Abstract
Quantitative and welltargeted design of modern alloys is extremely challenging due to their immense compositional space. When considering only 50 elements for compositional blending the number of possible alloys is practically infinite, as is the associated unexplored property realm. In this paper, we present a simple propertytargeted quantitative design approach for atomiclevel complexity in complex concentrated and highentropy alloys, based on quantummechanically derived atomiclevel pressure approximation. It allows identification of the best suited element mix for high solidsolution strengthening using the simple electronegativity difference among the constituent elements. This approach can be used for designing alloys with customized properties, such as a simple binary NiV solid solution whose yield strength exceeds that of the Cantor highentropy alloy by nearly a factor of two. This study provides general design rules that enable effective utilization of atomic level information to reduce the immense degrees of freedom in compositional space without sacrificing physicsrelated plausibility.
Introduction
For millennia metallurgists have explored mixtures of metallic and nonmetallic elements, socalled “alloys”, to produce materials with improved properties. Over the last century, the compositional complexity of alloys has dramatically increased in response to accelerating demands for component safety, efficiency, and resistance to harsh environments (Fig. 1a). Typical examples range from advanced automotive steels to the recently proposed complex concentrated alloys (CCAs) with multiprincipal elements at high concentrations^{1,2,3}. The immense composition space compels the design of a practically infinite number of alloys, causing the propertytargeted design of highperformance alloys to be extremely challenging. However, the local atomic environments in such alloys are chemically complex, and hence, not amenable to well established methods such as meanfield averaging. The advantage of many degrees of freedom is severely hindered by a lack of quantitative mixing rules, rendering alloy design an empirical trialanderror undertaking.
In this paper, we present a simple propertytargeted quantitative design approach for CCAs based on quantummechanically derived atomiclevel pressure approximation. This approach is inspired by the effective description of the complex atomic nature of amorphous alloys using atomiclevel pressure, which links the elemental information and the local atomic structure to various macroscopic properties such as glass transition and mechanical failure^{4}. We show that the dominant factor for solidsolution strengthening in single phase facecentered cubic (fcc) CCAs consisting of 3d transition metal elements (V, Cr, Mn, Fe, Co, and Ni) (3d CCAs) is the variation in the atomiclevel pressures originating from the charge transfer between neighboring elements. The material class includes large subgroups of important commercial alloys such as austenitic steels^{5,6}, Nibased superalloys^{7}, and the recently proposed highentropy alloys (HEAs)^{8,9,10,11}. We then numerically identify the relationship between configurational difference in the charge transfer and the macroscopic difference in the charge transfer among the elements, which enables connecting the atomiclevel fluctuation and complexity to the macroscopic property of the material. It allows establishment of a design recipe which uses elemental information (electronegativity) (Fig. 1b) without explicit electronic structure calculations as an efficient vehicle for more systematic and constitutive structureproperty design of CCAs.
Results
Atomiclevel pressure in complex concentrated alloys
The atomiclevel pressure originates from the misfit of an element with its surroundings in terms of its atomic size, electronic state and charge transfer (Supplementary Fig. 1). In conventional dilute alloys, most solutes are surrounded by solvent atoms, with little interaction among solutes (Fig. 1a). Thus, previous approaches for treating the bonding and misfit in dilute alloys assume a fixed atomiclevel pressure for a solute element^{12,13}. On the other hand, the local atomic environment in CCAs is chemically complex (Fig. 1a)^{14,15}; thus the atomiclevel pressure fluctuates depending on the specific local environment. We therefore apply the quantummechanically derived atomiclevel pressure approach. This approach allows calculation of the local electronic energy for finite deformations using the densityfunctional theory (DFT) technique and calculation of the local stress of an individual atom from numerical derivatives of this energy. This approximation was introduced previously^{16,17}, where it was calculated using the locally selfconsistent multiple scattering (LSMS) method^{18,19}. Details of the method are described in the Methods subsection “Firstprinciples calculations of atomiclevel pressure”.
