Twinning nucleation in Cu-8 at. % Al single crystals - CiteSeerX

PHILOSOPHICAL MAGAZINE A, 2002, VOL. 82, N O. 1, 167±191

Twinning nucleation in Cu±8 at.% Al single crystals M. Niewczasy Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7

and G. Saada Laboratoire d’Etude des Microstructures, CNRS±ONERA, 92322 Chatillon, France [Received 9 August 2000 and accepted in revised form 14 May 2001]

Abstract Transmission electron microscopy observations of the dislocation substructure in Cu±8 at.% Al single crystals deformed in tension have been carried out to elucidate the origin of the twinning dislocation which produces conjugate twinning in this alloy. It is argued that the `correct’ Heidenreich± Shockley dislocation is obtained in the reaction of the primary dislocation with the faulted (Frank) dipole. These interactions are expected to occur frequently during the deformation. Di erent con®gurations of the twinning sources are analysed with respect to the nucleation and the growth of the twin. Geometrical issues related to the pinning of the component dislocations at the nodes formed, the motion of the single twinning dislocation and the non-coherent twinning front and the passing barriers encountered during twin growth are analysed. It is shown that there are no geometrical and energetic obstacles to the development of a macroscopic twin by the pole mechanism from the source proposed here.

} 1. Introduction The nucleation of deformation twinning in fcc crystals is a heterogeneous process associated with the activation of …a=6†h112i Shockley partial dislocations. Twinning occurs at a critical stress of the order of ®=b (® is the stacking-fault energy and b the length of the Shockley partial dislocation); therefore it is usually observed after some plastic ¯ow when the ¯ow stress and the dislocation densities are high. Under the action of applied stress, the twinning dislocation expands, extends the stacking fault and produces a layer of coherent interface parallel to the close-packed {111} plane, the K1 plane. Other mechanisms are required to form new layers in the direction normal to the twinning plane in order to develop a macroscopic twin. Current dislocation models describing the nucleation and the growth of twins in speci®c crystal structures have been thoroughl y reviewed by Christian and Mahajan (1995). In the present work we examine closely the nucleation of twinning in Cu± 8 at.% Al single crystals deformed in tension at room temperature. Transmission electron microscopy (TEM) observations of the dislocation substructure developed y Author for correspondence. E-mail: [email protected] Philosophica l Magazin e A ISSN 0141±8610 print/ISSN 1460-699 2 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080 /0141861011006772 5

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in the crystals prior to the onset of twinning as well as in the twinned material have been made in order to elucidate the nucleation of the twin in this geometrically controlled system. The proposed dislocation model of the twin source builds on the properties of a special type of defect, namely faulted dipoles. It is appropriate to comment on their structure in the following. } 2. Faulted dipoles: literature review The formation and structure of faulted dipoles have been discussed by a number of researchers (for example Chiang et al. (1980)). Here, we brie¯y recall the basic geometry, origin and some relevant TEM contrast properties of these defects. The presence of faulted dipoles (known also as Frank dipoles) in the substructure of deformed materials has been reported in some early TEM observations, for example in Ni±Co single crystals by Mader (1963), in Cu single crystals by Basinski (1964) and Steeds (1966), in Ag by Moon and Robinson (1967), in Ge by Alexander (1968), and in Cu±2 at.% Al single crystals by Pande and Hazzledine (1971). Under TEM two-beam di racting conditions, faulted dipoles are recognized in the structure as faint dislocation lines running in h110i directions. Hirsch (1963) interpreted their contrast as narrow dipoles consisting of two Frank partial dislocations of opposite sign in the primary glide plane, bounded by a stacking-fault ribbon in the non-coplanar glide plane. It has been established that faulted dipoles are formed by transformation of primary 60° unfaulted dipoles, provided that the height of the primary dipole is lower than some critical value depending on the stacking fault energy (Steeds 1967, Carter and Holmes 1975, Wintner and Karnthaler 1978). In a geometrically consistent description this transformation can be achieved by dissociation of primary dislocations into Shockley and Frank partials and the subsequent annihilation of Shockley partials in the non-coplanar plane. In this way two families of faulted Frank loops aligned along two h110i directions, not parallel to the primary Burgers vector, can be formed in the primary glide plane (®gure 1 (a)). Each segment of Frank dislocation may dissociate into a Shockley and a stair-rod dislocation in the appropriate plane. Subsequent reaction between non-coplanar Shockley dislocations will lead to the ®nal characteristic `Z’ con®guration of the elongated dipole as shown in ®gure 1 (b). The equilibrium con®guration of the faulted dipole is determined by elastic interaction between the Shockley and the stair-rod dislocations aligned along the length of the dipole (Steeds 1967, Carter and Holmes 1975, Wintner and Karnthaler 1978). Depending upon the type of component dislocations, faulted dipoles can exist in two di erent con®gurations: `Z’ or `S’ shaped (Seeger 1964, Steeds 1967). Only Z-shaped vacancy-typ e faulted dipoles, that is with intrinsic stacking faults, have been identi®ed in the structure of deformed materials (Carter and Holmes 1975, Antonopoulos et al. 1976, Carter 1977, Winter et al. 1978). Under weak-beam di racting conditions the contrast comes from bounding Shockley dislocations running along the length of the dipole (Steeds 1967, Carter and Holmes 1975). Depending upon the direction of the operating re¯ection, di raction contrast is localized either òutside’ or ìnside’ the dipole. In the case of inside contrast, the faulted dipole parallel to the foil is recognized as a strong white line running along the h110i direction. Figures 1 (a) and (b) describe the geometry and the topology of the Frank and faulted dipole aligned along [011], at the intersection of (1111) and (1111) planes. Venables suggested that faulted loops belonging to the primary plane introduced by irradiation (Venables 1961) or by deformation (Venables 1964) may provide the `correct’ twinning dislocation to

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Figure 1. The geometry and the dislocation structure of (a) the Frank dipole and (b) the faulted dipole aligned along the [011] direction intersection of the …111† and …111 † planes.

produce conjugate twins in single crystals of Cu±Al alloys. This idea is developed in the present work. } 3. Experimental procedure Cu±8 at.% Al single crystals of dimensions 3 mm 3 mm 70 mm were grown by a modi®ed Bridgman technique. The initial crystallographic orientation of the tensile axis and of the lateral faces were [541], (12 23) and (11 11) respectively. The crystals were annealed under a vacuum of 10 6 Torr at 973 K for 72 h and then

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cooled to room temperature at a rate of 6 K h 1 . The crystals were deformed in tension at room temperature at a strain rate of approximatel y 10 4 s 1 . The thin foils were prepared by electropolishing in 80% phosphoric acid at room temperature, at a voltage of approximatel y 1.8±2 V; at steady state the rate of thinning was about 200 AÊ s 1 . The specimen was illuminated from behind by a laser beam and observed continuously, the polishing was stopped before perforation occurred when the laser light was just visible through the specimen. The dislocation substructure developed in the samples was examined under weak-beam di racting conditions using a Philips CM12 scanning transmission electron microscope operating at 120 kV. For high-resolution electron microscopy (HREM) observations a JEOL 2010 ®eld emission gun electron microscope operating at 200 kV was used. Twobeam and weak-beam images of chosen areas of the substructure were taken using low order h111i and/or h220i re¯ections available in the section cut parallel to the primary (1111) glide plane. The weak-beam di racting conditions were set at k · g>5 and at k · g>3 for {111} and {220} re¯ecting planes respectively. When necessary, TEM observations were carried out on other sections of the deformed crystal. Special caution was taken to control the sense of the foil normal. This was achieved by asymmetrical dishing of the slice with a pre-marked reference direction, spark cut from a given section; the orientation of the Thompson tetrahedron could always be correlated with the di raction pattern and there was no ambiguity in the sense and the sign of the operating g re¯ection. HREM observations were carried out in thinfoil sections for which the twinning plane was oriented edge on, that is the foil normal was [011], the intersection of the primary and the conjugate (twinning) plane. The lattice images were obtained by including low order symmetrical re¯ections (000, h111i and h200i) available in this orientation in the objective aperture. The defocus of the objective lens was adjusted to give the best resolution. The direct working magni®cations of HREM images were between 5 105 and 106 . HREM observations were carried out on partially twinned samples, so that the matrix could always be di erentiated from the twin phase. } 4. Experimental results Figure 2 shows the stereographi c projection of the matrix orientation used in TEM analysis, the initial [541] orientation of the tensile axis and the (11 11) and (12 23) orientations of the lateral faces of the Cu±8 at.% Al single crystals. Figure 3 shows a tensile stress±strain curve of the [541] Cu±8 at.% Al single crystal deformed at room temperature. The deformation occurs initially by dislocation glide on the [101](1111) primary slip system and the tensile axis of the sample rotates towards [101] around [1111], normal to the cross-plane (®gures 2 and 3). After 73% strain, the orientation of the tensile axis can be calculated to be approximatel y [734] assuming the single-glide formula l1 sin ¶1 ˆ l0 sin ¶0 , where l1 and l0 are the length of the sample after and before extension respectively and ¶1 and ¶0 are the angles between the direction of the tensile axis and the glide direction after and before extension respectively (Schmid and Boas 1968). This represents about 4° of the overshoot from a stable [211] orientation; the primary slip system is predominant up to the point where twinning deformation occurs. Twinning occurs on the conjugate (twinning) (1111) plane in the [121] direction (®gure 2) in the form of a LuÈders type front moving through the sample. The change in the dominant mode of deformation from dislocation glide to twinning occurs abruptly with a profound load drop, as shown in ®gure 3. At this point, the resolved stress on the [121](11 11) twin


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Figure 2. (100) stereographic projection, showing the initial [541] orientation of the tensile axis and the …1111† and …1223† orientations of the lateral faces of Cu±8 at.% Al single crystals used in the present work. [734] is the orientation of the tensile axis at the point of twinning calculated from the extension of the sample with the assumption that the primary slip system is operating only. [121] …1111† is the twinning system.

system is about 110 MPa and can be regarded as the critical stress for twin nucleation. The load drops and corresponding intensive acoustic e ects observed during twinning reveal the occurrence of relaxation processes in the substructure. TEM observations of the dislocation substructure were carried out at di erent stages of deformation, which are indicated by arrows in ®gure 3. Figure 4 shows a weak-beam analysis of the substructure in the parent crystal after 60% strain (about 200 MPa tensile stress) in a section parallel to the primary glide plane in the g ˆ ‰1111Š and g ˆ ‰1111Š re¯ections. Although the dislocation substructure at this stage of deformation exhibits complicated three-dimensional arrangement, one can easily identify the following elements.

(i) Primary dislocations with a total Burgers vector of (a/2)[101] are shown by arrows in ®gure 4. They are dissociated into partials bounded by a stacking fault.

