Torsional Detwinning Domino in Nanotwinned One-Dimensional

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Torsional Detwinning Domino in Nanotwinned One-dimensional Nanostructures Haofei Zhou,†,‡ Xiaoyan Li,† Ying Wang,§ Zishun Liu,§ Wei Yang,|| and Huajian Gao‡,*

†

Center of Advanced Mechanics and Materials, Applied Mechanics Laboratory,

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China ‡

§

School of Engineering, Brown University, Providence, RI 02912, USA

International Center for Applied Mechanics, State Key Laboratory for Strength and

Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China ||

Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China

*

Corresponding author. [email protected]

Abstract: How to maintain sustained deformation in one-dimensional nanostructures without localized failure is an important question for many applications of nanotechnology. Here we report a phenomenon of torsional detwinning domino which leads to giant rotational deformation without localized failure in nanotwinned one-dimensional metallic nanostructures. This mechanism is demonstrated in nanotwinned Cu nanorods via molecular dynamics simulations, where coherent twin boundaries are transformed into twist boundaries and then dissolved one by one, resulting in practically unlimited rotational deformation. This finding represents a fundamental advance in our understanding of deformation mechanisms in one-dimensional metallic nanostructures. 1

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Keywords: Detwinning, torsion, plastic deformation, one-dimensional nanostructures, atomistic simulation

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One-dimensional (1D) nanostructures

represent an

important class of

nanomaterials which play a critical role in a broad range of applications including nanoelectronics, high efficiency energy storage, ultrasensitive sensing and nanoelectromechanical devices.1,2 While extensive experimental and numerical studies have been performed on the mechanical properties and deformation mechanisms of 1D nanostructures under tension, compression and bending,3-11 little is known about the torsional deformation of 1D nanostructures.12-16 During the last decade, it has been widely demonstrated that coherent twin boundaries (TBs) can be exploited to enhance the mechanical and physical properties of materials,17-30 and recently, nanowires and nanopillars containing nano- and even angstrom-scaled twins have been synthesized in laboratories and shown to possess a number of interesting properties.31,32 These advances are calling for investigations aimed at the mechanical behaviors and deformation mechanisms of nanotwinned (nt) 1D nanostructures. Here we report a unique and surprising new deformation mechanism governed by a torsional detwinning domino in nt-1D nanostructures. In this phenomenon, the TBs are dissolved one-by-one under torsion, giving rise to sustained plastic deformation without localized failure. The reported torsional detwinning domino deviates from all previously reported detwinning mechanisms in which TB migration is involved. Successive slip of twinning partial dislocations causes TB migration, coarsening and annihilation of twins in various material systems including bulk and thin films,18,25,33-37 nanowires/nanopillars,4,31,32,38 or near crack tips.39-41 Detwinning via the migration of incoherent Σ3 {112} TBs is similar to coherent TB migration because both are governed by the collective glide of multiple twinning partial dislocations on neighboring {111} planes, as shown in previous experimental and computational 3


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studies.42 Recent experiments have shown that the nucleation and growth of second-generation twins within original twin grains, during which TB migration might be involved, could also stimulate detwinning in Mg.43 To demonstrate the torsional detwinning domino in nt-1D nanostructures, we performed a series of atomistic simulations on nt- and twin-free Cu nanorods of various diameters (i.e., D=10, 20 and 30 nm) at three different temperatures (i.e., T=1, 300 and 600 K). The height of each nanorod was chosen to be three times the diameter. In nt-nanorods, parallel Σ3 (111) TBs were embedded with uniform spacings of λ=0.63, 0.84, 1.05, 1.26, 1.68, 2.1, 3.15, 5.25, 10.5 nm. The TBs are aligned nearly perpendicular to the axial direction Z with a tilt angle less than ±5° (a slight tilt angle was to demonstrate the robustness of the reported phenomenon). The largest nanorod contains 5,600,000 atoms. The interatomic interactions were described by an embedded atom method potential.44 The time step was chosen as 1 fs. Before loading, the system was equilibrated for 200 ps during which the nanorod was allowed to relax along the axial and radial directions. Torsion with a constant surface shear strain rate of 3.33×107 s-1 was then applied to the nanorod until a total surface shear strain of 420% was reached, corresponding to a total simulation time of 126 ns. An initial velocity profile corresponding to the specified shear strain rate was imposed to all atoms at the onset of loading. Afterwards, ten layers of atoms at both ends were rotated at a constant rate around the axis while the remaining atoms were allowed to adjust their coordinates through NVT ensemble.45 Axial stress may arise during torsion due to the fixed boundary condition along the axis. In some cases, the nanorods were preloaded by a uniaxial tensile stress (via a simulation box stretching method) up to 0.5 GPa (at which no yielding occurs) before torsion to demonstrate that the axial stress is not essential to the emergence of detwinning domino. The 4


