Extended Dislocations in Plastically Deformed Metallic Nanoparticles

1. Introduction

Nanomaterials with excellent mechanical properties have attracted significant nanoelectromechanical systems (NEMS) research interest as sensors, transducers and actuators [1–7]. Growing demand for nanomaterials has stimulated the study of their deformation mechanisms, which is helpful in understanding and/or guiding the experimental work.

Due to the large surface-to-volume ratio of nanomaterials, their deformation defects are different to those of bulks. It has been reported that Shockley partial dislocations have smaller critical nucleation shear stress than that of other dislocations when the size of metallic materials is reduced to a nanoscale [8, 9]. Scientists have discovered that Shockley partial dislocations nucleated and slid the free surface of metallic nanowires or the grain boundary of nanocrystallines, leaving stacking faults or twins [8–10]. To our knowledge, however, studies on nucleation for single crystalline nanoparticles (NPs) are scarce. Recently, Sun et al. [11] investigated the deformation behaviour of sub-crystalline silver particles (10 nm) and found that liquidlike pseudoelasticity is driven by surface diffusion of NPs and external-loading media, instead of internal dislocations. The deformation temperature is important to induce the recoverable surface-diffusion process, which inspires us to investigate the deformation of NPs at low temperatures, where the temperature-dominated surface diffusion will not be particularly active. Although some advanced experimental methods such as in-situ Scanning Probe Microscopy (SPM) and Transmission Electron Microscope (TEM) have been performed to deform NPs, the theoretical study of their deformation mechanisms, which are inexpensive and save time, is still a good choice [4, 12]. Using an electron beam during normal imaging may accelerate the diffusion of atoms on free surfaces, which misleads the observation [13, 14]. In this circumstance, atomic simulation is still a good candidate for studying the deformation of NPs. In our previous work, the deformation mechanism of core/shell-type NPs was studied using atomic simulation and we found that twins are the main defects in face-centred-cubic (FCC) metallic materials resulting from the ultrahigh atomic stress in the core/shell boundary [15]. In the present work, the compressive deformation of FCC metallic NPs was investigated using molecular dynamics (MD) to reveal their deformation mechanism.

2. Simulation Model and Method

Spherical configurations of Cu and Ag NPs with a diameter of 6 nm were constructed initially. To obtain the deformation features of the aforementioned naked NPs, the core/shell of Cu/Ag NPs (core with a diameter of 6 nm and shell with a thickness of 1 nm) was constructed.

In the present simulation, the embedded-atom potentials for Cu, Ag, and Cu-Ag were chosen from refs. [16, 17] as they are calibrated using the ab initio values of stacking fault and twin formation energies. The properties were obtained by fitting the values at a temperature of 0. However, the prediction of the properties at other temperature is reliable, for example solubility energies (1052 K) and the enthalpy of mixing of liquid Cu–Ag solutions at 1423 K. Further, a simple Cu-Ag phase diagram can be predicted by current embedded-atom method (EAM) potential, which is comparable to experimental results. Many researchers have employed current potential to study the formation and properties of Cu–Ag nanostructures using an MD simulation method [18–25]. The success of investigations into the mechanical deformations of the Cu–Ag system particularly encourages this work.

After the initial construction, MD simulation was carried out to equilibrate the NPs at 200 K using a Nosé-Hoover thermostat [26, 27] to ensure the reconstruction of the atoms in the surface and interface. The time step was fixed at 0.001 picoseconds (ps) and the Velert-velocity integration algorithm was used. Free-boundary conditions were imposed in all three dimensions. In addition, LAMMPS software developed at Sandia National Laboratories [28, 29] was used in the simulation, and the visual atomic configuration was generated using VMD software [30].

Two types of uniaxial loading, dynamic and quasistatic, were chosen in this simulation. This excludes the possible problems of misleading results obtained from the single deformation mode. The dynamic loading at a strain rate of 10⁹ s⁻¹ was performed provided that the rigid atoms (pink in Figure 1) in the upper slab had a negative velocity along the z-direction, while the rigid atoms in the lower slab were fixed. In the case of quasistatic loading, the NPs were first strained based on the prescribed compressive strain increment of 0.5% along the z-direction and then relaxed for 20 ps to obtain a macroscopic equilibrium configuration.

