**A Review of the Serrated-Flow Phenomenon and Its Role in the Deformation Behavior of High-Entropy Alloys**

**Jamieson Brechtl 1, Shuying Chen 2, Chanho Lee 2, Yunzhu Shi 3, Rui Feng 2, Xie Xie 2, David Hamblin 2, Anne M. Coleman 2, Bradley Straka 2, Hugh Shortt 2, R. Jackson Spurling <sup>2</sup> and Peter K. Liaw 2,\***


Received: 10 July 2020; Accepted: 11 August 2020; Published: 13 August 2020

**Abstract:** High-entropy alloys (HEAs) are a novel class of alloys that have many desirable properties. The serrated flow that occurs in high-entropy alloys during mechanical deformation is an important phenomenon since it can lead to significant changes in the microstructure of the alloy. In this article, we review the recent findings on the serration behavior in a variety of high-entropy alloys. Relationships among the serrated flow behavior, composition, microstructure, and testing condition are explored. Importantly, the mechanical-testing type (compression/tension), testing temperature, applied strain rate, and serration type for certain high-entropy alloys are summarized. The literature reveals that the serrated flow can be affected by experimental conditions such as the strain rate and test temperature. Furthermore, this type of phenomenon has been successfully modeled and analyzed, using several different types of analytical methods, including the mean-field theory formalism and the complexity-analysis technique. Importantly, the results of the analyses show that the serrated flow in HEAs consists of complex dynamical behavior. It is anticipated that this review will provide some useful and clarifying information regarding the serrated-flow mechanisms in this material system. Finally, suggestions for future research directions in this field are proposed, such as the effects of irradiation, additives (such as C and Al), the presence of nanoparticles, and twinning on the serrated flow behavior in HEAs.

**Keywords:** high-entropy alloys; serrated flow; microstructure; data analysis

#### **1. Introduction**

#### *1.1. High-Entropy Alloys*

High-entropy alloys (HEAs) are an important class of materials that emerged in the earlier part of this century [1,2]. Background investigations on HEAs officially began circa 1996, which resulted in five publications on the subject in 2004 [1–5]. HEAs are a type of complex concentrated alloy that typically consists of five or more (some HEAs with four elements have also been reported) principal elements with atomic concentrations that vary from 5 to 35 atomic percent (at.%) [1,2,6–8]. This combination of elements leads to a high mixing entropy that favors disordered solid solutions at

elevated temperatures [9,10]. An HEA is also defined as having a configurational entropy (Δ*Sconf*) of 1.5*R* or more, where *R* is the ideal gas constant [11]. The configurational entropy of an alloy can be calculated with Equation (1) [12]:

$$
\Delta S\_{conf} = -R \sum\_{i}^{n} \frac{1}{n\_i} \text{Ln} \frac{1}{n\_i} \tag{1}
$$

where *R* is the ideal gas constant, and 1/*ni* is the atomic fraction of the ith element in the alloy. In the case of an equiatomic alloy, 1/*ni* can be written as *Xi*, where *Xi* equals the atomic percentage of the element, which gives Δ*Sconf* = *RLnXi*. Therefore, the primary elements in HEAs, in principle, can produce an alloy that has more than twice the configurational entropy, as compared to conventional alloys (0–0.7*R*) [13]. This relatively high entropy, as well as not having one primary element, results in several effects that account for the unique properties exhibited by HEAs, such as the high-entropy effect, severe lattice distortion, sluggish diffusion, and the cocktail effect [13,14].

It is commonly accepted that the high-entropy effect explains why HEAs often are observed to form only one or two primary phases, while the Gibbs's phase rule indicates that they could form five or more phases [13–16]. This effect also corresponds to a relatively lower Gibbs free energy of mixing for the solid-solution phases in the alloy, as determined by the following equation [13–15]:

$$
\Delta G\_{mix} = \Delta H\_{mix} - T\Delta S\_{mix} \tag{2}
$$

where Δ*Gmix*, Δ*Hmix*, *T*, and Δ*Smix* are the Gibbs free energy of mixing, the enthalpy of mixing, the temperature, and the entropy of mixing, respectively. Since the intermetallics consist of highly ordered structures, the increase in Δ*Smix* is often sufficient to lower the Δ*Gmix* of the solid-solution phases to below that of the Δ*Gmix* of the corresponding intermetallic phases in HEAs [13,15]. Consequently, solid-solution phases are more energetically favorable than intermetallic phases in many HEAs. However, there is still controversy regarding the true effect of high entropy on the solid-solution stability of HEAs. For instance, recent research suggests that the mutual solubility of constituent elements in the primary structure of the HEAs is the contributing factor for only one or two primary phases in the matrix [15]. Therefore, more investigative work is needed to determine the true mechanism for the high solid-solution stability of HEAs.

Another factor contributing to the unique properties of HEAs is known as the lattice distortion effect, which corresponds to the random distribution of elements in the matrix [17,18]. This random distribution leads to a large amount of lattice distortion due to a lower adherence to the Hume-Rothery rules than conventional alloys [13]. Figure 1 compares the lattice structure of a monoatomic metal and a multiatomic alloy with an HEA [14]. It is apparent from the figure that the lattice of the HEA is significantly more distorted, as compared to the other metals. The severe lattice distortion is thought to enhance the strength, hardness, and thermal and electrical resistance of HEAs by disrupting dislocation motion and increasing the mean free path of electrons and phonons [13,14,17].

Furthermore, the random distribution of elements (and lattice-potential energy) in the alloy may also be a contributing factor to the sluggish diffusion of atoms and vacancies to lattice sites of lower energies [13,19]. This sluggish diffusion effect results in the potential beneficial properties of slower grain growth [2], exceptional elevated-temperature stability [20], high-temperature strength [21], and a higher recrystallization temperature. However, there is still some debate regarding the validity of the sluggish-diffusion concept [18,22–25].

The last effect is the cocktail effect, which states that inter-element interactions give rise to unusual behaviors, as well as average composite properties (rule-of-mixtures) [26]. It should also be mentioned that the composite properties arise from the mutual interactions among the different elements, as well as the severe lattice distortion [27]. This phenomenon has been described as an overall effect that arises from the composition, structure, and microstructure that can result in enhanced properties [26,28]. For example, the cocktail effect has been cited as a contributing factor for the high magnetization and electrical resistance, high strength, and excellent plasticity of a FeCoNi(AlSi)0.2 HEA [29].

**Figure 1.** An illustration of the lattice distortion in an element, a conventional alloy, and a high-entropy alloy (adapted from Reference [14]).

In terms of their properties, HEAs exhibit good corrosion resistance [30–35], high fatigue resistance [36–41], high strength and fracture toughness [17,42–47], notable resistance to shear failure [48], decent irradiation stability [49–58], biocompatibility [59–61], good cryogenic tensile and fracture properties [46,62–65], and excellent wear resistance [66,67]. Figure 2 presents an Ashby plot that compares the fracture toughness and yield strength of HEAs with several other material systems [46,47]. As can be seen, the HEAs have fracture-toughness values that exceed almost every material on the graph. Furthermore, this alloy has comparable yield strength values with those of stainless steels. Figure 3 displays the endurance limit and ultimate tensile strength for HEAs and conventional alloys [68]. Except for the bulk metallic glass, the HEA exhibits comparatively higher values, as compared to the other alloy systems.

**Figure 2.** An Ashby plot that compares the fracture toughness and yield strength of HEAs with several other materials and bulk metallic glasses (reproduced from Reference [46] with permission).

**Figure 3.** Endurance limits of the Al0.5CoCrCuFeNi HEA as a function of the ultimate tensile strength, compared with other structural materials and bulk metallic glasses (BMGs) (reproduced from Reference [68] with permission).

#### *1.2. The Serrated-Flow Phenomenon*

The *Concise Oxford English Dictionary* defines a serration as "a tooth or point of a serrated edge or surface" [69]. A typical morphology of serrations is displayed in Figure 4a,b, which features the high-magnification light micrographs of the teeth for two species of shark [70]. The figure displays the two main types of serrations: large primary serrations and small, interspersed secondary serrations. Serrations have also been observed in different phenomena, such as the Barkhausen noise in magnetic materials [8,71–74] (see Figure 5), crackling noise during earthquakes [75–78], discrete strain bursts in nanocrystals [76], various economic indices [79–83], and neuronal avalanches that occur during the operation of the brain's network (see Figure 6) [84–92].

**Figure 4.** High-magnification light micrographs of the (**a**) primary and secondary serrations of the distal heel of the *Galeocerdo cuvier* shark tooth (scale bar of 2 mm), and (**b**) the primary serrations of the cutting edge of the *Prionace glauca* shark tooth (scale bar of 2 mm) (reproduced from Reference [70] with permission).

**Figure 5.** Magnetic Barkhausen noise, as illustrated by irreversible discontinuities in the magnetization, M, as the alternative current magnetic field, H, is varied (adapted from Reference [71]).

**Figure 6.** Electroencephalographic recordings during different brain states in humans, which include awake with eyes open, awake with eyes closed, sleep stage 2, sleep stage 4, rapid eye movement (REM) sleep state, and epileptic seizure (adapted from Reference [92]).

In addition to the above phenomena, serrations (or the serrated flow) can occur during the dynamic strain aging (DSA) of a material while undergoing mechanical testing. The serrated flow is of great engineering significance since it corresponds to substantial changes in the microstructure and can have deleterious effects on the mechanical behavior of a material [8,9,93–95]. For example, the plastic instabilities that occur during the serrated flow can lead to a loss in the ductility of the material, as well as an unacceptable surface quality [95]. The study of this type of behavior is important for the production of metal-alloy parts with smooth surfaces and elucidates the fundamental understanding of microscale plastic deformation [96]. The serrated-flow behavior that occurs during material deformation has been reported as far back as the 1970s [97]. With respect to the compression and tension testing, serrations are typically characterized by the fluctuations in the corresponding stress–strain graph [8]. Additionally, the serrated flow has also been observed in the form of pop-ins during nanoindentation testing, which are characterized by displacement bursts in the load vs. displacement graph. A variety of material systems have exhibited this type of behavior [8,9,98,99], including single and polycrystalline metals [100–103], Al alloys [8,104–115], Cu alloys [116], V alloys [117,118], steels [8,119–131], granular systems [132], bulk metallic glasses (BMGs) [8,97,132–151], and medium-entropy and high-entropy alloys [8,9,68,94,152–181]. It is thought that the standard interpretations can be applied to HEAs since a similar weakening effect is observed during the DSA, where solute atoms diffuse and lock dislocations, thereby immobilizing them [153,180].

A graph listing the number of journal publications that feature the serrated flow phenomenon in HEAs for years 2011 through July 2020 is displayed in Figure 7. The results featured in the graph were found via the Web of Science database. As can be seen, the number of articles continually increased per year. Before moving on, it should be mentioned that the numbers for 2020 are still ongoing.

**Figure 7.** The number of publications that discuss the serrated-flow phenomenon in high-entropy alloys for years ranging from 2011 to July 2020, as determined from the Web of Science publication database.

#### *1.3. Mechanisms of Serrated Flow*

Several mechanisms can contribute to the serrated flow in materials during mechanical deformation, including the dislocation pinning by solute atoms, mechanical twinning, grain boundaries, order–disorder phase transformations, phase transformations induced by stress and strain, and yielding across fracture surfaces in brittle materials [182–186]. In terms of dislocation pinning by solute atoms, both interstitial and substitutional solutes can participate in the dislocation-pinning process. With respect to interstitial solute pinning, Cottrell and Bilby provided one of the earliest theoretical explanations for this phenomenon [187]. In their work, they suggested that at sufficiently high temperatures interstitial solute atoms (such as C or N) diffuse toward dislocation cores, where they impede their motion [186]. As mentioned above, substitutional solutes can also participate in dislocation pinning. Here, substitutional solutes diffuse via a vacancy mechanism during plastic deformation [188]. Solute atoms that are larger than the parent metal diffuse below the dislocation line, where the strains are tensile [189]. On the other hand, solutes that are smaller than the matrix atoms migrate to sites above the dislocation line, where compressive strains occur. Consequently, these strain fields pin the dislocation in place. Once the stress becomes sufficiently large, the dislocation can break free and then migrate until it is pinned again. If the pinning and unpinning processes are repeated cyclically, it leads to a DSA effect [190].

Figure 8 presents an illustration of the pinning–unpinning cycle [152]. Dislocations are pinned by diffusion solute atoms (labeled I in the figure). As the stress increases, the dislocation remains pinned (section II) until a critical stress is reached, where there is sufficient energy for the dislocation to break free. The escape of the dislocation corresponds to the stress drop in the stress–strain curve (marked by III in the figure). After escaping the solute atoms, the dislocation moves until it is pinned again by a

migrating solute atom (designated IV in the figure). This pinning is accompanied by an increase in the stress, and the process then repeats.

**Figure 8.** An illustration of the serration process that occurs due to the pinning and unpinning of solute atoms by dislocations. Each numbered section corresponds to the different interactions between the solutes and dislocations during the serration event as follows: (**I**) the solute atoms pin a moving dislocation, which corresponds to the onset of the serration; (**II**) the dislocation remains pinned as the stress increases; (**III**) enough stress builds up, and the dislocation escapes; and (**IV**) the solute atoms catch and re-pin the dislocation (adapted from Reference [152]).

It is widely accepted that, in HEAs, any atom can act as a solute atom [1]. However, there are cases in which only certain atoms in the matrix can participate in dislocation locking. In fact, some investigations have revealed that only certain additives such as Al may hinder dislocation motion in HEAs during deformation [94,180,191]. For instance, Yasuda et al. compared the mechanical behavior of Al0.3CoCrFeNi and CoCrFeNi HEAs during DSA [191]. It was reported that the Al-containing HEA exhibited serrations, whereas the latter specimen did not. It was hypothesized that the creation of Al-containing solute atmospheres near a moving dislocation core results in an increase of the frictional stress on the dislocations, thus resulting in serrations. Niu et al. also found a similar trend when investigating the DSA in Al0.5CoCrFeNi and CoCrFeNi HEAs [94]. In a study of carbon-doped CoCrFeMnNi HEAs, serrated flows at room temperature (RT) occurred due to interactions between carbon impurities and dislocations, as well as an increase in the stacking fault (SF) energy due to the presence of the carbon [192].

In addition to solute atoms, nanoparticles and phase structures may also participate in the dislocation-locking process during the serrated flow in HEAs. For instance, Chen et al. reported that L12 particles could provide an obstacle for the mobile dislocations during compression in an Al0.5CoCrCuFeNi HEA [155]. Transmission electron microscopy (TEM) characterization revealed that L12 nanoparticles were present in the matrix for the samples tested at 500 and 600 ◦C. In another study [193], it has been reported that, in CoCrFeNiMnVx HEAs, sigma-phase particles and dendrite boundaries can provide a barrier to dislocation motion, resulting in serrations. Twinning, which occurs at cryogenic temperatures, is another mechanism that can lead to the serrated flow in HEAs by hindering dislocation motion at twin boundaries [8,9,122,123,159]. In Reference [159], it was found that, in the CoCrFeNi HEA, twinning at 4.2 K was accompanied by a phase transformation from face-centered cubic (FCC) to hexagonal close-packed (HCP) structures. In other studies [194,195], the lack of observed serrations during mechanical testing of FeCrCoNiMnV and FeNiCoCuV HEAs indicates that the presence of secondary phase precipitates within the solid solution grains may be required for the serrated flow to occur. However, another investigation on an FeCoNiCuCr HEA found that serrations did not occur, even though there were intermetallic compound precipitates present in the matrix [196].

In contrast to crystalline alloys such as HEAs and steels, the serrated flow in BMGs occurs via different mechanisms. Table 1 compares some of the different mechanisms responsible for the serrated flow in crystalline and amorphous alloys. These mechanisms include the flow defect unit and liquid-like site agglomeration, and shear band formation and propagation [197–201]. The activation and percolation of the flow units are typically caused by an applied stress or elevated temperature [202]. In terms of the shear bands, there is a general agreement that their initiation is caused by local structural softening, owing to free volume generation [203]. After the initiation of a shear band, it propagates until the stored elastic energy is sufficiently released, leading to its arrest [198].



Figure 9 displays an illustration of the serration process that occurs due to the flow defect accumulation and subsequent shear-band initiation and propagation in a BMG [197]. At the beginning of a serration event, the applied stress leads to stress concentrations that arise from the modulus difference between the flow units and the glassy matrix. Consequently, these stress concentrations activate the flow units (labeled I in the figure). With an increase in the applied stress, the liquid-like regions that are composed of the flow defects grow and coalesce with adjacent regions (labeled II in the figure). Once the applied stress reaches a critical value, which corresponds to the local peak in the stress (labeled III in the graph), a shear band is initiated in the coalesced liquid-like region. Consequently, the shear band activates and propagates, leading to a drop in the stress that corresponds to a dissipation of the stored elastic energy. Once a sufficient amount of elastic energy is dissipated, the shear band is arrested. This arrest is accompanied by a recovery of the liquid-like layer and subsequent restoration of the solid-like (glassy) matrix (labeled IV in the figure) in the shear plane [198]. Once the glassy matrix is fully restored, the process repeats.

#### *1.4. Types of Serrated Flow*

Serrations have been categorized into five distinct types, which have been labeled as A, B, C, D, and E [8,9,153,182,204]. Figure 10 displays the five types of serrations, and as can be observed, each type of serrations has distinct characteristics. Type-A serrations occur in a periodic fashion in which the curve rises above the general level of stress values before there is a sharp decrease in the stress. Type-B serrations typically fluctuate above the general level of stress with a higher frequency, as compared to Type-A serrations. Type-C serrations are characterized by yield drops that occur below the general level of stress. Furthermore, the elapsed time between stress drops is randomly distributed. In contrast to the serration types previously described, Type-D serrations consist of a stair-stepping pattern in the stress vs. strain graph. Finally, Type-E serrations are characterized by fluctuations that display an irregular pattern and occur with little or no working hardening during band propagation [182]. It should also be noted that combinations of serration types can occur during the serrated flow, such as A + B and B + C [204]. As is discussed later, in Section 1.5, the serration type is typically dependent on the applied strain rate or test temperature during either tension or compression

testing [8,9,133,153]. For instance, serrations have been observed to transition from Type-A to Type-C behavior when the testing temperature is increased [180].

**Figure 9.** An illustration of the serration process that occurs due to the flow defect accumulation in a BMG during compression. Each numbered section corresponds to the evolution of the flow unit and glassy matrix during the serration event as follows: (**I**) activation of flow units in liquid-like regions under an applied stress, (**II**) growth and coalescence of the liquid-like cites with adjacent cites with an increase in stress, (**III**) initiation of a shear band when a critical stress is reached, and (**IV**) a drop in the stress that is accompanied by shear-band arrest and subsequent restoration of the glassy matrix in the shear plane (adapted from Reference [197]).

**Figure 10.** The five types of serrations that can occur during mechanical testing (adapted from References [180,182].

#### *1.5. Factors A*ff*ecting Serrated Flow*

The serration behavior can be affected by factors such as the composition, temperature, irradiation damage, and strain rate. For example, Cai et al. [205] observed that, in Ni-based superalloys, the γ precipitates could lead to the aggregation of solute atoms around moving dislocations, thereby enhancing serrations in the material. Similarly, Chandravathi et al. found that the serration behavior in a 9Cr-1Mo steel was also born out of DSA, as the serrated flow was observed in all variations of microstructures, except in certain conditions where increased soaking temperatures above 1423 K led to elevated hardness due to the larger austenitic grain size [206]. Bayramin et al. determined that microstructure played a primary role in the serration behavior of a dual-phase low-carbon steel [207]. They found that the DSA was more pronounced in a sample with a higher concentration of the ferrite phase present. From the results, they hypothesized that the pronounced DSA was attributed to a dislocation density, which increased with increasing strain, such that the dislocation pinning was more prevalent in the ferrite microstructures. Such behavior has been observed in prior studies [207–209].

