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Description of the Computer Model

To better understand the apparent series behavior of hydrating cement pastes, a computer model was used to demonstrate that a tortuous pore phase is responsible for the observed series- like behavior. A model that computes impedance spectra from digital representations of microstructure in two or three dimensions has been developed [19]. Each discrete element (pixel) in an array is considered to be a single, homogeneous phase. The impedance properties of all the pixels are then summed to give an overall composite response for this microstructure. Garboczi and Bentz [20] have published a complete review of this type of microstructural representation of cement paste. The ability of this model to quantitatively reproduce experimental IS curves from a simulated microstructure of cement paste provides a powerful tool in establishing structure-property relationships, especially concerning AC electrical properties.

The microstructural development of cement paste during hydration was simulated using the Bentz-Garboczi C₃S hydration model [21]. Images used for this model are typically 100³ pixels, where each pixel is a cube of volume 1 cubic micrometer. The cement particle size distribution used in these simulations consists of ten different sizes, and was taken from a measured distribution of C₃S powder. This distribution was used in all the models presented here. The computer resources needed to implement the impedance computation on a 100³ image are significant, and require the use of a supercomputer for both memory and processing time. Since supercomputer resources were limited, smaller 50³ models, which can be executed on various high-performance workstations, were used to explore general trends and to establish initial phase parameters.

A measure of the connectivity of the pore phase (and other phases) is readily achieved in the simulated microstructures by using a "burning" algorithm [18]. The burning algorithm propogates a "fire" throughout the microstructural phase being assessed. If the fire reaches the opposite side from which it was started, the phase is percolated in that particular direction. This technique was employed on all the microstructures used in this study to determine if the pore fluid phase was connected in the direction of the applied voltage. For systems of both sizes (50³ and 100³) and several different water to cement ratios, the percolation threshold of pore fluid phase occurred when its volume fraction fell slightly below 20%. This is consistent with the 18% value determined from the same model previously using a different cement particle size distribution [18], implying that the capillary porosity percolation threshold is fairly insensitive to cement particle size distribution.

The impedance of the model is computed by solving the complex Laplace's equation for a random, complex conductor [3]. A node is placed in the middle of each pixel, and a RC (a resistor in parallel with a capacitor) element is connected between each node, with the value of R and C determined by the local complex conductivity for the phase occupying that pixel. Electrodes are added to either side of the microstructure image, and "wired" into the network. A single frequency voltage is then applied, and the complex Kirchoff's laws are solved for the network using a conjugate gradient method. Details are given in [19]. The total current in the network, divided by the applied voltage, then defines the impedance.

Before implementing the impedance simulation, it was necessary to estimate the impedance properties of the individual phases, in order to be able to define the local RC elements to be used. Once reasonable estimates were obtained, these parameters were modified to achieve the best fit with the experimental data. Especially important are the properties of the pore fluid and C-S-H, since these phases are the most abundant and are considered the most important in determining the electrical properties of cement paste.

The conductivity of pore fluid was determined experimentally by direct measurement after being extracted from cement paste, and is in the range of 1 − 10 S/m [5]. It is assumed that pore fluid from C3S pastes is not as conductive as that of portland cement paste because of the absence of alkali ions [5]. The conductivity of saturated CH (lime) solution was found to be 2 S/m⁻¹, which is taken in the model for simplicity as 1 S/m⁻¹. Since the number ultimately to be computed is the composite conductivity, normalized by the pore fluid conductivity, an overall multiplicative factor in the pore fluid conductivity will not affect results. Because the pore fluid is highly conductive, its dielectric constant is not directly measurable by the impedance technique, since not enough of an arc can be generated to analyze, as was mentioned above. The relative dielectric constant is then assumed to be that of water, k_r = E/E₀ = 80 [22]. Some experimental results have put an upper bound of approximately 150 on the value of k_r [2].

Previous results from a DC simulation [23] indicated that the conductivity of C-S-H was about 400 times less than pore fluid. The first estimate of the dielectric constant of C-S-H is based on measurements for synthetic C-S-H and completely hydrated samples that used silica fume to produce a material that is mostly C-S-H [2]. These measurements indicate that the relative dielectric constant of C-S-H is at least several times higher than that of pore fluid. Also, since C-S-H is probably a layered structure [1], with pore fluid residing to some extent in between the solid layers, it is reasonable to expect that its dielectric constant is larger than that of pore fluid, perhaps significantly larger, amplified due to a fluid-solid layering effect [3]. The subject of dielectric amplification as it occurs in C-S-H, and in cement paste itself, will be discussed in Part II [25] of this paper. Since both C₃S and CH are known to be insulators, they are assumed to have very low conductivities and low dielectric constants. Columns 2 and 3 of Table 1 show the initial parameters used in the model for a large number of the 50³ systems.