Abstract
Ingots of the bulk metallic glass (BMG), Zr_{64.13}Cu_{15.75}Ni_{10.12}Al_{10} in atomic percent (at. %), are compressed at slow strain rates. The deformation behavior is characterized by discrete, jerky stressdrop bursts (serrations). Here we present a quantitative theory for the serration behavior of BMGs, which is a critical issue for the understanding of the deformation characteristics of BMGs. The meanfield interaction model predicts the scaling behavior of the distribution, D(S), of avalanche sizes, S, in the experiments. D(S) follows a power law multiplied by an exponentiallydecaying scaling function. The size of the largest observed avalanche depends on experimental tuningparameters, such as either imposed strain rate or stress. Similar to crystalline materials, the plasticity of BMGs reflects tuned criticality showing remarkable quantitative agreement with the slip statistics of slowlycompressed nanocrystals. The results imply that materialevaluation methods based on slip statistics apply to both crystalline and BMG materials.
Introduction
In this study, we analyze and model slowlycompressed pillars of bulk metallic glasses (BMGs)^{1,2,3,4,5,6,7,8} (Figure 1). The pillar deformation proceeds via slips, observable through acousticemission measurements^{5} or steps (serrations) in the stressstrain curves (Figure 2). Here we present a quantitative model and theory for the serration statistics in BMGs, which is critical for the understanding of the deformation behavior of BMGs. We compare our experimental results on BMGs with the predictions of our model, which has previously shown good agreement in describing the slip statistics of nano and microcrystals^{9,10,11}. Furthermore, scaling collapses of the serration distributions at lower stresses predict the critical stress with roughly 5% accuracy. BMGs are noncrystalline amorphous alloys whose microstructures have no periodic longrange order^{1,2,3,4,5,6,7,8}. In monotoniccompression tests, BMGs deform by the intermittent nucleation, propagation and subsequent arrest of shear bands in highlylocalized regions of large compressive stresses^{12} (see the Supplementary Material). At a specific temperature and strain rate, a serrated plastic flow is usually observed in the compressive stressstrain curve after the yield point, marked by almostperiodicallyrecurring sudden stress drops with smaller stress drops during the loading intervals inbetween. Cumulated shear bands can be as large as the system itself, or two to three orders of magnitudes smaller^{13}. The slip sizes are broadly distributed^{14}.
Our model^{9,10,11} predicts a powerlaw distribution of slip sizes multiplied with an exponentiallydecaying cutoff function. The cutoff depends on experimentallytunable parameters, such as strain rate or stress. The model is a meanfield model with no explicit spatial dependence. Thus, it predicts that the long lengthscale behavior of the slip statistics should be universal and independent of microscopic structural details^{10}. In particular, it predicts that the statistics of the slip avalanches in slowlycompressed BMGs have the same scaling behavior as those observed for slowlycompressed crystalline materials. In the following, we test this hypothesis. We first describe the model and then show the experimental results and their comparison to the model predictions.
The model^{10} assumes that typical materials have weak spots and that a slowlyincreasing shear stress or a slow shear rate triggers weak spots to slip. Each weak spot is stuck until the local stress exceeds a random local failure stress. It then slips, thereby relaxing the local stress to a local (random) arrest stress. In crystals, the weak spots may be the location of dislocations and their slips correspond to dislocation slips. In BMGs, weak spots may be the locations of shear transformation zones (STZs), shear bands, liquidlike sites, or other relatively weak regions in the material^{12,15,16,17,18}.
Weak spots are elastically coupled, so that a slipping weak spot can trigger other weak spots to slip, creating a slip avalanche. At the slowest (“adiabatic”) driving rate, a slip avalanche finishes before the next one is started. Slip avalanches are detected as steps in strain (for slowlyincreasing stressboundary conditions) or as stress drops (for fixed strainrateboundary conditions). The elastic interaction between the weak spots is sufficiently longranged so that meanfield theory (MFT), which assumes infinite range interactions, correctly predicts the scaling behavior of the slip statistics on longlength scales^{10}. The MFT predictions agree with the slip statistics of slowlycompressed nanocrystals^{9}. Here we study whether MFT can also predict the slip statistics in BMGs.
The MFT model predicts many statistical distributions and quantities, such as the probability distribution of slip sizes and the power spectra of the acoustic emission^{10} and their dependence on experimentallytunable parameters, such as the applied strain rate and the stress.
For a low imposed strain rate, Ω and nearfailure stresses, the MFT predicts that the probabilitydistribution function (PDF) of the magnitudes, S, of the stressdrop avalanches, scales in the steady state, where the timeaveraged stress is constant, as^{11}
This scaling form is predicted to be universal, i.e., independent of the microscopic details, with κ = 1.5. The exponentiallydecaying cutoff is given by the universal scaling function, D′(SΩ^{λ}). It reflects that the maximum observed slip size, S_{max}, depends on the strain rate as S_{max} ~ Ω^{−λ}. In MFT, the universal exponent is λ = 2 in the steady state.
