Orientation dependence of the nucleation and growth of partial dislocations and possible twinning mechanisms in aluminum

Journal of the Mechanics and Physics of Solids 60 (2012) 277–294

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Orientation dependence of the nucleation and growth of partial dislocations and possible twinning mechanisms in aluminum Nitin P. Daphalapurkar n, K.T. Ramesh Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD, USA

a r t i c l e i n f o

abstract

Article history: Received 6 July 2011 Received in revised form 4 October 2011 Accepted 27 October 2011 Available online 2 November 2011

Plastic deformation in certain nanocrystalline metals (grain size less than 100 nm) or under certain conditions is associated with profuse activation of partial dislocations with grain boundaries acting as both the source and the sink. We use molecular dynamics (MD) simulations of pure aluminum to investigate the orientation dependence of deformation mechanisms, and show that two types of twins can be generated. Theoretical calculations are used to understand the effect of loading orientation on the activation of twins. The theoretical framework incorporates (a) a tensorial stress state, (b) dislocation loops and (c) non-singular core energies for accurate descriptions of criteria for partial dislocation activation. Both the theoretical predictions and the MD simulation demonstrate that twins should be possible (but rare) in nanocrystalline aluminum. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Nucleation Molecular dynamics Partial dislocations Twinning Nanocrystalline aluminum

1. Introduction Plastic deformation in the face-centered cubic (FCC) metal aluminum is generally dominated by slip through the movement of perfect dislocations because twinning is difficult (due to the high stacking fault energy (SFE) of Al). However, partial dislocations and twinning are observed in NC aluminum. Why and how are these new mechanisms nucleated? It is generally the case that nanocrystalline (NC) metals (grain size less than 100 nm) undergoing plastic deformation exhibit high flow stresses, and new deformation mechanisms may therefore be observed in such metals which were previously inaccessible in their coarse-grained counterparts. Specific observations of new mechanisms in NC FCC metals include (a) profuse activation of partial dislocations from grain boundaries and their propagation (Shan et al., 2004; Wang et al., 2002; Chen et al., 2003), (b) twinning in preferentially oriented grains of Al (Chen et al., 2003; Zhu et al., 2004), and (c) twinning through the so-called co-operative activation of partial dislocations (CAP; Li et al., 2011; Wu et al., 2008) as contrasted to monotonic activation of partial dislocations (MAP; e.g., Wu et al., 2008). A common requirement for all of the above observations is the nucleation of partial dislocations. We seek, therefore, to understand the criteria for homogeneous nucleation of partial dislocations (Tschopp and McDowell, 2008; Aubry et al., 2011), and we focus our studies on aluminum because NC aluminum has the potential to provide very high strength lightweight structural materials (Zhang et al., 2007). Once the nucleation and growth of partials has been understood, the implications for the development of deformation twins can be considered. In this work, we show that the deformation mechanism of twinning through CAP (and its relative probability compared to dislocation slip and traditional twinning through MAP) can be predicted based on the tensorial stress and the activation of partial dislocations.

n

Corresponding author. Tel.: þ1 410 516 8781; fax: þ1 410 516 4316. E-mail address: [email protected] (N.P. Daphalapurkar).

0022-5096/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2011.10.009

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There have been several studies in the literature that associate the emission of partial dislocations from grain boundaries (GBs) with the development of specific deformation mechanisms in nanocrystalline metals such as NC aluminum. While such approaches provide insight, they ignore some critical issues: the nucleation of partial dislocations at GBs is not a sufficient condition for the development of a specific deformation mechanism. Once partial dislocation nucleation has occurred, some finite mechanical stress is required within the interior of the grain for the partial dislocation to propagate away from the grain boundary. Swygenhoven and co-workers (Bitzek et al., 2008; Van Swygenhoven et al., 2006) showed that the elevated levels of stress due to arrangement of atoms at grain boundaries have a very short range (a fraction of atomic spacing). We thus need to account for the interaction of the partial dislocation with the current tensorial stress state within the grain in order to understand the propagation of the partial. Further, there is a curvature associated (Froseth et al., 2004) with the partial dislocations that activate from the grain boundaries. We show in this work that this curvature has a significant influence on the stress required for the growth of the partial dislocation loops and thus on the development of specific deformation mechanisms. The tensorial nature of the stress state results in the orientation dependence of the nucleation and propagation of partial dislocation loops, and therefore of potential twinning mechanisms vis-a-vis perfect dislocation slip. Other contributions in the literature have addressed the nucleation of twins from heterogeneities (Venables, 1964; Lagerlof et al., 2002; Tadmor and Hai, 2003; Kibey et al., 2007a; Liu and Xu, 2009) and as a result of dislocation reactions, along with the orientation dependence on twinning (Mahajan and Chin, 1973). Ogata et al. (2005) probed the energy landscape of deformation twinning in FCC metals Al and Cu using density functional theory (DFT) calculations and calculated an ideal shear stress for Al as 2.84 GPa. Tschopp and McDowell (2008) considered the stress required for homogeneous nucleation of dislocations in single crystal copper under uniaxial loading (as a function of crystallographic orientation). Using molecular dynamics (MD), they examined the influence of both Schmid and non-Schmid stresses under uniaxial compression and tension, and observed a tension–compression asymmetry in dislocation nucleation, including differences in the nucleation of trailing partial dislocations (for Cu). As Tschopp and McDowell (2008) point out, an understanding of the homogeneous nucleation of dislocations is a necessary prerequisite for the understanding of a number of fundamental problems in plasticity, including the onset of dislocation development during nanoindentation, the development of plasticity in nanoscale domains within which heterogeneous sources may be difficult to find, the provision of fundamental information to discrete dislocation dynamics simulations in source limited domains, and the development of criteria for the relative roles of dislocation nucleation and dislocation multiplication. Similarly, studies of the homogeneous nucleation of partial dislocations are important for the development of an understanding of the potential for twinning mechanisms. Specifically, in nanocrystalline Al the core of the dislocation is extended and as a result the partial dislocations play an important role in deformation twinning (Zhu et al., 2004). In this work we investigate the orientation dependence of the nucleation and growth of partial dislocations and the resulting twinning mechanisms in FCC metals, with a specific focus on aluminum (we focus on aluminum because of the current interest in developing very high strength aluminum through nanocrystalline and nanotwinned structures). We begin with molecular dynamics (MD) simulations of initially dislocation free crystals that demonstrate a number of specific mechanisms. We then establish a theoretical framework which considers the nucleation and growth of partial dislocation loops and which incorporates (a) the tensorial stress state, (b) dislocation loops, and (c) non-singular core energies. Some input to the theoretical formulation is obtained from the MD simulations. Our theoretical results suggest that two types of twins are possible in FCC metals, with probabilities determined by the orientation dependence of the specific mechanisms involved. We examine, in particular, the mechanism of twinning through co-operative activation of partial dislocations on the basis of theoretical considerations, and support these results through MD simulations. 2. Molecular dynamics simulations of nucleation of partial dislocations Molecular dynamics (MD) simulations are useful in interrogating the energy landscapes in a crystal structure that lead to specific deformation mechanisms, and also in identifying potential new deformation mechanisms. We focus on the relative likelihood of specific deformation mechanisms by examining the deformation of aluminum single crystals deformed under shear along specific crystallographic directions. The approach is shown schematically in Fig. 1(a): a cube of the material is deformed in shear. Since the majority of deformation mechanisms are restricted to the (111) glide planes in FCC metals, we restrict our examination to loading orientations within the glide plane, i.e., our shearing deformations are applied in the glide plane, but the direction of the applied shear is varied within the glide plane. Our objectives are to (1) identify the tensorial thresholds associated with the homogeneous nucleation of partial dislocations, (2) understand the partial dislocation reactions that can occur and the deformation mechanisms that can evolve, and (3) examine the mechanisms for the possible development of deformation twins. 2.1. Methodology Simulations are carried out using the LAMMPS (Plimpton, 1995) package. An embedded atom method potential is used for Al (Mishin et al., 1999) which captures the elastic constants, SF energy, and shear strengths of the realistic crystal structure (Mishin et al., 1999; Boyer et al., 2004). The specimen is a cube that is 16 nm on all sides [schematic in Fig. 1(a)], and the cube is large enough to avoid image effects (Cai et al., 2003). Periodic boundary conditions were used on all faces

