Introduction

Disconnection is the elementary carrier for phase transformation and twinning1,2. Disconnection has dislocation character with Burgers vector b and step character with height h, described as (b, h)2,3,4. The motion of disconnections on the habit plane not only accommodates shear displacement but also reorient crystal into new orientation or new phase1,2,3,4,5,6. A dislocation is subject to the same Peierls stress as it moves forward or backward when the glide plane holds a two-fold rotation symmetry about its normal axis7. This is referred to as symmetric resistance to dislocation motion. Dislocation motion accommodates shear displacement on slip plane, generating plastic deformation. Following the understanding to dislocation motion, it is widely recognized that a disconnection is also subject to symmetric resistance because it has dislocation character and the rational twin plane holds a two-fold rotation symmetry. Obviously, the step character associated with disconnections was totally ignored in evaluating the moving direction dependence of glide resistance.

Phase transformation is generally operated by two different disconnections (i.e. b1 and b2 on the left and right) with the same signed Burgers vector to relieve the coherency stress5,8 with a tilt wall left at the origin. Burgers vector of b1 and b2 affects their mobility, which could be revealed by the shape of a thin domain that does not fully traverse the grain1,9. If b1 and b2 have different Burgers vector, they are likely to manifest different mobilities since their core structures differ. Correspondingly, a thin domain will exhibit lath shape. Alternatively, if b1 and b2 have the same Burgers vector but the opposite line sense (i.e., a closed disconnection loop), a lenticular-shaped domain will be expected because of the same mobility of both disconnections. However, a lath-shaped domain is generally developed even if b1 and b2 have the same mobility, because misfit strain along the interface will be reduced in one direction but increase in another direction10. The accumulated misfit strain will impede the gliding of disconnection in one direction. A comprehensive description and discussion can be found in ref. 11.

Deformation twinning is accomplished through nucleation and motion of twinning disconnections (TDs)5,12,13. Corresponding to the mirror symmetry about twin plane, twinning does not create misfit strain along the twin plane (coherent twin boundary, CTB). Elementary TD (bT) and its complementary TD (bC) have opposite sign where bT-bC is a full lattice translation vector along the twinning shear direction12. When both bT and bC are activated, the same argument as the case of phase transformation is applied for the propagation and shape of twins. However, bC is rarely activated because it has a bigger Burgers vector, higher formation energy and lower mobility, compared with bT. Therefore, elementary TD is the only TD during twinning.

Twins are three-dimensional entities, as schematically illustrated in Fig. 1 for simplicity. bT (or (bT, h0)) and -bT (or (-bT, -h0)) glide toward the twinning shear (η1) and the opposite direction (-η1), and toward the lateral sides along the -λ and λ directions, achieving twin propagation/growth. Since normal K1 is a two-fold rotation axis for twin plane, the mobility of bT and -bT on ±K1 twin planes is generally treated to be the same, and the twin shall exhibit a lenticular shape with symmetric twin tips, as illustrated in Fig. 1a, b. However, bT and -bT on ±K1 twin planes may have different Peierls barrier. First, with steps character14,15,16, a TD core may relax on a slanted plane to form a coherency disclination17,18,19 and does not retain the two-fold symmetry about the rotation axis K1. Second, the glide of a TD is accompanied with atomic shuffling that may not retain the two-fold rotation symmetry3,20. Thus, twins may grow into lath shape with asymmetric tips, as illustrated in Fig. 1d, e.

Fig. 1: Schematics of three-dimensional (3D) twin shapes.
figure 1

a Lenticular-shaped and d lath-shaped twins. Lenticular-shaped twin exhibits symmetric twin shape in b BS and c DS view. Lath-shaped twin exhibits an asymmetry twin shape in e BS view and quasi-symmetric twin shape in f DS view.

