Fiber "Pull-Out" Simulations: Typical impedance curves for the single steel wire at various depths of submersion in tap water are shown in Fig. 4a. (Note that there is only a single bulk arc, due to the fact that the wire is essentially a part of the electrode to which it is attached.) Dramatic changes are seen in the first few mm of submersion; thereafter the changes are more gradual. This is better depicted in Fig. 4b, where we take the resistance at electrode/bulk intercept in Fig. 4a and subtract the resistance, approximately 600 Ω, of the volume of water between the fixed wire tip and the bottom copper gauze electrode. The experimental (circles) and model (solid line) results have both been normalized by their respective values at the point where the wire just touched the water surface. Both experiment and modeling results fall by as much as 90 % in the first 3 mm of submersion. Both sets of data are in good qualitative agreement with each other, although there are some quantitative differences. These differences are probably due to the fact that the depth of immersion was hard to measure for the experiment for small immersions (< 0.2 mm), because the diameter of the computational cell was about five times smaller than that of the experimental set-up, and because the one-pixel thick wire in the finite element computations did not precisely simulate the cylindrical wire used in the experiment.
Fig. 4. Results of the pull out/water submersion study corresponding to the cell in Fig. 3a. a) impedance spectra, and b) spreading resistance values (cusp values minus the known water resistance for the 61 mm water height). Pull-out can be viewed as the complement of submersion
Figure 5a shows a schematic view of the computational cell used. A replica was made of double the real system, giving a symmetric system upon which it was much easier to perform numerical computations. Once the overall resistance was computed, the resistance of the part of the wire extending between the water layers was subtracted, and the remaining resistance was divided by two, to give an equivalent resistance to that measured in experiment. A resolution of 0.1 mm/pixel was used, in order to be able to see the behavior of the impedance in the first few tenths of a millimeter immersion of the wire tip, where most of the resistance drop took place.
The results shown in Fig. 4b are a consequence of the "spreading resistance" effect at the wire tip. The spreading resistance of a solid in point contact with an infinitely conducting cylindrical electrode was first established by Holm [30] and placed on a firm mathematical footing by Newman [31]. For a planar point contact the spreading resistance (Rsp ) is:
Rsp =
(4  a) −1 | (2) |
where a is the radius of the cylindrical electrode and σ is the conductivity of the medium being contacted. Based upon the water conductivity and the radius of the wire in the present study, this value should be 33.3 k Ω, to be compared with the "just touching" experimental value of 44.5 k Ω (normalized to unity in Fig. 4b). This disagreement was probably due to the fact that the rate of change of resistance was extremely large in the first mm of submersion, which is precisely where the amount of submersion is difficult to gauge due to meniscus formation, thereby leading to fairly large experimental uncertainties in this region. After the wire has been submerged a few tenths of a millimeter, better accuracy is obtained for the amount of submersion, so that a better comparison with theory is with the predicted spreading resistance of an embedded hemispherical cap of radius, r [30]:
| Rsp = (2 π σ r ) −1 | (3) |
which would correspond to a value of 21.2 k Ω. The experimental value at 0.26 mm of submersion, or approximately the same as the wire radius, was 20.9 k Ω. Figure 5b shows a current map of the model results when the wire just touched the water surface, in which the lighter the gray scale, the higher the local current. Dramatic current concentration is seen in this figure near the wire tip.
Fig. 5. Showing (a) a schematic of the numerical cell used in the finite element simulation of the pull-out experiement, and (b) a gray scale image of the current density (current increases with lightness of gray scale) when the wire tip was just touching the water surface.
The results of the submersion experiment can be interpreted in terms of how a crack-bridging conductive fiber might contribute to overall composite conductivity. At DC or low AC frequencies, there is no electrical continuity across the crack because of the insulating surface impedance. At intermediate frequencies, however, this "coating" impedance (polarization or film impedance) becomes negligible, and the fiber then provides electrical continuity across the crack. Increasing fiber pull-out corresponds to decreasing submersion; maximum submersion means minimum pull-out. The flat parts of Fig. 4b show that there is little change in electrical continuity until most of the fiber has been pulled out of the matrix, assuming of course that the part of the fiber still in physical contact with the matrix also maintains electrical contact with the matrix as it is being pulled out.
