- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Thermodynamic Model for Thermomigration in Metals

Daniel J. Long*, Edmund Tarleton, Alan C. F. Cocks, and Felix Hofmann†

  • *Contact author: daniel.long@eng.ox.ac.uk
  • †Contact author: felix.hofmann@eng.ox.ac.uk
PDF

Phys. Rev. Lett. 137, 096301 – Published 24 August, 2026

DOI: https://doi.org/10.1103/196y-5h6h

Abstract

We investigate the mechanisms involved in the thermomigration of interstitial hydrogen in metals. Using irreversible thermodynamics, we develop a comprehensive mechanistic model to capture the controlling effects. Crucially, through validation against published experimental data, our results demonstrate that an electron-wind effect plays a significant role, particularly for materials in which the thermomigration direction matches the heat flux. These findings provide new insights into the factors that affect the localization of solutes in metals. Moreover, our results indicate that atomistic models may be inadequate for detailed thermomigration studies due to the omission of electronic effects.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (66)

  1. O. D. González and R. A. Oriani, Thermal diffusion of dissolved hydrogen isotopes in iron and nickel, Trans. Metall. Soc. AIME 233, 10 (1965).
  2. J. Lufrano and P. Sofronis, Enhanced hydrogen concentrations ahead of rounded notches and cracks—competition between plastic strain and hydrostatic stress, Acta Mater. 46, 1519 (1998).
  3. E. Martínez-Pañeda, C. F. Niordson, and R. P. Gangloff, Strain gradient plasticity-based modeling of hydrogen environment assisted cracking, Acta Mater. 117, 321 (2016).
  4. T. Depover, S. Hertelé, and K. Verbeken, The effect of hydrostatic stress on the hydrogen induced mechanical degradation of dual phase steel: A combined experimental and numerical approach, Eng. Fract. Mech. 221, 106704 (2019).
  5. E. Elmukashfi, E. Tarleton, and A. C. F. Cocks, A modelling framework for coupled hydrogen diffusion and mechanical behaviour of engineering components, Comput. Mech. 66, 189 (2020).
  6. H. Abdolvand, Progressive modelling and experimentation of hydrogen diffusion and precipitation in anisotropic polycrystals, Int. J. Plast. 116, 39 (2019).
  7. Y. Liu, S. El Chamaa, M. R. Wenman, C. M. Davies, and F. P. E. Dunne, Hydrogen concentration and hydrides in Zircaloy-4 during cyclic thermomechanical loading, Acta Mater. 221, 117368 (2021).
  8. T. Das, R. Chakrabarty, J. Song, and S. Yue, Understanding microstructural influences on hydrogen diffusion characteristics in martensitic steels using finite element analysis (FEA), Int. J. Hydrogen Energy 47, 1343 (2022).
  9. A. Tondro and H. Abdolvand, On the effects of texture and microstructure on hydrogen transport towards notch tips: A CPFE study, Int. J. Plast. 152, 103234 (2022).
  10. A. Tondro, M. Taherijam, and H. Abdolvand, Diffusion and redistribution of hydrogen atoms in the vicinity of localized deformation zones, Mech. Mater. 177, 104544 (2023).
  11. M. Malo, C. Moreno, A. Morono, and I. Penalva, Thermo-migration of hydrogen isotopes in EUROFER and AISI 316L steel, J. Nucl. Mater. 606, 155629 (2025).
  12. D. Dasgupta, S. Blondel, E. Martinez, D. Maroudas, and B. D. Wirth, Impact of soret effect on hydrogen and helium retention in PFC tungsten under ELM-like conditions, Nucl. Fusion 63, 076029 (2023).
  13. K. Hashizume, K. Ogushi, T. Otsuka, and T. Tanabe, Thermomigration of tritium in V-4Cr-4Ti alloy, J. Nucl. Mater. 417, 1175 (2011).
  14. P. Lapka, M. Seredynski, and A. Cwik, Preliminary study on supercritical hydrogen and bleed air heat exchanger for aircraft application, Proceedings of the Institution of Mechanical Engineers, Part G (Journal of Aerospace Engineering) 232, 2231 (2018).
  15. A. C. Patrao, I. Jonsson, C. Xisto, A. Lundbladh, and T. Gronstedt, Compact heat exchangers for hydrogen-fueled aero engine intercooling and recuperation, Appl. Therm. Eng. 243, 122538 (2024).
