Fig. 1 displays the impedance spectra in a Nyquist plot for a neat w/c=0.4 paste specimen and a typical c-FRC composite specimen, both at 28 days of hydration. The beginning of the large electrode arc is visible at the right in each spectrum. The intersection of this arc with the bulk arc(s) in Fig. 1 represents the true bulk resistance, as was confirmed by 4-point DC measurements. This point is denoted throughout this paper as RDC . Whereas one bulk arc is evident in the case of the neat paste, two arcs are apparent in the c-FRC spectrum. In other words, the presence of 1 wt% chopped steel fibers had little or no effect on the value of RDC, but the AC behavior was dramatically changed, so that the single bulk arc became divided into two separate arcs.
This behavior was successfully duplicated by the physical simulations of needles in tap water. These physical simulations neglect the random fiber arrangement and orientation, and focus only on the effects of fiber properties in the cement paste matrix and individual fiber geometry. As will be shown below, by neglecting the random fiber arrangement and orientation, the main features of the two-arc behavior were still captured, even though there will of course be some quantative differences with experiments at high fiber loadings. Therefore the principal features of the two bulk arc behavior are mainly only a function of individual fiber geometry and properties, and not of the random fiber arrangement and orientation.
Figure 2 shows Nyquist plots for the tap water alone (before and after adding needles) and with 304 stainless steel needles 25, 50, or 75 mm long. It can be seen that the single bulk arc in the no-needle (tap water only) case is subdivided into two arcs of varying size by the presence of the conductive needle. Furthermore, the value of R DC, the bulk resistance, at the intersection of the real axis with the electrode arc on the extreme right, does not change significantly with the addition of the needles. However, the value of the real resistance at the intersection between the two bulk arcs, denoted R int, does decrease as the length of the fibers increases. The longer fibers of course have a somewhat larger volume concentration. Since the value of RDC is fairly constant, and Rint decreases as the fiber length increases, the size of the righthand bulk arc increases, and the size of the lefthand bulk arc decreases, as the fiber length increases.
Figure 2: Experimental Nyquist plots for a single 0.5 mm diameter needle suspended in tap water, for three different needle lengths (25, 50, and 75 mm), and for the plain tap water both before and after adding the needles.
Additional physical simulations were conducted wherein an increasing number of parallel 50 mm long needles (in this case copper, 0.5 mm diameter) were suspended in tap water on an insulating mesh support framework. The minimum needle-to-needle separation was 5 mm. In Fig. 3, as the number of needles increases from 1 to 16, the lefthand bulk arc shrinks, the righthand bulk arc grows, and the intersection between the two arcs, Rint, decreases. This decrease can be understood simply.
Figure 3: Experimental Nyquist plots for a various numbers of 0.5 mm diameter, 50 mm long needles suspended in tap water, along with the plain tap water.
In Fig. 4, the real bulk resistance at the
intersection of the two bulk arcs, Rint,
normalized by the DC bulk resistance, RDC,
is taken from the data in Figs. 2 and 3 and plotted vs. the total length of the two tap water regions, L,
lying between the ends of the needles and the electrodes, where L is the
difference between the total sample length between the electrodes, 90 mm, and
the needle length. Also plotted (lower dashed line) is the resistance of just
this tap water region, R = 
L/A, where
is the tap water resistivity and A is the
cross-sectional area of the sample, normalized by the value at L=90 mm. The
upper solid line without data points is the computer simulation result
(described in detail later) for single needles of various lengths. The solid
curve with three data points is the single needle experimental result from
the needles were all 50 mm long. If the 50 mm needle region was a solid
highly conducting region, the resistance at the intersection, R
int, would be composed of only the resistance
of the two 20 mm tap water regions at either end of the fiber array that lie
between the needles and the electrodes. As more needles are added, the needle
region approaches this limit, and the multiple-needle data points approach the
R vs. L line as the number of needles increases.
Figure 4: The ratio Rint/R DC for various geometries, vs. the length of the needle-free section of the sample, from experiment and computer simulation.
