A Temperature–Stress Phase Diagram of Carbon-Supersaturated bcc-Iron, Exhibiting “Beyond-Zener” Ordering


Carbon ordering in supersaturated body-centered iron was investigated by means of atomic-scale simulations. Beyond the well-known Zener ordering of carbon atoms, our results reveal a first-order transition occurring upon temperature change when a single crystal is subjected to axial compression. This ordering produces orthorhombic martensite due to unequal carbon redistribution over the three octahedral interstitial sites. The resulting phase diagram is of the rare homotectoid type. Connection is made with the thermoelastic behavior of martensitic alloys, which proves to be similar to that of shape-memory alloys.

Graphic Abstract

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    W.L. Fink and E.D. Campbell, Influence of Heat Treatment and Carbon Content on the Structure of Pure Iron Carbon Alloys, Trans. Am. Soc. Steel Treat., 1926, 9, p 717

    Google Scholar 

  2. 2.

    L. Morsdorf, C.C. Tasan, D. Ponge, and D. Raabe, 3D Structural and Atomic-Scale Analysis of Lath Martensite: Effect of the Transformation Sequence, Acta Mater., 2015, 95, p 366-377. https://doi.org/10.1016/j.actamat.2015.05.023

    Article  Google Scholar 

  3. 3.

    P. Zhang, Y. Chen, W. Xiao, D. Ping, and X. Zhao, Twin Structure of the Lath Martensite in Low Carbon Steel, Prog. Nat. Sci. Mater. Int., 2016, 26, p 169-172. https://doi.org/10.1016/j.pnsc.2016.03.004

    Article  Google Scholar 

  4. 4.

    W. Zhang, Y.M. Jin, and A.G. Khachaturyan, Phase Field Microelasticity Modeling of Heterogeneous Nucleation and Growth in Martensitic Alloys, Acta Mater., 2007, 55, p 565-574. https://doi.org/10.1016/j.actamat.2006.08.050

    Article  Google Scholar 

  5. 5.

    A. Stormvinter, G. Miyamoto, T. Furuhara, P. Hedström, and A. Borgenstam, Effect of Carbon Content on Variant Pairing of Martensite in Fe-C Alloys, Acta Mater., 2012, 60, p 7265-7274. https://doi.org/10.1016/j.actamat.2012.09.046

    Article  Google Scholar 

  6. 6.

    H.K.D.H. Bhadeshia, Bainite in Steels: Theory and Practice, 3rd ed., Maney Publishing, Leeds, 2015

    Google Scholar 

  7. 7.

    J.H. Jang, H.K.D.H. Bhadeshia, and D.-W. Suh, Solubility of Carbon in Tetragonal Ferrite in Equilibrium with Austenite, Scr. Mater., 2013, 68, p 195-198. https://doi.org/10.1016/J.SCRIPTAMAT.2012.10.017

    Article  Google Scholar 

  8. 8.

    C. Garcia-Mateo, J.A. Jimenez, H.W. Yen, M.K. Miller, L. Morales-Rivas, M. Kuntz et al., Low Temperature Bainitic Ferrite: Evidence of Carbon Super-Saturation and Tetragonality, Acta Mater., 2015, 91, p 162-173. https://doi.org/10.1016/j.actamat.2015.03.018

    Article  Google Scholar 

  9. 9.

    C. Zener, Theory of Strain Interaction of Solute Atoms, Phys. Rev., 1948, 74, p 639-647. https://doi.org/10.1103/PhysRev.74.639

    Article  MATH  Google Scholar 

  10. 10.

    A.G. Khachaturyan and G.A. Shatalov, On the Theory of the Ordering of Carbon Atoms in a Martensite Crytal, Fiz. Met. Met., 1971, 32, p 1-9

    Google Scholar 

  11. 11.

