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The formation of stacking fault tetrahedra in Al and Cu
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Acta Materialia 59 (2011) 19–29 www.elsevier.com/locate/actamat
The formation of stacking fault tetrahedra in Al and Cu III. Growth by expanding ledges H. Wang a,⇑, D.S. Xu a, R. Yang a, P. Veyssie`re b a
Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, People’s Republic of China b LEM, CNRS-ONERA, BP 72, 92322 Chaˆtillon, France Received 10 February 2010; received in revised form 9 July 2010; accepted 27 July 2010 Available online 27 September 2010
Abstract Ledge expansion and the concomitant growth of a stacking fault tetrahedron (SFT) are investigated in Al and Cu by molecular dynamics (MD) simulations by addition of vacancy rods with selected lengths. Ledge expansion is largely governed by the site preference of vacancies on the SFT edges resulting in distinct stable ledged SFTs. Both edge- and corner-facing ledge configurations may be adopted. The growth of SFTs, especially large ones, is controlled by thermal agitation. The mobile part of the ledges consists of a dipole of Shockley partials generally oriented in the 60° mixed orientation that move in a thermally activated manner, reflecting a certain core reorganization of the Shockley dipole. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Stacking fault tetrahedron; Growth; Ledge mechanism; Molecular dynamics simulations; Shockley dipole
1. Introduction In Au essentially but also in other metals such as Ag, Ni and Cu and certain of their alloys stacking fault tetrahedron (SFT) edges amount to a few tens of nanometers [1–8]. After quenching and annealing edge lengths of the order of 100 nm, and thus containing at least 105 atoms, have been observed [9,10]. Whereas small sized tetrahedra may grow by successive absorption of vacancies at jogs on the tetrahedron faces [11–16] (see also Wang et al. Part II [26]), one may wonder if this mechanism is still operative for SFTs containing one to two orders of magnitude more vacancies than the several nanometer large SFTs generated by irradiation or plastic deformation [17,18]. As an alternative to the Silcox and Hirsch mechanism [1], SFT growth (and shrinkage) by expansion of ledges on SFT faces has been proposed [11,19], although little investigated. Experimental evidence is scarce [20]. Because the distance between certain segments of the ledges is of the order of the inner cut-off radius, SFT growth by ledge expansion cannot be studied by linear elas⇑ Corresponding author. Tel.: +86 24 23971946.
E-mail address: [email protected] (H. Wang).
ticity calculations of interactions between piecewise dislocations. Recently, ledges were observed by MD simulations of the growth of an initially defective triangular Frank loop [16,21] and upon SFT intersection by a moving dislocation [22–24]. An SFT, which is in general not perfect, may evolve in significantly distinct ways depending on where the next vacancy or vacancy cluster comes from. Several local configurational minima may be attained, such as the double SFT discussed in Wang et al. Parts I [25] and II [26]. Furthermore, a few additional vacancies may have segregated in the time needed for a vacancy to find its lowest energy position, which makes a systematic MD simulation coverage of SFT growth by vacancy absorption intractable in the absence of simplifying assumptions. Here we investigate the alternative ledge propagation process in relation to the site preferences determined in Wang et al. Part II [26]. 2. Method Because the simulations have already revealed an unexpected bulging upon the addition of the third vacancy on
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.07.045
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the SFT edge (insets in Fig. 4a and b in Wang et al. Part II [26]), we have forced a perfect SFT to grow by addition of rigid rods comprising a given number of vacancies along h1 1 0i, named full and partial rods depending upon whether or not the preset number of vacancies is enough to yield the next perfect SFT. For this purpose, rods containing 7 and 20 vacancies (dubbed 7-rod and 20-rod, respectively) were
added to an edge of a 21-SFT and a 190-SFT, respectively (see Fig. 1a and b). In addition, partial growth of the 190SFT was studied by adding three partial rods, 3-, 8- and 13-vacancies long. In the cross-section of the perfect SFT corner (Fig. 1c) the site labeled V is where the rod is placed (Fig. 1d). For a ledge to expand along the (1 1 1) SFT face whose trace on the cross section is embodied by the dashed
Fig. 1. (a) A 190-SFT with atoms colored according to their centro-symmetry parameters. (b) As (a) with an extra vacancy row added to the bottom facing one SFT edge. (c and d) Cross-sections of the stair-rod partial at the edge center (dashed lines in (a and b), respectively). Atoms with fcc centrosymmetry parameter (12) are not displayed in (a and b). All atomic configurations are visualized using AtomEye [43].
Fig. 2. The various growth stages at 300 K over 10 ps of a 21-SFT/7-rod combination in (a) Cu and (b) Al.
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arrow in Fig. 1d the location of V must be unique. The simulations were conducted at 0, 300 and 600 K in Cu and Al and at 900 K in Al. Initially, one of the partial rod’s extremities was either forced to the corner or centered on the SFT edge (see Fig. 4a0 and b0, respectively). 3. Results There are two possible configurations for a ledge formed by vacancy absorption. The ledge referred to as obtuse, Iledge or 109.5° moves towards the SFT corners, whereas the other termed acute, V-ledge or 70.5° moves towards the SFT edge [13]. 3.1. The growth of ledges by addition of full vacancy rods Growth is first investigated in configurations containing a 21-SFT and a 7-rod aged for 10 ps at 300 K in both Cu
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and Al, together with a 190-SFT and a 20-rod aged for 100 ps at 600 and 900 K in Cu and Al, respectively. Both the 21-SFT and 190-SFT grow by expansion of ledges (Figs. 2 and 3, respectively). In a given metal growth is largely independent of SFT size (compare Figs. 2a and 3a for Cu and Figs. 2b and 3b for Al). Perceivable differences between Cu and Al are found in the initial ledge forming stage. In Cu ledges nucleated at the two rod extremities (Figs. 2a1, a2, 3a1 and a2). In Al incipient ledge expansion is more homogeneous along the SFT edge than in Cu (see Fig. 3b2). SFT Growth behavior, however, rapidly becomes similar in the two metals (compare Figs. 2a4 and b4 or Fig. 3a3 and b3). The above material-dependent properties are consistent with the vacancy site preferences in Al and Cu reported in Wang et al. Part II (Figs. 3b and 6a in [26]). In Cu the binding of a vacancy is strongest in the middle of the SFT edge and weakest at the SFT corner, thus encouraging sideways
Fig. 3. Growth of a 190-SFT/20-rod ensemble at 900 K over 100 ps in (a) Cu and (b) Al.