Full size image

When viewing a twin from ‘bright side’ (BS, along λ direction), bT and -bT are edge type, a twin may grow a lenticular shape with symmetric twin tips (Fig. 1b) or a lath shape with asymmetric twin tips (Fig. 1e) depending on whether the TD relaxes into a coherency disclination on a slanted plane. When viewing a twin from ‘dark side’ (DS, along η1 direction), bT and -bT are screw type, the same Peierls stress is expected as it glides in the -λ and λ directions because of the two-fold symmetry about the rotation axis K121,22. Twins may grow a lenticular shape with symmetric tips in DS view, as shown in Fig. 1c, f. In addition, big step or facet associated with pileup of multiple TDs may form and influence propagation/growth of twins.

The aforementioned argument demands a comprehensive understanding and quantitative analysis of Peierls stress for TD or step as it moves in the ±η1 and ±λ directions. Moreover, 3D microscopy characterization of twins with focus on twin tips and twin shapes can provide direct evidence of Peierls stress’s effect on deformation twinning. In what follows, we conducted atomistic simulations and microscopy characterization of \(\left\{ {{{{\mathrm{10}}}}\bar 11} \right\}\) twins in Mg to examine our argument. \(\left\{ {{{{\mathrm{10}}}}\bar 12} \right\}\) twins12,14,15 in Mg alloys are also characterized and compared to \(\left\{ {{{{\mathrm{10}}}}\bar 11} \right\}\) twins.

Results

Glide resistance of twinning disconnection

The structures and energetics of elementary TDs associated with \(\left\{ {{{{\mathrm{10}}}}\bar 11} \right\}\) twinning, and their kinetics along different directions can be assessed by atomistic calculations14,23, which are also applied to studies of TDs associated with \(\left\{ {{{{\mathrm{10}}}}\bar 12} \right\}\) and \(\left\{ {{{{\mathrm{11}}}}\bar 22} \right\}\) twinning24,25. Figure 2a schematically shows TDs along CTBs (±K1 twin planes). For two TD loops on CTBs, four TDs (-bT, h0) and (bT, -h0), (bT, h0) and (-bT, -h0) are equivalent according to the crystallography of the twin. However, the core of TDs may not relax on twin plane. Figure 2b shows the atomic structures of a TD dipole on K1 twin plane in BS view. TDs are edge type and the step height h0 involves four atomic planes14,23. The core of two TDs relaxes on slanted planes to form a coherency disclination, which bonds \(\left\{ {\bar 1{{{\mathrm{011}}}}} \right\}\) pyramidal plane in twin and {0002} basal plane in matrix (referred to as Py1/B) for (bT, h0), or vice versa in matrix (referred to as B/Py1) for (-bT, -h0). Due to the lattice mismatch between pyramidal and basal planes, coherency stress develops in the core, tension in basal plane and compression in pyramidal plane for Mg3,26. Thus, the local stress field (details in Supplementary Note 1 and Supplementary Fig. 1) destroys the two-fold symmetry of the dislocation about the rotation axis K1. In comparison, Fig. 2c shows atomic structure of a TD dipole in DS view where TDs are screw type. Two TDs have planar-extended core structures. The two planes in twin and matrix have the same crystallographic index, thus zero coherency stress is developed. We further calculated the energy change (Fig. 2d) and the Peach-Koehler (P-K) resistance force7 (Fig. 2e) associated with the motion of (bT, h0) (right TDs in Fig. 2b, c). In the figure, x is the position of TD with respect to its initial position (Fig. 2b, c). The value l is the periodic distance along -η1 or λ directions. Therefore, x/l is the relative position of (bT, h0). An increase in relative position x/l indicates the motion of TD along -η1 or λ direction (causes twinning), and vice versa. The energy change profile (black curve in Fig. 2d) does not retain symmetry along η1 and -η1 directions. The P-K force (black curve in Fig. 2e) in -η1 direction is 0.44 N m−1, lower than that in η1 direction (0.53 N m−1). Thus, the difference in resistance force (0.09 N m−1) corresponds to the difference of 0.79 GPa in required resolved shear stress (RSS). In comparison, the energy change profile (red curve in Fig. 2d) is symmetric and the corresponding resistance force (red curve in Fig. 2e) is the same for TD motion in the λ and -λ directions. Atomistic simulation shows that single edge TD is easier to propagate along the -η1 direction than η1 direction while single screw TD along λ and -λ direction is subject to the same resistance force.

