# Unique universal scaling in nanoindentation pop-ins

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Aug 21, 2020

The results are consistent with the fact that the first pop-in is triggered by dislocation nucleation, which is driven by thermal activation16,23. In contrast, the second and subsequent pop-ins are dominated by the long- (indirect) and short-range (direct) dislocation interactions13,24,25 in the dislocation network that is formed during the previous and even during the current pop-in. They are more driven by mechanical forces and thus less driven by thermal activation than the first pop-in.

The distribution of the first pop-in shown in Fig. 2 appears to be a Gaussian. However, it should not be purely Gaussian because the occurrence probability of a thermally activated process is described by an Arrhenius equation based on the transition-state theory26. In our recent study, the occurrence probability of the first pop-in event was formulated as a function of Ppop-in23 (see Supplementary Note 6 for details)

$$p({P}^{{rm{pop}}{hbox{-}}{rm{in}}})=frac{k({P}^{{rm{pop}}{hbox{-}}{rm{in}}})exp left[-{dot{P}}^{-1}int_{0}^{{P}^{{rm{pop}}{hbox{-}}{rm{in}}}}k(P){rm{d}}Pright]}{int_{0}^{{P}_{{rm{c}}}^{{rm{pop}}{hbox{-}}{rm{in}}}}k({P}^{{rm{pop}}{hbox{-}}{rm{in}}})exp left[-{dot{P}}^{-1}int_{0}^{{P}^{{rm{pop}}{hbox{-}}{rm{in}}}}k(P){rm{d}}Pright]{rm{d}}{P}^{{rm{pop}}{hbox{-}}{rm{in}}}},$$

(2)

where k(Ppop-in) is the dislocation nucleation rate, and ({P}_{{rm{c}}}^{{rm{pop}}hbox{-}{rm{in}}}) is the maximum pop-in load at which the cumulative pop-in event probability approaches 1, and (dot{P}) is the loading rate. The concept of the model is basically the same as that of the model of Schuh and Lund27, while here the probability is more directly represented as the function of Ppop-in instead of the local shear stress at the dislocation nucleation point. Equation (2) can be rewritten as a function of the contact stress drop Δσ with the following assumption:

$$Delta sigma ={sigma }_{{rm{e}}}^{{rm{contact}}}-{sigma }_{{rm{c}}}^{{rm{contact}}}approx C{{P}^{{rm{pop}}hbox{-}{rm{in}}}}^{frac{1}{3}}-{sigma }_{{rm{c}}}^{{rm{contact}}},$$

(3)

where ({sigma }_{{rm{c}}}^{{rm{contact}}}) is the average contact stress at the end of the first pop-in; this assumption implies that the dislocation activity ceases under the contact stress, which is assumed to be constant here for a specific target material and surface orientation. Based on Hertz’s contact theory, the average contact stress ({sigma }_{{rm{e}}}^{{rm{contact}}}) immediately before the first pop-in under the load of Ppop-in is proportional to ({{P}^{{rm{pop}}hbox{-}{rm{in}}}}^{frac{1}{3}}): ({sigma }_{{rm{e}}}^{{rm{contact}}}=C{{P}^{{rm{pop}}hbox{-}{rm{in}}}}^{frac{1}{3}}), where C is a constant21. Equations (2) and (3) can be combined to formulate the occurrence probability of the first pop-in event as a function of the contact stress drop pσ). Then, the unknown parameters of Eqs. (2) and (3) were determined by fitting the equations to the experimental data for the first pop-in (see also Supplementary Note 6) as shown in Fig. 2a, c, e (solid line). The theory describes the experimental results very well. The first pop-in distribution was also plotted on a linear scale as shown in Supplementary Fig. 12 in Supplementary Note 7.