Temperature has also been reported to play an important role in the serration behavior of materials [8,117,153,210–214]. With an increase in temperature, the mobility of diffusing solute atoms increases, allowing them to pin dislocations at a faster rate. However, once a limiting temperature is exceeded, serrations do not occur due to factors such as a prohibitively high critical strain, smaller critical plastic strain for the termination of serrations, or exceedingly large atomic thermal vibrations, which prevent solutes from settling down the effective locking [8,153]. Conversely, a decrease in the temperature reduces the migration speed of the solute atoms, thereby diminishing their ability to lock dislocations. Once the temperature becomes low enough, the solutes can no longer catch and pin dislocations [215].

The increase in the pinning rate of dislocations (due to an increase in temperature) results in a significant change of the dynamical behavior of the stress and corresponding serrated flow curves (and serration type). For instance, Sakthivel et al. observed that when Hastelloy X underwent tension testing, the serration type changed with an increase in temperature [211]. More specifically, the serrated flow evolved from Type-A at 400 ◦C to Type-A + Type-B at 500–550 ◦C, followed by Type-B at 600 ◦C, and then finally to Type-C at 650 ◦C. It was also reported that the height of serrations in the stress–strain curve increased with an increase in the temperature. The critical plastic strain, which is the strain associated with the onset of serrations, was found to decrease with an increase in the test temperature.

It has also been reported that the combinations of temperature and irradiation damage can affect the serrated-flow behavior. Rowcliffe et al. reported that temperature and neutron irradiation led to a change in the serrated-flow behavior of the V-4Cr-4Ti alloy during tension [117]. Figure 11a,b presents the engineering stress vs. strain behavior for the unirradiated and irradiated alloys during tension at temperatures ranging from 20 to 850 ◦C and a strain rate of 10−<sup>3</sup> s−1. For the irradiation conditions, samples were bombarded by neutrons to doses of 0.5 displacements per atom (dpa) at temperatures ranging from 160 to 420 ◦C, and also to a dose of 0.1 dpa at 505 ◦C. It was observed that, for the unirradiated samples, pronounced serrations occurred for temperatures of 300–700 ◦C. Below 300 ◦C, serrations were not observed, since the interstitial solutes were relatively immobile. For temperatures above 700 ◦C, the magnitude and frequency of the serrations significantly declined, while at 800 ◦C, serrations were restricted to the Lüders region. At 850 ◦C, there were no observable serrations. As for the irradiated samples, the serrated flow was apparent for temperatures ≥420 ◦C. Furthermore, as compared to the unirradiated samples, there were significantly fewer and less pronounced serrations in the irradiated specimens. This reduction in the serrated flow may have been caused by a loss of pinning solute atoms that become trapped in either small dislocation loops (T ≤ 400 ◦C) or fairly coarse {0 0 1} defects (T > 420 ◦C) that formed during irradiation.

**Figure 11.** (**a**) Tensile curves for the unirradiated annealed V-4Cr-4Ti at a strain rate of 10−<sup>3</sup> s−<sup>1</sup> and (**b**) tensile curves for V-4Cr-4Ti irradiated to 0.5 dpa at 110–420 ◦C and to 0.1 dpa at 505 ◦C. The curves were offset on the strain axis for clarity (reproduced from Reference [117] with permission).

In addition to temperature and irradiation, strain rate can also affect the serrated-flow behavior [8,211,212,216,217]. Typically, the critical strain for the onset of serrations increases with an increase in the strain rate [8]. However, there are cases where the critical strain will also increase with a decreasing strain rate [217]. In this scenario, the increase in the critical strain is due to an increase in the strength of solute obstacles, which reduces the ability of solutes to migrate and pin dislocations. Robinson et al. reported the dependence of the serrated flow during Lüders-band formation on the strain rate (as well as temperature) in an Al-5Mg (at.%) alloy [212]. The results of this study can be observed in Figure 12, and as can be seen, Type-C serrations tended to occur at higher temperatures, while Types A and B were found at lower temperatures. Furthermore, the strain rate at which a transition in the serration type occurred, i.e., Type-A to Type-A + Type-B transition increased with an increase in the temperature.

**Figure 12.** Temperature and strain rate dependence of the serrated flow in the Al-5Mg (at.%) alloy that occurs during Lüders-band formation (reproduced with permission from Reference [8,212].

Similar to conventional crystalline alloys, the serrated flow in HEAs can be affected by factors such as temperature and strain rate [152,153,155,180,193]. For example, Type-C serrations have been typically observed at relatively lower strain rates and higher temperatures, while the reverse is true for Type-A serrations [152,153,155,180]. In an Al0.5CoCrCuFeNi HEA, for example, the transition from Types A to C serrations with increasing temperature was attributed to the increase in the migration speed of the solute atoms that pin dislocations [153]. In this scenario, the increased speed allows the solutes to immediately pin the dislocation after it breaks free. After pinning, the stress increase is characterized by a downward bending curve that is typical of Type-C serrations. Furthermore, the formation of ordered L12 nanoprecipitates and the emergence of the B2 phase in the alloy during testing at 600 ◦C were thought to play a role in altering the characteristic behavior of the serrated flow in this HEA [180]. Tsai et al. hypothesized that increasing the number of solute atoms in an HEA can increase the abundance and the atomic-size differences that enhances the pinning force in the lattice [181]. Such a change results in an increase in the range of temperatures in which the serrated flow can occur.

In BMGs, factors such as the strain rate, temperature, and pre-annealing can also affect the serrated-flow dynamics [133,145,150,151,218]. It has been reported that, during compression, an increase in the strain rate leads to the creation of free volumes [218]. The creation of free volumes is accompanied by the formation of shear bands that can better accommodate the applied strain during compression. Jiang et al. observed that, with a decrease in the temperature, the time interval between stress peaks during serrations significantly decreased [145]. It has been observed that, in a Zr52.5Cu17.9Ni14.6Al10Ti5 BMG, pre-annealing at 300 ◦C for one week led to more irregular serration behavior during compression testing [150]. The increase in the irregularity of the serrated flow was thought to be due to the annihilation of excess free volumes (and associated defects) during the thermally induced structural relaxation of the alloy [219,220].

#### **2. Modeling and Analytical Techniques**

In the past, several different types of modeling and analytical techniques have been used to help quantify the serration behavior in a great amount of alloy systems. These techniques include statistical [8,9,76,105,133–135,152–155,221,222], multifractal [180,216,221,223,224], and complexity [104,119,131,180,225] methods. The above methods have been used to model and analyze the serrated flow, since it is believed that serration curves contain scaling laws, self-similarity, and complex dynamics [180].

#### *2.1. Complexity Modeling and Analysis*

Entropy-based complexity measurements have been employed to analyze a host of different fluctuating phenomena, including financial-time series, biological signals, and time-dependent serrated-flow behavior during mechanical testing [119,226–228]. In the present context, entropy increases with the degree of disorder (or irregularity) and is the maximum for completely random systems [226]. With respect to the serration behavior in HEAs, higher values are attributed to more complex dynamical behavior during the serrated flow. For biological systems, the complexity of physiological signals is related to the ability of a living system to adjust to an ever-changing environment [227]. For instance, a relatively lower complexity measure has been attributed to pathological conditions, such as the congestive heart failure [226,227,229]. Applying the above ideas to material systems, the complexity of the serrated flow may be a measure of a material's ability to adapt to an applied load [151].

#### 2.1.1. Approximate Entropy Algorithm

One such method that has been used to gauge the complexity of a time series is known as the approximate entropy (ApEn) technique [230]. Lower ApEn values are assigned to more regular time series, while higher ApEn values are assigned to more irregular, less predictable time series [227]. This method has been widely used in physiology and medicine [226,231]. However, there are issues with this method, namely that the technique inherently includes a bias toward regularity (or regularity bias), as it will count a self-match of vectors [232].

The procedure used to calculate the ApEn is discussed below. Given a time series of size, *N*, one constructs the dataset, *X*(*i*) = [*x*(*i*), *x*(*i* + 1), ... , *x*(*i* + *m* − 1)] (m denotes the embedding dimension), consisting of *N* data samples from a given time series with 1 ≤ *i* ≤ *N* − *m* + 1 [233]. Next, one determines the number of given vectors, *n<sup>m</sup> <sup>i</sup>* (*r*), which are within a distance, *d*[*X*(*i*), *X*(*j*)] < *r* = 0.2σ, where σ is the standard deviation of *X* [154]. This distance is defined as follows [230]:

$$d[X(i), X(j)] = \max\_{k=1,2,\ldots,m} \left\{ \left| \mathbf{x}(i+k-1) - \mathbf{x}(j+k-1) \right| \right\} \tag{3}$$

Next we define the following:

$$\mathcal{O}^{m}(r) = \frac{1}{N - m + 1} \sum\_{i=1}^{N-m+1} \mathcal{C}\_{i}^{m}(r) \tag{4}$$

where *Cm <sup>i</sup>* (*r*) <sup>=</sup> *<sup>n</sup><sup>m</sup> <sup>i</sup>* (*r*)/(*<sup>N</sup>* <sup>−</sup> *<sup>m</sup>* <sup>+</sup> 1) [230], and <sup>Φ</sup>*m*(*r*) represents the average degree of self-correlation [153]. The approximate entropy, ApEn, can now be defined for a fixed *m* and *r* as follows:

$$ApEn(m, r, N) = \mathcal{O}^m(r) - \mathcal{O}^{m+1}(r) \tag{5}$$

where <sup>Φ</sup>*m*(*r*) <sup>−</sup> <sup>Φ</sup>*<sup>m</sup>* <sup>+</sup> 1(*r*) signifies the degree of randomness of the sequences, *<sup>X</sup>*(*i*) (*<sup>i</sup>* <sup>=</sup> 1, 2, ... ).

#### 2.1.2. Refined Composite Multiscale Entropy Methods

A technique that does not suffer from the issue of regularity bias is known as the multiscale-entropy (MSE) algorithm [227]. Similar to the ApEn technique, this method has been used to derive the representations of a system's dynamics on different time scales [234]. The MSE method has been used to analyze the complexity of the serration behavior in different material systems, including the low-carbon steel, Al-Mg alloy, and HT-9 steel [104,119,225].

Previous studies have observed that the microstructural composition can affect the complexity of serrations. For example, Sarkar et al. compared the serrated-flow behavior of a low-carbon steel and an Al-Mg alloy [225]. The results of the MSE analysis revealed that the carbon steel produced serrations that had significantly larger complexity values than the Al-Mg alloy. The authors surmised that the complexity values were related to the type of solute atoms in the matrix. In the present context, the carbon solutes in the steel are able to interact with both screw and edge dislocations, whereas the Al-substitutional solutes could only interact with edge dislocations [225]. As such, the higher variety of interactions exhibited by the low-carbon steel resulted in more intricate behavior during the serrated flow and, hence, higher complexity values as compared to the Al-Mg alloy. It has also been reported that an increase in the concentration of carbon impurities in steels led to an increase in the complexity of the serrated flow during tension [131]. In addition to the microstructure, the complexity of the serrated flow has also been linked to different serration types [104,180]. For instance, Type-A and Type-B serrations have been associated with a higher degree of complexity, as compared to Type-C [104].

Due to issues regarding the accuracy of the MSE algorithm when applied to smaller datasets (*N* < 750), alternative techniques were later developed [227,234,235]. One such algorithm includes the refined composite entropy multiscale entropy (RCMSE) method [235], which has been used to investigate various phenomena, such as the serrated flow, chaos, and noise, and cognitive tasks [131,151,178,180,235–237]. The advantage of the RCMSE technique was illustrated in Reference [235], where the RCMSE and MSE algorithms were applied to model and analyze the 1/*f* noise (*f* is the frequency of the generated noise). It was reported that the MSE technique had a non-zero probability of inducing undefined entropy values, whereas the RCMSE technique had negligible probability.

To begin the RCMSE analysis, the underlying trend of the stress vs. time data in the strain-hardening regime is first eliminated [119]. To perform this task, a moving average or a third-order polynomial is used to fit the data [238]. The fit is then subtracted from the original data. Subsequently, the coarse-grained time series, *y*<sup>τ</sup> *k*,*j* , is constructed by using the following equation [131]:

$$y\_{k,j}^{\tau} = \frac{1}{\tau} \sum\_{i=(j-1)\tau+k}^{j\tau+k-1} x\_i \quad \text{;} \quad 1 \le j \le \frac{N}{\tau} \qquad 1 \le k \le \tau \tag{6}$$

where *xi* is the ith point from the detrended time-series data, *N* is the total number of data points from the detrended time series, τ is the scale factor, and *k* is an indexing factor, which denotes at what data point to begin the averaging. To give the reader perspective, setting τ equal to 1 yields the original detrended time series. Once *y*<sup>τ</sup> *<sup>k</sup>*,*<sup>j</sup>* is determined, construct the time series, *<sup>y</sup>*<sup>τ</sup> *<sup>k</sup>* , which is represented as a vector for each τ [235]:

$$\mathbf{y}\_k^\pi = \begin{Bmatrix} y\_{k,1}^\pi & y\_{k,2}^\pi & \dots & y\_{k,M}^\pi \end{Bmatrix} \tag{7}$$

where each *y*<sup>τ</sup> *<sup>k</sup>*,*<sup>j</sup>* is determined from Equation (6), and *M* equals the integer below *N*/τ. Now determine the template vectors of dimension, *m*:

$$y\_{k,i}^{\tau,m} = \left\{ y\_{k,i}^{\tau} \mid y\_{k,i+1}^{\tau} \dots \dots \mid y\_{k,i+m-1}^{\tau} \right\} \quad ; \quad 1 \le i \le N-m \; ; 1 \le k \le \tau \tag{8}$$

Next, find n-matching sets of distinct template vectors for each *k*, using the following equation:

$$d\_{jl}^{\tau,m} = \|y\_j^{\tau,m} - y\_1^{\tau,m}\|\_{\infty} = \max\left\{ \left| y\_{1,j}^{\tau} - y\_{1,l}^{\tau} \right| \dots \left| y\_{i+m-1,j}^{\tau} - y\_{i+m-1,l}^{\tau} \right| \right\} < r \tag{9}$$

where *d*τ,*<sup>m</sup> jl* is the distance between two vectors, as calculated by the infinity norm [239], and *r* is typically chosen as 0.15 times the standard deviation of the data, which ensures that the sample entropy does not depend on the variance of the data [226,227,240]. As indicated by Equation (9), two vectors match when *d*τ,*<sup>m</sup> jl* is less than *r*. Now repeat the above process for template vectors of size *m* + 1. Next, determine the total number of matching vectors, *nm <sup>k</sup>*,τ, for *m* and *m* + 1 by summing from *k* = 1 to τ. Finally, the RCMSE (or sample entropy) value for the detrended time-series data can be determined by taking the ratio of the natural log for these two sums, using the following equation [235]:

$$R\text{CMSE}(X,\ \pi,\ m,\ r) = \text{Ln}\left(\frac{\sum\_{k=1}^{\tau} n\_{k,\tau}^{m}}{\sum\_{k=1}^{\tau} n\_{k,\tau}^{m+1}}\right) \tag{10}$$

Here, it should be mentioned that the sample entropy is only undefined when *n<sup>m</sup> <sup>k</sup>*,<sup>τ</sup> or *<sup>n</sup>m*+<sup>1</sup> *<sup>k</sup>*,<sup>τ</sup> , are zero for all *k*.

To illustrate the RCMSE method, the 1/*f* <sup>β</sup> noise was analyzed, using the RCMSE method in a similar fashion as was done in Reference [131]. This process was performed for β = 0, 1, and 2, which correspond to white, pink, and brown noise, respectively. Figure 13 features a graph of the mean Sample En. (sample entropy) vs. scale factor for the three types of noise plotted. As can be seen, the white and brown noise displayed decreasing and increasing trends, respectively. The sample-entropy values for the brown noise were greater than those for the white noise when τ > 22. This result indicates that the brown noise exhibits more complex behavior as compared to the white noise.

**Figure 13.** Mean sample entropy of the colored noise data and 1 ≤ τ ≤ 30 plotted with the colored noise plots for brown (β = 2), pink (or 1/*f* with β = 1), and white (β = 0) (adapted from Reference [131]).

The above findings also suggest that a random-walk signal with persistent behavior, such as the brown noise, contains higher spatiotemporal correlations, as compared to a noise-like signal that is inherent in the white noise [241]. As compared to the white and brown noise, the pink noise exhibits the greatest sample-entropy values for τ > 3. This result supports the theory that, as compared to the brown and white noise, the 1/*f* noise exhibits fluctuations that contain a greater degree of long-range correlations and are the most complex in nature [227]. Furthermore, the relatively higher complexity of the pink noise may also be attributed to its inherent mix of random-walk and noise-like characters.

#### *2.2. Multifractal Modeling and Analysis*

In contrast to the fractal approach, which can only characterize global scaling properties, the multifractal formalism can analyze not only the underlying fractal geometry but also the distribution of the underlying physical properties on the fractal support [180,242]. The multifractal method has been used to model and analyze the serrated flow in different material systems, including Al-Mg alloys [216,221,223,224,243–245], bulk-metallic glasses [246,247], and HEAs [180]. Specifically, this method has been used to examine the relationship between the serration type and its corresponding multifractal spectra. For instance, Lebyodkin et al. reported that the multifractal characteristics of the serrated flow in an Al-Mg alloy were affected by the phase composition of the material [242]. They also observed that the serrated deformation curves with an underlying multifractal and hierarchical structure results from the self-organization of dislocation motion. In another investigation, it was observed that the transition from Type-B to Type-A serrations was associated with a sharp increase in the multifractality of the stress-burst behavior [216].

In a similar fashion as the RCMSE method, the underlying trend associated with the strain hardening is omitted from the original signal. As discussed earlier, one can accomplish this task by subtracting away the moving average of the polynomial fit from the data. Next, determine the magnitude of the burst in the plastic activity, <sup>β</sup>*<sup>i</sup>* <sup>≈</sup> *d*σ *dt t*=*ti* . This step is accomplished by taking the absolute value of the numerical time derivative of the stress at each point in time. Now partition the detrended dataset into non-overlapping windows of size Δτ [244,245] and let the size of the window vary as (Δτ) <sup>2</sup>*<sup>j</sup>* for *j* = 1 ... *n*. For each window, calculate the normalized amplitude of the probability measure for the plastic bursts:

$$p\_i(\Delta \tau) = \frac{\left(\sum\_{j=1}^{M} \beta\_j\right)\_i}{\sum\_{k=1}^{P} \beta\_k} \tag{11}$$

where *M* is the total number of points in the *i*th window. Now compute the measure, μ*i*(Δτ, *q*):

$$\mu\_i(\Delta \tau, q) = \frac{p\_i^q(\Delta \tau)}{\sum\_j p\_j^q(\Delta \tau)}\tag{12}$$

where *q* is a real number. From μ*i*, determine the local singularity strength, α [216,248]:

$$\alpha = \lim\_{\Lambda \tau \to 0} \frac{\sum\_{i} \mu\_{i}(\Delta \tau, q) L n\left[p\_{i}(\Delta \tau)\right]}{L n\left(\Delta \tau\right)} \tag{13}$$

Since *pi*(Δτ)~Δτα, the singularity strength is an important quantity, since it can give a measure of the scaling relationship between the interval size and the probability measure of the dataset. Next, one can solve for the fractal dimension *f*(α) of the subset of intervals characterized by the singularity strength [180]:

$$f(\boldsymbol{\alpha}) = \lim\_{\Delta\boldsymbol{\tau} \to 0} \frac{\sum\_{i} \mu\_{i}(\Delta\boldsymbol{\tau}, q) L n \mu\_{i}(\Delta\boldsymbol{\tau})}{L n(\Delta\boldsymbol{\tau})} \tag{14}$$

It is important to note that, typically, the limits from Equations (13) and (14) cannot be calculated directly, and instead a linear-regression analysis must be implemented. A plot of *f*(α) vs. α is known as the multifractal spectra and is typically characterized by a parabolic shape. The width of the multifractal spectrum, which is defined here as Δ = α*max* − α*min*, measures the signal heterogeneity and is associated with the dynamical heterogeneity of the serrated flow [244].

