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Nanowelding of nickel and copper investigated using quasi-continuum simulations

  • Cheng-Da Wu
  • Te-Hua FangEmail author
  • Ying-Jhih Lin
  • Yu-Dong Jie
Original Paper
  • 156 Downloads

Abstract

The effects of contact interference, crystal orientation, and material type on the cold nanowelding mechanism and mechanics are studied using quasi-continuum simulations. These effects are investigated in terms of atomic trajectories, strain distribution, and the stress–strain curve. The simulation results show that welding using a contact interference of 0 nm leads to the formation of the fewest defects inside the material during cold welding, and that the number of defects increases with increasing contact interference. For the Ni–Ni welding pair, welding using a contact interference of 0 nm with the structural orientation [110] versus [001] has the largest ultimate strength and longest elongation during tensile testing due to the formation of fewer defects during welding. The welding quality obtained with the Ni–Ni welding pair is higher than that obtained with Ni–Cu and Cu–Cu welding pairs. During the tensile deformation process, dislocations nucleate from the free surface, propagate along the close-packed plane, and then terminate at the other free surface.

Keywords

Quasi-continuum method Metallic welding Contact interference Ultimate strength 

1 Introduction

The welding of nanomaterials has attracted a lot of attention due to its potential applications in micro- and nano-electromechanical systems (Hsu et al. 2013), transparent conductors (Kim et al. 2015; Hu et al. 2010), thin-film solar cells (Yang et al. 2011), wearable electronics (Hong et al. 2015), and cancer therapy (Huang et al. 2013). The nanoscale cold-welding technique, developed by Lu et al. (2010), is a mechanical approach for welding junctions at room temperature without solder, which minimizes the damage from high heat and stress effects. This technique is carried out using an Au nanowire (NW) pair (with NW diameters of 3–10 nm) and scanning probe microscopy. The Au NW pair is quickly cold-welded due to the strong adhesion interactions under low mechanical pressure. This promising bottom–up technique allows the construction, welding, and repair of NWs without introducing defects. The welding quality of cold welding is quite good (Lu et al. 2010), because the crystal orientation, strength, and electrical conductivity of the two NWs are the same.

A few studies have focused on the effects of contact extent and crystal orientation of metal NWs on the cold nanowelding mechanism and mechanics, which are important parameters for the determination of welding quality. Molecular dynamics (MD) simulation is a powerful tool for studying material interaction at the nanoscale and its mechanics (Wu et al. 2015, 2016a, b, 2017a, b). Atomic simulation avoids experimental noise and can reduce investigation cost. For instance, Pereira and da Silva (2011) simulated the cold-welding process of Au–Au, Ag–Ag, and Au–Ag NW pairs with NW diameters of 4.3 nm, and found that the welding region of NW pairs is almost free of defects and that the quality of the welding of the Ag–Au NW pair is good (good resistance to rupture). Wu et al. (2016a, b) investigated the effect of size of Au NW pairs on the cold-welding process. They found that, with a decrease in NW diameter, the alignment of NW pairs becomes more difficult due to structural instability. The elongation ability of welded NWs increases with increasing NW diameter. Welded NWs with smaller diameter have higher ratio of ultimate stress to original yield stress of NWs (Huang and Wu 2014), indicating a higher mechanical strength. Zhou et al. (2017) modeled the cold-welding process of Cu NWs. They found that the weld stress and ultimate stress significantly decreases with increasing temperature. When the temperature exceeds the molten temperature of the NW, the cold welding does not occur. Wang and Yi (2014) found that the mechanical strength of raw nanowire is always higher than that of the same-diameter welded one, and the difference gradually decreases with increasing diameter.

MD simulation is ideally suited for the study of nanosystem. However, the simulation of localized deformation and fracture in heterogeneous systems requires the use of relatively large systems and appropriate boundary conditions, which may result in system sizes difficult to access using direct MD due to the very time-consuming. The quasi-continuum (QC) method is a multi-scale combined MD approach (Tadmor et al. 1996a, b; Qiu et al. 2017; Fang et al. 2016) that mixed continuum and atomistic approach for simulating the mechanical response of the crystal materials. The previous QC simulation of the cold welding into T junctions (Wu et al. 2018) found that, at a small contact interference of 0–1 nm, the ultimate stress of welded pairs is independent of the magnitude of contact interference. Welding on the close-packed plane of both welded pairs leads to the highest ultimate stress.

The purpose of the present work is to study the effects of contact interference, crystal orientation, and material type on the head-to-head cold nanowelding mechanism and mechanics via QC simulation. The results are discussed in terms of atomic trajectories, strain distribution, and the stress–strain curve.