Figure 2a, b present introductory examples of the DFT calculated atomiclevel pressure values for each element in the FeNi CCA (Fig. 2a) and the Cantor HEA (equiatomic fcc CrMnFeCoNi) (Fig. 2b) plotted against the local atomic volume. The volume is defined as the Voronoicell volume associated with each atom. Other equiatomic 3d CCAs from the binary CoNi to the quaternary CrFeCoNi are displayed in Supplementary Fig. 2. The model contains 256 atoms in a supercell, and the atoms are randomly mixed without chemical shortrange order. Although the net atomiclevel pressure, i.e., the sum of all atomiclevel pressures, is zero, the individual atoms experience nonzero atomiclevel pressure values. Hence, all atoms in CCAs can act as dilatational or compressive sources which interact with defects.
Solidsolution strengthening versus atomiclevel pressure
Solidsolution strengthening is driven by fluctuations in the solutedislocation interaction energy^{20,21,22,23,24}. This energy is caused by the atomiclevel pressure through the interaction with the elastic field of a dislocation. As all atoms in the considered 3d CCAs have different pressures, the fluctuation in the interaction energy and the resultant solidsolution strength σ_{SS} originate from the configurational fluctuation of the atomiclevel pressure^{23,24}. If misfit volume in the previous solidsolution strengthening theory of CCAs^{23,24} is replaced by atomiclevel pressure P_{solute}, we obtain the following:
where Γ is the line tension, b is the Burger’s vector, and ΔP_{solute} denotes the ensemble standard deviation of P_{solute}. In the present 3d CCAs, differences in the moduli (~11%) and the Burger’s vectors (~1%) between the alloys^{25} are negligible compared to the difference in the ΔP_{solute} values (~44%). We thus reduce Eq. (1) to \(\sigma _{{\mathrm{SS}}}\propto \Delta P_{{\mathrm{solute}}}\) for the sake of simplicity of the parameterization.
Because the HallPetch effect is typically invariant with temperature and the Peierls friction stress is very small in fcc materials and is only weakly temperaturedependent, σ_{SS} essentially carries the temperaturedependent portion of the yield strength^{23,25}. In Fig. 2c, the ΔP_{solute} values of 3d CCAs are plotted against the reported temperaturedependent part of the yield strength at 0 K, i.e., solidsolution strength data σ_{SS}^{25}. The proportional relationship clearly reveals that solidsolution strengthening in these alloys indeed originates from the configurational fluctuation of the atomiclevel pressure.
Atomiclevel pressure versus charge transfer
We now discuss the origin of the atomiclevel pressure in 3d CCAs. From the viewpoint of elasticity, the atomic volume for each element is directly linked to its misfit volume and atomiclevel pressure^{20,21,22}; large elements should have positive/repulsive pressure and vice versa. Instead, the overall tendency of the present elementaveraged atomiclevel pressure in Fig. 2a, b does not show any apparent correlation with the atomic volume. Furthermore, the order of atomiclevel pressure of each element (Cr<Mn<Fe<Co<Ni) is opposite to the order of atomic sizes of the same elements in pure state (Cr>Mn>Fe>Co>Ni) (Fig. 2b, Supplementary Table 2). On the other hand, Fig. 3a shows a proportional relationship between charge transfer dQ that the solute atom experiences and atomiclevel pressure in 3d CCAs. This result is a clear indication that the atomiclevel pressure (and hence, the misfit volume) of 3d CCAs is driven by the charge transfer, not by the atomic size difference^{17}. Thus the relationship between the atomiclevel pressure and the charge transfer can be quantified as \(P_{s{\mathrm{olute}}} \propto {\mathrm{d}}Q\).
The main reason why the charge transfer is the dominant factor for the atomiclevel pressure in the 3d CCAs is that the differences in the atomic sizes among the elements considered here are relatively small. Figure 3b presents the average values of the final atomic radii R_{f,solute} measured by extended Xray absorption fine structure (EXAFS) and the pure atomic radii (Goldschmidt^{26}, Pauling^{27}) of the constituent elements in the 3d CCAs. We compare these values with the zero pressure atomic radius (Fig. 3c), an imaginary atomic size, before generating atomiclevel pressure \(R_{{\mathrm{zeroP}},\,{\mathrm{solute}}} = R_{{\mathrm{f}},{\mathrm{solute}}} \cdot \left( {1 + P_{{\mathrm{solute}}}/B_{{\mathrm{solute}},{\mathrm{cluster}}}} \right)^{1/3}\), where \(B_{{\mathrm{solute}},{\mathrm{cluster}}}\) is the bulk modulus of a cluster of atoms including the solute; see Supplementary Note 1. The differences among the zero pressure atomic radii are almost one to two orders of magnitude larger than the differences among the pure atomic radii and the differences among the average final atomic radii. This implies significant additional effects on the atomiclevel pressure, with the main cause being the charge transfer.
