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, sv, 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
sv = -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)=pv(r) for r<2r0 and p(2)(r)=p2 for r > 2r0 where
and the surface to volume ratio is sv = -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), limr
g(r)
0].
Invariably g(0) is taken as unity. A useful general
form for g is [34]
The resulting field is
characterized by a correlation length ,
domain scale
d and a cut-off scale rc.
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
where
i is a uniform deviate on [0, 2
)
and
is uniformly distributed on a unit sphere.
The magnitude of the wave vectors ki are distributed on
[0,
)
with a probability (spectral) density P(k). The density is related to g(r) by a Fourier transform [g(r) =
0
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
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
Berk [7] and Teubner [40] have shown that the two point correlation function is p(2)(r)=h(r) where
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, In, formed from the intersection of n primary models, has also been found useful. The statistical properties (p, sv 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].
Mod. | p | p(2)(r) | sv | hp |
N | h | h(r) | -4h'(0) | p |
I | h2 | h2(r) | -8hh'(0) | ![]() |
U | h(2-h) | [2h2+2h(r) | -8(1-h)h'(0) | ![]() |
-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
(
,
) =
( -
, 0.84), (-0.84,-0.25), (-0.25,0.25), (0.25,0.84) and (0.84,
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 (
= -
); (ii) a
symmetric two-cut field (
= -
) an (iii) an
asymmetric two-cut field. A concise way of expressing this is to take
Model | Standard | Asymmetric | Symmetric | |||
type | one-cut | two-cut | two-cut | |||
c=0 | c=1/2 | c=1 | ||||
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|
Normal (N) | -![]() |
-0.84 | -0.84 | -0.25 | -0.25 | 0.25 |
Intersection (I) | -![]() |
-0.13 | -1.09 | 0.22 | -0.59 | 0.59 |
Union (U) | -![]() |
-1.25 | -0.76 | -0.44 | -0.13 | 0.13 |