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Figure 1.

Simplified model of compressed NPs. Two layers (white) of the two slabs are where the atoms can move and the pink layer is where there is no movement of the particles. The rigid atoms in the upper slab were given a negative velocity along the z-direction, while the rigid atoms in the lower slab were fixed to compressed NPs.

3. Results and Discussion

Simulated compressive stress versus strain curves are shown in Figure 2. Interestingly, compressive stress increased in a zigzag fashion for the naked NPs (Figure 2a and b), which is unlikely to smooth the stress arising from the core/shell-type NPs (Figure 2c). Furthermore, the sawtooth feature, i.e., zigzag rising of loading, is independent of particle's shape and loading method. The absence of sawtooth behaviour for the core/shell Cu/Ag NPs can be understood by the existence of the core/shell interface. The experimental results show around 20 gigapascals (GPa) of atomic stress was generated at the interface region under loading [6]. The high pressure in the capsule was large enough to hide the system's energy variation originating from the dislocation movement.

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Figure 2.

Compressive stress versus strain curves of (a) Cu NPs; (b) and (c) Ag NPs with a spherical and icosahedral [31] shape respectively; and (d) Cu/Ag core/shell NPs. The strain rate of dynamic loading is 10⁹ s⁻¹. For quasistatic loading, the length of each strain step is 0.5% and relaxation time is 20 ps.

For the studied strain, the deformation process of spherical naked NPs can be divided into two stages. In stage 1 (strain < ∼40%), the loading was relatively smooth and the NPs still maintained their spherical shape; in stage 2 (strain > ∼40%), the loading increased in a zigzag fashion with strain, with the shape of NPs changing from a sphere to a cylinder. The entire deformation process can be found in the supplementary video (CuNP-compress.mpg). Specifically, the atomic structure evolution of the naked NPs in the deformation will be discussed later.

Figure 3 shows the evolution of a defect structure in naked NPs in deformation stage 1. The perfect FCC atoms and the surface atoms are not shown in the figures to discern defects in the NPs. At first, the leading Shockley partial dislocations of the Burgers vectors [1̄2̄1̄]/6 and [2̄11̄]/6 nucleate from the intersection of the slab and the free surface of NP. They then slide across the slip plane leaving two atomic layer of stacking faults (Figure 3b). In addition, two stacking faults intersect each other in the NP's interior. As deformation continues, the formed stacking faults gradually disappear from top to bottom due to the trailing partial dislocations of the Burgers vector [11̄2̄]/6 and [1̄1̄2̄]/6, as shown in Figure 3(c). In fact, the entire process includes splitting perfect dislocations of the Burgers vector [0̄1̄1]/2 and [̄10̄1]/2 into the leading and trailing partial dislocations (extended dislocation) to lower the total energy of the system. Therefore, the dislocations were successfully released onto the particle's surface resulting in the observed smooth loading in deformation stage 1 (Figure 2).

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Figure 3.

Dislocations in the NPs with a diameter of 6 nm: (a) initial configuration of NPs at 19% strain; (b) dislocations in the NPs at 19% strain; (c) dislocations in the NPs at 21% strain. Atoms are coloured according to common neighbour analysis (CNA). White indicates FCC atoms, red represents HCP atoms and green indicates defective atoms. Perfect FCC atoms and the surface in (b) and (c) are not shown to observe inner dislocations. The video (CuNP-compress-stage-1.mpg) also shows this process.

Figure 4 shows the zigzag compressive stress of naked NPs and the corresponding structure evolution in deformation stage 2. It is evident that defective atoms (red) periodically appear and disappear in the centre of the NP. A large number of defective atoms appear (top view) when the compressive stress increases to the local maximum value. On the other hand, only a small number of defective atoms remain in the interior part of the NP (bottom view) at the compressive stress of the local minimum value. It is worth noting that the sub-surface is composed of a compact defective atomic wall, which blocks the movement of the interior defect toward the free surface. Actually, the increasing compressive stress is accompanied by the potential energy rise, while the decreasing trend corresponds to the potential energy drop. Therefore, it can be inferred that the compressive deformation of metallic naked NPs is a repeated process of storing and releasing fault energy. A similar result is reported in atomic-scale friction, where a sawtooth friction force with stored and released energy was observed [5, 32].