The power-law distribution of the second and subsequent pop-ins is evidence of the catastrophic nature of these events, such as dislocation avalanches with a universal scaling nature. Usually many second and subsequent pop-ins at different indentation depths could be detected even in one load-displacement curve, such as 1 ≤ ni < 10, as shown in Fig. 1. Since we would use as much of them as possible for accelerating the second and subsequent pop-in sampling, we need a guarantee of the statistical independency of the pop-ins with respect to h. To confirm the statistical independency, we plotted 〈Δhh〉 vs. h and 〈Δσh〉 vs. h using all of the pop-in data (N × ni data), where 〈Δhh〉 and 〈Δσh〉 are the averages of the displacement burst Δh and the stress drop Δσ at a given indentation depth h, respectively, as shown in Supplemantary Figs. 22 and 23 in Supplementary Note 10. It is clearly seen that Δh shows a nice statistical independency within a wide range of indentation depth; (20.0–40.0) nm ≤ h ≤ (90.0–145.0) nm, while Δσ shows a weak but some indentation depth dependency as seen in Supplementary Fig. 23. Because of this reason, just in this case, we have decided to use Δh for the subsequent pop-in power-law analyses that were observed within the statistically independent displacement range (40.0 nm ≤ h ≤ 120.0 nm (Fe(100)), 40.0 nm ≤ h ≤ 95.0 nm (Fe(111)), and 20.0 nm ≤ h ≤ 145.0 nm (Cu(111))), even though Δσ can acceptably demonstrate the power law, as seen in Supplementary Fig. 10. We fitted the cumulative distribution of Δh with a power-law function: p(x) = αxβ as shown in Fig. 2, where α and β are constants. The scaling exponents (power-law exponents) were estimated to be β ~5.6 for Fe(100),  ~3.9 for Fe(111), and  ~6.4 for Cu(100). All the obtained βs values are much larger than those that are typical of the pillar-compression testing, regardless of BCC, FCC, or metallic glass and loading direction (crystal orientation) β = 1.0–1.84,5,6,7,8,9,10. Moreover, these are much higher than even those estimated in nanoindentation experiments and simulation for FCC Al and FCC Cu, β ~1.619,20.

The fundamental question, that is, why is there a difference in the power-law exponents between micropillar-compression and our nanoindentation testing is discussed later based on the analyses of the MD simulation results and a developed dislocation avalanche model. Here, another question arises: why is the obtained exponent for FCC Cu ~6.4 in our study much higher than the exponent ~1.6 reported in the nanoindentation experiments and simulation for the FCC metals? The difference is a result of a “first-subsequent mixed analysis” using all of the first and second and subsequent pop-in data analyzed together without separating the first pop-in. To confirm this, we plotted Δh for our BCC Fe and FCC Cu data without separating the first pop-in using the logarithmic binning (Supplementary Fig. 24 in Supplementary Note 11), that is, the same method as the papers19,20 used. Accordingly, we actually estimated the similar power-law exponent ~1.6 for FCC Cu using our experimental data (see Supplementary Fig. 24c in Supplementary Note 11).

Meanwhile, we would emphasize that especially for the BCC Fe cases, the first–subsequent mixed analysis does not show the power-law scaling at all because a clear separation does exist between the first and the second and subsequent pop-in distributions, as seen in Supplementary Fig. 24a, b. The straightforward reason for the separation is that BCC Fe tends to have a much larger first pop-in magnitude relative to the second and subsequent pop-in magnitudes compared with FCC Cu. This occurs because BCC metals typically have an almost 50% higher ideal shear strain than FCC metals28, which was defined as a necessary affine shear strain to reach a stress state exhibiting the ideal shear strength (critical shear stress in the dominant slip system)28,29. In other words, a perfect crystal maintains its mechanical stability up to the ideal shear strain at which dislocation nucleation can be triggered. Higher ideal shear strain results in a larger first pop-in depth h, and thus a larger pop-in load Ppop-in. This eventually results in a larger first pop-in magnitude because a larger elastic energy is stored immediately before the first pop-in.

The above-mentioned fact implies that the first pop-in separation is necessary to unveil the unique high-exponent universal-scaling nature hidden in the nanoindentation testing. The statistical independency plots (Supplementary Figs. 22 and 23) also demonstrate the fundamental difference between the first and second and subsequent pop-ins; the pop-ins occurring at h < 40.0 nm (Fe(100)), h < 40.0 nm (Fe(111)), and h < 20.0 nm (Cu(100)) are mostly the first pop-in showing a different trend from the following pop-ins, which is also confirmed in the plots of the actual load-displacement curves (Supplementary Figs. 3537). It is worth noting that even at high temperatures, such as at 373 K, high power-law exponent β = 5.0 was also observed (see Supplementary Note 12), whereas the power-law exponent slightly decreased with the increase in temperature; our dislocation avalanche model suggests the temperature dependency of the power-law exponents (see Supplementary Note 13).

