Repulsion leads to coupled dislocation motion and extended work hardening in bcc metals

K. Srivastava; D. Weygand; D. Caillard; P. Gumbsch

doi:10.1038/s41467-020-18774-1

Repulsion leads to coupled dislocation motion and extended work hardening in bcc metals

Srivastava, K.; Weygand, D.; Caillard, D.; Gumbsch, P. 2020-10-09 00:00:00 ARTICLE https://doi.org/10.1038/s41467-020-18774-1 OPEN Repulsion leads to coupled dislocation motion and extended work hardening in bcc metals 1,2 1 3 1,4 K. Srivastava , D. Weygand , D. Caillard & P. Gumbsch Work hardening in bcc single crystals at low homologous temperature shows a strong orientation-dependent hardening for high symmetry loading, which is not captured by clas- sical dislocation density based models. We demonstrate here that the high activation barrier for screw dislocation glide motion in tungsten results in repulsive interactions between screw dislocations, and triggers dislocation motion at applied loading conditions where it is not expected. In situ transmission electron microscopy and atomistically informed discrete dis- location dynamics simulations conﬁrm coupled dislocation motion and vanishing obstacle strength for repulsive screw dislocations, compatible with the kink pair mechanism of dis- location motion in the thermally activated (low temperature) regime. We implement this additional contribution to plastic strain in a modiﬁed crystal plasticity framework and show that it can explain the extended work hardening regime observed for [100] oriented tungsten single crystal. This may contribute to better understanding the increase in ductility of highly deformed bcc metals. 1 2 Institute for Applied Materials (IAM), Karlsruhe Institute of Technology (KIT), Straße am Forum 7, 76131 Karlsruhe, Germany. Research and Development, AG der Dillinger Hüttenwerke, Werkstraβe 1, 66763 Dillingen/Saar, Germany. CEMES-CNRS, 29 rue Jeanne Marvig, BP4347, F-31055 Toulouse Cedex 4 ✉ 4, France. Fraunhofer IWM, Wöhlerstr. 11, 79108 Freiburg, Germany. email: [email protected] NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications 1 1234567890():,; ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 etallic materials are used mainly in technical applica- components of the stress tensor inﬂuence their glide beha- 16,17 tions for their good formability, strength and toughness. viour . The complex core structure of screw dislocations and MThis favourable combination of materials properties its sensitivity to non-glide stresses is at the origin of this so-called 16,17 fundamentally relies on work hardening during plastic defor- non-Schmid behaviour . This core structure is believed to mation. The response of metals to a mechanical load by irre- make screw dislocations glide by the successive nucleation and versible plastic deformation occurs on a microscopic scale by the motion of pairs of kinks on an otherwise straight screw disloca- motion of dislocations, line defects of the crystalline structure. tion line and lead to the thermally activated deformation beha- Dislocations in general glide on densely packed crystallographic viour of bcc metals at low temperatures . Atomistic simulations 19–25 planes. Dislocation glide on such glide planes causes a relative and ﬁrst principle calculations have been used to investigate shift of the material above and below the glide plane in the the properties of screw dislocations and of kink-pair formation direction of a short lattice vector, the Burgers vector . The actual for bcc materials. The core structure is found to be non degen- plastic deformation is a consequence of the interplay between erate and compact. The complex screw dislocation core structure dislocation glide, dislocation multiplication and annihilation on also leads to the activity of unexpected glide systems . However, well-deﬁned slip systems, characterised by glide plane and Bur- the role of forest dislocations on work hardening in the low- gers vector. Dislocations move collectively upon straining the temperature regime is not understood. material and multiply. Dislocation multiplication leads to an Since screw dislocations determine the deformation behaviour increase in dislocation density. This in turn is believed to result in at low temperatures, we systematically investigate the mutual work hardening which manifests itself as an increase in the ﬂow interaction of screw dislocations in bcc metals using discrete stress of the material upon straining and gives e.g. a deep-drawn dislocation dynamics (DDD) simulations and in situ experiments. component its strength but also hinders further deformation. A Tungsten (W) is chosen as a representative of bcc metals. suitable heat treatment can often restore formability. Tungsten is elastically nearly isotropic and its mechanical prop- Work hardening is known to depend on the crystalline struc- erties are well studied and show extended work hardening ture of the metal, e.g. face centred cubic (fcc) metals like alumi- regimes for [111] and even more pronounced for [100] orienta- 14,15,27–29 nium or copper typically show stronger hardening than body tion . The investigation allows to identify a hitherto 2–5 centred cubic (bcc) metals like iron or tungsten . The long- overlooked mechanism based on the coupled motion of repulsive range interaction between dislocations on different glide planes oriented screw dislocation pairs in both simulations and experi- and with different character is caused by the elastic distortion that ments. The mechanisms’ origin is the internal shear stress dislocations introduce into the crystal lattice. Additionally, short- increase leading to screw dislocation glide on unexpected slip range interaction at the intersection of dislocations can lead to systems and thus a larger plastic deformation at the same mac- changes in the atomic conﬁguration of the dislocation core and to roscopic load occurs. This is a key aspect explaining the orien- 2,3,6–8 the formation of so-called dislocation junctions . The pro- tation dependence of the extended work hardening regimes, e.g. minent interpretation of work hardening as forest hardening is in tungsten. based on such interaction of dislocations on inclined slip systems hindering the motion of the mobile dislocations and thus leading 2,8,9 Results to an increase in the stress for further plastic deformation . Discrete dislocation dynamics simulations. DDD simulations Dislocation interactions may also result in multi-junctions are used to investigate the interaction of two screw dislocations involving more than two dislocations which in bcc metals have been shown to contribute signiﬁcantly to work hardening . on non-coplanar glide systems. All possible types of interaction 24,30 pairs are studied. An atomistically informed DDD code is All work hardening models intrinsically assume that the mutual interaction between dislocations always hinders disloca- employed here. Screw dislocation mobility is modelled by the 18,30 kink-pair formation mechanism , using atomistic values for tion motion either by repulsion on approach or by pinning at junctions, which prohibits the release and further motion of the the stress dependent activation enthalpy for kink-pair formation (details in Methods section) thus including also non-Schmid dislocation. Therefore work hardening models all include a contribution to the ﬂow stress which is inversely proportional to effect. While attractive interactions show the expected dislocation 31,32 reactions and act as obstacles by forming dislocation junc- the average dislocation spacing L which in turn is related to the pﬃﬃﬃ 4,8 tions, speciﬁc repulsively oriented screw dislocation pairs sur- dislocation density by L 1= ρ . prisingly show coupled gliding. Such coupled motion, where one While these implicit assumptions and the resulting work mobile screw dislocation (dislocation I), upon interaction with a hardening models appear plausible for fcc metals or bcc at high second repulsively oriented (immobile) screw dislocation (dis- temperatures T > T (T is the athermal temperature) where dis- c c location II), drives the second dislocation without slowing down. location glide is controlled only by the resolved shear stress on the glide plane, it is highly speculative for bcc metals at T < T whose This coupled motion occurs at constant applied stress. For room 5,11–13 temperature deformation, modelled here, the kink motion along deformation behaviour is known to be much more complex . A little understood example is the work hardening of tungsten the dislocation line (2 µm in length) occurs faster than the time 14,15 scale of kink-pair nucleation as long as the kinks see a positive single crystals at room temperature . At room temperature, for [100] tensile loading, an extended work-hardening regime is driving force. Figure 1a displays a schematic diagram of the simulation setup: observed with an initial ﬂow stress of about 250 MPa and with a work hardening that leads to ﬂow stresses of almost 1 GPa during two repulsively oriented screw dislocations, dislocation I belong- ing to glide systemðÞ 101½ 111 and dislocation II to glide system straining by 6–8% . Also a recent atomistically informed crystal plasticity model does not capture the initial hardening behaviour . ðÞ 011½111, are placed within a crystal subjected to tensile loading In bcc metals at low temperatures screw and non-screw dis- along the ½149 direction. The Schmid factors for dislocations I locations have a completely different response to the applied and II are m ≈ 0.5 and m ≈ 0.3. The nearest distance between I II stress σ . Non-screw dislocations can bend and their motion is the dislocations is initially about 50 nm. Dislocation I begins to app controlled exclusively by the resolved shear stress τ ¼ mσ glide once the resolved shear stress τ reaches about 1.8 GPa. res app res I The corresponding resolved shear stress on dislocation II is about on the glide plane, where m is the so-called Schmid factor. 1.2 GPa. The glide directions of the dislocations I and II are In contrast screw dislocations remain straight and additional 2 NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 ARTICLE marked as r and r and both are pointing to the left in Fig. 1a. The coupled motion of the two dislocations can be understood 1 2 Dislocation I reaches a velocity of about 1.05 nm/s, while based on this analysis of the stress distribution along the dislocation II is initially immobile (see Fig. 1b). dislocation line. Over the entire length of dislocation I (about 2 Figure 1b shows the velocity of the two dislocations as a µm) only a very short section of about 20 nm around the point of function of their nearest distance of approach. Dislocation I closest interaction experiences a signiﬁcantly reduced resolved approaches the immobile dislocation II causing it to glide once shear stress. Consequently, kink-pair nucleation is reduced very the nearest distance decreases to about 9.5 nm. From that point locally in this section, while all the rest of the dislocation line still on, the velocity of dislocation II increases rapidly to match the experiences high resolved shear stresses from the externally velocity of dislocation I. Their nearest distance then stabilises applied ﬁeld and kink-pair nucleation rate remains virtually around a value of d ≈ 7.3 nm. Both dislocations then glide unchanged. Therefore, the total velocity of dislocation I remains crit collectively. essentially constant. The situation of dislocation II is opposite: The stress states along both dislocation lines favour glide on once dislocation I has reached the critical distance, the kink-pair their initial habit planes. Both dislocations remain straight nucleation rate on dislocation II drastically