Abstract
Computational investigations of structural, chemical, and deformation behavior in highentropy alloys (HEAs), which possess notable mechanical strength, have been limited due to the absence of applicable force fields. To extend investigations, we propose a set of intermolecular potential parameters for a quinary AlCrCoFeNi alloy, using the available ternary Embedded Atom Method and LennardJones potential in classical moleculardynamics simulations. The simulation results are validated by a comparison to firstprinciples KorringaKohnRostoker (KKR)  Coherent Potential Approximation (CPA) [KKRCPA] calculations for the HEA structural properties (lattice constants and bulk moduli), relative stability, pair probabilities, and hightemperature shortrange ordering. The simulation (MD)derived properties are in quantitative agreement with KKRCPA calculations (firstprinciples) and experiments. We study Al_{x}CrCoFeNi for Al ranging from 0 ≤ x ≤2 mole fractions, and find that the HEA shows large chemical clustering over a wide temperature range for x < 0.5. At various temperatures highstrain compression promotes atomistic rearrangements in Al_{0.1}CrCoFeNi, resulting in a clusteringtoordering transition that is absent for tensile loading. Large fluctuations under stress, and at higher temperatures, are attributed to the thermoplastic instability in Al_{0.1}CrCoFeNi.
Introduction
Highentropy alloys (HEAs) are solid solutions^{1,2,3,4,5,6,7,8} consisting of five or more metallic elements in approximately equimolar ratios with elemental compositions typically between 5–35 atomic percent (at.%). They have attracted increasing attention, especially as structural materials, due to their remarkable mechanical strength and resistance to oxidation and fatigue at ambient and elevated temperatures^{3,4,5,9,10}. The compositional complexity does not automatically imply microstructural complexity due to the high mixing entropy. Often, an operational definition for HEAs is a solidsolution phase stabilized by the higher configurational entropy that increases with increasing temperature. With increasing the number of alloying elements (N), it should be noted, however, that the entropy increases as N ln N whereas interacting pairs that may drive chemical order increase as N^{2}; so at low temperatures any favorable ordering enthalpy could overcome the slower increasing entropy – a point often not given its due importance. Thus, investigations of the HEA structural, chemical, and relative stability properties arising from the unique compositionstructure relationship are alone intriguing, and deformation properties, in particular, have an important engineering role.
The deformation process in HEAs for quasistatic loading at low strain rates (10^{−2} to 10^{−4} s^{−1}) reveals the contributions of slip and twinning mechanisms^{11,12,13,14,15}. The deformation behavior at highstrain rates (<10^{2} s^{−1}), even for conventional alloys undergoing dynamic loading, is complex due to the localized strain accumulating along the adiabatic shear bands^{16,17}. Highstrain rates in a crystalline lattice can lead to amorphization in the shear band that arise during deformation^{18}, as shown by the formation of amorphous and nanocrystalline structures in the FeNiCr alloy. Knowledge of highstrain deformation is particularly important for applications where strain rates higher than ~10^{2} s^{−1} are encountered^{16,17,18,19,20,21,22}, such as hightemperature mechanical strength of lightweight armor materials, blast impact of debris on aircraft composite panels, satellites and spacecraft, as well as highspeed machining, all potentially relevant opportunities for employing HEAs. While research on HEAs is rapidly gaining interest, (1) the related literature on theoretical investigations is sparse and (2) experimental investigations are expensive and resource intensive. Additionally, because atomic arrangements in the alloy crystal significantly contribute to the material phase and deformation mechanics, we employ a combination of the computationallydemanding firstprinciples calculations and classical moleculardynamics (MD) simulations to explore the highstrain deformation in a model HEA.
Classical MD simulations are unable to provide quantitatively accurate predictions of mechanical deformation for these multielement alloys due to the lack of robust intermolecular potentials. Thus, earlier efforts in the literature have primarily resorted to computationallyexpensive and systemsizelimiting firstprinciples calculations, which are restricted in scope for understanding the deformation dynamics of very large systems. New potentials would enable a classical treatment of such alloys and allow us to examine much larger material domains consisting of several thousands of atoms.
