Next: Periodic and random
For the purposes of electrical conduction, the aggregate grains are simply inert obstacles to the flow of current. The basic model is then defined by two parameters: 1) the structure of the interfacial layer, and 2) the electrical contrast between this layer and the bulk cement paste. Clearly, both parameters will depend on the water:cement ratio as well as other properties of the bulk paste.
It is well-known that the aggregate-cement paste interfacial zone consists of a region of up to approximately 50 µm thick around each aggregate grain. Within this zone there is higher porosity, larger pores, and a higher volume fraction of calcium hydroxide [19,20]. This is the case for aggregate that is much less porous than the cement paste, like quartz or granite. In the case of lightweight aggregate, like a porous limestone, or aggregate that can react with water, like cement clinker aggregate, this situation can be reversed, with the interfacial zone actually denser than the bulk cement paste [21,26]. This paper concentrates on the case where the interfacial zone is less dense than the bulk cement paste, although the same models can also handle the case of denser interfacial zones.
In porous materials in general and cement paste in particular, the conductivity increases roughly as a power of the porosity. This behavior occurs when the pore space is filled with a conductive material, which is pore fluid in cement paste. The exponent in this power law generally lies between 1 and 3 and is, in many cases, quite close to the value 2 .
Figure 3 shows schematically how the capillary porosity does, and therefore how the conductivity might, vary across the interfacial zone, by assuming that the conductivity is proportional to the square of the porosity. In this paper we replace the variable conductivity interfacial zone region with a shell of fixed width and constant conductivity, for the sake of simplicity. There is more than one way to carry this out. One way is to average the conductivity by integrating across the interfacial zone, and then assume a fixed width shell around each aggregate grain in which the conductivity σs is higher than that of the bulk cement paste and is defined by:
where h is the assumed thickness of the interfacial zone in which the extra conductivity is concentrated and σ(x) is the actual conductivity as a function of x, the distance from the aggregate surface, as schematically shown in Fig. 3. All the other cement paste outside of the shell of thickness h is given the bulk cement paste value of conductivity, σp. For the schematic data shown in Fig. 3, the ratio σ(x = 0) / σp is about 14. Using h = 10,20, and 30 µm, in combination with Eq. (1), the value of σs / σp is calculated to be, respectively, 8, 5, and 4. This procedure has the advantage of forcing the chosen interfacial zone conductivity to be smaller as the width of the zone is made larger, so that the current around each grain should remain approximately fixed, and not depend sensitively on the assumed value of h. Other ways of computing this average are discussed in Ref. .
Figure 3: Schematic representation of how the capillary porosity and conductivity may vary across the interfacial zone. The horizontal dotted lines show the bulk cement paste values for both quantities.
Several authors have carried out experiments with planar or cylindrical aggregate shapes, so that the bulk cement paste and interfacial zone electrical flow paths are both one dimensional and in parallel. In this simple geometry, separate measurements of the bulk cement paste conductivity and the composite conductivity can be combined to extract a value for σs / σp , if a value for h is assumed. A range of values have been found, from about 10 , assuming h = 20 µm, to 12-15 , assuming h = 100 µm. This latter value for h seems much too large, since SEM investigations of the interfacial zone generally find that the porosity of the interfacial zone decreases to the bulk cement paste value by a distance of 30-50 µm from the aggregate grain surface [19,20].
Assuming a fixed thickness for the interfacial zone region, it is possible to compute separate values of porosity for this effective interfacial zone and for the bulk cement paste. Bourdette et. al.  have done this for two different mortars, 0.4 w/c and 0.5 w/c, with slightly varying size distributions and amounts of sand, both between 50 and 60 values of the interfacial zone (bulk cement paste) porosity, ranging from 46% (22%) to 30% (21%), for an assumed value of h=30 µm. Using the conductivity vs. capillary porosity curve for cement paste established in earlier computational work , these pairs of porosities correspond to ratios of conductivities ranging from σs / σp = 6-15, in reasonable agreement with the experimental values, and with the simple schematic data shown in Fig. 3. We note that the theoretical relation for bulk cement paste conductivity derived earlier  is, for capillary porosities larger than 18%, essentially quadratic in the capillary porosity, in agreement with the general ideas discussed above and displayed in Fig. 3.
Since there is no definitive value established experimentally for the value of σs / σp, we have chosen to allow this parameter to vary in the following computations, and studied the dependence of the composite conductivity, for a given sand content, on the value of σs / σp . However, based on the mercury intrusion and modelling results of Ref. , we have chosen h = 20 µm as the best value for the width of the interfacial zone. We emphasize that extracting the value of σs / σp from experiments will require an assumption for the value of h. That is another reason to investigate the composite conductivity as a function of a variable interfacial zone conductivity.
While the model we pursue in the remainder of this paper is highly simplified, we emphasize that its essential features could be specified in detail if more experimental data on the structure of mortars were available. In particular, we feel that nuclear magnetic resonance (NMR) studies would be of great value in determining the model's parameters. NMR may be particularly useful because the larger pores in the interfacial zone could be seen in relaxation studies as an independent contribution to the pore size distribution [30,31]. By contrast, in mercury intrusion the interfacial zone pores are detected at their correct size only if this zone percolates through the material. Before they are percolated, they will be classified as smaller bulk cement paste pores . Such experiments could also fix the relative weights of the interfacial zone and bulk cement paste porosities, thereby constraining the thickness of the layer and thus the electrical contrast. In principle, similar information is available directly from microscopy, but NMR has the advantage of being a non-destructive, non-invasive measurement.