Noncrystalline Solids— Three- Dimensional Imperfections

Some engineering materials lack the repetitive, crystalline structure. These noncrystalline, or amorphous, solids are imperfect in three dimensions. The two- dimensional schematic of Figure  4.19(a) shows the repetitive structure of a hypothetical crystalline oxide. Figure  4.19(b) shows a noncrystalline version of this material. The latter structure is referred to as the Zachariasen* model and, in a simple way, it illustrates the important features of oxide glass struc-tures. (Remember from Chapter 1 that glass generally refers to a noncrystalline material with a chemical composition comparable to that of a ceramic.) The building block of the crystal (the AO^3-₃ “triangle”) is retained in the glass; that is, short- range order (SRO) is retained. But long- range order (LRO)—that is, crystallinity— is lost in the glass. The Zachariasen model is the visual definition of the random network theory of glass structure, which is the analog of the point lattice associated with crystal structure.

Our first example of a noncrystalline solid was the traditional oxide glass because many oxides (especially the silicates) are easy to form in a noncrystalline state, which is the direct result of the complexity of the oxide crystal structures.

Rapidly cooling a liquid silicate or allowing a silicate vapor to condense on a cool substrate effectively “freezes in” the random stacking of silicate building blocks

or

G = ln N ln 2 + 1 = ln(8.04)

ln 2 + 1 = 4.01.

PRACTICE PROBLEM 4.5

In Example  4.5, we find the separation distance between disloca-tions for a 2° tilt boundary in aluminum. Repeat this calculation for (a) u = 1° and (b) u = 5°. (c) Plot the overall trend of D versus u over the range u = 0 to 5°.

PRACTICE PROBLEM 4.6

Find the grain- size number, G, for the case described in Example 4.6 if the micrograph was taken at a magnification of 300* rather than 100*.

*William Houlder Zachariasen (1906–1980), Norwegian- American physicist, spent most of his career working in x- ray crystallography. However, his description of glass structure in the early 1930s became a standard definition for the structure of this noncrystalline material.

(SiO⁴₄^- tetrahedra). Since many silicate glasses are made by rapidly cooling liq-uids, the term supercooled liquid is often used synonymously with glass. In fact, however, there is a distinction. The supercooled liquid is the material cooled just below the melting point, where it still behaves like a liquid (e.g., deforming by a viscous flow mechanism). The glass is the same material cooled to a sufficiently low temperature so that it has become a truly rigid solid (e.g., deforming by an elastic mechanism). The relationship of these various terms is illustrated in Figure 6.40.

The atomic mobility of the material at these low temperatures is insufficient for the theoretically more stable crystalline structures to form. Those semiconductors with structures similar to some ceramics can be made in amorphous forms also. There is an economic advantage to amorphous semiconductors compared with preparing high- quality single crystals. A disadvantage is the greater complexity of the elec-tronic properties. As discussed in Section 3.4, the complex polymeric structure of plastics causes a substantial fraction of their volume to be noncrystalline.

Perhaps the most intriguing noncrystalline solids are the newest members of the class, amorphous metals, also known as metallic glasses. Because metallic crystal structures are typically simple in nature, they can be formed quite easily.

It is necessary for liquid metals to be cooled very rapidly to prevent crystalliza-tion. Cooling rates of 1°C per microsecond are required in typical cases. This is an expensive process, but potentially worthwhile due to the unique properties of these materials. For example, the uniformity of the noncrystalline structure eliminates the grain- boundary structures associated with typical polycrystalline metals, which results in unusually high strengths and excellent corrosion resis-tance. Figure 4.20 illustrates a useful method for visualizing an amorphous metal

(a) (b)

FIGURE 4.19 Two- dimensional schematics give a comparison of (a) a crystalline oxide and (b) a noncrystalline oxide. The noncrystalline material retains short- range order (the triangularly coordinated building block), but loses long- range order (crystallinity). This illustration was also used to define glass in Chapter 1 (Figure 1.8).

FIGURE 4.20 Bernal model of an amorphous metal structure. The irregular stacking of atoms is represented as a

connected set of polyhedra. Each polyhedron is produced by drawing lines between the centers of adjacent atoms. Such polyhedra are irregular in shape and the stacking is not repetitive.

structure: the Bernal* model, which is produced by drawing lines between the centers of adjacent atoms. The resulting polyhedra are irregular in shape, and lack any repetitive stacking arrangement.

At this point, it may be unfair to continue to use the term imperfect as a general description of noncrystalline solids. The Zachariasen structure (Figure 4.19b) is uniformly and “perfectly” random. Imperfections such as chem-ical impurities, however, can be defined relative to the uniformly noncrystalline structure, as shown in Figure 4.21. Addition of Na⁺ ions to silicate glass substan-tially increases formability of the material in the supercooled liquid state (i.e., viscosity is reduced).

Finally, the state of the art in our understanding of the structure of noncrys-talline solids is represented by Figure 4.22, which shows the nonrandom arrange-ment of Ca²⁺ modifier ions in a CaO@SiO₂ glass. What we see in Figure 4.22 is, in fact, adjacent octahedra rather than Ca²⁺ ions. Each Ca²⁺ ion is coordinated by six O^2- ions in a perfect octahedral pattern. In turn, the octahedra tend to be arranged in a regular, edge- sharing fashion, which is in sharp contrast to the random distribution of Na⁺ ions implied in Figure  4.21. The evidence for

Si⁴⁺ O^2- Na⁺

FIGURE 4.21 A chemical impurity such as Na⁺ is a glass modifier, breaking up the random network and leaving nonbridging oxygen ions.

CaO₆

FIGURE 4.22 Schematic illustration of medium- range ordering in a CaO@SiO₂ glass. Edge- sharing CaO₆ octahedra have been identified by neutron- diffraction experiments. [Adapted from information in P. H. Gaskell et al., Nature 350, 675 (1991).]

*John Desmond Bernal (1901–1971), British physicist, was one of the pioneers in x- ray crystallog-raphy but is perhaps best remembered for his systematic descriptions of the irregular structure of liquids.

medium- range order in the study represented by Figure 4.22 confirms longstand-ing theories of a tendency for some structural order to occur in the medium range of a few nanometers, between the well- known short- range order of the silica tetrahedra and the long- range randomness of the irregular linkage of those tetrahedra. As a practical matter, the random network model of Figure  4.19b is an adequate description of vitreous SiO₂. Medium- range order such as that shown in Figure 4.22 is, however, likely to be present in common glasses contain-ing significant amounts of modifiers, such as Na₂O and CaO.

EXAMPLE 4.7

Randomization of atomic packing in amorphous metals (e.g., Figure  4.20) generally causes no more than a 1% drop in density compared with the crystalline structure of the same composition.

Calculate the APF of an amorphous, thin film of nickel whose density is 8.84 g/cm³.

SOLUTION

Appendix 1 indicates that the normal density for nickel (which would be in the crystalline state) is 8.91 g/cm³. The APF for the fcc metal structure is 0.74 (see Section 3.2). Therefore, the APF for this amor-phous nickel would be

APF = (0.74) * 8.84

8.91 = 0.734.

THE MATERIAL WORLD