Model brick layer

The effective terms ct" and ct1 may be broken down into core (co) and space charge (sc) contributions,285 q>L is the volume fraction of the grain boundaries (core and space charge zone), p = 1/3 and Pi = 2/3 in the ideal brick layer model. 

Hence, in simple cases each bulk layer, each grain boundary plane, and both electrodes of the brick layer model sample, can be represented by separate RC elements (Fig. 7b). The RC elements of the n bulk layers can be combined to a single RC element with the -fold resistance and the 1 / -fold capacitance of a single layer. The n — 1 grain boundary impedances can also be summed, as can the two electrode impedances, and hence the model sample corresponds to a series connection of three RC elements (Fig. 7c) with... 

In experimental impedance spectroscopic studies, however, several factors may complicate the interpretation of the spectra and a few of these complications will briefly be touched upon i) If high conductivities are considered (a > 10-3 S cm-1), then the corresponding relaxation frequencies are well above the measurement range of a conventional impedance set-up (frequencies up to ca. 10 MHz). Hence, processes with high conductivites cannot be separated by conventional impedance spectroscopy. ii) The assumption of a quasi-one-dimensional current flow, which is the basis of the above presented brick layer model, is often violated [203, 211-214]. Some complications due to multi-dimensional potential distributions will be discussed in Sec. 3.2.1. iii) Highly conductive regions perpendicular to the electrodes (e.g. highly... 

It is particularly interesting to test whether an analysis of a conventional macroscopic impedance measurement based on a brick layer model leads to the same results as obtained with microcontacts. The relaxation frequency of the grain boundary arc measured in the conventional impedance experiment (1.2 Hz) is similar to the mean value (3.0 Hz) deduced from the microelectrode experiments, although not identical. One possible reason for the moderate discrepancy is the inaccuracy with respect to the temperature measurement, which is somewhat difficult in the case of the microelectrode set-up. A temperature error of about 20 K could already explain the difference. 

It is worth comparing these locally obtained values with the effective conductivity creff of the same sample measured in a conventional setup. A measurement with macrosopic electrodes yields one semicircle in the complex impedance plane and an effective conductivity of 42 10 9 ft 1 cm-1. According to the brick layer model for... 

Figure 4.1.4. Brick layer model for a two-phase ceramic (a) Overall view, showing array of cubic grains, separated by flat grain boundaries, (b) Exploded view of a single cell, showing parallel electrical paths (i) through grains and grain boundaries, and (ii) along grain boundaries.

Case (i). For <7, Ogt, the brick layer model is equivalent to the series layer model but with a one-third weighting of the grain boundary resistance. This reflects the fact that grain boundaries in only one of the three orientations (i.e. normal to the current) have a blocking effect. The circuit equivalent of the brick layer model is, therefore, that of Figure 4.1.1 with parameters... 

A salient point in this work is that, by specifying a variation in conductivity with distance from the grain boundary core, conductivity profiles in both orientations can be integrated and a phenomenological description obtained that is consistent with the brick-layer model. On one hand, this is encouraging, because it shows... 

Brailsford and Hohnke [1983] have applied the Maxwell-Wagner model to grain boundaries in two-phase systems. Their microstructural model, shown in Figure 4.1.8fc, consists of a spherical grain of radius r2 surrounded by a shell of outer radius a, which represents the grain boundary and has a volume fraction Xi = 1 - (jjrif. The authors observe that for Xi —> 0 and 1/ 2 V i the effective medium model becomes identical to case (i) of the brick layer model, namely Eq. (6). Further, we have found that for x —> 0 and y/i xffz, it reduces to case (ii) of the brick layer model, namely Eq. (7). ... 

Figure 4.1.34. Arrhenius plots of the grain interior and grain boundary resistivities for two zirconia ceramics (a) Tetragonal zirconia ceramic (Zr02 3 mole % Y2O3) in which the lines have different slopes, as expected from the brick layer model, b) Partially stabilized ceramic (Zr02 6 mole % Y2O3) in which the slopes of the lines are similar, as expected for discrete grain boundary phase. (Courtesy of Silicates Industriels.)...

Choice of appropriate model IS is not a technique that can or should be apphed without prior knowledge of the system. Impedance spectra must be interpreted in the context of a model, be this a simple brick-layer model for a ceramic, or an advanced one based on electrode kinetics. When used in conjunction with electron microscopy, IS provides information about structure, and especially grain boundary structure. The microstructural information and the models derived from this are what make the conclusions of IS unequivocal. 

Fig.3 (a) Schematic diagram showing the ionic conduction in polycrystalline stabilized zirconia specimen with a resistive grain boxmdary and (b) its simplification by the brick layer model, (d electrode area, / specimen thickness, dgi average grain size, <%4 grain boundary thickness)... 

Under the assumption of a brick layer model and serial equivalent circuit, can be calculated by the following equation [8]... 

Impedance spectroscopy has been a useful tool for separating the grain-interior and grain-boundaiy contributions in many electroceramics. [48,49] In most analyses, the brick layer model, which assumes electrically and dimensionally homogeneous grain and grain boimdary, was used. [7,13,24-26] (Fig.3(b)) However, the distribution of the and Pgh values can become spatially uneven for the specimens under an electric field [50-53] or with a functionally graded composition. [54]... 

The very famous brick layer model is used to correlate the grain/grain boundary conductivity to the microstmcture of the specimen. The brick layer model assumes ceramic samples to consist of grains with high conductivity, which are separated by uniform grain boundaries. Positive space change potentials are obtained when... 

Realistic "brick layer model" represents system as "grains" (discon-... 

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