Cellular Ceramics / p3

3.2 Modeling Structure–Property Relationships in Random Cellular Materials 277

Fig. 6 Three-dimensional Gaussian random field (GRF) models. a) Single-cut, b) two-cut, c) open-cell intersection, and d) closed-cell union. After Ref. [4].

of seed-point distributions in tessellations and network models, the size and shape distribution in Boolean or sphere-pack models, and the choice of correlation functions in Gaussian level-set models. What types of measurable information might be used to choose a model for a particular material? If the goal is to develop structure– property relations, are there particular characteristics of structure that play a key role in determining properties? Even for “simple” properties like conductivity and elasticity there seems to be no simple answer to this question.

As for structure–property relationships there are two approaches to the problem of characterizing structure depending on whether the methods apply to arbitrary structures or particular classes of models. We will focus largely on the former methods, but mention the very detailed literature on the morphology of foams (see, for example, Refs. [29, 32]). This research focuses on well-defined cells with polyhedral facets. Key geometrical questions that have been answered include estimates of the number of cell faces, the surface area of cells, and their edge length. The results depend on the spatial distribution of cells, which may be randomly chosen or generated by using algorithms that mimic froth formation. The shape of cell struts and thickness of cell walls can also be predicted by using physical principles for idealized foam models [29].

Although the correlation between cell features, such as the average number of cell faces, and foam properties is not known, detailed morphological studies of equilibrated foams could be used to tune Voronoi tessellation models. For example, the random sequential adsorption (RSA) method of generating a sphere pack whose centers are subsequently used as the seed-point distribution of a tessellation results

278 Part 3 Structure

in a model in which the number of edges per polyhedral face is about 14.9, whereas equilibrated foams have a value of around 13.7 [29] and are more monodisperse. The random sequential adsorption algorithm does not provide very dense packings [32], which in turn leads to elongated and unusual cell shapes in the tessellation process. It is possible to model more monodisperse foams with a denser sphere pack. The tessellation shown in Fig. 4f is quite monodisperse and was obtained by removing 2 % of the spheres from a bcc array and “thermally” shaking the array until an isotropic system was obtained [26]. I conjecture that the average number of edges per cell of this model will be closer to 13.7. Note that there are many types of cellular solids and foams that do not satisfy the strict assumptions of polyhedral facets. In these cases more general characterization methods are necessary.

The most obvious way of characterizing a general porous or composite material is by using the hierarchy of correlation functions of one of the phases. The first-order function is the probability that a point lies within the phase, which is just the volume fraction or relative density. The second-order function is the probability that two randomly chosen points a distance r apart both lie in one phase, and so forth. The rate of decay and presence of peaks in the correlation function indicate the degree and nature of “correlation” in the composite. Examples of normalized autocorrelation functions are shown in Fig. 7. The data are for the openand closed-cell tessellations shown in Fig. 4e and f, based on a dense hard-sphere pack, while the third model is a closed-cell tessellation based on a Poisson distribution. The Fourier transform I(q) of the correlation function, which corresponds exactly with smallangle scattering intensity, is more revealing (see Fig. 8). The peak around q = 0.25 corresponds to a correlation at wavelength 2 p/q = 25, which is the average cell diameter of the models. If an uncorrelated (Poisson) seed distribution is used the subsidiary peaks are far weaker (Fig. 8). The correlation function clearly contains important

Fig. 7 The normalized two-point correlation does not appear to provide a strong signature of structure. The dashed and solid lines are for the openand closed-cell tessellations shown in Figs. 4e and f, respectively. The dot-dashed line is a closed-cell tessellation of a Poisson distribution.

3.2 Modeling Structure–Property Relationships in Random Cellular Materials 279

Fig. 8 The Fourier transform (scattering intensity) of the correlation functions for the open- (dashed) and closed-cell (solid) tessellations. The tessellation based on a completely random (Poisson) seed distribution (dot-dashed) only shows a scattering peak at the average cell size.

morphological information, but like most measures it does not appear to have a direct link with properties: the openand closed-cell models have similar scattering profiles but very different properties, while two closed-cell tessellations with quite different scattering profiles will have similar physical properties.

The implied weak dependence of properties on the correlation function is borne out by the rigorous property-prediction methods, which prove that the properties of isotropic materials do not explicitly depend on the two-point function [1, 3]. Indeed, rigorous prediction methods indicate that third-order functions are needed to distinguish between composites, but the differentiation between materials with quite different properties is not strong. For example the bounds are continuous over a percolation transition. This implies that “higher order” information is critical in determining properties, but even accurate measurement of the third-order function is difficult.

A range of other methods of characterizing microstructure have been identified. These include surface–surface and surface–volume correlation functions, which are important in flow problems [1], and the chord-length probability function which measures the probability that a line segment in a specified phase has given length. Given a three-dimensional structure it is also possible to define a pore size distribution. The actual definition of pore size is complicated by the generally irregular shape of pores and their interconnections. Other measures which have received attention are the lineal-path function and coarseness functions, which measure the variance of (say) porosity as a function of window size. These functions and interrelations between them have been reviewed by Torquato [1].

