Skip to main content
SpringerLink
Log in

Finding Critical Nucleus in Solid-State Transformations

  • Symposium: Solid-State Nucleation and Critical Nuclei during First Order Diffusional Phase Transformations
  • Published:
Metallurgical and Materials Transactions A Aims and scope Submit manuscript

Abstract

Based on the phase-field total free energy functional and free-end nudged elastic band (NEB) algorithm, a new methodology is developed for finding the saddle-point nucleus in solid-state transformations. Using cubic → tetragonal transformations in both two and three dimensions as examples, we show that the activation energy and critical nucleus configuration along the minimum energy path (MEP) can be determined accurately and efficiently using this new approach. When the elastic energy contribution is dominant, the nucleation process is found to be collective with the critical nucleus consisting of two twin-related variants. When the elastic energy contribution is relatively weak, the critical nucleus consists of a single variant, and the polytwinned structure develops during growth through a stress-induced autocatalytic process. A nontrivial two-variant critical nucleus configuration is observed at an intermediate level of the elastic energy contribution. This general method is applicable to any thermally activated process in anisotropic media, including nucleation of stacking faults and dislocation loops, voids and microcracks, and ferroelectric and ferromagnetic domains. It is able to treat nucleation events involving simultaneously displacive and diffusional components, and heterogeneous nucleation near pre-existing lattice defects.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. J.W. Cahn, J.E. Hilliard: J. Chem. Phys., 1959, vol. 31, pp. 688–99

    Article  CAS  Google Scholar 

  2. J.W. Cahn, and J.E. Hilliard: J. Chem. Phys., 1958, vol. 28, pp. 258–67

    Article  CAS  Google Scholar 

  3. J.B. Rundle, W. Klein: Phys. Rev. Lett., 1989, vol. 63, pp. 171–74

    Article  Google Scholar 

  4. J.R. Rice, G.E. Beltz: J. Mech. Phys. Solids, 1994, vol. 42, pp. 333–60

    Article  CAS  Google Scholar 

  5. R. Poduri, L.Q. Chen: Acta Mater., 1996, vol. 44, pp. 4253–59

    Article  CAS  Google Scholar 

  6. A.C.E. Reid, G.B. Olson, B. Moran: Phase Transitions, 1999, vol. 69, pp. 309–28

    Article  CAS  Google Scholar 

  7. A.G. Khachaturyan: Theory of Structural Transformations in Solids, John Wiley & Sons, New York, NY, 1983, pp. 198–212

    Google Scholar 

  8. J.R. Rice: J. Mech. Phys. Solids, 1992, vol. 40, pp. 239–71

    Article  CAS  Google Scholar 

  9. Y. Wang, L.Q. Chen, and A.G. Khachaturyan: in Computer Simulation in Materials Science—Nano/Meso/Macroscopic Space and Time Scales, H.O. Kirchner, K.P. Kubin, and V. Pontikis, eds., Kluwer Academic Publishers, Dordrecht/Boston/London, 1996, pp. 325–71

  10. Y. Wang, L.Q. Chen: Methods in Material Research, John Wiley & Sons. Inc, New York, NY, 2000 pp. 2a.3.1–2a.3.23

    Google Scholar 

  11. A. Karma: in Encyclopedia of Materials: Science and Technology, K.H.J. Buschow, R.W. Cahn, M.C. Flemings, B. Ilschner, E.J. Kramer, S. Mahajian, eds. Elsevier, Oxford, United Kingdom, 2001, vol. 7, pp. 6873–86

    Google Scholar 

  12. L.Q. Chen: Ann. Rev. Mater. Res., 2002, vol. 32, pp. 113–40

    Article  CAS  Google Scholar 

  13. Y.U. Wang, Y.M. Jin, A.G. Khachaturyan: in Handbook of Materials Modeling, Part B: Models, S. Yip, ed., Springer, New York, NY, 2005 pp. 2287–305

    Chapter  Google Scholar 

  14. Y.A. Chu, B. Moran, G.B. Olson: Metall. Mater. Trans. A, 2000, vol. 31A, pp. 1321–31

    Article  CAS  Google Scholar 

  15. L. Gránásy, T. Börzsönyi, T. Pusztai: Phys. Rev. Lett., 2003, vol. 88, p. 206105-1–4

    Google Scholar 

  16. G. Henkelman, H. Jonsson: J. Chem. Phys., 2000, vol. 113, pp. 9978–85

    Article  CAS  Google Scholar 

  17. C. Shen, J. Li, M.J. Mills, Y. Wang: in Integral Materials Modeling, G. Gottstein, ed., Wiley-VCH, Heidelberg, 2007, pp. 243–52

    Chapter  Google Scholar 

  18. E. Wigner: Trans. Faraday Soc., 1938, vol. 34, pp. 29–41

    Article  CAS  Google Scholar 

  19. H. Jonsson, G. Mills, and K.W. Jacobsen: in Classical and Quantum Dynamics in Condensed Phase Simulations, B.J. Berne, G. Ciccotti, and D.F. Coker, eds., World Scientific, NJ, 1998, pp. 385–404

