Let us assume the following relation is applicable for determining the lattice self — diffusivity value in copper (FCC, lattice parameter a0 = 0.3615 nm):

where b is the number of positions an atom can jump to, d is the dimension (if for one-dimensional flow, d = 1; for two-dimensional flow, d = 2; and for three­dimensional flow, d = 3), and other terms are already defined in Eq. (2.43).

Determine b, l, and DL, given that diffusion takes place along (110) direction at 500 °C (773 K) and Q = 209 kJ mol-1.

Solution

For an FCC along (110) direction, b = 12, and l is given by half of the face diagonal length (P2 a0/2), that is, 0.2566 nm or 2.566 x 10-8 cm.

Therefore, using the given equation, we obtain

12 2 DL = 2—3 (2.556 x 10-8 cm (1013 s-1) exp

= 9.83 x 10-14 cm2 s-1.

The activation energy for vacancy diffusion is composed of two terms, activation enthalpy (or energy) for migration and activation enthalpy for vacancy formation. Calculated and experimental activation energies for diffusion in gold and silver through vacancy mechanism are shown in Table 2.6. The activation energy for dif­fusion of self-interstitials appears in the form similar to the vacancy diffusion (i. e., both formation energy and migration energy are included). Although for interstitial impurity diffusion the formulation is almost the same, for vacancy diffusion, it does not involve any probability factor similar to the vacancy formation energy; instead, it contains only the migration energy term. That is why the substitutional diffusion (including self-diffusion) occurring through vacancy mechanism is much slower than the interstitial impurity diffusion. see the example in Figure 2.48

showing carbon (interstitial) diffusion in у-iron, Cr substitutional diffusion, and self-diffusion in у-iron. Substitutional impurity diffusion is also influenced by the atom size and charge effects of the impurities. Generally, oversized (compared to the host atom) substitutional impurities have a higher migration energy than that of the undersized substitutional impurities. Increase in the valence of the substitu­tional impurity atom has been found to reduce the activation energy. When solute- vacancy complexes are created, they would also affect the diffusion.

For a given crystal structure and bond type, <Q;elf /RTm is more or less constant, where Tm is the melting temperature (K). It has been found that most close-packed metals tend to possess a Qself/RTm of ~18. The activation energy for self-diffusion is proportional to the melting temperature. For example, Figure 2.49 gives the acti­vation energy for self-diffusion of various FCC metals plotted against their melting temperatures. Correlations by Sherby and Simnad [8] revealed the following rela­tion between the activation energy for self-diffusion, melting point, and valence:

Q self = R(K 0 + V )Tm, (2.47)

300

2000

Figure 2.49 Activation energy for lattice self-diffusion versus melting temperature for various FCC metals [8].

14 for BCC 16 for HCP 18 for FCC 20 for diamond structure

(2.48)

K о

While this formulation worked well for many metals, some metals such as Zr, Ti, Hf, U, and Pu were noted to deviate from the predictions.

Special Note Direct measurement of self-diffusivities is not easy because of the difficulty in tracking identical, individual atoms. That is why tracers (such as radioisotopes) that are chemically the same as the host atoms but detectable by analytical methods are often used. However, tracer diffusivity is quite close to the self-dif — fusivity, yet it is little less. It occurs because the tracer jumps could be correlated in that they can jump back to the sites wherefrom they originally started the jump. Because of these basically "wasted” jumps, the tracer diffusivity (Dtracer) values are less than the self-diffusivity (Dse|f), that is, Dtracer =f Dse|f, where f is the correlation factor. Based on exhaustive geometric principles, the correlation factors can be found out. For vacancy diffusion, the following are the correlation factors for different crystal structures (0.781 for FCC, 0.721 for HCP, and 0.655 for BCC). In general, we can see that the variation due to the correlation factors is mostly quite small since the accuracy with which diffusivity can be determined is limited. Furthermore, consideration of correlated jumps is required in the case of associated defects (such as the solute-vacancy complex).