Introduction

Recent research efforts have focused on intermittency in crystal plasticity. For example, acoustic emissions from stressed ice revealed discrete dislocation avalanches, whose magnitudes follow a power-law distribution1,2. Mechanical testing on microcrystals detected sudden strain bursts3,4, which also follow the power-law scaling between the number of bursts and their magnitudes4,5,6,7. Together, these experiments convincingly demonstrated criticality among relatively large and rapid dislocation events. On the other hand, the fundamental operating units in crystal plasticity are individual dislocations2,3,4. There is a long history of transmission electron microscopy (TEM) studies of dislocations and their interaction mechanisms8,9,10, which have also revealed complex dislocation patterns in deformed crystals, including dislocation cells11 and persistent slip bands12. However, how intermittent dislocation activities stem from complex dislocation patterns, how these activities impact the dislocation pattern formation, and how this trend leads to crystal slip, are all unresolved critical questions5,13,14.

To start addressing these questions, we followed the dislocation motions inside a slowly compressed nanopillars using the bright-field (BF) TEM, while simultaneously we measured the load and displacement using a pico-indenter (model PI 95, Hysitron, MN, USA). Together, these two measurements enabled us to detect a broad range of dislocation motions from few nm/s to hundreds of μm/s. For the material, we focused on high entropy alloys (HEAs), composed of five or more elements of near-equal molar percentages in nearly-random solutions15,16,17,18. HEAs are promising materials for their excellent thermomechanical properties19. Previous studies also demonstrated unexplained mechanical behaviors in HEAs, including sudden stress drops or “serrations” in the stress–strain curves of bulk samples20. The HEA of Al0.1CoCrFeNi studied here has a simple fcc structure, but because of the differences in atomic radii, the material exhibits severe lattice distortions17,21, providing a complex dislocation-pinning landscape.

Here, we show that dislocations accumulate and are stored in the HEA nanopillars in a manner opposite to mobile dislocation starvation6,22,23. Dislocation storage leads to the formation of dislocation bands and dislocation pileups in front of the band. The dislocation band propagates intermittently along the compression direction through dislocation slips. The avalanche mechanism that we identified is based on interactions between the dislocation pileup and the dislocation band, and between the pileup and dislocation pinning centers.

Results

Deformation of the HEA nanopillar

The experimental design to observe dislocation avalanches is shown in Fig. 1a using a TEM nanoindenter holder. A nanopillar is deformed by compression. The deformation process is simultaneously observed by electron imaging and the measurement of load and displacement, which are used to calculate the stress and strain in the nanopillar. An as-prepared nanopillar is imaged in Fig. 1b using the BF TEM. The composition and structure of the HEA sample were characterized by atom probe tomography (APT), electron imaging, and diffraction (see Supplementary Figs. 1–3, Supplementary Table 1, and Supplementary Notes 1 and 2). The HEA nanopillar was oriented along [647] and observed close to \(\left[ {1\bar 20} \right]\) (Fig. 1c). Along [647], the primary and secondary slip systems are \((1\bar 11)\)[011] and \((1\bar 11)\)[110], with Schmid factors of 0.4 and 0.36, respectively. Figure 2a shows the measured stress–time curve from the as-prepared nanopillar shown in Fig. 1b. The nanopillar deformation process can be divided into three stages with: (I) little or no stress drops, (II) medium stress drops, and (III) large stress drops. In Stage I, dislocations activated from the pillar top and quickly accumulated forming a band of high dislocation density, which then propagated like a wave toward the pillar bottom (Fig. 2b). Stage II is characterized by small dislocation avalanches, and the resulting stress drops in the stress–strain curve. This effect is shown in Fig. 2c where a group of dislocations jumped within the camera time resolution of 0.1 s, resulting in a stress drop of 0.8 MPa. In Stage III, dislocation avalanches led to large crystal slips, as evidenced by TEM imaging (Fig. 2d), by scanning electron microscopy (SEM) (Fig. 2e) and by the correlation between the measured stress–time curve and the captured video (Supplementary Movies 1 and 2).