    P.V. Chirkov, A.A. Mirzoev, and D.A. Mirzaev, Tetragonality and the Distribution of Carbon Atoms in the Fe-C Martensite: Molecular-Dynamics Simulation, Phys. Met. Metallogr., 2016, 117, p 34-41. https://doi.org/10.1134/S0031918X1601004X

    Article  Google Scholar 

  12. 12.

    K.A. Taylor and M. Cohen, Ageing of Ferrous Martensites, Prog. Mater Sci., 1992, 36, p 225-272

    Google Scholar 

  13. 13.

    G.V. Kurdjumov and A.G. Khachaturyan, Nature of Axial Ratio Anomalies of the Martensite Lattice and Mechanism of Diffusionless Gamma to Alpha Transformation, Acta Metall., 1975, 23, p 1077-1088. https://doi.org/10.1016/0036-9748(75)90354-3

    Article  Google Scholar 

  14. 14.

    Z. Fan, L. Xiao, Z. Jinxiu, K. Mokuang, and G. Zhenqi, Lattice-Parameter Variation with Carbon Content of Martensite. II. Long-Wavelength Theory of the Cubic-to-Tetragonal Transition, Phys. Rev. B., 1995, 52, p 9979-9987

    Article  Google Scholar 

  15. 15.

    A. Udyansky, J. von Pezold, A. Dick, and J. Neugebauer, Orientational Ordering of Interstitial Atoms and Martensite Formation in Dilute Fe-Based Solid Solutions, Phys. Rev. B., 2011, 83, p 184112. https://doi.org/10.1103/PhysRevB.83.184112

    Article  Google Scholar 

  16. 16.

    A. Udyansky, J. Von Pezold, V.N. Bugaev, M. Friák, and J. Neugebauer, Interplay Between Long-Range Elastic and Short-Range Chemical Interactions in Fe-C Martensite Formation, Phys. Rev. Condens. Matter Mater. Phys., 2009, 79, p 224112. https://doi.org/10.1103/physrevb.79.224112

    Article  Google Scholar 

  17. 17.

    R. Naraghi, M. Selleby, Stability of Fe-C Martensite-Effect of Zener-Ordering, in: P.C. and G.S. John Allison (Ed.), 1st World Congrress on Integrated Computational Materials and Engineering, TMS, pp. 235–240.

  18. 18.

    R. Naraghi, M. Selleby, and J. Ågren, Thermodynamics of Stable and Metastable Structures in Fe-C System, CALPHAD: Comput. Coupling Phase Diag. Thermochem., 2014, 46, p 148-158. https://doi.org/10.1016/j.calphad.2014.03.004

    Article  Google Scholar 

  19. 19.

    C.W. Sinclair, M. Perez, R.G.A. Veiga, and A. Weck, Molecular Dynamics Study of the Ordering of Carbon in Highly Supersaturated Alpha-Fe, Phys. Rev. B., 2010, 81, p 224204. https://doi.org/10.1103/PhysRevB.81.224204

    Article  Google Scholar 

  20. 20.

    O. Waseda, J. Morthomas, F. Ribeiro, P. Chantrenne, C.W. Sinclair, and M. Perez, Ordering of Carbon in Highly Supersaturated α-Fe, Model. Simul. Mater. Sci. Eng., 2019, 27, p 015005. https://doi.org/10.1088/1361-651X/aaef22

    Article  Google Scholar 

  21. 21.

    S. Djaziri, Y. Li, G.A. Nematollahi, B. Grabowski, S. Goto, C. Kirchlechner et al., Deformation-Induced Martensite: A New Paradigm for Exceptional Steels, Adv. Mater., 2016, 28, p 7753-7757. https://doi.org/10.1002/adma.201601526

    Article  Google Scholar 

  22. 22.

    D.V. Wilson, B. Russell, and J.D. Eshelby, Stress Induced Ordering and Strain-Ageing in Low Carbon Steels, Acta Metall., 1959, 7, p 628-631. https://doi.org/10.1016/0001-6160(59)90132-4

    Article  Google Scholar 

  23. 23.