There may be additional reasons for the predominant role of the charge transfer in the atomiclevel pressure which originates from the complex nature of CCAs. In metallic liquids and glasses, the local structure (the bond length and the coordination number) effectively changes to accommodate the size misfit effect, and the charge transfer predominantly causes the atomiclevel pressure^{17,28}. Although CCAs are topologically less complex compared to metallic glasses, fluctuation in local bond lengths^{15} and displacements of atoms from their ideal lattice positions^{14} may accommodate the size misfit effects. This relaxation via the lattice distortion may be a general property of CCAs and further studies are needed.
Electronegativity difference versus solidsolution strengthening
The significant role of charge transfer in determination of atomiclevel pressure demands a departure from the purely mechanical perspective and implies the need of using electronic structure calculations to predict the solidsolution strengthening. The most practical approach to design a new alloy, however, would be a design rule which does not require computationally expensive fullfield calculations but instead utilizes specific elemental information that sufficiently represents the underlying atomic information, such as atomic size, electronegativity, and valence electron concentration (VEC)^{29}, or experimentally accessible global average atomic properties^{30}. Following this idea, we propose a pathway to use such elemental information for predicting fluctuations of the atomiclevel pressure in CCAs.
Considering the relation between the atomiclevel pressure and charge transfer, the relationship in Eq. (1) can be described by the difference in charge transfer as
To understand the impact of each chemical element in detail, we decompose \(\Delta \left( {{\mathrm{d}}Q} \right)\) as
where \(\left\langle {{\mathrm{d}}Q} \right\rangle _X\) is the average of the charge transfers over the atoms of the element X, \(\Delta _X^2\left( {{\mathrm{d}}Q} \right)\) is the variance of the charge transfers for the element X, \(\left\langle \cdots \right\rangle _{{\mathrm{element}}}\) is the weighted average of an elementspecific quantity over the element, and \(\Delta _{{\mathrm{element}}}^2\left( \cdots \right)\) is the weighted variance of elementspecific quantity among the constituent elements (see details in Supplementary Note 2). The two terms on the righthand side reflect two different levels of fluctuations in CCAs. The first term \(\Delta _{{\mathrm{element}}}^2\left( {\left\langle {{\mathrm{d}}Q} \right\rangle _X} \right)\) reflects the macroscopic difference in charge transfer among the elements. The second term \(\left\langle {\Delta _X^2\left( {{\mathrm{d}}Q} \right)} \right\rangle _{{\mathrm{element}}}\) is the configurational difference in charge transfer caused by different local atomic environments averaged over all constituent elements.