2 Model and methodology

2.1 QC model

The physical model of the head-to-head cold nanowelding process consists of a metallic substrate pair, as shown in Fig. 1a. The dimensions of each substrate are 40 nm (X) × 20 nm (Y). The substrates have a tip with dimensions of 2 nm (X) × 4 nm (Y); the tip is the welding area. Periodic boundary condition is applied in the X- and Y-directions in the models. To study the effect of the structural orientation of the Ni–Ni welding pair, three structural orientations, namely [110] versus [001], \( [\bar{1}10] \) versus [111], and [111] versus \( [\bar{1}10] \), are set.
Fig. 1

a Schematic illustration of cold-welding QC model and b two tips on substrates

Two kinds of atom are set in the model, namely local and non-local atoms. Local atoms are used for continuum regions, such as the substrates, and undergo near-homogeneous deformation. The energies of these atoms are calculated from the local gradient of deformation based on the Cauchy–Born rule (Shenoy et al. 1999). Non-local atoms are set in areas where dramatic deformation would take place, such as the tips. The interaction of non-local atoms is described using the embedded-atom method (Onat and Durukanoğlu 2013), which reveals the atomistic nature. An enlarged view of the welding area is shown in Fig. 1b. An undesired ghost force arises due to the asymmetry in energy accumulation at the interface between the continuum and atomistic regions, leading to a deviation in the solution. A recalculation method (Miller and Tadmor 2009) is applied to correct the ghost force to improve the accuracy of the solution in the transition region. To avoid the early interaction, a separation distance of 1 nm is set between the tips. Cold nanowelding is performed by applying a constant displacement of 0.01 nm per step along the negative Z-direction on the top substrate, which forces it to approach the bottom substrate. Cold welding thus occurs due to the atomic diffusion and forming bonding between the two tips. To examine welding quality, the cold-welded pair is stretched until its breaking point.

