Skip to main content
SpringerLink
Log in

A New Analytical Method for Computing Concentration-Dependent Interdiffusion Coefficient in Binary Systems with Pre-existing Solute Concentration Gradient

  • Published:
Journal of Phase Equilibria and Diffusion Aims and scope Submit manuscript

Abstract

A new analytical method for computing concentration-dependent interdiffusion coefficients, when a non-uniform solute distribution pre-exists in a substrate prior to diffusion in a binary system, is developed from Fick’s laws of diffusion. The key concept of the new method is validated by experimental data reported in the literature. This new method addresses the limitations of previous analytical techniques such as Boltzmann–Matano, Saucer–Friese, and Sarafianos methods, which are restricted to a single solute concentration profile, with no non-uniform initial solute distribution. The analyses of numerically simulated concentration profiles show that previous standard analytical methods are erroneous when computing the interdiffusion coefficients in binary systems with significant non-uniform initial solute distributions. In contrast, the new analytical method can reliably compute the concentration-dependent interdiffusion coefficient operative between two isothermal concentration profiles obtained at different diffusion times. Practically, this method can be used to extract concentration-dependent interdiffusion coefficients from planar diffusion systems with non-uniform initial solute distribution caused by pre-diffusion or multi-staged diffusion processes.

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
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Q. Zhang, and J.-C. Zhao, Extracting Interdiffusion Coefficients from Binary Diffusion Couples Using Traditional Methods and a Forward-Simulation Method, Intermetallics, 2013, 34, p 132–141

    Article  Google Scholar 

  2. F. Sauer, and V. Freise, Diffusion in binären Gemischen mit Volumenänderung, Zeitschrift für Elektrochemie, 1962, 66, p 353–362

    Google Scholar 

  3. C. Wagner, The Evaluation of Data Obtained with Diffusion Couples of Binary Single-Phase and Multiphase Systems, Acta Metall., 1969, 17, p 99–107

    Article  Google Scholar 

  4. L.D. Hall, An Analytical Method of Calculating Variable Diffusion Coefficients, J. Chem. Phys., 1953, 21, p 87–89

    Article  ADS  Google Scholar 

  5. T. Ahmed, I.V. Belova, A.V. Evteev, E.V. Levchenko, and G.E. Murch, Comparison of the Sauer-Freise and Hall Methods for Obtaining Interdiffusion Coefficients in Binary Alloys, J. Phase Equilib. Diffus., 2015, 36, p 366–374

    Article  Google Scholar 

  6. N. Sarafianos, An Analytical Method of Calculating Variable Diffusion Coefficients, J. Mater. Sci., 1986, 21, p 2283–2288

    Article  ADS  Google Scholar 

  7. D. Ansel, I. Thibon, M. Boliveau, and J. Debuigne, Interdiffusion in the Body Cubic Centered β-phase of Ta–Ti Alloys, Acta Mater., 1998, 46, p 423–430

    Article  ADS  Google Scholar 

  8. G.L.E. Gall, and J. Debuigne, Acta Mater., 2000, 35, p 2297–2305

    Article  Google Scholar 

  9. L. Zhu, Q. Zhang, Z. Chen, C. Wei, and G. Cai, Measurement of Interdiffusion and Impurity Diffusion Coefficients in the bcc Phase of the Ti–X (X = Cr, Hf, Mo, Nb, V, Zr) Binary Systems Using Diffusion Multiples, J. Mater. Sci., 2017, 52, p 3255–3268

    Article  ADS  Google Scholar 

  10. S.K. Tang: University of Waterloo, 2009.

  11. J. Lienig and M. Thiele: in Fundamentals of Electromigration- Aware Integrated Circuit Design, Cham, Switzerland, 2018, pp. 13–26.

