Starting from 2002 until now, almost 400 papers have been published where the term “high-entropy alloys (HEAs)” is used (Figure 1), in most cases with the focus on microstructure and mechanical properties. The interest in these alloys has especially increased for the last 5 years. What are HEAs? Originally they have been defined as homogenous alloys composed of 5–13 major elements with concentrations between 5 and 30 at.%, and with simple average crystal structures such as cF4-Cu (fcc), cI2-W (bcc) or cP2-CsCl [1–3]. Their stability (at high enough temperatures) against phase separation and formation of intermetallic phases was explained by the high mixing entropy, which could be achieved if the number of elements was large enough and the stoichiometry close to equiatomic. This was demonstrated experimentally on a few examples and seemed to open up a wide new area for alloy design. The huge number of possible element combinations would make possible combinations of tunable properties that would be hardly possible in conventional alloys such as high hardness, strength and ductility, oxidation and wear resistance, magnetism, etc. [4, 5].
In contrast to the expectations, it was found that most element combinations studied experimentally lead to multiphase alloys, which were labeled “HEAs” subsequently anyway. In these cases the microstructure became an important factor for the properties to be studied in the same way as in conventional alloy design. For the development of functional materials is can be crucial to have homogenous single-phase materials. Therefore, it is of interest to screen all the materials labeled “HEAs” so far, whether or not they may be obtainable as single-phase materials. This is the intention of this paper. As we will see in the following that only a few element combinations can be classified as single-phase HEAs.
The formation of a particular solid phase is determined by both thermodynamic and kinetic factors . From the thermodynamic point of view, the phase with the lowest Gibbs free energy at given conditions will be formed if ΔGmix<0, with ΔGmix=ΔHmix TΔSmix. This is always the case for ∣ΔHmix∣<TΔSmix or ΔHmix<0. The mixing enthalpy, ΔHmix, can be seen as a measure of the long-range order of the system, which leads to the formation of intermetallic compounds when ΔHmix<0 or phase separation if ΔHmix>0. Nevertheless, in multicomponent systems the configurational mixing entropy ΔSconf increases with the number of constituents n, since
(x is the fraction of the i-th element), has its maximum value for the equiatomic composition and is a measure of the degree of randomness (disorder) of the system. At sufficiently high temperatures, ΔSconf favors the formation of simple solid solutions instead of intermetallic compounds. It should be noted, that the calculation of the configurational entropy of mixing is based on the assumption that each atom is distributed with equal probability, ignoring correlations (short-range order) and interactions between the atoms. Therefore, not only ΔSconf, but also mixing enthalpy and vibrational entropy can significantly contribute to the phase stability in equiatomic multi-component alloys . In thermodynamic equilibrium, enthalpy (order) and entropy (disorder) have to compromise providing a minimum of the free energy . In case of single-phase HEAs, the mixing enthalpy is in the range –15<ΔHmix<5 KJ/mol and ΔSconf ≥1.61R (for n=5) .
How to “predict” single-phase HEAs?
Until now, more than 25 possible multicomponent systems have been investigated for the existence of HEAs, based on different combinations of n=4–9 elements out of 16 (Figure 2). Two main research families can be identified: I – combinations of light transition metals with Al, and II – solid solutions of transition metals only (see Figure 2). Their selection and investigation have been done more or less by trial and error  since phase diagrams for such multi-component systems are not known. So far, partial vertical cross-sections of phase diagrams have been experimentally studied for the systems Cr-Fe-Co-Ni-Al-Cu , Cr-Fe-Co-Ni-Al-Mo , Cr-Fe-Co-Ni-Al-Nb , and Cr-Fe-Co-Ni-Cu-Ti  (note that many authors refer to a mixture of several fcc or bcc and cP2-CsCl type phases as fcc or bcc solid solutions) and simulated by the CALPHAD method for Cr-Fe-Co-Ni-Al , Cr-Fe-Co-Ni-Mn , and Cr-Nb-Ti-V-Zr , which are in agreement with experimental observations of [7, 16], and , respectively. Some new single-phase HEAs such as fcc (CoFeMnNi, CuNiPdPt, CuNiPdPtRh) and hcp (CoOsReRu) have been calculated . The accuracy of CALPHAD depends on the completeness of the databases used, i.e., the availability of thermodynamic and kinetic data for all edge (binary, ternary, etc.) phases . In this way, this approach can be used as a basis for the prediction of interactions between the components in multi-component systems; however, disagreements may be observed between the number of phases, their compositions and volume fractions . Ab initio molecular dynamic simulations for the refractory single-phase HEA TiZrHfNbTa shows the absence of short-range order and segregation in the liquid state that might indicate of formation of single-phase microstructure in solids .
