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Beyond the empirical characterization of pore and throat sizes, the pore geometry can be characterized in a rigorous way mathematically using correlation functions, which can be measured using image analysis. Since they are used for bounds and the reconstruction of images, topics which are covered later in this chapter, we shall review them in some detail.
In 2-D, we define a function f (i,j), where (i,j) indicates the location of a pixel in the M x N image, i = 1, ..., M and j = 1, ..., N, and f (i,j) = 0 for solids, and f (i,j) = 1 for pores. Then the first order pore correlation function is S1 = < f (i,j) > = φ, where
and where A = M x N is the number of pixels in the image. A similar definition holds in d dimensions. The second order correlation function, S2(x,y), is defined similarly, by
To write the above equation in this way assumes that the system is translation invariant, so that only the difference vector between two pixels matters, and not the absolute location of the two pixels. If the image is also isotropic, then with r = |(x,y)|, S2(x,y) = S2(r,θ) = S2(r), so that S2 is a function of distance only.
The value of S2(r) carries information about how far away different parts of the microstructure still "feel" each other. When r → 0, S2 → φ, since f (i,j)2 = f (i,j). For non-zero values of r, one can think of f (i + x, j + y) as a weighting probability factor for f (i,j). At a given value of (i,j), such that f (i,j) = 1, if there is a correlation in the system up to a distance rc, and if r = (x2 + y2 )1/2 < rc, then f (i + x, j + y) has a better than average ( > φ) chance of also having the value of 1. The overall integral will still be less than φ, however. As r → ∞, then there is no causal connection between the points (i,j) and (i + x, j + y), as long as there is no long-range order, so the probability associated with the pixel at (i + x, j + y) being equal to unity is just φ, independent of (i,j). Therefore S2 → φ2 in this limit. A simple physical way of understanding S2 is to think of it as the probability of finding two randomly selected points that are both in the pore space. This probability turns out to depend on the distance between the two points. Clearly, S2 = φ when r = 0 and decays to the value φ2 as r → ∞. The decay length is a measure of the pore size. Because digital images have a finite size (M x N), the actual evaluation of the two-point correlation function can be achieved using
A simple mathematical exercise is the case of overlapping spheres, where each identical sphere is randomly placed in 3-D without regard to any of the other spheres. The volume outside the solid spheres is the pore space. This case has been solved analytically [38,39]. If ρ is the number of overlapping spheres per unit volume, and R is the radius of the spheres, then
Figure 5 shows S2(r) plotted as a function of r, where R=1 is the radius of the spheres and ρ = 0.29, so that S2 (0) = φ ≈ 0.3. One can see that S2 decreases as r increases from zero, and is always monotonically decreasing. For systems where there are distinct grains, there are usually oscillations after the first large decrease in S2. In this exactly solvable case, S2 actually reaches the value φ2, as can be seen in eq. (6) and comparing with eq. (4). For any random isotropic pore space with smooth surfaces, the slope of the 2-point correlation function at r = 0 is given exactly by [40,41,42]:
S2 is therefore always a decreasing function of r for small r, because of this negative initial slope.
Figure 5: Two-point correlation function for overlapping spheres (exact theoretical result). The sphere radius is R=1, and the number density of spheres was 0.29, so the volume fraction of pore space, φ, which is the space surrounding the solid spheres, is approximately 0.3.
S2 (r) also contains additional information when fractal geometry is present . In the case where the material phase considered is a mass fractal, then S2 (r) rdm - 3 when r is in the fractal limit (less than the image size and greater than the pixel size), where dm is the mass fractal dimension (see section 2.1).
When the material phase is Euclidean, but its surface is fractal, with dimension ds, then the small r limit is given in terms of the surface fractal dimension: S2 (r) φ − B r3 − ds + ..., where B is a constant . When the material phase is a mass fractal with a fractal surface, then other mathematical forms must be considered .
Higher order correlation functions are defined similarly. Although in practice, 2-point functions are most used, 3-point functions are fairly common, but correlations past 3-point are rarely used. For an isotropic translationally invariant material, the 3-point correlation function, S3, is a function of three variables, r1, r2, and θ, where these can be thought of as defining a triangle with two sides of length r1 and r2, with θ being the angle between these two sides. S3 is then the probability of finding the three vertices of this triangle all in the pore phase .
When computing correlation functions from digital images, it is important to correctly handle certain technical isues such as converting to polar coordinates, especially at small r, and to consider the limitations of digital images, such as digitizing curved surfaces, which were mentioned earlier. Refs. [39,43,44] give explanations of the methods that must be used, the pitfalls of which to be wary, and the sources and magnitudes of possible errors.
Other than characterizing pore geometry, one of the principal uses of correlation functions is in the area of computing bounds for the effective properties of composite materials [45,46]. Bounds are analytical formulas that give, for some particular property like elastic moduli or electrical conductivity, the upper and lower limits for what the effective composite property can be.
Bounds are classified by their order. An nth-order bound usually includes information from the nth order correlation function [45,46]. Of course, if there are more than two phases, there will be more than one nth-order correlation function. However, the second order bounds for elastic moduli and electrical/thermal conductivity, commonly called the Hashin-Strichkman bounds , are unusual in that they do not explicitly contain information from SS2, the second-order correlation function, other than S2 (0) = φ. But the third-order bounds for these properties do have parameters computed from S3. The first-order bounds are simply the parallel (Voigt) upper bound and series (Reuss) lower bounds, for which the phases are arranged in a parallel or series microstructure. These use the same information as the second-order bounds, the volume fraction and properties of each phase, but are wider apart than the Hashin-Strichkman bounds.
As bounds incorporate higher and higher correlation functions, they are known to become tighter and tighter, increasing their usefulness at the expense of a great increase of computational difficulty. In fact, it is known that the isotropic and anisotropic electrical conductivity [45,47] and isotropic elastic moduli  of a random isotropic two-phase composite can be written down exactly in powers of the difference of the properties of the two phases. The coefficients in the power series are functions of all the correlation functions of any order for the composite. So in general, the properties of a porous material will depend on all order correlation functions.
Bounds are most useful for composite materials where none of the phases have zero properties. They are less useful for air-filled porous materials. This is because in the array of nth-order bounds, the lower bound always has something of a series character and the upper bound always has something of a parallel character. For air-filled porous materials, this means that the lower bound is always close to zero, because air approximates a zero property phase. So there is really only an upper bound for air-filled porous materials, which may or may not be very close to the actual effective properties. For a liquid-filled porous media, a meaningful lower bound can exist. However, for elastic properties, a zero shear modulus in the liquid phase causes both the lower shear modulus and Young's modulus bounds to be zero.
There has been much work in the past decade or so on bounding the permeability, which is a more difficult problem than that of bounding the effective electrical conductivity or elastic moduli [45,49,50,51,52]. Permeability is different from quantities like electrical conductivity and elastic moduli. The conductivity, for example, is defined at every point of the material, and the overall effective conductivity is found by solving Laplace's equation, eq. (9), for the composite and averaging over this solution and the microscopic conductivities. However, there is no microscopic permeability, because permeability cannot be defined at a point, even in the pore space, but is defined instead by averaging over solutions of the Navier-Stokes equations in a porous material.