Cellular Ceramics / p3

267

3.2

Modeling Structure–Property Relationships in Random Cellular Materials

Anthony P. Roberts

3.2.1

Introduction

Today new and advanced materials are constantly being developed. The central goal is to develop materials with superior properties to those currently available, or to achieve similar properties at a reduced cost or weight. Since materials synthesis and extensive empirical testing are both time-consuming and expensive, it is important to guide the process as far as possible by theoretical modeling. Composite and porous materials have found broad application in virtually every field of science and engineering because they offer combinations of properties not available in their constituent materials. A key problem is to understand and quantify the relationship between the internal structure of these materials and their properties. The resulting structure–property relationships are used for designing and improving materials, or conversely, for interpreting experimental relationships in terms of microstructural features.

Ideally, the aim is to construct a theory that employs general microstructural information to make accurate property predictions. Since the properties of advanced composites and porous materials are dependent on their complex internal structure, it has proved extremely difficult to develop accurate structure–property relationships in all but the simplest of cases. Two main problems must be solved in order to generate accurate structure–property theories. First, an appropriate model of microstructure is needed; second, the properties of the microstructure must be accurately evaluated. Often these are contradictory goals; a realistic structural model generally prohibits the use of analytical techniques to predict properties, and conversely, the requirement of making an accurate property prediction generally dictates an oversimplified model of structure.

This chapter surveys past results and recent progress in the related fields of measuring and modeling microstructure, and predicting the properties of random materials. Attention is restricted to the influence of the shape of the solid–pore morphology on material properties, rather than the role of the internal microstructure of the solid struts (which is taken as given). We also deal only with the rigidity (stiffness) of porous materials, rather than more complex mechanical properties such as failure stress.

Cellular Ceramics: Structure, Manufacturing, Properties and Applications.

Michael Scheffler, Paolo Colombo (Eds.)

Copyright 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-31320-6

268 Part 3 Structure

No single model or simulation can attempt to reproduce the rich variety of structures that can occur in porous materials in general, or even in the subset defined by cellular ceramics. A brief introduction is provided to the main classes of statistical models that have been used to represent porous materials. These include Boolean, level-set Gaussian, coalescing bubble, and Voronoi tessellation models. The statistical characterization of porous materials and how this information can be used to “tune” models so that they better mimic real materials are also discussed. This offers the possibility of statistically reconstructing three-dimensional models from experimentally measured two-dimensional images. Technical aspects of rigorously predicting composite properties have been reviewed in two recent monographs [1, 2]. A simplified summary of these theoretical results is provided for general porous materials, as well as results more directly relevant to cellular materials. Recent practical advances in computational theories which are pertinent to cellular ceramics are reviewed.

3.2.2

Theoretical Structure–Property Relations

The deformation of composite materials under stress is governed by the equations of elasticity. In principle it is therefore necessary to solve these equations within a representative element of a heterogeneous material to determine its macroscopic behavior [1–3]. All of the methods described below follow this approach, although other theoretical approaches bypass the elasticity equations altogether. The minimum solid area models, reviewed in Part 4, are an example. An example of cellular ceramic and a simple model of bubble growth and coalescence are shown in Fig. 1. To illustrate theories discussed below computational estimates of Young’s modulus as a function of density for this model of “expanding bubbles” are plotted in Fig. 2.

Fig. 1 a) An example of a CT scan of a cellular ceramic. Image courtesy of J. Zeschky, see also Ref. [50]. b) A simple “expandingbubbles” model generated by dilating spherical pores seeded at the centers of a dense hard sphere pack.

3.2 Modeling Structure–Property Relationships in Random Cellular Materials 269

Relative density

Fig. 2 A comparison of five theories with finite-element data for the expanding-bubbles model shown in Fig. 1. The predictions and bounds are described in the text. The solid line is a fit to the data based on Eq. (13).

The simplest approach to estimating the properties of a composite material is to use the rule of mixtures, which states that the Young’s modulus is

where p is the volume fraction of phase one and E1 (E2) is the modulus of phase 1 (2). This formula is exact for a material composed of slabs of each material, but only for stress parallel to the direction of the slabs. For a porous material p = / s corresponds to the relative density and we take E1 = Es (and E2 = 0). The rule predicts a relative modulus of E/Es = p, which significantly overestimates the effective modulus (see Fig. 2).

The best known theories for more realistic models apply to low concentrations of spherical inclusions in an otherwise homogeneous matrix [3]. For randomly distributed dilute spherical pores of concentration 1–p, the relative Young’s modulus is giv-

Here attention is restricted to the case where the Poisson ratio of the solid structure is ms = 0.2. Although algebraically quite complex, the result only differs by a few percent [4] if 0 < ms < 0.4, which is generally the case. Note that the result applies to a solid with a few remote bubbles, that is, the incipient stages of foam formation.

