The agreement between modeling (Fig. 7) and experiment (Fig. 6) is reasonable, considering the complexity of the system under study. It is important to note that the models were not simply equations fit to experimental data, but were derived from fundamental work, independent of the experimental data. In addition to modeling the overall conductivity, the models can also predict the properties of the constituent phases (see below).
The immediate and most important result of both the modeling and experimental data is that the ITZ does not seem to significantly enhance the overall electrical conductivity of the mortar (Figs. 6 and 7). This conclusion is based on a comparison of the normalized conductivities (mortar and paste, σmortar /σpaste ) with the (1−Vf,sand) 3/2 power law line. It would appear that the main effect of adding more sand is simply blocking and re-directing conductive flow, so that any effect of the ITZ is dominated by the effect of adding more aggregate.
The qualitative adherence to the (1−Vf,sand )3/2 law requires additional discussion, however. Both modeling and other experimental work have demonstrated that when aggregate is added to cement paste, several things occur. First, the ITZ forms and increases in volume fraction in proportion to aggregate surface area (assuming a fixed thickness of ITZ). Second, since there is a lower amount of cement and greater amount of water in the ITZ and because the total amount of water and cement in the mortar is conserved, there must be a higher amount of cement and a lower amount of water in the matrix. This reduces the water/cement ratio of the matrix paste. Figure 8 shows the results of this redistribution of cement and water, using the multi-scale model described above. The main utility of this model is the ability to carry out calculations like this. At the maximum amount of sand in the mortar, 50 % by volume, the matrix water/cement ratio is reduced to 0.36 from a nominal value of 0.4, which will have a significant effect on the value of matrix paste conductivity. In concrete, which often has a volume fraction of aggregates of 60 % or more, this effect will be even more pronounced. Therefore, as sand is added, the matrix paste conductivity is affected via the lowering of the water/cement ratio, as shown in Fig. 8. By closely examining the experimental data with the help of the multiscale model, the complex interplay between ITZ and matrix paste can be sorted out.
Figure 8: Effect of volume fraction of aggregate on matrix water/cement ratio.
Some indication that complex interactive processes are taking place with time and with the addition of sand can be seen in the data in Figs. 6 and 7. For example, the normalized mortar conductivity (σmortar/σpaste) starts near the B-H line, (1−Vf,sand)3/2 at one hour, deviates slightly above the line between 24 h and 72 h and approaches the line again subsequent to 336 h, for both the experimental and model data. Although this deviation is subtle, theoretical modeling can be used to provide insight as to what is happening to the paste contained in the ITZ and matrix. To quantify this, a factor, κ will be defined: κ = (σ mortar/σ paste)/(1−Vf,sand )3/2. This is the normalized mortar conductivity divided by the value of the B-H law at a given volume fraction of sand. When κ is equal to one, the normalized mortar conductivity (σmortar/σ paste) lies exactly on the (1−V f,sand)3/2 line (the thick line in Figs. 6 and 7).
As mentioned previously, two things happen when aggregate is added to paste. First, the ITZ forms around the aggregate particles, increasing the local conductivity in the ITZ because of the higher porosity, which tends to increase the overall mortar conductivity. Second, the water/cement ratio of the matrix paste is reduced, which tends to lower the mortar conductivity. Although it is not possible to experimentally separate these two effects, the multi-scale modeling results can be so used.
The κ values for the experimental mortar conductivity results are shown in Fig. 9 as a function of sand content (filled triangles). The open square symbols show the same value of κ for the experimental data, but with the mortar conductivity normalized by the matrix conductivity, as determined by the multi-scale model. As the contrast between the ITZ and matrix conductivities develops and the ITZ volume fraction increases with the sand content, this modified κ value also increases. In contrast, the open circle symbols show how the ratio of the matrix paste conductivity to the nominal paste conductivity decreases when the sand content increases (from the multi-scale model). Recall that the amount of sand governs the volume fraction of ITZ and therefore the reduction of the water/cement ratio of the matrix paste. The higher the ITZ volume fraction, the lower the water/cement ratio of the matrix paste. The net effect is a rough cancellation of positive (ITZ) and negative (matrix) contributions, to give the κ curve (triangles) shown in Fig. 9 (and Fig. 6).
Figure 9: Experimental κ using the nominal paste and bulk matrix paste normalizations, and the ratio of bulk matrix paste to nominal paste conductivity (see text for explanation) plotted versus volume fraction of sand. Note the opposite, but nearly equal conductivity contributions of the ITZ and matrix pastes. Data is for neat cement mortar 72 hours hydration (degree of hydration ≈ 0.63).