Fig. 1
figure 1

The design of nanopillar compression experiment. a illustrates the nanopillar compression experiment as performed in a transmission electron microscopy (TEM). The electron image of the nanopillar and the stress (τ)–strain (γ) curve are simultaneously recorded as functions of time. b shows a bright-field (BF) image of an as-prepared high entropy alloy (HEA) nanopillar, on which the in-situ TEM experiment was carried out (scale bar is 500 nm). The image contrast indicates a relatively uniform starting dislocation microstructure. c presents a schematic of the nanopillar orientation and the activated slip system, \(\left( {1\bar 11} \right)[0\overline {11} ]\), with a Frank–Read source under compression

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Fig. 2
figure 2

In-situ nanomechanical measurement and transmission electron microscopy (TEM) observation of slip avalanches. a shows the measured stress–time curve with color-coded deformation stages (I, II, and III). b–d present two captured images at each stage, one is before and one is after a nanopillar slip. The images are labeled from 1 to 6, and their recording time is labeled below the image. Regions of the images are highlighted by red and white arrows to mark the dislocation band and the dislocation pileup. e is a scanning electron microscopy (SEM) image of the deformed nanopillar, which confirms the large crystal slips. 1–2 and 3–4 are two difference images between the images of 1 and 2, 3, and 4, respectively; they are displayed in color. The color bar marks the difference intensity level in counts. The scale bar is 500 nm. For further details, see the text

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Avalanche size distribution

The magnitude of dislocation avalanches can be measured by the size of the nanopillar slip (S) recorded in the displacement–time curve (Fig. 3a). To examine the scaling behavior for the slip-size distributions and other statistical properties, we plot the Complementary Cumulative Distribution Function (CCDF), C(S,σ), in Fig. 3b. The CCDF gives the probability of measuring a slip event that is larger than the size, S, at the applied stress, σ. In the mean field theory (MFT) model7,24, in the absence of hardening and weakening, at a certain level of the applied stress, the slip size distribution D(S,σ) follows a power law multiplied with an exponentially decaying function (fS), which limits the avalanches to be smaller than a cut-off size, \(S_{\max }\sim \left( {\sigma _c - \sigma } \right)^{ - \frac{1}{\beta }}\).

$$D\left( {S,\sigma } \right)\sim S^{ - \kappa }f_S\left[ {S(\sigma _c - \sigma )^{\frac{1}{\beta }}} \right].$$
(1)
Fig. 3
figure 3

Characteristics of dislocation avalanches in the Al0.1CoCrFeNi nanopillars. a shows the characteristic stress– and displacement–time curves for the in-situ compression experiment and for a [647]-oriented nanopillar. b plots the stress-binned complementary cumulative distribution, C(S, σ), of slip sizes as a function of the stress level over the maximum stress using 145 events from seven [647]-oriented samples of ~512–655 nm in diameters, compressed at a strain rate of 1 × 10−3/s and with similar load–displacement characteristic. c shows the scaling collapse of the same data as well as the predicted scaling function (see the text for details). d, e show two different types of displacement jumps. The velocity is calculated by the ratio of the displacement jump, S, and duration t. f Slip velocity as a function of shear stress. g Stress drop versus slip size. The avalanche velocities in (e) are averaged over a rolling slip size window of 2 nm, where the slip size distribution within the window is represented by the vertical bar, which marks the minimum and the maximum slip size

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Here, S is the magnitude of the displacement jump of one slip avalanche, and σc is the failure stress, also called a critical stress. The MFT model predicts that κ = 1.5 and β = 0.5. The CCDF C(S,σ) is given by

$$C\left( {S,\sigma } \right)\sim {\int}_S^\infty {D\left( {S\prime ,\sigma } \right)dS\prime } \sim S^{ - \left( {\kappa - 1} \right)}g\left( {S\left( {\sigma _c - \sigma } \right)^{\frac{1}{\beta }}} \right),$$
(2)

where \(g\left( x \right) = x^{\kappa - 1}{\int}_x^\infty {e^{ - At}t^{ - \kappa }dt}\) is a universal scaling function7. By introducing f = (1 − σ/σc), the CCDF can also be evaluated as7,

$$C\left( {S,f} \right)\sim {\int}_S^\infty {D\left( {S\prime ,f} \right)dS\prime } \sim f^{\frac{{\left( {\kappa - 1} \right)}}{\beta }}g_f\left( {Sf^{\frac{1}{\beta }}} \right).$$
(3)

