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Incipient plasticity and dislocation nucleation of FeCoCrNiMn high-entropy alloy
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Acta Materialia 61 (2013) 2993–3001 www.elsevier.com/locate/actamat
Incipient plasticity and dislocation nucleation of FeCoCrNiMn high-entropy alloy C. Zhu a, Z.P. Lu b, T.G. Nieh a,⇑ b
a Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China
Received 4 December 2012; received in revised form 22 January 2013; accepted 27 January 2013 Available online 22 February 2013
Abstract Instrumented nanoindentation was conducted on a FeCoCrMnNi high-entropy alloy with a single face-centered cubic structure to characterize the nature of incipient plasticity. Experiments were carried out over loading rates of 25–2500 lN s1 and at temperatures ranging from 22 to 150 °C. The maximum shear stress required to initiate plasticity was found to be within 1/15 to 1/10 of the shear modulus and relatively insensitive to grain orientation. However, it was strongly dependent upon the temperature, indicating a thermally activated process. Using a statistical model developed previously, both the activation volume and activation energy were evaluated and further compared with existing dislocation nucleation models. A mechanism consisting of a heterogeneous dislocation nucleation process with vacancy-like defects (3 atoms) as the rate-limiting nuclei appeared to be dominant. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: High-entropy alloys; Nanoindentation; Pop-in; Activation volume and energy; Dislocation nucleation
1. Introduction Instrumented nanoindentation has been extensively used to study incipient plasticity in crystals in recent years [1–6]. Experimentally, the incipient plasticity is marked by a distinct displacement burst (or pop-in) on the load–displacement curve which typically occurs at an indentation depth of about 10–20 nm or less. Since the indented volume is small (of the order of nanometers), and thus probably devoid of dislocation [7], the pop-in event is therefore attributable to the nucleation of dislocations in the crystal [5,8–13]. The pop-in stress is generally of the order of 1/30l–1/5l, where l is the shear modulus, corresponding to the “ideal” or “theoretical” strength of the crystal [14,15]. The activation energy for the nucleation of dislocation was also observed to depend upon the crystalline structure and, specifically, is lower in face-centered cubic (fcc) metals than that in body-centered cubic (bcc) metals ⇑ Corresponding author. Tel.: +1 865 974 5328; fax: +1 865 974 4115.
E-mail address: [email protected] (T.G. Nieh).
since nucleation of partial dislocations is energetically favored in fcc structures in contrast to bcc where nucleation of full dislocations is favored [16]. Structural defects leading to dislocation nucleation have been discussed by Mason et al. [1]. Based on a stressassisted, thermally activated process, they proposed a statistical approach to analyze dislocation nucleation in Pt and successfully extracted the activation energy and volume of incipient plasticity during nanoindentation. From the obtained data, they concluded that heterogeneous rather than homogeneous dislocation nucleation was the prevalent process. In the past decade, high-entropy alloys (HEAs) have attracted much research interest because of their unusual structural properties [17–20]. Traditionally, in designing an alloy, the major component is selected based on a specific property requirement and other alloying components are added in small amounts to achieve secondary properties without sacrificing the primary one. By contrast, HEAs are multicomponent alloys containing several components (>5) in equal atomic proportions. However, it is of
1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2013.01.059
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particular interest to note that, despite containing a large number of components, HEAs actually exhibit a significant degree of mutual solubility to form a single fcc or bcc phase, instead of complex ordered intermetallics [21]. As a result of the different atomic sizes and chemical bonds of the constituent elements, HEAs possess a highly distorted lattice structure [22]. Dislocation lines in HEAs due to structural distortion are not straight, but wiggled. Thus elastic as well as plastic deformation in HEAs is expected non-traditional. In this study, we applied instrumented nanoindentation to study incipient plasticity in a FeCoCrNiMn HEA with a simple fcc structure. 2. Experimental The material used in this study was prepared by arcmelting a mixture of constituent metals (purity > 99 wt.%) with a nominal composition Fe20Co20Cr20Ni20Mn20 (at.%) in a Ti-guttered high-purity argon atmosphere. The ingot was remelted at least four times to ensure homogeneity before it was drop-cast into a mold. The as-cast ingot was further rolled by 70% reduction in thickness. Rectangular samples were sliced from the rolled plate, ground and polished to a mirror finish of 0.01 lm. The samples were then annealed in vacuum at 900 °C for 10 h to induce recrystallization and grain growth, and to remove any surface stress resulting from the mechanical polishing. Fig. 1a is an optical micrograph of the sample after the above processes, showing grain sizes of 30–50 lm. The crystal structure was identified by X-ray diffraction (XRD) to be a single fcc phase (Fig. 1b). Instrumented nanoindentation experiments were performed at room temperature using a Hysitron Triboindenter (Hysitron, Inc., Minneapolis, MN), with a Berkovich diamond tip with an effective tip radius of 233 nm determined using standard methods [23–25]. Indentations were made at 4 lm intervals, with a constant load of 200 lN and loading time of 5 s; the indentation depth was typically 30 nm. The 4 lm interval was chosen to avoid any overlap of plastic zones created by neighboring indentations while at the same time accommodating as many indentations as possible on a single grain. Eleven grains were randomly selected and 120 indentations on each grain were carried out. For nanoindentations under changing loading rates, a blunt Berkovich tip with an effective radius of 638 nm was used and loading rates of 25, 250 and 2500 lN s1 with a maximum load of 700 or 800 lN were applied. All the experiments were conducted under openloop loading functions. Nanoindentation tests at elevated temperatures were accomplished with an attached heating stage [26]. The same blunt Berkovich tip was employed, producing essentially spherical surface contacts at the depth probed in the present study. At least 16 indents at each temperature, i.e. 22, 50, 100 and 150 °C, were conducted. The loading function, 600 lN at peak load, was run under an open-loop condition consisting of a 3 s loading segment and a 3 s unloading
Fig. 1. (a) Optical microstructure of the annealed FeCoCrNiMn sample with an average grain size 30–50 lm. (b) XRD pattern indicating a single fcc structure.
