Los Alamos National Laboratory, Los Alamos NM 87545, USA (Received 22 December 2008; final version received 3 January 2009)

Transcription

1 Philosophical Magazine Vol. 89, No. 8, 11 March 2009, Molecular dynamic studies of the interaction of a/6h112i Shockley dislocations with stacking fault tetrahedra in copper. Part II: Intersection of stacking fault tetrahedra by moving twin boundaries M. Niewczas* and R.G. Hoagland Los Alamos National Laboratory, Los Alamos NM 87545, USA (Received 22 December 2008; final version received 3 January 2009) The interactions of moving twin boundaries with stacking fault tetrahedra (SFTs) have been studied by molecular dynamics. The results reveal a spectrum of processes occurring during these interactions. In general, they lead to damage of the parent SFT and formation of new defects in the twin lattice. The character of these defects depends on the nature of the twinning front, the size of the SFT and its orientation with respect to incoming dislocations. Typical structures that may be produced in the twin include product-sfts, free vacancies, planar stacking faults bounded by partial dislocations, mutually linked stacking faults on non-coplanar {111} T planes, small {111} T tetrahedra and their partial forms. Dislocation mechanisms involved in the formation of these defects are being analyzed. Keywords: stacking fault tetrahedra; Shockley dislocation; twinning transformation; atomic structure; defect structures; deformation twinning; dislocation interactions; lattice defects; molecular dynamic simulations; strengthening mechanisms; vacancies 1. Introduction Electron microscopy observations of the structure of irradiated and plastically deformed austenitic steels, available in the literature, reveal the presence of narrow, defect-free twins [1]. The results indicate that defect-free channels are formed inside the mechanical twins which propagate through an irradiated defect substructure. In fcc crystals, mechanical twinning is accomplished by movement of a/6h112i Shockley partial dislocations. The growth of a macroscopic twin requires that many layers of the parent crystal are swept by twinning dislocations, which interact with every element of the pre-existing substructure transforming it into a new configuration in the twin lattice [2,3]. The fundamental dislocation mechanism involved in the formation of defect free-zones during mechanical twinning must, therefore, differ from that of glide dislocations. The latter processes have been the focus of extensive studies [4 20], but the former have rarely been considered in literature. *Corresponding author. niewczas@mcmaster.ca ISSN print/issn online ß 2009 Taylor & Francis DOI: /

2 728 M. Niewczas and R.G. Hoagland Twinning occurs in various crystal systems. Serra et al. [21] have recently presented molecular dynamic studies of the interaction of a f10 12g incoherent twinning boundary with preexisting clusters of self-interstitial atoms and vacancies in hcp -zirconium. The data showed various interactions and end-products depending upon the initial orientation and location of the point defects with respect to the propagating interface. Some of these interactions include hardening effects induced by defect clusters, cluster drag due to the moving interface, total or partial absorption of clusters by the twin boundary, and the transformation of the clusters to a new orientation and shape inside the twin lattice. It was concluded that f10 12g twin boundaries act as defect sinks or recombination centers, which facilitate removal of radiation defects from regions of radiation damage. In part I of this work [22], we presented molecular dynamic studies of the interaction of Shockley dislocations in pure edge orientation with stacking fault tetrahedra in copper, and analyzed defects derived from homogeneously sheared SFTs. From these studies it was inferred that SFTs distorted by a twinning shear are unstable and transform to structures containing small product SFTs, rows of vacancies and secondary stacking faults bounded by partial dislocations. Strength of SFT as an obstacle for mobile Shockley dislocations depends upon intersection geometry and the cutting plane. An isolated Shockley may propagate through the SFT by shearing the apex, formation of a residual loop or through the reactions with dislocations at the tetrahedron base. The reactions at the base appear to be critical to defect stability. They involve formation of two glissile dislocations, which may glide, removing part of the stacking faults on tetrahedron faces and, thus, converting it to a less effective obstacle. The present work is a continuation of these studies and deals with the intersection of SFTs by moving twin boundaries during twin growth. Ledge-type interfaces and lenticular-type twin boundaries are considered here. 2. Computational method The computational procedure has been described in part I [22]. Simulations were carried out using a rectangular computational cell with dimensions nm containing 166,600 atoms and x, y, and z axes parallel to [121], ½111Š and ½101Š. Stacking fault tetrahedra (SFTs) in the size range nm, introduced to the lattice, were subjected to interactions with lenticular twins and with twins growing by ledge mechanism. Two distinct interaction geometries were considered. In edge-on interaction, the stacking fault tetrahedron has its apex-up and with respect to the positive direction of the z axis (Figure 1 in part I), a moving twin boundary intersects the defect from the right hand side and interacts first with the AD edge of the tetrahedron. In face-on interaction, the apex of the stacking fault tetrahedron points down and the twin boundary interacts first with the face of the defect. A vector field analysis of the atomic displacements in the dislocated lattice relative to the perfect crystal was used to study the details of the twin SFT intersections at the atomic level [23]. As in part I, we used the Hirth and Lothe convention for indexing crystallographic planes and directions [24]; the corners A, B, C, D of our basic Thomson tetrahedron correspond to the D, C, B, A corners of Hirth and Lothe standard tetrahedron.