#### *2.3. Mean-Field Theory and the Mean-Field Interaction Model*

During deformation, several materials, such as BMGs, HEAs, densely packed granular materials, rocks, and other composite materials, can exhibit jerky flows that occur via slip avalanches [76,134,153,249–252]. These slip avalanches typically exhibit a broad distribution of slip sizes, which can be described by relatively simple probability distribution function (pdf) models [253]. These mean-field interaction models, which are based on the mean-field theory (MFT), assume that a solid material has weak spots that are elastically coupled [132]. The type of weak spots will also depend on the kind of material. For instance, in crystalline materials, weak spots may comprise regions with dislocations that can undergo dislocation slip. In non-crystalline materials, on the other hand, weak spots may be regions that consist of shear bands, shear-transformation zones, or other relatively weak regions in the material [133,254,255]. This type of long-range interaction terminates once the stress at every point in the system is below its current failure stress [256].

The MFT model assumes that, during an applied shear stress, a weak spot can trigger other weak spots to slip, resulting in a slip avalanche [133]. Furthermore, the model assumes that the long length-scale behavior of the slip statistics is independent of the microscopic structural details of the material. The power of the MFT model lies in its ability to predict many statistical distributions and quantities. Typically, these models involve a power-law distribution of slip sizes multiplied with an exponentially decaying cutoff function [132–134,140,251,252,256]. As an example, the underlying equation for the MFT model has been written as follows [133]:

$$D(S, q) = S^{-\kappa} D' \{ S q^{\lambda} \} \tag{15}$$

where *S* is the avalanche-slip size; *q* is an experimentally tunable parameter, such as the applied stress or strain rate; *D* (*y*) is a universal scaling function; and κ and λ are universal power-law exponents. It is important to note that the maximum avalanche-slip size, *Smax*, is proportional to *q* as *q*λ. For a more comprehensive discussion on the MFT model, please see Reference [76].

Another important quantity is the corresponding complementary cumulative distribution function (CCDF), *C*(*S*, *q*), which gives the probability of observing an avalanche of size greater than *S*, is useful for systems with low numbers of avalanches [133,180]:

$$C(S,q) = \int\_S^{\infty} D(S',q)dS'\tag{16}$$

Importantly, this type of analysis is used by experimentalists and theorists to model the stress-drop behavior in multiple systems, including BMGs [133] and HEAs [8,9,32,153,180]. If we substitute Equation (15) into Equation (16) and integrate, we obtain the following:

$$\mathbb{C}(S,q) = q^{\lambda(\kappa-1)}\mathbb{C}'(Sq^{\lambda})\tag{17}$$

where *C* (*Sq*λ) is some other scaling function. Next, plot *C*(*S*, *q*)*q*−λ(<sup>κ</sup> <sup>−</sup> 1) vs. *Sq*<sup>λ</sup> and then tune the universal exponents until the curves lie on top of each other. This procedure will yield the correct values of the critical exponents.

This type of modeling and analysis has been proven to be a useful tool in the analysis of serrated flows in HEAs. Zhang et al. applied the MFT-modeling technique to examine the effect of temperature on the weak spot (dislocations) avalanche dynamics in HEAs [8]. Carroll et al. employed the MFT model to analyze the serrations that were exhibited by low-, medium-, and high-entropy alloys [153]. Results indicated that the serration-type change from Type-A to Type-B to Type-C with an increase in the temperature. It was also reported that an increase in the strain rate corresponded to a transition from Type-C to Type-B to Type-A serrations in the HEAs.

In addition to analyzing the stress-drop behavior of materials undergoing tension and compression testing, statistical analysis of the nanoindentation pop-in behavior has also been conducted. For this type of analysis, the cumulative probability distribution of the size of the displacement bursts, *S*, is analyzed. A displacement burst is characterized as a sudden displacement at a constant load during a plastic-deformation event. One such equation that can be used to analyze this type of behavior is written as follows [257]:

$$P(\succ S) = AS^{-\beta}e^{-\frac{S}{S\_c}} \tag{18}$$

where *A* is a normalization constant, β is a scaling exponent, and *Sc* is the cutoff value of *S*. It should be mentioned that, for pop-in sizes smaller than *Sc*, their behavior follows a power-law distribution. Once *S* exceeds the cutoff value, the magnitudes of the pop-ins decrease in an exponential fashion.

#### *2.4. Chaos Analysis*

Chaos, which is the exponential sensitivity to small perturbations, is a ubiquitous phenomenon in nature [258,259]. Furthermore, chaos is related to how a deterministic dynamical system can be potentially unpredictable due to an extreme sensitivity to initial conditions, which is also known as the "Butterfly Effect" [259]. Chaos has been observed in many different phenomena, including electric circuits [260–262], weather [263–265], physiological systems [266–268], financial markets [269–271], complex networks [272], the serrated flow in alloys [116,216,273], and ecological systems [274–276].

One of the earliest attempts to apply a chaotic mathematical model to the repeated yielding behavior was done by Ananthakrishna et al. [277]. Similar to the Lorenz system [278], their model consisted of a set of four time-dependent differential equations. The equations were based on the time-dependent behavior of the stress, the mobile density, the immobile density, and the density of structures with clouds of solute atoms. The results of the analysis indicated that the dynamical behavior of this system exhibited an infinite sequence of period-doubling bifurcations that eventually led to chaos. Subsequent works confirmed the presence of chaos in the serrated flow of CuAl and AlMg alloys during tension testing [116,216,273]. For the studies involving the AlMg alloy, the chaotic behavior was observed for strain rates below ~3 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup>−<sup>1</sup> and corresponded to Type-B serrations [216,273].

One such way to determine whether a dynamical system is exhibiting the chaotic behavior is to determine the largest Lyapunov exponent [279] for the given time series. A positive exponent signifies that the underlying dynamical behavior is chaotic. On the other hand, a negative value indicates that the system will evolve into a stable state [159]. The following discussion gives a basic recipe on how to calculate this value for the time-dependent serration data from an experiment. From a given stress-time series, one can define the reconstructed attractor *Y*(*t*, *m*) = [σ(*t*), σ(*t* + τ), ... , σ(*t* + [*m* − 1]τ)], where σ is the stress value at time *t* = [1, 2, ... *N* − (*m* − 1)τ], τ is an arbitrarily chosen delay time, *m* is the embedding dimension, and N is the number of data points in the time series [154,280].

To begin the analysis, one locates the nearest neighbor (in a Euclidean sense [280]) to the initial point, *Y*0(*t*0, *m*), and determines the distance between them. This distance is defined as *L(t*0) = |*Y*(*t*0, *m*) − *Y*0(*t*0, *m*)|. At some later time, which is denoted as *t*1, the initial length will become *L* (*t*1) = |*Y*(*t*1, *m*) − *Y*0(*t*1, *m*)|. However, if it is determined that *Y*0(*t*1, *m*) is not a nearest neighbor point of *Y*(*t*1, *m*), then a new point, *Y*1(*t*1, *m*), will be chosen. Next, determine *L*(*t*1) = |*Y*(*t*1, *m*) − *Y*1(*t*1, *m*)|. For each step, the angle between *L*(*ti*) and *L*(*ti*+1) is minimized to reduce the influence on the orbit evolution when a nearest neighbor point is chosen [159]. This process is repeated until the m-dimensional vector, *Y*(*ti*), reaches the end of the time series [154]. From the results, one can determine the largest Lyapunov exponent, λ<sup>1</sup> [280]:

$$\lambda\_1 = \frac{1}{t\_M - t\_0} \sum\_{k=1}^{M} \text{Log}\_2 \frac{L'(t\_k)}{L(t\_{k-1})} \tag{19}$$

where *M* is defined as the total number of the replacement steps, and *tM* is the final time point.

#### **3. Serration Studies in HEAs**

#### *3.1. Al0.5CoCrCuFeNi HEA*

Chen et al. performed multiple investigations on the serrated-flow behavior in an Al0.5CoCrCuFeNi HEA [152,154,155,157]. In Reference [152], they examined the effects of temperature on the serrated flow of the above alloy during compression tests. Here, tests were performed at temperatures of 400, 500, and 600 ◦C, using a strain rate of 5 <sup>×</sup> 10−5·s−1. Synchrotron X-ray diffraction (XRD) (Advanced Photon Source (APS), Argonne National Laboratory, Chicago, IL, USA) revealed that the alloy contained only an FCC phase after testing at 400 and 500 ◦C, whereas both FCC and body-centered-cubic (BCC) phases were present in the sample tested at 600 ◦C. Furthermore, it was reported that, with an increase in the test temperature, the critical strain for the onset of serrations decreased, whereas the Young's modulus increased. The increase in the Young's modulus with temperature was attributed to the presence of the BCC phase in the sample. It was thought that the decrease in the critical strain with increasing temperature was due to the increased mobility of solute atoms that can catch and pin dislocations, reducing the amount of strains that can occur before the onset of serrations.

In a subsequent study, the serration behavior of the above alloy was examined after being compressed at strain rates of 5 <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>s</sup><sup>−</sup>1, 2 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup><sup>−</sup>1, and 2 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup>−<sup>1</sup> and temperatures of 400, 500, and 600 ◦C [180]. Figure 14 displays the stress vs. strain curves of the experiments. Table 2 presents a list of the serration type for each of the experimental parameters, as defined above. The serration type was dependent on the test temperature, and to a smaller extent, the strain rate. For instance, Type-A serrations were primarily observed at 400 ◦C, Types A and B were seen at 500 ◦C, and Type-C serrations occurred at 600 ◦C.

**Figure 14.** Stress vs. strain graph for the Al0.5CoCrCuFeNi HEA samples tested at strain rates of (**a**) 2 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1, (**b**) 2 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup><sup>−</sup>1, and (**c**) 5 <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>s</sup>−<sup>1</sup> and temperatures of 400–600 ◦C (reproduced from Reference [180] with permission).

**Table 2.** Strain rates, test temperatures, and serration types for the Al0.5CoCrCuFeNi HEA that underwent compression testing (from Reference [180]).


The serration data were analyzed by using different analytical techniques, such as the RCMSE (complexity) and multifractal methods. The results of the complexity analysis can be observed in Figure 15a–c. For all of the strain rates, the sample-entropy values were the highest for the samples compressed at 500 ◦C, while they were the lowest for those compressed at 600 ◦C. This result indicates that, at 500 ◦C, the serrations exhibited the most complex dynamical behavior, while the opposite was true for the specimens compressed at 600 ◦C. Using the results from Table 2, we can surmise that, as compared to the Type-A and Type-B serrations, the Type-C serrations exhibited less complex behavior. This lower complexity was attributed to the repeated pinning and unpinning of dislocations [152],

which is characteristic of simple behavior. In contrast, the more complex behavior inherent in the Type-A and Type-B serrations corresponded to a couple of factors. Firstly, the relatively greater complexity of the serration behavior could be attributed to the presence of the deterministic chaos that is inherent in Type-B serrations. Secondly, the increased complexity could be related to the greater variety of defect interactions that occur during the serrated flow. These interactions include the solute atom–dislocation line, dislocation line–dislocation line, dislocation line–precipitate interactions, as well as the solute atom–solute atom interactions [180].

**Figure 15.** Sample entropy vs. scale factor for the Al0.5CoCrCuFeNi HEA samples tested at strain rates of (**a**) 2 <sup>×</sup> 10−<sup>3</sup> s<sup>−</sup>1, (**b**) 2 <sup>×</sup> 10−<sup>4</sup> s<sup>−</sup>1, and (**c**) 5 <sup>×</sup> 10−<sup>5</sup> s−<sup>1</sup> and temperatures of 400–600 ◦C, where the sample entropy was plotted for scale factors ranging from 1 to 20 (reproduced from Reference [180] with permission).

Figure 16a–c displays the multifractal spectra for the specimens tested at the prescribed temperatures and strain rates. As can be observed, all the spectra exhibited a parabolic shape for all the experimental conditions. Furthermore, the multifractality was generally found to increase with an increase in the temperature for each strain rate. Consequently, the Type-C serrations were accompanied by the widest multifractal spectra (greatest multifractality), as compared to the other serration types. There was also a sharp increase in the multifractality, and hence the dynamical heterogeneity, at 600 ◦C for a strain rate of 2 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1. This burst in the multifractality may represent a transition point between different serration types or dynamical behavior [244]. It should also be mentioned that, for the samples tested at 400 and 500 ◦C, where Type-A and Type-B serrations were observed, the multifractality of the serrated flow was markedly lower. The relatively lower multifractality of the serrations at these temperatures may correspond to serration dynamics that exhibit the self-organized criticality [216].

**Figure 16.** The multifractal spectra (f(α) vs. α) for the Al0.5CoCrCuFeNi HEA samples tested at strain rates of (**a**) 2 <sup>×</sup> 10−<sup>3</sup> s<sup>−</sup>1, (**b**) 2 <sup>×</sup> 10−<sup>4</sup> s<sup>−</sup>1, and (**c**) 5 <sup>×</sup> 10−<sup>5</sup> s−<sup>1</sup> and temperatures of 400–600 ◦C, where the sample entropy was plotted for scale factors ranging from 1 to 20. (Results reproduced from Reference [180] with permission.)

In another study [155], the Al0.5CoCrCuFeNi HEA was compression tested at temperatures ranging from RT to 700 ◦C and a strain rate of 5 <sup>×</sup> 10−<sup>5</sup> s−1. The subsequent TEM (see Figure 17a–d) and XRD characterization revealed that, after testing at a temperature of 600 ◦C, the matrix contained dislocations, fully ordered L12 particles, and both BCC and FCC phase structures in the alloy. It was also determined that, at 500 ◦C, there were the partially ordered L12 particles and an FCC phase present in the matrix. It was hypothesized that, during compression, the L12 particles can act as an obstacle for the moving dislocations [155]. This obstruction reduces the mobility of the dislocations enough such that mobile solute atoms can catch and pin them, resulting in the serrated flow. With regards to the serration dynamics, the less complex serrations (Type-C) that were observed at 600 ◦C may somehow be related to the presence of the fully ordered L12 nanoparticles and BCC phase in the matrix. On the other hand, the more complex serrations observed at 500 ◦C (Types A and B) were apparently associated with a matrix that contains partially ordered nanoparticles and a single FCC phase. However, the exact relation between the nanoparticles and serration dynamics are still not well understood and should therefore be the focus of future investigations.

**Figure 17.** (**a**,**b**) The dark and bright field TEM image of the Al0.5CoCrCuFeNi HEA deformed at 600 ◦C. (**c**,**d**) The high-resolution TEM (HRTEM) images orientating in the (001) zone axis and the corresponding electron-diffraction pattern. Zoom-in images show the structure of the ordered L12 phase (reproduced from Reference [155] with permission).

In addition to compression and tension testing, the nanoindentation [281] serration behavior of the Al0.5CoCrCuFeNi HEA has also been examined [157]. For their work, nanoindentations were performed, using a NanoTest Vantage (Micro Materials Ltd., Wrexham, UK) equipped with a diamond Berkovich indenter. For the indentations, the samples were indented to a maximum load of 100 mN, using a loading rate of 10 mN/s. To study the effect of temperature on the indentation deformation, the experiments were performed at RT and 200 ◦C. Figure 18 displays the nanoindentation load as a function of displacement. The magnified insets show that there was a noticeable stair-step pattern for both the RT and 200 ◦C conditions, which correspond to the serrated flow (displacement bursts). These serrations were thought to be caused by several factors, including the breakaway of dislocations from various obstacles, such as atoms or precipitates, dislocation multiplication, or the evolution of dislocation cells or tangles [282].

Furthermore, it was observed that, for the 200 ◦C nanoindentation, the sample exhibited larger displacement bursts, as compared to the RT condition. These relatively larger displacement bursts indicate that, at higher temperatures, there is a greater concentration of dislocations that can be activated during the nanoindentation deformation. It should also be noted that, as compared to the RT condition, the tip penetrated the sample at greater depths during nanoindentation at 200 ◦C. This result suggests that the sample exhibited softening with an increase in the temperature and was most likely a direct result of the promoted thermal activation of dislocations during deformation at higher temperatures.

**Figure 18.** Nanoindentation pop-in behavior of the Al0.5CoCrCuFeNi HEA tested at room temperature (RT) and 200 ◦C (reproduced from Reference [157] with permission).

In a later study, Chen et al. analyzed the nanoindentation-serration behavior of the Al0.5CoCrCuFeNi HEA, using multiple analytical techniques, including the chaos analysis and ApEn methods [154]. Here, indentations were performed, using a Berkovich tip at RT and 200 ◦C to a peak load of 100 mN. For each temperature condition, three different holding times, namely 5, 10, and 20 s, at the peak load were used. Figure 19 displays the associated depth vs. time data for the nanoindentation experiments with the prescribed temperatures and loading times. As can be observed, there was serrated flow in the holding regime of the data.

**Figure 19.** Depth-time curves for the nanoindentation of the Al0.5CoCrCuFeNi HEA with the holding times of 5, 10, and 20 s at RT and 200 ◦C (from Reference [154]).

The data corresponding to the serrated flow that occurred during the holding periods were analyzed by using methods such as the chaos and ApEn analytical techniques. Figure 20 displays the largest Lyapunov exponent, λ1, and the ApEn results as a function of holding time for times ranging from 5 to 20 s. For both the RT and 200 ◦C conditions, the λ<sup>1</sup> values are positive for all the loading times. This finding signifies that for all the experimental conditions, the serrated flow is associated with slip-band dynamics that exhibit chaotic behavior. The result also means that the serration dynamics are sensitive to initial conditions. Furthermore, the largest Lyapunov exponent increased with an increase in the temperature (for a given holding time), indicating that the serrated flow exhibited a greater degree of chaotic behavior at 200 ◦C. This increase in the chaotic dynamics at the elevated temperature was attributed to a greater mobility of dislocations. It was also observed that, for both temperatures, λ<sup>1</sup> attained a minimum at a holding time of 10 s, which indicates that the serrated-flow behavior was the least affected by initial conditions, as compared to the other holding times. In terms of the ApEn values, they exhibited a similar trend as λ<sup>1</sup> for the sample tested at 200 ◦C. Furthermore, both values attained a maximum for a holding time of 5 s. The authors hypothesized that the maximum values for the lower holding time corresponded to the increase in the number of interactions among the slip bands that are indicative of a more complex process. As for the room-temperature nanoindentation, a holding time of 10 s was accompanied by both a minimum λ<sup>1</sup> value and a maximum value for the ApEn. This result indicates that the serration behavior exhibited dynamical behavior that has both a relatively low sensitivity to initial conditions and also a greater degree of freedom [154].

**Figure 20.** The largest Lyapunov exponent, λ1, and approximate entropy, ApEn, for the holding times of 5, 10, and 20 s at RT and 200 ◦C (reproduced from Reference [154]).

