2.3 Atomic aspects of volume diffusion

This subsection show how the Brownian motion (or random walk) relates to Fick's classical laws.

Let us consider a system of migrating particles. The paths of particles belonging to time $t$ are represented by vectors $\vec{R}(t)$ . The projection on the Ox axis is denoted by $X$ and is equal to the sum of the projections $x_i$ of the elementary jump vectors denoted by $\vec{r}_i$ . Since the average value of $x_i$ is:

$$\left<x_i\right>=\lim_{N \rightarrow \infty}\frac{1}{N}\left[\sum_{i=1}^{N}x_i\right],$$ (2.10)

considering a particle making $n$ jumps (on average) during the time $t$ , it can be written:

$$\left<X\right>=n\lim_{N \rightarrow \infty}\frac{1}{N}\left[\sum_{i=1}^{N}x_i\right].$$ (2.11)

In the same way, the quadratic mean free path can be given by:

$$\left<X^2\right>=n\lim_{N \rightarrow \infty}\frac{1}{N}\left[\su... ...}^{N}x_i^2 + 2 \sum_{i=1}^{N}\sum_{\substack{j=1 \ i \ne j}}^{N}x_ix_j\right].$$ (2.12)

If the migration of particles is random (Brownian migration or random walk), the first Fick's equation is:

$$j_x=-\frac{\left< X^2 \right>}{2t} \frac{\partial \rho}{\partial x},$$ (2.13)

where the Brownian diffusion coefficient in the direction x is defined by the relation:

$$D_x=-\frac{\left< X^2 \right>}{2t}.$$ (2.14)

Replacing the quadratic mean free path by its expression 2.12 and neglecting the second term with double summation (the double product terms of this relation compensate each other for a random walk, since jumps in opposite directions have the same probability), one obtains:

$$D_x=\frac{n}{2t}\lim_{N\rightarrow\infty}\frac{1}{N}\left[\sum_{i=1}^N x_i^2\right]=\frac{\Gamma}{2}\left<x_i^2\right>,$$ (2.15)

where $\Gamma = n/t$ is the total jump frequency of atoms. Note that the Brownian diffusion coefficient is often approached by the diffusion coefficient of an isotope (or tracer) $D_x^*$ . We recall that, theoretically, $D_x^* = D_xf$ , where $f$ is a correlation factor. Its presence is necessary because the migration of the marked (or tracer) atoms is not always completely random, i.e. the second term with double summation in 2.12 cannot be neglected. In the case of self-diffusion $f (\leq1)$ is usually a numerical factor depending on the crystal structure and diffusion mechanism. For impurity or heterodiffusion (the tracer atoms are different from the atoms of the matrix), this factor can depend on the temperature as well.

In a crystal, the migration of atoms takes place by site-by-site. The positions of these sites are perfectly defined by the structure. If $\Gamma_s$ denotes the frequency along direction $s$ and $Z$ is the number of neighbouring sites, one can write:

$$\Gamma=\sum_{s=1}^Z \Gamma_s.$$ (2.16)

Since the proportion of jumps in a given direction is equal to $\Gamma_s/\Gamma$ ,

$$D_x=\frac{\Gamma}{2}\left<x^2\right> = \frac{\Gamma}{2} \lim_{N\r... ..._{s=1}^ZN \frac{\Gamma_s}{\Gamma} x_s^2 = \frac{1}{2}\sum_{s=1}^Z\Gamma_sx_s^2.$$ (2.17)

According to this expression, in the case of selfdiffusion by vacancy mechanism, it is possible to determine the diffusion coefficient along Ox if the lattice parameter is known.

In an isotrop crystal with body centred cubic (BCC) structure (or face centred cubic - FCC), the jumping lengths and frequencies are the same for the $8$ (FCC: $12$ ) directions [see Figure 2.2 and Figure 2.3]. Applying relation (2.17) for the direction Ox, one can write for both of the structure types:

$$D_x=\frac{1}{2}8\left(\frac{a}{2}\right)^2 = \Gamma_sa^2.$$ (2.18)

Figure 2.2: Elementary jump for vacancy mechanism in a BCC structure. ($\square$  vacancy; $\bullet$  atoms).

Figure 2.3: Elementary jump for vacancy mechanism in an FCC structure. Only the elementary jumps that have the projection on the axis Ox different from zero are marked by arrows ($\square$  vacancy; $\bullet$  atoms).