2.4 Interdiffusion

If we contact two materials, let's say $A$ and $B$ , the $A$ atoms start to enter into the $B$ material, whereas the $B$ atoms to the $A$ . This phenomenon is called interdiffusion. The mixing process continues until the system reaches its equilibrium state in which the atoms are distributed completely randomly if $A$ and $B$ forms completely miscible alloy. (see Fig. 2.4)

Figure 2.4: Interdiffusion process in completely miscible alloys. The mixing continues until the system reaches its equilibrium state, i.e. when the atoms are distributed completely randomly. The bottom panel shows the composition profile of the $A$ atoms. The right panel illustrates the scheme of the phase diagram.

In principle this kind of a process can be described by Fick's equations (see sec. 2.1), i.e. Boltzmann's transformation (see sec. 2.2) can be applied (until the boundary conditions are valid). Consequently, a plane with constant composition shifts proportionally to the square root of the time, which also means that the thickness of the diffusion zone is also $\propto \sqrt{t}$ . (see Fig. 2.1)

If the system has a tendency for phase separation, the intermixing stops when the initially pure materials cannot solve more impurity atoms, i.e. when the impurity concentration reaches the solubility limit. (see Fig. 2.5) In this state the impurity atoms are distributed completely randomly on both side of the interface but the composition is different.

Figure 2.5: Interdiffusion process in phase separating system. The intermixing stops when the initially pure materials cannot solve more impurity atoms, i.e. when the impurity concentration reaches the solubility limit. The bottom panel shows the composition profile of the $A$ atoms. The right panel illustrates the scheme of the phase diagram.

If the system favors ordering, the atoms are not distributed randomly but tend to form a well defined chemical order (stoichiometric order). For instance, if the stoichiometric composition is $A_{50\%}B_{50\%}$ and the lattice is BCC, an $A$ atom wants to have has only $B$ neighbors and vica-versa. Consequently, the atoms in the diffusion zone are ordered form the beginning of the process and the width of this ordered zone growths in time until the atoms are completely ordered in the whole sample. (see Fig. 2.6)

Figure 2.6: Interdiffusion process in ordering system. The atoms in the diffusion zone are ordered form the beginning of the process and the width of this ordered zone growths in time until the atoms are completely ordered in the whole sample. The bottom panel shows the composition profile of the $A$ atoms. In the ordered phase it has an oscillatory character. Since in this specific crystal structure and direction, pure atomic planes alternate. The right panel illustrates the scheme of the phase diagram.

2.4.1 Atomic fluxes

According to the Onsager's theorem, in an $A/B$ binary system, if the only driving force is the gradient of the chemical potential ($\mu_i$ ), the flux of $i$ ($A$ , $B$ ) atoms relative to the lattice planes can be given as:

$$\vec{j_i}=-L_{ii}\mathrm{grad}\mu_i$$ (2.19)

where $L_{ii}$ is the 'Onsager coefficient'. The chemical potential can be expressed in the following way:

$$\vec{\mu_i}=\mu_0+k_bT\ln \gamma_i c_i,$$ (2.20)

where $k_B$ is Bolzmann's constant and $T$ the absolute temperature. Moreover, $c_i$ and $\gamma_i$ are the atomic fraction and the thermodynamic activity coefficient, respectively. Combining equations (2.19) and (2.20) and using that $\rho_i\partial c_i/\partial\rho_i=c_i$ and $\rho_a+\rho_B=\rho=\mathrm{conts}$ , one obtains:

$$\vec{\j_i}=-\frac{k_BTL_{ii}}{\rho_i}\left[1+\frac{\partial\ln\ga... ..._i=-\frac{k_BTL_{ii}}{\rho_i}\Theta\mathrm{grad}\rho_i=-D_i\mathrm{grad}\rho_i,$$ (2.21)

where $\Theta$ is the thermodynamic factor. $D_i$ is the intrinsic diffusion coefficient which relates to the Brownian diffusion coefficient $D^i$ (see section 2.3) by $\theta$ :

$$D_i=\Theta D^i.$$ (2.22)

2.4.2 Interdiffusion coefficient

When two species of atoms intermingle, their rate of mixing depends on the diffusion rates of both species. For diffusion in an isolated system, an interdiffusion coefficient (or mutual diffusion coefficient or chemical diffusion coefficient) can be defined, which gives the rate at which the original concentration gradient disappears.

