Skip to main content
SpringerLink
Log in

Greater diffusion rate of carbon atoms from nonlinear migration in micro-cell and spatially heterogeneous stable states in FCC iron

  • Computation
  • Published:
Journal of Materials Science Aims and scope Submit manuscript

Abstract

The nonlinear migration of C atom between nearest interstitial sites in fcc iron has been studied in the paper. It reveals an optimum migration pathway in the preferred crowdion direction in its infancy; the pathway is finally along the tetrahedral direction. This nonlinear tendency can be intensified by a repulsive force from another near neighbor C atom in the crowdion direction. We introduce an interaction model based on Coulomb’s force, which indicates that this nonlinear migration mechanism is a foregone conclusion with a certain atomic geometric configuration. However, for a multiple-C system, we find that there exists a typical preferred orientation, which is a continuously stable structure maintained by the C atoms migrating in the crowdion direction. The first neighbor migrated structure is the main formation as a result of this preferential orientation. Based on this, we derive a novel diffusion equation by introducing a proportionality coefficient \( K_{\rho } \), which can be closely related to the C structures absorbed on the surface during carburization. The diffusion coefficient will increase dramatically with a small perturbation of \( K_{\rho } \). Moreover, a spatially heterogeneous occupation is introduced on a much larger scale. The results not only show that it is a relatively stable state but that the C diffusion coefficient is significantly larger than that of the other disorder-free C states. Meanwhile, it provides a repulsive force for nonlinear migration in the micro-cell. As a result, a greater diffusion rate will be achieved from both spatial heterogeneity and the nonlinear migration of C atoms.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9

Similar content being viewed by others

References

  1. Sarıkaya Y, Önal M (2012) High temperature carburizing of a stainless steel with uranium carbide. J Alloy Compd 542:253–256

    Article  Google Scholar 

  2. Bhadeshia HKDH (2010) A commentary on: “Diffusion of carbon in austenite with a discontinuity in composition”. Metall Mater Trans A 41(7):1605–1615

    Article  Google Scholar 

  3. Cheeseman JR, Frisch MJ, Devlin FJ, Stephens PJ (1996) Ab initio calculation of atomic axial tensors and vibrational rotational strengths using density functional theory. Chem Phys Lett 252(3–4):211–220

    Article  CAS  Google Scholar 

  4. Fuchs M, Scheffler M (1999) Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory. Comput Phys Commun 119(1):67–98

    Article  CAS  Google Scholar 

  5. Jiang DE, Carter EA (2005) Carbon atom adsorption on and diffusion into Fe(110) and Fe(100) from first principles. Phys Rev B 71(4):045402

    Article  Google Scholar 

  6. Jiang DE, Carter EA (2003) Carbon dissolution and diffusion in ferrite and austenite from first principles. Phys Rev B 67(21):214103

    Article  Google Scholar 

  7. Simonetti S, Pronsato ME, Brizuela G, Juan A (2007) The C–C pair in the vicinity of a bcc Fe bulk vacancy: electronic structure and bonding. Physica Status Solidi (b) 244(2):610–618

    Article  CAS  Google Scholar 

  8. Asaro R, Farkas D, Kulkarni Y (2008) The Soret effect in diffusion in crystals. Acta Mater 56(6):1243–1256

    Article  CAS  Google Scholar 

  9. Hepburn DJ, Ackland GJ (2008) Metallic-covalent interatomic potential for carbon in iron. Phys Rev B 78(16):165115

    Article  Google Scholar 

  10. Ruda M, Farkas D, Garcia G (2009) Atomistic simulations in the Fe–C system. Comput Mater Sci 45(2):550–560

    Article  CAS  Google Scholar 

  11. Liu X, Huo C, Li Y, Wang J, Jiao H (2012) Energetics of carbon deposition on Fe(100) and Fe(110) surfaces and subsurfaces. Surf Sci 606(7–8):733–739

    Article  CAS  Google Scholar 

  12. Domain C, Becquart CS, Foct J (2004) Ab initio study of foreign interstitial atom (C, N) interactions with intrinsic point defects in α-Fe. Phys Rev B 69(14):144112

