Vacancy flux

The vacancy flux and the corresponding lattice shift vanish if bA = bB. In agreement with the irreversible thermodynamics of binary systems i.e., if local equilibrium prevails), there is only one single independent kinetic coefficient, D, necessary for a unique description of the chemical interdiffusion process. Information about individual mobilities and diffusivities can be obtained only from additional knowledge about vL, which must include concepts of the crystal lattice and point defects. 

We begin with the simplest case. A vacancy flux j° (driven, for example, by inhomogeneous particle radiation) flows across a multicomponent crystal (k = 1,2,.., n) and the component fluxes are restricted to one sublattice. We assume no other coupling between the fluxes except the lattice site conservation, which means that we neglect cross terms in the formulation of SE fluxes. (An example of coupling by cross terms is analyzed in Section 8.4.) The steady state condition requires then that the velocities of all the components are the same, independent of which frame of reference has been chosen, that is,... 

A system which can easily be treated in this way is a single phase binary alloy. For preparation, however, let us consider an A crystal with a vacancy flux driven across it. In view of the fact that jA +y v = 0 in the steady state lattice system, the vacancy flux induces a counterflux of A, which shifts the whole crystal in the direction of the surface where the vacancy source is located. The shift velocity vb is yV... 

For the slab of a binary alloy (A, B) across which the vacancy flux jw = j° flows, we derive from Eqn. (8.3)... 

Eqn. (8.6) describes the steady state concentration profile of an (A, B) alloy which has been exposed to the stationary vacancy flux j°. The result is particularly simple if the mobilities, b are independent of composition, that is, if P = constant. From Eqn. (8.6), we infer that, depending on the ratio of the mobilities P, demixing can occur in two directions (either A or B can concentrate at the surface acting as the vacancy source). The demixing strength is proportional toy°-(l-p)/RT, and thus directly proportional to the vacancy flux density j°, and to the reciprocal of the absolute temperature, 1/71 For p = 1, there is no demixing. 

We proceed by considering a slab of an oxide crystal AO and assume that a cation vacancy flux is driven across it. In contrast to the single sublattice alloy discussed above, where the vacancies have been introduced into the lattice as an independent component, the vacancy flux j° in AO can be induced by different oxygen activities at the two opposite surfaces. At the oxidizing surface, the defect reaction is y 02 = Oq +V +2-h In semiconducting AO, the flux of ionized A vacancies is compen-... 

In the next step, we discuss the demixing of semiconducting oxide solid solutions (A, B)0 as illustrated in Figure 8-2. Instead of formulating the constant cation vacancy flux as in the steady state condition of Eqn. (8.1), let us express this condition explicitly and note that the frame connected to the oxide sample surface moves with the same velocity as the components so that the composition does not change with time... 

Equation (8.14) demonstrates once more that the cation flux caused by the oxygen potential gradient consists of two terms 1) the well known diffusional term, and 2) a drift term which is induced by the vacancy flux and weighted by the cation transference number. We note the equivalence of the formulations which led to Eqns. (8.2) and (8.14). Since vb = jv - Vm, we may express the drift term by the shift velocity vb of the crystal. Let us finally point out that this segregation and demixing effect is purely kinetic. Its magnitude depends on ft = bB/bA, the cation mobility ratio. It is in no way related to the thermodynamic stability (AC 0, AG go) of the component oxides AO and BO. This will become even clearer in the next section when we discuss the kinetic decomposition of stoichiometric compounds. 

If we could arrange the demixing experiment (Fig. 8-2) such that the vacancy flux (caused by the activity difference at opposite surfaces) remains constant and the crystal therefore shifts with constant velocity, we could calculate the time required to attain the steady state with... 

If we identify jy in Eqn. (8.66) with j° in Eqn. (8.3), we can use Eqn. (8.3) to calculate the (steady state) demixing when a stress driven vacancy flux flows across the solid solution (A,B)0. At a fixed oxygen potential, we obtain from the steady state condition... 

For the last part of Eqn. (9.10), we have assumed that Nhigh thermodynamic stability of the (spinel) product phase. A semi-quantitative, and by no means strict, discussion of the internal reaction kinetics is as follows. As long as div/v = 0 in the region S< < F, we can formulate the cation vacancy flux as... 