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Figure 3. The tensile stress±tensile strain characteristic for Cu±8 at.% Al single crystals with [541] initial orientation of the tensile axis, deformed at room temperature. The arrows indicate stages at which TEM observations were carried out.

(ii) A Lomer±Cottrell network formed by reaction between primary and conjugate dislocations is shown by small arrowheads in ®gure 4 (b). (iii) Faulted dipoles are indicated in ®gures 4 (a) and (b) by large arrowheads. There are two families of faulted dipoles present in the substructure. One family, aligned along the [011] (intersection of primary and conjugate plane), is visible only when the (1111) planes are in the re¯ecting position (®gure 4 (a)); hence the Burgers vector is …a=3†‰11 11Š The other family is aligned along ‰1110Š (intersection of the primary and the critical plane). These dipoles are visible only when imaged using the g ˆ ‰1 111Š re¯ection and therefore the Burgers vector of this family is …a=3†‰111Š. It is di cult to determine the density of faulted dipoles per unit volume of the crystal, since many of them are present in the form of dot-like loops. However, based on the TEM observations shown in ®gure 4, it is clear that faulted dipoles are prominent elements of the dislocation substructure. They are unevenly distributed in the crystal volume, being present in abundance within densely dislocated areas. At this stage there is a range of dipole lengths from 0.1 to 0:2 mm down to barely visible dots, that is presumably beyond the available resolution. A number of images of faulted dipoles aligned along both [011] and ‰11 10Š were examined under di erent weak-beam di racting conditions using a tilting technique


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Figure 4. Weak-beam observations of the dislocation substructure of the parent crystal after 60% strain (200 MPa) in the section parallel to the primary glide plane. Primary dislocations with a Burgers vector of …a=2†‰101Š are shown by arrows. Lomer± Cottrell dislocations are shown by small arrowheads in (b). Faulted dipoles aligned along the [011] direction intersection of the primary and the conjugate plane, and along the ‰110Š direction intersection of the primary and the critical plane are shown by large arrowheads in (a) and (b) respectively. Note that a large density of two families of faulted dipoles is produced during deformation.

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(Antonopoulos et al. 1976). All faulted dipoles analysed were identi®ed as being of vacancy type in a Z con®guration and their height was less than 100 AÊ . Similarly, Lomer±Cottrell dislocations were found to be present in a low-energy con®guration, that is involving an acute intersection with …a=6†h110i stair-rod dislocations. Further TEM studies were carried out on specimens deformed to 84% strain, that is to the middle of twinning (®gure 3). At this stage of deformation, the sample consists of the parent region consisting of the substructure developed up to the onset of twinning, that is about 73% strain, the twinned region consisting of the transformed substructure of the parent crystal, and the transient region associated with the front of the propagating twin. Figure 5 shows the weak-beam observations of the substructure of the matrix at the front of the propagating twin in a section parallel to the primary glide plane. A qualitatively new dislocation substructure is observed. Instead of a family of faulted dipoles aligned along [011], there is a large density of stacking fault tetrahedra or defects indicating disintegration of faulted dipoles, as shown in ®gure 5. Figure 6 shows the analysis of the dislocation content of this substructure at higher magni®cations in three {111} re¯ections. Primary dislocations shown by arrowheads in ®gure 6 (b) are visible with g ˆ ‰11 11Š and g ˆ ‰111Š re¯ections and are invisible with the g ˆ ‰1111Š re¯ection. Fully closed stacking-fault tetrahedra and truncated tetrahedra are shown by arrowheads in ®gure 6 (a). The analysis outlined in ®gure 6 indicates a possible reaction of primary dislocations with faulted dipoles, which may lead to di erent structural transformations of this defect. In the case of short dipoles these interactions may lead to the transformation of dipoles into stacking-faul t tetrahedra as shown in ®gures 5 and 6. For long dipoles the reaction will lead to unlocking one arm of the dipole owing to formation of an …a=6†‰121Š twinning dislocation which, as will be discussed later, is free to move in the conjugate plane and can produce one layer of twin under su cient stress. The twinning source, which may be obtained in this way, is shown at S in ®gure 6 (b). Valuable information about the propagation of the twin can be obtained using the HREM technique. Figure 7 shows two examples of the dislocation arrangemen t at the tip of a propagating twin commonly encountered in Cu±8 at.% Al crystals during the early stages of twinning deformation. Figure 7 (a) shows both a macroscopic twin of roughly 55 atomic planes in thickness and, below it, an arrangemen t of twinning dislocations forming a narrow lenticular twin. There are only one or two twinning dislocations at the tip of the twin, with the other partial dislocations positioned behind as shown by arrows. Figure 7 (b) shows twinning fronts advancing through the crystal, most probably generated from di erent sources. Twinning dislocations arranged one on top of the other on …1111† lattice planes, in the form of an incoherent twin boundary, approach each other from opposite directions (shown by arrows). If they operate on the same plane, they will annihilate and create a macroscopic twin. This leads to the formation of residual defects or ledges at the interface as shown by arrowheads in ®gure 7 (b). Further TEM observations were carried out on fully twinned crystals, that is at about 95% strain as indicated by the arrow in ®gure 3. The intrinsic feature of the twinned substructure shown in ®gure 8 is the presence of the unevenly distributed wide stacking faults one to three layers thick extending along the interface plane (twinning plane). Characteristic arrangements of two to three straight dislocations at the interface are indicated by arrowheads in the micrograph. Such arrangements of closely spaced straight dislocations were found only in the twinned substructure and it is clear that they are strictly related to the presence of stacking faults. Observed


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Figure 5. Weak-beam observations of the substructure of the matrix at the front of the propagating twin (approximately 73% strain) from the section parallel to the primary glide plane. A large density of stacking-fault tetrahedra and truncated tetrahedra (indicating disintegration of the faulted dipoles) is visible in the structure. The arrowheads in (b) indicate the probable sources of the twinning dislocation. See text for details.

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Figure 6. (a)±(c) Analysis of the substructure from ®gure 5 at higher magni®cations in three {111} re¯ections. Primary dislocations are shown by arrowheads in (b). Fully closed stacking-fault tetrahedra and truncated tetrahedra are shown by arrowhead symbols in (a). The analysis indicates that reaction between the primary …a=2†‰101Š dislocation and the …a=3†‰111Š faulted dipole leads to disintegration of the faulted dipoles. A twinning source is shown at S in (b). (d) Traces of all {111} planes on the observation plane. See text for details.

stacking faults on {111} planes, non-coplana r to the twinning plane and shown by arrows in ®gure 8, are probably related to the relaxation of the interfacial stresses. } 5. The origin of the twinning dislocation: twinning source The consequences that may arise after a primary dislocation reacts with a faulted dipole can be inferred from the analysis presented in ®gure 6. The sequence of events is shown in ®gure 9. The Frank dipole considered is su ciently long that the e ect of dissociation may safely be neglected. Therefore, for simplicity, dissociation of the primary …a=2†‰101Š dislocation and the Frank …a=3†‰11 11Š dipole is not considered. The …a=2†‰101Š dislocation gliding on the …1111† primary plane approache s a Frank (faulted) dipole, as shown in ®gure 9 (a). The interaction between the primary and Frank dislocations proceeds along [011] direction, the intersection of the pri-


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Figure 7. HREM observations of twin propagation in Cu±8 at.% Al single crystals during the early stages of twinning. (a) The parent crystal (matrix), a new twin of roughly 55 atomic planes of thickness and also a lenticular-shaped twinning front. Twinning dislocations forming this twinning front are shown by arrows. (b) Non-coherent twinning fronts generated from di erent sources are shown by arrows; residual defects or ledges at the interface are shown by arrowheads.

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Figure 8. Weak-beam observations of the twinned substructure in the section parallel to …001 †T plane in the twin. Trans-substructural stacking faults extended along the …111 † twinning plane are visible in the micrograph. Con®gurations of two to three straight dislocations at the interface are indicated by arrowheads. Stacking faults on other {111} planes, non-coplanar to the habit plane inside the twin, are indicated by arrows.

mary and the conjugate planes, and leads to the formation of an …a=6†‰121Š partial dislocation according to the reaction …a=2†‰101Š ‡ …a=3†‰1 111 1Š ˆ …a=6†‰121Š (®gure 9 (b)). The …a=6†‰121Š Heidenreich±Shockley dislocation produces the correct shear displacement to generate an intrinsic stacking fault on the …11 11† conjugate plane during deformation. This dislocation is pinned at two nodes N1 and N2 (®gure 9 (b)). Since it is bounded by a sessile Frank loop, the whole con®guration represents a source that can be activated under appropriate stress conditions. Depending upon the position of the primary dislocation with respect to the twinning dislocation, four di erent con®gurations of twinning source, shown in ®gures 9 (c)±( f ), can be considered. The con®gurations shown in ®gures 9 (c) and (d) are equivalent to the con®gurations shown in ®gures 9 (e) and ( f ) respectively. Because of the same natures of nodes formed, that is nodes at N1 and N2 are `true generating nodes’ in all cases (for the discussion of the type of dislocation nodes in pole models for twinning, see Sleeswyk (1974)), only two source con®gurations are geometrically independent. They di er in the position of the primary dislocation with respect to the faulted loop. Figures 10 and 11 show the initial stage of the operation of the two sources shown in ®gures 9 (c) and (d) in the perspective view on the …11 11† twinning plane. For clarity, only the Frank jog is represented in the diagrams.


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Figure 9. (a)±( f ) Sequential stages of the interaction of the primary …a=2†‰101Š dislocation with a …a=3†‰111Š Frank (faulted) dipole. (c)±( f ) Possible source con®gurations that can be obtained depending upon the orientation of the primary dislocation with respect to the faulted dipole. Note that, because the natures of the nodes formed at N1 and N2 are the same, the con®guration in (c) is geometrically equivalent to the con®guration in (e), whereas the con®guration in (d) is equivalent to the con®guration in ( f ). See text for details.

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Figure 10 (a) is a double-dislocation pole source with two primary dislocations of antiparallel Burgers vector (drawn as screw dislocations) anchored at nodes N 1 and N2 by a Frank dislocation. During each revolution, two opposite arms of the twinning dislocation operating on the same glide plane interact outside the nodes N1 and N2 and produce both a close twinning partial dislocation loop outside the nodes N1 and N2 and a segment of the twinning dislocation, between these nodes (®gure 10

Figure 10. (a) The initial stage of the operation of a double dislocation pole twinning source (source A (®gure 9 (c))) in the perspective view on the …111† twinning plane. (b) Twinning dislocation after the ®rst revolution is stacked one interplanar spacing above the Frank jog. This stress barrier may terminate further growth of the twin by the pole mechanism. Note that, neglecting the passing stress problem in (b), this source will develop half-space twinned crystal and the residual Frank dipole will be left at the twin±matrix interface.


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(b)). The outside loop can further expand under the applied stress, producing a stacking-faul t layer. The inner segment of the twinning dislocation can repeat the sweeping movement around the pole dislocations, nucleating another twinning loop on an adjacent …11 11† plane (®gure 10). However, in order to continue the helical movement around the primary dislocation poles, the Shockley dislocation will have to break away from the Frank loop after the ®rst complete revolution, as shown in ®gure 10 (b). Figure 11 shows a double dislocation pole source with two primary dislocations of the parallel Burgers vector (drawn as screw dislocations) anchored at nodes N1 and N2 by a Frank loop. Figure 11 (b) shows the stage wherein the opposite arms of Shockley dislocations must break away from each other in order to continue the

Figure 11. Di erent stages of the operation of a double dislocation pole twinning source (source B (®gure 9 (d))) in the perspective view on the …111† twinning plane. (a) The nucleation of the twinning dislocation. (b) In order to develop a macroscopic twin, the twinning dislocations have to break away from each other during the ®rst revolution. (c) Another stress barrier is associated with the Frank dislocation. A Frank dislocation and two twinning partial dislocations, after the ®rst complete revolution, will form a characteristic `triplet’ shown. (d) The arrangement of the twinning dislocations stack at the Frank dipole, forming the block of untransformed matrix inside the twinned crystal. Note that, neglecting the passing stress barriers in (b) and (c), this source will develop a full-space twinned crystal and the residual Frank loop will be immersed inside the twinned lattice. See text for details.

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spiralling movement and to cause the twin to grow. The twinning dislocation will also experience another energy barrier during its spiralling movement, coming from the Frank loop (Frank jog) as shown schematically in ®gure 11 (c). The twinning dislocation may in fact be stuck at the Frank loop during each revolution, creating a block of the untransforme d matrix between the nodes N1 and N 2 , as shown schematically in ®gure 11 (d). The growth of the twin will proceed outside the nodes N1 and N2 , that is outside the faulted dipole. In both cases considered above, the pole, the Frank and the twinning dislocations form `true generating nodes’ at N1 and N2 ; so the sense of spiralling of the pole dislocations is such that, apart from the passing stress barriers, there is no geometrical di culty in making the twin grow by the pole mechanism. The frequency of the interactions between primary dislocations and faulted dipoles which leads to the formation of the `correct’ twinning dislocations in the substructure is reasonably high. Figure 12 shows an example of the twinned structure, wherein the existence of these double-pole con®gurations is clearly visible. The TEM observations shown in ®gure 12 were carried out in a section with ‰5 511ŠT foil normal in the twin; faulted dipoles, formed in the matrix at the intersection of the primary and the conjugate (twinning) plane, are also parallel to the foil in the

Figure 12. Two-beam bright-®eld observations of the twinned substructure in Cu±8 at.% Al single crystals with the twinning plane in the re¯ecting position, g ˆ ‰111Š. Selected …a=2†‰020Š dislocations, transformed products of primary …a=2†‰101Š dislocations by twinning, are shown by arrows. Defects shown by arrowheads exhibit strong di raction contrast, which indicate that they are not simple …a=3†‰111Š Frank dipoles but composite defects. Conservation of the Burgers vector at the nodes indicates that the total Burgers vector of these defects has to be …a=2†‰020Š. Note the high density of defect con®gurations similar to these considered in ®gures 9 (d) and 11.


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twinned crystal. One can observe a high density of defects, indicated by arrowheads in ®gure 12, aligned along [011], the intersection of the plane of the foil and the twinning plane. They exhibit strong wide di raction contrast, clearly indicating defects more complex than single dislocations. These defects are composite Frank loops with the total Burgers vector of …a=2†‰0220ŠT . It is known (Basinski et al. 1997) that the …a=2†‰101Š Burgers vector of the primary dislocation is transformed by twinning to …a=2†‰02 20ŠT . There is a close correlation between the distribution of these defects in the twinned crystal and the distribution of long faulted dipoles in the matrix prior to twinning. Pole dislocations indicated by arrows in ®gure 12 have been identi®ed as …a=2†‰0220ŠT dislocations inside the twin. } 6. Discussion Several conditions imposed on the deformed material must be simultaneously satis®ed to generate twinned lattice by dislocation mechanisms. In the following section we discuss these conditions with reference to the twinning sources considered in } 5. 6.1. The origin of the twinning dislocation Structural observations presented in } 4 indicate that the …a=6†‰121Š Shockley dislocation required to produce twinning deformation in Cu±8 at.% Al single crystals is obtained in the reaction of the primary …a=2†‰101Š dislocation with one arm of a Frank (faulted) dipole. It is then mandatory that the glide of the primary dislocation occurs on exactly the same lattice plane as the appropriate Frank dislocation, so the reaction …a=2†‰101Š ‡ …a=3†‰11111Š ˆ …a=6†‰121ŠM proceeds along the length of the dipole, that is along the [011] direction, the intersection of the primary and the conjugate planes (®gure 9). The product of this reaction, an …a=6†‰121Š twinning dislocation, is free to move in the conjugate plane. On the contrary, the reaction of the same primary dislocation with the other arm of the dipole does not generate the twinning dislocation …a=2†‰101Š ‡ …a=3†‰1111Š ˆ …a=6†‰52 25ŠM ; thus, the requirement for the presence of a `correct’ dislocation to nucleate twinning is not satis®ed. As discussed in } 2, faulted dipoles are transformed from primary unfaulted dipoles and their height is usually less then 10 nm. Despite the relatively small dimensions of these defects, interactions of primary dislocations with faulted dipoles are expected to occur frequently during the deformation, since the substructure contains a large number of Frank±Read sources which produce dislocations on the same lattice planes as those occupied by faulted dipoles. The equilibrium con®guration of the faulted dipole is determined from the force balance between component dislocations, that is the net force per unit length of each dislocation is zero at equilibrium (Steeds 1967, Carter and Holmes 1975). Replacement of the Frank dislocation by the Heidenreich±Shockley dislocation in one arm of the dipole destabilizes the whole con®guration. This may lead either to the transformation of short dipoles into stacking fault tetrahedra as seen in ®gures 5 and 6, or to a twinning source in the case of long dipoles as shown in } 5. The estimated critical length of the faulted dipole which can act as a source for twinning dislocation in the present deformation conditions is around 13 nm (1 ˆ ¬·b=½ where ¬ 0:33, · 30 GPa, b 0:147 nm and ½ 110 MPa). There is a great preponderanc e of dipoles in the substructure of the parent crystal whose length exceeds this critical value (®gure 4). This agrees well with the observed length distribution of defects believed to act as sources of the twinning dislocation.

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6.2. Twinning source: geometry Let us now return to the geometrical di erences between two source con®gurations obtained from faulted dipoles. The con®guration shown in ®gure 10 (see also ®gure 9 (c) or (e)), from now on called source A, represents the situation wherein both arms of the primary dislocation are evenly positioned on the same side of the faulted loop. This con®guration can be de®ned as a double-pole dislocation source with two pole dislocations of antiparallel Burgers vector (®gure 10 (a)). It is similar to the model described by Song and Gray (SG) (1995). Di erences between the source A and the SG model are related to the origin of the twinning dislocation and also to the spatial development of the macroscopic twin. In the SG model, the twinning partial originates from the dissociation of a glide dislocation. This requires a mechanism of anchoring for the other component partial dislocation and it is di cult to envisage this process even in the dense dislocation substructure of the parent crystal prior to twinning. Venables (1961) came to the conclusion that twinning dislocations cannot be activated from glide dislocation at least in medium- and high-stacking-fault-energ y materials. Also, the SG mechanism would produce full-space twinned crystals with one layer of stacking fault (old matrix) terminated by the Heidenreich±Shockley dislocation, immersed in the fully twinned lattice. In our case, the twinning loop can wind up on one side of the Frank loop only (®gure 10 (b)); the source A produces half-space twinned crystals, with the residual Frank dipole being left at the matrix±twin interface. The con®guration shown in ®gure 11, from now on called source B, represents the situation wherein the poles of primary dislocation are positioned on the opposite side of the faulted dipole, forming a characteristic con®guration with an èxtended’ sessile jog. This case can be described as a double-pole source with two pole dislocations of the same Burgers vector. Source B is similar to the con®guration considered by Basinski, Szczerba and Embury (BSE) (1994), wherein the Frank dipole corresponds to the superjog in the BSE model. The di erences between source B and the double-pole source of BSE are again related to the origin of the twinning dislocation. In the latter model, the twinning dislocation originates from the dissociation of the perfect superjog. This initially produces two àntigenerating nodes’ of Cottrell±Bilby (1951) type (see also Sleeswyk (1974)) and, to allow for the growth of the twin, one must incorporate the node conversion mechanisms as suggested by BSE. The correct spiralling of the pole dislocation leading to growth of the twin by the pole mechanism in source B (also in source A), that is `true generating nodes’, is formed during the reaction between the primary dislocation and the faulted dipole. Thus, there is no necessity for node conversion to develop the twin from this source. The growth of the twin from source B proceeds by spiralling movement of the twinning dislocation around two poles (one arm winding up, and the other down). This produces a full space twinned crystal, the residual Frank loop being immersed inside the twinned lattice.

6.3. The growth of the twin: passing barriers As already mentioned, a Heidenreich±Shockley dislocation derived from a faulted dipole su ers classical passing stress problems during the initial stage of twin growth. Di erent energy barriers will be encountered during the operation of source A and source B respectively. They are considered in the following.


185

In order to continue the spiralling movement around the primary dislocation poles, the twinning Shockley in source A has to break away from the Frank jog (Frank dipole) as was shown in ®gure 10 (b). The Shockley dislocation is positioned on the adjacent …11 11† twinning plane above the Frank dislocation; so the passing stress is of the order of the theoretical strength of the material. Since this stress barrier is unlikely to be matched by any type of internal stress concentration, the twinning dislocation will be locked at the Frank jog after one revolution, without possibility of further movement. Source A will produce only one layer of twin and the doublet of Frank and Shockley dislocations may be left at the interface, as can also be inferred from ®gure 9. In the case of source B, two di erent situations must be considered. The barrier is encountered during the ®rst revolution when the spiralling arms of Shockley dislocation, outside the nodes N 1 and N2 (®gure 11 (b)), are separated by twice the interplanar spacing. The passing stress can be calculated from the following equation: ½ ˆ ·b=8º…1 ¸ †z where · is the shear modulus, b is the Burgers vector of the Heidenreich±Shockley dislocation, ¸ is the Poisson ratio and z is the separation between the passing dislocations (Haasen 1978). The estimation gives a stress of the order of ·=50 or about 600 MPa for Cu (· 30 GPa, b 1:47 AÊ, · 0:33 and z 4:17 AÊ). In most practical situations, the loop is not symmetrical and the passing dislocations are not parallel, which may reduce the e ective passing stress. Furthermore, the arms of the spiral sweep a large area, the moving dislocations may become jogged which, in some cases at least, decreases the passing stress. Finally, the kinetic energy of the moving dislocations may be large enough to allow for passing. The collision of the passing Shockley dislocations outside the nodes N1 and N2 , may proceed on the very small length in which case, the e ective passing energy is low. Standard calculations show that the energy di erence E per unit length between stable and unstable positions of an edge dislocation dipole is independent of its height. In our case, E ˆ ‰·b2 =12º…1 ¸ †Š…1 ln 2† 10 2 ·b2 . Here b is the magnitude of the Burgers vector of the perfect dislocation. The activation energy per unit length for passing between two Shockley dislocations is therefore of the order 0.05 eV per length b and can be acquired even from thermal ¯uctuations. For the particular case of twinning in Si, it was concluded by SG that thermal activation provides su cient energy to break the dipole of Shockley dislocations positioned on adjacent {111} planes. It is expected, however, that in many cases the above argument will not apply. Then, the straight segments of Shockley dislocations may be trapped, forming the characteristic dipole of two straight Shockley dislocations with associated stacking faults, similar to the con®gurations seen in ®gure 8. They subsequently may relax and form stacking faults on non-coplanar {111} planes. Another energy barrier for the operation of source B is encountered when two Shockley dislocations, after they break away from each other, approach a Frank jog (®gures 11 and 13). The passing stress for a single Shockley dislocation, the same as in source A, is of the order of the theoretical shear stress and therefore it is expected that the single twinning dislocations will be locked at the Frank dipole. However, contrary to source A, the twinning dislocation can continue its spiralling motion outside the nodes N1 and N 2 , whereas it will build up a stack of twinning dislocations at the Frank jog between the nodes N1 and N2 (®gures 11 (d) and 13). The propagatio n of the twinning front outside the nodes N1 and N2 may proceed by the lenticular type of twinning front (®gure 7 (a)) or as a wall of twinning partials one on

186


Figure 13. (a)±( f ) Cross-section through the Frank jog showing the sequence of the events during the operation of the twinning source derived from the faulted dipole. (d) The twinning dislocations stacked at the …a=3†‰111Š dislocation due to the stress barrier will form a non-coherent boundary. (e) This boundary can be rearranged to a con®guration in which the two Shockley dislocations and the Frank dislocation form an …a=2†‰020Š jog ……a=3†‰111Š ‡ …a=6†‰1121Š ‡ …a=6†‰1121Š ˆ …a=2†‰020ŠT †. ( f ) The boundary can be released from the sessile jog under su cient stress. The broken lines represent physical …111 † twinning planes on which Shockley dislocations operated. Note that there are planes inside the twinned material, which are not transformed by twinning dislocation.

top the other in the form of an incoherent twin boundary (®gure 7 (b)). The question arises as to whether the stress which builds up because of the stacking of a number of twinning dislocations between nodes N1 and N2 will eventually be su cient to release these dislocations from the Frank jog. To rationalize such a possibility, it should be noted that, with respect to the twin lattice, the triplet Frank and two Heidenreich±Shockley dislocations represent the …a=2†‰02 20ŠT dislocation ……a=3†‰1111Š ‡ …a=6†‰1121Š ‡ …a=6†‰1121Š ˆ …a=2†‰0220ŠT ; see also the BSE paper). The


187

boundary formed between nodes N 1 and N 2 can thus be rearranged to the con®guration shown in ®gure 13 (e), wherein the ®rst twinning Shockley of the boundary is now at a distance of two interplanar spacing from the …a=2†‰02 20ŠT jog. Since the stress necessary to tear the ®rst Shockley partial away from the jog is rather large, one expects the formation of an incoherent twin boundary as indicated in ®gure 13 ( f ). Such incoherent twin boundaries are commonly observed in the structure of the present alloy (®gure 7). The release of the incoherent twin boundaries from the sessile jogs represents an e cient relaxation process in the substructure. It is likely to be responsible for ¯ow instabilities (jerky ¯ow) and related acoustic e ects observed during twinning in a number of fcc metals and alloys. The problem reduces then to the estimation of the passing stress between the …a=2†‰02 20ŠT jog and the dislocation wall. This point deserves speci®c investigation (Mitchell and Hirth 1991, Kamat et al. 1996) and will be the subject of a further paper. Finally let us consider more closely the energetics of the interaction of two twinning dislocations with a Frank jog shown in ®gure 13 (c), that is the interaction energy when two Shockley dislocations are attempting to pass a Frank jog at the same time. It is shown in Appendix A that, in the case of a jog of pure edge character, the interaction energy of the triplet is zero. Thus, there is no energy barrier for twinning dislocations to pass the jog and to continue their spiralling movement provided that twinning dislocations are passing the jog at the same time. This is a quite general conclusion applicable also to some other models. Based on above argument, it can be shown for example that a source proposed by BSE does not su er an energy barrier and can develop a macroscopic twin by the pole mechanism. } 7. Summary In making this analysis, we do not claim to have found a universal process, which is the clue to mechanical twinning in fcc metals. We do think, however, that twinning is a heterogeneous process and that it is likely that various mechanisms may operate. We only wish to show, consistently with experimental observations, that the interactions between primary dislocations and faulted dipoles may act as a twinning sources in deformed [541] Cu±8 at.% Al single crystals. The twinning …a=6†‰121Š dislocation results from the reaction between primary …a=2†‰101Š dislocation and the Frank …a=3†‰1 111 1Š dislocation. However, not all these interactions will lead to twinning sources. Depending upon the orientation of the primary dislocation with respect to the dipole, two geometrically di erent con®gurations of the twinning source (de®ned as source A and source B respectively) may be formed in the lattice. They both consist of a `correct’ …a=6†‰121Š twinning dislocation mobile in the conjugate plane bounded by a Frank loop and two primary dislocations acting as pole dislocations. There are no geometrical di culties to impede production of a macroscopic twin by either source. However, because of the passing stress barrier, it is expected that only source B can operate during deformation. The growth of the twin proceeds by the motion of the Shockley partial around the primary poles. There are two types of stress barrier encountered during the operation of source B. The ®rst is associated with the operating twinning dislocations passing each other two interplanar spacing apart, outside the sessile jog. This barrier is of the order of 0.05 eV per unit length of Burgers vector and can be surmounted by the thermal ¯uctuations. The second stress barrier is associated with the sessile Frank jog. It is shown that this energy barrier is reduced to zero if two twinning dislocations pass the sessile jog simultaneously. Otherwise, twinning dislocations may stack at the Frank jog, produ-