Antonaglia et al. applied the mean-field theory to analyze the slip-avalanche statistics for the investigated conditions [167]. For the experiments, specimens with a length of 4 mm and diameter of 2 mm underwent uniaxial-compression tests at a strain rate of 4 <sup>×</sup> 10−<sup>4</sup> s−1, at temperatures of 7, 7.5, and 9 K. The test temperature was controlled by using liquid He. Figure 21a shows the stress vs. strain data resulting from the compression experiments, and the results of the analysis are displayed in Figure 21b. As can be seen, serrations were observed at all three test temperatures. Furthermore, it was found that the magnitude of the serrations significantly increases with an increase in the compression strain. It was also noted that a similar trend has previously been observed in BMGs [133,136,283]. The largest stress-drop size decreases as the temperature increases from 7 to 9 K. This decrease in the slip size for the above temperature range is in agreement with the model prediction, which states that the magnitude of the slip avalanche will be reduced as the temperature increases [284]. The reduction

in the avalanche size was related to the twinning phenomenon in the HEA at cryogenic temperatures, where an increase in the temperature increased the difficulty in which deformation twinning could be induced during compression.

**Figure 21.** (**a**) Compressive stress–strain curves for the Al0.5CoCrCuFeNi HEA at temperatures ranging from 7 to 9 K, at a strain rate of 4 <sup>×</sup> 10−<sup>4</sup> s<sup>−</sup>1, and (**b**) the CCDF for the Al0.5CoCrCuFeNi HEA (reproduced from Reference [167] with permission).

In a similar study, the serrated flow during uniaxial compression was analyzed by using the chaos formalism [165]. Here, samples were compressed at a strain rate of 4 <sup>×</sup> 10−<sup>4</sup> s−<sup>1</sup> and temperatures ranging from 4.2 to 9 K. It was found that the serrated flow exhibited similar behavior (7.5 and 9 K) to that observed by Antonaglia et al. [167]. The results of the analysis revealed that the largest Lyapunov exponent was negative at all the test temperatures. This result indicates that the serrated flow did not exhibit chaotic behavior and indicates that the slip dynamics are stable. It is interesting to note that these results are in contrast with those from Chen et al, where it was determined that the serrated flow during nanoindentation at RT and 200 ◦C exhibited chaotic behavior [154]. However, it is important to note that the deformation mechanisms associated with twinning at cryogenic temperatures are completely different from those which occur during nanoindentation pop-ins at temperatures greater than or equal to RT. Therefore, these findings suggest that the sensitivity of the serration behavior to initial conditions may depend on the underlying deformation mechanisms.

#### *3.2. Al0.1CoCrFeNi HEA*

Xia et al. studied the deformation mechanisms of an Al0.1CoCrFeNi HEA that was produced by vacuum-levitation methods [285]. For the experiment, specimens underwent uniaxial-compressive tests where samples were exposed to a strain of 2 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup>−<sup>1</sup> and temperatures of 77 K, 200 K, and 298 K. Results showed that the serrated flow was only observed in the sample compressed at 77 K.

To gain a better understanding of the underlying deformation mechanisms, TEM characterization was performed on the samples after testing (see Figure 22a–d). Figure 22a,b displays the microstructures of the specimens that were tested at 200 and 298 K, respectively. As can be seen, the sample that was compressed at 200 K had a greater dislocation density, as compared to the sample that was tested at 298 K. The results of the Burgers-vector analysis revealed that at these temperatures, the plastic deformation in the HEA occurred mainly by the planar slip of 1/2 < 110> type dislocations on {111}-type planes. This result indicates that deformation at these temperatures occurs only by the dislocation glide. Importantly, this finding is consistent with what is observed in other FCC solid solutions [286]. Figure 22c,d shows the TEM imaging results for the sample tested at 77 K. Here, the microstructure consisted of narrow deformation twins that had widths on the order of tens of nanometers. Furthermore, the TEM characterization indicated that nanoscale-deformation twins leading to the {111} <110> primary slip system were the primary deformation mechanisms responsible for the serrated flow during compression.

**Figure 22.** TEM bright field images of the Al0.1CoCrFeNi alloy tested at (**a**) 25 ◦C (298 K), which indicates that there is a tendency toward the dislocation-cell formation when the strain reaches 50%, (**b**) −73 ◦C (200 K), with the (111)-type slip plane, (**c**) −196 ◦C (77 K), which displays nanotwins, and (**d**) the corresponding high-resolution TEM image for (**c**) that displays the region where two deformation twins with a thickness of about 2 nm are located. The selected area electron diffraction patterns and fast Fourier transform (zone axis (110)) are also displayed (reproduced from Reference [285] with permission).

Hu et al. examined the serrations in the compressed nanopillars composed of the Al0.1CoCrFeNi HEA [160]. For the experiments, pillars with diameters of 500–700 nm underwent in situ compression tests, using direct electron imaging. The compression tests were performed by using a strain rate of ~1 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup>−<sup>1</sup> in a Hysitron PI95 picoindenter equipped with a 2 <sup>μ</sup>m flat punch diamond indenter in a JEOL 2010 LaB6 TEM (JEOL USA Inc., Boston, MA, USA) with an operating energy of 200 keV. The mechanical-deformation data were accumulated, using a data-acquisition rate of 500 Hz.

Figure 23 presents the stress vs. strain data for the in situ TEM compression test. As can be observed, there were multiple slip events that occurred during the experiment. Furthermore, the authors divided the deformation process into three stages that consisted of (1) little or no stress drops, (2) medium-sized stress-drops, and (3) relatively larger stress-drops. A comparison of the stress-time data and the three corresponding stages of the stress-drop behavior, as observed by the TEM and SEM, can be seen in Figure 24a–e. With respect to stage I, the stress drops corresponded to the wave-like propagation of dislocations from the top of the pillar to the bottom. As for Stage-II stress drops, they were attributed to small dislocation avalanches. Finally, stage III stress drops corresponded to dislocation avalanches that led to large crystal slips.

The CCDF analysis was performed on the serration behavior, and the results can be seen in Figure 25a,b. The data were modeled according to Equations (15) and (16), for which *<sup>D</sup>*(*S*, *<sup>q</sup>*) <sup>=</sup> *<sup>S</sup>*−(<sup>κ</sup> <sup>−</sup> 1)*g*[*S*(<sup>σ</sup> <sup>−</sup> <sup>σ</sup>*c*) <sup>1</sup>/β], where κ and β are critical exponents, σ is the applied stress, σ*<sup>c</sup>* is the failure stress, and *g* is a universal scaling function defined as *x*κ−<sup>1</sup> <sup>∞</sup> *<sup>x</sup> <sup>e</sup>*<sup>−</sup>*Att* <sup>−</sup>κ*dt* [222] with *A* = 1.2. These data were for samples with diameters ranging from ~512 to 655 nm. For Figure 25a, the CCDF, as a function of the stress level over the maximum stress, was plotted with respect to the slip size. Figure 25b displays the rescaled CCDFs in which the curves have collapsed onto one another. The critical exponents, κ and β, were determined by tuning them until the curves collapsed, and were found to be consistent with the predicted MFT values of 1.5 and 0.5, respectively [222].

**Figure 23.** The stress vs. strain curve for the in situ TEM nanopillar compression tests for the Al0.1CoCrFeNi HEA (reproduced from Reference [160]).

**Figure 24.** The in situ nanomechanical indentation and TEM results for the Al0.1CoCrFeNi HEA in terms of the (**a**) stress vs. time data (stages I, II, and III labeled in the figure), the TEM imaging results for the serrations that occurred during (**b**) stage I, (**c**) stage II, and (**d**) stage III (with color-coded deformation stages), and (**e**) the SEM image of the deformed nanopillar that confirms that large crystal-slip ensued during the experiment (reproduced from Reference [160]).

**Figure 25.** (**a**) The plots of the stress-binned CCDF of slip sizes as a function of the stress level over the maximum stress samples with diameters ranging from ~512 to 655 nm, compressed at a strain rate of <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup>−<sup>1</sup> and with similar load–displacement characteristic. (**b**) The scaling collapse of the same data with the predicted scaling function (reproduced from Reference [160]).

The results of the analysis led to the following important conclusions. Firstly, the distribution used in their analysis is predicted by a simple, coarse-grained model [222,256]. Secondly, the avalanche mechanism responsible for the observed serration behavior in the samples is based not only on the interactions between the dislocation bands and the dislocation pileups, but also the pileup and the dislocation pinning centers, as well. Thirdly, the applied stress is a critical tuning parameter due to the dependence of the slip avalanche on the stress level. Finally, the deformation behavior exhibited by the Al0.1CoCrFeNi HEA is characteristic of the tuned critical behavior rather than the self-organized criticality, indicating that the corresponding avalanche distribution is universal in nature.

#### *3.3. Al0.3CoCrFeNi HEA*

The effect of Al on the serrated flow has also been observed in an investigation involving the Al0.3CoCrFeNi HEA [191]. Here, Yasuda et al. examined the deformation behavior of the single-crystal CoCrFeNi and Al0.3CoCrFeNi HEAs during compression. For their experiment, they compressed samples at a strain rate of 1.7 <sup>×</sup> 10−<sup>4</sup> s−<sup>1</sup> for temperatures ranging from <sup>−</sup>180 to 1000 ◦C (93–1273 K). Figure 26 displays the results, and, as can be observed, the Al0.3CoCrFeNi HEA exhibited the serrated flow, while the other specimen did not when tested at 600 ◦C. The serrated flow that was observed in the DSA regime was thought to be related to Al-containing solute atmospheres that were created near a moving dislocation core, thus leading to an increase of the frictional stress on the dislocations. Similar to the findings reported by Niu et al. [94], this result indicates that Al atoms play an important role in the serrated-flow behavior that occurs in the HEA during DSA. This finding also suggests that not every atom can act as a solute that participates in dislocation locking that results in the serrated flow.

**Figure 26.** Stress vs. strain curves for the Al0.3CoCrFeNi and CoCrFeNi single crystals that were compressed at temperatures ranging from 93 to 1273 K (from −180 to 1000 ◦C) (reproduced from Reference [191] with permission).

Zhang et al. also examined the compression behavior of the Al0.3CoCrFeNi HEA [169]. The compression tests were performed on a Gleeble 3500 thermo-mechanical simulator (Dynamic Systems Inc., Poestenkill, NY, USA) at temperatures ranging from 400 to 900 ◦C at a strain rate of 10−<sup>3</sup> s<sup>−</sup>1. Figure 27a–d shows the true stress vs. true strain behavior for the samples tested at the above conditions. Serrations were observed in all of the deformation curves, and it was determined that the serration type was dependent on the test temperature. A summary of the serration type

for each of the experimental conditions is displayed in Table 3. As can be seen in the table, Type-C serrations were observed at 400, 500, and 600 ◦C. At 700 ◦C, the serrated flow exhibited Type-B + Type-C serrations, whereas only Type-B serrations could be observed in the deformation curve at 800–900 ◦C. The authors reported that the number of relatively larger stress drops decreases with respect to the test temperature. Furthermore, it was also found that Type-A serrations were not observed in any of the deformation curves. These results appear to contradict those of other studies [8] where serrations have been observed to evolve from Type-A to Type-B, and then to Type-C with increasing temperature. Importantly, it was hypothesized that Type-C serrations were related to the nucleation and growth of twins, while Type-B serrations corresponded to the pinning (by mobile solute atoms) and unpinning of moving dislocations that is inherent in the DSA model [94,191].

**Figure 27.** (**a**) True stress–strain curves of the Al0.3CoCrFeNi HEA tested at temperatures ranging from 400 to 900 ◦C and a strain rate of 10−<sup>3</sup> s<sup>−</sup>1, and the magnified regions of the serrations at temperatures of (**b**) 600 ◦C, (**c**) 700 ◦C, and (**d**) 800 ◦C (reproduced from Reference [169] with permission).

**Table 3.** Summary of the serration type exhibited by the Al0.3CoCrFeNi HEA during tension at a strain rate of 1 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup>−<sup>1</sup> and temperatures of 400–800 ◦C (from Reference [169]).


Qiang et al. investigated the nanoindentation-serration behavior of as-cast and torsionally deformed (nano-grained) Al0.3CrCoFeNi HEA samples [287]. Room-temperature nanoindentations

were performed by using a Hysitron Triboindenter TI950 (Bruker Corp., Santa Barbara, CA, USA) equipped with a Berkovich indenter. For the deformed samples, they underwent a compressive pressure of 10 GPa (RT) for 1, 3, and 10 rotations at one revolution per minute. For the nanoindentations, 15 indents were performed in which a loading rate of 250 μN/s was used. The samples were also examined, using different methods, such as XRD and Vickers-hardness tests. The XRD characterization revealed that all the samples contained a single-phase FCC structure. Vickers-hardness tests revealed that torsion led to a significant hardening of the alloy. The results also showed that the grain size was reduced from hundreds of microns in the as-cast state to tens of nanometers after 10 revolutions. It was also found that nanotwins and SFs were observed in the nanograins, despite the reported high SF energy.

In terms of mechanical behavior, it was surmised that, in the deformed nanograined HEA, the emission of Shockley partial dislocations at the grain boundaries and/or grain-boundary sliding may be the dominant deformation mechanism in the alloy. It was also reported that the as-cast HEA showed the slip-avalanche behavior, while the deformed samples did not exhibit any pronounced pop-ins (see Figure 28a). The lack of pop-ins exhibited by the deformed samples were attributed to the presence of affluent grain boundaries that act as extra mediators for plastic deformation and barriers for dislocation motion. The pop-ins that occurred in the as-cast sample exhibited relatively smaller pop-ins, as compared to alloys with a BCC lattice that displays only one pronounced pop-in [288,289]. This difference in the magnitude and the number of pop-ins was likely due to different dislocation-nucleation processes in the alloys, such as perfect dislocations in BCC crystals and partial dislocations in FCC crystals [289].

**Figure 28.** (**a**) Depth change of each pop-in event (Δh) as a function of indentation depth (h) and (**b**) the cumulative-probability distribution of the strain-burst size of the as-cast Al0.3CrCoFeNi HEA (reproduced from Reference [287] with permission).

Figure 28b presents the cumulative-probability distribution of the displacement burst size, *S*[*P*(> *S*) = *AS*−β*e* − *S Sc* ], for the as-cast Al0.3CrCoFeNi HEA sample. As previously discussed, the distribution represents the percentage of pop-in events with the displacement burst size being larger than a given value, *S*. As can be seen in the figure, the probability of observing a pop-in of a size, *S*, significantly decreases with an increase in the size, *S*. It was also reported that the parameters *Sc* and β were approximately 0.014 ± 0.000 and 0.15 ± 0.02, respectively, which indicates that slip avalanches did indeed occur in the as-cast sample during the nanoindentation deformation.

#### *3.4. Al0.5CoCrFeNi HEA*

Niu et al. examined the tension behavior of Al0.5CoCrFeNi and CoCrFeNi alloys [94]. For their investigation, samples underwent tension at strain rates of 10−<sup>3</sup> s−1, <sup>5</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup><sup>−</sup>1, and <sup>10</sup>−<sup>4</sup> <sup>s</sup><sup>−</sup>1, in addition to temperatures of 200–500 ◦C. Figure 29a–c presents the stress vs. strain curves for the

Al0.5CoCrFeNi HEA samples tested in the above conditions. As can be observed, the serrated flow consisted of Type-A, Type-B, Type-C, Type-A + Type-B, and Type B + Type-C serrations, which have been tabulated in Table 4. The results indicate that the serrations evolve from Type-A to Type-A + Type-B, and then to Type-B + Type-C, with a decrease in the strain rate and increase in the temperature. They also found that the serrated flow was a consequence of the DSA effect, and not strain-induced transformations or twinning. Statistical analysis was also performed on the serration behavior, and it was found that the distribution of the stress-drop magnitudes primarily exhibited the power-law behavior.

**Figure 29.** The tensile true stress–strain curves for the Al0.5CoCrFeNi HEA tested at temperatures of 200–500 ◦C and strain rates of (**a**) 10−<sup>3</sup> s−<sup>1</sup> where selected regions of the corresponding dynamic strain aging (DSA) curve are in (**a1**), (**a2**), and (**b**) 5 <sup>×</sup> 10−<sup>4</sup> s−<sup>1</sup> where selected regions of the corresponding DSA curve are in (**b1**), and (**c**) 10−<sup>4</sup> s−<sup>1</sup> where selected regions of the corresponding DSA curve are in (**c1**) and (**c2**) (reproduced from Reference [94] with permission).


**Table 4.** Strain rates, test temperatures, and serration types for the Al0.5CoCrCuFeNi HEA during tension testing (from Reference [94]).

Figure 30a,b shows a comparison of the stress vs. strain behavior for the Al0.5CoCrFeNi and CoCrFeNi alloys. As can be seen, the Al-containing HEA exhibited significantly more pronounced serrations, as compared to the CoCrFeNi alloy. From this result, the authors surmised that the Al played a significant role in the DSA phenomenon in the HEA and was attributed to the atomic-size mismatch between the Al and the rest of the matrix atoms. In addition to the above finding, the authors reported some other interesting results. For example, it was observed that the tensile strength and elongation of the material both decreased with increasing temperature, signifying an embrittlement of the alloy. They noted that similar trends were reported in other Al-containing HEAs [290,291], and thus concluded that the trend of embrittlement may be related to the DSA phenomenon in HEAs.

**Figure 30.** The engineering stress–strain curves for the (**a**) CoCrFeNi and (**b**) Al0.5CoCrFeNi HEAs during compression at a strain rate 10−<sup>3</sup> s−<sup>1</sup> and temperature of 400 ◦C (reproduced from Reference [94] with permission).

#### *3.5. Al0.7CoCrFeNi HEA*

Basu et al. investigated the nanoindentation-deformation behavior in a multi-phase Al0.7CrFeCoNi HEA that consisted of both BCC and FCC phases [292]. The energy-dispersive spectroscopy (EDS) characterization revealed that the BCC phases were composed of the (Ni, Al)-rich ordered B2 phase and (Fe, Cr)-rich disordered A2 phase. Figure 31a presents the load vs. displacement curve for four different indents that were performed inside the BCC HEA grain. Figure 31b presents a magnification of the indents, which features the elastic to plastic transitions, signifying a clear transition from Hertzian behavior [293]. The plastic regime of the nanoindentation curve that is displayed in Figure 31b consists of multiple pop-in events that are characteristic of the serrated flow in the alloy. The serrated flow was thought to be a consequence of the resistance provided by the interfacial-strengthening mechanisms between the soft A2 phase and the elastically stiffer B2 matrix against the dislocations.

**Figure 31.** The (**a**) load vs. displacement curves for four different indents that were made inside the body-centered-cubic (BCC) grain in the HEA over indentation depths of 0 to 500 nm, and the (**b**) highlighted region, which shows the data for two randomly selected indent curves. Here, the graph features the elastic to plastic transition and contains multiple pop-ins (reproduced from Reference [292] with permission).

