† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 51201015 and U1532262).
Five-fold twinned nanostructures are intrinsically strained or relaxed by extended defects to satisfy the space-filling requirement. Although both of metallic and semiconductor five-fold twinned nanostructures show inhomogeneity in their cross-sectional strain distribution, the evident strain concentration at twin boundaries in the semiconductor systems has been found in contrast to the metallic systems. Naturally, a problem is raised how the chemical bonding characteristics of various five-fold twinned nanosystems affects their strain-relieving defect structures. Here using three-dimensional (3D) electron diffraction mapping methodology, the intrinsic strain and the strain-relieving defects in a pentagonal Ag nanowire and a star-shaped boron carbide nanowire, both of them have basically equal radial twin-plane width about 30 nm, are non-destructively characterized. The non-uniform strain and defect distribution between the five single crystalline segments are found in both of the five-fold twinned nanowires. Diffraction intensity fine structure analysis for the boron carbide five-fold twinned nanowire indicates the presence of high-density of planar defects which are responsible for the accommodation of the intrinsic angular excess. However, for the Ag five-fold twinned nanowire, the star-disclination strain field is still present, although is partially relieved by the formation of localized stacking fault layers accompanied by partial dislocations. Energetic analysis suggests that the variety in the strain-relaxation ways for the two types of five-fold twinned nanowires could be ascribed to the large difference in shear modulus between the soft noble metal Ag and the superhard covalent compound boron carbide.
Up to now, five-fold twinning has been found in widespread materials including face-centered cubic (FCC) metals (such as Au, Ag, Cu), semiconductors (such as Si, Ge, diamond), covalent compounds, and even organic materials.[1–5] Nanoscale five-fold twinned structures have drawn more research interests for many decades due to their unusual symmetry[1,6] and the unique properties, such as anomalous strength,[7,8] tunable plasmonic properties,[9] and enhanced catalytic activity.[2,10]
Ideally, each of the five equally shaped single crystalline segments composing a five-fold twinned structure encloses an angle of 72°. However, in the known five-fold twinned structures an intrinsic angular misfit remains due to the incomplete 360° space-filling. For FCC metals or diamond-cubic materials, the dihedral angle between {111} twin planes is 70.35° which results in a well-known angular deficiency of 7.35° in the formation of five-fold twinning.[1] Superhard α-rhombohedral boron suboxide (B6O) nearly fits the space-filling requirement of five-fold twinning by remaining an angular deficiency about 1°.[5] But α-rhombohedral boron carbide (nominal atomic content B4C, space group
Recently, aberration-corrected electron microscopy investigation combined with first-principles simulation has indicated that semiconductor (diamond, silicon) five-fold twinned structures exhibit shear-modulus-dependent strain concentration at twin boundaries in contrast to the relatively smooth strain distribution in metallic five-fold twinned structures.[3] But it has been unclear how the chemical bonding characteristics influence the structural relaxation phenomena in five-fold twinned nanostructures and the related mechanism should be explored both experimentally and theoretically. Until now, the experimental studies on the strain-relaxation phenomena and the related mechanism of five-fold twinned structures were almost focus on the metallic objects.[15–20] In comparison, limited experimental investigations revealed the defect structures for the elastic strain relaxation in covalent five-fold twinned structures.[14,21–24]
In this work, the strain-relieving defects have been non-destructively revealed in two types of five-fold twinned nanowires (FTNWs) using 3D electron diffraction mapping methodology. One is a pentagonal Ag nanowire with diameter of 55 nm and the other is a star-shaped B4C nanowire with diameter of 92 nm, both of them have basically equal twin-plane width in the radial direction. The 3D electron diffraction mapping methodology allows to obtain the 3D intensity distribution of specific reflections diffracted from nanostructures[18,21] in a relatively small tilting range compared with real-space electron tomography.[25] Traditionally, the transverse structural observation of one-dimensional (1D) nanostructures in transmission electron microscope (TEM) required the cross-sectional sample preparation which inevitably introduces external strain or defects interfering the characterization of the intrinsic structures. In comparison, 3D electron diffraction mapping has shown a non-destructive nature in the characterization of cross-sectional strain and defects of 1D nanostructures.[18,21] By using this methodology, the non-uniform strain and defect distribution between the five single crystalline segments in two FTNWs have been found. For the 55-nm-diametered Ag nanowire, the inhomogeneous strain is partially relieved by introducing stacking fault layers accompanied by partially dislocations localized near the twin boundaries or in the middle of the single crystalline segments. In contrast, the internal strain in the 92-nm-diametered B4C nanowire with high shear modulus is almost totally relieved by the formation of high density of planar defects, such as stacking faults and microtwins, in each single crystalline segment. Energetic analysis suggested that the different strain relaxation phenomena in the two types of FTNWs are closely connected with their shear modulus. To understand the related mechanism behind further theoretical investigation by first-principle calculations is required.
