Defect Gibbs free energy

The papers of Wagner and Schottky contained the first statistical treatment of defect-containing crystals. The point defects were assumed to form an ideal solution in the sense that they are supposed not to interact with each other. The equilibrium number of intrinsic point defects was found by minimizing the Gibbs free energy with respect to the numbers of defects at constant pressure, temperature, and chemical composition. The equilibrium between the crystal of a binary compound and its components was recognized to be a statistical one instead of being uniquely fixed. 

In these equations gv is the change in Gibbs free energy on taking one atom from a normal lattice site to the surface of the crystal and (gt + gv) the change when an atom is taken from a normal lattice site to an interstitial site, both at constant temperature and pressure. cr denotes a site fraction of species r on its sublattice, and is the chemical potential of a normal lattice ion in the defect-free crystal. 

In considering the equilibrium of the crystal with a second phase the Gibbs chemical potentials are required and we therefore express these in terms of the defect chemical potentials so far discussed. The Gibbs free energy of the system is given by... 

We consider first the activity coefficients. The contribution of the defects, N cation vacancies and N divalent ions, to the Gibbs free energy of the doped crystal is... 

A linear variation of the Gibbs free energy of the defect process with T, of type... 

In Eq. (1.36), Nj is the equilibrium number of point defects, N is the total number of atomic sites per volume or mole, Ej is the activation energy for formation of the defect, is Boltzmann s constant (1.38 x 10 J/atom K), and T is absolute temperature. Equation (1.36) is an Arrhenius-type expression of which we will see a great deal in subsequent chapters. Many of these Arrhenius expressions can be derived from the Gibbs free energy, AG. 

At equilibrium, at constant T, the Gibbs free energy of the system must be a minimum with respect to changes in the number of defects Usi thus... 

In this chapter, we discuss classical non-stoichiometry derived from various kinds of point defects. To derive the phase rule, which is indispensable for the understanding of non-stoichiometry, the key points of thermodynamics are reviewed, and then the relationship between the phase rule, Gibbs free energy, and non-stoichiometry is discussed. The concentrations of point defects in thermal equilibrium for many types of defect structure are calculated by simple statistical thermodynamics. In Section 1.4 examples of non-stoichiometric compounds are shown referred to published papers. 

The translationally ordered state characteristic of crystallites that are large with respect to the wavelengths of X-rays represents perfect phases in the sense of thermodynamics, to which phase diagrams apply. Even this idealized state of matter cannot exist without deviations from perfect ordering, however, because of a requirement of thermodynamics (the entropy of the material at equilibrium must be nonzero for the Gibbs free energy to be minimized). Thus, the material will contain a number of deviations from the ideal arrangement, called defects."... 

Point defects where atoms are missing from lattice sites are called vacancies, or Schottky defects. Their nnmber density depends on the temperature and on the Gibbs free energy of formation of defects. 

Here gp is the Gibbs free energy to form a vacancy, k is the Boltzmann constant, and T is the temperature. Diffusion in a crystal lattice occurs by motion of atoms via jumps between these defects. For example, vacancy diffusion - the most common mechanism in close-packed lattices such as face-centered cubic fee) metals, occurs by the atom jumping into a neighboring vacancy. The diffusion coefficient, D, therefore will depend upon the probability that an atom is adjacent to a vacancy, and the probability that it has sufficient energy to make the jump over the energy barrier into the vacancy. The first of these probabilities is directly proportional to c,. and the... 

Here, c+ and c are the site fractions of the cation and anion vacancies, respectively, Ks is the equilibrium constant for the formation of the Schottky pair, and gs (= g+ + g, the sum of the individual defect formation energies) is the Gibbs free energy to form the pair. In the bulk of a pure crystal, the condition of electrical neutrality demands that the concentrations of each defect in the pair are equal that is ... 

Note that we are using the Gibbs free energy rather than the Helmholtz free energy at this point. For the defect solid, we must define free-energy in terms of the free-energy of the perfect solid. Go, as related to the free-energy of the defect solid, vis ... 

Wumbers of Defects as a Function of Gibbs Free Energy... 

You will note that when it is possible to plot as a function of G, the Gibbs free energy, a minimum occurs which is the number of defects at the given temperature for the given solid. Note also that the enthalpy of defects increases continuously as a function of Nd while both of the entropy-contributions decrease. The net effect is a minimum value for Gd, at which point we can get the number of intrinsic defects present. Thus, we have shown again that there is a specific number of intrinsic defects present in the soUd at any given temperature, this time by Thermodynamics. 

AGpd is the Gibbs free energy of radiation point defects... 

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