Reaching unconventionally large Hall-Petch coefficients in face-centered cubic high-entropy alloys

ABSTRACT The Hall-Petch strengthening coefficient of face-centered cubic alloys has the potential to increase. Herein, we experimentally determined an unconventionally large Hall-Petch coefficient equal to 1100 MPa·µm1/2 and a large lattice friction stress for novel Co0.95Cr0.8Fe0.25Ni1.8Mo0.475 high-entropy alloys (Mo0.475 HEAs). Detailed microstructural characterizations showed Mo segregation at grain boundaries (GBs) and no apparent nano-clustering in the matrix. With the increased Mo content, Mo segregating at GBs is an unexpected outcome. The unconventionally large Hall-Petch coefficient is ascribed to the newly generated effect due to Mo segregation at GBs, besides the factors in the grain interior, e.g. the increased solid-solution strengthening. This study is dedicated to enriching the diversity of Hall-Petch strengthening mechanisms for the development of high-strength materials. GRAPHICAL ABSTRACT

tensile strength [6].As well, medium-entropy alloys (MEAs) with three principal elements have attracted interest also because of the excellent properties [7,8].A typical MEA is CoNiV alloy with the single-phase microstructure, which demonstrates excellent mechanical properties at room temperature (e.g. with a tensile strength around 1 GPa and a ductility of 50%) owing to the severe lattice distortion and slip-band refinement during deformation [8].Nonetheless, the yield strength of FCC H/MEAs is still relatively low when compared with body-centered cubic (BCC) H/MEAs [9].The yield strength of micron and submicron polycrystalline materials can be described with the classical Hall-Petch relationship [10,11], where σ y is the yield strength, σ 0 is the intrinsic lattice friction stress, k y is the Hall-Petch coefficient, and d is the grain size.The Hall-Petch coefficient can represent the grain-size sensitivity of yield strength and the hardening potency by grain refinement of polycrystalline materials.
To enhance the Hall-Petch strengthening capability is of great significance for increasing the yield strength of FCC H/MEAs.A large Hall-Petch coefficient means the large average resistance of grain boundaries (GBs) to plastic strain transmission in polycrystalline materials.The resistance process consists of two steps, i.e. the stress/energy transmission across the GB and the activation of dislocation source.It was found by Takaki et al. [12] that the carbon addition and segregation at GBs in iron increased the Hall-Petch coefficient from 150 to 600 MPa•µm 1/2 .They suggested that the GBs was stabilized by carbon segregation, and stress/energy transmission became difficult.Besides, Akama et al. [13] found that the Ni addition to steel and Ni segregation at GBs gave rise to an increase of 133% in the Hall-Petch coefficient.Moreover, the activation of dislocation source can be influenced by the magnitudes of lattice friction stress, coherency stress, stacking fault energy (SFE), and elastic modulus.First, the Hall-Petch coefficient above 1000 MPa•µm 1/2 has been reported for many BCC metals including pure Cr, Ta, and Mo [14][15][16], which is due to the large lattice friction stress.The analogous effect due to the increased lattice friction stress is observed for HEAs.The multi-principal-element feature of HEAs endows themselves with severe lattice distortion and thus an increased intrinsic lattice friction stress [17,18], which inhibits the generation of dislocations.The k y values are 677, 621, and 854 MPa•µm 1/2 for CoCrFeNiMn HEA [19], Co 17.5 Cr 12.5 Fe 55 Ni 10 Mo 5 HEA [20], and CoNiV MEA [21], respectively.Second, to date, the Hall-Petch coefficient above 1000 MPa•µm 1/2 was reported only once for FCC alloys, i.e.CoCrFeNiAl 0.3 HEA [22].The unusually large Hall-Petch coefficient is due to the additional coherency stress caused by Ni-Al nano-clustering.In addition, it was reported that the stacking fault energy [23] or elastic modulus [24] of alloys dominated the variation in the k y values of specific alloy systems.
In this study, the Hall-Petch coefficients were determined for three types of FCC CoCrFeNi-based HEAs.The atomic compositions are CoCrFeNi, CoCrFeNiMo 0.2 , and Co 0.95 Cr 0.8 Fe 0.25 Ni 1.8 Mo 0.475 , which are referred to as Mo 0 , Mo 0.2 , Mo 0.475 HEAs, respectively.In the alloys, the Mo contents and solid-solution strengthening increase, which is favorable for an increase in Hall-Petch coefficients [13,21].In HEAs, lattice elastic strain energy increases as the solid-solution Mo concentration and lattice distortion increase.Increasingly high lattice energy relative to GB could promote elemental segregation at GBs.Before experimental characterization of GBs, Mo segregation and Ni depletion at GBs have been predicted in the Mo 0.475 HEA using the Hillert's parallel tangent law in Appendix A. Finally, we summarized and analyzed the Hall-Petch strengthening mechanisms in FCC H/MEAs.