To reveal how the chemical bonding characteristics of various five-fold twinned nanosystems affects their strain-relieving defect structures, two types of FTNWs, metallic and covalent, were selected in the study. It is well known that Ag FTNWs are among the most researched metallic five-fold twinned nanostructures due to their facile synthesis method[26] and the potential applications.[2,8] Thus commercial Ag FTNWs were used for the internal structure characterization in this study. Covalent five-fold twinned nanostructures are widely found in semiconductors (such as Si, Ge, diamond) and compounds. But Si or Ge five-fold twinned nanostructures are mostly formed in semiconductor thin films.[1] Usually, the synthetic Si nanowires have been found to have a single crystalline structure.[27–29] It has been found that α-rhombohedral boron-rich compounds, such as B6O and B4C, can form five-fold twinned nanowire structures. However, the angular misfit of B6O five-fold twinned structures is only about 1°,[5] which is much smaller than the 7.35° angular deficiency in the formation of Ag five-fold twinned structures. The big difference in the misfit angle can cause the difference in strain distribution and the related strain-relieving defect structures between B6O and Ag FTNWs. In contrast, B4C FTNWs have show a misfit angle about 5°,[4] which is more comparable to that of the Ag FTNWs. Thus the B4C FTNWs were chosen for the internal structure comparison study with the Ag FTNWs.
The B4C FTNWs as shown in Fig.
![]() | Fig. 1. Scanning electron microscope (SEM) images of (a) the sintered B4C nanowires and (b) the solution-synthesized Ag nanowires dispersed on a silicon wafer. |
The α-rhombohedral B4C nanowires synthesized in a solid-state reaction at 1100 °C have shown a star-shaped or truncated star-shaped cross-sectional morphology in scanning electron microscope (SEM).[4] The electron diffraction analysis for the B4C FTNWs (supporting information Fig.
For Ag FCC lattice, the primitive unit cell is rhombohedral structure as illustrated by black dashed lines in Fig.
As shown in Fig.
To precisely identify the orientation relationship between the five single crystalline segments of the FTNWs, two groups of reflections were reconstructed for each FTNW. For the Ag FTNW, the reciprocal volume containing (331) and
We carried out 3D electron diffraction mapping for a B4C FTNW with diameter of 92 nm as shown in Fig.
As the intensity distributions of specific reflections diffracted from a finite nano-sized crystallite are strongly affected by the nano-object shape, internal strain and defects, to uncover the cross-sectional morphology, relaxation mechanism and the related defect structures of the B4C FTNW, the 2D intensity distribution of (112) reflections in the reciprocal plane perpendicular to the nanowire twinning axis is extracted from the 3D data set shown in Fig.
As demonstrated in Fig.
![]() | Fig. 6. High-resolution TEM image of T4 segment oriented to [010] zone-axis showing irregular spaced stacking faults (SF). |
In Fig.
A Ag FTWN with diameter of 55 nm as shown in Fig.
To analyze the cross-sectional strain and defects of the Ag FTNW shown in Fig.
For each line-profile shown in Figs.
To identify the influence of the stacking faults on the intensity distribution of (331) reflections, we carried out a series of kinematic diffraction simulations according to the structural model of Ag single crystalline segment with a single layer of stacking fault. The kinematic diffraction simulation results indicate that the stacking fault layers with distance to their parallel twin plane in the range about from 5 nm to 17 nm, could induce evident streaking or peak splitting of (331) reflection along the direction perpendicular to the stacking fault layers (see supporting information, Fig.
