Defect Formation and Reaction Equations

The creation of antisite defects can occur during crystal growth, when atoms are misplaced on the surface of the growing crystal. Alternatively, they can be created by internal mechanisms once the crystal is formed, provided that sufficient energy is applied to allow for atom movement. 

Defects are often deliberately introduced into a solid in order to modify physical or chemical properties. However, defects do not occur in the balance of reactants expressed in traditional chemical equations, and so these important components are lost to the chemical accounting system that the equations represent. Fortunately, traditional chemical equations can be easily modified so as to include defect formation. The incorporation of defects into normal chemical equations allows a strict account of these important entities to be kept and at the same time facilitates the application of chemical thermodynamics to the system. In this sense it is possible to build up a defect chemistry in which the defects play a role analogous to that of the chemical atoms themselves. The Kroger-Vink notation allows this to be done provided the normal mles that apply to balanced chemical equations are preserved. 

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Reactions involving the creation, destruction, and elimination of defects can appear mysterious. In such cases it is useful to break the reaction down into hypothetical steps that can be represented by partial equations, rather akin to the half-reactions used to simplify redox reactions in chemistry. The complete defect formation equation is found by adding the partial equations together. The mles described above can be interpreted more flexibly in these partial equations but must be rigorously obeyed in the final equation. Finally, it is necessary to mention that a defect formation equation can often be written in terms of just structural (i.e., ionic) defects such as interstitials and vacancies or in terms of just electronic defects, electrons, and holes. Which of these alternatives is preferred will depend upon the physical properties of the solid. An insulator such as MgO is likely to utilize structural defects to compensate for the changes taking place, whereas a semiconducting transition-metal oxide with several easily accessible valence states is likely to prefer electronic compensation. 

At high temperatures the spinel MgAl204 can take in excess alumina to a composition of approximately 70 mol% A1203 (Fig. 4.5). (a) What are the possible formulas that fit the composition of this spinel Write the defect formation equation for the reaction if the excess A1 is (b) distributed over both magnesium and aluminum sites and (c) only over aluminum sites. Assume that there is no electronic compensation in the insulating oxide. 

In general, there is little problem in extending the concepts just outlined to more complex materials. The procedure is to write down the equations specifying the various equilibria point defect formation, electronic defect formation, the oxidation reaction, and the reduction reaction. These four equations, only three of which are independent, are augmented by the electroneutrality equation. Two examples will be sketched for the oxides Cr2C>3 and Ba2In2Os. 

Fick s second law states the conservation of the diffusing species i no i is produced (or annihilated) in the diffusion zone by chemical reaction. If, however, production (annihilation) occurs, we have to add a (local) reaction term r, to the generalized version of Fick s second law c, = —Vjj + fj. In Section 1.3.1, we introduced the kinetics of point defect production if regular SE s are thermally activated to become irregular SE s (i.e., point defects). These concepts and rate equations can immediately be used to formulate electron-hole formation and annihilation... 

Figure 56 uses the example of associate formation between the ionic defect O and the electronic defect h to emphasize that the strict treatment requires the solution of coupled diffusion-reaction relationships, describing the general (electro-)chemical reaction scheme with individual diffusion or rate constants as parameters (cf. Section VI. 2). Source terms (q) must be taken into account in the relevant continuity equations, e.g., for defect B that can be created by... 

When defect-dependent properties are to be interpreted, it is important that the defect concentrations as a function of temperature and the activities of the crystal components are known. For this purpose it is necessary to formulate reactions and write equations for the formation (or annihilation) of the defects. 

The defect formation may either occur internally in the solid or through reactions with the environment. In the following, the rules for formulating defect reactions will be described and applied to different defect stmcture systems, while in the next chapter, conditions for equilibrium and equations relating equilibrium defect concentrations to temperature, activities (partial pressures) of the components in a compound, impurity concentrations, etc., will be discussed. 

The law of mass action may be applied to reactions involving the formation and interaction of lattice defects which meet the criteria discussed in Section 1.1. Equation (3) is applicable when a crystal is in equilibrium with its ambient. At low concentrations of defects, concentrations may be used in place of activities in Eq. (3). In such cases, brackets usually are used to denote concentration, so that Eq. (2) may be written... 

