This section gives examples of different ways the spherical harmonic coefficients can be used to analyze the shape of the four kinds of coarse aggregates studied above. A recent paper by Masad et al. [17] contains other ways to use the spherical harmonic coefficients to quantitatively analyze the shape of particles. The most effective way to use the spherical harmonic coefficients to quantitatively analyze shape is an active topic of research.

Since real aggregates are not spherical, but a sphere is our most fundamental shape, in
some sense, how does one show the non-sphericity of real aggregates and the differences in
non-sphericity between aggregates? In sedimentary geology [18], a
simple way to show the non-sphericity of each kind of particle, as well as to show
differences between kinds and sizes of particles, is to make a plot of particle surface area
(S) vs. particle volume (V), which can both be computed from the spherical harmonic analysis
[1]. Figure 6 shows these plots for the
granite (GR) and Arizona (AZ and az) aggregates. The other aggregates gave similar results.
Clearly both kinds of aggregates lie above the theoretical line for spheres, S =
(36π)^{1/3} V^{2/3} ≈ 4.84 V^{0.67}, so that all four
kinds of aggregates are clearly non-spherical by this measure of shape.

Since the S,V data for all four kinds of aggregate seemed to cluster together well and
have the same character as the theoretical sphere curve, a function of the form S =
aV^{b} was fit to each set of data (Fig. 6). Values of a and
b, along with the R^{2} factor obtained in each case, are summarized in Table
6 for the four kinds of aggregates. The exponents agree very well
with the ideal value of 2/3, given that the uncertainties in the exponents are estimated to
be at least ± 0.03. The value of 2/3 comes from surface area being a 2-D quantity and
volume a 3-D quantity, hence 2/3 = 2 ¸
3. The dimensionless prefactors, however, seem to differ by more than the uncertainty, which is estimated to be about
± 0.5, based on the accuracy of the shape reconstruction process [1].

Consider other shapes, like cubes, ellipsoids, and rectangular parallelepipeds. Note that
for a cube, a highly faceted but equi−axed "rock," the surface area − volume
relation is exactly S= 6 V^{2/3}. In Table
6, all of the prefactors for the aggregates studied are
significantly greater than 6, the value for a cube. Since the surface area of ellipsoids is
a complicated elliptical function, we ignore this shape for the purposes of this exercise,
which is only to look at simple numerical examples [11,
19]. Next consider a rectangular parallelepiped, of edge lengths
p, q, and r. The volume is V = pqr, and the surface area is S = 2(pq + qr + pr). One cannot
obtain an analytic expression from these of the form S = aV^{b} , even though these
expressions are simple. However, many of these shapes can be simulated with the side lengths
being random variables, with the results then fit to the same numerical form, S =
aV^{b} . Two cases were tried, with all sides being randomly and evenly distributed
between an upper and a lower limit (lengths in millimeters). In the first case, 0.1
< p < 1.0, 0.2 < q < 1.0, and 0.3 < r < 1.0. In this case, the larger
particles will tend to be more equi−axed than the smaller particles. A good fit was obtained
to the numerical results, with a = 6.7, b = 0.60, and the goodness-of-fit parameter
R^{2} = 0.98. In the second case, 2 < p < 10, 2 < q < 10, and 2 <
r < 10. In this case, both the smallest and the largest particles would tend to be
equi−axed. A value of a = 7.2 and b = 0.64 was found (R^{2} = 0.99). Although
qualitative, these two cases indicate that how the shape depends on particle size can affect
the S vs. V curve. The value of the parameters in the S = aV^{b} relation could
perhaps be indicative of the shape character of the aggregates, and their meaning will be
further explored in future work.

Aggregate type |
Multiplicative factor "a" |
Exponent "b" |
R |

GR |
8.1 |
0.63 |
0.99 |

LS |
7.5 |
0.64 |
0.997 |

IN |
8.6 |
0.62 |
0.99 |

AZ |
9.1 |
0.61 |
0.98 |

Table 6: Results of fitting the surface area vs. volume plots for the
different kinds of aggregate types studied, where the fitted function was
S = aV^{b}. |
---|

Are there other measures of shape that will distinguish between different kinds of aggregates [20]? This is important to be able to do, especially for aggregates that are thought to perform differently due to shape differences. This is a matter of further research. Spherical harmonic techniques [1] will also allow the calculation of mean curvature, the moment of inertia, and the maximum linear extent of a particle in any three orthogonal directions [11], along with any other measure that can be defined on the surface. Some of these perhaps serve to further distinguish particle shape. Also, comparison of various quantities to their equivalent spherical counterparts can also highlight differences between particles.

Figure 6: Plot of surface area, as computed from the spherical harmonic
series for each rock, vs. the volume as taken from the original digital image, for the
granite (GR) and Arizona (AZ and az) aggregates studied. The solid line in each graph is the
analytical surface area vs. volume relationship for spheres, S = (36π)^{1/3}
V^{2/3} ≈ 4.84 V^{0.67}. The dashed line is a fit of the form S =
aV^{b} (see Table 4).

As an example of another shape measure, Fig. 7 shows plots of the ratio of the true particle surface area to the surface area of the equivalent sphere, vs. the diameter of the equivalent sphere, for the GR and LS aggregates. A value of one for this ratio is only obtained for a spherical particle, and since the sphere is the minimum surface area particle for a given volume, all values are above one and a higher value should indicate a larger non-sphericity. There are apparent qualitative differences between the two plots in Fig. 7, but this may not be statistically significant given the different number of points on each curve. The data are more randomly distributed than the points in Fig. 6. However, this may be because the surface area ratio is more sensitive to surface roughness and other features than to the basic shape of the particle. There seems to be a trend towards higher values of this dimensionless ratio for smaller equivalent spherical diameters, indicating that the smaller particles are more non-spherical than are the larger aggregates.

Figure 7: Plots showing the ratio of the true particle surface area to the surface area of the equivalent sphere, vs. the diameter of the equivalent sphere, for the GR and LS aggregates.

As was mentioned for the three reference rocks, the spherical harmonic coefficients can be used to compute another measure of particle shape, the L, W, and T parameters measured in three orthogonal directions. While it was certainly possible to compute these parameters for the AZ, GR, IN, and LS aggregates, it was not possible to remove the rocks from the cement paste matrix and perform direct measurements. Testing these computations against direct measurements on a larger set of standard rocks is the subject of another paper [11].

A final method concerns the moment of inertia tensor [1,
21], which relates the rotational response of the particle to an
applied torque, in a similar way that mass relates the translational response to an applied
force. If we take the trace of the moment of inertia tensor, divided by three, and divide by
the moment of inertia of the equivalent sphere, this ratio is a measure of the non-sphericity
of the particle, since it is equal to 1 for a spherical particle. One third times the trace
of the moment of inertia tensor for a sphere is 2/5 R^{2} = 1/10 D^{2},
where R is the radius and D is the diameter of a sphere. Figure
8 shows graphs of this quantity for the GR and LS aggregates. This
graph appears to show a greater qualitative difference between these two aggregates as
compared to the differences shown in Figs. 6 and
7 between GR and AZ. This may be because the moment of inertia of a
particle samples the non-sphericity more than does the volume or surface area, since three
lengths are implied in this parameter via the principal moments of inertia
[11]. There is a lot of noise in the graph, but just like in
Fig. 7, there seems to be a higher value of non-sphericity as the
particle diameter becomes smaller. The moment of inertia tensor is a subject of further
study [1, 21].

Figure 8: The ratio of the trace of the actual moment of inertia tensor to the moment of inertia of the equivalent diameter sphere for the GR and LS aggregates.