017).
  109. C. Varvenne and E. Clouet, Elastic dipoles of point defects from atomistic simulations, Phys. Rev. B 96, 224103 (2017).
  110. P.-W. Ma and S. L. Dudarev, Universality of point defect structure in body-centered cubic metals, Phys. Rev. Mater. 3, 013605 (2019).
  111. J. E. Arnold, Mace model repository, 2025, https://github.com/J19Arnold/AlTiCrMoW-MACE-OMAT-model.git.

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Phys. Rev. Materials 10, 073605– Published 21 July, 2026
  • Received 31 October 2025
  • Accepted 1 July 2026
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