Order! Order!
John C. Russ Professor Emeritus, Dept. of Materials Science and Engineering, North Carolina State Univ., 3002 Engineering Building 1 (EBI), 911 Partners Way, Raleigh, NC 27695
jruss@ncsu.edu
Abstract: Order, in the sense of arrangement, structure, coordination, or sequence, arises in many fields and at all scales. It is found with varying degrees of perfection in many if not most subjects for micros- copy, can take many forms, and can be challenging to recognize or to measure. A variety of examples are provided that are intended to make the reader more aware of the possibilities.
Keywords: Pattern, regularity, fractal dimension, orientation, neigh- bor distance
Introduction Humans like to say we crave order. We ask our children to
“keep their rooms in order,” order is demanded in courtrooms, and we definitely frown on disorderly conduct. But, in fact, perfect order is perfectly boring. In the sense of regularity, it means that observing one part of something means we know all about it, and there is nothing more to learn. Te painting Red Square by Kazi- mir Malevich (Figure 1a) is perhaps an extreme of perfect order. Intended as an avant-garde statement of the ultimate supremacy of pure feeling removed from any representational content, it does not cause the eye to linger or search for (nonexistent) details. Another abstract expressionist artist, Jackson Pollock, cre-
ated much more interesting paintings (Figure 1b) in which there are multiple colors and shapes. Te swirls and blobs are not ran- dom, although they may appear so at first. His paintings have in fact been shown to be fractal [1], one of the recurring ways that nature oſten organizes things. Tis makes Pollock’s work much more interesting to view than simply a collection of ran- domly placed and sized paint drops in random colors (that is, complete disorder). Groupings and structures that are interme- diate between perfect order and complete disorder are the most interesting. Te concepts of arrangement, proportion, and pat- tern arise in many fields, and not just those that involve aesthet- ics such as flower arranging (ikebana). Tis is especially true in applications that include microscopy, and several are worth examining to better understand the possible types of order and the available measurement tools.
Atomic Structure Atoms, and groups of atoms, gen-
erally form bonds that produce very regular three-dimensional arrange- ments. Even in liquids, there is generally some preferential local arrangement of atoms. Water, for example, has two hydrogen atoms bonded to the oxygen atom at the tetragonal angle of 104.5°, and the weak Van der Waals bonds from the dangling hydrogen atoms lead to a preferred arrangement of local molecules that is hexagonal. Tis
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accounts for the stability of the liquid, as other similar chemicals such as hydrogen sulfide are not bent, and are gases at room tem- perature. When water freezes, the bent molecules become lined up in the hexagonal 120° arrangement seen in snowflakes. Tere are 14 Bravais lattices that are possible for long-range
order. Other possibilities exist: non-periodic arrangements of regions with atomic lattices that cannot extend to fill space have been found in both man-made and natural materials (read Te Second Kind of Impossible by Paul Steinhardt for a highly entertaining and scientifically sound background). It is possible using the TEM or AFM to visualize the atomic positions (Fig- ure 2a, 2b). Te bonds between atoms are very strong, and sin- gle perfect crystals have important mechanical and electronic properties, but most materials are quite imperfect. Even a 0/0/0 diamond (highest quality cut, color, and clarity) has locations in the lattice where an atom is missing (vacancy) or some atom other than carbon, for example, hydrogen, has squeezed in among the atoms (interstitial). Modern solid-state electronics depend on highly perfect silicon single crystals that are inten- tionally doped with atoms that replace silicon (substitution) and modify the electronic structure to create semiconductors. In addition to these local defects in crystal structure, most
crystalline materials consist of multiple crystals that have dif- ferent orientations. Te places where these grains meet (Figure 2c) are grain boundaries that strongly affect the overall prop- erties (and make materials orders of magnitude weaker than a perfect single crystal). In addition, dislocations within the grains are atomic-scale offsets in the regular lattice that can shiſt position relatively easily and so cause deformation of the grain and cumulatively of the bulk material. Max von Laue used X-ray diffraction to reveal the periodic-
ity of atomic spacing in crystals (for which he received the 1914 Nobel Prize in physics). Electron diffraction is commonly used to reveal the structure of specimens in the TEM and can be used to measure the atomic spacing as shown in Figure 3.
Figure 1: Abstract paintings: (a) Kazimir Malevich, Red Square; (b) Jackson Pollock, Number 8. doi:10.1017/S1551929520000279
www.microscopy-today.com • 2020 March
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