Ingots with nominal compositions of CoCrFeNi, CoCrFeNiMo 0.2 , and Co 0.95 Cr 0.8 Fe 0.25 Ni 1.8 Mo 0.475 alloys were fabricated by high-frequency induction melting 4-5 types of pure metals.The as-cast ingots were subjected to homogenization annealing at 1473 K for 86.4 ks (1 d) in a vacuum furnace and hot rolling to 60% reduction.Immediately, they were moved to a furnace with N 2 gas protection for the solid-solution treatment at 1473 K for 10 min, before quenching in the water.Then, they were cold rolled to 85% reduction (CR85%); subsequent recrystallization annealing was conducted at 1373 K for different periods of time.In order to obtain the precipitate-free microstructure, the annealing temperature should be no lower than the solvus temperature of σ or µ phase when examining the thermodynamic phase diagrams (Figure S1).According to the calculated results in Figure A2, the segregation tendency of Mo is increased with the decreasing temperature in the FCC single phase zone.Thus, the temperature 1373 K, just ∼ 5 K higher than the solvus temperature in the Mo 0.2 and Mo 0.475 HEAs, was chosen as the annealing temperature.Since it is unlikely for GB segregation to occur in the Mo 0 HEA at 1373 K, as indicated in Appendix A2, the same annealing temperature was chosen for recrystallization.Electron backscatter diffraction (EBSD) scans were performed to measure the grain size of recrystallized specimens at a fine step size recommended by TSL OIM data collection software (version 7.3, EDAX), using a JEOL JSM-IT800 field emission scanning electron microscopy (SEM) operating at 15 kV.EBSD data were analyzed using TSL OIM analysis software.To be consistent with related studies [20][21][22]25,26], the annealing twin boundaries were not counted when determining grain sizes herein.The average grain size in one image was determined with an area fraction (f i ) weighted average method involving all grains, where A i is an individual grain area.Then, a mean value with error bounds was derived from three images of each specimen.
Flat tensile specimens of dimensions, 16 mm gauge length, ∼ 2 mm width, and ∼ 1.5 mm thickness were cut via electrical discharge machining, ground using SiC abrasive papers, and subjected to quasi-static tensile testing at a rate of 0.1 mm/min using an INSTRON 5969 Universal Testing Machine with a video extensometer (AVE2, Instron).Three tensile tests were conducted for each condition.A scanning transmission electron microscopy (STEM) foil vertical to a GB was fabricated by FIB microsampling technology.STEM imaging and energy dispersive X-ray spectroscopy (EDS) elemental distribution mapping around the GB were performed by using a double spherical aberration-corrected STEM (Titan 3 G2 60-300, FEI) at an accelerating voltage of 300 kV.Threedimensional atom probe (3DAP) specimens were prepared with focused ion beam (FIB, FEI Quanta 3D), and the measurements were conducted on a CAMECA LEAP-4000 HR at 50 K with 20% for pulse fraction and 200 kHz for pulse rate.The raw data were analyzed on the IVAS ver.3.6 software.
Figure 1 is the typical EBSD-Inverse Pole Figure maps showing the various grain sizes in the recrystallized specimens of three HEAs.Table 1 lists the specimen processing condition and measured grain size in each condition.As the annealing time prolonged, the grain size was increased evidently in the range of 25-110 µm.Figure 2(a-c) shows the typical tensile engineering stress-strain curves for three types of alloys, each alloy with three different grain sizes.All the alloy specimens exhibit desirable strength-ductility trade-off.In the comparable regime of grain sizes, the Mo 0.475 alloys exhibit higher tensile strengths than the Mo 0.2 and Mo 0 alloys.With the decreased grain size, the yield strength   concentration profiles from the horizontal EDS line scans in Figure 3(a).In addition to the situations of Mo and Ni atoms, the evident variation in concentration at the GB cannot be discerned from the profiles of Co, Cr, and Fe atoms.This result agrees with the predicted results of GB segregation in Appendix A2.Moreover, selected area electron diffraction (SAED) patterns obtained from [001] and [110] FCC zone axis in Figure 3(c,d) show no extra observable spots besides the typical SAED pattern of FCC metals, indicating the absence of any second phase/ordering in the Mo 0.475 alloy.Figure 4(a) shows the elemental distribution maps in the 3DAP tip reconstruction.No apparent chemical partitioning at the nanoscale can be identified in the solid solution of Mo 0.475 alloy.Figure 4(b) shows the observed frequency distributions in solid step lines for Co, Fe, and Mo, in agreement with the corresponding binomial distributions (theoretically random distribution) in dashed lines.The observed peak of Cr deviated slightly down right, and the observed peak of Ni deviated slightly down left.The deviation can be quantified using statistical  parameters such as Pearson correlation coefficient (µ).µ ranges from 0 to 1, with µ = 0 for the completely random solid solution.Although the µ values were slightly larger for Cr and Ni (0.0622 and 0.0727) than those for Co, Fe, and Mo (0.0352, 0.0269, and 0.0262), the µ values were still small.This implied that even there existed inhomogeneous distributions of Ni and Cr atoms, the extent was so low that Ni-Cr clustering cannot be discerned from partial radial distribution functions with respect to Ni atoms in the samples (Figure 4(c)).
