Grain boundary diffusion coefficient

File:GrainBoundaryDiffusion.png that adds a term for sideflow out of the boundary, given by the equation$a\backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; t\}+f(y,t)=aD\text{'}\{\backslash partial^2\; \backslash varphi\backslash over\backslash partial\; x^2\}$, where $D\text{'}$ is the diffusion coefficient, $2a$ is the boundary width, and $f(y,t)$ is the rate of sideflow.]]

The general way to measure grain boundary diffusion coefficients was suggested by Fisher.{{Cite journal |last=Fisher |first=J. C. |date=January 1951 |title=Calculation of Diffusion Penetration Curves for Surface and Grain Boundary Diffusion |url=http://aip.scitation.org/doi/10.1063/1.1699825 |journal=Journal of Applied Physics |language=en |volume=22 |issue=1 |pages=74–77 |doi=10.1063/1.1699825 |bibcode=1951JAP....22...74F |issn=0021-8979|url-access=subscription }} 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 $\backslash delta$, 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.

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

$\backslash frac\{\backslash partial\; c\_b\}\{\backslash partial\; t\}=D\_b\backslash left(\{\backslash partial^2\; c\_b\backslash over\backslash partial\; y^2\}\backslash right)+\backslash frac\{2D\}\{\backslash delta\}\backslash left(\backslash frac\{\backslash partial\; c\}\{\backslash partial\; x\}\backslash right)\_\{x=\backslash delta/2\}$

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.{{Cite journal |last=Whipple |first=R.T.P. |date=1954-12-01 |title=CXXXVIII. Concentration contours in grain boundary diffusion |url=https://doi.org/10.1080/14786441208561131 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=45 |issue=371 |pages=1225–1236 |doi=10.1080/14786441208561131 |issn=1941-5982|url-access=subscription }} The diffusion profile therefore can be depicted by the following equation.

$(dln\backslash bar\{c\}/dy^\{6/5\})^\{5/3\}=0.66(D\_1/t)^\{1/2\}(1/D\_b\backslash delta)$

To further determine $D\_b$, two common methods were used. The first is used for accurate determination of $D\_b\; \backslash delta$. The second technique is useful for comparing the relative $D\_b\; \backslash delta$ 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 $D\_b\; \backslash delta$.
• Method 2: To compare the length of penetration of a given concentration at the boundary $\backslash \; \backslash Delta\; y$ with the length of lattice penetration from the surface far from the boundary.

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• Kirkendall effect
• Phase transformations in solids
• Mass diffusivity

{{DEFAULTSORT:Grain Boundary Diffusion Coefficient}}

Category:Diffusion