Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3ⁿgrain boundaries

Abstract

Multiple twinning in cubic crystals is represented geometrically by a threedimensional fractal and algebraically by a groupoid. In this groupoid, the variant crystals are the objects, the misorientations between the variants are the operations, and the AE3 n operators are the different types of operations (expressed by sets of equivalent operations). A general formula gives the number of variants and the number of AE3 n operators for any twinning order. Different substructures of this groupoid (free group, semigroup) can be equivalently introduced to encode the operations with strings. For any coding substructure, the operators are expressed by sets of equivalent strings. The composition of two operators is determined without any matrix calculation by string concatenations. It is multivalued due to the groupoid structure. The composition table of the operators is used to identify the AE3 n grain boundaries and to reconstruct the twin related domains in the electron back-scattered diffraction maps.

Twin related domains in 3D microstructures of conventionally processed and grain boundary engineered materials

“…This could be studied by examining the twin ordering. In particular, Reed [14,17] and Cayron [18,27,36] have accomplished this by reconstructing the TRD twinning tree. Note that, as described in the literature, twinning tree construction is inherently ambiguous, as the designation of the central grain, or the root of the twinning tree is not unique.…”

Grain orientation statistics of grain-clusters and the propensity of multiple-twinning during grain boundary engineering

“…The proportion of twin-boundaries could be increased to more than 60% from about 40% by the treatment of GB-engineering [11,12,[17][18][19][20][21], because the face-centered-cubic law stacking-fault-energy metals, such as Ni-based alloys, austenitic stainless steel and brass, are prone to twinning [1,2] in the wake of migrating of the recrystallization front boundaries during annealing. The iterative process of twinning operations starting from a single nucleus is referred to as multiple-twinning [21][22][23][24][25], and the generated annealing twins can be reconstructed as a twinning tree, which is referred to as twin-chain [8,11,22,26,27]. The assembly of all grains in the tree is named grain-cluster or twin-related-domain [8,11,12,24,26].…”

Laser processing for thin film crystalline silicon solar cells

“…This texture is lost during the solidification toward the centre of the line and the scanning direction because of the twinning process and by the formation of new nuclei in the central part of the irradiated line. The weakening of the texture due to twinning is a common phenomenon in low stacking fault energy cubic materials18,19 .…”