Diffusion of vacancies

In most practical cases (and at moderate voltages) the high-field growth law can control film growth, say up to only a maximum of 10 nm, as at this thickness the field strength effects become even less important than film growth due to diffusion of vacancies or ions. 

Diffusion is based mainly on the diffusion of vacancies grain boundaries may act as sinks for these vacancies. This vacancy movement and annihilation cause the porosity of the powder compact to decrease during sintering. 

For the diffusion of vacancies on a face-centered cubic (f.c.c.) lattice with lattice constant a, let the probability of first- and second-nearest-neighbor jumps be p and 1 — p, respectively. At what value of p will the contributions to diffusion of first- and second-nearest-neighbor jumps be the same Solution. There is no correlation and, using Eq. 7.29,... 

The diffusion of vacancies is uncorrelated for the same reasons given above for diffusion of the interstitial atoms. After each jump, a vacancy will have the possibility of jumping into any one of its 12 nearest-neighbor sites with equal probability. 

In a general way, fatigue failure may be interpreted in terms of the enhancement of the diffusion of vacancies to grain boundaries. If sufficient vacancies arrive there, the cohesion between grains is reduced and the metal s strength lost. 

Diffusion of vacancies in metal surfaces theory and experiment... 

Two alternative approaches exist. The first one involves significantly lowering the temperature to values where the diffusion of vacancies can be observed with a technique like STM. At lower temperatures a surface vacancy can then be artificially created by ion bombardment or direct removal of an atom by the tip. This approach has been applied successfully to several semiconductor surfaces [29-31]. For metal surfaces, although vacancy creation at a step by direct tip manipulation of the surface has been demonstrated [32], to our knowledge, no studies have been published where the diffusion of artificially created vacancies in a terrace has successfully been measured. The second approach involves the addition of small amounts of appropriate impurities that serve as tracer atoms in the first layer of the surface [20-24]. The presence and passage of a surface vacancy is indirectly revealed by the motion of these embedded atoms. If one seeks to measure both the formation energy and the diffusion barrier of surface vacancies explicitly, a combination of these two approaches is needed. 

The first qualitative observation of vacancy-induced motion of embedded atoms was published in 1997 by Flores et al. [20], Using STM, an unusual, low mobility of embedded Mn atoms in Cu(0 0 1) was observed. Flores et al. argued that this could only be consistent with a vacancy-mediated diffusion mechanism. Upper and lower limits for the jump rate were established in the low-coverage limit and reasonable agreement was obtained between the experimentally observed diffusion coefficient and a theoretical estimate based on vacancy-mediated diffusion. That same year it was proposed that the diffusion of vacancies is the dominant mechanism in the decay of adatom islands on Cu(00 1) [36], which was also backed up by ab initio calculations [37]. After that, studies were performed on the vacancy-mediated diffusion of embedded In atoms [21-23] and Pd atoms [24] in the same surface. The deployment of a high-speed variable temperature STM in the case of embedded In and an atom-tracker STM in the case of Pd, allowed for a detailed quantitative investigation of the vacancy-mediated diffusion process by examining in detail both the jump frequency as well as the displacement statistics. Experimental details of both setups have been published elsewhere [34,35]. A review of the quantitative results from these studies is presented in the next subsections. 

Hole-nucleation rupture of foam bilayers was considered on the basis of formation of nucleus-holes from molecular vacancies existing in the film in Section 3.4.4. The experimentally determined parameters of film rupture along with the hole-nucleation theory of rupture of amphiphile bilayers of Kashchiev-Exerowa [300,301,354,402] made it possible to evaluate the coefficient of lateral diffusion of vacancies in foam bilayer. 

Despite some positive results of experimental verification of earlier macroscopic--fiow models, diffusion of vacancies is, nowadays, predominantly thought of as the controlling transport mechanism in the sintering of solid oxides. Solution of the problem depends on the determination of diffusion paths. 

The rate-determining mechanism involved in initial as well as final stages of solid--state sintering is, according to current ideas, boundary or lattice diffusion of vacancies from sources at the boundary. The main arguments in favour of the diffusion mechanism are as follows ... 

Several workers [10,11] describe the sintering mechanism in oxide powders as a diffusion of vacancies in the ceramic. The grain growth corresponds to a reduction of surface free energy. 

An extended work, experimental as well as theoretical, on mechanisms in thin microscopic foam lamellas and macroscopic foams has been published by Krugljakov Exerowa (1990). In the theory of Exerowa Kashiev (1986) thin-film stability is explained in terms of lateral diffusion of vacancies in a lattice like adsorption layers. As an example, selected experimental results of Exerowa et al. (1983) are shown in Fig. 3.18. 

Densification occurs by the bulk diffusion of vacancies away from the cylindrical pore channels toward the grain boundaries (curved arrows in Fig. 10.14/ ). 

During the intermediate stage of sintering, the porosity is eliminated by the diffusion of vacancies from porous areas to grain boundaries, free surfaces, or dislocations. The uniformity of particle packing and lack of agglomerates are important for the achievement of rapid densification. 

One final note In Chap. 7 it was stated that it was of no consequence whether the flux of atoms or defects were considered. To illustrate this important notion once again, it is worthwhile to derive an expression for the creep rate based on the flux of defects. Substituting Eq. (12.11) in the appropriate flux equation for the diffusion of vacancies, i.e.. 

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