3 Philosophical Magazine 729 (a) (b) Figure 1. Modeling of the interaction of a twin growing by a ledge mechanism and a SFT. (a) The ledge is formed by a single Shockley dislocation gliding along the matrix twin interface on ð1 11Þ plane. (b) The ledge is formed by two Shockley dislocation on opposite sides of the SFT interacting with the defect under the applied stress (see text for details). 3. Results 3.1. Interaction between twins growing by the ledge mechanism and SFT The interaction of SFT with twins growing by the ledge mechanism has been modeled using similar methods to those outlined in Section 4.2 of part I. In the first approach, a single Shockley dislocation formed a ledge one atomic layer high and, by applying shear stress to the lattice, was forced to intercept the SFT sequentially, plane after plane, as shown schematically in Figure 1a. In the second method, two Shockley dislocations of opposite-sign were forced to interact with the SFT from both sides sequentially, plane after plane, along the defect height (Figure 1b). As before, both methods yield the same result,

4 730 M. Niewczas and R.G. Hoagland but the second approach required lower applied stress to accomplish the reaction between the propagating ledge and the stacking fault tetrahedron, due to the attraction between the two opposite-sign Shockley dislocations. Depending on the size of the tetrahedron and its orientation with respect to the growing twin, different defects are produced. These are considered below for a twin growing from the bottom to the top part of the lattice Apex-up configuration Figure 2a shows disregistry analysis of a model containing a small stacking fault tetrahedron, revealing details of the atomic movements on ð111þ planes during twin growth. The crystal containing the SFT was used as a reference lattice in calculating the (a) layer 4 layer 3 layer 2 layer 1 layer 0 (base) layer 1 Figure 2. (Color online). Formation of a truncated SFT as a result of the interaction of the twin, growing by a ledge mechanism with an SFT in the apex-up configuration containing 36 vacancies. (a) Disregistry analysis of the ð1 11Þ layers during twin growth. (b) Structure of the SFT partially incorporated into the twin lattice, revealed by the common neighbor method. (c) Structure of the SFT, restored in the twin, after the collapse of the apex, revealed by the common neighbor method. (d) Crystallographic relationship between the parent SFT (ABCD) and the truncated product (A T B T C T ) formed inside the twin lattice.

5 Philosophical Magazine 731 (b) parent (c) parent twin twin (d) Figure 2. Continued. disregistry plot. When the twin ledge coincides with the base of the tetrahedron (Figure 1a), the Shockley dislocation reacts with the stair-rods along the edges AB, AC and BC. These reactions, discussed in Section of part I, yield new a/6[211] and a/6[112] Shockley partials along the AB and AC edges and a/6[101] and a/6[121] dislocations along the BC edge of the tetrahedron. An arrow-free triangular loop, visible in Figure 2a (column 1, base layer), indicates that neither the Shockley dislocation nor the other dislocation, formed as a result of the dislocation reactions, penetrate the area inside the tetrahedron. The layer below the base plane (column 1, layer 1) is filled up with the same vectors, indicating that the Shockley dislocation has previously swept through this plane. Finally, it is seen that the layer above the base plane shows almost no atomic movements, indicating that no dislocation traversed this plane. When a Shockley dislocation interacts with a tetrahedron on the adjacent plane above its base (layer 1, column 2), the dislocation penetrates the area inside the defect. This process is associated with significant rearrangement of atoms within the base of the tetrahedron (base layer, column 2). As a result, a new stacking fault tetrahedron-like defect is being restored in the twinned part of the crystal. During further thickening of the twin, Shockley partials propagating on layer 2 then on layer 3 produce stable loops around the stacking fault tetrahedron (column 3 and column 4 in Figure 2a). When the next Shockley dislocation moves on the plane above and intercepts the tetrahedron, the remaining apex part of the defect located in the parent crystal collapses, transforming it into a truncated stacking fault tetrahedron, which remains stable in the twin lattice. The newly formed stacking fault tetrahedron is rotated 60 around the ½1 11Š axis normal to the twinning plane with respect to the parent defect from which it originated. Figures 2b and c show the

