Effects of coherency stress and vacancy sources/sinks on interdiffusion across coherent multilayer interfaces - Part I: Theory

Haibo Wan, Yao Shen, Xuejun Jin, Youxing Chen, Jian Sun

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A phase field model corresponding to vacancy-mediated interdiffusion in coherent multilayers with completely miscible constituents is developed to explore the effects of several factors on interdiffusion across coherent multilayer interfaces, such as: (1) the dependence of diffusion potentials and mobilities on coherency stress; (2) the dependence of diffusion potentials and mobilities on composition; (3) the elastic constant inhomogeneity resulting from a inhomogeneous composition distribution; and (4) the properties of vacancy sources/sinks. The Gibbs free energy of the system consists of chemical and elastic energies. The gradient energy is neglected as the multilayers under consideration can be chemically well approximated by an ideal substitutional solution model. Elastic energy is a function of the stress-free strain and inhomogeneous elastic moduli distributions, while the stress is solved by anisotropic phase field microelasticity theory. The diffusion potentials are obtained straightforwardly as functional derivatives of the free energy with respect to composition and are in keeping with previous derivations that involved many mathematical manipulations or quite advanced theories. The diffusion mobilities are affected by the stress through modification of the vacancy formation and migration energies. Two limiting cases of vacancy sources/sinks are taken into account: ideal vacancy sources/sinks are uniformly and densely distributed, or not present at all, so the vacancy concentration is in equilibrium all the times, as determined by the local stress and composition in the former case, but deviates from the equilibrium concentration in the latter. The model can be conveniently extended to consider the non-ideal activity of vacancy sources/sinks by introducing a general kinetic relation for the vacancy creation rate.

Original languageEnglish (US)
Pages (from-to)2528-2538
Number of pages11
JournalActa Materialia
Issue number6-7
StatePublished - Apr 2012

Bibliographical note

Funding Information:
The research was supported by the National Science Foundation of China under Project Nos. 50971090 , 50601018 and 50890174 . Critical comments and helpful discussions with Dr. J Svoboda are gratefully acknowledged. Helpful discussions with Dr. Jian Wang, Prof. Yunzhi Wang, Prof. Max Lagally and Prof. Hanchen Huang are also acknowledged. The authors are also grateful to Dr. Jian Wang for critical review of the manuscript. The authors thank the reviewer for critical comments.


  • Coherency stress
  • Multilayers
  • Phase field models
  • Substitutional diffusion
  • Vacancies


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