Multiscale Simulation of Surface Defect Influence in Nanoindentation by a Quasi-Continuum Method

2.2.1 Nanoindentation without defect

Nanoindentation without defect is necessary to be studied for comparison, and the load-displacement curve of the nanoindentation on a defect-free surface is shown in Figure 2, where load (N/m) is presented as per unit length of indenter in the out-of-plane direction. It can be easily found out that the load curve gradually increases in the loading process (OA segment), indicating the elastic stage of thin films. The load first reaches the maximum value of 15.14 N/m at the load step 0.48 nm (point A) and then suddenly drops to the minimum value of 7.67 N/m at point B.

To find out the reason of such abrupt load decline in AB segment, the atoms structure and out-of-plane displacements have been probed and shown in Figure 3, where the step of 0.48 and 0.50 nm is, respectively, corresponding to the point A and point B in Figure 2.

Through Figure 3, we can make a conclusion here that the load reaches the critical value for dislocation emission at point A, which indicates the onset of the plastic stage. After that two Shockley partial dislocations are emitted at point B. Therefore the nanohardness of Al thin film without defect is 16.24 GPa, calculated by equation [33]: H=PmaxA, where P_max is the maximum value of load and A is the contact area of the indenter.

Due to the indenter width 0.932 nm in this simulation, the yield load is obtained approximately 15.14 N/m, which is smaller than 24.7 N/m acquired by Tadmor and Miller [31] with the indenter width 2.5 nm. It is reasonable that the paper [34] shows that as the indenter width decreases, the yield load would decline because of the requirement decline of the necessary strain energy.

2.2.2 Distance effect of the surface pit

Figure 4 shows the nanohardness with various distances cases of the surface pit defects. Compared to the defect-free situation shown as a red horizontal, it indicates that the nanohardness of pitted surface has been declined. That is because a discontinuity at the boundary and the structure may lead to the reduction of stain energy storage when indenting. Additionally, when the adjacent distance between the pit and indenter (d) increases, the nanohardness increases in a wave that goes up in a period of three atoms (donated by circle in Figure 4) and finally tends to the case of nanoindentation on a defect-free surface.

To make a further probe, it is well-known that many physical properties mostly depend on the stacking patterns of atoms, such as cleavage, electronic band structure, and optical transparency [35]. Based on this simulation, the periodic arrangement of atoms “ABCABC” on {1 1 1} atomic close-packed planes of face-centered cubic metal (the illustration as shown in Figure 4) is exactly corresponding to increasing distances d on [1 1 1] direction. That is to say, when the pit moves each atom in the [1 1 1] direction away from the indenter, the strain energy at the pit surface on the {1 1 1} stacking fault energies (SFE) changes because of the proximity of the pit [36]. Consequently, such wave pattern associated with a cycle of three atoms is closely related to the crystal structure of periodic atom arrangement on {1 1 1} atomic close-packed planes of FCC metal.

A further discussion has been made in order to figure out the spatial extent of surface pit influence on nanohardness. It can be found out from Figure 4 that when the distance between the pit and indenter increases, the nanohardness gradually close to the nanohardness of defect-free case (16.24 GPa). If we set 1.5% to determine whether the nanohardness influence exists, it can be found out that when the adjacent distance (d) goes 16 atomic spacing far away from the indenter, there is almost no effect on nanohardness (as shown in Figure 4). Moreover, it can be predicted that each material has its critical value of such spatial extent of surface pit influence on nanohardness, which might have great significance to the size design of thin film in microchips without obvious reduction of the hardness.

However, it can be easily found out that the first three distances cases, respectively, d = 1d₀, d = 2d₀, and d = 3d₀, do not match such wave pattern well. Atomic structure and corresponding strain distribution of these three cases have been further carried out to explain such unusual phenomenon.

Von Mises strain distribution and a strain comparison before and after the notch propagation in the distance cases of, respectively, 1d₀, 2d₀, and 3d₀, are shown in Figure 5. It can be easily found out that a notch formed at the left side of surface pit in the distance cases of 1d₀ and 2d₀, which actually induces serious damage to the structure of materials and severe strain concentration (as shown in Figure 5A–D), while it does not if the distance d equals 3d₀ (as shown in Figure 5E and F). That is to say, due to the great reduction of the nanohardness in the cases of 1d₀ and 2d₀, the first three distance cases in nanohardness curve as shown in Figure 4 will not match the wave pattern that goes up in a period of three atoms.