The exact way in which these data points approach the R vs. L line is of interest. One might think that resistance of the needle region would decrease simply inversely with the cross-sectional area of the needles, as more and more of this region becomes occupied by highly conducting material. However, the resistance of the needle region decreases much faster than the needle cross-sectional area, because current is concentrated into the needles. The needles have a much greater lowering effect on the resistance than just their cross-sectional area would imply. However, the needles do not lower the resistance as much as if the entire needle region were highly conducting, as only part of the needle region is made up of highly conducting material.
The effect of the needles at the Rint point can then be thought of in two ways, either by comparing their effect to the matrix of tap water without needles, or by comparing their effect to having the needle region totally filled with highly conducting material. The first point of view is discussed more fully in the Appendix. The second point of view is often used in the IS literature, and is associated with a concept called the spreading resistance. This concept will be used in the rest of this discussion.
To understand the idea of spreading resistance, consider a unit cube of conducting material with electrodes covering opposite faces. When current is applied, the resistance is just that of the entire block, where current is uniform within the block. Now suppose one electrode were to be replaced by a wire whose tip just touched the cube face. The measured resistance is now much greater, since all the current, which begins fairly uniformly at the flat electrode face, has to bunch together at the wire tip in order to pass out of the material. The difference between the two measured resistances is called the spreading resistance. Therefore, in Fig. 4, the differences between the upper curves and data points, compared to the lower dashed curve, are just the various spreading resistances, which are different for different needle geometries. The spreading resistance is due to current-bunching in the vicinity of the wire tip and inhomogenous current density in the bulk. These concepts are developed further below.
Using pixel-based computer simulations, the mechanisms of the two-bulk arc behavior were tested and elucidated. The impedance diagrams of Figs. 2 and 3 were successfully simulated using the pixel network and the thin layer on the needles, as described above. The computer simulation results are shown in Figs. 5 and 6, which correspond to Figs. 2 and 3, respectively. The resulting agreement with the experimental Nyquist plots was quite good. Furthermore, the simulated ratio of Rint to RDC, in both the single and multiple needle cases, agreed well with the experimental values.
Figure 5 shows the computer simulation results for single needles of different lengths. The qualitative aspects of the experimental and computer simulation plots can now be readily explained, bearing in mind that the volume fraction of the single needles is very small, certainly less than 1%. It has been shown [17, 18] that long, thin needle-like inclusions produce little effect on the bulk conductivity when they are insulating compared to the matrix, but have great effect when they are highly conducting compared to the matrix. Once the aspect ratio gets beyond 10 or so, all needles have essentially the same effect on the conductivity when they are insulating. But when the needles are highly conducting, then the effect of the needle on the bulk conductivity is very sensitive to the aspect ratio [18] (see Appendix).
Figure 5: Computer-simulated Nyquist plots, in units where RDC for the plain tap water-filled cell is unity, for a single 0.5 mm diameter needle suspended in tap water, for three different needle lengths (25, 50, and 75 mm).
Figure 6: Computer-simulated Nyquist plots for a various numbers of 0.5 mm diameter, 50 mm long needles suspended in tap water, in units where RDC for the plain tap water-filled cell is unity.
In Fig. 5 (model) and Fig. 2 (experiment), therefore, the DC resistance at the low frequency end of the arcs on the right of the graph, RDC, is found for the case when the needles are insulating, because the polarization layer is insulating at this frequency. Because insulating needle-like inclusions have little effect on the conductivity, the resistance at this point is essentially just that of the tap water. The needle length does not matter very much at this point [17,18]. As the frequency is increased, however, the impedance of the "coating" on the needle goes to zero and a low frequency arc (on the right) is traced out until the imaginary impedance touches the real axis. At this point, the real resistance, Rint, is that of the matrix plus a highly conducting needle. Now the effect of the highly conducting needles is quite sensitive to the needle length. Since these needles are cylindrical wires, with the same diameter, the aspect ratio varies linearly with the length. The value of Rint then moves left to lower resistance as the needle length increases.
