The equation (2.8) (see the topic above “Fick’s Second Law (III): Diffusion in a non-Steady State”) should only be applied when a semi-infinite solid diffusion1 with a constant source of solute atoms on its surface is equal to CS and these atoms or molecules to be diffused are already in the solid at concentration C0. In the case of a solid that initially free of the atoms or molecules to be diffused and whose surface is kept in a constant concentration of atoms, the following boundary condition should be assumed:
a- While x=0 , we have CS for any t.
The use of this boundary condition in the solution of Fick’s Second Law, equation (2.7), generates the solution:
\[ C(x,t) = C _ {S}[1-erf(\frac{x}{2\sqrt{Dt}})] \tag{2.28} \]
Figure 2.2.22 presents the graph of variation of solute in concentration C with a distance x for this situation.
On the other hand, if the concentration of solute on the surface is maintained constant and is equal to 0 and the solid initially has a solute concentration of, the following boundary conditions should be assumed:
a- For x = 0, then C = 0 for any t.
b- For x > 0, then C = C0 when t = 0.
The application of this boundary condition in the solution from Fick’s Second Law, equation (2.7), results in the following solution:
\[ C(x,t) = C _ {0}erf(\frac{x}{2\sqrt{Dt}}) \tag{2.29} \]
Figure 2.2.22 shows the variation of the concentration of a solute C with a distance of x, for the case of diffusion in a semi-infinite solid, where the surface is maintained without solute for different amounts of time.
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.
