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PF-LBM Modelling of Dendritic Growth and Motion in an Undercooled Melt of Fe-C Binary Alloy

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Abstract

In the present study, a two-dimensional phase field model coupled with Lattice Boltzmann method (PF-LBM) is proposed to predict the dendritic growth and motion in the melt of Fe-C binary alloy, where the phase field method (PF) is used to calculate the dendritic growth, including the phase field and the concentration field, and the lattice Boltzmann method (LBM) is used to calculate the flow field. The dendrite motion is determined by Newton’s Second Law and tracked by Lagrangian point in a Cartesian coordinate system. Later, the model validations were performed with the benchmark of a solid particle settlement in a stagnant fluid and particle motion in a shear flow, and the results show that the present model is capable of predicting the solid particle motion in the fluid flow. Finally, the model is adopted to investigate the dendritic growth and motion in a forced fluid flow (laminar flow or rotational flow), and the dendrite settlement in a terrestrial environment. The results show that when the forced fluid flow is a laminar flow, the free dendrite would be driven to translate, and the relative velocity between the dendrite and flow fluid decreases, resulting in weak influence of fluid flow on the dendritic growth. When the forced fluid flow is a rotational fluid flow, the dendrite would centrifugally rotate on the domain center with a counterclockwise self-spinning, and the rotation radius becomes larger and larger. For the case of dendrite settlement in a terrestrial environment, the relative movement between the dendrite and melt promotes the downward branch growth, but inhibits the upward branch growth, and two vortices form at the wake region of dendrite. Therefore, the settling dendrite shows a significant asymmetrical morphology.

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References

  1. A. Ludwig, M. Wu and A. Kharicha: Metall. Mater. Trans. A, 2015, vol. 46, pp. 4854-4867

    Article  CAS  Google Scholar 

  2. H. Combeau, M. Založnik, S. Hans and P. E. Richy: Metall. Mater. Trans. B, 2009, vol. 40, pp. 289-304.

    Article  CAS  Google Scholar 

  3. G. Lesoult, Mater. Sci. Eng. A, 2005, vol. 413-414, pp. 19-29.

    Article  CAS  Google Scholar 

  4. M. C. Flemings: ISIJ Int., 2000, vol. 40, pp. 833-841.

    Article  CAS  Google Scholar 

  5. T. Schenk, H. NguyenThi, J. Gastaldi, G. Reinhart, V. Cristiglio, N. Mangelinck-Noël, H. Klein, J. Härtwig, B. Grushko, B. Billia and J. Baruchel: J. Cryst. Growth, 2005, 275, 201-208.

    Article  CAS  Google Scholar 

  6. H. Yasuda, Y. Yamamoto, N. Nakatsuka, M. Yoshiya, T. Nagira, A. Sugiyama, I. Ohnaka, K. Uesugi and K. Umetani: Int. J. Cast Metal Res., 2013, vol. 22, pp. 15-21.

    Article  CAS  Google Scholar 

  7. H. Yasuda, T. Nagira, M. Yoshiya, N. Nakatsuka, A. Sugiyama, K. Uesugi and K. Umetani: ISIJ Int., 2011, vol. 51, pp. 402–408.

