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Equilibrium Multi-precipitate Configurations

Abstract

In this paper, we formulate a phase-field model for the computation of equilibrium configurations of multiple phases that arise out of a solid-state precipitate reaction, in the presence of coherency stresses. Here, we utilize the phase-field framework to minimize the sum of the elastic and the interfacial energies for a given volume of the precipitates using an extension of the volume-preserved Allen–Cahn algorithm (Garcke et al. in Math Models Methods Appl Sci 18(08):1347–1381, 2008; Bhadak et al. in Metall Mater Trans A 49A(11):5705–5726, 2018). Using this technique, we investigate the precipitate organization for three solid-state reactions. The first is the classical two-phase precipitate reaction that leads to the formation of core–shell microstructures, where we clarify the influence of elasticity on the formation of such clusters. Following this, we investigate two symmetry-breaking transitions (cubic to tetragonal) and (hexagonal to orthorhombic), that lead to the formation of multi-variant clusters where we study the organization of the precipitates as a function of the elastic properties of the precipitate and the matrix.

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References

  1. A. Maheshwari, A.J. Ardell, Scr. Metall. Mater. vol. 26, p. 347 (1992)

    CAS  Google Scholar 

  2. A. Ardell: Metall. Mater. Trans. B 1970, vol. 1 (2), pp. 525–534.

    Google Scholar 

  3. P. Rastogi, A. J. Ardell: Acta Metallurgica 1971, vol. 19 (4), pp. 321–330.

    CAS  Google Scholar 

  4. T. M. Pollock, S. Tin: Journal of propulsion and power 2006, vol. 22 (2), pp. 361–374.

    CAS  Google Scholar 

  5. S. K. Makineni, S. Sugathan, S. Meher, R. Banerjee, S. Bhattacharya, S. Kumar, K. Chattopadhyay: Scientific reports 2017, vol. 7 (1), pp. 1–9.

    Google Scholar 

  6. E. Marquis, D. N. Seidman: Acta materialia 2001, vol. 49 (11), pp. 1909–1919.

    CAS  Google Scholar 

  7. A. Tolley, V. Radmilovic, U. Dahmen: Scripta materialia 2005, vol. 52 (7), pp. 621–625.

    CAS  Google Scholar 

  8. R. A. Karnesky, D. C. Dunand, D. N. Seidman: Acta Materialia 2009, vol. 57 (14), pp. 4022–4031.

    CAS  Google Scholar 

  9. C. Monachon, M. E. Krug, D. N. Seidman, D. C. Dunand: Acta Materialia 2011, vol. 59 (9), pp. 3398–3409.

    CAS  Google Scholar 

  10. V. Radmilovic, A. Tolley, E. Marquis, M. Rossell, Z. Lee, U. Dahmen: Scripta Materialia 2008, vol. 58 (7), pp. 529–532.

    CAS  Google Scholar 

  11. V. Radmilovic, C. Ophus, E. A. Marquis, M. D. Rossell, A. Tolley, A. Gautam, M. Asta, U. Dahmen: Nature materials 2011, vol. 10 (9), pp. 710–715.

    CAS  Google Scholar 

  12. B. Chen, L. Pan, R. Wang, G. Liu, P. Cheng, L. Xiao, J. Sun: Materials Science and Engineering: A 2011, vol. 530, pp. 607–617.

    CAS  Google Scholar 

  13. M. Krug, D. C. Dunand, D. N. Seidman: Acta Materialia 2011, vol. 59 (4), pp. 1700–1715.

    CAS  Google Scholar 

  14. D. Erdeniz, W. Nasim, J. Malik, A. R. Yost, S. Park, A. De Luca, N. Q. Vo, I. Karaman, B. Mansoor, D. N. Seidman, et al.: Acta Materialia 2017, vol. 124, pp. 501–512.

    CAS  Google Scholar 

  15. B. Jiang, D. Yi, X. Yi, F. Zheng, H. Wang, B. Wang, H. Liu, Z. Hu: Materials Characterization 2018, vol. 141, pp. 248–259.

    CAS  Google Scholar 

  16. M. E. Van Dalen, T. Gyger, D. C. Dunand, D. N. Seidman: Acta Materialia 2011, vol. 59 (20), pp. 7615–7626.

    Google Scholar 

  17. Z. Zhang, M. Li, D. Pan, H. Su, X. Du, P. Li, Y. Wu, et al.: Materials and Design 2016, vol. 90, pp. 1077–1084.

    Google Scholar 

  18. A. De Luca, D. C. Dunand, D. N. Seidman: Acta Materialia 2018, vol. 144, pp. 80–91.

    Google Scholar 

  19. Z. Mao, W. Chen, D. N. Seidman, C. Wolverton: Acta Materialia 2011, vol. 59 (8), pp. 3012–3023.

    CAS  Google Scholar 

  20. J. Cahn: The Journal of Chemical physics 1977, vol. 66 (8), pp. 3667–3672.

    CAS  Google Scholar 

  21. J. K. Lee: Materials Science and Engineering A 1997, vol. 238 (1), pp. 1–12.

    Google Scholar 

  22. J.K. Lee, J. Mater. 140:37–40 (1997)

    Google Scholar 

  23. A. Devaux, L. Naze, R. Molins, A. Pineau, A. Organista, J. Guedou, J. Uginet, P. Heritier: Materials Science and Engineering: A 2008, vol. 486 (1–2), pp. 117–122.

