Probing Stochastic Nano-Scale Inelastic Events in Stressed Amorphous Metal

One fundamental yet longstanding issue in materials science is how local inelasticity arises within an amorphous structure before yielding occurs. Although many possible scenarios were postulated or predicted by theories and simulations,however, direct experimental evidence has been lacking today due to the lack of a sensitive way to detect nano-scale inelasticity. Through the carefully designed microcompression method as coupled with the state-of-art nano-scale electric resistance measurement, we here unfold a stochastic inelastic deformation process in a Zr-based metallic glass, which takes place via the recurrence of two types of short-lived inelastic events causing structural damage and recovery, respectively, prior to yielding. Our current findings reveal that these stochastic events not only self-organize into sub-critical events due to elastic coupling, but also compete with each other in a way that enables the whole amorphous structure to self-heal as well as to sustain local damage.

control. A load function, consisting of loading, holding and unloading segments, was applied, onto which a constant voltage of 5 V was superimposed to extract the electric properties of the metallic glass in situ [ Fig. 1(b)]. Figure 1(c) shows the typical displacement (h) and electric current (I) data, as a function of time (t), measured simultaneously from the microcompression test. It should be emphasized here that, due to the surface roughness effect and the settling of the nanoindenter on the micropillar, there is a period of 1-2 seconds before an electric Ohm's contact could be established (See 'Electric Contact Analysis in Microcompression' in the SI). Therefore, for the sake of simplicity, only the data collected after the settling time were chosen for the data analyses. After considering both the geometric and size change induced by the samll deformation in the tapered micropillar, a simple relation can be derived Dr r~1 Ohm's Law in Microcompression' in the SI), which relates the electric resistivity r of the metallic glass to the measured nominal or engineering strain e, the Poisson's ratio n(<1/3) and the electric current I.
In the above relation, D represents an incremental change of a physical quantity and e 5 h/H 0 , where H 0 denotes the initial height of a micropillar. Here it should be noted that, due to the ultrahigh sensitivity of our testing system, the noise-to-signal ratio for the calculated Dr/r is very low and about 1 3 10 24 (see 'Error Analysis' in the SI).
With the above equation, the relative change in the electric resistivity (Dr/r) of the metallic glass can be extracted from the measured displacement (h) and current (I) data. Figure 2(a) shows the typical curves of Dr/r and h versus time t as the micropillar was compressed to a peak stress of ,2000 MPa. Before reaching this peak stress, a few pop-in events can be witnessed on the h-t curve. Note that, for the sake of consistency, a displacement pop-in event is herein defined as a displacement jump greater than ,2 nm in size. In accord with the pop-in events, a few pronounced resistivity spikes can be also observed on the Dr/r -t curve. As seen in the inset of Fig. 2(a), these resistivity spikes, ranging from 0.001 to 0.1 in magnitude, correlate very well with the pop-in strains. As pop-in events originate from shear banding in metallic glasses [4][5][6] , therefore, the resistivity spike corresponding to the electric resistivity increase signifies a deformation-induced 'damage' in the metallic glass. Most interestingly, besides these pronounced resistivity spikes occurring concomitantly with displacement pop-in, many secondary spikes also appear before or after a typical pop-in event. As shown in Figs. 2(b)-(d), parts of these secondary spikes point upwards (p-type spikes), indicating the electric resistivity increase in the metallic glass like those at displacement pop-in; while parts of them point downwards (n-type spikes), indicative of the electric resistivity reduction. Note that the smallest secondary spikes possess a magnitude jDr/rj greater than 5 3 10 24 , which is about fivefold of the maximum experimental error that could occur to the derived resistivity change. As such, we believe that these secondary spikes are the manifestation of real structural alteration caused by the local inelastic events in the metallic glass.