Figure 3d shows an application of this approach in terms of the probability distribution of charge transfers in FeNi in the inset of Fig. 3a. \(\Delta _{{\mathrm{element}}}\left( {\left\langle {{\mathrm{d}}Q} \right\rangle _X} \right)\) is 0.093, \(\sqrt {\left\langle {\Delta _X^2\left( {{\mathrm{d}}Q} \right)} \right\rangle _{{\mathrm{element}}}}\) is 0.046, and Δ(dQ) is 0.104. The ratio between \(\Delta _{{\mathrm{element}}}\left( {\left\langle {{\mathrm{d}}Q} \right\rangle _X} \right)\) and \(\sqrt {\left\langle {\Delta _X^2\left( {{\mathrm{d}}Q} \right)} \right\rangle _{{\mathrm{element}}}}\) is about 2.0. If we assume that the distributions of Fe and Ni have an isosceles triangle shape and are congruent to each other, the ratio between \(\Delta _{{\mathrm{element}}}\left( {\left\langle {{\mathrm{d}}Q} \right\rangle _X} \right)\) and \(\sqrt {\left\langle {\Delta _X^2\left( {{\mathrm{d}}Q} \right)} \right\rangle _{{\mathrm{element}}}}\) becomes \(\sqrt 6\) (∼2.45). Indeed, the ratios for all considered 3d CCAs were between 2.02 and 2.41 (Fig. 3e and Supplementary Fig. 3). This implies that the stronger the macroscopic difference in charge transfer is, the greater is the difference (fluctuation) of charge transfer from the variation in local atomic environments. This result makes it possible to predict a configurational fluctuation of charge transfer, and hence the atomiclevel pressure, using the average charge transfer of each element dQ_{X}, which can be simply approximated by the local electronegativity difference \(\chi _X  \chi _{{\mathrm{element}}}\), where χ_{X} is the electronegativity of element X (Supplementary Table 2), and χ_{element} is the weighted average electronegativity over the element^{31} (Supplementary Fig. 4). Consequently, the solidsolution strengthening effect in 3d CCAs can be rationalized by the electronegativity difference among the constituting elements, \(\chi = \sqrt {\mathop {\sum }\limits_X c_X\left( {\chi _X  \left\langle \chi \right\rangle _{{\mathrm{element}}}} \right)^2}\), where c_{X} is the composition, (Fig. 1b) as
We use Allen’s scale as it reflects Fermi energies of d elements^{32}. Additional discussion to predict = σ_{SS} is presented in Supplementary Note 3.
Electronegativity difference versus ideal mixing entropy in 3d CCAs
Until now, we have assumed a random chemical mixture of elements, whereas in reality there may be some chemical shortrange order among elements. Although minor, the effects of the shortrange order on the yield strength would also be proportional to Δχ, as (1) the shortrange ordering may be favored in alloys with high Δχ due to the large electronic interaction between elements, and (2) its strengthening effect is caused by the resistance to randomization of atomic configurations (breaking shortrange orders), which occurs during dislocation gliding^{33,34}.
Figure 4a shows a diagram of the relationship between Δχ and the ideal mixing entropy ΔS_{mix} (\(=  R\mathop {\sum }\limits_X c_X{\mathrm{ln}}c_X\), where c_{X} is the composition of element X and R is the gas constant) for chemical complexity of 3d CCAs. We calculated Δχ and ΔS_{mix} with all possible combinations of 3d transition metal elements (V, Cr, Mn, Fe, Co, Ni). The number of combinations is 53130 when the compositional interval is 5 at.%. We added V here due to the observation that the charge transfer is also the main contributor to atomiclevel pressure in Vcontaining CCAs (Supplementary Fig. 5). Among these alloys, we selected alloys with VEC larger than or equal to 7.5 for their high potency of forming fcc solidsolution alloys^{35}, bringing the number of configurations down to 27262. Several important CCA examples are listed in Supplementary Table 3.
There are several interesting points in diagram Fig. 4a. First, in spite of its lower chemical complexity, CrCoNi (Δχ = 0.100) has a higher solidsolution strengthening than CrMnFeCoNi (Δχ = 0.080). This is consistent with a previous observation that CrCoNi exhibits higher yield strength than CrMnFeCoNi^{11}. Second, FeMn, basis for many commercially used ironmanganese steels, is located at a very low position (Δχ ≈ 0.020). This implies good potential to develop a 3d CCA with very high strength if we optimize the plasticity mechanisms of CCA with high Δχ. Third, mixing entropy, one of the key properties of HEAs (a subclass of CCAs), is not strongly correlated with the electronegativity difference and thus with high solidsolution strengthening per se. Furthermore, there is a region in which the mixing entropy should be decreased to obtain greater solidsolution strengthening effects. Mixing entropy does not include information about the difference among the constituting elements, i.e., differences in local atomic configurations. On the other hand, atomiclevel pressure includes the overall information of interatomic potential energies between the central atom and the surrounding environments. Thus, the atomiclevel pressure is a relevant parameter to describe the complexity of CCAs, connecting the local atomic information, such as the structure and the chemistry, to the statistical mechanical theories, and thereby to the macroscopic properties.
