In Materials Science, most of the experimental measurements have been performed by placing two blocks of material in direct contact, making them a diffusion pair, then measuring the composition, distance, time, and temperature. The sample is then cut on a plane as shown in Figure 2.2.5 ( see post 2.2.4), and the composition can be determined on a refined scale via X-ray diffractometry, hardness tests, or, if the sample is radioactive, by various techniques in counting radioactivity. Repeating for various temperatures, it’s possible to determine the diffusion coefficient D at these temperatures, through the relationship shown in Figure 2.2.5. The diffusion coefficient D of many substances follows the Arrhenius Equation:
\[D = D _ {0} e ( – \frac{Q}{RT}) \tag{2.10}\]
\( D _ {0}, Q \), are, respectively, the pre-exponential constant independent of temperature and the energy of activation for diffusion, R is the gas constant and T is the absolute temperature (K). Using the natural logarithms of Eq. (2.2.10), we get:
\[ lnD = – \frac{Q}{R}(\frac{1}{T}) + lnD _ {0} \tag{2.11} \]
Using decimal logarithms in Eq. (2.11), we get:
\[ logD = – \frac{Q}{2,3R}(\frac{1}{T}) + logD _ {0} \tag{2.12} \]
Equations 2.11 and 2.12 assume the equation forms of a straight line:
\[ y=ax+b \tag{2.13} \]
In this way, for example, if the value of ln D was plotted as a function of 1/T, the activation energy could be calculated assuming that the angular coefficient of the straight line is – Q / R. The pre-exponential constant can be determined extrapolating a line to the intersection with the ordered axis. The value read, the linear coefficient of the line, is ln D0.
Figure 2.2.7: Carbon diffusion in a steel sample (after carburizing): darker region around the grains.