    Google Scholar 

  24. A. Niang, B. Viguier, J. Lacaze: Materials characterization 2010, vol. 61 (5), pp. 525–534.

    CAS  Google Scholar 

  25. C. Zhang, L. Yu, H. Wang, Materials, 12(13), 2096 (2019).

    CAS  Google Scholar 

  26. R. Sundararaman, P. Mukhopadhyay, S. Banerjee: Metallurgical and Materials Transactions A 1992, vol. 23 (7), pp. 2015–2028.

    CAS  Google Scholar 

  27. J. He, H. Han, S. Fukuyama, K. Yokogawa: Acta Materialia 1998, vol. 46 (1), pp. 215–223.

    CAS  Google Scholar 

  28. R. Cozar, A. Pineau: Metallurgical Transactions 1973, vol. 4 (1), pp. 47–59.

    CAS  Google Scholar 

  29. A. Detor, R. DiDomizio, N. Zhou, R. Shi, Y. Wang, D. Mcallister, M. Mills: Metallurgical and Materials Transactions A 2018, vol. 49 (3), pp. 708–717.

    CAS  Google Scholar 

  30. P. Phillips, D. Mcallister, Y. Gao, D. Lv, R. Williams, B. Peterson, Y. Wang, M. Mills: Applied Physics Letters, 100(21), 211913 2012.

    Google Scholar 

  31. R. Shi, D. P. McAllister, N. Zhou, A. J. Detor, R. DiDomizio, M. J. Mills, Y. Wang: Acta Materialia 2019, vol. 164, pp. 220–236.

    CAS  Google Scholar 

  32. C. Leroux, A. Loiseau, D. Broddin, G. Vantendeloo: Philosophical Magazine B 1991, vol. 64 (1), pp. 57–82.

    CAS  Google Scholar 

  33. D. Schryvers, G. Van Tendeloo, J. Van Landuyt, S. Amelinckx: Meccanica 1995, vol. 30 (5), pp. 433–438.

    Google Scholar 

  34. A. H. Heuer: Journal of the American Ceramic Society 1987, vol. 70 (10), pp. 689–698.

    CAS  Google Scholar 

  35. Y. Wang, H.-Y. Wang, L.-Q. Chen, A. G. Khachaturyan: Journal of the American Ceramic Society 1995, vol. 78 (3), pp. 657–661.

    CAS  Google Scholar 

  36. K. Muraleedharan, D. Banerjee, S. Banerjee, S. Lele: Philosophical magazine A 1995, vol. 71 (5), pp. 1011–1036.

    CAS  Google Scholar 

  37. S. Nag, Y. Zheng, R. E. Williams, A. Devaraj, A. Boyne, Y. Wang, P. C. Collins, G. B. Viswanathan, J. Tiley, B. C. Muddle, et al.: Acta Materialia 2012, vol. 60 (18), pp. 6247–6256.

    CAS  Google Scholar 

  38. N. Miyano, T. Norimura, T. Inaba, K. Ameyama: Materials transactions 2006, vol. 47 (2), pp. 341–347.

    CAS  Google Scholar 

  39. S. Balachandran, A. Kashiwar, A. Choudhury, D. Banerjee, R. Shi, Y. Wang: Acta Materialia 2016, vol. 106, pp. 374–387.

    CAS  Google Scholar 

  40. Y. Wen, Y. Wang, L.-Q. Chen: Acta materialia 1999, vol. 47 (17), pp. 4375–4386.

    CAS  Google Scholar 

  41. Y. Wen, Y. Wang, L. Bendersky, L.-Q. Chen: Acta materialia 2000, vol. 48 (16), pp. 4125–4135.

    CAS  Google Scholar 

  42. Y. Wen, Y. Wang, L.-Q. Chen: Acta materialia 2001, vol. 49 (1), pp. 13–20.

    CAS  Google Scholar 

  43. S. Bhattacharyya, T. Abinandanan: Acta Materialia 2009, vol. 57 (3), pp. 646–656.

    CAS  Google Scholar 

  44. B. Bhadak, R. Sankarasubramanian, A. Choudhury: Metall. Mater. Trans. A 2018, vol. 49A (11), pp. 5705–5726.

    Google Scholar 

  45. H. Garcke, B. Nestler, B. Stinner, F. Wendler: Math. Models and Methods in Applied Sciences 2008, vol. 18 (08), pp. 1347–1381.

    CAS  Google Scholar 

  46. J. Lee: Scripta Metallurgica et Materialia 1995, vol. 32 (4), pp. 559–564.

    CAS  Google Scholar 

  47. W. Johnson, J. Cahn: Acta metallurgica 1984, vol. 32 (11), pp. 1925–1933.

    CAS  Google Scholar 

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Acknowledgments

We thank Prof. T.A. Abinandanan (IISc) and Prof. S. Bhattacharya (IITH) for many valuable inputs and stimulating discussions on the subject. We express our appreciation to Boeing-India for the support under the Project PC36032.

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Correspondence to Bhalchandra Bhadak.

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Manuscript submitted January 1, 2020.

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Bhadak, B., Singh, R.K. & Choudhury, A. Equilibrium Multi-precipitate Configurations. Metall Mater Trans A 51, 5414–5431 (2020). https://doi.org/10.1007/s11661-020-05903-0

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  • DOI: https://doi.org/10.1007/s11661-020-05903-0