Skip to main content

Role of Solid–Solid Interfacial Energy Anisotropy in the Formation of Broken Lamellar Structures in Eutectic Systems

Abstract

Eutectic solidification gives rise to a wide range of microstructures. A commonly observed morphology is the periodic arrangement of lamellar plates with well-defined orientations of the solid–solid interface in a given eutectic grain. It is typically believed that this form of morphology develops due to the presence of solid–solid interfacial energy anisotropy. In this paper, we provide evidence using phase-field simulations where our focus is on alloys where the minority phase fraction is low. Our aim is to establish the role of solid–solid interfacial energy anisotropy in the stabilization of broken lamellar structures in such systems in contrast to the formation of a rod microstructure. In this regard, we conduct phase-field simulations for different strengths of anisotropy in both constrained and extended settings, using which we clarify the mechanisms by which a lamellar arrangement gets stabilized in the presence of anisotropy in the solid–solid interfacial energy.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

References

  1. W. Kurz, D.J. Fisher: Fundamentals of Solidification, vol 1. Trans Tech Publications, Aedermannsdorf (1986)

    Google Scholar 

  2. Dantzig J.A., Rappaz M. (2009) Solidification. EPFL Press, Lausanne

    Google Scholar 

  3. G.A. Chadwick: Prog. Mater. Sci. 1963, vol. 12, pp. 99–182.

    Google Scholar 

  4. J. Hunt, K. Jackson, Trans. Metall. Soc. AIME 236(6), 843–52 (1966)

    CAS  Google Scholar 

  5. R. Elliott, Eutectic Solidification Processing: Crystalline and Glassy Alloys. Elsevier, Amsterdam (2013)

    Google Scholar 

  6. M. Croker, R. Fidler, R. Smith, Proc. R. Soc. Lond. A 335, 15–37 (1973)

    CAS  Google Scholar 

  7. K. Kassner, C. Misbah, Phys. Rev. A 44, 6513 (1991)

    CAS  Google Scholar 

  8. U. Hecht, L. Gránásy, T. Pusztai, B. Böttger, M. Apel, V. Witusiewicz, L. Ratke, J. De Wilde, L. Froyen, D. Camel, et al.: Mater. Sci. Eng. R 2004, vol. 46, pp. 1–49.

    Google Scholar 

  9. A. Karma, M. Plapp: JOM 2004, vol. 56, pp. 28–32.

    CAS  Google Scholar 

  10. H. Walker, S. Liu, J. Lee, R. Trivedi: Metall. Mater. Trans. A 2007, vol. 38, pp. 1417–25.

    Google Scholar 

  11. A. Parisi, M. Plapp: Acta Mater. 2008, vol. 56, pp. 1348–57.

    CAS  Google Scholar 

  12. M. Perrut, S. Akamatsu, S. Bottin-Rousseau, G. Faivre: Phys. Rev. E 2009, vol. 79, pp. 032602.

    Google Scholar 

  13. M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz, R. Trivedi: Acta Mater. 2009, vol. 57, pp. 941–71.

    CAS  Google Scholar 

  14. S. Akamatsu, M. Plapp: Curr. Opin. Solid State Mater. Sci. 2016, vol. 20, pp. 46–54.

    CAS  Google Scholar 

  15. R. Contieri, C. Rios, M. Zanotello, R. Caram: Mater. Charact. 2008, vol. 59, pp. 693–99.

    CAS  Google Scholar 

  16. A. Dennstedt, L. Ratke: Trans. Indian Inst. Met. 2012, vol. 65, pp. 777–82.

    CAS  Google Scholar 

  17. A. Choudhury: Trans. Indian Inst. Met. 2015, vol. 68, pp. 1137–43.

    CAS  Google Scholar 

  18. J. Hötzer, M. Jainta, P. Steinmetz, B. Nestler, A. Dennstedt, A. Genau, M. Bauer, H. Köstler, U. Rüde: Acta Mater. 2015, vol. 93, pp. 194–204.

    Google Scholar 

  19. J. Hötzer, P. Steinmetz, M. Jainta, S. Schulz, M. Kellner, B. Nestler, A. Genau, A. Dennstedt, M. Bauer, H. Köstler, et al.: Acta Mater. 2016, vol. 106, pp. 249–59.

