Statistical models of microstructure

We focus on two classes of statistical models for which it is possible to evaluate the microstructure parameters ($\zeta$, $\eta$) that occur in the above rigorous theories. Some of the models are based on spheroidal inclusions, and should be relevant to the effective medium theories. The models are also known to mimic some realistic materials (see e.g. (Torquato 1991) and (Roberts and Knackstedt).

The most well known class of statistical models are generated using the 'Boolean' scheme, in which a model is generated by placing objects at random (uncorrelated) points in space. Since the objects are uncorrelated, they can overlap. We consider an overlapping solid sphere model, and its inverse, the overlapping spherical pore or 'swiss-cheese' model (see figure 1). The latter is obtained by creating pores in a solid matrix. These models have a long history (Serra, 1988), and the microstructure parameters $\zeta$ and $\eta$ have been evaluated (Torquato, 1991),(Helte, 1995). For comparison with the SCM and DM, we also considered oblate spheroidal pores with an aspect ratio of four (see figure 1). The correlation functions of this model have not been computed to our knowledge.

Figure 1: Boolean models of porous media.
(a) Overlapping solid spheres, (b) spherical pores and (c) oblate spheriodal pores (aspect ratio four).

These Boolean models have percolation thresholds, when the objects are considered to be one phase and the background a second phase. For the overlapping sphere model, starting with a matrix and adding spheres, the sphere phase percolates at a volume fraction of about 0.29 (Garboczi et al. , 1995), while the matrix, which starts out continuous, loses continuity above a sphere volume fraction of about 0.97 (Torquato, 1991). When the random objects are prolate or oblate spheroids, the percolation threshold of the objects has been computed Garboczi et al. , 1995), but not the matrix, which is a much harder computational problem. The percolation threshold for an oblate object of aspect ratio four (width divided by thickness), is about 0.2, which is less than that for spheres. The 'isoperimetric' theorem conjectures that for Boolean models with a convex Euclidean shape, the sphere gives the largest volume fraction of overlapping objects at the percolation threshold (Garboczi et al. , 1995).

It is possible to generate other realistic microstructure models using the level-cut Gaussian random field (GRF) scheme. One starts with a Gaussian random field y(r) which assigns a (spatially correlated) random number to each point in space. A two-phase solid-pore model can be defined by letting the region in space where − ∞ < y(r) < β be solid, while the remainder corresponds to the pore-space (Fig. 2a). The solid fraction has a percolation threshold of about 0.13 (Roberts & Teubner, 1995). Since the model is symmetric the pores become continuous at about 0.87. An interesting 'two-cut' GRF model  (Berk, 1987) can be generated by defining the solid phase to lie in the region − β < y(r) < β (Fig. 2b). The solid phase of the two-cut model remains connected at all volume fractions, i.e. the percolation threshold is zero. This is because the solid walls of the microstructure do not separate into pieces as the volume fraction decreases, but instead become thinner. Open- and closed-cell models can be obtained from the two-cut version by forming the intersection (Fig. 2c) and union (Fig. 2d) sets of two statistically independent two-cut GRF models (Roberts, 1997). By construction, the percolation threshold of the open- and closed-cell models is also zero. It is important to note that in no way can these random field models be considered as dispersions. However, at dilute porosities, the pores do become disconnected.

Figure 2: Three-Dimensional Gaussian random field (GRF) models.
(a) single-cut, (b) two-cut, (c) open-cell intersection (d) closed-cell union.

The random fields on which the models are based can be entirely specified by the field-field correlation function G(r₁,r₂) y (r₁) y (r₂) where · denotes a volume average. We only consider isotropic and stationary random fields, in which case G(r₁, r₂) = g(r) with r = |r₁ - r₂|. We employ the function

$$g(r)= \exp\left( -\frac r\xi \right) \left( 1+ \frac r\xi \right) \frac{\sin (2\pi r/d) }{ (2\pi r/d)}.$$ (3.1)

The parameter d controls the position of the maximum in the correlation function, which roughly governs the cell size, while the values of the correlation length $\xi$ and parameter deffect the properties (e.g. roughness) of the pore-solid interface in the level-cut GRF model. We choose the length scales as $\xi=\frac6\pi \mu{\rm m}=1.91 \mu{\rm m}$ and $d=\sqrt{6} \mu{\rm m}=2.45 \mu{\rm m}$, which corresponds to a surface area to total volume ratio of 1 (µm)² / (µm)³ when β = 0 for the single-cut GRF model.

