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## Periodic model

We first present the results for the bcc periodic model described in Section 2.2. Figure 7 shows the normalized conductivity of the composite, σ / σp , as a function of σs / σp. The interfacial zone cement paste percolates in this model, and so plays a strong role in the overall conductivity, as can be seen from the graph. Three

Figure 7: Composite conductivity for the bcc model is plotted vs. the interfacial zone conductivity. [Both are normalized by bulk paste conductivity.] The solid dots are the numerical data, the solid line is the Pade approximant explained in the text.

points are worthy of note. First, the conductivity at σs / σp = 1 is that which would be obtained if the interfacial zone cement paste had the same porosity and therefore the same conductivity as the bulk cement paste. The presence of the insulating sand grains in this case reduces the overall normalized conductivity from 1 to 0.35. This is consistent with a 3/2 power law found in suspensions of spheres. In this case, (0.46)3/2 = 0.31 [24,25,34]. Second, as we noted in connection with Fig. 6, the composite conductivity can be viewed as the result of a competition between the insulating sand grains and the interfacial shells. Figure 7 shows that when σs / σp ≈ 6, the composite conductivity first achieves a value equal to the matrix conductivity, so for this microstructure, this value of σs / σp causes the greater interfacial zone conductivity to cancel out the effect of the insulating sand grains. Third, the Pade approximant provides an excellent fit to the computed data points in Fig. 7, so that this analytical curve could be used to accurately predict the composite conductivity at other values of σs / σp that were not numerically computed. [Here the four parameters F, Λ, f, and λ were computed directly from the finite element solutions for two limiting cases, σs = σp and σp = 0].

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