Here, \(g_f\left( x \right) = {\int}_x^\infty {e^{ - At}t^{ - {\it{\kappa }}}dt}\) is another scaling function whose form was given previously7. With this explicit function for the slip-size distribution, a scaling collapse of the experimental slip avalanche size distribution can be used to verify the exponent values predicted by the MFT model. The log–log scaled CCDFs in Fig. 3b include 145 slip events from seven HEA nanopillars, extracted directly from the displacement–time signal (an example is shown in Fig. 3a, for details about the analysis, see Supplementary Fig. 4 and Supplementary Note 3). The measured slip sizes range from several to hundreds of nm, which were sorted into 4 categories based on the value of σ when the event occurred, from 0.6 to 1 times σm (where σm = σc is initially taken to be the maximum observed stress in a nanopillar). In Fig. 3c, we plot \(C\left( {S,f} \right)f^{\frac{{\kappa - 1}}{\beta }}\) versus \(Sf^{\frac{1}{\beta }}\) and we find that the rescaled CCDFs binned over all stress levels collapse onto each other, whose shape follows a universal scaling function as gf(x) with A = 1.2. The critical exponents (σm, β, and κ) shown in the figure were tuned for the collapse. This process yielded exponents that are consistent with  the predicted MFT model values, κ = 1.5 and β = 0.57.

Avalanche velocity and stress drop

In addition to the size, we also determined the avalanche velocity and stress drop from the recorded displacement– and stress–time curves, respectively. The velocity varies as shown in Fig. 3d, e. Figure 3f shows the correlation between the avalanche velocity and the applied stress, while Fig. 3g plots the stress drops and slip sizes from 4 nanopillars of different diameters. The nanopillar slips are only observed with the resolved shear stress (RSS), τ, greater than 172 MPa (Fig. 3f). Above this stress level, the avalanche velocity tends to rise with the RSS. However, the velocity is broadly distributed, as indicated by the wide range of velocities above and below the average values (Fig. 3f). The stress drop depends linearly on the observed slip size; the slope of the curve depends on the diameter of the nanopillar (Fig. 3g).

Dislocation activities during an avalanche

Figure 4 examines the dislocation activities that led to the avalanche at 16.7 s in Fig. 2a using the difference image, which is obtained by subtracting the “after” frame from the “before” frame for two consecutive video frames. The positive and negative intensities in the difference image correlate with the creation and annihilation of dislocations, as is illustrated by examples shown in Fig. 2b, c. Image 1 of Fig. 4a captures the early stage of the dislocation band formation, where the bow-out of several dislocations was observed, and Image 2 reveals the formation of dislocations in front of the dislocation band, which become a dislocation pileup in Image 3. The avalanche captured in Image 5 of Fig. 4a resulted in a pillar displacement of 3.88 ± 0.17 nm at 16.7 s. The avalanche was accompanied by dislocation activities that spanned the entire pillar, including a jump of the dislocation band. It also created a new dislocation array, as marked by the arrow in Image 5 of Fig. 4a. Intermittent dislocation activity also occurred before and after the dislocation avalanche (Images 4 and 6 in Fig. 4a), resembling a “foreshock” and an “aftershock” of an earthquake. In both cases, we observed dislocation movements in front of the dislocation band, giving rise to the nanopillar slips of 0.78 ± 0.16 and 1.01 ± 0.17 nm, respectively.

