Dislocation self-organization under single slip straining and dipole properties

Materials Science and Engineering A 483–484 (2008) 191–194

Dislocation self-organization under single slip straining and dipole properties Yu-Lung Chiu a , Patrick Veyssi`ere b,∗ a

Department of Chemical and Materials Engineering, The University of Auckland, Private Bag 92019, Auckland City, New Zealand b LEM, CNRS-ONERA, BP 72, 92322 Chatillon, France Received 4 June 2006; received in revised form 3 January 2007; accepted 17 January 2007

Abstract Spontaneous microstructural organization under single slip is investigated by transmission electron microscopy. The formation and the structure of dislocation entanglements are analyzed on three types of fcc-based systems, Al, Cu and TiAl, all deformed by 1 1 0 {1 1 1} slip. Differences are found that depend on stacking fault energy and lattice friction. The importance of dipolar configurations is outlined. Selected properties of dipoles are analyzed theoretically under isotropic and anisotropic elasticity in cubic systems. At variance from screw and near-screw dipoles, the stress-free equilibrium angle of an edge dipole is little dependent on the material’s elastic anisotropy. In Cu, for instance, a screw dipole is at equilibrium at around 59◦ from the slip plane, and this angle is unchanged over a range of dislocation characters of approximately ±20◦ . On the other hand, given a dipole height, the passing stress is a maximum in the screw orientation. It is, however, not a minimum in the edge orientation. Static and dynamic dipole properties are but little affected by dissociation down to a dipole height of the order of a few times the dissociation distance. © 2007 Elsevier B.V. All rights reserved. Keywords: Dislocation self-organization; Single slip; Dipoles; Passing stress

1. Introduction Often concealed by other mechanisms such as junction formation, dipolar interactions are best manifested under single slip conditions and/or in certain low-symmetry crystals. At temperatures low enough to hinder atom transport, deformation microstructures exhibit a more or less pronounced tendency towards self-patterning forming tangles of dipolar and multipolar configurations (an early account of this property can be found in Ref. [1]). Although it is not yet clear how dislocations all with the same Burgers vector manage to self-organize forming braids, sometimes under rather regular patterns, it makes no doubt that dislocation dipoles play a substantial role in this property [2]. The characteristics of dislocation bundles built up in the early stages of monotonic straining differ from one material to another even within the same crystal structure. At low temperatures, dipole formation makes use of cross-slip exclusively. The

∗

Corresponding author. E-mail address: [email protected] (P. Veyssi`ere).

0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.01.169

stacking fault energy influences dislocation organization rather markedly not only via cross-slip processes but also indirectly because of dipole transformations forming sessile faulted debris. Substantial microstructural differences are expected between crystallographically similar systems deformed under comparable conditions depending, of course, on stacking fault energy but also on lattice friction and possibly on elastic anisotropy. The object of the present, brief contribution is twofold. The first part is aimed at pointing out changes in both overall organization and properties of dipolar configurations, depending on material’s parameters. The second part addresses certain static and dynamical properties of dipoles derived from isotropic and anisotropic elasticity in relations with microstructural observations. 2. Experimental results To apprehend the influence of the above-mentioned material’s parameters in dislocation self-organization, three comparable single-crystalline materials, Cu, Al and TiAl, have been deformed under similar conditions. In brief, the samples were oriented to force single slip operation to 4% of resolved shear strain slip and at a temperature low enough to hamper bulk diffu-

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sion. In Cu and Al the stacking fault energies are moderate and high, respectively. Both deform via dislocations with Burgers vectors of 1/21 1 0 not subject to detectable resistance from the lattice. TiAl is a slightly more complex material. Ordered under a fcc-based structure and tetragonal (c/a = 1.02), it is comprised of Ti and Al planes that alternate in the c = c[0 0 1] direction. This latter property defines unit translations, hence Burgers vectors of the 1/21 1 0] and 0 1 1] types (in the h k l] notation, permutations are restricted to the first two indices). If uniaxially loaded in the vicinity of the [0 2 1] orientation, TiAl single crystals deform by 1/21 1 0] dislocations [3], dubbed ‘ordinary’ dislocations. It is well known that ordinary dislocations exhibit a slightly extended core. They are submitted to a rather large lattice friction, up to 100 MPa, as attested by the presence of cusped dislocation in large numbers [4]. The samples were deformed under comparable conditions and thin foils for transmission electron microscopy (TEM) were sectioned as close as possible to the operating slip plane; deformation procedures and sample preparation are described elsewhere [5]. For lack of space, the present experimental results will be restricted to three examples, one per crystal investigated. As illustrated in Fig. 1(a), self-organization in Al results in inhomogeneous tangles comprised of essentially edge dipolar configurations interconnected by segmented dislocations oriented on average parallel to the screw direction (Fig. 1(b)). Segmentation results from the intersection of the extremities of prismatic loops by mobile dislocations [6,7], a direct evidence of the so-called collinear junction [8]. The configuration shown in Fig. 1(a) actually represents one heterogeneous single tangle. Not shown is the property that the tangles are several tens of micrometers apart separated by crystal regions remarkably exempt of dislocation debris, therefore indicating that in Al there is no obstacle to the sweeping of loops towards pre-existing agglomerates [2]. Unexpected and specific to Al is the extreme