2.2 QC methodology

The QC code was downloaded from a website (www.qcmethod.com) and developed by our research team. The following gives a description of the QC methodology (Shenoy et al. 1999; Tadmor et al. 1996). The system is built up of a number of atoms. A given atom in the reference configuration is specified by a triplet of integers \( l_{{}} = \left( {l_{1} ,l_{2} ,l_{3} } \right) \). The position of the atom in the reference configuration is then given as follows:
$$ X\left( {l_{\alpha } } \right) = \sum\limits_{\alpha = 1}^{3} {l_{\alpha } B_{\alpha }^{k} + R^{k} } , $$
(1)
where \( B_{\alpha }^{k} \) is the αth Bravais lattice vector associated with the kth grain, and Rk is the position vector of a reference atom in the kth grain, which serves as the origin for the atoms in the kth grain. Coincident with the nodes of a finite-element mesh, the deformed position of the ith atom may be obtained by interpolation form:
$$ x_{i} = \sum\limits_{\alpha = 1}^{R} {N_{\alpha } \left( {X_{i} } \right)x_{\alpha } } , $$
(2)
where R is the number of the representative atoms, and Nα and xα are the finite-element shape function and the position of the representative atom α, respectively.
The total atomic energy is given by the function:
$$ E_{\text{tot}} = \sum\limits_{i = 1}^{N} {E_{i} } , $$
(3)
where N is the total number of atoms, Ei is the well-defined atomic site energy of the ith atom and calculated from the embedded-atom method (Onat and Durukanoğlu 2013) as follows:
$$ E_{i} = \frac{1}{2}\sum\limits_{j} {\phi \left( {r_{ij} } \right) + F\left( {\rho {}_{i}} \right)} , $$
(4)
in which \( \phi (r_{ij} ) \) is a pair potential between atom i and j, \( r_{ij} = \left| {x_{i} - x_{j} } \right| \) is the inter-atomic distance, \( F(\rho_{i} ) \) is an electron-density dependent on embedding energy, and \( \rho_{i} = \sum\nolimits_{j} {f_{i} (r_{ij} )} \) is the density contribution of atom i from each of the neighbors.
The potential function of the system under consideration is defined as follows:
$$ \varPi = E_{\text{tot}} - \sum\limits_{i = 1}^{N} {f_{i} \cdot u_{i} } , $$
(5)
where ui is the displacement of the ith atom, and fi is the external force acting on the ith atom.
The QC method formulates approximation strategies that preserve the essential details of the problem while reducing degrees of freedom. A subset of R atoms (R ≪ N) are selected from the system of N atoms as representative atoms. The displacements of the representative atoms are considered as the relevant degrees of freedom of the system, and the reduced total energy is given by summing only over the representative atoms with weights as follows:
$$ \prod = \sum\limits_{\alpha = 1}^{R} {n_{\alpha } E_{\alpha } \left( {u_{1} ,u_{2} , \ldots ,u_{R} } \right) - \sum\limits_{\alpha = 1}^{R} {n_{\alpha } f_{\alpha } \cdot u_{\alpha } } } , $$
(6)
where nα is the number of atoms represented by representative atom a and satisfies \( N = \sum\nolimits_{\alpha = 1}^{R} {n_{\alpha } } . \).
In the approximation method, the formulations are not completely compatible, so that non-physical forces, namely so-called the ghost force, arise on atoms in transition zones between different regions. Ghost forces do not come from a potential, and as a result, they are not symmetrical. To avoid the ghost forces, the forces acting on atoms in the transition zones have to be corrected. Because of it, the total energy is modified as follows:
$$ \prod = \sum\limits_{\alpha = 1}^{R} n_{\alpha } E_{\alpha } \left( {u_{1} ,u_{2} , \ldots ,u_{R} } \right) - \sum\limits_{\alpha = 1}^{R} n_{\alpha } f_{\alpha } \cdot u_{\alpha } - \sum\limits_{\alpha = 1}^{R} {f_{\alpha }^{\text{G}} \cdot u_{\alpha } } , $$
(7)
where \( f_{i}^{\text{G}} \) is the ghost force acting on atom i.
There are two kinds of atoms in the system, namely local and non-local atoms. Non-local atoms lie near defects, and the deformation fields are changing rapidly in the region around it, so that its neighboring atoms experience completely different environments. Local atoms are further away from defects, and the neighbors experience environments nearly identical to that of local atoms. The energies of non-local atoms are computed by an explicit consideration of all its neighbors, and the energies of local atoms are computed from the local deformation gradients using Cauchy–Born rule (Shenoy et al. 1999). Therefore, the total energy of the system can be written as follows:
$$ \prod = \sum\limits_{\alpha = 1}^{{R_{\text{L}} }} n_{\alpha } E_{\alpha }^{\text{loc}} \left( {F_{1} ,F_{2} , \ldots ,F_{M} } \right) - \sum\limits_{\beta = 1}^{{R_{\text{NL}} }} E_{\beta } \left( {u_{1} ,u_{2} , \ldots ,u_{R} } \right) - \sum\limits_{\gamma = 1}^{R} {f_{\gamma }^{\text{G}} \cdot u_{\gamma } } , $$
(8)
where \( E_{\alpha }^{\text{loc}} \) is the energy of local atom α, and M and Fi are the number of elements and the local deformation gradient of element i associated with local atom α. Eβ is the energy of the non-local atom β. RL and RNL are the total numbers of local and non-local atoms, respectively, and R = RL + RNL.
The principle of minimum potential energy states that a system is at equilibrium when its potential energy is minimum. It demands that the gradient of the potential energy is zero, namely
$$ \frac{\partial \varPi }{{\partial u_{\alpha } }} = 0. $$
(9)

Displacements of atoms are achieved by iteratively solving a series of non-linear equations, and then, other results may be obtained.

3 Results and discussion

3.1 Effect of contact interference

The term contact interference refers to the separation extent between tips on a substrate pair during cold welding. Three interference values (δ) (0, − 0.5, and − 1.0 nm) are used for cold welding of a Ni–Ni pair, respectively. An interference of 0 nm represents complete contact of the two tips. Overlap occurs when negative interference values are used; it increases with increasing negative interference. Figure 2 shows the variation of longitudinal stress versus strain for the Ni–Ni pair for various interference values during cold welding and tensile deformation processes. The stress–strain curves show an approximate Young’s modulus value. Welding using an interference of 0 nm leads to the largest ultimate strength and the longest elongation before breaking. For interference values of − 0.5 and − 1.0 nm, the ultimate strength values are similar. The strains at the ultimate strength are 23.5, 17.8, and 33.1% for interference values of 0, − 0.5, and − 1.0 nm, respectively, indicating an unclear relationship between the strain at ultimate strength and contact interference. The letters “a”–”f” in Fig. 2 represent welded pairs at various strain values from cold welding to tensile deformation processes, which are visualized as snapshots (a)–(f) in Fig. 3.
Fig. 2

Variation of longitudinal stress versus strain for welding using contact interference values of 0, − 0.5, and − 1.0 nm during cold-welding and tensile deformation processes

