On the correct interpretation of compression experiments of micropillars produced by a focused ion beam

2.1 Motivation

In order to model for the first time, qualitatively as well as quantitatively, the intermittent plastic behavior during nickel (Ni) micropillar compression experiments [6], a stochasticity-enhanced gradient plasticity model was implemented with a cellular automaton [13], with the simulation results being in very close agreement to the experimental ones. In this work [13], in order for the simulation results to model the experimental ones, instead of the Young’s modulus of Ni (~200 GPa) for the elastic part of the stress-strain graphs, a much lower value was used, corresponding to the initial slope depicted in the reported experimental stress-strain graphs. It should also be noted here that the slopes of the elastic part of all the experimental measurements in [6] should be the same, irrespective of the pillar’s size. The authors of [6] do not present an explanation for this difference of the slopes of the elastic regions with the actual modulus of elasticity of Ni other than misalignment between the pillar and the indenter flat punch.

The aforementioned discrepancy provided motivation to propose a new interpretation of the compression measurements. It is thought that in the aforementioned works [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], the differences between the measured and the actual modulus values are not caused by errors in the respective measurements, but the main cause is the way that the reported stress-strain curves are calculated from the load-displacement measurements during micropillar compression. This new interpretation of pillar compression measurements is described in the next section.

It should be also noted, however, that this discrepancy between elastic moduli and the slopes of the elastic part in the stress-strain curves has concerned researchers before. Greer et al. [14] employed the Sneddon correction to account for sink-in effects, assuming that the pillar is pushing as a nanoindenter into the substrate, by also satisfying the criterion of having an initial slope equal to the elastic modulus. The pillar and substrate were treated like two distinct materials and a comparison between measured stiffness of FIB-produced pillars and the theoretically predicted values showed satisfactory agreement.

Fei et al. [15] state that issues of tapering, misalignment, and sink-in phenomena affect measurements, and to this cause, produce a finite element method to assess the accuracy of the microcompression measuring technique, taking into account the Sneddon correction. The total displacement was measured by microcompression experiments as the combined effect of the indenter, pillar, and substrate strain. It was concluded that the Sneddon correction can tackle sink-in phenomena, even for thick substrates, regardless of the substrate material, as the finite element method (FEM) produces negligible substrate strains. Still, the microcompression method is not deemed suitable for measuring the elastic modulus. This is attributed to the stress state not being uniform because of tapering angles. Another issue mentioned was that the Sneddon correction requires a half-space, which is not always valid for microcompression tests.

2.2 Theoretical stress-strain calculation

In every micropillar compression experiment, the measured quantities are the load P, with which the flat punch indenter pushes the upper surface of the pillar, and the deformation h of the pillar, while the stress and strain quantities are calculated from P and h, respectively. In Figure 1, a pillar compression configuration is depicted. A small material volume of thickness H is used for producing micropillars of varying sizes through the FIB technique. The region marked as A_c is the specimen part that is cut-off by the FIB in order for pillars with different diameters and heights, e.g. the ones marked with A₁ and A₂ in Figure 1, to be left protruding from the volume marked as A_s, which is acting as a “substrate” to the pillars.

Figure 1:

Schematic of a micropillar compression test.

In all the aforementioned works [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], the stress is calculated by dividing the load P exerted on the pillar’s upper surface by this surface, while the strain is calculated by dividing h with the respective pillar height, e.g. H_p. It should be noted that this calculation provides the local strain and stress values at each point of the “pillar” protruding from the specimen’s surface.

As can be seen from Figure 1, the material volume being compressed by the indenter flat punch is not the pillar of height H_p, but a pillar of height H_p+H_s, marked with dashed lines. It can also be seen that the load P is exerted at the upper surface of the material on an area equal to the pillar’s surface, but at the bottom of the material on an area equal to the surface area of the whole sample. This means that there is a distribution of local stresses along the sample height, as there are two very different areas of load application, i.e. the area πDp2 at the top, with D_P denoting the diameter of the pillar at hand, and at the bottom of the sample the area A of the “substrate”. In addition, there is a distribution of local strains along the compressed specimen height. This comes from the fact that the elastic moduli of the “pillar” and the “substrate” are the same; thus, at least during elastic loading, one should expect that

with ε_p denoting the local strain at each point of the “pillar” (the strain calculated and reported in the aforementioned works) [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and ε_s denoting the local strain at each point of the “substrate”, while σ_p and σ_s are the local stress values at the “pillar” and “substrate”, respectively, calculated as

In the next section, two examples of applying the proposed formulation when calculating the stress-strain response of micropillars under compression are given.

3.1 Compression of Ni micropillars

A number of pure Ni micropillars were subjected to microcompression tests using a flat diamond indenter tip in [6]. All of them were derived from a single crystal Ni specimen using FIB and featured diameters between 1 and 40 μm and aspect ratios between 2:1 and 3:1. Figure 2 shows the resulting engineering stress-strain graphs from [6].

