Thermodynamics of the Point Defect

What we describe as the energy state for a given set of conditions is actually the average of that of a Boltzmann population. The discipline best suited for handling such a system is statistics, hence the name. The approach used for manipulating molecular populations in Statistical Mechanics is quite involved, and we shall touch very briefly on the mathematics Involved. We will first describe each of these approaches separately and then a combined version. Hopefully, this will aid in your understanding of the two methods of determining the effect of the point defect upon the properties of the solid. 

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Of necessity, we cannot be exhaustive, and there are many treatises which deal solely with the thermodynamics of the point defect. 

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. 

Analyses of the defect chemistry and thermodynamics of non-stoichiometric phases that are predominately ionic in nature (i.e. halides and oxides) are most often made using quasi-chemical reactions. The concentrations of the point defects are considered to be low, and defect-defect interactions as such are most often disregarded, although defect clusters often are incorporated. The resulting mass action equations give the relationship between the concentrations of point defects and partial pressure or chemical activity of the species involved in the defect reactions. 

Let us now discuss some details of practical relevance. From the Gibbs phase rule, it is evident that crystals consisting of only one component (A) become nonvariant by the predetermination of two thermodynamic variables, which for practical reasons are chosen to be Pand T. In these one-component systems, it is easy to recognize the (isobanc) concentration dependence of the point defects on temperature. From the definition of the vacancy chemical potential for sufficiently small vacancy mole fractions Nv, namely //v = /A (P, T) + RT- In Vv, together with the condition of equilibrium with the crystal s inerL surroundings (gas, vacuum), one directly finds... 

Alan Allnatt s research interests at Western Ontario have been concerned with the statistical mechanics of the transport of matter through crystals. His earliest work centered on obtaining methods for calculating the equilibrium distributions and thermodynamic properties of the point defects (vacancies, interstitials, solutes) that make transport possible. He first studied dilute systems, so the methods could be largely analytical. The methods for ionic crystals,... 

Samples presently available are rich in extended defects, dislocations and grain boundaries, and the thermodynamics of growth at low temperatures suggests high densities of at least some of the point defects. 

When we consider continuous scale growth, we can expect that the mobile species from the metal (cations diffusing out) will be supplied by alloy grain-boundaries, bulk defects, and dislocations. These diffusivities are quite different from each other D(bulk) D(dislocation) < D(grain boundary) < D(surface). Therefore, we expect the formation of voids around the alloy grain boundaries and dislocations as the scale continues to grow. The chief concerns, here, is How can we prepare an inert state (kinetically and thermodynamically) for the point defects, for the grain boundaries, and especially for the dislocations in the alloy substrate ... 

We begin by examining what continuum mechanics might tell us about the structure and energetics of point defects. In this context, the point defect is seen as an elastic disturbance in the otherwise unperturbed elastic continuum. The properties of this disturbance can be rather easily evaluated by treating the medium within the setting of isotropic linear elasticity. Once we have determined the fields of the point defect we may in turn evaluate its energy and thereby the thermodynamic likelihood of its existence. 

For the thermodynamic treatmerrt of the point defect equilibria, one has to take into accoimt the electroneutrality condition... 

Defect chemistiy is a chemistry within the solid state that is analogous to the long-familiar chemistiy in the liquid phase, and arises from departures from the ideal crystal structure which are thermodynamically unavoidable, the point defects. While defect chemistry foundations were established over 60 years ago, this area of chemistiy enables one to date to... 

Crystal structure, crystal defects and chemical reactions. Most chemical reactions of interest to materials scientists involve at least one reactant in the solid state examples inelude surfaee oxidation, internal oxidation, the photographie process, electrochemieal reaetions in the solid state. All of these are critieally dependent on crystal defects, point defects in particular, and the thermodynamics of these point defeets, especially in ionic compounds, are far more complex than they are in single-component metals. I have spaee only for a superficial overview. 

