Grain boundary diffusion coefficient
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The grain boundary diffusion coefficient is the diffusion coefficient of a diffusant along a grain boundary in a polycrystalline solid.cite-ref-1[1] It is a physical constant denoted D b {\displaystyle D_{b}} , 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 D b {\displaystyle D_{b}} is the same in both types of samples. However, at temperatures below 700 °C, the values of D b {\displaystyle D_{b}} with polycrystal silver consistently lie above the values of D b {\displaystyle D_{b}} with a single crystal.cite-ref-2[2]
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Measurement
The general way to measure grain boundary diffusion coefficients was suggested by Fisher.cite-ref-3[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 δ δ {\displaystyle \delta } , the length is y {\displaystyle 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.
∂ ∂ c ∂ ∂ t = D ( ∂ ∂ 2 c ∂ ∂ x 2 + ∂ ∂ 2 c ∂ ∂ y 2 ) {\displaystyle {\frac {\partial c}{\partial t}}=D\left({\partial ^{2}c \over \partial x^{2}}+{\partial ^{2}c \over \partial y^{2}}\right)} where | x | > δ δ / 2 {\displaystyle |x|>\delta /2}
∂ ∂ c b ∂ ∂ t = D b ( ∂ ∂ 2 c b ∂ ∂ y 2 ) + 2 D δ δ ( ∂ ∂ c ∂ ∂ x ) x = δ δ / 2 {\displaystyle {\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 ) {\displaystyle c(x,y,t)} is the volume concentration of the diffusing atoms and c b ( y , t ) {\displaystyle 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.cite-ref-4[4] The diffusion profile therefore can be depicted by the following equation.
( d l n c ¯ ¯ / d y 6 / 5 ) 5 / 3 = 0.66 ( D 1 / t ) 1 / 2 ( 1 / D b δ δ ) {\displaystyle (dln{\bar {c}}/dy^{6/5})^{5/3}=0.66(D_{1}/t)^{1/2}(1/D_{b}\delta )}
To further determine D b {\displaystyle D_{b}} , two common methods were used. The first is used for accurate determination of D b δ δ {\displaystyle D_{b}\delta } . The second technique is useful for comparing the relative D b δ δ {\displaystyle 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 ) {\displaystyle c(y)} . Then we used the above formula that developed by Whipple to get D b δ δ {\displaystyle D_{b}\delta } .
• Method 2: To compare the length of penetration of a given concentration at the boundary Δ Δ y {\displaystyle \ \Delta y} with the length of lattice penetration from the surface far from the boundary.
References
cite-note-11. ↑ P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965.
See also