6 732 M. Niewczas and R.G. Hoagland configuration of the stacking fault tetrahedron before and after collapse, as obtained by the common neighbor method. Detailed analysis of the internal structure of the truncated stacking fault tetrahedron reveals the presence of vacancies inside the defect, most likely introduced during apex collapse. The diagrammatic representation illustrating the crystallographic relationship of the parent stacking fault tetrahedron (ABCD) and its truncated product (A T B T C T ) is shown in Figure 2d. Twins growing by the ledge mechanism cause extensive damage to larger SFTs in ABCD configuration and yield defects different from those discussed above. Figure 3 shows the structure of a stacking fault tetrahedron initially containing 300 vacancies at various stages of twin growth. Figure 3a corresponds to the stage where the twin extends from the bottom part of the lattice up to the plane containing the base of the stacking fault tetrahedron, marked by arrows. After reacting with the stair-rod dislocations along the AB and AC edges of the tetrahedron, the twinning partial bows around the tetrahedron (Figure 3b). In the next step, the oppositely signed bowed sections of the propagating Shockley are annihilated, thus restoring a segment of the Shockley partial between the B and C nodes reacting with the stair-rod along the BC edge and the continuous dislocation line, which propagates further on its glide plane (Figure 3c). Partial dislocations a/6[211] and a/6[112], formed in reactions between the propagating Shockley ledge and the stair-rods along AB and AC, glide on ABD and ACD faces, removing part of the fault area from these planes together with the segment of the stair-rod dislocation along AD (Figure 3d). This, in effect, opens the tetrahedron and makes it easier for subsequent movement of the ledge on adjacent planes. As the twin thickens, the part of the tetrahedron incorporated in the twin lattice is being destroyed and the stacking fault tetrahedron is transformed into an object consisting of dislocations aligned along the former BC edge and stacking faults on four {111} T planes bounded by appropriate partial dislocations, a process forming mutually linked defect structures inside the twin. Figure 4a shows progress in this transformation, after the twin has thickened by a few ð1 11Þ layers. The position of the matrix twin interface is marked by arrows in Figure 4a. Figures 4b and c show a perspective view on the parent tetrahedron in the matrix and the defects restored in the twin lattice, revealing development of the imperfections mentioned above. Relevant crystallographic planes in parent and twin parts of the stacking fault tetrahedron are marked on the figure Apex-down configuration The interaction of the growing twin with an apex-down stacking fault tetrahedron yields structures different from those observed previously. Disregistry analysis shows that the propagating ledge shears the stacking fault tetrahedron atoms, starting from the first layer containing the apex atom of the defect. Consequently, the stacking fault tetrahedron becomes penetrable by the ledge moving on the adjacent plane, shearing the atoms on this layer to their new positions. This process occurs in tetrahedra of all sizes. As the twin expands and the propagating Shockley intercepts the parent stacking fault tetrahedron on adjacent layers, a new tetrahedron-like defect is restored in the twin lattice. This is illustrated in Figure 5, showing perspective views of two stacking fault tetrahedra, the first containing 36 vacancies (Figures 5a and b) almost fully incorporated into the twin lattice, and the second, larger tetrahedron with 300 vacancies (Figures 5c and d) with its apex in the twin. The atoms in Figure 5 are displayed according to their potential energy, and only

7 Philosophical Magazine 733 (a) (b) (c) Figure 3. (Color online). Interaction of a twin growing by a ledge mechanism with a SFT containing 300 vacancies in apex-up configuration. (a) Snapshot of the simulation cell at the stage where the matrix twin interface coincides with the base of the SFT. (b) Top view of the SFT and the passing Shockley partial bowing out around the defect. (c) Top view of the SFT a few picoseconds after the Shockley partial wound around, reacted with the stair-rod along the BC edge, and restored the continuity of the propagating ledge. (d) Perspective view of the SFT after reaction of the Shockley ledge with the stair-rod dislocations at the base. a/6[211] and a/6[112] partials formed along AB and AC edges glide on ABD and ACD faces, removing a large part of the stacking fault from these planes., (d) higher energy atoms are imaged (see energy scale enclosed). The twinned part of the parent stacking fault tetrahedron constitutes a new tetrahedron with faces extended on three available {100} T planes and its base in the matrix twin interface (ð1 11Þ plane). The {100} T stacking fault tetrahedron is stable only to a height of five ð1 11Þ layers, including its base. As the twin intercepts a stacking fault tetrahedron on the adjacent plane