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279

E [211] D [321]

Trailing partial for [110]

C [110] [112]

OC

OA



B [231]

OE O

θ

A [121]

Leading partial for [110] [211] Fig. 1. (a) Schematic of the specimen geometry and the applied shear strain. (b) Three consecutive (111) atomic layers of an FCC crystal structure along with the Burgers vector OC ½110=6 of a perfect dislocation and the corresponding Burgers vectors of an extended dislocation OA ½121=6 (leading partial) and OE ½211=6 (trailing partial). (c) Shearing direction (yellow arrow) within the glide plane with respect to the leading partial OA and the orientation space under consideration 0 o y o 601. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(which require remapping velocities of all atoms when they cross the boundaries). The cube was oriented such that the (111) glide plane observes the maximum resolved shear stress. The crystal structure at 300 K is generated by creating an ˚ randomly initializing the velocities of the atoms corresponding to the FCC lattice structure with lattice constant 4.05 A, temperature of 300 K (velocity command in LAMMPS) at constant volume. This was followed by a relaxation step in which the volume of the cube is isotropically dilated so as to get a pressure of about 0.1 MPa while the temperature is maintained at 300 K (NPT ensemble in LAMMPS) by timestepping through the molecular dynamics governing equation using a time step Dt ¼ 0:5 fs. Constant strain-rate shearing is conducted by modifying the shape of the MD cell with a shear strain rate of 1 ns  1. Simulations were conducted at 300 K temperature maintained using Nose/Hoover temperature thermostat under constant volume conditions. We consider an initially homogeneous atomic structure in these single crystal simulations because our objective is to understand the tensorial thresholds associated with the homogeneous nucleation of partial dislocations and the development of the associated deformation mechanisms. We recognize that in the presence of heterogeneities (such as free-surfaces, grain boundaries and crack-tips) a very wide range of stress states can exist which may encourage activation of a wide range of deformation mechanisms and produce dependencies on multiple factors such as the available activation energy and transition time (Warner et al., 2007). Because our simulations generate an initially uniform stress field, one can better characterize the state of stress under which the fundamental mechanisms develop. An overall tensorial stress r was extracted from the MD simulations using the Virial theorem (Clausius, 1870; Maxwell, 1974), with the stress components averaged both temporally (over 1 ps) and spatially (over the entire volume of the sample). Shear strain (XY-component) is measured as the ratio of the shear along the X-axis to the perpendicular box length along Y; shear is applied to the simulation box which envelops the system of atoms and is initially orthogonal. The shear increases according to the engineering strain rate as, g_ Dt. Any atom that crosses a periodic boundary has its components of velocity modified by an amount equal to the difference in velocities between the two ‘‘crossing’’ boundaries. The constant shear strain rate in the specimen under simple shear is maintained by modifying the X component of velocity of atoms while the motion (position) of the atoms naturally tracks the shape of the box. Our simulation approach is most easily visualized as follows. Consider the cube of Fig. 1(a), with the sides of the cube defined by the orthogonal triad fe1 ,e2 ,e3 g. The (e1 ,e3 ) plane is constrained to be the (111) plane in the FCC crystal structure. Shearing is developed by modifying the shape of the MD cell in the direction e1 as shown by the thick arrow. We explore the orientation space relative to the glide plane in the crystal by changing the shearing direction relative to the crystallographic axes, i.e., we reconstruct the MD cell so that the shearing direction e1 explores different directions within the (111) plane. The atomic arrangement in the (111) plane is shown in Fig. 1(b), with the Burgers vector corresponding to the perfect dislocation shown by the vector OC, that for the corresponding leading partial shown by OA, and that for the corresponding trailing partial shown by OE. Using the symmetry of the system, we consider only the portion of the orientation space within the glide plane that corresponds to the angular sector AOE; within this angular sector, the net perfect Burgers vector is ½110. This orientation space is shown schematically in Fig. 1(c). The partials of interest (shown in Fig. 1(b)) have Burgers vectors that correspond to directions OA ½121 and OE ½211 in Fig. 1(c) (the corresponding leading and trailing partial dislocations would be ½121=6 and ½211=6, respectively). We now let the direction of applied shear rate vary from y ¼ 01 to 601 through 111, 301, and 491, where y is the angle made by the applied shear strain rate direction with ½121 within the (111) glide plane. The five specific orientations that we examine in these simulations are denoted by OA, OB, OC, OD and OE in Fig. 1(c), and correspond to the specific crystallographic directions ½121, ½231, ½110, ½321, and ½211. The first of these (OA) corresponds to shearing along the direction of the Burgers vector of the leading partial dislocation, the third (OC) corresponds to shearing along the direction of the Burgers vector of the perfect dislocation, and the last (OE) corresponds to shearing along the direction of the Burgers vector of the trailing partial dislocation.

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We examine the evolution of the tensorial stress needed to sustain the deformations and the specific deformation mechanisms that develop in each case.

2.2. Observations from MD simulations The tensorial stress state required to develop a specific net shear strain rate along a given shearing orientation evolves with increasing strain during the simulations, and we show part of this evolution in the shear stress versus shear strain curves of Fig. 2(a). The ‘‘shear stress’’ in this figure is the specific component s21 defined in terms of the cube (so that, e.g., the shear stress s21 in the case of shearing along the ½211 direction is the stress component on the face with normal [111] along the direction ½211). The shear strain shown in the figure is the corresponding shear strain (g21 ). The five curves shown in Fig. 2(a) correspond to the five directions of applied shear strain in Fig. 1(c), namely OA ½121 (in red), OB ½231 (in yellow), OC ½110 (in black), OD ½321 (in blue), and OE ½211 (in green); the orientation set is shown again for clarity in Fig. 2(a). In all five cases, these shear stress vs. shear strain curves are characterized by an initial nonlinear rise in the shear stress with increasing shear strain until the stress reaches a peak. The peak stress in each case is always found to correspond to the nucleation of a leading partial dislocation, and we therefore characterize these peak shear stresses as the nucleation stresses for the orientation-dependent homogeneous nucleation of the partial dislocation. These observations are consistent with the results of Ryu et al. (2011) on homogeneous dislocation nucleation in single crystal copper. Ryu et al. (2011) present simulation results on pure shear for one specific case of orientation along OA ½121 and for varying temperatures from 0 to 600 K. In our study, however, we keep the temperature at 300 K and vary the orientation of shear with respect to the lattice. The partial dislocation was observed to nucleate at the lowest peak stress (2.34 GPa) during shearing along the direction OA. The highest nucleation stress (4.3 GPa) is that for loading along OE. Following the peak, the shear stress falls rapidly to a certain value which corresponds to the stress required for dislocation motion on the slip plane. This stress level is much smaller than the nucleation stress.

5.0

D [321] C [110]

An

ti-t dir winn ec tio ing n

E [211]

B [231] O

A [121] Twinning direction

σXY, Shear Stress (GPa)

4.5

[211] OE [321] OD [110] OC [231] OB [121] OA

4.0 3.5 3.0 2.5 2.0

LP

1.5 1.0

TP

0.5

Twinned 0

F.C.C.

