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On Evaluation of the Gibbs–Thomson Effect and Selection of Nucleus Size for the Kampmann–Wagner Numerical Model

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Abstract

For the Kampmann–Wagner numerical (KWN) model, to accurately describe the microstructure evolution in a solid-state precipitation reaction requires a good evaluation of the Gibbs–Thomson effect and an appropriate nucleus size to be used. In the present work, it is found that there exists a big discrepancy between the nucleation thermodynamics and growth thermodynamics when adopting the widely used Gibbs–Thomson equation to evaluate the Gibbs–Thomson effect. This discrepancy can reduce the computational accuracy of the KWN model. When adopting the numerical method, “the equivalence of nucleation thermodynamics and growth thermodynamics” can be achieved. But due to the relatively large value of the critical size, some waves can be formed in the PSD function if the nucleus size is not carefully selected. Based on the analysis of formation mechanism of waves in the PSD function, it is suggested that, when adopting the numerical method to evaluate the Gibbs–Thomson effect, the most appropriate nucleus size should be the one derived from Zeldovich’s theory.

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References

  1. D.L. McDowell and R.A. LeSar: MRS Bull., 2016, vol. 41, pp. 587–93.

    Article  Google Scholar 

  2. A. Deschamps and C.R. Hutchinson: Acta Mater., 2021, vol. 220, p. 117338.

    Article  CAS  Google Scholar 

  3. M. Perez, M. Dumont, and D. Acevedo-Reyes: Acta Mater., 2008, vol. 56, pp. 2119–32.

    Article  CAS  Google Scholar 

  4. R. Wagner, and R. Kampmann: in Materials science and technology: a comprehensive treatment, R.W. Cahn ed., John Wiley, Weinheim, 1991, pp. 381–94.

  5. Q. Zhang, S.K. Makineni, J.E. Allison, and J.C. Zhao: Scr. Mater., 2019, vol. 160, pp. 70–74.

    Article  CAS  Google Scholar 

  6. J.P. Sanhueza, D. Rojas, O. Prat, J. Garcia, R. Espinoza, C. Montalba, and M.F. Melendrez: Mater. Chem. Phys., 2017, vol. 200, pp. 342–53.

    Article  CAS  Google Scholar 

  7. W. Cao, S.L. Chen, F. Zhang, K. Wu, Y. Yang, Y.A. Chang, R. Schmid-Fetzer, and W.A. Oates: Calphad, 2009, vol. 33, pp. 328–42.

    Article  CAS  Google Scholar 

  8. R. Schmid-Fetzer and F. Zhang: Calphad, 2018, vol. 61, pp. 246–63.

    Article  CAS  Google Scholar 

  9. C. Zhang, W. Cao, S.L. Chen, J. Zhu, F. Zhang, A.A. Luo, and R. Schmid-Fetzer: JOM, 2014, vol. 66, pp. 389–96.

    Article  CAS  Google Scholar 

  10. O.R. Myhr and Ø. Grong: Acta Mater., 2000, vol. 48, pp. 1605–15.

    Article  CAS  Google Scholar 

  11. J.D. Robson, M.J. Jones, and P.B. Prangnell: Acta Mater., 2003, vol. 51, pp. 1453–68.

    Article  CAS  Google Scholar 

  12. J.D. Robson: Acta Mater., 2004, vol. 52, pp. 4669–76.

    Article  CAS  Google Scholar 

  13. D. den Ouden, F.J. Vermolen, L. Zhao, C. Vuik, and J. Sietsma: Comp. Mater. Sci., 2011, vol. 50, pp. 2397–2410.

    Article  Google Scholar 

  14. D. den Ouden, L. Zhao, C. Vuik, J. Sietsma, and F.J. Vermolen: Comp. Mater. Sci., 2013, vol. 79, pp. 933–43.

    Article  Google Scholar 

  15. Q. Du, W.J. Poole, and M.A. Wells: Acta Mater., 2012, vol. 60, pp. 3830–39.

    Article  CAS  Google Scholar 

  16. Q. Du, K. Tang, C.D. Marioara, S.J. Andersen, B. Holmedal, and R. Holmestad: Acta Mater., 2017, vol. 122, pp. 178–86.

    Article  CAS  Google Scholar 

  17. Q. Shu, V.V. Visuri, T. Alatarvas, and T. Fabritius: Metall. Mater. Trans. B, 2020, vol. 51, pp. 2905–16.

    Article  Google Scholar 

  18. J.Z. Zhao, L. Ratke, and B. Feuerbacher: Model. Simul. Mater. Sci. Eng., 1998, vol. 6, pp. 123–39.

    Article  CAS  Google Scholar 

  19. J.Z. Zhao, H.L. Li, Q.L. Wang, and J. He: Comp. Mater. Sci., 2008, vol. 44, pp. 400–03.

    Article  CAS  Google Scholar 

  20. J.Z. Zhao, T. Ahmed, H.X. Jiang, J. He, and Q. Sun: Acta Metall. Sin. (Engl. Lett.), 2017, vol. 30, pp. 1–28.

    Article  CAS  Google Scholar 

  21. Q.L. Wang and J.Z. Zhao: Mater. Sci. Eng. A, 2010, vol. 528, pp. 268–72.

    Article  Google Scholar 

  22. J.Z. Zhao, Q.L. Wang, H.L. Li, and J. He: Metall. Mater. Trans. A, 2011, vol. 42A, pp. 3200–07.

    Article  Google Scholar 

  23. Q.L. Wang and H. Xiong: Metall. Mater. Trans. A, 2021, vol. 52A, pp. 4987–96.

    Article  Google Scholar 

  24. H.B. Aaron, D. Fainstein, and G.R. Kotler: J. Appl. Phys., 1970, vol. 41, pp. 4404–10.

    Article  Google Scholar 

  25. I.V. Markov: Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth, and Epitaxy, 3rd ed. World Scientific, New Jersey, 2016.

