Statistical models of microstructure

The two basic models we employ to describe two-phase composite microstructure are the overlapping sphere model and the level-cut Gaussian random field (GRF) model. In this section we review the statistical properties of these models which are useful for reconstructing composites. The simplest, and most common, quantities used to characterize random microstructure are p, the volume fraction of phase 1, s_v, the surface area to total volume ratio and p⁽²⁾(r), the two-point correlation function (or (r) [p⁽²⁾(r) - p² ]/[p - p²] the auto-correlation function). p⁽²⁾(r) represents the probability that two points a distance r apart lie in phase 1. Here we only consider isotropic materials where p⁽²⁾ does not depend on direction. Also note that p = p⁽²⁾(0) and s_v = -4dp⁽²⁾(0)/dr.

Realizations of the overlapping sphere model [46] are generated by randomly placing spheres (of radii r₀) into a matrix. The correlation function of the phase exterior to the spheres (fraction p) is p⁽²⁾(r)=p^v(r) for r<2r₀ and p⁽²⁾(r)=p² for r > 2r₀ where

$$v(r)=1+\frac34 \left(\frac r{r_0}\right) -\frac1{16}\left(\frac r{r_0}\right)^3,$$

(1)

and the surface to volume ratio is s_v = -3p ln p /r₀. With modification it is also possible to incorporate poly-dispersed and/or hollow spheres. The overlapping sphere model is the most well characterized of a wider class called Boolean models, which have been recently reviewed by Stoyan et al. [39].

The internal interfaces of a different class of composites can be modeled by the iso-surfaces (or level-cuts) of a stationary correlated Gaussian random field (GRF) y(r) (so called because the value of the field at randomly chosen points in space is Gaussian- distributed). Moreover, if r is fixed, the distribution over an ensemble will also be Gaussian. Correlations in the field are governed by the field-field correlation function g(r) = (y(0)y(r)) which can be specified subject to certain constraints [|g(r)| < g(0), lim_r g(r) 0]. Invariably g(0) is taken as unity. A useful general form for g is [34]

$$g(r)=\frac{e^{-r/\xi}-(r_c/\xi)e^{-r/r_c}}{1-(r_c/\xi)} \frac{\sin 2\pi r /d}{2\pi r /d}. \end{displaymath} img21.gif$$ (2)

The resulting field is characterized by a correlation length , domain scale d and a cut-off scale r_c. The cut-off scale is necessary to ensure 1 ­ g(r) ~ r² as r 0; fractal iso-surfaces are generated if 1 - g(r) ~ r. There are many algorithmic methods of generating random fields. A straight forward method is to sum N (~1000) sinusoids with random phase and wave-vector

$$y(\mbox{\boldmath$r$})=\sqrt{\frac{2}{N}}\sum_{i=1}^{N} \cos... ...{\mbox{\boldmath$k$}}_i \cdot {\mbox{\boldmath$r$}} + \phi_i), \end{displaymath} img26.gif$$ (3)

where $\phi_i$ _i is a uniform deviate on [0, 2) and $\hat{\mbox{\boldmath$k$}}_i$ is uniformly distributed on a unit sphere. The magnitude of the wave vectors k_i are distributed on [0, $[0,\infty)$) with a probability (spectral) density P(k). The density is related to g(r) by a Fourier transform [g(r) = ₀$\INT_0^\infty P(k)\sin kr (kr)^{-1} dk$ P (k sin kr(kr)^-1 dk]. Note that P(k)>0 specifies an additional constraint on g(r). Although this formulation of a GRF is intuitive, the Fast Fourier Transform method is more efficient [1,37].

Following Berk [7] one can define a composite with phase 1 occupying the region in space where $\alpha\leq y(\mbox{\boldmath$r$}) \leq \beta$ y(r) and phase 2 occupying the remainder. The statistics of the material are completely determined by the specification of the level-cut parameters and the function g(r) (or P(k)). The volume fraction of phase 1 is

$$p=h=p_\beta-p_\alpha\;\;\;\;{\rm where}\;\;\;\; p_\gamma=(2\... ..._{-\infty}^\gamma e^{-t^2/2} dt\;, \;\;\gamma = \alpha, \beta. \end{displaymath} img33.gif$$ (4)

Berk [7] and Teubner [40] have shown that the two point correlation function is p⁽²⁾(r)=h(r) where

The auxiliary variables h and h(r) are needed below. The singularity at t=1 can be removed with the substitution t = sin $t=\sin\theta$. The specific surface is s_v=-4h'(0) where

$$-h'(0) = \frac{\sqrt{2}}{2\pi}\left( e^{-\frac12 {\alpha^2}}... ...2 \beta^2} \right)\sqrt{\frac{4\pi^2}{6d^2}+\frac{1}{2r_c\xi}} \end{displaymath} img37.gif$$ (6)

with g given by Eqn. (2).

