Order! Order!
Figure 15: Arrangements of fibers: (a) regular orthogonal weave; (b) irregular orthogonal weave; (c) regular weave including variations in angle; (d) irregular predomi- nantly orthogonal weave; (e) partially aligned bonded nylon fibers; (f) nonwoven “random” fibers in paper.
(for example, filters). A fully 3D arrangement of “randomized” fibers (Figure 16) generally shows strong preferred orientations and can- not achieve high density because the fibers interfere with each other. With either straight or flexible fibers, this increases pore space and fluid retention, important for absorption of flu- ids (for example, diapers). Randomness may be treated as either
Figure 16: A partially “randomized” 3D array of straight fibers with its orientation distribution func- tion showing the degree of partial order.
complex patterns but are still primarily regular with rep- etitious structural arrangement. Nonwoven materials are assuming increasing importance because of their properties. Nonwovens such as paper may have partial isotropy, substan- tial anisotropy, or approach randomness. It is important to understand that extreme anisotropy may not produce ran- domness in local areas. Arrangements, even locally, in which fibers are predominantly aligned may produce higher den- sity (lower permeability and porosity) and weaker transverse strength. Fiber orientation, determined using analysis of images from light or electron microscopy in 2D or microCT in 3D can provide measurements of local fiber orientations, which are typically represented by polar plots of the orienta- tion distribution function. In three dimensions things become more complicated.
Even if the third (thickness) dimension is relatively small compared to lateral extent, the orientation of fibers can vary greatly and this strongly affects properties such as permeability
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a particular form of order or the absence of order. Questions of randomness are very dif- ficult to answer [15]. For example, biochemists have puzzled whether proteins are the result of historic stepwise construction or are random sequences of individual peptides that have been selected by evolution. Intuition oſten fails in attempting to define or detect true random- ness in a sequence, although at any point it may be impossible to predict the next value [16]. Statistical analysis shows that each of the digits in the value of pi (3.1415926…) has the same overall frequency of occurrence, but there is no pattern of regularity, repeats, or order in the sequence. Yet it can hardly be called ran- dom since it is entirely determined by a simple mathematical constant.
Conclusion Tere are a variety of methods that can be used to char-
acterize the regularity of patterns, including diffraction and various measurement techniques. Tese are applicable at any scale, from atomic to real-world dimensions. As arrangements depart from perfect order, the measurements that can mean- ingfully describe them become increasingly difficult and vary from one application to another. Finally, determining that a pattern is truly random is extremely difficult; the statistical tests required are complex and not very satisfying. Relying on human vision to assess a degree of order or randomness is gen- erally insufficient.
References [1] RP Taylor et al., Nature 399(6735) (1999) 422. [2] American Society for Testing and Materials, ASTM E112 Standard Test Methods for Determining Average Grain
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