Physically, the resistivity, r, of a metallic glass can be related to its amorphous structure through the Ziman's model 18 , which is: where u is the atomic volume, e the electric charge of a 'free' electron, k F the Fermi wave vector, v F the Fermi velocity, S(q) the structure factor, and m(q) the pseudo-potential which can be replaced by a scattering matrix when the above model is applied to metallic glasses. In the 1980s, Kelton and Spaepen 19 showed that, upon external stimulation, the resistivity change, Dr, of a metallic glass can be directly related to the variation in its atomic packing density Dj through Dr 5 CDj. Here C is the pre-factor that depends mainly  on the structural factor S(q) of the metallic glass (See 'Ziman's Model' in the SI). Since the change in the structural factor of metallic glasses is insignificant during elastic deformation and thermal annealing 19 , we here assume C to be a constant. In such a case, the electricity fluctuation as we measured can be solely ascribed to the density fluctuation. In what follows, let us discuss the possible relationship between the electricity and density of metallic glasses. According to Mooij 19,20 , the change in the electric resistance Dr of a metallic glass is correlated with the change in its density under thermal annealing. However, this correlation depends on the initial thermal expansion coefficient a of the metallic glass. In general, it was found that electricity decreases (Dr , 0) with thermal annealing if a , 5 3 10 25 K 21 or otherwise increases with thermal annealing. On the other hand, based on the data from 13 types of metallic glasses with different glass transition temperatures T g , Kato and co-workers 21 discovered that aT g , 8.24 3 10 23 for metallic glasses. As the T g of Zr-based metallic glasses are around 700 K, the corresponding a should be about 1.2 3 10 25 K 21 . This suggests that the electric resistance of Zr-based metallic glasses should decrease with thermal annealing or structural ordering, which is consistent with the previous result reported by Bai and co-workers 22 . Along this line of reasoning, we can therefore ascribe the increase in Dr/r of the Zr-based metallic glass to structural disordering while the decrease to structural ordering. In other words, the recurrence of these p-and n-type spikes indi-cates that the atomic structure of the metallic-glass micropillar was repeatedly experiencing damage and recovery even though the applied stress was far below the yielding point. . Note that, within the data scattering , 1 nm, there are no observable pop-in events on the h-t curves, agreeing with the 'elastic' character of the overall deformation in the metallic glass. In sharp contrast, the corresponding curves of Dr/r versus t display many spike-like features. Compared to those prominent spikes at displacement pop-in, these spikes occurring in the 'elastic' regime appear much shorter and last only for a short period of time. Due to the limit of our data acquisition rate (,10 3 points per second), the measured duration times Dt d for these secondary spikes range from ,1 to ,1.5 ms and appear to be independent of the spike heights and the applied stresses (See 'Spike Lasting Time' in the SI). By comparison, a pop-in event generally lasts much longer, the duration of which scales with the pop-in strain and can reach up to ,8 ms. This behavior suggests that, unlike the pop-in or shear banding events which are size dependent 4,5 , the local inelastic events behind the secondary spikes behave in a stochastic manner, which may be scale independent. To further characterize the stochasticity of these secondary spikes, we performed a statistical analysis to calculate the probability for the occurrence of the secondary spikes. For this purpose, the total number of the secondary spikes with the same magnitude jDr/rj is added up irrespective of the spike sign and the timing when the spike is spotted. The inset of Fig. 3(a) displays histogram of these secondary spikes, which was obtained from the microcompression test with the peak stress of 740 MPa. Interestingly, the statistical distributions of these secondary spikes exhibit the similar trend regardless of the peak stress [ Fig. 3(a)]. Over the range of the spike magnitude from 10 24 to 10 22 , these distributions can be fitted very well to the empirical where a is the power-law exponent characterizing the stochastic behavior of the spikes having a magnitude , x c , in which x c is a critical magnitude above which the spike frequency exhibits exponential decaying, as shown in Fig. 3. Through the nonlinear data fitting, a is obtained to be 1.5 6 0.1 (See 'Estimation of Power Law Exponent' in the SI) with x c , 10 23 . Now let us discuss the underpinning physics behind our experimental observations. In the metallic-glass literature, it is a longstanding notion that metallic glasses are dynamically heterogeneous, which can be viewed as a 'mixture' composed of loosely-(liquid-like) and densely-packed (solid-like) atoms 8-10,23-28 , as illustrated in Fig. 3(b) or (c). Upon mechanical loading, inelastic events, such as shear transformation events, may occur in the regions centered in these liquid-like atoms, the behavior of which resembles viscous 'flow units' [29][30][31] accommodating inelastic deformation through local con-figurational rearrangements, whilst the solid-like ones behave elastically upholding the integrity of the amorphous structure. Within the framework of the classic shear-transformation-zone (STZ) model 32 , it was postulated that these inelasticity events could only lead to structural disordering. However, our current results clearly indicate that structural disordering is only one possible outcome of these inelastic events while the other possible outcome is structural ordering or densification. Furthermore, our results also show that the occurrence of these inelastic results is stochastic, which supports the assumption of stochastic shear transformation as we recently proposed in developing a mean-field anelastic model for MGs 29,33,34 .