increases near the throughout the entire simulation. No cross slip is triggered (see point of closest approach until the dislocation velocity reaches the Methods) due to the mutual interaction. The maximal resolved velocity of dislocation I. Thereafter the two dislocations show shear stresses along the dislocations due to their mutual elastic coupled glide, slightly oscillating around the critical distance due interaction at the point of nearest distance is shown in Fig. 2a. to the different orientations of the elementary kink steps on the The distribution of the negative (positive) additional shear stress two dislocations. acting on dislocation I (dislocation II) along the dislocation line for the distance d is shown in Fig. 2b). The curves are crit asymmetric with respect to the nearest point of interaction as the In situ transmission electron microscopy. To mimic this sce- respective resolved shear stresses are shown. nario experimentally, repulsively oriented screw dislocations in ab Driving dislocation I: (101)[111] Dislocation II r Driven dislocation II: (011)[111] 1,5 Dislocation I 0,5 010 20 Nearest distance [nm] Fig. 1 Simulation setup and coupled motion. a Schematic view of the setup with two repulsively oriented screw dislocations I and II with Burgers vectors b and b respectively. The indicated directions r and r are parallel to the Peach–Kohler force due to the external loading resolved in the respective 1 2 1 2 primary glide planes. Only repulsive pairs, where the glide directions r and r have a positive scalar product have to be considered. The location A and B 1 2 marked on the dislocation line are the nearest points between the two dislocations deﬁning the nearest distance d. b The variation of the velocity of dislocations I and II with Burgers vectors [111] and ½111 is plotted versus the minimal distance of approach. ab Driving dislocation I: (101)[111] Driving dislocation I: (101)[111] Driven dislocation II: (011)[111] Driven dislocation II: (011)[111] From externally applied stress –200 From mutual interaction 0 –400 –600 –500 0 1020304050 –50 0 50 Nearest distance [nm] Relative position along dislocation [a ] Fig. 2 Stress distribution on repulsively interacting dislocations. a Resolved shear stress from externally applied stress on dislocation I and II (dashed line) and shear stress due to mutual dislocation interaction (line) at the closest point of interaction (nearest distance); b resolved shear stress due to the repulsive dislocation interaction along dislocation I and II. Dislocations are at distance d . The relative positions are given in multiples of the lattice crit constant a and measured from the closest point of interaction. NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications 3 Resolved shear stresses on glide plane [MPa] Velocity (nm/s) Contribution of mutual interaction to resolved shear stress [MPa] ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 thin ﬁlms of a tungsten singe crystal are investigated by TEM. leaving dislocation II again immobile with respect to the ﬁxed Thin ﬁlm specimens of tungsten suitable for in situ deformation point ʻz’, between (e) and (f). The velocity of dislocation Ia experiments have been prepared and strained in a transmission remains almost constant close to 25 nm/s during the whole electron microscope (TEM, see Methods for a description of the process. Then a second dislocation Ib arrives and pushes again method and Caillard and Supplementary Figs. 1, 2 for corre- dislocation II in (f) and (g). The image (h) which is the difference sponding observations in Fe and Nb). The foil plane is ð117Þ, the between (a) and (d) shows the starting positions in dark and the local tensile axis is [521], and the sample thickness is 300 nm. ﬁnal ones in bright. The velocity of dislocation II ﬂuctuates Under such conditions, repulsively oriented dislocation pairs around the one of the driving dislocation Ia or Ib. The could be identiﬁed: Dislocations of type I with Burgers vector ﬂuctuations are more pronounced than for the DDD results. 1=2½111 and length 360 nm are the most mobile in the (101) During this coupled motion, the point of closest distance moves along the direction [010] of intersection between the two plane, in agreement with a high Schmid factor of 0.485. Dis- slip planes. Since this direction is almost within the plane of the locations of type II with Burgers vector ½[111] later move on the TEM-foil, long distances of coupled motion can be observed. The ð101Þ plane, but initially are immobile due to the lower Schmid minimum distance during the coupled motion has been measured factor of 0.458. in projection, and corrected from perspective effects. At 300 K, Figure 3i shows the schematic of the TEM sample geometry. this distance ﬂuctuates a little bit, but remains equal to about 24 The slip planes of the studied dislocations intersect the two free surfaces along the directions noted slip traces, and intersect each nm when coupled motion takes place. Similar observations of collective glide have been made a T = 200 K, but the correspond- other along the direction noted node path. Dislocations are represented by the straight lines marked I and II, gliding to the ing critical distance is unfortunately too small to be measured. left on the respective slip planes 1 and 2. Figure 3a–g shows the TEM observation of these dislocations for different times during the deformation at T = 300 K (see also the Supplementary Discussion Movie). In the TEM sample, the ﬁrst respectively second driving Both experiments and simulation show coupled motion of dislocation is labelled dislocation Ia respectively dislocation Ib. repulsively oriented screw dislocation pairs. In both cases screw Dislocation Ia approaches dislocation II between (a) and (b). dislocations remain straight and no cross slip is observed. The Note that dislocation II is immobile with respect to the ﬁxed DDD simulations suggest that this behaviour is a direct con- point ʻx’. Coupled motion takes place in (c) and (d) (see ﬁxed sequence of dislocation motion by the kink-pair mechanism. points ʻx’ and ʻy’) until (e) where dislocation Ia has moved away Kink-pair nucleation occurs locally and therefore depends on the Fig. 3 In situ TEM observation of coupled glide: experimental observation of dislocation glide at different time steps. Reference points are indicated by ʻx, y, z’. a Dislocation Ia approaches the still dislocation II. b–d Dislocation Ia pushes dislocation II: their point of closest distance moves along the direction ʻnode path’ until it emerges at the bottom surface in (e). f, g Another dislocation Ib pushes dislocation II. h Difference-image showing the motion between (b) (black dislocations) and (d) (bright dislocations). i Scheme corresponding to (h). The video sequence can be downloaded as a Supplementary Movie. 4 NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 ARTICLE 18,30 stress state along the entire screw dislocation line . If the into account for kink-pair nucleation in bulk. However, this is critical resolved shear-stress (CRSS) is reached upon raising the very different from the restoring forces due to line tension effects applied stress, dislocations of type I move while dislocations of observed in fcc metals. type II should and do remain initially immobile in experiment The observed collective glide of dislocations has signiﬁcant and simulation. Once the two dislocations get close enough, the consequences on the evolution of the plastic strain rate in bcc region of closest encounter on dislocation II experiences resolved metals in the thermally activated regime. Classical phenomen- shear stresses which are higher than the stresses on dislocation I. ological work hardening relations have been proposed with a ﬂow This has the consequence that kink pairs on dislocation II are stress inversely proportional to the average dislocation spacing L pﬃﬃﬃ generated in this region of closest encounter at a rate (per unit which can be calculated from the dislocation density as 1= ρ. length) somewhat higher than on dislocation I, where nucleation Dislocations are considered there as mutual obstacles which leads occurs along the entire line. The kinks generated at this location to so-called forest hardening. Reﬁned models consider an effec- of closest encounter move along the dislocation line as long as tive hardening between slip systems by speciﬁc hardening coef- 32–34 there is a positive driving force which is true for both experiments ﬁcients and simulation. Dislocation I could in principle slow down The situation for bcc metals in the thermally activated regime slightly because part of the total length of the dislocation along differs: the screw dislocations in the repulsive pairs remain which kink-pair nucleation may occur is now at lower stresses straight. Our results show that the high effective ʻstiffness’ of (c.f. Fig. 2b). However, already at our line length of 2 µm, this screw dislocations leads to a coupled glide of dislocation slowing down is too small to be observed in the simulations. ensembles, here exemplarily studies at pairs or groups of screw Eventually the total kink generation rates along both dislocations dislocations (see Supplementary Note 1 and Supplementary match and both dislocations move at the same velocity. This Fig. 5). Within the range of validity of the kink-pair nucleation implies that these repulsively oriented screw dislocations should mechanism for dislocation motion, the mutual interaction of never cross each other as long as both are able to move. Motion of these repulsively oriented dislocation pairs is limited to modifying a primary screw dislocation can activate certain repulsively the local stress state in the zone of closest approach. Therefore it oriented screw dislocation (detailed in Supplementary Discus- locally enhances the kink-pair nucleation rate. sion) even if it is located on a slip plane with a very low resolved During coupled motion the second dislocation (slip system 2) shear stress. contributes to the total plastic shear rate on slip system 2, due to Even quantitatively, the detailed TEM investigations and the dislocation glide on slip system 1, even though macroscopically simulations can well be related to each other. The smaller dif- the resolved shear stress is below its CRSS. In order to take such a ferences in Schmid factor in the experimental setup compared to phenomenon into account we propose to modify the equations the simulations make dislocation II move somewhat earlier, i.e. at for the plastic shear rate evolution by adding an interaction term, an about 3 times larger critical distance, than in the simulations. expressing this coupling for the well accepted strain rate based 18,35 The TEM investigation also shows clearly that the mechanism is formulation repeatable for dislocation II as dislocation Ib moving on a glide ΔH ðσÞ α α plane parallel to that of dislocation Ia takes over the role of the γ _ ¼ γ _ exp ; ð1Þ std 0 k T driving dislocation as dislocation Ia moves out of the observed area of the thin ﬁlm. During in situ experiments at lower tem- where γ _ denoted the standard plastic shear strain rate normally std perature much shorter critical distances between the coupled 36,37 used in crystal plasticity (CP) formulations , ΔH (σ) is the dislocations are observed which are beyond the limit of resolu- stress dependent activation enthalpy for system α, k is the tion. The smaller critical distance between the coupled pairs leads Boltzmann constant and T is the temperature. to larger interaction stresses, as required for activating screw The coupled motion of speciﬁc screw dislocation pairs is dislocation motion at lower temperatures. captured by adding a shear strain rate γ _ coupl Since our observations can be explained by dislocation motion based on kink-pair nucleation alone, this coupling mechanism α α α α γ _ ¼ γ _ þ γ _ ¼ γ _ þ f K γ _ ; tot std coupl std coupl αβ std ð2Þ between two repulsively oriented screw dislocations is likely to β ≠ α be observed in all bcc metals and other materials where kink-pair generation governs dislocation motion. In Supplementary Figs. 1, where γ _ is then the total plastic shear strain rate. The coupling tot 2 additional TEM evidence for the coupling in Fe and Nb at T= matrix K parametrises the coupled motion between slip system αβ 95 K is presented, conﬁrming the universality of the phenomenon. α and β and f is a scaling factor between 0 and 1. The terms in coupl The in situ observations and DDD results are both obtained for the matrix K depend on the crystallographic orientation of the αβ straight dislocations, reaching from surface to surface. In bulk two slip systems and the local stress state as several conditions material, dislocations are often anchored in networks which leads (see also Supplementary Note 2) have to be fulﬁlled. The condi- to an increase in line length during motion. One could speculate tions are (i) repulsive orientation; (ii) the resolved Peach–Koehler that this might possibly suppress screw dislocation motion and force due to the stress state on both dislocations has a positive hence the coupling mechanism. However, the increase of length scalar product and (iii) include only repulsive dislocation pairs corresponds to the nucleation of new kink pairs, which takes which can glide over extended distances (see also Supplementary place anyway, even in case of straight screws ending at the sur- Fig. 6). faces. These kinks subsequently either accumulate at the dis- In this analysis, only {110} slip systems are considered (Sup- location extremities in the case of bulk material or disappear at plementary Table 1), as suggested by atomistic simulations .In the surfaces. This increase of length is actually a restoring force, experimental observations for [110] loading slip activity on {112} but already included in the kink-pair nucleation process. There- systems is reported too . Therefore the focus in the following is fore, the only ingredient, which has not been taken into account on [100] and [111] loading directions, where {110} slip planes are for mimicking bulk behaviour, is the elastic repulsion between reported. kinks accumulated at the screw extremities—forming a mixed This coupling has important consequences on the plastic ﬂow: dislocation (curved section with non-screw character). This for a given stress state, the coupling enhances the plastic strain repulsion is part of the internal stress due to elastic interactions without requiring an additional applied stress and thus the strain- with all neighbouring dislocations, which of course must be taken hardening rate ∂σ=∂ϵ is effectively lowered. NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 stress σ opposing to the applied stress, namely by a stress σ − int σ . Here we show that in a 3-D model the velocity of dislocations int is determined by the highest internal stress acting in the direction of the applied stress along the dislocation line, namely by a stress σ + σ . This strongly increases the average dislocation velocity in int 600 comparison with classical models, not only for the driven dis- locations studied in this article, but also for all dislocations. We also obtain the surprising result that the dislocation velocity tends [100] Beardmore [111] Beardmore to increase with increasing internal stress, i.e. with increasing [100] CP (std) deformation. Depending on the loading state, the number of [100] CP (coupl) active repulsive dislocation pairs varies and is highest for the [111] CP (std) [100] orientation, explaining the observed extended work hard- [111] CP (coupl) ening behaviour of tungsten. This strain-softening effect compensates for other classical 0.00 0.02 0.04 0.06 0.08 0.10 0.12 strain-hardening ones and may also accounts for a lower global [–] strain-hardening of bcc metals. These effects must be taken into Fig. 4 Deformation behaviour of tungsten single crystals for the [100] account in macroscopic models like CP. The relevance for the and [111] loading direction. The experimental curves are taken from coupled motion increases with dislocation density, as the density Beardmore and Hull . For the [100] direction an extended work-hardening of possible pairs increases too. For very high dislocation densities, regime is observed; crystal plasticity simulation results without (CP(std)) the internal stresses ﬂuctuate over short distances, thus triggering and including the coupling mechanism (CP (coup)) are shown for both possibly the coupled motion. Therefore this may also lead to the orientations. (A detailed comparison is included in Supplementary 38–41 enhanced ductility observed for heavily cold rolled tungsten . Discussion). The coupling mechanism leads to an extended work hardening regime for [100] representing well the experimental ﬁndings, while for the [111] orientation the deformation behaviour is only slightly changed. Methods Discrete dislocation dynamics model and setup. The following setup for the pillar is taken: the size of the box is 2 μm with an aspect ratio of 1:2:1. The DDD The coupling mechanism is a key to rationalise the extended model used to simulate the screw dislocation motion is described in Srivastava 30,42 work hardening regime of tungsten single crystals at low homo- et al. . The mobility of the screw dislocation is governed by an Arrhenius law logous temperature for [100] loading direction shown in Fig. 4. and accounts for the inﬂuence of the entire stress tensor on the activation energy 14 15 of screw dislocations as against a pure shear stress based formulation in lit- Beardmore and Hull and Argon and Maloof observed for the 43–45 erature . The model effectively takes into account changes in the dislocation [100] loading direction a gradual ʻroundish’ strain-hardening core structure due to the applied stress by including non-Schmid terms in the curve in contrast to [111] which display a classical hardening activation enthalpy. In the DDD model the effective kink-pair nucleation rate for behaviour after yielding. the screw dislocation section is calculated on all three possible glide planes for a screw dislocation. In principle, a screw dislocation may glide on different glide For the [100] loading direction a total of 12 coupled pairs exist, planes and therefore split into distinct sectors depending on the local stress state. where both dislocations have the same non-zero Schmid factor. This is not the case for the current setups. For all non-screw orientations phonon For the [111] only three coupled pairs exist. Details on the pair drag limited glide is assumed as for fcc metals. counting are given in Supplementary Note 2 and Supplementary Tables 2–5. Thus, the additional plastic strain at a given stress Details to the repulsive pairs in the DDD setup. The dislocations are labelled as level due to the coupled pairs is most pronounced for loading I and II depending on their respective role. Dislocation I is mobile due to the along [100] direction. The value of f was set to 0.5. Physically coupl externally applied loading and drives dislocation II, once a critical minimal distance a value of 1 would mean, that each driving dislocation triggers a is reached. ʻrepulsive’ event. For the chosen value of 0.5 every other dis- The dislocation I belongs to the slip systemðÞ n ; b :ðÞ 101½111 and dislocation I I location triggers such an event. Furthermore, in case of [100] II belongs to the slip systemðÞ n ; b :ðÞ 011½111. Both dislocations have screw II II orientation. The glide planes intersect along parallel to the ½111 direction and the loading direction, a closed chain of coupled pairs is available, direction of shortest distance between the two screw dislocations is along the½ 110 which further increases the plastic strain rate. The inclusion of the direction. In order to be repulsively oriented, the line direction of dislocation I is mechanism in the CP model (Supplementary Discussion and chosen parallel to its Burgers vector, while for dislocation II the line direction is Supplementary Fig. 7) has a rather modest effect on the stress antiparallel to its Burgers vector. strain curve from the CP model for [111] loading. For the [100] loading direction, the coupling changes the initial hardening (CP Details on the local dislocation loading. A uniaxial loading on the sample is (std) curve) quite drastically and leads to an excellent agreement −1 applied, until dislocation I reaches a velocity of about 1 nm s . During further with the experimental results (Fig. 4; stress strain curves marked simulation the externally applied load is kept constant. The Schmid factors on by CP(coupl)). dislocation I respectively II are 0.5, respectively, 0.3. In conclusion, a new mechanism inﬂuencing the work hard- The minimum activation enthalpy of kink-pair nucleation of dislocation II occurs at the position of nearest approach where the total interaction is strongest. ening in bcc metals in the kink-pair regime, based on the The velocity of dislocation II is dominated by the kink pairs nucleated in the repulsive interactions between non-coplanar slip systems in bcc interaction zone which then spread out along the dislocation line. As mentioned in metals, is presented. The repulsively oriented forest screws, rather the main part, the overall direction of the Peach–Koehler force would drive both than obstructing the motion of an incoming screw dislocation, the dislocation in a similar direction. Therefore once, kink pairs are nucleated, the stress level on the dislocation line outside the interaction zone drives these pairs glide collectively at no additional applied stress. The present work along the dislocation line. On dislocation II the peak resolved shear stress exceeds shows that the interaction zone is limited to a very small region of the one of dislocation I by about 50 MPa leading to the same effective velocity the interacting dislocations. These results deeply change the required for coupled motion. This used velocity law implicitly assumes also that 18,30,24 concept of internal stresses, at least in bcc metals or materials kink collision is unlikely . For dislocation II this is obviously true, as the zone of interaction, where the kink-pair nucleation rate is drastically increased, is where the kink-pair mechanism govern dislocation motion. In 2- extremely small and both kinks will glide easily to the opposite sides of this zone. D models it is generally assumed that moving dislocations are Outside this zone of interaction (nearest approach), kink-pair nucleation is very subjected to a uniform stress along their line, and that this stress unlikely and therefore this assumption justiﬁed. For dislocation I the length oscillates in the direction of motion. Under such conditions, the dependency of the mobility law is supported by observations on Fe at room average dislocation velocity is determined by the highest internal temperature . 