Fig. 1. A low magnification TEM view of a single crystal of Al deformed in single slip at 77 K and sectioned parallel to its slip plane. (b) A magnified view of the region boxed in (a).

Fig. 2. A low magnification TEM view of a single crystal of Cu deformed to 2% in single slip at 77 K. The defect distribution in the background is comprised of a significant fraction of rectilinear features elongated along either one of the two arrows marking ±60◦ orientations with respect to the Burgers vector, thus typical of faulted dipoles.

variety of loop shapes which suggests that certain markedly rounded debris have resulted from the merging and subsequent reorganization of several loops possibly by pipe-diffusion [5]. The situation in Cu is somewhat different from the above in Al under several respects. The main difference resides in the distribution of faulted dipoles all over the sample, including in between the braids. Difficult to observe under dynamical contrast conditions, faulted dipoles can be seen in the weak-beam view of Fig. 2 as rectilinear rods parallel to either one of the two arrows marked ‘±60◦ ’. Although the dense wall located below the arrow labeled ‘g × b’ (bottom right-hand side of Fig. 2) contains a substantial number of edge dipolar configurations, its main orientation is at 60◦ from b, which suggests that dislocations have accumulated on faulted dipoles or arrays of these forming bundles. In Al where faulted dipoles do not form, the mean orientation of bundles is edge (Fig. 1(a)). The present observations are consistent with only a small fraction of the dislocations, if any, having actually disappeared during thin foil preparation [5]. In our opinion, the scarcity (in Al) and the quasi-total lack (in Cu) of long, meandering dislocations results: (i) from the large free-flight distance of these relative to the foil lateral dimensions and (ii) from the fact that a resolved shear strain of 4%, as given to the present samples, corresponds to the passage of about 20 mobile dislocations in the crystal volume sampled for the TEM investigations. Altogether these properties make it somewhat unlikely indeed that a mobile dislocation could have been retained within the area investigated (about 80 ␮m × 80 ␮m). The low temperature deformation microstructures in Al and Cu are remarkably distinct from that encountered in TiAl (Fig. 3). The frequency of meandering dislocation is substantially higher in TiAl than in Cu and Al, which is fully consistent with a free-flight distance significantly less in TiAl than in Al and Cu. Whereas TiAl exhibits the same kind of debris as those observed in high stacking fault energy Al, its deformation microstructure manifests a moderate tendency towards entanglement. The density of prismatic loops throughout the sample is generally small relative to Al and Cu, although some dense loop agglomerates have been reported [6,7,9]. This is in turn consis-

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lines. Dipole properties are conveniently accounted for by the effective shear stress τ eff defined by τ eff = (τ xy be + τ zy bs )/b where b = [be 0 bs ] = b[sin θ 0 cos θ]. Under isotropic elasticity the effective shear stress is written τ eff = −

μb ξ (ξ 2 [(ν − 2) + ν cos 2θ] 4π(1 − ν)h (ξ 2 + 1)2

+ [(ν − 2) cos 2θ + ν]) where ξ = x/h. From the zeros of this expression that correspond with a stable equilibrium, it can be easily shown that the stressfree equilibrium angle αs = 90◦ is not restricted to screw dipoles but that up to a critical character given by   ν 1 , θc = cos−1 2 2−ν

Fig. 3. TiAl single crystal deformed in compression to 2% at room temperature along [1 5 3]. (b) A magnified view of the area boxed in (a), showing the distribution of prismatic loops and the interaction of these with an individual dislocation (left) and a dipole (upper right).