To evaluate the bounds it is necessary to derive the two- and three-point correlation functions of the models. This has been done for the single (Roberts & Teubner, 1995) and two-cut models (Roberts & Knackstedt, 1996), although the results are quite cumbersome and not repeated here. The results can be generalised for the open-cell and closed-cell models as follows. In the usual way, we define an indicator function Θ (r) that is unity in the solid and zero in the pore space. The volume fraction, and correlation functions can be defined by $p=\langle \Theta({\bf r}) \rangle$, $p^{(2)}_{ij}=p^{(2)}(r_{ij})= \langle\Theta({\bf r}_i)\Theta({\bf r}_j)\rangle$, and $p^{(3)}_{ijk}=p^{(3)}(r_{ij},r_{ik},r_{jk})= \langle \Theta({\bf r}_i) \Theta({\bf r}_j) \Theta({\bf r}_k) \rangle$where $r_{ij}=\vert{\bf r}_i-{\bf r}_j\vert$. The fact that the correlation functions only depend on the distance between points reflects our restriction to statistical stationarity and isotropy. Now suppose we have two independent, but statistically identical, random materials having indicator functions Φ(r) and Ψ(r) with volume fraction q, and correlation functions q ⁽²⁾_{i j} and q⁽³⁾_ijk. If we form the intersection set Θ (r) = Φ (r) Ψ (r) the volume fraction is just p = Φ (r) Ψ (r) = Φ (r)Ψ(r) = q². Similarly, the correlation functions are p⁽²⁾_ij = (q⁽²⁾_ij)² and p⁽³⁾_ijk = (q⁽³⁾_ijk)². These equations can be used to calculate the correlation functions of the open-cell GRF model, which is defined by the intersection of two two-cut GRF models. For the closed-cell GRF model the union set is formed by taking Θ (r) = Φ (r) + Ψ (r) − Φ (r) Ψ (r). In this case, the volume fraction is p = q(2 − q) and the correlation functions are

p⁽²⁾_ij = 2q² + 2(1 − 2q) q⁽²⁾_ij + (q⁽²⁾_ij)² (3.2)

p⁽³⁾_ijk = 2(q⁽²⁾_ij + q⁽²⁾_ik + q⁽²⁾_jk) (q + q⁽³⁾_ijk) + 2(1 − 3q) q⁽³⁾_ijk − (q⁽³⁾_ijk)²

− 2(q⁽²⁾_ijq⁽²⁾_ik + q⁽²⁾_ik q⁽²⁾_jk + q⁽²⁾_jk q⁽²⁾_ij). (3.3)

The use of intersection and random sets to extend other types of microstructural models has also been employed by (Jeulin and Savary 1997).

We used the quadrature method of (Roberts and Knackstedt 1996) to evaluate $\zeta$ and $\eta$ for the models described above (excluding the oblate pore model). The results, presented in Table 1, have not been previously reported for the open- and closed-cell GRF models. Application of the parameters is not restricted to porous materials. Data in the table can be used to bound the elastic moduli and electrical or thermal conductivity of composite (i.e. non-porous) materials (Torquato, 1991). The parameters $\zeta$ and $\eta$ have also been evaluated for several other models (Torquato, 1991), (Jeulin & Savary, 1997).

Table 1: The microstructure parameters which appear in three-point bounds and expansions of the bulk and shear moduli, and the electrical conductivity. The results for solid spheres were previously calculated by (Torquato 1991). Since the spherical pores are the 'inverse' of solid spheres the parameters are related by $\zeta '=1-\zeta$' = 1 − , ' = 1 − $\eta '= 1-\eta$ for p' = 1 − p.
	single-cut GRF		two-cut GRF		open-cell GRF		closed-cell GRF		solid spheres		spherical pores
p	$\zeta$	$\eta$	$\zeta$	$\eta$	$\zeta$	$\eta$	$\zeta$	$\eta$	$\zeta$	$\eta$	$\zeta$	$\eta$
0.05	0.11	0.07	0.89	0.73	0.48	0.34			0.03	0.04	0.40	0.30
0.10	0.16	0.12	0.84	0.65	0.51	0.36	0.89	0.73	0.06	0.07	0.44	0.34
0.15	0.20	0.16	0.82	0.61	0.53	0.39	0.86	0.69	0.09	0.11	0.48	0.38
0.20	0.25	0.21	0.80	0.59	0.56	0.42	0.84	0.67	0.11	0.15	0.52	0.42
0.30	0.33	0.30	0.79	0.60	0.61	0.48	0.82	0.65	0.17	0.22	0.58	0.49
0.40	0.41	0.40	0.79	0.63	0.66	0.55	0.81	0.66	0.23	0.29	0.65	0.56
0.50	0.50	0.50	0.80	0.67	0.70	0.62	0.80	0.68	0.29	0.37	0.71	0.63
0.60	0.58	0.60	0.82	0.72	0.75	0.69	0.81	0.72	0.35	0.44	0.77	0.71
0.70	0.67	0.70	0.84	0.78	0.80	0.76	0.81	0.76	0.42	0.51	0.83	0.78
0.80	0.75	0.79	0.87	0.84	0.85	0.83	0.82	0.81	0.48	0.58	0.89	0.85
0.90	0.84	0.88	0.90	0.91	0.89	0.90	0.84	0.86	0.56	0.66	0.94	0.93
0.95	0.88	0.92	0.91	0.93	0.91	0.92	0.85	0.89	0.60	0.70	0.97	0.96