Next: Three Dimensional Model
Figure 1 shows a plot of keff versus time for a 0.4 w/c cement paste. The dashed line represents data obtained from impedance arcs that were very difficult to fit since they were barely measurable. The dashed line data are also probably not reliable because they were obtained at high frequencies (greater than 3 MHz) at which induction effects occur due to the experimental apparatus. The solid line is computed from data obtained from larger bulk arcs occurring at lower, more reliable frequencies. The effective dielectric constant is seen to decrease from a very large value near 90,000 and eventually level off at a value near 4000, at times longer than 120 hours.
Figure 1: Experimental results for change in the effective dielectric constant, keff, as a function of reaction time for a 0.4 water:cement ratio cement paste.
Figure 2 shows a plot of keff versus water:cement ratio for a single portland cement after long reaction times (11 months). It can be seen that the effective dielectric constant increases with water to cement ratio, and also that the values are still large, ranging from 4000 to above 12000 for the highest w/c pastes.
Figure 2: Experimental results for the effective dielectric
constant, keff of cement pastes hydrated for
11 months versus the water:cement ratio.
Figure 2: Experimental results for the effective dielectric constant, keff of cement pastes hydrated for 11 months versus the water:cement ratio.
These high values are difficult to explain, especially since the material phase in the microstructure that has the highest dielectric constant would have thought to have been pore fluid, with a dielectric constant near 80 . However, separate dielectric measurements of synthetic calcium silicate hydrate (C-S-H) and of pastes containing mostly C-S-H, indicate that C-S-H has an effective dielectric constant significantly higher than pore fluid, perhaps on the order of εC-S-H = 1000 . This does not, however, explain the observed phenomena, for two reasons.
First, this value of εC-S-H is not high enough to explain the effective dielectric constant of 90,000 for pastes at very early reaction times when only a small percentage of C-S-H is present. And second, if C-S-H were responsible for the high dielectric constant of the paste, the value for the paste should increase, not decrease, with reaction time, since more C-S-H is being produced, replacing the much lower dielectric constant water and cement.
The dielectric response of cement paste has been attributed [9,15] to electrical double layers that form at the interfaces between pore fluid and product and/or reactant surfaces. Calculations by Xie et. al.  indicate that a double layer (Stern layer) in the cement system should have a dielectric constant of 17.4, which is significantly less than observed values of bulk cement paste. However, double layers are certainly present on cement particles after mixing with water, yet the bulk arc does not appear until after a significant amount of reaction product has formed (about 12 hours). Possibly, as the hydration reaction proceeds, the surfaces available to form double layers increase dramatically, which could explain the delayed appearance of the bulk arc. But this does not explain why the dielectric constant of bulk paste decreases as hydration proceeds, a process that is producing more and more free surface.
An alternative explanation of the dielectric response of cement paste is that the arrangement of phases within the microstructure provides an amplification mechanism that gives rise to the high values of keff observed. We define "amplification mechanism" as a geometrical arrangement of material phases that results in an effective dielectric constant that is much higher than the value for any individual material phase. As an example, we note that this phenomenon is observed in composite materials composed of an insulating and a conducting phase, where the conducting phase is very close to, but just below, its percolation threshold [16,17,18]. The complex geometry that gives rise to this amplification mechanism is "the existence of many almost pure conducting channels which stretch across the entire length of the system and are blocked off only by very thin barriers. Every channel of this type contributes an abnormally large capacitance, and all of these are connected in parallel" .
A simpler geometry that also gives rise to dielectric amplification by forcing current across thin layers of insulating material is shown in Figure 3 (after Feynman et. al. ). This figure shows a two dimensional schematic of a three dimensional perfect parallel plate capacitor into which a conductive plate has been placed. The overall capacitance is affected by the thickness of the inserted plate as follows:
|C||=||keff ε0 A/D = ks ε0 A/(D> − b)|
|=||ksε0 A/d = ks ε0 (A/D)(D/d)||(1)|
from which it follows that:
|keff = ksD/d||(2)|
Figure 3: Schematic diagram of a parallel plate capacitor (after Feynman et al. ) with a crossbar thicknes, t of 1 pixel.Here keff is the true composite effective dielectric constant, ks is the dielectric constant of the material that is in the space d; εo is the permittivity of free space; A is the area of the electrodes; and D, d, and b are as shown in the figure. It is clear that as b becomes larger, d becomes smaller, and the capacitance increases. This phenomenon is observed in grain boundary ceramics in which conductive grains are isolated by insulating grain boundaries . The composite dielectric constant is proportional to D/d, where D is the grain size, and d is the grain boundary thickness. The larger the grain size (conductor) and the thinner the grain boundary (insulator), the greater the amplification. In these materials, the high conductivity material phase is disconnected.
A similar mechanism may be responsible for the observed dielectric response in cement pastes. As the microstructure develops, relatively large (~ 20 µm) capillary pores, filled with highly conductive pore fluid, are reduced in size as reaction products fill space. Following the model shown in Figure 3, the conductive pores would correspond to the conductive plate sandwiched between the much more insulating layers of reaction product. As hydration proceeds, capillary pore size D tends to decrease, and the reaction product thickness d tends to increase. Thus, the overall capacitance, starting out high, would then be systematically reduced as the hydration reaction proceeded. However, the isolation, or disconnection, of the conductive pore phase, as pictured in Fig. 3 and found in grain boundary ceramics , is not necessary to produce the observed phenomena. We next use computer simulation modelling to explore the possibility that highly tortuous pathways of continuous conductive pore fluid channel the current into regions where it is forced to drop across thin layers of insulating material, thus producing a high value of keff.