Mathematically characterizing the shape of particles is an old and not completely satisfactory activity. Many empirical and analytical classifications of shape, though often based only on two dimensional (2-D) properties, exist in the literature [1,2,3,4,5]. The differential geometers have derived many tools for characterizing surfaces, such as mean curvature and Gaussian curvature, along with precise definitions of usual quantities like volume and surface area [6]. Physicists have contributed the concept of the moment of inertia tensor [7]. There are also ways to indirectly characterize shape, via the effect of inclusion shape on composite properties when there is a low volume fraction of inclusions in a matrix [8,9,10,11,12]. These effects are often used by polymer scientists and composite material theorists under the name of transport property virial coefficients or dilute limits. The same kind of shape-based calculations are also found in many other areas of science and engineering, as the problem of particle shape is ubiquitous [8]. These concepts provide a powerful mathematical "tool-kit"with which to characterize the shape of particles. However, applying these mathematical techniques to real, complex, random shapes is a non-trivial task, and much less attention has been paid to this undertaking than to analytical shapes like ellipsoids, torii, cubes, etc. There has of course been a great deal of attention paid to fractal particles [13]. But the kind of particles found in concrete, random but relatively compact particles, seem to have had less attention paid to them in current research.
In the context of aggregates used in concrete, why does one want to mathematically characterize shape? There are at least three reasons. The first is to simply be able to classify different aggregates from different sources, so that their shape differences can be quantified. The second is to be able to quantitatively relate true 3-D aggregate shape characteristics to performance properties. Anecdotal evidence that, for example, "more angular aggregates" give different concrete workability is not enough information on which to base a shape optimization of aggregates in concrete. If the shape of particles in a given aggregate type can be mathematically and quantitatively represented in 3-D, inclusion shape-composite property relationships can be developed in a meaningful way.
The third reason to mathematically characterize shape is so that real, random particles can be successfully incorporated into computational models. The random structure of concrete makes it very difficult to be able to model a representative piece of concrete using digital-image-based models, as has been successfully done at the cement particle and paste level [14]. Too many voxels are required, mainly because of the presence of the interfacial transition zone (ITZ) and the wide size range of aggregates present [15]. For example, a representative size sample of a concrete with a maximum aggregate size of about 10 mm should be a cube with a side length of least 50 mm. To adequately resolve the ITZ as well as all the aggregate requires that the minimum voxel size should be at most about 5 µm on an edge, since the ITZ is about 20 µm in width. The minimum sample size in voxels will then be 10,0003 or 1012. Storing the model with one byte per voxel would require a computer memory load of 1 TByte, which is uncomfortably large for being able to handle even one model per computer!
To overcome this difficulty, continuum models, called hard-core soft-shell models [16], have been used. These store only the position, orientation, size, and shape information of each particle. If a particle is a sphere, its shape is known and only its diameter need be stored, along with the coordinates of its center. For an ellipsoidal particle, its center, the length of its three semi-axes, and a 3-D vector denoting its orientation have to be stored. Any regular geometric particle can be handled in a similar way. In addition, one must be able to decide if two particles overlap, so that a reasonable concrete model with non-overlapping aggregates can be generated. For Euclidean particles such as ellipsoids, this can be readily done through fairly simple mathematical functions since the geometry is completely characterized by the orientation and length of the three semi-axes [17,18,19]. However, real aggregates are not smooth and regular particles, but have random shapes. To be able to insert random particles into these kinds of continuum models requires that each particle be characterized by a limited set of numbers, much fewer than required by digital techniques, where the location of each voxel needs to be stored.
Motivated by the above three factors, the focus of this paper is how mathematical shape analysis of real particles can be successfully performed by acquiring real shapes via x-ray tomography and by analyzing them using spherical harmonic functions. Each particle can be reduced to a limited set of numbers, the coefficients of the spherical harmonic expansion, which fully characterize the particle shape at the resolution of the original image. Continuum models using real aggregates can then be built almost as readily as those involving shapes like spheres and ellipsoids, once the problem of determining overlap for these random particles is overcome.
The kind of spherical harmonic technique described in this paper has been used previously for approximating molecular orbital surfaces [20,21,22] of proteins and other molecules. It has also been used recently, in a very similar way to that which this paper describes, to characterize the shape of a really big rock - an asteroid [23]. A different form of spherical harmonic mathematical techniques is also used in geodesy in the study of the shape of the earth and the shape of satellite orbits [24], an extra-large length-scale shape characterization problem. Similar mathematics as described herein have been used to analyze the shape of aggregates in serial sections [25]. The novel feature of the method used here is that it starts from a 3-D digital image acquired from x-ray tomography. The special features of the spherical harmonic analysis caused by this source of original images will be seen in later sections of the paper. The intention of this paper is to serve as a guide to those desiring to carry out shape analysis using spherical harmonic function techniques. As such, many mathematical details are included in the main text and in appendices to enable practical use to be made of the paper.
The whole procedure of mathematically analyzing shape is far easier to visualize in 2-D than in 3-D. A similar kind of analysis, for 2-D representations of aggregates used in concrete and other shapes, has been carried out previously [26,27,28]. We review and extend this 2-D formulation here, in order to give a framework of understanding and to introduce concepts, before going into the full 3-D analysis.