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Typical impedance results for a 7 day old 20% volume fraction steel ball bearing/OPC composite are compared to the results for a plain OPC specimen of identical age in Fig. 3. In each case, the value of RDC value is that obtained from 4-point dc measurements, corrected for the longer inter-electrode spacing of the 2-point vs. the 4-point geometries. There is good agreement between RDC and the impedance intersection at ≈1700 Ωbetween the single bulk arc and the electrode arc for the plain OPC sample. Similarly, the value of RDC for the composite is consistent with the impedance cusp at ≈2250 Ω between the bulk features (to the left) and the electrode arc (to the right). It should be pointed out that there is often strong convolution between the rightmost bulk arc and the electrode arc, as exhibited here and in Fig. 4(b) (described below). This can prevent an accurate evaluation of the dc resistance, based upon impedance measurements alone. For conductive particle/fiber composites, it is therefore advisable to make independent 4-point dc resistance measurements, as was done in the present work.
Fig. 3. Typical Nyquist plots for a 7 day 20% volume fraction steel ball bearing/OPC composite and plain OPC with Rcusp and RDC labeled. The numbers corresponding to darkened points indicate log frequency (Hz) values. The minimum −Im(Z) for the steel and plain samples occurs at a log frequency (Hz) value of ≈4.
Fig. 4. Trends by volume fraction of (a) glass ball bearing/OPC composites
and (b) steel ball bearing/OPC composites. Four-point dc values are shown for
comparison on the Re(Z) axis. The numbers corresponding to darkened points
indicate log frequency (Hz) values. The minimum−Im(Z) for the samples
occurs at a log frequency (Hz) value of ≈4.
Figure 4(a) and (b) display impedance spectra for plain OPC and composites at 5% volume fraction increments up to 20% for insulating (glass) and conductive (steel) ball composites, respectively. As illustrated in Fig. 1, the addition of glass beads does not significantly alter the shape of the impedance spectra, but shifts them progressively along the real impedance axis in Fig. 4(a) (increasing resistivity). Also shown are the dc resistances, which agree well with the bulk-electrode cusp values in the impedance spectra to within an estimated ±10% uncertainty. This is based upon an approximate 5% uncertainty in inter-electrode spacing in each experiment and is consistent with experimental reproducibility.
Fig. 5. Normalized conductivity of steel and glass ball bearing/OPC composites plotted with Meredith and Tobias formulas.
With the addition of steel ball bearings, the spectra of Fig.
4(b) consistently show the dual-arc bulk response
of Fig. 1, as described above. The
bulk-electrode cusp point, when clearly
observed, is consistent with the 4-point dc resistance value, and increases
monotonically with the volume fraction of ball bearings.
At the same time, the bulk cusp resistance (between the two bulk arcs) decreases
monotonically with increasing volume fraction of ball bearings.
Figure 5 displays normalized conductivity values, σ(composite)/σ(matrix), vs. volume fraction of particles (steel or glass balls) in 7 day samples. The upper line in Fig. 5 is derived from the cusp resistance (the intersection of the two bulk arcs) for the steel ball bearing composites, with the matrix value coming from the dc resistance of OPC paste. There are three sets of data on the lower line. Two of them are for glass bead composites, one employing the bulk-electrode cusp value from the impedance plots vs. RDC for OPC paste and the other using RDC values exclusively. There is good agreement between the two approaches. The third set of data is the value of RDC from 4-point dc measurements for steel ball composites. Of great interest is the fact of the agreement between the dc resistance values of the steel and glass ball composites. The model lines in the two cases will be discussed further below.
The behavior in Fig. 5 provides confirmation of the freqency-switchable coating model [6]. Figure 6 shows a conceptual model for current flow at dc (Fig. 6(a)) vs. that at the cusp frequency under ac excitation (Fig. 6(b)). The highly resistive oxide film surrounding a steel ball bearing electrically isolates it from the matrix at dc and low ac frequencies, and the ball behaves as if it were an insulating sphere. The balls not only reduce the volume fraction of matrix (the conducting phase) but cause current to be constricted to regions between particles. Hence, resistance increases (Fig. 4(b)) and conductivity decreases (Fig. 5). At the cusp frequency, however, displacement currents have shorted out the oxide film, and the ball becomes a short-circuit path for current flow through the matrix. Composite resistance decreases (Fig. 4(b)) and conductivity increases dramatically (Fig. 5).
Fig. 6. Conceptual model for the current flow (a) at dc and (b) at the cusp frequency under ac excitation.
The absence of a percolation threshold, even for the conductive sphere composites under ac excitation, requires comment. Due to the oxide film, we do not anticipate a dc percolation threshold under fields up to the breakdown strength of the oxide film surrounding the particles. However, the oxide film resistance is no longer in effect at the cusp frequency in the ac experiments. The absence of an ac percolation threshold with loading levels up to 42% volume fraction of steel ball bearings is noteworthy. The reason for this may be that cement particles (median particle size ≈10 µm) completely coat the balls during processing. At closest approach, adjacent particles will be separated by at least 10 µm. Later, hydration products replace the parent cement particles in the inter-particle space. For this reason, the steel particle/OPC system provides a unique opportunity to study mixing law behavior for the electrical properties of such composites, without the onset of percolation.