Through the analysis of the intensity mapping results of B4C FTNW with diameter of 92 nm and Ag FTNW with diameter of 55 nm, it can be confirmed that for both of the FTNWs, the star-disclintion-core induced elastic strain could be relieved by introducing defect structures. For the B4C FTNW with diameter of 92 nm, the evident flares of (112) reflections along the direction perpendicular to the twin planes indicate the presence of high density of planar defects (such as stacking fault and micro-twin) in each single crystalline segment. The elastic strain energy is diminished through the formation of the high density of planar defects in the 92-nm-diametered B4C FTNW. However, for the Ag FTNW with diameter of 55 nm, the combination of the intensity fine structure analysis and the kinematic electron diffraction simulation indicates that the star-disclination induced elastic strain in some segments (T1, T2, T3, and T4) could be partially relieved by the formation of stacking fault layers accompanied by partial dislocations near the twin plane, or in the middle of the single crystalline segments.
In this study, the two types of FTNWs have different cross-sectional morphology. As shown in Fig.
![]() | Fig. 9. Cross-sectional geometry of (a) B4C FTNW and (b) Ag FTNW free of internal strain and defects. |
For FCC five-fold twinned nanostructures, such as Ag FTNWs, there exists an intrinsic angular deficiency (Θ) of 7.35° to be accommodated as shown in Fig.
For the Ag FTNW with radius about 30 nm, the reported experimentally measured Poisson’s ratio is 0.191 (ν = 0.191).[34] The experimentally measured average Young’s modulus (E) for Ag FTNW with diameter about 55 nm is 120 GPa,[8] then we can approximately evaluate the shear modulus for such Ag FTWN to be about 50 GPa according to the relationship G = E/2(1 + ν). Thus we approximately estimate the elastic strain energy (in a unit length) of the Ag star-disclination cylinder with radius of 29 nm to be about 1.7× 10−8 J/m (R = 29 nm, G = 50 GPa, Θ = 7.35°, ν = 0.191). Therefore, as shown in Fig.
![]() | Fig. 10. The calculated elastic strain energies of different FTNW models with star-disclination core. |
Similarly, we can estimate the magnitude of the elastic strain energy (in a unit length) of B4C FTNW with radius of 30 nm. Because there is not reported experimental result about the Poisson’s ratio of B4C nanowires, we use the value ν = 0.18 for B4C bulk material[35] as an approximation. The reported experimentally measured Young’s modulus of B4C nanowires is 428 GPa,[36] then the approximate shear modulus for B4C nanowire is about 181 GPa according to the relationship G = E/2(1 + ν). Due to the angular misfit (Θ) for B4C FTNWs in the range from 4.55° to 5.25°, the estimated elastic strain energy (in a unit length) of B4C FTNW with radius of 30 nm is in the range from 2.5 × 10−8 J/m to 3.3× 10−8 J/m. Taking account of the star-shaped cross-section morphology of the B4C FTNW in this study, its corresponding theoretical elastic strain energy (in a unit length) could be larger than the above estimated lower limit of 2.5 × 10−8 J/m, because that the estimation according to the star-disclination model is only considering the inner cylinder part with radius of 30 nm as illustrated in Fig.
The angular-misfit induced elastic strain energy is the driving force for the structural relaxation of five-fold twinned nanostructures. The above energetic analysis indicates that the elastic strain energy for B4C FTNW with diameter of 92 nm is more than 1.5 times larger than that of the Ag FTNW with diameter of 55 nm. Thus in an energetic view, the accumulated elastic strain energy of the 92-nm-diametered B4C FTNW is prone to be relieved compared with that of the 55-nm-diametered Ag FTNW. According to the theoretical equation for the evaluation of disclination induced elastic strain energy, GΘ2R2/16π (1 – ν),[11] even with the same radius, the star-disclination induced elastic strain energy of B4C FTNW (G = 181 GPa, ν = 0.18, Θ = 4.55°∼ 5.25°) is about 30% ∼ 80% larger than that of the Ag FTNW (G = 50 GPa, ν = 0.191, Θ = 7.35°). Due to the larger shear modulus (G = 181 GPa) of B4C nanowires, the elastic strain energy of B4C FTNWs increases with the nanowire radius more rapidly than that of Ag FTNWs.