    Google Scholar 

  26. S. Shahandeh and S. Nategh: Mater. Sci. Eng. A, 2007, vol. 443, pp. 178–84.

    Article  Google Scholar 

  27. M. Hillert: Phase Equilibria, Phase Diagrams and Phase Transformations, 2nd ed. Cambridge University Press, Cambridge, 2008.

    Google Scholar 

  28. M. Qian: Metall. Mater. Trans. A, 2002, vol. 33A, pp. 1283–87.

    Article  CAS  Google Scholar 

  29. M. Perez: Scr. Mater., 2005, vol. 52, pp. 709–12.

    Article  CAS  Google Scholar 

  30. Q. Du, M. Perez, W.J. Poole, and M. Wells: Scr. Mater., 2012, vol. 66, pp. 419–22.

    Article  CAS  Google Scholar 

  31. B. Rheingans and E.J. Mittemeijer: Calphad, 2015, vol. 50, pp. 49–58.

    Article  CAS  Google Scholar 

  32. B. Rheingans and E.J. Mittemeijer: Metall. Mater. Trans. A, 2015, vol. 46A, pp. 3423–39.

    Article  Google Scholar 

  33. R.W. Hyland Jr.: Metall. Trans. A, 1992, vol. 23, pp. 1947–55.

    Article  Google Scholar 

  34. C. Watanabe, T. Kondo, and R. Monzen: Metall. Mater. Trans. A, 2004, vol. 35A, pp. 3003–08.

    Article  CAS  Google Scholar 

  35. E. Clouet, A. Barbu, L. Lae, and G. Martin: Acta Mater., 2005, vol. 53, pp. 2313–15.

    Article  CAS  Google Scholar 

  36. E. Clouet and M. Nastar: Phys. Rev. B, 2007, vol. 75, p. 132102.

    Article  Google Scholar 

  37. G.M. Novotny and A.J. Ardell: Mater. Sci. Eng. A, 2001, vol. 318, pp. 144–54.

    Article  Google Scholar 

  38. E.A. Marquis and D.N. Seidman: Acta Mater., 2001, vol. 49, pp. 1909–19.

    Article  CAS  Google Scholar 

  39. J. Røyset and N. Ryum: Int. Mater. Rev., 2005, vol. 50, pp. 19–44.

    Article  Google Scholar 

  40. S. Iwamura and Y. Miura: Acta Mater., 2004, vol. 52, pp. 591–600.

    Article  CAS  Google Scholar 

  41. H.I. Aaronson, M. Enomoto, and J.K. Lee: Mechanisms of Diffusional Phase Transformations in Metals and Alloys, CRC Press, Boca Raton, 2010.

    Google Scholar 

  42. A. Harten: J. Comput. Phys., 1983, vol. 49, pp. 357–93.

    Article  Google Scholar 

  43. P.K. Sweby: SIAM J. Numer. Anal., 1984, vol. 21, pp. 995–1011.

    Article  Google Scholar 

  44. R.J. LeVeque: SIAM J. Numer. Anal., 1996, vol. 33, pp. 627–65.

    Article  Google Scholar 

  45. R.J. LeVeque: Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2003.

    Google Scholar 

  46. S.V. Patankar: Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.

    Google Scholar 

  47. S.I. Fujikawa: Defect Diffus. Forum, 1997, vol. 143–47, pp. 115–20.

    Article  Google Scholar 

  48. J.L. Murray: J. Phase Equilibria, 1998, vol. 19, pp. 380–84.

    Article  CAS  Google Scholar 

  49. H. Bo, L.B. Liu, and Z.P. Jin: J. Alloys Compd., 2010, vol. 490, pp. 318–25.

    Article  CAS  Google Scholar 

  50. H. Bo, L.B. Liu, J.L. Hu, and Z.P. Jin: Comput. Mater. Sci., 2017, vol. 133, pp. 82–92.

    Article  CAS  Google Scholar 

  51. A.T. Dinsdale, SGTE Pure elements (unary) database, v4.4, 2001.

  52. M. Perez, M. Dumont, and D. Acevedo-Reyes: Acta Mater., 2009, vol. 57, p. 1318.

    Article  CAS  Google Scholar 

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Acknowledgment

The authors gratefully acknowledge the financial support from Natural Science Foundation of Hebei Province (E2023203125).

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On behalf of all authors, the corresponding author states that there is no conflict of interest.

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  1. National Engineering Research Center for Equipment and Technology of Cold Rolled Strip, Yanshan University, Qinhuangdao, 066004, P.R. China

    Qing-Liang Wang & Hao-Ran Liu

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  1. Qing-Liang Wang
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  2. Hao-Ran Liu
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Wang, QL., Liu, HR. On Evaluation of the Gibbs–Thomson Effect and Selection of Nucleus Size for the Kampmann–Wagner Numerical Model. Metall Mater Trans A (2024). https://doi.org/10.1007/s11661-024-07380-1

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