Using x-ray computed tomography and microtomography to generate spherical-harmonic-based mathematical representations, we have fully characterized the complete three-dimensional shape of large (6 mm – 19 mm) and microfine (< 100 mm) granitic rock particles. The morphological features of the rock characterizations were accurate down to several times the voxel size of the tomographic images, since it takes several voxels to adequately represent a morphological feature. Since the mathematical representation of the complete particle was accurate, any property that could be computed for a regular particle could also be computed for these rocks. In particular, any integral over the volume or surface of a particle could be numerically computed. This fact was invaluable for computing various moments of the volume and surface. It is important to note that these techniques can be applied to any kind of irregular particles whose size causes them to lie in the range of tens of micrometers to tens of millimeters, a factor of 1000 in size. These techniques allow irregular particles to be treated mathematically in the same way regular particles are treated.
This paper assembled the results of measurements of volume, density, surface area, and dimension performed on two sets of irregular 3-D particles. For the larger particles, traditional volumetric measurement using Archimedes’ principle was compared to the value of volume obtained from the spherical harmonic-based mathematical reconstruction obtained from x-ray CT scan data. There was excellent agreement between these two sets of data. Multiple imaging, employing Cauchy’s theorem, was used to check the value of surface area obtained from the spherical harmonic-based mathematical reconstruction obtained from x-ray CT scan data. There was good agreement, within experimental uncertainty, between these two sets of data, although there was likely a tradeoff between the lower bound nature of the multiple projection images and the higher resolution of these images (0.04 mm/voxel) compared to the resolution of the CT data (0.06 mm/voxel). The higher resolution would tend to increase the measured surface area by including more surface detail while the lower bound nature of the measurement, since the rocks were not convex, would tend to give a lower estimate. This comparison between CT-SH and the measured surface area and volume served the purpose of validating the CT-SH data and calculations. The CT data were then able to provide estimates of the variation in density within a single particle. Key to this was being able to linearly relate the average gray level of the CT images to traditional density measurements of each particle. The variation in density of the component minerals was anticipated but numerical information on the mixtures to be found in actual commercial samples was hitherto unavailable. The difference in the distribution of internal density between the 0.5 and the 0.75 rocks was not anticipated and is not fully understood. However, six rocks of each class do not give very good statistics, so no conclusions can be drawn from this difference.
It is straightforward to relate the dimensions of a regular object to its surface area and volume. This relation between physical dimensions and geometric properties is most, but not all, of what is meant by “particle shape.” Relating dimensions and geometrical properties for irregular particles is always done through equivalent shape models. We have introduced several three-parameter models, since the one-parameter models like the equivalent volume sphere are inadequate. These three-parameter models were based on the box (rectangular parallelepiped) and tri-axial ellipsoid. The geometrical properties that were used to create these models by equating the values for an irregular particle to that of the equivalent shape were the externally measured or computed length, width, and thickness; the principal moments of volume (analogous to the principal moments of inertia); and the absolute first moments of volume. Three choices of geometrical properties and two choices of equivalent shapes resulted in six three-parameter equivalent shape models. These models were developed for both the 12 larger test rocks and for the 300+ microfine rocks.
All six models were used to compute both surface area and volume using the analytic formulas for the volume and surface area of tri-axial ellipsoids and boxes. Plotting predicted volume and surface area vs. measured volume and surface area resulted in excellent linear correlations in all cases. All linear correlations approximately went through the origin, as they should, and some linear relations had a slope of nearly unity (within experimental uncertainty), indicating that the volume and/or surface area of the effective shape was nearly equal to that of the real particle. This was found for the 12 test rocks to be the AFM-box model, for the surface area, and the PMV-box model, for the volume. For the microfine aggregates, the two models that gave slopes closest to unity were the PMV-box model, for the surface area, and the PMV-box model, for the volume. These slopes being very close to unity means that somehow the equivalent shape models closely match the real particles. These particular models need to be studied more, and for different particles, to see if they still give slopes that are close to unity.
The remarkable similarities found in Table 9 between the shape distributions for the 12 test rocks and for the micro-fine aggregates seem to indicate that the microfines and the large particles, crushed from the same source, have nearly identical shapes, at least in a statistical sense. This is a new result, and implies a sort of self-similarity in the rock-crushing process, as more and more crushing results in the same kind of shapes. It was not surprising then to see strong similarities between Figs. 6-9 and Figs. 10-13, and Tables 7 and 10. One must note that the material used was a fairly homogeneous granitic rock, so different results might be expected for a more heterogeneous material that might have mineral phase segregation according to particle size. In view of this consideration, it would be very interesting to characterize the mineralogy of the microfine aggregates using individual particle scanning electron microscopic and X-ray microprobe techniques like those used for portland cement powders (Bentz and Stutzman, 1994).
It would also be interesting to perform this same analysis on rocks from another source to see if other rock types can be characterized in this way, and if the coefficients found in the model correlations are robust between different types of aggregates. Our guess is that these three-parameter models will perform similarly well among different particle types, as long as the particles are not too different from those studied. But the validity of these three-parameter models might go far beyond the types of particles that we have studied, extending even to particles with strong anisotropy. These results should be of great use when modeling particulate materials – this was one of the motivations for undertaking the research.
This paper made extensive use of X-ray CT data. In fact, the moment-derived dimensions for the three-parameter models would have been impossible without this data, or data like it, for instance from LADAR. While CT devices are currently relatively expensive and few in number, the results provide so much useful information that wider use should see a reduction in cost. The advantages offered will offset cost because each new set of data will increase the statistical confidence in the results so obtained. But in the interim, this paper showed that that for a modest investment of time, larger rock particles upon which one can readily measure L, W, and T could be characterized quite well using three-parameter equivalent shape models generated from these simple measurements.
When rocks like these are used in composites like concrete, or indeed when any particle is used as a second phase filler in a matrix, the change in composite properties due to the introduction of a second phase depends on the shape of the particulate inclusion and on the property contrast between inclusion and matrix. This change in composite properties can be handled analytically for particle shapes like spheres and ellipsoids. Recent work has started to quantify these changes for rectangular parallelepipeds and irregular shapes like those considered in this paper (Garboczi et al., 2005a; Garboczi and Douglas, 2005b).
Finally, in a more speculative vein: the whole problem of particle packing is not understood very well theoretically for anything besides spherical or ellipsoidal particles. This problem has been explored using digital image (voxel) techniques (Jia, 2001), but using continuum particles as represented by spherical harmonic series may be able to add insight. Usually, continuum approaches involve less memory, more computational time and more complicated algorithms, but are not limited by digital resolution.
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