Figure 5(a) shows the k y -lattice friction shear stress (LFSS, τ 0 ) data for Mo 0 , Mo 0.2 , and Mo 0.475 HEAs and a variety of FCC metals and alloys cited from references [21,22,[27][28][29][30].It can be seen that the Mo 0.475 alloy has an unconventionally large k y value among FCC metals and alloys.After a regression analysis was performed, it was found that the k y value of alloys monotonically increases with the increasing τ 0 value.This indicated a strong positive correlation between the LFSS and Hall-Petch coefficient.Further, it is noted that the k y value of Mo 0.475 alloy has a significant upward deviation from the regressed dashed line.Whereas the τ 0 value of Mo 0.475 HEA is 49 MPa smaller than that of CoNiV alloy, the k y value of Mo 0.475 HEA is 246 MPa•µm 1/2 larger than that of CoNiV alloy.The large k y value of Mo 0.475 HEA cannot be completely ascribed to the large LFSS.In addition to the LFSS, the stacking fault energy [23] and shear modulus [24] of alloys are other two key factors determining the magnitude of Hall-Petch coefficient.It is expressed as below: where G is the shear modulus, b is the magnitude of dislocation Burgers vector, and γ SFE is the SFE.The shear modulus and dislocation Burgers vector of CoNiV alloy were taken from Ref. [8].The shear moduli of Mo 0 , Mo 0.2 and Mo 0.475 alloys are derived from the elastic moduli (E = 184, 209, and 222 GPa, respectively) obtained from the elastic stage in the stress-strain curves (G = E/2(1+ν), ν is the Poisson's ratio for CoCr-FeNi alloys [25]), and their dislocation Burgers vectors are derived from the lattice constants measured by us [33].The SFE of CoNiV MEA in Ref. [34] was adopted, and the SFEs of Mo 0 , Mo 0.2 , Mo 0.475 HEAs were calculated with ab initio method, as presented in Appendix B. By substituting the experimentally derived τ 0 values (τ 0 = σ 0 /q, q is the Taylor factor equal to 3.06 for FCC metals) into Eq.2, the k y value is calculated for Mo 0 , Mo 0.  larger than its theoretically estimated value that is derived from the LFSS, SFE, and G.The unusually large k y value of Mo 0.475 HEA cannot be ascribed to the LFSS, SFE, and G.
According to the classical dislocation pile-up model by Hall [10] and Armstrong [35], when the concentrated stress due to dislocation pile-up at GBs is larger than a critical stress, the Frank-Read dislocation source in an adjacent grain is activated and micro-yielding occurs.Liu et al. [21] suggested that the generation of Frank-Read dislocation sources is a process related to dislocation cross-slip.The increased LFSS and decreased SFE can weaken the cross-slip ability of dislocations, and inhibit the dislocation nucleation.Meanwhile, it should be noted that the activation of dislocation sources across GBs inevitably should overcome the GB barrier in order to transfer the stress/energy into the neighboring grain.In fact, the GB acts as a physical barrier during microyielding process.With Mo segregation, the GB energy is decreased and the GB is stabilized [36].The enhanced GB stability further hinders stress/energy transmission for micro-yielding.
Moreover, local Ni-Al nano-clustering gave rise to remarkably large Hall-Petch coefficients of 1014 and 1244 MPa•µm 1/2 in CoCrFeNiAl 0.3 and CoFeNiAl 0.3 HEAs, respectively [22].This is similar to the k y value of Mo 0.475 HEA, as they cannot be completely explained using high LFSS, high G, and low SFE.Taking into account the GB-segregation-induced and nanoclustering-induced Hall-Petch strengthening, Eq. 2 could be modified, as below: where k GB is a function of GB energy (ϒ GB ) and correlates negatively with ϒ GB , and k cluster is a resolved k y term stemming from nano-clustering.Figure 5(c) shows the resolved Hall-Petch coefficients (k SFE , k LFSS , k GB , and k cluster ) for six types of H/MEAs.Among them, four H/MEAs exhibit k y values above 800 MPa•µm 1/2 , and three HEAs exhibit k y values above 1000 MPa•µm 1/2 .The large k y value of CoNiV MEA mainly originates from a large k LFSS value, which is 1.6-3.0times of k LFSS values of the other alloys.Nevertheless, as the k y value of CoNiV MEA is composed of only two resolved terms, its k y value does not exceed that of Mo 0.475 , CoCrFeNiAl 0.3 , and CoFeNiAl 0.3 HEAs with three resolved terms.Likewise, if the four effects could be simultaneously achieved, new breakthroughs can be realized.The simultaneous addition of V, Mo, and Al into CoFeNi-based alloys is a promising endeavor.V-induced large lattice distortion generates a k LFSS value of 717 MPa•µm 1/2 , Mo segregation at GBs produces an additional k GB value of 326 MPa•µm 1/2 , and Ni-Al-clustering creates an additional k cluster value of 759 MPa•µm 1/2 .These three terms sum up to 1802 MPa•µm 1/2 .Plus a k SFE value of 200 MPa•µm 1/2 at least, the total k y value will exceed 2000 MPa•µm 1/2 , which is unconventionally large for metallic materials.
In summary, Mo segregation enhanced the role of GBs as a barrier against stress/energy transmission just before yielding.In this way, it imparted an additional 326 MPa•µm 1/2 to the k y value of polycrystalline materials.We