2.2.3 Critical load for dislocation emission with initial surface pit

As is known to all that the pit influences the nanohardness is actually through the way of affecting nucleation and emission of the dislocation. It is necessary and significant to make a further probe on the critical load for elastic-to-plastic transition in the case of nanoindentation on the pitted surface according to the formula of defect-free model, where the formula to calculate the critical load for dislocation emission is carried out by Tadmor [31], shown as Eq. (1):

where Pcr is the critical load of dislocation emission, h is the depth of dislocation dipole emitted down beneath the indenter, a is the half width of indenter, γ111 is the (1 1 1) surface energy of Al single crystal, and k is the slope of the load-displacement curve during the elastic section.

The critical load calculated by the simulation results of QC method and by Eq. (1) of dislocation theory has been displayed, respectively, as “QC data” and “theory load” in Table 2. It is necessary to note that the simulation data in the case of 1d₀ and 2d₀ is not suitable to be taken into account because of the notch propagation. The differential of the critical data between QC method and dislocation theory is fluctuant as the distance (d) changes. Based on the result that the nanohardness goes up in a period of three atoms, the critical load for elastic-to-plastic is also in such periodicity. Thus, the correction form (set as Δ) might be reasonably defined as the following:

where A(d) is the hardness reduction due to the surface pit and B·Sin(d) is specially set for the periodic arrangement of atoms. It is well recognized that the critical load of elastic-to-plastic transition will decrease [37], when the pit size (D, H as shown in Figure 1) increases. We use a dimensionless factor Da1⋅Ha1(dividing by the crystallographic lattice constant) to express the size influence of the pit, which has already been demonstrated reasonable in published article [24]. If the distance between the indenter and the pit is infinitely large, the influence on nanohardness can be almost ignored, and if the pit size increases, the rate of nanohardness change decreases with the distance variation. According to the function property and calculation formula designed by Tadmor [31], we take the form of ln1+d0dd0D⋅h0Hto express the distance effect of surface pit, considering that it is relevantly reasonable. Moreover, the affection of surface pit is closely due to the material property such as Burgers vector b→, shear modulus μ, and Poisson ν. It is reasonable to apply μb4π1−νto express the influence of material property based on Eq. (1). In addition, the atomic periodical arrangement is actually three atoms “ABCABC” on {1 1 1} atomic close-packed planes of FCC metal. Namely, a form of Sin2π3d0⋅d+φis proper to express the periodicity of atom arrangement. Besides, we know that the unit of correction term (Δ) is exactly N/m. So, the correction can be defined as the following based on the discussion above:

where α, β, and φare three constants that need to be matched and fitted. Based on the simulation data in Table 2, these three constants α, β, and φcan be acquired by calculation approximately 32, 215, and −π3, respectively. So the critical load for the first dislocation emission of Al film has been revised with initial surface pit as follows:

Figure 6 shows the comparison of the critical load for dislocation emission of Al thin film in different distance cases calculated by the theoretical formula before and after modification. Though there is no parameter d in Eq. (1), the curves with blocks are calculated by depth h corresponding each distance case in this simulation. The simulation QC data is closer to the theoretical results which are calculated by Eq. (4) after modification. That is to say, such modification to the theoretical formula is justified as the pit size and the distance between the pit and indenter have both been taken into account.

The modified formula displays the decreasing trend of nanohardness as the distance between the pit and indenter increases, which quite agrees with the experimental results of nanoindentation on the surface step with different distances [14]. Further, such study might be referential to the research of material properties with defects, especially in microchips and MEMS.

3.2.1 Width effect on the yield load due to the pit defect

As is known to all that the yield load of materials is one of the most important factors of the material properties, however it can be obviously affected by defects such as surface pit defect. Normally, the yield load can be easily obtained from the first peak load in the load-displacement curve, which suggests onset of the elastic-to-plastic transition. In this chapter, we have taken 10 different widths of the pit from D = 1d₀ to 10d₀ with a fixed height H = 5h₀, in order to investigate the width effect of surface pit defect on the yield load. The change of the yield load curve as pit width has been revealed in Figure 9. Generally, a reduction tendency of the yield load of Al thin film with the pit defect displays result from more and more serious destruction to the atomic structure by the increase of the pit width. Further, the yield load experiences an extremely slow reduction when the pit width increases from D = 1d₀ to 7d₀; after that it obviously drops from 14.8 to 14.24 N/m when the pit width reaches 7d₀. Then, the yield load decreases slowly again.

