Thermodynamic point defect model

The p0 dependence of oxygen nonstoichiometry (8) was determined by using coulometric titration. The data were analyzed using a simple point defect model and thermodynamic quantities were calculated. From this model, the standard enthalpy for oxidation (AH0f) and disproportionation (A77D) were determined to be -140.7 and 228.7 kJ/mol, respectively. The mobilities of the electron holes, electrons, and oxygen ions were calculated from the conductivity data using the defect concentrations determined from the stoichiometry and point defect model. 

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... 

Defect thermodynamics, as outlined in this chapter, is to a large extent thermodynamics of dilute solutions. In this situation, the theoretical calculation of individual defect energies and defect entropies can be helpful. Numerical methods for their calculation are available, see [A. R. Allnatt, A. B. Lidiard (1993)]. If point defects interact, idealized models are necessary in order to find the relations between defect concentrations and thermodynamic variables, in particular the component potentials. We have briefly discussed the ideal pair (cluster) approach and its phenomenological extension by a series expansion formalism, which corresponds to the virial coefficient expansion for gases. 

The statistical thermodynamic approach, along lines already indicated, has been more tractable and suggestive. Models have been based on the Fowler-Guggenheim treatment of localized monolayers, in which account is taken of energy terms arising from interaction between point defects in nearest neighbor... 

HEYNE As you stated a lot of confusion exists due to different points of view from which solid electrolyte problems are approached. I should like to emphasize that such confusions could be considerably reduced if thermodynamic arguments and model considerations would always be clearly separated. For instance splitting up of a components chemical potential into either an electrochemical potential plus an electrostatic potential [usual in normal electrochemistry) or into an ion electrochemical pot. plus an electron chem. pot. (= Fermi level), is completely arbitrary [and unnecessary) from a purely thermodynamic point of view. As soon as we split in one way or another we must be aware, and clearly state, that we use a certain model such as for instance the band picture of a semi-conductor, or the defect structure of a solid electrolyte. 

Equations (14.14) and (14.18) can be used as starting point for generating equations describing O2 and H2 permeation within single-phase perovskite membranes. Key to these equations is the nature of the boundary conditions at the feed/membrane and permeate/membrane surfaces. To this aim, one needs to address appropriate defect point thermodynamics to establish equilibrium and surface exchange relations for all potential species that can play a role during permeation. As a general rule, the law of mass action can be used to predict the concentration of ionic vacancies, protons, electrons, and electron holes in the membrane. Below we describe a series of models that can be deduced for ID steady-state permeation within perovskite and extensively other MIEC membranes. 

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