γxy = 0.112 (Lp)

γxy = 0.115

LP : Leading partial nucleates TP : Twinning partial nucleates

0.02

0.04 0.06 0.08 0.1 0.12 γXY, Shear Strain (GPa)

0.14

Core of a partial dislocation (red)

H.C.P. (blue)

γxy = 0.114 (Twp)

γxy = 0.117 (twinned)

Fig. 2. (a) Shear orientation cases used for MD simulation and the resulting stress–strain response in shear. (b) Representative MD simulation results for shearing along OA. Results show snapshots at different shear strains (gxy ) and a twin being developed within a portion of the specimen. Atoms are colored according to their structural configuration, FCC green, HCP blue and cores of the partial dislocations are colored red. At gxy ¼ 0:112 a leading partial dislocation nucleates which eventually extends over the length of the specimen. At gxy ¼ 0:114 the very first twinning partial dislocation is observed. This twinning partial propagates over the length of the specimen and would lead to a two-layer stacking fault. At gxy ¼ 0:115 a number of twinning partials are seen propagating over the specimen length and the thickness of the twin increases. Finally, a snapshot at gxy ¼ 0:114 shows a fully developed twin. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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281

11

0

In all the cases a leading partial dislocation loop was observed to nucleate in the specimen at the peak in the shear stress, after which the partial propagates outwards, i.e., the size of the loop increases on the (111) glide plane. Subsequently, another partial dislocation activates, and (depending on the direction of the applied shear) it is either a trailing partial dislocation (with Burgers vector 16½211) on the same glide plane as the leading partial dislocation or a twinning partial (with Burgers vector 16½121) on the adjacent [111] (glide) plane. The distance over which the leading partial dislocation loop propagates before activation of the second partial dislocation changes with shearing orientation: it is largest for shearing along OA (½121, parallel to the Burgers vector of the leading partial), and decreases continuously as the shearing orientation sweeps over OB, OC, OD and OE. We now examine the specific mechanisms that are activated along each shearing orientation. The simplest case to consider is that for shearing along OC (the perfect dislocation direction, ½110). As in all cases considered, the leading partial is first nucleated at the time corresponding to the peak stress, and this is followed immediately by a drop in the shear stress and the propagation of unloading stress waves away from the nucleation site. Multiple nucleation events of this type occur within the simulation domain. For shearing along this particular orientation (OC), a trailing partial is observed to nucleate immediately after the leading partial, and the resulting extended dislocation propagates out along the (111) glide plane. The separation between the two partials is small (even at these very high stresses in the MD simulations) because of the high stacking fault energy (SFE) of aluminum. Examination of any one of these extended dislocation loops shows that the separation between partials is a strong function of orientation in the glide plane (Copley and Kear, 1968; Lu et al., 2002; Byun, 2003). Once nucleated, the extended dislocation sweeps across the sample; numerous such slip events are observed (Fig. 3), but no twinning is observed in this case.

E [211] D [321]

C [110]

B [231] O

(FLp-FTp) >> 0

A [121]

Leading partial for [110] Twinned region

OA: “Soft” twin, for shearing along OA. Viewing along [101]

Twin boundary

OC: Dislocation core Stacking fault Slip, for shearing along OC. Viewing along [111] Twin boundary

OE: “Hard” twin, for shearing along OE. Viewing along [211]

Fig. 3. Orientation dependent deformation mechanism map in aluminum. (a) Different shearing directions within the orientation space lead to different values of ðF Lp F Tp Þ. (b) ‘‘Soft twins’’ were observed when sheared along OA, ‘‘hard twins’’ for shearing along OE, and slip through nucleation and propagation of extended dislocations for shearing along OC. Atoms are colored according to their structural configuration, FCC green, HCP blue, and cores of the partial dislocations are colored red. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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A completely different mechanism is observed for shearing along the ½121 direction (direction OA in Fig. 1(c)). In this case, the nucleation of the leading partial is immediately followed by the nucleation of a twinning partial on the adjacent glide plane instead of the nucleation of a trailing partial on the same glide plane. Since the twinning partial has the same Burgers vector as the prior leading partial, we can view this nucleation event as the preferential nucleation of the twinning partial at the site of the prior nucleation of the leading partial (in preference to the nucleation of a new leading partial at a different point in the crystal). This nucleation process is dynamic, with the local state being modified immediately through the release of unloading waves, and resulted in the fall in the global stress. As the deformation continues, additional twinning partials are nucleated and propagate through the system, resulting in the development of a rapidly thickening twin. At first glance, this result seems surprising, since aluminum is known to be a high stacking fault energy metal and so twinning is expected to be difficult. From our simulations, with this particular loading orientation, it is not possible to nucleate extended dislocations—instead, one nucleates twins. However, this is true for only a very small part of the orientation space, as we show et seq. (i.e., the fact that twins can be nucleated with one specific loading orientation does not imply that twins will be easily developed in the material). Further, these simulations are of homogeneous nucleation of twins: the simulations involve very high shear stresses (approximately 10% of the shear modulus) and very high shear rates, and there are no pre-existing dislocations in the system. An initial dislocation structure will almost always exist in experimental samples, and then the question is not one of the preferential nucleations of twins over perfect dislocations, but preferential slip of existing dislocations over nucleation of twins. Thus while our simulation results do demonstrate that twinning is possible in aluminum given the right conditions, they should not be taken to imply that twinning is easy in aluminum. Fig. 3 shows snapshots of the dislocations and twins obtained from MD simulations for loading along OA, OC, and OE orientations. A common neighbor analysis (Honeycutt and Andersen, 1987) coloring scheme has been adopted for atoms based on the local crystal structure. The atoms colored blue are in an HCP structure and represent a stacking fault (SF). The atoms not associated with either the host FCC or HCP structure are colored red; atoms in the baseline FCC structure are not shown. In our simulations, the red atoms are mainly those belonging to the cores of partial dislocations and atoms which form clusters associated with thermal noise. When the shearing orientation is changed from the OA direction to the OB direction, i.e., ½231, extended twins are again observed rather than extended dislocations, but slightly higher stresses are required to activate this deformation mechanism. However, this mechanism is no longer favored as the shearing direction rotates into OC, as noted previously: extended perfect dislocations are observed instead (i.e., a trailing partial is nucleated instead of a twinning partial). As the shearing orientation continues to rotate into the OD and then the OE i.e., ½211 direction, however, we see the development of an intermediate mechanism (Fig. 3): the nucleation of small twins that grow through a cooperative activation mechanism. This mechanism is activated only after much higher shear stresses, almost twice as large as that needed to nucleate partials for loading along OA, and so we will refer to the resulting twins as ‘‘hard twins’’ in contrast to the ‘‘soft twins’’ generated at the lower stresses along OA. These hard twins are characterized by twin tips, are terminated by partial dislocations and have a thickness of at least two glide planes. Note that a pair of adjacent stacking faults is sometimes called a microtwin in the literature. However, in an FCC system a twin cannot be defined until at least three adjacent stacking faults have been nucleated, and so in the strict sense we should define a microtwin as a set of at least three adjacent stacking faults. Some of these hard twin configurations do grow to be microtwins: Fig. 4 shows one such hard twin configuration consisting of 3–4 glide planes (4–5 atomic planes), and bounded by twin boundaries (blue colored atoms). This hard twinned configuration was developed through simultaneous and co-operative activation of two different partial dislocations of Burgers vectors OA 16½121 and OF 16½112 on adjacent glide planes. It is this co-operative activation of partial dislocations (sometimes called the CAP mechanism, Li et al., 2011) that allows partial dislocation activation without going over the relatively high anti-twinning barrier. There is a net attractive force of interaction between these two partials because of their very different Burgers vectors and hence they propagate simultaneously. Further increase in the shear strain in the specimen leads to activation of alternating OA and OF twinning partial dislocations and a twinned configuration. This type of ‘‘hard twin’’ has not been observed before in DFT simulations because of the displacement constraints set in such simulations (Kibey et al., 2007b), where particularly in the anti-twinning direction these constraints would result in a higher estimate for stress. The specific shearing orientations that we have chosen to examine provide a full sampling of the entire orientation space, given the crystal symmetry. Within the subspace that we have examined, extended twins are observed for shearing along OA, extended dislocations are observed for shearing along OC, and very limited twinning followed by dislocation motion is observed for shearing along OE. The intermediate orientations represent mixtures of these deformation modes. Examining the full orientation space, therefore, one observes that dislocation slip is the dominant mechanism that is activated in most of the orientation space, with small subregions within which the twinning mechanism is preferentially activated, and an even smaller region where extended twins may be activated. The shearing deformations considered in this work make it possible to explore the oriented loading space and observe the resulting deformation mechanisms (perfect dislocations, extended dislocations, leading, trailing and twinning partials and subsequent twinning through multiple mechanisms) under constrained volume conditions. Existing results in the literature (e.g., Warner et al., 2007) have suggested that slip and twinning may act as competing mechanisms to each other. Our MD simulations results are consistent with this view, but also demonstrate the importance of loading

