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Role of interfacial energy and liquid diffusivities on pattern formation during thin-film three-phase eutectic growth

Abstract

In this paper, we investigate three-phase eutectic growth during thin-film directional solidification of a model symmetric ternary eutectic alloy. In contrast to two-phase eutectics that have only a single possibility, \(\alpha \beta \alpha \beta \ldots \), as the growth pattern, during three-phase eutectic growth infinite possibilities exist. Here, we explore the possible existence of pattern selection influenced by the change in the solid–solid interfacial energies and the diffusivities. We begin the study by estimating the undercooling vs. spacing variation for the simplest possible configurations of pattern lengths 3 and 4, where phase-field simulations are utilized to quantify the influence of the solid–solid interfacial energy and the contrast in the component diffusivities. Subsequently, extended simulations consisting of multiple periods of \(\alpha \beta \delta \ldots \) as well as \(\alpha \beta \delta \beta \ldots \) are carried out for assessing the stability of the configurations to long-wavelength perturbations in spacing. Thereafter, growth competition among the simplest patterns is investigated through phase-field simulations of coupled growth of configurations of the type \(\left[ \alpha \beta \delta \right] _m\left[ \alpha \beta \delta \beta \right] _n\), (m, n) being the respective number of periods. Finally, pattern selection is studied by initializing with random initial configurations and classifying the emerging patterns based on the phase sequences. The principal finding is that, while there is no strong phase selection, between the solid–solid interfacial energies and the contrast in the component diffusivities, we find the latter to strongly influence pattern morphology.

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References

  1. Akamatsu S, Plapp M (2016) Eutectic and peritectic solidification patterns. Curr Opin Solid State Materi Sci 20(1):46–54

    CAS  Article  Google Scholar 

  2. Zener C (1946) Kinetics of the decomposition of austenite. Trans Aime 167:550–595

    Google Scholar 

  3. Tiller WA (1958) Liquid metals and solidification. ASM, Metals Park, OH, p 276

  4. Hillert M, Steinhauser H (1960) The structure of white cast iron. Jemont Ann 144:520–559

    CAS  Google Scholar 

  5. Jackson KA, Hunt JD (1966) Lamellar and rod eutectic growth. Trans Metallurg Soc AIME 236:1129–1142

    CAS  Google Scholar 

  6. Langer JS (1980) Eutectic solidification and marginal stability. Phys Rev Lett 44(15):1023

    CAS  Article  Google Scholar 

  7. Akamatsu S, Plapp M, Faivre G, Karma A (2002) Pattern stability and trijunction motion in eutectic solidification. Phys Rev E 66(3):030501

    CAS  Article  Google Scholar 

  8. Akamatsu S, Faivre G, Plapp M, Karma A (2004) Overstability of lamellar eutectic growth below the minimum-undercooling spacing. Metallurg Mater Trans A 35(6):1815–1828

    Article  Google Scholar 

  9. Plapp M, Karma A (2002) Eutectic colony formation: a phase-field study. Phys Rev E 66(6):061608

    Article  CAS  Google Scholar 

  10. Datye V, Langer JS (1981) Stability of thin lamellar eutectic growth. Phys Rev B 24(8):4155

    CAS  Article  Google Scholar 

  11. Kassner K, Misbah C (1991) Spontaneous parity-breaking transition in directional growth of lamellar eutectic structures. Phys Rev A 44(10):6533

    CAS  Article  Google Scholar 

  12. Faivre G, Mergy J (1992) Tilt bifurcation and dynamical selection by tilt domains in thin-film lamellar eutectic growth: experimental evidence of a tilt bifurcation. Phys Rev A 45(10):7320

    CAS  Article  Google Scholar 

  13. Karma A, Sarkissian A (1996) Morphological instabilities of lamellar eutectics. Metallurg Mater Trans A 27(3):635–656

    Article  Google Scholar 

  14. Ginibre M, Akamatsu S, Faivre G (1997) Experimental determination of the stability diagram of a lamellar eutectic growth front. Phys Rev A 56(1):780

    CAS  Google Scholar 

  15. Parisi A, Plapp M (2008) Stability of lamellar eutectic growth. Acta Materialia 56(6):1348–1357

    CAS  Article  Google Scholar 

  16. McCartney DG, Hunt JD, Jordan RM (1980) The structures expected in a simple ternary eutectic system: part 1. theory. Metallurg Trans A 11(8):1243–1249