Let us briefly outline the main concepts of a (linear) stability analysis and refer to the situation illustrated in Figure 11-7. If we artificially keep the moving boundary morphologically stable, we can immediately calculate the steady state vacancy flux, /v, across the crystal. The boundary velocity relative to the laboratory reference system (crystal lattice) is... 

Finally, let us briefly point out some essential features of the stability analysis for a more general transport problem. It can be exemplified by the moving a//9 phase boundary in the ternary system of Figure 11-12. Referring to Figure 11-7 and Eqn. (11.10), it was a single independent (vacancy) flux that caused the motion of the boundary. In the case of two or more independent components, we have to formulate the transport equation (Fick s second law) for each component, both in the a- and /9-phase. Each of the fluxes jf couples at the boundary b with jf, i = A,B,... (see, for example, Eqn. (11.2)). Furthermore, in the bulk, the fluxes are also coupled (e.g., by electroneutrality or site conservation). 

To solve the vacancy flux equation between dislocations of opposite sign we have to know the dislocation geometry (distance and orientation) in the lattice as the boundary condition. If we consider as a zeroth order approach only the average distance, a, between the dislocations, even this quantity depends on the applied stress and the functioning of dislocation multiplication. Nevertheless, since about l/b2 vacancies are needed for a climb shift of unit length, we may conclude from Eqn. (14.28) and the vacancy flux that the steady-state climb velocity, >d, of a dislocation with edge character is... 

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]. 

A net vacancy flux develops in a direction opposite that of the fastest-diffusing species (species 1 in Fig. 3.3). Nonequilibrium vacancy concentrations would develop in the diffusion zone if they were not eliminated by dislocation climb. However, under usual conditions it is expected that a sufficient density of dislocations will be present to maintain the vacancy concentration near equilibrium [8]. It can therefore be assumed, to a good approximation, that fiy = 0, and therefore Vpv = 0 everywhere in the diffusion zone. Using Eqs. 3.7 and 3.8 with Vpy = 0 yields... 

Another type of motion of crystal/vapor interfaces occurs when a supersaturation of vacancies anneals out by diffusing to the surface where they are destroyed. In this process, the surface acts as a sink for the incoming vacancy flux and the surface moves inward toward the crystal as the vacancies are destroyed. This may be regarded as a form of crystal dissolution, and the kinetics again depend upon the type of surface that is involved. 

The class of creep mechanisms of interest here are those that are mediated by stress-biased diffusion. If we are to consider the vacancy flux in a given grain within a material that is subjected to an applied stress, it is argued that the vacancy formation energy differs in different parts of the grain, and hence that there should be a gradient in the vacancy concentration leading to an associated flux. This... 

In Kukushkin [39] it was demonstrated that stresses in the crystal generate a vacancy flux, which, in the case under consideration, is proportional to the radial component of the elastic stress tensor. This vacancy flux can be represented by the following expression ... 

The second component of the vacancy flux is related to the temperature gradient. In Stark [40] it was shown that, in the presence of a temperature gradient in the crystal, there arise an atom flux and a counter vacancy flux, which can be written in the form ... 

The total vacancy flux in the crystal under the given conditions can be written as ... 

Atomistically, both mechanisms entail the diffusion of ions from the grain boundary region toward the neck area, for which the driving force is the curvature-induced vacancy concentration. Because there are more vacancies in the neck area than in the region between the grains, a vacancy flux develops away from the pore surface into the grain boundary area, where the vacancies are eventually annihilated. Needless to say, an equal atomic flux will diffuse in the opposite direction, filling the pores. 

Fig. 3.11. Model for elimination of porosity. Vacancy flux by a) grain boundary diffusion, (b) volume diffusion.

Because the atomic flux in sintering is equal and opposite to the vacancy flux, there is ... 

The cylindrical pores along the edges enclose each face of the tetrakaidecahedron, as shown in Fig. 5.18a. Because the vacancy flux from the pores terminates on the faces of the boundary, as shown in Fig. 5.18b, it can be assumed the diffusion is radial from a circular vacancy source while the shape effects on the comer of the tetrakaidecahedron is neglected [27]. To remain the boundary to be flat, the vacancy flux per unit area of the boundary should be the same over the whole boundary. The diffusion flux field can be treated as that of the temperature distribution in a surface-cooled and electrically heated cylindrical conductor. The flux per unit length of the cylinder can be expressed as... 

In terms of vacancy movement, the vacancy flux, J ac. is expressed as... 

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