    J.Y. Yan and A.V. Ruban, Configurational Thermodynamics of C in Body-Centered Cubic/Tetragonal Fe: A Combined Computational Study, Comput. Mater. Sci., 2018, 147, p 293-303. https://doi.org/10.1016/j.commatsci.2018.02.024

    Article  Google Scholar 

  24. 24.

    A.V. Ruban, Self-Trapping of Carbon Atoms in Alpha′-Fe During the Martensitic Transformation: A Qualitative Picture from ab Initio Calculations, Phys. Rev. B Condens. Matter Mater. Phys., 2014, 90, p 144106. https://doi.org/10.1103/physrevb.90.144106

    Article  Google Scholar 

  25. 25.

    P. Maugis, Nonlinear Elastic Behavior of Iron-Carbon Alloys at the Nanoscale, Comput. Mater. Sci., 2019, 159, p 460-469. https://doi.org/10.1016/J.COMMATSCI.2018.12.024

    Article  Google Scholar 

  26. 26.

    M.A. Shtremel and F.F. Satdarova, Influence of Stresses on Order in Interstitial Solutions, Fiz. Met. Met., 1972, 34, p 699-708

    Google Scholar 

  27. 27.

    P. Chirkov, A. Mirzoev, and D. Mirzaev, Carbon Ordering in Martensite Lattice Under External Stress: Thermodynamic Theory and Molecular Dynamics Simulation, Phys. Status Solidi., 2018, 1700665, p 1700665. https://doi.org/10.1002/pssb.201700665

    Article  Google Scholar 

  28. 28.

    P.V. Chirkov, A.A. Mirzoev, and D.A. Mirzaev, Role of Stresses and Temperature in the Z Ordering of Carbon Atoms in the Martensite Lattice, Phys. Met. Metallogr., 2016, 117, p 1138-1143. https://doi.org/10.1134/S0031918X16110041

    Article  Google Scholar 

  29. 29.

    P. Maugis, Ferrite, Martensite and Supercritical Iron: A Coherent Elastochemical Theory of Stress-Induced Carbon Ordering in Steel, Acta Mater., 2018, 158, p 454-465. https://doi.org/10.1016/J.ACTAMAT.2018.08.001

    Article  Google Scholar 

  30. 30.

    P. Maugis, F. Danoix, H. Zapolsky, S. Cazottes, and M. Gouné, Temperature Hysteresis of the Order-Disorder Transition in Carbon-Supersaturated α-Fe, Phys. Rev. B., 2017, 96, p 214104. https://doi.org/10.1103/PhysRevB.96.214104

    Article  Google Scholar 

  31. 31.

    R.W. Balluffi, Introduction to Elasticity Theory for Crystal Defects, Cambridge University Press, Cambridge, 2012, https://doi.org/10.1017/CBO9780511998379

    Google Scholar 

  32. 32.

    P. Maugis, S. Chentouf, and D. Connétable, Stress-Controlled Carbon Diffusion Channeling in bct-Iron: A Mean-Field Theory, J. Alloys Compd., 2018, 769, p 1121-1131. https://doi.org/10.1016/J.JALLCOM.2018.08.060

    Article  Google Scholar 

  33. 33.

    S. Chentouf, S. Cazottes, F. Danoix, M. Goune, H. Zapolsky, and P. Maugis, Effect of Interstitial Carbon Distribution and Nickel Substitution on the Tetragonality of Martensite: A First-Principles Study, Intermetallics, 2017, 89, p 92-99. https://doi.org/10.1016/j.intermet.2017.05.022

    Article  Google Scholar 

  34. 34.

    L.D. Landau and E.M. Lifshitz, Statistical Physics, 2nd ed., Pergamon Press, New York, 1969, https://doi.org/10.1016/0368-3265(59)90121-5

    Google Scholar 

  35. 35.

