Previous: Description of the
Figure 7 shows a comparison of the normalized conductivity data presented in Figure 3 with results from a number of 1003 models. The models include several different water to cement ratios and degrees of reaction. The curve shown is actually a fit to many different degrees of hydration and w/c ratios, and has been shown to describe well the simulation results using the given choice of parameters . Although the model overpredicts the conductivity at lower volume fractions of porosity, it shows similar trends to experimental values, in agreement with earlier simulations . The important point is that in the model, the capillary porosity has not lost long-range connectivity in any of the microstructures until below about 20%. Therefore the transition from parallel to series behavior seen in the experimental results is obtainable in a tortuous, yet connected, conductive phase. The results from the model can be adjusted to better match the experimental curves in Figure 7 by decreasing the (unknown) conductivity of the C-S-H phase with respect to that of the (known) pore fluid conductivity. However, it is possible that the overprediction of conductivity is at least partially an artifact of the finite resolution of the digital- image model. Since the distribution of porosity in the model is limited to quantized units of 1 cubic micrometer, the same amount of porosity in the model will be distributed in fewer, larger pores than probably exist in cement paste. It is known that capillary pores can be smaller than 1 micrometer . Simple models have shown this latter type of distribution to have a somewhat higher conductivity at the same total porosity .
Figure 7: Comparison of normalized conductivity for different w/c ratios
experiment (0.35, 0.5) and simulation (various).
Even though the model parameters listed in columns 2 and 3 of Table 1 provide a good fit to the normalized zero-frequency conductivity data shown in Figure 7, they must be modified slightly to obtain a better rounded fit to the impedance response at all frequencies. To achieve the best overall fit with experimental results, systems of similar water to cement ratios and degrees of reaction were compared. Since model and experimental geometries are different, the following three aspects of the IS curves were used to compare and modify the model values: normalized conductivity; dielectric constant as measured by EQUIVCRT , a commercial impedance spectra analysis software package; and the arc peak frequency, which is the frequency at which the imaginary part of the impedance takes its maximum value. Each of these parameters is sample-geometry independent .
Figure 8b shows a simulated curve from a 1003 pixel model whose parameters have been modified to match closely the accompanying experimental curve in Fig. 8a. The simulated spectrum goes to the origin at high frequencies as predicted, and the shape of the arc at high frequencies (left side of the curve) shows the presence of another arc, which, however, is not well-separated from the main arc for this fairly high degree of reaction paste. The model parameters used in Figure 8 are given in columns 4 and 5 of Table 1. Fig. 8b does not show an electrode arc. It is possible to simulate such an arc, by placing an RC element between the simulated electrode and microstructure , but the data that make up the electrode arc are not relevant to the response of the microstructure. To make certain of this, a few cases were run with and without the electrode arc. It was found that omission of the electrode arc had a negligible effect on the response of the microstructure and the computed bulk resistance. This result was also found previously in 2-D models . Electrode arcs were thus not usually simulated, in order to conserve limited supercomputer resources.
Figure 8a: Impedance arc from experiment.
Figure 8b: Impedance arc from simulation.
Figure 8b: Impedance arc from simulation.
The impedance curves displayed in Figures 9 (a) and (b) were generated by the model and represent the evolution of a single 1003 system as the hydration reaction proceeds. The phase parameters used for these simulations are given in the fourth and fifth columns of Table 1. The shape of Figure 9 (b) exhibits the offset resistance that would be observed if the right hand side of the arc were to be fit with an equivalent circuit at experimentally observable frequencies. The simulated spectra, as in Fig. 8b, also go to the origin at high frequency. At smaller degrees of hydration (earlier hydration times), the two-arc behavior is clearly displayed. As the amount of pore fluid is decreased, the higher frequency arc diminishes, leading to the identification of the low frequency arc as being due primarily due to C-S-H, and the diminishing higher frequency arc being due primarily to the remaining pore fluid. The model indicates that there is then no true offset resistance, but rather a second arc corresponding to the pore fluid phase. As the pore fluid is replaced with more resistive reaction products, the bulk resistance increases, and the pore fluid arc is less noticeable. Since it is known that the pore fluid is percolated in the first three of the four microstructures analyzed in Fig. 9 ( = 0.229, 0.314, and 0.505), it is clear that the observed behavior in the impedance curves is a result of tortuous, but connected, pore fluid phase that is intertwined with continuous conduction paths of C-S-H.
Figure 9a: Simulated impedance curves for a single 1003 system with a water to cement ratio of 0.4; = 0.229, 0.314.
Figure 9b: Simulated impedance curves for a single 1003 system with a water to cement ratio of 0.4; = 0.505, 0.650. Note change of scale between (a) and (b).
In general, the behavior of the simulated IS curves is the result of a composite response of the microstructure, which depends only on the properties of the individual phases and their arrangement within the microstructure. The response is not dependent on any special interaction or interfaces between the phases, because reasonable agreement between experiment and simulation was obtained without any such interactions or interfaces being simulated. The origin of the bulk arc has been attributed by other researchers [12,28] to electrical double layers that form on surfaces that develop in the microstructure of cement paste. This hypothesis has significant ramifications with respect to the effective dielectric constant of the bulk arc, and will be discussed further in Part II  of this paper.