Acoustic emission spectra and statistics of dislocation movements in Fe₄₀Mn₄₀Co₁₀Cr₁₀ high entropy alloys

There has been a boom of research on “high-entropy alloys” (HEAs) and their derivatives.^1–7 HEAs are different from other alloys, as they are composed of multiple principal elements unlike conventional solutions (e.g., 316L stainless steel which we will use here for comparison) which contain few host metals and small concentrations of dopants. All HEAs possess a high degree of compositional inhomogeneity.^2,3 Furthermore, the heterogeneity is often enhanced by local lattice distortions and local chemical order.^2,8 All these effects generate a rugged energy landscape, raise the activation barriers which govern dislocation movements, and influence dislocation pathways in slip, faulting, and twinning.⁹ 

Specifically, the heterogeneity in HEAs alters the dislocation slip mode. Recent molecular dynamics simulations on HEAs⁹ reveal that moving dislocations can adopt an unusual “nanoscale segment de-trapping” (NSD) mode in the presence of chemical complexity and abundant inhomogeneity. Dislocation lines split into segments that rest in energetically favorable locations and others struggling to escape from energetically unfavorable positions. Under applied stresses, the forward motion of the dislocation is, therefore, non-uniform and associated with the stick-slip movement of several local segments.^3,9 However, there is no direct evidence at a macroscopic temporal and spatial scale to show the difference between the dynamics of dislocation movements in HEAs and conventional solution alloys, as we show in this paper.

Although some kinetics of dislocation motion is easily captured by molecular dynamics simulations, the spatial and temporal scale is too small to represent the bulk sample. In addition, it is also difficult to do so by classic observation. Although in situ transmission electron microscopy (TEM) can capture the dislocation movements, TEM observations inevitably involve thin film effects^4,10,11 and, thus, cannot represent the dislocation behavior inside the bulk sample. We are using acoustic emission (AE) in this study, where AE stems from elastic waves generated by sudden structural changes within the material, which is an efficient way to monitor the deformation mechanisms on a macroscopic spatial and temporal scale.¹² Over the last few years, AE research has progressed to comprehend the underlying mechanisms in metals and alloys, such as dislocation motion in single crystals,^13–15 detwinning/twinning in porous Ti–Ni alloy,¹⁶ porous collapses in Mg–Ho,¹⁷ and crack propagation in steel.¹⁸ Recent results show that the avalanche mechanisms in metals and alloys, e.g., the dislocation movements, dislocation entanglements, detwinning–twining processes, and martensitic transformations, can be distinguished by a detailed profile analysis of AE signals.^19–21 We are using this technique for the analysis of and the comparison between one HEA and one typical stainless steel and focus on the mobility of dislocations as reflected by the AE signals which are mainly influenced by their pinning/depinning behavior. These AE signals stem from collisions between dislocations or from interactions between dislocations and defects. The simple creep of dislocations emits not enough noise to be measurable in an AE experiment.

In order to identify AE from dislocations, we have measured the relevant AE in HEA and 316L stainless steel and compared the normalized wave profiles of these signals. We demonstrate in this paper that the dynamical properties of dislocation (e.g., correlations energy∼amplitude, energy∼duration, amplitude∼duration, and the distribution of AE signals duration) are rather similar in both 316L stainless steel (hereafter denoted as 316L SS) and Fe₄₀Mn₄₀Co₁₀Cr₁₀ HEA, while the dislocation AE signals in Fe₄₀Mn₄₀Co₁₀Cr₁₀ HEA show long duration and a bias toward shorter waiting times for subsequent dislocation movements. The similarity between the AE profiles in 316L SS and HEA nevertheless indicates that despite strong similarities in their dislocation pinning/depinning behavior, the complex energy landscape in HEA leads to other, more subtle AE differences from “ordinary” metals.