  16. H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (John Wiley and Sons, New York, 1985).
  17. D. Guedes, A. Oudriss, S. Frappart, G. Courlit, S. Cohendoz, P. Girault, J. Creus, J. Bouhattate, A. Metsue, F. Thebault, L. Delattre, D. Koschel, and X. Feaugas, The influence of hydrostatic stress states on the hydrogen solubility in martensitic steels, Scr. Mater. 84–85, 23 (2014).
  18. J. O’M Bogkris, W. Beck, M. A. Genshaw, P. K. Subramanyan, and F. S. Williams, The effect of stress on the chemical potential of hydrogen in iron and steel, Acta Metall. 19, 1209 (1971).
  19. D. T. Peterson and H. M. Herro, Partial molar volumes of hydrogen and deuterium in niobium, vanadium, and tantalum, Metall. Trans. A 14, 17 (1983).
  20. B. F. Kammenzind, B. M. Berquist, R. Bajaj, P. H. Kreyns, and D. G. Franklin, The long range migration of hydrogen through Zircaloy in response to tensile and compressive stress gradients, Technical Report, Bettis Atomic Power Lab., West Mifflin, PA (United States), 1998.
  21. B. Gault, A. Chiaramonti, O. Cojocaru-Miredin, P. Stender, R. Dubosq, C. Freysoldt, S. K. Makineni, T. Li, M. Moody, and J. M. Cairney, Atom probe tomography, Nat. Rev. Methods Primers 1, 51 (2021).
  22. M. S. Veshchunov, V. E. Shestak, and V. D. Ozrin, A new model of hydrogen redistribution in Zr alloy claddings during waterside corrosion in a temperature gradient, J. Nucl. Mater. 472, 65 (2016).
  23. M. Sugisaki, H. Furuya, H. f. Sekiya, and K. Hashizume, Thermal diffusion of tritium and protium in alpha phase of zirconium, Fusion Technol. 14, 723 (1988).
  24. H. Maki and M. Sato, Thermal diffusion of hydrogen in Zircaloy-2 containing hydrogen beyond terminal solid solubility, J. Nucl. Sci. Technol. 12, 637 (1975).
  25. F. Nagase and T. Fuketa, Investigation of hydride rim effect on failure of Zircaloy-4 cladding with tube burst test, J. Nucl. Sci. Technol. 42, 58 (2005).
  26. W. Beck, P. Blanpain, T. Fuketa, A. Gorzel, Z. Hozer, K. Kamimura, Y.-H. Koo, D. Maertens, O. Nechaeva, M. Petit et al., Nuclear Fuel Safety Criteria Technical Review, OECD Nuclear Energy Agency, 2012.
  27. D. Kamerman, M. Bachhav, T. Yao, X. Pu, and J. Burns, Formation and characterization of hydride rim structures in Zircaloy-4 nuclear fuel cladding tubes, J. Nucl. Mater. 586, 154675 (2023).
  28. A. Sawatzky, Hydrogen in Zircaloy-2: Its distribution and heat of transport, J. Nucl. Mater. 2, 321 (1960).
  29. J. B. J. Chapman, M. R. Gilbert, and D. Nguyen-Manh, Ab initio investigation into the stability of hydrogen isotopes (protium, deuterium, and tritium) in α-Fe and dilute FeCr alloys, Phys. Rev. Mater. 9, 035401 (2025).
  30. R. Kirchheim and A. Pundt, 25—hydrogen in metals, in Physical Metallurgy (Fifth Edition), edited by David E. Laughlin and Kazuhiro Hono (Elsevier, Oxford, 2014), pp. 2597–2705.
  31. P. L. de Andres, J. Sanchez, and A. Ridruejo, Hydrogen in alpha-iron: Role of phonons in the diffusion of interstitials at high temperature, Sci. Rep. 9, 12127 (2019).
  32. D. G. Fedorov, Partitioning of the vibrational free energy, J. Phys. Chem. Lett. 12, 6628 (2021).
  33. A. Campero and J. A. D. Ponce, Relationship between the atomic structure and electrochemistry. 1. Electric force, standard reduction potential, and standard reaction Gibbs free energy, ACS Omega 5, 12046 (2020).
  34. M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis (Cambridge University Press, Cambridge, England, 2007).
  35. Z. Zhang, J. Peeters, V. Popovich, and C. Ayas, Combined effects of stress and temperature on hydrogen diffusion in non-hydride forming alloys applied in gas turbines, Int. J. Hydrogen Energy 47, 30687 (2022).

  36. We note that, in Zhang et al. [35], there is an error in this equation, as the concentration, CL, is used as the argument of the natural logarithm. Here we use the correct flux definition.