8 734 M. Niewczas and R.G. Hoagland (b) (a) (c) Figure 4. (Color online). Transformation of the parent SFT by a ledge-growing twin. (a) Snapshot of the lattice after partial transformation of the parent SFT; position of the interface is shown by arrows. (b) and (c) Perspective views of the partially transformed SFT in (a), showing the parent part of the defect and the defects restored in the twin lattice; all atoms are displayed according to their potential energy. beyond its fifth layer, the {100} tetrahedron collapses. This process occurs by nucleating a partial dislocation loop on ð1 1 1Þ T associated with the removal of the entire stacking fault from (010) T and part of the faults from ð100þ T and ð001þ T. As a result, the {100} T tetrahedron transforms to a defect consisting of a partial dislocation loop on ð1 1 1Þ T and a partial tetrahedron with faces on ð100þ T and ð001þ T, bordered by a stacking fault on ð1 1 1Þ T and the matrix twin interface on ð111þ T (Figures 6a and b). Geometrically, the stacking fault tetrahedron part of the total defect is similar to the distorted stacking fault tetrahedron produced by the homogeneous twin shear shown in Figure 2d and discussed in Section 3.2 of part I. During further twin growth, the SFT in the parent is successively being destroyed, supplying vacancies to the defects formed in the twin lattice. The SFT part with faces on ð100þ T and ð001þ T is continuously restored, preserving the constant height of about four or

9 Philosophical Magazine 735 (a) (b) (c) Figure 5. (Color online). Structure of the SFTs restored in the twin lattice in apex-down configurations after interaction with the twin growing by a ledge mechanism, displayed according to the potential energy of the atoms. (a) and (b) Two views of a{100} T SFT formed in the twin lattice from the parent defect containing initially 36 vacancies. (c) and (d) Two views of the SFT containing initially 300 vacancies with its apex part incorporated into the twin as a {100} T SFT. five layers during twin growth. New vacancies supplied by the parent SFT are absorbed by the stacking fault on ð 1 1 1Þ T allowing the partial dislocation loop to expand and increase the fault area (Figures 6c and d). (d) 3.2. Interaction of a lenticular twin with SFT These interactions have been modeled by allowing Shockley dislocations to intercept the stacking fault tetrahedron sequentially on ð111þ planes so that it mimics the cutting of the defect by a lenticular-shaped twining front consisting of a leading Shockley dislocation with other dislocations gliding behind on adjacent planes. In practice, these simulations were carried out by placing the first Shockley partial such that it intercepted the stacking fault tetrahedron near the middle part of the defect. After this dislocation passed, the next Shockley was introduced on an adjacent plane. This was repeated for the third, fourth and, if necessary, more dislocations. The order in which twinning dislocations interacted with

10 736 M. Niewczas and R.G. Hoagland (a) (b) tetrahedron (c) Figure 6. (Color online). Destruction of a large SFT in apex-down configuration by a growing twin. (a) and (b) Different views of the parent SFT partially incorporated into the twin lattice, showing collapse of the {100} T tetrahedron to a structure consisting of a partial dislocation loop on ð111þ T and a partial tetrahedron with faces on ð 100Þ T and (001) T. (c) and (d) Views of the parent SFT during further twin growth, showing continuous destruction of the parent SFT and growth of the partial dislocation loop on (111) T in the twin lattice. the stacking fault tetrahedron is shown in Figure 7a for the particular case of a stacking fault tetrahedron of edge length of 2.3 nm, containing 36 vacancies. Figure 7b shows disregistry plots of the atoms on ð1 11Þ layers during different stages of twin growth, obtained with the reference lattice containing 36 vacancies SFT. The analysis reveals that Shockley partials gliding on layers 3, 4 and 2, respectively, formed stable loops around the stacking fault tetrahedron (column 1, Figure 7b). The next twinning dislocation, intersecting the stacking fault tetrahedron on layer 1, also forms a loop, but the stress concentration, which developed as a result of loop stacking, induced collapse of the upper loop on layer 4 and reshuffling of the atoms inside the tetrahedron (column 2, Figure 7b). The next Shockley dislocation gliding on layer 5 cut the tetrahedron through, but the defect still upheld stable loops on layers 1, 2 and 3 (column 3, Figure 7b). When a subsequent twinning dislocation arrived on the plane coinciding with the base plane of the tetrahedron (layer 0), the reactions occurring between incoming Shockley partials and stair-rods destabilized the previously built loop structure on layer 1, 2 and 3, eventually (d)

