Previous: Two-Phase Interpenetrating Random
A computer algorithm has been developed that can accurately compute the impedance spectrum of a simulated microstructure in two or three dimensions. Model inputs include: 1) a digital representation of microstructure and 2) the electrical properties of each individual phase of the microstructure. The individual phase electrical properties can be read from a table of experimentally known values or can be simulated using a fitted circuit.
The calculation scheme has been shown to be very accurate for composite systems with known solutions, which provides a basis for extending the model to more complicated multi-phase composites whose solution are not known analytically.
The overlapping sphere, interpenetrating phase composite model demonstrated how non- circular, multiple arc behavior can appear even when both phases of a random two-phase composite are fully percolated. Thus what is usually referred to as "series-like behavior" can result even when it is not possible to characterize the microstructure as being a series combination of phases in any way. Also, the dangers in inferring microstructure from DC characteristics alone were clearly seen in Fig. 7, which showed that both the Maxwell-Wagner and self-consistent effective medium theories gave accurate (within a few percent) predictions of the DC resistivity. The percolation aspects of the microstructures implied by these two equations are much different, however. The additional use of finite frequency data showed that the self-consistent theory gave a reasonably good prediction for the overall character of the impedance curve, while the Maxwell- Wagner equation badly misrepresented the shape of the impedance curve.
While this approach has been developed to study the impedance response of cement-based materials, it is generally applicable to any heterogeneous material whose microstructure can be represented by a two- or three-dimensional digital image, if the electrical properties of the individual phases of the microstructure are known. Perfect bonding between phases is usually assumed, although interface impedances can be readily handled by the algorithm. A direct digital image representation of microstructure that can be compared with accurate numerical impedances allows quantitative microstructure-property relationships to be developed.
The ability to directly compute the impedance curve of any microstructure allows better microstructural inferences from experimental IS curves, removing dependence on overly simple resistor-capacitor circuit models. Such circuit models are often very useful, and even quantitatively accurate in some instances, but may hinder progress in using IS in the study of the complex microstructure of random composite materials [34,35].
Current computing capabilities of machines that are generally available to the average academic user limit three-dimensional models to sizes not much larger than 1003, although that limit will only improve as larger memory and higher speed computers become more widely available to the materials science community. These improvements in computing power will make possible the use of higher resolution digital images, which will in turn result in more accurate numerical predictions of the complex electrical properties of materials with intricate random microstructures.