    Google Scholar 

  20. P. Steinmetz, J. Hötzer, M. Kellner, A. Dennstedt, B. Nestler: Comput. Mater. Sci. 2016, vol. 117, pp. 205–14.

    CAS  Google Scholar 

  21. K. A. Jackson, J. D. Hunt: Trans. Metall. Soc. AIME 1966, vol. 236, pp. 1129–42.

    CAS  Google Scholar 

  22. M. Taylor, R. Fidler, R. Smith: J. Cryst. Growth 1968, vol. 3, pp. 666–73.

    Google Scholar 

  23. T. Digges, R. Tauber: Metall. Trans. 1971, vol. 2, pp. 1683–89.

    CAS  Google Scholar 

  24. H. Kerr, M. Lewis: J. Cryst. Growth 1972, vol. 15, pp. 117–25.

    CAS  Google Scholar 

  25. T. Digges Jr, R. Tauber: J. Cryst. Growth 1971, vol. 8, pp. 132–34.

    CAS  Google Scholar 

  26. M. Savas, L. Clapham, R. Smith: J. Mater. Sci. 1990, vol. 25, pp. 909–13.

    CAS  Google Scholar 

  27. H. Kerr, W. Winegard: Can. Metall. Q. 1967, vol. 6, pp. 67–70.

    CAS  Google Scholar 

  28. M. Taylor, R. Fidler, R. Smith: Metall. Trans. 1971, vol. 2, pp. 1793–98.

    CAS  Google Scholar 

  29. M. Şahin, E. Çadirli: J. Mater. Sci. Mater. Electron. 2012, vol. 23, pp. 484–92.

    Google Scholar 

  30. J. Bromley, F. Vnuk, R. Smith: J. Mater. Sci. 1983, vol. 18, pp. 3143–53.

    CAS  Google Scholar 

  31. G. Piatti, G. Pellegrini: Journal of Materials Science 1976, vol. 11 (5), pp. 913–924.

    CAS  Google Scholar 

  32. G. Beghi, G. Piatti, K. Street: J. Mater. Sci. 1971, vol. 6, pp. 118–25.

    CAS  Google Scholar 

  33. M. Notis, D. Shah, S. Young, C. Graham: IEEE Trans. Magn. 1979, vol. 15, pp. 957–66.

    Google Scholar 

  34. M. Sahoo, G. Delamore, R. Smith: J. Mater. Sci. 1980, vol. 15, pp. 1097–1103.

    CAS  Google Scholar 

  35. G. Nishimura, R. Fidler, M. Taylor, R. Smith: Can. Metall. Q. 1969, vol. 8, pp. 319–22.

    CAS  Google Scholar 

  36. P. Taylor, H. Kerr, W. Winegard: Can. Metall. Q. 1964, vol. 3, pp. 235–37.

    CAS  Google Scholar 

  37. D. Jaffrey, G. Chadwick: Metall. Trans. 1970, vol. 1, pp. 3389–96.

    CAS  Google Scholar 

  38. F. Vnuk, M. Sahoo, D. Baragar, R. Smith: J. Mater. Sci. 1980, vol. 15, pp. 2573–83.

    CAS  Google Scholar 

  39. H. Kaya, M. Gündüz, E. Çadirli, O. Uzun: J. Mater. Sci. 2004, vol. 39, pp. 6571–76.

    CAS  Google Scholar 

  40. H. Kaya, E. Çadırlı, M. Gündüz: J. Mater. Eng. Perform. 2003, vol. 12, pp. 456–69.

    CAS  Google Scholar 

  41. Y. Goto, M. Kurosaki, H. Esaka: J. Jpn Inst. Met. 2011, vol. 75, pp. 392–97.

    CAS  Google Scholar 

  42. M. Şahin, F. Karakurt: Physica B 2018, vol. 545, pp. 48–54.

    Google Scholar 

  43. K. Sharma, R. Rai: Thermochim. Acta 2012, vol. 535, pp. 66–70.

    CAS  Google Scholar 

  44. B. Caroli, C. Caroli, G. Faivre, J. Mergy: J. Cryst. Growth, vol. 118, pp. 135–50 (1992)

    CAS  Google Scholar 

  45. A. Valance, C. Misbah, D. Temkin, K. Kassner: Phys. Rev. E, vol. 48, pp. 1924 (1993)

    CAS  Google Scholar 

  46. S. Bottin-Rousseau, M. Şerefolu, S. Akamatsu, G. Faivre: IOP Conf. Ser. Mater. Sci. Eng. 27, 012088 (2012)

    Google Scholar 

  47. S. Akamatsu, S. Bottin-Rousseau, M. Şerefoğlu, G. Faivre: Acta Mater. 2012, vol. 60, pp. 3199–3205.

    CAS  Google Scholar 

  48. S. Akamatsu, S. Bottin-Rousseau, M. Şerefoğlu, G. Faivre: Acta Mater. 2012, vol. 60, pp. 3206–14.

    CAS  Google Scholar 

  49. S. Ghosh, A. Choudhury, M. Plapp, S. Bottin-Rousseau, G. Faivre, S. Akamatsu: Phys. Rev. E, 91, 022407 (2015).

    Google Scholar 

  50. S. Ghosh, M. Plapp: Acta Mater. 2017, vol. 140, pp. 140–48.

    CAS  Google Scholar 

  51. L. Rátkai, G. I. Tóth, L. Környei, T. Pusztai, L. Gránásy: J. Mater. Sci. 2017, vol. 52, pp. 5544–5558.

    Google Scholar 

  52. A. Choudhury, B. Nestler: Phys. Rev. E, 85, 021602 (2012).

    Google Scholar 

  53. M. Plapp: Phys. Rev. E, 84, 031601 (2011).

    Google Scholar 

  54. R. Kobayashi: Physica D 1993, vol. 63, pp. 410–23.

    Google Scholar 

  55. J. J. Eggleston, G. B. McFadden, P. W. Voorhees: Physica D 2001, vol. 150, pp. 91–103.

    CAS  Google Scholar 

  56. P. Mathis: J. Indian Inst. Sci. 2016, vol. 96, pp. 179–98.

    Google Scholar 

  57. I. Ansara, A. Dinsdale, M. Rand (1998) Cost 507: Thermochemical Database for Light Metal Alloys. Vol. 2, Office for Official Publications of the European Communities, Luxembourg.