When the two species in an interdiffusion experiment have unequal intrinsic diffusion coefficients, there is a net atom flux across any plane in the diffusion zone. Thus, more atoms will be on one side of the interface after diffusion which results a net volume transport. This is equivalent to the creation of a non-uniform stress-free strain: on one side of the diffusion zone contractions, while on the other side extractions will arise. The stress field related to this stress-free strain contributes to the atomic fluxes across the driving force $-\Omega\mathrm{grad} p$ , and could cause a plastic deformation (by creep or by dislocation glide) as well. The plastic flow obviously relaxes the stress developed and results in a complex feed-back effect. The description of the interdiffusion process then depends on the ratio of the relaxation time of plastic flow, $\tau$ , and the time of diffusion $t$ .

If $t \gg \tau$ the relaxation of stress can be considered to be fast and almost complete. In this case the stress gradient as a driving force can be neglected. However, the relaxation of stresses is equivalent to a convective transport in the diffusion zone: e.g. for vacancy mechanism, expansion as well as contractions on different sides of the diffusion zone can be realized by annihilation and creation of vacancies at edge dislocations. From experimental point of view, if there is no change in the lateral dimensions of the specimens, a marker wire introduced originally at the interface appears to move toward one end of the diffusion couple. This effect, which was first observed by Kirkendall is called the Kirkendall shift (see Fig. 2.7).

Figure 2.7: Kirkendall shift. (a) During the diffusion, $A$ atoms diffuse to the right across the marker plane (designed by the dashed line) while $B$ atoms diffuse to the left. (b) If A diffuses faster than $B$ , more atoms are on the right of the markers after diffusion than before. Also fewer atoms are on the left. This expands the crystal volume on the right and shrinks that on the left. If the plane containing the markers is held in a fixed position, the crystal moves to the right by a distance $x_K$ . The hollow circles indicate the region with a surplus of vacancies, where porosity may be found. The plus signs indicate the region where extra planes are added. (c) If the crystal is then moved into alignment with diagram (a), it appears that the wires have moved to the left a distance $x_K$ .

The marker wire is assumed to identify a given lattice plane. The flux $\vec{j_i}$ of $i$ ($A$ , $B$ ) species with respect to a given lattice plane (in the lattice frame) can be expressed in terms of intrinsic diffusion coefficients as in equation (2.21). If the plane is moving with velocity $\vec{v}$ with respect to the ends of the diffusion couple (in the laboratory frame of reference), the flux of $\vec{j'_i}$ $i$ ($A$ , $B$ ) species with respect to the ends of the diffusion couple is:

$$\vec{j'_i}(\vec{r},t)=\vec{j_i}(\vec{r},t)+\rho_i(\vec{r},t)\vec{v}(\vec{r},t).$$ (2.23)

In a two component crystal of constant dimensions and atom density (the number of lattice sites is conserved, i.e. $\partial(\rho_a+\rho_B)/\partial t=0$ ), it is necessarily true that $\vec{j'_A}=\vec{j'_B}$ and $\partial c_A/\partial x=-\partial c_B/\partial x)$ . Thus the total atom flux $\vec{j'_t}$ with respect to the ends of the couple is:

$$\vec{j'_t}=\vec{j'_A}+\vec{j'_B}=-\left(D_A-D_B\right)\mathrm{grad}\rho_A+\rho \vec{v}=0.$$ (2.24)

The Kirkendall velocity can then be written in the following form:

$$\vec{v}=\frac{1}{\rho}\left(D_A-D_B\right)\mathrm{grad}\rho_A.$$ (2.25)

Therefore, using equations (2.4), (2.23) and (2.24) the fluxes of $A$ and $B$ atoms can be expressed by:

$$\vec{j'_A}=-\vec{j'_B}\frac{1}{\rho}\left(\rho_B D_A + \rho_A D_B\right)\mathrm{grad}\rho_A.$$ (2.26)

Consequently, the interdiffusion can be characterised by only one diffusion coefficient defined in the following way:

$$\tilde{D}:=\frac{1}{\rho}\left(\rho_B D_A + \rho_A D_B\right)=c_B D_A + c_A D_B,$$ (2.27)

where $c_i = \rho_i/\rho$ ($i = A,B$ ) is the atomic fraction. Equation (2.27) is called as Darken's formula.

On the other hand, if $t \ll \tau$ (but $t$ is long enough for the development of the stress gradient) then we can be in a second limit, when practically there is no stress relaxation at all ( $\vec{v} \approx 0$ ). An additional term proportional to the stress (pressure) gradient should be added to the right hand side of equation (2.4) and it can be shown that the mixing process is controlled by:

$$\tilde{D}_{NP}:=\frac{D_AD_B}{c_AD_A+c_AD_B}.$$ (2.28)

Here the index NP indicates that this is the so-called Nernst-Planck limit. After an initial transient period the pressure gradient developed makes the two fluxes equal, i.e. the volume transport will be determined by the slower intrinsic diffusion coefficient (series coupling of currents) in contrast to the Darken's limit (parallel coupling), where the chemical diffusion coefficient is determined by the faster one.