In the extreme prolate limit, the aspect ratio a/b diverges, while in the extreme oblate limit, the inverse aspect ratio, b/a, diverges. In these limits, the above shape functions reduce to the following limiting forms:
| [σ]∞ → (a/b)2 [9 ln (a/b) ] −1 | (prolate)(17) |
| [σ]∞> → ( 8 b ) / ( 9 π a ) (oblate) | (18) |
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| [ σ ] φ ~ 1 | (24) |
| φ = 1 − e −nVp | (25) |
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| C → ( 2 / π ) ( b / a )1/3 (oblate) | (28) |
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Figure 1 shows a graphical comparison of the shape functionals described in Sections 2.1-2.6. We observe the widely different variation of these functionals with particle aspect ratio. The very different magnitudes of [σ]∞ and −[σ]o for needle-like particles is especially notable. It is also observed that despite the many differences in the qualitative variation of these functionals with particle asymmetry, all these properties exhibit absolute minima for the sphere. In many of the cases presented, this result has been proven for all objects of finite fixed volume, as was indicated above.
Figure 1: Various shape functionals (defined in the text) plotted vs. aspect ratio for ellipsoids of revolution. Notice that the minimum occurs in each case for the sphere.