11 Philosophical Magazine 737 (a) (b) layer 6 layer 5 layer 4 layer 3 layer 2 layer 1 layer 0 (base) Figure 7. (Color online). Transformation of the parent SFT of an edge length of 2.3 nm during interaction with a lenticular twin. (a) Diagram showing the order in which twinning dislocations intercept the defect. (b) Disregistry analysis of the ð1 11Þ layers during different stages of the twin SFT interaction. (c) SFTs restored in the twin lattice. (d) Two SFTs with an edge length of 1.5 nm sharing a common edge; a truncated SFT with {100} T faces shares its base with one SFT. allowing penetration of the base of the stacking fault tetrahedron, utilizing new dislocation loops nucleating inside the defect. Subsequent Shockley dislocation of the lenticular twinning front arriving on layer 6 (column 4, Figure 7b) induced further atomic reshuffling along the glide plane, leading to restoration of a defect inside the twinned lattice, as shown

12 738 M. Niewczas and R.G. Hoagland (c) (d) Figure 7. Continued. in Figure 7c. The new defect formed consists of two full SFTs with edge length of approximately 1.5 nm, sharing a common edge and a smaller truncated tetrahedron on {100} T, sharing a base with another tetrahedron. Figure 7d shows the crystallographic structure of three tetrahedra and their relationship to their parent stacking fault tetrahedron. It is interesting to note that atomic rearrangements taking place in the crystal lattice, leading to the restoration of new SFTs in the twin, occur by dislocation mechanisms, as will be discussed later in this work. Interactions of non-coherent twinning fronts, containing different numbers of twinning dislocations with SFT, have also been examined. Discussion of these processes is beyond the scope of the present work and will be the subject of a separate publication. 4. Discussion 4.1. Interactions of lenticular twin and SFTs With the exception of possibly the first two apex layers, a SFT appears to be a strong impenetrable obstacle to propagating Shockley dislocations, at least under the stresses considered here. The twinning partial passes the defect by the Orowan mechanism, leaving a residual loop around it. In contrast to the reactions occurring at the base of an SFT, the formation of a loop is reversible and the defect recovers its original structure after the loop disappears. The strength of the SFT depends on its cross-section; it presents a stronger obstacle for face-on (apex-down) than for edge-on (apex-up) interaction. This reflects the combined effect of elastic interactions between gliding Shockley dislocation and stair-rods, and formation of bow-out configuration during the passing event. The stress concentration produced by a single loop is not sufficient to shear the SFT, in agreement with other results in the literature [13,17,19]. It has been observed that an SFT is able to support a few partial loops stacked one on top of the other; the exact number of loops supported depends on the cross-section of the SFT and the applied stress. For small SFTs, we have observed that the tetrahedron is stable under the stress produced by three partial loops and an external stress of a few hundred MPa. The addition of a fourth loop usually promotes either tunneling of one loop through the smallest cross-section of the defect or the collapse of all loops (e.g. Figure 7b). Collapse of the loops appears to be critical for the stability of the SFT because it initiates decomposition and transformation

13 Philosophical Magazine 739 (a) distance along Z (Å) (b) distance along X (Å) Figure 8. (Color online). Analysis of atomic movements along the basal plane after the collapse of loops for an SFT with 36 vacancies. (a) Disregistry vector plot showing the domains produced by new dislocations nucleating inside the defect on the basal plane. (b) Diagrammatic representation of the domains in (a). of the defect to a new structure in the twin lattice in the form of a smaller size product SFTs on {111} T and partial SFT on {100} T (Figure 7a). At the critical point of the transformation, collapse of the loops occurs by a mechanism that relies on new dislocations nucleated along the cut plane inside the SFT and reactions between these dislocations (Figure 7b). In the case of the collapse of a 36-vacancy SFT, the most pronounced atom reshuffling occurs on the basal plane of the defect; therefore, in the following we discuss this process in greater detail. Figure 8a shows disregistry analysis along the basal plane of the parent SFT after collapse of loops. Six different domains, characterized by their individual shear vectors, can be clearly differentiated. The disregistry plot was obtained with reference to a parent lattice containing a perfect SFT; the actual (effective) shear vector of a given domain is obtained by addition of a domain vector from the disregistry plot in Figure 8a and a shear vector of the reference lattice, i.e. a/6[121] M. Indexing with respect to the matrix coordinates, one obtains the following shear vectors for a given domain: a 6 ½121Š M þ 0 ¼ a 6 ½121Š M in domain I a 6 ½121Š M þ a 6 ½ 112Š M ¼ a 2 ½011Š M in domain II a 6 ½121Š M þ a 6 ½242Š M ¼ a 6 ½363Š M in domain III a 6 ½121Š M þ a 6 ½21 1Š M ¼ a 2 ½110Š M in domain IV

