Singular filament

A. T. Winfree and S. H. Strogatz, Singular filaments organize chemical waves in three dimensions. I. Simple waves , Physica, 8D, 35 (1983) II. Twisted waves , ibid., 9D, 65 (1983) III. Knotted waves , ibid., 10D, 333 (1983). 

Figure 6.11 Simulation of a scroll ring. (Reprinted from Winfree, A. T. Strogatz, S. H. 1983. Singular Filaments Organize Chemical Waves in Three Dimensions. II. Twisted Waves, Physica 9D, 65-80, with kind permission from Elsevier Science, The Netherlands.)...

The thermoplastic textiles that are to be utilized can be applied as mono- or co-extruded oriented films, or filament fleeces, as given in Figure 22.1. In filamentary products, the singular filaments are subjected to stretching during the spinning process. Contrastingly, the tapes for woven fabrics are cut out of extruded films and are subsequently stretched and then further processed to textile structures. 

It is natural to introduce a new coordinate system s,p, q) using the scroll filament as one axis and the normal and binormal directions as the other two (x, y,z)R(s) -I- pN(s) -I- qB s). This representation is locally orthogonal, but if the curve R has nontrivial curvature, the representation is unique only in a neighborhood of the centerline R. For distances more than the radius of curvature away from the curve R, the coordinate system has singularities. 

Clusters, grains, lamellar structures, or filaments with dimensions smaller than 10 nm can be considered as nanostructured materials. These systems present a high ratio of surface area and volume with singular properties. In catalysis, changes in the surface and bulk electronic structure, caused by nanostructuring, may influence the reactivity, causing it to differ substantially from the conventional systems. Therefore, structure-sensitive reactions are strongly influenced by particle sizes... 

Materials which such a complex structure cannot of course exhibit a simple rheological behavior, and relatively easy arguments may be produced to explain—so far qualitatively—how this complex structure affects most of the flow singularities of rubber compound. As illustrated in Figure 5.17, typical nonlinear effects observed with CB filled compounds appear as logical consequences of such a soft three-dimensional network of complex rubber-aggregate entities with connective filaments. 

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