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ledge expansion. Conversely, site preference in Al favors SFT apexes, but only slightly, and hence the rather homogeneous ledge motion. 3.2. Ledge expansion from partial rods At 0 K in Cu a rod placed at the edge center is stable (not shown), whereas, depending on the number of vacancies in the rod, the same forced to the corner undergoes a perceivable structural relaxation. Below 3 vacancies the rod is stable. The 3-rod relaxes into the same ledge as in Fig. 4a1. Beyond 3 vacancies the relaxed configurations exhibit a half grown ledge connected to the rod (see the inset between Fig. 4a2 and a3). The barrier that opposes either growth from the central part of the SFT edge or further ledge expansion is, however, overcome upon aging at 600 K (Fig. 4b1–b3), again consistent with the relative stabilities of the corner and central configurations in Cu reported in Wang et al. Part II [26]. The snapshots selected in Fig. 5 reflect the growth processes that yield the final configurations in Fig. 4a3 and b3 that are adopted in Cu by a 190-SFT combined with a 13vacancy rod. Fig. 5a1–a8 shows that, when placed at the SFT corner the rod tends to generate ledges that expand horizontally to the left, while vacancies located in the central part of the SFT edge will not expand until they are met by the ledge front, at which stage they proceed downward forming a symmetrical, ledged 193-SFT. The final configuration is entirely consistent with the “V-ledge” analyzed by Kuhlmann-Wilsdorf [13]. Vacancy absorption causes the ledge to move towards an SFT edge. The situation differs significantly when the vacancy rod is initially forced to its central, minimum energy position, thus pre-
venting ledges from nucleating at corners (Fig. 5b1–b8). In this case the 3-rod and 8-rod emit one wedged dipole of Shockley partials (Fig. 4b1 and b2), probably because both rods are too short to relax otherwise (the dipolar Shockley partial configuration is shown schematically in (c) and discussed in Section 4.3). In contrast, the 13-rod emits one wedged Shockley partial at each of its extremities, while its median part remains unaffected. This behavior again reflects the property shown in Fig. 6a of Wang et al. Part II [26], that the further a vacancy lies away from the SFT corner, the more stable it is. Because of differences in binding energy along the rod, thermal activation at 600 K encourages the nucleation of a Shockley dipole at the rod extremities. The asymmetrical ledge development in Fig. 5b originates from the fact that the rod extremities are closer to the upper than to the bottom SFT corner, so that the lateral edge of the upper Shockley partial reacts with the neighboring SFT edge before the lower Shockley partial can do so (Fig. 5b3 and b5). The property that ledge expansion is influenced by vacancy site preference is manifest in both cases by the strong central pinning in Fig. 5a3–a5 and by the central trailing cusp in Fig. 5b4–b7). The simulations were performed in Al for the same rod sizes on a 190-SFT at 600 K over 50 ps. The final configurations after quenching consist of an acute ledge when the rod is forced at the corner. They show a pronounced propensity to forming a fragmented ledge when the rod is forced to the middle of the SFT edge (Fig. 6). It is worth emphasizing that within the same aging period of 50 ps the ledge expands to the maximum possible surface in Cu (Fig. 4) but not, however, in Al (Fig. 6), indicating that the motion of the Shockley dipole is relatively more difficult in Al than Cu.
Fig. 4. A vacancy rod is added to the corner (a0) and the center (b0) of a 190-SFT edge in Cu and relaxed at 600 K during 50 ps. The atoms in (a0 and b0) are colored according to their coordination number. (a1–a3) The relaxed configurations of a 3-, 8- and 13-rod laid at the SFT corner. The inset between (a2 and a3) is obtained after relaxation at 0 K of the same initial configuration as for (a3). (b1–b3) As (a1–a3), respectively, with the rods centered on the SFT edge.
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3.3. The effects of additional vacancies Following the work of Osetsky et al. on Cu, conducted, however, under a different potential [27], the ledge properties were investigated by addition of individual vacancies. After the introduction of vacancies from the apex of a 190-SFT at 0 K, no perceivable changes were found in Cu until the third vacancy was added, forming a ledge analogous to that shown in Fig. 4a1. The following vacancies provoke little relaxation of the ledge–rod configurations, which are all similar to the inset in Fig. 4. It is only with the last one of the full vacancy rod, the 20th vacancy, that another localized ledge comparable with that located at the other apex is created (e.g. the two corners in Fig. 7a). In Al no such relaxation is found after vacancy addition at 0 K. On the other hand, after the introduction of vacancies to a fully grown ledge on a 203-SFT (Fig. 7b),
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imperfect SFTs behave similarly in Cu and Al. Relaxation at 0 K does not encourage ledge growth and configurations such as that exemplified in Fig. 7b are stable. Subsequent aging at 600 K favors ledge growth, which is complete in Cu within 20 ps (Fig. 7c), while it is incomplete in Al (similar to, for example, Fig. 5a7). In agreement with previous results (Section 3.4 in Martinez et al. [27]), the present MD simulations at 0 K show that adding vacancies to a pre-existing SFT in Cu is not enough to stimulate the growth of a perfect SFT to the next perfect SFT because of various intermediate metastable states. We also confirm that SFT growth is strongly size dependent. A difference lies in the fact that SFT reconstruction is observed in Martinez et al. [27] at 0 K upon addition of 12 and 14 vacancies to a 91-SFT and 136-SFT, respectively, compared with the addition of three vacancies to the 190-SFT in the present investigation (Fig. 4a1).
Fig. 5. The growth of the relaxed configurations in Fig. 4a3 and b3 (13-rod at 600 K in Cu) showing two distinct growth directions. Whereas the surface of the untransformed stacking fault differs between (a8 and b8), the number of vacancies (i.e. 7) needed to generate the next perfect 210-SFT is the same.
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4. Discussion The true role played by the ledge mechanism in SFT growth is relatively difficult to apprehend because the initial configuration is preset, which biases its own evolution. Several general features are, however, worth discussing. 4.1. The operative growth mechanism Numerous studies have been devoted to the stability of SFTs versus that of other vacancy clusters utilizing either energy calculations under linear elasticity [28–31] or MD approaches under a variety of atomic potentials [32–35]. Because the detailed growth mechanism is experimentally unknown and in view of the variety of diffusion routes towards a cluster, one is led to making use of simplifying assumptions. As far as elasticity calculations are concerned, jog nucleation and subsequent ledge expansion are not tractable under the approximation of linear elasticity. On the other hand, the study of the relative stability of SFTs versus faulted or perfect loops has been restricted to the Silcox and Hirsch mechanism (see, for example, section 10–4 in Hirth and Lothe [36]), while this mechanism is confirmed neither experimentally nor numerically. The wellaccepted result that beyond a critical size perfect SFTs
should form depending on the metal properties (stacking fault energy and elastic constants) relies on a comparison between the energies of various configurations. This comparison in turn postulates that vacancy segregation engenders perfectly triangular Frank loops [36], which have never been observed experimentally while hexagonal Frank loops are profuse. Deviations from the perfectly triangular Frank loop are sufficient to explain the well-documented presence of truncated SFTs [37] as is clearly illustrated by the numerical simulations of Kadoyoshi et al. (Fig. 3 in Kadoyoshi et al. [21]), Martinez et al. ([27], Fig. 3) and Poletaev and Starostenkov [16]. On the other hand, the differences between the present work on ledge expansion and the analysis of SFT growth by vacancy addition in Osetsky et al. [38] mostly stem from how the vacancies are forced to the SFT. In Osetsky et al. [38] Cu vacancies were added one by one from one vertex to the next, then statically relaxed and forced to stay in metastable positions. Similarly in Part II of this series [26] vacancies were initially placed at their most favored sites and relaxed at various temperatures in order to reveal their possible path join the SFT. In other words, and as far as numerical approaches are concerned, the final SFT and associated growth properties are to a large extent determined by the postulated initial configuration. In spite of the considerable effort devoted
Fig. 6. As Fig. 4 in Al. The ledges in (a2, b2, a3 and b3) are not fully expanded after 50 ps relaxation at 600 K.