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common neighbor analysis (CNA) method46 was adopted to identify defects that arise during deformation. Colors are assigned to atoms according to their local crystal order: blue stands for perfect atoms, cyan for atoms in stacking faults or TBs, and pink for atoms near vacancies, dislocation cores or a surface. Figure 1(a) shows the typical surface morphology of a twisted nt-nanorod with D=20 nm and λ=1.05 nm. It is seen that the surface becomes rough with numerous slip steps. The average depth of the slip steps are on the order of 1 nm. The roughness is associated with the depletion of TBs (i.e. detwinning) during torsion, as indicated by Fig. 1(b). For clarity, fcc atoms are set to be transparent. It is clear that all twin planes underneath the rough surface have been dissolved, leaving an “infected” region filled with various types of lattice defects. Similar plastic deformation patterns were observed in all simulated nt-nanorods. The detwinning mechanism will be described shortly. For comparison, the deformation patterns of a twisted single-crystalline twin-free counterpart were also displayed in Fig. 1(c). The surface of the twin-free nanorod after torsion is featured by a few slip ledges, which have much larger scales (about 5 nm in depth) and are distributed much more narrowly than those observed in Fig. 1(a). This implies that the plastic deformation is localized to several atomic planes in the twin-free samples. The “infected” region is filled with a few stacking fault ribbons and dislocation lines, as shown in Fig. 1(d). Figure 2(a) shows the surface shear-stress vs strain relations of nt (red curve, with D=20 nm and λ=1.05 nm) and single-crystalline (blue curve) nanorods during torsion. The shear stress and shear strain at the surface are given by τ=2T/πR3 and ε=κR, respectively, where T is the torque, κ the twist per unit length, and R the radius of the nanorod. The applied torque is calculated based on the elasticity theory as,

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T=

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Ω N σ yzi xi − σ xzi y i ) ( ∑ L i =1

(1)

i i where σ xzi and σ yz are the shear stress components of atom i, xi and y the

Cartesian coordinates of atom i, Ω the atomic volume, L the sample height and N the total number of atoms in the simulated sample. It is observed from Fig. 2(a) that both samples responded cyclically to the external load with a period of about 17.5% strain, corresponding to a twist angle of 60°. In each cycle of the stress-strain curve of the twin-free samples, a number of dislocations nucleated from the surface and then reacted with each other, leading to dislocation multiplication and formation of complex dislocation networks on the (111) plane. These mechanisms are similar to those reported in previous atomistic simulations of single-crystalline Au nanowire.13,14 In contrast to the twin-free samples, it is seen that each cycle on the stress-strain curve of the nt-samples corresponds to the depletion of one TB, as evidenced by Fig. 1(b). To further understand the twist-induced deformation mechanism, we extracted the atomic-level Lagrangian strain tensor η from the twisted configurations and calculated the local von Mises shear invariant defined by η Mises =

1 2 Tr ( η − ηm I ) , 2

{

}

where ηm is the hydrostatic part of the strain tensor, I stands for the unit tensor, and Tr(·) denotes the trace of a second-order tensor.47 Figures 2(b)-(g) show the shear strain contours on the axial sections of the twisted nanorods, where atoms are colored according to their ηMises values. The twist-induced strain gradient along the radial direction is clearly identified. For the nt-nanorod, detwinning started at a TB near the middle of the nanorod, which was transformed into a twist boundary to accommodate a certain amount of plastic strain. Further deformation caused the twisted boundary to disappear and the original TB was dissolved. The same detwinning process then

