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A model that relates the three-dimensional microstructure of cement paste to impedance has already been described in Part I and elsewhere [10,20,21]. The electrical parameters assigned to each phase are given in Table 1. The simululated impedance data from the model can be analyzed in the same way as the experimental data was, and the value of k_{eff} determined. A plot of k_{eff} versus reaction time* [22] for a simulated 0.4 w/c cement paste system is shown in Figure 4, analogous to the plot in Figure 1. Three of the five values displayed are at least a factor of two above 500, the highest value of k assigned to any phase (C-S-H) in the model. Therefore, an amplification mechanism must be present.
Phase | Conductivity (S/m) | k_{r} |
Pore fluid | 1.0 | 80 |
C-S-H | 0.01 | 500 |
C_{3}S | 3.6 x 10^{−8} | 1 |
CH | 3.6 x 10^{−8} | 1 |
Table 1: Phase parameters used in three dimensional impedance simulations. The DC conductivity is σ, and the relative dielectric constant is k_{r}. |
Figure 4: 3-D model results for the change in the effective dielectric constant as a function of reaction time for a 0.4 water:cement ratio model system. Reaction time is calculated from degree of reaction and published kinetic relationships [22].
It is crucial to note that the first three data points in Figure 4, corresponding to the shortest reaction times, come from model systems in which the conductive pore phase is known directly from the model to be still percolated in the direction of the applied field [23]. Thus, isolation of the conductive phase (as in Figure 3) is not required to produce amplification of the low frequency effective dielectric constant.
The model is not, however, capable of reproducing the full range of amplification that is observed experimentally, probably due to the limited resolution of its digital image-based format. The smallest unit of material that may be represented is 1 pixel, and therefore to achieve an amplification of 100x would roughly require a pore diameter of 100 pixels (D=100, d = 1 in equation (2)). This degree of resolution is not currently possible.
In Figure 5, a plot of k_{eff} versus water:cement ratio for model systems displays a trend similar to that observed in Figure 2. All three data points in Figure 5 are for models in which approximately the same fraction of cement initially present had been reacted, about 0.73, but with different initial w/c values. Therefore, the volume fractions of pore fluid were different, increasing with increased w/c, whereas the volume fraction of C-S-H decreased with higher w/c. One might then expect the value of k_{eff} to be lower for higher w/c, because a smaller fraction of the high dielectric constant phase (C-S-H) was present. Since this is clearly not the case, again dielectric amplification by the microstructure must be taking place. The greater amplification corresponding to the higher water:cement ratios may be correlated with a higher value for D (equation (2)), rather than for d, since the latter is likely to be independent of water:cement ratio, especially for older pastes. In other words, d is related to the amount of product that has formed, while D is dependent on the amount of pore fluid (capillary porosity) that is present at any given time.
The three dimensional computer models have thus shown that the high value of k_{eff} of the bulk arc of cement paste may be the result of microstructural amplification. The same trends were observed with respect to reaction time and water:cement ratio as seen experimentally. It was also shown that disconnection of the conductive phase (pore fluid) is not necessary to produce this amplification.
Reaction time is calculated from degree of reaction and published kinetic relationships [22].