Order! Order!
Figure 5: Macroscopic man-made objects that show uniform regularity as described in the text.
Precipitate particles are generally self- avoiding, since their formation depletes the surrounding matrix of some element(s). Particulates that tend to cluster or agglomerate because of some attractive property, such as static electri- cal charge on their surfaces, may cause difficulties in application to substrates. Nonuniformity must be controlled in spraying, electrodeposition, 3D printing, painting, and even the forming of particle board. Uniformity of size and dispersal of second-phase regions in mixed polymers is important for their properties [3]. A molecular strategy produces self-avoid- ance in patterning axons and dendrites in both vertebrates and invertebrates [4,5]. Partial regularity characterizes the spacing of leopard spots and zebra stripes [6]. Topological analysis of clusters and cluster boundaries arises in fields from astronomy to genetics [7]. All of these, and more, are cases in which uniformity is rarely perfect but must be monitored and characterized. Another aspect of partial order
Figure 6: Examples of neighbor distance with the measured ratio of mean value to the expected value for a random distribution calculated from area and number: (a) precipitate particles in a magnesium alloy (self-avoiding); (b) deposited PLGA nanoparticles (random); (c) oil droplets in an emulsion (clustered).
involves the need to find repetitions of a pattern that may be partially obscured or have minor variations, and may occur in non-regular positions. Cross-correlation is a useful tool for this, as illustrated in Figure 8. Te process can be envisioned as sliding the target pattern over the search area and calculating the degree of match. Te result shows peaks that locate and measure the degree of matches.
Fractal Structures Another way that many natural
Figure 7: Military formation vs. a flock of birds.
structures are formed involves fractal geometry. Te principle is a self-similar repetitive arrangement at many different dimensions. In erosion or turbulence, for example, processes operate in similar ways at both large and small scales. Figure 9 illustrates a strictly repetitive geomet- ric operation that produces a Sierpinski gasket: the initial triangle has its central triangular section removed, leaving four identical smaller triangles. Te same pro- cedure is applied to each of them, and this is repeated ad infinitum leaving as an end limit a structure with an infinite bound- ary length and zero area. Branching structures are a common
Figure 8: Cross-correlation analysis to identify structural patterns: (a) SEM image of a leaf surface; (b) target (enlarged to show pixels); (c) results showing peaks at matched (red) and partially matched (green) locations.
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phenomenon, and these can also be cre- ated by a repetitive scaling as shown in Figure 10a. Small changes in the origi- nal pattern produce different results [8].
www.microscopy-today.com • 2020 March
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