Kirkendall planes

Flux-Induced Instahility and Bifurcations of Kirkendall Planes 163... 

From the intuitive viewpoint, it seems strange that the markers initially smear over the whole concentration range of the diffusion couple (located in quite a narrow region, though), and then gather into one plane which corresponds to only one fixed (constant) composition this plane is generally referred to as the Kirkendall plane. It is accepted [31-35] that the (K-planes is a plane of initial contact moving at parabolic dependence... 

Flux-Induced Instability and Bifiircations of Kirkendall Planes 1165... 

The Kirkendall effect is a well-known phenomenon resulting from the difference in intrinsic diffusivities of chemical constituents of substitution solid solutions (non-reciprocal diffusion). Many textbooks provide a detailed and quantitative treatment of this important phenomenon (Philibert, 1991) schematized in Fig. 2.2 for a homogeneous diffusion couple of constituents A and B forming a continuous substitutional solid solution. In Fig. 2.2a, the initial position of the contact surface is marked by fixed inert markers that define the origin, also named the Matano plane (M), of the reference frame centred on the mass or the number of moles this reference frame is conunonly used to define interdiffusion processes and the unique interdiffusion coefficient that permits the characterization of the transport of A and B. Moving inert markers determine the actual position of the initial contact surface (Fig. 2.2b) and therefore visualize the drift of lattice planes within the diffusion zone. They mark the origin, also named the Kirkendall plane, of the lattice-fixed frame of reference that permits the definition of the different intrinsic diffusion coefficients of the A and B constituents. Relative to the Matano plane, this drift of lattice planes is equivalent to a translation of the diffusion couple without apparent deformation and without the action of any externally applied stress or, in other words, to a rigid body translation of the phase lattice. 

One specific aspect of the Kirkendall effect in two-phase systems, emphasized in Section 2.2.2, resides in the need to define one Kirkendall plane, i.e. one lattice-fixed frame of reference, for both phases of such a system. For a diffusion couple between two constituents forming one or several compounds, the Kirkendall planes of all the phases, the reacting phases as well as the product phases, coincide with the initial contact surface at f = 0. These planes tend to separate as the product phases grow and the reacting phases recede. Thus, the behaviour of inert markers located at the initial contact surface is difficult to determine as they can be attached to each of these possible Kirkendall planes. Figure 2.9 shows that 10 pm tungsten wires used as inert markers in a Ti-Ni diffusion couple were broken longitudinally and split in several pieces that mark the Kirkendall planes of the different phases (Bastin and Rieck, 1974). This observation provides an impressive illustration... 

Evidence concerning the identity of the mobile species can be obtained from observation [406,411—413] of the dispositions of product phases and phase boundaries relative to inert and immobile markers implanted at the plane of original contact between reactant surfaces. Movement of the markers themselves is known as the Kirkendall effect [414], Carter [415] has used pores in the material as markers. Product layer thickness has alternatively been determined by the decrease in intensity of the X-ray fluorescence from a suitable element which occurs in the underlying reactant but not in the intervening product layers [416]. 

Several points are to be noted. Firstly, pores and changes of sample dimension have been observed at and near interdiffusion zones [R. Busch, V. Ruth (1991)]. Pore formation is witness to a certain point defect supersaturation and indicates that sinks and sources for point defects are not sufficiently effective to maintain local defect equilibrium. Secondly, it is not necessary to assume a vacancy mechanism for atomic motion in order to invoke a Kirkendall effect. Finally, external observers would still see a marker movement (markers connected by lattice planes) in spite of bA = bB (no Kirkendall effect) if Vm depends on composition. The consequences of a variable molar volume for the determination of diffusion coefficients in binary systems have been thoroughly discussed (F. Sauer, V. Freise (1962) C. Wagner (1969) H. Schmalzried (1981)]. 

In the Kirkendall effect, the difference in the fluxes of the two substitutional species requires a net flux of vacancies. The net vacancy flux requires continuous net vacancy generation on one side of the markers and vacancy destruction on the other side (mechanisms of vacancy generation are discussed in Section 11.4). Vacancy creation and destruction can occur by means of dislocation climb and is illustrated in Fig. 3.36 for edge dislocations. Vacancy destruction occurs when atoms from the extra planes associated with these dislocations fill the incoming vacancies and the extra planes shrink (i.e., the dislocations climb as on the left side in Fig. 3.36 toward which the marker is moving). Creation occurs by the reverse process, where the extra planes expand as atoms are added to them in order to form vacancies, as on the right side of Fig. 3.36. This contraction and expansion causes a mass flow that is revealed by the motion of embedded inert markers, as indicated in Fig. 3.3 [4]. 

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],... 

In diffusional processes, such as the classic Kirkendall effect of interdiffusion in a bulk diffusion couple of A and B, the atomic flux of A is not equal to the opposite flux of B. If we assume that A diffuses into B faster than B diffuses into A, we might expect that there will be a compressive stress in B, since there are more A atoms diffusing into it than B atoms diffusing out of it. However, in Darken s analysis of interdiffusion, there is no stress generated in either A or B. But Darken has made a key assumption that vacancy concentration is in equilibrium everywhere in the sample. To achieve vacancy equilibrium, we must assume that lattice sites can be created and/or annihilated in both A and B, as needed. Hence, provided that the lattice sites in B can be added to accommodate the incoming A atoms, there is no stress. The addition of a large number of lattice sites implies an increase in lattice planes if we assume that the mechanism of vacancy creation and/or annihilation is by dislocation climb mechanism. It further implies that lattice planes can migrate. 

The simplest and classical treatment of the Kirkendall effect in binary homogeneous systems assumes that the differences between the intrinsic diffusion fluxes of the two substitutional constituents are compensated by the action of local vacancy sinks and sources that maintains the system in local equilibrium, i.e. in states that can be completely defined by the knowledge of appropriate state variables to permit the calculation of pertinent state functions such as, for example, the chemical potential of system constituents. The drift of lattice planes is one important characteristic of the Kirkendall effect in stress-free homogeneous systems and is a consequence of the action of these vacancy sources and/or sinks distributed along the diffusion zone. As the system remains in local equilibrium by the action of vacancy sinks and sources, the vacancy concentration or molar fraction remains constant and equal to its equilibrium value within the entire diffusion couple. Therefore, no effective gradient of vacancy concentration is established in the diffusion zone. However, the local action of vacancy sinks or sources along the diffusion direction is formally equivalent to a vacancy flux Jy related to the required local density of vacancy sources or sinks ps equal to the flux divergence,... 

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. 

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