Skip to main content
SpringerLink
Log in

Using the phase-field crystal method in the multi-scale modeling of microstructure evolution

  • Dendrite Growth Processes
  • Overview
  • Published:
JOM Aims and scope Submit manuscript

Abstract

The phase-field-crystal method is a new modeling technique that incorporates the periodic nature of a crystal lattice by considering a free energy functional that is minimized by periodic density fields. This simple approach naturally incorporates elastic and plastic deformations and multiple crystal orientations and can be used to study a host of important material processing phenomena, including grain growth, dendritic and eutectic solidification, and epitaxial growth. This paper reviews the phase-field-crystal formalism and its use in modeling of microstructure evolution in pure and binary alloy systems.

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

Similar content being viewed by others

References

  1. J.B. Collins and H. Levine, Phys. Rev. B, 31 (1985), p. 6119.

    Article  CAS  Google Scholar 

  2. J. S. Langer, Directions in Condensed Matter Physics (Singapore: World Scientific, 1986), p. 164.

    Google Scholar 

  3. B. Grossman et al., Phys. Rev. Lett., 71 (1993), p. 3323.

    Article  Google Scholar 

  4. K.R. Elder et al., Phys. Rev. Lett., 72 (1994), p. 677.

    Article  CAS  Google Scholar 

  5. K.R. Elder et al., Phys. Rev. E, 61 (2000), p. 6705.

    Article  Google Scholar 

  6. J.A. Warren and W.J. Boettinger, Acta Metall. Mater. A, 43 (1995), p. 689.

    Article  CAS  Google Scholar 

  7. J.W. Cahn and J.E. Hilliard, J. Chem. Phys., 28 (1958), p. 258.

    Article  CAS  Google Scholar 

  8. H.E. Cook, Acta Metall., 18 (1970), p. 297.

    Article  CAS  Google Scholar 

  9. S.M. Allen and J.W. Cahn, Acta Metall., 23 (1975), p. 1017.

    Article  CAS  Google Scholar 

  10. S.M. Allen and J.W. Cahn, Acta Metall., 24 (1976), p. 425.

    Article  CAS  Google Scholar 

  11. S.M. Allen and J.W. Cahn, Acta Metall., 27 (1979), p. 1085.

    Article  CAS  Google Scholar 

  12. S. Hu and L. Chen, Acta Metall., 49 (2001), p. 463.

    CAS  Google Scholar 

  13. A. Karma and W.-J. Rappel, Phys. Rev. E, 53 (1996), p. 3017.

    Article  Google Scholar 

  14. A. Karma, Phys. Rev. Lett., 87 (2001), p. 115701.

    Article  CAS  Google Scholar 

  15. K.R. Elder et al., Phys. Rev. E, 64 (2001), p. 21604.

    Article  CAS  Google Scholar 

  16. R. Folch and M. Plapp, Phys. Rev. E, 72 (2005), p. 011602.

    Article  CAS  Google Scholar 

  17. N. Provatas et al., International Journal of Modern Physics B, 19 (2005), p. 4525.

    Article  CAS  Google Scholar 

  18. N. Provatas, J. Dantzig, and N. Goldenfeld, Phys. Rev. Lett., 80 (1998), p. 3308.

    Article  CAS  Google Scholar 

  19. N. Provatas, J. Dantzig, and N. Goldenfeld, J. Comp. Phys., 148 (1999), p. 265.

    Article  Google Scholar 

  20. N. Provatas et al., Phys. Rev. Lett., 82 (1999), p. 4496.

    Article  CAS  Google Scholar 

  21. N. Provatas et al., Phys. Rev. Lett., 91 (2003), p. 155502.

    Article  Google Scholar 

  22. C. Lan, Y. Chang, and C. Shih, Acta Mater., 51 (2003), p. 1857.

    Article  CAS  Google Scholar 

  23. M. Greenwood, M. Haataja, and N. Provatas, Phys. Rev. Lett., 93 (2004), p. 246101.

    Article  Google Scholar 