Thermodynamic considerations imply that all crystals must contain a certain number of defects at nonzero temperatures (0 K). Defects are important because they are much more abundant at surfaces than in bulk, and in oxides they are usually responsible for many of the catalytic and chemical properties.15 Bulk defects may be classified either as point defects or as extended defects such as line defects and planar defects. Examples of point defects in crystals are Frenkel (vacancy plus interstitial of the same type) and Schottky (balancing pairs of vacancies) types of defects. On oxide surfaces, the point defects can be cation or anion vacancies or adatoms. Measurements of the electronic structure of a variety of oxide surfaces have shown that the predominant type of defect formed when samples are heated are oxygen vacancies.16 Hence, most of the surface models of... 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

At r > Tr, the relaxation of a non-equilibrium surface morphology by surface diffusion can be described by Eq. 1 the thermodynamic driving force for smoothing smoothing is the surface stiffness E and the kinetics of the smoothing is determined by the concentration and mobility of the surface point defects that provide the mass transport, e.g. adatoms. At r < Tr, on the other hand, me must consider a more microscopic description of the dynamics that is based on the thermodynamics of the interactions between steps, and the kinetics of step motion [17]. 

The question raised by Anderson (1970,1971) and Anderson et al (1973) as to whether anion point defects are eliminated completely by the creation of extended CS plane defects, is a very important one. This is because anion point defects can be hardly eliminated totally because apart from statistical thermodynamics considerations they must be involved in diffusion process. Oxygen isotope exchange experiments indeed suggest that oxygen diffuses readily by vacancy mechanism. In many oxides it is difficult to compare small anion deficiency with the extent of extended defects and in doped complex oxides there is a very real discrepancy between the area of CS plane present which defines the number of oxygen sites eliminated and the oxygen deficit in the sample (Anderson 1970, Anderson et al 1973). We attempt to address these issues and elucidate the role of anion point defects in oxides in oxidation catalysis (chapter 3). 

More generally, co is independent of the external gas pressure k is the Boltzmann constant (1.38 x 10 erg deg ) and T is the temperature in Kelvin. Furthermore, the equilibrium between co and a collapsed CS plane fault is maintained by exchange at dislocations bounding the CS planes. Clearly, this equilibrium cannot be maintained except by the nucleation of a dislocation loop and such a process requires a supersaturation of vacancies and CS planes eliminate supersaturation of anion vacancies (Gai 1981, Gai et al 1982). Thus we introduce the concept of supersaturation of oxygen point defects in the reacting catalytic oxides, which contributes to the driving force for the nucleation of CS planes. From thermodynamics. 

We wish here to obtain the thermodynamic equations defining the liquidus surface of a solid solution, (At BB)2, ). It is assumed that the A and atoms occupy the sites of one sublattice of the structure and the C atoms the sites of a second sublattice. For the specific systems considered here Sb and play the role of C in the general formula above. It is also assumed that the composition variable is confined to values near unity so that the site fractions of atomic point defects is always small compared to unity. This apparently is the case for the solid solutions in the two systems considered. Then it can be shown theoretically (Brebrick, 1979), as well as experimentally for (Hgj CdJ2-yTe)l(s) (Schwartz et al, 1981 Tung et al., 1981b), that the sum of the chemical potentials of A and C and that of and C in the solid are independent of the composition variable y ... 

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. 

Defect thermodynamics provide the guidelines for the solution of this practical problem. In Chapter 2, the basic ideas on how to influence point defect concentrations by doping with (heterovalent) additions were presented. Due to the electroneutrality condition and the laws of mass action, we can control the point defect... 

These assumptions, however, oversimplify the problem. The parent (A,B)0 phase between the surface and the reaction front coexists with the precipitated (A, B)304 particles. These particles are thus located within the oxygen potential gradient. They vary in composition as a function of ( ) since they coexist with (A,B)0 (AT0<1 see Fig. 9-3). In the Af region, the point defect thermodynamics therefore become very complex [F. Schneider, H. Schmalzried (1990)]. Furthermore, Dv is not constant since it is the chemical diffusion coefficient and as such it contains the thermodynamic factor /v = (0/iV/01ncv). In most cases, one cannot quantify these considerations because the point defect thermodynamics are not available. A parabolic rate law for the internal oxidation processes of oxide solid solutions is expected, however, if the boundary conditions at the surface (reaction front ( F) become time-independent. This expectation is often verified by experimental observations [K. Ostyn, et al. (1984) H. Schmalzried, M. Backhaus-Ricoult (1993)]. 

