Dislocations Frenkel

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

Note that we can use the same statistical mechanical approach to calculate SchottslQi" pairs, Frenkel pairs, divancies (which are associated vacancies), impurity-vacancy complexes, and line dislocation-point defect complexes. 

If the missing ion has not been completely removed as in a Schottky defect, but only dislocated to a nearby interstitial site, the result is called a Frenkel defect (Fig. 

The resulting equilibrium concentrations of these point defects (vacancies and interstitials) are the consequence of a compromise between the ordering interaction energy and the entropy contribution of disorder (point defects, in this case). To be sure, the importance of Frenkel s basic work for the further development of solid state kinetics can hardly be overstated. From here on one knew that, in a crystal, the concentration of irregular structure elements (in thermal equilibrium) is a function of state. Therefore the conductivity of an ionic crystal, for example, which is caused by mobile, point defects, is a well defined physical property. However, contributions to the conductivity due to dislocations, grain boundaries, and other non-equilibrium defects can sometimes be quite significant. 

If a crystal lattice were perfect (i.e., contained no defects), slip would have to occur by the simultaneous movement of an entire plane of atoms over another plane of atoms. Frenkel (1926) developed a simple model that calculates the yield stress in a perfect crystal as G/2tt, where G is the shear modulus of the material. Actual yield stresses are two to four orders of magnitude lower than this. The concept of a dislocation was later introduced to explain the discrepancy between the measured and theoretical shear strengths of material. The dislocation... 

In atomic or molecular sohds, common types of point defects are the absence of an atom or molecule from its expected position at a regular lattice site (a vacancy), or the presence of an atom or molecule in a position which is not on the regular lattice (an interstitial). In ionic solids, these point defects occur in two main combinations. These are Schottky defects, in which there are equal numbers of cation and anion vacancies within the crystal, and Frenkel defects, in which there are cation vacancies associated with an equal number of "missing" cations located at non-lattice, interstitial positions. Both are illustrated in Figure 1.1. Point defects are also found in association with altervalent impurities, dislocations, etc., and combinations of vacancies with electrons or positive holes give rise to various types of colour centres (see below). 

Frenkel and Kontorova were not the hrst ones to use the model that is now associated with their names. However, unlike Dehlinger, who suggested the model [99], they succeeded in solving some aspects of the continuum approximation. Like the PT model, the FK model was first used to describe dislocations in crystals. Many of the recent applications are concerned with the motion of an elastic object over (or in) ordered [100] or disordered [101] structures. The FK model and generalizations thereof are also increasingly used to understand the friction between two solid bodies. 

Electron microscopic studies in the 1940s proved that supported catalysts possess a crystalline structure, dispelling earlier conjecture of amorphicity. However, practical catalysts are never uniform, exhibiting particle size distribution, lattice defects (Frenkel or Schottky), and dislocations. The following questions then arise Are all lattice surfaces equally active Do surface clusters of particles and surface atoms have comparable activity Does catalytic activity depend on particle size Is there an optimal particle size or distribution These questions remain, in general, still unanswered. However, in recent electrocatalytic studies concern about these effects is shown, following similar concern in conventional heterogeneous catalysis. 

The condition of electrical neutrality will not apply at the surface of a crystal, and since g+ is not equal to g there will be an excess of one of the defects. This effect, which is referred to in the early literature as the Frenkel-Lehovec space charge layer-results in an electric potential at the surface of the crystal [23-25]. In this instance, the surface will not simply be the external surface but will also include internal surfaces such as grain boundaries and dislocations. The effect decays away in moving from the surface to the bulk, and can be treated by classical Debye-Hiickel theory [26-29]. This leads to a Debye screening length, Lp, given by... 

Franc FC, Read J (1952) Crystal growth and dislocations Advances in Physics, v. 1,1 91-109 Freeman KH, Hayes JM, Trendel J-M, Albrecht P (1990) Evidence from carbon isotope measurements for diverse origins of sedimentary hydrocarbons. Nature 343 254-256 Frenkel Yal, Pines D (1945) Kinetic theory of liquids. Leningrad, Isdatelstvo AN SSSR, p 592 (in Russian)... 

In Schottky defects, the expelled atom passes to the surface of the metal or to dislocations or grain boundaries (i.e areas of free high energy). Thermal vibration may also initiate a Frenkel vacancy where the atom is expelled into an interstitial position. However, the numbers of interstitials and vacancies in metals are independent. A further kind of imperfection, a crowdion, is thought to be formed in body centered cubic (bcc) structures, owing to an additional atom being forced into a close-packed <111> row of atoms. 

Frenkel, Jacov Il ich was born 10 February 1894 in Rostov-on-Don, Russia and died 23 January 1952 in Leningrad, Russia. He worked on dislocations, twinning, and much more. 

On the other hand materials deform plastically only when subjected to shear stress. According to Frenkel analysis, strength (yield stress) of an ideal crystalline solid is proportional to its elastic shear modulus [28,29]. The strength of a real crystal is controlled by lattice defects, such as dislocations or point defects, and is significantly smaller then that of an ideal crystal. Nevertheless, the shear stress needed for dislocation motion (Peierls stress) or multiplication (Frank-Read source) and thus for plastic deformation is also proportional to the elastic shear modulus of a deformed material. Recently Teter argued that in many hardness tests one measures plastic deformation which is closely linked to deformation of a shear character [17]. He compared Vickers hardness data to the bulk and shear... 

Here, B denotes the place where a lattice can form, for example, the grain boundary, a surface or a dislocation. Hence, unlike the Frenkel defect, the Schottky defect creates new lattice sites. Since Mm and Oq are equivalent to Mb and Ob, respectively, Eq. (12.3) can also be written as... 

TEM images have revealed the following. Irradiation at 100 °C (lluence 8.3 X 10 n/m ) did not induce observable dislocation loops and the defects produced are isolated point defects or small clusters. This means that irradiation-induced hardening in a sample irradiated at 100 °C, which had no clear dislocation loops, is due mainly to point defects, i.e., interstitials and vacancies. Probably Frenkel pairs are formed in this manner. At 470 °C with a fluence of 2.4 x lO, dislocation loops were also found by TEM observations... 

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... 

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