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spread one after the other into the neighboring TBs under increasing torsion, as shown in Figs. 2(b)-(d). Plastic deformation was accumulated and delocalized through the consecutive depletion of TBs in the “infected” region outlined by a dashed box in Fig. 2(d). This is in contrast to the highly localized plastic deformation in the single-crystalline nanorod, as shown in Figs. 2(e)-(g). More interestingly, only one TB is depleted in the nt-nanorod for every 60° of torsion, which may be referred to as a single detwinning event. Every detwinning event sets off a subsequent detwinning event on a neighboring TB, leading to a detwinning domino propagating along the nanorod. Each peak in the stress-strain curves shown in Fig. 2(a) corresponds to dislocation nucleation on a TB. A significant difference between the two curves in Fig. 2(a) is that the peak stress remains at about 2.0 GPa throughout the torsion for the nt-nanorod, while it decreases gradually from as high as 3.2 GPa in the first cycle to less than 1.3 GPa at the end for the single-crystalline sample. The reduction in peak stresses of the twin-free sample implies that dislocations there can nucleate and multiply with ease due to progressive strain localization and accumulation of defects. However, the peak stresses of the nt-sample remain at a nearly constant stress level associated with the onset of individual detwinning events and the subsequent strain delocalization. Additional simulations on nt-nanorods with different TB spacings revealed that the critical stress required for surface dislocation nucleation that leads to the onset of detwinning increases if the TB spacing is reduced, following the usual “smaller is stronger” rule widely observed in small scale materials. The reader is referred to the Supplementary Information for more details (see Supplementary Figure 1 and associated discussions). We proceed to explore the physical nature of the detwinning domino in the 7


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nt-nanorod. First imagine a typical nt-structure with a stacking sequence of CABCBAC. Figure 3(a) shows a diagram of three consecutive (111) layers. If all atomic layers above the twin plane C are rigidly twisted by an angle of 60°, the resulting configuration is shown in Fig. 3(b). It is seen that the mirror symmetry across the twin plane C has been removed. The final stacking sequence is CABCABC, recovering the perfect fcc structure. Thus the detwinning domino essentially originates from the rotational symmetry of {111} planes in fcc metals. To reveal further mechanistic details of the detwinning domino at the atomic scale, we provide a series of simulation snapshots in Figs. 4(a)-(i) that capture the structural evolutions of a TB at various strain levels. Initially, a partial dislocation nucleates from the free surface and glides along the TB, as shown in Fig. 4(a). This dislocation is then trapped at the center due to the stress gradient in the radial direction. Meanwhile, another partial dislocation nucleates from the free surface, as shown in Fig. 4(b). More partial dislocations emit from the free surface and react with each other, leading to the formation of dislocation junctions indicated by the dashed lines in Fig. 4(c). Each arm of the junction is an extended dislocation consisting of leading and trailing partial dislocations separated by a stacking fault. Similar dislocation structures have been reported in atomistic

simulations of a

single-crystalline Au nanowire under torsion and computational studies on the structure of (111) twist boundaries in fcc metals.13,48 As more dislocations interact with the pre-existing dislocation structures, a complex dislocation network becomes visible in Fig. 4(d). Crystallographically, such dislocation network corresponds to a high-angle twist boundary. The partial dislocations in the network are geometrically necessary to accommodate the lattice mismatch between the original TB and the adjacent (111) plane. A twist from 0° to 30° gradually dismantles the coherency of the 8


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TB, which results in a fully disordered configuration shown in Fig. 4(e). Interestingly, when the twist angle increases from 30° to 60°, the dislocation network gradually coarsens as dislocations progressively escape from the free surface, as shown in Figs. 4(f)-(h). Such phenomenon is accompanied by a significant drop in the applied shear stress after yielding, as evidenced by Fig. 2(a). The final configuration after twisted by 60° is just a perfect fcc structure. It is noted that each detwinning event is mediated by the formation and annihilation of a twist boundary, and intense dislocation interactions occur during the detwinning process, leading to the formation of various types of defects, such as vacancies, stacking fault tetrahedrons (SFTs) and even extrinsic stacking faults (ESFs), as shown in Fig. 4(i). To understand the phenomenon of detwinning domino, it will be helpful to clarify two questions. First, why does detwinning occur at different TBs, rather than localize in the same plane? The second question is why detwinning occurs sequentially, rather than randomly at any twin planes. To address these questions, we have analyzed the distribution of shear stress σθz in the twisted nt-nanorod and found significant stress concentration at the junctions of TBs and the free surface (see Supplementary Figure 2). More importantly, the maximum σθz occurs at the outermost TB of the nt region, which is found to be 3.56GPa, about 0.5GPa higher than that on the rest of the TBs. This maximum σθz induces detwinning always at the edge of the nt region, leading to the observed sequential detwinning. The torsional detwinning domino in a nt-1D nanostructure departs from all previously reported detwinning mechanisms in that it does not require TB migration. Instead, detwinning is realized through sequential transformation (into a twist boundary) and annihilation of TBs with a rotational period of 60°, which contributes to the delay of shear localization in the nanostructure, as indicated by Fig. 1. The 9