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Figure 4.

The atomic structure of NPs in the sawtooth stress-strain curves. Atoms are coloured according to common neighbour analysis (CNA). White stands for FCC atoms, red for defective atoms, green for the free surface atoms. Perfect FCC atoms and the surface are not shown to display the inner defects. The cartoon (CuNP-compress-stage-2.mpg) was also prepared to represent the periodic process of accumulation and release of internal defects.

To understand the sawtooth loading clearly, the defective structure of naked NPs in stage 2 is analysed and shown in Figure 5. Under local maximum loading, numerous stacking faults intersect each other in the contact area between the slab and NP (Figure 5a). It is unlikely in stage 1 where the stacking fault intersects the interior of the NP (Figure 3). The stacking fault configuration can be attributed to the movement of two leading partial dislocations of the Burgers vector [112]/6 and [11̄2]/6. It is believed that the end of the former dislocation is the nucleation site of the latter. Until the leading dislocations occupied all active sites, the trailing partial dislocations of the Burgers vector [̄121]/6 and [2̄1̄1]/6 slides along foregoing pathway driven by loading (Figure 5b). The sliding of the leading and trailing dislocations along the same atomic layer is deemed equivalent to a perfect dislocation (Figure 5c). Therefore, the asynchronous movement of leading and trailing partial dislocations constitutes an extended dislocation, resulting in the sawtooth loading.

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Figure 5.

Disappearing process of stacking faults for interpreting sawtooth loading of NPs: (a) folded stacking faults at the strain of 54% with local maximum loading; (b) defects begin to disappear at the strain of 55%; and (c) enlarged view of a small region from (b) showing the stacking faults disappearing

The aforementioned analysis indicates that the extended dislocation was the main defect that was formed in metallic naked NPs during compression stages 1 and 2. In stage 1, the nucleation and annihilation of the leading and trailing dislocations can be observed simultaneously (Figure 3); while in stage 2, they appear separately: first, the leading partial dislocations nucleate and slide along the slip plane to form stacking faults, which corresponds to the load rising; second, the counterpart nucleates to annihilate stacking faults, which corresponds to the load reduction. This asynchronous movement can be understood by comparing the size of NPs with the equilibrium width (d₀) of the extended dislocation, which is:

d 0 = G b 2 8 π γ 2 - υ 1 - υ ( 1 - 2 υ cos 2 φ 2 - υ )

(1)

Where G is the shear modulus; b is the Burgers vector of the Shockley partial dislocation; γ is the stacking fault energy per unit area; v is the Poisson ratio; and φ= 0° and 90° for the screw and edge dislocations, respectively.

Using Eq. (1), for crystal Cu, d₀ is calculated to be approximately 2.0–5.2 nm with an average value of 3.6 nm. This simple estimation gives a reasonable explanation to our simulation results. For instance, during deformation stage 1, the extended dislocation forms by the synchronous movement of two partial dislocations, while in deformation stage 2, the extended dislocation forms by asynchronous movement of the two partial dislocations. This is because the size of the NPs is lower than d₀. For crystal Ag, the average value of d₀ is 14.8 nm, which means the size of NPs in stage 1 (diameter: 6 nm) is much less than d₀. However, the obvious sawtooth loading is found in stage 2. This can be explained as the sliding of dislocations in deformation stage 1 to the free surface to release energy, but in case of deformation stage 2, the contact area between the slab and the NP was sufficient to stop the movement of dislocations then store the stacking energy.

4. Conclusion

Molecular dynamic simulations were performed to predict and investigate the compressive deformation behaviour of crystalline metallic NPs. Sawtooth stress-strain curves, a repeated process of storing and releasing of fault energy, were found by simulation. This can be attributed to the asynchronous movement between the leading and trailing partial dislocation consisting of extended dislocation. The equilibrium width (d₀) of the extended dislocation was compared with the size of the NPs. NPs smaller than d₀ ensured asynchronous movement between partial dislocations. Extended dislocation and its movement in nanoscale materials were observed in this study, which may be useful for applying nanomaterials in the nanomechanical field.