Grain boundary diffusion coefficient explained
The grain boundary diffusion coefficient is the diffusion coefficient of a diffusant along a grain boundary in a polycrystalline solid.[1] It is a physical constant denoted
, and it is important in understanding how grain boundaries affect
atomic diffusivity. Grain boundary diffusion is a commonly observed route for solute migration in polycrystalline materials. It dominates the
effective diffusion rate at lower temperatures in metals and metal alloys. Take the apparent self-diffusion coefficient for single-crystal and polycrystal silver, for example. At high temperatures, the coefficient
is the same in both types of samples. However, at temperatures below 700 °C, the values of
with polycrystal silver consistently lie above the values of
with a single crystal.
[2] Measurement
The general way to measure grain boundary diffusion coefficients was suggested by Fisher.[3] 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
, the length is
, 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.
=D\left({\partial2c\over\partialx2}+{\partial2c\over\partialy2}\right)
where
cb\over\partial
\right)x=\delta/2
where
is the volume concentration of the diffusing atoms and
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.[4] The diffusion profile therefore can be depicted by the following equation.
(dln\bar{c}/dy6/5)5/3
(1/Db\delta)
To further determine
, two common methods were used. The first is used for accurate determination of
. The second technique is useful for comparing the relative
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,
. Then we used the above formula that developed by Whipple to get
.
- Method 2: To compare the length of penetration of a given concentration at the boundary
with the length of lattice penetration from the surface far from the boundary.
See also
Notes and References
- P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965.
- Book: Shewmon, Paul . 2016 . Diffusion in Solids . 10.1007/978-3-319-48206-4. 2016diso.book.....S . 978-3-319-48564-5 . 137442988 .
- Fisher . J. C. . January 1951 . Calculation of Diffusion Penetration Curves for Surface and Grain Boundary Diffusion . Journal of Applied Physics . en . 22 . 1 . 74–77 . 10.1063/1.1699825 . 1951JAP....22...74F . 0021-8979.
- 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 . 45 . 371 . 1225–1236 . 10.1080/14786441208561131 . 1941-5982.