6 NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications [MPa] NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 ARTICLE Stresses acting on the dislocations. Coupled motion occurs only if both dis- 2. Mott, N. F. F. CXVII. A theory of work-hardening of metal crystals. Philos. locations remain in their initial habit plane, therefore the question of cross slip has Mag. Ser. 7, 1151–1178 (1952). to be addressed: cross slip occurs only if the activation energy of glide for screw 3. Šesták, B. & Seeger, A. The relationship between the work-hardening of B.C.C. dislocation is minimum on a plane other than the habit plane. Due to symmetry in and F.C.C. metals. Phys. Status Solidi 43, 433–444 (1971). bcc metals, this limits the angle χ between the habit plane and the maximum 4. Nabarro, F. R. N., Basinski, Z. S., Holt, D. B. D., Basynski, Z. S. & Holt, D. B. resolved shear stress plane (MRSSP) to be within −30° ≤ χ ≤ 30°. Supplementary D. The plasticity of pure single crystals. Adv. Phys. 13, 193–323 (1964). Fig. 3a shows that the MRSSP angle χ of both screw dislocation remains within this 5. Butler, B. G. et al. Mechanisms of deformation and ductility in tungsten—a range and therefore the lowest activation enthalpy is on the corresponding glide review. Int. J. Refract. Met. Hard Mater. 75, 248–261 (2018). plane of the dislocation. 6. Seeger, A., Diehl, J., Mader, S. & Rebstock, H. Work-hardening and work- Supplementary Figure 3b shows that the corresponding activation enthalpy for softening of face-centred cubic metal crystals. Philos. Mag. 2, 323–350 the driven dislocation II in the interaction zone is reduced signiﬁcantly below the (1957). activation enthalpy of the driving dislocation I because it needs to nucleate all its 7. Hartley, C. S. & Hirth, J. P. Interaction of nonparallel, noncoplanar kinks in the short interaction zone. dislocations. Acta Met. 13,79–88 (1965). 8. Taylor, G. I. The mechanism of plastic deformation of crystals. Part I. Theoretical. Proc. R. Soc. A Math Phys. Eng. Sci. 145, 362–387 (1934). Screw dislocation velocity. The local screw dislocation velocity is given by: 9. Cereceda, D. et al. Unraveling the temperature dependence of the yield ba L ΔHðÞ σ v ¼ ν exp ; strength in single-crystal tungsten using atomistically-informed crystal ð3Þ l k T plasticity calculations. Int. J. Plast. 78, 242–265 (2016). 10. Bulatov, V. V. et al. Dislocation multi-junctions and strain hardening. Nature where v is the Debye frequency, b magnitude of the Burgers vector, a is kink D 0 pﬃﬃﬃﬃﬃﬃﬃ 440, 1174–1178 (2006). height on 110 systems (a ¼ a 2=3) and a is the lattice constant. 11. Christian, J. W. Some surprising features of the plastic deformation of body- L is the screw dislocation length and l is the critical length for the nucleation of 18 centered cubic metals and alloys. Metall. Trans. A 14, 1237–1256 (1983). the kink-pair, which is calculated from the activation volume as l . c a b 12. Taylor, G. Thermally-activated deformation of BCC metals and alloys. Prog. Mater. Sci. 36,29–61 (1992). Experimental setup. In situ straining experiments were carried out in a JEOL 13. Caillard, D. Geometry and kinetics of glide of screw dislocations in tungsten 2010HC transmission electron microscope working at 200 kV, using the room- between 95K and 573K. Acta Mater. 161,21–34 (2018). temperature and the low-temperature straining holder designed by GATAN. The 14. 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Structure of dislocation cores in metallic materials and its impact on classical extinction rules using several diffraction conditions, and the slip planes their plastic behaviour. Prog. Mater. Sci. 36,1–27 (1992). were deduced from the directions and separation distances of the slip traces left by 18. Dorn, E. & Rajnak, S. Nucleation of kink pairs and the Peierls’ mechanism of the moving dislocations at the two surfaces. plastic deformation. Trans. AIME 230, 1052–1064 (1964). The local direction of the tensile axis can slightly deviate from the imposed one 19. Gröger, R., Bailey, A. G. & Vitek, V. Multiscale modeling of plastic by several degrees in the foil plane. However, it can be determined with a pretty deformation of molybdenum and tungsten: I. Atomistic studies of the core good accuracy in samples with rounded holes and containing no cracks, on the structure and glide of 1/2〈111〉 screw dislocations at 0K. Acta Mater. 56, basis of ﬁnite element calculations. Local Schmid factors can then be determined 5401–5411 (2008). with an accuracy of a few percent. 20. Weinberger, C. R., Boyce, B. L. & Battaile, C. C. Slip planes in bcc transition The local shear-stress intensity can be deduced from the critical widths of metals. Int. Mater. Rev. 58, 296–314 (2013). expanding screw dipoles, using elasticity models. Supplementary Figure 4 shows a screw dipole formed at 300 K in less than one frame of the video. After determining 21. Chen, Z. M., Mrovec, M. & Gumbsch, P. Atomistic aspects of 1/2 <111> screw dislocation behavior in α-iron and the derivation of microscopic yield the dipole plane on the basis of the slip trace direction at its emergence point (noted tr.), the critical width corrected from perspective effects has been estimated criterion. Model. Simul. Mater. Sci. Eng. 21, 055023 (2013). as 85 nm in (c). Similar measurements on about ten expanding dipoles yield values 22. Dezerald, L., Proville, L., Ventelon, L., Willaime, F. & Rodney, D. First- comprised between 65 and 90 nm, corresponding to a local stress of 640 ± 100 principles prediction of kink-pair activation enthalpy on screw dislocations in MPa, at 300 K. This stress value is consistent with macroscopic values at 10% bcc transition metals: V, Nb, Ta, Mo, W, and Fe. Phys. Rev. B 91, 094105 strain, or measured by the so-called incremental-straining-temperature-lowering (2015). tests, namely after the exhaustion of the mobile non-screw segments . 23. Dezerald, L., Rodney, D., Clouet, E., Ventelon, L. & Willaime, F. Plastic Stress measurements during in situ relaxation tests show a 14% decrease of local anisotropy and dislocation trajectory in BCC metals. Nat. Commun. 7, 11695 stress τ when the velocity v of dislocations decreases by a factor of 7, which (2016). Δ ln v corresponds to a stress-velocity dependence 15. 24. Weygand, D., Mrovec, M., Hochrainer, T. & Gumbsch, P. Multiscale Δ ln τ TEM images are showing the screw dislocation coupling phenomenon in Fe and simulation of plasticity in bcc metals. Annu. Rev. Mater. Res. 45, 369–390 Nb at T = 95 K are shown in Supplementary Figs. 1, 2. The observations conﬁrm (2015). that the coupling is a mechanism present in crystal structures, where (screw) 25. Kraych, A. et al. Non-glide effects and dislocation core ﬁelds in BCC metals. dislocation have to overcome a barrier by the kink-pair mechanism in order npj Comput. Mater. 5, 109 (2019). to glide. 26. Marichal, C. et al. Origin of anomalous slip in tungsten. Phys. Rev. Lett. 113, 25501 (2014). 27. Brunner, D. Temperature dependence of the plastic ﬂow of high-purity Data availability tungsten single crystals. Int. J. Mater. Res. 101, 1003–1013 (2010). The datasets generated during or analyzed during the current study are available from 28. Brunner, D. Stress-relaxation tests in the work-hardening regime of tungsten the corresponding author on reasonable request. single crystals below 300K. Mater. Sci. Eng. A 483–484, 525–528 (2008). 29. Schmitt, N. J. Experimentelle Untersuchung des Verformungs- und Code availability Bruchverhaltens von ein-, bi- und polykristallinem Wolfram auf der Mikroskala Access to the code will be provided at the host institution of the corresponding author (KIT, Karlsruhe, 2016). https://doi.org/10.5445/IR/1000062471 upon reasonable request. The occurrence of the coupling mechanism does not depend on 30. Srivastava, K., Gröger, R., Weygand, D. & Gumbsch, P. Dislocation motion in code details. The basic principles are described under methods. tungsten: atomistic input to discrete dislocation simulations. Int. J. Plast. 47, 126–142 (2013). 31. Madec, R. & Kubin, L. P. Second-order junctions and strain hardening in bcc Received: 7 February 2020; Accepted: 14 September 2020; and fcc crystals. Scr. Mater. 58, 767–770 (2008). 32. Queyreau, S., Monnet, G. & Devincre, B. Slip systems interactions in α-iron determined by dislocation dynamics simulations. Int. J. Plast. 25, 361–377 (2009). 33. Franciosi, P. The concepts of latent hardening and strain hardening in metallic single crystals. Acta Met. 33, 1601–1612 (1985). References 34. Madec, R., Devincre, B., Kubin, L., Hoc, T. & Rodney, D. The role of collinear 1. Hirth, J. P. & Lothe, J. Theory of Dislocations (Wiley, 1992). interaction in dislocation-induced hardening. Science 301, 1879–1882 (2003). NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 35. Rice, J. R. R. Inelastic constitutive relations for solids: an internal-variable Author contributions theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 K.S. and D.W. conceived the simulation tools and conducted the simulations and ana- (1971). lyzed and interpreted the data; wrote and edited the paper. D.C. conceived, designed, 36. Asaro, R. J. Micromechanics of crystals and polycrystals. Adv. Appl. Mech. 23, conducted and analyzed the experiments; wrote and edited the paper. P.G. provided the 1–115 (1983). sample; contributed to the conception of the work; interpretation of the data, wrote and 37. Wu, T.-Y., Bassani, J. L. & Laird, C. Latent hardening in single crystals I. edited the paper. theory and experiments. Proc. R. Soc. A Math. Phys. Eng. Sci. 435,1–19 (1991). Funding 38. Wei, Q. & Kecskes, L. J. J. Effect of low-temperature rolling on the Open Access funding enabled and organized by Projekt DEAL. tensile behavior of commercially pure tungsten. Mater. Sci. Eng. A 491,62–69 (2008). 39. He, B. B. et al. High dislocation density-induced large ductility in deformed Competing interests and partitioned steels. Science 357, 1029–1032 (2017). The authors declare no competing interests. 40. Reiser, J. et al. Ductilisation of tungsten (W): on the increase of strength and room-temperature tensile ductility through cold-rolling. Int. J. Refract. Met. Additional information Hard Mater. 64, 261–278 (2017). Supplementary information is available for this paper at https://doi.org/10.1038/s41467- 41. Bonnekoh, C. et al. The brittle-to-ductile transition in cold rolled tungsten 020-18774-1. plates: impact of crystallographic texture, grain size and dislocation density on the transition temperature. Int. J. Refract. Met. Hard Mater. 78, 146–163 Correspondence and requests for materials should be addressed to D.W. (2019). 42. Srivastava, K. Atomistically-informed discrete dislocation dynamics modeling of Peer review information Nature Communications thanks Brady Butler and the other plastic ﬂow in body-centered cubic metals (Karlsruhe Institute of Technology, anonymous reviewer(s) for their contribution to the peer review of this work. Peer 2014). https://doi.org/10.5445/IR/1000042367 reviewer reports are available. 43. Tang, M., Kubin, L. P. & Canova, G. R. Dislocation mobility and the mechanical response of b.c.c. single crystals: a mesoscopic approach. Acta Reprints and permission information is available at http://www.nature.com/reprints Mater. 46, 3221–3235 (1998). 44. Chaussidon, J., Robertson, C., Rodney, D. & Fivel, M. Dislocation dynamics Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in simulations of plasticity in Fe laths at low temperature. Acta Mater. 56, published maps and institutional afﬁliations. 5466–5476 (2008). 45. Po, G. et al. A. phenomenological dislocation mobility law for bcc metals. Acta Mater. 119, 123–135 (2016). 46. Caillard, D. Kinetics of dislocations in pure Fe. Part I. In situ Open Access This article is licensed under a Creative Commons straining experiments at room temperature. Acta Mater. 58, 3493–3503 Attribution 4.0 International License, which permits use, sharing, (2010). adaptation, distribution and reproduction in any medium or format, as long as you give 47. Brunner, D. & Glebovsky, V. Analysis of ﬂow-stress measurements of high- appropriate credit to the original author(s) and the source, provide a link to the Creative purity tungsten single crystals. Mater. Lett. 44, 144–152 (2000). Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory Acknowledgements regulation or exceeds the permitted use, you will need to obtain permission directly from P.G. acknowledges ﬁnancial support from German Research Foundation (DFG) through the copyright holder. To view a copy of this license, visit http://creativecommons.org/ project grant Gu367/30. D.W. acknowledges ﬁnancial support for the research group licenses/by/4.0/. FOR1650 Dislocation based plasticity funded by the German Research Foundation (DFG) under contract numbers WE3544/5-2. We acknowledge support by the KIT- Publication Fund of the Karlsruhe Institute of Technology. © The Author(s) 2020 8 NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nature Communications Springer Journals http://www.deepdyve.com/lp/springer-journals/repulsion-leads-to-coupled-dislocation-motion-and-extended-work-U5WIecdN4b