Assuming a probability p(t,t) for the net transition of local structural configurations along the stress direction at time t and shear stress t, it can be proved that p(t,t) satisfies the following dynamic equation in order to retain the total free energy of the system 29,33 : where v is the attempt frequency; DG is the energy barrier and V is the activation volume of the local inelastic event. For a constant shear stress t at load hold, we obtain: where t R is the average relaxation time of these inelastic events, which equals t R~1 2v e { DG kT . From Eq. (3), it can be inferred that, when t/t R goes to infinity, the 'memory' of the MG about the stochastic inelastic events at t 5 0 will die out while the probably p(t,t) for the net transition along the stress direction will approach a steady-state value p(t) 5tV/kT. Note that Eq.(2) was derived for the simple case of a constant activation volume. This simplification can be eliminated by taking into account the possible distribution of the activation volume in a metallic glass. Thus, the real probability for the net configurational transition at the given size V should be p(t)f(V)dV, where f is the probability density function of the activation volume for an inelastic event. If the 'liquid-like cell' model of Cohen and Grest 23 is followed, it can be shown that f V ð Þ~p p LLR is the volume fraction of the liquid-like cells, V f the activation volume averaged over all liquid-like cells while V c as the smallest activation volume. In such a case, the following cumulative probability can be obtained for the constant stress activation of the local inelastic events with their activation volumes greater than V: With Eq. (4), we consider a general case that each spike event as detected from the micropillar involves a series of different sized activations. As such, the probability for these spatially isolated activation events decay exponentially when the smallest activation size involved grows, as indicated by Fig. 4(a).
To verify the above thinking, let us follow the recent analysis of Krisponeit et al. 35 , in which the total activation volume or slip as involved in an inelastic deformation event is correlated with the incubation or waiting time Dt int ahead of the event. Experimentally, Dt int can be measured as the time interval between two consecutive spike events of the same kind. Figures 4 (b)-(c) display the histograms of Dt int obtained for the holding stresses of 296 MPa and 592 MPa, respectively. Consistent with our theory, the general trend of the waiting times Dt int clearly shows the distribution of an exponential decay. Meanwhile, as the holding stress is doubled, it appears that longer waiting times could be detected, which also agrees with our prediction. However, it needs to point out that this  agreement is only in a qualitative manner. Rather than the strong linear stress dependence, the stress dependence of the accumulative probability, as seen from our data, is very weak. A similar behavior could be observed at other holding stresses. This weak stress dependence suggests that the simple mean-field anelastic model we proposed needs further development. One possible way of improvement is to include a term of elastic coupling into the mean-field modeling. In such a case, we can study how the variation of the stress field is coupled with the variation of the local properties, which may explain the weak stress dependence shown by our experimental data. The related theoretical work is under progress and will be discussed in our future work. Despite the weakness of the mean-field modeling, it is still worth mentioning that, with the mean-field model and Dt d , 1 ms, we can estimate an average activation energy of ,0.5 eV for these local inelastic events, which agrees with the previous results obtained from nanoindentation 29,34 (See 'The Estimation of Activation Energy' in the SI).
Next, we would like to further address the following questions: in what cooperative way these chain-like inelastic events occur? And even if the occurrence of these events is stochastic in nature, are their outcomes, such as the structural relaxation and damage, also stochastic? In principle, these local inelastic events may occur in an isolated manner as pictured in the previous mean-field theories 15,32 , or cooperatively through different mechanisms. As illustrated in Fig. 3(b), one possible mechanism of the cooperative activation is through elastic coupling: as one inelastic event takes place, it alters the elastic stress field nearly, thus triggering another inelastic event and so on and so forth, until eventually an avalanche event occurs; alternatively, the other possible mechanism involves de-caging or break-down of a local elastic environment, which leads to the coales- cence of the neighboring liquid-like regions [ Fig. 3(c)]. In the literature, theoretical models have been developed to account for the former case of the cooperative motion 7 . According to Dahmen et al. 7 , should elastic coupling govern the cooperative motion, the distribution of the avalanche magnitude is of a power law with the exponent of ,1.5 7 . Comparing our experimental result with this theoretical prediction, it appears that the secondary spikes, particularly for those with a small magnitude (,10 23 ), is likely to be caused by the avalanches triggered through elastic coupling. Nevertheless, we envision that the latter mechanism due to the de-caging effect is also feasible for the high-magnitude spikes; however, the probability for its occurrence must be very low.