tent with the lattice friction opposing the sweeping of loops by mobile dislocations. As this has been demonstrated in the case of Ni [10,11], it is instructive to characterize the distribution of prismatic loops in bundles. In the small dipole height limit, however, there are certain experimental precautions that must be taken in conducting TEM measurements [12]. It is, in particular, essential to perform dipole image simulations which require a precise knowledge of the configurations expected, and that in turn depends on the elastic properties of the material under investigation. This was one of the main reasons why the work presented below was initiated. Another objective was to investigate the effects of dislocation character on the dynamical behavior of dipoles as this is regarded as a basic ingredient in the mechanisms contributing to entanglement. 3. Dipole properties It is well known that in isotropic solids the habit plane of dipolar dislocations at equilibrium under no applied stress, makes an angle αs = 90◦ and αe = 45◦ to the slip plane in the screw and edge orientations, respectively, and that the passing stresses in the screw and edge orientations are given by τ ps = μb/4πh and τ pe = μb/8π(1 − ν)h, respectively (μ is the shear modulus, ν the Poisson ratio, b the Burgers vector modulus and h, also dubbed the dipole height, is the distance between the two partners’ respective slip planes). To the authors’ knowledge, comparatively little is documented on dipoles properties in the mixed orientation, on their dependence upon elastic anisotropy and on the influence of dislocation dissociation. We consider a referential Oxyz such that Oy is perpendicular to the slip plane and Oz parallel to the dislocation

it is also the equilibrium angle of mixed dipoles. The angle θ c , which depends only on the Poisson ratio, is quite large, typically in the range 35–41◦ . Beyond θ c the dipole equilibrium angle decreases gradually down to the angle of αe = 45◦ expected in the edge orientation. Dipole properties at equilibrium are strongly affected by elastic anisotropy (Fig. 4). Degenerated in isotropic media (αs = ±90◦ ), the stress-free equilibrium angle αs in anisotropic crystals deviates from the isotropic solution, sometimes rather markedly, and the solution is no longer degenerated. For instance, αs equals 59◦ in Cu, about 61◦ in Ag and Au, 65◦ in Ni, 76◦ in Ge and Si and 85.5◦ in Al. The stress-free equilibrium angles of screw dipoles in Cu, Ag and Au correspond with a dipole habit plane not far from a {1 1 1} cross-slip plane. This result differs from Yoo’s findings on cross-slip properties in Ni3 Al [13] in that Yoo’s analysis referred to two 1/21 1 0 partials with the same sign (and connected by an antiphase boundary). With increasing the edge component, α remains constant and non-degenerated up to some critical angle, θ c . It subsequently assumes two distinct, asymmetrical values while decreasing to near the isotropic solution αe = ±45◦ in the edge orientation. In Cu for instance, αs = 59◦ up to a critical dipole character θ c ≈ 21◦ , and beyond an angle θ * ≈ 41.7◦ , which is almost independent of elastic anisotropy (Fig. 4), the dipole may adopt either one of two asymmetrical equilibrium configurations at −83◦ and +49◦ , respectively. The two equilibrium angles become symmetrical at about ±43◦ upon approaching

Fig. 4. Orientation dependencies of the stress-free equilibrium dipolar angles calculated under elastic anisotropy.

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Fig. 5. Orientation dependence of the passing stress in Cu for a dipole height of 55 nm calculated under elastic anisotropy.

the edge orientation. Dipolar configurations at equilibrium are strongly affected by dissociation but only for dipole heights less than four to six times the stress-free dissociation width of an isolated dislocation [14]. In the most general case of a rectilinear dipole with given character θ, the passing stress, τ p (θ), is calculated as the maximum of |τ eff |. Since in real deformation conditions dipoles are forced to meander throughout a forest of obstacles, the passing stress is actually the stress required to destroy the weakest portion of the dipole, hence the minimum of τ p (θ). It is from this portion that the rest of the dipole can be gradually unzipped through line tension effects. The passing stress in the edge orientation, τ pe (90◦ ), is never a minimum except for ν nearing 1/4. One finds numerically that given a dipole height and for Poisson ratios slightly larger than 1/4 and up to 1/2, τ p (θ) shows a minimum in a mixed dipole orientation thus defining the actual true passing stress of a curved dipole. For ν ≈ 0.35, the minimum occurs at θ ≈ 60◦ , but it is so flat that between 30◦ and 90◦ in character, a dipole will oppose a roughly constant resistance to an applied stress. In isotropic solids, for instance, and in the case of undissociated dislocations (or equivalently of large dipole heights), τ p peaks in the screw orientation at which it exhibits a magnitude at least 33% higher than it does in the 30◦ to 90◦ character range. This latter property, which is illustrated in Fig. 5 in the case of Cu, is worth considering in analyzing constrained deformation conditions as encountered in Cu cycled to saturation. It is indeed hypothesized that the critical stage defining the materials saturation strength in fatigue is related to the lengthening of hairpin-like configurations delimited by two walls of edge multipolar features and to the passing of screw dipoles, and there is an on-going debate as to how one should model this [15,16]. The present results indicate that the passing is certainly eased if dipoles can undergo limited reorientations (Fig. 5). In Cu, dipole

