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# Electrical conductivity and microstructure

The dc electrical conductivity of a complicated composite material, like cement paste, depends on microstructure in several ways. In general, for an insulating material containing pores that are filled with a conductor, the bulk conductivity σ, averaged over the sample dimensions, can be written [3] as σ = β φ σo, where φ is the porosity, σo is the conductivity of the material filling the pores, and β is a dimensionless factor summarizing parameters like pore connectivity and tortuosity. The factor of porosity corrects for the cross-sectional area available for flow. In porous sandstone rocks, for example, where much work has been carried out in relating microstructure and electrical conductivity, the dimensionless quantity F = σo / σ, called the formation factor, has been defined [4]. For the relatively low porosities encountered in sandstones, a power law in φ is often found to be a reasonable description of experimental values of F [4]. This general relationship is usually called Archie's law [5].

The dc electrical properties of sandstone rocks can be described electrically by a two-phase composite model, where the sand grains are insulating and the pore fluid, which is usually a brine, is a conductor [6]. There can be complications due to double layers in the clay minerals, which can form interstitial phases between sand grains, but these are usually minor. Cement paste, however, is a more complicated composite conductor because its microstructure and the conductivity of its pore fluid are interrelated and time-dependent. The time-dependent d.c. electrical conductivity of cement paste may be described as σ(t) = Γ(t) σo(t), where the relative conductivity Γ(t) is defined by Γ(t) = σ(t) / σo(t) = β(t) φ(t) . The dependence on time t is explicitly shown. By this choice of variables, we have separated the bulk conductivity into a microstructure-dependent part, Γ, and a pore fluid part, σo. By simultaneously measuring σ and σo, we can monitor microstructure via Γ, and hydration chemistry via σo. Ref. [1] focused on σo, while this present work focuses on Γ.

Atkinson [7] made careful measurements of σ on cement pastes where he tried to replace the pore fluid with a very high molarity electrolyte, so that even if replacement was not totally complete, the electrolyte conductivity would swamp the conductivity of the original pore fluid. This was done so that a known value of σo could be used for the determination of Γ. Atkinson and Nickerson did the electrical work as a test of the Nernst-Einstein relation [2], and found reasonable agreement with the equivalent diffusion measurements. Systematic measurements of Γ throughout the hydration process were not carried out.

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