Before presenting the main points of the paper, the mathematical background will be briefly reviewed [1]. The spherical harmonic mathematical analysis of particle shape relies on eq. (1), which states that any sufficiently smooth function r(θ, φ), where θ and φ are the azimuthal and polar angles of 3-D spherical coordinates, can be written as a series of spherical harmonic functions [1], where the Ynm are the complex spherical harmonic functions and the anm are complex coefficients.
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| (1) |
Strictly speaking, the series in eq. (1) becomes exact only as N →∞. However, like 2-D Fourier series, a finite value of N is usually found to give an adequate approximation of a given function, within some specified uncertainty limit. There are other mathematical approaches to analyzing particle shape that have been extensively discussed [9].
In the case of aggregates used in concrete, the function r(θ,φ) gives the distance from the centroid (equal to the center of mass for a uniform density object) to a given point on the surface of the particle, in a direction specified by the two angles θ (latitude) and φ (longitude). Using a numerically determined r(θ, φ) function from a 3-D image, in this case derived from x-ray CT, one can determine the first N = 20 or so coefficients, which are usually enough to satisfactorily represent the image [1]. In the cases studied in this paper, the values of N range from N = 20 to N = 40, with most cases having N about 20. Once these coefficients are obtained, one can recreate approximations for the r(θ, φ) function, which determines the shape of the particle. Storing up to n = N coefficients requires (N + 1)2 complex numbers.
Figure 1 shows how a given particle is represented using increasing numbers of spherical harmonic coefficients. From left to right and top to bottom, the even values of N used in eq. (1) increase from N = 0 to N = 22. The particle has been made to rotate slightly with increasing N, in order to show more of the particle's surface. The N = 0 value corresponds to an equivalent sphere, with a radius averaged over the entire particle. Using all the coefficients up to N = 2 gives an effective ellipsoid [10], but larger values of N are absolutely necessary to recreate fine details of the shape. However, the geometrical interpretation of these values of N > 2 is not as straightforward.