Next: Correlation functions and Up: Geometrical and topological Previous: Geometrical and topological

Stereology, mathematical morphology, and fractal analysis

There are several bodies of mathematical knowledge and techniques that have been developed and that are actively used to mathematically analyze and characterize the microstructure of porous materials, or indeed any material.

Stereology is the mathematical science of predicting 3-D quantities by measuring 2-D quantities. There are books available on this topic [19,20], with many new articles produced each year. Stereology combined with image analysis [21,22] can be a powerful tool for inferring quantities like φ and s_p that are the same in 2-D as in 3-D. Stereology cannot, however, analyze quantities that change between dimensions, like percolation quantities (see Section 2.3). Mathematical morphology is a related and powerful tool for studying images of porous materials [23].

Another body of knowledge available for analyzing and characterizing random materials is that of fractal mathematics. Making use of the techniques of stereology, mathematical morphology, and image analysis, questions of fractal geometry [24,25,26,27,28] can be explored in digital images of porous materials, whether these images are 2-D or 3-D. For an object to be fractal, it must display scale invariance over a range of length scales. A given digital image must have enough resolution so that it can display a reasonable range of length scales, in order for its potential fractal character can be analyzed. A rough rule of thumb is that scale-invariance must be displayed over at least one order of magnitude of length scale, in order for an object to be considered to have fractal character. Therefore the image must contain at least that much resolution. In a digital image of a porous material, the size of the image is L x L, and the pixel length is p. It is clear then that we must have L >> 10 p, since looking at length scales too close to the digital resolution will bring in the digital "graininess", and looking at length scales too close to L will bring in finite size effects.

Assuming that the image is adequate in terms of length scales and resolution, one way to examine the possible fractal nature of an object in a digital image is to measure how the object fills Euclidean space as a function of the size of the region being examined. Such a property is characterized by the mass fractal dimension d_m. In the case of a digital image, we can count the number of pixels that are contained within a given radius. The number of points M(r) ∝ r^dm . Certainly, the range of r over which this relationship could hold would be for p < r < L. If the object did fill space uniformly, then d_m = d, the Euclidean dimension. As an example of fractal objects, objects built up in threee dimensions by diffusion limited aggregation or percolation networks at the percolation threshold have d_m 2.54.

Another way of determining d_m is to construct a grid which covers the digital image, of box size l. It would be easiest, in the case of a digital image, to choose l to be an integer number times p. By counting the number of boxes that included part of the object, as a function of grid spacing l, one obtains the box dimension, d _b = d_m, from the relation M(l) l^−db. Once again, the range of grid spacing that would produce such a relation would be between the pixel size p and the image size L.

A surface can be rough in a way such that it can also be characterized by a fractal dimension, this time a surface fractal dimension. In a 2-D digital image, a "compass" of opening t can used to step around the surface (perimeter) and measure its apparent length S(t). If the surface fractal dimension is d_s, then S(t) t^−ds. A grid method can also be used, similar to the determination of the mass fractal dimension, which is defined for a 2-D or 3-D digital image. One counts how many grid boxes have surface within them, S(l), for various grid sizes l. If the surface is fractal, then S(l) l^−ds, where d_s is again the fractal dimension of the surface.

Experimentally, one can directly determine the fractal dimension by use of small angle scattering, whether neutron or X-Ray scattering. Further details can be found in the chapter by Sinha in this book.

For porous materials, the pore space itself, if it has pores over a wide range of length scales, can be a mass fractal [29,30,31,32]. If the pore surface is very rough, which would be the case for a high surface area material, then the pore surface could form a surface fractal [29,30,31,32]. Studies of fractal geometry have been carried out for rocks [25,33], aerogels [34], and cement-based materials [35]. The transport properties of fractal pore spaces have also been studied theoretically [36,37].