An ingot of Fe₄₀Mn₄₀Co₁₀Cr₁₀ HEA (hereafter, named as FeMnCoCr HEA) was prepared by electromagnetic levitation melting under a high-purity argon atmosphere. All raw materials had a purity higher than 99% (wt. %). The as-cast ingot was first homogenized at 1373 K for 6h and hot forged to a 30 × 40 × 850 mm³ square bar, followed by water quenching. Dog-bone shaped samples with the gauge dimensions of 12 × 4 × 2 mm³ were taken from the annealed bar. An ingot cold rolled 316L SS sample (the same material was used in Ref. 20) was cut by a wire-saw with the same sample size as the FeMnCoCr HEA. The composition of the 316L stainless steel is 0.02C–16.4Cr–10.5Ni–1.4Mn–2.1Mo–0.5Si (wt. %). Uniaxial tensile experiments were performed at room temperature with a rate of 10⁻² mm/min on an Instron 5969 Universal Testing System. Electron backscatter diffraction (EBSD) measurements were conducted to characterize the evolution of the microstructure at different strains. The average grain size of both samples was 50 μm.

AE signals during the stretching process were monitored by a piezoelectric sensor (Vallen-Systeme GmbH) with a frequency range of 200–800 kHz. The recording signals were first pre-amplified by 40 dB and then transferred for waveform analysis using an AMSY-6 AE-measurement system (Vallen-Systeme GmbH) with a frequency range of 95–850 kHz. An AE signal is defined as a “burst” in the noise spectrum. A threshold of 23.3 dB was used to filter out the background noise. The threshold was determined by a tension experiment where the metal sample was replaced by rubber. The detectors were attached to the rubber sample.^17,20 This experiment allowed us to measure the noise of the experimental arrangement without the AE effect of the metal sample. For details, see Refs. 17 and 20.

The engineering stress–strain curve in Fig. 1 of FeMnCoCr HEA shows good mechanical properties. The ultimate tensile strength is higher than 500 MPa and the elongation is over 60% [red line in Fig. 1(a)]. The sample emits acoustic signals, AE, during stretching [Fig. 1(a)] which were separated into two populations, as shown in yellow and purple in Fig. 1. The two jerk populations differ in their energy exponents which becomes obvious when the two jerk populations are analyzed separately using the maximum likelihood method (ML).²² For FeMnCoCr HEA, both populations are power law distributed with two different exponents, namely, ɛ = 1.7 (yellow) and ɛ = 2.0 [purple in Fig. 1(b)]. A similar exponent was already found for dislocation movements in 316L SS with ɛ = 1.6,^19,20 which is the typical field-integrated mean-field value in AE.¹² 

We first proceed to a more detailed analysis of these AE signals. The correlation between their energy and amplitude [E(A)], amplitude and duration [A(D)], and energy and duration [E(D)] of FeMnCoCr HEA are shown in Figs. 2(a)–2(c). They follow the predicted power law correlations for long durations. Pairwise analysis of the three AE parameters energy (E), amplitude (A), and duration (D) shows multiple curves for the same abscissa (Fig. 2). The analysis of two branches is schematically shown in the inset of Fig. 2(a). We first draw a line with a slope of 0.5 in the log–log plot of E(A) and then project the data points in the E(A) curve to this line. The distribution of the projected points follows a double Gaussian distribution, in which each Gaussian distribution represents one branch of the AE signals. This phenomenon is called “multi-branching”²³ and is clearly seen in FeMnCoCr HEA. Similar multi-branch scaling was identified in 316L SS,^19,20 which helped to classify dislocation movements, detwining–twinning, and martensitic transformation during deformation. The reader is referred to this analysis for further Refs.19, 20, and 23, here we focus on the FeMnCoCr HEA.

Multi-branching implies multivalued functions like E = SA^219,20,23 with an exponent near 2 for all curves^19,20,23 while the prefactor S depends on the actual AE species which emits the signal. Multi-branching also indicates that several mechanisms (populations) with different values of S appear in the same material. In addition, the exponents are expected from theory to be between 2 and 3 depending on the AE wave profiles.²⁴ The exponents are defined at the limit of large amplitudes, while the small amplitudes are indeed different for the purple line (detwinning/twining). Here, the curves are bent which means that the power law is not fully obeyed for small amplitudes. The physical reason is, most likely, that “mild” avalanches with different waveforms overlap with the “wild” avalanches which we consider here. We do not see any deviations from the power law dependence with an exponent equal to 2 for the large amplitudes.

We then separate the two populations in FeMnCoCr HEA as shown in Fig. 1(a) where signals shown in yellow are mainly centered close to the yield point, while the signals in purple are more widely distributed. The observed multivalued E(A), A(D), and E(D) correlations in FeMnCoCr HEA (Fig. 2) provide us with an additional fingerprint for the detection of the coincidence between dislocation movement and detwinning/twinning during the deformation of the FeMnCoCr HEA.

