Lattice‐Distortion‐Enhanced Yield Strength in a Refractory High‐Entropy Alloy

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High strength is an important property for a variety of metallic-material applications.

To meet the demand for structural materials that exhibit excellent mechanical properties, there are tremendous efforts in the design and development of new advanced materials with unique strengthening mechanisms as well as strengths that exceed conventional metallic alloys. [1][2][3][4][5][6][7][8] The mechanical properties of materials can be strongly related to their microstructures. Thus, the establishment of a link between the adjusted microstructure and the improved yield strength is a critical issue in materials science and engineering. [9][10][11][12] Hence, there are many research activities with the goal of enhancing the strength through tuning the alloy compositions and thermomechanical treatments, which promotes the dislocation pile-up due to an increase in the interfacial density. [3,13] Although these well-developed concepts of alloydesign and heat-treatment strategies can efficiently induce sufficient stress fields, which enhance the strength of a given material, it cannot be applied to solid-solution phases with large grain sizes that only rely on the solid-solution strengthening mechanism. [14,15] The dislocations usually travel through the solvent lattice in conventional single-phase materials, where their mobility is hindered by the lattice friction caused by solute atoms. However, due to the limited number of solute atoms in the matrix, single-phase solid-solution alloys tend to exhibit the short-range lattice friction when plastic yielding is initiated. [16,17] Despite their simple microstructures, high-entropy alloys (HEAs) have demonstrated outstanding mechanical properties. [3,6,[18][19][20][21] The original definition of HEAs is a single-phase solid-solution alloy, which contains multi-principal elements generally with equi-molar or near-equi-molar atomic ratios, based on the new alloy-design strategy with the high configurational entropy. [6,18] This equi-molar composition, in turn, enables to implement a unique solid-solution strengthening effect in the HEAs. Specifically, a large number of homogeneously-distributed solute atoms with different atomic sizes and characteristics lead to distorted lattices with long-range lattice friction. [5,14,22] This long-range lattice friction is accompanied by a decrease in the mobility of dislocations in HEAs. Hence, the lattice distortion leads to an increase in the achievable yield strength of the solid-solution alloys without an appreciable change in the microstructure. In fact, Maresca and Curtin [23] have recently established an analytical solid-solution strengthening model in HEAs at elevated temperatures, considering the type of dominant mobile dislocations. Deviating from the traditional perspective of screw-dislocation dominance in conventional BCC alloys, the random field solutes in BCC refractory HEAs have been found to create intrinsic large-energy barriers for the thermally-activated edge glide. Hence, edge dislocations could play a critical role to achieve high yield strengths in refractory HEAs. Furthermore, there have been substantial efforts to establish the concrete relationship between the inherent lattice distortions and their effects on yield strengths in HEAs. [5,14,19,24] However, two major challenges have been encountered while demonstrating the unique solid-solution-strengthening mechanism in HEAs during experimental verifications. [24,25] Firstly, few HEA compositions have been experimentally validated as consisting of a single solid-solution phase with chemical homogeneity. [3,15,26] Secondly, methods for quantitative and qualitative measurements of lattice distortions in HEAs have not been clearly established. [24,25,27,28] Expanding upon our previous study of a quaternary single body-centered-cubic (BCC) solid-solution phase refractory NbTaTiV HEA, [14] we have designed a NbTaTiVZr HEA to demonstrate the relationship between the lattice distortion and yield strength. The evolution of the phase and microstructure of the NbTaTiVZr HEA, from the as-cast to the thermallytreated conditions, were carefully investigated, using focused experiments coupled with theoretical approaches. Based on the calculation of phase-diagrams (CALPHAD) modeling, the single BCC solid-solution phase with a chemical homogeneity was achieved through systematic heat-treatment processes, which show the same microstructure as the NbTaTiV refractory HEA. [14] The lattice distortions of the NbTaTiV and NbTaTiVZr refractory HEAs were quantitatively calculated and measured by first-principles calculations, synchrotron Xray diffraction (XRD)/neutron diffraction, atom-probe-tomography (APT), transmissionelectron-microscopy (TEM), and scanning-transmission-electron-microscopy (STEM) techniques. The degree of lattice distortion was considerably higher after the addition of Zr into the NbTaTiV HEA, which results in the significant enhancement of the yield strength with comparable plasticity.

The prediction of phase evolution in both the non-equilibrium and equilibrium states, including the possible chemical distributions for the NbTaTiVZr alloy, was performed, employing the CALPHAD modeling (Figure S1, Supporting information). The solidification path in a non-equilibrium state was determined, using the Scheil-Gulliver models, [29,30] which presumes the equilibrium mixing in the liquid state without the consideration of diffusion in the solid state. The CALPHAD modeling anticipates the presence of a single BCC solidsolution at the end of the whole non-equilibrium solidification (Figure S1a, Supporting information). In fact, the actual solidification path will be intermediate between Scheil-Gulliver and equilibrium models. [29] The theoretical prediction anticipates the likely-chemical concentration related to Ta and Zr due to continuous cooling: Elements, Ta and Nb, enrich the dendrites at earlier stages, whereas elements, Zr, Ti, and V, segregate in the interdendritic region at later stages.

