Skip to main content
SpringerLink
Log in

Calculation and Use of Variable Diffusivity for the Analysis of Transient Liquid Phase Bonding Behavior

  • Original Research Article
  • Published:
Metallurgical and Materials Transactions A Aims and scope Submit manuscript

Abstract

A numerical forward simulation technique is used to calculate the concentration-dependent interdiffusion coefficient \(\left( {D = F\left( c \right)} \right)\) during transient liquid phase (TLP) bonding, without the assumption of initial step-function of concentration in space that renders conventional methods like the Boltzmann–Matano, Sauer–Freise, Wagner, and Hall methods unsuitable for this purpose. In contrast to what is commonly assumed, the experimental results confirm that \(D = F\left( c \right)\) can indeed change with time during TLP bonding, which affirms that the use of a straight line of best fit to represent the relationship between isothermal solidification width and square root of time during TLP bonding, as commonly done in the literature is only valid, provided that \(D = F\left( c \right)\) is verified to be time-independent. Furthermore, anomalous reduction of diffusivity with the increase in temperature during TLP bonding is observed and reported for the first time in this work, which causes the extent of isothermal solidification to reduce with the increase in bonding temperature. The observed reduction in diffusivity with the increase in temperature, in contradiction to what is normally expected based on the common diffusion Arrhenius relationship, is due to the nature of concentration dependence of diffusivity and reduction of solubility with the increase in temperature. Therefore, the common simplistic approach of assuming that interdiffusion coefficient is constant and independent of concentration during TLP bonding can have significant consequences in relation to proper understanding and explanation of TLP bonding behavior.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Z. Mao, T. Niu, Q. Wang, Y. Nie, J. Yan, J. Deng, and L. Yue: Mater. Lett., 2020, vol. 268, p. 127483.