There are two main approximate methods of extending Eq. (2) to finite concentrations of inclusions. The first method has been called the self-consistent or effectivemedium approximation (E.M.A.) [3, 5, 6]. For general ms and phase properties the formula cannot be expressed in closed form, but for hollow inclusions and a solid Poisson’s ratio of ms = 0.2 the result is given again by Eq. (2), which is therefore expected to be useful outside the strict dilute limit. However, as can be seen in

270 Part 3 Structure

Fig. 2, the result is inaccurate for the expanding-bubbles model with relative densities below p = 0.8. The second common generalization of the dilute result to finite porosity is provided by the differential effective medium (D.E.M.) theory (see Ref. [7] for a review). Again the result is only expressible in closed form if ms = 0.2 and the inclusion phase is hollow. This gives

The differential result provides a very good approximation for the expanding-bub- bles model for relative densities p > 0.3 (see Fig. 2). For intermediate densities, this is probably more fortuitous than an indication of a good model: At low relative densities the actual microstructure of the bubble-growth model is quite different from that “built-in” [1] to the differential result [8].

To go beyond the spherical-inclusion approximation two relevant strands of literature have emerged for porous materials. The fully general approach incorporates arbitrary microstructural information, typified by variational bounding methods. An alternative approach focuses on periodic structures.

The rigorous bounding methods are dealt with in several comprehensive reviews [1–3, 9]. For high-contrast (e.g., porous) composites the bounds tend to be so far apart that they lack predictive power. Indeed, for porous materials the lower bound is zero, but there is evidence that the upper bound Eu can sometimes provide a reasonable approximation [1], as well as a useful theoretical upper limit. The most commonly used result is the Hashin–Shtrikman bound (H.S. bnd) for isotropic materials. For porous solids the bound simplifies to

The result is only weakly dependent on ms via the constant C, and C = 1 for ms = 0.2. Note that the rule of mixtures exceeds the bound (which does not apply to anisotropic materials). A key theoretical advantage of the bounding approach is that it can be extended to incorporate arbitrary microstructural information. For example the three-point (3-p) bounds of Milton and Phan-Thien [10] for ms = 0.2 are of the form of Eq. (4) with [4]

The parameters g and f involve multidimensional integrals of the three-point correlation functions of the microstructure. These have been tabulated for a wide range of materials [4, 9, 11, 12]. Although progress has been made for many realistic classes of models, it is quite a challenging task to actually derive the correlation functions for a given statistical model (e.g., no results exist for Voronoi tessellations). The improvement of the three-point bounds over the Hashin–Shtrikman bounds is shown in Fig. 2. The three-point bound provides a reasonable approximation for p ‡ 0.4, but significantly overestimates the modulus of the bubble model for lower densities. Note that the bound has been evaluated for uncorrelated pore cen-

3.2 Modeling Structure–Property Relationships in Random Cellular Materials 271

ters, which does not exactly match the expanding-bubbles model. Nevertheless, the bound for the bubble model is expected to be slightly greater than the bound for uncorrelated pores, because the struts are more uniform in the former model (and hence provide greater stiffness). In general it has been shown that the three-point bound does not provide accurate predictions for low-density (p < 0.4) porous materials [4].

The most effective approach for low-density cellular materials has been to model the properties of periodic arrays by approximating the behavior of single cells with thin-beam theories. The results provide qualitative predictions of foam properties and illustrate the basic mechanisms of deformation in cellular materials. First consider a simple cubic array of uniformly spaced intersecting aligned struts. From elementary considerations, the Young’s modulus is E/Es » p/3 for uniaxial compression along a strut axis. The linear dependence of modulus on density is typical of model foams that contain straight-through struts that traverse the extent of the sample; longitudinal compression or tension is the only mode of deformation [13–15]. Since most foams do not contain straight-through struts, beam bending comes into play [14–17]. In this case a quadratic dependence of the modulus on density is observed. The most commonly used result for open-cell foams is [18]

where the prefactor C » 1 and Poisson’s ratio have been empirically determined. This semi-empirical formula broadly describes data obtained for many different types of foams.

Zhu et al. [15] and Warren and Kraynik [19] derived analytic results for an opencell tetrakaidecahedral model packed in a body-centered cubic array. The model is shown in Fig. 3a. The results of Zhu et al. for Young’s modulus and Poisson’s ratio

Fig. 3 Models of cellular solids. a) The periodic tetrakaidecahedral model [28]. b) and c) are node-bond models with coordination numbers (average number of connections per node) of 5.5 and 12, respectively [42].