    Article  CAS  Google Scholar 

  8. C. R. Swaminathan and V. R. Voller: Metall. Trans. B, 1992, vol. 23, pp. 651-664.

    Article  CAS  Google Scholar 

  9. J. Caldwell and C.-C. Chan: Appl. Math. Model., 2000, vol. 24, pp. 45-53.

    Article  Google Scholar 

  10. V. R. Voller: Int. J Heat Mass Trans., 2008, vol. 51, pp. 823-834.

    Article  CAS  Google Scholar 

  11. J. A. Spittle and S. G. R. Brown: J. Mater. Sci., 1995, vol. 30, pp. 3989-3994.

    Article  CAS  Google Scholar 

  12. Y. H. Shin and C. P. Hong: ISIJ Int., 2002, vol. 42, pp. 359-367.

    Article  CAS  Google Scholar 

  13. M. F. Zhu, S. Y. Lee and C. P. Hong: Phys. Rev. E, 2004, vol. 69, pp. 061610.

    Article  CAS  Google Scholar 

  14. M. F. Zhu, T. Dai, S. Y. Lee and C. P. Hong: Comput. Math. Appl., 2008, vol. 55, pp. 1620-1628.

    Article  Google Scholar 

  15. S. Luo, M. Zhu and S. Louhenkilpi: ISIJ Int., 2012, vol. 52, pp. 823-830.

    Article  CAS  Google Scholar 

  16. T. M. Rodgers, J. D. Madison and V. Tikare: Comput. Mater. Sci., 2017, vol. 135, pp. 78-89.

    Article  Google Scholar 

  17. A. M. Ferrenberg and R. H. Swendsen: Phys. Rev. Lett., 1998, vol. 61, pp. 2635-2638.

    Article  Google Scholar 

  18. Y. Saito and T. Ueta: Phys. Rev. A, 1989, vol. 40, pp. 3408-3419.

    Article  CAS  Google Scholar 

  19. D. Juric and G. Tryggvason: J. Comput. Phys., 1996, vol. 123, pp. 127-148.

    Article  CAS  Google Scholar 

  20. M. Nakagawa, Y. Natsume and K. Ohsasa: ISIJ Int., 2006, vol. 46, pp. 909-913.

    Article  CAS  Google Scholar 

  21. Y. Yang, J. W. Garvin and H. S. Udaykumar: Int. J Heat Mass Trans., 2005, vol 48, pp. 5270-5283.

    Article  CAS  Google Scholar 

  22. S.Pan, and M. Zhu: Acta Metall., 2010, vol. 58, pp. 340-352.

    CAS  Google Scholar 

  23. H. S. Udaykumar, S. Marella and S. Krishnan: Int. J Heat Mass Trans., 2003, vol. 46, pp. 2615-2627.

    Article  Google Scholar 

  24. J. H. Jeong, N. Goldenfeld and J. A. Dantzig: Phys. Rev. E, 2001, vol. 64, pp. 041602.

    Article  CAS  Google Scholar 

  25. T. Suzuki, M. Ode, S. G. Kim and W. T. Kim: J. Cryst. Growth, 2002, 237-239, 125-131.

    Article  Google Scholar 

  26. S. G. Kim, W. T. Kim and T. Suzuki: Phys. Rev. E, 1999, vol. 60, pp. 7186-7197.

    Article  CAS  Google Scholar 

  27. S. G. Kim, W. T. Kim and T. Suzuki: Phys. Rev. E, 1998, vol. 58, pp. 3316-3323.

    Article  CAS  Google Scholar 

  28. W. J. Boettinger, J. A. Warren, C. Beckermann and A. Karma: Rev. Mater. Res., 2002, vol. 32, pp. 163-194.

    Article  CAS  Google Scholar 

  29. C. Beckermann, H. J. Diepers, I. Steinbach, A. Karma and X. Tong: J. Comput. Phys., 1999, vol. 154, pp. 468-496.

    Article  CAS  Google Scholar 

  30. S. Luo and M. Y. Zhu: Comput. Mater. Sci., 2013, vol. 71, pp. 10-18.

    Article  Google Scholar 

  31. W. Wang, S. Luo and M. Zhu: Comput. Mater. Sci., 2014, vol. 95, pp. 136-148.

    Article  CAS  Google Scholar 

  32. D. Chatterjee and S. Chakraborty: Phys Lett. A, 2006, 351, 359-367.

    Article  CAS  Google Scholar 

  33. D. Sun, M. Zhu, S. Pan and D. Raabe: Acta Mater., 2009, vol. 57, pp. 1755-1767.

    Article  CAS  Google Scholar 

  34. B. Jelinek, M. Eshraghi, S. Felicelli and J. F. Peters: Comput. Phys Commun., 2014, 185, 939-947.

    Article  CAS  Google Scholar 

  35. 35. A. Zhang, J. Du, Z. Guo, Q. Wang and S. Xiong: Metall. Mater. Trans. B, 2018, vol. 49, pp. 3603-3615.

    Article  CAS  Google Scholar 

  36. J. Zhang: Microfluid. Nanofluid., 2011,vol. 10, pp. 1-28.

    Article  Google Scholar 

  37. R. H. H. Abadi,, M.H. Rahimian and A. Fakhari: J. Comput. Phys., 2018, vol. 374, pp. 668-691.

    Article  CAS  Google Scholar 

  38. H. Liang, Z. H. Chai, B. C. Shi, Z. L. Guo and T. Zhang: Phys. Rev. E, 2014, 90, 063311.

    Article  CAS  Google Scholar 

  39. P. M. Raja, S. Sarkara, S. Chakrabortya, G. Phanikumarb, P. Duttaa and K. Chattopadhyay: Int. J Heat Fluid Fl., 2002, vol. 23, pp. 298-307.

