Coincidence-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... 

Figure 4.9. As the rotation begins, the atoms above and below the plane no longer coincide. However, at some specific rotation angles a firaction of the atoms will be brought back into coincidence. Figure 4.11 (b) shows the position of the atoms after a rotation of 36.9°. The five numbered atoms (a fifth of the total) now coincide. This behavior can be described in terms of the coincident-site lattice (CSL). This is the lattice formed by the points of coincidence. The CSL is described by S, which is the inverse of the fraction of the coincident sites. Therefore the CSL for the twist boundary in Figure 4.11(b) is a S = 5 boundary. The complete notation for this boundary is S = 5, 36.9°/[001].

Points being elements of both lattices. The density of sites of coincidence is denoted by the E-nomenclature In the case of a En boundary, 1/n gives the fraction of lattice points that form the coincidence site lattice. 

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).

Fig. S A coincident site boundary based on a 35.9° tilt about (100) in a simple cubic lattice. One of every five lattice sites is coincident to both boundaries, = 1 /5 or E = 5. (Ref [25], from Physical Metallurgy Principles, 3rd edition by Reed-Hill/Abbaschian, 1992. Reprinted with permission of Brooks/Cole, a division ofThomson Learning www.thomsonrights.com, Fax 800 730-2215.)...

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