Furthermore, we investigated the distributions of the first and subsequent pop-in magnitudes in the above-mentioned displacement-controlled MD simulations at 5 K on the BCC Fe(100) and Fe(111). In the MD simulations, a force drop19 and “fictitious” displacement burst were employed for measuring the pop-in magnitude (for definition details of the force drop and fictitious displacement burst and sampling algorithm of these data from the MD load-displacement curve, see Supplementary Note 14). The results are given in Supplementary Note 9. In both the force drop and the fictitious displacement burst plots, we can see the universal power-law scaling distribution in the subsequent pop-ins with high power-law exponents, which agree reasonably well with the experimental power-law exponents; however, large-scale events should be truncated in MD because of the model size limitation.

The high exponents and the materials and surface-orientation dependencies may be attributed to the unique boundary condition and stress and dislocation distributions of nanoindentation, which are different from the uniaxial-loading pillar-compression testing. We recall that the displacement burst results from the motion of dislocation ensembles, and the magnitude of the burst is proportional to the total migration distance of dislocations. The driving force of dislocation (more exactly it is a “dislocation segment”; however, simply, “dislocation” is used hereafter for simplicity) in a material is the total stress exerted on the dislocation, which originates from both the application of an external force to the target material and the stress field produced by other dislocations. The fundamental difference between these testing methods is the force-applying geometry and the resulting stress field.

In micropillar load-controlled compression testing (to a single crystal of pure metal), background stress distribution should be uniform over a slip plane because the exerted stress distribution on a slip plane originated from applying an external force. Hence, once dislocation starts to move, the dislocation motion can only be suppressed and then terminated by dislocation–dislocation interactions. Thus, dislocation multiplication is the major termination mechanism of the displacement burst (dislocation avalanche) in micropillar compression testing30, while dislocation escape from the free surface is another minor termination mechanism. Note that in extremely small nanopillar testing, the dominant termination mechanism is reversed. In this case, dislocation nucleation from the free surface, and then passing through the entire sample and escaping from the free surface, can be the major mechanism because of the small probability of dislocation multiplication, and thus the small possibility of dislocation–dislocation interaction31,32.

However, in nanoindentation testing, the background stress distribution produced by the indenter is not uniform15,33. Although a very high local background stress field (the level of which is comparable with the ideal shear strength23,33) is created in the local region near the indenter tip, the stress level rapidly decreases with an increase in the distance from the indenter tip23,33. Thus, the starvation of the driving force owing to the lack of background stress in remote fields far from the indenter tip is one of the reasons for the termination of the displacement burst, in addition to the dislocation–dislocation interactions owing to dislocation multiplication, as we directly show using the MD results as follows.

To observe the dislocation activities and the corresponding stress distribution change and atomic motion during the first and subsequent pop-ins in the displacement-controlled MD (during the fictitious displacement bursts), we visualized the dislocation pattern immediately before and after the pop-ins, together with von Mises stress-invariant distribution, the change in the von Mises atomic-strain invariant34, and the atomic displacement along the indenter axis (for the first pop-in, see Fig. 3 for BCC Fe (100), Supplementary Fig. 29 for BCC Fe (111), and Supplementary Fig. 30 for FCC Cu (100); for the subsequent pop-ins, see Fig. 4 for BCC Fe (100), Supplementary Fig. 31 for BCC Fe (111), and Supplementary Fig. 32 for FCC Cu (100) (see also movies in Supplementary Movie 1)). Note that the atomic-strain-invariant visualization allows us to directly observe the atoms contributing to the plastic deformation (displacement burst) produced within a pop-in. Moreover, together with dislocation pattern visualizations before and after the pop-in, the history of dislocation motion during the pop-in can also be determined. The atomic displacement along the indenter axis allows us to observe regions that are contributing to the indenter displacement.