Hulbert [77] discusses the consequences of the relatively large concentrations of lattice imperfections, including, perhaps, metastable phases and structural deformations, which may be present at the commencement of reaction but later diminish in concentration and importance. If it is assumed [475] that the rate of defect removal is inversely proportional to time (the Tammann treatment) and this effect is incorporated in the Valensi [470]—Carter [474] approach it is found that eqn. (12) is modified by replacement of t by In t. This equation is obeyed [77] by many spinel formation reactions. Zuravlev et al. [476] introduced the postulate that the rate of interface advance under diffusion control was also proportional to the amount of unreacted substance present and, assuming a contracting sphere (radius r) model... 

However, if the molecules of 5 had R alkyl chains longer than Me, the steric hindrance prevented 100% substitution and IR examinations indicated a 50% less derivatization. Moreover, XPS analysis showed that the surface is partly modified by substitution of hydrogen by halogen . In the case of 5 with X = I and to some extent X = Br, the formation of X radicals (besides 12) in a secondary reaction was reported . They participate in reactions analogous to equations 21 and 22b, but with X instead of 12, and attach to the Si surface improving the electronic passivation of the surface at defect sites, sterically inaccessible to 12. A possibility that surface dangling bonds may also appear in the charged states was discussed as well . 

Partial pressure of oxygen controls the nature of defects and nonstoichiometry in metal oxides. The defects responsible for nonstoichiometry and the corresponding oxidation or reduction of cations can be described in terms of quasichemical defect reactions. Let us consider the example of transition metal monoxides, M, 0 (M = Mn, Fe, Co, Ni), which exhibit metal-deficient nonstoichiometry. For the formation of metal vacancies in M, 0, the following equations can be written ... 

The kinetic equations are useful as a fitting procedure although their basis - the homogeneous system - in general does not exist. Thus they cannot deal with segregation and island formation which is frequently observed [27]. Computer simulations incorporate fluctuation and correlation effects and thus are able to deal with segregation effects but so far the reaction systems under study are oversimplified and contain only few aspects of a real system. The use of computer simulations for the study of surface reactions is also limited because of the large amount of computer time which is needed. Especially MC simulations need so much computer time that complicated aspects (e.g., the dependence of the results on the distribution of surface defects) in practice cannot be studied. For this reason CA models have been developed which run very fast on parallel computers and enable to study more complex aspects of real reaction systems. Some examples of CA models which were studied in the past years are the NH3 formation [4] and the problem of the universality class [18]. However, CA models are limited to systems which are suited for the description by a purely parallel ansatz. 

Figure 2. Predicted enthalpy of formation of TiH2 from Ti + H2 as a function of the excess volume according to the universal law and the Birch-Murnaghan 2° order equation. The effect leads to an increase in enthalpy of formation for TiH2 when such regions are introduced. The Ti hydriding reaction illustrates the fact that not all materials will benefit from the introduction of high energy structural defects (Eq. (7) and Table 1).

The extent of the reactions indicated by Equations 1 and 2 or the molar sulfate-to-sulfur ratio is 2.4 zb 0.2 when rock pyrite is used and 1.4 zb 0.4 for sedimentary pyrite found in the coals used in this work. Although both materials are FeS2 of the same crystal structure, differences in reacivity have been documented which are attributed to impurities and crystal defects peculiar to the various possible modes of formation (7). For coal, no significant variation in this ratio was found with ferric ion concentration, acid concentration, coal, or reaction time. The results for each coal are found in Table II. 

Surface Superbasic Sites of One-electron Donor Character. - The reaction of alkali metal with anionic vacancies on the oxide surfaces (equation 1) leads to the creation of colour centres of F type. The transfer of one electron from the alkali metal atom to an anionic vacancy is the reason for the formation of these defects. The largest quantities of this type of active centre are obtained by evaporation of the alkali metal onto an oxide surface calcined at about 1023 K, at which temperature the largest quantity of anionic vacancies is formed. Oxide surfaces calcined at such high temperatures contain only a small quantity of OH groups ca. 0.5 OH per 100 for MgO and 0.8 OH per 100 for AI2O3), so their role in the reaction is small and the action of alkali metal leads selectively to the creation of defects of the electron in anionic vacancy type. The evidence for such a reaction mechanism is the occurrence of specific colours in the oxide. Magnesium oxide after deposition by evaporation of sodium, potassium, or a caesium turns blue, alumina after sodium evaporation becomes a navy blue in colour, and silica after sodium evaporation becomes violet-brown in colour. ... 

