Predicting the macroscopic properties of composite or porous materials with random microstructures is an important problem in a range of fields [6,19,41,38]. There now exist large-scale computational methods for calculating the properties of composites given a digital representation of their microstructure; eg. permeability [4,2], conductivity [3,37] and elastic moduli [18,29]. A critical problem is obtaining an accurate three-dimensional (3D) description of this microstructure [4,15,48].
For particular materials it may be possible to simulate microstructure formation from first principles. Generally this relies on detailed knowledge of the physics and chemistry of the system, with accurate modeling of each material requiring a significant amount of research. Three-dimensional models have also been directly reconstructed from samples by combining digitized serial sections obtained by scanning electron microscopy [26], or using the relatively new technique of x-ray microtomography [16]. In the absence of sophisticated experimental facilities, or a sufficiently detailed description of the microstructure formation (for computer simulation), a third alternative is to employ a statistical model of the microstructure. This procedure has been termed "statistical reconstruction" since the statistical properties of the model are matched to those of a two-dimensional (2D) image [30,3,4,34]. Statistical reconstruction is a promising method of producing 3D models, but there remain outstanding theoretical questions regarding its application. First, what is the most appropriate statistical information (in a 2D image) for reconstructing a 3D image, and second, is this information sufficient to produce a useful model? In this paper we address these questions, and test the method against experimental data.
Modeling a composite and numerically estimating its macroscopic properties is a complex procedure. This could be avoided if accurate analytical structure-property relations could be theoretically or empirically obtained. Many studies have focussed on this problem [19]. In general, the results are reasonable for a particular class of composites or porous media. The self-consistent (or effective medium) method of Hill [22] and Budiansky [11] and its generalization by Christensen and Lo [13] is one of the most common for particulate media [19]. No analogous results are available for non-particulate composites. A promising alternative to direct property prediction has been the development of analytical rigorous bounds (reviewed by Willis [47], Hashin [19] and Torquato [41]). There is a whole hierarchy of these bounds, each set tighter than the next, but depending on higher and higher order correlation functions of the microstructure. The original Hashin and Shtrikman [20] bounds that have been widely used by experimentalists implicitly depend on the two-point correlation function of the microstructure, although the only quantities appearing in the formulas are the individual properties of each phase and their volume fractions. To go beyond these bounds to higher-order, more restrictive (i.e., narrower) bounds, it is necessary that detailed information be known about the composite in the form of three-point or higher statistical correlation functions [5,28], which do appear explicitly in the relevant formulas. Evaluation of even the three point function is a formidable task, so use of these bounds has in the past been restricted to composites with spherical inclusions. It is now possible to evaluate the bounds for non-particulate composites [37], and it is interesting to compare the results with experimental and numerical data. If the properties of each phase are not too dissimilar the bounds are quite restrictive and can be used for predictive purposes [20]. Sometimes experimental properties closely follow one or the other of the bounds, so that the upper or lower bound often provides a reasonable prediction of the actual property even when the phases have very different properties [41,35]. It is useful to test this observation.
In this study we test a generalized version [34] of Quiblier's [30] statistical reconstruction procedure on a well-characterized silver-tungsten composite. Computational estimates of the Young's moduli are compared to experimental measurements. The composite is bi-continuous (both phases are macroscopically connected) and therefore has a non-particulate character. As such the microstructure is broadly representative of that observed in open-cell foams (such as aerogels), polymer blends, porous rocks, and cement-based materials. By comparing our computations of the moduli to the results of the self-consistent method we can test its utility for non-particulate media. An advantage of the reconstruction procedure we use is that it provides the statistical correlation functions necessary for evaluating the three-point bounds. Comparison of the Young's modulus to the bounds therefore allows us to determine the bounds' range of application for predictive purposes.