Matano interface

We now have to make the definition of x more explicit and decide where to locate the origin (the Matano interface) for the conservation condition above to hold. In the laboratory, one would use an arbitrary coordinate X, such as the distance to one end of the experimental device, then... 

Figure 8.12 The Boltzmann-Matano technique. Initially, concentration is C0 to the left of the initial interface located at x0, C, to the right. The hatched areas between C0 and C, must be equal, which defines the position of the Matano interface. The framed area represents the numerator of equation (8.4.9). The diffusion coefficient is computed from the same equation.

We adopted a second method called the Boltzmann-Matano interface method [see Eq. (5)], which is more accurate for the purely diffusive process. Notice that Ma et al. successfully employed this method, since the spreading profiles they measured exhibit little mass buildup at the front and negligible dewetting. 

Figure 2.3-10 illustrates the integral and derivative that are to be computed from the exTCrimental curve to calculate the diffusion coefficient. The z 0 plane is known as the Matano interface and must be. located on the profile by applying the condition... 

The differential quotient (dx/dc) is taken at the point where the concentration is equal to c. Furthermore, it is evident that all material which has diffused out of the semi-infinite space X < 0 must now be found in the semi-infinite space x > 0. Therefore, the coordinate of the so-called Matano interface = 0 can be calculated by means of the equation ... 

The practical analysis of a diffusion experiment in order to determine the concentration dependent chemical diffusion coefficient D (c) is carried out as follows. First of all, the Matano interface Xn = 0 is determined graphically using eq. (5-58). This defines a coordinate system. Then, for every concentration c, the slope (dx/dc) of the experimental curve x (c) and the... 

This formula permits us to calculate the concentration dependent diffusion coefficient D (c) without the necessity of determining the position of the origin of the coordinate system, since the integrals in eq. (5-59) are no longer dependent upon the position of the origin. In part (a) of Fig. 5-9, the calculation of the concentration dependent diffusion coefficient according to eq. (5-59) is illustrated. In Fig. 5-9 part (b), the method of determining the Matano interface is shown. 

Only the differential dx appears in eq. (7-12). Therefore, as stated above, B(N2) can be determined in a coordinate system with an arbitrary origin, and it is not necessary to determine the position of the Matano interface. If (A a) is known as a function of the mole fraction, and if the diffusion profile N2 (x) has been measured, then eq. (7-12) can be used to determine the chemical diffusion coefficient (iVa). This could be done graphically, for instance, by a modification of the procedure illustrated in Fig. 5-8. 

The meaning of the right-hand-side of eq. (7-26) now becomes clear. Finally, it may be noted from eq. (7-26) that d jdt becomes equal to zero when the diffusion currentsin the a- and j5-phases at the phase boundary x = ( are equal at all times. The Matano interface is then coincident with the phase boundary. In all other cases, the phase boundary moves away from the Matano plane with velocity /21 as has just been calculated. 

To compare the experimental composition profiles with the calculated profiles, the experimental Matano interface was determined by an averaging process. For three of the elements with large concentration differences (Cr, Re, and Ti), the Matano interface was independently located. With respect to the average position, the coordinates for Cr, Re, and Ti profiles for the 100 h treatment were -15 pm, -1 pm and +17 pm, respectively. These variations are small compared to the large diffusion distances. 

Figure 4. (a) Predicted location of wliere maximum pore density is expected for Rene-N4/Rene-N5 at 1293 °C. (b) Back scatter image of Rene-N4/Rene-N5 diffusion couple after 100 h at 1293 °C. Thin white Ime indicates position of microprobe scan. The dashed white line corresponds to the Matano interface. 

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