Figure 2 .
Figure 2. Typical engineering stress-strain curves of (a) Mo 0 , (b) Mo 0.2 , and (c) Mo 0.475 alloys with different grain sizes.(d) Plot of yield stress vs reciprocal of the square root of grain size (i.e.σ y vs d −1/2 ), and the linear fitting curve based on the Hall-Petch formulation.

2 .
Figure 3(b) shows the integrated atomic Table Three groups of lattice friction stress (σ 0 ) and Hall-Petch coefficient (k y ) along with the coefficient of determination (R 2 ) in the linear fitting.A mean value with error bounds is calculated for each alloy.

Figure 3 .
Figure 3. (a) High-magnification HAADF-STEM image and corresponding EDS mapping of each element, centering on a vertically positioned grain boundary.(b) Atomic concentration profiles obtained by vertically integrating horizontal EDS line scans in (a).The SAED pattern obtained from (c) [001] and (d) [110] FCC zone axis.

Figure 4 .
Figure 4. (a) Reconstruction of the 3DAP tip.(b) Observed and binomial frequency distributions of five alloy elemental concentrations.(c) Partial radial distribution functions with center ion of Ni and bin width of 0.1 nm.The concentrations are normalized to the measured bulk concentrations; a value greater than unity indicates clustering.

2 ,
Mo 0.475 HEAs, and CoNiV MEA.Their experimental and calculated k y data are indicated in Figure 5(b).While the experimental and calculated k y values for Mo 0 , Mo 0.2 , and CoNiV alloys are close, the experimental k y value of Mo 0.475 HEA is 326 MPa•µm 1/2

Figure 5 .
Figure 5. (a) Hall-Petch coefficient (k y ) with respect to lattice friction shear stress (LFSS, τ 0 ) for Mo 0 , Mo 0.2 , and Mo 0.475 HEAs and a variety of FCC metals and alloys cited from references.The data with * in the top right corner are adopted from Ref. [31], and those with # are from Ref. [32].A regressed dashed line is indicated.(b) The experimental and calculated k y data for Mo 0 , Mo 0.2 , Mo 0.475 HEAs, and CoNiV MEA.(c) Resolved k y values with regards to SFE, LFSS, GB, and nano-clustering for six H/MEAs.
found a new Hall-Petch strengthening mechanism for CoCrFeNi-based HEAs, i.e.Mo segregation at GBs.The GB segregation and increased solid-solution strengthening contributed to an unconventionally large Hall-Petch coefficient of Mo 0.475 HEA.In conjunction with the existing Hall-Petch strengthening mechanisms of nano-clustering, high LFSS, high G, and low SFE, we further analyzed the potential pathways to increasing the Hall-Petch coefficients of FCC HEAs to the level of 2000 MPa•µm 1/2 .

Table 1 .
Specimen processing condition and measured grain size in each condition for three types of alloys.