Vacancy source

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... 

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. 

Equations 3.71 and 3.72 can be further developed in terms of the self-diffusivity using the atomistic models for diffusion described in Chapters 7 and 8. The resulting formulation allows for simple kinetic models of processes such as dislocation climb, surface smoothing, and diffusional creep that include the operation of vacancy sources and sinks (see Eqs. 13.3, 14.48, and 16.31). 

If the vacancies are subsaturated, the dislocation tends to produce vacancies and therefore acts as a vacancy source. In that case, Eq. 11.5 will still hold, but fiy will be negative and the climb force and climb direction will be reversed. Equation 11.5 also holds for interstitial point defects, but the sign of will be reversed. 

Climbing Dislocations as Sinks for Excess Quenched-in Vacancies. Dislocations are generally the most important vacancy sources that act to maintain the vacancy concentration in thermal equilibrium as the temperature of a crystal changes. In the following, we analyze the rate at which the usual dislocation network in a... 

Volume diffusion. In this model, the vacancy source is the pore surface of radius pp and the sink is the spherical surface of radius / , where R s> Pp (Fig. 10.15a). By solving the flux equation, subject to the appropriate boundary conditions, it can be shown that (see App. lOD)... 

In the simplest model for representing the elimination of porosity in a solid. Fig. 3.11, volume diffusion takes place by means of lattice defects, in particular, vacancies. Pores play the role of vacancy sources, and grain boundaries act as vacancy sinks. The key point in this mechanism is that a... 

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... 

So, the concentration profile evolution between vacancy sources/sinks (that is, in microscopic scale, d J) is determined by the NG coefficient, Dng- On the other... 

Let us consider the interdiffusion process in a coarsened spatial scale, regarding vacancies sources/sinks as continuously distributed in an alloy. We take account of their action as a sink/source term in the equation for vacancy redistribution ... 

Having applied the continuity equation (taking into consideration vacancies sources and sinks), in the linear approximation one obtains... 

For the sake of simplicity, let us consider a one-component hollow nanosheU where vacancy sources/sinks are provided only by internal and external surfaces. At such an assumption, inside the nanoshell the law of conservation of vacancies is fiilfUled... 

Zero in Equation 7.124 was put just to remind about the absence or ineffectiveness of vacancy sources and sinks inside the shell. 

In this case, the strain rate is determined by the rate of emission or absorption of vacancies. Figure 11.5 shows the example of two edge dislocations pinned at two obstacles. Dislocation 1 has to absorb vacancies to climb dislocation 2 needs to emit them. Thus, vacancies can be transported from one dislocation to the other, with one dislocation acting as vacancy source, the other as vacancy sink. The vacancy current density, j, determines the rate of deformation. This quantity can be estimated. 

Fig. 11.5. Pile-up of dislocations at obstacles and vacancy diffusion. Dislocation 1 is a vacancy sink, dislocation 2 a vacancy source...

The positive climb process requires long-range transport of matter and consumes vacancies. Continuous deformation will hence lead to a depletion of vacancies and eventually limit the effectiveness of the mechanism. Roitsch et al. [50] showed that besides the metadislocation loops, a second set of dislocations is involved in the deformation mechanism, which compensates the depletion of vacancies. The second set of dislocations consists of loops on (0 01) habit planes with [010] Burgers vectors [51]. These dislocations move by pure negative climb, that is, the expansion of the loops is connected to a production of vacancies, and hence the systems acts as a vacancy source. The interaction of the two loop systems in the form of vacancy exchange ensures continuous operation of the deformation mechanism. 

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. 

Concentration profiles calculated for interdiffusion between two solid solutions a and p to illustrate the reversible role of a given Interface as a vacancy source or sink (a) interdiffusion between pure A constituent and saturated P solid solution (b) interdiffusion between saturated a solid solution and pure B constituent (van Loo, 1990). 

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