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[112] bhtw

O

E [211]

Original configuration

B1 C1

B1 C1

C1 A2

C1 B2

A2 B2

B2 A2

B2 C2

A2 C2

C2 A3

C2 A3

283

A [121] bLp

bl bhtw bl Orig I

II

III

B1 A1 C1 B1 C1 B1 A1 C1 A2 C2 B2 B2 B2 A2 A2 A2 C2 C2 C2 C2 A3 A3 A3 A3

Before twinning

After twinning (step II)

Fig. 4. ‘‘Hard twin’’ mechanism in FCC aluminum. (a) Schematic of the relative displacement between atomic layers A2 (on top)–C2 (on bottom). Original configuration shows the untwinned layers. Step I: Layer B2 undergoes a displacement relative to C2 through bLp ¼ ½121=6 (red arrow). Step II: Layer A2 undergoes a displacement relative to B2 through bhtw ¼ ½112=6 (green arrow). (b) Direction of shear loading OE (yellow arrow) favorable for ‘‘hard twin’’ mechanism with respect to the Burgers vectors involved, y ¼ 601. (c) Table shows the change in the stacking sequence when the lattice accommodates bLp (I), bhtw (II) and again a bLp (III) leading to a twinning configuration. (d) MD simulation results: column on the left shows the configuration of atoms before twinning comprising six (111) atom layers (B1 to A3), each snapshot within its rows depicts two adjacent glide planes. Atoms in FCC and HCP configuration are colored green and blue, respectively, while those belonging to the core of the partial dislocation are colored red. The red colored atoms within layers A2 and B2 are the deregistered atoms that eventually lead to a ‘‘hard twin’’. The twinned configuration shows twin boundaries C1–C2 and the twin matrix A2–B2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

orientation and the potential richness of phenomena that can be activated even in high stacking fault energy materials like aluminum. This is important in problems such as the dynamic deformation of nanocrystalline aluminum (Li et al., 2009), where high stresses may be sustained in any case. In the next section we consider the potential mechanisms that can develop in this system from a theoretical viewpoint, and seek to understand the likelihood of observation of these phenomena.

3. Theoretical considerations: slip versus twinning We seek to develop a theoretical understanding of the stresses required to develop specific deformation mechanisms as a function of shearing orientation, so as to explain the results from our MD simulations, and so as to provide insight into the likelihood of observation of these mechanisms. This latter objective is driven by the fact that the total volume of material examined at these length scales within experimental methods such as TEM is extremely small, and some sense of the probability of observation is needed to determine the importance of a given mechanism for each material. Such insights are also useful in up-scaling our results to dislocation dynamics codes or phase field models at the mesoscale. There are several fairly sophisticated models for dislocation nucleation in the literature, e.g., Xu and Argon (2000), Lu et al. (2001), and Aubry et al. (2011), that provide a basis for this analysis. Most of these consider the transition from the homogeneous state (without the dislocation) to the dislocated state (the state after the nucleation has occurred, with a dislocation loop), and approach the problem in terms of an energy path or in terms of a bifurcation condition. Continuum models based on the concepts of generalized stacking fault energies, often using the Frenkel approximation for movement along the sliding plane, can be used to estimate the nucleation stress (in a manner analogous to that used by Rice and Beltz, 1994). Correlations between such models and atomistic simulations of nucleation (Xu and Argon, 2000; de Koning et al., 2003) have suggested that the nucleation can be modeled in terms of a non-constant Burgers vector that may evolve

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during the process (Aubry et al., 2011), and the continuum models then generate stress levels comparable with the atomistic simulations. Rather than attempt to rigorously model the process of nucleation itself, we consider the stress required to propagate an existing dislocation loop of finite (but small) size, and ask how this propagation stress (sometimes referred to as an activation stress within the literature) depends on the orientation of the shearing deformations of the crystal. We then compare these propagation stresses to the stresses developed in our atomistic simulations, and seek to develop some insights into the likelihood of activation of specific deformation mechanisms within the material. We consider the expansion of existing dislocation loops of the various types (perfect, leading, trailing and twinning partials), and combinations of such loops to develop specific dislocation slip and twinning mechanisms. Our analysis of these activation stresses accounts for the curvature of the partials, the interaction between partials, nonlinear elastic effects, and the effect of the normal stresses. We limit our analysis to circular dislocation loops because of the complexity associated with deriving the closed-form solutions for elliptical loops. One rationale for our approach to consideration of propagation stresses is based on the concept that in the nanocrystalline materials of eventual interest, local elevated stresses at grain boundaries are typically sufficient to nucleate partial dislocation loops, but the question of observable deformation mechanisms relates to the availability of sufficient stress within the grains in order to activate these already nucleated partial dislocation loops. We begin by considering the stresses needed to propagate a leading partial dislocation loop, and then examine the stresses needed to develop either a trailing partial dislocation on the same glide plane or a twinning partial dislocation on the adjacent glide plane. In each case we consider the entire range of loading orientations, and thereby develop a sense of the relative ease of activation of the dislocation slip and twinning mechanisms in each orientation (Christian and Mahajan, 1995 provide an excellent review of such mechanisms). The following simple theoretical analysis does not account for the inertial terms associated with the dislocation motion; such inertial effects are discussed later in the paper. 3.1. Growth of a (leading) partial dislocation loop Consider the slow growth of a dislocation loop of radius rLp with the Burgers vector corresponding to a leading partial and gliding on a (111) plane as shown in Fig. 5(a). Since this is a leading partial dislocation, the slipped area represents a stacking fault (colored green in the figure). An externally applied stress field provides the necessary force to increase the radius of the loop from rLp to r Lp þdr Lp . The energetics of this static problem can be written as

dEtotal ¼ dEstrain þ dEfault dW ext

ð1Þ

where we account for the strain energy Estrain in the dislocation core, the stacking fault energy Efault and the work Wext done by the external stresses. For the strain energy of the partial dislocation loop, we use the theoretical description recently developed by Cai et al. (2006), which gives the strain energy of a circular dislocation loop of radius r as     mb2 2n 8r 1 ð2Þ Estrain ¼ 2pr ln 2 þ a 2 8p 1n where a is the radius of the partial dislocation core over which the Burgers vector is allowed to spread, b is the magnitude of the Burgers vector b that is in the plane of the loop, m ¼ 26:5 GPa is the shear modulus and n ¼ 0:347 is Poisson’s ratio for Al. For the case of p this ffiffiffi (leading partial) dislocation loop of radius rLp, we obtain the strain energy EstrainLp by setting r ¼rLp and b ¼ 9bLp 9 ¼ a0 = 6, where a0 ¼4.1 A˚ is the lattice constant of Al at 300 K and approximately 1 bar pressure. We note here that this refined definition of the strain energy of the loop is different from that used by Zhu et al. (2004). For the