Stacking faults are of common occurrence in Ag FTNWs[1,6,12,26,32] and B4C FTNWs,[14,21,37] due to their low stacking fault energy. The stacking fault energy of B4C has been reported to be 75 mJ/m2.[38] Ag metals have lower stacking fault energy. The earlier experimental publication about Ag stacking fault energy is 22 mJ/m2,[39] which is larger than the theoretical result predicted by molecular dynamics simulation.[40] But to our knowledge, there are no reported results about Ag five-fold twinned structures with high density of planar defects. The reported cross-sectional TEM observation of thin Ag FTNWs with diameter less than 100 nm indicated that their internal defects, such as partial dislocations and stacking faults, are mostly localized near the twin boundaries.[12,17,26,41,42] This is consistent with our experimental analysis results of 3D intensity mapping and cross-sectional high-resolution observation for the Ag FTNWs with diameter of several tens nanometers. As the Ag FTNWs are growing thicker than few micrometers, the elastic strain could be relieved by opening gaps on one side of the nanowire side faces.[15] Thus we believe that for Ag five-fold twinned structures, the formation of high density of planar defects is not a preferential way to compensate the intrinsic angular deficiency. The disclination induced lattice distortion combined with stacking fault layers and partial dislocations is one of the favorable way for the angular misfit accommodation in Ag FTNWs with diameter less than 100 nm.
Because of the superhard nature for covalent compound B4C, the presence of partial disclination in B4C FTNWs could induce higher internal strain. Thus the disclination strain field can only exist in the nanometer-sized core of B4C FTNWs.[14] For the B4C FTNW with diameter of 92 nm, the high density of planar defects is the channel for the disclination strain relaxation. In addition, the reported finite element analysis has indicated that the formation of re-entrant surfaces for the five-fold twinned nanostructures can also relieve the elastic strain energy due to the disclination.[43] Therefore, the star-shaped cross-section morphology with five re-entrant groves of the B4C FTNWs is also ascribed to the elimination of the elastic strain energy.
The above analysis about the intensity fine structure of the B4C FTNW has indicated the different cross-sectional size of the five single crystalline segments. This can also be confirmed by the peak intensity variation of the five (112) reflections. As shown in Fig.
We have carried out 3D electron diffraction mapping for a 92-nm-diametered star-shaped B4C FTNW and a 55-nm-diametered pentagonal Ag FTNW, both of them have similar radial twin-plane width about 30 nm, to investigate their internal structure details. The analysis of the geometric relationship, intensity distributions and fine structures of the reflection spots in the reconstructed reciprocal volume acquired by the 3D electron diffraction mapping, revealed the inhomogeneous nature of both of the FTNWs.
Due to the different lattice structures, B4C FTNWs and Ag FTNWs demonstrate different angular-misfit natures. Ag FTNWs consisting of five single FCC crystalline segments remain an intrinsic angular deficiency of 7.35°. But for α-rhombohedral B4C FTNWs, to form 360° space-filling there exists an angular excess in the range from 4.55° to 5.25° according to the various α lattice parameters. The intrinsic angular misfit for the five-fold twinned nanostructures could induce inhomogeneous lattice distortion around the star-disclination core which coincides with the twinning axis. With the increasing radius of five-fold twinned nanostructures, the elastic strain field could be relieved by defect structures.
For the B4C FTNW with diameter of 92 nm in this study, quantitative analysis of the strong flares of (112) reflections along the directions perpendicular to the twin planes indicated the presence of high density of planar defects (such as stacking faults or microtwins) in five single crystalline segments which is a way for the elastic strain relaxation. In addition, the twining-axis shift due to the different sizes between the five single crystalline segments and the re-entrant grooves in the star-shaped cross-section can also relieve the star-disclination-induced elastic strain in the B4C FTNW. However, for the pentagonal Ag FTNW with diameter of 55 nm, the elastic strain is partially relieved by the formation of less localized stacking-fault layers accompanied by partial dislocations, as well as the twining-axis shift.
Energetic analysis indicated that the variety of the strain relaxation ways of the two types of FTNWs is ascribed to the difference in chemical bonding characteristics and the related mechanical properties between the soft noble metal Ag and the superhard covalent compound B4C. Because of the high shear modulus of B4C FTNWs, the star-disclination-induced elastic strain field can only exist in a few-nanometer-sized core. With the increase of the radius, the fast-increasing elastic strain energy in the B4C FTNWs must be relieved through introducing low-energy defect structures, such as stacking faults and microtwins, to minimize the system energy. In comparison, Ag FTNWs have lower shear modulus. Our study confirmed that for pentagonal Ag FTNWs with diameter about 55 nm or even more, the inhomogeneous lattice distortion could coexist with localized stacking fault layers and partial dislocations to compensate the intrinsic angular deficiency of 7.35°.
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