segment, and remained unchanged throughout all the tests. The temperature was monitored and controlled using a J-type thermocouple in direct contact with the specimen surface. The test system was allowed about 1 h to thermally stabilize at the desired test temperature. For successive indentations at a fixed temperature, the tip was maintained in contact with the sample surface in transition from one indentation to the next at a 2 lN set-point load to ensure thermal stability. The temperature variation during the entire test was within ±0.2 K. 3. Results 3.1. Nanoindentation tests at room temperature Typical load–displacement (P–h) curves at shallow indentation are presented in Fig. 2a. Each curve is displaced along the x-axis and only the loading portions are shown for clarity. The P–h curves are from three different grains and represent the general trend observed for all other grains. These P–h curves are observed to exhibit a displacement burst (or pop-in) at nearly constant load. The deformation before pop-in is elastic, as confirmed by
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gives a fitting coefficient of 3.65 with a standard deviation of 0.09 and a corresponding reduced modulus of 179 ± 4 GPa. An alternative method to obtain the reduced modulus is through statistical fitting. In this case, a collection of P–h pairs at the pop-in of 120 indentations from a single grain (grain #10) are compiled in a P–h3/2 plot, according to Eq. (1), as shown in Fig. 3. As indicated in the figure, the popin load ranges from 50 to 110 lN and the pop-in depth from 6.2 to 9.6 nm. The slope of the red curve in Fig. 3 yields a fitting constant of 3.70 with a standard deviation of 0.03. Given the radius R, the reduced modulus was readily calculated through Eq. (1) as 182 ± 1 GPa, which is quite consistent with the value of 179 ± 4 GPa estimated from the individual curve in Fig. 2b. Similar procedures were also carried out on another 10 grains we randomly selected for nanoindentation measurements and the resultant fitting constants are listed in Table 1. A comparison between the statistical fitting constant, a1, and the average constant from the 120 individual fittings, a2, for each grain is shown in Table 1; the two methods essentially produce consistent results. With the reduced modulus obtained from each grain, the maximum shear stress can be subsequently calculated using the following equations [28]: 1=3 6PE2r pm ¼ ; ð3Þ p 3 R2 smax ¼ 0:31pm : Fig. 2. (a) Typical load–displacement (P–h) curves from three different grains. Curves are offset from the origin for clarity. The pop-in events are marked by the black arrows. (b) Hertzian fitting by Eq. (1) for one of the P–h curves from (a) yields a fitting coefficient 3.65 and a reduced modulus of 179 GPa.
a reversible loading–unloading behavior slightly below the pop-in load, and can be described by Hertzian elastic theory [27]. According to the theory, the elastic response from a sample surface to a spherical contact follows the relationship: 4 P ¼ Er R1=2 h3=2 ; 3
ð4Þ
Here pm is the mean pressure at the pop-in and smax is the maximum shear stress, which is located directly beneath the bottom of the contact at a depth of 0.48 of the contact radius. The maximum shear stress (smax) at the pop-in in Fig. 2b was thus calculated as 5.6 GPa, which corresponds to 0.075l, where l is the shear modulus of the alloy (74 GPa). (Note: l is not available for the FeCoCrMnNi alloy and we estimate it from the Young’s modulus and a Poisson’s ratio of 0.3, assuming an isotropic solid.)
ð1Þ
where P is the applied indenting load, R is the tip radius of the indenter, h is the penetration depth measured from the sample free surface up to the bottom of the contact, and Er is the reduced modulus of the indenter–sample combination, which is derived from the equation: 1 1 m2i 1 m2s ¼ þ : Er Ei Es
ð2Þ
Here mi (=0.07) and Ei (=1141 GPa) are the Young’s modulus and Poisson’s ratio of the diamond indenter, and ms, Es are the Poisson’s ratio and Young’s modulus of the sample, respectively. The Hertzian fitting in Eq. (1) for one of the representative curves in Fig. 2a is shown in Fig. 2b, which
Fig. 3. Statistics of 120 P–h3/2 pairs at pop-in. The fitting yields a fitting coefficient 3.70 and a reduced modulus of 182 GPa.
3.70 ± 0.03 3.73 ± 0.20 3.45 ± 0.03 3.50 ± 0.24
Performing the above procedure over the remaining 119 indentations on grain #10, the cumulative probability of pop-in events as a function of normalized shear stress is plotted in Fig. 4a; the maximum shear stress is in the range of 1/14l–1/11l. Extending the analysis to other grains, a plot of cumulative probability of pop-ins for all 11 grains is presented in Fig. 4b. The figure shows that there is little difference from grain to grain and the normalized shear stress for the onset of plasticity lies in the range 1/15l–1/ 10l. Given that all test conditions are identical, the result suggests that there is only slight crystalline anisotropy in the current alloy. 3.2. Nanoindentation tests at elevated temperatures Prior to high-temperature testing, we performed a series of experiments at room temperature to evaluate the rate dependence on pop-in load at loading rates of 25–2500 lN s1. The data was then analyzed following the methodology proposed by Schuh et al. [1,29]. In the analysis, the cumulative probability, F(P), is correlated with the indentation pop-in load, P, via the equation:
3.22 ± 0.03 3.31 ± 0.20 a1 a2
3.13 ± 0.04 3.19 ± 0.26
3.12 ± 0.03 3.26 ± 0.22
3.01 ± 0.03 3.15 ± 0.26
3.14 ± 0.03 3.24 ± 0.19
3.46 ± 0.03 3.48 ± 0.26
3.64 ± 0.03 3.72 ± 0.22
3.72 ± 0.03 3.71 ± 0.27
10 9 8 7 6 5 4 3 2 1 Grain#
Table 1 Comparison for 11 different grains between statistically fitting constant a1 as graphically shown in Fig. 3 and a2 obtained from individual fitting as shown in Fig. 2b.
3.50 ± 0.03 3.53 ± 0.24
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11
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Fig. 4. (a) The plot of cumulative probability of pop-in vs. normalized shear stress shows that the shear stress for onset of yielding is in the range of 1/14–1/11l, where l is the shear modulus of the sample. (b) A similar plot for all grains shows little difference and the normalized shear stress is in the range 1/15–1/10l.
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ln½ lnð1 F Þ ¼ aP 1=3 þ b;
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ð5Þ
where the parameter b is a weak function of the pop-in load P as compared to the first term on the right-hand side in the equation. The parameter a in Eq. (5) is correlated with the activation volume V through: 2=3 p 3R V¼ kT a: ð6Þ 0:47 4Er By plotting ln[ln(1 F)] vs. P1/3, as shown in Fig. 5, experimental data were found to fall onto approximately linear lines for data sets acquired at three different loading rates. The slope of the fitted curve determines the a value, ˚ 3 with a from which the activation volume value of 34 A 3 ˚ standard deviation of 7 A is deduced following Eq. (6). The current alloy has an fcc crystal structure with a lattice ˚ (Fig. 1b); thus, V is about 3=4 the volparameter of 3.61 A ume of a unit cell, or the activation volume contains 3 atoms (V 0.75a3 = 3O, where O is the atomic volume). The typical P–h curves obtained at four different temperatures are shown in Fig. 6a. A significant decrease in pop-in load is observed as the temperature increases from 22 to 150 °C. The average pop-in loads are 423 ± 19, 390 ± 18, 306 ± 18 and 260 ± 25 lN at temperatures of 22, 50, 100 and 150 °C, respectively; the pop-in load drops nearly 30% from 22 to 150 °C. The load–displacement curve prior to pop-in for four temperatures fits well with the Hertzian equation and the fittings are depicted in Fig. 6b. Based on the fitting, the reduced modulus at 22, 50, 100 and 150 °C is calculated to be 184 ± 4, 181 ± 4, 180 ± 7 and 178 ± 9 GPa, respectively, exhibiting a slight decrease with increasing temperature. These load–temperature data are subsequently used to extract the activation enthalpy. We, again, adopted the method developed by Mason et al. [1]. The temperature and pop-in load at a specific F(P) are accordingly correlated through the equation:
Fig. 6. (a) Typical P–h curves at temperatures of 22, 50, 100 and 150 °C. The average load at pop-in drops nearly one-third as the temperature increases from 22 to 150 °C. (b) Hertzian fitting curves for the four temperatures are shown.
Fig. 7. Graphical demonstration of Eq. (7) to extract the activation enthalpy for the pop-in. The three fitted lines converged at T = 0 K to a common y-intercept (0.101 ± 0.001 N1/3), which scales with enthalpy H via Eq. (7). Fig. 5. Extracting activation volume from experimental data using Eq. (5). Experimental data are obtained from 120 datum points for each loading rate at 22 °C. The solid lines in red are the best linear fits.
P
1=3
2=3 p 3R H ; ¼ ckT þ 0:47 4Er V
ð7Þ
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where c is a complex function whose form is not of particular importance and H is the activation enthalpy. The fitting of experimental data is presented in Fig. 7 (details as to how to construct such a plot are provided in Ref. [1]); the intercept with the vertical axis determines the ratio of H/V. With this intercept and the activation volume V measured from Eq. (6), we now obtain a value of 1.72 ± 0.35 eV for the activation energy H. It is noted that a typical variation of 5% from both the tip radius R (638 ± 30 nm) and reduced modulus is expected to yield a deviation of only 6–7% in H and V, which is insignificant in comparison with that from the fitting. 4. Discussion 4.1. Effect of surface oxide It has been recognized that pop-in can be triggered by the breakthrough of a passivated thin oxide layer on the sample surface [4,30,31]. In such a case, the pop-in depth at high temperatures would be larger than that at low temperatures since a higher-temperature exposure would result in more oxidization. In addition, in the current study, samples were held at the test temperature for more than 60 min before testing to ensure thermal equilibrium. However, as indicated in Fig. 6, the pop-in depth obviously decreases with increasing temperatures and, specifically, the average pop-in depth at 150 °C is about 3 nm shallower than that measured at room temperature. We can, therefore, conclude that our observed pop-in is not caused by the breakthrough of oxide film but is an intrinsic behavior of the material. There are other theories on the nature of incipient plasticity. In the following, we will conduct both qualitative and quantitative assessments of these theories based on our experimental results, and evaluate their applicability.
extended by Michalske and Houston [36] and Chiu and Ngan [8]. In the model, the Gibbs free energy at 0 K for the formation of a dislocation loop in the presence of applied shear stress is expressed as: DG ¼ 2prW pr2 bsa ;
ð8Þ
where r, W and b are, respectively, the radius, the line energy and the Burgers vector of the loop, and sa is the applied shear stress acting on the slip plane where the dislocation loop forms. The first term on the right-hand side of Eq. (8) is the formation energy of a loop and the second term gives the external work done by the applied shear stress sa. Here, we exclude the stacking fault energy appearing in the original equation since it is relatively small [36] in comparison with the formation energy. The line energy W, resulting from the lattice strain in the vicinity of dislocation boundary is: 2 m lb2 4r W¼ ln 2 ; ð9Þ 2ð1 mÞ 4p ro where m is the Poisson’s ratio, l is the shear modulus, and ro is the dislocation core radius approximately equal to Burgers vector b. To form a loop with the critical radius rc, two criteria must be met: the formation energy is completely provided by the external work at 0 K, and the applied shear stress sa reaches a critical value sc: sc ¼
lb ; 2prc
ð10Þ
where sc is the resolved shear stress of smax on the active slip system and is usually taken as half smax [8] whose value at 0 K can be extrapolated from that at high temperatures (Fig. 5). From Eq. (4) and with P as the average pop-in load, smax is computed as 5.85 ± 0.1, 5.64 ± 0.12, 5.18 ± 0.17 and 4.90 ± 0.24 GPa at 22, 50, 100 and 150 °C, respectively. The variation of smax with respect to
4.2. Homogeneous dislocation nucleation It has been suggested that the very first pop-in during nanoindentation is associated with a homogeneous dislocation nucleation mechanism [8–11,32–34]. According to this hypothesis, the pop-in involves a cooperative motion of many atoms to form a critical-sized dislocation loop, and, consequently, the activation energy would be on the order of several eV or even higher [8,35–37]. A large activation energy would minimize the possibility of dislocation nucleation assisted by thermal fluctuation, especially at a temperature as low as the 0.2–0.3Tm used in the present study. This seems to contradict our experimental observation that there is a strong temperature dependence of the pop-in load. To further elucidate this conclusion, we carry out a quantitative evaluation of the homogeneous dislocation nucleation model and compare it with our experimental results. The homogeneous dislocation nucleation model was initially proposed by Hirth and Lothe [38] and recently
Fig. 8. The variation of maximum shear stress as a function of temperature results in an intercept with the vertical axis at 8 GPa, which is the strength of the alloy at 0 K at the onset of plasticity.
C. Zhu et al. / Acta Materialia 61 (2013) 2993–3001
temperature is plotted in Fig. 8. The fitted line intercepts with the vertical axis (at 0 K) at 8 GPa, or 1/9.25l. This is the ideal shear strength of the alloy at 0 K (in the absence of thermal energy) assuming the modulus remains constant at low temperatures. As sc 0.5smax, according Eq. (10), we obtain rc 3b. After substituting rc into Eqs. (8) and (9), the formation energy at 0 K in the absence of stress is calculated to be 8.5 eV, which substantially exceeds 1.72 eV, the experimental value. The Gibbs free energy at finite temperature, e.g. at room temperature, can be also estimated from Eqs. (8)–(10), given the experimental measured value sc = l/27. The DG is calculated to be 5 eV (196kT), which is well beyond thermal fluctuation and, as a result, within our testing range temperature would essentially have no effect on the pop-in behavior. The above calculation excludes, again, homogeneous dislocation nucleation as the possible mechanism. 4.3. Vacancy-mediated heterogeneous dislocation nucleation Both activation enthalpy H and activation volume V are indicative of an atomic-scale event. The fact that an activation volume is of the order of 3O suggests a vacancy or vacancy cluster related process as rate limiting for the onset of yielding. Several authors [39–41] have proposed that vacancies, if supersaturated in a solution, might form clusters and then collapse to make dislocation loops. For example, Davis and Hirth [40] applied this theory to Al and found the external stress enhanced the nucleation rate and the stress required must be of the same order of magnitude as the theoretical strength (l/2p) in order to produce a discernible nucleation rate. Mason et al. [1] also discussed the possibility of vacancy-related nucleation in single-crystal Pt since the activation volume was found to be as small as 0.5b3; however, they questioned this view because the activation enthalpy (0.28 eV) derived from their experiment was significantly lower than that measured by the diffusion method (1.5 eV). According to the transition-state theory [42,43], the nucleation rate at which the critical event (e.g. formation of the critical nucleus) occurs is governed by the Gibbs free energy: DG nc ¼ n0 m exp ; ð11Þ kT and DG ¼ H s V ;
ð12Þ
where n0 is the number of available vacancy sites in the stressed volume, m is the pre-exponential factor-attempting frequency on the order of Debye frequency, H is the activation enthalpy or activation energy, s is the applied shear stress, V is the activation volume. DG is also scaled with kT [44] and can be written as: DG ¼ kkT ;
ð13Þ
2999
where k is the scaling coefficient. The n0 in Eq. (11) can be estimated in the following way. The vacancy concentration at thermal equilibrium at room temperature is assumed to be 106 and the stressed volume is scaled with the contact radius a by Ka3, where K approximates to be p [1]. The vacancy sites n0 is, then, expressed as: n0 ¼
pa3 ðstressed volumeÞ 4 ðatoms=per unit cellÞ ðvolume per unit cellÞ
a30
106 ðone vacancy per million atomsÞ;
ð14Þ
where a0 (nm) is the lattice parameter, 4 represents the number of atoms in a fcc unit cell, and 4ðpa3 =a30 Þ includes all atom sites in the stressed volume. According to Hertzian theory, the contact radius can be estimated as: pffiffiffiffiffiffiffiffiffi a ¼ h R ðnmÞ; ð15Þ where h is the average pop-in depth and R is the tip radius. Combining Eqs. (14) and (15), n0 is estimated to be of the order of 100 in the stressed volume. It is noted that in the above calculation, we adopted the generally accepted room temperature equilibrated vacancy concentration, 106 (one vacancy per million atoms), for our estimate. In reality, the test samples were annealed at 900 °C for 10 h and then furnace cooled within a few minutes to room temperature at which the actual (supersaturated) vacancy concentration would be higher than the equilibrium. Therefore, our calculation is an underestimation. An accurate calculation is hindered by the lack of the vacancy formation energy. It is also worth noting that the vacancy concentration, even if this is assumed to be at equilibrium and uniform in the unstressed state, would become nonuniform in the stressed region because of the stress gradient that is driving the migration of vacancy, as will be discussed below in Eq. (18). However, for simple discussion and easy estimation, as a first approximation we assume in the present paper that the vacancy concentration is evenly distributed under the stress. Assuming H and V are independent of temperature and stress, and inserting the measured H and V values into Eq. (12), k is calculated through Eq. (13) to be 17.5, 17.8, 18.5 and 18 at 22, 50, 100 and 150 °C, respectively. It is noted in Eq. (13) that k is a scaling factor and its physical origin can be viewed as a measure of thermal fluctuation, which helps to trigger the nucleation of dislocations in concert with the applied stress, although the exact physical process resulting in the final value of k (18 here) is still unclear. We also calculated the k value for pure Pt using data ˚ 3, T = 298 K) from (H = 0.28 eV, s = 4.4 GPa, V = 9.7 A Manson et al. [1] and found it was 1. Thus, the k value appears to be material dependent. However, the fact that k is nearly constant suggests a similar partition of thermal fluctuation in the final dislocation nucleation process within the temperature range in our study. Combining Eqs. (12) and (13), and the stress-temperature correlation can be expressed as:
3000
s¼
C. Zhu et al. / Acta Materialia 61 (2013) 2993–3001
kk H Tþ : V V
ð16Þ
Substituting the average k = 18 into Eq. (16), the following equation is readily derived: s ¼ 0:00716T þ 7:93 ðGPaÞ;
ð17Þ
which is in excellent agreement with our experimental data in Fig. 8, suggesting the robustness of the measured H and V. In the case of vacancy-mediated nucleation, the activation energy is the vacancy migration energy, Fm, and the stress bias responsible for migration would be the pressure gradient, rrp , under the indenter. Eq. (11) is therefore transformed to: F m jrrp jbV nc ¼ n0 m exp : ð18Þ kT Fm is not available for HEAs. However, the migration energies for fcc metals such as Cu, Ag, Au, Ni, Pd and Pt are found to be 1.3, 1.1, 0.9, 1.6, 1.4 and 1.5 eV, respectively [45]. Our measured Fm = 1.72 ± 0.35 eV for the current fcc FeCoCrMnNi falls in the right range, though is slightly larger. The fcc crystal structure in HEAs is noted to be somewhat different from that in conventional fcc alloys [22]. It is distorted because of the different sizes as well as chemistry of the constituent atoms. Lacking a major diffusing element, vacancy migration in HEAs is expected to involve cooperative motion of several atoms in order to maintain proper composition portioning, rather than the conventional one-to-one atom–vacancy exchange. This collective atomic motion may explain the slightly higher activation enthalpy as compared to that in the conventional fcc metals and an activation volume (3 atoms) that is larger than a single atom. 5. Conclusion Instrumented nanoindentation experiments were performed on a HEA FeCoCrNiMn at different temperatures (22–150 °C) and loading rates (25–2500 lN s1) to examine the nature of incipient yielding. Experimental data were analyzed and further compared with existing models for dislocation nucleation. The following conclusions are drawn. 1. Indentation pop-in, which marks the onset of yielding, is observed at every temperature and loading rate conducted in this study. 2. The observed pop-in phenomenon is not caused by the breakthrough of the oxide layer of the surface but is an intrinsic behavior of the material. 3. The maximum shear stress required to induce incipient plasticity is 1/10l–1/15l, where l is the shear modulus of the alloy, and appears to depend only slightly on the grain orientation.
4. The pop-in load decreases with increasing temperature and is a strong function of temperature; it drops about 30% from 22 to 150 °C, indicating that the pop-in event is a thermally activated process. 5. The extracted activation energy and volume for the ˚ 3, onset of plasticity are 1.72 ± 0.35 eV and 34 ± 7 A respectively. Both values favor a vacancy or vacancylike mediated heterogeneous dislocation nucleation as the mechanism for the onset of plasticity. 6. The higher activation energy and volume in the current FeCoCrNiMn alloy in comparison with that obtained from conventional fcc metals suggest that vacancy migration in HEAs is not a traditional process of direct atom–vacancy exchange, but involves the cooperative motion of several atoms. Acknowledgements This work was supported by the National Science Foundation under Contract DMR-0905979. Instrumentation for the nanoindentation work was jointly funded by the Tennessee Agricultural Experiment Station and UT College of Engineering. Z.P.L. acknowledges the financial support from the National Natural Science Foundation of China under the Grant Nos. 50901006, 51010001 and 51001009, the Program of Introducing Talents of Discipline to Universities under the Grant No. B07003, and the Program for Changjiang Scholars and Innovative Research Team in University. References [1] Mason JK, Lund AC, Schuh CA. Phys Rev B 2006;73:054102. [2] Bei H, Gao YF, Shim S, George EP, Pharr GM. Phys Rev B 2008;77:060103. [3] Biener MM, Biener J, Hodge AM, Hamza AV. Phys Rev B 2007;76:165422. [4] Gerberich WW, Nelson JC, Lilleodden ET, Anderson P, Wyrobek JT. Acta Mater 1996;44:3585. [5] Page TF, Oliver WC, McHargue CJ. J Mater Res 1992;7:450. [6] Shibutani Y, Tsuru T, Koyama A. Acta Mater 2007;55:1813. [7] Shim S, Bei H, George EP, Pharr GM. Scripta Mater 2008;59:1095. [8] Chiu YL, Ngan AHW. Acta Mater 2002;50:1599. [9] Kelchner CL, Plimpton SJ, Hamilton JC. Phys Rev B 1998;58:11085. [10] Kiely JD, Houston JE. Phys Rev B 1998;57:12588. [11] Lorenz D, Zeckzer A, Hilpert U, Grau P, Johansen H, Leipner HS. Phys Rev B 2003;67:172101. [12] Wang W, Jiang CB, Lu K. Acta Mater 2003;51:6169. [13] Zuo L, Ngan AHW, Zheng GP. Phys Rev Lett 2005;94:095501. [14] Cottrell A. Dislocation and plastic flow in crystals. Oxford: Clarendon Press; 1953. [15] Ogata S, Li J, Hirosaki N, Shibutani Y, Yip S. Phys Rev B 2004;70:104104. [16] Wang L, Bei H, Li TL, Gao YF, George EP, Nieh TG. Scripta Mater 2011;65:179. [17] Yeh JW, Chen YL, Lin SJ, Chen SK. Mater Sci Forum 2007;560:9. [18] Yeh JW. Ann Chim – Sci Mater 2006;31:633. [19] Yeh JW, Chen SK, Lin SJ, Gan JY, Chin TS, Shun TT, et al. Adv Eng Mater 2004;6:299. [20] Senkov ON, Wilks GB, Miracle DB, Chuang CP, Liaw PK. Intermetallics 2010;18:1758. [21] Cantor B, Chang I, Knight P, Vincent A. Mater Sci Eng A 2004;375:213.
C. Zhu et al. / Acta Materialia 61 (2013) 2993–3001 [22] Yeh J-W, Lin S-J, Chin T-S, Gan J-Y, Chen S-K, Shun T-T, et al. Metall Mater Trans A 2004;35:2533. [23] Schuh CA, Lund AC. J Mater Res 2004;19:2152. [24] Bei H, Lu ZP, George EP. Phys Rev Lett 2004;93:125504. [25] Wang L, Lu ZP, Nieh TG. Scripta Mater 2011;65:759. [26] Song SX, Jang JSC, Huang JC, Nieh TG. Intermetallics 2010;18:702. [27] Hertz H. Miscellaneous papers: Macmillan; 1896. [28] Johnson KL. Contact mechanics. Cambrige: Cambridge University Press; 1987. [29] Schuh CA, Mason JK, Lund AC. Nat Mater 2005;4:617. [30] Bahr DF, Kramer DE, Gerberich WW. Acta Mater 1998;46:3605. [31] Gerberich WW, Venkataraman SK, Huang H, Harvey SE, Kohlstedt DL. Acta Metall Mater 1995;43:1569. [32] Lilleodden ET, Zimmerman JA, Foiles SM, Nix WD. J Mech Phys Solids 2003;51:901.
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Acta Materialia 61 (2013) 2993–3001 www.elsevier.com/locate/actamat
Incipient plasticity and dislocation nucleation of FeCoCrNiMn high-entropy alloy C. Zhu a, Z.P. Lu b, T.G. Nieh a,⇑ b
a Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China
Received 4 December 2012; received in revised form 22 January 2013; accepted 27 January 2013 Available online 22 February 2013
Abstract Instrumented nanoindentation was conducted on a FeCoCrMnNi high-entropy alloy with a single face-centered cubic structure to characterize the nature of incipient plasticity. Experiments were carried out over loading rates of 25–2500 lN s1 and at temperatures ranging from 22 to 150 °C. The maximum shear stress required to initiate plasticity was found to be within 1/15 to 1/10 of the shear modulus and relatively insensitive to grain orientation. However, it was strongly dependent upon the temperature, indicating a thermally activated process. Using a statistical model developed previously, both the activation volume and activation energy were evaluated and further compared with existing dislocation nucleation models. A mechanism consisting of a heterogeneous dislocation nucleation process with vacancy-like defects (3 atoms) as the rate-limiting nuclei appeared to be dominant. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: High-entropy alloys; Nanoindentation; Pop-in; Activation volume and energy; Dislocation nucleation
1. Introduction Instrumented nanoindentation has been extensively used to study incipient plasticity in crystals in recent years [1–6]. Experimentally, the incipient plasticity is marked by a distinct displacement burst (or pop-in) on the load–displacement curve which typically occurs at an indentation depth of about 10–20 nm or less. Since the indented volume is small (of the order of nanometers), and thus probably devoid of dislocation [7], the pop-in event is therefore attributable to the nucleation of dislocations in the crystal [5,8–13]. The pop-in stress is generally of the order of 1/30l–1/5l, where l is the shear modulus, corresponding to the “ideal” or “theoretical” strength of the crystal [14,15]. The activation energy for the nucleation of dislocation was also observed to depend upon the crystalline structure and, specifically, is lower in face-centered cubic (fcc) metals than that in body-centered cubic (bcc) metals ⇑ Corresponding author. Tel.: +1 865 974 5328; fax: +1 865 974 4115.
E-mail address: [email protected] (T.G. Nieh).
since nucleation of partial dislocations is energetically favored in fcc structures in contrast to bcc where nucleation of full dislocations is favored [16]. Structural defects leading to dislocation nucleation have been discussed by Mason et al. [1]. Based on a stressassisted, thermally activated process, they proposed a statistical approach to analyze dislocation nucleation in Pt and successfully extracted the activation energy and volume of incipient plasticity during nanoindentation. From the obtained data, they concluded that heterogeneous rather than homogeneous dislocation nucleation was the prevalent process. In the past decade, high-entropy alloys (HEAs) have attracted much research interest because of their unusual structural properties [17–20]. Traditionally, in designing an alloy, the major component is selected based on a specific property requirement and other alloying components are added in small amounts to achieve secondary properties without sacrificing the primary one. By contrast, HEAs are multicomponent alloys containing several components (>5) in equal atomic proportions. However, it is of
1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2013.01.059
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particular interest to note that, despite containing a large number of components, HEAs actually exhibit a significant degree of mutual solubility to form a single fcc or bcc phase, instead of complex ordered intermetallics [21]. As a result of the different atomic sizes and chemical bonds of the constituent elements, HEAs possess a highly distorted lattice structure [22]. Dislocation lines in HEAs due to structural distortion are not straight, but wiggled. Thus elastic as well as plastic deformation in HEAs is expected non-traditional. In this study, we applied instrumented nanoindentation to study incipient plasticity in a FeCoCrNiMn HEA with a simple fcc structure. 2. Experimental The material used in this study was prepared by arcmelting a mixture of constituent metals (purity > 99 wt.%) with a nominal composition Fe20Co20Cr20Ni20Mn20 (at.%) in a Ti-guttered high-purity argon atmosphere. The ingot was remelted at least four times to ensure homogeneity before it was drop-cast into a mold. The as-cast ingot was further rolled by 70% reduction in thickness. Rectangular samples were sliced from the rolled plate, ground and polished to a mirror finish of 0.01 lm. The samples were then annealed in vacuum at 900 °C for 10 h to induce recrystallization and grain growth, and to remove any surface stress resulting from the mechanical polishing. Fig. 1a is an optical micrograph of the sample after the above processes, showing grain sizes of 30–50 lm. The crystal structure was identified by X-ray diffraction (XRD) to be a single fcc phase (Fig. 1b). Instrumented nanoindentation experiments were performed at room temperature using a Hysitron Triboindenter (Hysitron, Inc., Minneapolis, MN), with a Berkovich diamond tip with an effective tip radius of 233 nm determined using standard methods [23–25]. Indentations were made at 4 lm intervals, with a constant load of 200 lN and loading time of 5 s; the indentation depth was typically 30 nm. The 4 lm interval was chosen to avoid any overlap of plastic zones created by neighboring indentations while at the same time accommodating as many indentations as possible on a single grain. Eleven grains were randomly selected and 120 indentations on each grain were carried out. For nanoindentations under changing loading rates, a blunt Berkovich tip with an effective radius of 638 nm was used and loading rates of 25, 250 and 2500 lN s1 with a maximum load of 700 or 800 lN were applied. All the experiments were conducted under openloop loading functions. Nanoindentation tests at elevated temperatures were accomplished with an attached heating stage [26]. The same blunt Berkovich tip was employed, producing essentially spherical surface contacts at the depth probed in the present study. At least 16 indents at each temperature, i.e. 22, 50, 100 and 150 °C, were conducted. The loading function, 600 lN at peak load, was run under an open-loop condition consisting of a 3 s loading segment and a 3 s unloading
Fig. 1. (a) Optical microstructure of the annealed FeCoCrNiMn sample with an average grain size 30–50 lm. (b) XRD pattern indicating a single fcc structure.
segment, and remained unchanged throughout all the tests. The temperature was monitored and controlled using a J-type thermocouple in direct contact with the specimen surface. The test system was allowed about 1 h to thermally stabilize at the desired test temperature. For successive indentations at a fixed temperature, the tip was maintained in contact with the sample surface in transition from one indentation to the next at a 2 lN set-point load to ensure thermal stability. The temperature variation during the entire test was within ±0.2 K. 3. Results 3.1. Nanoindentation tests at room temperature Typical load–displacement (P–h) curves at shallow indentation are presented in Fig. 2a. Each curve is displaced along the x-axis and only the loading portions are shown for clarity. The P–h curves are from three different grains and represent the general trend observed for all other grains. These P–h curves are observed to exhibit a displacement burst (or pop-in) at nearly constant load. The deformation before pop-in is elastic, as confirmed by
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gives a fitting coefficient of 3.65 with a standard deviation of 0.09 and a corresponding reduced modulus of 179 ± 4 GPa. An alternative method to obtain the reduced modulus is through statistical fitting. In this case, a collection of P–h pairs at the pop-in of 120 indentations from a single grain (grain #10) are compiled in a P–h3/2 plot, according to Eq. (1), as shown in Fig. 3. As indicated in the figure, the popin load ranges from 50 to 110 lN and the pop-in depth from 6.2 to 9.6 nm. The slope of the red curve in Fig. 3 yields a fitting constant of 3.70 with a standard deviation of 0.03. Given the radius R, the reduced modulus was readily calculated through Eq. (1) as 182 ± 1 GPa, which is quite consistent with the value of 179 ± 4 GPa estimated from the individual curve in Fig. 2b. Similar procedures were also carried out on another 10 grains we randomly selected for nanoindentation measurements and the resultant fitting constants are listed in Table 1. A comparison between the statistical fitting constant, a1, and the average constant from the 120 individual fittings, a2, for each grain is shown in Table 1; the two methods essentially produce consistent results. With the reduced modulus obtained from each grain, the maximum shear stress can be subsequently calculated using the following equations [28]: 1=3 6PE2r pm ¼ ; ð3Þ p 3 R2 smax ¼ 0:31pm : Fig. 2. (a) Typical load–displacement (P–h) curves from three different grains. Curves are offset from the origin for clarity. The pop-in events are marked by the black arrows. (b) Hertzian fitting by Eq. (1) for one of the P–h curves from (a) yields a fitting coefficient 3.65 and a reduced modulus of 179 GPa.
a reversible loading–unloading behavior slightly below the pop-in load, and can be described by Hertzian elastic theory [27]. According to the theory, the elastic response from a sample surface to a spherical contact follows the relationship: 4 P ¼ Er R1=2 h3=2 ; 3
ð4Þ
Here pm is the mean pressure at the pop-in and smax is the maximum shear stress, which is located directly beneath the bottom of the contact at a depth of 0.48 of the contact radius. The maximum shear stress (smax) at the pop-in in Fig. 2b was thus calculated as 5.6 GPa, which corresponds to 0.075l, where l is the shear modulus of the alloy (74 GPa). (Note: l is not available for the FeCoCrMnNi alloy and we estimate it from the Young’s modulus and a Poisson’s ratio of 0.3, assuming an isotropic solid.)
ð1Þ
where P is the applied indenting load, R is the tip radius of the indenter, h is the penetration depth measured from the sample free surface up to the bottom of the contact, and Er is the reduced modulus of the indenter–sample combination, which is derived from the equation: 1 1 m2i 1 m2s ¼ þ : Er Ei Es
ð2Þ
Here mi (=0.07) and Ei (=1141 GPa) are the Young’s modulus and Poisson’s ratio of the diamond indenter, and ms, Es are the Poisson’s ratio and Young’s modulus of the sample, respectively. The Hertzian fitting in Eq. (1) for one of the representative curves in Fig. 2a is shown in Fig. 2b, which
Fig. 3. Statistics of 120 P–h3/2 pairs at pop-in. The fitting yields a fitting coefficient 3.70 and a reduced modulus of 182 GPa.
3.70 ± 0.03 3.73 ± 0.20 3.45 ± 0.03 3.50 ± 0.24
Performing the above procedure over the remaining 119 indentations on grain #10, the cumulative probability of pop-in events as a function of normalized shear stress is plotted in Fig. 4a; the maximum shear stress is in the range of 1/14l–1/11l. Extending the analysis to other grains, a plot of cumulative probability of pop-ins for all 11 grains is presented in Fig. 4b. The figure shows that there is little difference from grain to grain and the normalized shear stress for the onset of plasticity lies in the range 1/15l–1/ 10l. Given that all test conditions are identical, the result suggests that there is only slight crystalline anisotropy in the current alloy. 3.2. Nanoindentation tests at elevated temperatures Prior to high-temperature testing, we performed a series of experiments at room temperature to evaluate the rate dependence on pop-in load at loading rates of 25–2500 lN s1. The data was then analyzed following the methodology proposed by Schuh et al. [1,29]. In the analysis, the cumulative probability, F(P), is correlated with the indentation pop-in load, P, via the equation:
3.22 ± 0.03 3.31 ± 0.20 a1 a2
3.13 ± 0.04 3.19 ± 0.26
3.12 ± 0.03 3.26 ± 0.22
3.01 ± 0.03 3.15 ± 0.26
3.14 ± 0.03 3.24 ± 0.19
3.46 ± 0.03 3.48 ± 0.26
3.64 ± 0.03 3.72 ± 0.22
3.72 ± 0.03 3.71 ± 0.27
10 9 8 7 6 5 4 3 2 1 Grain#
Table 1 Comparison for 11 different grains between statistically fitting constant a1 as graphically shown in Fig. 3 and a2 obtained from individual fitting as shown in Fig. 2b.
3.50 ± 0.03 3.53 ± 0.24
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Fig. 4. (a) The plot of cumulative probability of pop-in vs. normalized shear stress shows that the shear stress for onset of yielding is in the range of 1/14–1/11l, where l is the shear modulus of the sample. (b) A similar plot for all grains shows little difference and the normalized shear stress is in the range 1/15–1/10l.
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ln½ lnð1 F Þ ¼ aP 1=3 þ b;
2997
ð5Þ
where the parameter b is a weak function of the pop-in load P as compared to the first term on the right-hand side in the equation. The parameter a in Eq. (5) is correlated with the activation volume V through: 2=3 p 3R V¼ kT a: ð6Þ 0:47 4Er By plotting ln[ln(1 F)] vs. P1/3, as shown in Fig. 5, experimental data were found to fall onto approximately linear lines for data sets acquired at three different loading rates. The slope of the fitted curve determines the a value, ˚ 3 with a from which the activation volume value of 34 A 3 ˚ standard deviation of 7 A is deduced following Eq. (6). The current alloy has an fcc crystal structure with a lattice ˚ (Fig. 1b); thus, V is about 3=4 the volparameter of 3.61 A ume of a unit cell, or the activation volume contains 3 atoms (V 0.75a3 = 3O, where O is the atomic volume). The typical P–h curves obtained at four different temperatures are shown in Fig. 6a. A significant decrease in pop-in load is observed as the temperature increases from 22 to 150 °C. The average pop-in loads are 423 ± 19, 390 ± 18, 306 ± 18 and 260 ± 25 lN at temperatures of 22, 50, 100 and 150 °C, respectively; the pop-in load drops nearly 30% from 22 to 150 °C. The load–displacement curve prior to pop-in for four temperatures fits well with the Hertzian equation and the fittings are depicted in Fig. 6b. Based on the fitting, the reduced modulus at 22, 50, 100 and 150 °C is calculated to be 184 ± 4, 181 ± 4, 180 ± 7 and 178 ± 9 GPa, respectively, exhibiting a slight decrease with increasing temperature. These load–temperature data are subsequently used to extract the activation enthalpy. We, again, adopted the method developed by Mason et al. [1]. The temperature and pop-in load at a specific F(P) are accordingly correlated through the equation:
Fig. 6. (a) Typical P–h curves at temperatures of 22, 50, 100 and 150 °C. The average load at pop-in drops nearly one-third as the temperature increases from 22 to 150 °C. (b) Hertzian fitting curves for the four temperatures are shown.
Fig. 7. Graphical demonstration of Eq. (7) to extract the activation enthalpy for the pop-in. The three fitted lines converged at T = 0 K to a common y-intercept (0.101 ± 0.001 N1/3), which scales with enthalpy H via Eq. (7). Fig. 5. Extracting activation volume from experimental data using Eq. (5). Experimental data are obtained from 120 datum points for each loading rate at 22 °C. The solid lines in red are the best linear fits.
P
1=3
2=3 p 3R H ; ¼ ckT þ 0:47 4Er V
ð7Þ
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where c is a complex function whose form is not of particular importance and H is the activation enthalpy. The fitting of experimental data is presented in Fig. 7 (details as to how to construct such a plot are provided in Ref. [1]); the intercept with the vertical axis determines the ratio of H/V. With this intercept and the activation volume V measured from Eq. (6), we now obtain a value of 1.72 ± 0.35 eV for the activation energy H. It is noted that a typical variation of 5% from both the tip radius R (638 ± 30 nm) and reduced modulus is expected to yield a deviation of only 6–7% in H and V, which is insignificant in comparison with that from the fitting. 4. Discussion 4.1. Effect of surface oxide It has been recognized that pop-in can be triggered by the breakthrough of a passivated thin oxide layer on the sample surface [4,30,31]. In such a case, the pop-in depth at high temperatures would be larger than that at low temperatures since a higher-temperature exposure would result in more oxidization. In addition, in the current study, samples were held at the test temperature for more than 60 min before testing to ensure thermal equilibrium. However, as indicated in Fig. 6, the pop-in depth obviously decreases with increasing temperatures and, specifically, the average pop-in depth at 150 °C is about 3 nm shallower than that measured at room temperature. We can, therefore, conclude that our observed pop-in is not caused by the breakthrough of oxide film but is an intrinsic behavior of the material. There are other theories on the nature of incipient plasticity. In the following, we will conduct both qualitative and quantitative assessments of these theories based on our experimental results, and evaluate their applicability.
extended by Michalske and Houston [36] and Chiu and Ngan [8]. In the model, the Gibbs free energy at 0 K for the formation of a dislocation loop in the presence of applied shear stress is expressed as: DG ¼ 2prW pr2 bsa ;
ð8Þ
where r, W and b are, respectively, the radius, the line energy and the Burgers vector of the loop, and sa is the applied shear stress acting on the slip plane where the dislocation loop forms. The first term on the right-hand side of Eq. (8) is the formation energy of a loop and the second term gives the external work done by the applied shear stress sa. Here, we exclude the stacking fault energy appearing in the original equation since it is relatively small [36] in comparison with the formation energy. The line energy W, resulting from the lattice strain in the vicinity of dislocation boundary is: 2 m lb2 4r W¼ ln 2 ; ð9Þ 2ð1 mÞ 4p ro where m is the Poisson’s ratio, l is the shear modulus, and ro is the dislocation core radius approximately equal to Burgers vector b. To form a loop with the critical radius rc, two criteria must be met: the formation energy is completely provided by the external work at 0 K, and the applied shear stress sa reaches a critical value sc: sc ¼
lb ; 2prc
ð10Þ
where sc is the resolved shear stress of smax on the active slip system and is usually taken as half smax [8] whose value at 0 K can be extrapolated from that at high temperatures (Fig. 5). From Eq. (4) and with P as the average pop-in load, smax is computed as 5.85 ± 0.1, 5.64 ± 0.12, 5.18 ± 0.17 and 4.90 ± 0.24 GPa at 22, 50, 100 and 150 °C, respectively. The variation of smax with respect to
4.2. Homogeneous dislocation nucleation It has been suggested that the very first pop-in during nanoindentation is associated with a homogeneous dislocation nucleation mechanism [8–11,32–34]. According to this hypothesis, the pop-in involves a cooperative motion of many atoms to form a critical-sized dislocation loop, and, consequently, the activation energy would be on the order of several eV or even higher [8,35–37]. A large activation energy would minimize the possibility of dislocation nucleation assisted by thermal fluctuation, especially at a temperature as low as the 0.2–0.3Tm used in the present study. This seems to contradict our experimental observation that there is a strong temperature dependence of the pop-in load. To further elucidate this conclusion, we carry out a quantitative evaluation of the homogeneous dislocation nucleation model and compare it with our experimental results. The homogeneous dislocation nucleation model was initially proposed by Hirth and Lothe [38] and recently
Fig. 8. The variation of maximum shear stress as a function of temperature results in an intercept with the vertical axis at 8 GPa, which is the strength of the alloy at 0 K at the onset of plasticity.