  24. C. Lan and C. Shih, J. Cryst. Growth, 264 (2004), p. 472.

    Article  CAS  Google Scholar 

  25. Y.U. Wang et al., Appl. Phys. Lett., vol. 78, 2001, pp. 2324–2326.

    Article  CAS  Google Scholar 

  26. Y.U. Wang et al., Philos. Mag., 81 (2001), p. 385.

    Article  CAS  Google Scholar 

  27. Y.U. Wang et al., Acta Mater., volume number? (2001), p. 1847.

  28. Y.M. Jin and A.G. Khachaturyan, Philos. Mag. Lett., 81 (2001), p. 607.

    Article  CAS  Google Scholar 

  29. M. Haataja et al., Phys. Rev. B, 65 (2002), p. 165414.

    Article  Google Scholar 

  30. L.Q. Chen and A.G. Khachaturyan, Script. Metall. et Mater., 25 (1991), p. 61.

    Article  CAS  Google Scholar 

  31. Y. Wang and A. Khachaturyan, Acta. Mater., 43 (1995), p. 1837.

    Article  CAS  Google Scholar 

  32. Y. Wang and A. Khachaturyan, Acta. Mater., 45 (1997), p. 759.

    Article  CAS  Google Scholar 

  33. L.-Q. Chen and W. Yang, Phys. Rev. B, 50 (1994), p. 15752.

    Article  CAS  Google Scholar 

  34. B. Morin et al., Phys. Rev. Lett., 75 (1995), p. 2156.

    Article  CAS  Google Scholar 

  35. J.A. Warren, R. Kobayashi, and W.C. Carter, J. Cryst. Growth, 211 (2000), p. 18.

    Article  CAS  Google Scholar 

  36. R. Kobayashi, J.A. Warren, and W.C. Carter, Physica D., 140D (2000), p. 141.

    Article  Google Scholar 

  37. J.A. Warren, W.C. Carter, and R. Kobayashi, Physica (Amsterdam), 261A (1998), p. 159.

    Google Scholar 

  38. J.A. Warren et al., Acta Mater., 51 (2003), p. 6035.

    Article  CAS  Google Scholar 

  39. L. Granasy et al., Phys. Chem. Glasses, 45 (2004), p. 107.

    CAS  Google Scholar 

  40. L. Granasy, T. Pusztai, and J.A. Warren, J. Phys.: Condens. Matter, 16 (2004), p. R1205.

    Article  CAS  Google Scholar 

  41. K.R. Elder et al., Phys. Rev. Lett., 88 (2002), pp. 245701:1–245701:4.

    Article  Google Scholar 

  42. K.R. Elder and M. Grant, Phys. Rev. E, 70 (2004), p. 051605.

    Article  CAS  Google Scholar 

  43. J. Berry, M. Grant, and K.R. Elder, Phys. Rev. E, 73 (2006), p. 031609.

    Article  CAS  Google Scholar 

  44. C.V. Achim et al., Phys. Rev. E, 74 (2006), p. 021104.

    Article  CAS  Google Scholar 

  45. P. Stefanovic, M. Haataja, and N. Provatas, Phys. Rev. Lett., 96 (2006), p. 225504.

    Article  Google Scholar 

  46. S. Majaniemi and M. Grant, Phys. Rev. B, 75 (2007), p. 054301.

    Article  Google Scholar 

  47. N. Goldenfeld, B.P. Athreya, and J.A. Dantzig, Phys. Rev. E, 72 (2005), p. 020601(R).

    Article  Google Scholar 

  48. N. Goldenfeld et al., J. Stat. Phys. (to be published in 2007).

  49. K. Elder et al., Phys. Rev. B, 75 (2007), p. 064107.

    Article  Google Scholar 

  50. W.T. Read and W. Shockley, Phys. Rev., 78 (1950), p. 275.

    Article  CAS  Google Scholar 

  51. K. Aust and B. Chalmers, Metal Interfaces (Metals Park, OH: American Society of Metals, 1952), p. 153.

    Google Scholar 

  52. N. Gjostein and F. Rhines, Acta Metall., 7 (1959), p. 319.

    Article  CAS  Google Scholar 

  53. R.J. Asaro and W.A. Tiller, Metall. Trans., 3 (1972), p. 1789.

    CAS  Google Scholar 

  54. M. Grinfeld, J. Nonlin. Sci., 3 (1993), p. 35.

    Article  Google Scholar 

  55. M. Grinfeld, Dokl. Akad. Nauk SSSR, 290 (1986), p. 1358.

    Google Scholar 

  56. M. Grinfeld, Sov. Phys. Dokl., 31 (1986), p. 831.