Out of the more than 100 reported multi-component alloys (Table 1), only a few with approximately equiatomic compositions can be called single-phase solid solutions (at least in the as-cast form): CrFeCoNi , CrFeCoNiAl0.3 , CrFeCoNiMn (the first single-phase HEA discovered) [7, 62], Cr2Mn27Fe40Co5Ni26 , CrFeCoNiMnAl0.2 , NbMoTaW , VNbMoTaW  and TiZrHfNbTa [81, 90]. Based on only X-ray powder diffraction (XPD) data, CrFeCoNiPd and CrFeCoNiPd2  are also considered to be single-phase alloys. It should be noted that standard XPD cannot provide information on the phase content of an alloy with accuracy better than ∼5%. The formation of intermetallic compounds or of small precipitates at the grain boundaries, as well as inhomogeneities in the composition can be detected only with techniques, such as high-resolution transmission electron microscopy (HRTEM) or 3D atom-probe tomography.
Systems of light transition metals (TM) with Al
None of the reported equiatomic multicomponent alloys of light transitions metals with Al are single-phase solid solutions (except foils of CrFeCoNiAlCu  and CrFeCoNiAlCuTiV  synthesized by splat cooling). Their common feature is the eutectic type of interaction between the components. For the systems CrFeCoNiAlx [16, 23] and CrFeCoNiMnAlx  (x=0–3) single-phase HEAs with cF4-Cu structure type forms with very low Al content (x≤0.3), whereas with high Al content (x≥1), phases with cI2-W and cP2-CsCl type structures result.
Analyses of the binary phase diagrams of the systems TM-Al show that transition metals, for example Fe, can dissolve up to ∼45 at.% of Al at high temperature (1310°C), whereas the solubility of TM in Al is negligible ∼0 at.% . It also should be mentioned that Al has a larger atomic radius than transition metals and for some of them a lower electronegativity. It easily forms many intermetallic compounds with TM and has a strong tendency towards covalent bonding. According to thermodynamic calculations of binary systems  the strongest interaction is between Ni and Al which leads to formation of stable compound NiAl (cP2-CsCl type structure), whereas the weakest between Cr and Al.
Systems of transition metals only
In equiatomic light-TM single-phase HEAs (CrFeCoNi, CrFeCoNiMn), phases with cF4-Cu type structure have been observed, whereas for refractory metals (NbMoTaW, VNbMoTaW, TiZrHfNbTa) cI2-W type phases result. Although the constituting elements of some of these single-phase HEAs have different structure types at ambient conditions, their high-temperature allotropes (except Cr) have the same as the resulting single-phase HEA structure. Consequently, these HEAs are single-phase solid solutions based on the structure type of one element. As data of binary phase diagrams are most comprehensibly available, a guideline based on binary phase diagrams was proposed, which thus can be practically applied to multi-component alloy systems. According to this approach single-phase solid solutions are assumed in systems, which are characterized by complete solubility or at least have a large solid solution range between their binary pairs . Thermodynamic calculations in  and  support the conclusion that trends observed in the binary systems are also seen in multicomponent alloys.
Another approach, based on the Bozzolo-Ferrante-Smith method for alloys, was discussed in . The concept of critical concentrations for each element (limits of solubility) was proposed as an indicator of single-phase HEA formation. For the system V-Nb-Mo-Ta-W it was shown that the transition point to the high-entropy regime was reached for the equiatomic composition in agreement with the experimental observations .
The probability of the formation of fcc single-phase HEAs in the systems MnFeCoNiM and MnFeCoNiSmM (M – an other metallic element) based on a thermodynamic sublattice model and first-principles calculations was examined in . This approach can give an idea whether particular elements prefer to be equally distributed on the Wyckoff positions or have a tendency to occupy only certain sites. However, these theoretical calculations were compared only with a few experimentally studied single-phase HEAs, where the predictive power of this method seems limited.