14 740 M. Niewczas and R.G. Hoagland a 6 ½121Š M þ a 6 ½330Š M ¼ a 6 ½451Š M in domain V a 6 ½121Š M þ a 6 ½033Š M ¼ a 6 ½154Š M in domain VI The domains define the areas swept by one or more dislocations of appropriate Burgers vector nucleated during loop collapse. The dislocation lines are located at the domain border (Figure 8b) and their Burgers vectors can be obtained by considering shear vectors of neighboring domains. Although detailed analysis of this dislocation network is beyond the scope of this work, by inspecting the shear vectors of domains in Figure 8b, one can conclude that both partial and perfect dislocations participate in the transformation and in restoring the new product SFTs in the twin lattice, the dislocation reactions involved, however, are quite complex Transformation of the SFTs during thickening of a twin It was shown in Section 3.1 that a twin growing by the ledge mechanism introduces substantial damage to the SFT structure. For a fixed direction of twin growth, the transformation products are different for the apex-up and apex-down SFT configurations. In the following, we discuss these two interaction geometries in more details Apex-up configuration Figure 9 shows disregistry plots for a partially transformed SFT, discussed previously with reference to Figures 4a c, along six Shockley dislocation ð111þ glide planes. The plots show that Shockley dislocations react with stair-rods along the base edges of the parent SFT and do not enter into the interior of the defect (Figure 9a). After these reactions have taken place, newly formed Shockley dislocations move slightly in the ABD and ACD faces of the SFT, removing part of the stacking faults on these planes and part of the stair-rod dislocation along the AD edge (Figure 4). Thus, the defect becomes partially open for twinning dislocations gliding on adjacent planes, which penetrate it and produce homogeneous stacking faults along their glide planes (Figures 9b d). Figure 9e shows the stage of the twin growth where the position of the matrix twin interface coincides with the fourth plane above the basal plane of the defect. The disregistry graph (Figure 9e) reveals both the penetration of the SFT and the formation of extended nodes visible as two untransformed islands in the corners of the SFT at the BCD face of the tetrahedron. The disregistry plot also reveals that, in the plane above the interface (Figure 9f), some reshuffling of the atoms occurs inside the SFT ahead of the growing twin, preceding the penetration of the defect by the propagating ledge. The growing twin transforms BCD, the K 2 plane of the parent tetrahedron, to K T 2. The main aim of this part of the discussion is to determine the Burgers vector of the dislocations bordering the stacking fault on the K T 2 plane in the twin lattice. Figure 10 shows disregistry analysis of the screw and edge components of the Burgers vector of dislocations at the bottom plane of the partially transformed SFT shown in Figure 4. The disregistry was determined along the ð111þ T plane, coinciding with the base of the transformed defect, and also along (010) T at the intersection of the basal plane and

15 Philosophical Magazine 741 distance along Z (Å) distance along Z (Å) distance along Z (Å) distance along Z (Å) distance along X (Å) distance along X (Å) distance along X (Å) distance along X (Å) the K T 2 plane of the new defect in the twin. The graph clearly reveals the presence of stairrod and Shockley dislocations in pure-edge orientations (disregistry 0.85 and 1.48 A ) with exact positions at x ¼ 27 nm and x ¼ 43 nm shown by vertical arrows. It can be seen that the data closely match the disregistry vector plot shown in Figure 9a, where these two dislocations are also visible. Disregistry along the (010) T plane also reveals the presence of an edge dislocation with a disregistry value of about 1 Å, which arises from projection of the Burgers vector of Shockley dislocations entering the node A onto (010) T (see, for example, Figure 4). An important implication arising from the above analysis is that a a/3[111] M Frank dislocation, bordering an intrinsic stacking fault on K 2 (111) M, does not change its character after the twinning transformation, i.e. it inherits the a/3[111] T Burgers vector and an intrinsic stacking fault on the K T 2 (111) T plane. It follows that the fault-like structure distance along Z (Å) distance along Z (Å) distance along X (Å) distance along X (Å) Figure 9. (Color online). Disregistry vector analysis along six ð111þ T glide planes of a partial dislocation traversing the defect. The glide planes are marked as 0, þ1, þ2 planes, etc. on the bottom left corners of the figures; the 0 plane defines the base of the SFT.