An important goal of research in characterizing microstructure is to improve the match between model and real materials. This matching has been termed “statistical reconstruction”, where one tunes a model to match measured information. Two

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avenues have been pursued in this field: model-based and model-independent approaches. Both methods can be based on two-dimensional information, which allows a three-dimensional model to be built up from a two-dimensional image (provided the model is isotropic). Model-based approaches use quantitative statistical measures to select a statistical model with appropriate morphological parameters to match structure. Examples include matching the correlation functions of level sets of Gaussian random fields [37, 43] to an experimentally measured correlation function. A problem with this approach [44] is that the two-point function is practically nonunique, that is, many models were able to reproduce a given correlation function. To solve this problem the chord distribution was successfully employed [44] to distinguish between several candidate models. More recent work uses integral geometry (volume fraction, surface area, integral mean curvature, and genus) as a basis for choosing the parameters of polydisperse Boolean models [30]. The latter models also provided good predictions of material properties.

As noted above, model-independent methods can be used to “reconstruct” materials [45]. To match microstructure these models flip voxels from one phase to another until the statistical properties of the digital model match those of the composite. Two advantages of this approach are that first, the microstructure does not have to be “close” to a model (as in the model-based approach), and second the microstructure measures employed (e.g., the correlation functions) need not be available in a simple form. Apart from being computationally intensive, a disadvantage of not imposing some structure via a model is that fine details, such as thin struts, which tend not to be strongly emphasized by arbitrary microstructure measures (such as chord distributions and correlation functions) may not appear in the digital reconstruction.

The possibility of modeling a given composite, given statistical information about its microstructure, is a relatively new field. While there has been excellent progress, the generality and robustness of the method are not yet known. Near the percolation threshold, where just a few small connections dictate the properties, it is unlikely that statistical methods will be able to reconstruct accurate models for structure– property predictions. On the other hand, materials are not generally produced at or near this threshold. In part the requirements for modeling are reduced by the availability of micro-CT scans, but for low-contrast and/or nonporous materials, as well as materials with submicrometer structure, modeling will remain a key component of understanding properties. Moreover, once a model is chosen, parameters of the model, such as porosity and surface area, can be varied to study the influence of microstructure on properties and how they may be optimized for a given application.

3.2.4

Computational Structure–Property Relations

As discussed in Section 3.2.3 considerable effort is required to generate models which quantitatively match the statistical properties of a given material. An expedient alternative is to simply choose representative parameters for each model, and

3.2 Modeling Structure–Property Relationships in Random Cellular Materials 281

calculate theoretical structure–property relationships for each “class” of models. This approach has been undertaken for a range of materials [4, 26, 42, 46]. The elastic properties of the digitized models have been calculated by the finite-element method [27]. Instead of tabulating the results at discrete intervals simple two-para- meter curves were fitted to the data. For mediumto high-density solids the equation

was used to describe the data. Note that p0 is a fitting parameter and does not necessarily correspond to a geometrical percolation threshold. For example, for the expanding-bubbles model shown in Fig. 1b, m = 1.61 and p0 = 0.088 for volume fractions p ‡ 0.12. The formula would not be accurate if extrapolated to lower fractions.

At lower densities data for solids which remain interconnected are well described by a the power law

which can be used to extrapolate the data below the computed range. The fitting parameters for the models shown in Figs. 3–6 are reported in Tab. 1. In all cases the solid Poisson’s ratio of ms = 0.2 was used.

Table 1 Simple structure–property relations for different model porous materials. For p > pmin, the data can be described by E/Es = [(p–p0)/(1–p0)]m to within a few percent. For the last seven models (which do not have a finite percolation threshold), the data can be extrapolated by using the power law E/Es = Cpn for p < pmax. The Poisson’s ratio can be approximately described by Eq. (15) with parameters m1 and p1. The values of m1 and p1 for the last four models are estimated by extrapolating data obtained at one Poisson’s ratio (ms = 0.2).

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In general it is expected that the Young’s modulus of a porous material will depend on volume fraction, microstructure, and Poisson’s ratio of the solid phase ms. However, it has been recently proven [47–49] that E is independent of ms in two dimensions, and that E is practically independent (to within a few percent) of ms in three dimensions [4]. In contrast, the bulk and shear moduli do depend quite significantly on ms. To calculate the bulk and shear moduli it is therefore necessary to estimate the Poisson ratio of the porous material. The following simple formula has been found to provide a rough fit of the data [42]

The fitting parameters are listed in Tab. 1. Figure 9 shows the typical “flow-dia- gram” behavior of the Poisson’s ratio which tends to become independent of Poisson’s ratio of the solid phase as the density decreases [4]. This is because the microstructure dictates the amount of macroscopic lateral expansion of a body subjected to uniaxial stress at low densities. To see this consider the extreme case of diamondshaped cell made of four struts which is uniaxially loaded from corner to corner. The unloaded corners will bow out and thus cause a much more significant lateral expansion than the contribution of the “internal” expansion of the struts.