    Article  CAS  Google Scholar 

  2. O.B. Dehkordi and A.M. Hadian: Ceram. Int., 2016, vol. 42, pp. 1705–12.

    Article  CAS  Google Scholar 

  3. Y. Tian, H. Fang, N. Ren, Y. Zhao, B. Chen, F. Wu, and K. Paik: J. Alloys Compd., 2020, vol. 828, p. 154468.

    Article  CAS  Google Scholar 

  4. S. Vazirian and A. Farzadi: J. Alloys Compd., 2020, vol. 829, p. 154510.

    Article  CAS  Google Scholar 

  5. A.G. Bigvand and O.A. Ojo: Metall. Mater. Trans. A, 2014, vol. 45A, pp. 1670–74.

    Article  CAS  Google Scholar 

  6. A. Ghoneim and O.A. Ojo: Metall. Mater. Trans. A, 2012, vol. 43A, pp. 900–11.

    Article  CAS  Google Scholar 

  7. O.A. Ojo, O.A. Olatunji, and M.C. Chaturvedi: Philos. Mag., 2017, vol. 97(11), pp. 419–28.

    CAS  Google Scholar 

  8. D. Nagy and X. Huang: J. Eng. Gas Turb. Power., 2009, vol. 131, pp. 1–7.

    Article  CAS  Google Scholar 

  9. L.O. Osoba and O.A. Ojo: Metall. Mater. Trans. A, 2013, vol. 44A, pp. 4020–24.

    Article  CAS  Google Scholar 

  10. P. Davies, A. Johal, H. Davies, and S. Marchisio: Int. J. Adv. Manuf. Technol., 2019, vol. 103, pp. 441–52.

    Article  Google Scholar 

  11. Y. Fang, X. Jiang, D. Mo, D. Zhu, and Z. Luo: Int. J. Adv. Manuf. Technol., 2019, vol. 102, pp. 2845–63.

    Article  Google Scholar 

  12. S. Piegert, B. Laux, and J. Rosler: IOP Conf. Ser. Mater. Sci. Eng., 2012, vol. 33, pp. 1–2.

    Article  CAS  Google Scholar 

  13. S.D. Nelson, S. Liu, and S. Kottilingam: Weld World., 2014, vol. 58, pp. 593–600.

    Article  CAS  Google Scholar 

  14. O.A. Idowu, N.L. Richards, and M.C. Chaturvedi: Mater. Sci. Eng. A, 2005, vol. 397, pp. 98–112.

    Article  CAS  Google Scholar 

  15. S.D. Duvall and W.A. Owczarski: Weld. J., 1967, vol. 46(9), pp. 423–32.

    Google Scholar 

  16. M.H. Haafkens and G.H. Matthey: Weld. J., 1982, vol. 61(25), pp. 25–30.

    CAS  Google Scholar 

  17. O.A. Ojo, N.L. Richards, and M.C. Chaturvedi: Scripta. Mater., 2004, vol. 50, pp. 641–46.

    Article  CAS  Google Scholar 

  18. W.F. Gale and D.A. Butts: Sci. Technol. Weld. Join., 2004, vol. 9, pp. 283–300.

    Article  CAS  Google Scholar 

  19. A. Malekan, M. Farvizi, S.E. Mirsalehi, N. Saito, and K. Nakashima: Mater. Sci. Eng. A, 2019, vol. 755A, pp. 37–49.

    Article  CAS  Google Scholar 

  20. T. Lin, H. Li, P. He, H. Wei, L. Li, and J. Feng: Intermetallics, 2013, vol. 37, pp. 59–64.

    Article  CAS  Google Scholar 

  21. S.S.S. Afghahi, A. Ekrami, S. Farahany, and A. Jahangiri: Philos. Mag., 2014, vol. 94(11), pp. 1166–76.

    Article  CAS  Google Scholar 

  22. N. Sheng, J. Liu, T. Jin, X. Sun, and Z. Hu: Philos. Mag., 2014, vol. 94(11), pp. 1219–34.

    Article  CAS  Google Scholar 

  23. S.D. Duvall, W.A. Owczarski, and D.F. Paulonis: Weld. J., 1974, vol. 53(4), pp. 203–14.

    CAS  Google Scholar 

  24. I. Tuah-Poku, M. Dollar, and T. Massalski: Metall. Mater. Trans. A, 1988, vol. 19A, pp. 675–86.

    Article  CAS  Google Scholar 

  25. W.D. MacDonald and T.W. Eagar: The Metal Science of Joining, TMS, Warrendale, PA, 1992, pp. 93–100.

  26. Y. Ye, G. Zou, W. Long, Q. Jia, H. Bai, A. Wu, and L. Liu: Sci. Technol. Weld. Join., 2019, vol. 24(1), pp. 52–62.

    Article  CAS  Google Scholar 

  27. M.K. Dinkel, P. Heinz, F. Pyczak, A. Volek, M. Ott, E. Affeldt, A. Vossberg, M. Goken, and R.F. Singer: TMS (The Minerals, Metals & Materials Society) Conference, Superalloys, 2008.

  28. J.T. Adu: Transient Liquid Phase Bonding of Aerospace Single-Crystal Rene-N5 Superalloy, M.Sc. diss., University of Manitoba, 2019.

  29. M. Pouranvari, A. Ekrami, and A.H. Kokabi: J. Alloys Compd., 2009, vol. 406, pp. 270–75.

    Article  CAS  Google Scholar 

  30. D.M. Turriff and S.F. Corbin: Metall. Mater. Trans. A, 2008, vol. 39A, pp. 28–38.

    Article  CAS  Google Scholar 

  31. Y. Zhou and T. North: Z. Für. Met., 1994, vol. 85(11), pp. 775–80.

    CAS  Google Scholar 

  32. B. Binesh and A.J. Gharehbagh: J. Mater. Sci. Technol., 2016, vol. 32(11), pp. 1137–51.

    Article  CAS  Google Scholar 

  33. M. Pouranvari, A. Ghasemi, and A. Salmasi: Metall. Mater. Trans. A, 2022, vol. 53A, pp. 126–35.

    Article  CAS  Google Scholar 

  34. E.D. Moreau and S.F. Corbin: Metall. Mater. Trans. A, 2022, vol. 51, pp. 2882–92.

    Article  CAS  Google Scholar 

  35. M. L. Kuntz: Quantifying Isothermal Solidification Kinetics during Transient Liquid Phase Bonding using Differential Scanning Calorimetry, PhD thesis, University of Waterloo, Canada, 2006, p. 214.

  36. C. Sinclair, G. Purdy, and J.E. Morral: Metall. Mater. Trans. A, 2000, vol. 31A, pp. 1187–92.

    Article  CAS  Google Scholar 

  37. J. Li, P. Agyakwa, and C.M. Johnson: Acta. Mater., 2010, vol. 59, pp. 3429–43.

    Article  CAS  Google Scholar 

  38. D.E. Michael: Analyses of Isothermal Solidification Kinetics During Transient Liquid Phase Bonding of Silver with Aluminium and Copper Filler Metals, M.Sc. diss., University of Manitoba, 2018.

  39. A. Ghoneim: Experimental and Theoretical Studies of Transient Liquid Phase Bonding of Nickel Based Materials, M.Sc. diss., University of Manitoba, 2008.