The corresponding complementary cumulative distribution function (CCDF), C(S,Ω), which gives the probability of observing an avalanche of size greater than S, is useful for systems with low numbers of avalanches:
Here, C′(SΩ^{λ}) is another universal scaling function, λ(κ − 1) = 1 in MFT^{10,11} in the steady state and u ≡ SΩ^{λ}.
Likewise, for the lowest (“adiabatic”) strain rate, the distribution of stressdrop avalanches is predicted to follow a modified power law as a function of applied stress^{9,10}. For f ≡ (1 − τ/τ_{C}), where τ is the applied stress and τ_{C} is the critical (failure) stress, the model predicts^{9,10}:
where D″ is a universal scaling function and σ = 0.5 in MFT. The corresponding CCDF scales as^{9}:
Again C″(x) is a universal scaling function^{9}. The corresponding distributions were extracted from the experiments for slowlycompressed ingots of BMGs (see the Methods Section) and compared to the model predictions.
Widom scaling collapses^{19} of the experimental stressdrop size distributions yield the critical exponents, κ and λ, for the strainratevaried distributions and κ and 1/σ for the stressbinned distributions. In Figures 3 and 4, we plot C(S,Ω)Ω^{−λ(κ−1)} versus SΩ^{λ} for the strainratevaried distributions and C(S,f)f^{−}^{(κ−1)/σ} versus Sf^{1/σ} for the stressbinned distributions, respectively. The critical exponents (τ, λ, κ and σ) and τ_{C} are tuned until the curves lie on top of each other, thereby yielding the correct values of these critical exponents and τ_{C}. The collapses themselves describe the scaling functions, C′(SΩ^{λ}) and C″(Sf^{1/σ}). Error bars for the exponents indicate the range of exponents that give approximately the same quality collapse.
Results
The morphology of a lateral surface after compressive fracture is described in Figure 1(a). Multiple primary shear bands can be found, denoted by the short white arrows and their slip direction is indicated by the long white arrow. With a closer look at the adjacent region of the fracture plane, which is marked by a rectangular in Figure 1(a), secondary shear bands can be located by the short white arrows in Figure 1(b). Furthermore, intensive interactions of shear bands appear in the lowerright part of the figure. The shearband initiation, propagation and arrest, including the interaction between different shear bands, are expected to contribute to the serration events and these processes are closely related to the characteristics in deformation, such as the stress drop in the stressstrain curve.
The complementary cumulative distribution functions (CCDFs) of stressdrop magnitudes were extracted from the stresstime curves shown in Figure 2. First, CCDFs are constructed, taking stress drops from the entirety of each sample's stresstime curve. The CCDFs for three different strain rates are shown in the main body of Figure 3. The axes were rescaled by changing κ and λ until the distributions lie on top of each other^{9}. For this collapse, it was found that κ = 1.42 ± 0.20 and λ = 0.22 ± 0.02. The collapse function in Figure 3 is the scaling function, C′(x), of Equation (2). Plugging this information into Equation (2) then predicts the scaling behavior of the slipavalanchesize distribution for other strain rates as well. Note that for the higher strain rates, the samples break before they reach the steady state – Figure 2 shows that the stress versus time plots have no flat region for strain rates of 2 × 10^{−4} and 1 × 10^{−3} s^{−1}.
The second collapse, shown in Figure 4, was performed on the most slowlystrained sample using CCDFs from stress bins near the critical failure stress. Nearly all of the stress drops occurred above 92.0% of the highest average stress achieved in the sample. Three partitions of average stresses were chosen – 94.0–96.0%, 96.0–97.0% and 97.0–97.6%, where the percentages indicate the percents of the maximum stress on the sample, 1,980 MPa. Avalanches were not sampled from higher than 97.6% of the maximum stress, because near the critical stress, the avalanche sizes are cut off by the finite system size, i.e., the avalanches “feel” the boundaries of the sample. In this region, finitesize corrections to the infinite system predictions of the theoretical model become nonnegligible^{19}. Therefore, it is preferable to keep the stress bins close, but not too close to the critical stress.
The second collapse yielded exponents of κ = 1.40 ± 0.28, 1/σ = 1.85 ± 0.20 and a relative critical stress ratio of τ_{C/}τ_{Max} = 1.05 ± 0.01. The error bars reflect statistical fluctuations resulting from the finite number of avalanches per sample. The parameter, τ_{C/}τ_{Max}, indicates the critical stress as a fraction of the maximum achieved stress, τ_{Max} = 1,980 MPa. The fitted critical stress, τ_{C} and the measured maximum applied stress, τ_{Max}, are slightly different in that the critical stress is the applied stress at which an infinite system is (by extrapolation) expected to yield in an infinitelylarge avalanche. However, in a finite sample, one finds a samplespanning avalanche at the maximum stress, which is below the critical stress.