Here, we derive relevant forcefield parameters for a model quinary Al_{0.1}CrCoFeNi HEA shown in Fig. 1, using the available ternary Embedded Atom Method (EAM) and LennardJones (LJ) potentials. Structural properties (e.g., lattice constant, ‘a_{0}’, and bulk modulus, ‘B’) are calculated for our model HEA using MD simulations; we then validate our forcefield potentials by the comparison with available experiments and new firstprinciples KorringaKohnRostoker (KKR)  Coherent Potential Approximation (CPA) electronicstructure predictions. Besides structural properties, the KKRCPA directly predicts the global and local relative stability of phases in HEAs, the WarrenCowley shortrange order (SRO) in the most stable, hightemperature solidsolution phase, as well as the underlying electronic origin for the chemical clustering or ordering behavior – pair by pair. Subsequently, our new forcefield potential is utilized for calculating properties of Al_{0.1}CrCoFeNi, in particular, the pair correlation function [g(r) or g_{α−β} (r)] that helps identify ordering and clustering mechanisms. The roles of temperature and quenching rates on the clusteringordering characteristics in Al_{0.1}CrCoFeNi are also investigated. Finally, the stressstrain analysis is performed under two different loading conditions (tension and compression) across a range of temperatures varying from cryogenic (77 K), room temperature (300 K), and at an elevated temperature (700 and 1,000 K). Our results show that clustering dominates over ordering tendencies in the Al_{0.1}CrCoFeNi. However, highstrain compression can induce a clusteringtoordering transition in these multielement materials.
Results and Discussion
The response of atoms to cluster or order in alloys offers deep insights to the microstructural behavior. This trend is particularly important for multicomponent alloys, because the ordering can drive the system to various microstructural states, such as crystalline or intermetallic solids, from the disordered FCC (A1), BCC (A2), or HCP (A3) conditions. In Al_{x}CrCoFeNi, the Al content can influence the final structural configuration^{6,23,24,25,26,27} due to the larger atom size, compared to other constituents, which considerably changes the lattice ordering. When the mole fraction of Al <0.3, a singlephase A1 is observed, and a singlephase A2 is found for Al >1.17, while a twophase (A1 + A2) for 0.3 ≤ Al ≤1.17 is reported^{24,26}. Using the methods and approximations for accurate estimates of the formation enthalpy and relative energy of A1 and A2 phases^{28}, the firstprinciples KKRCPA results in Fig. 2 for Al of 0 < x < 2 mole fractions predict a similar global stability found in the experiment, i.e., A1 for x < 0.5, A1 + A2 for 0.5 ≤ Al ≤1.25, and A2 for Al > 1.25. While the A1 phase is more stable than A2 for smaller x, the positive formation enthalpies of A1 and A2 phases imply that clustering is expected in the HEA. Our findings are in agreement with experimental observations^{24} and semiempirical CALPHAD calculations^{29}, especially for x = 0.1. Hence, for subsequent simulations, we consider only the A1 Al_{x}CrCoFeNi with x = 0.10. In MD calculations, Al_{0.1}CrCoFeNi is quenched from the melt phase, starting at 2,200 K. This arrangement ensures that the solidsolution alloy formed at 300 K is through the hightemperature melt phase at 2,200 K, as the experimental^{24} melting temperature reported for CrCoFeNi and Al_{0.3}CrCoFeNi are around 1,690 K and 1,655 K, respectively.
We also calculate the WarrenCowley shortrange order (SRO) parameters, α_{αβ} (k), for all αβ pairs and for selected alloy compositions, with focus on Al_{0.1}CrCoFeNi, using the thermodynamic linearresponse based on the KKRCPA method and its charge selfconsistent potentials and densities. This fundamental approach identifies uniquely the chemical modes driving SRO through the chemical interchange energies, (k;T) – the thermodynamic cost to swap α and β atom types at two distinct sites, as reflected in the Fourier wavevector, k. When (k;T) > 0 and has a maximum at a particular, k = k_{0}, for a specific αβ pair, it defines the unstable chemical mode and the dominant pair correlations (k_{0} = 0 indicates clustering, i.e., long wavelength “order”, and k_{0} ≠ 0 dictates finitewavelength ordering with the k_{0} periodicity). While multiple pairs can have a peak at k_{0}, a typical one will be dominant and other pairs will show correlations due to probability sum rules^{28}. As shown in Fig. 3, the AlNi pair in (k;T) possesses the strongest interchangeenergy fluctuations at a longwavelength given by k = [000] (a Γpoint for the A1 lattice). These energies are manifested in the observable α_{αβ} (k) through the AlNi pair with a peak at k = [000] (i.e., AlAl or NiNi pairs clusters in real space), and a weak AlCr nearestneighbor ordering at k = [001] (Xpoint). The diverging AlNi pair of α_{αβ} (k = [000]) at the spinodal temperature (T_{sp}) indicates the absolute instability of the alloy that forces clustering in Al_{0.1}CrCoFeNi, manifest in the AlAl and NiNi pairs, for example, and that can be compared to MD simulations from new force fields. The calculated spinodal temperature, T_{sp} = 840 K, shows good agreement with the experimentallyobserved transition temperature of 813–823 K for Al_{x}CrCoFeNi, 0 ≤ x ≤ 0.3^{24}.