  12. B. Chao, S.H. Chae, X. Zhang, K.H. Lu, J. Im, and P.S. Ho, Investigation of Diffusion and Electromigration Parameters for Cu–Sn Intermetallic Compounds in Pb-free Solders Using Simulated Annealing, Acta Mater., 2007, 55, p 2805–2814

    Article  ADS  Google Scholar 

  13. M. Liu, H. Fu, C. Xu, W. Xiao, Q. Peng, H. Yamagata, and C. Ma, Precipitation Kinetics and Hardening Mechanism in Al-Si Solid Solutions Processed by High Pressure Solution Treatment, Mater. Sci. Eng. A, 2018, 712, p 757–764

    Article  Google Scholar 

  14. C. Zhong, F. Liu, Y. Wu, J. Le, L. Liu, M. He, J. Zhu, and W. Hu, Protective Diffusion Coatings on Magnesium Alloys: A Review of Recent Developments, J. Alloys Compd., 2012, 520, p 11–21

    Article  Google Scholar 

  15. W. Chen, Q. Li, and L. Zhang, Resistance of Magnesium Alloys to Corrosion Fatigue for Biodegradable Implant Applications: Current Status and Challenges, Materials (Basel), 2017, 10, p 1–11

    Google Scholar 

  16. E. Kirkendall, L. Thomassen, and C. Uethegrove, Rates of Diffusion of Copper and Zinc in Alpha Brass, Trans. Am. Inst. Min. Metall. Eng., 1939, 133, p 186–203

    Google Scholar 

  17. E.O. Kirkendall, Diffusion of Zinc in Alpha Brass, Trans. Am. Inst. Min. Metall. Eng., 1942, 147, p 104–109

    Google Scholar 

  18. C. Matano, On the Relation Between the Diffusion-Coefficients and Concentrations of Solid Metals, J. Phys., 1933, 8, p 109–111

    Google Scholar 

  19. T. Ahmed, I.V. Belova, and G.E. Murch, Finite Difference Solution of the Diffusion Equation and Calculation of the Interdiffusion Coefficient Using the Sauer–Freise and Hall Methods in Binary Systems, Proc. Eng., 2015, 105, p 570–575

    Article  Google Scholar 

  20. S. Santra, and A. Paul, Role of the Molar Volume on Estimated Diffusion Coefficients, Metall. Mater. Trans. A, 2015, 46, p 3887–3899

    Article  Google Scholar 

  21. M.T. Modes: in Electrochemistry and Corrosion Science, 2006, p. 124.

  22. O. Karabelchtchikova, and R.D. Sisson, Carbon Diffusion in Steels: A Numerical Analysis Based on Direct Integration of the Flux, J. Phase Equilib. Diffus., 2006, 27, p 598–604

    Article  Google Scholar 

  23. O. Karabelchtchikova, and R.D. Sisson, Fundamentals of Mass Transfer in Gas Carburizing, Defect Diffus. Forum, 2007, 266, p 171–80

  24. A.D. Smigelskas, and E.O. Kirkendall, Zinc Diffusion in Alpha Brass, Trans. AIME, 1947, 171, p 130–142

    Google Scholar 

  25. S. Gasparin, J. Berger, D. Dutykh, and N. Mendes, Stable Explicit Schemes for Simulation of Nonlinear Moisture Transfer in Porous Materials, J. Build. Perform. Simul., 2018, 11, p 129–144

    Article  Google Scholar 

  26. O. Olaye, and O.A. Ojo, Leapfrog/Dufort–Frankel Explicit Scheme for Diffusion-Controlled Moving Interphase Boundary Problems with Variable Diffusion Coefficient and Solute Conservation, Model. Simul. Mater. Sci. Eng., 2019, 28, p 1–24

    Google Scholar 

  27. Y. Zhou, W.F. Gale, and T.H. North, Modelling of Transient Liquid Phase Bonding, Int. Mater. Rev., 1995, 40, p 181–196

    Article  Google Scholar 

Download references

Acknowledgments

The authors also thank the NSERC of Canada for providing the financial support for the project.

Author information

Authors and Affiliations

  1. University of Manitoba, Winnipeg, MB, R3T5V6, Canada

    O. Olaye & O. A. Ojo

Authors
  1. O. Olaye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. A. Ojo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to O. Olaye or O. A. Ojo.

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

Olaye, O., Ojo, O.A. A New Analytical Method for Computing Concentration-Dependent Interdiffusion Coefficient in Binary Systems with Pre-existing Solute Concentration Gradient. J. Phase Equilib. Diffus. 42, 303–314 (2021). https://doi.org/10.1007/s11669-021-00883-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11669-021-00883-z

Keywords

Search

Navigation