The collapse function in the inset of Figure 4 is the scaling function, C″(x), of Equation (4). Plugging the extracted exponents, κ and 1/σ, the critical stress, τ_{C} and the scaling function, C″(x), into Equation (4) then predicts the scaling behavior of the slipavalanchesize distribution for other stress windows as well.
Discussion

1
We first compare the results on BMGs or amorphous materials to model predictions and crystal plasticity. The values of 1.42 ± 0.20 and 1.40 ± 0.28 for the magnitude scaling exponent, κ, in Figures 3 and 4, respectively, agree, within error bars, with the model prediction of κ = 3/2^{10} using the meanfield theory. Within error bars, the values of 1.85 ± 0.20 for the exponent, 1/σ, in Figure 4, also agrees with the MFT prediction of 1/σ = 2. As expected, the experimental exponent, λ = 0.22, deviates from the value that the MFT predicts for the steady state, because at the higher shear rates, the samples do not reach the steady state. 234, 181 and 189 avalanches were collected for the samples at strain rates of 5 × 10^{−5}, 2 × 10^{−4} and 1 × 10^{−3}s^{−1}, respectively.
The stressbinned data contained 38, 28 and 26 avalanches in the bins with the average avalanche stresses at 95.2%, 96.5% and 97.3% of the maximum stress bins, respectively, in Figure 4. Stress bins were chosen to be small, because the BMG samples did not exhibit the avalanche behavior until about 92.0% of the maximum stress was achieved, originating from BMGs' high strength but limited ductility^{20,21}. As explained above, finitesize effects affect the distribution of avalanches close to the critical fracture stress. Thus, stress bins must be kept a few percents below the maximum stress^{9}. Note that the stressbinned data collapsed in the same way, with the same exponents, as stressbinned data on slowlycompressed nanocrystals^{9}, illustrating the similar slip statistics of crystals and amorphous materials.

2
Our MFT model assumes the presence of elasticallycoupled weak spots. This assumption applies to BMGs for the following reasons. Recent studies reveal that structural heterogeneities exist in BMGs, which are mainly composed of solidlike (denselypacked) sites and liquidlike (anelastic) sites or socalled weak spots^{22,23,24,25}. Using the amplitudemodulation dynamic atomic force microscopy (AMAFM), one can easily find two types of sites from the local energydissipation map^{24,25}. It should be pointed out that the correlation length of the heterogeneity equals ~ 2 nm, which is in good agreement with the size of STZs^{24,26,27}. Furthermore, STZ dynamic simulations, based on the kinetic Monte Carlo method, have been performed to study the interaction between STZs during deformation in BMGs^{28}. The results clearly show that at low stress levels, these STZs will behave separately, which corresponds to the elasticdeformation mode. When the stress exceeds a certain value, the activation of one STZ will induce the subsequent STZ activation in its immediate neighborhood, i.e., the slip of one weak spot triggers other weak spots to slip. These experimental and simulation results strongly support our model assumptions.

3
The observed scaling behavior and the scaling collapse in the inset of Figure 3 reflect the tuned criticality. Ren et al.^{29} find that the elasticenergy density released in avalancheslip events of BMGs follows a powerlaw distribution for a high strain rate (2.5 × 10^{−2} s^{−1}), which they interpret as a signature of selforganized criticality (SOC). However, both their and our avalanchesize distributions reflect distinct strainrate dependence. As shown in Figure 3, we observe avalanchesize distributions whose exponential cutoff moves to larger sizes with decreasing strain rate. The data of Ren et al.^{29} also show evidence of broader distributions (larger average stressdrop sizes) with decreasing strain rates, which may signify scaling with respect to strain rate. In other words, the strain rate here is a tuning parameter of the tuned critical point of BMGs^{9,19}. Tuned criticality is fundamentally different from SOC, which always exhibits pure powerlaw scaling, without the need for parameter tuning to a critical point^{30}. In contrast, the plastic flow in BMGs exhibits tuned criticality with a critical point at low strain rates and at nearfailure stresses, similar to crystals. For both crystals^{9} and BMGs, the tuning parameters (strain rate and/or stress) must, thus, be tuned to their critical values in order to observe the powerlaw scaling behavior^{9,19}.
Conclusions
In conclusion, we have presented new analysis and modeling of the dependence of avalanche statistics in BMGs on the applied strain rate and the stress. We obtained the first scaling collapse of the slipavalanche statistics in BMGs. We have shown that the distribution of avalanche sizes varies with strain rate and applied stress, which indicates that both the strain rate and the applied stress are critical tuning parameters. Because we observe that the criticality is tuned, we conclude that the avalanche distributions reflect an ordinary (tuned) critical point rather than selforganized criticality in amorphous solid deformation.
