The literature on Solid state diffusion consider the experiments done by Ernest Kirkendall in the 1940’s, as essential to the advance in comprehension of diffusion in solids. Kirkendall showed inter-diffusion in solids and that the essential mechanism for migration is the presence of vacancies in the solids.
Before Kirkendall’s work, the japanese scientist C. Matano, in 1933, had already proposed a technique for analyzing data of diffusion experiments. This technique is based on a solution to the equation 2.7 (Fick’s Second Law), that was proposed by L. Boltzmann in 1894. The Boltzmann solution states that the coefficient of diffusion D is a function of concentration and time. For a specific composition, the diffusion coefficient can also vary with the position along the material (x), if there is a concentration gradient in the material. This requires a solution of Fick’s Second Law in the following form:
\[\frac{\partial C _ {0}}{\partial t}= \frac{\partial }{\partial x}\overline{D}(C _ {0})\frac{\partial C _ {0}}{\partial x} (2.30)\]
the solution for the equation (2.30) isn’t trivial, being tied to graphical integration for the determination of the coefficient of diffusion \( \overline{D} \).
Consider the semi-infinite diffusion pair that makes up a block made up of two alloys of differing materials soldered together as in Figure 2.2.23. Several wires of a material insoluble in the two alloys are placed between the two alloys.
These wires are the marker material in the diffusion process experiment and as was discussed in the “Kirkendall Effect” topic, the markers move during the diffusion.
After the experiment and chemical analysis of the slices of material, the first step is to trace a concentration curve as a function of distance along the bar, measured from an adequate point of reference (which could be, for example, an end of the diffusion pair). As a second step, choose the traversal section of the bar that is equal to the total flow of both atomic species in diffusion (A and B, for example). This section, known as the Matano Interface, finds itself in the position where the areas M and N in Figure 2.2.23 are equal.
Once the Matano interface2 is located, it serves as the origin of x coordinate (the distances to the right of the interface will be considered positive and to the left, negative). The Boltzmann solution for Fick’s Second Law, for a coordinate system as defined, will be:
\[ \overline{D} = – \frac{1}{2t} \frac{\partial x}{\partial C _ {0}} \int_{C _ {1}}^{C _ {0}} xd C _ {0} \tag {2.31} \]
where t is the duration of diffusion, C 0 is the concentration in atomic units at a distance of x of the Matano interface and C1 is the concentration on one side of the diffusion pair, on a point away from the interface, where the composition is constant and isn’t affected by the process of diffusion.
1Diffusion is considered semi-infinite when the diffusing atoms travel only a portion of the distance in the direction of diffusion. The bars are much larger than the distance that the atoms that diffused could reach, even with large experimental times (days or months). In general, the distances involved in diffusion experiments are on the order of micrometers or millimeters.
2There is evidence that it would stay in the soldered region (Reed-Hill, R E, Abbaschian, R. “Physical metallurgy principles.” 4th ed. (2008).).