Fig. 2: Structures and kinetics of single TD.
figure 2

a Schematics showing TDs along CTBs. b Atomic structure of a pure edge TD dipole in BS view. c Atomic structure of a pure screw TD dipole in DS view. d The change of potential energy associated with the migration of an edge TD and a screw TD. e The resistance force associated with the migration of an edge TD and a screw TD.

Full size image

Multiple TDs (bT, h0) piling up on specific crystal plane3 can be treated as a super-disconnection (nbT, nh0) which often relaxes into a low energy facet, described by “A/B” facet (“A” and “B” represents the crystal plane in twin and matrix) and recently named as coherency disclination17. For example, as demonstrated by experimental observation19,27,28 and MD simulations25,29, P/B and B/P facets of \(\left\{ {{{{\mathrm{10}}}}\bar 12} \right\}\) twin bonds the prismatic and basal interfaces and Py1/B and B/Py1 of \(\left\{ {{{{\mathrm{10}}}}\bar 11} \right\}\) twin bonds pyramidal and basal planes. These A/B facets have disclination character and can glide as entities30. A disclination can be decomposed into three components (wedge, twist, and misfit components in Supplementary Note 1 and Supplementary Fig. 1a). Types of disclinations with different components and their corresponding stress fields have implication on the symmetric/asymmetric kinetics of disclinations (details in Supplementary Note 1 and Supplementary Fig. 1b–g).

We further investigated the mobility of facets along twinning and detwinning directions which depends on facet index. Prediction of the facets starts with the coherent dichromatic complex (CDC). In BS view (Fig. 3a), there are three possible interfaces, which are {0002}M/T\({{{\mathrm{||}}}}\left\{ {\bar 1{{{\mathrm{011}}}}} \right\}_{{{{\mathrm{T/M}}}}}\) (B/Py1) interface, \(\left\{ {{{{\mathrm{10}}}}\bar 10} \right\}_{{{{\mathrm{M/T}}}}}{{{\mathrm{||}}}}\left\{ {{{{\mathrm{10}}}}\bar 13} \right\}_{{{{\mathrm{T/M}}}}}\) (Pr1/Py2) interface, and conjugate TB \(\left\{ {\bar 1{{{\mathrm{013}}}}} \right\}_{{{{\mathrm{M/T}}}}}{{{\mathrm{||}}}}\left\{ {\bar 1{{{\mathrm{013}}}}} \right\}_{{{{\mathrm{T/M}}}}}\) (K2/K2). In DS view (Fig. 3b), \(\left\{ {1\bar 2{{{\mathrm{10}}}}} \right\}_{{{{\mathrm{M/T}}}}}{{{\mathrm{||}}}}\left\{ {\bar 12\bar 10} \right\}_{{{{\mathrm{T/M}}}}}\) (Pr2/Pr2) interface, \(\left\{ {{{{\mathrm{02}}}}\bar 21} \right\}_{{{{\mathrm{M/T}}}}}{{{\mathrm{||}}}}\left\{ {{{{\mathrm{02}}}}\bar 2\bar 1} \right\}_{{{{\mathrm{T/M}}}}}\) (Py3/Py3) interface, and \(\left\{ {\bar 4{{{\mathrm{223}}}}} \right\}_{{{{\mathrm{M/T}}}}}{{{\mathrm{||}}}}\left\{ {4\bar 2\bar 2\bar 3} \right\}_{{{{\mathrm{T/M}}}}}\) (Py4/Py4) interface are possible. Using atomistic simulations, we calculated formation energies, and analyzed kinetics of these possible facets. In BS view, facets in Fig. 3c–e are stable. The line energy associated with B/Py1 or Py1/B and K2/K2 facets are almost the same and lower than that associated with Pr1/Py2 or Py2/Pr1 facet (Supplementary Note 2 and Supplementary Fig. 2a). K2/K2 facet first transforms to discrete TDs and then migrates. Therefore, B/Py1 or Py1/B facet is favored based on both energetic and kinetic consideration in BS view. In contrast, the facets in DS view in Fig. 3f–h, manifest only Pr2/Pr2 facet is stable (Supplementary Note 1 and Supplementary Fig. 2b), and other facets degenerate into discrete TDs.