According to the Hume-Rothery rules, solid solutions can be formed in systems containing elements, whose atomic radii differ by <≈15% and which have a similar electronegativity (or electron concentration, e/a ratio). For a visualization of these rules, see, for instance, the Darken-Gurry map  of possible solid solutions of other elements in Ta . For single-phase HEAs, these limits are narrower. By analyzing different compositions, the following was proposed as necessary condition:
atomic size difference δ≤6.6% , where ci and ri are the concentration and the atomic radius of the i-th element, and the average atomic radius, respectively;
valence electron concentration VEC≥8 for fcc solid solutions and VEC<6.87 for bcc phases , where
parameter Ω≥1.1 , defined as with the melting temperature of the i-th element, ΔSmix and ΔHmix the entropy and enthalpy of mixing predicted from Miedema theory , respectively.
It should be noted that many multiphase multicomponent alloys also satisfy these criteria.
Crystal structure of single-phase HEAs and the “coloring problem”
All single-phase HEAs known so far have simple average crystal structures of either the cF4-Cu ( ), the cI2-W ( ) type or its ordered variant cP2-CsCl ( ). Single-crystalline CrFeCoNi and CrFeCoNiAl0.3 have been already synthesized [87, 93], however, no structural data are available from X-ray single-crystal diffraction experiments. Therefore, many questions are still open: What is the local atomic structure of HEA? How are the different atoms distributed on the lattice? Is there a short-range ordering or clustering? Which correlations are present between atoms? What kinds of clusters are there? All these questions refer to the “coloring problem” of HEAs . Each element will tend to occupy the position that minimizes its site energy as well as the bond energies, which depend on preferred coordinations, interaction potentials and atomic volumes of the different elements. To experimentally determine the elemental distribution, several complimentary methods have to be applied. Single-crystal diffraction analyses have to be carried out with X-rays and/or neutrons depending on the scattering contrasts of the elements. In addition to the Bragg reflections, which reflect the average structure, the diffuse intensities have to be taken into account as well, in order to get information of the actual local structure. Further information can be obtained from atom probe measurements and electron microscopic investigations.
As it has already been mentioned above, family I of single-phase HEAs contains light transition metals and Al. This element selection resembles that of Heusler phases (a typical representative is Cu2MnAl), which are ordered variants of the cI2-W-type (2×2×2 supercells). In Figure 3, some cI2-W-type derivative structures are illustrated. Locally, in bcc HEAs, related atomic arrangements may be found.
In order to study the variation of local atomic correlations with temperature in refractory single-phase HEA MoNbTaW, a combination of Monte Carlo and molecular dynamics methods has been applied . The simulations show that a phase transition from the disordered cI2-W type structure to the ordered cP2-CsCl one takes place between 327 and 527°C; this is similar to other results , whereas favored/disfavored atom pairs in these two works differ.
The electronic structures and basic bulk properties of CrFeCoNi, CrFeCoNiCu, CrFeCoNiCuTix, and CrFeCoNiTi alloys at static conditions (0 K), which were assumed to be single-phase paramagnetic random fcc solid solutions, were investigated with DFT calculations using the exact muffin-tin orbitals (EMTO) method in combination with the coherent potential approximation (CPA) in . Calculated and experimental values of atomic volumes for these HEAs are in relatively good agreement, whereas the Young’s moduli differ (except for CrFeCoNiTi). A possible reason can be the assumed conditions used in the simulations, as only CrFeCoNi is a single-phase solid solution according to the experimental observations.
Synthesis and mechanical properties of single-phase HEAs
Single-phase HEAs can be synthesized like any other conventional alloy [3, 106]. So far, the samples were prepared in most cases by arc or induction melting , splat quenching [2, 39] or by the Bridgman technique . It should be noted that some of the alloys were analyzed just in their as-cast state, which conserves the structure of the primary solidified phase, if the kinetics allows. The as-cast phase can undergo phase transformations (ordering, spinodal decomposition/chemical unmixing, precipitation, segregation, nanoparticles formation etc.) during annealing, what can have strong influence on the physical properties of an alloy. In contrast, proper annealing followed by rapid quenching leads to the thermodynamically most stable phase(s) at the annealing temperature.