The variation of the cohesive energies (eV/atom) with the lattice constant (in Å) is calculated at 0 K, shown in Fig. 4, and compares firstprinciples and MD predictions with experiments. We predict the lattice parameter (a_{0}) from KKRCPA as 3.45 Å through local density approximation (LDA), 3.51 Å from the generalized gradient approximation (GGA), and 3.57 Å from the GGA corrected (GGAc) for thermal expansion from zeropoint phonon contributions from Grüneison theory, which is in agreement with roomtemperature measurements (3.57 Å)^{30}. The MD simulations find 3.55 Å at 0 K, in reasonable agreement with KKR and experiment. KKRCPA also yield improved structural properties (e.g., lattice constant, ‘a_{0}’, and bulk modulus, ‘B’) using GGAc, i.e., a_{0} = 3.57 Å and B = 1.58 MPa at 300 K. MD simulations^{31,32} finds B by evaluating the curvature of the energy curve (Fig. 4) at a_{0} (3.55 Å). For the validation of the proposed EAMLJ forcefield, before analyzing structural and deformation behavior in Al_{0.1}CrCoFeNi, a detailed quantitative comparison for ‘a_{0}’ and ‘B’ for Al_{x}CrCoFeNi at x = 0.1, 1.0 and 1.5 mole fraction are given in Table 1.
Structural pair correlation, g(r), in the realspace, as shown in Fig. 5[a–e], are derived from the MD simulations by histograms to identify (un)favorable neighboring atomic pairs constituting the alloy to predict the chemical mechanism (clustering or ordering) in Al_{0.1}CrCoFeNi. We observe the strong affinity between the like pairs, i.e., AlAl, CrCr, CoCo, FeFe, and NiNi. While the AlAl pair correlation dominates all the like pairs, no significant contribution is noted from unlike pairs except AlCr [Fig. 5a]. The strong affinity of like pairs reveals clustering in Al_{0.1}CrCoFeNi, in qualitatively good agreement with linearresponse results (Figs 3 and 5). In Fig. 6, we illustrate pair probabilities calculated from the linearresponse (firstprinciples) and forcefield (MD) up to the 10^{th} shell in the realspace. The probability of finding unlike pairs at neighboring sites decreases with increasing the shell size. The difference in the AlNi probability is attributed to the fact that the “size effect”, which includes local site displacements is not included in the (k;T) calculations.
From MD simulations, the AlAl pair correlations and trends at 500 K, 700 K, and 1,000 K in Fig. 7 are similar to the pair correlations found at 300 K. There is a marginal increase in the peak, g(r), of the AlAl pairs in Fig. 7 while going from 300 to 500 K, which is attributed to an initial increase in the size of the AlAl cluster size. However, as a temperature rise induces disorder in the atomistic rearrangement, this correlation decreases above 500 K in the simulated HEA.