    Article  Google Scholar 

  17. McCartney DG, Jordan RM, Hunt JD (1980) The structures expected in a simple ternary eutectic system: part II. The Al–Ag–Cu ternary system. Metallurg Trans A 11(8):1251–1257

    Article  Google Scholar 

  18. Böyük U, Maraşlı N, Kaya H, Çadırlı E, Keşlioğlu K (2009) Directional solidification of Al–Cu–Ag alloy. Appl Phys A 95(3):923–932

    Article  CAS  Google Scholar 

  19. Genau A, Ratke L (2012) Morphological characterization of the Al-Ag-Cu ternary eutectic. Int J Mater Res 103(4):469–475

    CAS  Article  Google Scholar 

  20. Dennstedt A, Ratke L (2012) Microstructures of directionally solidified Al–Ag–Cu ternary eutectics. Trans Indian Inst Metals 65(6):777–782

    CAS  Article  Google Scholar 

  21. Dennstedt A, Helfen L, Steinmetz P, Nestler B, Ratke L (2016) 3D synchrotron imaging of a directionally solidified ternary eutectic. Metallurg Mater Trans A 47(3):981–984

    CAS  Article  Google Scholar 

  22. Steinmetz P, Hötzer J, Kellner M, Genau A, Nestler B (2018) Study of pattern selection in 3D phase-field simulations during the directional solidification of ternary eutectic Al-Ag-Cu. Comput Mater Sci 148:131–140

    CAS  Article  Google Scholar 

  23. Ruggiero MA, Rutter JW (1997) Origin of microstructure in the 332 K eutectic of the Bi-In-Sn system. Mater Sci Technol 13(1):5–11

    CAS  Article  Google Scholar 

  24. Apel M, Böttger B, Witusiewicz V, Hecht U, Steinbach I (2004) Lamellar pattern formation during 2D-directional solidification of ternary eutectic alloys. Solidif Crystal 271–279

  25. Rex S, Böttger B, Witusiewicz V, Hecht U (2005) Transient eutectic solidification in In-Bi-Sn: two-dimensional experiments and numerical simulation. Mater Sci Eng A 413:249–254

    Article  CAS  Google Scholar 

  26. Witusiewicz VT, Hecht U, Rex S, Apel M (2005) In situ observation of microstructure evolution in low-melting Bi-In-Sn alloys by light microscopy. Acta Materialia 53(13):3663–3669

    CAS  Article  Google Scholar 

  27. Bottin-Rousseau S, Şerefoğlu M, Yücetürk S, Faivre G, Akamatsu S (2016) Stability of three-phase ternary-eutectic growth patterns in thin sample. Acta Materialia 109:259–266

    CAS  Article  Google Scholar 

  28. Mohagheghi S, Şerefoğlu M (2017) Dynamics of spacing adjustment and recovery mechanisms of ABAC-type growth pattern in ternary eutectic systems. J Cryst Growth 470:66–74

    CAS  Article  Google Scholar 

  29. Mohagheghi S, Şerefoğlu M (2018) Quasi-isotropic and locked grain growth dynamics in a three-phase eutectic system. Acta Materialia 151:432–442

    CAS  Article  Google Scholar 

  30. Mohagheghi S, Hecht U, Bottin-Rousseau S, Akamatsu S, Faivre G, Şerefoğlu M (2019) Effects of interphase boundary anisotropy on the three-phase growth dynamics in the \(\beta \) (In)-In2Bi-\(\gamma \) (Sn) ternary-eutectic system. IOP Conf Ser Mater Sci Eng 529(1):012010

    CAS  Article  Google Scholar 

  31. Böyük U, Maraşlı N (2009) The microstructure parameters and microhardness of directionally solidified Sn-Ag-Cu eutectic alloy. J Alloys Compd 485(1–2):264–269

    Article  CAS  Google Scholar 

  32. Sturz L, Witusiewicz VT, Hecht U, Rex S (2004) Organic alloy systems suitable for the investigation of regular binary and ternary eutectic growth. J Cryst Growth 270(1–2):273–282

    CAS  Article  Google Scholar 

  33. Witusiewicz VT, Sturz L, Hecht U, Rex S (2006) Phase equilibria and eutectic growth in quaternary organic alloys amino-methyl-propanediol-(D) camphor-neopentylglycol-succinonitrile (AMPD-DC-NPG-SCN). J Cryst Growth 297(1):117–132

    CAS  Article  Google Scholar 

  34. Himemiya T, Umeda T (1999) Three-phase planar eutectic growth models for a ternary eutectic system. Mater Trans JIM 40(7):665–674