#### *3.6. AlxNbTiMoV HEA*

Chen et al. investigated the compressive behavior of a BCC AlxNbTiMoV HEA [179]. Here, the molar ratio, x, of Al ranged from 0 to 1.5. For their experiments, samples underwent compression tests at RT and strain rates of 5 <sup>×</sup> 10−<sup>5</sup> s−1, 5 <sup>×</sup> 10−<sup>4</sup> s−1, 5 <sup>×</sup> 10−<sup>3</sup> s−1, and 5 <sup>×</sup> 10−<sup>2</sup> s−1. Figure 32 displays the compressive engineering stress–strain curves for the alloy that was compressed at a strain rate of 5 <sup>×</sup> 10−<sup>4</sup> s−1. As can be seen, there were observable serrations for x = 0, 0.25, and 0.5 around the yielding point. The authors surmised that the serrated flow was a consequence of the interactions between moving dislocations and Cottrell atmospheres [294]. In this context, the serrations occur due to the drag effect caused by Cottrell atmospheres that surround a dislocation and lock it in place. It was found that the yield strength of the alloy increased with an increase in x for this range of molar ratios. Furthermore, these values of x corresponded to atomic-size differences and lattice parameters that were greater than 3.22% and 3.191 Å, respectively. This result suggests that in the Al-containing HEA, there is an atomic size disparity and lattice volume, which, when exceeded, prohibits dislocation locking from occurring. As for the samples that did not contain Al (x = 0), serrations were observed for all the given strain rates. Furthermore, the amplitudes of the serrations were relatively larger for the intermediate strain rates of 5 <sup>×</sup> 10−<sup>4</sup> s−<sup>1</sup> and 5 <sup>×</sup> 10−<sup>3</sup> s−1, while they were barely noticeable for the lowest strain rate of 5 <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>s</sup><sup>−</sup>1.

**Figure 32.** The engineering stress vs. engineering strain for the AlxNbTiMoV (x = 0, 0.2, 0.5, 0.75, 1, and 1.5) HEAs (reproduced from Reference [179]).

It is interesting to note that the presence of serrations in the sample which does not contain Al is in contrast with the results as reported by Yasuda et al., where serrations were observed in AlxCoCrFeNi HEA when x = 0 [191]. The serrations in the NbTiMoV alloy (i.e., x = 0) may be due to various reasons. For instance, the presence of interstitial impurities, such as C, may pin dislocations [295]. On the other hand, the observed serrations may be due to the other constituent elements pinning the dislocations, which supports the theory that any atom in the matrix can act as a solute in HEAs. Therefore, more work is required to elucidate the effect of impurities and additives, such as Al and C, on the serrated-flow behavior in HEAs.

#### *3.7. Al5Cr12Fe35Mn28Ni20 HEA*

The serration behavior was examined in an Al5Cr12Fe35Mn28Ni20 HEA that was tension tested at temperatures of 300 and 400 ◦C and a strain rate of 1 <sup>×</sup> 10−<sup>4</sup> s−<sup>1</sup> [167]. Here, the gauge section of the specimens was 3 mm in diameter and 12 mm in length. The stress vs. displacement and the corresponding CCDF curve are displayed in Figure 33a,b. It is apparent from the graph that the samples exhibited serrations at both temperatures. Moreover, the magnitudes of the serrations were found to increase as the sample deformed. From the results of the statistical analysis, as displayed in Figure 33b, the magnitude of the largest stress drop decreased with an increase in the temperature. Importantly, this result is consistent with the reported trend predicted by the simple mean-field model [222,252,256]. The authors hypothesized that the relatively smaller slip sizes at the higher temperature is attributed to the increased thermal-vibration energy of the pinning solute atoms. More specifically, the pinning effect of the solute atoms on the dislocations is reduced at higher temperatures, since the solute atoms have a greater tendency to "shake away" from their low-energy sites for pinning [167,187].

#### *3.8. Ag0.5CoCrCuFeNi HEA*

Laktionova et al. investigated the compression behavior of the Ag0.5CoCrCuFeNi HEA at temperatures ranging from 4.2 to 300 K (from −269 to 27 ◦C) [296]. Here, cylindrical samples with a diameter of 2 mm and length of 4 mm were undergoing compression testing at a strain rate of <sup>4</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup><sup>−</sup>1. The loading of the sample was terminated when the strain reached values of ~20–30%. To perform the testing at temperatures below 77 K, samples were cooled with helium vapor, while the nitrogen vapor was used for temperatures ranging from 77 to 300 K. Figure 34 displays the stress–strain curves for the samples tested at the prescribed conditions. As can be observed, the samples exhibited the serrated flow at testing temperatures of 4.2 and 7.5 K, while serrations were not observed at any of the higher-temperature conditions. Furthermore, the strain at which the serrations commenced increased with the temperature. In particular, serrations began to appear at strains of 1.2% and 4.5% for the samples tested at 4.2 and 7.5 K, respectively. It was also reported that the stress-drop magnitude increased with an increase in the strain. For example, at 4.2 K, the magnitude increased from 20 MPa at a strain of ~2% to 67 MPa for a strain of roughly 23%.

#### *3.9. CoCrFeMnNi HEA (Cantor Alloy)*

Carroll et al. examined the serration behavior in multiple alloys, including CoCrFeMnNi HEA with CoCrFeNi HEA, CoFeNi medium-entropy alloy, CoNi low-entropy alloy (LEA), and pure Ni [153]. The above samples had configurational entropies ranging from 0R for the pure Ni to 1.61R for the CoCrFeMnNi HEA. Here, the specimens underwent tension tests with strain rates and test temperatures ranging from 1 <sup>×</sup> 10−<sup>5</sup> to 1 <sup>×</sup> 10−<sup>2</sup> s−<sup>1</sup> and 300 to 700 ◦C, respectively. Figure 35 displays the stress–strain curves for the CoCrFeMnNi HEA tested at a strain rate of 1 <sup>×</sup> 10−<sup>4</sup> s−<sup>1</sup> for temperatures of 300–600 ◦C [181]. The observed serrated flow consisted of Type-A, Type-B, and Type-C serrations, where the serration type depended upon the temperature. Table 5 displays a summary of the serration type and the corresponding temperature for the tests performed at strain rates of <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> to <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>2</sup> <sup>s</sup>−<sup>1</sup> and 300 to 600 ◦C. For the sample tested at the highest strain rate, only Type-A serrations were observed. Furthermore, the sample that was tested at 1 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup>−<sup>1</sup> exhibited serrations

that were Types A and C at the highest and lowest temperatures, respectively. For the intermediate temperatures of 400 and 500 ◦C, Type-B serrations were observed. It should be noted that a similar trend was observed in Al0.5CoCrCuFeNi HEA [180], as discussed previously.

**Figure 33.** (**a**) Stress vs. displacement curves for the Al5Cr12Fe35Mn28Ni20 HEA tension tested at temperatures of 573 and 673 K, at a strain rate of 1 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup>−<sup>1</sup> and (**b**) the results of the CCDF analysis for the serration events for the Al5Cr12Fe35Mn28Ni20 HEA in tension experiments under a constant strain rate of 1 <sup>×</sup> <sup>10</sup>-4 s-1 (reproduced from Reference [167] with permission).

**Figure 34.** The stress–strain curves for the Ag0.5CoCrCuFeNi HEA that was undergoing tension testing at temperatures ranging from 4.2 to 300 K, at a strain rate of 4 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup><sup>−</sup>1. The inset displays a close-up of the serrated flow that occurred in the specimens tested at 4.2 and 7.5 K. (Reproduced from Reference [296] with permission).

**Figure 35.** Stress vs. strain curves for the CoCrFeMnNi HEA tension tested at a strain rate of <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup>−<sup>1</sup> and temperatures of 300–600 ◦C (reproduced from Reference [181]).


**Table 5.** Summary of the serration type exhibited by the CoCrFeMnNi HEA during tension at strain rates from 1 <sup>×</sup> <sup>10</sup>−<sup>4</sup> to 1 <sup>×</sup> <sup>10</sup>−<sup>2</sup> <sup>s</sup>−<sup>1</sup> and temperatures of 300–600 ◦C (from Reference [153]).

It was found that, for a strain rate of 1 <sup>×</sup> 10−<sup>4</sup> s−1, the pure Ni and CoNi alloy did not exhibit serrations at any of the prescribed temperatures. Furthermore, the temperature range for which serrations were observed increased with an increasing chemical complexity of the alloy, such that the CoCrFeMnNi HEA was had the largest range of temperatures. From here, the authors suggested that the structure of the HEA prevents thermal vibrational from destroying the pinning effect, thereby allowing mobile solute atoms to pin the moving dislocations at lower temperatures.

Fu et al. examined the deformation behavior of a CoCrFeMnNi HEA that underwent homogenization, cold-rolling, and recrystallization [163]. The samples were homogenized at 1100 ◦C, for 24 h, in vacuum, and then subsequently cold-rolled, which led to a reduction in thickness of 40%. After cold-rolling, the sheets were recrystallized at 900 ◦C for 1 h. After the samples were fabricated, they underwent tension testing at temperatures ranging from RT to 800 ◦C and strain rates from 1.0 <sup>×</sup> <sup>10</sup>−<sup>5</sup> to 5.0 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1. The engineering stress vs. strain data for the samples tested at a strain rate of 3.0 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup>−<sup>1</sup> and temperatures ranging from RT to 800 ◦C can be observed in Figure 36a,b. As can be seen in the figures, the serrations are well defined for temperatures ranging from 300 to 600 ◦C. Furthermore, the serration type was dependent on the test temperature (see Figure 36b). At 300 ◦C, Type-A serrations were observed in the graph, while Types A + B were seen in the sample tested at 400 ◦C. On the other hand, Type-B serrations occurred in the sample that was tested at 550 ◦C, while Type-C serrations were observed at 600 ◦C.

It was also found that the critical plastic strain for the onset of serrations was significantly affected by both the temperature and the applied strain rate. For example, at a strain rate of 3.0 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup><sup>−</sup>1, the critical strain decreased monotonically with respect to the temperature. In contrast, the critical strain increased with an increase in the strain rate. From the critical strain, it was determined that, for the above strain rate, there were two temperature regimes for the activation energy of the serrated flow. For the first region, which corresponded to test temperatures of 300–500 ◦C, the activation energy was 116 kJ/mol, and the solute pinning of dislocations was controlled by pipe diffusion. As for the second region, which occurred at temperatures ranging from 500 to 600 ◦C, the activation energy for serrations was 296 kJ/mol. In this region, the pinning-and-unpinning process was dominated by a cooperative lattice diffusion mechanism in which Ni was the rate-controlling constituent.

It should be noted that serrations were not observed for temperatures either below 300 ◦C or above 600 ◦C. For the lower-temperature conditions, the lack of observable serrations is most likely a consequence of diffusing solute atoms that are too slow to catch and pin mobile dislocations [153]. At higher temperatures, it is thought that the thermal vibration of atoms is large enough as to prohibit the effective locking of dislocations [155,297].

**Figure 36.** (**a**) Engineering stress–strain curves for the CoCrFeMnNi HEA performed at temperatures ranging from RT to 800 ◦C and a strain rate of 3 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup>−<sup>1</sup> and (**b**) the partially enlarged segments for the curves that exhibited serrations (reproduced from Reference [163] with permission).

Figure 37a,b displays the engineering stress vs. strain curves for the samples tested at strain rates ranging of 1.0 <sup>×</sup> 10−5–5.0 <sup>×</sup> 10−<sup>3</sup> s−<sup>1</sup> at a temperature of 500 ◦C. From the figures, it is apparent that the serration type is dependent on the strain rate. At the lowest strain rate, namely 1.0 <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>s</sup><sup>−</sup>1, the sample exhibited Type-B + Type-C serrations, while Type-A serrations were observed at the highest strain rate. At intermediate strain rates, the stress-drop behavior exhibited Type-A <sup>+</sup> Type-B (1.0 <sup>×</sup> <sup>10</sup><sup>−</sup>5) and Type-B (1.0 <sup>×</sup> 10−4) serrated flow. Table 6 features the serration type for the temperature and strain-rate conditions of the experiments.

**Figure 37.** (**a**) The engineering stress vs. engineering strain curves for the sample tested at strain rates ranging from 1.0 <sup>×</sup> <sup>10</sup>−<sup>5</sup> to 5.0 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1, at a temperature of 500 ◦C. (**b**) The enlarged regions of (**a**) for engineering strain values ranging from 0.34 to 0.38 (reproduced from Reference [163] with permission).

**Table 6.** Summary of the serration type exhibited by the CoCrFeMnNi HEA during tension at strain rates of 1 <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>s</sup><sup>−</sup>1–5 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup>−<sup>1</sup> and temperatures of 300–600 ◦C (from Reference [163]).


In a later study, Fu et al. performed a similar study on the CoCrFeMnNi HEA, but this time they examined samples that had been cold rolled such that the sample thicknesses were reduced by 20%, 30%, and 40% [168]. Furthermore, the sheets were recrystallized at both 900 and 1000 ◦C (1 h) for each cold-rolling condition. XRD revealed that the alloy retained a simple FCC solid-solution phase after cold rolling and recrystallization. Samples undergoing quasi-static tensile tests were performed at temperatures ranging from RT to 800 ◦C. The fractured surfaces of the samples were examined by using SEM.

Figure 38a–c features the engineering stress vs. engineering strain for the HEA samples that were tested at temperatures ranging from RT to 800 ◦C and rolling conditions of 20%, 30%, and 40% reduced thicknesses. In the sample recrystallized at 900 ◦C for 1 h, the serrated flow was observed at the intermediate temperatures of 400 and 600 ◦C for all the rolling conditions. Interestingly, there was a discontinuous gap in the serrated flow for the samples tested at 600 ◦C. Similar to other studies involving the same alloy [153,163], the serrations appear to be of Types A and B at 400 ◦C, while they were characteristic of Type-C behavior at 600 ◦C. Moreover, for the samples tested at 600 ◦C, the discontinuous feature in the serrated flow was the most pronounced in the sample that had a thickness reduction of 40%. No serrations were observed in the samples tested at RT or 800 ◦C. The lack of observed serrations at 800 ◦C was attributed to the softening of the alloy at the higher temperature. As for the lack of observable serrations at RT, this trend was again due to the relatively low speed of the moving solutes, thus rendering them unable to reach and pin dislocations [163].

**Figure 38.** Graphs of the engineering stress–strain behavior for the recrystallized (900 ◦C, 1 h) and cold-rolled CoCrFeMnNi HEA with a rolling ratio of (**a**) 40%, (**b**) 30%, and (**c**) 20% (reproduced from Reference [168] with permission).

The SEM characterization revealed massive annular striation patterns located around the dimples on the fractured surface for the specimens tested at 400 and 600 ◦C. These striation patterns were attributed to a decrease in the ductility of the alloy. In contrast, no such striations were observed in the sample tested at 800 ◦C.

Wang et al. examined the deformation behavior of a spark-plasma-sintered CoCrFeMnNi HEA during high-strain-rate compression [166]. The results of the XRD characterization determined that the samples had a simple FCC structure. For the mechanical testing, samples were subjected to room-temperature impact tests, using a split-Hopkinson pressure bar system and strain rates ranging from 1 <sup>×</sup> 103 to 3 <sup>×</sup> 103 s−1. During dynamic deformation, these strain rates amounted to loading rates that varied between 1200 and 2800 s<sup>−</sup>1. Figure 39 displays the true stress vs. true strain behavior for the specimens, and as can be seen, there are observable serrations in the graph. Furthermore, these serrations appear comprise Type-A serrations. It was also determined that the serration behavior in the HEA is sensitive to changes in the applied-loading rate. Additionally, the general temperature of the sample approached values as high as 1300 K during testing. In this scenario, local hotspots in the matrix can weaken intergranular bonding, leading to the formation of cracks and microvoids in the severe-deformation zone.

**Figure 39.** Compressive true stress–strain curves of the CoCrFeMnNi HEA at loading rates that varied from 1200 to 2800 s−<sup>1</sup> (reproduced from Reference [166] with permission).

From the results, the authors hypothesized the following process involved in initializing the serrated flow. During the initial stages of deformation, dislocations form and then subsequently accumulate along the grain boundaries that become elongated during compression. As the deformation continues, the grains become more elongated, leading to a further decrease in the width of the grains. Meanwhile, the generated thermal hotspots weaken the intergranular bonding and ultimately lead to an eventual collapse of the grain boundaries. Consequently, numerous microvoids form along the grain boundaries, and shear bands are generated in the specimen, resulting in the observed serrated flow.

Before moving on, it is important to note that the split Hopkinson pressure bar test depends on the one-dimensional wave propagation in the bar [298]. The relatively smooth nature of the stress fluctuations, as can be seen in Figure 39, is indicative of poor pulse shaping during the test. This result suggests that the fluctuations may be due to machine noise rather than an actual physical metallurgical phenomenon.

Tirunilai et al. compared the deformation behavior of the CoCrFeMnNi HEA and pure Cu at cryogenic temperatures [299]. Here, tensile tests were performed at RT (295 K), 77, 8, and 4.2 K. For the experiments performed at 4.2 K, the samples were placed in a liquid He bath, whereas samples were placed in a vacuum cryostat when tested at 8 K. To examine the serration behavior in the HEA, tensile tests were performed at mean plastic strain rates of 6 <sup>×</sup> <sup>10</sup><sup>−</sup>5, 3 <sup>×</sup> <sup>10</sup><sup>−</sup>4, and 1 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1. A sample rate of 10 Hz was used for the data acquisition rate.

Figure 40a features a plot of the engineering stress–strain curve for the HEA tested at RT, 77 K, and 4.2 K and strain rate of 3 <sup>×</sup> 10−<sup>4</sup> s−1. As can be seen, there were pronounced serrated flows at 4.2 K whereas no apparent serrations were present at temperatures of 77 K and RT. Furthermore, deformation twinning was observed in the samples tested at 4.2 and 77 K. It was also found that the ductility in the sample deformed at 4.2 K remained relatively high.

**Figure 40.** The stress–strain curves for the (**a**) CoCrFeMnNi HEA under tensile load at RT, 77, and 4.2 K. (**b**) CoCrFeMnNi HEA and pure Cu at 4.2 K (reproduced from Reference [299] with permission).

Interestingly, the authors reported that such serrated flow behavior was significantly less pronounced in pure Cu when tested at a strain rate of 3 <sup>×</sup> 10−<sup>4</sup> s−<sup>1</sup> at 4.2 K, as shown in Figure 40b. For instance, the serration amplitude was ~150 MPa in the HEA, whereas it was only 5 MPa in the Cu metal. The significantly more pronounced serrations observed in the HEA (as compared to the pure Cu) were attributed to the impact of the solid solution in the alloy. It was also suggested that the discrepancy in the serration behavior may be a result of the differences between the properties of the material systems, such as yield strength, ultimate tensile strength, work-hardening, thermal conductivity, and heat capacity.

Figure 41 features the stress vs. strain data for the samples that were tested at 8 K and strain rates ranging from 6 <sup>×</sup> 10−<sup>5</sup> to 1 <sup>×</sup> 10−<sup>3</sup> s−1. As can be observed, the characteristics of the serrated flow were affected by the strain rate. For example, the time between stress-drops apparently decreased with an increase in the strain rate. The insets of the figure suggest that, for strain rates of <sup>6</sup> <sup>×</sup> <sup>10</sup>−<sup>5</sup> and <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup><sup>−</sup>1, the serrated flow consisted of Type-B serrations. On the other hand, the serrations were reminiscent of Type-D behavior at the highest strain rate.

Guo et al. examined the effects of carbon impurities on the deformation behavior of a CoCrFeMnNi HEA [192]. Here, the undoped and doped (0.93 atomic % C) samples were subjected to tension testing at RT and a strain rate of 1.6 <sup>×</sup> 10−<sup>3</sup> s−1. Figure 42a,b compares the true stress vs. strain of the carbon-doped and undoped HEA. As can be observed, serrations are clearly present in the deformation curve for the doped sample. This result is in contrast with the undoped specimen where serrations were absent. A magnified view of the serrated-flow behavior in the doped sample is displayed in Figure 42b. Here, the serrations exhibit a stair-step type pattern. The authors reported that the serrated flow exhibited Type-A serrations, although they could have instead been Type-D serrations (see Figure 10) [182].

**Figure 41.** The engineering stress vs. strain curves for the CoCrFeMnNi HEA compressed at strain rates ranging from 6 <sup>×</sup> <sup>10</sup>−<sup>5</sup> to 1 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup>−<sup>1</sup> at 8 K (reproduced from Reference [299] with permission).