Fig. 4
figure 4

The dislocation avalanche and dislocation pileup. a Dislocation activities leading to and after an avalanche are revealed by the difference of two consecutive images for events marked as 1–6 at the specified starting time. The color bar marks the intensity level in counts (maximum counts in an image is 255). The scale bar is 500 nm. b Post-mortem transmission electron microscopy (TEM) image of a deformed high entropy alloy (HEA) nanopillar recorded in bright field under the two-beam diffraction condition (inset). The nanopillar was thinned a few degrees off the normal direction of [1−20] by focused ion beam (FIB). The Thompson’s tetrahedron shows the orientation of the primary slip plane (BCD, colored) relative to the electron beam. The scale bar is 100 nm. c A schematic illustration of dislocation characters and distribution in the sample of (b) compiled from images taken under different two-beam conditions. Blue dashed lines in (b) indicate the edge of the original nanopillar

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The formation of dislocation bands

The initial yielding of the HEA nanopillars is accompanied by the dislocation multiplication through the Frank–Read or single-arm dislocation sources inside the nanopillars (see Image 1 of Fig. 4a, Supplementary Fig. 5, and Supplementary Movies 3 and 4). This situation differs from elemental fcc metals, for which dislocation starvation and strengthening are observed when the initial dislocation density is below a diameter-dependent critical value25, and the mobile dislocations escape the crystal due to the image forces exerted by the surface6,22,23. The accumulation and storage of dislocations in the HEA form the dislocation bands, or bundles, with the slowdown of dislocations. Post-mortem TEM characterization shows that the dislocations in the band consist of primary dislocations with b = 1/2[011] and coplanar secondary dislocations with b = 1/2[110], and the dislocation band is composed of mostly primary edge dislocations (Fig. 4b, c). The secondary dislocations are formed in the vicinity of the dislocation band. A rotation is also generated by the formation of a dislocation band (amounts to 3.2° in Fig. 4b). Dislocation bundles seen by ex-situ TEM have been reported before in the deformed fcc single crystals oriented for the single slip26. These dislocations are known to arrange approximately in planes parallel to the slip plane. The stability of the structure under a constant applied stress is achieved through dislocation dipoles, which cancel the long-range stress. Previous studies have also identified “glide zones” in the primary slip plane near the bundles, which act as the dislocation source26,27. However, the dislocation microstructure in deformed single crystals of elemental metals is on the scale of tens to hundreds of microns. In the HEA nanopillars, we see multiple dislocation bands formed by the accumulation of dislocations. This trend has never been seen before on such a small scale.

Dislocation pileup

The experimental result in Fig. 4b confirms the dislocation pileup in front of the band. The creation of a dislocation array also indicates the operation and shut off of dislocation sources, as well the presence of obstacles that impede the front dislocation motion. Figure 5 shows the formation of a dislocation pileup in-situ. Under the applied shear stress, the mobilized part of the front dislocation bowed out (Fig. 5a). When the threshold stress was reached in Fig. 5b, the Frank–Read source(s) operated and generated a dislocation pileup, as well as a jump of the dislocation band. The restricted motion of dislocations in the pileup produced a back stress to the following dislocations and opposed the Frank–Read source operation. In the past, the existence of the dislocation pileup during crystal slip, and the related long-range internal stresses have been the subject of intense debate; evidence for their existence were only found under the special conditions of low-temperature cooling and dislocation pinning by neutron irradiation26. The stress produced by the pileup in the front is proportional to the number of piled-up dislocations. Thus, a rise and fall of the local stress is expected as the pileup forms and breaks away. This trend is indeed observed by the in-situ electron diffraction (Supplementary Fig. 6 and Supplementary Note 4).

Fig. 5
figure 5

The formation of dislocation pileup in front of a dislocation band. a shows the bowing out of the mobile part of a dislocation under the applied stress. b captures the Frank–Read dislocation source operation, which generated a dislocation pileup that also resulted in a sudden movement of the dislocation band, as evidenced in the difference image

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The flow of plastic deformation

To correlate intermittent dislocation activities with the laminar flow of plastic deformation and work-hardening, Fig. 6 shows a spatiotemporal map of dislocation-band propagation together with the stress–time curve. The spatiotemporal map was obtained by plotting the density of line pixels (DLP) along the pillar length direction. The line pixels represent the line features recorded in the diffraction contrast images. They are obtained by applying a line filter to the recorded images (details provided in the Methods section and in Supplementary Figs. 7, 8). Figure 6 shows the formation and propagation of multiple dislocation bands for the nanopillar measured in Fig. 2. The dislocations first activated near the nanopillar top (Supplementary Fig. 9), when the stress was below the yield strength (64 MPa) of the bulk Al0.1CoCrFeNi sample28. This activation led to the 1st band formation, which then propagated through the nanopillar at the average velocity of ~70 nm/s and up to the distance of ~900 nm. The 2nd band started at a position lower than the 1st band and propagated by ~ 600 nm. The flows of 1st and 2nd bands are relatively smooth in Stage I, but become strongly intermittent in Stages II and III. In Stage III, additional bands formed and propagated through the nanopillar. The width of the dislocation bands also changed with time.