Fu and Chung [32] attempted to measure the conductivity between a single stainless steel fiber partially embedded in a cement body and external electrodes on that body during debonding and pull-out. Unfortunately, the contact resistance became immeasurably large beyond the maximum shear stress, i.e., during the actual pull-out process. It is conceivable that there was no longer any electrical contact between the fiber and the matrix beyond this point, as proposed by Fu and Chung [32]. However, since only DC measurements were performed, their resistances were probably dominated by the resistance of the passive oxide film on the steel surface. This is known to be quite large, especially so for stainless steel [33]. Based on the present work, we would suspect that physical contact was always maintained, while electrical contact was not seen due to the insulating film impedance on the surface of the fiber. The present work suggests that such work needs to be repeated at frequencies sufficient to overcome this interfacial impedance.
Fiber Debonding Simulations: The results of impedance studies of a single 50 mm long wire (insulated, 0.5 mm diameter) suspended equidistant from the electrodes (90 mm apart) on the axis of a 25 x 25 x 100 mm tap water cell are shown in Fig. 6a. The completely coated wire exhibits a single bulk arc, virtually indistinguishable from the cell without the wire, as expected (due to the insulation). As the first 2 mm of insulation is stripped from the ends of the wire, dual arc behavior is obtained. As progressively larger amounts of the ends are exposed, the value of (RDC − RCUSP ) or γ increases to its maximum value for the bare wire. Numerical simulations (Fig. 6b ) are in excellent agreement with the experimental spectra. If we compute the value of γ at a given degree of tip exposure, and normalize it by the maximum value of γ, obtained for the bare wire, and plot this ratio vs. the degree of tip exposure, the plot in Fig. 6c is obtained. Again, the modeling data points agree well with the experimental data points. The solid line is a spline fit to both sets of data. It is interesting to note that more than 80% of the ultimate change in R CUSP (or γ) occurs within the first 20% of tip exposure.
One can think of tip exposure as the reverse of debonding, assuming of course that debonding means the loss of electrical contact between the fiber and the matrix sheath surrounding it, which may not be the case. In the present work debonding is simulated by the presence of increasing lengths of impermeable polymer coating on the middle of the wire. The fraction of the wire covered with polymer is equivalent to the fraction debonded in a debonding situation. Similar to the pull-out situation above, debonding proceeds from right to left in Fig. 6c, from the bare wire (100 % electrical contact) to the fully coated wire (no electrical contact). These results suggest that in the early stages of debonding, assuming debonding is equivalent to loss of electrical contact, changes in RCUSP (and γ) will be minor. Only when the fiber is debonded more that about 40% will major changes in composite conductivity take place.
Fig. 6. Results of the debonding/tip exposure study corresponding to the configuration in Fig. 2a. a) experimental impedance spectra vs. extent of wire tip exposure, b) computed spectra vs. extent of wire tip exposure, and c) normalized plot of γ for a bare wire. Debonding can be viewed as the reverse of tip exposure.
It should be stressed that these changes should be measured at the bulk cusp frequency. The only work to date on single-fiber debonding was the DC resistance study of Fu and Chung [32], mentioned previously. For untreated and acetone-treated steel fibers the contact resistance was high and there were no detectable changes up to the maximum pull-out stress. For acid-treated fibers, however, the contact resistance was significantly smaller and there was approximately an order of magnitude increase in contact resistance up to the maximum stress, beyond which contact was suddenly lost. These results are consistent with progressive debonding taking place along the fiber. By repeating such studies at the bulk cusp frequency in Fig. 6a, it should be possible to eliminate surface impedances and isolate fiber-matrix continuity effects, during both debonding and subsequent pull-out.
Fiber Orientation Studies: The experimental and computed spectra vs. wire orientation angle are displayed in Figs. 7a and b, respectively. Note that only the low frequency arc is shown for the numerical computations. In both instances, the wire was 25 mm long (0.5 mm diameter) on the axis of a tap-water cell of cross-section 25 mm x 25 mm and equidistant from the measurement electrodes placed 31 mm apart. The angle, θ, is between the wire and the direction of the applied field. When the wire is perpendicular to the field ( θ = 90º), there is no low frequency arc, and the single "bulk" arc is indistinguishable from that of the tap water alone. As θ decreases to zero, where the wire is parallel to the applied field, the maximum value of (RDC − R CUSP) or γ is obtained. In Fig. 7c the ratio γ ( θ ) / γ ( θ = 90º ) is plotted vs. angle. There is excellent agreement between numerical and experimental results.