Propertytargeted quantitative design of 3d CCAs
In order to validate the benefits of the approach as summarized in Fig. 4a, we developed a binary NiV solid solution as a test case with the aim of achieving very high solidsolution strengthening, guided by the electronegativity difference. The eutectoid point (Ni: 63.2 at.%, V: 36.8 at.%) is chosen to obtain a single fcc phase solid solution with high stability (Supplementary Fig. 6a). Through casting, homogenization and cold rolling followed by recrystallization annealing (Supplementary Fig. 6b), we obtained a single fcc phase with the average grain size (diameter) of 8.1 µm (the inset in Fig. 4b). Figure 4b shows the representative tensile stressstrain curve of the developed NiV deformed at a strain rate of \(1 \times 10^{  3}s^{  1}\) at room temperature. To show the substantial improvement in properties obtained by such a simple binary test alloy, the curves for two other CCAs (CrCoNi^{10} and CrMnFeCoNi^{8}) with similar grain sizes are also presented. The yield strength of the NiV solid solution alloy is about 750 MPa and the ultimate tensile strength is close to 1200 MPa, notably outperforming the previous CCAs. Δχ of NiV is 0.169, which is much larger than those of CrCoNi and CrMnFeCoNi. This clear result again confirms our hypothesis that an alloy with large electronegativity difference has a large strengthening effect, i.e., large complexity of local environments even with its relatively low mixing entropy.
Discussion
Here we established an efficient strategy for a propertytargeted quantitative atomiclevel complexity design of 3d CCAs based on the atomiclevel pressure approximation. In CCAs, the local atomic environment is chemically complex. For example, the elementresolved mean lattice distortions of CrMnFeCoNi HEA are small on average, namely <0.5%, but their local fluctuations (i.e., standard deviations) caused by the fluctuation in local environments are an order of magnitude larger^{15}. This complexity problem is a general challenge in many disordered solutions when aiming at quantitatively predicting their properties, as it requires complicated electronic structure calculations to obtain atomically resolved unit quantities of individual atoms. We thus have devised a suite of quantitative parameters which (a) can reduce the complexity problem inherent in CCAs and HEAs; and (b) can be related to several important properties, such as the mechanical response. As a result, we demonstrated that the dominant factor for the solidsolution strengthening in 3d CCAs is the variation in the atomiclevel pressures originating from charge transfer among neighboring elements. Moreover, we developed a design recipe which uses readily accessible electronegativity values to obtain the solidsolution strength in 3d CCAs, which include large subgroups of important commercial alloys.
Prediction of the solidsolution strength does not provide a full prediction of all micromechanical strengthening mechanisms of CCAs. For instance, 2/3 of the yield strength of CrMnFeCoNi (360 MPa for the sample with a grain size of 4.4 µm) is caused by grain boundary strengthening^{36}. The remaining portion of the yield strength—equivalent to its temperaturedependent portion, mainly stemming from solidsolution strengthening— is 125 MPa at room temperature. Incidentally, the atomiclevel pressure is not limited to solidsolution strengthening analysis alone. Recently, a proportional relationship between the mean squared atomic displacement (and hence, lattice distortion) and grain boundary strengthening has been proposed by comparing NiCoCr and NiCoV CCAs^{37} where a significant role of grain boundary strengthening was observed for NiCoV (grain boundary strengthening: 365 MPa, temperaturedependent portion: 383 MPa). This finding shows that the atomiclevel pressure (and stress) influence grain boundary strengthening as well, considering the fact that lattice distortion originates from atomiclevel pressure (and stress). Furthermore, similar to solid solution strengthening, thermal activation barriers and activation volumes experienced by gliding dislocations also originate from the configurational fluctuation of the atomiclevel pressure^{38}. This means that the atomiclevel pressure concept is a fundamental material parameter which may help understanding and quantifying also several other deformation mechanisms. Therefore, further studies are required to explore quantitative relationships between atomiclevel pressure, charge transfer, and various deformation mechanisms.