Fig. 3

Snapshots of Ni–Ni pair from cold-welding to tensile deformation processes. aε = 2.5% and δ = 0 nm, bε = − 4% and δ = − 0.5 nm, cε = − 12.5% and δ = − 1.0 nm, dε = 12.8% and δ = − 0.5 nm, eε = 20.6% and δ = − 1.0 nm, and fε = 23.5% and δ = 0 nm. Atoms in snapshots are color-coded according to magnitude of their equivalent Von Mises strain values

Figure 3a–f shows snapshots of the Ni–Ni pair from cold-welding to tensile deformation processes. The atoms in the snapshots are color-coded according to the magnitude of their equivalent Von Mises strain values. Slight tensile deformation occurs in the middle of the welded tip at a strain of 2.5% for welding using an interference value of 0 nm during tensile testing, with a few atoms becoming disordered, as shown in Fig. 3a. Figure 3b, c shows snapshots of the welded pair during the cold-welding process at strains of − 4 and − 12.5%, respectively. Dislocations nucleate from the free surface, propagate along the close-packed plane, and then terminate at the other free surface. The number of dislocations significantly increases with increasing interference, relaxing the contact pressure.

Figure 3d–f shows snapshots of the welded pair at the ultimate strength during tensile testing for interference values of − 0.5, − 1.0, and 0 nm, respectively. For interference values of − 0.5 and − 1.0 nm, the main deformation appears in the middle of the welded pair, with minor deformation around the shoulders of the pair, as shown in Fig. 3d, e. For an interference of − 1.0 nm, which leads to the largest elongation at ultimate strength, slight necking occurs in the middle of the welding pair and at its shoulders, as shown in Fig. 3e. Relatively uniform tensile deformation appears for an interference of 0 nm due to a good reconstruction process with the formation of a few defects (Pereira and Da Silva 2011) at the joining interface during cold welding, as shown in Fig. 3f. The reconstruction process is driven by surface energy due to the high surface-to-volume ratio (Hyde et al. 2005; Diao et al. 2003) of the tips.

3.2 Effect of structural orientation of welding pair

A contact interference of 0 nm is used in subsequent simulations due to its good welding quality. To study the effect of the structural orientation of the Ni–Ni welding pair, three structural orientations, namely [110] versus [001], \( [\bar{1}10] \) versus [111], and [111] versus \( [\bar{1}10] \), are set. Figure 4 shows the variation of longitudinal stress versus strain for the Ni–Ni pair with various structural orientations during cold-welding and tensile deformation processes. The welding pair with the structural orientation [111] versus \( [\bar{1}10] \) has the highest Young’s modulus and the earliest breaking point. The welding pair with the structural orientation [110] versus [001] has the largest ultimate strength and elongation, and that with \( [\bar{1}10] \) versus [111] has the lowest ultimate strength, indicating a significant dependence of welding quality on the structural orientation of the welding pair. The letters “a”–”f” in Fig. 4 represent welded pairs at various strain values, which are visualized as snapshots (a)–(f) in Fig. 5.
Fig. 4

Variation of longitudinal stress versus strain for welding using structural orientations of [110] versus [001], \( [\bar{1}10] \) versus [111], and [111] versus \( [\bar{1}10] \), during cold-welding and tensile deformation processes

Fig. 5

Snapshots of Ni–Ni pair from cold-welding to tensile deformation processes. aε = 23.5% and [110] versus [001], bε = 20.8% and \( [\bar{1}10] \) versus [111], cε = 17.8% and [111] versus \( [\bar{1}10] \), dε = 26% and [110] versus [001], eε = 23.5% and \( [\bar{1}10] \) versus [111], and fε = 19.4% and [111] versus \( [\bar{1}10] \). Atoms in snapshots are color-coded according to magnitude of their equivalent Von Mises strain values

Figure 5a–f shows snapshots of the Ni–Ni pair with various structural orientations during cold-welding and tensile deformation processes. Figure 5a–c shows snapshots of the Ni–Ni pair with structural orientations of [110] versus [001], \( [\bar{1}10] \) versus [111], and [111] versus \( [\bar{1}10] \), when they reach the strain at ultimate strength, respectively. The pair with the structural orientation [110] versus [001] has the largest ultimate strength and elongation, and a few dislocations appear at its middle. The distribution of dislocations gradually expands to the outside for the structural orientation [111] versus \( [\bar{1}10] \). A relatively uniform deformation appears around the welding pair for the structural orientation \( [\bar{1}10] \) versus [111]. Figure 5d–f shows snapshots of the Ni–Ni pair with structural orientations of [110] versus [001], \( [\bar{1}10] \) versus [111], and [111] versus \( [\bar{1}10] \), right before necking, respectively, corresponding to strains of 26, 23.5, and 19.4%. Necking occurs in the middle of the welding pair for structural orientations [110] versus [001] and [111] versus \( [\bar{1}10] \), and occurs at the bottom for \( [\bar{1}10] \) versus [111].

