Using several models

From the foregoing discussion it is clear that a model more general than a single level-cut random field is needed to reproduce the random isotropic microstructures seen in many composite materials. At present there is no one model that can achieve this. Instead it has been proposed that a number of morphologically distinct models (N, I and U of Sec. 2) be incorporated [34]. This is very simple to do by using the relevant formula for p⁽²⁾_mod (r) (Table 1) in Eqn. (8). It was found that many of these models were able to match any given p⁽²⁾_expt(r) (providing further evidence that the two-point correlation function does not provide sufficient information for a useful reconstruction). The problem then becomes how to choose the best model. Clearly higher-order statistical properties need to be taken into account.

The quantities p and p⁽²⁾(r) are the first and second of an infinite hierarchy of correlation functions, the three-point function p⁽³⁾(r,s,t) representing the probability that three points, distances r, s and t apart fall in phase 1, and so forth. Two random composites can only be said to be statistically identical if their N-th order correlation functions ( N = 1,2,3,4...) ( are identical [48]. Therefore the exact statistical reconstruction of a composite requires matching all of the correlation functions². An obvious method of choosing the best of several models is then to compare p⁽³⁾ of each model to experimental data. As well as being memory and time intensive [8] it has been shown [34] that p⁽³⁾, like p⁽²⁾, may not contain the relevant morphological information (ie. it does not provide a strong signature of microstructure). A comparative study of several high-order statistical quantities found that the simplest and most discriminating signature of microstructure was the chord-distribution functions for each phase ^(j) (r) (j = 1,2). $\rho^{(j)}(r)$ ^(j)(r) is the probability that a randomly chosen chord in phase j has length r. A chord is defined as any line-segment which lies entirely in phase j with end points at the phase interface. Like p⁽²⁾ the chord functions are the same whether measured from a two or three dimensional element of the microstructure.

The chord-functions can be employed in a reconstruction algorithm as follows. First the morphological parameters [the length scales in Eqns. (1) and (2)] of each model (overlapping spheres, or the the various level-cut GRF's) are chosen to fit p⁽²⁾ _expt. We have found that most models are able to provide a reasonable fit of p⁽²⁾_expt (r) (e.g. Ep⁽²⁾<0.1). If this is not the case the model is unlikely to provide a useful reconstruction and may be rejected. Second, of the remaining candidates, the model that best reproduces the experimental chord functions ^(j)_expt is selected as the best reconstruction. We quantify the error by a normalised least square sum;

$$[E\rho^{(j)}]^2 = \frac{\sum_{i=1}^{M} [\rho^{(j)}_{\rm rec}(... ...expt}(r_i)]^2} {\sum_{i=1}^{M} [\rho^{(j)}_{\rm expt}(r_i)]^2}$$ (9)

where $\rho^{(j)}_{\rm {rec}}$ ^(j)_rec are the measured chord-distributions of the reconstruction for phase j=1,2 (at M points). The final reconstruction thus has approximately the same chord functions as the experimental image as well as sharing the low-order quantities p and p⁽²⁾(r).

A limitation of this `model-based' technique is that one of the of models tested must, for some choice of its morphological parameters, be able to approximately reproduce the experimental microstructure. For example, it would be unlikely that a model derived from the iso-surfaces of a random field would be able to mimic the highly structured morphology of randomly packed hard spheres. In such a case we would expect that none of the model chord functions would reproduce the experimental data. At present there have not been a sufficient number of studies to provide numerical criteria on E $E\rho^{(j)}$ ^(j) for acceptability. The general approach we have outlined is not restricted to the models given in Sec. 2. Ultimately it would be useful to incorporate poly-disperse overlapping spheres and other Boolean models such as those based on Poisson and Voronoi polyhedra. The latter models have proved useful in the analysis of mineralogical materials [25] and flow in porous filters [10]. There may also prove to be more useful discriminants of microstructure than the chord-functions.

To conclude this section we note that a recent study [49] has considered a `model independent' scheme based on sequentially moving filled pixels (representing the target phase) on a grid so that the reconstruction reproduces statistical properties of the original image. The method differs from ours in that numerical estimates of p⁽²⁾_rec replace p⁽²⁾_mod in Eqn. (8), and Eqns. (8) and (9) are coupled. The authors also employ the lineal path distribution function L^(j)(r) in Eqn. (9) which is related to the pore-chord functions by $\rho^{(j)}(r)= \frac14 s_v d^2L^{(j)}(r)/dr^2$ ^(j)(r) = 1/4 s_vd² L^(j) (r) /dr² [43].