To determine the deformation modes, we characterize the evolution of the microstructure by EBSD for the initial sample, a sample with 40% strain, and a fractured sample (Fig. 3). Figures 3(a)–3(c) show the evolution of the twin patterns and the geometrically necessary dislocation (GND) 2D maps for three different deformation states of the material. The initial sample of FeMnCoCr HEA contains 57.8% growth twins, the twin concentration reduces to 12.7% after 40% deformation, and increases to 47.6% at the fracture point. The EBSD analysis of twins demonstrates that a detwinning–twinning process occurred during the entire deformation. The calculations of the EBSD GND maps are based on the quantification of the crystal orientation spread within individual grains that arise because of dislocation accumulation during heterogeneous plastic deformation.^25,26 Figure 3(a) shows a very small local mis-orientation value which proves that the as-received material was fully recrystallized. It indicates a low dislocation density for the initial sample (with average GND density value²⁷ of around 4 × 10¹³ m⁻²). When the sample is deformed to 40% plastic strain [Fig. 3(b)], the average dislocation density is 2.1 × 10¹⁴ m⁻² and increases to 3.5 × 10¹⁴ m⁻² near the failure point [Fig. 3(c)]. We can, hence, conclude that the avalanche mechanism in FeMnCoCr HEAs is related to dislocation movements and detwinning/twinning.

We are now able to distinguish between dislocation and twinning movements in FeMnCoCr HEA. Avalanche dynamics of dislocations are usually power law distributed with exponents of 1.6 for nickel,²⁸ molybdenum,²⁹ steel,^19,20 and a dislocation slip model.³⁰ This exponent coincides with the expected field-integrated mean-field value 1.6.³⁰ Power law exponents for detwinning/twinning avalanches are usually greater, ca. 2 in Ti–Ni alloy¹⁶ and >2 in low Ni content-316L steel.¹⁹ Based on these classifications, we identify the purple AE signals (for FeMnCoCr HEA) in Fig. 1 as related to detwinning/twinning with an exponent ɛ ∼ 2.0, while the AE signals in yellow (for FeMnCoCr HEA) in Figs. 1 and 2 as related to dislocation movements with an exponent ɛ = 1.7. The exponent ɛ = 1.7 is similar to that of dislocation movements (including nucleation and growth) in 316L SS with ɛ = 1.6,²⁰ so we conclude that the dislocation AE signals of FeMnCoCr HEA is attributed to nucleation and growth of dislocations, not to dislocation entanglement with an exponent ɛ = 1.4.¹⁹. Previous studies have shown that the slip size distribution in HEA nano pillars, such as Al_0.1CoCrFeNi³¹ and Al_0.3CoCrFeNi,³² follow a power law with an exponential decay function for single crystals.³³ This is not seen in FeMnCoCr HEA which follows a power law distribution without any noticeable exponential cut-off effects.

Based on these results, we can now compare AE signals of dislocation movements in FeMnCoCr HEA and 316L SS, where the E(A), A(D), and E(D) scaling of dislocations in Figs. 2(d)–2(f). Figure 2(d) shows that the E ∼ A² curves of FeMnCoCr HEA and 316L SS overlap. Figures 2(e) and 2(f) show that the signals are more abundant in FeMnCoCr HEA than in 316L SS. For signals with short durations (region I, duration <20 μs) and long durations (region III, duration >3000 μs), only one event in regime I and no events in regime III have been found in 316L SS. The duration probability density function (PDF), in Fig. 4, is defined as the number of events in a pre-determined duration range per bin width, divided by the total number of events. The PDFs of AE duration of FeMnCoCr HEA and 316L SS in Fig. 4 show that very short and very long durations (regions I and III) occur only in FeMnCoCr HEA. In contrast, the curves are identical in middle region II.