    H. Okamoto, A Two-Peak Miscibility Gap, J. Phase Equilibria, 1993, 14, p 336-339. https://doi.org/10.1007/BF02668230

    Article  Google Scholar 

  36. 36.

    J.L. Meijering, Thermodynamic Analysis and Synthesis of Phase Diagrams, Physica, 1981, 103B, p 123-130

    Google Scholar 

  37. 37.

    T. Nishizawa, Progress of CALPHAD, Mater. Trans., JIM, 1992, 33, p 713-722

    Article  Google Scholar 

  38. 38.

    S. Djaziri, Y.J. Li, A. Nematollahi, C. Kirchlechner, B. Grabowski, S. Goto et al., Deformation-Induced Martensite in Severely Cold-Drawn Pearlitic Steel: A New Mechanism at Play, Dusseldorf, 2016, https://doi.org/10.13140/RG.2.2.28444.28806

    Article  Google Scholar 

  39. 39.

    R. Rementeria, J.A. Jimenez, S.Y.P. Allain, G. Geandier, J.D. Poplawsky, W. Guo et al., Quantitative Assessment of Carbon Allocation Anomalies in Low Temperature Bainite, Acta Mater., 2017, 133, p 333-345. https://doi.org/10.1016/j.actamat.2017.05.048

    Article  Google Scholar 

  40. 40.

    M.J. Genderen, M. Isac, A. Böttger, and E.J. Mittemeijer, Aging and Tempering Behavior of Iron-Nickel-Carbon and Iron-Carbon Martensite, Metall. Mater. Trans. A, 1997, 28, p 545-561. https://doi.org/10.1007/s11661-997-0042-5

    Article  Google Scholar 

  41. 41.

    L. Cheng, N. Van Der Pers, A. Böttger, T.H. de Keijser, and E.J. Mittemeijer, Lattice Changes of Iron-Nitrogen Marteniste on Aging at Room Temperature, Metall. Trans. A, 1991, 22A, p 1957-1967

    Article  Google Scholar 

  42. 42.

    V.G. Veeraraghavan and P.G. Winchell, 200, 020, and 002 X-Ray Peaks in Tempered Fe-18 Ni-C Martensites, Metall. Trans. A, 1975, 6, p 701-705. https://doi.org/10.1007/BF02672289

    Article  Google Scholar 

  43. 43.

    P.C. Chen and P.G. Winchell, Martensite Lattice Changes During Tempering, Metall. Trans. A, 1980, 11, p 1333-1339. https://doi.org/10.1007/BF02653487

    Article  Google Scholar 

  44. 44.

    A.G. Khachaturyan, Theory of Structural Transformations in Solids, Dover Publications, New York, 2008

    Google Scholar 

  45. 45.

    J.D. Eshelby, The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems, Proc. R. Soc. A., 1957, 241, p 376-396. https://doi.org/10.1007/1-4020-4499-2_18

    MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, Cambridge University Press, Cambridge, 2007

    Google Scholar 

  47. 47.

    H.K.D.H. Bhadeshia and R.W.K. Honeycombe, Steels: Microstructure and Properties, 3rd ed., E. Arnold, London New York, 2006

    Google Scholar 

  48. 48.

    K. Bhattacharya, Microstructure of Martensite, Oxford University Press, Oxford, 2003

    Google Scholar 

Download references


This work was supported by the French Agence Nationale de la Recherche (Contract C-TRAM ANR-18-CE92-0021). The author thanks J. M. Joubert and P. Benigni for fruitful discussions on the topology of phase diagrams.

Author information



Corresponding author

Correspondence to P. Maugis.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Maugis, P. A Temperature–Stress Phase Diagram of Carbon-Supersaturated bcc-Iron, Exhibiting “Beyond-Zener” Ordering. J. Phase Equilib. Diffus. 41, 269–275 (2020). https://doi.org/10.1007/s11669-020-00816-2

Download citation


  • iron–carbon
  • long-range ordering
  • solid solution
  • stress effect