Frenkel voiding

We use the term Kirkendall effect to include both Kirkendall shift and Kirkendall voiding (or Frenkel voiding). We present a detailed analysis of the interaction of the Kirkendall effect and the inverse Kirkendall effect in nanoscale... 

There is no Kirkendall shift during phase formation, i.e. all vacancy fluxes go to the formation of Kirkendall (or Frenkel) voids instead of being annihilated by internal sinks hence, they do not causing the lattice shift. 

Some very interesting ideas concerning the relationship between free-volume formation and the energy of one mole of hole formation were developed in detail by Kanig42. Kanig introduced some improvements to the definition of free-volume, On the basis of Frenkel s ideas43 he divided the free-volume into two parts, one of which is determined only by the thermal vibrations of atoms in the lattice of a real crystal while the other is connected with inherent free-volume, i.e. voids and holes. It is the latter that makes possible the exchange of particles, i.e. the very existence of the liquid state. He introduced some new definitions of fractions of free-volume ... 

The concept of a zero-dimensional intrinsic point defect was first introduced in 1926 by the Russian physicist Jacov Il ich Frenkel (1894-1952), who postulated the existence of vacancies, or unoccupied lattice sites, in alkali-halide crystals (Frenkel, 1926). Vacancies are predominant in ionic solids when the anions and cations are similar in size, and in metals when there is very little room to accommodate interstitial atoms, as in closed packed stmctures. The interstitial is the second type of point defect. Interstitial sites are the small voids between lattice sites. These are more likely to be occupied by small atoms, or, if there is a pronounced polarization, to the lattice. In this way, there is little dismption to the stmcture. Another type of intrinsic point defect is the anti-site atom (an atom residing on the wrong sublattice). 

Given the diversity of relevant applications, it is not surprising that the characterization of voids in disordered systems has an appreciable history, which can be traced back to primitive hole theories of the liquid state (Frenkel, 1955 Ono and Kondo, 1960). While the early theories offer an admittedly rudimentary lattice description of voids, recent computational advances permit an exact (and highly efficient) characterization of the continuum void geometry present in particle packings in two (Rintoul and Torquato, 1995) and three dimensions (Sastry et al., 1997a). 

Cohen and Turnbull s critical free-volume fluctuations picture of selfdiffusion in dense liquids is similar to the vacancy model of self-diffusion in crystals. However, in crystals individual vacancies exist and retain their identity over long periods of time, whereas in liquids the corresponding voids are ephemeral. The free volume is distributed statistically so that at any given instance there is a certain concentration of molecule-sized voids in the liquid. However, each such void is short-lived, being created and dying in continual free-volume fluctuations. The Frenkel hole theory of liquids ignores this ephemeral, statistical character of the free volume. 

Finally, the occurrence of different types of defects can serve as a guide in the identification of the H migration mechanism. The defects can be (or H502 ) defects that occur either as (a) thermally activated (Frenkel) excess sites, such as are present in ionic cpd materials, (b) voids in a quasi-liquid sublattiee , (e) proton defeets in the same sense as oxonium defects in a rigid sublattice or (d) proton voids occurring in a quasi-liquid proton sublattice. 

Note DFT/PaSD with void model, self-consistent model of a mixture of voids, cylindrical and slit-shaped pores, self-consistent regularization with respect to both PoSD (fy(i p)) and PaSD ((l)( z)) with the model of voids, (5 ), Aw=Sgg j-/(5 ) - 1, f)pHH e fractal dimension with Frenkel-Halsey-Hill equation accounting for adsorbate surface tension effects (Quantachrome Instruments software), Ag j is the gelatin adsorption in mg per gram of silica. 

The defects which disrupt the regular patterns of crystals, can be classified into point defects (zero-dimensional), line defects (1-dimensional), planar (2-dimensional) and bulk defects (3-dimensional). Point defects are imperfections of the crystal lattice having dimensions of the order of the atomic size. The formation of point defects in solids was predicted by Frenkel [40], At high temperatures, the thermal motion of atoms becomes more intensive and some of atoms obtain energies sufficient to leave their lattice sites and occupy interstitial positions. In this case, a vacancy and an interstitial atom, the so-called Frenkel pair, appear simultaneously. A way to create only vacancies has been shown later by Wagner and Schottky [41] atoms leave their lattice sites and occupy free positions on the surface or at internal imperfections of the crystal (voids, grain boundaries, dislocations). Such vacancies are often called Schottky defects (Fig. 6.3). This mechanism dominates in solids with close-packed lattices where the formation of vacancies requires considerably smaller energies than that of interstitials. In ionic compounds also there are defects of two types, Frenkel and Schottky disorder. In the first case there are equal numbers of cation vacancies... 

The Kirkendall effect is commonly accompanied by the Frenkel effect, the void formation in the diffusion zone. In foreign literature, the Frenkel effect is often referred to as Kirkendall voiding, which is rather confusing, as Kirkendall and Frenkel effects are competitive vacancies annihilating at the dislocation kinks and causing the Kirkendall shift, cannot be used for Kirkendall voiding, and vice versa. 

In 2004, void formation in spherical samples was rediscovered at the nanolevel and discussed in [4-6]. HoUow nanosheUs of cobalt and iron oxides and sulfides have been obtained by means of reaction of metaUic nanopowders with oxygen or sulfur. Contrary to [2, 3], these results have been explained by the Frenkel effect - out-diffusion of metal through the formation of a spherical layer of the compound is faster than in-diffusion of oxygen or sulfur through the same phase. This inequahty of fluxes generates the inward flux of vacancies, meeting inside and forming the void in the internal part of the system. 

Mackenzie and Shuttleworth (MS) analyzed [24] the shrinkage rate of a spherical shell according to Frenkel s rnethod. The shell, shown in Fig. 6, can be used to represent the densification of a body containing spherical pores. The dimensions of the shell are chosen so that the central void occupies the same volume fraction as the pores in the sintering body. This is a much more elegant treatment than the analysis of the coalescence of spheres, because the shell remains spherical as it shrinks. Exact expressions can be written for the change in surface area and the energy dissipated in viscous flow as the shell contracts. The result is... 

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