Grain boundary diffusion coefficient – Wikipedia

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[edit]

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.

${displaystyle {frac {partial c}{partial t}}=Dleft({partial ^{2}c over partial x^{2}}+{partial ^{2}c over partial y^{2}}right)}$

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[edit]

See also[edit]