Effects of Voids in Tensile Single-Crystal Cu Nanobeams

Abstract

Molecular dynamic simulations of defect nanosized beams of single-crystal Cu, loaded in displacement controlled tension until rupture, have been performed. The defects are square-shaped, through-the-thickness voids of different sizes, placed centrally in the beams. Three different cross section sizes and two different crystallographic orientations are investigated. As expected, the sizes of the beam cross section and the void as well as the crystal orientation strongly influence both the elastic and the plastic behaviors of the beams. It was seen that the strain at plastic initiation increases with beam cross section size as well as with decreasing void size. It is further observed that the void deformed in different ways depending on cross section and void size. Sometimes void closure, leading to necking of the beam cross section followed by rupture occurred. In other cases, the void elongated leading to that the two ligaments above and below the void ruptured independently.

1. Introduction

Nanotechnology provides applications in an increasing number of engineering fields, with tailor manufactured products for everyday use. Mobile phones, medical sensors and solar cells are examples of well-established application areas. However, since it is an experimentally verified fact that nanosized structures respond differently on mechanical loading than macroscopic structures of the same material, design and dimensioning of nanocomponents lack a solid ground corresponding to traditional dimensioning handbook rules at the macroscale.

The reason for traditional engineering dimensioning rules becoming obsolete at small enough metric scales is that, with decreasing structural size, the number of surface close atoms as compared to number of bulk atoms increases and, at some point, no longer is negligible. Electron redistribution close to the surfaces will leave the surface close atoms in energy states deviating from those of bulk atoms. This influences the interatomic bonding forces and, thereby, the material response to mechanical loading as discussed by e.g., [1, 2, 3]. This effect is accentuated if the atoms are placed at, or close to, corners and edges of the structure, e.g., [4]. As a consequence, the material properties will vary with size for small enough structures, and the effects become obvious below about 50–100 nm.

Also, the crystallographic orientation is of uttermost importance for the material properties, e.g., [5, 6]. The crystallographic orientation imposes anisotropy in the structure and also decides the surface topology which influences the mechanical properties. The orientation further sets the preferred slip plane directions and, thereby, influences the plasticity development.

Here, the tensile response of single-crystal nanosized Cu beams, holding square-shaped, through-the-thickness voids, will be investigated with respect to elastic and plastic behavior and eventual size dependence in the mechanical response. The rational for this investigation is that even if defect-free structures might be intended at manufacturing, defects will always be present to some extent. If their presence influences the mechanical response of the structure, the product functionality might be at risk.

To this end, 3D single-crystal nanosized Cu beams of different cross section sizes, holding through-the-thickness voids of different sizes, will be investigated using the molecular dynamics free-ware large-scale atomic/molecular massively parallel simulator (LAMMPS), see [7]. The loading will be displacement controlled tension along the length direction, and two different crystallographic orientations will be considered and compared.

2. Geometry and boundary conditions

The structures considered are single-crystal face centered cubic (fcc) Cu beams of length L = 100a₀ and square cross section with three different side lengths s = 6Na₀, N = 1, 2, 3, with a₀ = 3.615 Å denoting the lattice parameter for Cu. Each beam holds a symmetrically placed, square-shaped through-the-thickness void of width w and height h, cf. Figure 1(a) where a centrally placed coordinate system (x, y, z) is introduced. The sensitivity to voids will be investigated by varying the width w, keeping the relative void height h = s/3 constant.

Figure 1.

(a) Beam configuration and coordinate system. (b) and (c) Crystallographic orientations.

Two different crystallographic orientations are considered to determine its influence on the mechanical response. For the first orientation, referred to as the [100]-orientation, the coordinates (x, y, z) coincide with crystallographic directions according to: x = [100], y = [010] and z = [001], cf. Figure 1(b). For the second, referred to as the [110]-orientation, x = [110], y = [−110] and z = [001], cf. Figure 1(c).

