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# Part III Chapter 5. Conductivity

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This chapter covers the computation of conductivity for general models. By "conductivity" is meant electrical conductivity (DC and AC), or thermal conductivity. These computations give quantitative results for microstructure- property relationships. Some of the systems studied are pure models, while for others there is accompanying experimental evidence, sometimes for cement-based systems.

This section discusses modelling of electrical conductivity and fluid flow in two dimensions. The relationship between the electrical conductivity, fluid permeability, and various length scales, including lambda, the Katz-Thompson parameter, and the hydraulic radius, is studied.

This section describes the principles behind the A.C. impedance spectroscopy code used in the work described in Part I Chapter 5. Details of actually using the code are contained in Part II Chapter 2, which contains the manual for all such programs.

This section presents a study of the polarizability (intrinsic conductivity) and the intrinsic viscosity for a very wide range of shapes. It is found that for a very wide range of shapes, the intrinsic conductivity, in the conducting limit, is proportional to the intrinsic vioscosity in the vanishing shear limit.

This section extends the work of the previous section to the polarizability of objects with a general conductivity compared to the matrix. It is shown that a simple Pade approximant, incorporating knowledge of the intrinsic conductivity in various limits, describes well the intrinsic conductivity of the object for any value of its conductivity. The approach is also shown to work for intrinsic viscosity and intrinsic elastic moduli.

This section describes the "two-arc" behavior found in some conductive fiber-reinforced cement-based composites. The behavior found, where the presence of an insulating layer on the fibers causes another arc in the impedance spectra to be seen, is generic to composites with a weakly conducting matrix and non-percolating conductive fibers with a frequency-switchable layer. This means that the fibers act like insulating inclusions at low frequency and highly conductive inclusions at higher frequencies.

The "two-arc" behavior found in some conductive fiber-reinforced cement-based composites is also clearly seen in composites with spherical inclusions. The inclusions are insulating at low frequency and highly conductive at higher frequencies. Different mixing laws are shown to work well at these different frequencies to predict D.C. conductivity as a function of volume fraction of inclusions.

### Go to Part III Chapter 6: Fluid flow

References

(1) N.S. Martys and E.J. Garboczi, Physical Review B 46, 6080-6090 (1992).
(2) R.T. Coverdale, E.J. Garboczi, and H.M. Jennings, Computational Materials Science 3, 465-474 (1995).
(3) J.F. Douglas and E.J. Garboczi, Advances in Chemical Physics 91, 85-153 (1995).
(4) E.J. Garboczi and J.F. Douglas, Physical Review E 53, 6169-6180 (1996).
(5) J.M. Torrents, T.O. Mason, A. Peled, S.P. Shah, Journal of Materials Science, 36 (16), 4003-4012 (2001).
(6) M.A. Campo, L.Y. Woo, T.O. Mason and E.J. Garboczi, Journal of Electroceramics, 9 (1), 49-56 (2002).