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  • Print publication year: 2010
  • Online publication date: July 2014

5 - Applications of the diffusion equation

Summary

Introduction

When the initial and boundary conditions are given, many solutions of the diffusion equation have been obtained in the literature and textbooks, and the solutions have been applied to many technical problems [1–6]. In this chapter, we shall select a few thin-film problems to illustrate the application of Fick's first and second laws.

Darken's analysis of the Kirkendall effect of interdiffusion in a bulk diffusion couple will be presented [3–5]. Why do we include the bulk diffusion behavior in a book on thin-film reliability? This is because interdiffusion is one of the most common behaviors in materials, whether it is in bulk or in thin films. More importantly, it will help us to understand the mechanism of failure in thin films. In Darken's analysis of interdiffsuion, specifically, no void formation and no stress are assumed, hence there is no failure in the microstructure. It is a constant volume kinetic process. While Darken's analysis explains the Kirkendall shift (or lattice shift) by assuming vacancy to be equilibrium everywhere in the sample, it does not allow Kirkendall (or Frenkel) void formation. Only when vacancies are supersaturated will voids nucleate and form. Thus, Kirkendall (or Frenkel) void formation will require a different condition from that in Kirkendall shift, so we assume that Kirkendall shift is missing in void formation.

[1] D., Turnbull, “Phase changes,” Solid State Physics 3 (1965) 225.
[2] J. W., Christian, The Theory of Transformation in Metals and Alloys (Pergamon Press, New York, 1965).
[3] D. A., Porter and K. E., Easterling, Phase Transformations in Metals and Alloys (Chapman and Hall, London, 1992).
[4] P. G., Shewmon, Transformations in Metals (Indo American Books, Delhi, 2006).
[5] P. G., Shewmon, Diffusion in Solids, 2nd edn (TMS, Warrendale, PA, 1989).
[6] A. P., Sutton and R. W., Balluffi, Interfaces in Crystalline Materials (Oxford University Press, Oxford, 1995).
[7] V. V., Slezov, Chapter 4 in Kinetics of First-order Phase Transitions (Wiley-VCH, Weinheim, 2009).
[8] A. M., Gusak and K. N., Tu, “Kinetic theory of flux-driven ripening,” Phys. Rev. B66, (2002) 115403.