Aside from the mechanical properties, the phase stability of CCAs is usually discussed in terms of mixing enthalpy and lattice strain energy of the alloy system^{3}. The Miedema model^{31}, a widely accepted fundamental theory to predict the mixing enthalpy, includes charge transfer as an important parameter. On the other hand, the lattice strain energy of CCAs has been discussed with regard to the difference in the atomic sizes of the constituent elements in their pure states. As the lattice strain energy also originates from the atomiclevel pressure, we suggest that charge transfer dominates not only the mixing enthalpy but also the lattice strain energy, thereby affecting phase stability. Therefore, the proposed electronictomacro coupling approach using the concept of atomiclevel pressure provides a fundamental perspective on the atomiclevel complexity of disordered metallic materials. It enables computation of the local energy landscape and provides a means to condense it into theoretical models without sacrificing physicsrelated plausibility or capability for trend analysis, thereby helping to meet increasing needs to customize complexityinduced properties.
Methods
Specimen preparation
The samples were produced by arc melting method using metallurgical ingredients above 99.9% purity under Tigettered highpurity Argon atmosphere. The alloy button was remelted more than five times to improve the compositional homogeneity. In cases where the alloy included Mn or Cr, the elements were placed and thus partially sealed below the other constituents to minimize evaporation.
The EXAFS samples (Ni, CoNi, FeNi, MnCoNi, MnFeNi, CrCoNi, FeCoNi, CrMnCoNi, MnFeCoNi, CrFeCoNi, CrMnFeCoNi) were then suction casted into a watercooled copper mold with a rectangular cavity (=12 mm width × 4 mm thickness × 50 mm length). The suction casted alloys were homogenized at 1050–1200 °C for 24 h in an Ar atmosphere and eventually quenched in water. The homogenized samples underwent cold rolling to a total rolling reduction ratio of 70–85% followed by annealing above the recrystallization temperature in an Ar atmosphere followed by water quenching. Homogenization and recrystallization annealing conditions are the same as outlined in the ref. ^{25}, except CrMnFeCoNi (recrystallization annealing at 1000 °C for 1 h). The bulk samples were mechanically ground into a 15 µmthick ribbon, with SiC abrasive paper down to P4000.
The binary NiV alloy was then suction casted into a watercooled copper mold with rectangular shape cavity (=15 mm width × 6 mm thickness × 50 mm length). The suction casted alloy was cold rolled to 30% thickness reduction to destroy the eutectoid structure established during the solidification process. The cold rolled sample was then wrapped in Ta foil and homogenized at 1075 °C for 45 h inside a quartz tube under a high vacuum condition. The homogenized sample was cold rolled to 60% thickness reduction and then underwent recrystallization annealing at 920 °C for 3 min.
Analysis
EXAFS measurements were carried out on the 7D beamline of the Pohang Accelerator Laboratory (PLSII, 3.0 GeV, Pohang, Korea). The measurement conditions are fully described in the ref. ^{15}. The obtained datasets of 3d CCAs were aligned and processed to avoid instrumental background and absorption from other edges using the Athena in the IFEFFIT 1.2.11d suite of software^{39}. All processes were conducted in the same conditions: preedge range with energy of −150 to −30 eV, normalization range with energy of 50–500 eV, and spline range with wave numbers of 0–11 Å^{−1}. The structural parameters of the first peak were obtained through the fits of the absorption data to single scattering paths with wave numbers of 3–10.5 Å^{−1} and interatomic distances of 1–3 Å for each element (Cr, Mn, Fe, Co, Ni). In order to decrease the number of variables, we assumed (1) homogeneous compositional distribution, i.e., the coordination number of each bond type is 12/n, where n is the number of elements, and (2) uniform bond length around the fitted element (Supplementary Fig. 7a). As a result, the final average atomic radius R_{f,solute} of each element was obtained. The EXAFS modeling Rfactors range from 0.001 to 0.003 for all fittings. The sequence of sizes of the obtained average atomic radii (Fig. 3b) matches well with the sequence of theoretical mean atomic radii (Fig. 2a, b, and Supplementary Fig. 1) obtained by DFT calculation. We also quantitatively confirmed this in^{15}. The average atomic radius \(\bar R_{{\mathrm{EXAFS}}} = \mathop {\sum }\limits_n c_nR_{{\mathrm{f}},{\mathrm{solute }}{\it{n}}}\) matches well with the reported average atomic radius measured using Xray diffraction by the relation \(\bar R_{{\mathrm{XRD}}} = a/\sqrt 2\) (Supplementary Fig. 7b)^{23}. Although there are small differences between \(\bar R_{{\mathrm{EXAFS}}}\) and \(\bar R_{{\mathrm{XRD}}}\), they come from the narrow EXAFSnormalization, spline, and fitting range used here, due to the similar energy range of the elements. As the processing conditions are unified in whole samples, the small differences in the absolute values are negligible.
