Grain boundary diffusion coefficient

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

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.

${\displaystyle {\frac {\partial c}{\partial t}}=D\left({\partial ^{2}c \over \partial x^{2}}+{\partial ^{2}c \over \partial y^{2}}\right)}$ 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.

• Kirkendall effect
• Phase transformations in solids
• Mass diffusivity