Many more models (for which p⁽²⁾(r) can be simply evaluated) can be formed from the intersection and union sets of the overlapping sphere and level-cut GRF models. Here we define a few representative models which have been shown to be applicable to composite and porous media. A normal model (N) corresponds to Berk's formulation. Models can also be formed from the intersection (I) and union (U) of two statistically identical level-cut GRF's. Another model, I_n, formed from the intersection of n primary models, has also been found useful. The statistical properties (p, s_v and p⁽²⁾) of each model are given in terms of the properties of Berk's model [Eqns. (4), (5) and (6)] in Table 1 [34].

Table 1: The volume fraction p, two-point correlation function p₂(r) and surface to volume ratio s_v of models N, I, U and I_n in terms of the properties [h, h(r) and h'(0)] of Berk's two-level cut Gaussian random field model. The formula h_p is used for calculating the level-cut parameters (see Table 2).

Mod.	p	p(2)(r)	sv	hp
N	h	h(r)	-4h'(0)	p
I	h2	h2(r)	-8hh'(0)	$\sqrt{p}$
U	h(2-h)	[2h2+2h(r)	-8(1-h)h'(0)	$1-\sqrt{1-p}$
		-4hh(r)+h2(r)]
In	hn	hn(r)	-4nhn-1h'(0)	p1/n

Since the volume fraction of the models is a function of both level-cut parameters (, ) there is a continuum of choices which correspond to a given volume fraction. For example ( $(\alpha,\beta)$, ) = ( -$(-\infty,0.84)$, 0.84), (-0.84,-0.25), (-0.25,0.25), (0.25,0.84) and (0.84, $(0.84,\infty)$ in Berk's model all correspond to p=20%. The final two choices are statistically identical to the first two and therefore provide nothing new. We note that a small change in these parameters will only slightly alter the microstructure, so as a compromise between simplicity and generality it is suggested only three distinct cases be considered: (i) the common single-cut field ( $\alpha=-\infty$ = -); (ii) a symmetric two-cut field ( = - ) an (iii) an asymmetric two-cut field. A concise way of expressing this is to take

where h_p for models N, I, U and I_n is given in Table 1 and c [0,1]. Setting c=0, 1 and ½ gives cases (i), (ii) and (iii) respectively. The implicit formula for finding ( $(\alpha,\beta)$ , ) from p and p $p_\beta$ is shown in Eqn. (4). As an example the results of the calculation for nine different models (N, I and U at c=0, ½ and 1) at volume fraction 20% is shown in Table 2. The model N (c=0) is the single level cut GRF previously used by Quiblier [30] and Teubner [40]. Model I (c=1) has been used to model aerogels [33] and model N (c=1) is Berk's [7] two level cut model of microemulsions.

Table 2: The level-cut parameters $\alpha$ and $\beta$ for different Gaussian random field models are calculated by solving Eqn. (4) where p $p_\alpha=\frac c2 - \frac c2 h_p$ = c/2 - c/2h_p and p $p_\beta=\frac c2 + (1-\frac c2) h_p$ = c/2 + (1-c/2)h_p. h_p is shown in Table 1 for each model and c = 0, ½, 1. This table shows the results of the calculation for volume fraction p=20%.

Model	Standard		Asymmetric		Symmetric
type	one-cut		two-cut		two-cut
	c=0		c=1/2		c=1
	$\alpha$	$\beta$	$\alpha$	$\beta$	$\alpha$	$\beta$
Normal (N)	-$\infty$	-0.84	-0.84	-0.25	-0.25	0.25
Intersection (I)	-$\infty$	-0.13	-1.09	0.22	-0.59	0.59
Union (U)	-$\infty$	-1.25	-0.76	-0.44	-0.13	0.13