0.900. Note that Poisson’s ratio depends on orientation. The notation m12 corresponds to expansion measured in the h010i or h001i directions. Note that the foam is relatively stiff under uniform compression, with the bulk modulus given by K/Ks = 1/9·p, and m12 fi 0.5 as p fi 0.

Several methods have been proposed to derive analytic predictions for random isotropic foams. A typical result, which performs an isotropic average of randomly

who noted that the results are equivalent to those of Gent and Thomas [20]. In the low-density limit, the same results were derived for a rotationally averaged simple cubic structure [14]. The absence of bending in these models is indicated by the linear dependence of Young’s modulus on density.

Warren and Kraynik [14] derived analytic results for the properties of a foam comprised of isotropically oriented tetrahedrally arranged struts. The geometry can be visualized as a node located at the center of a tetrahedron with equilateral faces, in which the four struts (separated by an angle of 109.5 ) connect the central node to

model, beam bending is the primary mode of deformation for uniaxial compression. However, Eq. (9) imply K/Ks = 1/9·p, that is, bending is not activated under pure compression. Like the tetrakaidecahedral model, Poisson’s ratio of the model therefore tends to 0.5 at low densities.

Christensen [13] derived a result for a closed-cell material comprised of randomly

where the subscript “s” indicates the solid phase. The linear dependence (n = 1) of modulus on density is typical for cellular materials with straight-through elements. In this case, cell-wall stretching is the only mechanism of deformation.

Analysis of more complex closed-cell foams is very difficult, but computational results [21–25] have been obtained for the closed-cell tetrakaidecahedral foam shown in Figure 3a. Simone and Gibson [25] recently found that Young’s modulus is nearly equal (within 10 %) for loading in the h100i, h111i, and h110i directions. For the density range 0.05 < p < 0.20, their results for the h100i direction were fitted with the formula

3.2 Modeling Structure–Property Relationships in Random Cellular Materials 273

which is consistent with the result E100/Es » 0.33 p obtained by Renz and Ehrenstein

[24]. For the case where the face thickness is 5 % of the edge thickness, Mills and Zhu [23] found E100/Es » 0.06 p1.06 in the density range 0.015 < p <0.1.

For closed-cell materials Gibson and Ashby proposed the semi-empirical formula

where u is the fraction of solid mass in the cell-edges (the remaining fraction 1–u is in the cell faces). The first term of Eq. (12) accounts for deformation in the cell edges. Note that the case u = 1 corresponds to the semi-empirical formula for opencell solids (i.e., Eq. 6). The second term corresponds to stretching deformation in the cell faces. The result provides good agreement with data for closed-cell foams when 0.6 £ u £ 0.8 [18]. However, some foams have u < 0.5 [23], in which case Eq. (12) violates the Hashin–Shtrikman upper bound [26]. The idea of partitioning stiffness into contributions from the “open-cell” skeleton and “closed-cell” faces was shown not to work for materials with an increasing number of missing faces [26]. This is because the interaction between microstructure is inherently nonlinear.

The above models generally assume a relatively simple microstructure in order to “solve” the equations of elasticity; that is, they prioritize solution of the second problem of property prediction over the first. The bounding methods incorporate arbitrary microstructure, but are limited to particular classes of models for which highorder correlation functions are available and are not guaranteed to provide an accurate predictive method. An alternative which has been made viable by the increasing speed of computers is to use the finite element method to estimate properties of realistic models [27]. This leads to new questions about designing accurate models, but also allows other pertinent questions to be tackled. For example, how do the various theories for porous/composite materials perform and how does the effect of disorder alter the properties of solids. The latter means the effect of randomness, irregular or missing struts, and missing faces. Moreover, the properties of noncellular porous materials such as aerogels and bone can also be studied by this method.

3.2.3

Modeling and Measuring Structure

There are three broad approaches to modeling random materials. The first is to simulate material formation from first principles. Generally, detailed geometrical, physical, and chemical elements of the relevant processes must be accurately modeled, making simulation quite complex. Whole fields of research are devoted to some of the many classes of porous and composite materials. For example, determining minimal surface energy configurations of “single cells” of periodic foams [28, 29] is a significant undertaking. Other well-known examples for which detailed simula-

274 Part 3 Structure

tions are undertaken include spinodal decomposition, bone growth, aerogel formation, and sphere packing. Since these processes are all relevant to modeling porous and cellular ceramics it is impossible to develop a unified simulation scheme for all materials of interest. A second approach is to employ and adapt a variety of realistic statistical models that appear to mimic the structure observed in actual materials. A quantitative correspondence between the model and real structures can be made by choosing the model to match statistical information about the material. For example, the model may be chosen to match the volume fraction and correlation function (equivalently, small-angle scattering profile) of a sample. A third method that has recently become viable is to directly compute the properties of a digital model obtained from a micro computer tomography scan [30]. This section focuses on the second approach, as random models are particularly convenient for generating computational structure–property relationships.

