X-ray tomography and particle acquisition

X-ray computed tomography (CT) [32] offers a nondestructive technique for visualizing features in the interior of opaque solid objects to obtain digital information on their 3-D geometry and topology. In the case of concrete, the aggregates used in the concrete can be visualized inside a concrete cylinder.

The aggregate images to be shown in this paper were taken from an x-ray tomograph of a real concrete sample captured using an X-ray CT system located at the Turner Fairbank Highway Research Center [33]. Concrete prisms having a 75 mm x 75 mm cross sectional area, a water/cement mass ratio of 0.5, with quartz sand used as the fine aggregate and limestone used as the coarse aggregate, were made and used. The gray scale image that comes from the tomographic process was thresholded to a black and white image by recovering the known volume of aggregates contained in the sample.

Figure 6 shows a 270³ pixel portion of the final result, cut out of the original image (this size was chosen only for convenience, as much larger samples can be handled). Aggregates (high density) in Fig. 6 appear white, while the matrix, consisting of cement paste and unresolved fine aggregate particles, appears black. The large flat areas on the aggregates showing on the faces of the cube are from the cut through the sample, and are not part of the real image. The voxels are cubes with dimensions of approximately 0.4 mm per side. The physical size of the concrete sample shown in Fig. 6 is then about 108 mm x 108 mm x 108 mm. The image shown in Fig. 6 represents preliminary work at a fairly coarse resolution, but is adequate for the purpose of illustrating the mathematical algorithms described in subsequent sections. Much higher resolution, of about 20µm - 40 µm per voxel side, is possible.

Figure 6: A 270³ voxel piece of a tomographic image of a concrete material.

Given that a 3-D multi-aggregate image has been obtained, one can proceed to extract individual particles. Ideally, when carrying out this procedure routinely, the image should be taken of a system with a fairly low volume percent of aggregate, say 20 %, so that on the average, most particles are not near each other. The image in Fig. 6, however, was of a real concrete at a practical aggregate volume percentage (around 60 %). Because of this fact, when the 3-D image was made, many particles appeared to be in contact. This was because of the fairly coarse resolution of the image, which would make many close but not touching contacts to appear as real contacts. Converting the gray scale image to a binary image can also cause some artificial particle contacts. This situation was handled with a simple erosion and dilation algorithm [34], which broke apart the tenuously connected aggregates, without significantly changing their size or shape.

A "burning" algorithm was used to identify single particles. It is quite analogous to the algorithm used in percolation studies, both in digital, pixel-based models [35], and in continuum model studies [12], for determining the connectivity of extended clusters. Imagine a 3-D cube of pixels, like that shown in Fig. 6, where each pixel is labeled either matrix (1) or particle (2). Assume that enough precautions have been taken, either physically, by keeping the volume fraction of aggregate low, or numerically, using some kind of erosion/dilation routine, that no particle is touching any other particle. We will also stay away from the boundaries and so not allow any artificially "sliced" particles to be identified. Scan through the image until a pixel is found that has label "2." This corresponds to the single black pixel shown in Fig. 7, which shows a model particle in 2-D. Now find all nearest neighbors (back-front, left-right, up-down) of this pixel that also have label "2." In Fig. 7, the first round of this neighbor identification will come up with those pixels labelled "a." Save the locations of these pixels, and then find all neighbors of the "a" pixels that also have the same label ("b" in Fig. 7). Iterate this process until no more pixels of label "2" can be found. The collection of pixels (voxels in 3-D) found constitutes a single particle.

Figure 7: A 2-D illustration of how the burning algorithm for particle identification works. The black pixel is found first, then the "a" neighbors, then the "b" neighbors, and finally the "i" neighbors, completing the particle.

The main image is now systematically examined for particles, which when found are stored in a simple database. The coordinates of each voxel, relative to the center of mass of the particle in which it was found, are stored. Any particle can easily be regenerated by placing its center of mass at any location in a 3-D digital image. These coordinates are used to generate a surface function of the 3-D particle, which can be analyzed with spherical harmonic functions, as is described in the next section.