Fig. 6
figure 6

Plastic flow in the nanopillar. The map shows the density of line pixels (DLP) distribution along the pillar versus time, overlapped with the stress–time curve. An example of the DLP measurement is provided in Supplementary Figure 8

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Figure 6 also shows that the dislocation avalanche at 16.8 s occurred after the propagation of the band 1 stopped. Before the avalanche, only small dislocation movements and small slips are observed, as evidenced in Image 4 of Fig. 4a. In the HEA nanopillars, the large pillar slips occurred in Stage III. The slip sizes involved in the large displacement jumps reached about 102 nm. A notable stress drop is also associated with the slip, which corresponds to the stress released by the escaping dislocations. The large stress drops are on the order of 102 MPa.

Discussion

The avalanche mechanism that we identified for HEAs is based on interactions between the dislocation pileup and the dislocation band, and between the pileup and dislocation pinning centers. Unlike the slip of individual dislocations, which can be approximated as elastic strings of constant stiffness, the stiffness of a pileup is dependent on the long-range interactions between dislocations29. Our statistical analysis shows the slip avalanche’s dependence on the stress level, which indicates that the applied stress is a critical tuning parameter. Deformation of HEAs thus exhibits the tuned critical behavior rather than self-organized criticality. This result agrees with recent findings on bulk metallic glasses30 and elemental metal micropillars7. Thus, while the avalanche mechanisms are materials specific, the avalanche distribution is universal. The distribution used in our analysis is predicted by a simple, coarse-grained, model7,24. Other power-law distributions of different exponents and cutoff functions have been predicted31,32,33 or observed34. However, a quantitative differentiation between these distributions requires a much larger slip-size range and sample size than what are measured here in the HEA nanopillars35.

Lastly, we note that the demonstration of the less observable dislocation processes during the quiescent periods between avalanches was made possible by the low dislocation mobility in the HEA. To put it in perspective, the dislocation motion in elemental metals is generally very fast in single crystals of pure metals (for pure Cu, the fast velocity was measured in the range of ~1–102 m/s36). The slip velocities, measured in the micropillars of several metals and Si using the nanoindentation techniques, are considerably less. Nonetheless, they are still quite large, from 103 to 106 nm/s37. In these measurements, the dislocations involved were not observed. Thus, it was not possible to pin down the mechanism behind the dislocation motions. In a TEM, it is possible to observe and measure the speeds of individual dislocations. Previously, Kiener et al.38 reported the dislocation speed for the individual dislocation motion at ~2.5 μm/s in the irradiated copper, where dislocations were slowed down by irradiation defects. Here, we detected two types of dislocation motions: dislocation avalanches with the slip velocities reaching 104 nm/s (Fig. 3f), and the slow dislocation speeds at ~101 nm/s, involving a few dislocations in front of the dislocation band even under the shear stress ~200 MPa (Supplementary Fig. 10). A critical question is thus what slows down the dislocations? A major factor that distinguishes conventional metals and the HEAs is the severe lattice distortions in HEAs, due to the atomic size variations of the constituent elements and the variations in the binding energy between atom pairs. The severe lattice distortions are expected to lead to solid-solution hardening, however, the exact nature of lattice distortions in HEAs is still unknown for the lack of suitable probes17. Our study of the HEA structure has demonstrated an inhomogeneous strain field and low angle boundaries (Supplementary Figs. 3, 1). Local clusters of nm size are observed in the measured strain maps. The lattice distortions increase the lattice friction stress, also known as the Peierls–Nabarro (P–N) stress. A dislocation line only moves when sufficiently high stress is provided to overcome the P–N stress28. Additionally, the low-angle grain boundaries are also well-known as strong dislocation barriers, which hinder the dislocation motion. Previous studies have also demonstrated that HEAs have the low stacking fault energy (SFE)17,18,39. The fcc metals with the low SFE cannot easily cross slip, and hence the slip is localized on a limited set of slip planes. Together, the low SFE and the strong pinning centers contribute to the low dislocation mobility in the HEA that is critical to the formation of dislocation pileups. Thus, understanding the nature of inhomogeneous strain fields and their contribution to the plastic instability in HEAs will help the design of new and better multicomponent alloys.

