Grain boundary diffusion coefficient

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 ${\displaystyle D_{b}}$, 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 ${\displaystyle D_{b}}$ is the same in both types of samples. However, at temperatures below 700 °C, the values of ${\displaystyle D_{b}}$ with polycrystal silver consistently lie above the values of ${\displaystyle D_{b}}$ with a single crystal.^[2]

Measurement

A model of grain boundary diffusion developed by JC Fisher in 1953. This solution can then be modeled via a modified differential solution to Fick's Second Law that adds a term for sideflow out of the boundary, given by the equation${\displaystyle a{\frac {\partial \varphi }{\partial t}}+f(y,t)=aD'{\partial ^{2}\varphi \over \partial x^{2}}}$, where ${\displaystyle D'}$ is the diffusion coefficient, ${\displaystyle 2a}$ is the boundary width, and ${\displaystyle f(y,t)}$ is the rate of sideflow.

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 ${\displaystyle \delta }$, the length is ${\displaystyle y}$, 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.

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

${\displaystyle {\frac {\partial c_{b}}{\partial t}}=D_{b}\left({\partial ^{2}c_{b} \over \partial y^{2}}\right)+{\frac {2D}{\delta }}\left({\frac {\partial c}{\partial x}}\right)_{x=\delta /2}}$

where ${\displaystyle c(x,y,t)}$ is the volume concentration of the diffusing atoms and ${\displaystyle c_{b}(y,t)}$ 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.^[4] The diffusion profile therefore can be depicted by the following equation.

${\displaystyle (dln{\bar {c}}/dy^{6/5})^{5/3}=0.66(D_{1}/t)^{1/2}(1/D_{b}\delta )}$

To further determine ${\displaystyle D_{b}}$, two common methods were used. The first is used for accurate determination of ${\displaystyle D_{b}\delta }$. The second technique is useful for comparing the relative ${\displaystyle D_{b}\delta }$ 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, ${\displaystyle c(y)}$. Then we used the above formula that developed by Whipple to get ${\displaystyle D_{b}\delta }$.
• Method 2: To compare the length of penetration of a given concentration at the boundary ${\displaystyle \ \Delta y}$ with the length of lattice penetration from the surface far from the boundary.

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. Bibcode:2016diso.book.....S. doi:10.1007/978-3-319-48206-4. ISBN 978-3-319-48564-5. S2CID 137442988.
3. Fisher, J. C. (January 1951). "Calculation of Diffusion Penetration Curves for Surface and Grain Boundary Diffusion". Journal of Applied Physics. 22 (1): 74–77. Bibcode:1951JAP....22...74F. doi:10.1063/1.1699825. ISSN 0021-8979.
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. 45 (371): 1225–1236. doi:10.1080/14786441208561131. ISSN 1941-5982.

See also

• Kirkendall effect
• Phase transformations in solids
• Mass diffusivity