The engineering stress versus engineering strain curves for Al_{0.1}CrCoFeNi under dynamic compressive and tensile loadings at different temperatures from 77 K to 1,000 K are illustrated in Fig. 8[a,b], respectively. For compressive loading, a peak flow stress of ~125 MPa is recorded at 77 and 300 K, while it is ~90 MPa at 700 and 1,000 K. Above the flow stress, the thermoplastic instability in the shear band causes thermal softening that dominates the stressstrain regime till an engineering strain of 0.6%. Thermal softening is more profound at elevated temperatures (700 K and above), as evident by the drastic drop in the flow stress at lower temperatures (77 K and 300 K). A consistent flow stress is observed until strain hardening. Strain hardening at various temperatures occurs at approximately similar strains, and beyond it the flow stress abruptly increases. The stressstrain curves for compressive loading show a marked evidence of strain hardening. The wide spectrum of thermoplastic instability fluctuations at elevated temperatures is dissimilar to the instability found in shear bands of alloys under compressive loading conditions. In the case of tensile deformation [Fig. 8b], beyond the ultimate tensile stress (or critical flow stress) of ~75 MPa, a plastic flow occurs across the temperature range considered (77 K to 1,000 K). No strain hardening is observed under tensile deformation. Under compressive loading the clustering to ordering transition occurs with the increase in unlike pair correlations (e.g., AlCo andAlCr) and decrease in like pair correlations (e.g., AlAl), Fig. 9.
This prediction suggests that the quenched Al_{0.1}CrCoFeNi evolves from a phaseseparated to an ordered phase. However, for the corresponding tensile strain case, the clustering phase still dominates, as it induces a plastic flow condition, similar to a wiredrawing process. Thus, the transition to ordering state among the different elements of the Al_{0.1}CrCoFeNi is possible only for the high compressive strain condition. At higher temperatures, we find large fluctuations in stress due to compressive loading. This thermoplastic instability in Al_{0.1}CrCoFeNi leads to shearinduced disordering similar to the structural changes arising in bulk metallic glasses (BMGs)^{33,34}. Literature also suggests that mechanical alloying and shock compression of the FeCu alloy system results in the A1A2 transformation^{35}. Compression at the highstrain rate (10^{10} s^{−1}) in our investigation is identical to an shock compression treatment^{35}. The structural transformation of the A2 Fe into A3/A1 Fe phases under uniaxial compression has been observed in an earlier MD simulation^{36}. Thus, the results of mechanical alloying and shock compression are essentially in accord with those from highstrain rate quenching and compression analysis, as presented here for Al_{0.1}CrCoFeNi.
Summary
In this report, we establish a forcefield for a quinary AlCrCoFeNi alloy and validate it by comparing structural properties, relative stability, and pair probabilities with firstprinciples calculations and limited experimental results. After the validation, we employ the new potential to investigate the largescale deformation characteristics of Al_{0.1}CrCoFeNi using atomistic MD simulations. Thus, we address the absence of welldeveloped forcefield functions for highentropy alloys and provide the EAMLJ parameters to define the self and crossinteractions of the participating elements in the alloy.
In Al_{0.1}CrCoFeNi, the electronicstructurebased linearresponse calculations predict clustering driven by AlNi pair correlations (forcing AlAl and NiNi clustering behavior). The MD simulations are in agreement with the clustering trend, as given by the large clustering domains of like pairs dominated by AlAl. Although an increase in temperature reduces the clustering strength of like pairs, AlAl correlations still dominate. The HEA shows a clusteringtoordering transition under compressive loading, which is attributed to atomistic rearrangements at high strains. Corresponding pair correlations further corroborates the ordering behavior in the strained alloy. This investigation provides further motivation for the experimental exploration of hightemperature compressive thermoplastic instability in Al_{0.1}CrCoFeNi.
Methods
Electronicstructure
We investigate the phase stability of the solidsolution phase of the metallic Al_{x}CrCoFeNi alloys (x = 0.0 – 2.0 molefraction) using the KorringaKohnRostoker (KKR) Green’s function method in combination with the screened Coherent Potential Approximation (CPA)^{37,38}. The scalarrelativistic approximation is used for the valence electrons (no spinorbit terms). The exchangecorrelation functionals used are the von BarthHedin^{39} local density approximation (LDA) as parameterized by Moruzzi, Janak, and Williams^{40} and generalized gradient approximation (GGA) by PerdewBurkeErnzerhof revised for solids (PBEsol)^{41}. A variational definition^{42} of the potential zero (v_{0}) is utilized to yield the kinetic energies and dispersion that approach those of fullpotential methods^{42,43}. Potentials, charge densities, and total energies are obtained, using a complexenergy GaussLegendre semicircular contour with 24 points, and Brilliounzone integrations use a special kpoint method^{44} with a 20 × 20 × 20 mesh. Charge selfconsistency is accelerated using the modified Broyden’s second method convergence technique^{45}. Electronic properties and total energies are evaluated using the Voronoi polyhedra (VP) integration^{46} for sphericallyaveraged radial functions in the sitecentered, sphericalharmonic (Y_{L}) basis, where L = (l, m) is a composite angular quantum number referring to orbital (l) and spin (m). With L_{max} = 3, we include s, p, d, and fsymmetries in the basis. To account for the thermal expansion at the finite temperature from phonons on the lattice constant and bulk modulus, we include the zeropoint energy via the Grüneisen model^{40}.