Fig. 3: Structures and kinetics of twin facets.
figure 3

CDC associated with \(\left\{ {{{{\mathrm{10}}}}\bar 11} \right\}\) twinning a viewed along λ direction predicts the B/Py1 or Py1/B, Pr1/Py2 or Py2/Pr1 and K2/K2 interfaces, and b viewed along -η1 direction predicts the Pr2/Pr2, Py3/Py3 and Py4/Py4 interfaces. Relaxed structures containing c K2/K2 facet, d Py1/B facet, and e Pr1/Py2 facet in BS view. Relaxed structures containing f Pr2/Pr2 facet, g Py3/Py3 facet, and h Py4/Py4 facet in DS view. i The change of potential energy associated with the migration of a Py1/B or a Pr2/Pr2 facet with respect to the number of edge or screw TDs on the facet. j Difference in resistance force associated with the migration of a Py1/B or Pr2/Pr2 facet along twinning and detwinning directions with respect to number of edge or screw TDs on the facet. Schematics showing terraces on CTBs consisting of k CTBs, A/B and B/A facets in BS view and l CTBs and A/A facets in DS view.

Full size image

The energy change and the resistance force associated with the motion of the 8-layer thick Py1/B and Pr2/Pr2 facets with two TDs (Supplementary Note 2 and Supplementary Fig. 2c, d) share similar features to those shown in Fig. 2d, e with single TD, indicating asymmetric kinetics of Py1/B facets and symmetric kinetics of Pr2/Pr2 facets. As shown in Fig. 3i, when more TDs are involved, the kinetic barrier associated with migration of the Py1/B facet with edge TDs greatly increases while the kinetic barrier associated with migration of the Pr2/Pr2 facet with screw TDs slightly increases. Figure 3j exhibits the difference in resistance force with respect to the number of TDs on a facet. For motion of screw TDs, the difference in force is nearly zero. For motion of edge TDs along, the difference in force associated with motion of an 8-layer thick Py1/B facet (with two TDs) increases to 0.20 N m−1 from 0.09 N m−1 of a 4-layer step (with single TD). Correspondingly, the difference in required RSS is 0.88 GPa for an 8-layer thick Py1/B facet and 0.79 GPa for a 4-layer Py1/B step. Thus, the asymmetric kinetics of edge TDs becomes more obvious when more TDs pile up to form longer facet.

The symmetric/asymmetric kinetics of facets eventually affects the shape of twins. Figure 3k, l schematically shows a twin embedded in the matrix in the coordinate that y-axis is along K1 direction and x-axis is the migration direction of steps/facets. On ±K1 planes, terraces that consist of CTBs and facets may form. Based on crystallography associated with compound twins, a twin orientation is obtained by a 180° rotation about either K1 or η1 direction. If x-axis is along -η1 direction (Fig. 3k), atomistic simulations suggest that long Py1/B facets have higher mobility during twinning than that during detwinning. Consequently, during twinning, A/B facets migrate faster than B/A facets, causing lath-shaped twins with asymmetric twin tips. In comparison, if x-axis is along λ direction (Fig. 3l), planes in matrix and twin have the same index (A/A type facet), atom displacements during motion of such facet along λ and -λ directions are equivalent. Moreover, we may speculate that A/A type facets that contain wedge and/or twist disclination components also have the same mobility when migrating in the opposite directions because the core structures retain the two-fold symmetry about the rotation axis K1 (i.e., Supplementary Fig. 1c, f). In this scenario, twins may exhibit lenticular shape with symmetric twin tips in lateral DS view while exhibit lath-shape with asymmetric twin tips in forward BS view. These speculations can be validated by TEM characterization of twin facets and twin shape in different directions.