Fig. 9 Typical behavior of Poisson’s ratio of a porous material as a function of density and Poisson’s ratio of the solid skeleton. The lines represent a best fit to Equation (15). Finite-element data is for the open-cell GRF model shown in Fig. 6c.

In general most models [4] have a Poisson’s ratio of around m = 0.2 at low densities, with the interesting exception of the random open-cell Voronoi tessellation, which has a ratio of m » 0.5 [42]. Physically, this interesting behavior corresponds to a material which preserves volume under uniaxial compression; the compression is matched by lateral expansion. Equivalently, the material has much higher bulk modulus than shear modulus. The computational results are well predicted by the War-

3.2 Modeling Structure–Property Relationships in Random Cellular Materials 283

ren–Kraynik model (Eq. 9). Essentially, the bending,mode of deformation is not activated when a near-tetrahedral joint is uniformly loaded, but under shear or uniaxial loading, bending is activated. This implies K Q p, but E Q p2, so that m = 1/2 – E/6K fi 0.5 as p fi 0. To my knowledge this behavior has not been observed in real foams, although the prediction is robust to variations in seed distribution [42] and even to missing struts. Around 15 % of the struts must be deleted to reduce the Poisson’s ratio to around 0.35 [42].

Examples of comparisons between computational theories and experimental data are shown in Figs. 10–12. Figure 10 plots data for SiOC ceramic foam [34] against results for the open-cell tessellation and the expanding-bubbles model. The latter is used because of the close similarity between SiOC and the model, at least at intermediate volume fractions. For relative density p > 0.2 both models are consistent with the data, while for 0.1 < p < 0.2 the open-cell tessellation provides a much better match. This indicates that SiOC has a more structurally efficient morphology than that provided by the expanding-bubbles model.

Relative density

Fig. 10 A comparison of the expanding bubble model (–––) and open-cell (– – –) Voronoi tessellation model with data from SiOC foams ( ) [34].

Figure 11 compares data from open-cell foams with various computational theories. The models are able to span the data observed in real systems, although it is also necessary to check that the geometry of the models is correct by using micrographs or micro-CT images. Data for a closed-cell glass foam are shown in Fig. 12. The closed-cell Voronoi tessellation provides an excellent match with the data. For this review I also computed the theoretical elastic stiffness of a ceramic foam/Mg alloy composite studied by Zeschky et al. [50]. The foam ceramic support (fraction p = 0.18) has a morphology similar to that shown in Fig. 1 and is infiltrated with magnesium. Under the authors’ assumption that the moduli of the ceramic and metal are Es = 190 GPa and Em = 37 GPa, the expanding-bubbles model this gave a prediction of 51 GPa, while for the open-cell tessellation the stiffness was slightly

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3.2.5

Summary

A summary of structure–property relationships for porous materials has been provided. The two equally important aspects of the problem are the development of realistic microstructural models and accurate estimation of their properties. Analytical theories have been derived for particular microstructures in highand low-den- sity limits. These theories are useful for demonstrating the effect of particular variations of structure on properties. For example, the dilute-pore (i.e., high-density) theories can be extended to ellipsoidal inclusions, while the thin-beam theories for lowdensity periodic cells can elucidate the role of strut shape in properties. A disadvantage of simple structural models is that they do not accurately describe many materials; most cellular solids are not composed of periodic arrays of well-defined beams connected at nodes. Moreover, it is not clear how accurate these theories are for intermediate densities. Thus, if disagreement between theory and experiment is found, it is not clear if the microstructural model is at fault, or the approximation used to estimate its properties.

Computational theories for statistical models avoid this problem; deviations between experimental data and prediction definitely correspond to a difference in structure, which can be informative. In this approach it is implicitly assumed that material microstructure can be divided into a number of recognizable “classes”, and that the structure–property relationships among members of the same class will be similar. For example, there are many different types of sintered materials, but it is hypothesized that their properties can be estimated by the structure–property relationship of overlapping solid spheres (see Fig. 5a). Similarly I conjecture that foams made up of networks of tetrahedral elements can be effectively modeled by the Voronoi tessellation with a seed-point distribution given by the centers of random hard spheres. Of course there will be differences depending on which seed distribution is employed to generate the tessellation, or whether the model has been allowed to equilibrate in some way. However, it is assumed that these differences are minor compared to those found between classes of models. The models can be refined and adapted to study variants of these problems. For example, the effect of missing cell faces has been studied [4], and it is possible to model polydisperse foams by using generalized tessellation schemes [51].

Methods of making the models more accurate have been discussed. These “statistical reconstruction” procedures rely on matching statistical properties of the model to those of the real material. Since many statistical properties of isotropic materials can be measured in two-dimensions it is possible to reconstruct a three-dimensional medium from an image. If three-dimensional information is available, from microCT scans, for example, then other measures such as integral geometric techniques can be employed [52]. If a micro-CT image is available and only a single property estimate is required, then the finite-element techniques can be used directly on the image [53].

This chapter dealt with the linear-elastic properties of heterogeneous materials. Other macroscopic properties such as conductivity, diffusivity, and fluid permeabil-