Brouwer Diagrams and Frozen-In Profiles

In many ceramics, intrinsic and extrinsic disorder, as well as the disorder due to nonstoichiometry, have to be considered. Independent of the dopant level, the mass action laws of intrinsic disorder, of the e-h equilibrium and of the reaction with the surrounding phase are valid in thermodynamic equilibrium. Together with the electroneutrality equation 

For broad partial pressure regimes, however, only two mobile defects, or one mobile defect and the dopant, play a role in the electroneutrality equation (Eq. (11)). In theses cases simple partial pressure dependences, namely power laws with concentrations being proportional top 02) , can be calculated (see e.g. Refs. [57, 80, 89, 90]). One example of such a simplified situation has been examined above (Eq. (10)). 

Considering different pairs of majority defects, all relationships between defect concentrations and partial pressure can be constructed from simplified situations, and this leads to so-called Brouwer diagrams. Figs. 2a and 3a show such Brouwer diagrams for a pure oxide MO with Schottky disorder, and for a Schottky-disordered oxide with a negative dopant. (Please notice that the exact curves calculated from the complete electroneutrality equation (Eq. (11)) exhibit smooth transitions rather than sharp bends.) 

It should be emphasized that the kinetics of the equilibration process after a change of the surrounding phase (e.g. a p(O2) change) requires the movement of defects and can be rather sluggish, particularly at lower temperatures. Therefore, non-equilibrated ionic solids with composition gradients can easily occur, and often the preparation conditions rather than the actual surroundings determine the defect concentrations (frozen-in compositions). On the other hand, internal defect reactions 

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