Fig. 7. (a) The configuration obtained by successive addition of 20 vacancies onto the edge of a 190-SFT in Cu and relaxation at 0 K. (b) The configuration relaxed at 0 K after introduction of one additional vacancy next to the fully grown ledge of a 203-SFT in Cu. (c) As (b) after 600 K aging for 20 ps resulting in full ledge zipping.
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so far to this complex matter, a sufficiently reliable description of the actual growth process is not available. In this context, some early experimental investigations of SFTs in quenched and annealed gold [20,39] are worth keeping in mind. They indeed show that nucleation of vacancy clusters is completed in the very early stages of annealing and that growth occurs by successive absorption of vacancies thus supporting the ledge rather than the Silcox and Hirsch mechanism. 4.2. 70.5° versus 109.5° ledges In spite of a pronounced growth asymmetry, the ledge engendered by rods forced to an SFT apex (Fig. 5a) moves towards an edge (acute ledge), whereas the ledge resulting from either a rod placed at the middle of an edge (Fig. 5b) or a full rod (Figs. 2 and 3) moves towards an SFT corner (obtuse ledge). The origin of this property can be understood by inspection of the dislocation-equivalent configuration of a vacancy absorbed on an SFT edge. We regard jog nucleation along the SFT edge (Fig. 8) as the first move of the fault in plane d to plane d0 . The jog
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may adopt two alternative configurations. In one (Fig. 8a or, equivalently, Fig. 8b) it subsequently expands sideways by glide of two Shockley partials with distinct Burgers vectors (dA and Cd) to reach both extremities, forming a full V-ledge (Fig. 8c). This obtuse (109.5°) configuration often found in models [11,12,36] never forms in MD simulations (Figs. 4 and 5), except of course when an SFT is cut by a moving dislocation [40,41]. The detailed description of the obtuse ledge is nevertheless useful to the understanding of the growth processes described hereafter. Once the first ledge has fully expanded to the SFT edges (not shown) the upper stacking fault ribbon is bordered by a dipole of ±db sessile stair-rod partial dislocations. A modified form of this ledge is shown in Figs. 17–20 of Hirth and Lothe [36] after the dA Shockley partial has merged with edge AB or, alternatively, as a configuration possibly adopted by a vacancy nucleated at an apex (see Fig. 10). The mobile parts are comprised of two pairs of Shockley dipoles ±Cd and ±Cb bordering an undefected surface. These partials are glissile parallel to edge AC in planes d and b, respectively, where they should not show an orientation preference and, hence, the equivalence between Fig. 8a and b.
Fig. 8. Dislocation-equivalent SFT configurations resulting from the absorption of one vacancy (the proportions are not respected on purpose). The triangle denoted (d0 ) embodies the SFT face {1 1 1} plane of the next perfect SFT. In this figure, as well as in the following schemes, grey figures are stacking fault free parallelograms. (a) Centered jog with two pairs of glissile Shockley partial dipoles glissile sidewise to form a ledge with Burgers vector db along edge AC. (b) A variant of (a) illustrating that the Shockley partials can adopt any orientation in their own slip plane. (c) The configuration resulting from expansion of the jog up to the SFT edges and the absorption of a second vacancy. (d) A jog again centered on edge AC, however, not glissile along the direction of AC. This jog, which would generate a V-ledge if formed on edge AB instead of edge AC, is actually that shown in (a). (e) The configuration after this jog has fully expanded up to the other SFT edge. (f) Ledge growth by absorption of a second vacancy.
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After the ledge generated by the first vacancy has fully expanded along the SFT edge a second vacancy is added onto this ledge, resulting in transport of the upper ±bd stair-rod dipole one step above (Fig. 8c). An alternative jog configuration is presented in Fig. 8d, where, at variance with Fig. 8a, the Shockley partials ±Bd and ±Bc are now glissile parallel to AB in planes parallel to d and c in Fig. 8e trailing a stair-rod quadrupole (±cd) (there is an equivalent configuration glissile parallel to CB). Fig. 8f shows the ledge fully extended between two SFT edges, a configuration clearly unfavorable compared, for example, with the same assuming it were nucleated on AB (see the bottom ledge along AC in Figs. 17–19 of Hirth and Lothe [36]). Similarly to Fig. 8c, Fig. 8f represents the next growth step of this hypothetical ledge after absorption of a second vacancy. It is worth noting that if operative up to apex C, the mechanism yields an acute ledge. Conversely, the nucleation of jogs onto the other side until the step merges with edge AB yields an obtuse ledge. In so far as jog nucleation at the middle part of an edge is concerned, a variant of Fig. 8d is shown in Fig. 9a where the lateral faces of the jog are not mutually parallel but parallel to the SFT faces a and c. In this case the glide motion of the upper loop results in a sessile triangular jog (Fig. 9b). Restricted to one vacancy, this configuration makes little sense, but the same formed on a rod several vacancies long is remarkably similar to the triangles generated by the simulations (see, for example, Figs. 4b1, b2, 5a3, b2 and b5). We now consider jog nucleation at an SFT apex (Fig. 10). In the dislocation-equivalent description absorption of the first vacancy at A again makes no sense, as shown by Fig. 10a. It is only after the second vacancy is absorbed that the configuration resembles the expected 70°5 ledge (Fig. 10a), similar to Figs. 17–19 in Hirth and Lothe [36]. The growth mechanism is shown in Fig. 10c with the absorption of a third vacancy. The configuration contains a loop whose edges consist of two pairs of dipolar Shockley partials (±Ca and ±Cd) glissile in distinct planes parallel to a and d, respectively, transporting the ledge stair-rod da by a zipping mechanism analogous to Fig. 8c. The numerical simulations presented in Wang
et al. Part II [26] have shown that the most favored sites for the first vacancy is located at the SFT apex, however, not at site 1 but at site 2 (Fig. 3a of Wang et al. Part II [26]). The most likely site for the second vacancy is A (Fig. 4b of Wang et al. Part II [26]), fairly consistent with Fig. 8c. Some structural transformations resulting from vacancy absorption by a relatively small 66-SFT have recently been investigated by MD simulations using a model potential for fcc crystals [16]. They show several significant differences from the present results. In particular, (i) the final ledge configuration, facing either a corner or an edge, is concluded to depend on the number of added vacancies relative to the initial number of vacancies, a property dubbed ‘step sign change’ in Poletaev and Starostenkov [16], and (ii) in the course of the simulations the same ledge may reorient itself, changing the corner that it faces with an apparently moderate activation energy. In Poletaev and Starostenkov [16] the SFT is engendered from a platelet of vacancies with a jogged triangular shape which upon relaxation first collapses into a Frank loop, then into an SFT by a mechanism a` la Silcox and Hirsch. We believe that the initial set-up is largely responsible for the step sign change. The jogged triangular platelet is actually equivalent to a vacancy rod placed along one edge of the triangle, in a corner position, before the SFT is formed. The origin of property (ii) is unclear as it involves a considerable structural rearrangement under no clear driving force and may originate from the atomic potentials utilized in Poletaev and Starostenkov [16]. 4.3. The core structure of the mobile dislocations bordering the ledge Common to Al and Cu is the property that the ledges show a preferred orientation along h1 1 0i close packed directions at 60° from the initial vacancy rod. A possible explanation can be drawn from the evolution of an asymmetrical jog (Fig. 9). However, the sequences shown in Fig. 5a3–a5 and Fig. 5b2–b4, where triangular ledges move back and forth, together with the blocked configurations in
Fig. 9. (a) An asymmetrical variant of Fig. 8d. (b) The same fully expanded by glide. (c) Dislocation-equivalent sketch of segmentation of the Shockley partial dipoles observed during MD simulations of SFT growth (see, for example, Fig. 3). (d) A schematic representation of the structure of wedged Shockley dipoles consistent with the simulated configurations of, for example Fig. 3a5 and b6.
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Fig. 6, suggest a core controlled behavior. Fig. 9c shows that whereas the lateral edges of the dislocations bordering the surface step are stair-rod in nature, the inner portion comprises a glissile dipole of Shockley partials (i.e. ±Cd, see Fig. 9d). It is interesting to compare the kinetics of SFT growth observed in Al and Cu after a full vacancy rod is added to a 190-SFT in order to form the next perfect 210-SFT. In both cases growth takes place by a ledge-like process with a markedly trailing central portion. Our results on site preference in Cu would therefore offer a reasonable explanation for the preferred 60° orientation. However, the fact that a similar trailing behavior is observed in Al (Fig. 3b6), where site preference should encourage the propagation of the central part at the expense of the lateral segments, rules out this interpretation. We believe that ledge expansion is controlled by the length of the Shockley partial dipole. For simplicity, we consider the growth of an SFT with edge length L via (i) two equilateral triangles with edge length, LL = aL, (ii) a central equilateral triangle and (iii) a trapezium with equal surfaces all (Fig. 11). We consider symmetrical configurations, i.e. the Burgers vector of the Shockley dipole is Bd and the ledges propagate in plane d. In configurations (i) and (ii) the bordering Shockley dipoles are 60° in character (self-energy proportional to 4 m=4ð1 mÞ), but pure edge in configuration (iii) (self-energy proportional to 1=ð1 mÞ). Hence, growth via a central equilateral triangle p necessitates a length of dipolar Shockley partials 2 as large as that required for the two lateral triangles, consistent with the behavior observed for Al and Cu. However, growth pffiffiffiffiffiffiffiffiffi via the trapezium necessitates a length equal to 4ð 14a2 1Þ LL so that, with m ½ and for a 6 ½, the trapeað4mÞ zoidal ledge should always be favored, clearly at variance with Fig. 3 (a similar conclusion is reached with a Burgers vector that makes the configuration asymmetrical, e.g. Cd in plane d).
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In order to explain the observed segmentation one is led to the hypothesis that the ±60° Shockley dipoles reorganize themselves at the atomic scale to form a stable core relative to all other dipole orientations. Formally, instead of the dipolar ±Bd Shockley partials bordering a non-faulted strip, there is a possibility in the ±60° orientations that, driven by strong attractive forces, each ±Bd partial splits into ±(Bc + cd) or ±(Ba + ad) in planes parallel to c and a, respectively, and that the Shockley partials mutually annihilate by glide in the appropriate {1 1 1} plane. MD simulations indicate that ledge expansion is significantly impeded by lattice friction, suggesting in turn some complex rearrangement similar to what was found for edge dipoles in fcc metals [42]. The effects of lattice friction are best viewed in dynamic sequences where the ledges do not move continuously but stop at times. Motion takes place by kink nucleation and propagation along the 60° portions (Fig. 3a4 and b5) with occasional backward motion. Fig. 12 shows snapshots taken every 2.5 ps in Al annealed at 900 K (see Fig. 3b4)
Fig. 11. Three possible ledge configurations segmented along the h1 1 0i crystallographic directions.
Fig. 10. Configurations resulting from the absorption of vacancies at the SFT apex A. (a) One vacancy. (b) Two vacancies (an alternative view is shown at site B). (c) The growth mechanism of an acute ledge.
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Fig. 12. Snapshots of the ledge growth in Al at 900 K selected around Fig. 3b4, showing the back and forth motion of ledge kinks in places. Arrows indicate growth directions.
where kink motion occurs backwards in Fig. 12b and forward in Fig. 12c on the right-hand side Shockley dipole portion, then on the left-hand side (Fig. 12c–e). 5. Conclusions Highly dependent on the method chosen for its investigation, the actual process of SFT growth is largely unknown. Here rods containing selected numbers of vacancies were forced on the edge of SFTs of various sizes in Al and Cu, chosen for their significantly distinct stacking fault energies. At the atomic level the ledge mechanism exhibits the following properties. 1. Ledge nucleation and the early stage of ledge expansion are both material and temperature dependent. In Cu vacancy rods prefer to aggregate at the middle of SFT edges, while in Al they aggregate at SFT vertexes. 2. Ledge transformation depends critically on the material and, within a given material, on rod location. 3. The ledge growth rate increases with temperature. It is faster in Cu than in Al at the same temperature. 4. Depending on temperature and on the number of vacancies in the rod relative to that stored in the SFTs, ledge expansion yields different configurations, including edge- and corner-facing ledges 5. The Shockley partial dipole that borders a ledge is highly segmented along h1 1 0i directions, reflecting several kinds of core reorganization, consistent with the observed dependence of ledge expansion upon temperature.
Acknowledgements The support of the Ministry of Science and Technology of China under Grant No. 2006CB605104 and the Natural Science Foundation of China under Grant No. 50911130367 and No. 50631030 is gratefully acknowledged. Patrick Veyssie`re should like to thank the Institute of Metal Research, Chinese Academy of Sciences for financial support and outstanding hospitality. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.actamat.2010. 07.045.
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Acta Materialia 59 (2011) 19–29 www.elsevier.com/locate/actamat
The formation of stacking fault tetrahedra in Al and Cu III. Growth by expanding ledges H. Wang a,⇑, D.S. Xu a, R. Yang a, P. Veyssie`re b a
Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, People’s Republic of China b LEM, CNRS-ONERA, BP 72, 92322 Chaˆtillon, France Received 10 February 2010; received in revised form 9 July 2010; accepted 27 July 2010 Available online 27 September 2010
Abstract Ledge expansion and the concomitant growth of a stacking fault tetrahedron (SFT) are investigated in Al and Cu by molecular dynamics (MD) simulations by addition of vacancy rods with selected lengths. Ledge expansion is largely governed by the site preference of vacancies on the SFT edges resulting in distinct stable ledged SFTs. Both edge- and corner-facing ledge configurations may be adopted. The growth of SFTs, especially large ones, is controlled by thermal agitation. The mobile part of the ledges consists of a dipole of Shockley partials generally oriented in the 60° mixed orientation that move in a thermally activated manner, reflecting a certain core reorganization of the Shockley dipole. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Stacking fault tetrahedron; Growth; Ledge mechanism; Molecular dynamics simulations; Shockley dipole
1. Introduction In Au essentially but also in other metals such as Ag, Ni and Cu and certain of their alloys stacking fault tetrahedron (SFT) edges amount to a few tens of nanometers [1–8]. After quenching and annealing edge lengths of the order of 100 nm, and thus containing at least 105 atoms, have been observed [9,10]. Whereas small sized tetrahedra may grow by successive absorption of vacancies at jogs on the tetrahedron faces [11–16] (see also Wang et al. Part II [26]), one may wonder if this mechanism is still operative for SFTs containing one to two orders of magnitude more vacancies than the several nanometer large SFTs generated by irradiation or plastic deformation [17,18]. As an alternative to the Silcox and Hirsch mechanism [1], SFT growth (and shrinkage) by expansion of ledges on SFT faces has been proposed [11,19], although little investigated. Experimental evidence is scarce [20]. Because the distance between certain segments of the ledges is of the order of the inner cut-off radius, SFT growth by ledge expansion cannot be studied by linear elas⇑ Corresponding author. Tel.: +86 24 23971946.
E-mail address: [email protected] (H. Wang).
ticity calculations of interactions between piecewise dislocations. Recently, ledges were observed by MD simulations of the growth of an initially defective triangular Frank loop [16,21] and upon SFT intersection by a moving dislocation [22–24]. An SFT, which is in general not perfect, may evolve in significantly distinct ways depending on where the next vacancy or vacancy cluster comes from. Several local configurational minima may be attained, such as the double SFT discussed in Wang et al. Parts I [25] and II [26]. Furthermore, a few additional vacancies may have segregated in the time needed for a vacancy to find its lowest energy position, which makes a systematic MD simulation coverage of SFT growth by vacancy absorption intractable in the absence of simplifying assumptions. Here we investigate the alternative ledge propagation process in relation to the site preferences determined in Wang et al. Part II [26]. 2. Method Because the simulations have already revealed an unexpected bulging upon the addition of the third vacancy on
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.07.045
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the SFT edge (insets in Fig. 4a and b in Wang et al. Part II [26]), we have forced a perfect SFT to grow by addition of rigid rods comprising a given number of vacancies along h1 1 0i, named full and partial rods depending upon whether or not the preset number of vacancies is enough to yield the next perfect SFT. For this purpose, rods containing 7 and 20 vacancies (dubbed 7-rod and 20-rod, respectively) were
added to an edge of a 21-SFT and a 190-SFT, respectively (see Fig. 1a and b). In addition, partial growth of the 190SFT was studied by adding three partial rods, 3-, 8- and 13-vacancies long. In the cross-section of the perfect SFT corner (Fig. 1c) the site labeled V is where the rod is placed (Fig. 1d). For a ledge to expand along the (1 1 1) SFT face whose trace on the cross section is embodied by the dashed
Fig. 1. (a) A 190-SFT with atoms colored according to their centro-symmetry parameters. (b) As (a) with an extra vacancy row added to the bottom facing one SFT edge. (c and d) Cross-sections of the stair-rod partial at the edge center (dashed lines in (a and b), respectively). Atoms with fcc centrosymmetry parameter (12) are not displayed in (a and b). All atomic configurations are visualized using AtomEye [43].
Fig. 2. The various growth stages at 300 K over 10 ps of a 21-SFT/7-rod combination in (a) Cu and (b) Al.
H. Wang et al. / Acta Materialia 59 (2011) 19–29
arrow in Fig. 1d the location of V must be unique. The simulations were conducted at 0, 300 and 600 K in Cu and Al and at 900 K in Al. Initially, one of the partial rod’s extremities was either forced to the corner or centered on the SFT edge (see Fig. 4a0 and b0, respectively). 3. Results There are two possible configurations for a ledge formed by vacancy absorption. The ledge referred to as obtuse, Iledge or 109.5° moves towards the SFT corners, whereas the other termed acute, V-ledge or 70.5° moves towards the SFT edge [13]. 3.1. The growth of ledges by addition of full vacancy rods Growth is first investigated in configurations containing a 21-SFT and a 7-rod aged for 10 ps at 300 K in both Cu
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and Al, together with a 190-SFT and a 20-rod aged for 100 ps at 600 and 900 K in Cu and Al, respectively. Both the 21-SFT and 190-SFT grow by expansion of ledges (Figs. 2 and 3, respectively). In a given metal growth is largely independent of SFT size (compare Figs. 2a and 3a for Cu and Figs. 2b and 3b for Al). Perceivable differences between Cu and Al are found in the initial ledge forming stage. In Cu ledges nucleated at the two rod extremities (Figs. 2a1, a2, 3a1 and a2). In Al incipient ledge expansion is more homogeneous along the SFT edge than in Cu (see Fig. 3b2). SFT Growth behavior, however, rapidly becomes similar in the two metals (compare Figs. 2a4 and b4 or Fig. 3a3 and b3). The above material-dependent properties are consistent with the vacancy site preferences in Al and Cu reported in Wang et al. Part II (Figs. 3b and 6a in [26]). In Cu the binding of a vacancy is strongest in the middle of the SFT edge and weakest at the SFT corner, thus encouraging sideways
Fig. 3. Growth of a 190-SFT/20-rod ensemble at 900 K over 100 ps in (a) Cu and (b) Al.
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ledge expansion. Conversely, site preference in Al favors SFT apexes, but only slightly, and hence the rather homogeneous ledge motion. 3.2. Ledge expansion from partial rods At 0 K in Cu a rod placed at the edge center is stable (not shown), whereas, depending on the number of vacancies in the rod, the same forced to the corner undergoes a perceivable structural relaxation. Below 3 vacancies the rod is stable. The 3-rod relaxes into the same ledge as in Fig. 4a1. Beyond 3 vacancies the relaxed configurations exhibit a half grown ledge connected to the rod (see the inset between Fig. 4a2 and a3). The barrier that opposes either growth from the central part of the SFT edge or further ledge expansion is, however, overcome upon aging at 600 K (Fig. 4b1–b3), again consistent with the relative stabilities of the corner and central configurations in Cu reported in Wang et al. Part II [26]. The snapshots selected in Fig. 5 reflect the growth processes that yield the final configurations in Fig. 4a3 and b3 that are adopted in Cu by a 190-SFT combined with a 13vacancy rod. Fig. 5a1–a8 shows that, when placed at the SFT corner the rod tends to generate ledges that expand horizontally to the left, while vacancies located in the central part of the SFT edge will not expand until they are met by the ledge front, at which stage they proceed downward forming a symmetrical, ledged 193-SFT. The final configuration is entirely consistent with the “V-ledge” analyzed by Kuhlmann-Wilsdorf [13]. Vacancy absorption causes the ledge to move towards an SFT edge. The situation differs significantly when the vacancy rod is initially forced to its central, minimum energy position, thus pre-
venting ledges from nucleating at corners (Fig. 5b1–b8). In this case the 3-rod and 8-rod emit one wedged dipole of Shockley partials (Fig. 4b1 and b2), probably because both rods are too short to relax otherwise (the dipolar Shockley partial configuration is shown schematically in (c) and discussed in Section 4.3). In contrast, the 13-rod emits one wedged Shockley partial at each of its extremities, while its median part remains unaffected. This behavior again reflects the property shown in Fig. 6a of Wang et al. Part II [26], that the further a vacancy lies away from the SFT corner, the more stable it is. Because of differences in binding energy along the rod, thermal activation at 600 K encourages the nucleation of a Shockley dipole at the rod extremities. The asymmetrical ledge development in Fig. 5b originates from the fact that the rod extremities are closer to the upper than to the bottom SFT corner, so that the lateral edge of the upper Shockley partial reacts with the neighboring SFT edge before the lower Shockley partial can do so (Fig. 5b3 and b5). The property that ledge expansion is influenced by vacancy site preference is manifest in both cases by the strong central pinning in Fig. 5a3–a5 and by the central trailing cusp in Fig. 5b4–b7). The simulations were performed in Al for the same rod sizes on a 190-SFT at 600 K over 50 ps. The final configurations after quenching consist of an acute ledge when the rod is forced at the corner. They show a pronounced propensity to forming a fragmented ledge when the rod is forced to the middle of the SFT edge (Fig. 6). It is worth emphasizing that within the same aging period of 50 ps the ledge expands to the maximum possible surface in Cu (Fig. 4) but not, however, in Al (Fig. 6), indicating that the motion of the Shockley dipole is relatively more difficult in Al than Cu.
Fig. 4. A vacancy rod is added to the corner (a0) and the center (b0) of a 190-SFT edge in Cu and relaxed at 600 K during 50 ps. The atoms in (a0 and b0) are colored according to their coordination number. (a1–a3) The relaxed configurations of a 3-, 8- and 13-rod laid at the SFT corner. The inset between (a2 and a3) is obtained after relaxation at 0 K of the same initial configuration as for (a3). (b1–b3) As (a1–a3), respectively, with the rods centered on the SFT edge.
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3.3. The effects of additional vacancies Following the work of Osetsky et al. on Cu, conducted, however, under a different potential [27], the ledge properties were investigated by addition of individual vacancies. After the introduction of vacancies from the apex of a 190-SFT at 0 K, no perceivable changes were found in Cu until the third vacancy was added, forming a ledge analogous to that shown in Fig. 4a1. The following vacancies provoke little relaxation of the ledge–rod configurations, which are all similar to the inset in Fig. 4. It is only with the last one of the full vacancy rod, the 20th vacancy, that another localized ledge comparable with that located at the other apex is created (e.g. the two corners in Fig. 7a). In Al no such relaxation is found after vacancy addition at 0 K. On the other hand, after the introduction of vacancies to a fully grown ledge on a 203-SFT (Fig. 7b),
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imperfect SFTs behave similarly in Cu and Al. Relaxation at 0 K does not encourage ledge growth and configurations such as that exemplified in Fig. 7b are stable. Subsequent aging at 600 K favors ledge growth, which is complete in Cu within 20 ps (Fig. 7c), while it is incomplete in Al (similar to, for example, Fig. 5a7). In agreement with previous results (Section 3.4 in Martinez et al. [27]), the present MD simulations at 0 K show that adding vacancies to a pre-existing SFT in Cu is not enough to stimulate the growth of a perfect SFT to the next perfect SFT because of various intermediate metastable states. We also confirm that SFT growth is strongly size dependent. A difference lies in the fact that SFT reconstruction is observed in Martinez et al. [27] at 0 K upon addition of 12 and 14 vacancies to a 91-SFT and 136-SFT, respectively, compared with the addition of three vacancies to the 190-SFT in the present investigation (Fig. 4a1).
Fig. 5. The growth of the relaxed configurations in Fig. 4a3 and b3 (13-rod at 600 K in Cu) showing two distinct growth directions. Whereas the surface of the untransformed stacking fault differs between (a8 and b8), the number of vacancies (i.e. 7) needed to generate the next perfect 210-SFT is the same.
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4. Discussion The true role played by the ledge mechanism in SFT growth is relatively difficult to apprehend because the initial configuration is preset, which biases its own evolution. Several general features are, however, worth discussing. 4.1. The operative growth mechanism Numerous studies have been devoted to the stability of SFTs versus that of other vacancy clusters utilizing either energy calculations under linear elasticity [28–31] or MD approaches under a variety of atomic potentials [32–35]. Because the detailed growth mechanism is experimentally unknown and in view of the variety of diffusion routes towards a cluster, one is led to making use of simplifying assumptions. As far as elasticity calculations are concerned, jog nucleation and subsequent ledge expansion are not tractable under the approximation of linear elasticity. On the other hand, the study of the relative stability of SFTs versus faulted or perfect loops has been restricted to the Silcox and Hirsch mechanism (see, for example, section 10–4 in Hirth and Lothe [36]), while this mechanism is confirmed neither experimentally nor numerically. The wellaccepted result that beyond a critical size perfect SFTs
should form depending on the metal properties (stacking fault energy and elastic constants) relies on a comparison between the energies of various configurations. This comparison in turn postulates that vacancy segregation engenders perfectly triangular Frank loops [36], which have never been observed experimentally while hexagonal Frank loops are profuse. Deviations from the perfectly triangular Frank loop are sufficient to explain the well-documented presence of truncated SFTs [37] as is clearly illustrated by the numerical simulations of Kadoyoshi et al. (Fig. 3 in Kadoyoshi et al. [21]), Martinez et al. ([27], Fig. 3) and Poletaev and Starostenkov [16]. On the other hand, the differences between the present work on ledge expansion and the analysis of SFT growth by vacancy addition in Osetsky et al. [38] mostly stem from how the vacancies are forced to the SFT. In Osetsky et al. [38] Cu vacancies were added one by one from one vertex to the next, then statically relaxed and forced to stay in metastable positions. Similarly in Part II of this series [26] vacancies were initially placed at their most favored sites and relaxed at various temperatures in order to reveal their possible path join the SFT. In other words, and as far as numerical approaches are concerned, the final SFT and associated growth properties are to a large extent determined by the postulated initial configuration. In spite of the considerable effort devoted
Fig. 6. As Fig. 4 in Al. The ledges in (a2, b2, a3 and b3) are not fully expanded after 50 ps relaxation at 600 K.
Fig. 7. (a) The configuration obtained by successive addition of 20 vacancies onto the edge of a 190-SFT in Cu and relaxation at 0 K. (b) The configuration relaxed at 0 K after introduction of one additional vacancy next to the fully grown ledge of a 203-SFT in Cu. (c) As (b) after 600 K aging for 20 ps resulting in full ledge zipping.
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so far to this complex matter, a sufficiently reliable description of the actual growth process is not available. In this context, some early experimental investigations of SFTs in quenched and annealed gold [20,39] are worth keeping in mind. They indeed show that nucleation of vacancy clusters is completed in the very early stages of annealing and that growth occurs by successive absorption of vacancies thus supporting the ledge rather than the Silcox and Hirsch mechanism. 4.2. 70.5° versus 109.5° ledges In spite of a pronounced growth asymmetry, the ledge engendered by rods forced to an SFT apex (Fig. 5a) moves towards an edge (acute ledge), whereas the ledge resulting from either a rod placed at the middle of an edge (Fig. 5b) or a full rod (Figs. 2 and 3) moves towards an SFT corner (obtuse ledge). The origin of this property can be understood by inspection of the dislocation-equivalent configuration of a vacancy absorbed on an SFT edge. We regard jog nucleation along the SFT edge (Fig. 8) as the first move of the fault in plane d to plane d0 . The jog
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may adopt two alternative configurations. In one (Fig. 8a or, equivalently, Fig. 8b) it subsequently expands sideways by glide of two Shockley partials with distinct Burgers vectors (dA and Cd) to reach both extremities, forming a full V-ledge (Fig. 8c). This obtuse (109.5°) configuration often found in models [11,12,36] never forms in MD simulations (Figs. 4 and 5), except of course when an SFT is cut by a moving dislocation [40,41]. The detailed description of the obtuse ledge is nevertheless useful to the understanding of the growth processes described hereafter. Once the first ledge has fully expanded to the SFT edges (not shown) the upper stacking fault ribbon is bordered by a dipole of ±db sessile stair-rod partial dislocations. A modified form of this ledge is shown in Figs. 17–20 of Hirth and Lothe [36] after the dA Shockley partial has merged with edge AB or, alternatively, as a configuration possibly adopted by a vacancy nucleated at an apex (see Fig. 10). The mobile parts are comprised of two pairs of Shockley dipoles ±Cd and ±Cb bordering an undefected surface. These partials are glissile parallel to edge AC in planes d and b, respectively, where they should not show an orientation preference and, hence, the equivalence between Fig. 8a and b.
Fig. 8. Dislocation-equivalent SFT configurations resulting from the absorption of one vacancy (the proportions are not respected on purpose). The triangle denoted (d0 ) embodies the SFT face {1 1 1} plane of the next perfect SFT. In this figure, as well as in the following schemes, grey figures are stacking fault free parallelograms. (a) Centered jog with two pairs of glissile Shockley partial dipoles glissile sidewise to form a ledge with Burgers vector db along edge AC. (b) A variant of (a) illustrating that the Shockley partials can adopt any orientation in their own slip plane. (c) The configuration resulting from expansion of the jog up to the SFT edges and the absorption of a second vacancy. (d) A jog again centered on edge AC, however, not glissile along the direction of AC. This jog, which would generate a V-ledge if formed on edge AB instead of edge AC, is actually that shown in (a). (e) The configuration after this jog has fully expanded up to the other SFT edge. (f) Ledge growth by absorption of a second vacancy.
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After the ledge generated by the first vacancy has fully expanded along the SFT edge a second vacancy is added onto this ledge, resulting in transport of the upper ±bd stair-rod dipole one step above (Fig. 8c). An alternative jog configuration is presented in Fig. 8d, where, at variance with Fig. 8a, the Shockley partials ±Bd and ±Bc are now glissile parallel to AB in planes parallel to d and c in Fig. 8e trailing a stair-rod quadrupole (±cd) (there is an equivalent configuration glissile parallel to CB). Fig. 8f shows the ledge fully extended between two SFT edges, a configuration clearly unfavorable compared, for example, with the same assuming it were nucleated on AB (see the bottom ledge along AC in Figs. 17–19 of Hirth and Lothe [36]). Similarly to Fig. 8c, Fig. 8f represents the next growth step of this hypothetical ledge after absorption of a second vacancy. It is worth noting that if operative up to apex C, the mechanism yields an acute ledge. Conversely, the nucleation of jogs onto the other side until the step merges with edge AB yields an obtuse ledge. In so far as jog nucleation at the middle part of an edge is concerned, a variant of Fig. 8d is shown in Fig. 9a where the lateral faces of the jog are not mutually parallel but parallel to the SFT faces a and c. In this case the glide motion of the upper loop results in a sessile triangular jog (Fig. 9b). Restricted to one vacancy, this configuration makes little sense, but the same formed on a rod several vacancies long is remarkably similar to the triangles generated by the simulations (see, for example, Figs. 4b1, b2, 5a3, b2 and b5). We now consider jog nucleation at an SFT apex (Fig. 10). In the dislocation-equivalent description absorption of the first vacancy at A again makes no sense, as shown by Fig. 10a. It is only after the second vacancy is absorbed that the configuration resembles the expected 70°5 ledge (Fig. 10a), similar to Figs. 17–19 in Hirth and Lothe [36]. The growth mechanism is shown in Fig. 10c with the absorption of a third vacancy. The configuration contains a loop whose edges consist of two pairs of dipolar Shockley partials (±Ca and ±Cd) glissile in distinct planes parallel to a and d, respectively, transporting the ledge stair-rod da by a zipping mechanism analogous to Fig. 8c. The numerical simulations presented in Wang
et al. Part II [26] have shown that the most favored sites for the first vacancy is located at the SFT apex, however, not at site 1 but at site 2 (Fig. 3a of Wang et al. Part II [26]). The most likely site for the second vacancy is A (Fig. 4b of Wang et al. Part II [26]), fairly consistent with Fig. 8c. Some structural transformations resulting from vacancy absorption by a relatively small 66-SFT have recently been investigated by MD simulations using a model potential for fcc crystals [16]. They show several significant differences from the present results. In particular, (i) the final ledge configuration, facing either a corner or an edge, is concluded to depend on the number of added vacancies relative to the initial number of vacancies, a property dubbed ‘step sign change’ in Poletaev and Starostenkov [16], and (ii) in the course of the simulations the same ledge may reorient itself, changing the corner that it faces with an apparently moderate activation energy. In Poletaev and Starostenkov [16] the SFT is engendered from a platelet of vacancies with a jogged triangular shape which upon relaxation first collapses into a Frank loop, then into an SFT by a mechanism a` la Silcox and Hirsch. We believe that the initial set-up is largely responsible for the step sign change. The jogged triangular platelet is actually equivalent to a vacancy rod placed along one edge of the triangle, in a corner position, before the SFT is formed. The origin of property (ii) is unclear as it involves a considerable structural rearrangement under no clear driving force and may originate from the atomic potentials utilized in Poletaev and Starostenkov [16]. 4.3. The core structure of the mobile dislocations bordering the ledge Common to Al and Cu is the property that the ledges show a preferred orientation along h1 1 0i close packed directions at 60° from the initial vacancy rod. A possible explanation can be drawn from the evolution of an asymmetrical jog (Fig. 9). However, the sequences shown in Fig. 5a3–a5 and Fig. 5b2–b4, where triangular ledges move back and forth, together with the blocked configurations in
Fig. 9. (a) An asymmetrical variant of Fig. 8d. (b) The same fully expanded by glide. (c) Dislocation-equivalent sketch of segmentation of the Shockley partial dipoles observed during MD simulations of SFT growth (see, for example, Fig. 3). (d) A schematic representation of the structure of wedged Shockley dipoles consistent with the simulated configurations of, for example Fig. 3a5 and b6.
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Fig. 6, suggest a core controlled behavior. Fig. 9c shows that whereas the lateral edges of the dislocations bordering the surface step are stair-rod in nature, the inner portion comprises a glissile dipole of Shockley partials (i.e. ±Cd, see Fig. 9d). It is interesting to compare the kinetics of SFT growth observed in Al and Cu after a full vacancy rod is added to a 190-SFT in order to form the next perfect 210-SFT. In both cases growth takes place by a ledge-like process with a markedly trailing central portion. Our results on site preference in Cu would therefore offer a reasonable explanation for the preferred 60° orientation. However, the fact that a similar trailing behavior is observed in Al (Fig. 3b6), where site preference should encourage the propagation of the central part at the expense of the lateral segments, rules out this interpretation. We believe that ledge expansion is controlled by the length of the Shockley partial dipole. For simplicity, we consider the growth of an SFT with edge length L via (i) two equilateral triangles with edge length, LL = aL, (ii) a central equilateral triangle and (iii) a trapezium with equal surfaces all (Fig. 11). We consider symmetrical configurations, i.e. the Burgers vector of the Shockley dipole is Bd and the ledges propagate in plane d. In configurations (i) and (ii) the bordering Shockley dipoles are 60° in character (self-energy proportional to 4 m=4ð1 mÞ), but pure edge in configuration (iii) (self-energy proportional to 1=ð1 mÞ). Hence, growth via a central equilateral triangle p necessitates a length of dipolar Shockley partials 2 as large as that required for the two lateral triangles, consistent with the behavior observed for Al and Cu. However, growth pffiffiffiffiffiffiffiffiffi via the trapezium necessitates a length equal to 4ð 14a2 1Þ LL so that, with m ½ and for a 6 ½, the trapeað4mÞ zoidal ledge should always be favored, clearly at variance with Fig. 3 (a similar conclusion is reached with a Burgers vector that makes the configuration asymmetrical, e.g. Cd in plane d).
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In order to explain the observed segmentation one is led to the hypothesis that the ±60° Shockley dipoles reorganize themselves at the atomic scale to form a stable core relative to all other dipole orientations. Formally, instead of the dipolar ±Bd Shockley partials bordering a non-faulted strip, there is a possibility in the ±60° orientations that, driven by strong attractive forces, each ±Bd partial splits into ±(Bc + cd) or ±(Ba + ad) in planes parallel to c and a, respectively, and that the Shockley partials mutually annihilate by glide in the appropriate {1 1 1} plane. MD simulations indicate that ledge expansion is significantly impeded by lattice friction, suggesting in turn some complex rearrangement similar to what was found for edge dipoles in fcc metals [42]. The effects of lattice friction are best viewed in dynamic sequences where the ledges do not move continuously but stop at times. Motion takes place by kink nucleation and propagation along the 60° portions (Fig. 3a4 and b5) with occasional backward motion. Fig. 12 shows snapshots taken every 2.5 ps in Al annealed at 900 K (see Fig. 3b4)
Fig. 11. Three possible ledge configurations segmented along the h1 1 0i crystallographic directions.
Fig. 10. Configurations resulting from the absorption of vacancies at the SFT apex A. (a) One vacancy. (b) Two vacancies (an alternative view is shown at site B). (c) The growth mechanism of an acute ledge.
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H. Wang et al. / Acta Materialia 59 (2011) 19–29
Fig. 12. Snapshots of the ledge growth in Al at 900 K selected around Fig. 3b4, showing the back and forth motion of ledge kinks in places. Arrows indicate growth directions.
where kink motion occurs backwards in Fig. 12b and forward in Fig. 12c on the right-hand side Shockley dipole portion, then on the left-hand side (Fig. 12c–e). 5. Conclusions Highly dependent on the method chosen for its investigation, the actual process of SFT growth is largely unknown. Here rods containing selected numbers of vacancies were forced on the edge of SFTs of various sizes in Al and Cu, chosen for their significantly distinct stacking fault energies. At the atomic level the ledge mechanism exhibits the following properties. 1. Ledge nucleation and the early stage of ledge expansion are both material and temperature dependent. In Cu vacancy rods prefer to aggregate at the middle of SFT edges, while in Al they aggregate at SFT vertexes. 2. Ledge transformation depends critically on the material and, within a given material, on rod location. 3. The ledge growth rate increases with temperature. It is faster in Cu than in Al at the same temperature. 4. Depending on temperature and on the number of vacancies in the rod relative to that stored in the SFTs, ledge expansion yields different configurations, including edge- and corner-facing ledges 5. The Shockley partial dipole that borders a ledge is highly segmented along h1 1 0i directions, reflecting several kinds of core reorganization, consistent with the observed dependence of ledge expansion upon temperature.
Acknowledgements The support of the Ministry of Science and Technology of China under Grant No. 2006CB605104 and the Natural Science Foundation of China under Grant No. 50911130367 and No. 50631030 is gratefully acknowledged. Patrick Veyssie`re should like to thank the Institute of Metal Research, Chinese Academy of Sciences for financial support and outstanding hospitality. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.actamat.2010. 07.045.
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