2.5 Composition dependent diffusion coefficient - Diffusion asymmetry

Usually the tracer (or Brownian) diffusion coefficient of a diffusion specie is different on the two sides of a diffusion couple (diffusion asymmetry). (see Fig. 2.8) This originates from the difference in the bonding strength in the different matrixes. Therefore, the jump probabilities of the atoms or the diffusion coefficient depend on the local environment, i.e. on the local composition.

Figure 2.8: Diffusion asymmetry: diffusion coefficient of a diffusion specie is different on the two sides of a diffusion couple.

The interface in a diffusion couple is usually not atomically sharp but more or less diffused. This means that the composition changes, consequently the diffusivity also changes accordingly. Therefore the (tracer) diffusion coefficients are composition dependent.(see Fig. 2.9) (for this reason the 2.7 form of Fick's second law is not advised)

Figure 2.9: The interface in a diffusion couple is usually not atomically sharp. The composition changes, consequently the diffusivity also changes accordingly.

The composition dependence of the diffusion coefficient ($D$ ) or the jump frequency ($\Gamma$ ) can be considered in both continuum and atomic models, for instance, in the following form: $D = D_0 \exp(mc)$ or $\Gamma = \Gamma_0 \exp(mc)$ , where $D_0$ and $\Gamma_0$ are composition independent pre-exponential factors, $c$ is the composition of one of the constituents (e.g. of $A$ ) and $m$ is a parameter determining the strength of the composition dependence. It is worth introducing an $m' = m \log_{10}e$ ($e$ is the base of natural logarithm) parameter, since thus $\log_{10}(D/D_0) = \log_{10}(\Gamma/\Gamma_0) = m'c$ , i.e. its value gives in orders of magnitude the ratio of the jump probabilities or diffusion coefficients in the pure $A$ and $B$ matrixes. (see Fig. 2.10) This gives the possibility to determined $m'$ experimentally from the self and impurity diffusion coefficients (and can also be called as diffusion asymmetry parameter).[2] Note that, $m'$ can also be derived from the difference in the $A-A$ and $B-B$ pair interaction energies (see e.g. Ref. [2]). If $m'$ is positive/negative, the jumps are faster/slower in the $B$ matrix. For instance, $m' = 4$ means that the jumps are $10,000$ times faster in the $B$ matrix than in $A$ . The ratio of the diffusion coefficients is usually several orders of magnitude ($4-7$ ). The diffusion asymmetry is seldom considered despite that in nature usually different atoms have different mobility in different materials.

Figure 2.10: The parameter $m'$ quantifies the strength of the diffusion asymmetry. It gives in orders of magnitude the difference between the diffusion coefficients in the pure $A$ and $B$ matrixes.

The animation in Fig. 2.11 illustrates how the diffusivity of an $A$ ($B$ ) atom changes while it diffuse from the $A$ ($B$ ) matrix to the $B$ ($A$ ).

Figure 2.11: The parameter $m'$ quantifies the strength of the diffusion asymmetry. It gives in orders of magnitude the difference between the diffusion coefficients in the pure $A$ and $B$ matrixes.

Remark: Usually we, suppose that the diffusion coefficient has exponential composition dependence. However, it is not obligatory, any kind of function may be assumed, it does not alter the general conclusions.