Grain boundary diffusion coefficient

The grain boundary diffusion coefficient is the diffusion coefficient of a diffusant along a grain boundary in a polycrystalline solid. It is a physical constant denoted $$D_b$$, 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 $$D_b$$ is the same in both types of samples. However, at temperatures below 700 °C, the values of $$D_b$$ with polycrystal silver consistently lie above the values of $$D_b$$ with a single crystal.

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.