The phase structure of the recrystallized NiV alloy sample was identified at room temperature using highenergy Xray diffraction (HEXRD) performed at sector 6IDD of the advanced photon source (APS) at the Argonne National Laboratory, Chicago, USA. HEXRD patterns were collected in the transmission mode. The wavelength of the Xray beam used was 0.123686 Å and the distance between sample and 2D detector was 394.6497 mm. 2D image collected by a 2D detector was converted into 1D pattern for final data analysis using the Fit2D software.
The microstructure of the recrystallized NiV alloy was characterized using a Hitachi SU70 field emission scanning electron microscopy (SEM) with energydispersive Xray spectroscopy (EDS) and electron backscattered diffraction (EBSD). The chemical uniformity was examined using EDS (XMAX50, Horiba) at the microscopic scale. EBSD measurement was performed with a Hikari camera and the TSL OIM datacollection software. The EBSD scan step size was 75 nm and a tolerance angle of 5° was used for grain identification.
Rectangular dogbone shaped tensile specimens, with a gauge length of 10 mm, were machined from the recrystallized sheet sample by electrical discharge machining (EDM). Oxidation layer formed during EDM cutting was removed by mechanical grinding using SiC paper. Both sides of the specimens were also ground resulting in a final thickness of ~1.6 mm and a gauge width of ~2.6 mm. Uniaxial tensile tests were carried out (Instron 5967, Instron, Norwood, USA) at an engineering strain rate of 10^{3} s^{−1}. The strain evolution during the tensile test was measured by AVE camera. In total, 5 samples were tested to confirm reproducibility.
Firstprinciples calculations of atomiclevel pressure
Classically, the Eshelby inclusion model^{40} has been utilized to elaborate the concept of atomiclevel pressure. In order to reflect atomistic and electronic effects, however, we applied here a quantummechanically derived atomiclevel pressure approximation. The key elements that define the atomic stresses are (i) decomposition of the system’s energy into atomic contributions and (ii) observing the change of the atomic energy in response to cell distortions.
There are different approaches to decompose the energy, each of which can be used to define atomiclevel stresses^{41,42,43,44}. In this work we used the Voronoi tessellation as implemented in the locally selfconsistent multiple scattering (LSMS) method^{18,19}. In the LSMS method, the total volume is decomposed into Voronoi polyhedron around each atom and the energy is calculated within each polyhedron. The total energy of the system is then given as the sum of atomic energies. We define the energy per atom as:
where \(n^i\left( {\it{\epsilon }} \right)\) is the local density of states on site i at energy ∈, Ω_{i} is the volume of space associated with site i, V_{KS}(r) is the Kohn–Sham potential, ρ(r) electron density, ϕ Poisson potential, ρ_{C} charge density, ∈_{XC} local exchange correlation energy, and \({\boldsymbol{E}}_{{\boldsymbol{Z}}_{\boldsymbol{i}}}\) electrostatic field due to nuclei. The first two terms in the equation constitute site kinetic energy, the third term is the electrostatic energy due to the electronic density, the fourth term is the exchangecorrelation energy and the last term is the electrostatic energy due to nuclei, each corresponding to a single atomic volume^{16}. It should be noted that this energy decomposition allows full relaxation of the electron density under cell distortions minimizing the energy of the system in the spirit of the BornOppenheimer principle, and thus differs from the affine transformation in the scaling equations introduced by Nielsen and Martin^{45,46}. The LSMS method calculates the electron density within each atomic site, and the Poisson equation is then solved with periodic boundary condition to obtain the Hartree potential. The effective KohnSham potential is obtained by adding the exchangecorrelation potential and the cycle is repeated until a selfconsistent result is achieved. The exchangecorrelation energy was treated in the local approximation using the functional of Von Barth and Hedin^{47}. The atomic energy was calculated within the Voronoi polyhedral for each site. The LSMS takes into account the multiple scattering contributions from atoms only in the local interaction zone (LIZ) to obtain the electron density on that site. In our case, the LIZ radius is set at 14.8a_{0}, and the maximum angular moment is l = 3.
