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

, 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

is the same in both types of samples.

However, at temperatures below 700 °C, the values of

with polycrystal silver consistently lie above the values of

with a single crystal.

[2] 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

, the length is

, 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.

is the volume concentration of the diffusing atoms and

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.

, two common methods were used.

The first is used for accurate determination of

The second technique is useful for comparing the relative

of different boundaries.