It is necessary and significant to compare the nanoindentation on a stepped surface with H = 5h₀ as shown in Figure 7(b), where the pit width can be treated as infinitely large. The result is approximately 14.23 N/m, very close to the yield load in the case of D = 10d₀ (the red point in Figure 9). This implies that the yield load of Al thin film nearly equals the yield load value in the case of stepped surface when the pit width increases to 10d₀.

3.2.2 Height effect on the yield load due to the pit defect

We have taken 10 different heights of the pit from H = 1h₀ to 10h₀, with a fixed width D = 5d₀, in order to investigate the height effect of surface pit defect on the yield load as shown in Figure 7(c). The change of the yield load curve as pit height has been revealed in Figure 10. Similarly, the yield load experiences an extremely slow reduction when the pit height increases from H = 1h₀ to 5h₀; after that it obviously drops from 14.79 to 14.14 N/m when the pit width reaches 6d₀. Then, the yield load decreases slowly again.

In the same way, it is also necessary and significant to compare the nanoindentation on a stepped surface with H = 10h₀ as shown in Figure 7(d). The result is approximately 13.75 N/m (the red point in Figure 10), which is quite near the yield load 13.93 N/m in the case of H = 10h₀.

3.2.3 The investigation of dislocation nucleation and the estimation of Peierls stress

We have carried out a further probe of atomic snapshot and corresponding out-of-plane displacement plot, in order to explain the obvious drops of yield load (D = 7d₀ to 8d₀ segment in Figure 9, H = 5h₀ to 6h₀ segment in Figure 10). Taking the cases of D = 1d₀ in width effect simulation and H = 1h₀ in height effect simulation, for example, it can be easily found out from nucleated dislocations and UZ contours displayed in Figure 11 that two dissociated <1 1 0> edge dislocations are emitted beneath the indenter after nucleation during the thin film yielding. Moreover, the dislocated structure along with the out-of-plane displacements experienced by the atoms has also been displayed in Figure 11, with the dimension 0.1 nm, where a fingerprint of the dislocations has been clearly shown between the partials in the stacking fault regions. According to the structure of FCC metal, it can be easily found out that the dislocations are composed of 1/6 <1 1 2> Shockley partials. On the left,

and on the right,

Comparing all these width and height effect cases in this simulation, we find that emission depth of dislocations changes due to the size of the pit. Take Figure 11(a) and (b), for example, the dislocation dipole similarly travels into bulk after nucleation at the load step of 0.5 nm; however, its center of the emission depth settles, respectively, at the depth of 5.2 and 6.08 nm. It might be predicted that the yield load of thin film in macroscopy corresponds to the emission depth of dislocation in microscopy.

Considering that Peierls stress is exactly the resisting force during the dislocation movement due to the lattice structure, all these emission depths of dislocations have been adopted as an equilibrium distance to further calculate the Peierls stress predicted by the EAM potential [38]. Except the lattice friction, there are two forces acting on the dislocation: (i) the Peach-Koehler force (F_PK) due to the indenter stress field driving the dislocation into bulk and (ii) the image force (F_I) pulling the dislocation up to the surface. The dislocation, which is forced by the sum of these two forces, escapes the attractive region and propagates into the bulk and is finally stopped by lattice friction. Consequently, the force on the dislocation will be balanced at the equilibrium depth by the lattice friction force that is due to the Peierls stress (σp) [31].

Shear stress beneath the indenter is necessary to be further obtained to calculate the Peach-Koehler force. In this simulation, the rectangular indenter is frictionless, applying to an elastic thin film occupying the lower half-plane. When y < 0, the shear stress in bipolar coordinates is [39]

where P is the indentation load. According to the coordinate system of 2a indentation contact (the width of indenter is 2a), as shown in Figure 12, at a depth h beneath the right indenter tip, there is r=a2+h2, r1=h, r2=4a2+h2, θ=−tan−1h/a, θ1=−π/2, and θ2=−tan−1h/2a. The Peach-Koehler force is

where σis the applied stress tensor, b is the Burgers vector, and ℓis the dislocation line vector.