At the first pop-in, vast dislocations were nucleated and spread in a fan-like pattern from the indenter tip (Fig. 3a, b). The dislocation behavior has been observed in a nanoindentation experiment using transmission electron microscopy (TEM)13; eventually, a local high-density dislocation field was formed in the near field of the indenter tip. At the same time, the stress distribution beneath the indenter immediately contracted and decreased (Fig. 3c, d), and then the indentation load dropped (Fig. 3e) with the generation of a plastic strain (Fig. 3f) and atomic displacement (Fig. 3g) mostly just beneath the indenter tip.

Then, during the subsequent pop-in, the dislocation field expanded out, and simultaneously, new dislocations were formed at the local high-stress field near the indenter tip (Fig. 4a, b). The dislocations reduced the total stress distribution beneath the indenter (Fig. 4c, d). At the same time, some dislocations existing in the remote fields (indicated by a purple arrow in Fig. 4a, b) traveled further because of an additional pushing force, which originated from the near-field dislocation motions (=a reactive force to the backstress force acting on the near-field dislocations from remote field dislocations) (dislocation motion cascade). These dislocation activities generated a certain amount of plastic strain (Fig. 4f) and atomic displacements (Fig. 4g), which contributed to indenter displacement. However, this pushing force does not increase forever because the following dislocations also lose the driving force for these motions when they enter the remote fields with a lower background stress. Moreover, the background stress distribution itself decreases under the constant pop-in load because of a decrease in the contact stress (stress drop) between the indenter and the target material due to the increase in the contact area with the progress of the pop-in. Thus, the dislocations existing in the remote fields will be stopped, as seen in Fig. 4 and in Supplementary Figs. 31 and 32. Meanwhile, the near-field dislocations exhibited a vigorous activity with generating plastic strain to accommodate the indenter tip motion, which was nucleated in the subsequent pop-in at the very local high-stress field beneath the indenter. These dislocations directly interact with each other owing to the high local dislocation density, and most of them seem to become immobile immediately after generating a certain amount of plastic strain and indenter displacement with the reduction in the remote field stress. At this stage, the near-field dislocations can no longer contribute to the generation of further plastic strain and thus the indenter displacement either through its motion or by pushing other dislocations. Note that some of the near-field dislocations (indicated by the orange arrow in Fig. 4a, b) escaped out to the remote fields with the generation of a certain indenter displacement, but these were also stopped eventually when the backstress from the remote field dislocations, pushing force from the near-field dislocations, and the background stress were balanced.

All the above-mentioned unique termination mechanisms of the dislocation motion in nanoindentation testing originate from the nonuniform stress and dislocation distributions with a rapid decay with respect to the distance from the indenter tip, in addition to the dislocation–dislocation direct interaction in the near field. Hence, the pop-in owing to the dislocation avalanche is fundamentally restricted by the unique stress and dislocation distributions of nanoindentation testing by restricting the dislocation motion, which is not formed in the micropillar-compression testing. The additional restriction is the reason for high power-law exponents, i.e., the probability of large-scale events becomes significantly small.

On the basis of the above-mentioned discussion and MD observations concerning the dislocation activities in the unique nanoindentation stress field, we would propose a dislocation avalanche model in the unique nanoindentation stress field (see Supplementary Note 13). The model can successfully explain the origin of the power-law exponents in the second and subsequent pop-ins, which is related to the materials’ intrinsic properties, temperature, and the surface property; however, further studies are necessary for the quantitative determination of the power-law exponents. Moreover, the model can also explain the reason for the difference in the power-law exponent between the micropillar compression and nanoindentation testing.

In summary, the distribution of pop-in magnitude transitions from Gaussian-like for the first pop-in to power-law-like for the second and subsequent pop-ins, as demonstrated by nanoindentation testing. The Gaussian-like distribution of the first pop-in was consistent with the theoretical distribution based on the thermal activation theory of dislocation nucleation. Thus, the data indicate that the first pop-in is dominated by a thermal activation process of dislocation nucleation, as has been reported in many past studies. More importantly, the second and subsequent pop-ins are dominated by dislocation avalanches, which follow power-law statistics. The power-law exponent for this distribution was much larger than that for uniaxial pillar-compression testing. The difference may be attributed to the spatial inhomogeneities in the stress and dislocation density. Thus, it can be concluded that inhomogeneous mechanical deformation belongs to a different universal-scaling class than uniaxial deformation.