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Figure 6 shows a schematic of a model microstructure referred to here as the "I" model because the C-S-H phase forms a capital "I". The model represents an area between two reacting cement grains that are coated with C-S-H. Just enough product has formed to connect the two grains by a very thin layer of C-S-H. Continued reaction is simulated by thickening the connecting crossbar. The impedance is computed by applying a voltage in the direction shown on the figure. Figure 7 shows the impedance curves for different thickness, t, values. Although it is overshadowed by the low frequency arc, a second, higher frequency arc does exist for each thickness, implying that this microstructural arrangement has a series character [10]. The blow-up of the region (0-5, 0-5) illustrates this second arc behavior more clearly. Indeed the middle section of the I model is a series arrangement of C-S-H and pore fluid, although the upper and lower sections are arranged in parallel. As the thickness of the crossbar increases from one to six pixels, the composite resistance also increases as expected. The effective dielectric constant of the larger, lower frequency arc decreases with increasing t as shown in Figure 8. As the thickness increases from 1 to 2, the amplification ratio, D/d, of the middle section is reduced from 20/1 to 20/2, and k_{eff} also drops by a factor of two. However, the value of k_{eff} of the composite when t = 1 is not 20 x ε_{C-S-H} = 20,000, but closer to 10 x ε_{C-S-H} = 10,000, since the middle section, from which comes the dielectric amplification, represents only 50% of the volume (area) of the composite. The value of k_{eff} in Fig. 8 roughly follows a 1/t power law for small t, as would be expected from the above analysis.
Figure 6: Schematic of the I model, shown with a crossbar thickness, t, of two pixels.
Analyzing the current distribution in the model is useful to show the effect of microstructural geometry on the impedance properties. Figure 9 shows a map of DC current intensity for the I model when t = 1. Figure 9 (a) represents the microstructure, where the blue phase is pore fluid and the red phase is C-S-H. Fig. 9 (b) shows a gray scale image of the current intensity at each pixel where white = high current and black = low current, with various scales of gray for intermediate current values. It is apparent that some of the current is going around the crossbar and through some of the "bulk" C-S-H.
Figure 9a: I model with a crossbar thickness of 1. Blue=pore fluid, red=C-S-H.
Figure 9b: Corresponding current intensity plot of system in Fig. 9a, where white represents the highest value of current.
Figures 10 (a) and (b) show the effective conductivity (σ_{eff}) distribution functions for each phase of the I model (t = 1). The effective conductivity for a given pixel is calculated by dividing the actual local current by the macroscopic field that has been applied to the total sample. The difference between the effective conductivity and the actual material conductivity for a given pixel is then a measure of the influence of the pixel's local environment on the current going through it. The effective conductivity distribution function is then just a histogram showing what volume fraction of material has a given effective conductivity. More formally, the total average conductivity σ_{comp} of a multi-phase composite is defined by [12]:
which can then be rewritten for a two-phase composite as
resulting in the equation
where σ_{eff}(1) and σ_{eff}(2) and are the average value of the effective conductivity for phases 1 and 2, respectively, V_{1} = c_{1} V, V_{2} = c_{2} V are the partial volumes of each phase, V_{1} + V_{2} = V is the total sample volume, and ε_{o} is the original applied field. The values of the average effective conductivity for each phase are shown in Table 2 for each crossbar thickness. For each thickness in the I model, the effective conductivity of the C-S-H phase is increased above its assigned value of 0.01 (Fig. 10 (a)), and the effective conductivity of the pore phase is decreased significantly from its assigned value of 1 (Fig. 10 (b)). The composite resistance, R is equal to the inverse of σ_{eff} for this square, two dimensional geometry, and is also listed in Table 2.
t (pixels) |
Effective conductivity (C-S-H) |
Effective conductivity (pore fluid) |
V_{f} (C-S-H) |
V_{f} (pore fluid) |
Composite conductivity | R |
1 | 0.01737 | 0.18539 | 0.525 | 0.475 | 0.097 | 10.290 |
2 | 0.01729 | 0.10952 | 0.550 | 0.450 | 0.059 | 17.009 |
3 | 0.01680 | 0.07951 | 0.575 | 0.425 | 0.043 | 23.014 |
4 | 0.01625 | 0.06297 | 0.600 | 0.400 | 0.035 | 28.622 |
5 | 0.01569 | 0.05233 | 0.625 | 0.375 | 0.029 | 33.979 |
6 | 0.01514 | 0.04478 | 0.650 | 0.350 | 0.026 | 39.194 |
Table 2: Effective conductivity data for the I model. σ_{eff} is the effective conductivity for a given phase, as defined in eqs. (3)-(5), V_{f} is the volume fraction of a phase, σ_{comp} is the composite DC conductivity, and R = 1/σ_{comp} = the bulk resistance. |
For the I model, we have found that the frequency associated with the peak of the low frequency arc (ω_{o}) is the same for each thickness value. This implies that k_{eff} is a function of R via the following equation which defines k_{eff} :
ω_{o} = σ / (k_{eff} ε_{o}) | (6) |
where σ is the conductivity of the low frequency arc. As t increases, both σ and k_{eff} must decrease by the same factor in order to leave the value of ω_{o} unchanged.