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eISSN: 2041-1723
DOI: 10.1038/s41467-020-18774-1
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Abstract

ARTICLE https://doi.org/10.1038/s41467-020-18774-1 OPEN Repulsion leads to coupled dislocation motion and extended work hardening in bcc metals 1,2 1 3 1,4 K. Srivastava , D. Weygand , D. Caillard & P. Gumbsch Work hardening in bcc single crystals at low homologous temperature shows a strong orientation-dependent hardening for high symmetry loading, which is not captured by clas- sical dislocation density based models. We demonstrate here that the high activation barrier for screw dislocation glide motion in tungsten results in repulsive interactions between screw dislocations, and triggers dislocation motion at applied loading conditions where it is not expected. In situ transmission electron microscopy and atomistically informed discrete dis- location dynamics simulations conﬁrm coupled dislocation motion and vanishing obstacle strength for repulsive screw dislocations, compatible with the kink pair mechanism of dis- location motion in the thermally activated (low temperature) regime. We implement this additional contribution to plastic strain in a modiﬁed crystal plasticity framework and show that it can explain the extended work hardening regime observed for [100] oriented tungsten single crystal. This may contribute to better understanding the increase in ductility of highly deformed bcc metals. 1 2 Institute for Applied Materials (IAM), Karlsruhe Institute of Technology (KIT), Straße am Forum 7, 76131 Karlsruhe, Germany. Research and Development, AG der Dillinger Hüttenwerke, Werkstraβe 1, 66763 Dillingen/Saar, Germany. CEMES-CNRS, 29 rue Jeanne Marvig, BP4347, F-31055 Toulouse Cedex 4 ✉ 4, France. Fraunhofer IWM, Wöhlerstr. 11, 79108 Freiburg, Germany. email: [email protected] NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications 1 1234567890():,; ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 etallic materials are used mainly in technical applica- components of the stress tensor inﬂuence their glide beha- 16,17 tions for their good formability, strength and toughness. viour . The complex core structure of screw dislocations and MThis favourable combination of materials properties its sensitivity to non-glide stresses is at the origin of this so-called 16,17 fundamentally relies on work hardening during plastic defor- non-Schmid behaviour . This core structure is believed to mation. The response of metals to a mechanical load by irre- make screw dislocations glide by the successive nucleation and versible plastic deformation occurs on a microscopic scale by the motion of pairs of kinks on an otherwise straight screw disloca- motion of dislocations, line defects of the crystalline structure. tion line and lead to the thermally activated deformation beha- Dislocations in general glide on densely packed crystallographic viour of bcc metals at low temperatures . Atomistic simulations 19–25 planes. Dislocation glide on such glide planes causes a relative and ﬁrst principle calculations have been used to investigate shift of the material above and below the glide plane in the the properties of screw dislocations and of kink-pair formation direction of a short lattice vector, the Burgers vector . The actual for bcc materials. The core structure is found to be non degen- plastic deformation is a consequence of the interplay between erate and compact. The complex screw dislocation core structure dislocation glide, dislocation multiplication and annihilation on also leads to the activity of unexpected glide systems . However, well-deﬁned slip systems, characterised by glide plane and Bur- the role of forest dislocations on work hardening in the low- gers vector. Dislocations move collectively upon straining the temperature regime is not understood. material and multiply. Dislocation multiplication leads to an Since screw dislocations determine the deformation behaviour increase in dislocation density. This in turn is believed to result in at low temperatures, we systematically investigate the mutual work hardening which manifests itself as an increase in the ﬂow interaction of screw dislocations in bcc metals using discrete stress of the material upon straining and gives e.g. a deep-drawn dislocation dynamics (DDD) simulations and in situ experiments. component its strength but also hinders further deformation. A Tungsten (W) is chosen as a representative of bcc metals. suitable heat treatment can often restore formability. Tungsten is elastically nearly isotropic and its mechanical prop- Work hardening is known to depend on the crystalline struc- erties are well studied and show extended work hardening ture of the metal, e.g. face centred cubic (fcc) metals like alumi- regimes for [111] and even more pronounced for [100] orienta- 14,15,27–29 nium or copper typically show stronger hardening than body tion . The investigation allows to identify a hitherto 2–5 centred cubic (bcc) metals like iron or tungsten . The long- overlooked mechanism based on the coupled motion of repulsive range interaction between dislocations on different glide planes oriented screw dislocation pairs in both simulations and experi- and with different character is caused by the elastic distortion that ments. The mechanisms’ origin is the internal shear stress dislocations introduce into the crystal lattice. Additionally, short- increase leading to screw dislocation glide on unexpected slip range interaction at the intersection of dislocations can lead to systems and thus a larger plastic deformation at the same mac- changes in the atomic conﬁguration of the dislocation core and to roscopic load occurs. This is a key aspect explaining the orien- 2,3,6–8 the formation of so-called dislocation junctions . The pro- tation dependence of the extended work hardening regimes, e.g. minent interpretation of work hardening as forest hardening is in tungsten. based on such interaction of dislocations on inclined slip systems hindering the motion of the mobile dislocations and thus leading 2,8,9 Results to an increase in the stress for further plastic deformation . Discrete dislocation dynamics simulations. DDD simulations Dislocation interactions may also result in multi-junctions are used to investigate the interaction of two screw dislocations involving more than two dislocations which in bcc metals have been shown to contribute signiﬁcantly to work hardening . on non-coplanar glide systems. All possible types of interaction 24,30 pairs are studied. An atomistically informed DDD code is All work hardening models intrinsically assume that the mutual interaction between dislocations always hinders disloca- employed here. Screw dislocation mobility is modelled by the 18,30 kink-pair formation mechanism , using atomistic values for tion motion either by repulsion on approach or by pinning at junctions, which prohibits the release and further motion of the the stress dependent activation enthalpy for kink-pair formation (details in Methods section) thus including also non-Schmid dislocation. Therefore work hardening models all include a contribution to the ﬂow stress which is inversely proportional to effect. While attractive interactions show the expected dislocation 31,32 reactions and act as obstacles by forming dislocation junc- the average dislocation spacing L which in turn is related to the pﬃﬃﬃ 4,8 tions, speciﬁc repulsively oriented screw dislocation pairs sur- dislocation density by L 1= ρ . prisingly show coupled gliding. Such coupled motion, where one While these implicit assumptions and the resulting work mobile screw dislocation (dislocation I), upon interaction with a hardening models appear plausible for fcc metals or bcc at high second repulsively oriented (immobile) screw dislocation (dis- temperatures T > T (T is the athermal temperature) where dis- c c location II), drives the second dislocation without slowing down. location glide is controlled only by the resolved shear stress on the glide plane, it is highly speculative for bcc metals at T < T whose This coupled motion occurs at constant applied stress. For room 5,11–13 temperature deformation, modelled here, the kink motion along deformation behaviour is known to be much more complex . A little understood example is the work hardening of tungsten the dislocation line (2 µm in length) occurs faster than the time 14,15 scale of kink-pair nucleation as long as the kinks see a positive single crystals at room temperature . At room temperature, for [100] tensile loading, an extended work-hardening regime is driving force. Figure 1a displays a schematic diagram of the simulation setup: observed with an initial ﬂow stress of about 250 MPa and with a work hardening that leads to ﬂow stresses of almost 1 GPa during two repulsively oriented screw dislocations, dislocation I belong- ing to glide systemðÞ 101½ 111 and dislocation II to glide system straining by 6–8% . Also a recent atomistically informed crystal plasticity model does not capture the initial hardening behaviour . ðÞ 011½111, are placed within a crystal subjected to tensile loading In bcc metals at low temperatures screw and non-screw dis- along the ½149 direction. The Schmid factors for dislocations I locations have a completely different response to the applied and II are m ≈ 0.5 and m ≈ 0.3. The nearest distance between I II stress σ . Non-screw dislocations can bend and their motion is the dislocations is initially about 50 nm. Dislocation I begins to app controlled exclusively by the resolved shear stress τ ¼ mσ glide once the resolved shear stress τ reaches about 1.8 GPa. res app res I The corresponding resolved shear stress on dislocation II is about on the glide plane, where m is the so-called Schmid factor. 