Figure 2:

Engineering stress-strain graphs of pure Ni micropillar compression [6]. The slope of the straight line provides a modulus value of 50 GPa.

The initial slope of the curves (defined by a straight line) has a value of 50 GPa, which is significantly lower than the elastic modulus of bulk Ni (~78 GPa [6]). It is also evident that specimens of different diameters exhibit different initial slopes. While the authors include a detailed analysis on the form of the elicited curves, there is no comment on the slope of the graph at the elastic part. However, misalignment issues can be excluded due to the inclusion of a goniometer sample stage.

To this cause, these curves are recalculated using Eq. (3), thus, taking into account not only the contact area between the indenter tip and micropillar but also the area on the bottom of the substrate where load is also applied (3×3 mm [6]). The height of the pillar is also taken into account (7.4 mm substrate height plus specimen height [6]). The corrected graph is presented in Figure 3. It is noted that the same color code was used for the differently sized specimens as in the original paper [6].

Figure 3:

Recalculated engineering stress-strain graphs of pure Ni micropillar compression using Eq. 3. The slope of the straight line provides a modulus value of 200 GPa.

The recalculated graph exhibits stress-strain responses with initial slopes of about 200 GPa, a value very close to the elastic modulus of bulk Ni. Also, the slopes no longer exhibit divergence initially.

It should be noted that the corrected graph presents the curves in reverse order, i.e. pillars of greater size exhibit higher yield stresses, a fact that, at first glance, contradicts the “smaller is stronger” concept. But this is not the case actually. In order to understand this, one should take into account that the FIB method produces pillars by removing the substrate material around a designated area, instead of depositing on it. As a result, in order to produce larger pillars, the “effective” volume of the “substrate” material, i.e. H_sA_s in Figure 1, is getting smaller. Thus, it is actually the specimen with the larger pillars that include a smaller “effective” volume, and not vice versa, as it can be mistakenly thought at first glance.

3.2 Compression of metallic glasses micropillars

Stress-strain curves have been reported for size-dependent microcompression tests on palladium-silicon (PdSi) metallic glass columns [16]. More specifically, a 5.5 μm-thick Pd₇₇Si₂₃ metallic glass thin film was fabricated by Argon ion-beam sputtering onto a Si-(100) substrate. Micro and nanocolumns were cut with FIB, with diameters ranging between 200 nm and 2000 nm at a fixed aspect ratio of height/diameter ~3 [16].

In their experimental measurements, shown in Figure 4, the authors attribute the low slope of the initial part of the stress-strain curves for very small samples to the inevitable top-rounding originating from the FIB-milling procedure [17], having, as a result, small samples (<300 nm) deform plastically at lower engineering stress. The authors also note an unexpected change of elastic slope as a function of sample size, not due to size-dependent elasticity, as well as an intermittent-to-smooth flow behavior.

Figure 4:

Engineering stress-strain graphs during Pd₇₇Si₂₃ metallic glass micropillar compression [16]. The slope of the straight line provides a modulus value of 50 GPa.

We apply below the proposed formulation to the experiments of [16]. Using Eq. (3) with H=5.5 μm and various values for H_p, the stress-strain graph of Figure 4 is now transformed to the one shown in Figure 5.

Figure 5:

Recalculated engineering stress-strain graphs during Pd₇₇Si₂₃ metallic glass micropillar compression using Eq. (3). The slope of the straight line provides a modulus value of 90 GPa.

Comparing the graphs shown in Figures 4 and 5, it can clearly be seen that in contrast to the reported results [16], the recalculated results show that all pillars, irrespective of diameter, have the same initial slope, calculated as 90 GPa, which is very close to the modulus of elasticity of the metallic glass measured as 86 GPa using nanoindentation [16].

Another study conducted by Dubach et al. [18] included microcompression of 0.3, 1, and 3 μm diameter samples made of a zirconium (Zr)-based metallic glass (Zr_41.2Ti_13.8Cu_12.5-Ni₁₀Be_22.5). The total specimen height that was taken into account for the correction was 3 mm and the aspect ratios (height/diameter) varied from 2 to 2.5. Slight tapering of 2°–4° is mentioned. The obtained results are presented in Figure 6.

Figure 6:

Engineering stress-strain graphs during Zr-based metallic glass micropillar compression [18]. The slope of the straight line provides a modulus value of 40–46 GPa.

Taking into account the height of the substrate and Eq. 3, the recalculated graph is presented in Figure 7. It is evident that now the initial slopes are very similar and feature values of 90 GPa. The bulk elastic modulus of the above-mentioned metallic glass is considered to be 96 GPa by the authors of [18], as mentioned in [19].

Figure 7:

Recalculated engineering stress-strain graphs during Zr-based metallic glass micropillar compression using Eq. (3). The slope of the straight line provides a modulus value of 90 GPa.