Microscopy evidence of symmetric/asymmetric twinning

Microscopy characterization shows a lath-shaped \(\left( {{{{\mathrm{10}}}}\bar 11} \right)\) twin domain in BS view (Fig. 4). For simplicity, the boundaries toward ±K1 and ±η1 directions are denoted. Two purple points, as marked the apexes along K1 in Fig. 4a, do not lie on the same axis. Furthermore, the twin tips as shown in Fig. 4b, c exhibit asymmetry shape about ±η1 direction. Figure 4d, magnified high-resolution TEM image from Fig. 4b, manifesting CTBs (white line), K2/K2 (green line), Py1/B and B/Py1 (yellow line) facets. More Py1/B facets on -K1 (bottom) than B/Py1 facets on K1 (top), explaining the greater deviation of the -K1 boundary from twin plane. In contrast, more Py1/B facets on K1 than B/Py1 facets on -K1, resulting a greater deviation of the K1 boundary from on the other side of the same twin domain (Fig. 4c, e). Both asymmetry along K1 axis and twin tip along η1 axis are highlighted in Fig. 4f. The experimental evidence suggests that Py1/B facets have higher mobility than B/Py1 facets during twinning.

Fig. 4: TEM micrographs of a \(\left( {{{{\mathrm{10}}}}\bar 11} \right)\) twin in \(\left[ {1\bar 2{{{\mathrm{10}}}}} \right]\) (BS) view.
figure 4

a A lath-shaped twin domain in low magnification image. The inset is the SADP confirming the \(\left[ {1\bar 2{{{\mathrm{10}}}}} \right]\) observation direction and \(\left( {{{{\mathrm{10}}}}\bar 11} \right)\) twin orientation. The HAADF image showing the b η1 (\(\left[ {{{{\mathrm{10}}}}\bar 1\bar 2} \right]\) direction) twin tip and c -η1 (\(\left[ {\bar 1{{{\mathrm{012}}}}} \right]\) direction twin tips enclosed by the green rectangles in a respectively. d, e High magnification images of the regions enclosed by the orange rectangles in b and c showing asymmetric twin tips comprising CTBs, K2/K2, B/Py1 and Py1/B facets. The insets are the FFT of the regions enclosed by the red and blue squares, confirming the matrix and twin orientation. f Schematic showing the geometric shape of \(\left( {{{{\mathrm{10}}}}\bar 11} \right)\) twins in BS view.

Full size image

In DS view, twin and matrix are identical, where TDs is hard to distinguish at atomic level. However, twin boundaries can be roughly identified at low magnification with aid of local contrast differences, and an overall symmetric twin tip is observed (Supplementary Note 3 and Supplementary Fig. 3). Since characterization of twin facets at atomic resolution remains challenging, alternatively, we characterized A/A type \(\left( {1\bar 1{{{\mathrm{01}}}}} \right)_{{{{\mathrm{T/M}}}}}||\left( {1\bar 1{{{\mathrm{01}}}}} \right)_{{{{\mathrm{M/T}}}}}\) (Py1/Py1) facets which can be found in \(\left[ {0\bar 1{{{\mathrm{11}}}}} \right]\) view (Supplementary Note 3 and Fig. 4). The twin as shown in Fig. 5a exhibits a lenticular-shaped domain. Two purple points mark the ±K1 apexes lie on the same axis. This indicates \(\left( {{{{\mathrm{10}}}}\bar 11} \right)\) twin can symmetrically propagate along non ±η1 direction (here: ±\(\left[ {\bar 56\bar 14} \right]\)). The twin tip in Fig. 5b is symmetric about \(\left[ {\bar 56\bar 14} \right]\) direction. HRTEM in Fig. 5c, d displays two regions magnified from Fig. 5b with similar numbers and heights of Py1/Py1 facets on ±K1 boundaries, indicating a symmetric kinetics of the A/A type Py1/Py1 facets along ±\(\left[ {\bar 56\bar 14} \right]\) directions.