The analyses on the wave profiles for dislocations in FeMnCoCr HEA and 316L SS [Figs. 5(a) and 5(b)] also show that the duration of dislocation movements in FeMnCoCr HEA is longer than in 316L SS. For example, the duration of dislocations in FeMnCoCr HEA [Fig. 5(a)] at amplitude A = 200 μV is about three times longer than dislocations in 316L SS [Fig. 5(b)]. The difference between dislocation–movement signals of FeMnCoCr HEA and 316L SS is equally seen in the statistical distribution of normalized amplitude squared (A²) waveforms [Fig. 5(c)]. In this analysis, all maximum amplitudes for each acoustic signal are normalized to unity. All A² curves decrease after a maximum amplitude near the beginning of each time sequence in a roughly exponential fashion. A² of dislocation movement in FeMnCoCr HEA [yellow points in Fig. 5(c)] shows a much slower decay compared with the dislocations in 316L SS [red points in Fig. 5(c)], indicating that dislocation movements last longer during plastic deformation in FeMnCoCr HEA than in 316L SS.

Further differences are seen in the waiting time distributions¹⁷ in Fig. 6. All waiting times are power law distributed for short waiting times (<0.5 s for 316L SS and <0.1 s for FeMnCoCr HEA) and long waiting times (>20 s for 316L SS and FeMnCoCr HEA). The slope of the power law is 0.89 (<0.5 s region) and 1.09 (>20 s region) for 316L SS, which is close to the un-rescaled value of unity.³⁴ In contrast, the slope of the power law for waiting times in FeMnCoCr HEA is higher with 1.3 (<0.1 s region) and 1.42(>10 s region). Such two power law regimes with different slopes for short and long times are commonly observed for creep³⁴ and force scenarios.^35–38 The theoretical background^39,40 is based on the interference of exponential waiting times and background signals with the avalanches measured in the experiment. Short waiting times show exponents of 1 – ν while long waiting times show 2 + ξ with ν, ξ < 1.³⁶ The difference between the two materials indicates that the steeper slope of the waiting time in FeMnCoCr HEA favors short waiting times and disfavors long waiting times. This is true for the immediate aftershocks and for the continuum region of long waiting times.⁴¹ Nevertheless, the probability distribution of aftershocks, as expressed by the Omori law,³⁶ is very similar for both materials (Fig. 7). The Omori exponent is −0.98 and randomization is weak even after >10 s. While the Omori law dependences in Fig. 7 are the same for both materials, the actual waiting times between avalanches are biased against shorter times in HEA. Dephasing of avalanches leads to deviations of avalanches probabilities at typical times of greater than 1 s. This is often related to the mixing of aftershocks with new mainshocks. The deviations of the Omori behavior in our samples are much weaker than usually observed (e.g., Refs. 36, 42, and 43).

We noted that recent work implied that the duration of the AE avalanches can be distorted if the true duration time is comparable to the attenuation time of acoustic waves in the medium. This wave profile model is based on single avalanche behavior. In contrast, AE signals of dislocation movement are generated by various overlapping local avalanches^18,19,23 so it was not possible to define the “average” waveform with sufficient accuracy. Improved mathematic models for multiple scaling are urgently needed.

1. Acoustic Emission Spectra of Fe₄₀Mn₄₀Co₁₀Cr₁₀ HEA show two branches of the energy ∼ amplitude correlation, whereby one avalanche process relates to the movement of dislocations and the other to detwinning/twinning processes. The energy exponent of dislocation movements is 1.7 for both materials which demonstrate that dislocation movements in HEAs are very similar to other metals when observed by AE.

2. Differences emerge from dislocation related AE signals in Fe₄₀Mn₄₀Co₁₀Cr₁₀ HEA because they show systematically longer durations and a bias toward faster waiting times for subsequent dislocation movements compared with 316L stainless steel.

3. The aftershock rates for dislocation–movement signals are nevertheless very similar in 316L stainless steel and in Fe₄₀Mn₄₀Co₁₀Cr₁₀ HEA so that no differentiation is possible based on the Omori law.

This work was supported by the National Natural Science Foundation of China (NNSFC) (No. 51931004) and 111 Project 2.0 (No. BP2018008). E.K.H.S. is grateful to EPSRC (No. EP/P024904/1) and H2020 Marie-Sklodowska-Curie Actions (No. 861153).

The authors have no conflicts to disclose.

Yan Chen: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Ke Tang: Investigation (equal); Methodology (equal). Boyuan Gou: Methodology (equal); Validation (equal); Visualization (equal). Feng Jiang: Methodology (equal). Xiangdong Ding: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Ekhard K. H. Salje: Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.