3.3 Effect of metallic material of welding pair

Three metallic welding pairs, namely Ni–Ni, Ni–Cu, and Cu–Cu, are used with a contact interference of 0 nm to study the effect of material type. Figure 6 shows the variation of longitudinal stress versus strain for three welding pairs during cold-welding and tensile deformation processes. The magnitude of Young’s modulus for the tested welding pairs follows the order: Ni–Ni > Ni–Cu > Cu–Cu, and that of ultimate strength follows the order: Ni–Ni > Cu–Cu > Ni–Cu. The Ni–Ni welding pair has the best welding quality in terms of ultimate strength and elongation before breaking, whereas the Ni–Cu welding pair has the worst quality in terms of ultimate strength and strain at ultimate strength. The Cu–Cu welding pair has the highest ultimate strength and the largest strain at ultimate strength; however, it breaks right after reaching its ultimate strength. The strains at ultimate strength are 23.5, 13.3, and 30.0% for Ni–Ni, Ni–Cu, and Cu–Cu pairs, respectively. The letters “a”–”f” in Fig. 6 represent welded pairs at various strain values, which are visualized as snapshots (a)–(f) in Fig. 7.
Fig. 6

Variation of longitudinal stress versus strain for Ni–Ni, Ni–Cu, and Cu–Cu welding pairs during cold-welding and tensile deformation processes

Fig. 7

Snapshots of Ni–Ni, Ni–Cu, and Cu–Cu pairs from cold welding to tensile deformation processes. aε = 23.5% for Ni–Ni pair, bε = 13.3% for Ni–Cu pair, cε = 30.0% for Cu–Cu pair, dε = 36.5% for Ni–Ni pair, eε = 16.4% for Ni–Cu pair, and fε = 30.6% for Cu–Cu pair. Atoms in snapshots are color-coded according to magnitude of their equivalent Von Mises strain values (unit: Angstrom)

Figure 7a–c shows snapshots of the three tested welding pairs when they reach the strain at ultimate strength, respectively. For the Ni–Ni welding pair, a few dislocations form at its middle, as shown in Fig. 7a, as the main deformation area; uniform deformation occurs elsewhere. Large deformation occurs in the joining area and in part of the Cu substrate for the Ni–Cu welding pair due to the lattice mismatch at the joining interface, as shown in Fig. 7b. Regarding the bonding energy of atoms, that of Ni–Ni interaction is higher than those of Ni–Cu and Cu–Cu interactions; therefore, the Ni–Cu interface is weaker under loading. The Cu–Cu welding pair has the highest number of atoms with high strain, as shown in Fig. 7c, indicating uniform deformation of the welding pair. The enlarged snapshot in Fig. 7c shows an asymmetrical joining interface for the Cu–Cu pair.

Figure 7d, e shows snapshots of Ni–Ni and Ni–Cu welding pairs right before necking, respectively. This interface will quickly become a breaking source with an increase in strain, as shown in Fig. 7f. A previous study of the cold-welding process with the Au–Ag NW pair (Pereira and Da Silva 2011) showed good welding quality, including a good reconstruction process and mechanical strength, due to the very similar lattice parameters and the same structural orientation of the two metals during the cold-welding process.

4 Conclusion

QC simulations were used to investigate the effects of contact interference, crystal orientation, and welding material type on the cold-welding mechanism and mechanics. The following conclusions were obtained:
  1. 1.

    For the Ni–Ni welding pair, a contact interference of 0 nm leads to the largest ultimate strength and the longest elongation.

     
  2. 2.

    During the tensile deformation process, dislocations nucleate from the free surface, propagate along the close-packed plane, and then terminate at the other free surface.

     
  3. 3.

    The Ni–Ni welding pair with the structural orientation [110] versus [001] has the largest ultimate strength and elongation.

     
  4. 4.

    The Ni–Ni welding pair has a better welding quality than those of Ni–Cu and Cu–Cu welding pairs.

     

Notes

Acknowledgements

This work was supported by the Ministry of Science and Technology, Taiwan under Grants MOST 106-2221-E-151-015-MY3, MOST 106-2221-E-151-026-MY3, and MOST 106-2221-E-033-023.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringChung Yuan Christian UniversityTaoyuanTaiwan
  2. 2.Department of Mechanical EngineeringNational Kaohsiung University of Science and TechnologyKaohsiungTaiwan

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