A beam is built from the repetition of fcc Cu unit cells, and to mimic clamped end boundary conditions, all atoms within four unit cells at each end of the beam are restricted from movements in the y- and z-directions. During the relaxation step, one atom is fixed in all directions. The atoms in between the clamped ends are free to move in all directions without constraints. The beams are loaded in tension along the x-axis until final rupture. The load is displacement controlled through applying a constant velocity, v_end, in the +x- and –x-directions to all atoms within the clamped beam ends.

3. Molecular dynamic simulations

3.1. Simulation procedure

For the simulations, the molecular dynamics free-ware LAMMPS has been employed and the atomic images are produced using OVITO, developed by [8].

In the simulations, an NVT-ensemble, with constant number of particles N, constant volume V and absolute temperature T, kept at a constant temperature of 0.01 K through a Nosé-Hoover thermostat as found in [9] is employed. Before load application, relaxation of the atomic ensemble in order to reach the equilibrium state is imposed during 5000 time steps, with each time step equal to 5 fs. This gives the relaxation time equal to 25 ps, which was found to be sufficiently long to reach equilibrium with good accuracy as judged from the variations in axial stress with time. Thereafter, a constant velocity of v_end = a₀/200/ps is imposed in the +x- and −x-directions at all atoms within the clamped ends using a constant time step of 5 fs.

3.2. Interatomic potential

To calculate the atomic interactions, an embedded atom method (EAM) potential according to [10, 11] is employed. The potential has one pair-wise repulsive and one N-body attractive part and the potential energy $E_i$ of atom i is given by:

$$E_i=f\left(\sum _{j\ne i}\rho \left(r^{\mathit{ij}}\right)\right)+\frac{1}{2}\sum _{j\ne i}\phi \left(r^{\mathit{ij}}\right)$$E1

where $r^{\mathit{ij}}$ is the distance between atoms i and j, φ is a pair-wise potential function, ρ is the contribution to the electron charge density from atom j at the location of atom i and f is an embedding function that represents the energy required to place atom i into the electron cloud. For the present study, the potential file Cu_u3.eam, provided by LAMMPS and developed by [12], is used.

3.3. Centrosymmetry parameter

The results are evaluated and illustrated using the centrosymmetry parameter, CSP, as defined by [13]. The CSP is a measure of the deviation from a perfect lattice configuration, and for an atom, the CSP is defined according to

$$\mathit{CSP}={\displaystyle \sum _{i=1}^{N/2}}{\left|{\mathbf{R}}_i+{\mathbf{R}}_{i+N/2}\right|}^2$$E2

Here, N is the number of nearest neighbors in the lattice surrounding the atom, equal to 12 for an fcc lattice. The vectors R_i and R_i+N/2 are the vector pairs of opposite nearest-neighbors to the atom. For a perfect lattice, the CSP, through the definition (Eq. (2)), becomes CSP < 3. Since the CSP of an atom is a measure of the positions of the atoms nearest neighbor pairs, both crystallographic orientation and structure geometry are of importance. For the present crystallographic orientations and beam geometries, CSP values for atoms located at surfaces, edges and corners are shown in Table 1. In the present investigation, the CSP reached values up to 60 for atoms situated at or close to corners and edges. Values in the interval between 9 and 21 are found for atoms affected by local defects such as voids, partial dislocations or stacking faults.

Lattice structure	CSP
Ideal fcc structure	CSP < 3
Fault sites	3 ≤ CSP < 9
Surface atoms [100]	9 ≤ CSP < 21
Surface atoms [110]	9 ≤ CSP < 25
Edge and corner atoms [100]	CSP ≥ 21
Edge and corner atoms [110]	CSP ≥ 25

Table 1.

CSP values for present geometries and orientations.

As an illustration of the placements of atoms with CSP > 21, a beam with orientation [100] is shown at two different strain levels in Figure 2. In the figure, each individual atom is shown as a filled circle, with color according to the CSP value. In Figure 2, all atoms with CSP ≤ 21 are colored red; the rest, found at edges and at corners, have their CSP in the interval 21 < CSP ≤ 60. Figure 2(a) shows the situation directly after relaxation, at zero axial load, with high CSP values along the edges of the beam, and Figure 2(b) at an axial strain of ε_x = 0.1, where also edges that have emerged through slip events attain high CSP values.