A/iAg as a function of time with a single and spatially fixed sensor at , or one can determine D with several sensors as a function of the coordinate if at a given time [K.D. Becker, et al. (1983)]. An interesting result of such a determination of D is its dependence on non-stoichiometry. Since >Ag = DAg d (pAg/R T)/d In 3, and >Ag is constant in structurally or heavily Frenkel disordered material (<5 1), DAg(S) directly reflects the (normalized) thermodynamic factor, d(pAg/R T)/ In 3, as a function of composition, that is, the non-stoichiometry 3. From Section 2.3 we know that the thermodynamic factor of compounds is given as the derivative of a point defect titration curve in which nAg is plotted as a function of In 3. At S = 0, the thermodynamic factor has a maximum. For 0-Ag2S at T = 176 °C, one sees from the quoted diffusion measurements that at stoichiometric composition (3 = 0), the thermodynamic factor may be as large as to 102-103. 

If macroscopic thermodynamics are applied to materials containing a popnlation of defects, particnlarly nonstoichio-metric compounds, the defects themselves do not enter into the thermodynamic expressions in an exphcit way. However, it is possible to construct a statistical thermodynamic formahsm that will predict the shape of the free energy-temperatnre-composition curve for any phase containing defects. The simplest approach is to assnme that the point defects are noninteracting species, distributed at random in the crystal, and that the defect energies are constant and not a ftmction either of concentration or of temperatnre. In this case, reaction eqnations similar to those described above, eqnations (6) and (7), can be used within a normal thermodynamic framework to deduce the way in which defect populations respond to changes in external variables. 

A proper description of electronic defects in terms of simple point defect chemistry is even more complicated as the d electrons of the transition metals and their compounds are intermediate between localized and delocalized behaviour. Recent analysis of the redox thermodynamics of Lao.8Sro,2Co03. based upon data from coulometric titration measurements supports itinerant behaviour of the electronic charge carriers in this compound [172]. The analysis was based on the partial molar enthalpy and entropy of the oxygen incorporation reaction, which can be evaluated from changes in emf with temperature at different oxygen (non-)stoichiometries. The experimental value of the partial molar entropy (free formation entropy) of oxygen incorporation, Asq, could be... 

Only the point defects are thermodynamically equilibrium defects. All others, which sometimes are referred to as biographical defects , depend on the prehistory of each sample - that is, on the method and conditions of preparation, heat treatments, and so on. Therefore, a thermodynamic approach can be applied basically for the point defects. 

Normally, the point defects are expected to be in a local or global equilibrium state when the thermodynamic approach can be used. Within the framework of this approach, the defects and their simplest associates are treated as chemical species [9,15-19]. Therefore, the chemical potential of each structural element (p ), which may correspond to atoms (ions) in their regular positions or defects, and the Gibbs energy change for any process involving the i-type species (AG), can be written as... 

The rules for quasi-chemical reactions are the same as for the normal chemical reactions, namely mass balance and electroneutrahty conditions one extra requirement appears, however, for crystalline solids where the ratio of sites in the crystal structure should be constant and should satisfy to stoichiometric formula. This means that if, for instance, in the AB2 crystal one site for A atom is formed, then automatically two B-sites appear as well, regardless of their occupancy. It should be noted that the point defects and/or the processes of their formation can also classified into two groups, namely stoichiometric and nonstoichiometric. The first type of process does not disturb the stoichiometric ratio of components constituting the crystal, which is a closed thermodynamic system the second type leads to nonstoichiometric compounds by exchanging components between the... 

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