Finally, let us discuss whether the outcome of the inelastic events, i.e. structural damage versus relaxation, is stochastic. As seen in Figs. 2(a)-(d), it is evident that the inelastic events can cause either structural damage (p-type spike) or relaxation (n-type spike). While this finding contradicts the original STZ model 32 , which assumed that the local inelastic events are the source of structural disordering (damage), however, it supports the two-state STZ model later developed by Falk and Langer 10 , in which it was postulated that a local inelastic event has two opposite effects: it can cause either jamming (relaxation) or disordering (damage). Furthermore, we have shown that the magnitude of these local inelastic events follows a power-law distribution conforming to the behavior of self-organized or tuned criticality 7,36,37 . However, it needs to point out the outcome of these inelastic event is not 'stochastic' at all, which clearly shows the stress dependence and also the stress-rate dependence. To avoid the complexity in counting the magnitude of the spike event, Figure 5 only displays the difference between the numbers of the pand n-type spikes occurring under different stress conditions. In doing so, we are only interested to know the relative frequency of the two types of events. Interestingly, it can be seen that more p-type spikes tend to occur during loading, signifying the damaging of the amorphous structure, while more n-type spikes tend to occur during unloading, signifying the recovery of the amorphous structure. More interestingly, the numbers of the p-and n-type spikes tend to be equal at load hold, indicative of the situation of a dynamic equilibrium. These findings indicate that, through the competition of the different local inelastic events, the metallic glass can sustain local damage as well as heal itself through structural relaxation. To our knowledge, this is the first experimental report of this important 'selfhealing' mechanism as mediated through the activation of local inelastic events, which has many important implications and is worth further research.
To conclude, it is demonstrated in this Letter that, with the stateof-art NanoECR technique, the local inelastic events hidden, otherwise, in the overall elastic response of the metallic glass could be revealed. Due to the elastic coupling, these local inelastic events take place cooperatively, exhibiting a self-organized behavior as seen in their magnitude distribution. In contrast to the classic model, it is also shown that the effects of these inelastic events are twofold: it can either damage or heal the amorphous structure. Depending on the applied stress and stress rate, the competition of the local events can lead to overall damage or recovery of the amorphous structure. These intriguing and multi-scaled deformation characters as seen in the metallic glass are parallel to those already known phenomena observed for earthquakes, sand pile avalanches, dislocation slip dynamics and a variety of other dynamic systems 36,38 , implying the universality of the underlying deformation dynamics across the general field of materials science and condensed matter physics.

Methods
The Zr-based metallic-glass micropillars were prepared by FEI Quanta 3D 200 FIB/ SEM dual beam system. Following the ion-milling method detailed in Ref. 17, the constant voltage of 30 keV, the current density of Gallium ion beams ranging from 7 nA to 100 pA was used to carve out the micropillars. The micropillars have the top diameters ranging from 1.5 , 2.2 mm and the aspect ratios from 2.1-2.8. Same as the literature results, the taper angle is well controlled below 3u. The nanoECR tests were subsequently conducted on the micropillars with load control using a 10-mm flat-end diamond punch. The loading, holding and unloading times were all fixed at 10 seconds and a constant voltage of 15 V was applied throughout the whole microcompression tests. The peak load was started from 500 mN and then increased to 1000 mN, 1500 mN, 2000 mN and all the way till 6000 mN with same testing parameters. Below 6000 mN, no obvious displacement bursts could be observed. To correlate the pop-in strain and the resistivity spike, the load was further increased to 7000 mN to trigger prominent pop-in events. The similar experiments were repeated on other micropillars to ensure data repeatability.