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

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 equationa ∂ φ ∂ t + f ( y , t ) = a D ′ ∂ 2 φ ∂ x 2 {\displaystyle a{\frac {\partial \varphi }{\partial t}}+f(y,t)=aD'{\partial ^{2}\varphi \over \partial x^{2}}}, where D ′ {\displaystyle D'} is the diffusion coefficient, 2 a {\displaystyle 2a} is the boundary width, and f ( y , t ) {\displaystyle f(y,t)} is the rate of sideflow.

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

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

∂ c b ∂ t = D b ( ∂ 2 c b ∂ y 2 ) + 2 D δ ( ∂ c ∂ x ) x = δ / 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 c ( x , y , t ) {\displaystyle c(x,y,t)} is the volume concentration of the diffusing atoms and c b ( y , t ) {\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. The diffusion profile therefore can be depicted by the following equation.

( d l n c ¯ / d y 6 / 5 ) 5 / 3 = 0.66 ( D 1 / t ) 1 / 2 ( 1 / D b δ ) {\displaystyle (dln{\bar {c}}/dy^{6/5})^{5/3}=0.66(D_{1}/t)^{1/2}(1/D_{b}\delta )}

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

See also