The most common models of random cellular solids are generated by Voronoi tessellation of distributions of “seed points” in space. Around each seed is a region of space that is closer to that seed than any other. This region defines the cell of a Voronoi (or Dirichlet) tessellation [31]. Placing a solid wall at each face of these cells gives a closed-cell tessellation. An open-cell tessellation results if only the edges where two cell-walls intersect are defined as solid. For several different random (e.g., Poisson) distributions of seed points, the average number of faces per cell falls in the range 13.7–15.5 [32].

The Voronoi tessellation can also be obtained [31] by allowing spherical bubbles to grow with uniform velocity from each of the seed points. Where two bubbles touch, growth is halted at the contact point, but allowed to continue elsewhere. In this respect the tessellation is similar to the actual process of liquid-foam formation [33]. Of course, physical constraints such as minimization of surface energy will also play an important role in shaping the foam. Depending on the properties of the liquid and the processing conditions, the resultant solid foam will be comprised of open and/or closed cells. Figure 4 shows a summary of this process.

The amount of order in the Voronoi tessellation depends on the order in the seed points. If regular arrays are used, ordered anisotropic foams will result. Indeed the open-cell models used by Warren and Kraynik [19], Zhu et al. [15] and Ko [16] turn out to be equivalent to Voronoi tessellations of the body-centred cubic (bcc, Fig. 3a), face-centred cubic, and hexagonal close-packed lattices. If a purely random (Poisson) distribution of points is used, highly irregular isotropic foams containing a wide size distribution of large and small cells will result.

It is worth noting that the tessellation of the bcc array (the tetrakaidecahedral cell model discussed above) is a reasonable approximation to the foam introduced by Lord Kelvin [19, 22, 28]. The cells of the Kelvin foam are uniformly shaped, fill space, and satisfy Plateau’s law of foam equilibrium (three faces meet at angles of 120 , and four struts join at 109.5 ). For this to be true, the faces and edges are slightly curved [28], unlike those of the tetrakaidecahedral cell model.

While the importance of the openand closed-cell tessellations is clear, a related model appears relevant to certain types of cellular ceramics. Figure 1 shows a porous ceramic in which the bubbles have coalesced to an open-cellular structure with cir-

276 Part 3 Structure

range of other statistical models that have been studied. The Boolean models involve placing solid objects or pores at uncorrelated points in space. Boolean models [1, 35] have been studied in great detail because it is relatively easy to determine statistical measures of the models. It is possible to generate other realistic microstructure models by using the level-cut Gaussian random field (GRF) scheme [36–38]. One starts with a Gaussian random field y(r) which assigns a (spatially correlated) random number to each point in space. The random field can be generated by a number of methods, including summing sinusoids with random coefficients. The distribution of coefficients entirely determines the statistical properties of the resultant field (see for example, Refs. [37–39]). A two-phase solid–pore model can be defined by letting the region in space where –¥ < y(r) < b be solid, while the remainder corresponds to the pore space (Fig. 6a). This type of structure is reminiscent of that generated in spinodal decomposition. An interesting two-cut GRF model [38] can be generated by defining the solid phase to lie in the region –b < y(r) < b (Fig. 6b). This type of structure, observed in microemulsions, has formed the basis for biomimetic silica materials [40]. Openand closed-cell models can be obtained from the two-cut version by forming the intersection (Fig. 6c) and union (Fig. 6d) sets of two statistically independent two-cut GRF models [41]. The open-cell model has been used to model aerogels [41], while the closed cell model is similar to the “wavy” pores seen in some foamed polymers [26].

Fig. 5 Boolean models of porous media. a) Overlapping solid spheres, b) spherical pores, and c) oblate spheroidal pores (aspect ratio four). After Ref. [4].

Figure 3 shows two “node-bond” network models which were studied to determine the influence of coordination number on properties [42]. The nodes are defined as the centers of hard spheres. Two nodes are connected by a strut if they are closer than a specified distance. If a value of 1.1 times the original sphere diameter is used a loosely connected structure results. The average number of connections that a node has is 5.5. If the interconnection distance is increased to 1.5 the coordination number increases to 12.

While statistical models capture particular qualitative features of many different materials it is natural to ask how well a model is able to mimic a given microstructure. Since there are many degrees of freedom in these models, it is possible to tune them to some extent to match measured information; for example, the specification