**Figure 42.** (**a**) True stress–true strain curves for CoCrFeMnNi HEA tested with and without carbon doping. The (**b**) enlarged image of the true stress–true strain curve and strain hardening rate–true strain curve in the strain range of 17.5–33.5% for the carbon-doped HEA. (Reproduced from Reference [192] with permission.)

The occurrence of the serrated flow in the HEA was attributed to the interplay among the carbon, SFs, and moving dislocations. The sharp increase in the stress during a serration event corresponds to the alteration of the short-range order by the dislocations that is accompanied by the migration of C from octahedral sites to tetrahedral sites and SFs. Since it is energetically unfavorable for the C to remain in the SF, the C will hop back into the octahedral site. Consequently, the SF energy is decreased, thus resulting in the accelerated plastic deformation that is characterized by the plateau region between successive serrations.

#### *3.10. CoCuFeNiTi HEA*

Samal et al. examined the serrated flow in an equiatomic CoCuFeNiTi HEA [162]. Here, cylindrical samples underwent isothermal hot-compression tests at temperatures ranging from 800 to 1000 ◦C and strain rates between 10−<sup>3</sup> and 10−<sup>1</sup> s<sup>−</sup>1. Figure 43a,b presents the true stress vs. true strain for the samples compressed at strain rates of 10−<sup>1</sup> and 10−<sup>3</sup> s<sup>−</sup>1, respectively. For both strain rates, the amplitude of the serrations increased with strain rate (at a given temperature) and a decrease in temperature (at a given strain rate). However, for the sample compressed at a strain rate of 10−<sup>3</sup> s<sup>−</sup>1, the serrations were only prominent for lower temperatures. The authors surmised that the decrease in the size of serrations at higher temperatures was a consequence of the increased thermal-vibration energy required to pin solute atoms. Such a hypothesis has also been discussed in a previous investigation [153]. Furthermore, they suggested that the increase in the serration amplitude at the lower strain rates was due to an increase in the time allotted for atoms to lock dislocations. Table 7 displays a summary of the serration types observed and their corresponding temperatures and strain rates. As can be seen, the serrations type was dependent on the experimental conditions. From the table, it is evident that, at the higher strain rate, the serration type evolved from Types A to A + B with temperature, whereas the serration type transitioned from Type-A + Type-B to Type-B at the lower strain rate.

**Figure 43.** True stress–true strain plots at different temperatures and strain rates of (**a**) 10−<sup>1</sup> and (**b**) 10−<sup>3</sup> s−<sup>1</sup> (reproduced from Reference [162] with permission).


**Table 7.** Summary of the serration type exhibited by the CoCuFeNiTi HEA during compression at strain rates of 1 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>s</sup><sup>−</sup>1–5 <sup>×</sup> <sup>10</sup>−<sup>1</sup> <sup>s</sup>−<sup>1</sup> and temperatures of 800–1000 ◦C (from Reference [162]).

#### *3.11. CoCuFeMnNi HEA*

Sonkusare et al. investigated the deformation behavior of a single-phase FCC CoCuFeMnNi HEA [300]. For their experiments, the HEA samples underwent RT compression tests at strain rates of 10−<sup>3</sup> and 3 <sup>×</sup> 10<sup>3</sup> s−1. To achieve the higher strain rate, a gas gun (6 kg/cm2) was used to propel a 300 mm long striker bar into the sample. Figure 44 displays the true stress vs. true strain plot. From the graph it is apparent that the serrated flow occurred in the sample that was tested at the higher strain rate, while there were no observable serrations present in the deformation behavior of the sample compressed at a strain rate of 10−<sup>3</sup> s<sup>−</sup>1. Electron backscatter diffraction (EBSD) characterization revealed some interesting results. For instance, lenticular deformation twins were observed in the sample tested at the higher strain rate, suggesting that deformation twinning was likely the mechanism responsible for the serrated flow at the higher strain rate condition. Moreover, as compared to the lower strain-rate condition (10−<sup>3</sup> s−1), the microstructure for the sample tested at the higher strain rate consisted of relatively smaller-sized grains, a higher percentage of higher angle grain boundaries, and a smaller density of geometrically necessary dislocations.

**Figure 44.** The true stress vs. true strain for the CoCuFeMnNi HEA compression tested at strain rates of 10−<sup>3</sup> <sup>s</sup>−<sup>1</sup> and 3 <sup>×</sup> <sup>10</sup><sup>3</sup> <sup>s</sup>−<sup>1</sup> (reproduced from Reference [300] with permission).

#### *3.12. CoCrFeNi HEA*

The deformation behavior, during tension, was examined in a CoCrFeNi HEA for temperatures ranging from 4.2 to 293 K [159]. For the experiments, samples were tested at a strain rate of 10−<sup>3</sup> s<sup>−</sup>1. To cool the samples, liquid nitrogen (77 and 200 K), and liquid helium (4.2 K) were used. Figure 45a–d shows the results of the microstructural characterization, as determined by optical microscopy, XRD, and TEM. As can be observed from Figure 45a, the as-cast sample contained nearly equiaxed grains. Subsequent analysis determined that the grains consisted of a mean size of about 13 μm. The results of the XRD characterization, as displayed in Figure 45b, indicates that the as-cast sample consisted of an FCC structure. Figure 45c,d presents the TEM results, which reveals that both dislocations and {111} twins were present in the microstructure of the as-fabricated material.

**Figure 45.** The results of the (**a**) optical microscopy imaging, (**b**) XRD characterization, (**c**) TEM image, and (**d**) selected area electron diffraction results for the CoCrFeNi HEA (reproduced from Reference [159] with permission).

Figure 46 displays the engineering stress vs. engineering strain for the samples tested at temperatures ranging from 4.2 to 293 K for a strain rate of 10−<sup>3</sup> s−1. It is evident from the figure that the serrated flow was only observed in the samples deformed at temperatures of 4.2 and 20 K. Upon analysis, the largest Lyapunov exponent for both conditions was found to be positive (0.05 for 4.2 K, and 0.001 for 20 K), indicating that the serration behavior was chaotic (unstable dynamics). Interestingly, TEM characterization revealed that the sample tested at 4.2 K underwent an FCC to HCP phase transformation during tension testing. Moreover, many high-density defects were present, such as nano-twinning, SFs, and the HCP stacking. From the above results, the authors concluded that the unstable behavior exhibited by the sample tested at 4.2 K was a consequence of the FCC to HCP transformation.

**Figure 46.** The engineering stress vs. engineering strain curves for the CoCrFeNi HEA tested at temperatures ranging from 4.2 to 293 K and a strain rate of 10−<sup>3</sup> s<sup>−</sup>1. The inset displays a magnification of the serrated flow behavior from the specimen tested at 4.2 K. The inset at the bottom-right portion of the figure displays a photograph of the dog-bone-shaped samples, before and after tensile tests. (Reproduced from [159] with permission.)

#### *3.13. HfNbTaTiZr HEA*

Chen et al. investigated the deformation behavior of a BCC HfNbTaTiZr HEA [301]. TEM and atomic probe tomography confirmed that the alloy did not contain any secondary phases. They performed tensile tests, under the displacement control, using a strain rate of <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>s</sup><sup>−</sup>1. During testing, samples were exposed to temperatures of 77, 298, 573, and 673 K. It was observed that, similar to conventional alloys, the yield and flow stress decrease, while the ductility increases with an increase in the temperature. Figure 47 presents the engineering stress–strain curve for the HfNbTaTiZr HEA that was tested at temperatures ranging from 77 to 673 K. As indicated by the graph, visible serrations were only exhibited by the sample that was tested at 673 K. However, it was determined that strain hardening was attributed to both forest hardening and DSA hardening in the specimens tested at 573 and 673 K. The authors surmised that there were two reasons for why DSA was likely a strengthening mechanism in the sample tested at 573 K (despite there being no visible serrations). Firstly, during deformation, the sample tested at the above temperature displayed similar hardening characteristics as the sample tested at 673 K. Secondly, serrations may not always be visible during the DSA [302].

**Figure 47.** The engineering stress vs. engineering strain for the BCC HfNbTaTiZr HEA that was tested at temperatures ranging from 77 to 673 K, at a strain rate of 1 <sup>×</sup> 10−<sup>4</sup> s−<sup>1</sup> (reproduced from Reference [301] with permission).

#### **4. Summary and Future Directions**

Table 8 and Figure 48a–e both display the reported serration types for different kinds of HEAs. From the results, a few trends can be observed. In terms of the experimental conditions, serrations were reported for strain rates and temperatures ranging from <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>5</sup> <sup>s</sup>−<sup>1</sup> to <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>1</sup> <sup>s</sup>−<sup>1</sup> and RT to 1100 ◦C, respectively. The samples exhibited serration Type-A, Type-B, and Type-C serrations, as well as serration type combinations, such as Type-A + Type-B and Type-B + Type-C. On the other hand, none of the HEAs were reported to display Type-A + Type-C, Type-D, or Type-E serrations. The greatest variety of different strain rates were observed for the Type-A serrations. Of the serration types that were observed, Type-B serrations were the most common serration displayed by the HEAs.

**Table 8.** Reported serration type, as found from the literature, for Type-A, Type-B, Type-C, Type-A + Type-B, and Type-B + Type-C for different HEAs tested in a range of strain rates and temperatures. RT: room temperature.



**Table 8.** *Cont.*

The data also reveal some important aspects about the serrated-flow phenomena in HEAs. For instance, out of all the alloys featured in Figure 48a–e and Table 8, the CoCrFeMnNi HEA was the only material to have exhibited all the reported serration types. Furthermore, the carbon-doped CoCrFeMnNi HEA was the only alloy reported to display serrations at RT. This effect of additives was also observed in [94], where Al was found to play a substantial role in the serration behavior of an Al0.5CoCrFeNi HEA. These findings, therefore, highlight the significance of additives to the serrated-flow process in HEAs.

Figure 49a,b displays a hypothetical schematic for the extent of serrations in HEAs as a function of the temperature and strain rate, as based on the reviewed literature. As can be seen in Figure 49a, there are two major temperature ranges where the serrated flow is observed. The first region, which corresponds to temperatures ranging from 4.2 to 77 K (cryogenic), represents the serrations that occur primarily due to twinning mechanisms. The findings from previous investigations suggest that the extent of the serrations may increase with a decrease in the temperature, which is due to an increase in the twinning mechanisms [166]. As for temperatures below 4.2 K, the authors are unaware of any investigations which have observed the serrated flow in this range, and this should therefore be the subject of future work. However, for temperatures between 77 K and RT serrations do not occur. The absence of serrations in this temperature regime is likely due to the absence of both twinning and dislocation pinning [153]. Serrated flow returns at temperatures between RT and 1100 and is primarily caused by the solute pinning of dislocations [152,180,192,303]. The extent of the serrations initially increases due to an increase in the migration speed of the solute atoms, which is accompanied by an acceleration in the rate of dislocation pinning [152]. However, after a certain temperature is exceeded, the enhanced thermal vibration of the solute atoms reduces their ability to pin dislocations, thereby decreasing the extent of serrations. Once the temperature rises above a certain value (1100 ◦C in the

case of the reported HEAs), the thermal vibrations completely overcome the effect of solute atom pinning of dislocations, and therefore no serrated flow is observed [153].

**Figure 48.** A graphical representation of the reported results that are listed in Table 8 for serration Types (**a**) A, (**b**) B, (**c**) C, (**d**) A + B, and (**e**) B + C.

**Figure 49.** *Cont.*

**Figure 49.** Basic schematics of the extent of serrations as a function of (**a**) temperature and (**b**) strain rate.

A similar graph depicting the extent of serrations as a function of the strain rate is presented in Figure 49b. For the lower strain rate regime (10<sup>−</sup>5–1 s−1), dislocation pinning is the primary mechanism underlying the serrated flow. For this regime, the extent of the serrations decreases with an increase in the strain rate, which is perhaps due to the increased propagation speed of dislocations [153]. Once the strain rate exceeds 1 s<sup>−</sup>1, the dislocation velocity is such that no solute atmosphere can be formed [305] and no serrations occur. At strain rates greater than 10<sup>3</sup> s−1, twinning appears to be the primary mechanism [166]. Based on the above study, the extent of the serrations appears to increase with strain rate due to the increase in the twinning. Between these two strain rate regions no, serrations have been reported to occur. The lack of observable serrations during plastic deformation is due to the absence of both twinning and dislocation pinning. It should be noted that, for strain rates less than 10−<sup>5</sup> s−<sup>1</sup> and above 10<sup>3</sup> s<sup>−</sup>1, there is little available in the literature as to whether the serrated flow will occur in these ranges, and this should therefore be the subject of future investigations.

Based on the literature reviewed for the present work, the following important points can be summarized about the serrated-flow phenomenon in HEAs:


Although much has been accomplished regarding the study of the serrated flow phenomena in HEAs, there is still considerable work to be done. Future work could investigate phenomena such as the effects of irradiation displacement damage on the deformation behavior of HEAs. Currently, there are not many investigations that have thoroughly examined how particle irradiation affects the serration behavior in HEAs. One example of such future studies could examine how neutron irradiation and thermal annealing influence the serrated flow in HEAs during either tension or compression. This study would be significant, since it would explore how irradiation-induced changes in the microstructure, such as the introduction of dislocation loops or precipitates into the matrix during particle bombardment, could affect the serrated-flow dynamics. Another important experiment could involve the micropillar compression or nanoindentation testing of samples irradiated by ions at a variety of temperatures and irradiation doses. Here, the avalanche statistics of the as-cast and irradiated specimens could be modeled and analyzed according to the MFT formalism for different compression rates or irradiation conditions. It is expected that such studies would provide the substantial insight into the effects of irradiation on the plastic-deformation behavior of HEAs.

Future investigations could also involve a more comprehensive study on the effects of additives, such as C and Al, as well as the presence of nanoparticles, on the serrated flow in HEAs. For these studies, the effects of additive concentration on the serration behavior in a wide array of HEAs would be investigated. Importantly, these studies would examine how the concentration of impurities would affect such factors as the temperature range or strain rate at which serrations would occur. Furthermore, perhaps it will be determined whether there exists a minimum required concentration for the serrated flow to occur. Machine-learning methods could also be employed to elucidate the relationship among the additive concentration, microstructure (such as dislocation dynamics), and serrated-flow behavior [306–310]. Consequently, a greater understanding of solute atom-dislocation pinning dynamics in HEAs would be achieved.

Additionally, more work is needed to better examine the twinning-induced serration phenomena in HEAs. These experiments could involve mechanically testing (compression and/or tension) HEA specimens at temperatures ranging from cryogenic to elevated temperatures and strain rates of greater than 103 s−1. It would also be important to examine whether there is a minimum temperature or maximum strain rate at which the serrations would occur. For these experiments, the microstructure could be characterized by using techniques such as TEM and EBSD, while the serrated flow behavior could be analyzed and modeled by using different analytical techniques. The results of such experiments would provide fundamental insight on the relation between twinning and the serration behavior in HEAs.

Finally, there should be more studies that would model and analyze the serrated-flow behavior in HEAs (during tension and compression), using alternative methods, such as the complexity-analysis, multifractal-analysis, and the chaos-analysis techniques. Moreover, theoretical models covering a wide range of scales, including first principles, molecular dynamics, dislocation dynamics, finite-element methods, and crystal plasticity, could be used to quantify, predict, and simulate the effects of composition, temperature, strain rate, additive, irradiation, and environment on serrated flows of HEAs. The results of these types of analyses and modeling would be paired with advanced microstructural-characterization techniques, such as in situ TEM, XRD, and neutron diffraction. The combination of the microstructural and analytical/modeling results would likely yield unexpected and illuminating findings regarding

the link between the microstructural behavior and the dynamics of plastic deformation during the serrated flow.

#### **5. Conclusions**

In this review, a clear summary of the serrated flow phenomenon, as it occurs in HEAs, was provided. Importantly, this review includes several key findings as revealed by the literature. For instance, additives, such as C and Al, can lead to the occurrence of the serrated flows in HEAs. In terms of the microstructure, serrations typically occur due to either twinning mechanisms (at cryogenic temperatures) or dislocation pinning by solute atoms at temperatures of RT and above. Interestingly, such behavior is not typically seen in conventional crystalline alloys. It has also been found that the serrated flow is significantly affected by experimental conditions, such as the strain rate and test temperature. In HEAs, several different serration types have been observed, including A, B, C, or combinations of serrations, such as A + B and B + C. It is interesting to note that the serration Type-A + Type-C, Type-D, and Type-E has not been reported to occur in this alloy system. Several different types of analytical methods, including the mean-field theory formalism and the complexity-analysis technique, have been successfully applied to examine the serrated flow in HEAs. The results of the analyses indicated that the serrated flow in HEAs consist of complex dynamical behavior. For example, serrations have exhibited chaotic fluctuations that were attributed to a phase transformation (FCC to HCP phases) at cryogenic temperatures. Lastly, further exploratory work is needed to gain a more fundamental perspective on the serrated-flow phenomenon in HEAs. Some areas in need of additional or more comprehensive research to analyze the serrated flow in HEAs have been identified as the analysis of irradiation effects, additives, such as C and Al, the presence of nanoparticles, twinning, and the use of alternate experimental methods, such as in situ TEM.

**Author Contributions:** Conceptualization, J.B. and S.C.; investigation, J.B., X.X., D.H., B.S., and C.L.; methodology, J.B., X.X., and C.F.; project administration, P.K.L.; supervision, J.B. and P.K.L.; validation, Y.S., A.M.C., and R.F.; resources, J.B.; writing—original draft, all authors; review and editing, C.F., E.I., F.G.M., P.D., R.J.S., and H.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** S.C., C.L., R.F., X.X., and P.K.L. very much appreciate the supports of (1) the US National Science Foundation (DMR1611180 and 1809640) and (2) the US Army Research Office (W911NF-13-1-0438 and W911NF-19-2-0049), with J. Yang, G. Shiflet, D. Farkas, M. P. Bakas, D. M. Stepp, and S. Mathaudhu as program managers.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Activation Volume and Energy for Dislocation Nucleation in Multi-Principal Element Alloys**

#### **Sanghita Mridha, Maryam Sadeghilaridjani and Sundeep Mukherjee \***

Department of Materials Science and Engineering, University of North Texas, Denton, TX 76203, USA; sanghitamridha@my.unt.edu (S.M.); Maryam.Sadeghilaridjani@unt.edu (M.S.)

**\*** Correspondence: sundeep.mukherjee@unt.edu; Tel.: +1-940-565-4170; Fax: +1-940-565-2944

Received: 25 January 2019; Accepted: 21 February 2019; Published: 23 February 2019

**Abstract:** Incipient plasticity in multi-principal element alloys, CoCrNi, CoCrFeMnNi, and Al0.1CoCrFeNi was evaluated by nano-indentation and compared with pure Ni. The tests were performed at a loading rate of 70 μN/s in the temperature range of 298 K to 473 K. The activation energy and activation volume were determined using a statistical approach of analyzing the "pop-in" load marking incipient plasticity. The CoCrFeMnNi and Al0.1CoCrFeNi multi-principal element alloys showed two times higher activation volume and energy compared to CoCrNi and pure Ni, suggesting complex cooperative motion of atoms for deformation in the five component systems. The small calculated values of activation energy and activation volume indicate heterogeneous dislocation nucleation at point defects like vacancy and hot-spot.