The image force applying to the dislocation dipole with width d = 2a at the emission depth h can be displayed as follows:

Peierls stress of all these cases of pit size has been calculated and plotted based on the discussion above, where Figures 13 and 14, respectively, show Peierls stress of width effect and height effect. In the case of width effect, Peierls stress maintains around the value of 100 MPa with tiny fluctuation from D = 1d₀ to 7d₀; after that the Peierls stress displays a sudden obvious dropdown to about 70 MPa, which is quite similar compared with yield load curve in Figure 9. In the case of height effect, Peierls stress maintains around the value of 70 MPa with tiny fluctuation from H = 1h₀ to 5h₀; after that the Peierls stress displays a sudden obvious dropdown to about 50 MPa, which is also greatly in accordance with the yield load curve in Figure 10. The conclusion can be drawn that such obvious decline of yield load (D = 7d₀ to 8d₀ segment in Figure 9, H = 5h₀ to 6h₀ segment in Figure 10) is closely related to the severe reduction of the Peierls stress, suggesting that it is reasonable and effective to explain the variation of yield load through the Peierls stress.

3.2.4 Size coefficient

We make a further probe on the difference of turning point between width effect and height effect, corresponding to D = 7d₀ in the width effect simulation and H = 5h₀ in the height effect simulation. It can be predicted that the influence degree of width factor is different from the height factor. It is necessary to quantify the size effect of surface pit defect to explain the reason of these differences. It can also be easily recognized that if the distance between the pit and the indenter decreases, the influence would be much more severe on the hardness and yield load. Namely, controlling the same influence of the pit on the nanohardness, the larger size of the pit is required when the pit goes far away from the indenter. Therefore, we defined a size coefficient α as the following, which is dimensionless in order to explain the size effect of surface pit defect:

where “L*” is the characteristic length of the pit, such as D in the width effect simulation or H in the height effect simulation and “d*” is the distance between the center of the indenter and the left boundary of the pit, namely, 6d₀ in this simulation.

According to the simulation result of width effect, the critical width value to make a sudden obvious drop of yield load is 7d₀ (as shown the point D = 7d₀ in Figure 9). Consequently, the size coefficient α is approximately 1.17 (L∗d∗=Dd∗=7d06d0=76). When α reaches approximately 2 (L∗d∗=Dd∗=10d06d0=1.7), as shown in the point D = 10d₀ in Figure 9, the yield load of thin film with surface pit defect nearly equals the one of nanoindentation with surface step as shown in the red point in Figure 9.

According to the simulation result of height effect, the critical height value to make a sudden obvious drop of yield load is 5h₀ (as shown the point H = 5h₀ in Figure 10). Consequently, the size coefficient α is approximately 0.51 (L∗d∗=Hd∗=5h06d0=0.51). When α reaches approximately 1 (L∗d∗=Hd∗=10h06d0=1.02), as shown in the point H = 10h₀ in Figure 10, the yield load of thin film with surface pit defect nearly equals the one of nanoindentation with surface step as shown in the red point in Figure 10.

By contrast, the size coefficient of height is approximately half of the one of width to boost the sudden decline of yield load, implying that the height of the pit has a greater influence on the yield load than the width.

Moreover, the change of yield load of thin film as the pit area has been plotted and shown in Figure 15, where we can easily find out that the slope of yield load curve by height increasing is larger than the one by width increasing. That is to say, the height increasing makes the yield load decrease faster. Besides, the yield load by width increasing is smaller than the one by height increasing during 5h₀d₀ to 25h₀d₀ segment of the pit area, which results from the height of the pit in width increase curve (as shown red curve in Figure 15) is larger than the other one (black curve). If the area increases over than 25h₀d₀, the yield load by height increase is smaller than the one by the width increase. It can be well explained that the height of the pit in the curve of height increase goes up over 6h₀, while the one in the curve of width increase still maintains 5h₀, suggesting that the height of the pit has played a more important role on yield load.