Figure 2.

Red atoms: CSP ≤ 21, nonred atoms: 21 < CSP ≤ 60. [100]-orientation and s = 6a₀. (a) ${\epsilon }_x=0$ and (b) ${\epsilon }_x=0.1$.

4. Results and discussion

4.1. Recorded strain levels

Beams of geometry according to Figure 1, holding voids with aspect ratios w/h = 1, 2, 3, 4, are loaded under displacement controlled tension until final rupture. After an initially elastic phase, plasticity will appear through slip along close-packed {111} planes. For each case, the strain at plastic initiation, ε_i, the strain ε_mc at eventual closure of the mid-section of the void, the strain at eventual total void closure, ε_c and the strain at rupture were determined through monitoring the instantaneous CSP values for all atoms during the loading process. If the beam ruptures due to necking as a result of void closure, this strain is denoted ε_f1. In case the void does not close, but instead expands so that the ligaments above and below the void rupture independently, these strains are denoted ε_f1 and ε_f2. Recorded strains are plotted for each beam size and orientation in Figure 3, and the values are given in Table 2.

Figure 3.

Strains at plastic initiation, ε_i, at closure of the mid-section of the void, ε_mc, at void closure ε_c and at ruptures ε_f1 and ε_f2. (a), (c) and (e): [100] orientation. (b), (d) and (f): [110] orientation. Values from Table 2.

w/h, [100] s = 6a0	εi	εcm	εc	εf1	εf2
1	0.0353	—	—	0.0978	0.1087
2	0.0136	0.0707	0.0734	0.1495	—
3	0.0082	0.0217	0.0245	0.1033	—
4	0.0108	0.0109	0.0162	0.1223	—
w/h, [110] s = 6a0
1	0.0109	—	0.0299	0.0951	—
2	0.0082	—	0.0217	0.1196	—
3	0.0082	—	0.0163	0.1223	—
4	0.0082	—	0.0162	0.1223	—
w/h, [100] s = 12a0
1	0.0571	—	—	0.2011	0.2310
2	0.0408	—	—	0.2826	0.3043
3	0.0326	—	—	0.2609	0.2609
4	0.0300	—	—	0.1984	0.2609
w/h, [110] s = 12a0
1	0.0353	—	—	0.1440	0.1821
2	0.0299	0.2065	—	0.2636	0.2880
3	0.0245	0.2120	—	0.2962	0.2962
4	0.0245	0.2418	—	0.2826	0.3261
w/h, [100] s = 18a0
1	0.0625	—	—	0.3777	0.4293
2	0.0571	—	—	0.2853	0.3125
3	0.0462	—	—	0.2717	0.3451
4	0.0435	—	—	0.2690	0.3206
w/h, [110] s = 18a0
1	0.0380	—	—	0.6603	—
2	0.0326	—	—	0.7470	—
3	0.0326	—	—	0.3451	0.3750
4	0.0299	—	—	0.4620	0.4701

Table 2.

Strains at plastic initiation, ε_i, at closure of the mid-section of the void, ε_mc, at void closure ε_c and at ruptures ε_f1 and ε_f2. Graphs in Figure 3.

4.2. Elastic response

In Figure 4, all strains at plastic initiation, taken from Table 2, have been merged for comparison; in Figure 4(a), for the [100] orientation and in Figure 4(b), for the [110] orientation. From simulations of solid beams, it was observed that the strain at plastic initiation, ε_i, is, in practice, independent on cross section size for each orientation as also concluded in e.g., [14], where pure metric scaling effects were investigated for solid single-crystal Cu beams. In [14], it was found that ε_i ≈ 0.094 for the [100] orientation and ε_i ≈ 0.068 for the [110] orientation, so that the [110] orientation yields first, with the ratio between initiation strains about 0.7. The values for solid beams are included in Figure 4 as circles at w/h = 0. As seen in Figure 4, ε_i tends to increase with beam cross section size as well as with decreasing ratio w/h for both orientations. It can also be noted that ε_i for the solid beams is markedly higher than for the voided beams, and in all cases, the [110] orientation initiates first.

