Anisotropic Deformation in the Compressions of Single Crystalline Copper Nanoparticles

Atomistic simulations are performed to probe the anisotropic deformation in the compressions of face-centred-cubic metallic nanoparticles. In the elastic regime, the compressive load-depth behaviors can be characterized by the classical Hertzian model or flat punch model, depending on the surface configuration beneath indenter. On the onset of plasticity, atomic-scale surface steps serve as the source of heterogeneous dislocation in nanoparticle, which is distinct from indenting bulk materials. Under [111] compression, the gliding of jogged dislocation takes over the dominant plastic deformation. The plasticity is governed by nucleation and exhaustion of extended dislocation ribbons in [110] compression. Twin boundary migration mainly sustain the plastic deformation under [112] compression. This study is helpful to extract the mechanical properties of metallic nanoparticles and understand their anisotropic deformation behaviors.


Introduction
Single crystal nanoparticles play increasing important roles in a wide variety of fields such as fuel cells, energetic materials and high performance catalyst. Among these applications, some crucial functions are closely in coordination with the mechanical properties of nanoparticles. It is well known that the mechanical properties of such surface-confined materials are drastically different from their macro-scale counterparts. For example, both the Young's modulus and hardness of single crystal nanoparticles are a bit larger than those of their bulk counterparts [1,2]. Extensive efforts have been devoted to investigate the unique properties of nanoparticles.
Anisotropy is one inherent characteristic of crystalline materials, and strongly affects their elastic and plastic deformation. For bulk single crystal materials, the reduced Young's modulus, the location of nucleation sites and the nucleation stress vary with different lattice orientations under indentation [3]. In incipient plasticity, dislocation nucleation depends largely on the available slip systems [5]. Anisotropic effects become even more prominent in low dimensional nanostructures such as nanowires and nanopillars. For example, lattice orientation in axial direction significantly affects the yield stress of gold nanowires [6]. The exhibiting tension-compression asymmetry in nanopillars depends on crystallographic orientation [7]. The plastic deformation of nanowires under torsion can be either homogeneous or heterogeneous, depending on the wire orientation [8]. In the case of silicon nanowires under uniaxial tension, the fracture mechanism would switch from brittle to ductile for varying axial crystallographic orientation [9]. The prominent anisotropy in low dimension materials is closely related to the intrinsic crystal structure and the extrinsic surface morphology.
Free surfaces usually serve as preferential dislocation nucleation sites. Changing surface structure by removing weakly bound atoms produces a striking rise in yield strength [10].
For nanoparticles, such factors as surface facets, geometric profile and internal twin boundaries contribute to their unique properties [11]. Extensive experiments and simulations were performed to investigate the deformation mechanisms in nanoparticles. For example, phase transition was observed in single-crystal silicon nanoparticles during uniaxial compression [13]. The in situ TEM indentation of silver nanoparticle revealed reversible dislocation plasticity in nanoparticles [14].
During uniaxial compression, lateral free surface strengthened defect-free gold nanoparticles by draining dislocations from particles [15]. For uniaxial [001] compression of spherical copper nanoparticles, deformation twinning dominated the severe plastic deformation [16]. Microstructural evolution of tin dioxide nanoparticles under compression exhibited the formation of shear bands, twinning and stacking faults [17].
Up to now, there is still a lack of systematic investigation of anisotropic behaviors of face-centred-cubic (fcc) metallic nanoparticles. In the present study, we conduct molecular dynamics (MD) simulations to illuminate both the elastic characterization and plastic deformation mechanisms of copper nanoparticles, aiming to present a landscape of anisotropic deformation of low dimensional nanostructures.

MD simulations are conducted using the open-source simulator LAMMPS developed by Sandia
National Laboratories. The embedded atom method (EAM) model is utilized to describe the atomic interaction of copper atoms. According to the EAM model, the total energy U of a system containing N atoms is expressed as is the pair-wise interaction energy related to the interatomic distance rij between atoms pairs i and j. In the present study, we use an EAM potential for copper parameterized by Mishin et al [18], which was constructed by fitting both experimental and ab-initio computational data, for instance, cohesive energy, bulk modulus, elastic constants, intrinsic stacking fault energy, and vacancy formation and migration energies. This potential has been widely used to investigate mechanical properties and deformation mechanisms in different nanostructures [19].
The simulation model of uniaxially compressing a nanoparticle is depicted in Figure 1. One single crystal copper nanoparticle with radius of 10 nm is carved out of bulk defect-free single crystal copper, and contains about ~ 0.35 million atoms. Then the particle is placed between two rigid planar indenters. A repulsive potential is utilized to model the frictionless compression as , where K is a specified constant representing the rigidity of the planar indenter. Compression Loading procedure is implemented within the framework of canonical (NVT) ensembles. Before compression, the as-carved spherical nanoparticle firstly is performed structure relaxation using the conjugate gradient method, and then is equilibrated at 10 K for about 20 ps to relief the internal stress. When compression is conducted, the top and bottom planar indenters are simultaneously move towards the center of nanoparticle with a speed of ~ 0.1 Å/ps, and the compression depth δ is denoted by the displacement of one indenter. During the loading process, the temperature is controlled at 10 K using a Nosé-Hoover thermostat, and the time step of velocity-Verlet integration is chosen as 2.0 fs.
To identify the characteristics of nucleated defects inside the particles and visualize their evolution processes, the local crystal structure of each atom is computed based on the common neighbor analysis (CNA) [21]. Atomic configurations and defect structures are visualized using Ovito [22] and Paraview [23]. To reveal the mechanical response of nanoparticles, the variations of contact area and averaged contact stress are examined. Contact area is determined by the Delaunay triangulation algorithm, and the averaged contact stress is defined as the loading force divided by contact area.