Methods

Sample preparation

The Al0.1CoCrFeNi HEA with a targeted composition of 2.44 at% for Al and 24.4 at% for Co, Cr, Fe, and Ni was fabricated by the Sophisticated Alloys Inc. (Butler, PA) using vacuum induction melting. The as-cast samples were hot isostatic pressed at 1100 °C for 1 h under a 207 MPa ultra-high-purity argon pressure to reduce porosity. The composition and chemical homogeneity were checked by the APT analysis (Supplementary Fig. 1 and Supplementary Movie 5). A crystal grain of hundreds μm in size, and [647] orientation was identified by electron backscattered diffraction (EBSD) and selected to produce HEA nanopillars. Nanopillar samples with diameters of 500–700 nm on a base of ~3 times wide for lift-out were fabricated and attached to the Mo lift-out grids, using a dual beam focused ion beam (FIB) and a three-step process involving the milling currents of 40 and 7 pA, 30 kV Ga ions, in the last two steps to reduce the milling damage. The milling damage was evaluated by electron diffraction, and the surface modification is <4 nm thick (Supplementary Fig. 11). The composition uniformity of the pillars was confirmed by X-ray energy dispersive spectroscopy (EDS). The pillar had a taper angle of 2.5° and aspect ratios (diameter:height) of ~1:3 to minimize the lateral constraint. This geometry provides a stable imaging condition during the compression test (the orientation of a similar nanopillar was monitored by diffraction, see Supplementary Fig. 12).

In-situ compression test

In situ compression tests of the HEA nanopillars were performed, using a picoindenter (Hysitron PI95, Hysitron, MN) equipped with a 2-μm flat punch diamond indenter in a JEOL 2010 LaB6 TEM (JEOLUSA, Boston), which was operated at 200 keV. This indenter was operated with the feedback loop on. The instrument was calibrated using a flat region of the load–time curve, where we measured the standard deviation of the load and displacement at 0.1 μN and 0.56 nm, respectively. The compression tests were performed at the displacement rates of 0.5–1.5 nm/s, resulting in a strain rate of ~1 × 10−3/s. The load, displacement, and time data were read out at 500 Hz. The stress as the pillar is compressed is peaked at the pillar top and decreases gradually towards the pillar bottom (Supplementary Fig. 13). The stated shear stress in the text is an average number calculated by τ = σcosλcosϕ, where σ = F/A with F for the applied force and A for the cross-sectional area of the nanopillar, and cosλcosϕ = 0.4. A video of each test was recorded under the image condition of Supplementary Fig. 14, using a charge-coupled device (CCD) camera (Orius CCD Camera, Gatan, Pleasanton, CA) with a resolution of 720 × 480 pixels and readout at 10 frames per second. The pillar slip was measured by the nanoindenter and by fitting the intensity profile of the indenter edge as recorded in the TEM image. (The edge position can be determined with a standard deviation of 0.26 nm.) To contrast the dislocation activities in the HEA nanopillars, we also observed the dislocation activities in a compressed Cu nanopillar (Supplementary Fig. 15 and Supplementary Movie 6).

Measurement of dislocation activities

We measured the dislocation activities by quantifying the dislocation line contrast in the recorded images, using a line detection algorithm. The recorded BF-TEM images were transformed into binary images, where the contrast of a dislocation yields a line pixel with the value of 1 (examples are shown in Supplementary Fig. 7). By counting the number of line pixels along the pillar diameter and the total number of line pixels, we gained the information about the dislocation bands as well as the intermittent dislocation activities. Using the same method, we also quantified the initial dislocation density of a HEA nanopillar (Supplementary Fig. 16).