Next: Simulation of Impedance
Impedance spectroscopy (IS) is a useful, non-destructive tool for analyzing many properties of electroceramic materials . In this technique, a small, single-frequency AC field is applied to a sample, and the amplitude and phase of the resulting current measured. The amplitude of the AC signal is chosen to be small enough to assume a linear response of the material. Usually the impedance, the ratio of the applied voltage to the resulting current, is computed and analyzed.
Recently, IS has been applied to analyze the microstructure of cement paste via its impedance response [2,3,4,5,6]. Most experimental techniques for analyzing pore structure, like scanning electron microscopy or mercury intrusion porosimetry , require drying and/or high vacuum. Because the properties and microstructure of cement paste are sensitive to the current moisture state and history of the material, these kind of microstructural analyses are often difficult to interpret, since removal of moisture can significantly change the microstructure. The development of a non-destructive, in situ microstructural analysis technique like IS, therefore, is proving to be useful for studies of the development of microstructure in hydrating cement pastes.
Interpreting the results of IS experiments to give information on microstructure, especially for complex materials like cement-based materials, requires some kind of theoretical model. Most experimental results are interpreted in terms of series or parallel combinations of resistors and capacitors and special elements like the "constant phase element" (CPE) . The choice of parameters in these models are not always unique, resulting in some degree of inherent ambiguity. Also, it is often incorrect to describe the complex topology of real microstructures by using simple series and parallel ideas. This has been the reason for using more complicated elements like the CPE element. These elements, however, while they can give reasonable fits to IS data, are not easily related to microstructural features.
The approach taken in this work has been to first represent microstructure via digital image-based models [8,9] or actual 2-D micrographs. X-ray tomography can also be used to give experimentally-determined 3-D microstructures . Therefore the starting point is an actual or simulated microstructure that is numerically stored and thus can be numerically analyzed. Various physical properties of these 2-D or 3-D microstructural images can then be computed using exact finite-difference or finite element algorithms. "Exact" means that the correct properties are computed for the given microstructure and choice of individual phase properties, within the limit of the finite resolution of microstructural features in the digital image. In this way the arrangement of phases within the microstructure can be correlated directly with observed bulk properties. An overall review of this modeling approach for cement-based materials has been presented by Garboczi and Bentz [11,12]. The approach is readily generalized to other materials .
The computational scheme described in this paper is a continuation of previous work on the simulation of IS experiments . The previous work used the Y − algorithm , which is limited to two dimensions but is very efficient. This algorithm directly computes the admittance of a heterogeneous microstructure, without actually solving Laplace's equation for the system. In most cases, however, microstructure must be represented in three dimensions to accurately represent complicated geometries and especially percolation phenomena. The ideas behind the Y − algorithm  have been generalized and extended to develop the three dimensional Fogelholm algorithm . The Fogelholm algorithm becomes extremely slow for systems where the highly conducting phase is far from a percolation threshold [16,17]. Since most materials of interest are not necessarily near to such a percolation threshold, we have developed a new algorithm, based on a different approach.
This new algorithm uses a finite difference approach and a conjugate gradient algorithm to solve the AC electrical equations. This algorithm is the complex analog of a previous algorithm that solved DC composite electrical problems , and may be applied to either two- or three-dimensional images. The method vectorizes well, and so runs very efficiently on modern vector supercomputers. This algorithm allows IS experiments to be simulated on arbitrary three dimensional systems.