In the computer simulation technique used, all local current density information is available as well as the global average current or admittance. This has been used, in the 50 mm long, single needle case, to demonstrate the current-bunching ideas and lend support to the fact that the needle acts differently at different frequencies. At frequencies corresponding to the RDC and Rint points on the arc, the current magnitudes along a longitudinal slice that included the needle were saved. These currents were converted to gray scale images using a gray scale table that brought out the detail in how the current densities varied spatially. The results are shown in Fig. 7. Figure 7(top) shows the local currents at a frequency when the needle is insulating, and Fig. 7(bottom) shows the currents at a frequency when the needle is highly conducting. In Fig. 7(top), the current is uniform and is almost undisturbed by the presence of the needle. However, the presence of the highly conducting needle in Fig. 7(bottom) clearly alters the initially uniform current density. The needle draws current into itself, causing much higher currents near the tips and depleting current in regions surrounding the midpoint of the needle, since most of the current is now going through the needle instead of the tap water matrix.
Figure 7(bottom) also aids in visualizing the two contributions to overall resistance at Rint, when the wire is conducting. Towards the outer ends (external electrodes) there are bulk contributions due to the tap water not unlike the low frequency (insulating wire) situation in Fig. 7(top). These are in series with spreading resistances as current bunches to the tips of the wire. Roughly speaking, we may associate the former (bulk) contribution with the lower line in Fig. 4 and the latter (spreading resistance) contribution with the difference between the upper and lower lines in the same plot. Note that as the number of wires increases from 1 to 16 in Fig. 4, the value of Rint approaches the lower line, i.e., the value corresponding to that of the outer tap water regions. Since the spreading resistances of adjacent wires act roughly in parallel, neglecting interaction effects, their contribution becomes less and less significant as their number increases.
Figure 7: Grey scale images of a longitudinal section of the computer simulation cell, in the plane of the needle, for the 50 mm needle—(a) at DC frequencies (RDC in Fig. 5), (b) at a frequency where the needle is highly conducting (Rint on Fig. 5). White is high current density, dark gray is low current density. The gray scale was chosen so as to bring out the detail of the current distributions.
A simplified equivalent circuit based upon this interpretation is presented for a single fiber in cement paste in Fig. 8. In addition to the overall bulk cement paste RC element (upper path), there is a parallel path consisting of outer paste RC elements (between the external electrodes and the spreading resistance regions) and spreading resistance RC elements. Note that the fiber/coating has been represented by a frequency-activated switch. The switch is actually an additional RC element with parameters as described above in the simulation section. Its representation as a frequency-activated switch is useful for the sake of discussion. (The actual fiber resistance is ignored since it is negligibly small.) The time constants of the bulk paste, outer paste, and spreading resistance elements are assumed to be identical. At DC and low frequency AC, the bottom path is essentially an open circuit due to the high "coating" resistance, and the circuit behaves as if the fibers (bottom path) are absent; the DC resistance/capacitance is that of the paste matrix (upper path). As frequency increases, however, the impedance of the "coating" RC element in the bottom path goes to zero and the switch is thrown. The resistance now measured is predominantly the lower path, or the sum of outer paste plus spreading resistance contributions.
Figure 8: Simplified equivalent circuit model for the needle/tap water system. Rb, Cb are bulk paste parameters, Rbo and Cbo are outer paste parameters (between the electrodes and the spreading resistance regions when the wires are conducting), and Rsp and C sp are the spreading impedance parameters.
Based upon physical and computer simulations, the frequency-switchable coating model is capable of interpreting the general features of the IS spectra of c-FRC, which are dual bulk arcs with fibers vs. single arcs without fibers. The DC resistance (low frequency bulk intercept in Nyquist plots) is largely unaffected by the presence of a low volume fraction of conductive fibers. This is due to the fact that their polarization resistances/double layer capacitances and/or oxide film coatings serve to completely insulate them under such conditions. As the frequency increases, however, the coating/polarization impedance disappears and the fibers then act as short-circuit paths in the composite. The residual resistance R int, the diameter of the high frequency bulk arc, has contributions from fiber-to-electrode and fiber-to-fiber bulk resistance plus an important spreading resistance component. The fiber aspect ratio, which was a negligible factor in DC resistivity, now plays a major role.
These results provide the basis for an ongoing study of the impedance response of fiber-reinforced composites. Subsequent work will address the issues of fiber orientation, fiber-to-fiber distances (i.e., fiber volume fraction), fiber/matrix debonding, fiber pull-out, and how these factors influence the impedance spectra of cement-based composites with conductive fiber additions. The effects of deformation/damage of the fiber/matrix on the IS spectra will also be explored.