In most cases, the mechanical properties of single-phase HEAs were investigated by compression, Vickers hardness or tensile tests. The light-TM based single-phase HEAs are less hard (for example, CrFeCoNi 135 HV ) than the refractory ones (NbMoTaW 454 HV ), but they are more ductile (plastic strain 75% and engineering strain 7% for CrFeCoNi and NbMoTaW, respectively).
Solid solution effect
High hardness and strength of single-phase HEAs are assumed to originate from severe elastic distortions of the crystal lattice caused by the presence of n elements with different atomic sizes. For small amounts of foreign atoms this is called the solid solution strengthening effect. The strengthening originates in the interaction of the strain field around dislocations with the lattice strains, which increase the activation energy for dislocation movement. Differences in chemical interaction and bond strength (covalent contributions) between different elements in the single-phase HEAs can further modify the potential energy surface of a dislocation gliding plane . However, in equiatomic single-phase HEAs, there are no clearly defined solute or solvent atoms. The mechanism of dislocations movement in these “corrugated structures” is unclear . No quantitative models for the description of strengthening by lattice distortions are provided yet.
Influence of thermo-mechanical treatment on mechanical properties of single-phase HEAs
Single-phase HEAs are potentially very valuable materials, which can have a variety of different tunable properties, depending on the kind and concentration of constituting elements (the so-called ‘cocktail effect’) . For instance, materials that possess superior mechanical properties at elevated temperatures are in great demand for the aerospace industry. NbMoTaW, VNbMoTaW, and TiZrNbHfTa are examples for HEAs, which possess a high-temperature strength and thermal resistance comparable to superalloys [80, 82]. As it can be seen from Figure 4, the yield strength values for these single-phase HEAs are higher than for Haynes 230, and at T>800°C even higher than Inconel 718, with rather weak decrease of yield strength up to 1600°C. This strong resistance to softening at high temperatures is probably due to the sluggish diffusion  through the strained lattices of single-phase HEAs.
Alloys that possess both high strength and ductility are also promising for low-temperature structural applications . From first-principle electronic structure calculations it was shown that the stacking fault energy decreases with increasing number of components in the alloy (∼100 mJ/m2 and <15 mJ/m2 for NiAl and CrFeCoNiMn, respectively), and approximately on the same order of some conventional austenitic stainless steels (18 mJ/m2 for AISI 304L). The alloy Cr26Fe20Co20Ni14Mn20 with optimal value of stacking fault energy 7.7 mJ/m2 was designed by varying Ni/Cr concentration.
It is well known that mechanical properties are very sensitive to the microstructure and therewith to the preparation conditions. For example, annealing temperature and time, cold or hot deformation (rolling), and grain size play an important role [16, 21, 22]. As with other alloys, the grain size has a major influence on tensile strength, i.e., the Hall-Petch effect, as it was shown for CrMnFeCoNi .
The hardness of CrFeCoNiAl0.25 amounts to 113 HV , quite similar to that of the pure metals. After deformation by rolling to 50% reduction in thickness, its hardness was significantly improved to 325–346 HV. No stress-induced phase transition was observed under rolling. Cold rolling increases the strength of the alloy Cr2Mn27Fe40Co5Ni26  due to work hardening, whereas it decreases the ductility, what may be explained by a decreasing amount of strain hardening with increasing deformation. The dislocation-slip deformation mechanism is proposed for this alloy: with applying higher strains more slip planes are activated and the dislocation density increases; the planar dislocation structure is reorganized into dislocation cells .
Tensile tests of single-phase HEAs with compositions CrFeCoNi and CrFeCoNiMn over a wide temperature range show an increase in ductility and strength with an decrease in temperature . Closer investigations indicate an increasing importance of the twinning deformation mechanism at lower temperature. This was explained by a more pronounced effect of the thermal activation of the dislocation mobility due to the higher amount of obstacles on the gliding planes, in comparison to alloys with a lower number of alloying elements. On a smaller scale, this effect can already be observed in binary alloys, e.g., Cu-Al, Cu-Ge, Cu-Mn [112–114]. In contrast, it seems that the stacking-fault energy does not change much with temperature, and therefore can be ruled out as the cause for the difference in the deformation mechanism.
The reduction of crystal defects such as dislocations and grain boundaries can improve the strength and ductility of alloys . Indeed, micron- and nanometer-sized pillars of NbMoTaW exhibit higher strength and ductility than the respective bulk materials. The properties of the HEA pillars are better than of the pillars of any of the pure elements. Furthermore, the size effect for the properties resulted smaller for the single-phase HEA than for the pure metals. The better mechanical characteristics are explained by the high amount of local atomic distortion typical for HEAs.
