Thermodynamics crystal/lattice defects

This is the first book devoted to the theoretical modelling of refractory carbides and nitrides and alloys based on them. It makes use of computational methods to calculate their spectroscopic, electric, magnetic, superconducting, thermodynamical and mechanical properties. Calculated results on the electronic band structure of ideal binary transition-metal carbides and nitrides are presented, and the influences of crystal lattice defects, vacancies and impurities are studied in detail. Data available on chemical bonding and the properties of multi-component carbide- and nitride-based alloys, as well as their surface electronic structure, are described, and compared with those of bulk crystals. 

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 rate (or kinetics) and form of a corrosion reaction will be affected by a variety of factors associated with the metal and the metal surface (which can range from a planar outer surface to the surface within pits or fine cracks), and the environment. Thus heterogeneities in a metal (see Section 1.3) may have a marked effect on the kinetics of a reaction without affecting the thermodynamics of the system there is no reason to believe that a perfect single crystal of pure zinc completely free from lattic defects (a hypothetical concept) would not corrode when immersed in hydrochloric acid, but it would probably corrode at a significantly slower rate than polycrystalline pure zinc, although there is no thermodynamic difference between these two forms of zinc. Furthermore, although heavy metal impurities in zinc will affect the rate of reaction they cannot alter the final position of equilibrium. 

When in solid solution in the solid state, an impurity will alter the crystallinity by introducing impurity defects into the crystal lattice, thereby changing the thermodynamic and other physical properties of the solid, including the solubility and dissolution rate [2,37]. Prolonged equilibration of the solid state with the saturated solution, however, usually leads to recrystallization of the solute and to a consequent return of the crystallinity and the measured solubility of the solid state to that of the pure, highly crystalline solid. 

There is a third law of thermodynamics. It can be stated in the following way The entropy of a perfect crystal at 0 K is zero. A perfect crystal is one with no lattice defects. The third law gives rise to the concept of absolute entropy. There will be no further mention of the third law in this book. 

The vacancy flux and the corresponding lattice shift vanish if bA = bB. In agreement with the irreversible thermodynamics of binary systems i.e., if local equilibrium prevails), there is only one single independent kinetic coefficient, D, necessary for a unique description of the chemical interdiffusion process. Information about individual mobilities and diffusivities can be obtained only from additional knowledge about vL, which must include concepts of the crystal lattice and point defects. 

The appropriate thermodynamic and statistical-mechanical formalism for the application of molecular simulation to the study of point defects has been given only recently, by Swope and Andersen [90]. These workers identified the number of lattice sites M as a key thermodynamic variable in the characterization of these systems. A real solid phase is free to adopt a value for M that minimizes the system free energy, because it can in principle create or destroy lattice sites through the migration of molecules to and from the surface of the crystal. The resulting bulk crystal can thus disconnect the molecule number N from the lattice-site number M, and thereby achieve an equilibrium of lattice defects in the form of vacancies and interstitials. 

Many industrial crystallization processes, by necessity, push crystal growth rates into a regime where defect formation becomes unavoidable and the routes for impurity incorporation are numerous. Since dislocations, inclusions, and other crystal lattice imperfections enhance the uptake of impurities during crystallization, achieving high purity crystals requires elimination of impurity incorporation and carry-over by both thermodynamic and non-thermodynamic mechanisms. Very generally, the impurity content in crystals can be considered as the sum of all of these contributions... 

These observations concerning ionic transport can be explained on the basis of defect chemistry and crystal stmctme of the solid materials. The ideal crystal is in fact an abstract concept that is used in crystallographic descriptions. The lattice of a real crystal always contains imperfections. A snitable classiflcation of crystalline defects can be achieved by first considering point defects and then proceeding to one- and higher-dimensional defects. Point defects are atomic defects whose effects are limited only to their immediate snrronndings. They exist in a state of complete thermodynamic equilibrinm. Examples are ionic vacancies in the regnlar crystal lattice, or interstitial atoms or ions. 

An ideal crystal consists of a perfectly ordered arrangement of atoms, ions, or molecules. However, in any real crystal, at temperatures above absolute zero, there are always imperfections or defects in the crystal lattice, as discussed in Chapter 5. This chapter will deal with defects whose distribution and concentration in the lattice are governed by the laws of thermodynamics. In pure crystals such defects are called native defects. The existence of native defects in a lattice arises from a tendency of a crystal to increase its entropy or degree of disorder. As defects are introduced into a crystal, the entropy AS will increase. The number of defects will be limited, however, by the enthalpy necessary to form the defects, AH. The actual number of defects present at any temperature is that which gives a minimum in the free energy G of the crystal according to the relation... 

At the beginning of the century, nobody knew that a small proportion of atoms in a crystal are routinely missing, even less that this was not a mailer of accident but of thermodynamic equilibrium. The recognition in the 1920s that such vacancies had to exist in equilibrium was due to a school of statistical thermodynamicians such as the Russian Frenkel and the Germans Jost, Wagner and Schollky. That, moreover, as we know now, is only one kind of point defect an atom removed for whatever reason from its lattice site can be inserted into a small gap in the crystal structure, and then it becomes an interstitial . Moreover, in insulating crystals a point defect is apt to be associated with a local excess or deficiency of electrons. 

The formation of the combination of defects may be described as a chemical reaction and thermodynamic equilibrium conditions may be applied. The chemical notations of Kroger-Vink, Schottky, and defect structure elements (DSEs) are used [3, 11]. The chemical reactions have to balance the chemical species, lattice sites, and charges. An unoccupied lattice site is considered to be a chemical species (V) it is quite common that specific crystal structures are only found in the presence of a certain number of vacancies [12]. The Kroger-Vink notation makes use of the chemical element followed by the lattice site of this element as subscript and the charge relative to the ideal undisturbed lattice as superscript. An example is the formation of interstitial metal M ions and metal M ion vacancies, e.g., in silver halides ... 

Lattice Vacancies and Interstitials Defects such as lattice vacancies and interstitials fall into two main categories intrinsic defects, which are present in pure crystal at thermodynamic equilibrium, and extrinsic defects, which are created when a foreign atom is inserted into the lattice. 

The notion of point defects in an otherwise perfect crystal dates from the classical papers by Frenkel88 and by Schottky and Wagner.75 86 The perfect lattice is thermodynamically unstable with respect to a lattice in which a certain number of atoms are removed from normal lattice sites to the surface (vacancy disorder) or in which a certain number of atoms are transferred from the surface to interstitial positions inside the crystal (interstitial disorder). These forms of disorder can occur in many elemental solids and compounds. The formation of equal numbers of vacant lattice sites in both M and X sublattices of a compound M0Xft is called Schottky disorder. In compounds in which M and X occupy different sublattices in the perfect crystal there is also the possibility of antistructure disorder in which small numbers of M and X atoms are interchanged. These three sorts of disorder can be combined to give three hybrid types of disorder in crystalline compounds. The most important of these is Frenkel disorder, in which equal numbers of vacancies and interstitials of the same kind of atom are formed in a compound. The possibility of Schottky-antistructure disorder (in which a vacancy is formed by... 

---