Twin rLp

rLp

rLp

rTwp rTp

Circular loop of a leading partial dislocation

Circular loop of an extended dislocation

Microtwin/ 2-layer stacking fault

Fig. 5. Equilibrium configurations of (a) a leading partial dislocation, (b) trailing partial (along with the leading partial) dislocation of an extended dislocation, and (c) a twinning partial dislocation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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second term in Eq. (1), we note that a stacking fault is generated when the leading partial dislocation sweeps over the glide plane, and Efault is the energy associated with creating this stacking fault: Z Efault ¼ gs da ¼ pr2Lp gs ð3Þ As

where gs is the stacking fault energy (a characteristic property) of the material, and As is the area of the stacking fault. This stacking fault energy cost is part of the energy budget associated with the growth of a leading partial dislocation. For Al we use gs ¼ 146 mJ=m2 (Mishin et al., 1999). Finally, the last term in Eq. (1) is the work Wext done by the external stress in increasing the radius of the loop. Under equilibrium conditions this stress will be equal to the stress required to sustain a dislocation. This work term is given by Z Z ½ðb  rÞ  n  ½n  ðb  nÞ W ext ¼ 2p Fg  dr ¼ 2p  dr ð4Þ 9b  n9 where Fg is the glide component of the Peach–Koehler force (see Hirth and Lothe, 1992) responsible for increasing the size of the dislocation loop within the glide plane, which depends on the tensorial stress state r, the perfect Burgers vector b and the dislocation line vector n as shown. For the leading partial dislocation, we have b ¼ bLp . Note that the integral in Eq. (4) is an integral over the entire circular loop. Considering the partial derivatives of the energies with respect to rLp, we obtain the forces on the leading partial dislocation loop, which must be in equilibrium: @Estrain @Efault @W ext þ  ¼0 @r Lp @r Lp @r Lp

ð5Þ

Working through the algebra, we can use Eq. (5) to develop an expression for the stress required to grow a dislocation loop of radius rLp as a function of the loading orientation y. The tensorial stress state enters into Eq. (5) through Eq. (4), and in general there may be a variety of tensorial stress states that are consistent with a dislocation loop of specific size (i.e., the single equation (5) cannot determine the entire stress state). The simplest approximation one can make is to assume that the only non-zero stress component is sXY , in which case we can use Eq. (5) to compute the orientation dependence of the shear stress required to propagate a leading partial dislocation loop of given size rLp. In order to avoid the specific nucleation problem, we assume for all of our calculations that the initial radius r Lp0 of the dislocation loop is five times the core radius of a partial, where the core radius of a partial dislocation itself is set to a ¼ 9bLp 9 (note also that in our 300 K MD simulations the size of the typical deregistered atomic cluster associated with thermal noise was comparable to 59bLp 9); with such a large initial loop size, we may neglect the effects of an evolving core (Aubry et al., 2011). Our computed stresses are the stresses that will sustain a leading partial dislocation loop of radius r Lp ¼ 59bLp 9, and we compare the orientation dependence of these finite loop stresses with the nucleation stresses obtained from the MD simulations (note that a leading partial dislocation is always the first partial nucleated in the simulations at all orientations). The corresponding theoretically predicted orientation dependence of the shear stress for the unconstrained simple shearing assumption is shown in Fig. 6 as the open squares (the orientation angle plotted on the abscissa is obtained as one sweeps from OA to OE in Fig. 1). The computed nucleation shear stresses computed from the full MD simulations are also shown in Fig. 6 as the closed triangles. Our theoretical estimates of the shear stress are remarkably close to the computed shear stresses given the lack of sophistication of the model, capturing both the approximate magnitude of the stress and the approximate orientation dependence. The magnitude of the theoretically derived stress required is obviously a function of the assumed size of the leading partial loop, and in this sense our choice of the loop radius appears to have

Fig. 6. Stress for activating a leading partial dislocation.

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been fortuitous; overall, the model captures the trends observed in the simulation fairly well. Note that, the continuum theoretical model (CTM) predictions do not capture the nucleation stresses, but rather describe the growth stresses for an assumed initial dislocation configuration. We have essentially set up our initial loop radius to match the MD result for nucleation at the orientation y ¼ 0. Our results then show that the orientation dependence in the MD results (on nucleation) is consistent with the orientation dependence predicted by the CTM results. This is as much as we can get from the comparison. We note that the MD simulations are also handling other complexities such as thermal contributions, inertial effects, and evolving dislocation cores. Note, however, that the stress state in the MD simulations is fully multiaxial due to the constrained volume conditions, while the purely analytical results presented in Fig. 6 as the open squares only account for shear stresses. We can build a hybrid model as follows. For any given shearing orientation, the MD simulations provide the full tensorial stress state. We can incorporate the evolving tensorial stress state (extending indefinitely along the loading path) into the theoretical model, and ask the following question: At what point on that path in stress space is Eq. (5) satisfied? This will give us a theoretical estimate of the propagation stress along a given shearing orientation, which we can compare with the computed peak stress in the MD simulations. The predictions of this hybrid approach are also presented in Fig. 6 as the closed squares, and correspond remarkably well with the MD results. This correspondence demonstrates that for this material, the ‘‘non-Schmid’’ stresses do have a perceptible (if weak) effect on the stress for propagation of the leading partial. Our calculations suggest that (for the specific case of aluminum) these normal stress effects can be neglected, and the simple shear calculations performed instead, with resulting errors of the order of only 10%. We note here that McDowell and co-workers considered the effects of stress components normal to the glide plane on the activation stress for leading partial dislocations for Cu (Tschopp and McDowell, 2008), and observed much stronger normal stress effects for that material, probably as a result of the very different generalized stacking fault energy for copper as compared to aluminum. The difference between the analytical (multiaxial) and MD results for the OE orientation (corresponding to the y ¼ 601 angle) may be related to the much more complex dislocation mechanism that is developed at these high angles in the MD simulations. We note here that our analytical calculations that account for the multiaxial stress effects include explicitly the component of the Burgers vector perpendicular to the glide plane, in addition to the component of the partial Burgers vector that is in the glide plane. The atomic motion corresponding to the partial dislocation requires that there be a component of the displacement that is perpendicular to the glide plane (the stacking fault area may also contribute, but this contribution is difficult to quantify). We estimate this normal component of the displacement (8:768  103 9bLp 9 for Al) from the change in dimensions of the box (normal to the glide plane) in MD simulations of the aluminum under pure shear configurations (Ogata et al., 2002), with all but the sXY components relaxed by appropriately changing the volume of the box as the crystal is strained. Now that we understand the orientation dependence of the stress required to propagate a leading partial dislocation loop, we consider the development of two competing mechanisms: the development of an extended dislocation through the subsequent propagation of a trailing partial dislocation loop and the development of a twin through the successive subsequent propagations of twinning partial dislocations (note that in both cases we assume that a leading partial dislocation loop has already been activated). We seek to understand the likelihood of twinning in aluminum by examining the relative ease with which these two mechanisms are activated. 3.2. Stress thresholds for an extended dislocation An extended dislocation will be developed if a trailing partial dislocation loop is nucleated after the initial leading partial dislocation loop, as shown in Fig. 5(b). The total energy of the dissociated dislocation configuration can be written as

dEtotal ¼ dEleading þ dEtrailing þ dESFE þ dEint dW ext

ð6Þ

where the first and the second terms on the right hand side are the strain energies (Eq. (2)) of the leading and trailing partial dislocation loops, respectively; the third term is the fault energy associated with the stacking fault ESFE ¼ pðr 2Lp r 2Tp Þ; the fourth term is the energy due to interaction between the dislocation loops, and the fifth term is the external work done by the applied stress field. The expressions for the strain energies of the two loops are identical to those given in Eq. (2), with the loop radii defined by rLp and rTp for the leading and trailing partials respectively (Fig. 5(b)). The contribution of the stacking fault (SF) to the total energy of the system will obviously depend on the width rs of the stacking fault (the separation of the leading and trailing partials). An expression for the SF width can be obtained from the equilibrium positions of the partials, which perceive a repulsive interaction force that is balanced by the attractive force associated with the SF energy in the presence of stress (Hirth and Lothe, 1992). In these terms, the SF width of an extended dislocation subjected to an externally applied stress field is (Hirth and Lothe, 1992) r s ¼ A=½gs 0:5ðF Lp F Tp Þ