Figure 4.

Strains at plastic initiation ε_i. (a) [100] orientation and (b) [110] orientation.

The higher initiation strain for solid beams is expected since a void acts as a local stress raiser, weakening the structure.

Another observation is that the initiation strain for the two thicker beams, with s = 12a₀ and s = 18a₀, is relatively close in comparison with the thinnest with s = 6a₀, which initiates at markedly lower strains for both orientations. This indicates that a limiting value is approached with increasing s. A comparison of the initiation strains between solid beams, ε_isolid, and beams holding the smallest voids with w/h = 1 shows that, for both orientations, ε_i/ε_isolid ≈ 0.3 for s = 6a₀ and ε_i/ε_isolid ≈ 0.6 for s = 12a₀ and s = 18a₀ so that the void influence has decreased. Even so, the influence from a void, even if small, is always present.

4.3. Atomic arrangements during plastic deformation for the [100] orientation

Starting with the [100] orientation, with recorded strains in Table 2 and plotted in Figure 3(a), (c) and (e), it is seen that for the smallest cross section, s = 6a₀ Figure 3(a), the void closes in all cases except for w/h = 1. For the other ratios of w/h, still for s = 6a₀, the voids close first at the middle of the void at strain ε_cm, thus forming two separate voids. Final void closure appears shortly after, at ε_c, where after rupture occurs at ε_f1. The different scenarios are shown in Figure 5.

Figure 5.

Events during deformation of beams with s = 6a₀ and (a)–(c) w/h = 1 and (d)–(f) w/h = 3. [100] orientation. Coded in the CSP. The states ε_x = 0 are taken directly after relaxation.

In the case s = 6a₀ and w/h = 1, the void expands and the two ligaments above and below the void rupture independently at ε_f1 and ε_f2. Snapshots of the events during deformation for this case are shown in Figure 5(a)–(c), coded in the CSP. The state ε_x = 0 is taken directly after relaxation. Activated {111} slip planes are seen to appear after plastic initiation and spread along the entire beam.

An example of a sequence of events during void closure is seen in Figure 5(d)–(f) for s = 6a₀ and w/h = 3. As seen, the void closes first at the center of the void, forming two separate voids which both close shortly after formation. This leads to that the initially voided but now healed cross-sectional part necks, and final rupture occurs at the strain ε_f1, cf. Figure 3(a). Also, here the {111} slip planes are activated, but localization of the plasticity to the formerly voided region, followed by rupture, occurs before the plasticity has reached the beam ends. Instead, elastic regions remain away from the necking region.

For the larger beam cross sections, s = 12a₀ and s = 18a₀, no void closure occurs. Instead, the voids expand and the ligaments above and below the void rupture independently at ε_f1 and ε_f2. The series of events are thus similar to what is seen in Figure 5(a)–(c).

4.4. Atomic arrangements during plastic deformation for the [110] orientation

For the [110] orientation and s = 6a₀, Figure 3(b), total closure of the voids occurs for all w/h where after the now solid center cross section ruptures at strain ε_f1. The voids are filled, atom plane by atom plane, from one side of the void to the other like a zipper, through slip along {111} planes. The case s = 6a₀, w/h = 1 is illustrated in Figure 5(a)–(c), where the configuration directly after relaxation, at ε_x = 0, is included. As seen, the plasticity in this case is localized to the void vicinity, similar to the case of void closure for the [100] orientation and s = 6a₀ in Figure 5(d)–(f).

A comparison between the corresponding relaxed states for the [100]- and the [110] orientations, at ε_x = 0, shows the impact of crystallographic orientation. The [110] orientation creates a more roughened surface as compared to the [100] orientation. Comparing Figures 5(a) and 6(a), with the thinnest ligaments surrounding the voids and thus the weakest cross sections, it is seen that the deformation of the void in the relaxed state is much more obvious for the [110] orientation. The void seems to bulge due to the lower constraints on the atoms in combination with the orientations of the preferred {111} slip planes for the [110] orientation.

Figure 6.