The ratio of the conductivity of the ITZ and matrix pastes (σITZ/σmatrix) was predicted using the multi-scale models. 37-39 This ratio, or contrast, is shown in Fig. 10 as a function of degree of hydration for a single volume fraction of sand (Vf,sand = 20 %). This plot shows that there is a marked dependence of this ratio on degree of hydration. There are several important features to note in the plot. First, the initial and final ratios are quite low (σITZ/ σmatrix ≈ 2). However, when the degree of hydration is in the range of 0.5 to 0.8 (equivalent to approximately 72 to 336 h of hydration at room temperature), the ratio shows a maximum value of approximately seven.
Figure 10: Ratio of contrast between ITZ and matrix pastes determined by multi-scale models (volume fraction of sand = 20 %).
This behavior can be understood by examining the conductivity versus porosity curve for cement pastes, shown in Fig. 11. This curve was generated from an equation originally developed by Garboczi et al.,24 has been verified experimentally, and is part of the multi-scale model. 37-39 The shape of the curve reflects the fact that in addition to the porosity decreasing with increasing hydration, there are also significant changes in the connectivity of the capillary pore structure. If it is assumed that the only difference between the ITZ and matrix paste is in initial porosity and that they track at the same rate along the same conductivity versus porosity curve, the general shape of the σITZ/σmatrix curve can be explained. This is demonstrated graphically in Fig. 11 by the three sets of vertical and horizontal lines. The vertical lines differ by constant porosity, and the distances between the horizontal lines represent the corresponding differences in conductivities. Early and late in the hydration, the horizontal lines are spaced relatively closely, while near the middle of hydration, the horizontal lines have a significant gap indicating there is a relatively large difference in conductivity. This interpretation is borne out in both experiment and modeling by plotting (σmortar/σmatrix)/(1−Vf,sand )3/2 versus the degree of hydration in Fig. 12. This term is similar to κ, except that it is normalized by the conductivity of the matrix, rather than of a paste of water/cement = 0.4. This reflects the increased conductivity of the ITZ as well as the reduced conductivity of the matrix as more aggregate is added. In Fig. 12, we can see that both the model and experimental results start at approximately unity, increase to a maximum value, then decrease again to a value close to unity. Recall that a value of one indicates the mortar behaves as if there was no influence of the ITZ.
Figure 11: Electrical conductivity of cement paste versus porosity determined by microstructural model and verified by experiment.
Figures 10 and 12 clearly show that it is only during the middle stages of hydration that the ratio of ITZ to matrix paste conductivity is significant (≈ 7). At a degree of hydration equal to approximately 0.7 (equal to about 7 days of hydration at room temperature), the ratio is much smaller (≈ 2), and the ITZ plays a relatively minor role in the overall mortar conductivity. Previous studies of a mortar assuming fixed conductivities for both matrix and ITZ paste 10, 41 showed that a value of σITZ/σmatrix ≈ 6 was necessary for the increased conductivity of the ITZ to overcome the decreased conductivity due to the insulating aggregate particles. Since the σITZ /σ matrix ratio in the mortar in the present study only temporarily achieves that value and remains well below that value after a few hundred hours, the insulating effect of the aggregate wins out over the increased conductivity of the ITZ. The lowered conductivity of the matrix paste, due to the redistribution of cement and water, also contributes to negating the effect of the ITZ.
Figure 12: Model and experimental (σmortar/σmatrix)/(1−Vf,sand ) 3/2 factor versus degree of hydration for 20 volume % sand mortar.
To overcome this redistribution effect, which was not considered in the previous studies mentioned,10 , 41 it is estimated that a value for the σITZ /σmatrix ratio on the order of 10-20 would be required to overcome the effect of the aggregate particles. Such a high value for the conductivity ratio is not realistic given the conductivity versus porosity dependence in Fig. 11. Since the highest value seen for σITZ /σmatrix is less than 10, it is not likely that the ITZ will ever significantly enhance the conductivity of cement mortar.
It is interesting to compare the present results to some older experimental data by Whittington et al.36 In this very careful paper, the conductivity of cement paste, mortar, and concrete was examined as a function of time and of aggregate volume fraction. The role of ITZ was not considered, and the AC frequency used to remove polarization effects appeared to be fixed, so there is no way of telling if the actual DC bulk resistance was measured or not. Also, only time was used as a variable, not degree of hydration. Keeping in mind these limitations, some comparisons can be made. For concrete, Whittington et al. found that conductivity, normalized by the conductivity of the cement paste from which it was made, approximately followed a power low in the quantity (1−Vf,sand ), but with a power of 1.2 rather than 1.5. They also found fluctuations in this ratio over time, but it is impossible to tell from their graphs whether this variation was similar to what was found in the present work. Aside from the emphasis on the ITZ in the present work, the ideas of Whittington et al. 36 have provided a basis for current thinking about concrete conductivity.