    CAS  Article  Google Scholar 

  35. Choudhury A, Plapp M, Nestler B (2011) Theoretical and numerical study of lamellar eutectic three-phase growth in ternary alloys. Phys Rev E 83(5):051608

    Article  CAS  Google Scholar 

  36. Hötzer J, Steinmetz P, Jainta M, Schulz S, Kellner M, Nestler B, Genau A, Dennstedt A, Bauer M, Köstler H et al (2016) Phase-field simulations of spiral growth during directional ternary eutectic solidification. Acta Materialia 106:249–259

    Article  CAS  Google Scholar 

  37. Lahiri A, Choudhury A (2017) Revisiting jackson-hunt calculations: Unified theoretical analysis for generic multi-phase growth in a multi-component system. Acta Materialia 133:316–332

    CAS  Article  Google Scholar 

  38. Hecht U, Gránásy L, Pusztai T, Böttger B, Apel M, Witusiewicz V, Ratke L, De Wilde J, Froyen L, Camel D et al (2004) Multiphase solidification in multicomponent alloys. Materi Sci Eng R Rep 46(1–2):1–49

    Google Scholar 

  39. Choudhury A, Nestler B (2012) Grand-potential formulation for multicomponent phase transformations combined with thin-interface asymptotics of the double-obstacle potential. Phys Rev E 85(2):021602

    Article  CAS  Google Scholar 

  40. Plapp M (2011) Unified derivation of phase-field models for alloy solidification from a grand-potential functional. Phys Rev E 84(3):031601

    Article  CAS  Google Scholar 

  41. Choudhury A (2015) Pattern-formation during self-organization in three-phase eutectic solidification. Trans Indian Inst Metals 68(6):1137–1143

    CAS  Article  Google Scholar 

  42. Bauer M, Hötzer J, Jainta M, Steinmetz P, Berghoff M, Schornbaum F, Godenschwager C, Köstler H, Nestler B, Rüde U (2015) Massively parallel phase-field simulations for ternary eutectic directional solidification. In: Proceedings of the international conference for high performance computing, networking, storage and analysis, pp 1–12

  43. Choudhury A, Yabansu YC, Kalidindi SR, Dennstedt A (2016) Quantification and classification of microstructures in ternary eutectic alloys using 2-point spatial correlations and principal component analyses. Acta Materialia 110:131–141

    CAS  Article  Google Scholar 

  44. Steinmetz P, Hötzer J, Kellner M, Dennstedt A, Nestler B (2016) Large-scale phase-field simulations of ternary eutectic microstructure evolution. Comput Mater Sci 117:205–214

    CAS  Article  Google Scholar 

  45. Choudhury A (2012) Quantitative phase-field model for phase transformations in multi-component alloys, vol Band 21. KIT Scientific Publishing

  46. Khanna S, Aramanda SK, Choudhury A (2020) Role of solid-solid interfacial energy anisotropy in the formation of broken lamellar structures in eutectic systems. Metallurg Mater Trans A 51(12):6327–6345

    CAS  Article  Google Scholar 

  47. Choudhury A, Kellner M, Nestler B (2015) A method for coupling the phase-field model based on a grand-potential formalism to thermodynamic databases. Curr Opin Solid State Materi Sci 19(5):287–300

    CAS  Article  Google Scholar 

  48. Vondrous A, Selzer M, Hötzer J, Nestler B (2014) Parallel computing for phase-field models. Int J High Perform Comput Appl 28(1):61–72

    Article  Google Scholar 

  49. Folch R, Plapp M (2005) Quantitative phase-field modeling of two-phase growth. Phys Rev E 72(1):011602

    CAS  Article  Google Scholar 

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Acknowledgements

The authors would like to thank DST-SERB, India 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.

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Correspondence to Abhik Choudhury.

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This study was funded by DST-SERB, India, through the project (DSTO1679). The authors declare that they have no conflict of interest.

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All the data required for the reproduction of the results in the paper are already mentioned in the paper. The phase-field code used for the generation of the results cannot be shared at this point of time; however all the details of the model formulation are present in the paper from which the code can be created.

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Khanna, S., Choudhury, A. Role of interfacial energy and liquid diffusivities on pattern formation during thin-film three-phase eutectic growth. J Mater Sci 56, 17646–17664 (2021). https://doi.org/10.1007/s10853-021-06434-8

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  • DOI: https://doi.org/10.1007/s10853-021-06434-8