Region 3. Ai relatively low oxygen chemical potential, the process of Equation (3.33) dominates, and the reaction in Equation (3.32) is suppressed. The crystal lattice releases oxygen atoms, and this oxygen loss is accompanied by the formation of oxygen vacancies and electrons as dominating defects. The electroneutrality condition can, therefore, be approximated by the following equation ... 

Ideal D-inodcl calculations fail to account simultaneously for the extents of retention and byproduct formation (.Section 7.2.9). The equivalence of equations (7.2.S) and (7.53) shows that this defect in D-modcl calculations can be repaired by assigning the initial encounter of R- and Mg/ a higher reaction probability than later ones, in the freckles model with geminate reaction (Section 7.3.11). 

The relationship between fatigue life and carbonyl content can be explained as follows according to the photooxidation mechanism of PE, carbonyl groups result mainly from a Norrish type II reaction, i.e., for each carbonyl formation, there is a scission of a segment of a molecule chain. Such scission creates a defect in the structure which can grow and propagate into a microcrack under application of a load. Under cyclic loading, it is understandable that the number of cycles the sample can sustain will be directly related to the number of defects (such as microcracks, microvoids. ..), as is clearly described by equation (2). ... 

The concentration of defects can be derived from statistical thermodynamics point of view, but it is more convenient treat the formation of defects as a chemical reaction, so that equilibrium constant of mass action can be applied. For a general reaction, in which the reactants A and B lead to products C and D, the equation is given by ... 

The subscript defect refers to the site of repeatable growth. These equations can be compared to eq. (4-19) which is applicable for a homogeneous defect reaction in ionic crystals, and to eq. (6-6) which is applicable for an inhomogeneous defect reaction in ionic crystals. In these equations, in contrast to eq. (7-1), the pairwise formation of defects is evident. 

Despite the requirement of the constancy in the ratio of regular sites, the total number of regular lattice sites may change in a defect reaction, and therefore, the defect equation may include the formation or annihilation of lattice sites as long as the proper ratios are maintained. 

Although the above reaction is possible, it is more likely that Fe203 (s) formed as a result of electrochemical oxidation at a sufficiently high potential followed by interaction of Fe " by OH ions, the latter being formed by cathodic reaction 02-F2H20-F4e 40H. Iron oxidizes as Fe(s) - aq)+3e. As suggested by equation (1), the oxide formation is increased by increasing the activity of OH ions. When a surface film of an oxide or hydroxide develops, corrosion is either eliminated or retarded. The passive films maybe as thin as 2-10 nm, and they offer a limited electronic conductivity, and behave like semiconductors with metallic properties rather than the properties shown by bulk oxides. The films also allow a limited amount of conductivity of cations because of lattice defects and a slow anodic dissolution. Because of film formation, there is a reduction in the current density for cathodic reduction and an increase in the current density for an anodic polarization. The above frictors lead to retardation of corrosion. 

It is clear that the experimental curves, measured for solid-state reactions under thermoanalytical study, cannot be perfectly tied with the conventionally derived kinetic model functions (cf. previous table lO.I.), thus making impossible the full specification of any real process due to the complexity involved. The resultant description based on the so-called apparent kinetic parameters, deviates from the true portrayal and the associated true kinetic values, which is also a trivial mathematical consequence of the straight application of basic kinetic equation. Therefore, it was found useful to introduce a kind of pervasive des-cription by means of a simple empirical function, h(a), containing the smallest possible number of constant. It provides some flexibility, sufficient to match mathematically the real course of a process as closely as possible. In such case, the kinetic model of a heterogeneous reaction is assumed as a distorted case of a simpler (ideal) instance of homogeneous kinetic prototype f(a) (1-a)" [3,523,524]. It is mathematically treated by the introduction of a multiplying function a(a), i.e., h(a) =f(a) a(a), for which we coined the term [523] accommodation function and which is accountable for a certain defect state (imperfection, nonideality, error in the same sense as was treated the role of interface, e.g., during the new phase formation). 

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