A meanfield theoretical approach predicts the experimentallyachieved values for the critical scaling exponents. The strainrate scaling exponent, λ, differs from the meanfield prediction for the steady state because at the higher strain rates, the samples are not in the steady state. Most importantly, we find that the critical exponents, κ and σ and the scaling forms are consistent within error bars with the predictions of our MFT. The critical exponents, κ and σ, also agree within error bars with recent experiments on nanocrystal plasticity^{9}. Note that the exponent, λ, is yet to be determined for crystal plasticity. The present result suggests that the model's predictions and interpretation of serrations as slip avalanches of weak spots apply to both crystals and amorphous materials, irrespective of the microscopic details and structures. This observation implies that the same evaluation methods (using the slipavalanche statistics and acoustic emission below the failure stress) can be employed to predict quantities, such as the critical stress, in both crystalline and BMG materials. Moreover, from the slipsize distributions at the lower strain rates, or at lower stresses, we can predict the serration statistics at higher strain rates or at higher stresses, respectively (see Figures 3 and 4, respectively).
Methods
Ingots of an amorphous Zr_{64.13}Cu_{15.75}Ni_{10.12}Al_{10} (nominal atomic percents) BMG were prepared by arcmelting the alloy mixture of Zr, Cu, Ni and Al with purity higher than 99.9 weight percent in a Tigettered highpurity argon atmosphere. The melting and solicitation processes are repeated at least five times to achieve chemical homogeneity. Then the melted mixture is suction cast into a watercooled copper mold to form a cylindrical cast rod, 60 mm in length and 2 mm in diameter^{13,14}. The cast rods were then cut into cylindrical bars with 4 mm in length. The two compression faces of each bar were then carefully polished to be parallel to each other. The sample was uniaxially compressed at 298 K (room temperature) using a computercontrolled MTS 809 materials testing machine at a constant strain rate. Three strain rates, 5 × 10^{−5} s^{−1}, 2 × 10^{−4} s^{−1} and 1 × 10^{−3} s^{−1} were employed in the compression experiments, with a dataacquisition rate of 33 Hz. Figure 1 shows images taken by scanning electron microscopy of the lateral surfaces of one of the compressivelyfractured samples at a strain rate of 5 × 10^{−5} s^{−1}. The fractograph clearly indicates the multiple shear bands along which the sample deformed.
The slowest strain rate, closest to the theoretical adiabatic limit, was selected and examined at different values of applied stresses to determine the stress dependence of the distribution of stressdrop avalanches. The sample compressed at 5 × 10^{−5} s^{−1} exhibits many avalanche events at stresses above 1,800 MPa, which is 92.0% of the maximum attained stress for this sample. The stressdrop avalanches were extracted for values of averagestress intervals of 94.0–96.0%, 96.0–97.0% and 97.0–97.6%.
The complementary cumulative distribution function (CCDF) of stress drops as a function of their magnitudes is constructed numerically for each different strain rate^{9}. CCDFs are also constructed for each stress bin of the adiabaticallycompressed sample.
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Acknowledgements
We thank Nir Friedman, Michael LeBlanc, Tyler Earnest and Braden Brinkman for helpful conversation. We gratefully acknowledge the support of the US National Science Foundation (NSF) through grants DMR 1005209, DMS 1069224 (KAD and JA), DMR0909037, CMMI0900271 and CMMI1100080, the Department of Energy (DOE), NEUP 00119262, DEFE0008855 (PKL and XX) with Drs. Curry, Huber, Cooper, Finotello, Ardell, Taleff, Cedro, Jensen, Tan and Lesica as contract monitors. KAD and PKL thank DOE for the support through project DEFE0011194 with the project manager, Dr. Markovich. JWQ acknowledges the financial support of the National Natural Science Foundation of China (No. 51101110) and the Youth Science Foundation of Shanxi Province, China (No. 20120210181). PKL very much appreciates the support of the U.S. Army Research Office project (W911NF1310438) with the program manager, Dr. Mathaudhu.
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J.A. prepared Figures 2–4. X.X. and P.K.L. prepared Figure 1. J.Q., X.X., Y.Z. and P.K.L. collected the experimental data that was analyzed and compared to the model predictions by J.A., J.T.U. and K.A.D. All authors reviewed the manuscript.
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Supplementary Information
Tuned Critical Avalanche Scaling in Bulk Metallic Glasses
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Antonaglia, J., Xie, X., Schwarz, G. et al. Tuned Critical Avalanche Scaling in Bulk Metallic Glasses. Sci Rep 4, 4382 (2014). https://doi.org/10.1038/srep04382
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DOI: https://doi.org/10.1038/srep04382
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