There have been several outstanding reviews of mixing laws and effective media theories for electrocomposites [9−12]. In the dilute limit, all such equations should reduce to Maxwell´s equation:
 |
(1) |
where σ is the conductivity of the composite, σm is the
conductivity of the matrix phase, 
is the ratio of particle conductivity to that
of the matrix, 
is the volume fraction of particles, and higher-order terms are
neglected. In the case of conducting spheres,
= ∞ and the first-order
coefficient, otherwise known as the "intrinsic conductivity"
[13], is 3. With insulating spheres
( = 0) the intrinsic conductivity is −3/2.
Other models extend calculations beyond the dilute range. The Maxwell-Wagner equation (also known as the Maxwell-Garnett equation [12] or Wiener´s rule [11]) based on the well-known Clausius-Mossoti equation, is given by [11]:
 |
(2) |
where σp is the particle conductivity. This model is formally equivalent to the Hashin-Shtrikman lower bound (conductive particles) and upper bound (insulating particles) [11, 14] and will be referred to as the "MW-HS" equation(s).
Bruggeman´s asymmetric (BA) medium theory [
11] for conducting spheres is given
by:
 | (3) |
wheras for insulating spheres the BA equation is:
 |
(4) |
Zuzovsky and Brenner performed calculations for the effective conductivity
of a simple cubic array of spheres embedded in a matrix vs. volume fraction of spheres [15]:
 | (5) |
Finally, Meredith and Tobias [10] extended Fricke’s treatment [16] of ellipsoidal particles within a Clausius-Mosotti framework by mixing half of the spheres at a given volume fraction, calculating the composite conductivity, and using this as the matrix for a new composite made with the addition of the other half of the spheres. The resulting equations are:
 | (6) |
for conducting spheres and
 | (7) |
for insulating spheres.
The various mixing laws and effective media theories are plotted vs. particle volume fraction in Fig. 7 for the case of conductive spheres, with the experimental data of the present study superimposed. The Meredith-Tobias equation exhibits the best agreement with the experimental results. The Zuzovsky-Brenner model comes close; however, this model is for an ordered array of particles, which is clearly not the case in the present work. The Bruggeman asymmetric medium equation exhibits a markedly higher conductivity. This is not unexpected, since the BA model is most applicable for a wide range of particle sizes as opposed to the single size in the present work. As expected, all the models approach the Maxwell line for volume fractions less than 0.1, the oft-quoted upper limit for the dilute regime. The slope of 3 in this regime is exactly the intrinsic conductivity of conducting spheres [13].
Fig. 7. Various mixing laws and effective media theories plotted vs. particle volume fraction for conductive spheres, with the experimental data superimposed.
The Meredith-Tobias equation can be factored to determine the higher-order coefficients in the virial expansion of Eq. (1). The resulting equation is:
 | (8) |
and the first few terms are given in Table 1. The second order coefficient is of interest, since this has been independently calculated to be 4.51 [17, 18], in excellent agreement with the value of 4.5 obtained by factoring the Meredith-Tobias equation.
The corresponding plot in the insulating particle regime was inconclusive and has not been shown. There is considerable scatter in the experimental data and the various mixing laws/effective media equations are quite similar. Although we cannot conclusively determine which model is best for the insulating particle regime, it is interesting to note that the Meredith-Tobias model factors to:
 |
(9) |
where the second-order coefficient of 0.5625 is in good agreement with the calculated value of 0.588 [17, 18], and higher-order terms are negligibly small (see Table 1). The first-order coefficient of −3/2 is consistent with the known intrinsic conductivity of insulating spheres [13].
| Table 1. Values of the coefficients in the factored Meredith-Tobias equation: σ/σm
= 1 + a + b2 +
c3 + . . . . | ||
|---|---|---|
| Power of
|
Conductive particle
coefficients |
Insulating particle coefficients |
| 1 | a = 3 | a = −1.5 |
| 2 | b = 4.5 | b = +0.5625 |
| 3 | c = 5.25 | c = −0.09375 |
| 4 | d = 5.625 | d = +0.03516 |
| 5 | <e = 5.8125 | e = −0.00586 |
Fig. 8. Symmetric I-V curve where current was applied in pulses from 50 mA to −50 mA, waiting 20 s between each current application.
In recent work we extended 4-point resistance measurements of steel fiber-reinforced composites to higher current densities [19]. The resulting I-V plots looked virtually identical to Fig. 8, which was measured in the present work by increasing applied current to ±50 mA. It was found that the increased slope above the threshold voltage (in the high current or low resistance regime) corresponded to Rcusp in the corresponding impedance diagrams. Similarly, the central slope (low current or high resistance regime) agrees with the bulk-electrode intersection in impedance diagrams; RDC is determined from this slope. It was argued that at the threshold voltage, local fields are sufficient to drive interfacial reactions into active or trans-passive corrosion regimes; the oxide film resistance is eliminated. It would seem that steel inclusions can be rendered “conductive” by means of increased frequency (in ac measurements) or by increased field strength (in dc measurements).
There are problems, however, with high-field dc measurements in cement-based composites. First, corrosion products are anticipated on particle surfaces if high dc currents are applied for any period of time [20]. Second, such currents can lead to joule heating effects [21]. The data in Fig. 8 were taken with 20 s delays between individual points to allow for thermal relaxation. If this procedure was not employed, the threshold voltages under negative vs. positive bias differed significantly, and the outer (high current) regimes had different slopes. Finally, the resistances in the high-current regime were found to be significantly higher than Rcusp in impedance measurements. One proposed explanation is that high local current densities (between particles) can result in permanent microstructural changes in the matrix [21]. In contrast, impedance spectroscopy is non-destructive and the only reliable means of assessing composite electrical properties in the "conductive" particle regime.
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