1.2 GPa. The glide directions of the dislocations I and II are In contrast screw dislocations remain straight and additional 2 NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 ARTICLE marked as r and r and both are pointing to the left in Fig. 1a. The coupled motion of the two dislocations can be understood 1 2 Dislocation I reaches a velocity of about 1.05 nm/s, while based on this analysis of the stress distribution along the dislocation II is initially immobile (see Fig. 1b). dislocation line. Over the entire length of dislocation I (about 2 Figure 1b shows the velocity of the two dislocations as a µm) only a very short section of about 20 nm around the point of function of their nearest distance of approach. Dislocation I closest interaction experiences a signiﬁcantly reduced resolved approaches the immobile dislocation II causing it to glide once shear stress. Consequently, kink-pair nucleation is reduced very the nearest distance decreases to about 9.5 nm. From that point locally in this section, while all the rest of the dislocation line still on, the velocity of dislocation II increases rapidly to match the experiences high resolved shear stresses from the externally velocity of dislocation I. Their nearest distance then stabilises applied ﬁeld and kink-pair nucleation rate remains virtually around a value of d ≈ 7.3 nm. Both dislocations then glide unchanged. Therefore, the total velocity of dislocation I remains crit collectively. essentially constant. The situation of dislocation II is opposite: The stress states along both dislocation lines favour glide on once dislocation I has reached the critical distance, the kink-pair their initial habit planes. Both dislocations remain straight nucleation rate on dislocation II drastically increases near the throughout the entire simulation. No cross slip is triggered (see point of closest approach until the dislocation velocity reaches the Methods) due to the mutual interaction. The maximal resolved velocity of dislocation I. Thereafter the two dislocations show shear stresses along the dislocations due to their mutual elastic coupled glide, slightly oscillating around the critical distance due interaction at the point of nearest distance is shown in Fig. 2a. to the different orientations of the elementary kink steps on the The distribution of the negative (positive) additional shear stress two dislocations. acting on dislocation I (dislocation II) along the dislocation line for the distance d is shown in Fig. 2b). The curves are crit asymmetric with respect to the nearest point of interaction as the In situ transmission electron microscopy. To mimic this sce- respective resolved shear stresses are shown. nario experimentally, repulsively oriented screw dislocations in ab Driving dislocation I: (101)[111] Dislocation II r Driven dislocation II: (011)[111] 1,5 Dislocation I 0,5 010 20 Nearest distance [nm] Fig. 1 Simulation setup and coupled motion. a Schematic view of the setup with two repulsively oriented screw dislocations I and II with Burgers vectors b and b respectively. The indicated directions r and r are parallel to the Peach–Kohler force due to the external loading resolved in the respective 1 2 1 2 primary glide planes. Only repulsive pairs, where the glide directions r and r have a positive scalar product have to be considered. The location A and B 1 2 marked on the dislocation line are the nearest points between the two dislocations deﬁning the nearest distance d. b The variation of the velocity of dislocations I and II with Burgers vectors [111] and ½111 is plotted versus the minimal distance of approach. ab Driving dislocation I: (101)[111] Driving dislocation I: (101)[111] Driven dislocation II: (011)[111] Driven dislocation II: (011)[111] From externally applied stress –200 From mutual interaction 0 –400 –600 –500 0 1020304050 –50 0 50 Nearest distance [nm] Relative position along dislocation [a ] Fig. 2 Stress distribution on repulsively interacting dislocations. a Resolved shear stress from externally applied stress on dislocation I and II (dashed line) and shear stress due to mutual dislocation interaction (line) at the closest point of interaction (nearest distance); b resolved shear stress due to the repulsive dislocation interaction along dislocation I and II. Dislocations are at distance d . The relative positions are given in multiples of the lattice crit constant a and measured from the closest point of interaction. NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications 3 Resolved shear stresses on glide plane [MPa] Velocity (nm/s) Contribution of mutual interaction to resolved shear stress [MPa] ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 thin ﬁlms of a tungsten singe crystal are investigated by TEM. leaving dislocation II again immobile with respect to the ﬁxed Thin ﬁlm specimens of tungsten suitable for in situ deformation point ʻz’, between (e) and (f). The velocity of dislocation Ia experiments have been prepared and strained in a transmission remains almost constant close to 25 nm/s during the whole electron microscope (TEM, see Methods for a description of the process. Then a second dislocation Ib arrives and pushes again method and Caillard and Supplementary Figs. 1, 2 for corre- dislocation II in (f) and (g). The image (h) which is the difference sponding observations in Fe and Nb). The foil plane is ð117Þ, the between (a) and (d) shows the starting positions in dark and the local tensile axis is [521], and the sample thickness is 300 nm. ﬁnal ones in bright. The velocity of dislocation II ﬂuctuates Under such conditions, repulsively oriented dislocation pairs around the one of the driving dislocation Ia or Ib. The could be identiﬁed: Dislocations of type I with Burgers vector ﬂuctuations are more pronounced than for the DDD results. 1=2½111 and length 360 nm are the most mobile in the (101) During this coupled motion, the point of closest distance moves along the direction [010] of intersection between the two plane, in agreement with a high Schmid factor of 0.485. Dis- slip planes. Since this direction is almost within the plane of the locations of type II with Burgers vector ½[111] later move on the TEM-foil, long distances of coupled motion can be observed. The ð101Þ plane, but initially are immobile due to the lower Schmid minimum distance during the coupled motion has been measured factor of 0.458. in projection, and corrected from perspective effects. At 300 K, Figure 3i shows the schematic of the TEM sample geometry. this distance ﬂuctuates a little bit, but remains equal to about 24 The slip planes of the studied dislocations intersect the two free surfaces along the directions noted slip traces, and intersect each nm when coupled motion takes place. Similar observations of collective glide have been made a T = 200 K, but the correspond- other along the direction noted node path. Dislocations are represented by the straight lines marked I and II, gliding to the ing critical distance is unfortunately too small to be measured. left on the respective slip planes 1 and 2. Figure 3a–g shows the TEM observation of these dislocations for different times during the deformation at T = 300 K (see also the Supplementary Discussion Movie). In the TEM sample, the ﬁrst respectively second driving Both experiments and simulation show coupled motion of dislocation is labelled dislocation Ia respectively dislocation Ib. repulsively oriented screw dislocation pairs. In both cases screw Dislocation Ia approaches dislocation II between (a) and (b). dislocations remain straight and no cross slip is observed. The Note that dislocation II is immobile with respect to the ﬁxed DDD simulations suggest that this behaviour is a direct con- point ʻx’. Coupled motion takes place in (c) and (d) (see ﬁxed sequence of dislocation motion by the kink-pair mechanism. points ʻx’ and ʻy’) until (e) where dislocation Ia has moved away Kink-pair nucleation occurs locally and therefore depends on the Fig. 3 In situ TEM observation of coupled glide: experimental observation of dislocation glide at different time steps. Reference points are indicated by ʻx, y, z’. a Dislocation Ia approaches the still dislocation II. b–d Dislocation Ia pushes dislocation II: their point of closest distance moves along the direction ʻnode path’ until it emerges at the bottom surface in (e). f, g Another dislocation Ib pushes dislocation II. h Difference-image showing the motion between (b) (black dislocations) and (d) (bright dislocations). i Scheme corresponding to (h). The video sequence can be downloaded as a Supplementary Movie. 4 NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 ARTICLE 18,30 stress state along the entire screw dislocation line . If the into account for kink-pair nucleation in bulk. However, this is critical resolved shear-stress (CRSS) is reached upon raising the very different from the restoring forces due to line tension effects applied stress, dislocations of type I move while dislocations of observed in fcc metals. type II should and do remain initially immobile in experiment The observed collective glide of dislocations has signiﬁcant and simulation. Once the two dislocations get close enough, the consequences on the evolution of the plastic strain rate in bcc region of closest encounter on dislocation II experiences resolved metals in the thermally activated regime. Classical phenomen- shear stresses which are higher than the stresses on dislocation I. ological work hardening relations have been proposed with a ﬂow This has the consequence that kink pairs on dislocation II are stress inversely proportional to the average dislocation spacing L pﬃﬃﬃ generated in this region of closest encounter at a rate (per unit which can be calculated from the dislocation density as 1= ρ. length) somewhat higher than on dislocation I, where nucleation Dislocations are considered there as mutual obstacles which leads occurs along the entire line. The kinks generated at this location to so-called forest hardening. Reﬁned models consider an effec- of closest encounter move along the dislocation line as long as tive hardening between slip systems by speciﬁc hardening coef- 32–34 there is a positive driving force which is true for both experiments ﬁcients and simulation. Dislocation I could in principle slow down The situation for bcc metals in the thermally activated regime slightly because part of the total length of the dislocation along differs: the screw dislocations in the repulsive pairs remain which kink-pair nucleation may occur is now at lower stresses straight. Our results show that the high effective ʻstiffness’ of (c.f. Fig. 2b). However, already at our line length of 2 µm, this screw dislocations leads to a coupled glide of dislocation slowing down is too small to be observed in the simulations. ensembles, here exemplarily studies at pairs or groups of screw Eventually the total kink generation rates along both dislocations dislocations (see Supplementary Note 1 and Supplementary match and both dislocations move at the same velocity. This Fig. 5). Within the range of validity of the kink-pair nucleation implies that these repulsively oriented screw dislocations should mechanism for dislocation motion, the mutual interaction of never cross each other as long as both are able to move. Motion of these repulsively oriented dislocation pairs is limited to modifying a primary screw dislocation can activate certain repulsively the local stress state in the zone of closest approach. Therefore it oriented screw dislocation (detailed in Supplementary Discus- locally enhances the kink-pair nucleation rate. sion) even if it is located on a slip plane with a very low resolved During coupled motion the second dislocation (slip system 2) shear stress. contributes to the total plastic shear rate on slip system 2, due to Even quantitatively, the detailed TEM investigations and the dislocation glide on slip system 1, even though macroscopically simulations can well be related to each other. The smaller dif- the resolved shear stress is below its CRSS. In order to take such a ferences in Schmid factor in the experimental setup compared to phenomenon into account we propose to modify the equations the simulations make dislocation II move somewhat earlier, i.e. at for the plastic shear rate evolution by adding an interaction term, an about 3 times larger critical distance, than in the simulations. expressing this coupling for the well accepted strain rate based 18,35 The TEM investigation also shows clearly that the mechanism is formulation repeatable for dislocation II as dislocation Ib moving on a glide ΔH ðσÞ α α plane parallel to that of dislocation Ia takes over the role of the γ _ ¼ γ _ exp ; ð1Þ std 0 k T driving dislocation as dislocation Ia moves out of the observed area of the thin ﬁlm. During