    Article  Google Scholar 

  13. Becquart CS, Domain C, Foct J (2005) Ab initio calculations of some atomic and point defect interactions involving C and N in Fe. Philos Mag 85(4–7):533–540

    Article  CAS  Google Scholar 

  14. Ohnuma T, Soneda N, Iwasawa M (2009) First-principles calculations of vacancy–solute element interactions in body-centered cubic iron. Acta Mater 57(20):5947–5955

    Article  CAS  Google Scholar 

  15. Hepburn DJ, Ferguson D, Gardner S, Ackland GJ (2013) First-principles study of helium, carbon, and nitrogen in austenite, dilute austenitic iron alloys, and nickel. Phys Rev B 88(2):024115

    Article  Google Scholar 

  16. Bhadeshia HKDH (2004) Carbon–carbon interactions in iron. J Mater Sci Technol 39(12):3949–3955

    CAS  Google Scholar 

  17. Barouh C, Schuler T, Fu C-C, Jourdan T (2015) Predicting vacancy-mediated diffusion of interstitial solutes in α-Fe. Phys Rev B 92(10):104102

    Article  Google Scholar 

  18. Tapasa K, Barashev A, Bacon D, Osetsky Y (2007) Computer simulation of carbon diffusion and vacancy–carbon interaction in α-iron. Acta Mater 55(1):1–11

    Article  CAS  Google Scholar 

  19. Forst CJ, Slycke J, Van Vliet KJ, Yip S (2006) Point defect concentrations in metastable Fe–C alloys. Phys Rev Lett 96(17):175501

    Article  Google Scholar 

  20. Paxton AT, Elsässer C (2013) Analysis of a carbon dimer bound to a vacancy in iron using density functional theory and a tight binding model. Phys Rev B 87(22):224110

    Article  Google Scholar 

  21. Liu P, Xing W, Cheng X, Li D, Li Y, Chen X-Q (2014) Effects of dilute substitutional solutes on interstitial carbon in α-Fe: interactions and associated carbon diffusion from first-principles calculations. Phys Rev B 90(2):024103

    Article  Google Scholar 

  22. Qiao L, Zhang X, Wang S, Yu S, Hu X, Wang L, Zeng Y, Zheng W (2014) First-principles investigations on the adsorption and diffusion of carbon atoms on the surface and in the subsurface of Co (111) related to the growth of graphene. RSC Adv 4(65):34237–34243

    Article  CAS  Google Scholar 

  23. Hu X, Björkman T, Lipsanen H, Sun L, Krasheninnikov AV (2015) Solubility of boron, carbon, and nitrogen in transition metals: getting insight into trends from first-principles calculations. J Phys Chem Lett 6(16):3263–3268

    Article  CAS  Google Scholar 

  24. Aguiar-Hualde J-M, Magnin Y, Amara H, Bichara C (2017) Probing the role of carbon solubility in transition metal catalyzing single-walled carbon nanotubes growth. Carbon 120:226–232

    Article  CAS  Google Scholar 

  25. Cao Y, Ernst F, Michal GM (2003) Colossal carbon supersaturation in austenitic stainless steels carburized at low temperature. Acta Mater 51(14):4171–4181

    Article  CAS  Google Scholar 

  26. Michal G, Ernst F, Kahn H, Cao Y, Oba F, Agarwal N, Heuer A (2006) Carbon supersaturation due to paraequilibrium carburization: stainless steels with greatly improved mechanical properties. Acta Mater 54(6):1597–1606

    Article  CAS  Google Scholar 

  27. Michal GM, Ernst F, Heuer AH (2006) Carbon paraequilibrium in austenitic stainless steel. Metall Mater Trans A 37(6):1819–1824

    Article  Google Scholar 

  28. Ernst F, Avishai A, Kahn H, Gu X, Michal GM, Heuer AH (2009) Enhanced carbon diffusion in austenitic stainless steel carburized at low temperature. Metall Mater Trans A 40(8):1768–1780

    Article  Google Scholar 

  29. Gu X, Michal GM, Ernst F, Kahn H, Heuer AH (2014) Concentration-dependent carbon diffusivity in austenite. Metall Mater Trans A 45(9):3790–3799

