Point Defects in Solid Solutions

The observed uptake of water by quartz under conditions of high temperature, pressure, and water fugacity indicates that the diffusivity and/or solubility of water-related point defects in solid solution are much lower than are required by the Griggs model (Gerretsen et al. 1989 Kronenberg, Kirby, and Rossman 1986 Rovetta, Holloway, and Blacic 1986). 

The fourth and final chapter looks at the question of point defects in solid solutions that are slightly or highly concentrated. The role of doping of insulating and semiconductive ionic materials is discussed, as is the description of models of gas dissolution in solids. The chapter finishes with an examination of the methods used to calculate the equilibrium constants of point-defect creation. 

An example of the type of data associated with solution hardening it is the mission of our models to explain was shown in fig. 8.2(a). For our present purposes, there are questions to be posed of both a qualitative and quantitative character. On the qualitative side, we would like to know how the presence of foreign atoms dissolved in the matrix can have the effect of strengthening a material. In particular, how can we reconcile what we know about point defects in solids with the elastic model of dislocation-obstacle interaction presented in section 11.6.2. From a more quantitative perspective, we are particularly interested in the question of to what extent the experimental data permit a scaling description of the hardening effect (i.e. r oc c") and in addition, to what extent statistical superposition of the presumed elastic interactions between dislocations and impurities provides for such scaling laws. 

Charged defects in solids can interact with one another in an analogous way to the interactions between ions (or between ions and electrons) in a solution. In the solid-state situation, the crystal may be viewed as a neutral medium into which the charged defects are dissolved. This similarity between solution chemical interactions and defect interactions in the solid state has resulted in the field of defect chemistry, which provides basic methods for studying the effects of point defects in solids. The methods are normally applicable to fairly low defect concentrations. Generally, a broad distinction is made between intrinsic defects that are thermally gena-ated in pure compounds and extrinsic defects produced by external influences such as impurities and gaseous atmospheres. References 2 and 3 provide a detailed discussion of point defects and defect chemistry in metal oxides. 

An effect which is frequently encountered in oxide catalysts is that of promoters on the activity. An example of this is the small addition of lidrium oxide, Li20 which promotes, or increases, the catalytic activity of dre alkaline earth oxide BaO. Although little is known about the exact role of lithium on the surface structure of BaO, it would seem plausible that this effect is due to the introduction of more oxygen vacancies on the surface. This effect is well known in the chemistry of solid oxides. For example, the addition of lithium oxide to nickel oxide, in which a solid solution is formed, causes an increase in the concentration of dre major point defect which is the Ni + ion. Since the valency of dre cation in dre alkaline earth oxides can only take the value two the incorporation of lithium oxide in solid solution can only lead to oxygen vacaircy formation. Schematic equations for the two processes are... 

The preceding paragraphs illustrate that analogies between point defects in a crystal and solute molecules in a solution have been used previously but in a fairly elementary way. However, the implications of the existence of such analogies in the formulation of the statistical mechanics of interacting defects has not been considered in detail apart from an early paper by Mayer,69 who was interested primarily in the relation of defect interactions, to the solid-liquid phase transition in crystals with short-range forces. The... 

Another defect problem to which the ion-pair theory of electrolyte solutions has been applied is that of interactions to acceptor and donor impurities in solid solution in germanium and silicon. Reiss73>74 pointed out certain difficulties in the Fuoss formulation. His kinetic approach to the problem gave results numerically very similar to that of the Fuoss theory. A novel aspect of this method was that the negative ions were treated as randomly distributed but immobile while the positive ions could move freely. 

Although the theory of solutions has been widely used in formulating problems of defects in solids the problems encountered differ in certain respects. The most obvious point is that defects are restricted to discrete lattice sites, whereas the ions in a solution can occupy any position in the fluid. Sometimes no allowance is made for this fact. For example, it has not been demonstrated that at very low concentrations, in the absence of ion-pair effects, the activity coefficients are identical with those of the Debye-Hiickel theory. It can be plausibly argued51 that at sufficiently low concentrations the effect of discreteness is likely to be negligible, but clearly in developing a theory for any but the lowest concentrations the effect should be investigated. A second point... 

The concepts required for a quantitative treatment of the reactivity of solids were now clear, except for one important issue. According to the foregoing, point defect energies should be on the same order as lattice energies. Since the distribution of point defects in the crystal conforms to Boltzmann statistics, one was able to estimate their concentrations. It was found that the calculated defect concentrations were orders of magnitude too small and therefore could not explain the experimentally observed effects which depended on defect concentrations (e.g., conductivity, excess volume, optical absorption). Jost [W. Jost (1933)] provided the correct solution to this problem. Analogous to the fact that NaCl can be dissolved in H20... 

We have discussed point defects in elements (A) and in nearly stoichiometric compounds having narrow ranges of homogeneity. Let us extend this discussion to the point defect thermodynamics of alloys and nonmetallic solid solutions. This topic is of particular interest in view of the kinetics of transport processes in those solid solutions which predominate in metallurgy and ceramics. Diffusion processes are governed by the concentrations and mobilities of point defects and, although in inhomogeneous crystals the components may not be in equilibrium, point defects are normally very close to local equilibrium. 

The doped semiconductor materials can often be considered as well-characterized, diluted solid solutions. Here, the solutes are referred to as point defects, for instance, oxygen vacancies in TiC - phase, denoted as Vq, or boron atoms in silicon, substituting Si at Si sites, Bj etc. See also -> defects in solids, -+ Kroger-Vink notation of defects. The atoms present at interstitial positions are also point defects. Under stable (or metastable) thermodynamic equilibrium in a diluted state, - chemical potentials of point defects can be defined as follows ... 

