To produce the data shown in Fig. 1, samples were prepared from Type I portland cement at a water-to-cement ratio of 0.4 by weight. Steel fibers (average length 2 mm, diameter of 30 µm, 1 wt%) were dry mixed with the cement for 1 minute in a Hobart planetary mixer. The water was then added, and mixing continued for 3 minutes at low speed, following by 1 minute of hand mixing to ensure homogenization, and 3 more minutes of machine mixing. Specimens were cast in rectangular polycarbonate containers (25 x 25 x 100 mm) with C-1018 plain carbon steel electrodes cast in place with a separation of 90 mm. These samples were used to establish the basic frequency-dependent behavior of c-FRC, which was then further investigated using model systems, which is the main emphasis of this paper. To confirm the true bulk resistance of the c-FRC specimens, 4-point DC measurements were also conducted. By correcting for the differing geometries in 2-point AC vs. 4-point DC measurements, the two results could be compared.
The model systems, or physical simulations, were carried out using polycarbonate containers of the identical size and shape as those used for the c-FRC samples, but employing tap water as the electrolyte. The conductivity of tap water was appropriate to simulate the conductivity of mature cement-based specimens. "Needles" of 304 steel or copper wire (0.5 mm diameter; 25, 50, 75 mm long) were positioned along the axis of the specimen container equidistant from the electrodes, suspended on insulating supports. Impedance measurements with and without the support structures (in the absence of needles) showed that these did not contribute to the spectra obtained.
Impedance measurements were carried out using a personal computer-controlled frequency response analyzer (Solartron 1260 with Z60 control software, Schlumberger, Cambridge, UK)* over the frequency range of 0.1 Hz to 10 MHz (10 points per decade). The excitation amplitude was varied from 25 mV to 1.0 V, with no obvious change in bulk spectral features. Spectra were analyzed with the aid of the software called "Equivalent Circuit" [15].
A FORTRAN 77 finite-difference numerical program, called ac3d.f, was used to carry out the pixel-based computer simulations. This program can be found at http://ciks.cbt. nist.gov/garboczi/, Chapter 2, along with a manual in HTML or hard copy format [16]. This program was designed to compute the finite frequency properties of random materials, whose microstructure can be represented by a 3-D digital image. However, the program can also be effectively used for non-random, but analytically intractable, geometries, as in the present case.
Pixels in a 3-D digital image were used to construct a representation of the experimental set-up. The length scale used was 0.5 mm/pixel, so that the model was approximately 50 x 50 x 180 pixels in size, matching the 25 x 25 x 90 mm sample (electrode-electrode) dimensions. The needles were one pixel or 0.5 mm in width, and were suspended in the middle of the sample chamber as in the experimental arrangement.
In the computation process, there was a finite difference node in the middle of each pixel. As part of the computation, bonds were assigned between each pair of nodes, reflecting the conductivities of the materials in each pair of nodes. The electrodes were taken to be highly conductive, as was the needle. No polarization layer was taken to be on the electrodes, so that the electrode arc was not included or computed. A polarization layer was taken on the surfaces of the needles. It is well known that high electrode resistances and capacitances arise due to charge transfer polarization/double layer formation and/or oxide film formation, e.g., the well-known passive oxide film which forms on steel in high pH solutions [10]. It was assumed that similar "coating" elements form on needles/fibers, as they are also metallic and embedded in an ionic conductor. The admittance of this layer on the needles was taken to be that of the electrodes (fitted from the experimental electrode arc), but adjusted for the differences in size and shape between the needle surface and the parallel plate electrodes. In this way, only the resistance and the conductance of the layer were specified, so that the unknown conductivity, dielectric constant, and thickness of the layer did not need to be individually determined. The water pixels had the correct admittance for the tap water used (~0.03 S/m). The needle itself had a bulk DC conductivity approximately 109 times that of the tap water.
Once the model geometry and admittances were set up, a uniform electric field was applied along the needle direction, as was done experimentally, and a conjugate gradient method was used to find the solution to the complex conductivity problem. The frequency was systematically varied, similar to experiment, and the Nyquist plot determined.