Kirkendall shift

The Kirkendall effect is commonly accompanied by the Frenkel effect, the void formation in the diffusion zone. In foreign literature, the Frenkel effect is often referred to as Kirkendall voiding, which is rather confusing, as Kirkendall and Frenkel effects are competitive vacancies annihilating at the dislocation kinks and causing the Kirkendall shift, cannot be used for Kirkendall voiding, and vice versa. 

Substituting Equation 6.123 in the constraint of Equation 6.116, we obtain the velocity of Kirkendall shift. 

An interesting point with the second case (thermomigration-driven Kirkendall shift) is that there is no void formation or hillock extrusion at all. 

In both the cases, back stress and Kirkendall shift, the equation for redistribution of volume fractions is of the type... 

In the hollow nanosheU formation, we have the opposite phenomenon - an almost pure Frenkel effect with a fully or partially suppressed Kirkendall shift Most probably, this happens due to peculiarities of interdiflusion in nanosystems (1) there is little place for dislocations in a nanoparticle so that K-sinks are just lacking (2) a common Kirkendall shift in radial direction (simultaneous for all atoms) leads to tangential deformation and corresponding stresses in spherical or cyhndrical samples, which are large in the case of nanosystems, and suppress the shift. 

As mentioned above, in a hollow nanoshell formation, we have an almost pure Frenkel effect with fuUy or partially suppressed Kirkendall shift In [10], we analyzed the formation of a hollow compound nanoshell. We demonstrated that the Gibbs-Thomson effect, leading to the shrinkage of ready compound shells, should influence the formation stage as well. Sometimes it may even suppress the nanoshell formation. 

We use the term Kirkendall effect to include both Kirkendall shift and Kirkendall voiding (or Frenkel voiding). We present a detailed analysis of the interaction of the Kirkendall effect and the inverse Kirkendall effect in nanoscale... 

There is no Kirkendall shift during phase formation, i.e. all vacancy fluxes go to the formation of Kirkendall (or Frenkel) voids instead of being annihilated by internal sinks hence, they do not causing the lattice shift. 

Note that we use the fluxes in the lattice reference frame and intrinsic diffusivities (instead of interdiffusivity) according to our basic approximation of no Kirkendall shift. We neglect the correlation factors of Manning s vacancy wind terms [28] since they can change results only quantitatively. Both tracer diffusivities are proportional to vacancy concentration. Here we use one more approximation that we treat only one effective vacancy concentration, without distinguishing between sublattices in the compound. We take... 

This means that oxygen is lost to the gas phase at the spinel-Fe203 interface and picked up again at the spinel-MgO interface, the amount of this oxygen being of that in the spinel formed. Thus there is a Kirkendall shift of markers which approaches of the thickness of the spinel layer formed. 

If, within the diffusion zone, there is no active vacancy source or sink, then no drift of lattice planes could occur and the difference in the diffusion fluxes of substitutional chemical species would result in vacancy supersaturation and build-up of local stress states within the diffusion zone. Return to local equilibrium in a stress-free state could be achieved by the nucleation of pores leading to the well-known Kirkendall porosity (Fig. 2.2d). All intermediate situations are possible depending on local stress states and the density, distribution and efficiency of vacancy sources or sinks. However, it should be emphasized that complete Kirkendall shift would occur only in stress-free systems in local equihbrium. Therefore, all obstacles to the free relative displacement of lattice planes would lead to local non-equilibrium. Such a situation corresponds to the build-up of stress states that modify the conditions of local equilibrium and the action of vacancy sources or sinks these stress states must therefore be taken into account to define and analyse these local conditions and their spatial and temporal evolutions. 

Let us analyze these results one step further and ask about a quantitative measure of the Kirkendall effect. This effect had been detected by placing inert markers in the interdiffusion zone. Thus, the lattice shift was believed to be observable for an external observer. If we assume that Vm does not depend on concentration and local defect equilibrium is established, the lattice site number density remains constant during interdiffusion. Let us designate rv as the production (annihilation) rate of the vacancies. We can derive from cA+cB+cv = l/Vm and jA +/ B +./v = 0 that... 

The intrinsic diffusion coefficients, Dk and DB, of a binary alloy A-B express the diffusion of the components A and B relative to the lattice planes [7], Therefore, during interdiffusion, a net flux of atoms across any lattice plane is present, where, normally, the diffusion rates of the diffusing particles A and B are different. Subsequently, this interdiffusion process provokes the shift of lattice planes with respect to a fixed axis of the sample, result which is named the Kirkendall effect [9],... 

Fig. 2.13. Result of a Kirkendall marker experiment carried out by Rutherford backscattering spectroscopy. The upper part shows the marker shifts (A m and A ) for the tungsten marker and the Zr edge from the backscattering spectrum. The lower part compares the experimental data with the expected behavior of the marker shift when only one species moves (dashed lines). Within experimental error, it is found that only Ni moves (see [2.46] for details)...

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