Brouwer diagrams

The type of disorder may be determined by conductivity measurements of electronic and ionic defects as a function of the activity of the neutral mobile component [3]. The data are commonly plotted as Brouwer diagrams of the logarithm of the concentration of all species as a function of the logarithm of the activity of the neutral mobile component. The slope is fitted to the assumption of a specific defect-type model. 

What information does a Brouwer diagram display ... 

Figure 7.9 Brouwer diagram for a phase MX in which Schottky defects are the main point defect type (a) initial points on the diagram, (b) variation of defect concentrations in the near-stoichiometric region, (c) extension to show variation of defect concentrations in the high partial pressure region, (d) extension to show variation of defect concentrations in the low partial pressure region, and (e) complete diagram.

BROUWER DIAGRAMS ELECTRONIC DEFECTS 7.6.1 Electronic Defects... 

A Brouwer diagram is drawn up for one temperature and is an isothermal representation of the situation in the phase. A number of parameters determine the way in which any diagram changes with temperature. 

Frenkel and Schottky defect equilibria are temperature sensitive and at higher temperatures defect concentrations rise, so that values of Ks and Kv, increase with temperature. The same is true of the intrinsic electrons and holes present, and Kc also increases with temperature. This implies that the defect concentrations in the central part of a Brouwer diagram will move upward at higher temperatures with respect to that at lower temperatures, and the whole diagram will be shifted vertically. 

The approximations inherent in Brouwer diagrams can be bypassed by writing the appropriate electroneutrality equation as a polynomial equation and then solving this numerically using a computer. (This is not always a computationally trivial task.) To illustrate this method, the examples given in Sections 7.5 and 7.6, the MX system, will be rewritten in this form. 

Defect populations and physical properties such as electronic conductivity can be altered and controlled by manipulation of the surrounding atmosphere. To specify the exact electronic conductivity of such a material, it is necessary to specify its chemical composition, the defect types and populations present, the temperature of the crystal, and the surrounding partial pressures of all the constituents. Brouwer diagrams display the defect concentrations present in a solid as a function of the partial pressure of one of the components. Because the defect populations control such properties as electronic and ionic conductivity, it is generally easy to determine how these vary as the partial pressure varies. 

An increase in temperature will cause the general form of a Brouwer diagram to ... 

Using the information in Section 7.7.1, sketch the Brouwer diagram for Cr203-The oxide is an insulator under normal conditions, so assume that Schottky defects dominate. 

The original description of Brouwer diagrams is well worth consulting. It is ... 

Information on polynomial forms of Brouwer diagrams for Cr203 is in ... 

Brouwer diagrams plot the defect concentrations in a solid as a function of the partial pressure of the components of the material and are a convenient way of displaying electronic properties (Sections 7.6-7.9). These can be readily extended to include the effects of doping by acceptors or donors. 

The four Eqs. [(8.3)—(8.6)] are simplified using chemical and physical intuition and appropriate approximations to the electroneutrality Eqs. (8.7) and (8.10). Brouwer diagrams similar to those given in the previous chapter can then be constructed. However, by far the simplest way to describe these equilibria is by way of polynomials. This is because the polynomial appropriate for the doped system is simply the polynomial equation for the undoped system, together with one extra term, to account for the donors or acceptors present. For example, following the procedure described in Section 7.9, and using the electroneutrality equation for donors, Eq. (8.9), the polynomial appropriate to donor doping is ... 

The discussion of Brouwer diagrams in this and the previous chapter make it clear that nonstoichiometric solids have an ionic and electronic component to the defect structure. In many solids one or the other of these dominates conductivity, so that materials can be loosely classified as insulators and ionic conductors or semiconductors with electronic conductivity. However, from a device point of view, especially for applications in fuel cells, batteries, electrochromic devices, and membranes for gas separation or hydrocarbon oxidation, there is considerable interest in materials in which the ionic and electronic contributions to the total conductivity are roughly equal. 

Brouwer Diagram Representation of Mixed Proton Conductivity... 

A number of factors must be taken into account when the diagrammatic representation of mixed proton conductivity is attempted. The behavior of the solid depends upon the temperature, the dopant concentration, the partial pressure of oxygen, and the partial pressure of hydrogen or water vapor. Schematic representation of defect concentrations in mixed proton conductors on a Brouwer diagram therefore requires a four-dimensional depiction. A three-dimensional plot can be constructed if two variables, often temperature and dopant concentration, are fixed (Fig. 8.18a). It is often clearer to use two-dimensional sections of such a plot, constructed with three variables fixed (Fig. 8.18h-8.18<7). 

The Brouwer diagram approach can be illustrated with reference to the perovskite structure oxide system BaYbvPr VC>3, which has been explored as a potential cathode material for use in solid oxide fuel cells. The parent phase... 

Figure 8.18 Schematic representation of defect concentrations in mixed proton conductors on a Brouwer diagram (a) three-dimensional plot with two variables fixed (b)-(d) two-dimensional plots with three variables fixed.

These equations then allow the Brouwer diagrams to be constructed (Fig. 8.19). 

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