16 742 M. Niewczas and R.G. Hoagland Figure 10. Edge and screw components of the disregistry for the partially transformed SFT shown in Figures 4 and 9. The disregistry is plotted along the ð111þ T plane inside the twin lattice corresponding to the basal plane of the parent SFT and along ð010þ T at the intersection of former basal plane of the SFT and the K T 2 plane of the transformed defect. The graph reveals the presence of stair-rod and Shockley dislocations in pure-edge orientation with exact positions indicated by vertical arrows corresponding to the model coordinates x ¼ 27 nm and x ¼ 43 nm, respectively. Note that the data match exactly the disregistry vector plot shown in Figure 9a, where these two dislocations are also visible. restored inside the twin lattice is bounded by stair-rod a/6h100i T dislocations at the intersection of {111} T planes, and a/6h112i T Shockley dislocations gliding away and extending the stacking faults on appropriate {111} T. The above analysis suggests the following picture of the interaction of a ledge-growing twin, with SFTs in the apex-up configuration depicted diagrammatically in Figure 11. The process is the same in the case of a twin growing from the top to the bottom of an apex-down tetrahedron. The transformation of the SFT begins with reactions between the propagating ledge and the stair-rod dislocations at the base of the tetrahedron. The stairrod dislocations along the AB, AC and BC edges are replaced with a/6[211] M, a/6[112] M and a/3[111] M partials, the first two dislocations are mobile in ABD and ACD faces (Figure 11a). The Frank dislocation dissociates immediately into partial dislocations (Figures 9a and 10), this detail is omitted in Figure 11 for clarity. As the twin thickens and the new ledge propagates on adjacent planes, a/6[211] M and a/6[112] M Shockley partials are pressed against edges BD and CD and react with the stair-rods according to the equations: a 6 ½211Š M þ a 6 ½011Š M ¼ a 3 ½111Š M along BD a 6 ½112Š M þ a 6 ½110Š M ¼ a 3 ½111Š M along CD:

17 Philosophical Magazine 743 Figure 11. Diagrammatic representation of the interactions of a ledge-type twinning front with the SFT in apex-up configuration. (a) a/6[121] ledge dislocation reacts with stair-rods dislocations in the tetrahedron base, forming new a/6[211] and a/6[112] Shockley and a/3 [111] Frank dislocation. (b) Ledge gliding on adjacent plane forces a/6[211] and a/6[112] Shockleys to react with segments of the stair-rods along BD and CD edges. (c) Shockley loop is closing behind the SFT. (d) Configuration of the partially transformed SFT after the twin has grown by a few ð1 11Þ layers. This restores a Frank dislocation along BC and part of the BD and CD edges incorporated into the twin lattice, indexed with respect to twin coordinates as a/3[111] T (Figure 11b). The propagating ledge is held up against the stacking fault on the BCD face of the SFT between nodes N 1 and N 2 (Figure 11c), but the arms of the twinning dislocation move outside the nodes, wind around the SFT and are partially annihilated. This restores the continuity of the dislocation line at the propagating ledge and produces a segment of Shockley dislocation of opposite sign on the other side of the stacking fault on the BCD face between nodes N 1 and N 2. Two opposite Shockley dislocations between nodes N 1 and N 2 annihilate immediately, producing a homogenous interface along the glide plane of the ledge. This elementary process is repeated on every plane during thickening of the twin, resulting in the transformation of the stacking fault from K 2 (111) M onto the K T 2 (111) T plane and incorporating bordering Frank dislocation into the twin lattice (Figure 9d). If such a structure lowers its energy by developing extended nodes N 1 and N 2 at the interface (Figure 9e), and if the segments of Frank dislocation dissociate inside the twin on the new {111} T planes, the resulting structure will consist of mutually linked stacking faults on ð1 11Þ, ð111þ T, ð1 11Þ T and ð1 1 1Þ T bounded by a/6h110i T stair-rod dislocations at the plane intersections and a/6h112i T Shockley partials gliding away and extending the stacking faults, as shown in Figures 4b and c. Further details of the transformation of Frank loops on the K 2 plane are available elsewhere [23].