The first principles calculation of atomiclevel stresses was performed in two stages employing two different DFT based methods: (i) the projected augmented wave method (PAW)^{48} as implemented in Vienna abinitio simulation package (VASP)^{49,50} was used for geometry optimization of the CCAs, and (ii) the LSMS method is applied to compute the atomiclevel stresses as outlined above.
The disordered environment was simulated using the supercell method, based on conventional cubic special quasirandom structures (SQS)^{51} with 256 atoms included except for NiV (108 atoms). We used the plane wave energy cutoff of 400 eV and Γcentered 2 × 2 × 2 MonkhorstPack grids^{52} for the Brillouin zone integration. The exchangecorrelation is treated in the generalized gradient approximation (GGA), parametrized by Perdew et al.^{53} Structural optimizations were performed using the quasiNewton algorithm and the MethfesselPaxton smearing scheme (with smearing 0.1 eV). The equilibrium volumes of the SQS structures were first optimized till the pressure vanishes, followed by atomic relaxation until the HellmannFeynman forces were lower than 0.005 eV Å^{−1}. The cubic cell shape was kept throughout our calculations. All supercell calculations, except for CrCoNi, were performed based on collinear magnetic states.
Next, the optimized supercell structures were subject to affine deformations, in which the volume of the supercell is changed by about 1% and atomic energies were calculated using the LSMS method. In general, the atomiclevel stress tensor is defined as:^{49}
where α and β are Cartesian corrdinates, Ω_{i} is the atomic volume at site i, and \(f_{ij}^\alpha\) and \(r_{ij}^\beta\) are the force and distance between atoms i and j. The atomiclevel hydrostatic pressure is then given as:
Under the volume strain we applied to the supercell, the atomiclevel pressure can be calculated for each atom as the negative of derivative of the energy with respect to the atomic volume. Accordingly, we calculated the energy of each atom at two different volumes and calculated the derivative by dividing the difference in energy at two different strains by the volume strain.
The electron density was integrated within the Voronoi polyhedron associated with each lattice site to calculate the local charge transfer.
Data availability
The data that support the findings of this study are available from the corresponding authors, at d.raabe@mpie.de, egami@utk.edu, or espark@snu.ac.kr, upon reasonable request.
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Acknowledgements
This work was supported by the National Research Foundation of Korea grant funded by the Korean government (Ministry of Science and ICT) (NRF2018M3A7B8060601), the Ministry of Trade, Industry & Energy (MOTIE, Korea) under Industrial Technology Innovation Program (No. 10076474) and the Korea Polar Research Institute under contract Polar Academic Program (PD16010). F.K., Y.I. and D.R. gratefully acknowledge funding from the Deutsche Forschungsgemeinschaft (SPP 2006) and F.K. gratefully acknowledges funding from NWO/ STW (VIDI grant 15707). K.O. and T.E. are supported by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Part of the research of K.O. was supported by National University of Mongolia through grant FELLOWSHIP GRANTP2018 for his visiting scholar project. Part of this research used resources of the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory, supported by the Office of Science of the Department of Energy under contract DEAC0500OR22725, and the Advanced Photon Source, operated for the DOE Office of Science by Argonne National Laboratory under contract No. DEAC0206CH11357. The authors gratefully acknowledge fruitful discussions with Prof. William Curtin, Dr. Gerard Paul Leyson, Dr. Duancheng Ma, and Ms. So Yeon Kim.
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H.S.O., E.S.P., T.E., D.R., C.C.T. and F.K. designed the research project. H.S.O., S.J.K., W.H.R. and K.N.Y. performed the experiments (supervised by E.S.P.). K.O. and S.M. conducted the simulation (supervised by T.E.). H.S.O., F.K., K.O., D.R., T.E. and E.S.P. wrote the paper with comments and input from C.C.T. and Y.I. All authors contributed to the discussion of the results, and commented on the manuscript.
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Oh, H.S., Kim, S.J., Odbadrakh, K. et al. Engineering atomiclevel complexity in highentropy and complex concentrated alloys. Nat Commun 10, 2090 (2019). https://doi.org/10.1038/s41467019100127
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