Grain boundary diffusion coefficient

Measurement

The general way to measure grain boundary diffusion coefficients was suggested by Fisher. In the Fisher model, a grain boundary is represented as a thin layer of high-diffusivity uniform and isotropic slab embedded in a low-diffusivity isotropic crystal. Suppose that the thickness of the slab is \delta, the length is y, and the depth is a unit length, the diffusion process can be described as the following formula. The first equation represents diffusion in the volume, while the second shows diffusion along the grain boundary, respectively.

\frac{\partial c}{\partial t}=D\left({\partial^2 c\over\partial x^2}+{\partial^2 c\over\partial y^2}\right) where |x|\delta/2

\frac{\partial c_b}{\partial t}=D_b\left({\partial^2 c_b\over\partial y^2}\right)+\frac{2D}{\delta}\left(\frac{\partial c}{\partial x}\right)_{x=\delta/2}

where c(x, y, t) is the volume concentration of the diffusing atoms and c_b(y, t) is their concentration in the grain boundary.

To solve the equation, Whipple introduced an exact analytical solution. He assumed a constant surface composition, and used a Fourier–Laplace transform to obtain a solution in integral form. The diffusion profile therefore can be depicted by the following equation.

(dln\bar{c}/dy^{6/5})^{5/3}=0.66(D_1/t)^{1/2}(1/D_b\delta)

To further determine D_b , two common methods were used. The first is used for accurate determination of D_b \delta. The second technique is useful for comparing the relative D_b \delta of different boundaries.

• Method 1: Suppose the slab was cut into a series of thin slices parallel to the sample surface, we measure the distribution of in-diffused solute in the slices, c(y). Then we used the above formula that developed by Whipple to get D_b \delta.
• Method 2: To compare the length of penetration of a given concentration at the boundary \ \Delta y with the length of lattice penetration from the surface far from the boundary.

References

References

1. P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965.
2. Shewmon, Paul. (2016). "Diffusion in Solids".
3. Fisher, J. C.. (January 1951). "Calculation of Diffusion Penetration Curves for Surface and Grain Boundary Diffusion". Journal of Applied Physics.
4. Whipple, R.T.P.. (1954-12-01). "CXXXVIII. Concentration contours in grain boundary diffusion". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.