On the small angle twist sub-grain boundaries in Ti3AlC2

Tilt-dominated grain boundaries have been investigated in depth in the deformation of MAX phases. In stark contrast, another important type of grain boundaries, twist grain boundaries, have long been overlooked. Here, we report on the observation of small angle twist sub-grain boundaries in a typical MAX phase Ti3AlC2 compressed at 1200 °C, which comprise hexagonal screw dislocation networks formed by basal dislocation reactions. By first-principles investigations on atomic-scale deformation and general stacking fault energy landscapes, it is unequivocally demonstrated that the twist sub-grain boundaries are most likely located between Al and Ti4f (Ti located at the 4f Wyckoff sites of P63/mmc) layers, with breaking of the weakly bonded Al–Ti4f. The twist angle increases with the increase of deformation and is estimated to be around 0.5° for a deformation of 26%. This work may shed light on sub-grain boundaries of MAX phases, and provide fundamental information for future atomic-scale simulations.

Scientific RepoRts | 6:23943 | DOI: 10.1038/srep23943 morphology. According to the g·b values in Table 1, the Burgers vectors of the dislocation segments out of contrast in Fig. 1b- by a, b and c, Fig. 1e), respectively. For the sake of clarity, the dislocation segments in contrast are illustrated by solid lines and those out of contrast are marked with black dotted lines, as shown in Fig. 1b-d. White arrows represent the Burgers vectors of the dislocations out of contrast. Evidently, the Burgers vectors are parallel to the dislocation segments, indicating that the segments are screw type. The dislocation reaction at the nodes is a + b → c. Figure 1f schematically illustrates the formation of HSDNs. The a-and b-type dislocation arrays encounter each other and react to form c-type segments. Under interaction forces, the final equilibrium configuration of HSDNs is established 16 . Theoretically, the initial a-and b-type dislocation arrays can be screw, edge or mixed type 16 , provided equation (1) is satisfied: , and react to form c-type (red) dislocation segments (② ). Dislocation configurations evolve to be HSDNs (③ ). Theoretically, the initial a-and b-type dislocation arrays can be screw, edge or mixed type. Nevertheless, irrespective of the nature of the initial dislocation arrays, the dislocation segments in the final equilibrium dislocation configuration are screw type. For the sake of simplicity, only the case for screw dislocation arrays is considered.
where b i is the Burgers vector and ξ i is the direction vector of dislocation line. Since a 2 + b 2 > c 2 , the above-mentioned reaction is energetically favorable. However, not all the crossing aand b-type segments can react to form c-type segments. As marked by the black arrows in Fig. 1a, a long a-type segment fails to react with b-type segments although they intersect at several positions, wherein the intersection angles are approximately 90°. This is more apparent in Supplementary Fig. S1. According to Hirth et al., crossing dislocations that are a few degrees from being orthogonal cannot react 16 . Therefore, orthogons (Supplementary Fig. S1) or pentagons (Fig. 1a) instead of hexagons are formed because of unfavorable dislocation line directions. It is worth noting that, apart from a + b → c (marked by the black circle in Supplementary Fig. S1), the reaction a + c → b (marked by the red circle in Supplementary Fig. S1) contributes to the formation of HSDNs as well.
The plane of twist sub-grain boundaries. With the formation of HSDNs, SATGBs are established on the basal plane where the networks are located. Then, a scientific question naturally arises: which basal atomic plane is the most likely boundary plane? We address this issue via studies on atomic-scale deformation (illustrations are presented in Supplementary Fig. S2) and general stacking fault energy (GSFE) landscapes. The crystal structure of Ti 3 AlC 2 comprises edge-sharing Ti 6 C octahedron layers bonded by C-Ti2a and C-Ti4f, and Ti 6 Al triangular prism layers bonded by Al-Ti4f (Fig. 2a, Ti2a and Ti4f denote the Ti atoms located at the 2a and 4f Wyckoff sites of P6 3 /mmc, respectively). Figure 2b plots the changes in bond length for Al-Ti4f, C-Ti2a and C-Ti4f against applied tensile strains along [0001]. It can be seen that the changes in C-Ti2a and C-Ti4f are negligible, and most of the strains are accommodated by the elongation of Al-Ti4f. Specifically, the stretches of Al-Ti4f are 6∼ 10 and 9∼ 66 times those of C-Ti2a and C-Ti4f, respectively. Similar features can be identified in the hydrostatic compression (Fig. 2c), where the contractions of Al-Ti4f are 2∼ 3 times those of C-Ti2a and C-Ti4f. Therefore, Al-Ti4f is the weakest bond in Ti 3 AlC 2 , and shear is believed to occur most easily therein.
To further quantitatively confirm this from an energetic point view, GSFE was calculated (see the Methods and Supplementary Fig. S3 for details). The GSFE of Al-Ti4f is significantly lower than those of C-Ti4f and C-Ti2a (Fig. 3a), and the local maximum (USF) of Al-Ti4f is only 18.9% and 12.5% of those of C-Ti2a and C-Ti4f, respectively (Table 2). Besides, the ideal shear strength (maximum restoring force, defined as the maximum slope of the GSFE curve) of Al-Ti4f is only 19.2% and 12.2% of those of C-Ti2a and C-Ti4f, respectively (Table 2), which is consistent with the trend of interplanar spacing. Thereby, the screw dislocations are believed to be initiated by breaking the Al-Ti4f bonds 21 . Further, the SATGBs are most probably located between the Al and Ti4f atomic layers.

