Planar defects – two-dimensional imperfections

What you must remember:

1) point defects and linear defects are acknowledgements that crystalline materials cannot be made flaw-free.

2) These imperfections exists in the interior of each of these materials.

3) There are various forms of planar defects.

Fig.4.14 shows a twin boundary which separates 2 crystalline regions that are, structurally, mirror images of each other.

Fig.4.14 A twin boundary separates 2 crystalline regions that are, structurally, mirror images of each other.

This highly symmetrical discontinuity in structure can be produced by deformation (e.g., in bcc and hcp metals) and by annealing (e.g., in fcc metals). A more detailed picture of atomic-scale surface geometry is shown in Fig. 4.16. This Hirth-Pound model of a crystal surface has elaborate ledge systems rather than atomically smooth planes.

The most important planar defect for our consideration in this introductory course occurs at the grain boundary, the region between 2 adjacent single crystals, or grains.

Fig. 4.15. Simple view of the surface of a crystalline material

Fig. 4.16. A more detailed model of the elaborate ledge like structure of the surface of a crystalline material. Each cube represents a single atom.

In the most common planar defect, the grains meeting at the boundary have different orientations. Aside from the electronic industry, most practical engineering materials are polycrystalline rather than in the form of single crystals.

Fig. 4.17 Typical optical micrograph of a grain structure, 100x.

The grain boundaries (Fig.4.17) have been lightly etched with chemical solutions so, that they reflect light differently from the polished grains, thereby giving a distinctive contrast.

Fig. 4.18 shows a usually simple grain boundary produced when 2 adjacent grains are tilted only a few degrees relative to each other. This tilt boundary is accommodated by a few isolated edge dislocations.

Most grain boundaries involve adjacent grains at some arbitrary and rather large disorientation angle. The grain boundary structure in this general case is considerably more complex than that shown in Fig. 4.18. Now we have improved understanding for the nature of the structure of high-angle grain boundary (by electron microscopy and computer modeling).

Explanation N1:

1. A central component in the analysis of grain boundary structure is the concept of the coincident site lattice (CSL). Fig. 4.19.

2. A high-angle tilt boundary (q = 36.9⁰) between 2 simple square lattices is shown in Fig. 4.19a. This specific tilt angle has been found to occur frequently in grain boundary structure in real materials. The reason for its stability is an especially high degree of registry between the 2 adjacent crystal lattices in the vicinity of the boundary region. (Remember: that a number of atoms along the boundary are common to each adjacent lattice).

Fig.4.18. Simple grain boundary structure. This is termed a tilt boundary because it is formed when two adjacent crystalline grains are tilted relative to each other by a few degrees (θ). The resulting structure is equivalent to isolated edge dislocations separated by the distance b/ θ, where b is the length of the Burgers vector, b.

3. This correspondence at the boundary has been quantified in terms of the CSL.

(a) (b)

Fig.4.19. (a) A high-angle (q = 36.9⁰) grain boundary between 2 square lattice grains can be represented by a coincidence site lattice, as shown in (b).

4. Fig.4.19b shows that, by extending the lattice grid of the crystalline grain on the left, one in five of the atoms of the grain on the right is coincident with that lattice.

5. The fraction of coincident sites in the adjacent grain can be represented by the symbol S^-1 = 1/5 or S = 5, leading to the label for the structure in Fig. 4.19a as a “S5 boundary”.

6. The geometry of the overlap of the 2 lattices also indicates why the particular angle of (q = 36.9⁰) arises. One can demonstrate that q = 2 tan^-1 (1/3).

Explanation N2:

Another indication of the regularity of certain high-angle grain boundary structures is given in Fig. 4.21, which illustrates a S5 boundary in an fcc metal.

This is a three-dimensional projection with the open circles and closed circles representing atoms on two different, adjacent planes (each parallel to the plane of this page. The crystalline grains can be considered to be composed completely of tetrahedra and octahedra.

1. Polyhedra formed by drawing straight lines between adjacent atoms in the grain boundary region are irregular in shape due to the misorientation angle but reappear at regular intervals due to the crystallinity of each grain.

Fig.4.20. A S5 boundary in an fcc metal, in which the [100] directions of two adjacent fcc grains are oriented at 36.9^oto each other.

2. Low-angle model (Fig. 4.18) serves as a useful analogy for the high-angle case. Specifically a grain boundary between 2 grains at some arbitrary, high angle will tend to consist of regions of good correspondence (with local boundary rotation to form a S n structure, where n is a relatively low number) separated by grain boundary dislocations (GBD), linear defects within the boundary plane.

3. The GBD associated with high-angle boundaries tend to be secondary in that they have Burgers vectors different from those found in the bulk material (primary dislocations).

With atomic-scale structure in mind, we can return to the microstructure view of grain structures (e.g., Fig.4.17). In describing microstructures, it is useful to have a simple index of grain size. A frequently used parameter standardized by American Society for Testing and Materials (ASTM) is the grain-size number, G, defined by

N = 2 ^G-1 (1)

where N is the number of grains observed in an area of 1 in.² (=645 mm²) on a photomicrograph taken at a magnification of 100 times (100 x), as shown in Fig. 4.21.

Fig.4.21. Specimen for the calculation of the grain size number, G, 100x. The material is a low-carbon steel similar to that shown in Figure 4-17.

The calculation of G follows.

There are 21 grains within the field of view and 22 grains cut by the circumference, giving 21 + 22/2 = 32 grains

in a circular area with diameter = 2.25 in. The area density of grains is N = 32 grains / [ p(2.25/2)² ]in.² = 8.04 grains / in.²

From equation (1)

N = 2 ^G-1 or

G = ln N / ln 2 + 1 = ln (8.04)/ln 2 + 1 = 4.01

Although the grain- size number is a useful indicator of average grain size, it has the disadvantage of being somewhat indirect. It would be useful to obtain an average value of grain diameter from a microstructural section. A simple indicator is to count the number of grains intersected per unit length n_L, of a random line drawn across a micrograph. The average grain size is roughly indicated by the inverse of n_L, corrected for the magnification, M, of the micrograph.

Remember:

1) you must consider that the random line cutting across the micrograph (in itself, a random plane cutting through the microstructure) will not tend, on average, to go along the maximum diameter of a given grain.

2) Even for a microstructure of uniform size grains, a given planar slice (micrograph) will show various size grain sections (e.g., Fig. 4.22), and a random line would indicate a range of segment lengths defined by grain boundary intersections.

3) In general, then, the true average grain diameter, d, is given by

d = C/ n_LM (2)

where C is some constant greater than 1. Extensive analysis of the statistics of grain structures has led to various theoretical values for the constant, C. For typical microstructures, a value of C =1.5 is adequate.

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