Fig. 5: TEM micrographs of a \(\left( {{{{\mathrm{10}}}}\bar 11} \right)\) twin in \(\left[ {0\bar 1{{{\mathrm{11}}}}} \right]\) view.
figure 5

a A lenticular-shaped twin domain in low magnification image. b The HAADF image showing the twin tip enclosed by the orange rectangle in a. c, d High magnification images of the regions enclosed by the white rectangles in b showing Py1/Py1 facets on ±K1 boundaries. The insets are the FFT of the regions enclosed by the red and blue squares, confirming the matrix and twin orientation. e Schematic showing the geometric shape of \(\left( {{{{\mathrm{10}}}}\bar 11} \right)\) twin in \(\left[ {0\bar 1{{{\mathrm{11}}}}} \right]\) view.

Full size image

Experimental characterizations well demonstrate asymmetric propagation of TDs. During twinning, mobility of Py1/B facets (A/B type) is higher than that of B/Py1 facets. As a result, more Py1/B facets are observed near both twin tips. The outcome is a lath-shaped twin which is not symmetric about neither K1 nor η1 direction. Meanwhile, the distribution of Py1/Py1 facets (A/A type) on ±K1 boundaries should be nearly symmetric about both K1 and η1 direction, thus, a lenticular-shaped twin can be observed. Moreover, microscopy characterization of \(\left\{ {10\bar 12} \right\}\) twin shows lenticular-shaped domain in both BS and DS views (Supplementary Note 4 and Fig. S5), with B/P (A/B type) facets22. This discrepancy is attributed to the small difference in Peierls barriers for the elementary TD in \(\left\{ {10\bar 12} \right\}\) twins. Resistance forces for the glide of elementary TD in the η1 and -η1 direction on K1 plane are 0.876 × 10−2 N m−1 and 1.280 × 10−2 N m−1 for \(\left\{ {10\bar 12} \right\}\) twinning. Correspondingly, for edge \(\left\{ {10\bar 12} \right\}\) and \(\left\{ {10\bar 11} \right\}\) TDs, the difference in required RSS is 0.082 GPa for edge \(\left\{ {10\bar 12} \right\}\) TD, which is far smaller than 0.79 GPa for edge \(\left\{ {10\bar 11} \right\}\) TD.

Discussion

The glide resistance to TDs with both dislocation and step character is mainly accounted for based on the dislocation character, and widely recognized to be symmetric, i.e., the same for the forward and backward motion. Here, we for the first time reveal the influence of step character on the core structure of TDs. Correspondingly, a TD can be described with A/A type or A/B type coherency disclination. We further demonstrate the A/B type TD glides with asymmetric resistance in the forward and backward motion, while the A/A type TD glides with same resistance. With this knowledge, we successfully account for the formation of asymmetric and symmetric twin tips of \(\left\{ {10\bar 11} \right\}\) and \(\left\{ {10\bar 12} \right\}\) twins in Mg alloys.

Our work enriches the fundamental understanding of twinning mechanisms, may help develop the strategy of controlling twinning in hexagonal metals via tailoring core structure of TDs via solutes31,32,33. More importantly, the findings of symmetric and asymmetric glide resistance to A/A type and A/B type TDs can be directly incorporated into multiscale materials modeling tools. For example, phase filed model can predict 3D twin growth by integrating mobilities of twin facets calculated by atomistic simulations34. The mobilities of A/B and B/A twin facets, i.e., B/P and P/B facets of \(\left\{ {10\bar 12} \right\}\) twinning, are treated the same. Our findings suggest that mobilities of A/B and B/A twin facets should be evaluated separately for twinning other than \(\left\{ {10\bar 12} \right\}\) twinning. In addition, grain boundary dynamics are largely controlled by the formation and motion of disconnections along with the GB35. The symmetric/asymmetric mobilities can be directly incorporated into the available continuum models for grain boundary migration based on disconnections36,37.

Methods

Atomistic simulations

Molecular static/dynamics (MS/MD) simulations were conducted for Mg with the empirical interatomic potential developed by Liu et al.30. The construction of models containing steps/facets in BS view starts with an 80 × 80 × 1.60 nm bicrystal with \(\left( {\bar 1{{{\mathrm{011}}}}} \right)\) twin orientation in the coordinate that x-axis is along -η1 (in \(\left[ {{{{\mathrm{10}}}}\bar 12} \right]\) direction), y-axis is along K1 (normal to \(\left( {\bar 1{{{\mathrm{011}}}}} \right)\) plane), and z-axis is along -λ (in \(\left[ {1\bar 2{{{\mathrm{10}}}}} \right]\) direction). To create a 4n-layer step/facet, n pure edge TDs with Burgers vector (−0.133, 0, 0) nm are introduced on every four \(\left( {\bar 1{{{\mathrm{011}}}}} \right)\) layers by applying anisotropic Barnett-Lothe solutions38 followed by atoms shuffles. Similarly, an 80 × 80 × 2.352 nm bicrystal with \(\left( {\bar 1{{{\mathrm{011}}}}} \right)\) twinning orientation in the coordinate that x-axis is along λ (in \(\left[ {\bar 12\bar 10} \right]\) direction), y-axis is along K1 (normal to \(\left( {\bar 1{{{\mathrm{011}}}}} \right)\) plane), and z-axis is along -η1 (in \(\left[ {1{{{\mathrm{0}}}}\bar 1{{{\mathrm{2}}}}} \right]\) direction) is adopted to construct models containing steps/facets in DS view. After introducing n pure screw TDs with Burgers vector (0, 0, −0.133) nm and corresponding shuffling, a 4n-layer step/facet is constructed. To relax steps/facets in both BS and DS views, dynamic quenching is performed until the maximum force is <5 pN with fixed boundaries in x- and y-direction and periodic boundary condition in z-direction.

The energy barrier associated with the motion of TDs is calculated by Nudge Elastic Band (NEB) method39 with 39 intermediate states. The displacement between the steps/facets in the initial and final configurations is defined as l, which is along -η1 direction with magnitude 1.18 nm for edge TDs and along λ direction with magnitude 0.32 nm for screw TDs. The NEB calculations are converged until the difference in the total energy within 10 iterative steps is <10−3 eV. Peach-Koehler force7 is derived by differentiating the energy curve with respect the moving distance within the range from 0 to l/2. The RSS required to overcome the barrier is estimated through dividing the resistance force by the magnitude of Burgers vector of TDs.

Materials preparation and characterization

The Mg-0.2 at. % Gd ingot specimens were used for the present investigation. The specimens were firstly extruded at 300 °C. The as-extruded specimens were compressed by 8% at room temperature along the extrusion direction. For high-annular dark-field scanning transmission election microscope (HAADF-STEM) characterization, sequential aging (275 °C for 1 h) was carried out for twin boundary Gd atom segregation. Details of HAADF characterization method is described in ref. 19. For \(1\bar 2{{{\mathrm{10}}}}\) and \(0\bar 1{{{\mathrm{11}}}}\) direction observation, perforation of 3 mm disk specimen (former polished to a thickness of ~40 μm) by ion milling was carried out with low-angle (3°) and low energy (3 keV) ion-beam. High-resolution transmission electron microscope (HRTEM) and HAADF-STEM were carried out on a JEOL-2100F electron microscope with a voltage of 200 kV. The lattice constant (a = b = 3.21 Å, c = 5.21 Å, α = β = 90°, γ = 120°) is selected for pole figure and diffraction pattern analysis of \(\left\{ {{{{\mathrm{10}}}}\bar 11} \right\}\) twin in Mg.