Discussion
MAX phases are well recognized to be of bonding anisotropy 3,22-25 : M 6 X octahedron layers are strongly covalently bonded, while adjacent M-X slabs are relatively weakly coupled by M-A metallic bonds. For Ti 3 AlC 2 , the ideal shear strength with breaking Al-Ti4f is only 19.2% and 12.2% of those with breaking C-Ti2a and C-Ti4f, respectively (Table 2), indicating that the most probable shear plane is the one between the Al and Ti4f layers 21 . Schematically, the shear of Ti 6 Al triangular prisms is illustrated in Fig. 3b. Therefore, the SATGBs observed in this study are believed to be there. As a characteristic parameter of the twist boundary, the twist angle, θ, can be estimated by equation (2) 16,20 : h where b and l h are the lengths of the Burgers vector and hexagon edge, respectively. The calculated twist angle in Fig. 1a is approximately 0.26°. Figure 4a-c present the typical TEM morphologies of the HSDNs in the specimens with various deformations (4%, 14% and 26%). As the deformation proceeds, new dislocations intersect and react with the as-formed hexagonal networks. Consequently, small hexagonal dislocation cells form in large cells, and the average cell size diminishes. Statistical data of l h and θ are plotted in Fig. 4d against the strain. It can be seen that the twist angle scales with the applied strains. For the sample deformed by 26%, the twist angle is about 0.5°. Notably, the HSDNs can be observed in other slowly deformed MAX phases (like Ti 2 AlC, Nb 4 AlC 3 and etc., Supplementary Fig. S4). The formation of HSDNs/SATGBs is generic to low-energy dislocation configurations in slowly deformed MAX phases. For MAX phases, the collective behavior of basal dislocations includes not only the previously reported accumulation of dislocations vertical and parallel to the basal plane 9 , but also the formation of HSDNs/SATGBs. Dislocations are the carriers of plastic deformation. Their mutual interactions and reactions bring about work hardening. The formation of HSDNs contributes to previously identified strain hardening of Ti 3 AlC 2 11 , giving rise to SATGBs. Since the twist angle is very small (around 0.5°), the contribution of SATGBs to the plastic deformation of Ti 3 AlC 2 is quite limited. Sub-grains have been reported in the tensile and compressive creep of MAX phases [26][27][28] . Since the SATGBs are formed in compression with remarkably low strain rates, it is inferred that they likely contribute to the creep at high temperature.
With the formation of low energy dislocation structures, misorientations at grain and sub-grain scale are established. The misorientation angle increases with the strain in metals, linearly 29,30 or in a power law (with an exponent of 1/2) 31,32 . Our work indicates that the twist angle scales with the deformation in a roughly linear manner in the investigated strain and temperature ranges.
In summary, small angle twist sub-grain boundaries (around 0.5°) have been ubiquitously observed in uniaxially compressed Ti 3 AlC 2 . The twist sub-grain boundaries predominantly comprise hexagonal screw dislocation networks that result from basal dislocation reactions. The grain boundary plane is believed to be between the relatively weakly bonded Al and Ti4f layers. In addition, it is unambiguously demonstrated that the twist angle scales with the deformation. This work may shed light on the formation of low-energy dislocation configurations and its evolution with the deformation of MAX phases.   First-principles calculations. The deformation at atomic scale was modelled with the CASTEP module 34 .
Interactions of electrons with ion cores were represented by Vanderbilt-type ultrasoft pseudopotential 37 . The Broyden-Fletcher-Goldfarb-Shanno minimization method was used for geometry optimization, where the plane-wave cut-off energy and the Brillouin zone sampling were fixed at 450 eV and 5 × 5 × 2 Monkhorst-Pack-point meshes 38 , respectively. Differences in total energy, maximum ionic Hellmann-Feynman force, maximum ionic displacement and maximum stress were converged to 5 × 10 −6 eV/atom, 0.01 eV/Å, 5 × 10 −4 Å and 0.02 GPa, respectively. The fully optimized structure was used for uniaxial tension and hydrostatic compression. The deformation modes are illustrated in Supplementary Fig. S2. To ensure a uniaxial deformation along [0001], the lattice parameters perpendicular to the applied strain, as well as the internal coordinates of atoms, were fully relaxed until the stresses were converged to 0.02 GPa. For the GSFE calculation, a supercell with twenty-five atomic layers with Al layers at the center and surfaces was constructed using the fully optimized Ti 3 AlC 2 unit cell, where a vacuum slab of 15 Å in thickness was symmetrically added (see Supplementary Fig. S3). The disregistry on the plane of interest was introduced by rigidly shifting all the atoms above the target plane relative to those below that plane along < > 2110 . Without relaxing the supercell further, total energies of faulted structures were calculated. Then, the energy difference between the faulted and unfaulted structures gives the GSFE.