  20. G. Mills, H. Jonsson: Phys. Rev. Lett., 1994, vol. 72, pp. 1124–27

    Article  CAS  Google Scholar 

  21. G. Mills, H. Jonsson, G.K. Schenter: Surf. Sci., 1995, vol. 324, pp. 305–37

    Article  CAS  Google Scholar 

  22. G. Henkelman, B.P. Uberuaga, H. Jonsson: J. Chem. Phys., 2000, vol. 113, pp. 9901–04

    Article  CAS  Google Scholar 

  23. T. Zhu, J. Li, A. Samanta, H.G. Kim, S. Suresh: Proc. Nat. Acad. Sci. USA, 2007, vol. 104, pp. 3031–36

    Article  CAS  Google Scholar 

  24. J. Li, P.A. Gordon, and T. Zhu: unpublished research, 2007

  25. G.B. McFadden, A.A. Wheeler, R.J. Braun, S.R. Coriell, R.F. Sekerka: Phys. Rev. E, 1993, vol. 48, pp. 2016–24

    Article  CAS  Google Scholar 

  26. M.R. Sorensen, A.F. Voter: J. Chem. Phys., 2000, vol. 112, pp. 9599–606

    Article  CAS  Google Scholar 

  27. Y. Wang, A.G. Khachaturyan: Acta Metall. Mater., 1997, vol. 45, pp. 759–73

    CAS  Google Scholar 

  28. K.C. Russell: in Phase Transformations, H.I. Aaronson, ed., ASM, Metals Park, OH, 1970, pp. 219–68

    Google Scholar 

  29. H.I. Aaronson, J.K. Lee: in Lectures on the Theory of Phase Transformations, H.I. Aaronson, ed., TMS, Warrendale, PA, 1999, pp. 165–229

    Google Scholar 

  30. J. Li: MRS Bull., 2007, vol. 32, pp. 151–59

    CAS  Google Scholar 

  31. S.N. Luo, L.Q. Zheng, A. Strachan, D.C. Swift: J. Chem. Phys., 2007, vol. 126, pp. 034505-1–7

    Google Scholar 

  32. Y. Wang, L.Q. Chen, A.G. Khachaturyan: in Solid-Solid Phase Transformations, W.C. Johnson, J.M. Howe, D.E. Laughlin, W.A. Soffa, eds., TMS, Warrendale, PA, 1994, pp. 245–65

    Google Scholar 

  33. Y. Wang, H.Y. Wang, L.Q. Chen, A.G. Khachaturyan: J. Am. Ceram. Soc., 1995, vol. 78, pp. 657–61

    Article  CAS  Google Scholar 

  34. Y. Le Bouar, A. Loiseau, A.G. Khachaturyan: Acta Mater., 1998, vol. 46, pp. 2777–88

    Article  CAS  Google Scholar 

  35. Y.H. Wen, Y. Wang, L.Q. Chen: Philos. Mag. A, 2000, vol. 80, pp. 1967–82

    Article  CAS  Google Scholar 

  36. H.Y. Wang, Y.Z. Wang, T. Tsakalakos, S. Semenovskaya, A.G. Khachaturyan: Philos. Mag. A, 1996, vol. 74, pp. 1407–20

    Article  CAS  Google Scholar 

  37. K. Binder: Rep. Progr. Phys., 1987, vol. 50, pp. 783–859

    Article  CAS  Google Scholar 

  38. J.P. Simmons, C. Shen, Y. Wang: Scripta Mater., 2000, vol. 43, pp. 935–42

    Article  CAS  Google Scholar 

  39. C. Shen, J.P. Simmons, Y. Wang: Acta Mater, 2006, vol. 54, pp. 5617–30

    Article  CAS  Google Scholar 

  40. C. Shen, J.P. Simmons, Y. Wang: Acta Mater, 2007, vol. 55, pp. 1457–66

    Article  CAS  Google Scholar 

Download references

Acknowledgments

We acknowledge support by the AFOSR MEANSII (Grant No. FA9550-05-1-0135) and ONR D3D (Grant No. N00014-05-1-0504) programs and the Ohio Supercomputer Center. The work of JL is also supported by Grant No. NSF DMR-0502711.

Author information

Authors and Affiliations

  1. Department of Materials Science and Engineering, The Ohio State University, Columbus, OH, 43210, USA

    Chen Shen (Research Associate), Ju Li (Professor) & Yunzhi Wang (Professor)

Authors
  1. Chen Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ju Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yunzhi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunzhi Wang.

Additional information

This article is based on a presentation given in the symposium entitled “Solid-State Nucleation and Critical Nuclei during First Order Diffusional Phase Transformations,” which occurred October 15–19, 2006 during the MS&T meeting in Cincinnati, Ohio under the auspices of the TMS/ASM Phase Transformations Committee.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shen, C., Li, J. & Wang, Y. Finding Critical Nucleus in Solid-State Transformations. Metall Mater Trans A 39, 976–983 (2008). https://doi.org/10.1007/s11661-007-9302-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11661-007-9302-7

Keywords

Search

Navigation