Vacancy concentration gradient

The vacancy concentration gradient that results from the curvature and stress is AC/x = (1.017 - 0.8)/x = 0.217/,x, which is greater than either alone. The effect of stress, however, is much more significant. 

Figure 12.3 (a) Vacancy concentration gradients that develop as a result of stress gradients. The vacancy concentrations are higher below the tensile surface. Curved arrows denote direction of acancy fluxes, h) Schematic of a grain of diameter J subjected simultaneously to a tensile and a compressive stress. Curved arrows denote direction of atomic fluxes, (c) Shape of grain after creep has occurred. 

Figure 5.3. Evolution of the vacancy concentration gradient according to the x coordinates and time t...

Figure 10.3. Evolution of vacancy concentration gradient within the Wagner hypothesis...

Matter transport through diffusion can be described in terms of the flux of atoms or the counterflow of vacancies. The approach based on the counterflow of vacancies driven by a vacancy concentration gradient has been used predominantly in the early development of sintering theory, which is discussed in this subsection first. More general approach based on the flux of atoms driven by a chemical potential gradient will be presented later. The following discussion is started with the mechanism of grain boundary diffusion. 

Hence, the question arises as to how the system can implement the law of conservation of matter if lattice motion and vacancy sinks/sources functioning encounter difliculties or become impossible. The answer is simple - internal forces will appear in the system that will level the fluxes without displacement of lattice. If one excludes ionic crystals, there are two, known to us, types of such balancing forces - stress gradient and nonequilibrium vacancy concentration gradient For the result to be obtained, the type of such unclear force is of secondary importance, while the result is significant. 

Apparently, in the absence of any acting vacancy sinks/sources, the flux of vacancies toward a more mobile component will lead to their accumulation on this component s side and depletion on the side of a less active component. As a result, a nonuniform distribution of nonequilibrium vacancies will evolve. Correspondingly, a vacancy concentration gradient will appear, and it must influence both vacancies and atoms fluxes. 

Since the vertical gradient is what makes the torque rotate the grain, it is important to correlate the vacancy concentration gradient to the stress gradient. Since the stress is in equihbrium with the vacancy concentration,... 

Then, the product of vacancy flux density and atomic volume is determined by a simple equation, taking into account the vacancy concentration gradient caused by... 

Thus, there is a vacancy concentration gradient between dislocation 1 and 2. It is determined by the difference of the two densities and by the distance I between the dislocations. In a material containing several obstacles, I is proportional to the mean distance of the obstacles. This gradient causes diffusion of vacancies from dislocation 2 to dislocation 1. In this argument, we assumed that the vacancy concentration at both dislocations can still be described by using the Boltzmann equation which is valid only in thermal equilibrium. This is a valid assumption provided that the energy tV is small compared to the enthalpy of vacancy formation Qy. 

The vacancy concentration gradient dnjdx can be assumed as constant in the case of small fluctuations in concentration ... 

The vacancy concentration gradient can be estimated, according to equation (11.10), as (ni — n2)/d, with the grain size d replacing the dislocation distance. Thus, a vacancy current density... 

The two spheres of the Frenkel s model use the concept of viscous flow of atoms that relates the vacancy diffusion coefficient D the volume of the atom or vacancy O, and vacancy concentration gradient per unit area of the material (dQ/dx), as shown in the following equation [41] ... 

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