  37. G. R. Longhurst, The Soret effect and its implications for fusion reactors, J. Nucl. Mater. 131, 61 (1985).
  38. L. Onsager, Reciprocal relations in irreversible processes. I, Phys. Rev. 37, 405 (1931).
  39. L. Onsager, Reciprocal relations in irreversible processes. II, Phys. Rev. 38, 2265 (1931).
  40. See Supplemental Material at http://link.aps.org/supplemental/10.1103/196y-5h6h for equivalent formulations of heat flux in irreversible thermodynamics and the resulting particle and heat flux equations.
  41. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics North-Holland Series in Physics (North-Holland, Amsterdam, 1962).
  42. J. R. G. Da Silva, S. W. Stafford, and Rex B. McLellan, The thermodynamics of the hydrogen-iron system, J. Less Common Metals 49, 407 (1976).
  43. J. R. Bates and D. H. Andrews, Fundamental frequencies, interatomic forces and molecular properties, Proc. Natl. Acad. Sci. U.S.A. 14, 124 (1928).
  44. H. Wipf, Electro-and thermotransport of hydrogen in metals, in Hydrogen in Metals II: Application-Oriented Properties (Springer, New York, 2005), pp. 273–304.
  45. K. Wirtz, Zur kinetischen theorie der thermodiffusion im kristallgitter, Z. Phys. 44, 221 (1943).
  46. S. Zhang, S. Zhu, F. Meng, and S. Ogata, Hydrogen and vacancy concentrations in alpha-iron under high hydrogen gas pressure and external stress: A first-principles neural network simulation study, Int. J. Hydrogen Energy 90, 246 (2024).
  47. V. B. Fiks, On the thermodiffusion mechanism in fluids, Sov. Phys.-Solid State 3, 724 (1961).
  48. H. B. Huntington, Driving forces for thermal mass transport, J. Phys. Chem. Solids 29, 1641 (1968).
  49. R. A. Oriani, Thermomigration in solid metals, J. Phys. Chem. Solids 30, 339 (1969).
  50. B. A. Kolesov, How the vibrational frequency varies with temperature, J. Raman Spectrosc. 48, 323 (2017).
  51. A. Juan and R. Hoffmann, Hydrogen on the Fe(110) surface and near bulk BCC Fe vacancies: A comparative bonding study, Surface Sci. 421, 1 (1999).
  52. Y. Itsumi and D. E. Ellis, Electronic bonding characteristics of hydrogen in BCC iron: Part I. Interstitials, J. Mater. Res. 11, 2206 (1996).
  53. M. H. Weng, H.-T. Chen, Y.-C. Wang, S.-P. Ju, J.-G. Chang, and M.-C. Lin, Kinetics and mechanisms for the adsorption, dissociation, and diffusion of hydrogen in Ni and Ni/YSZ slabs: A DFT study, Langmuir 28, 5596 (2012).
  54. M. Perrut, Thermodynamic modeling by the calphad method and its applications to innovative materials, Aerospace Lab 9, 1 (2015).
  55. A. M. Baro and W. Erley, The chemisorption of hydrogen on a (110) iron crystal studied by vibrational spectroscopy (EELS), Surf. Sci. Lett. 112, L759 (1981).
  56. F. Hochard, H. Jobic, J. Massardier, and A. J. Renouprez, Gas phase hydrogenation of acetonitrile on raney nickel catalysts: Reactive hydrogen, J. Mol. Catal. A: Chem. 95, 165 (1995).
  57. R. A. Secco, Thermal conductivity and seebeck coefficient of Fe and Fe-Si alloys: Implications for variable Lorenz number, Phys. Earth Planet. Interiors 265, 23 (2017).
  58. S. Haupt, F. Edler, M. Bartel, and H.-F. Pernau, Van der Pauw device used to investigate the thermoelectric power factor, Rev. Sci. Instrum. 91, 115102 (2020).
  59. D. Wagenknecht, I. Turek, and K. Carva, Ab initio calculations of temperature dependent resistivity for transition metals, WDS 15, 42 (2015).
  60. R. Singh, M. Ahmad, A. Caceres, R. Eybel, and M. Medraj, The influence of defects on hydrogen-induced electrical resistivity changes in Fe-based systems: A first-principles study, Int. J. Hydrogen Energy 126, 429 (2025).
  61. C. Papastaikoudis, B. Lengeler, and W. Jager, Electrical resistivity of hydrogen in nickel, J. Phys. F 13, 2257 (1983).
  62. C. J. Smithells, E. A. Brandes, and G. B. Brook, Smithells Metals Reference Book, 7th ed. (Butterworth Heinemann, Oxford, 1992).
  63. R. A. Oriani, Reply to “On permeation techniques for electrotransport studies on metal-hydrogen systems,” Scr. Metall. 10, 817 (1976).
  64. R. A. Oriani and O. D. Gonzalez, Electromigration of hydrogen isotopes dissolved in alpha iron and in nickel, Trans. Metall. Soc. AIME 239, 1041 (1967).
  65. K. Zhang, B. Holmedal, T. Mánik, and A. Saai, Assessment of advanced Taylor models, the Taylor factor and yield-surface exponent for FCC metals, Int. J. Plast. 114, 144 (2019).
  66. S. Kang, P.-H. Huang, V. Petrov, A. Manera, T. Ahn, B. Kammenzind, and A. T. Motta, Determination of the hydrogen heat of transport in Zircaloy-4, J. Nucl. Mater. 573, 154122 (2023).

Outline

Information

Phys. Rev. Lett. 137, 096301– Published 24 August, 2026
  • Received 8 August 2025
  • Revised 14 May 2026
  • Accepted 14 July 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation