Indentation pop-in behavior of CoCrFeNiAl0.3 high-entropy alloy

The pop-in behavior of face-centered cubic (FCC) CoCrFeNiAl0.3 high-entropy alloy (HEA) under different loading rates was investigated by nanoindentation with spherical indenter. Experimental results show the distinct three-stage feature of the first pop-in, namely the elastic stage before pop-in, linear plastic stage and deceleration plastic stage for the pop-in evolution. Rate effect was analysed and critical shear stress required for the first pop-in event is obtained. A rheological model was proposed to characterize the pop-in behavior of CoCrFeNiAl0.3 HEA under different loading rates, and the related model parameters are given.


Introduction
Since first put forward in 2004, high-entropy alloys (HEAs) have attracted much attention in the field of material science and engineering [1,2]. Unlike traditional alloys based on one or two major elements, HEAs typically contain five or more principal elements with molar ratios ranging from 5 to 35 at% [2][3][4]. Generally, HEAs can form simple disordered solid solutions, namely a face-centered-cubic (FCC), a body centered-cubic (BCC) or hexagonal-closed-packed (HCP) structures [5]. Among them, the CoCrFeNiAl 0.3 HEA with face-centered-cubic structure has good impact resistance, high corrosion resistance, high temperature softening resistance, and no obvious ductile-brittle transition with the decrease of temperature [6][7][8][9]. Due to its unique structural characteristics and excellent properties, such high-entropy alloy has a good application prospect in aerospace and superconductivity. However, as a new kind of structural material with great potential value, the study on its deformation mechanism is still not complete.
Serrated flow, manifesting as instinct displacement bursts on the load-depth curve, is a plastic phenomenon that is useful for revealing deformation mechanism in metals and alloys. Till now, there have been many studies on the related serrated flow behavior of HEAs. Niu et al [10] conducted macro-tensile tests on FCC-HEAs under high temperature, showing the serration behavior. While Chen et al [11] showed the serration behavior of Al 0.5 CoCrFeNi HEA in the macro-compression test at low strain rate, and the influence of temperature on the mechanical properties and serration behavior was also analysed. With the rapid development of micro-nano manufacturing field, nanoindentation technique has become a standard method to characterize the mechanical properties of related functional materials and structures at micron and sub-micron scales, which mainly focus on the research on creep behavior [12], residual stress [13], hardening exponent [14] and size effect [15], etc. Moreover, serration behavior (pop-in) of metals or alloys can be effectively characterized by nanoindentation, including the influence of temperature and strain rate, as well as the microscopic mechanism [16]. Chen et al [17] studied the serration behavior based on indentation load-displacement curves of HEAs at room temperature and 200°C, respectively, it was found that the plastic zone grew faster and the serration behavior was more obvious at high temperature. In addition, Mason et al [18] and Jiao et al [19] thought that the serration behavior is strongly dependent on the indentation loading rate, and with increasing loading rate the serration behavior becomes less obvious or even disappears.
However, current study on the serration behavior of HEAs is still incomplete, especially on the single pop-in event, which restricts its development and application as emerging structural material. In this study, the first pop-in event in the serration behavior of CoCrFeNiAl 0.3 HEA was investigated at room temperature under different loading rates using nanoindentation. The purpose was to analyze the deformation characteristic of the first pop-in, and establish related mechanical model to characterize the pop-in behavior, which will lay a foundation for the further study on plastic behavior of HEAs to some extent.

Experimental procedure
In this research, the sample preparation was accomplished by arc-melting (99.99%) cylindrical rod CoCrFeNiAl 0.3 HEA (99.99% purity) with a hybrid of pure elements under a high purified argon environment. The alloy rod with a diameter of 6 mm was used for nanoindentation tests, surface of the tested specimen was polished with varid-size grained SiC sandpapers from coarse to fine (#800, 1000, 1200, 1500, 2000, 2500, 3000) and the diamond polishing slurry. Indentation tests were performed by adopting the Agilent Nano-indenter G200 test system, with force and displacement resolutions of 50 nN and 0.01 nm, respectively. The spherical diamond indenter with an effective radius of 5 μm and cylindrical diamond indenter with a diameter of 5.5 μm were applied, and the thermal drift rate decreased to 0.05 nm s −1 in the process of indentation test.
For the spherical indentation tests to probe the serration behavior of CoCrFeNiAl 0.3 HEA, the maximum indentation load was set to 30 mN. The three different loading rates were 60, 150 and 375 μN s −1 , respectively, and under each loading rate the indentation test was repeated about 70 times, among which the indentation data with obvious pop-in event were adopted. The indentation holding time and unloading time were 10 s and 2.5 s accordingly. Also, for the cylindrical indentation tests to obtain elastic modulus of CoCrFeNiAl 0.3 HEA, the maximum indentation load was set to 200 mN, and indentation loading time, holding time and unloading time were 5 s, 10 s and 2.5 s respectively. Additionally, the crystal structure was analyzed by x-ray diffraction (XRD) and the corresponding result is shown in figure 1, which shows the main presence of a typical FCC crystal structure in CoCrFeNiAl 0.3 HEA.