Point (microscopic) defects in contrast from the macroscopic are compatible with the atomic distances between the neighboring atoms. The initial cause of appearance of the point defects in the first place is the local energy fluctuations, owing to the temperature fluctuations. Point defects can be divided into Frenkel defects and Schottky defects, and these often occur in ionic crystals. The former are due to misplacement of ions and vacancies. Charges are balanced in the whole crystal despite the presence of interstitial or extra ions and vacancies. If an atom leaves its site in the lattice (thereby creating a vacancy) and then moves to the surface of the crystal, it becomes a Schottky defect. On the other hand, an atom that vacates its position in the lattice and transfers to an interstitial position in the crystal is known as a Frenkel defect. The formation of a Frenkel defect therefore produces two defects within the lattice—a vacancy and the interstitial defect—while the formation of a Schottky defect leaves only one defect within the lattice, that is, a vacancy. Aside from the formation of Schottky and Frenkel defects, there is a third mechanism by which an intrinsic point defect may be formed, that is, the movement of a surface atom into an interstitial site. Considering the electroneutrality condition for the stoichiometric solid solution, the ratio of mole parts of the anion and cation vacancies is simply defined by the valence of atoms (ions). Therefore, for solid solution M X, the ratio of the anion vacancies is equal to mJn. 

We consider solid solutions here because we can think of them as being formed by distributing a large number of point defects in a host crystal. As always, we must balance charge and be sure that the size of the impurity (guest) ion is appropriate to fit into the available site. If the impurity ions are incorporated in regular crystal sites the resulting phase is a substitutional solid solution. In an interstitial solid solution the impurity atoms occupy interstices in the crystal structure. The rules for substitutional solid solutions (the Hume-Rothery rules) can be summarized as follows. Note that the last two requirements are really very closely tied to the first two. 

Complete structural characterization of a material involves not only the elemental composition for major components and a study of the crystal structure, but also the impurity content (impurities in solid solution and/or additional phases) and stoichiometry. Noncrystalline materials can display unique behavior, and noncrystalline second phases can alter properties. Both the long-range order and crystal imperfection or defects must be defined. For example, the structural details which influence properties of oxides include the impurity and dopant content, nonstoichiometry, and the oxidation states of cations and anions. These variables also influence the point-defect structure, which in turn influences chemical reactivity, and electrical, magnetic, catalytic, and optical properties. 

Model vacancy coalescence is a simplified model for the analysis of individual parameters of process of the formation microvoids. Detailed calculations are presented in the articles (V.I. Talanin I.E. Talanin, 2010c). The fundamental interaction between impiuities and intrinsic point defects upon crystal cooling imder certain thermal conditions (T < 1423 K) leads to impurity depletion and the formation of a supersatiuated solid solution of intrinsic point defects. The decay of this supersatiuated solid solution causes the coagulation of intrinsic point defects in the form of microvoids. 

The first two types of stmcture elements are normal elements of the solid, while the others are native point defects. In general, a given solid contains several types of defects, which will be as many components in the thermodynamic sense of the term, and will form a solution with the normal elements. In practice, the problem boils down to the superposition of the equilibria of a base of the vector space. Usually, the defects are very dilute in comparison to the normal elements, so that they can be considered to be solvents with constant activity and the activities of the defects can be considered equal to their site fractions. 

P. P. Fedorov, Association of point defects in non stoichiometric Mi xRxp2+x fluorite-type solid solutions. Bull. Soc. Cat. Cien., 12, 349-381 (1991). 

Impurity point defects are found in solid solutions, of which there are two types substitutional and interstitial. For the substitutional type, solute or impm-ity atoms replace or substitute for the host atoms (Figure 4.2). Several featmes of the solute and solvent atoms determine the degree to which the former dissolves in the latter. These are expressed as four Hume-Rothery rules, as follows ... 

It is seen that the transformation of one solid into another is not really different from precipitation starting from a liquid solution the application of the methods of quasi-chemistiy allows the use of the same formalism. It will however be necessary to be careful with the connection of the two modes of description in particular, the energy of formation of the composed gas phase or pure liquid is eiqiressed starting from the elements, whereas the energy of formation of a point defect in a solid is expressed conqiared with the ideal solid. 

In this chapter shock modification of powders (their specific area, x-ray diffraction lines, and point defects) measurements via analytical electron microscopy, magnetization and Mossbauer spectroscopy shock activation of catalysis, solution, solid-state chemical reactions, sintering, and structural transformations enhanced solid-state reactivity. 

Now, suppose that we have a solid solution of two (2) elemental solids. Would the point defects be the same, or not An easy way to visualize such point defects is shown in the following diagram, given as 3.1.3. on the next page. It is well to note here that homogeneous lattices usually involve metals or solid solutions of metals (alloys) in contrast to heterogeneous lattices which involve compounds such as ZnS. 

It is important that the copper is in the monovalent state and incorporated into the silver hahde crystals as an impurity. Because the Cu+ has the same valence as the Ag+, some Cu+ will replace Ag+ in the AgX crystal, to form a dilute solid solution Cu Agi- X (Fig. 2.6d). The defects in this material are substitutional CuAg point defects and cation Frenkel defects. These crystallites are precipitated in the complete absence of light, after which a finished glass blank will look clear because the silver hahde grains are so small that they do not scatter light. 

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