  40. A. Ghanbar, D.E. Michael, and O.A. Ojo: Philos. Mag., 2019, vol. 99(17), pp. 2169–84.

    Article  CAS  Google Scholar 

  41. O.C. Afolabi and O.A. Ojo: Philos. Mag., 2021, vol. 101(9), pp. 1081–96.

    Article  CAS  Google Scholar 

  42. H. Mehrer: Interdiffusion and Kirkendall Effect. in Diffusion in Solids, vol. 155. Springer, Berlin, 2007, pp. 161–77.

  43. T. Ahmed, I.V. Belova, and G.E. Murch: Procedia Eng., 2015, vol. 105, pp. 570–75.

    Article  CAS  Google Scholar 

  44. S. Santra and A. Paul: Metall. Mater. Trans. A, 2015, vol. 46A, pp. 3887–99.

    Article  CAS  Google Scholar 

  45. L.D. Hall: J. Chem. Phys., 1953, vol. 87, pp. 87–89.

    Article  Google Scholar 

  46. M. Reisner, M. Oberkofler, S. Elgeti, M. Balden, T. Hoschen, M. Mayer, and T.F. Silva: Nucl. Mater. Energy, 2019, vol. 19, pp. 189–94.

    Article  Google Scholar 

  47. M. Okugawa and H. Numakura: Metall. Mater. Trans. A, 2015, vol. 46A, pp. 3813–14.

    Article  CAS  Google Scholar 

  48. T. Ahmed, I.V. Belova, A.V. Evteev, E.V. Levchenko, and G.E. Murch: J. Phase Equilib. Diffus., 2015, vol. 36, pp. 366–74.

    Article  CAS  Google Scholar 

  49. L. Zhu, Q. Zhang, Z. Chen, C. Wei, and G. Cai: J. Mater. Sci., 2017, vol. 52, pp. 3255–68.

    Article  CAS  Google Scholar 

  50. Q. Zhang, Z. Chen, W. Zhong, and J.C. Zhao: Scripta. Mater., 2017, vol. 128, pp. 32–35.

    Article  CAS  Google Scholar 

  51. O. Olaye and O. Ojo: Metall Mater. Trans., 2020, vol. 51(12), pp. 6482–97.

    Article  CAS  Google Scholar 

  52. W. MacDonald and T. Eagar: Metall. Mater. Trans. A, 1998, vol. 29A, pp. 315–25.

    Article  CAS  Google Scholar 

  53. R. Asthana: J. Colloid Interface Sci., 1993, vol. 158, pp. 146–51.

    Article  CAS  Google Scholar 

  54. D. Beke, I. Szabo, Z. Erdelyi, and G. Opposits: Mater. Sci. Eng., 2004, vol. 387, pp. 4–10.

    Article  CAS  Google Scholar 

  55. R.K. Jain and R.V. Overstraeten: Phys. Status Solidi A, 1974, vol. 25(1), pp. 125–30.

    Article  CAS  Google Scholar 

  56. G.B. Stephenson: Acta Metall., 1988, vol. 36(10), pp. 2663–83.

    Article  CAS  Google Scholar 

  57. I. Daruka, I.A. Szabo, D.L. Beke, C.S. Cserhati, A. Kodentsov, and F.J.J. Van Loo: Acta Mater., 1996, vol. 44(12), pp. 4981–93.

    Article  CAS  Google Scholar 

  58. G. Opposits, S. Szabo, I. Szabo, D. Beke, and Z. Guba: Scripta Mater., 1998, vol. 39(7), pp. 977–83.

    Article  CAS  Google Scholar 

  59. I. Belova and G. Murch: Philos. Mag., 2005, vol. 85(11), pp. 1191–1203.

    Article  CAS  Google Scholar 

  60. S. Donald and P. Gordon: Metall. Mater. Trans., 1977, vol. 8(10), pp. 1531–41.

    Article  Google Scholar 

  61. T.K. Kim: Korean J. Anesthesiol., 2015, vol. 68, pp. 540–46.

    Article  Google Scholar 

  62. M. Danielewski, B. Wierzba, R. Bachorczyk-Nagy, and M. Pietrzyk: J. Phase Equilib. Diffus., 2006, vol. 27(6), pp. 691–98.

    Article  CAS  Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge financial support by the NSERC of Canada.

Funding

This work was supported by the Natural Sciences and Engineering Research Council of Canada.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6, Canada

    O. C. Afolabi, H. Sada, O. Olaye & O. A. Ojo

Authors
  1. O. C. Afolabi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Sada
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. Olaye
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. O. A. Ojo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to O. C. Afolabi or O. A. Ojo.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Afolabi, O.C., Sada, H., Olaye, O. et al. Calculation and Use of Variable Diffusivity for the Analysis of Transient Liquid Phase Bonding Behavior. Metall Mater Trans A 53, 3392–3403 (2022). https://doi.org/10.1007/s11661-022-06755-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11661-022-06755-6

Search

Navigation