ð7Þ

where gs is the SF energy, A¼

m 2p

ðbLp  nÞðbTp  nÞ þ

m 2pð1nÞ

ðbLp  nÞ  ðbTp  nÞ

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depends on the Burgers vectors, the dislocation line vector and the elastic properties, and FLp and FTp are the Peach– Koehler forces per unit length acting on the leading and the trailing partial dislocations, respectively. Note that the stacking fault energy may itself be a weak function of the stress (Tschopp et al., 2008), but we ignore this for the predominantly deviatoric/shearing problems of interest here. Copley and Kear (1968) showed that the presence of an applied stress field can lead to the varying separation of partials, and that this is different than the width developed under no stress conditions. The SF width that has been observed in coarse-grained aluminum is approximately 0.55 nm because of the relatively high SFE. However, wide SF ribbons with average widths as large as 3.5 nm have been reported in NC Al (Liao et al., 2003). Theoretical predictions of stacking fault widths (considering the applied stress) have been demonstrated to have good agreement with those observed from DFT calculations in Al (Lu et al., 2002). The fact that the local stress field can have a significant influence on SF width, particularly in high SFE materials like aluminum, is known (Froseth et al., 2004; Hirth and Lothe, 1992) but is sometimes ignored in such discussions. A specific example of this influence is the so-called Escaig stress (the resolved shear stress along the edge components of the partials) (Bonneville et al., 1988) which is often used to consider when the constriction of extended dislocations can allow screws to cross-slip (Lu et al., 2002). The tensorial stress r in the sample will contribute to the Peach–Koehler (PK) forces FLp and FTp on the leading and trailing partial dislocations, respectively, through expressions of the form F ¼ ½ðbp  rÞ  n

ð8Þ

where r is the local (tensorial) stress, bp ¼ bLp and bp ¼ bTp are the Burgers vectors for the leading and trailing partial dislocations, respectively, and x is the line vector. By inspection of Eq. (7), we see that we can define an apparent stacking fault energy EAFE as EAFE ¼ gs 0:5ðF Lp F Tp Þ. If the stress state is such that the PK force difference ðF Lp F Tp Þ ¼ 0, the SF widths from Eq. (7) would be the same as the SF widths that would exist without any external stress (for very high SFE materials like aluminum, the SF width would then be negligible). However, for finite non-zero values of the PK force difference, one should expect to see a variety of SF widths, depending on the local PK force difference. To first order, the magnitude of the PK force difference depends linearly on the applied stress, and since higher applied stresses are needed to deform NC metals, one should expect to see a wider range of SF widths in an NC material than in its coarse-grained counterpart. This is in fact observed in the case of NC aluminum (Liao et al., 2003). More important, however, is the orientation dependence of the PK force difference within the orientation space of Fig. 3. For shearing along OC ½110, the direction of the Burgers vector of the perfect dislocation, there will be an equal contribution of force towards the leading and trailing partials. For loading along OA ½121 and OB ½231, a larger contribution of the applied load will be available for the leading partial, so that we have ðF Lp F Tp ÞOA 4ðF Lp F Tp ÞOB 40. On the other hand, for loading along OD ½321, and OE ½211, a relatively smaller component of the force is available for the leading partial, and we have ðF Lp F Tp ÞOE o ðF Lp F Tp ÞOD o0. As a consequence, the influence of the apparent stacking fault energy on the configuration of the extended dislocation loop is strongly orientation dependent (which is also, of course, why the computed SF width in the MD simulations varies along the loop when extended dislocations are observed). This orientation dependence is captured within Eq. (6). The fourth term on the right side of Eq. (6) is the interaction energy Eint between two co-axial dislocation loops of different radii. The force of interaction on one dislocation in the presence of the stress field of another dislocation is computed using a Peach–Koehler approach, Eq. (8). This is somewhat complicated for the case of two coaxial dislocation loops; e.g., the interaction force on the trailing partial is obtained by integrating over the circumference of the trailing partial loop the PK force due to the stress field of the leading partial dislocation, and is given by Z 2p @Eint =@r Tp ¼ ½ðbp Tp  rLp ðfÞÞ  nLp ðfÞ df ð9Þ f¼0

The full expressions for the stress field of a circular dislocation loop are obtained from the work of Langdon (2000). The final expression for the force of interaction on the circular loop of trailing partial dislocation in the presence of a leading partial dislocation loop is

p

1 3 4pr 2Tp ð1 þ nÞ½z2 þ ðr Tp r Lp Þ2 3=2 ½z2 þ ðr Tp þ r Lp Þ2 2 " ! r Lp  K 4r Tp ½z2 þ ðr Tp þr Lp Þ2 ½r 2Tp ½z4 r 4Tp ð2þ nÞz4 n þ 5z2 r 2Lp z2 þ ðr Lp r Tp Þ2 # 2

@Eint =@r Tp ¼ 2pr Tp 9bp 9 G cos

2z2 nr 2Lp þ2r 4Lp nr 4Lp þr 2Tp ð3z2 2z2 n4r 2Lp þ 2nr 2Lp Þ þ E 4r Tp

r Lp z2 þ ðr Tp r Lp Þ2

! ½r 2Tp ½r 6Tp ð1þ nÞ þr 4Tp ð5z2 3z2 n6r 2Lp þ3nr 2Lp Þ

ð10Þ þ r 2Tp ð4z4 3z4 n10z2 r 2Lp þ 2z2 nr 2Lp þ 6r 4Lp 3nr 4Lp Þ þ ðz2 þr 2Lp Þðz4 z4 n þ 7z2 r 2Lp 2r 4Lp þ nr 4Lp Þ R p=2 R p=2 2 0:5 2 0:5 where KðmÞ ¼ 0 ð1msin aÞ da and EðmÞ ¼ 0 ð1msin aÞ da are the complete elliptic integral of the first kind and second kind, respectively, z is the perpendicular distance between respective glide planes of the loops; since in this case we are dealing with co-planar dislocations, z¼0. The expression for interaction force on the leading partial

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dislocation, @Eint =@r Lp , is given by swapping rLp and rTp on the right side of Eq. (10). Note that, the interaction forces on the leading and the trailing partial dislocations are calculated from the stress field, rather than deriving them from the energy which can get complicated. In addition, the resulting interaction energies calculated from the interaction forces on rLp and rTp are equal and consistent with each other. Finally, the last term in Eq. (6) is identical in the form to that in Eq. (4) but is applied to the leading and trailing partial dislocation loops under consideration. Once all of these energetic contributions to Eq. (6) are known, we consider the stresses required to sustain the extended dislocation configuration shown in Fig. 5(b). Differentiating Eq. (6) with respect to rLp and rTp gives two partial differential equations that must be simultaneously satisfied. In order to derive consistent values of the radii of the leading (rLp) and the trailing (rTp) partials with the state of stress, we solve these two equations simultaneously under the condition that the stress in the system driving the two partials is the same. In the light of the previous results on the leading partial dislocation, we consider only the simple shear stress problem rather than the full multiaxial stress problem, recognizing that this may be a different approximation for the extended dislocation. Our solution approach is as follows. We seek a minimum size of rLp that would give a single loop radius for rTp. However if this rTp is less than the minimum loop size of 59bTp 9 (for the reasons articulated in Section 3.1), we then examine solutions for larger rLp. In such cases we obtain two solutions for rTp, and define the solution with larger rTp as the stable solution because this solution corresponds to the case where additional stress is needed to grow the loop. However, to arrive at this stable configuration the trailing partial dislocation first requires a larger stress for nucleation (denoted t in Fig. 7). For each value of the loading orientation y, we determine the nucleation shear stress t that is consistent with the pair of equations derived from Eq. (6). In this way we are able to compute (as a function of loading orientation) the stress required to develop an extended dislocation loop. We examined the dislocation loop developed in MD simulations to the extract values of stacking fault widths. In MD simulations, the partials do not resolve the stacking faults over some segments, and note that these are also the segments where the continuum analysis is likely invalid. The theoretical estimates of the effective stacking fault widths are compared with MD simulations, shown in Table 1, and the values agree reasonably well. The shear stresses (black solid line) required to develop the extended dislocation loop (i.e., a leading partial dislocation followed by a trailing partial) are presented in Fig. 8 as a function of shearing orientation, superimposed on the previously computed shear stresses needed to propagate just the leading partial dislocation (red squares). It is apparent that for all of the orientations of interest, it takes less stress to propagate a trailing partial immediately after the leading partial (i.e., the available shear stress will result in the immediate activation of a trailing partial dislocation loop once the leading partial dislocation loop has been activated).