Results and discussion
The representative spherical indentation loading curves with pop-in events under different loading rates are shown in figure 2(a), for clarity, the loading curves under loading rates of 150 μN s −1 and 375 μN s −1 are parallelly moved along the horizontal axis. It can be seen from figure 2(a) that, at the lower loading rate the loading curve shows more frequent pop-in events, while the discontinuous feature becomes inconspicuous with increasing loading rate, which indicates the strong dependence of pop-in events on loading rate. Pop-in events at the nanoscale is related to the activation of individual dislocation motion [19,20], and the change of pop-in events from low to high loading rate mainly depends on the adjustment between slip band and applied strain. When the loading rate is low, a single slip band will have sufficient time to accommodate the applied strain, which triggers the sudden strain change and results in displacement burst (pop-in event). However, under high loading rate, a single slip band can not adapt to the rapid applied strain, then the multiple slip bands will simultaneously operate to accommodate, resulting in the fewer pop-in events.
Also from figure 2(a), the first pop-in event under each loading rate occurs at almost constant load within the indentation depth range of 100 nm to 150 nm. The first 'pop-in', often considered as the beginning of the plastic deformation, is the research emphasis of this study. The indentation deformation prior to the first pop-in event can be identified as the Hertz elastic contact behavior, based on which the indentation load P and indentation depth h comply to the following relationship [21,22]: where R is tip radius of spherical indenter, E r is the reduced modulus. Based on equation (1), the curve fitting result of indentation elastic stage prior to the first pop-in event is shown in figure 2(b), showing the good fitting precision and the good applicability of equation (1).
Then, 210 sets of -P h 3 2 / data are provided in figure 2(c), in which the P and h are respectively the maximum indentation elastic load and depth prior to the first pop-in event, namely corresponding to the indentation load and depth at the beginning of first pop-in. Depth for the first pop-in event occurs from 76 nm to 160 nm, and corresponding load range is from 5.2 mN to 16.7 mN, just as shown in figure 2(c). The linear fitting result of the all -P h 3 2 / data is = P h 7.87 3 2 / with a standard deviation of 0.029, combining with equation (1) we can obtain the reduced modulus as E r =83.5±2.4 GPa. For the cylindrical indenter tests, the contact area is constant and equal to the cross-sectional area. The elastic unloading process is theoretically linear and can be used for determining the reduced modulus E r [23,24], namely, where P and h are the indentation load and depth during unloading respectively, a is the contact radius equal to the radius of the indenter (a=2.75 μm for this study). Based on equation (1) and unloading data ( figure 2(d)) of five tests, the average reduced modulus is obtained as 86.0 GPa. The obtained reduced modulus from spherical indentation test is nearly identical to that from cylindrical indentation test, indicating the effectiveness of obtaining reduced modulus through spherical indentation test result prior to the first pop-in event.
Generally, the onset of plastic deformation in nanoindentation is related to the first pop-in event on the indentation load-depth curve [22,[25][26][27][28]. When the maximum shear stress under the indenter exceeds the critical value, dislocation nucleation will be caused and the pop-in event occurs. According to Hertz theory and the corresponding reduced modulus, the maximum shear stress under spherical indenter and shear modulus of where E i =1141 GPa and ν i =0.07 are the elastic modulus and Poisson's ratio of diamond indenter respectively, ν s and E s are the Poisson's ratio and elastic modulus of CoCrFeNiAl 0.3 HEA, and the Poisson's ratio ν s is set to be 0.26 [26]. Here, the E r and P in equation (3) are corresponding to each single spherical indentation result, then 210 sets of maximum shear stress t , max elastic modulus E s and shear modulus G can be independently obtained based on equations (3)-(5).
It is generally recognized that the pop-in event can be attributed to the nucleation of one or more dislocations or the activation of a pre-existing dislocation source [30], then the reason for the initiation of plastic deformation in nanoindentation is determined by the difference between the indented volume and average dislocation spacing during plastic yielding.
The 210 sets of indentation data with obvious first pop-in event under three different loading rates were used to plot the typical cumulative probability distribution of the normalized maximum shear stress for the first occurrence of pop-in ( figure 3). The maximum shear stress for first pop-in is in the range of 1.7 GPa to 3.1 GPa, which corresponds to the range of 1/15 G to 1/10 G, indicating that the first pop-in event occurs by a dislocation nucleation mechanism [15,31].
In order to further investigate the deformation characteristics of the first pop-in event, statistics of 70 different indentation data under each loading rate is shown in figure 3 separately, as well as the linear fitting result which represents the general trend of cumulative probability with t G. max / Considering that the difference in the composition of the grains and intragrains may affect the local properties of HEAs in micron levels, the data points on the fitted line were representatively chosen for subsequent analyses. The typical relation between indentation depth and time round the first pop-in event is shown in figure 4(a), showing the three-stage characteristics. Due to different solute-dislocation interactions, it is suggested that dislocation motion in HEAs is determined by convertible lattice friction, triggering the stick-slip dislocation dynamics [10,30,32], based on  figure 4(b). Further, the model can be divided into three parts, which corresponds to the (I) elastic stage before the pop-in, (II) linear increased stage of indentation plastic deformation for pop-in and (III) deceleration stage in the latter part of pop-in, respectively.
The proposed model can be described by the following expression [33,34]: where e is the strain, s is the stress, E 0 and E 1 are the elastic modulus of different Hooke Solids, h 1 and h 2 are the viscous coefficient of different Newtonian fluids, s y is the yield strength of St Venant body, t is the indentation time and t 1 is a time constant, In nanoindentation test, the indentation stress s and strain e i can be expressed as s = H and e = h h i i n respectively, where H is indentation hardness, h is indentation depth and h in is the maximum indentation elastic depth prior to the first pop-in event. Then, equation (6) can be rewritten as: where H 0 is the indentation hardness at the maximum indentation elastic depth prior to the first pop-in event.
Further, by substituting 1 2 / indentation depth round the first pop-in event can be given as follows: Then, the chosen data points on the fitted line in figure 3 were used for curve fitting based on equation (8). For the sake of comparison analysis, indentation time is zeroed to the beginning of the first pop-in event, and related fitting parameters are shown in table 1. Further, by calculating the area of the load-depth curve before the first pop-in (the inset in figure 5), the indentation elastic deformation energy under each loading rate before the first pop-in is obtained as shown in figure 5.
It can be seen that the indentation depth-time curve before the first pop-in event (stage I) in figure 4(a) corresponds to the part I of proposed rheological model in figure 4(b), with increasing loading rate, the average value of fitted parameters h 0 (table 1) increases and more elastic deformation energy is stored ( figure 5). When the maximum shear stress under indenter exceeds the critical value (equation (3)), the first pop-in occurs with the release of indentation elastic energy stored in stage I, corresponding to the stage II and III in figure 4(a). For stage II, the plastic deformation increases rapidly, which corresponds to the part II in figure 4(b), the fitted parameter h 1 is relatively larger under higher loading rate. For stage III, due to that most of stored elastic energy has been released during stage II, the increasing rate of indentation plastic deformation gradually decrease and maximum indentation plastic depth tends to be the fitted parameter h , 2 corresponding to the part III. Under higher loading rate, the fitted parameter h 2 and j are relatively larger, while the parameter t 1 is smaller. Further, with the deepening of first pop-in event and continuous releasing of stored elastic energy, the critical shear stress under indenter needed for the next pop-in cannot be satisfied, then further indentation elastic deformation restarts.

Conclusions
In this study, nanoindentation tests were performed to investigate the pop-in behavior of CoCrFeNiAl 0.3 HEA at different loading rates. The pop-in event show strong dependence on loading rate, with the increasing indentation loading rate the serration behaviors gradually weaken, indentation deformation stage prior to the first pop-in event can be identified as purely elastic, which satisfies the Hertz elastic contact relation. Critical shear stress for the first pop-in under different loading rates is in the range of 1/15 G to 1/10 G, implying the dislocation nucleation mechanism governed serration behavior. Proposed rheological model can effectively characterize the three-stage feature of the first pop-in under different loading rates, related influencing factor of obtained model parameters are analyzed and discussed.