C. Zhu et al. / Acta Materialia 61 (2013) 2993–3001
temperature is plotted in Fig. 8. The fitted line intercepts with the vertical axis (at 0 K) at 8 GPa, or 1/9.25l. This is the ideal shear strength of the alloy at 0 K (in the absence of thermal energy) assuming the modulus remains constant at low temperatures. As sc 0.5smax, according Eq. (10), we obtain rc 3b. After substituting rc into Eqs. (8) and (9), the formation energy at 0 K in the absence of stress is calculated to be 8.5 eV, which substantially exceeds 1.72 eV, the experimental value. The Gibbs free energy at finite temperature, e.g. at room temperature, can be also estimated from Eqs. (8)–(10), given the experimental measured value sc = l/27. The DG is calculated to be 5 eV (196kT), which is well beyond thermal fluctuation and, as a result, within our testing range temperature would essentially have no effect on the pop-in behavior. The above calculation excludes, again, homogeneous dislocation nucleation as the possible mechanism. 4.3. Vacancy-mediated heterogeneous dislocation nucleation Both activation enthalpy H and activation volume V are indicative of an atomic-scale event. The fact that an activation volume is of the order of 3O suggests a vacancy or vacancy cluster related process as rate limiting for the onset of yielding. Several authors [39–41] have proposed that vacancies, if supersaturated in a solution, might form clusters and then collapse to make dislocation loops. For example, Davis and Hirth [40] applied this theory to Al and found the external stress enhanced the nucleation rate and the stress required must be of the same order of magnitude as the theoretical strength (l/2p) in order to produce a discernible nucleation rate. Mason et al. [1] also discussed the possibility of vacancy-related nucleation in single-crystal Pt since the activation volume was found to be as small as 0.5b3; however, they questioned this view because the activation enthalpy (0.28 eV) derived from their experiment was significantly lower than that measured by the diffusion method (1.5 eV). According to the transition-state theory [42,43], the nucleation rate at which the critical event (e.g. formation of the critical nucleus) occurs is governed by the Gibbs free energy: DG nc ¼ n0 m exp ; ð11Þ kT and DG ¼ H s V ;
ð12Þ
where n0 is the number of available vacancy sites in the stressed volume, m is the pre-exponential factor-attempting frequency on the order of Debye frequency, H is the activation enthalpy or activation energy, s is the applied shear stress, V is the activation volume. DG is also scaled with kT [44] and can be written as: DG ¼ kkT ;
ð13Þ
2999
where k is the scaling coefficient. The n0 in Eq. (11) can be estimated in the following way. The vacancy concentration at thermal equilibrium at room temperature is assumed to be 106 and the stressed volume is scaled with the contact radius a by Ka3, where K approximates to be p [1]. The vacancy sites n0 is, then, expressed as: n0 ¼
pa3 ðstressed volumeÞ 4 ðatoms=per unit cellÞ ðvolume per unit cellÞ
a30
106 ðone vacancy per million atomsÞ;
ð14Þ
where a0 (nm) is the lattice parameter, 4 represents the number of atoms in a fcc unit cell, and 4ðpa3 =a30 Þ includes all atom sites in the stressed volume. According to Hertzian theory, the contact radius can be estimated as: pffiffiffiffiffiffiffiffiffi a ¼ h R ðnmÞ; ð15Þ where h is the average pop-in depth and R is the tip radius. Combining Eqs. (14) and (15), n0 is estimated to be of the order of 100 in the stressed volume. It is noted that in the above calculation, we adopted the generally accepted room temperature equilibrated vacancy concentration, 106 (one vacancy per million atoms), for our estimate. In reality, the test samples were annealed at 900 °C for 10 h and then furnace cooled within a few minutes to room temperature at which the actual (supersaturated) vacancy concentration would be higher than the equilibrium. Therefore, our calculation is an underestimation. An accurate calculation is hindered by the lack of the vacancy formation energy. It is also worth noting that the vacancy concentration, even if this is assumed to be at equilibrium and uniform in the unstressed state, would become nonuniform in the stressed region because of the stress gradient that is driving the migration of vacancy, as will be discussed below in Eq. (18). However, for simple discussion and easy estimation, as a first approximation we assume in the present paper that the vacancy concentration is evenly distributed under the stress. Assuming H and V are independent of temperature and stress, and inserting the measured H and V values into Eq. (12), k is calculated through Eq. (13) to be 17.5, 17.8, 18.5 and 18 at 22, 50, 100 and 150 °C, respectively. It is noted in Eq. (13) that k is a scaling factor and its physical origin can be viewed as a measure of thermal fluctuation, which helps to trigger the nucleation of dislocations in concert with the applied stress, although the exact physical process resulting in the final value of k (18 here) is still unclear. We also calculated the k value for pure Pt using data ˚ 3, T = 298 K) from (H = 0.28 eV, s = 4.4 GPa, V = 9.7 A Manson et al. [1] and found it was 1. Thus, the k value appears to be material dependent. However, the fact that k is nearly constant suggests a similar partition of thermal fluctuation in the final dislocation nucleation process within the temperature range in our study. Combining Eqs. (12) and (13), and the stress-temperature correlation can be expressed as:
3000
s¼
C. Zhu et al. / Acta Materialia 61 (2013) 2993–3001
kk H Tþ : V V
ð16Þ
Substituting the average k = 18 into Eq. (16), the following equation is readily derived: s ¼ 0:00716T þ 7:93 ðGPaÞ;
ð17Þ
which is in excellent agreement with our experimental data in Fig. 8, suggesting the robustness of the measured H and V. In the case of vacancy-mediated nucleation, the activation energy is the vacancy migration energy, Fm, and the stress bias responsible for migration would be the pressure gradient, rrp , under the indenter. Eq. (11) is therefore transformed to: F m jrrp jbV nc ¼ n0 m exp : ð18Þ kT Fm is not available for HEAs. However, the migration energies for fcc metals such as Cu, Ag, Au, Ni, Pd and Pt are found to be 1.3, 1.1, 0.9, 1.6, 1.4 and 1.5 eV, respectively [45]. Our measured Fm = 1.72 ± 0.35 eV for the current fcc FeCoCrMnNi falls in the right range, though is slightly larger. The fcc crystal structure in HEAs is noted to be somewhat different from that in conventional fcc alloys [22]. It is distorted because of the different sizes as well as chemistry of the constituent atoms. Lacking a major diffusing element, vacancy migration in HEAs is expected to involve cooperative motion of several atoms in order to maintain proper composition portioning, rather than the conventional one-to-one atom–vacancy exchange. This collective atomic motion may explain the slightly higher activation enthalpy as compared to that in the conventional fcc metals and an activation volume (3 atoms) that is larger than a single atom. 5. Conclusion Instrumented nanoindentation experiments were performed on a HEA FeCoCrNiMn at different temperatures (22–150 °C) and loading rates (25–2500 lN s1) to examine the nature of incipient yielding. Experimental data were analyzed and further compared with existing models for dislocation nucleation. The following conclusions are drawn. 1. Indentation pop-in, which marks the onset of yielding, is observed at every temperature and loading rate conducted in this study. 2. The observed pop-in phenomenon is not caused by the breakthrough of the oxide layer of the surface but is an intrinsic behavior of the material. 3. The maximum shear stress required to induce incipient plasticity is 1/10l–1/15l, where l is the shear modulus of the alloy, and appears to depend only slightly on the grain orientation.
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