Fig. 7. Stress for developing a stable extended dislocation configuration.

Table 1 Values of effective stacking fault widths (nm), comparison of theoretical estimates with the results from MD simulations. Loading orientation

Theoretical

MD simulations

OB OC OD

0.96 0.633 0.474

1.64 7 0.21 0.625 7 0.15 0.42 7 0.15

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Fig. 8. Orientation dependent activation stress for a trailing and a twinning partial dislocation. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

3.3. Stress thresholds for twinning: propagating a twinning partial dislocation We now consider the alternative scenario, where the leading partial dislocation loop is followed by a twinning partial dislocation with the same Burgers vector bTwp ¼ bLp on an adjacent glide plane rather than a trailing partial dislocation with a different Burgers vector bTp on the original glide plane. This is shown schematically in Fig. 5(c), where a twinning partial dislocation loop with radius rTwp is created in the presence of a leading partial loop of radius rLp. The energetics of this configuration can be written as

dEtotal ¼ dEleading þ dEtwinning þ dEint þ dEisf þ dEtwin dW ext

ð11Þ

The first two terms on the right side of Eq. (11) are the strain energies (Eq. (2)) of the leading and twinning partials, respectively, the third term is the energy due to interaction between pffiffiffithese two loops as defined in the last section, with an exception that z takes the value of inter-planar separation, z ¼ a0 = 3. The fourth and the fifth terms are the fault energies associated with the intrinsic stacking fault and the twin bounded by two twin boundaries, respectively. The intrinsic stacking fault energy can be written as Z Eisf ¼ gs da ¼ pðr2Lp r2Twp Þgs ð12Þ As

where rTwp is the radius of the twinning partial dislocation, and As is the area of the intrinsic stacking fault (colored green in Fig. 5(c)). The fault energy associated with the twin bounded by two twin boundaries is Z Etwin ¼ 2 gtb da ¼ 2pr 2Twp gtb ð13Þ At

where gtb is the fault energy associated with the twin boundary, rTwp is the radius of the twinning partial dislocation, and At is the area of the twin boundary (colored orange in Fig. 5(c)). For Al we use gtb ¼ 76 mJ=m2 (Mishin et al., 1999). The sixth term is identical in the form to that in Eq. (4) but is applied to the leading and twinning partial dislocation loops under consideration. Once again, differentiating Eq. (11) with respect to rL and rtwp gives two partial differential equations that must be simultaneously satisfied. In these equations, the derivative of the interaction energy gives the force on each partial due to the stress field of the other partial, and these forces are again computed using the explicit expressions developed by Langdon (2000) for the stress fields of circular dislocation loops. Note that in this case the two partial dislocation pffiffiffiloops are on different glide planes, and so the computations include the effect of the (111) interplanar distance z ¼ a0 = 3. We again consider only the simple shear stress components based on the earlier discussion. Our solution approach is the same as that used in the last section. In this way we are able to compute (as a function of loading orientation) the stress required to develop this twinning configuration. The resulting shear stresses for the development of the twinned configuration are plotted in Fig. 8 as a function of shearing orientation, superimposed on the previously computed shear stresses needed to propagate the leading partial dislocation (red squares) and the extended dislocation (solid line). For the conditions considered here, the stress required for developing the twinning partial dislocation is lower than that required for a leading partial dislocation but generally larger than that for the extended dislocation. For the very small orientation domain between OA and OB (Fig. 1), the shear stresses associated with microtwin development are lower than those associated with the development of extended dislocations, suggesting that twinning should be occasionally (but rarely) observed in aluminum at sufficiently high stresses.

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Note that an entirely different configuration of partial dislocations was developed in the MD simulations (Fig. 3) in terms of the ‘‘hard twin’’ configuration observed at the shearing orientation OE. This hard twinned configuration was developed through the simultaneous and co-operative activation of two different partial dislocations of Burgers vectors OA 1 1 6½121 and OF 6½112 on adjacent glide planes. We can attempt to model this analytically using the same approach as in Eq. (11), but with the twinning partial having the Burgers vector corresponding to OF: bhtw ¼ 16½112 rather than that corresponding to OA. This results in a significant difference in the interaction energy, leading to an attractive force for this ‘‘hard twin’’ configuration instead of the repulsive force that is developed in the discussion of the first part of this section (which corresponds to a ‘‘soft twin’’ configuration by our earlier definition). The analysis of this ‘‘hard twin’’ configuration is not presented here because the solutions suggest that these two partials would be very close together, so much so that the assumed stress fields and overlapping dislocation energies are not reasonably described by the elasticity theory. In the next section we discuss our results in the context of competing deformation mechanisms in aluminum, and compare our results with the experimental evidence in the literature. 4. Discussion The summary of the stresses required to develop specific deformation mechanisms in Fig. 8 also provides some insights into the relative likelihood of observing these mechanisms in experiments. Our results show that for the vast majority of the orientation space within a single crystal subjected to shearing deformations, it is favorable to activate a trailing partial immediately after the leading partial dislocation has been activated, leading to the traditional dislocation slip mechanism, with generally small stacking fault widths. For a very small range of orientations (0 o y o101 from the ½121 direction), it appears to be easier to activate a twinning partial dislocation leading to a twin. However, the orientation space available for twinning in case of Al is much smaller compared to that available for slip. This indicates that it should be difficult, but not impossible, to find twins in Al. The orientation dependence of deformation mechanism, slip or twinning, is attributed to the crystalline anisotropy, i.e., the trailing 16/110S and the twinning 16/112S partial Burgers vectors are oriented differently within the glide plane. When shear is applied along a certain direction within the glide plane, to the first order or approximation, the activation of a partial dislocation with the maximum resolved shear stress is typically favored. For twinning to occur the leading and the twinning partials have three possible /112S directions along which they can activate. However, there are two possible /110S directions for activation of a trailing partial in each case following the activation of a leading partial. Thus, in the orientation space within the glide plane, the probability of activating a trailing partial should naturally be higher than a twinning partial. Specific material properties, in addition, would alter the orientation space available for twinning. Note that, our analytical results are specific to aluminum, and care should be taken in attempting to use this treatment to other FCC metals. Specifically, our analytical model uses an isotropic material assumption for the single crystal and assigns an equivalent linear elastic material property (Hirth and Lothe, 1992, p. 837). This isotropic approximation is justified for aluminum, but not for copper or nickel, for example. If the Al crystal is sheared along the appropriate orientations, some twins should be generated, provided the local stresses are sufficiently high to activate this mechanism. In any polycrystalline mass under arbitrary load, there is always a small probability that some of the crystals are properly oriented for the twinning deformation mechanism to be activated. Real samples in the experiments are of course never as idealized as in our calculations: samples are typically polycrystalline and contain impurities, many pre-existing dislocations and other heterogeneities. Further, the samples are rarely subjected to stresses of the order of several GPa and rates similar to those in the MD simulations. However, sufficiently high stresses will commonly be generated in nanocrystalline materials, and so the likelihood of observing deformation twins should be higher in nanocrystalline aluminum than coarse-grained aluminum. It is also true that local stresses may be elevated above background levels because of heterogeneities such as grain boundaries (indeed, MD simulations by Swygenhoven and co-workers (Van Swygenhoven et al., 2006; Van Swygenhoven and Derlet, 2008) suggest that there may be numerous partial dislocation nuclei at the grain boundary). These expectations from the model are consistent with the experimental observations to date. Shear loading experiments using a modified Kolsky bar for compression–torsion have demonstrated (Li et al., 2009) that extended twinning (what we have called ‘‘soft twins’’) is possible in NC Al at room temperature under high strain rate conditions. Fig. 9(a) shows an NC Al thin film (thickness 400 nm) used in the experiments, with an average grain size of 50 nm in the plane of the film and 200 nm in the thickness direction. The strain rates observed in these experiments were of the order of 5  105 s  1. TEM observation of specimens after shearing showed long twins (Fig. 9(b)) from the grain boundary and wide stacking faults (Fig. 9(c)). Similar twins have also been seen in the scratch experiments of Chen et al. (2003). Stacking faults covering two layers (like the configuration in Fig. 5(c)) have been observed in preferentially oriented grains (with maximum resolved shear stress along /121S) in coarse grained aluminum specimens deformed through ECAP (Han et al., 2008). Our MD results suggest that for y ¼ 601 it is energetically more favorable to nucleate a combination of partials of type OA and OF alternating each other rather than nucleation of the solitary OA leading partial. The resulting twins generally are much smaller, grow with greater difficulty, can terminate in twin tips rather than the grain boundary and require higher stresses. These ‘‘hard twin’’ configurations observed in the MD simulations have their experimental parallel in the so-called bundle and CAP mechanisms described by Wang et al. (2007) and Li et al. (2011). Related observations were