in situ experiments at lower tem- where γ _ denoted the standard plastic shear strain rate normally std perature much shorter critical distances between the coupled 36,37 used in crystal plasticity (CP) formulations , ΔH (σ) is the dislocations are observed which are beyond the limit of resolu- stress dependent activation enthalpy for system α, k is the tion. The smaller critical distance between the coupled pairs leads Boltzmann constant and T is the temperature. to larger interaction stresses, as required for activating screw The coupled motion of speciﬁc screw dislocation pairs is dislocation motion at lower temperatures. captured by adding a shear strain rate γ _ coupl Since our observations can be explained by dislocation motion based on kink-pair nucleation alone, this coupling mechanism α α α α γ _ ¼ γ _ þ γ _ ¼ γ _ þ f K γ _ ; tot std coupl std coupl αβ std ð2Þ between two repulsively oriented screw dislocations is likely to β ≠ α be observed in all bcc metals and other materials where kink-pair generation governs dislocation motion. In Supplementary Figs. 1, where γ _ is then the total plastic shear strain rate. The coupling tot 2 additional TEM evidence for the coupling in Fe and Nb at T= matrix K parametrises the coupled motion between slip system αβ 95 K is presented, conﬁrming the universality of the phenomenon. α and β and f is a scaling factor between 0 and 1. The terms in coupl The in situ observations and DDD results are both obtained for the matrix K depend on the crystallographic orientation of the αβ straight dislocations, reaching from surface to surface. In bulk two slip systems and the local stress state as several conditions material, dislocations are often anchored in networks which leads (see also Supplementary Note 2) have to be fulﬁlled. The condi- to an increase in line length during motion. One could speculate tions are (i) repulsive orientation; (ii) the resolved Peach–Koehler that this might possibly suppress screw dislocation motion and force due to the stress state on both dislocations has a positive hence the coupling mechanism. However, the increase of length scalar product and (iii) include only repulsive dislocation pairs corresponds to the nucleation of new kink pairs, which takes which can glide over extended distances (see also Supplementary place anyway, even in case of straight screws ending at the sur- Fig. 6). faces. These kinks subsequently either accumulate at the dis- In this analysis, only {110} slip systems are considered (Sup- location extremities in the case of bulk material or disappear at plementary Table 1), as suggested by atomistic simulations .In the surfaces. This increase of length is actually a restoring force, experimental observations for [110] loading slip activity on {112} but already included in the kink-pair nucleation process. There- systems is reported too . Therefore the focus in the following is fore, the only ingredient, which has not been taken into account on [100] and [111] loading directions, where {110} slip planes are for mimicking bulk behaviour, is the elastic repulsion between reported. kinks accumulated at the screw extremities—forming a mixed This coupling has important consequences on the plastic ﬂow: dislocation (curved section with non-screw character). This for a given stress state, the coupling enhances the plastic strain repulsion is part of the internal stress due to elastic interactions without requiring an additional applied stress and thus the strain- with all neighbouring dislocations, which of course must be taken hardening rate ∂σ=∂ϵ is effectively lowered. NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 stress σ opposing to the applied stress, namely by a stress σ − int σ . Here we show that in a 3-D model the velocity of dislocations int is determined by the highest internal stress acting in the direction of the applied stress along the dislocation line, namely by a stress σ + σ . This strongly increases the average dislocation velocity in int 600 comparison with classical models, not only for the driven dis- locations studied in this article, but also for all dislocations. We also obtain the surprising result that the dislocation velocity tends [100] Beardmore [111] Beardmore to increase with increasing internal stress, i.e. with increasing [100] CP (std) deformation. Depending on the loading state, the number of [100] CP (coupl) active repulsive dislocation pairs varies and is highest for the [111] CP (std) [100] orientation, explaining the observed extended work hard- [111] CP (coupl) ening behaviour of tungsten. This strain-softening effect compensates for other classical 0.00 0.02 0.04 0.06 0.08 0.10 0.12 strain-hardening ones and may also accounts for a lower global [–] strain-hardening of bcc metals. These effects must be taken into Fig. 4 Deformation behaviour of tungsten single crystals for the [100] account in macroscopic models like CP. The relevance for the and [111] loading direction. The experimental curves are taken from coupled motion increases with dislocation density, as the density Beardmore and Hull . For the [100] direction an extended work-hardening of possible pairs increases too. For very high dislocation densities, regime is observed; crystal plasticity simulation results without (CP(std)) the internal stresses ﬂuctuate over short distances, thus triggering and including the coupling mechanism (CP (coup)) are shown for both possibly the coupled motion. Therefore this may also lead to the orientations. (A detailed comparison is included in Supplementary 38–41 enhanced ductility observed for heavily cold rolled tungsten . Discussion). The coupling mechanism leads to an extended work hardening regime for [100] representing well the experimental ﬁndings, while for the [111] orientation the deformation behaviour is only slightly changed. Methods Discrete dislocation dynamics model and setup. The following setup for the pillar is taken: the size of the box is 2 μm with an aspect ratio of 1:2:1. The DDD The coupling mechanism is a key to rationalise the extended model used to simulate the screw dislocation motion is described in Srivastava 30,42 work hardening regime of tungsten single crystals at low homo- et al. . The mobility of the screw dislocation is governed by an Arrhenius law logous temperature for [100] loading direction shown in Fig. 4. and accounts for the inﬂuence of the entire stress tensor on the activation energy 14 15 of screw dislocations as against a pure shear stress based formulation in lit- Beardmore and Hull and Argon and Maloof observed for the 43–45 erature . The model effectively takes into account changes in the dislocation [100] loading direction a gradual ʻroundish’ strain-hardening core structure due to the applied stress by including non-Schmid terms in the curve in contrast to [111] which display a classical hardening activation enthalpy. In the DDD model the effective kink-pair nucleation rate for behaviour after yielding. the screw dislocation section is calculated on all three possible glide planes for a screw dislocation. In principle, a screw dislocation may glide on different glide For the [100] loading direction a total of 12 coupled pairs exist, planes and therefore split into distinct sectors depending on the local stress state. where both dislocations have the same non-zero Schmid factor. This is not the case for the current setups. For all non-screw orientations phonon For the [111] only three coupled pairs exist. Details on the pair drag limited glide is assumed as for fcc metals. counting are given in Supplementary Note 2 and Supplementary Tables 2–5. Thus, the additional plastic strain at a given stress Details to the repulsive pairs in the DDD setup. The dislocations are labelled as level due to the coupled pairs is most pronounced for loading I and II depending on their respective role. Dislocation I is mobile due to the along [100] direction. The value of f was set to 0.5. Physically coupl externally applied loading and drives dislocation II, once a critical minimal distance a value of 1 would mean, that each driving dislocation triggers a is reached. ʻrepulsive’ event. For the chosen value of 0.5 every other dis- The dislocation I belongs to the slip systemðÞ n ; b :ðÞ 101½111 and dislocation I I location triggers such an event. Furthermore, in case of [100] II belongs to the slip systemðÞ n ; b :ðÞ 011½111. Both dislocations have screw II II orientation. The glide planes intersect along parallel to the ½111 direction and the loading direction, a closed chain of coupled pairs is available, direction of shortest distance between the two screw dislocations is along the½ 110 which further increases the plastic strain rate. The inclusion of the direction. In order to be repulsively oriented, the line direction of dislocation I is mechanism in the CP model (Supplementary Discussion and chosen parallel to its Burgers vector, while for dislocation II the line direction is Supplementary Fig. 7) has a rather modest effect on the stress antiparallel to its Burgers vector. strain curve from the CP model for [111] loading. For the [100] loading direction, the coupling changes the initial hardening (CP Details on the local dislocation loading. A uniaxial loading on the sample is (std) curve) quite drastically and leads to an excellent agreement −1 applied, until dislocation I reaches a velocity of about 1 nm s . During further with the experimental results (Fig. 4; stress strain curves marked simulation the externally applied load is kept constant. The Schmid factors on by CP(coupl)). dislocation I respectively II are 0.5, respectively, 0.3. In conclusion, a new mechanism inﬂuencing the work hard- The minimum activation enthalpy of kink-pair nucleation of dislocation II occurs at the position of nearest approach where the total interaction is strongest. ening in bcc metals in the kink-pair regime, based on the The velocity of dislocation II is dominated by the kink pairs nucleated in the repulsive interactions between non-coplanar slip systems in bcc interaction zone which then spread out along the dislocation line. As mentioned in metals, is presented. The repulsively oriented forest screws, rather the main part, the overall direction of the Peach–Koehler force would drive both than obstructing the motion of an incoming screw dislocation, the dislocation in a similar direction. Therefore once, kink pairs are nucleated, the stress level on the dislocation line outside the interaction zone drives these pairs glide collectively at no additional applied stress. The present work along the dislocation line. On dislocation II the peak resolved shear stress exceeds shows that the interaction zone is limited to a very small region of the one of dislocation I by about 50 MPa leading to the same effective velocity the interacting dislocations. These results deeply change the required for coupled motion. This used velocity law implicitly assumes also that 18,30,24 concept of internal stresses, at least in bcc metals or materials kink collision is unlikely . For dislocation II this is obviously true, as the zone of interaction, where the kink-pair nucleation rate is drastically increased, is where the kink-pair mechanism govern dislocation motion. In 2- extremely small and both kinks will glide easily to the opposite sides of this zone. D models it is generally assumed that moving dislocations are Outside this zone of interaction (nearest approach), kink-pair nucleation is very subjected to a uniform stress along their line, and that this stress unlikely and therefore this assumption justiﬁed. For dislocation I the length oscillates in the direction of motion. Under such conditions, the dependency of the mobility law is supported by observations on Fe at room average dislocation velocity is determined by the highest internal temperature . 