    Google Scholar 

  58. A. Choudhury, M. Kellner, B. Nestler: Curr. Opin. Solid State Mater. Sci. 2015, vol. 19, pp. 287–300.

    CAS  Google Scholar 

  59. J.-O. Andersson, T. Helander, L. Höglund, P. Shi, B. Sundman: CALPHAD 2002, vol. 26, pp. 273–312.

    CAS  Google Scholar 

  60. J. Pstruś: J. Min. Metall. B 2017, vol. 53, pp. 309–18.

    Google Scholar 

  61. The HDF Group: Hierarchical Data Format, Version 5. http://www.hdfgroup.org/HDF5/1997-NNNN.

  62. J. Ahrens, B. Geveci, C. Law: The Visualization Handbook, vol. 717. Academic, New York (2005)

    Google Scholar 

  63. Ayachit, U., The Paraview Guide: A Parallel Visualization Application. Kitware, Inc., New York (2015)

    Google Scholar 

  64. J. D. Hunter, Comput. Sci. Eng., 9, 90 (2007).

    Google Scholar 

  65. S. Van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, T. Yu: PeerJ 2014, vol. 2, pp. e453.

    Google Scholar 

  66. D. Wheeler, D. Brough, T. Fast, S. Kalidindi, and A. Reid: PyMKS: materials knowledge system in Python, 2014

  67. D. B. Brough, D. Wheeler, S. R. Kalidindi: Integr. Mater. Manuf. Innov. 2017, vol. 6, pp. 36–53.

    Google Scholar 

  68. A. Parisi, M. Plapp: Europhys. Lett., 90, 26010 (2010).

    Google Scholar 

  69. A. Vondrous, M. Selzer, J. Hötzer, B. Nestler, Int. J. High Perform. Comput. Appl. 28, 61–72 (2014)

    Google Scholar 

  70. B. Saatçi, N. Maraşlı, M. Gündüz: Thermochim. Acta 2007, vol. 454, pp. 128–134.

    Google Scholar 

  71. J. Hoshen, R. Kopelman: Phys. Rev. B, 14, 3438 (1976).

    CAS  Google Scholar 

Download references

Acknowledgments

The authors would like to thank DST-SERB for funding through the Project (DSTO1679). SK would like to thank SERC and TUE-CMS, IISc, for providing access to high-performance computational resources, including the use of the SahasraT (Cray XC40) machine at SERC. The authors would also like to thank Prof. Mathis Plapp for insightful discussions during the course of the work.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Abhik Choudhury.

Additional information

Publisher's Note

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

Manuscript submitted June 7, 2020.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (avi 9956 KB)

Supplementary material 2 (avi 4109 KB)

Supplementary material 3 (pdf 14131 KB)

Appendices

Appendices

A. Non-dimensionalization of Simulation Parameters

The parameters used in the simulation are obtained through non-dimensionalization of the physical parameters. The procedure is enlisted below, where the asterisked values are defined as: \(T^{*} = 471.7,\) \(f^{*} = \dfrac{RT^{*}}{V_{\text {m}}},\) \(l^{*} = \dfrac{\sigma }{f^{*}}\) and \(t^{*} = \dfrac{{l^{*}}^{2}}{D}.\) Here, the values of molar volume \(V_{\text {m}} = 1.6\times 10^{-5}\,{\hbox {m}}^{3},\) interface energy \(\sigma = 0.104\,{\hbox {J}}\,{\hbox {m}}^{-2}\) and diffusivity \(D = 3.5\times 10^{-9}\,{\hbox {m}}^{2}\,{\hbox {s}}^{-1}\) are obtained from References 40, 60. Dividing the dimensional parameters with the corresponding asterisked (*) variables gives their non-dimensional values.

B. Numerical Consistency and Grid Resolution

The simulations have been checked for numerical accuracy against different grid resolutions (dx) and interface widths as shown in Figure AI.

Fig. AI
figure 17

Numerical consistency and grid resolution, checked with different interface widths

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khanna, S., Aramanda, S.K. & Choudhury, A. Role of Solid–Solid Interfacial Energy Anisotropy in the Formation of Broken Lamellar Structures in Eutectic Systems. Metall Mater Trans A 51, 6327–6345 (2020). https://doi.org/10.1007/s11661-020-05995-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11661-020-05995-8