The diffusion coefficient of vacancies is an important parameter for microstructural evolution, for it deter­mines the rate of mutual recombination of PDs. Migrating vacancies can also meet solute or impurity atoms and form immobile complexes, which can then dissociate. In quasi-equilibrium, when the rates of complex formation and dissociation events are equal to each other:

zv+C’Y C’s = n-Cvs [103]

Here, Cvs and Cs are the concentrations of com­plexes and solute atoms, respectively, C and Cv are the concentrations of free (unpaired) solute atoms and vacancies, respectively, v+ and v- are the frequencies of complex formation and dissociation events, respectively, and z is a geometrical factor, which is of the order of the coordination number for complexes with a short-range (first-nearest neigh­bor) interaction and unity for long-range interac­tions. The binding energy of the complex, £{(,, is defined from v-/v+ = exp(b£jbs). The solute con­centration is generally much higher than that of vacancies, hence

C0 « Cs

cv = Cv — Cvs [104]

Substituting these into eqn [103], one obtains aCyCs exp(bEys)

1 + aCs exp(bEjbs)

The total vacancy concentration is, therefore,

Cv = Cv + Cvs = Cv [1 + aCsexp(bEbs)] [106]

The effective diffusion coefficient of vacancies may be defined as

Dv

1 + a Cs exp(b£[bs) exp [-b(Evm + Ebs)]

While the vacancy concentration is approximately equal to

Cv « CvaCsexp^) [108]

The vacancy flux is, thus, equal to that in the absence of impurities,

Df Cv = DvCv [109]

which is supported by the measurements of the self-diffusion energy, which is almost independent of the presence of impurities. The main conclusion is that the total vacancy flux does not depend on the presence of impurity atoms. However, impurity trapping may affect the recombination rate and hence Cv may be increased.