Effective diffusion coefficient

The effective diffusion coefficient (also referred to as the apparent diffusion coefficient) of a diffusant in atomic diffusion of solid polycrystalline materials like metal alloys is often represented as a weighted average of the grain boundary diffusion coefficient and the lattice diffusion coefficient.[1] Diffusion along both the grain boundary and in the lattice may be modeled with an Arrhenius equation. The ratio of the grain boundary diffusion activation energy over the lattice diffusion activation energy is usually 0.4–0.6, so as temperature is lowered, the grain boundary diffusion component increases.[1] Increasing temperature often allows for increased grain size, and the lattice diffusion component increases with increasing temperature, so often at 0.8Tmelt (of an alloy), the grain boundary component can be neglected.
Modeling

The effective diffusion coefficient can be modeled using Hart's equation when lattice diffusion is dominant (type A kinetics):

\( {\displaystyle D_{\text{eff}}=fD_{\text{gb}}+(1-f)D_{\ell }} \)

where

\( {\displaystyle D_{\text{eff}}={}} \) effective diffusion coefficient
\( {\displaystyle D_{\text{gb}}={}} \) grain boundary diffusion coefficient
\( {\displaystyle D_{\ell }={}} \) lattice diffusion coefficient
\( {\displaystyle f={\frac {q\delta }{d}}} \)
\( {\displaystyle q={}} \) value based on grain shape, 1 for parallel grains, 3 for square grains
\( {\displaystyle d={}} \) average grain size
\( {\displaystyle \delta ={}} \) grain boundary width, often assumed to be 0.5 nm

Grain boundary diffusion is significant in face-centered cubic metals below about 0.8 Tmelt (Absolute). Line dislocations and other crystalline defects can become significant below ~0.4 Tmelt in FCC metals.
References

P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965.