Transmission electron microscopy (TEM) studies of some possible single-phase HEAs
In order to check experimentally existence of single-phase HEAs in light-transition metal systems the samples with the nominal compositions CrFeCoNi, CrFeCoNiAl0.3, PdFeCoNi and masses of 0.7 g each were synthesized by arc-melting from powders of pure metals (Cr: 99.996%, Fe: 99.998%, Co: 99.995%, Ni: 99.999%, Pd: 99.99% Al: 99.999%). The weight losses were less than 1%. The prealloys were placed in Al2O3 crucibles, then closed under Ar atmosphere in Ta ampoules and welded. The samples CrFeCoNi, CrFeCoNiAl0.3 were homogenization at 1300°C and PdFeCoNi at 1100°C for 1 week and subsequently quenched in water. From X-ray powder diffraction patterns and BSE images fcc lattice and single-phase microstructure for the synthesized samples was confirmed. TEM specimens were prepared by the standard mechanical grinding, dimpling and ion milling for perforation (PIPS 2). TEM experiments were performed on FEI Tecnai F30 machine operated at 300 kV.
TEM studies of the sample CrFeCoNiAl0.3 confirmed its single-crystalline nature and fcc structure from electron diffraction pattern acquired from 800 nm large areas. On a two beam bright field image acquired close to the [0–11] zone axis with the g=111 as a second strong reflection (Figure 5A), the heterogeneous morphology is observed due to inclusions seen as dark speckles (approximately 20 nm size or less) homogeneously distributed in the matrix. The same features were observed for the CrFeCoNi alloy. The chemical nature of the speckles was confirmed by HAADF-STEM, an atomic number sensitive imaging technique (Figure 5B), where the speckles look bright upon dark background contrast of the matrix. Most likely it might be concluded that the onset of the phase decomposition has already taken place. The Matrix (bright region) and speckles (dark region) are fully coherent phases with the same structure and very similar unit-cell parameters like it is reported for the binary alloy in the system Cr-Fe , and what was found in the as-cast alloy CrFeCoNiAl . The elemental content analyses of the small particles are not reliable since particles are too small to give a decent signal-to-noise ratio in the STEM-EDX line-profile. The compositional variations could be confirmed with 3D atom probe tomography, indicating that the material is not single-phase.
An analysis of the respective binary phase diagrams shows that the phase separation may be caused by the presence of Cr in the samples. Cr forms a continuous solid solution with Fe, whereas it can dissolve only up to 30 at.% of Ni . Therefore, in another experiment Cr was replaced by Pd, which forms continuous solid solutions with three other elements Fe, Co, Ni. However, the same heterogeneous morphology was observed as for the sample with Cr (Figure 6).
Conclusions and open questions
It is still an open question, how many different single-phase HEAs do exist in thermodynamic equilibrium, and in which compositional and temperature ranges. As it was shown in  for the single-phase HEA CrFeCoNiMn, the replacement of just one element at a time in the alloy by a neighboring element from the periodic table (Cr by Mo or V, Fe by V, Co by Ti, Ni by Cu) leads to a multiple-phase microstructure. Is there an optimum (with regard to the size of the compositional and temperature stability range) or maximum number of constituents before phase separation takes place? So far, the maximum number of constituents reported for single-phase HEAs was n=6 . What are the ranges of tolerance of the crystal lattice before phase separation takes place? According to the “confusion principle” : the more constituent elements of the alloy, the lower the chance for them to select one out of all possible crystal structures and, therefore, the greater the probability of glass formation followed by phase separation if thermally equilibrated.
The average crystal structures of the so far reported single-phase HEAs belong to one of the three simple types: cF4-Cu, cI2-W or cP2-CsCl. Are there any other structure types prone to the formation of HEAs? How are the elements distributed on the atomic sites of the structures? Is there any kind of short-range clustering and/or ordering? How distorted are the structures?
Single-phase HEAs have been reported to excel by a variety of properties, especially mechanical ones. What is the origin of special properties such as high hardness in combination with significant ductility? It is essential to understand the kind of short-range order present, if any, in order to control and tune interesting properties directly.
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