188


cing a block of untransformed matrix inside the twinned material. The stress that builds in this way may be occasionally released, producing load drops and acoustic e ects, which are observed during twinning. The direct consequences of the geometry of the proposed source, the arrangement of twinning dislocations at the Frank dipole and the propagation of such twinning front through the lattice are long layers of untransformed matrix left inside the twinned material. With respect to the twin lattice these layers are identi®ed as stacking faults extended along the interface. This conclusion is con®rmed by TEM observations of the substructure of the twinned crystal, which shows a large density of stacking faults extended along the twinning plane. Both the expected source con®guration and the pro®le of the expected twinning front produced by the proposed source are compatible with the TEM observations. In the real dislocated crystal, there will be a number of twinning sources activated in di erent parts of the sample. The twinning fronts produced by these sources will add up to form a thicker propagating incoherent twin boundary. If these boundaries approach each other from opposite directions, they produce a homogeneous slab of twinned material. It has not escaped our attention that the problem of the passing stress is not completely solved. However, some indications that passing of Frank jogs by twinning dislocations occurs in practice can be inferred from in-situ observations of deformation experiments carried out on thin foils of ®-TiAl alloy, by Farenc et al. (1993a,b). Although the exact distance between sessile and passing dislocations was not determined, the experiment of Farenc et al. showed that, under the stress conditions achieved during the deformation of thin foils in the electron microscope, Shockley dislocations can be unlocked from the sessile Frank dislocations. The passing of Frank jogs by twinning dislocations was the mechanism controlling the growth of the twin. ACKNOWLEDGEMENTS We are grateful to the late Professor Z.S.Basinski, who inspired us to work on this subject and to Dr M.S.Szczerba for stimulating discussions. The ®nancial support of the Natural Sciences and Engineering Research Council (Canada) is gratefully acknowledged.

APPENDIX A In order to calculate conveniently the interaction energy between one Frank dislocation, bF ˆ …a=3†‰1111Š, and one Shockley dislocation, bS ˆ …a=6†‰121Š, parallel to [011], we use the framework depicted in ®gure A 1. In this framework, p a a p bF ˆ p ‰010Š; bS ˆ ‰ 6 0 3 2Š; l ˆ ‰001Š; r ˆ ‰xh0Š: 3 12 The energy of interaction per unit length can be calculated from the following formula (Nabarro 1952): ³ ´ ³ ´ r r ·…bF · l†…bS · l† · Iˆ ln ‰…bF l†…bS l†Š ln 2º ra 2º…1 ¸ † ra …A1† £ ¤£ ¤ · …bF l† · r …bS l† · r ; 2º…1 ¸ †r2


189

(a)

(b) Figure A 1. The framework used in the calculation of the interaction energy between Frank jog and the Shockley dislocation: (a) coordinate system; (b) coordinates of the …a=6†‰121Š Shockley dislocation at S and …a=3†‰111Š Frank dislocation at F.

where ra is the cut-o radius. In our framework, we have ¡ ¢ bF · l ˆ 0;

¡

p ¢ a 2 ˆ bS · l ; 4

Since …bF · l†…bS · l† ˆ 0 and …bF Iˆ

¡

bF

l† · …bS

· 2º…1

¸

†r2

£

p ¢ a 3 ˆ l ‰100Š; 3

¡

¤£ l† · r …bS

¤ l† · r :

…bF

bS

p ¢ a 6 ˆ l ‰010Š: 12

l† ˆ 0, equation (A 1) reduces to

On the other hand, ¡ bF

p ¢ a 3 l · rˆ x 3

and

¡ bS

p ¢ a 6 l · rˆ h: 12

…A2†

190


Figure A 2. Passage of an …a=3†‰111Š Frank jog at F by two …a=6†‰121Š Shockley dislocations at S for which the interaction energy is zero. In this framework dislocation lines are along the Z axis, the Frank dislocation is at (0, 0) and the Shockley dislocations are at (x; h) and (x; h) respectively.

The interaction energy per unit length between two dislocations may be written Iˆ

xh ·a2 : 2º…1 ¸ † x2 ‡ h2

…A3†

The interaction energy depends upon the sign of h; hence, in the situation in ®gure A 2, where the Shockley dislocations are at (x; h) and (x; h), the total interaction energy of the Frank dislocation with two Shockley dislocations is zero.

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