Next: Description of the
The conductivity of cement paste may be described by a simple effective medium equation :
|σ = σ0 β Vf,||(1)|
where σ is the composite conductivity, σo is the conductivity of the capillary porosity, by far the most conductive phase in cement paste, Vf is the volume fraction of the capillary porosity, and β is a microstructural parameter related mainly to the tortuosity and connectivity of the capillary porosity, and to a lesser extent on the tortuosity and connectivity of the C-S-H phase. Normalizing the composite conductivity σ by σo, leaves βVf, which is a measure of the influence of the microstructure on the composite conductivity. The parameter β is an empirical constant related to the arrangement and relative amounts and conductivities of the conductive phases, and can only be analytically calculated for simple microstructures. For example, Figure 2 shows a plot of normalized conductivity, σ / σo, versus volume fraction of the less conductive phase, for two simple two-phase microstructural arrangements. In these examples, the conductivity of the two phases differ by factor of 10. As can be seen in the figure, β for the parallel case is 1, while for a series arrangement, β drops to a low value as soon as the lower conductive phase is present. It is also important to note that for any two conducting-phase material, the conductivity of the parallel and series arrangements are strict upper and lower bounds for the true composite conductivity .
Figure 2: Showing the composite conductivity of parallel (dashed line) and series (solid line) composite microstructures. The conductivity has been normalized by the conductivity of the more conducting phase. The two phase conductivities differ by a factor of 10. The x-axis is the volume fraction of the lower conductivity phase.
Figure 3 is a semi-log plot of the parallel and series bounds for conductivities differing by a factor of 1000, along with conductivities for two cement pastes with different water to cement ratios and various degrees of reaction, normalized by their pore fluid conductivities. If the amount of "parallel" or "series" character of the material is defined by where a conductivity data point lies between the parallel and series bounds, then it appears that the cement paste microstructure is making a transition from having a mostly parallel character at high volume fractions of porosity (low values of 1 − capillary porosity), to having a mostly series character at low volume fractions of porosity (high values of 1 − capillary porosity). The capillary pore space is clearly losing connectivity and volume, as it is filled in with reaction products, thus leading to the observed drop in composite conductivity. This transition is pointing to an eventual de-percolation of the capillary porosity, but de-percolation, or loss of connectivity is neither observed by other techniques , nor predicted by percolation theory  or computer simulation  until a volume fraction of capillary porosity near 20%. A disconnected capillary pore space would of course display series character as current flows through the (always) connected C-S-H and through the isolated capillary pore bubbles.
Figure 3: Normalized conductivity data for two ordinary portland cements with different water to cement ratios. The solid lines are the parallel and series bounds shown in Fig. 2. The x-axis is 1 − capillary porosity.
A series character in the microstructure at porosity volume fractions greater than 20% may also be indicated by the full Nyquist plot, according to the following argument. Figure 4 shows the impedance spectrum of Figure 1, with the bulk arc fitted by a semicircle whose center is depressed below the axis. The parameters n and Q come from fitting the arc with a "constant phase element", which becomes a perfect capacitor in parallel with a perfect resistor when n = 1 [2,3]. The capacitance of a CPE element is Q(iω)n, with n < 1 usually, and Q called an "effective capacitance". Fig. 4 shows that the fitted arc, when extended to frequencies higher than measured, does not extend to zero impedance, but rather to some real resistance greater than zero, defined as an "offset resistance" .
Figure 4: Impedance arc analysis used to determine effective dielectric constant, bulk resistance, and offset resistance. The parameters n, Q and ω are explained in the text.
This behavior is unexpected, because the conductive pore phase is almost certainly well connected for a sample with this initial water:cement ratio (0.45) and age of reaction (14 days). The impedance of a pure capacitor is inversely proportional to frequency:
where ω is the applied angular frequency, C is the capacitance of the capacitor, and i is the square root of −1. Thus, for an ideal connected phase, Zcapacitor approaches zero as the applied frequency becomes infinite. Since the C-S-H phase is always connected, from a few hours of hydration on, and since the C-S-H phase has a capacitive character , this implies that the arc must eventually go to the origin. Therefore, there must be a second arc, unseen by experiment because its frequency is beyond experimental limits.
The models usually used to analyze impedance spectra are simple series and parallel combinations of resistors and capacitors . Series behavior results in two arcs as seen in Figure 5, where two RC circuits arranged in parallel and series are shown, along with their impedance spectra. Thus, according to this analysis, the electrical properties of cement paste are behaving as if the conducting phases were assembled in series, which would imply that the capillary pore space is disconnected. Because of the low resistance of freshly mixed pastes, arising from the high conductivity of the pore fluid, an impedance arc cannot be observed at very early times, as most of the arc is at frequencies too high to be observed experimentally. However, enough of an arc to be analyzed can be observed at times that are still early enough so that the capillary pore space is almost certainly still percolated . Even at early hydration times, where the volume fraction of capillary porosity can be 40 resistance may be present.
Figure 5a: Impedance response for equivalent circuits arranged in parallel.
Figure 5b: Impedance response for equivalent circuits arranged in series.
Both aspects of the impedance spectrum described above indicate that the phases within the composite are arranged to some extent in a series manner. In other words, the microstructure must have some series character. Since the pore fluid is the most conductive phase in the microstructure, the series behavior implies, according to the usual models used to analyze IS, that the pore fluid is disconnected at volume fractions for which most other information indicates otherwise.
However, it has recently been shown that two-arc behavior, for a composite with two conducting phases that can differ greatly in conductivity, can be manifested when either both phases are connected, or when the lower conductivity phase is disconnected . This perhaps unexpected result can analytically be seen to be true by considering the dilute (small inclusion volume fraction c2) limit of the Maxwell-Garnett equation, for a composite made up of a matrix material and spherical inclusions :
where σ is the composite conductivity, σ1 is the complex matrix conductivity, and σ2 is the complex inclusion conductivity. When the inclusion volume fraction is small, eq. (3) gives the exact behavior of the composite conductivity. When the phase parameters are chosen so that the inclusions have a significantly different peak frequency response than the matrix, the exact conductivity, displayed in a Nyquist plot, displays a two arc behavior . An example is shown in Fig. 6, with the conductivities and volume fractions given in the caption. The peak frequencies for both arcs (in HZ) are marked on the figure, and the axes are in arbitrary resistivity units. The main current paths will of course go around the isolated inclusions, since they have a lower conductivity than the matrix, but there will always be some current going from the matrix, through the spherical inclusions, and back into the matrix, since this is a shorter distance than going around the inclusions. Any current which flows across a phase interface means that some series behavior will be displayed, and so two arcs will be observed if the peak frequencies for the two phases differ sufficiently. This simple analytical example shows how tortuosity (non-straight line current paths) can produce series-like two-arc behavior.
Figure 6: Graph of eq. (3) , plotting the imaginary part of the impedance vs. the real part. The phase complex conductivities were chosen to be σ1 = 1.0 + 100i, σ2 = 0.001 + 5000i, and the volume fraction of phase 2 was chosen to be 0.1. The units are such that εo = 1, so that the low-frequency dielectric constant of phases 1 and 2 are 100 and 5000, respectively. The peak frequencies (in Hz) are marked for both arcs, reflecting the units of the problem.