The calculated equilibrium phase diagram for the NbTaTiVZr HEA is shown in Figure S1b, Supporting information. The database predicts a liquidus (T liq ) temperature of 1,949 °C and a solidus (T sol ) temperature of 1,581 °C. The primary BCC solid-solution phase starts to decompose at 942 °C (T dec ). Note that at temperatures lower than 942 °C, the primary BCC phase will undergo phase decomposition: the formation of another BCC phase that is rich in hexagonal-close-packed (HCP) metals (Ti and Zr). The ratio of the temperature range, where the BCC phase is stable beyond the solidus temperature [i.e., ( T sol -T dec )

T sol ], is calculated to be 0.40 for the NbTaTiVZr HEA, agreeing with the suggestion made by Gao et al., [31] who proposed that a ratio greater than 0.30 typically favors a single-phase solid-solution in the ascast condition.

The scanning-electron-microscopy (SEM) back-scattered-electrons (BSE) image of the homogenized NbTaTiV [14] and NbTaTiVZr refractory HEAs are illustrated in Figures 1a,b.

The simple solid-solution microstructures are achieved after the homogenized thermal ,f. Note that the comprehensive microstructural evolutions from the as-cast to homogenized conditions for NbTaTiVZr, which were obtained by systematic heat treatments, are described in the Supporting information.

To provide the detailed atomic-scale chemical composition for the homogenized NbTaTiV and NbTaTiVZr HEAs, APT studies were employed, as illustrated in Figures 1g,h.

A one-dimensional concentration profile and atom maps for the homogenized NbTaTiV and

NbTaTiVZr samples indicate a uniform distribution of alloying elements with nearly-equiatomic percentage along 146 nm (NbTaTiV) and 67 nm (NbTaTiVZr)-length cylindrical regions. With the nano-scale atomic distributions, further measurements of the atomicdistribution randomness were quantified, using a frequency-distribution-analysis (FDA), [15,32] as presented in Figures 1g,h. The results of the FDA analysis for homogenized NbTaTiV and NbTaTiVZr HEAs reveal that the experimentally-observed data (dots) overlap extremely well with the binomial distributions (solid lines), implying excellent homogeneity of the elemental distribution in both HEAs. Furthermore, the experimentally-observed data and the binomial distribution can be quantitatively studied by the Pearson coefficient (μ). [14] The ranges of μ values are between 0 and 1, where 0 indicates a solid solution with a perfectly-random elemental distribution, and 1 represents a complete association of the elements, such as intermetallic compounds embedded in a matrix. The μ values for each element in the homogenized NbTaTiV and NbTaTiVZr samples are close to 0, indicating a significant random elemental distribution in the single BCC solid-solution phase.

The compressive engineering stress-strain curves for the as-cast and homogenizationtreated NbTaTiVZr HEAs at room temperature (RT) with a strain rate of 1 × 10

-3 s -1 are exhibited in Figure 2. The as-cast and homogenized alloys exhibit high yield strength (σ y ) values of 1,352 MPa and 1,518 MPa, and comparable compressive plastic strains (ɛ p ) of 19.8 % and 16.7 %, respectively. Similar to the mechanical properties of the NbTaTiV refractory alloy (the inset in Figure 2), the yield strength was slightly increased after the homogenization treatment. The small increase of yield strength after homogenization treatment for both HEAs maybe arise from the different extents of lattice distortions between as-cast and homogenized samples. The lattice distortions are caused by the random distribution of the constitutive elements that vary appreciably in size. Hence, it is expected that the extent of the latticedistortion effect is maximized when all elements are homogenously distributed. Based on the configurational-entropy approach, the chemical inhomogeneity in the as-cast sample is expected to reduce the lattice-distortion effect, as compared to the fully-homogeneous sample, which determines the extent of strengthening in the HEAs. The detailed values of the mechanical properties are summarized in Figure 2. With the addition of the element, Zr, in the NbTaTiV alloy, there is a pronounced enhancement of the yield strength from 1,239 ± 94

MPa to 1,518 ± 109 MPa with comparable plasticity, relative to the NbTaTiV alloy. In fact, both alloys show the single solid-solution phase formation, a simple feature of microstructures with similar grain sizes [200 -400 µm (NbTaTiV) and 350 -500 µm (NbTaTiVZr)] [14] and random atomic distributions.

