The experimental portion of this study consisted of measuring the electrical properties of mortars with impedance spectroscopy (IS), an in-situ, non-destructive technique.30, 31 IS separates the bulk and electrode responses of the material being measured, and uses low voltages (≈ 0.1 V/cm), so that the samples are not disrupted by the high fields sometimes necessary in DC measurements. 32 Although both capacitive and resistive information is measured with IS, only the DC resistance was used in this study. Measuring the AC response is still necessary, however, in order to be able to separate the electrode and bulk responses and get the true bulk conductivity. For the IS measurements, a Hewlett-Packard 4192A frequency response analyzer (FRA)* was used in the range of 5 Hz to 13 MHz, with 20 data points collected per frequency decade. The impedance spectra were corrected for parasitic lead effects.33 The DC resistance of each sample was taken at the real impedance "cusp" between the bulk and electrode arcs.
Mortar samples were made from a type I OPC (LaFarge) with a water/cement ratio of 0.40 and a silica sand conforming to ASTM C778. Sand volume fraction (Vf,sand) varied from 0.0 to 0.5, and conductivity was measured from time of mixing to 772 h of hydration time. The dry components were mixed in a Hobart planetary mixer for one minute, after which water was added and the slurry was mixed for an additional 20 minutes (no superplasticizer was added). After mixing, the samples were cast into molds (25 mm x 25 mm x100 mm) and vibrated under vacuum to reduce the volume of entrapped air. Flat stainless steel electrodes (19 mm x 40 mm, 19 mm x 25 mm embedded area) were embedded near the ends of the samples, with an inter-electrode distance of 85 mm. No bleeding was observed in any of the samples. The specimens were stored and measured in a 100 % relative humidity chamber at room temperature (see Fig. 4). Temperature increases during the early stages of hydration, although not quantified, were not significant after 24 h, when our first impedance measurements were made. To measure the degree of hydration, additional samples were prepared for loss-on-ignition (LOI) measurements. The samples were crushed, dried at 105 ºC, weighed, ignited at 1050 ºC for 90 minutes, and weighed again. LOI was calculated by dividing the mass loss by the ignited mass. The uncertainty in LOI measurements was estimated to be 0.01 g; the scatter at long hydration times is attributable to inadvertent differences in drying. The results indicate that the mortars and pastes hydrated at the same rate (see Fig. 5).
Figure 4: Schematic of experimental set-up indicating sample dimensions and electrode location.
Figure 5: Loss on ignition (g/g) versus hydration time for various mortars and paste used in this study. Mortars are labeled by their sand volume fraction.
The measured mortar conductivities (σ mortar ) were normalized by the of conductivity of a paste (σ paste ) with the same degree of hydration and water/cement ratio. (It is important to distinguish between the definition of paste and matrix paste. Matrix paste is one of three phases found in mortars, and resides between the aggregate particles (and ITZ paste see Fig. 2). "Paste," on the other hand, is defined as the single phase found in neat cement paste.) The raw conductivities for the mortars and paste at various ages are given in Table I and shown graphically in Fig. 6. Based upon error estimates for cross-sectional area and inter-electrode spacing, the uncertainty in measured conductivity is estimated at 5 %. Because the mortars and pastes hydrate at the same rate, it is sufficient to normalize by pastes of the same age (see Fig. 5). In Fig. 6, the normalized values (σ mortar /σ paste ), at a given degree of hydration, were plotted as a function of the volume fraction of sand for several degrees of hydration (hydration times), and were compared to the line of (1-Vf,sand) 3/2 , which is the well-established Bruggeman-Hanai (B-H) law for the case of spherical, non-conductive particles in a conductive matrix. 34 This model assumes that the conductivity of the matrix is constant as non-conductive particles are added, so that the effect of the particles is to dilute the matrix and re-direct conduction around themselves, making the conduction paths more tortuous. This by itself is a complicated many-body process. In the dilute limit of a low volume fraction of particles, it is known that the B-H law is exact. For general volume fractions of particles, this model has also been found to be accurate.35 The derivation of this equation also seems to follow concrete microstructure more closely than other approximations, hence is to be preferred for this reason as well. 34, 36 Also, there is no percolation threshold for the matrix or inclusions in this equation. The matrix is always connected, and the inclusions are always disconnected, as is the case in real concrete. Some of the other effective medium equations have different percolation thresholds. Therefore, if the matrix paste conductivity stays constant as more sand is added, and the effect of the ITZ on overall conductivity is negligible, then the experimental data should follow the model line. The data will go below the line if the ITZ regions have a negative effect on conductivity, and above the line if the ITZ regions have a positive effect on conductivity. If the presence of the ITZ regions also has an effect on the matrix, as was implied by Fig. 1, then a more careful analysis of the effect of the ITZ regions on mortar conductivity must be made, using a multi-scale model, as is described next.
Figure 6: Experimental mortar conductivities normalized by conductivity of paste (water/cement ratio = 0.4) versus volume fraction of sand.
Table I: Paste and mortar conductivities (Ohm−cm) −1 measured by impedance spectroscopy at various times and volume fractions of sand.