**Keywords:** multi-principal element alloys; dislocation nucleation; activation volume; activation energy; nano-indentation; high/medium entropy alloys

#### **1. Introduction**

Traditional alloy design employs the addition of alloying elements to a principal constituent to enhance its properties via multi-phase complex microstructure. Multi-principal element alloys (MPEAs) represent a new generation of material system consisting of several elements in equimolar or near equimolar proportions. Despite the complex chemistry, high configurational entropy may favor formation of a simple solid solution instead of complex intermetallic compounds in these alloys, which are also referred to as high entropy alloys (HEAs) [1]. Certain range in atomic size difference, electronegativity difference, and mixing entropy/enthalpy of the constituent elements result in multiple concentrated phases in the microstructure [2–5].

The mechanical properties of MPEAs have been widely studied in recent years over a range of temperatures for potential applications in nuclear and aerospace industries. These include tensile/compressive properties, fatigue behavior, and fracture toughness [6–12]. It has been hypothesized that MPEAs have a distorted lattice structure which makes dislocation movement difficult and non-traditional in these alloy systems [1,6–8,13–18]. To understand dislocation nucleation and the activation process, nano-indentation technique has been widely used. The displacement burst (pop-in) in the nano-indentation load versus contact depth curve is attributed to dislocation nucleation. The "pop-in" represents the transition from elastic to plastic deformation in materials and typically happens at a small indentation depth of <100 nm [19–26]. The stress associated with the pop-in is approximately equal to the theoretical strength of the material and happens in small indented volume with a very low probability of having pre-existing dislocations. Therefore, many researchers have attributed the pop-in event to dislocation nucleation [21,25,27–33], and others have proposed vacancy assisted mechanisms [34]. It has been demonstrated that the pop-in during nano-indentation is time and temperature dependent, a kinetically limiting process that requires activation energy. It occurs in

thermally and mechanically favorable positions, which makes it a probabilistic event, i.e., the load at which the pop-in takes place is distributed over a range rather than occurring at a fixed value each time, and this distribution is temperature and time dependent.

Nano-indentation technique has been extensively used to study the local scale elastic-plastic deformation behavior of metals/alloys and to determine the critical shear stress required for dislocation nucleation [19–26]. However, there are few reports and limited understanding of the effect of intrinsic length-scale and submicron plasticity in MPEAs. Wang et al. [35] and Zhu et al. [36] have investigated the activation volume and energy for the onset of plastic deformation in two body centered cubic (BCC) HEAs (of compositions NbTaTiZr and NbMoTaTiZr) and a face centered cubic (FCC) HEA (of composition CoCrFeMnNi) by nano-indentation. They reported a larger activation volume for HEAs compared to conventional metals, attributing it to complex nucleation mechanism. Still, there is limited understanding of the dependence of dislocation nucleation on alloy composition and microstructure, and there are multiple discrepancies in the published studies [2,36].

Here, incipient plasticity and dislocation nucleation were studied using nano-indentation for three different multi-principal element alloys, CoCrNi, CoCrFeMnNi, and Al0.1CoCrFeNi. All the three alloys showed a single-phase FCC microstructure without any secondary phases. The three-component CoCrNi alloy showed higher strength than five-component CoCrFeMnNi [7,37] and was chosen to study the effect of the number of constituents. A comparison was made with a pure metal, Ni, to understand solid solution strengthening effects. In addition, Al0.1CoCrFeNi alloy was chosen because it is a model FCC system for comparison with the results of CoCrFeMnNi. The present study paves the way for a fundamental understanding of small-scale deformation mechanisms in multi-principal element alloys as a function of alloy chemistry and microstructure.

#### **2. Experimental**

Alloys with nominal composition of CoCrNi, CoCrFeMnNi, and Al0.1CoCrFeNi (in mole fractions) were prepared by arc melting the constituent pure elements (purity > 99.99%) in Ti-gettered Ar atmosphere. As-cast alloys were rolled up to a 70% reduction in thickness. After rolling, they were annealed at 1173 K for 20 h to obtain equiaxed dislocation-free grains and to remove residual stress from processing. The annealed samples were polished with SiC abrasive paper followed by microfiber cloth with 1 μm diamond suspension. The polished samples were then put in vibromet with 0.04 μm colloidal silica suspension to produce a surface finish equivalent in texture to an electro-polished sample. Crystal structure and phase identification were done using Rigaku III Ultima X-ray diffractometer (XRD Rigaku Corporation, Tokyo, Japan) with Cu-Kα radiation with wavelength of 1.54 Å. Scanning electron microscopy (SEM) was done using FEI Quanta ESEM (FEI Company, Hillsboro, OR, USA) to analyze the grain size and microstructure of the alloys.

Nano-indentation (Bruker, Minneapolis, MN, USA) was performed using a Berkovich diamond probe. The pop-in behavior of alloys was evaluated with a maximum load of 350 μN at 298 K, 373 K, and 473 K. One hundred indentations were made at each temperature with 10 μm distance between two indents to avoid overlap of their plastic zones. The indentations were done in load control mode with a constant loading rate of 70 μN/s. Elevated temperature indentations were performed using XSol600 heating stage with bottom and top heating to avoid any thermal gradients. To prevent oxidation of the sample, a continuous flow of Ar + 5% H2 gas mixture was maintained with a flow rate of 1 lit/min.

#### **3. Results and Discussion**

Figure 1a–c show the XRD patterns for the three alloys studied, which confirmed that they formed a simple FCC solid solution and were free of any secondary phases as reported previously [38–40]. The backscattered SEM images of the alloys are shown in Figure 2a–c. All three alloys showed equiaxed microstructure with an average grain size of ~30–60 μm. They also showed a significant number of annealing twins, which indicates low stacking fault energy in these alloys.

**Figure 1.** X-ray diffraction (XRD) patterns of multi-principal (**a**) CoCrNi, (**b**) CoCrFeMnNi, and (**c**) Al0.1CoCrFeNi alloys.

**Figure 2.** Backscattered scanning electron microscopy (SEM) images of (**a**) CoCrNi, (**b**) CoCrFeMnNi, and (**c**) Al0.1CoCrFeNi show equiaxed grains after annealing with grain size ~30–60 μm.

Deformation during nano-indentation is initially elastic and follows the Hertzian equation [34].

$$P = 4/3E\_rR^{\frac{1}{2}}h^{\frac{3}{2}} \tag{1}$$

where, *P* is the load applied by the indenter, *R* is radius of the tip, and *h* is the indentation depth, *Er* is the reduced modulus of the indenter-sample combination and is defined by: <sup>1</sup> *Er* <sup>=</sup> <sup>1</sup>−*v*<sup>2</sup> *<sup>E</sup>* <sup>+</sup> <sup>1</sup>−*v*<sup>2</sup> *i Ei* . *E*, *ν*, *Ei* = 1141 GPa, and *ν<sup>i</sup>* = 0.07 are the Young's modulus and Poisson's ratio of the sample and diamond indenter, respectively. Figure 3a shows the typical load-displacement curve obtained from nano-indentation along with the Hertzian fit. The reduced modulus of the alloys was obtained from the fitting coefficient of the elastic part and values are summarized in Table 1 and are similar to earlier reports [41,42]. The initial part of the load-displacement curve followed Hertzian contact theory and subsequently deviated due to plastic deformation or elastic-to-plastic transition [34]. The load and displacement at the deviation point was determined for each indent. Figure 3b shows the temperature dependence of the pop-in load for the CoCrFeMnNi alloy. The load at first pop-in was found to decrease with increasing temperature; other alloys (CoCrNi and Al0.1CoCrFeNi) and pure Ni showed a similar trend. This thermal softening has also been reported in several previous studies [34,36].

**Table 1.** Reduced modulus, activation volume, and activation energy of present face-centered cubic (FCC) alloys.

**Figure 3.** (**a**) Representative load-displacement curve of CoCrFeMnNi alloy at room temperature along with the Hertzian; (**b**) load-displacement curve of CoCrFeMnNi alloy at different temperatures along with the Hertzian fitting for the elastic section.

Figure 4 shows the statistical distribution of the pop-in load at room temperature (298 K) and 473 K for Ni, CoCrNi, CoCrFeMnNi, and Al0.1CoCrFeNi. The experimentally obtained data at 373 K followed a similar trend and has not been included in the figure for clarity. It is also evident from Figure 4 that the pop-in load decreased with increasing temperature. The temperature dependence of displacement burst in multi-principal alloys indicates a stress-biased thermally activated mechanism similar to pure metals [34]. The shear stress beneath the indenter at pop-in is evaluated using Hertzian contact theory as [34]:

$$p\_m = \left(\frac{6PE\_r^2}{\pi^3 R^2}\right)^{1/3} \tag{2}$$

$$
\pi\_m = 0.31 p\_m \tag{3}
$$

where, *pm* is the mean contact pressure and *τ<sup>m</sup>* is the maximum shear stress at pop-in, which was in the range of ~ <sup>μ</sup> <sup>16</sup> – <sup>μ</sup> <sup>10</sup> (where μ is the shear modulus) at room temperature for all alloys. As the ideal shear strength of a metal also lies in a similar range ( <sup>μ</sup> <sup>30</sup> – <sup>μ</sup> <sup>5</sup> ) [36], it may be concluded that the pop-in event involves dislocation nucleation. Using the probabilistic distribution of pop-in stress, an analytical solution has been proposed to evaluate the activation volume and activation energy associated with the displacement burst [34]. Nucleation of dislocation involves an activation barrier which may be overcome by mechanical work done by the indenter or by the combined effect of thermal and mechanical work. The nucleation rate per unit volume, . *n* is represented by [34]:

$$
\dot{m} = \eta \exp\left(-\frac{\varepsilon - \sigma V}{kT}\right) \tag{4}
$$

where η is a pre-exponential frequency factor, is the activation energy barrier for the process, and σ*V* represents the stress bias. In the present condition, the stress bias is the maximum shear stress beneath the indenter. The activation volume of the process may be expressed as [34]:

$$V = \frac{\pi}{0.47} \left(\frac{3R}{4E\_R}\right)^{2/3} kT \alpha \tag{5}$$

where, <sup>α</sup> is a time-independent parameter and is obtained from the slope of ln[−ln(1 − *<sup>F</sup>*)] versus *<sup>P</sup>*1/3 plot (as shown in Figure 5), and *F* is the cumulative probability of the event (as shown in Figure 4). Figure <sup>5</sup> shows the ln[−ln(1 − *<sup>F</sup>*)] versus *<sup>P</sup>*1/3 for Ni and the three MPEAs at room temperature and 473 K. Using averaged α value obtained from the slope of the curves and Equation (5), the activation volume was calculated. Activation volume for the three alloys and Ni are included in Table 1.

**Figure 4.** Cumulative fraction of experiment loads at the pop-in events plotted at 298 K and 473 K for (**a**) Ni, (**b**) CoCrNi, (**c**) CoCrFeMnNi, and (**d**) Al0.1CoCrFeNi alloys.

**Figure 5.** Plot of ln[−ln(1 <sup>−</sup> *<sup>F</sup>*)] vs. *<sup>P</sup>*1/3 at 298 K and 473 K to determine the activation volume from experimental data analysis for (**a**) Ni, (**b**) CoCrNi, (**c**) CoCrFeMnNi, and (**d**) Al0.1CoCrFeNi.

Equation (4) indicates that also plays an important role in the nucleation process. The activation enthalpy or may be evaluated from the pop-in load and activation volume using the equation [34]:

$$P^{1/3} = \gamma kT + \frac{\pi}{0.47} \left(\frac{3R}{4E\_R}\right)^{2/3} \frac{\varepsilon}{V} \tag{6}$$

Figure 6 shows the *P*1/3 vs. temperature (*T*) curves for pure Ni and the MPEAs at three different cumulative frequencies (*F*). *P*1/3 vs. *T* shows a linear relation with slope γ*k* and the intercept at 0 K in *y*-axis representing the second part of Equation (6). With the obtained *P*1/3 at 0 K and the activation volume, was evaluated. This represents the energy required for nucleation at 0 K, i.e., all the work done is mechanical work, and there is no thermal work. The activation energy for pure Ni and the MPEAs are summarized in Table 1.

The average activation volume for pop-in in Ni was determined to be 4.12 Å3, which falls within a similar range for FCC-Pt [34] and hexagonal close packed (HCP)-Mg [43]. The activation volume and activation energy evaluated for MPEAs using first-order shear bias statistical model were in the range of 4 to 9 Å3 and 0.23 to 0.50 eV, respectively. These values are much lower than that required for homogenous dislocation nucleation which is higher than 1 eV. The discrepancy in activation enthalpy between the experimentally measured value and that expected for homogenous nucleation has been explained by the presence of subcritical loops and thermal fluctuations, while other studies have ruled out these possibilities [44–46]. Vacancy migration during nano-indentation has been proposed as another possible mechanism [2]. But the self-diffusion rate is very slow at room temperature, and vacancy migration energy lies in the range of 1–2 eV. The obtained activation enthalpy for the three MPEAs is much lower than the vacancy migration energy. In addition, due to lower diffusivity in equi-atomic multi-component alloys than pure metals, vacancy migration is more difficult in these alloys. Several simulation studies have shown that surface ledges may act as potential sites for heterogeneous dislocation nucleation by reducing the activation energy [47,48]. Another proposed mechanism for heterogeneous dislocation nucleation involves hot-spots. At high temperatures, thermal vibrations generate asymmetry which create hot-spot defects below the surface (in the bulk) and act as potential sites for dislocation nucleation, whereas at 0 K the dislocation nucleates at surface defects [45,49]. In the current study, it was observed that the displacement at pop-in decreased with increasing temperature. The average pop-in depth for CoCrFeMnNi alloy corresponding to 298 K, 373 K, and 473 K was 8 nm, 5 nm, and 3 nm, respectively. A similar trend was observed for the other two alloys, CoCrNi and Al0.1CoCrFeNi. The decreasing trend of pop-in depth with increasing temperature supports the hot-spot mechanism because with increasing temperature the fraction of hot-spot defects increases. This leads to lowering of the depth and load for pop-in. Vacancy clusters and impurity atoms have also been proposed as potential sites for dislocation nucleation [34]. Since the alloys were air cooled after annealing, there is a possibility of having vacancy concentration higher than the equilibrium concentration.

**Figure 6.** Plot of *P*1/3 vs. *T* to evaluate the activation enthalpy of the first burst for (**a**) Ni, (**b**) CoCrNi, (**c**) CoCrFeMnNi, and (**d**) Al0.1CoCrFeNi alloys.

The activation volume and energy of incipient plasticity for CoCrFeMnNi and Al0.1CoCrFeNi were roughly two times that of pure Ni. This indicates that in the case of MPEAs, the cooperative motion of several atoms is necessary for deformation in comparison with one to one atomic motion in pure metals. The activation volume and energy of the three-component CoCrNi alloy was marginally higher than pure Ni (14% higher) but significantly lower compared to the five-component alloys (~50% lower). This suggests that the dislocation mechanism in this medium entropy alloy (CoCrNi) may be similar to pure metals. The activation volume for dislocation nucleation in BCC TiZNbTa

and TiZrNbTaMo HEAs were determined to be ~30 Å3 [35] which is more than that of the current studied MPEAs. It should also be noted that the obtained activation volume and activation energy for CoCrFeMnNi in the current work is different compared to that earlier reported by Zhu et al. for the same composition [36]. This may be attributed to the sample preparation conditions and the resulting surface state.

#### **4. Conclusions**

Shallow depth nano-indentation was performed for multi-principal five-component Al0.1CoCrFeNi and CoCrFeMnNi alloys and three-component CoCrNi alloy to elucidate the mechanism associated with the onset of plasticity in these alloys. Nano-indentation was performed at three different temperatures of 298 K, 373 K, and 473 K. The experimentally obtained data were analyzed within the framework of first-order shear bias statistical model, and the activation volume and activation enthalpy of the process were calculated. The following can be concluded from the obtained experimental and analytical results:


**Author Contributions:** Conceptualization, S.M. (Sundeep Mukherjee); methodology, S.M. (Sanghita Mridha) and S.M. (Sundeep Mukherjee); validation, S.M. (Sanghita Mridha) and M.S.; formal analysis, M.S.; investigation, S.M. (Sanghita Mridha) and M.S.; resources, S.M. (Sundeep Mukherjee); data curation, S.M. (Sanghita Mridha) and M.S.; writing—original draft preparation, S.M. (Sanghita Mridha) and M.S.; writing—review and editing, M.S. and S.M. (Sundeep Mukherjee); visualization, M.S.; supervision, S.M. (Sundeep Mukherjee); project administration, S.M. (Sundeep Mukherjee).

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **High-Temperature Nano-Indentation Creep Behavior of Multi-Principal Element Alloys under Static and Dynamic Loads**

#### **Maryam Sadeghilaridjani and Sundeep Mukherjee \***

Department of Materials Science and Engineering, University of North Texas, Denton, TX 76203, USA; Maryam.Sadeghilaridjani@unt.edu

**\*** Correspondence: sundeep.mukherjee@unt.edu; Tel.: +1-940-565-4170; Fax: 1-940-565-2944

Received: 14 January 2020; Accepted: 9 February 2020; Published: 13 February 2020

**Abstract:** Creep is a serious concern reducing the efficiency and service life of components in various structural applications. Multi-principal element alloys are attractive as a new generation of structural materials due to their desirable elevated temperature mechanical properties. Here, time-dependent plastic deformation behavior of two multi-principal element alloys, CoCrNi and CoCrFeMnNi, was investigated using nano-indentation technique over the temperature range of 298 K to 573 K under static and dynamic loads with applied load up to 1000 mN. The stress exponent was determined to be in the range of 15 to 135 indicating dislocation creep as the dominant mechanism. The activation volume was ~25*b*<sup>3</sup> for both CoCrNi and CoCrFeMnNi alloys, which is in the range indicating dislocation glide. The stress exponent increased with increasing indentation depth due to higher density and entanglement of dislocations, and decreased with increasing temperature owing to thermally activated dislocations. The results for the two multi-principal element alloys were compared with pure Ni. CoCrNi showed the smallest creep displacement and the highest activation energy among the three systems studied indicating its superior creep resistance.

**Keywords:** multi-principal element alloys; creep; nano-indentation; stress exponent; activation volume; activation energy

#### **1. Introduction**

Material degradation related to creep is a serious concern reducing the efficiency and service life of components in different structural applications. Therefore, there is strong demand for materials with inherently high creep resistance. Multi-principal element alloys (MPEAs) have attracted much attention as a new generation of structural materials with excellent mechanical properties including creep resistance [1–3]. MPEAs are composed of several elements in equiatomic or near-equiatomic proportion and despite complex chemistry, they usually form a simple single- or multi-phase solid solution [1–3]. The presence of different sized atoms in the unit cell leads to high degree of lattice strain in MPEAs [1] and results in high hardness and strength, good fracture toughness at cryogenic and elevated temperatures, high fatigue resistance, good wear, erosion and corrosion resistance [3–7]. Local lattice distortion, nano-clustering, and inhomogeneity at microstructural length scale in MPEAs [8] may significantly affect their mechanical properties including creep resistance. Therefore, probing the small-scale deformation behavior and local creep processes in MPEAs is critical in establishing their application worthiness. Nano-indentation technique has been widely used to characterize the small-scale deformation behavior of materials [9–15]. However, there are limited reports on the creep behavior of MPEAs using nano-indentation, and majority of the studies are at room temperature with a maximum load of 100 mN [16–22]. There are no reports on the deformation behavior of MPEAs as a function of temperature comparing static and dynamic loads. This is important in wide ranging applications where components are subject to both static and cyclic loads such as aerospace, nuclear, automotive, and oil and gas industries [12].