    Article  Google Scholar 

  40. Chakraborty N, Chakraborty S (2007) Int J Heat Mass Trans 50(9):1805-1822.

    Article  CAS  Google Scholar 

  41. W. Miller, I. Rasin and S. Succi: Physica A, 2006, vol. 362, pp. 78-83.

    Article  CAS  Google Scholar 

  42. M. Do-Quang and G. Amberg: J. Comput. Phys., 2008, vol. 227, pp. 1772-1789.

    Article  Google Scholar 

  43. S. Karagadde, A. Bhattacharya, G. Tomar and P. Dutta: J. Comput. Phys., 2012, vol. 231, pp. 3987-4000.

    Article  Google Scholar 

  44. D. Medvedev, F. Varnik and I. Steinbach: Procedia Comput. Sci., 2013, vol. 18, pp. 2512-2520.

    Article  Google Scholar 

  45. R. Rojas, T. Takaki and M. Ohno: J. Comput. Phys., 2015, vol. 298, pp. 29-40.

    Article  CAS  Google Scholar 

  46. T. Takaki, R. Sato, R. Rojas, M. Ohno and Y. Shibuta: Comput. Mater. Sci., 2018, vol. 147, pp. 124-131.

    Article  CAS  Google Scholar 

  47. X. B. Qi, Y. Chen, X. H. Kang, D. Z. Li and T. Z. Gong: Sci. Rep., 2017, vol. 7, pp. 45770.

    Article  Google Scholar 

  48. L. Rátkai, T. Pusztai and L. Gránásy: npj Compu. Mater., 2019, vol. 5, pp. 113.

    Article  Google Scholar 

  49. E. G. Flekkøy: Phys. Rev. E, 1993, vol. 47, pp. 4247-4257.

    Article  Google Scholar 

  50. O. Behrend, R. Harris and P. B. Warren: Phys. Rev. E, 1994, vol. 50, pp. 4586-4595.

    Article  CAS  Google Scholar 

  51. J. Feng, H. H. Hu and D. D. Joseph: J. Fluid Mech., 1994, vol. 277, pp. 271-301.

    Article  Google Scholar 

  52. L. Liu, S. Pian, Z. Zhang, Y. Bao, R. Li and H. Chen: Comput. Mater. Sci., 2018, vol. 146, pp. 9-17.

    Article  CAS  Google Scholar 

  53. X. H. Wang and Y. Li: Metall. Mater. Trans. B, 2015, vol. 46, pp. 800-812.

    Article  CAS  Google Scholar 

  54. M. Ode, T. Suzuki, S. G. Kim and W. T. Kim: Sci. Technol. Adv. Mater., 2000, vol. 1, pp. 43-49.

    Article  CAS  Google Scholar 

  55. S. Okano, T. Nishimura, H. Ooi and T. Chino: J-STAGE, 1975, vol. 61, pp. 2982-2990.

    Google Scholar 

  56. K. Murakami, H. Aihara and T. Okamoto: Acta Metall., 1984, vol. 32, pp. 933-939.

    Article  CAS  Google Scholar 

  57. W. Kurz, B. Giovanola and R. Trivedi: Acta Metall., 1986, vol. 34, pp. 823-830.

    Article  CAS  Google Scholar 

  58. 58. P. Bouissou and P. Pelcé: Phys. Rev. A, 1989, vol. 40, pp. 6673-6680.

    Article  CAS  Google Scholar 

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Acknowledgments

The authors gratefully acknowledge the financial support of National Key Research and Development Plan (Grant Nos. 2017YFB0304100, 2016YFB0300105), National Natural Science of China (Grant Nos. 51674072, 51704151, 51804067) and Fundamental Research Funds for the Central Universities (Grant Nos. N182504014, N170708020, N172503013).

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Authors and Affiliations

  1. Key Laboratory for Ecological Metallurgy of Multimetallic Ores (Ministry of Education), Northeastern University, Shenyang, 110819, Liaoning, P.R. China

    Sen Luo, Peng Wang, Weiling Wang & Miaoyong Zhu

  2. School of Metallurgy, Northeastern University, Shenyang, 110819, Liaoning, P.R. China

    Sen Luo, Peng Wang, Weiling Wang & Miaoyong Zhu

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  1. Sen Luo
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  2. Peng Wang
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Manuscript submitted February 21, 2020.

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Luo, S., Wang, P., Wang, W. et al. PF-LBM Modelling of Dendritic Growth and Motion in an Undercooled Melt of Fe-C Binary Alloy. Metall Mater Trans B 51, 2268–2284 (2020). https://doi.org/10.1007/s11663-020-01925-6

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