6 NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications [MPa] NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 ARTICLE Stresses acting on the dislocations. Coupled motion occurs only if both dis- 2. Mott, N. F. F. CXVII. A theory of work-hardening of metal crystals. Philos. locations remain in their initial habit plane, therefore the question of cross slip has Mag. Ser. 7, 1151–1178 (1952). to be addressed: cross slip occurs only if the activation energy of glide for screw 3. Šesták, B. & Seeger, A. The relationship between the work-hardening of B.C.C. dislocation is minimum on a plane other than the habit plane. Due to symmetry in and F.C.C. metals. Phys. Status Solidi 43, 433–444 (1971). bcc metals, this limits the angle χ between the habit plane and the maximum 4. Nabarro, F. R. N., Basinski, Z. S., Holt, D. B. D., Basynski, Z. S. & Holt, D. B. resolved shear stress plane (MRSSP) to be within −30° ≤ χ ≤ 30°. Supplementary D. The plasticity of pure single crystals. Adv. Phys. 13, 193–323 (1964). Fig. 3a shows that the MRSSP angle χ of both screw dislocation remains within this 5. Butler, B. G. et al. Mechanisms of deformation and ductility in tungsten—a range and therefore the lowest activation enthalpy is on the corresponding glide review. Int. J. Refract. Met. Hard Mater. 75, 248–261 (2018). plane of the dislocation. 6. Seeger, A., Diehl, J., Mader, S. & Rebstock, H. Work-hardening and work- Supplementary Figure 3b shows that the corresponding activation enthalpy for softening of face-centred cubic metal crystals. Philos. Mag. 2, 323–350 the driven dislocation II in the interaction zone is reduced signiﬁcantly below the (1957). activation enthalpy of the driving dislocation I because it needs to nucleate all its 7. Hartley, C. S. & Hirth, J. P. Interaction of nonparallel, noncoplanar kinks in the short interaction zone. dislocations. Acta Met. 13,79–88 (1965). 8. Taylor, G. I. The mechanism of plastic deformation of crystals. Part I. Theoretical. Proc. R. Soc. A Math Phys. Eng. Sci. 145, 362–387 (1934). Screw dislocation velocity. The local screw dislocation velocity is given by: 9. Cereceda, D. et al. Unraveling the temperature dependence of the yield ba L ΔHðÞ σ v ¼ ν exp ; strength in single-crystal tungsten using atomistically-informed crystal ð3Þ l k T plasticity calculations. Int. J. Plast. 78, 242–265 (2016). 10. Bulatov, V. V. et al. Dislocation multi-junctions and strain hardening. 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Local Schmid factors can then be determined 5401–5411 (2008). with an accuracy of a few percent. 20. Weinberger, C. R., Boyce, B. L. & Battaile, C. C. Slip planes in bcc transition The local shear-stress intensity can be deduced from the critical widths of metals. Int. Mater. Rev. 58, 296–314 (2013). expanding screw dipoles, using elasticity models. Supplementary Figure 4 shows a screw dipole formed at 300 K in less than one frame of the video. After determining 21. Chen, Z. M., Mrovec, M. & Gumbsch, P. Atomistic aspects of 1/2 <111> screw dislocation behavior in α-iron and the derivation of microscopic yield the dipole plane on the basis of the slip trace direction at its emergence point (noted tr.), the critical width corrected from perspective effects has been estimated criterion. Model. Simul. Mater. Sci. Eng. 21, 055023 (2013). as 85 nm in (c). Similar measurements on about ten expanding dipoles yield values 22. Dezerald, L., Proville, L., Ventelon, L., Willaime, F. & Rodney, D. First- comprised between 65 and 90 nm, corresponding to a local stress of 640 ± 100 principles prediction of kink-pair activation enthalpy on screw dislocations in MPa, at 300 K. This stress value is consistent with macroscopic values at 10% bcc transition metals: V, Nb, Ta, Mo, W, and Fe. Phys. Rev. B 91, 094105 strain, or measured by the so-called incremental-straining-temperature-lowering (2015). tests, namely after the exhaustion of the mobile non-screw segments . 23. Dezerald, L., Rodney, D., Clouet, E., Ventelon, L. & Willaime, F. Plastic Stress measurements during in situ relaxation tests show a 14% decrease of local anisotropy and dislocation trajectory in BCC metals. Nat. Commun. 7, 11695 stress τ when the velocity v of dislocations decreases by a factor of 7, which (2016). Δ ln v corresponds to a stress-velocity dependence 15. 24. Weygand, D., Mrovec, M., Hochrainer, T. & Gumbsch, P. Multiscale Δ ln τ TEM images are showing the screw dislocation coupling phenomenon in Fe and simulation of plasticity in bcc metals. Annu. Rev. Mater. Res. 45, 369–390 Nb at T = 95 K are shown in Supplementary Figs. 1, 2. The observations conﬁrm (2015). that the coupling is a mechanism present in crystal structures, where (screw) 25. Kraych, A. et al. Non-glide effects and dislocation core ﬁelds in BCC metals. dislocation have to overcome a barrier by the kink-pair mechanism in order npj Comput. Mater. 5, 109 (2019). to glide. 26. Marichal, C. et al. Origin of anomalous slip in tungsten. Phys. Rev. Lett. 113, 25501 (2014). 27. Brunner, D. Temperature dependence of the plastic ﬂow of high-purity Data availability tungsten single crystals. Int. J. Mater. Res. 101, 1003–1013 (2010). The datasets generated during or analyzed during the current study are available from 28. Brunner, D. Stress-relaxation tests in the work-hardening regime of tungsten the corresponding author on reasonable request. single crystals below 300K. Mater. Sci. Eng. A 483–484, 525–528 (2008). 29. Schmitt, N. J. Experimentelle Untersuchung des Verformungs- und Code availability Bruchverhaltens von ein-, bi- und polykristallinem Wolfram auf der Mikroskala Access to the code will be provided at the host institution of the corresponding author (KIT, Karlsruhe, 2016). https://doi.org/10.5445/IR/1000062471 upon reasonable request. The occurrence of the coupling mechanism does not depend on 30. Srivastava, K., Gröger, R., Weygand, D. & Gumbsch, P. Dislocation motion in code details. The basic principles are described under methods. tungsten: atomistic input to discrete dislocation simulations. Int. J. Plast. 47, 126–142 (2013). 31. Madec, R. & Kubin, L. P. Second-order junctions and strain hardening in bcc Received: 7 February 2020; Accepted: 14 September 2020; and fcc crystals. Scr. Mater. 58, 767–770 (2008). 32. Queyreau, S., Monnet, G. & Devincre, B. Slip systems interactions in α-iron determined by dislocation dynamics simulations. Int. J. Plast. 25, 361–377 (2009). 33. Franciosi, P. The concepts of latent hardening and strain hardening in metallic single crystals. Acta Met. 33, 1601–1612 (1985). References 34. Madec, R., Devincre, B., Kubin, L., Hoc, T. & Rodney, D. The role of collinear 1. Hirth, J. P. & Lothe, J. Theory of Dislocations (Wiley, 1992). interaction in dislocation-induced hardening. Science 301, 1879–1882 (2003). NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18774-1 35. Rice, J. R. R. Inelastic constitutive relations for solids: an internal-variable Author contributions theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 K.S. and D.W. conceived the simulation tools and conducted the simulations and ana- (1971). lyzed and interpreted the data; wrote and edited the paper. D.C. conceived, designed, 36. Asaro, R. J. Micromechanics of crystals and polycrystals. Adv. Appl. Mech. 23, conducted and analyzed the experiments; wrote and edited the paper. P.G. provided the 1–115 (1983). sample; contributed to the conception of the work; interpretation of the data, wrote and 37. Wu, T.-Y., Bassani, J. L. & Laird, C. Latent hardening in single crystals I. edited the paper. theory and experiments. Proc. R. Soc. A Math. Phys. Eng. Sci. 435,1–19 (1991). Funding 38. Wei, Q. & Kecskes, L. J. J. Effect of low-temperature rolling on the Open Access funding enabled and organized by Projekt DEAL. tensile behavior of commercially pure tungsten. Mater. Sci. Eng. A 491,62–69 (2008). 39. He, B. B. et al. High dislocation density-induced large ductility in deformed Competing interests and partitioned steels. Science 357, 1029–1032 (2017). The authors declare no competing interests. 40. Reiser, J. et al. Ductilisation of tungsten (W): on the increase of strength and room-temperature tensile ductility through cold-rolling. Int. J. Refract. Met. Additional information Hard Mater. 64, 261–278 (2017). Supplementary information is available for this paper at https://doi.org/10.1038/s41467- 41. Bonnekoh, C. et al. The brittle-to-ductile transition in cold rolled tungsten 020-18774-1. plates: impact of crystallographic texture, grain size and dislocation density on the transition temperature. Int. J. Refract. Met. Hard Mater. 78, 146–163 Correspondence and requests for materials should be addressed to D.W. (2019). 42. Srivastava, K. Atomistically-informed discrete dislocation dynamics modeling of Peer review information Nature Communications thanks Brady Butler and the other plastic ﬂow in body-centered cubic metals (Karlsruhe Institute of Technology, anonymous reviewer(s) for their contribution to the peer review of this work. Peer 2014). https://doi.org/10.5445/IR/1000042367 reviewer reports are available. 43. Tang, M., Kubin, L. P. & Canova, G. R. Dislocation mobility and the mechanical response of b.c.c. single crystals: a mesoscopic approach. Acta Reprints and permission information is available at http://www.nature.com/reprints Mater. 46, 3221–3235 (1998). 44. Chaussidon, J., Robertson, C., Rodney, D. & Fivel, M. Dislocation dynamics Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in simulations of plasticity in Fe laths at low temperature. Acta Mater. 56, published maps and institutional afﬁliations. 5466–5476 (2008). 45. Po, G. et al. A. phenomenological dislocation mobility law for bcc metals. Acta Mater. 119, 123–135 (2016). 46. Caillard, D. Kinetics of dislocations in pure Fe. Part I. In situ Open Access This article is licensed under a Creative Commons straining experiments at room temperature. Acta Mater. 58, 3493–3503 Attribution 4.0 International License, which permits use, sharing, (2010). adaptation, distribution and reproduction in any medium or format, as long as you give 47. Brunner, D. & Glebovsky, V. Analysis of ﬂow-stress measurements of high- appropriate credit to the original author(s) and the source, provide a link to the Creative purity tungsten single crystals. Mater. Lett. 44, 144–152 (2000). Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory Acknowledgements regulation or exceeds the permitted use, you will need to obtain permission directly from P.G. acknowledges ﬁnancial support from German Research Foundation (DFG) through the copyright holder. To view a copy of this license, visit http://creativecommons.org/ project grant Gu367/30. D.W. acknowledges ﬁnancial support for the research group licenses/by/4.0/. FOR1650 Dislocation based plasticity funded by the German Research Foundation (DFG) under contract numbers WE3544/5-2. We acknowledge support by the KIT- Publication Fund of the Karlsruhe Institute of Technology. © The Author(s) 2020 8 NATURE COMMUNICATIONS | (2020) 11:5098 | https://doi.org/10.1038/s41467-020-18774-1 | www.nature.com/naturecommunications

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Repulsion leads to coupled dislocation motion and extended work hardening in bcc metals

Repulsion leads to coupled dislocation motion and extended work hardening in bcc metals

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Repulsion leads to coupled dislocation motion and extended work hardening in bcc metals

Repulsion leads to coupled dislocation motion and extended work hardening in bcc metals

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