Thermodynamic linearresponse
The KKRCPA grand potential (or free energy) is analytically expanded to secondorder in compositional fluctuations for an arbitrary Ncomponent alloy. The firstorder terms vanish identically in a homogeneouslyrandom (reference) state, whereas the secondorder term gives the symmetric thermodynamic functional^{47} (k;T), quantifying chemical interchange energies and being analytically related to the atomic shortrange order, SRO^{47,48}. The KKRCPA potentials, charge densities, and scattering matrices for a given solid solution are used to evaluate the linearresponse expressions. (k;T) includes all electronic structure, charge screening and transfer^{47}, and it is no more costly to evaluate a binary alloy, as it is a HEA. Hence, the SRO (cluster or ordering) and its origin are related directly to the electronic structure, and provide insight into the competing effects, such as bandfilling, atomicsize, Fermisurface nesting, and chargetransfer. Notably, at a fixed composition, assuming that site charges vary little with SRO, Pettifor’s force theorem is applicable and, then only the bandenergy variations for (k;T) survive^{49,50}, and doublecounting and exchangecorrelation terms vanish^{47}, simplifying the linearresponse expressions. (k;T) is evaluated on a logarithmic frequency mesh containing all Matsubara poles, such that the response functions can be interpolated to the correct poles, i.e., temperature^{51,52,53}. (k;T) is weakly temperature dependent from a Fermi factor, while (k) strongly depends on temperature as the point entropy is analytically included, and as such it diverges at the spinodal temperature T_{sp} for a specific maximum wavevector^{46}. (k;T) is formulated in a “host” picture for the computational expediency. Then it is converted to an “offdiagonal” representation for the ease of comparison to the experiment^{40}. The eigenvectors of the (k;T) chemical stability matrix just above T_{sp} reveal the ordering/clustering instability reflected in the SRO.
Moleculardynamics (MD) simulations
The highlyparallelized Largescale Atomistic Molecular Massively Parallelized Simulator^{54} (LAMMPS) package is used for MD, while the visual analysis and postprocessing of molecular trajectories is performed with Visual Molecular Dynamics^{55} (VMD). The MD simulations for investigating the structure of the Al_{0.1}CrCoFeNi HEA^{23,56} have previously employed LJ potentials that cannot offer reliable predictions for such multicomponent systems. Here, we assimilate the EAM/alloy potential parameters for Al, Co, Fe and Ni from the EAM database^{57,58} to model the elemental cross and selfinteractions. Also, the cross interactions of NiCr and FeCr and the selfinteractions of CrCr are described with the EAM/alloy potential^{57}. Only the AlCr and CoCr cross interactions are modeled, using the LennardJones (LJ) potential^{23,56,57}, due to the lack of the available EAM/alloy parameters for these interactions. The details of the functional forms of these atomic interactions are available in the literature^{54,58,59}. We employ the LennardJones potential (LJ) in the 126 form given by:
Here, σ is the distance where V_{ij}(σ) = 0, and ε is the well depth of the LJ potential. The parameters for σ and ε considered in the present work are discussed in Table 2. These LJ parameters have been previously employed for different MD studies^{56,60}. The LorentzBerthelot mixing rule is used to describe the cross interactions between AlCr and CoCr, such that for species, i and j, we have and . We note that there is <2% deviation in the cohesive energy and negligible change in the system density with the variation in the cutoff radius for all the crossinteractions (AlCr and CrCo) described by the LJparameters in the present work. Thus, we employ a cutoff radius of r_{cutoff} = 10 Å in all our MD simulations to obtain accurate results in a reasonable amount of computational time.
