Order! Order!
Figure 2: Atomic lattices: a) TEM image of a gold nanoparticle; b) AFM image of the surface of graphite; (c) TEM image of a grain boundary in gold foil.
“grain size” is usually done by super- imposing a circular grid and count- ing intersections with the boundaries, which measures the surface area of grain boundaries per unit volume [2]. Tis method is useful for an equilib- rium grain structure, but depending on heat treatment and mechanical deformation a duplex arrangement of different populations of grains may develop, in which case the “grain size” number can be misleading. Rolling the metal to a thin sheet (Figure 4b) or drawing it into a wire squeezes the grains into a distorted shape with
many more dislocations, which tangle and increase the hard- ness but decrease the ductility. Counting the number of inter- sections of a grid of lines with the boundaries as a function of direction is a commonly used method of assessing the degree of elongation and anisotropy. Some macroscopic man-made objects show uniform regular-
ity. A brick wall (Figure 5a) or hexagonal bathroom tile floor have minor variations that do not attract much attention. A parquet floor (Figure 5b) uses different wood grains and orientations that create more interest, and the elaborate tiling in the Alhambra (Fig- ure 5c) are regular and periodic but with enough variations and symmetries to engage the viewer. Even large man-made structures oſten exhibit regularity and symmetry along with complexity and unique design elements. Many cities have a regular array of streets (sometimes with unimaginative names like 42nd Street and 6th Avenue), which is convenient but not interesting. One way to measure the tendency toward regular or self-
Figure 3: Measuring atomic spacings using a selected area diffraction pattern (d-spacing • radius = constant).
Many common materials, especially metals, are produced
with controlled microstructures that seek to optimize the grain structure and dislocation density to provide the desired mechanical properties. Figure 4a shows the grain structure of a metal as it appears when polished and etched to reveal the grain boundaries. Generally the smaller and more uniform the grains the better the strength and ductility. Measuring the
avoiding spacing and its opposite, clustering, is to measure the neighbor distances. For a random distribution, such as a view of stars in the night sky (in spite of the human desire to impose order in the form of imagined constellations), the mean nearest neighbor distance is just 0.5•(Area/Number)1/2
. As illustrated
in Figure 6, a self-avoiding arrangement has a greater mean neighbor distance, giving a ratio to the value for random spac- ing greater than 1, while a clustered one has a smaller ratio value. Fourier transforms (or diffraction spots) may also be useful if the intensity profile of the peaks can be measured to represent the variations in spacing.
Partial Order A marching column of soldiers is
expected to show a perfect order with regular spacing, but in nature things are less perfect (Figure 7). Still, in a flock of birds or school of fish there is a degree of regularity that becomes apparent upon careful examination. Rather than a global regularity, each individual seeks to maintain the same approximate dis- tances from nearby neighbors as they move in coordination. Te characterization of partial
Figure 4: Metal grain structures: (a) equiaxed; (b) elongated by cold-rolling. 2020 March •
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order is important in many materials that are examined microscopically.
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