Results and Discussions
Compression of nanoparticle clearly demonstrates that the elastic response, initial dislocation nucleation and the following defects evolution vary with different loading directions. These orientation dependent features originate from the intrinsic crystallographic structure, surface configurations and the activated slip systems. In this section, we will analyze these behaviors in details. It is manifested in recent studies that the flat punch model should be utilized when surface step is compressed [24,25]. In this model, compressive load F is expressed as a function of compression depth δ by

[111] Compression
where a and E * are the contact radius and the reduced modulus, respectively. In Figure 2(a), we use Eq.
(3) to fit the loading curve in the elastic stage, and the fitted reduced modulus is ~ 201.2 GPa. In the [111] nanoindentation of bulk copper, the reduced modulus is extracted as ~193 GPa [5]. It is noted that nanoparticle have a slightly larger elastic modulus than that of bulk material, which may be attributed to surface effects. Based on this model, the theoretical predictions of contact stress and contact area are plotted in Figure 2(b), which coincide well with the MD computational results in elastic stage.
From yielding point, dislocation initiates inside the nanoparticle. The slip systems available for [111] compression is depicted in Figure 3(a), and initial surface morphology and surface steps are displayed in Figure 3(b). The outmost surface facet has a hexagonal shape, whose edges are preferential sites for dislocation nucleation. When yielding occurs, heterogeneous dislocations nucleate around surface steps as depicted in Figure 3 surface, similar to the case in metallic thin film [31].
In addition to the surface, single-arm source is another important dislocation source in confined volume structures [31]. Figure 4 depicts typical single-arm dislocation sources inside the deformed nanoparticle. In Figure 4 of bulk material, extended dislocations are not distorted heavily [5], and prismatic loops usually emanate from contact zone [28].

[110] Compression
Under [110] compression as shown in Figure 5(a), the loading curve increases continuously up to the yielding point, following a power-law function rather than the linear relation of [111] compression. Figure 5(b) demonstrates the evolution of contact stress and contact area with respect to compression depth. Before yielding point, three local peaks appear on the contact stress curve, and meanwhile stepwise increase emerges on the contact area curve. In this case, three (110) atom layers are flattened before yielding. Each time when one atom layer is flattened, contact area increase suddenly, leading to the drop of contact stress. When multiple atom layers are involved in contact, the classical Hertzian model can be used to capture the elastic response [24]. In this model, the load F as a function of compression depth δ is given as where E * is the reduced modulus of the nanoparticle, R is the radius of nanoparticle. is ~ 163 GPa [5]. Nanoparticle has a larger modulus than that of bulk material, similar to [111] compression. Figure 5(b) illustrates the comparisons between the simulation data and the Hertzian prediction. It is seen that the overall elastic behaviors can be approximately characterized by the Hertzian model.
In the case of [110] compression, Figure 6(a) depicts two types of <111>{110} available slip systems. Figure 6(b) shows different views of the initial surface configuration under indenter. Initial dislocation firstly nucleates beneath surface steps, as shown in Figure 6(c). In this stage, slip system of type I in Figure 6(a) is activated. Initial dislocations nucleate and glide in the {111} slip planes parallel to the compressive direction. Following the nucleation and propagation of leading partial dislocations, trailing partial dislocations also nucleate from contact surface, forming extended dislocation ribbons. These extended dislocation segments on four adjacent slip planes connect with each other, and compose a prismatic loop, which moves along its glide prism and then emanates from contact region, as shown in Figure 6(d) ~ (h). The prismatic loop is highlighted in Figure 6(i), and the intersection between neighboring extended ribbons is stair-rod dislocation.
After releasing the prismatic loop, dislocations are prone to nucleate on slip system II, shown in  compression, load increase in elastic stage follows a power-law function. However, the transition from elastic stage to plastic stage is more smooth, and no evident load-drop can be observed. Figure   8 This results in the migration of TBs toward center. When two TBs from the top and bottom regions get close to each other, dislocations begin to nucleate at contact surface, and glide on slip planes inclined to TBs, as show in Figure 9(j) ~ (l). The existing TBs serve as barriers to these new dislocations. A few dislocations may slip across the TBs, and make TBs defective [34]. Under [112] compression, owing to the orientation of the activated <110>{111}slip system, migration of TBs mainly sustains the plastic deformation.

Conclusions
We compression, deformation twinning is the major plastic deformation mechanism. This work reveals new atomic-scale features of mechanical response of metallic fcc nanoparticles.  Color scheme is the same as in Figure 3)