A FCC crystal lattice of 2.5 nm × 2.5 nm × 2.5 nm composed of randomlydistributed Al, Cr, Co, Fe and Ni atoms, shown in Fig. 1, is constructed with the elemental composition as described in Table 3. The simulation cell contains 62,500 atoms, and periodic boundary conditions are imposed in all directions. Energy minimization is carried out, using the steepest descent algorithm, with the energy and force tolerance set to 10^{−15} units (stopping tolerance for energy is unitless while force has a unit of eV/Å), to obtain the geometricallyoptimized lattice configuration for the randomlyarranged Al_{0.1}CrCoFeNi. The optimized structure is initialized at 2,200 K under an isothermalisobaric (NPT) ensemble at a pressure of 0 MPa for 90 picoseconds (ps) to melt the alloy using equilibrium MD simulations. This step is followed by rapid quenching of the alloy under the NPT ensemble at 0 MPa with two different cooling rates of 21.11 K/ps and 5.42 K/ps, respectively, to reach the desired temperatures between 77 K and 1,500 K. We employ the NoséHoover thermostat and barostat, each with a coupling time of 0.001 ps. Next, the structure is allowed to equilibrate for further 90 ps. A time step of 0.001 ps is maintained in all our MD simulations.
As shown in Fig. 10[a,b], we observe no significant change in g(r) for different atomic pairs with varying quenching rates at a fixed temperature (300 K). Thus, for all simulations, in the present study, a quench rate of 21.11 K/ps is maintained. The quenched HEA is, then, further equilibrated under the NPT and NVT (canonical) ensembles successively. The pressure and temperature constraints each with the coupling time of 0.001 ps are imposed by the NoséHoover thermostat and barostat, for a total time of 1 ns, followed by the NVT ensemble, for further 2 ns. Finally, the entire system is simulated in the absence of thermodynamic constraints for further 1 ns under the NVE (microcanonical) ensemble to ensure that we obtain an equilibrated structure. Next, tensile and compressive loadings of the alloy are performed independently at desired temperatures. The simulation cell is deformed in the xdirection of <100> with a strain rate of 10^{10} s^{−1}, for the engineering strain of 0.9%, while lateral boundaries are controlled using the NPT equations of motion to zero pressure. Higher strain rates, ~10^{10} s^{−1}, are chosen to provide predictions within reasonable computational wallclock times due to length and time scale limitations in MD simulations. This trend restricts the use of experimentallyrealizable high strain rates, ~10^{2} s^{−1}. Nevertheless, deformation characteristics observed in our investigation are representative of those observed for experimentallyapplied high strain rates. The atomic structures are analyzed from the molecular trajectories by pair correlations of different elements in the neighborhood of the chosen species.
Additional Information
How to cite this article: Sharma, A. et al. Atomistic clusteringordering and high strain deformation of Al_{0.1}CrCoFeNi highentropy alloy. Sci. Rep. 6, 31028; doi: 10.1038/srep31028 (2016).
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Acknowledgements
AS thanks the College of Engineering for the Graduate Research Initiative (GRI) fellowship. The U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division supported Ames Laboratory’s work (firstprinciples calculations by PS and DDJ). Ames Laboratory is operated for the U.S. DOE by Iowa State University under contract No. DEAC0207CH11358. PKL acknowledges the DOE, Office of Fossil Energy, National Energy Technology Laboratory, grant DEFE001194 and National Science Foundation (NSF), grant DMR1611180, with the program managers, J. Mullen and D. Farkas, respectively. The work at Iowa State University was partially supported by the Office of Naval Research under ONR grant N000141612548 with Dr. David Shifler as the program officer. Computing (AS and GB) was partially supported by the HPC@ISU equipment at Iowa State University, some of which was purchased by NSF funding under Major Research Instrumentation (MRI) grant number CNS 1229081 and CRI grant number 1205413.
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A.S. designed the MD study and performed the MD simulations. P.S. and D.D.J. carried out firstprinciples calculations of both structural and chemical shortrange order parameters for the hightemperature HEA. A.S., P.S., D.D.J. and G.B. analyzed the results and wrote the manuscript. P.K.L. reviewed the results and assisted in the manuscript preparation. G.B. and D.D.J. coordinated the project.
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Sharma, A., Singh, P., Johnson, D. et al. Atomistic clusteringordering and highstrain deformation of an Al_{0.1}CrCoFeNi highentropy alloy. Sci Rep 6, 31028 (2016). https://doi.org/10.1038/srep31028
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