    Article  CAS  Google Scholar 

  30. Fernandes FAP, Christiansen TL, Winther G, Somers MAJ (2017) Measurement and tailoring of residual stress in expanded austenite on austenitic stainless steel. Mater Sci Eng A 701:167–173

    Article  CAS  Google Scholar 

  31. Li W, Guo W, Zhu X, Jin X, Li X, Dong H (2017) The effect of applied compressive stress on the diffusion of carbon in carbon supersaturated S-phase layer. Surf Coat Technol 331:1–6

    Article  CAS  Google Scholar 

  32. Peng Y, Gong J, Chen C, Liu Z, Jiang Y (2018) Numerical analysis of stress gradient and traps effects on carbon diffusion in AISI 316L during low temperature gas phase carburization. Metals 8(4):214

    Article  Google Scholar 

  33. Jespersen FN, Hattel JH, Somers MAJ (2016) Modelling the evolution of composition-and stress-depth profiles in austenitic stainless steels during low-temperature nitriding. Modell Simul Mater Sci Eng 24(2):025003

    Article  Google Scholar 

  34. Brink BK, Ståhl K, Christiansen TL, Oddershede J, Winther G, Somers MAJ (2017) On the elusive crystal structure of expanded austenite. Scripta Mater 131:59–62

    Article  CAS  Google Scholar 

  35. Jang JH, Bhadeshia HKDH, Suh D-W (2013) Solubility of carbon in tetragonal ferrite in equilibrium with austenite. Scripta Mater 68(3–4):195–198

    Article  CAS  Google Scholar 

  36. Cermak J, Kral L (2014) Carbon diffusion in carbon-supersaturated ferrite and austenite. J Alloy Compd 586:129–135

    Article  CAS  Google Scholar 

  37. Enomoto M, Hayashi K (2015) Simulation of the growth of austenite during continuous heating in low carbon iron alloys. J Mater Sci 50(20):6786–6793. https://doi.org/10.1007/s10853-015-9234-3

    Article  CAS  Google Scholar 

  38. Wang J, Li Z, Wang D, Qiu S, Ernst F (2017) Thermal stability of low-temperature-carburized austenitic stainless steel. Acta Mater 128:235–240

    Article  CAS  Google Scholar 

  39. Tao Q, Wang J, Fu L, Chen Z, Shen C, Zhang D, Sun Z (2017) Ultrahigh hardness of carbon steel surface realized by novel solid carburizing with rapid diffusion of carbon nanostructures. J Mater Sci Technol 33(10):1210–1218. https://doi.org/10.1016/j.jmst.2017.04.022

    Article  Google Scholar 

  40. Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136(3B):B864–B871

    Article  Google Scholar 

  41. Sham LJ, Schlüter M (1983) Density-functional theory of the energy gap. Phys Rev Lett 51(20):1888–1891

    Article  Google Scholar 

  42. Segall MD, Lindan PJ, Probert MA, Pickard CJ, Hasnip PJ, Clark SJ, Payne MC (2002) First-principles simulation: ideas, illustrations and the CASTEP code. J Phys Condens Matter 14(11):2717–2744

    Article  CAS  Google Scholar 

  43. Clark SJ, Segall MD, Pickard CJ, Hasnip PJ, Probert MI, Refson K, Payne MC (2005) First principles methods using CASTEP. Zeitschrift für Kristallographie Cryst Mater 220(5/6):567–570

    CAS  Google Scholar 

  44. Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77(18):3865–3868

    Article  CAS  Google Scholar 

  45. Monkhorst HJ, Pack JD (1976) Special points for Brillouin-zone integrations. Phys Rev B 13(12):5188–5192

    Article  Google Scholar 

  46. Fischer TH, Almlof J (1992) General methods for geometry and wave function optimization. J Phys Chem 96(24):9768–9774

    Article  CAS  Google Scholar 

  47. Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117(1):1–19

    Article  CAS  Google Scholar 

  48. Lee BJ (2006) A modified embedded-atom method interatomic potential for the Fe–C system. Acta Mater 54(3):701–711

    Article  Google Scholar 

  49. Bonny G, Terentyev D, Pasianot RC, Poncé S, Bakaev A (2011) Interatomic potential to study plasticity in stainless steels: the FeNiCr model alloy. Modell Simul Mater Sci Eng 19(8):085008

