Grain coincident site lattice boundaries

The grain boundary energy 7gb should be proportional to . For small values of high coincidence occurs and the number of broken bonds can be minimized. = 1 corresponds to complete coincidence of the ideal crystal. Experimentally it was found that the correlation between 7Gb and is not that simple due to volume expansions or translations at the grain boundaries. A principal problem of the coincident site lattice model is that, even arbitrarily small variations of the lattice orientation lead mathematically to a complete loss of coincidence. This is physically not reasonable because an arbitrarily small deviation should have a small effect. This problem was solved by the O-lattice theory [343], For a comprehensive treatment of solid-solid interfaces and grain boundaries, see Refs. [344,345],... 

V. Randle, The Role of the Coincidence Site Lattice in Grain Boundary Engineering, The Institute of Materials, London, 1996. 

Fig. 9.41. Schematic of several representative grain boundaries with structures described by coincident site lattice model. This set of boundaries corresponds to a (001) rotation axis, and the atomic-level geometries have not been relaxed (courtesy of D. Pawaskar). The filled circles correspond to those sites (coincident sites) that are common to both lattices.

The atomic level structure of grain boundaries has been an important issue for the past several decades. In cubic materials geometrical constructs of periodic grain boundaries can be obtained for certain misorientation axis-angle combinations that are associated with coincident site lattices (CSLs). The CSLs are formed by the coincident sites of two hypothetically interpenetrating crystal lattices, where S is the reciprocal density of CSL sites. Much of the discussion of grain boundary structure and properties has revolved around the description of grain boundary structures in terms of the CSL, the displacement-shift complete (DSC) and the 0-lattice [10.10, 10.11]. 

Another possibility for obtaining CSLs exists by approximating the crystal structure by pseudocubic or tetragonal unit cells. This approach of applying the CSL to non-cubic systems has been discussed in the literature by means of the constrained coincident site lattice [10.12] which has, among others, also been applied to YBCO grain boundaries. 

When any deviation from the perfect crystal requires any of the macroscopic DOFs to be mediated, the related boundary can be termed the macroscopic grain boundary. Typical examples are general grain boundaries and special grain boundaries such as coincident site lattice (CSL) boundaries and twins. However, when only the micro-... 

Lin, P., Palumbo, G., Harase, (., and Aust K.T. (1996) Coincidence site lattice (CSL) grain boundaries and Goss texture... 

In the case of high-angle grain boundaries (see Fig. 5.21) both the extended dislocation models and coincidence site lattice models have proven worthwhile. The... 

Fig. 9.39. Interpenetrating lattices used to consider grain boundary structure (courtesy of D. Pawaskar). Open and filled circles correspond to the two host lattices and filled squares correspond to those atoms that are common to both lattices (i.e. the lattice of coincident sites).

An example of such a boundary is shown in Figure 10.7. Here, the special orientational relationship is that a fraction of the total lattice sites between the two grains coincides periodically. 

Coincidence grain boundary in a simple cubic lattice that corresponds to a 36.9° rotation about a <100> direction. One fifth of the atoms (dark circles) in the boundary have common sites. 

For example, if one-third of the A (or B) crystal lattice sites are coincidence points belonging to both the A and B lattices, then E = 1 / = 3. The value of also gives the ratio between the areas enclosed by the CSL unit cell and crystal unit cell. The value of E is a function of the lattice types and grain misorientation. The two grains need not have the same crystal structure or unit cell parameters. Hence, they need not be related by a rigid body rotation. The boundary plane intersects the CSL and will have the same periodicity as that portion of the CSL along which the intersection occurs (Lalena and Cleary, 2005). 

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