    Google Scholar 

  57. J.W. Matthews and A.E. Blakeslee, J. Cryst. Growth, 27 (1974), p. 118.

    CAS  Google Scholar 

  58. J.W. Matthews, J. Vac. Sci. Technol., 12 (1975), p. 126.

    Article  CAS  Google Scholar 

  59. Y. Bolkhovityanov et al., J. Appl. Phys., 79 (1960), p. 7636.

    Article  Google Scholar 

  60. A. Rockett and C. Kiely, Phys. Rev. B, 44 (1991), p. 1154.

    Article  Google Scholar 

  61. T. Anan, K. Nishi, and S. Sugou, Appl. Phys. Lett., 60 (1992), p. 3159.

    Article  CAS  Google Scholar 

  62. M. Ogasawara et al., J. Appl. Phys., 84 (1998), p. 4775.

    Article  CAS  Google Scholar 

  63. K. Lu, W. Wei, and J. Wang, Scripta Metall. et Mater., 24 (1990), p. 2319.

    Article  CAS  Google Scholar 

  64. A. Chokshi et al., Scripta Metall., 23 (1989), p. 1679.

    Article  CAS  Google Scholar 

  65. J. Schiøtz, F. Di Tolla, and K. Jacobsen, Nature, 391 (1998), p. 561.

    Article  Google Scholar 

  66. J. Schiøtz et al., Phys. Rev. B, 60 (1999), p. 11971.

    Article  Google Scholar 

  67. T. Yamasaki et al., Nanostruc. Mater., 10 (1998), p. 375.

    Article  CAS  Google Scholar 

  68. R. Peterson, Stress Concentration Design Factors (New York: Wiley & Sons., Inc., 1953).

    Google Scholar 

  69. B.P. Athreya et al., Phys. Rev. E (submitted 2007).

  70. M.J. Aziz and W.J. Boettinger, Acta Metall. Mater., 42 (1994), p. 257.

    Google Scholar 

  71. N.A. Ahmad et al., Phys. Rev. E, 58 (1998), p. 3436.

    Article  CAS  Google Scholar 

  72. J. Fan, M. Haataja, and N. Provatas, Phys. Rev. E, 74 (2006), p. 031602.

    Article  Google Scholar 

  73. M. Haataja et al., App. Phys. Lett., 87 (2005), p. 251901.

    Article  Google Scholar 

  74. M. Haataja and F. Léonard, Phys. Rev. B, 69 (2004), p. 081201.

    Article  Google Scholar 

  75. R.R. Bhat and P.P. Rao, Z. Metallkd., 75 (1994), p. 237.

    Google Scholar 

  76. S. Spooner and B. Lefevre, Metall. Trans. A, 11A (1975), p. 1085.

    Google Scholar 

  77. J.T. Plewes, Metall. Trans. A, 6A (1975), p. 537.

    CAS  Google Scholar 

  78. F. Helmi and L. Zsoldos, Scr. Metall., 11 (1977), p. 899.

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Materials Science and Engineering, McMaster University, Hamilton, ON, Canada, L8S-4L7

    N. Provatas & P. Stefanovic

  2. Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Il, 61801, USA

    J. A. Dantzig & B. Athreya

  3. Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois, 61801, USA

    P. Chan & N. Goldenfeld

  4. Department of Physics, Oakland University, Rochester, MI, 48309-4487, USA

    K. R. Elder

Authors
  1. N. Provatas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. Dantzig
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Athreya
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. P. Stefanovic
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. N. Goldenfeld
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. K. R. Elder
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Provatas, N., Dantzig, J.A., Athreya, B. et al. Using the phase-field crystal method in the multi-scale modeling of microstructure evolution. JOM 59, 83–90 (2007). https://doi.org/10.1007/s11837-007-0095-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11837-007-0095-3

Keywords

Search

Navigation