Packed bed models of resistivity for conduction probes

A model for the bulk effective resistivity of a dilute suspension (disperse phase) of noninteracting conducting spheres (not necessarily mono-dispersed) of material resistivity 9id and void fraction ad suspended in a continuous medium of material resistivity 9ic was derived by Maxwell (1954). His result is 

Holm (1967) identifies the contact resistance between particles of clean metal to be the result of current constriction at the point of contact. This geometric constriction together with the volume and surface resistivities integrated over the remaining volume and surface of a particle constitute the total resistance measured between two contacts located at the poles of the particle. In addition, if a thin film exists between the particle contacts, the tunnel effect provides a current independent of the film resistivity. 

By integrating the uniform surface resistivity y over the surface of a spherical particle, Johnson and Melcher (1975) give the total resistance of a single particle of radius r with the small contact cap radius a at opposite poles of the particle through which the current enters and leaves as 

The second equation is the bulk effective resistivity due to particle surface resistivity for a cubic array of mono-dispersed particles with the direction of the electric field aligned with the poles and volume conduction neglected. The constriction resistance is included in the integration of Equations 3.8. These equations are a weak function of the particle geometry. 

Holm (1967) gives the resistance for the volume resistivity of a single particle of radius r measured between opposite pole caps of diameter a and uniform material resistivity 91 as 1 

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