Here, we report the nano-indentation creep behavior of two MPEAs, namely CoCrNi and CoCrFeMnNi. These were chosen as model alloys with excellent mechanical properties [23] and a single-phase face centered cubic (FCC) microstructure [10,24]. Nano-indentation creep was studied under static and dynamic loads as a function of temperature and peak load. Stress exponent, activation volume, and activation energy of the alloys were evaluated. Nickel (Ni) was used as the reference metal with FCC crystal structure for comparison with the two MPEAs.

#### **2. Experimental**

Alloys with nominal compositions of CoCrNi and CoCrFeMnNi in equimolar ratios were prepared by arc-melting high purity elements (>99.9%) in a Ti-gettered argon atmosphere. To ensure chemical homogeneity, the ingots were flipped and remelted several times. As-cast alloys and Ni were then rolled up to 70% reduction in thickness followed by annealing at 1173 K for 20 h to get equiaxed grains with low dislocation density. The annealed samples were polished with silicon carbide papers followed by diamond suspension to a mirror finish for microstructural characterization and nano-mechanical tests. Rigaku III Ultima X-ray diffractometer (XRD, Rigaku Corporation, Tokyo, Japan) with 1.54 Å wavelength Cu-Kα radiation was used for crystal structure and phase characterization of the alloys. Microstructure, grain size, and grain orientation of the alloys were characterized by scanning electron microscopy (SEM) using FEI Quanta ESEM (FEI Company, Hillsboro, OR, USA) and electron backscatter diffraction (EBSD) technique.

Nano-indentation creep tests were done using a TI Premier Triboindenter (Bruker, Minneapolis, MN, USA) equipped with XSol600 heating stage for heating the samples up to 600 ◦C. To avoid oxidation, the tests were performed in a mixture of Ar + 5% H2 gas environment. A Berkovich sapphire tip was used in all creep tests. The initial tip calibration was done with a standard fused quartz reference sample. For each material, we adopted two different types of tests: (i) static constant load and (ii) dynamic mechanical analyze (DMA). The static creep tests were done by ramping the load to 500 mN and 1000 mN at temperatures of 298 K, 423 K, and 573 K and then held at maximum load for 120 s to determine creep response followed by unloading. The aim was to compare creep performance of the selected alloys as a function of temperature at two different loads. High load was used for the static creep tests to minimize the effect of surface and avoid indentation size effect (ISE). In nano-DMA tests, the selected loads were 50 mN, 100 mN, 500 mN and 1000 mN to study the ISE on dynamic creep behavior at the two temperatures of 298 K and 423 K. The frequency and amplitude were set to 100 Hz and 10% of the peak load, respectively. In all tests, a high loading rate of 20 mN/s was chosen to minimize plastic deformation during the loading segment so creep primarily occurred during the dwell time. Prior to each creep test at elevated temperature, the sample was held at the prescribed set point for at least 20 min for temperature stabilization and the indenter tip was maintained close to the sample to reduce temperature gradient. The thermal drift was automatically corrected by the Triboindenter software and was between 0.05–0.1 nm/s during testing. The effect of drift was also minimized by using large indentation depth and a short hold time [25]. At least 16 indents were done for each condition and the distance between each indent was larger than 100 μm to avoid interaction of their plastic zones.

#### **3. Results**

Figure 1 summarizes the microstructural characterization for CoCrNi, CoCrFeMnNi, and pure Ni. The backscattered SEM images of the homogenized samples are shown in Figure 1a,c,e with the insets showing X-ray diffraction patterns for the alloys. All the alloys showed single-phase FCC crystal structure without any secondary phases or precipitates. Figure 1b,d,f show the EBSD images of CoCrNi, CoCrFeMnNi, and Ni, respectively, indicating that all the alloys had equiaxed grains with the inset showing the grain size distribution. The grain sizes for CoCrNi were in the range of 5–50 μm (average

≈ 19 μm) and that for CoCrFeMnNi were in the range of 5–55 μm (average ≈ 22 μm). Ni showed a grain size distribution of 5–65 μm with an average of 30 μm. Significant numbers of annealing twins were observed in the microstructures indicating low stacking fault energy in the MPEAs.

**Figure 1.** Backscattered SEM images with XRD patterns as insets for (**a**) CoCrNi, (**c**) CoCrFeMnNi, and (**e**) pure Ni. EBSD maps with grain size distribution as insets for (**b**) CoCrNi, (**d**) CoCrFeMnNi, and (**f**) pure Ni.

Creep behavior under static load was evaluated as a function of temperature at two different loads. The load and displacement data recorded during nano-indentation creep of CoCrFeMnNi MPEA is shown in Figure 2a,b as a function of temperature. Figure 2a shows the results at 500 mN load while Figure 2b shows the data at 1000 mN. For each test, an array of 4 × 4 indents was made with 100 μm distance between each indent, therefore covering grains with several different orientations. The plots for Ni and CoCrNi were very similar and not included here. At the same load, indentation depth in CoCrFeMnNi sample increased with increasing temperature from 298 K to 423 K, indicating reduction in hardness. The corresponding change in displacement during hold time was used to analyze the time-dependent deformation behavior of the alloys. This is plotted in Figure 2c,d for CoCrFeMnNi at temperatures of 298 K, 423 K, and 573 K and peak load of 500 mN and 1000 mN, respectively. The creep displacement increased rapidly with time in the beginning followed by slowing down of the rate of increase. Fitting of the experimental data using an empirical relation for creep displacement is shown

and elaborated later in the discussion section. Figure 2e,f show the magnitude of creep displacement as a function of temperature for Ni, CoCrNi, and CoCrFeMnNi alloys at the loads of 500 mN and 1000 mN, respectively. The maximum creep displacements were in the range of 50 nm to 400 nm, depending on the alloy, holding load, and temperature. Each data point in Figure 2e,f is the average of at least 16 indents with the error bar including possible influence of crystal orientation. Increase in temperature and peak load increased creep displacement for pure Ni since it is a thermally activated process and diffusion is enhanced at elevated temperature and high load. However, for the MPEAs, distinctly different behavior was seen compared to pure Ni. At 500 mN load, the creep displacement in the case of CoCrNi first increased from 298 K to 423 K and then decreased with further increase of temperature to 573 K. For CoCrFeMnNi, the creep displacement decreased with increasing temperature. This may be attributed to generation of partial dislocations at these intermediate temperatures and/or solute drag effects which restrict dislocation mobility [26–28]. Overall, CoCrNi showed the smallest creep displacement among the three systems for the load and temperature range investigated.

**Figure 2.** Typical load-displacement plot after nano-indentation creep tests for CoCrFeMnNi alloy as a function of temperature under applied load of (**a**) 500 mN and (**b**) 1000 mN. Creep displacement versus hold time for CoCrFeMnNi alloy at different temperatures under load of (**c**) 500 mN and (**d**) 1000 mN. Total creep displacement as a function of temperature for Ni, CoCrNi, and CoCrFeMnNi alloys at (**e**) 500 mN and (**f**) 1000 mN.

The local time-dependent plastic deformation of CoCrNi and CoCrFeMnNi MPEAs was studied under dynamic load and compared with pure Ni. The samples were loaded to 50 mN, 100 mN, 500 mN, and 1000 mN and held for 120 s under dynamic loads. In each test, the amplitude of load was fixed at 10% of the peak load. Figure 3a,b show the representative creep displacement versus holding time at various loads for CoCrFeMnNi at 298 K and 423 K. The maximum creep depth increased with increasing load from 50 mN to 1000 mN with similar behavior observed for Ni and CoCrNi. Figure 3c,d show the magnitude of maximum creep displacement as a function of load for all studied alloys at 298 K and 423 K. Increasing peak load led to increase in creep displacement due to dislocation activation. Total creep displacement was found to be in the order of CoCrNi < Ni < CoCrFeMnNi, which was in agreement with the data obtained from static loading. The effect of surface oxidation on the measured creep displacement is expected to be minimal due to the use of Ar + 5% H2 gas environment and higher loads. There was no change in surface appearance of the samples after the high-temperature tests indicating minimal oxidation.

**Figure 3.** Representative creep displacement versus hold time for CoCrFeMnNi alloy as a function of loads at (**a**) 298 K and (**b**) 423 K. Maximum creep displacement for the three alloys at (**c**) 298 K and (**d**) 423 K showing larger creep depth with increase in load.

#### **4. Discussion**

Creep displacement, *h*, is a function of time, *t*, and follows the empirical relation [18]:

$$h(t) = h\_0 + a(t - t\_0)^p + kt \tag{1}$$

where *h*<sup>0</sup> and *t*<sup>0</sup> are the initial depth and time during creep period and *a*, *p*, and *k* are fitting constants. The dashed lines in Figure 2c,d represent the fitting curves corresponding to Equation (1) with

correlation coefficient *R*<sup>2</sup> > 0.95. For self-similar indentation probe similar to Berkovich used in the current study, indentation strain rate . ε and hardness (*H*) were obtained as [18]:

$$
\dot{\varepsilon} = \frac{1}{h} \frac{dh}{dt} \tag{2}
$$

and

$$H = \frac{P}{24.5h\_c^2} \tag{3}$$

where *dh dt* is the first derivative of the displacement-time curve (i.e., Equation (1)) with respect to *t*, *P* is applied load, *hc* is contact depth given by *hc* = *h*max − 0.75*P*/*S* for Berkovich indenter, and *h*max and *S* are maximum penetration depth and material stiffness, respectively. The hardness measured from Equation (3) was divided by 0.9 for sink-in and 1.02 for pile-up [29,30]. The susceptibility for pile-up or sink-in in nano-indentation tests can be found by the ratio of final indentation depth to the maximum indentation depth (*hf*/*h*max) with *hf*/*h*max ≥ 0.7 indicating pile-up and *hf*/*h*max < 0.7 indicating sink-in [31]. Since plastic deformation is a thermally activated process, stress exponent (reciprocal of strain rate sensitivity) provides valuable insight into creep deformation mechanism [18]. The creep stress exponent was calculated for the two MPEAs and pure Ni and its dependence on load and temperature was evaluated.

The creep stress exponent (denoted by *<sup>n</sup>*) was calculated from the slope of ln . ε versus ln *H* curves [18]. Figure 4 shows the creep stress exponent for the two MPEAs and pure Ni under static load to evaluate the effect of temperature and under dynamic (DMA) mode to study the effect of peak load or indentation depth. Static test analysis was done at two different loads of 500 mN and 1000 mN as shown in Figure 4a,b, respectively. Dynamic test analysis was done at two different temperatures of 298 K and 423 K as shown in Figure 4c,d, respectively. The *n* value for each condition was obtained from an average of 16 independent indentations. The average *n* values for the three systems were in the range of 15 to 135 in static tests. Creep stress exponent defines the creep mechanism. Typically, *n* = 1 is associated with diffusion creep (by lattice or grain boundary diffusion), *n* = 2 with grain boundary sliding, and *n* > 3 with dislocation creep [23]. Therefore, for all the current alloys, deformation was dominated by dislocation creep (climb or glide). The stress exponent for Ni dropped from 135 at 298 K to 50–60 at 573 K, likely due to thermally activated dislocations at elevated temperature. In a similar study, the stress exponent was found to be in the range of 20–60 for IN-718 at room temperature and decreased to 8–18 at 923 K [32]. The magnitude of stress exponent depends on the balance between generation and annihilation of dislocations [33]. Larger value of stress exponent is typically obtained when more dislocations are generated and involved during deformation [33]. The decrease in *n* value with increasing temperature may be due to enhancement in dislocation diffusivity and thermal recovery at higher temperature, so annihilation competes with dislocation generation. The negligible change in stress exponent as a function of temperature for CoCrFeMnNi HEA may be attributed to dislocation glide dictated deformation, which is thermally insensitive [34]. Since glide does not require interatomic diffusion (unlike climb), it is the dominant mechanism of deformation even at low temperature and is thermally insensitive [34]. Transmission electron microscopy (TEM) studies of MPEAs showed that in the early stages of plastic deformation, glide of dislocations on {111} planes was the dominant mechanism [35]. Due to low stacking fault energy and partial dislocations in MPEAs, cross slip is hindered, and plasticity occurs by dislocation glide [35]. The temperature independent stress exponent for CoCrFeMnNi may be attributed to sluggish diffusion [1,2], which would favor glide over climb dominated deformation. For pure Ni, deformation was climb dominated over the temperature range studied. For CoCrNi, the sharp drop in creep stress exponent is likely from change in climb dominated creep at low temperatures (similar to pure Ni) to glide dominated creep at high temperatures (similar to CoCrFeMnNi). In summary, the stress exponent for CoCrNi was close to that of pure Ni at room temperature but became similar to CoCrFeMnNi MPEA at the higher temperature. Recent atomistic modeling suggests that the CoCrNi system shows partial chemical ordering at the atomic scale at

room temperature while becoming random solid solution at higher temperatures [36]. Partial local ordering may act as barrier for dislocation movement explaining the much higher stress exponent for CoCrNi at 298 K compared to CoCrFeMnNi. Figure 4c,d show the variation in stress exponent for the dynamic tests. As the peak load increased from 50 mN to 1000 mN, the *n* value increased at both the temperatures of 298 K and 423 K, which is attributed to ISE [37,38]. During the loading process, dislocations are generated in the plastic deformation region beneath the indenter and their density is directly proportional to the load or depth [39]. At low load, dislocation generation rate is slow, and the stress exponent is low. At high load, dislocation generation rate is fast, and they interact and entangle with one another leading to increase in the stress exponent with value up to 100 for pure Ni [19,40]. The observed ISE of *n* may also be related to the mobility and diffusion of dislocations and how far they are from the free surface [41]. At smaller depth, dislocations are closer to the free surface and therefore have higher mobility and diffusion leading to lower *n*.

**Figure 4.** Stress exponent versus temperature from static test for Ni, CoCrNi and CoCrFeMnNi alloys at (**a**) 500 mN and (**b**) 1000 mN showing decrease of stress exponent with increasing temperature for Ni and negligible dependence of stress exponent on temperature for CoCrFeMnNi HEA. Stress exponent versus load from dynamic test for all alloys at (**c**) 298 K and (**d**) 423 K showing indentation size effect for stress exponent.

The high values of stress exponent obtained from nano-indentation of the three alloys may be attributed to the complex stress state underneath the indenter [16,21,38]. The stress exponent is a complex function of microstructure, mobile dislocation density and/or the activation area underneath the indenter [42]. The *n* value for Ni was significantly larger than those of MPEAs, even though the rate-controlling mechanism was the same for all three systems studied (i.e., dislocation dominated deformation mechanism) with almost identical microstructure. From Figure 4, it was concluded that the stress exponent determined from dynamic tests was slightly lower than that in static test. This may be because of oscillatory load which resulted in better propagation of dislocations and reduction of *n*.

For further insight into the mechanism of creep, the activation volume (*V\**) for the three systems was evaluated as [43]:

$$V^\* = \frac{3 \cdot \sqrt{3} \cdot k \cdot T \cdot n}{H} \tag{4}$$

where *k* is Boltzmann constant, *H* is hardness, *n* is stress exponent and *T* is temperature. The activation volume depends on the stress exponent and hardness. The obtained activation volume for the three systems averaged over the three temperatures (298 K, 423 K, and 573 K) and two loads (500 mN and 1000 mN) were 2.2 <sup>±</sup> 0.1 nm3 (~143*b*3), 0.4 <sup>±</sup> 0.11 nm3 (~25*b*3) and 0.43 <sup>±</sup> 0.1 nm3 (~26*b*3) for Ni, CoCrNi, and CoCrFeMnNi, respectively. Here, the Burgers vector (*b*) for the FCC alloys was calculated as *b* = 1/2*a*<sup>0</sup> [110], where *a*<sup>0</sup> is the lattice parameter with values of 0.352 nm, 0.3567 nm and 0.3597 nm for Ni, CoCrNi, and CoCrFeMnNi, respectively [35,44]. *V\** in the range of 100*b*3–1000*b*<sup>3</sup> is associated with dislocation interaction mechanism and in the range of 10*b*3–100*b*<sup>3</sup> with dislocation glide [45]. Therefore, the obtained activation volume for Ni was in the range for dislocation interaction while the lower value of *V\** for the two MPEAs indicates dislocation glide mechanism. Activation energy for dislocation nucleation is higher for a MPEA compared to pure metal [10]. Therefore, lower dislocation density in MPEAs favors dislocation glide over dislocation-dislocation interaction. In similar studies, CoCrFeMnNi homogenized sample showed activation volume of 0.2–1.02 nm3 using stress relaxation tests in the temperature range of 873 K to 1073 K [46] and activation volume of 52*b*<sup>3</sup> and 209*b*<sup>3</sup> were reported for coarse grained Ni and Al from nano-indentation tests, respectively [11]. In addition, Ni-based superalloy (IN-718) showed activation volume in the range of 0.05 to 0.1 nm3 within temperature range of 298 K to 923 K in tensile creep experiment [32].

The temperature dependence of indentation creep rate may be expressed as a power-law relation as [47]:

$$\dot{\varepsilon} = A \sigma^n \exp\left(-\frac{\mathcal{Q}}{RT}\right) \approx A H^n \exp\left(-\frac{\mathcal{Q}}{RT}\right) \tag{5}$$

where *A* is a material constant, *R* is universal gas constant, and *Q* is the creep activation energy. The slope of ln(. ε/*Hn*) versus 1/*T* curve gives the value of −*Q*/*R* [48,49] as shown in Figure 5 for the three studied systems. The average strain rate and hardness at each temperature over the holding time was selected for analysis. The creep activation energy of Ni, CoCrNi, and CoCrFeMnNi were found to be 200 ± 20 kJ/mol, 400 ± 100 kJ/mol and 25 ± 5 kJ/mol, respectively. The creep activation energy of CoCrNi was estimated to be 400 kJ/mol, while that of creep resistant Ni-based superalloy such as CMSX-2 and IN-X750 calculated from uniaxial tensile test was 230 kJ/mol and 306 kJ/mol, respectively [50,51]. CoCrNi showed the lowest creep displacement and highest activation energy supporting its superior creep resistance. Hardness or strength has a pronounced effect on creep resistance as previously reported [16,18]. CoCrNi showed the highest hardness compared to the other two systems and thus better creep resistance. However, the hardness of Ni and CoCrFeMnNi were similar and we attributed the lower creep resistance of CoCrFeMnNi alloy to its lower stacking fault energy (SFE) compared to Ni [52]. In confined volume deformation as in nano-indentation, lower SFE may result in increased dislocation generation and plastic flow, thus reducing creep resistance during holding time [52].

**Figure 5.** ln(. ε/*Hn*) versus 1000/*T* with slope giving the activation energy (*Q*) for Ni, CoCrNi, and CoCrFeMnNi alloys.

#### **5. Conclusions**

In summary, nano-indentation creep tests for CoCrNi and CoCrFeMnNi MPEAs were performed under static and dynamic (DMA) loads at 298 K, 423 K, and 573 K. Stress exponent was calculated from steady-state creep. The creep behavior of the alloys was compared in terms of stress exponent, activation volume, and activation energy. The following conclusions may be drawn:


**Author Contributions:** Conceptualization, M.S. and S.M.; methodology, M.S; validation, M.S. and S.M.; formal analysis, M.S.; investigation, M.S.; data curation, M.S. and S.M.; writing—original draft preparation, M.S.; writing—review and editing, S.M.; visualization, M.S. and S.M.; supervision, S.M.; Project administration, S.M.; All authors have read and agreed to the published version of the manuscript.

**Funding:** Sundeep Mukherjee acknowledges funding from the National Science Foundation (NSF) under Grant Number 1762545 (CMMI) for part of this work. Any opinions, findings, and conclusions expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

**Acknowledgments:** The authors thank the Materials Research Facility (MRF) at University of North Texas for access to the characterization equipment used in this study.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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