The yield strength of HEAs is usually determined by the intrinsic yield strength of alloys with several strengthening mechanisms, [33] as expressed by the following equation:

where ( σ 0.2 ) mix is the estimated intrinsic yield strength, using the rule of mixutre, ∆ σ S is the solid-solution strengthening, ∆ σ G is the grain-boundary strengthening, and ∆ σ P is precipitation strengthening. The general grain-boundary-strengthening model relies on the average size of the grain, which can be expressed by the classical Hall-Petch strengthening relationship as follows: [34,35]

where σ 0 is the lattice-friction stress, k is the strengthening coefficient of the HEAs, and d is the average grain size. The present two refractory HEAs indicate a pretty large average grain size of ≈ 300 µm and ≈ 425 µm for NbTaTiV and NbTaTiVZr, respectively. Thus, values of the effect of the grain-size distribution (k d -0.5 ¿ on yield strength can be small for both HEAs.

In fact, both HEAs contain relatively-large amounts of interstitial contents [1.1 at. % (atomic percent) of O and 0.4 at. % of N for NbTaTiV and 0.8 at. % of O and 0.2 at. % of N for NbTaTiVZr, respectively], which were observed from the APT mass spectra analyses (see Figure S3 in the Supporting information for the representative mass spectra of each sample).

These contained interstitial elements may also contribute to strengthening. Hence, the yield strength of both HEAs could be enhanced by interstitial strengthening. However, the amounts of O and N elements in the NbTaTiV and NbTaTiVZr HEAs are about the same within the uncertainty of APT. Therefore, the improved yield strength after the addition of Zr elements in the NbTaTiV HEA is not due to interstitial strengthening. Hence, solid-solution strengthening can be the dominant strengthening mechanism in both homogenized refractory HEAs. It can also be concluded that the addition of the equi-atomic percentage of the Zr element promotes the effect of solid-solution strengthening, which results in the enhancement of the yield strength of approximately ≈ 300 MPa.

The enhancement of the solid-solution strengthening by lattice distortion is significantly associated with the variation of atomic sizes as well as bond lengths of the HEA, which stems from the chemical interactions among the composed multi-principal elements. spread of the distributions is noticeable with the addition of the Zr element to the quaternary system, which indicates an increase in the lattice distortion. The atomic-specie pairs in Figure 3b suggests that the distribution of the first nearest neighbor (1NN) bond lengths is ordered by an increasing average atomic radii. From the above results, it is apparent that there is a clear relationship between the relative atomic sizes and the range of 1NN bond lengths. Regardless of the type of atomic pairs (homo/heteroatomic pairs), when the atomic pairs contain relatively-smaller atoms, such as V-V, Ti-V, Nb-V, and Ta-V, they exhibit the shortest 1NN bond lengths. Moreover, these groups of atomic specie pairs also have a very large range of the calculated 1NN bond length with the addition of the Zr element in the NbTaTiV alloy. The atomic specie pairs containing Zr, Ta-Ta, and Nb-Ta, which have larger atomic radii, show larger average 1NN bond lengths and the smallest range of 1NN bond lengths. The addition of Zr, an element with a larger atomic radius than all the other constituent elements, results in an increase in the range of 1NN bond lengths for all the atomic-specie pairs in the NbTaTiVZr system, except for the Nb-Ta and Nb-Nb pairs. These results indicate that the addition of Zr increases the lattice distortion, which is reflected in the increase of the range of 1NN bond lengths. Furthermore, the atomic pairs involving V are the most affected from the Zr addition, in which the minimum bond lengths decrease, and the range of bond lengths increases. The experimental results of phase and microstructural evolutions (synchrotron XRD, SEM-BSE, EBSD, and APT), mechanical properties, and theoretical calculations of strengthening for the NbTaTiV and NbTaTiVZr alloys could sufficiently characterize not only the single BCC phase formation with the homogeneous elemental distributions of both alloys, but also the enhancement of the yield strength by the lattice distortion after the addition of the equi-atomic percentage of the Zr element, as predicted by first-principles calculations. However, it is still necessary to quantitatively estimate the lattice distortions via the Previously, the total lattice distortion for the homogenized NbTaTiV alloy was quantitatively determined by theoretical and experimental approaches. [14] The lattice-distortion factor,

, is simply formulated from the Debye-Waller temperature factor, [36][37][38]

, [14,28] where the amplitude of the deviation displacement of atoms (u D ) was substituted instead of the thermal vibrational amplitude (u T ). In this case, the u D is a dominant effective term to determine the lattice-distortion factor, M D , which is expressed as:

where d i eff is the effective interatomic distance of the ith element, and d is average interatomic distance of n elements, which are determined from the lattice constant of the incorporated elements associated with their respective crystal structure, and n is the number of constituent elements. The detailed descriptions of the quantitative estimation of the lattice distortion, u D , are explianed and schematically illustrated in Figure S4, Supporting information.