Events during deformation of beams with (a)–(c): s = 6a₀, w/h = 1, and (d)–(f): s = 12a_0,w/h = 2. [110] orientation. Coded in the CSP. The states ε_x = 0 are taken directly after relaxation.

Increasing the cross section size to s = 12a₀, Figure 3(d), displays a different deformation pattern. In all cases, the void is expanding, so that two ligaments rupture individually above and below the growing void at strains ε_f1 and ε_f2. But for all cases apart from w/h = 1, the initial voids first close at the center, thus forming two separate voids. One of these will grow until the ligaments above and below it rupture at strains ε_f1 and ε_f2. This is illustrated in Figure 6(d)–(f) for w/h = 2.

For s = 18a₀ in the [110] orientation with curves in Figure 3(f), lastly, another phenomenon appears for w/h = 1 and w/h = 2. For these cases, initiation of plasticity as well as rupture occurs away from the void as illustrated in Figure 7 for s = 18a₀ and w/h = 1. For the larger w/h, w/h = 3 and w/h = 4, the voids elongate and the ligaments rupture individually over the void similar to the events in Figure 5(a)–(c).

Figure 7.

Events during deformation of beams with s = 18a₀ and w/h = 1. [110] orientation. Coded in the CSP. At (a) ε_x = 0, taken directly after relaxation, (b) ε_x = 0.405 and (c) ε_x = 0.649.

5. Summary

Molecular dynamic simulations of beams of single-crystal Cu of dimensions 100a₀ × s × s, s = 6Na₀, N = 1, 2, 3, and with a₀ denoting the lattice parameter of Cu, have been loaded in displacement controlled tension until rupture. Each beam holds a centrally placed, rectangular through-the-thickness defect of extension along the beam length direction equal to w and of height h = s/3. Aspect ratios w/h = 1, 2, 3, 4 were considered and the loading was applied along two crystallographic directions, the [100]- or the [110] direction, cf. Figure 1. The deformation development was monitored continuously and the strains at plastic initiation, at void closure and at rupture were recorded. The result is given in Table 2 and visualized in Figure 3.

By studying Figure 3 and Table 2, some general trends about the deformation behavior can be drawn. First in the case of total closure of the beams for s = 6a₀ for both crystallographic orientations, it is observed that the closure strain ε_c steadily decreases as the void width w increases. It is also observed that the strain at rupture is in the same range for both orientations for different values of s and, for both orientations, the values increase dramatically in magnitude with increasing s. Finally, it can be observed that no general trend regarding the influence of w/h on the failure strain can be drawn. In some cases, the strain increases, and in some cases, it decreases with increasing value of w/h.

It was concluded that geometrical features such as beam size and crystallographic orientation played a crucial role for the mechanical behavior. Plasticity develops through slip along closed packed {111} planes, and the [110] orientation always initiates plasticity first. Further, the strain at plastic initiation increases with beam cross section size as well as with decreasing ratio w/h for both orientations.

Studying the deformation pattern, it was found that the plasticity developed and the void deformed in different ways depending on cross section size, void aspect ratio and crystal orientation. As regards the events that lead to final rupture of the beams, different scenarios were observed.

In some cases, the void elongated and the two beam ligaments, above and below the void, eventually necked and ruptured independently. In such cases, the plasticity, through slip along {111} planes before the last ligament rupture, tended to extend away from the regions near the void and could sometimes reach the beam ends.

In the cases where closure of the voids occurred, the strain at closure decreased with increasing w/h. Also, it was observed that the strain at failure was relatively independent of crystallographic orientation and that it increased with increasing cross section size.

Sometimes the void first closes at the center, forming two separate voids. Then, two scenarios are possible. One is that the two voids both eventually close, followed by necking and rupture of the now healed cross section. In these cases, the plasticity localizes to the vicinity of the neck and leaves regions away from the neck elastic. The other possible scenario is that one of the created voids start to elongate and the ligaments above and below this void neck and rupture independently.

There were also cases where failure did not occur in the vicinity of the void; instead rupture occurred near one beam end after that the plasticity had spread over the entire beam.