rotations of 30◦ and 45◦ indeed decrease the maximum passing stress, that in the screw orientation, by 15 and 22%, respectively (the maximum stress reduction is of 25% for a rotation of approximately 70◦ ). It is worth emphasizing that published micrographs of deformation microstructures in Cu, pinned under stress by neutron irradiation, provide conspicuous evidence that dislocation with characters between 30◦ and 45◦ are relatively profuse [17]. Depending on local point-pinning debris distribution (see Ref. [14]), TEM reveals that the shape adopted by mobile dislocations is significantly affected. It is therefore quite unlikely that the approach of dipolar dislocations dominantly occurs according to an idealized screw dipole model. In real conditions, the passing would rather occur where this is the easiest, that is, where dislocations are substantially deviated from the screw direction. Under these conditions, the passing stress derived from the height of the screw dipoles left in the material, is thought to be overestimated by 15–20%. This correction is significant in view of the numbers usually confronted in the models [18]. It is noted in addition that the above-mentioned micrographs (i.e. Fig. 7.23 in Ref. [17]) contain piled-up configurations suggesting that the passing may be further assisted by stress concentration due to the grouping of three to four dislocations of the same sign, not infrequent in this figure. Finally, a remark should be made regarding the debris clearly left in the channels [14] but whose contribution to strength is usually neglected [15]. References [1] D. Kuhlmann-Wilsdorf, H.G.F. Wilsdorf, in: G. Thomas, J. Washburn (Eds.), Electron Microscopy and the Strength of Crystals, Interscience Publishers, New York, 1963, pp. 575–604. [2] J. Kratochv`ıl, M. Saxlov`a, Scr. Metall. Mater. 26 (1992) 113–116. [3] H. Inui, M. Matsumoro, D.-H. Wu, M. Yamaguchi, Philos. Mag. A 75 (1997) 395–423. [4] F. Gr´egori, University of Paris VI, 1999. [5] P. Veyssi`ere, Y.-L. Chiu, M. Niewczas, Z. Metallkd. 97 (2006) 189–199. [6] P. Veyssi`ere, F. Gr´egori, Philos. Mag. A 82 (2002) 567–577. [7] P. Veyssi`ere, F. Gr´egori, Philos. Mag. A 82 (2002) 579–590. [8] R. Madec, B. Devincre, L.P. Kubin, T. Hoc, D. Rodney, Science 301 (2003) 1879–1882. [9] F. Gr´egori, P. Veyssi`ere, Philos. Mag. A 82 (2002) 553–566. [10] J. Bretschneider, C. Holste, W. Kleinert, Mater. Sci. Eng. A 191 (1995) 61–72. [11] J. Bretschneider, C. Holste, B. Tipppelt, Acta Mater. 45 (1997) 3775–3783. [12] P. Veyssi`ere, J. Mater. Sci. 41 (2006) 2691–2702. [13] M.H. Yoo, Scr. Mater. 20 (1986) 915–920. [14] P. Veyssi`ere, Y.-L. Chiu, Philos. Mag. 87 (2007) 3351–3372. [15] L.M. Brown, Philos. Mag. 86 (2006) 4055–4068. [16] H. Mughrabi, F. Pschenitzka, Philos. Mag. 85 (2005) 3029–3045. [17] J.C. Grosskreutz, H. Mughrabi, in: A.S. Argon (Ed.), Constitutive Equations in Plasticity, MIT Press, Cambridge, Massachusetts, 1975, pp. 252–326. [18] K.W. Schwartz, H. Mughrabi, Philos. Mag. Lett. 86 (2006) 773–785.