According to the mathematical understanding of the lattice distortion, as discussed above, the d i eff for NbTaTiV and NbTaTiVZr HEAs were determined from the references, [39,40] considering the change of the atomic volume of the ith element dissolving into the jth element, ∆ V ij , [39,40] via the following Equation:

where f j is the atomic fraction of the jth element (0.25 for NbTaTiV and 0.2 for NbTaTiVZr), S4.

Based on the evolution of lattice distortions and hardness as a function of additional elements, it is clearly found that there is a negligible amount of distorted lattices in the pure Nb element, and the experimentally-determined values of the average lattice distortions, u D , for the single, binary, ternary, quaternary and quinary alloys in Nb-Ta-Ti-V-Zr refractory system are in excellent agreement with the theoretically-calculated parameters, exhibiting the gradual increase of the lattice-distortion values with the increase of additional elements. Especially, the HEA compositions, which consist of four and five elements, such as NbTaTiV and NbTaTiVZr, exhibit significant larger values of lattice-distortion factors, compared to single, binary, and ternary system.

Based on the macro-scale measurements of the mean-field lattice distortion by employing synchrotron and neutron diffraction, the locally-distorted lattice was physically 3.287 Å (NbTaTiVZr), respectively. Note that the spacing of the lattice-plane in the crystals can be measured using FFT of HRTEM with the resolution greater than 0.001 Å, employing FFT of HRTEM. [41] The calculated local lattice distortions, u D , are 0.120 Å and 0.183 Å, which are considerably close to the theoretically-and experimentally (neutron/synchrotron diffractions)-determined values of the average lattice distortions, as listed in Table 1 With the measured interatomic distances, d hkl i , and average of d hkl i from 2D-Gaussian fitting, the values for the lattice distortion in the two investigated HEAs were calculated by the following equation, which is adopted from Equation (3).

where d hkl i is the atomic distance of the i th pair along the [hkl] direction, as marked in Figures 3g,h, and d 5), are 0.112 Å and 0.161 Å, respectively, which is in good agreement with those results found from the other measurement approaches (Table 1). Note that the uncertainty of the latticedistortion factor is correlated to [2(n-1) ]

, where n is the count of measurements. [42,43] The number of spacings was measured from over 500 pairs on the both STEM images of two HEAs. Hence, the uncertainty of the lattice-distortion factor is lower than 3%, e.g., the value of uncertainty is smaller than 0.003 Å. The detailed values of the lattice distortions, which are calculated and measured by various methods, are summarized in Table 1. The slightly-lower values for the distorted lattices in both alloys, which are obtained from the HAADF-STEM method, are mainly due to different acquisition techniques used in determining the interatomic distances (see the Supporting information). From the consistently-calculated/measured results above, it can be concluded that the distorted lattices are not localized but uniformly distributed in both alloy systems. Therefore, the solid-solution strengthening, which stems from the distorted lattices, plays a critical role in significantly-improving the yield strength after the addition of the Zr element. However, more studies are necessary to provide the new insights into the respective contribution between the lattice distortion and short-range order, [44,45] using the various techniques, such as energy-filtered nanobeam diffraction and firstprinciples calculations, to affecting mechanical properties of HEAs.

With the aim of enhancing the lattice-distortion effect in HEAs, a single-phase BCC solid-solution NbTaTiVZr refractory HEA was designed and developed using thermodynamic modeling and experimental techniques. The detailed phases, microstructural evolutions, and elemental homogeneity were characterized, using the synchrotron XRD, SEM-BSE, and APT. 25 distribution of interatomic distances of 1NN bonds in the SQSs. The blue boxes correspond to the NbTaTiV, and the orange boxes represent the NbTaTiVZr. The box indicates the range from the first and third quantiles, the whisker extends to 1.5 times the interquartile ranges, and the dots represent outliers beyond the whiskers. Median and mean lines are represented by green solid lines and red dashed lines, respectively. c,d) HRTEM image and Fourier transformation of NbTaTiV and NbTaTiVZr alloys indicate that the simple BCC structures with the lattice constants are 3.216 Å (NbTaTiV) and 3.278 Å (NbTaTiVZr), respectively. e,f) The highmagnification HRTEM image for NbTaTiV and NbTaTiVZr HEAs, measuring the interatomic distances. g,h) The STEM-HAADF images of NbTaTiV with the [1 10] zone axis and NbTaTiVZr with the [ 001] zone axis. i,j) The 2D Gauss-fitted images, based on the STEM-HAADF data, providing the atomic positions of the NbTaTiV and NbTaTiVZr HEAs. 29