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Pore connectivity

Given a digital image of a porous material, it is simple to assess the connectivity of any phase. Usually, we want to know this for the pore phase, as the solid phase must be connected in order to have mechanical integrity of the sample. A simple method to use on a digital image is called a "burning algorithm" [53,54]. In 2-D, only one phase at a time in a porous material can be percolated [53,55]. In 3-D, several phases can simultaneously percolate. This fact reduces, but does not eliminate, the usefulness of the burning algorithm in 2-D.

The burning algorithm is a way of identifying all members of a cluster of connected pixels that span the image. Starting on one side of an image, "burn" one pore pixel by setting its gray scale to another number that is not in the existing range, e.g. not in the range 0-255. Then any pore pixel that touches this pixel is also set to the same number. Continue this process until there are no more "unburned" pore pixels left that are touching the last burned pixels. The process is similar to classifying all pixels of a certain gray value as being combustible, and then touching a match to one of them. If the "fire" burns from side of the image to the opposite side, then the burned pixels are said to form a spanning cluster, or percolate. This process can be repeated by starting the fire at any unburned pixel in order to identify all connected clusters, and all non-spanning clusters as well. This is an efficient way to determine if the pore space percolates through the digital image.

In performing the burning algorithm, one issue to consider is which pixels constitute a neighboring pixel for propagation of the "fire." The most common case is to consider the immediate nearest neighbors (4 in 2-D, 6 in 3-D). Alternately, the second nearest neighbors (4 in 2-D), or the second and third nearest neighbors (20 in 3-D) can also be considered. The connectivity of a phase in a digital image with square or cubic pixels has this degree of uncertainty. We note, however, that using only the first nearest neighbors in 2-D resulted in good agreement of percolation thresholds, determined on digital images, with their continuum counterparts [14]. Different numerical techniques for discretizing continuum equations on a digital image have natural definitions of connectivity connected with them, as we shall see in the next section below.

It is important to note that percolation thresholds are usually larger in 2-D than in 3-D. For instance, if one carries out site percolation on a square lattice digital image, considering only nearest neighbor connections, where a random fraction x of the pixels are white and (1-x) are black, then the white pixels will percolate only when x x_c, where x_c = 0.59 in 2-D, but x_c = 0.31 in a 3-D simple cubic lattice [55]. If the pore space of a real material followed these site percolation statistics, and had a porosity of 35%, it would have a percolated pore space, but a 2-D slice, seen in the microscope, would appear to have a disconnected pore space, as a porosity of 35% is much less than x_c = 0.59. So it is incorrect to study 3-D percolation quantities using 2-D images. Stereology breaks down in this instance, as the connectivity in 3-D and 2-D is fundamentally different. Real pore spaces, which generally have different kinds of percolation statistics, will also show this kind of difference. Since the connectivity of the pore space has a critical influence on transport properties such as permeability [56], its quantification can be critical for understanding microstructure/transport property relationships.