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stress-strain curve shown in Fig. 2 captures the cyclic nature of detwinning domino. The sequential depletion of TBs leads to sustained rotational plastic deformation. Take a nanorod containing a total of 60 TBs and with R/L=1/6 for instance, the complete depletion of these coherent interfaces via the “domino” detwinning mechanism gives rise to ~1000% of plastic strain. Thus the operation of a detwinning domino could lead to practically unlimited rotational deformation in a 1D nanostructure.In summary, we have revealed a new and unique phenomenon of torsional detwinning domino which may sustain rotational deformation in a nt-1D nanostructure. In this process, one TB is depleted for every 60° of twisting, and the process is accumulative under a constant peak stress until all TBs in the structure are removed. The structural evolution of each TB during detwinning is governed by the formation of a twist boundary, which is featured by a complex dislocation network. Detailed analysis revealed that a network of geometrically necessary dislocations consisting of numerous dislocation triple junctions tangle together to accommodate the lattice mismatch across the twisted TB. This detwinning domino is unique and interesting because it does not require the conventional detwinning mechanism by TB migration. In addition, the domino effect leads to sustained rotational deformation without localized failure. This mechanism may be exploited in designing materials and devices based on 1D nanostructures with unusual mechanical properties.

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Supporting Information. Discussion about the size effect of the torsional detwinning domino in Cu nt-nanorods.

Acknowledgements The authors gratefully acknowledge financial support by the NSF (under Grant No. CMMI-1161749), the NSFC (Grants No. 11321202, 11372152, and 51420105001) and the MRSEC Program (Award No. DMR-0520651) at Brown University. We also thank computational support from NICS (Grant No. MS090046).

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Figure captions Figure 1. Deformation patterns of nt (a-b) and single-crystalline (c-d) nanorods after being loaded by torsion to 420% of surface shear strain. (a) Surface morphology of twisted nt sample. (b) Configuration of twisted nt sample in the longitudinal section. (c) Surface morphology of twisted single-crystalline sample. (d) Configuration of twisted single-crystalline sample in the longitudinal section.

Figure 2. Comparisons of deformation responses and behaviors between nt and single crystalline nanorods. (a) Stress-strain relations of nt (top) and single-crystalline (bottom) nanorods under torsion. (b-d) Atomic shear strain contours of nt nanorods during twisting. (e-g) Atomic shear strain contours of single-crystalline nanorods during twisting.

Figure 3. Physical mechanism of twist-induced detwinning. (a) Schematic of three adjacent (111) atomic layers arranged in a twinned structure. (b) The same stack of atomic layers arranged in a perfect fcc structure after a twist of 60° is imposed on the layers above the twin plane C.

Figure 4. Structural evolution of a twin plane during a twist-induced detwinning event. (a-b) Nucleation of partial dislocations from the free surface. (c-d) Formation of dislocation junctions due to interactions among partial dislocations. (e) Atomic configuration of TB after twisting by ~30°. (f-g) Coarsening of dislocation network as dislocations escape from the surface. (h) Depletion of the TB after twisting by 60°. (i) Abundant residual lattice defects in the “infected” region after a series of detwinning events. 16


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Deformation patterns of nt (a-b) and single-crystalline (c-d) nanorods after being loaded by torsion to 420% of surface shear strain. (a) Surface morphology of twisted nt sample. (b) Configuration of twisted nt sample in the longitudinal section. (c) Surface morphology of twisted single-crystalline sample. (d) Configuration of twisted single-crystalline sample in the longitudinal section. 83x55mm (300 x 300 DPI)


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Comparisons of deformation responses and behaviors between nt and single crystalline nanorods. (a) Stress-strain relations of nt (top) and single-crystalline (bottom) nanorods under torsion. (b-d) Atomic shear strain contours of nt nanorods during twisting. (e-g) Atomic shear strain contours of single-crystalline nanorods during twisting. 168x170mm (300 x 300 DPI)


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Physical mechanism of twist-induced detwinning. (a) Schematic of three adjacent (111) atomic layers arranged in a twinned structure. (b) The same stack of atomic layers arranged in a perfect fcc structure after a twist of 60° is imposed on the layers above the twin plane C. 82x43mm (300 x 300 DPI)


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Structural evolution of a twin plane during a twist-induced detwinning event. (a-b) Nucleation of partial dislocations from the free surface. (c-d) Formation of dislocation junctions due to interactions among partial dislocations. (e) Atomic configuration of TB after twisting by ~30°. (f-g) Coarsening of dislocation network as dislocations escape from the surface. (h) Depletion of the TB after twisting by 60°. (i) Abundant residual lattice defects in the “infected” region after a series of detwinning events. 163x129mm (300 x 300 DPI)


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265x190mm (300 x 300 DPI)


Torsional Detwinning Domino in Nanotwinned One-Dimensional

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