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Disordered Models

In Fig. 8 we present the random walk simulation results for the disordered four grain size model together with the predictions of the SC-EMT and the D-EMT. We note that, as in Fig. 7, the overall shape of the curves is concave down. The simulation data could at most form a straight line. This would be the case if the two conducting phases, interfacial zone and bulk cement paste, were exactly in parallel. Then the overall conductivity would be given by a simple linear combination of the two phase conductivities, and as σ_s increased, the overall conductivity would increase linearly. Since the microstructure is such that these two cement paste phases are not in parallel, the curve must be sub-linear, or concave down. As σ_s / σ_p → ∞, the curve will eventually become straight as predicted by Eq. (18).

Figure 8: Composite conductivity for the random model is plotted vs. the interfacial zone conductivity. [Both are normalized by bulk paste conductivity.] The solid dots are the random walk data; also shown are the SC and D-EMT results.

The data in Fig. 8 indicate that to achieve an overall conductivity that is equal to the bulk cement paste conductivity, the value of σ_s / σ_p must be equal to approximately 8. This is higher than the corresponding value obtained in the bcc model, due to the greater tortuosity and smaller volume fraction of the interfacial zone in the present case. Increasing σ_s / σ_p has therefore a somewhat smaller effect on the overall conductivity. At σ_s / σ_p = 20, σ / σ_p ≈ 1.8, which is significantly less than the corresponding bcc value for the same reasons. We have attempted to fit the simulation data shown on Fig. 8 with a Pade approximant (not shown) similar to that in Fig. 7. The fit is less satisfactory than in the bcc case but is no worse than that offered by the two EMT curves. Again, this is an indication of the greater complexity of the random four grain size model.

Consider next the behavior of the conductivity as a function of the sand volume fraction [Fig. 9]. In Fig. 8 we saw that, at a sand volume fraction of 55%, a value of σ_s / σ_p ≈ 8 was required to make the composite conductivity equal to the bulk cement paste conductivity. This is remarkably close to the dilute limit result σ_s / σ_p ≈ 8.26 found in Fig. 6 and is in agreement with the EMT predictions discussed earlier. Viewing σ / σ_p as a function of sand concentration, these results imply that for σ_s / σ_p 8.26, σ / σ_p must start out with negative slope and remain less than unity. By contrast, for σ_s / σ_p 8.26, the curve starts out with positive slope and is always greater than unity. Because the dilute limit defines the essential features of the overall curve, the SC-EMT and D-EMT should correctly predict the essential structure of the composite conductivity. This is clearly evident in Fig. 9.

Figure 9: Composite conductivities (calculated by random walk simulations) for the random model are shown as a function of sand concentration for four values of the interfacial zone conductivity. Also shown are the predictions of the SC and D-EMT calculations. [Normalization is as in Figs. 7 and 8].

Figure 9 shows computed conductivity data for the random mortar model as a function of sand volume fraction, for σ_s / σ_p = 100, 20, 5, and 1. The sand size distribution was preserved at every volume fraction. The curves for σ_s / σ_p = 100 and 20 are concave up, with the σ_s / σ_p = 100 results showing clear evidence of the interfacial zone percolation threshold. The σ_s / σ_p = 5 and 1 curves have negative initial slopes, and remain below one, as expected. The σ_s / σ_p = 1 curve roughly follows a 3/2 power law in the total cement paste volume fraction, as would be expected since there is no difference between interfacial zone and bulk cement paste in this case.

Two features of the SC-EMT and D-EMT predictions are worth noting. First, for σ_s / σ_p 5, the SC-EMT is always above the D-EMT prediction, while for σ_s / σ_p 5, this situation is reversed. For very large values of the interfacial shell zone conductivity, the SC-EMT is clearly much above the data while the D-EMT is clearly well below the data. This is due to the fact that the SC-EMT predicts the interfacial zone percolation point at a lower volume fraction than is realistic, while the D-EMT predicts too high of a sand volume fraction at percolation. Second, there are several ways to take the limit of EMT when σ_s → σ_p. One can take h → 0, or take σ_s → σ_p; these limits give different results. For example, the D-EMT gives the well-known 1.5 power law when h → 0, but does not when σ_s → σ_p with h 0. Since the data clearly follows this power law in this limit, we employ the h → 0, limit for both the SC and D-EMT to describe the σ_s = σ_p system.