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Chapter 1
Grain Boundary Structures and Defects 1
1.1. Equilibrium structure of grain boundaries
A grain boundary is an interface between two crystals of the same structure. The mechanical properties of industrial materials are driven, not only by the properties of their component crystals, but also by those of the boundaries between those crystals, in particular the structure and chemical composition of the boundaries. The structural materials are generally polycrystalline and their mechanical properties are directly linked not only to the grain size but also to the grain boundaries. Moreover, since materials used in the electronics industry are to be as free from defects as possible, then it is no less true that complex manufacturing processes introduce stresses which are most often released by defects, among them dislocations, twins and also grain boundaries.
These grain boundary āobjectsā are therefore encountered in numerous materials; their structures and their mechanical, chemical and electrical properties have for decades been the subject of in-depth studies and they are becoming of fundamental importance in the new so-called nanomaterials. Before describing the defects found in grain boundaries during mechanical or chemical processes, we will, in this section, present three approaches which have been used in order to describe equilibrium boundaries; the purely geometric approach, the dislocation approach, historically the first, and finally the structural unit approach originally based on energy calculations. Here, two reference works are recommended: [PRI 06] and [SUT 95].
1.1.1. Geometric description and elements of bicrystaiography
1.1.1.1. Degrees of freedom
A grain boundary is defined geometrically using nine parameters, or degrees of freedom: six parameters define the interface operation, which links two adjacent crystals, and three define the interface plane. Of these nine parameters, five are said to be macroscopic and four are said to be microscopic. In cubic systems, where the operation of grain orientation is always rotational, the three macroscopic parameters defining this operation are the angle of rotation Īø and the directional cosines of the rotation axis [u v w]; the final two macroscopic parameters are those which define the orientation of the interface plane given by its normal. The four microscopic parameters are, on the one hand, three parameters which define the translation between grains (translation within the boundary plane and expansion perpendicular to the boundary) and, on the other, the parameter which enables the interface to be positioned along the normal to the boundary plane. These microscopic parameters are, in fact, āenergyā parameters, generally defined from the calculated atomic structure which is the most energetically stable and/or experimentally observed using electron microscopy. In non-cubic materials, the interface operation is not a simple rotation, but is often accompanied by a deformation.
1.1.1.2. Tilt, twist, mixed, symmetrical and asymmetrical grain boundaries
In the interests of simplification, this terminology is used to define grain boundaries according to the relative orientations of the rotation axis [u v w] with respect to the boundary plane {h k l}.
A grain boundary is said to be a tilt boundary if the axis [u v w] is contained within the boundary plane, a twist boundary if the axis [u v w] is perpendicular to the plane and mixed if the axis is inclined. Boundaries are said to be symmetrical if the plane between two grains, for instance 1 and 2, can be defined by the expression {h k l}1= {h k l}2, otherwise they are said to be asymmetrical.
1.1.1.3. Bicrystallography and coincidence
The concept of coincidence and the definition of a coincidence index were proposed by G. Friedel in 1920 and 1926 [FRI 20, FRI 26] in order to describe twins. Bollmann [BOL 70] developed this concept towards a more complete geometric description for grain boundaries, by using the notion of bicrystallography, where two lattices are interpenetrated: a bicrystal is obtained by the interface operation, then the positioning of the interface and finally by suppression, within each grain, of atoms from the other grain. Away from the interface, grain atoms occupy their position of equilibrium in the crystal. For some interface operations, the lattice nodes of the two crystals are coincident (Figure 1.1). We therefore define the CSL (coincidence site lattice) which is in fact the intersection of translation subsets of the two crystal lattices. The coincidence is characterized by a coincidence index Ī£ which is equal to the ratio between the volume of the coincidence unit cell and the volume of the primitive unit cell of the crystal. Ī£ is a whole number (an odd number for cubic materials) which varies discontinuously with Īø. A description of coincidences based on the indices Ī£ in cubic materials has been provided by Mykura [MYK 80]. A 3D coincidence only exists for cubic or hexagonal materials for a particular c/a ratio. In other cases, most grain boundaries are described as āclose to specific orientationsā [GRI 89].
1.1.1.4. DSC lattice
Boundaries encountered within materials are not necessarily coincidence boundaries; the deviation from the coincidence has been geometrically defined by Bollmann [BOL 70] by introducing the DSC (displacement symmetry conserving) lattice, which leaves the CSL invariant. This lattice is the combination of translation subsets of the two crystals; it enables the Burgers vectors of perfect boundary dislocations to be defined, i.e. those which leave the boundary structure unchanged.
The higher the coincidence index, the shorter the elementary vectors of the DSC lattice. Figure 1.2 shows a projection along the rotation axis 011 of the DSC lattice associated with the CSL lattice of the Ī£ = 3 boundary in the face-centered cubic material from Figure 1.1b. Dislocations which are not associated with a Burgers vector belonging to the DSC lattice are named āpartialā and they are situated between two boundary sections of different structures.
1.1.1.5. Limit of the geometric approach
The concepts of tilt and twist or symmetry and asymmetry are highly relative. Indeed, a tilt boundary can also be described as a twist boundary through the symmetrical operations of the crystals. The description chosen is generally the one which, from all possible descriptions, corresponds to the smallest rotation. The description of a boundary also depends on the extent to which it is observed: a micron-scale linear boundary may turn out to be multifaceted if observed over an atomic scale.
It should also be noted that a step perpendicular to the rotation axis in a tilt grain boundary corresponds to a facet of twist character (Figure 1.3). In the image obtained by high r...
