Physics:Grain boundary diffusion coefficient

Short description: Diffusion coefficient of a diffusant along a grain boundary

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 [math]\displaystyle{ D_b }[/math], 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 [math]\displaystyle{ D_b }[/math] is the same in both types of samples. However, at temperatures below 700 °C, the values of [math]\displaystyle{ D_b }[/math] with polycrystal silver consistently lie above the values of [math]\displaystyle{ D_b }[/math] 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 equation[math]\displaystyle{ a\frac{\partial \varphi}{\partial t}+f(y,t)=aD'{\partial^2 \varphi\over\partial x^2} }[/math], where [math]\displaystyle{ D' }[/math] is the diffusion coefficient, [math]\displaystyle{ 2a }[/math] is the boundary width, and [math]\displaystyle{ f(y,t) }[/math] 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 [math]\displaystyle{ \delta }[/math], the length is [math]\displaystyle{ y }[/math], 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.

[math]\displaystyle{ \frac{\partial c}{\partial t}=D\left({\partial^2 c\over\partial x^2}+{\partial^2 c\over\partial y^2}\right) }[/math] where [math]\displaystyle{ |x|\gt \delta/2 }[/math]

[math]\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} }[/math]

where [math]\displaystyle{ c(x, y, t) }[/math] is the volume concentration of the diffusing atoms and [math]\displaystyle{ c_b(y, t) }[/math] 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.

[math]\displaystyle{ (dln\bar{c}/dy^{6/5})^{5/3}=0.66(D_1/t)^{1/2}(1/D_b\delta) }[/math]

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

See also

• Kirkendall effect
• Phase transformations in solids
• Mass diffusivity