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Fig. 9. Postmortem TEM observation of twins and wide stacking faults observed in nanocrystalline aluminum film at room temperature at a strain rate of about 5  105 s  1 under a compression–torsion Kolsky bar. (a) As-received microstructure, (b) twins, and (c) wide stacking faults.

made by Wu et al. (2008), who observed an overall zero macroscopic strain (no shape change) associated with the twinned grain in nanocrystalline Al, Ni and Cu. They hypothesized the existence of bundles of different partials which simultaneously nucleate and propagate together leading to twin formation. However, our MD results show that once the combination of partials is nucleated on the adjacent glide planes and propagated over the entire specimen length, it is generally easier to develop the trailing partial dislocation after a hard twin configuration is generated and the specimen eventually undergoes a multilayer dislocation slip. This also suggests that these configurations should be rarely seen under postmortem TEM observations, and only under those conditions that generate very high stresses (as in nanocrystalline systems). As always, it is important to recognize that the MD simulations presented here represent highly idealized conditions and extremely high rates of deformation, and so very large stresses are predicted. Our theoretical approach also represents the highly idealized condition of single dislocation loops, but ignores the inertial terms and does not account for very high rates of loading (and should therefore underpredict the stresses developed in our MD simulations). The fact that our results are still able to capture some of the general mechanisms observed in the experiments is an indication, in part, of the wide variety of heterogeneities, stress raisers and stress states present in real nanocrystalline materials. The most recent very high shear rate experiments of Li et al. (2009) on nanocrystalline aluminum represent the experimental stress and strain rate states that are closest to those in the MD simulations, but there is still a gap in strain rate of several orders of magnitude. In a related vein, the stress-dependent and orientation-dependent apparent stacking fault energy EASF may be important in nanocrystalline metals because of the presence of stress gradients. In general, the difference ðF Lp F Tp Þ itself will be strongly dependent on stress gradients. Thus one expects the likelihood of twinning to be higher in materials with strong stress gradients. This is the case with nanocrystalline materials (Froseth et al., 2004), and the effect is accentuated by the large stresses that are developed during plastic deformations. These ideas are also consistent with the observation that twinning in nanocrystalline aluminum is often observed near grain corners (Chen et al., 2003) and other heterogeneities which provide strong gradients.

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5. Summary MD simulations of the simple shear of single crystals of aluminum are used to demonstrate that there is a strong orientation-dependence of the deformation mechanisms that are possible in aluminum. We show that deformation twins are favored for specific loading orientations even in this high stacking fault energy material, but that the associated stress levels are so large that this mechanism is only likely to be observed under the high stress conditions developed within nanocrystalline aluminum. Our simulations also demonstrate that because of the crystal structure and the energy landscape, there are two somewhat different deformation modes (in terms of the activation of partial dislocations) that can lead to deformation twins, resulting in both the potential for relatively large twins under some conditions and very small twins under other conditions. We also demonstrate that much of the character of these deformation mechanisms can be explained from a theoretical perspective by accounting explicitly for the orientation-dependent interaction of partial dislocation loops, and the theoretical analysis explains the relative difficulty of observing deformation twins in polycrystalline aluminum.

Acknowledgments This work was performed under the auspices of the Center for Advanced Metallic and Ceramic Systems (CAMCS) at the Johns Hopkins University. The authors acknowledge support from the Army Research Laboratory under the ARMAC-RTP Cooperative Agreement numbers DAAD19-01-2-0003 and W911NF-06-2-0006. The authors also thank Professor Evan Ma, Dr. Bin Li, and Dr. Buyang Cao at Johns Hopkins University for helpful discussions. Appendix A. Interaction energy between partial dislocation loops The expressions to find the interaction energy from the trailing and the leading partial dislocations (PD) of an extended dislocation configuration (radii, rLp and rTp, are for the leading and trailing PD, respectively) are as follows. Consider a pair of coaxial dislocations. For the (inner) trailing PD loop of radius ri (with a fixed outer loop radius, r o ¼ r Lp ) Z 2p dEintTp ¼ F PK Tp ðfÞr i df f¼0

and EintTp ¼

Z

r i ¼ rTp ri ¼ 0:0019bp 9

dEintTp dri

where F PK Tp ðfÞ ¼ 9rLp ðfÞ  bTp  xTp ðfÞ9. rLp is the stress field due to the leading PD loop. Explicit expressions for stress field of a circular dislocation loop are given by Langdon (2000). bTp is the Burgers vector of the trailing PD and xTp is the line vector. Note that rLp and xTp depend on the location of a particular point on the inner loop and are defined using an angle, f, where the reference f ¼ 0 is along bLp . Similarly, for the (outer) leading PD loop of radius ro (with a fixed inner loop radius, r i ¼ r Tp ) Z 2p dEintLp ¼ F PK Lp ðfÞr o df f¼0

and EintLp ¼

Z

r o -1

r o ¼ rLp

dEintLp dr o

where F PK Lp ðfÞ ¼ 9rTp ðfÞ  bLp  xLp ðfÞ9. Results from the above calculation are tabulated in Tables 2 and 3. Table 2 shows numerical values of the interaction energy (in eV) for the case of slip (between the trailing and the leading PD), and Table 3 shows those for the case of

Table 2 Interaction energies between the leading and the trailing partial dislocations for the case of slip. Loading orientation

˚ rTp (A)

˚ rLp (A)

dEint (eV)

OA OB OC OD OE

13.83 10.7 8.369 8.369 8.369

26.65 21.7 15.4 12.7 11.8

37.11 2.693 2.382 3.121 3.512

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293

Table 3 Interaction energies between the leading and the twinning partial dislocations for the case of twinning. Loading orientation

˚ rTwp (A)

˚ rLp (A)

dEint (eV)

OA OB OC OD OE

29.3 28.6 29 28.84 28.4

61.5 61.5 61.5 61.5 61.5

14.04 13.312 13.756 13.559 13.109

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