Investigation of Local Bulk Properties

Electrical bulk properties of ionic solids can be rather inhomogeneous (Sec. 3.1). In the following it is shown that microelectrodes are a very useful tool to gain spatially resolved information on the conductivity of such inhomogeneous solids. Let us first consider the case of a spherical microelectrode (radius rme) atop a sample with homogeneous bulk conductivity Ubuik- The bulk resistance R between the microelectrode and a hemispherical counter-electrode of radius rce (Fig. 12a) can be calculated by integrating the infinitesimal resistances of hemispherical shells according to 

For a circular microelectrode the calculation of the resistance between the microelectrode and the counter-electrode is more complicated [48-50], but again a very simple relation, namely 

From Eqs. (31) and (32) it can be calculated that 75% of the total resistance between a hemispherical microelectrode and a counter-electrode comes from a hemisphere with a radius of only 2dme and a semi-ellipsoid of a similar size can be defined for a circular microelectrode [49]. For that reason, the measured resistance reflects the electrical properties of a very small sample region. Microelectrodes can therefore serve as local conductivity probes on inhomogeneous solids as long as conductivity gradients occur on a much larger length scale than dme (Va/a dm J) the conductivity in the 75% hemisphere is almost constant and the local conductivity obtained with circular microelectrodes reads 

The repetition of microelectrode measurements on many different locations yields conductivity maps of the surface-near regions of inhomogeneous solids. Measure- 

For an extended counter-electrode which is far off the hemispherical microelectrode C = 7rr/me bulk (38) 

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