Point defects theory

Calculation of the oxygen vacancy concentration at the interconnector surface On the basis of the point defect theory, the oxygen vacancy concentration (mole fraction) 8 on the fuel and air side surfaces of the interconnector are calculated [34], In an equilibrium state, the formation of the oxygen vacancy can be described as follows using Kroger-Vink notation [35] ... 

With binary compounds, an additional source of impurity levels can arise from non-stoichiometry. Thus, many oxides are either cation or anion deficient the former, such as Ni(1 x)0 (x 0.01) are p-type whereas the latter, such as TiO(2 x) (x 0.01) are n-type. The extent of non-stoichiometry varies with the partial pressure of 02 in a manner that can usually be calculated from point-defect theory, provided that x remains small. 

Chazeau L, Cavaille JY, Canova G et al (1999a) Viscoelastic properties of plasticized PVC reinforced with cellulose whiskers. J Appl Polym Sci 71 1797-1808 Chazeau L, Cavaille JY, Terech P (1999b) Mechanical behaviour above Tg of a plasticised PVC reinforced with cellulose whiskers a SANS structural study. Polymer 40 5333-5344 Chazeau L, Paillet M, Cavaille JY (1999c) Plasticized PVC reinforced with cellulose whiskers. I. Linear viscoelastic behavior analyzed through the quasi-point defect theory. J Polym Sci Part B Polym Phys 37 2151-2164... 

Chazeau, L., PaiUet, M., CavaiUe, J.Y., 1999c. Plasticized PVC reinforced with ceUulose whiskers. I. Linear viscoelastic behavior analyzed through the quasi-point defect theory. Journal of Polymer Science Polymer Physics 37, 2151—2164. 

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. 

The effect of different types of comonomers on varies. VDC—MA copolymers mote closely obey Flory s melting-point depression theory than do copolymers with VC or AN. Studies have shown that, for the copolymers of VDC with MA, Flory s theory needs modification to include both lamella thickness and surface free energy (69). The VDC—VC and VDC—AN copolymers typically have severe composition drift, therefore most of the comonomer units do not belong to crystallizing chains. Hence, they neither enter the crystal as defects nor cause lamellar thickness to decrease, so the depression of the melting temperature is less than expected. 

The beginnings of the enormous field of solid-state physics were concisely set out in a fascinating series of recollections by some of the pioneers at a Royal Society Symposium (Mott 1980), with the participation of a number of professional historians of science, and in much greater detail in a large, impressive book by a number of historians (Hoddeson et al. 1992), dealing in depth with such histories as the roots of solid-state physics in the years before quantum mechanics, the quantum theory of metals and band theory, point defects and colour centres, magnetism, mechanical behaviour of solids, semiconductor physics and critical statistical theory. 

As also discussed in Chapter 4, dislocations also give rise to diffuse scatter from the near-core region. Point defects of all types also give diffuse scatter. Although we do not yet have complete theories to describe this at present, we can model the diffuse scatter quite well by a Lorentzian function of the form ... 

The importance of interactions amongst point defects, at even fairly low defect concentrations, was recognized several years ago. Although one has to take into account the actual defect structure and modifications of short-range order to be able to describe the properties of solids fully, it has been found useful to represent all the processes involved in the intrinsic defect equilibria in a crystal (with a low concentration of defects), as well as its equilibrium with its external environment, by a set of coupled quasichemical reactions. These equilibrium reactions are then handled by the law of mass action. The free energy and equilibrium constants for each process can be obtained if we know the enthalpies and entropies of the reactions from theory or... 

Two reasons are responsible, for the greater complexity of chemical reactions 1) atomic particles change their chemical identity during reaction and 2) rate laws are nonlinear in most cases. Can the kinetic concepts of fluids be used for the kinetics of chemical processes in solids Instead of dealing with the kinetic gas theory, we have to deal with point, defect thermodynamics and point defect motion. Transport theory has to be introduced in an analogous way as in fluid systems, but adapted to the restrictions of the crystalline state. The same is true for (homogeneous) chemical reactions in the solid state. Processes across interfaces are of great... 

Chemical solid state processes are dependent upon the mobility of the individual atomic structure elements. In a solid which is in thermal equilibrium, this mobility is normally attained by the exchange of atoms (ions) with vacant lattice sites (i.e., vacancies). Vacancies are point defects which exist in well defined concentrations in thermal equilibrium, as do other kinds of point defects such as interstitial atoms. We refer to them as irregular structure elements. Kinetic parameters such as rate constants and transport coefficients are thus directly related to the number and kind of irregular structure elements (point defects) or, in more general terms, to atomic disorder. A quantitative kinetic theory therefore requires a quantitative understanding of the behavior of point defects as a function of the (local) thermodynamic parameters of the system (such as T, P, and composition, i.e., the fraction of chemical components). This understanding is provided by statistical thermodynamics and has been cast in a useful form for application to solid state chemical kinetics as the so-called point defect thermodynamics. 

In 1937, dost presented in his book on diffusion and chemical reactions in solids [W. lost (1937)] the first overview and quantitative discussion of solid state reaction kinetics based on the Frenkel-Wagner-Sehottky point defect thermodynamics and linear transport theory. Although metallic systems were included in the discussion, the main body of this monograph was concerned with ionic crystals. There was good reason for this preferential elaboration on kinetic concepts with ionic crystals. Firstly, one can exert, forces on the structure elements of ionic crystals by the application of an electrical field. Secondly, a current of 1 mA over a duration of 1 s (= 1 mC, easy to measure, at that time) corresponds to only 1(K8 moles of transported matter in the form of ions. Seen in retrospect, it is amazing how fast the understanding of diffusion and of chemical reactions in the solid state took place after the fundamental and appropriate concepts were established at about 1930, especially in metallurgy, ceramics, and related areas. 

Statistical thermodynamics can provide explicit expressions for the phenomenological Gibbs energy functions discussed in the previous section. The statistical theory of point defects has been well covered in the literature [A. R. Allnatt, A. B. Lidiard (1993)]. Therefore, we introduce its basic framework essentially for completeness, for a better atomic understanding of the driving forces in kinetic theory, and also in order to point out the subtleties arising from the constraints due to the structural conditions of crystallography. 

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