SURFACE AREA

 We continue investigating particle geometry by considering surface area (SA). Mandelbrot (Mandelbrot, 1967) showed that the perimeter and area of a highly irregular 2-D object are not constants but are a function of scale. This is also true for highly irregular 3-D objects and perhaps explains some of the difficulties in efforts to define, measure and to compare surface properties. Now that tools are available to measure surface geometry of rocks directly, it may be necessary to define SA as either the value at a certain scale or as a relationship between scale and SA. This is a topic beyond the scope of this paper.

 The SA of an individual particle can be calculated from the CT data using spherical harmonic techniques. The relevant equations, which involve derivatives of the spherical harmonic functions, are given in detail in (Garboczi, 2002).  The SA of a closed body can also be estimated from the properties of multiple projections of the body. These projections are often termed shadows or silhouettes. The body’s silhouette is projected onto a 2-D surface, using a light source from a given direction, and the area of the 2-D projection measured. Cauchy (Cauchy, 1850) proved, for convex bodies, that the true surface area is equal to four times the average of the projected areas, where the average is a summation over all possible projections. In practice, only a finite number of projected areas are used. Presumably the larger the number of projections, the more accurate is the result, as long as the direction of the projections is chosen randomly or according to some quadrature scheme to eliminate bias. Clearly, few aggregate particles are truly convex, as was discussed earlier. For the non-convex case, Underwood (Underwood, 1972) proved that the Cauchy value is a lower bound. The projected area method has been investigated experimentally (Bodziony et. al., 1975, 1976; Umhauer and Gutsch, 1997; Lau et al, 2002). Theoretical distributions of area versus direction of view have been studied (Vickers et al., 1998), but only for regular bodies.

 Lau measured the areas of multiple projections of the 12 test rocks (Lau et al., 2002; Taylor et. al., 2005). In the study by Lau, over 1500 images were processed by hand. The equipment was then modified, the number of views per rock was increased, and the processing and analysis of the images was automated. The silhouettes of each rock were obtained by digital camera, binarized, and then analyzed using a commercial image processing program. The results are shown in Table 5, where they are compared with the results from the CT-based calculation. Since the projected area results are lower bounds, they should be lower than the CT results. However, inspection of Table 5 shows that this is not always true. This may indicate the influence of scale. The resolution of these particular CT scans were about 0.06 mm per voxel, so any surface details of about 0.1 mm or smaller are not properly represented. The projected area scans have an effective scale of about 0.04 mm per pixel, so that they probe slightly smaller texture on the surface and so gain SA compared to the CT scan data. Despite this point, or perhaps because of a cancellation between the lower bound nature of the multiple projected area SA and the higher resolution therefore higher SA images used, there is good agreement seen in Table 5 between CT calculation and projected area measurement, certainly well within the standard deviation of the measurements. This comparison serves as another check on the accuracy of the CT data and the spherical harmonic reconstruction technique, as well as illustrating the effectiveness of the multiple projected area measurement method.

Table 5. Measured and calculated surface areas for the 12 test rocks. The standard deviations over the 65 views for the multiple projections are given as a percentage of the mean value.

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