Also plotted in Fig. 7c is the function, cos
2θ. This
relationship is readily derived by considering the symmetries of the problem.
Take the z-axis along the wire direction, and the applied field in the y-z
plane with an angle θ between the field and
the wire direction, 
=
Eo (0,sin( θ ),cos( θ ) ). Using these
axes, the combination of matrix plus wire has rotational symmetry around the z-axis, and, since the wire is centered in the sample cell, it also has a mirror plane (the x-y plane at the wire center) with respect to the z-axis. Therefore, the overall conductivity tensor of the system has the form:
 | (4) |
At low frequency, these three components are all comparable and approximately equal to the matrix conductivity. At intermediate frequency, when Rcusp is observed, the xx and yy tensor components are little changed, but now the zz component is much larger.
The current resulting from the conductivity tensor in eq. (4) is
In the experimental set-up, the
current along the direction of the applied field is measured, or
(normalized by Eo2
). Since σ
yy does not change very much with frequency
(perpendicular to wire), while σ
zz increases greatly with frequency (parallel
to wire), the size of the arc should be proportional to cos
2( θ ), as seen in
Fig. 7c.
Fig. 7.
Result of the wire rotation study corresponding to the cell in Fig. 3b. a) experimental impedance spectra vs. angle, b) computed spectra vs. angle, and c) the cos2θ dependence of the ratio of γ ( θ ) to γ (θ = 90º).The most important result of these simulations is that oriented short conducting fiber composites will exhibit the maximum value of γ (or relative size of the low frequency arc) when measured in the direction of the fibers. In a perfectly oriented composite, there should be no measurable low frequency arc ( γ = 0 ) when measured perpendicular to the fiber direction.
Composite Studies: We are now in a position to interpret the
behavior of actual short conductive fiber composites. The impedance spectrum
in Fig. 1 is for a 28 day old random steel
fiber/cement-matrix composite. The bulk arc subdivision is given by γ
~0.33. In contrast, Fig.
8 displays impedance spectra for a 46 day old extruded carbon fiber/cement
matrix composite. The electrode arcs are not shown in
Fig. 8. Furthermore, the data have been normalized by R
DC in each case, which are listed in the figure caption.
The 
= 90 (zero shift) behavior is
similar to Fig. 1, with
γ = 0.4. The fact that it is not zero
(compare with Fig. 7) is due to the fact that the
orientation in the plane of the sample is not perfect. On the other hand, as
θ is decreased, γ increases to a value of 0.8 at θ ~ 18º (24 mm shift) consistent with the trend in
Fig. 7c and also with prior observations
[24-26], i.e., that subdivision of the bulk arc in
highly oriented composites is small (or even zero) when measured perpendicular
to the fiber direction, and maximum (approaching 100%) in the plane (or
direction) of the fibers.
Fig. 8. Experimental impedance spectra vs. orientation angle (shift of electrodes in Fig. 3c) for an extruded carbon fiber composite (0.5 vol % fibers). Electrode arcs are omitted and the data have been normalized by the DC resistance in each case−zero shift (1.9 kΩ ), 6 mm shift (3.7 kΩ ), 12 mm shift (8.2 kΩ ), and 24 mm shift (27 kΩ ).
It should be stressed that other factors (number density of fibers, fiber aspect ratio, distribution of fiber lengths, distribution of fiber orientations, etc.) will also play a role in determining bulk arc subdivision, just as they do in determining the percolation threshold. For example, the fiber content (number density) was much higher in the studies cited in the Introduction [24-26], undoubtedly contributing to the higher values of γ. Furthermore, Gerhardt [24] and Wang et al. [25] reported large decreases in the DC resistance when measuring parallel vs. perpendicular to the fiber direction. Even after correcting for changes in geometry, the DC conductivity is clearly being altered to a large extent by the presence of the fibers. Below the percolation threshold, we predict there should be only minor changes in DC conductivity due to fibers, assuming that the fibers are insulated by their interfacial impedances. What these results suggest is that there must be a high degree of fiber-fiber contact in the direction of preferred orientation, albeit not yet fully percolating. The aspects of fiber number density, fiber aspect ratio, distribution of fiber lengths and orientation, etc., are the subjects of ongoing research.
General Discussion: The necessary conditions for the unique impedance behavior of short conductive fiber composites documented in the present study are: