Twin Boundaries merely as Intrinsically Kinematic Barriers for Screw Dislocation Motion in FCC Metals

Metals with nanoscale twins have shown ultrahigh strength and excellent ductility, attributed to the role of twin boundaries (TBs) as strong barriers for the motion of lattice dislocations. Though observed in both experiments and simulations, the barrier effect of TBs is rarely studied quantitatively. Here, with atomistic simulations and continuum based anisotropic bicrystal models, we find that the long-range interaction force between coherent TBs and screw dislocations is negligible. Further simulations of the pileup behavior of screw dislocations in front of TBs suggest that screw dislocations can be blocked kinematically by TBs due to the change of slip plane, leading to the pileup of subsequent dislocations with the elastic repulsion actually from the pinned dislocation in front of the TB. Our results well explain the experimental observations that the variation of yield strength with twin thickness for ultrafine-grained copper follows the Hall-Petch relationship.


Energy curved with a small cell in the coupled analysis
In the coupled analysis, the core atomic region is simulated with two different cell sizes to check the dependence of the results on the size of the core region. The results calculated with the larger cells are shown in Fig. 1 in the main text, and the results from the smaller cells are shown in Fig. S1. The results calculated by using a smaller region are similar to those with a region twice larger in the and directions. This indicates that the deformation at the outer boundaries of the core atomic region is small enough to satisfy the linear elasticity assumption and the coupling method is valid.

Differences of displacements and stresses between simulation results and theoretical solutions
Due to the severe distortion near the dislocation core, the stresses in the core region calculated by atomistic simulation deviate a lot from the solutions derived with the classical anisotropic elastic theory. However, it is expected that these two methods can give consistent stresses in the region far from the dislocation core and thus a coupling atomistic -continuum method can be adopted to predict reasonably the dislocation -twin boundary interaction. To ensure the accuracy of the coupling method, it is necessary to check the consistency of the stresses across the interface between the atomic region II and the continuum region I. An alternative way is to check the consistency between the far-field stresses in region II calculated by using atomistic simulation and those derived with the classical elasticity theory. For ease of observation, only the differences of the stresses between the atomistic results and the analytical references in the region far away from the dislocation core are presented in Fig. S2. It can be observed that the maximum differences are located near the TB.
According to the elastic theory, should be continuous and is expected to be discontinuous across the TB. However, atomically sharp jump of the stresses is seen near the TB in atomistic simulations as shown in the detailed plots of the atomic stresses near the TB in Fig. S3.  It can be clearly seen that the differences of the stresses are typically less than 0.1 MPa near the interface between the atomic region and the continuum region.
The magnitudes of these differences are small enough to ensure a good accuracy of the coupling method.

Construction of end states for the NEB analysis
The Nudged Elastic Band (NEB) calculation requires two end states (i.e., the first and the last states) along the minimum energy path (MEP) as inputs. The first state is a dislocation-free cell and the last state is constructed as follows.
First, as shown in Fig. S5, an initial screw dislocation is introduced at the position (i.e., the approximate position of the dislocation core) by applying a displacement field, which is discontinuous across the slip plane, to the region <    It can be seen that, as the dislocation moves away from its original equilibrium position, the Peach-Koehler force originating from the fixed boundaries, which may be mistaken as a huge resistance from the twin boundary if a nanotwinned sample was considered, increases dramatically. It can also be seen from the formula that as the height ℎ increases, the Peach-Koehler force decreases, and this is consistent with the results of the NEB calculations in the main text.

Simulation method to
where ̅ = / and is the distance from the dislocation to the free boundary.
The force exerted on a dislocation from another one with a distance away from it is, 4 in which ̅ = / . From Fig. 3a in the main text, i.e., the pile-up configuration of 5 screw dislocations caused by the TB, we can see that: (1) the distance from any dislocation to the left/right boundary is larger than the model height ; (2) the largest distance between two neighboring dislocations is less than /2. Thus we have ̅ < 0.5 and ̅ > and it can be easily verified that, = sinh ̅ sinh 2 ̅ < sinh 0.5 sinh 2 0.0086.
This indicates that the contribution from the image dislocation is much less than that from the nearest neighbor dislocation, so the plate can be seen as infinitely long along the -axis and it makes sense to take the solution in the case of infinite length as the reference.