18 744 M. Niewczas and R.G. Hoagland Apex-down configuration When intersected at its apex, a SFT represents a weak obstacle for propagating Shockley dislocations. During the initial stage of twin growth, the tetrahedron is sheared by moving ledges, resulting in displacement of all its atoms to their new positions. However, the sheared apex is not transformed to the distorted configuration as one might expect during homogeneous shear, but to a stacking fault tetrahedron extended on {100} T planes (Section and Figure 5). Figure 12 shows schematically the geometrical relations between an ABCD parent tetrahedron, an apex part IJKD as it would be if distorted by homogeneous shear, and the actual {100} T tetrahedron, IGKL formed in the twin lattice. The restored IGKL tetrahedron has exactly half of the volume of the distorted tetrahedron IJKD produced by homogeneous shear and, therefore, lower specific energy per unit area. The formation of this defect is associated with the transport of excess vacancies along the propagating ledges to other places in the lattice. As the twin grows, a=6½110š M, a=6½011š M and a=6½101š M segments of the stair-rod dislocations along ID, JD, KD edges of the parent tetrahedron are incorporated into the twin. They preserve their Burgers vector but change their line direction and are aligned along three h100i T, i.e. IL, JL, KL edges of the {100} T tetrahedron (Figure 12). As a result, they all inherit {110} T glide planes and, therefore, are sessile in the twin lattice. It is energetically unfavorable for a {100} T SFT to grow beyond a certain size, corresponding to a height of approximately four or five ð111þ layers. The collapse of this structure during subsequent thickening of the twin occurs abruptly by the nucleation of a partial dislocation loop bounding the stacking fault on (111) T, which accommodates all of the excess vacancies from the collapsed part of {100} T tetrahedron (Figure 6). Formation of such a structure is governed by core reactions and the energy of the system, and not by the principles of shear transformation predicted by geometrical approaches. Figure 12. Geometrical representation of an ABCD tetrahedron with its apex incorporated into the twin as f100g T SFT. (a) IJKD indicates the apex of the parent ABCD tetrahedron; IJKD is the apex distorted by a homogeneous twinning shear; IJKL is the apex incorporated into the twin as a f100g T tetrahedron. (b) Geometrical relations between the distorted IJKD tetrahedron and f100g T IJKL tetrahedron restored in the twin.

19 Philosophical Magazine Summary We have considered interactions of moving twin boundaries with stacking fault tetrahedra expected to occur during onset of twining deformation in copper. The results obtained reveal a spectrum of processes associated with these interactions, leading to damage of the parent SFTs followed by production of new defects in the twin lattice. The main results can be summarized as follows: (1) A twin growing by a ledge mechanism produces substantial damage to the SFTs. The products of these reactions depend upon the direction of twin growth with respect to the SFT, but in general they lead to production of qualitatively different defect structures in the twin lattice. (a) The transformation of a SFT by a twin growing from its base towards its apex results in damage to the parent defect and formation of a structure consisting of stacking faults on four {111} T planes bounded by a=6h110i T stair-rod and a=6h112i T Shockley dislocations. (b) Interaction of the same SFT with a twin growing in the opposite direction produces a residual {100} T tetrahedron attached to a planar stacking fault on {111} T. The interaction involves the following stages: (i) damage to the apex portion of the parent SFT, (ii) formation of a new {100} T tetrahedron, (iii) growth of the {100} T tetrahedron to a height of about four or five lattice planes, (iv) collapse of the {100} T tetrahedron, (v) nucleation of a partial dislocation loop on (111) T and its growth. The transport of excess vacancies occurs along the ledges of the propagating twin to the stacking fault on (111) T. (2) Interaction of lenticular twins with small-sized SFTs leads to formation of product SFTs in the twin lattice rotated by 60 with respect to the parent defect. During restoration of such a structure, the parent SFTs initially show a certain resistance to the propagating front, then a few partial dislocation loops become stacked up against the defect, causing its collapse. The reshuffling of atoms taking place during SFT collapse and the formation of new defects in the twin is facilitated by nucleation of new dislocation loops inside the SFT and their mutual reactions. Acknowledgements This work was supported by the LANL Directed Research and Development Program. MN would like to thank the Los Alamos National Laboratory for this financial support during sabbatical leave. We also appreciate the helpful discussions with Professor John Hirth. References [1] C. Bailat, F. Groschel and M. Victoria, J. Nucl. Mater. 276 (2000) p.283. [2] Z.S. Basinski, M.S. Szczerba, M. Niewczas et al., Rev. Metall. 94 (1997) p [3] M. Niewczas, Dislocations and twinning in face centered cubic crystals, indislocations in Solids, Vol. 13, F.R.N. Nabarro and J.P. Hirth eds., Elsevier/North-Holland, Amsterdam, 2007, p.263.

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