    Article  Google Scholar 

  50. Lee SJ, Matlock DK, Van Tyne CJ (2013) Comparison of two finite element simulation codes used to model the carburizing of steel. Comput Mater Sci 68:47–54

    Article  CAS  Google Scholar 

  51. Eto Y, Umemoto M, Yoshida M (2015) Mechanism of abnormal surface carbon content reduction in Nb-bearing case hardening steel gas carburized after machining. ISIJ Int 55(1):227–234

    Article  CAS  Google Scholar 

  52. Kim DW, Cho HH, Lee WB, Cho KT, Cho YG, Kim SJ, Han HN (2016) A finite element simulation for carburizing heat treatment of automotive gear ring incorporating transformation plasticity. Mater Des 99:243–253

    Article  CAS  Google Scholar 

  53. Dal’Maz Silva W, Dulcy J, Ghanbaja J, Redjaïmia A, Michel G, Thibault S, Belmonte T (2017) Carbonitriding of low alloy steels: mechanical and metallurgical responses. Mater Sci Eng A 693:225–232

    Article  Google Scholar 

  54. Veiga RGA, Becquart CS, Perez M (2014) Comments on “Atomistic modeling of an Fe system with a small concentration of C”. Comput Mater Sci 82:118–121

    Article  CAS  Google Scholar 

  55. Restrepo OA, Mousseau N, El-Mellouhi F, Bouhali O, Trochet M, Becquart CS (2016) Diffusion properties of Fe–C systems studied by using kinetic activation–relaxation technique. Comput Mater Sci 112:96–106

    Article  CAS  Google Scholar 

  56. Restrepo OA, Becquart CS, El-Mellouhi F, Bouhali O, Mousseau N (2017) Diffusion mechanisms of C in 100, 110 and 111 Fe surfaces studied using kinetic activation-relaxation technique. Acta Mater 136:303–314

    Article  CAS  Google Scholar 

  57. Sahputra IH, Chakrabarty A, Restrepo O, Bouhali O, Mousseau N, Becquart CS, El-Mellouhi F (2017) Carbon adsorption on and diffusion through the Fe(110) surface and in bulk: developing a new strategy for the use of empirical potentials in complex material set-ups. Physica Status Solidi (b) 254(2):1600408

    Article  Google Scholar 

  58. Sheppard D, Terrell R, Henkelman G (2008) Optimization methods for finding minimum energy paths. J Chem Phys 128(13):134106

    Article  Google Scholar 

  59. Becquart CS, Raulot J, Bencteux G (2007) Atomistic modeling of an Fe system with a small concentration of C. Comput Mater Sci 40(1):119–129

    Article  CAS  Google Scholar 

  60. Liu WJ, Brimacombe JK, Hawbolt EB (1991) Influence of composition on the diffusivity of carbon in steels—I. Non-alloyed austenite. Acta Metallurgica Et Materialia 39(10):2373–2380

    Article  CAS  Google Scholar 

  61. McLellan RB (1996) The diffusion of heavy interstitial atoms in the absence of a particle density gradient. Acta Mater 44(10):4181–4185

    Article  CAS  Google Scholar 

  62. Tibbetts GG (1980) Diffusivity of carbon in iron and steels at high temperatures. J Appl Phys 51(9):4813–4816

    Article  CAS  Google Scholar 

Download references

Acknowledgements

This research is financially supported by the National Natural Science Foundation of China (No. 51641109), the Fundamental Research Funds for the Central Universities (No. 2017XKZD08).

Author information

Authors and Affiliations

  1. School of Materials Science and Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    Jian Wang, Qing Tao, Wei Lai, Chengjin Shen & Zhi Sun

  2. School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Liming Fu & Aidang Shan

Authors
  1. Jian Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qing Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liming Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Chengjin Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Zhi Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Aidang Shan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qing Tao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, J., Tao, Q., Fu, L. et al. Greater diffusion rate of carbon atoms from nonlinear migration in micro-cell and spatially heterogeneous stable states in FCC iron. J Mater Sci 53, 15952–15968 (2018). https://doi.org/10.1007/s10853-018-2755-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10853-018-2755-9

Keywords

Search

Navigation