Revisiting Stability Criteria in Ball-Milled High-Entropy Alloys: Do Hume–Rothery and Thermodynamic Rules Equally Apply?

Revisiting Stability Criteria in Ball-Milled High-Entropy Alloys: Do Hume–Rothery and Thermodynamic Rules Equally Apply? Javier S. Blázquez,* Alejandro F. Manchón-Gordón, Antonio Vidal-Crespo, Rafael Caballero-Flores, Jhon J. Ipus, and Clara F. Conde 1. Introduction Progress in Materials Science has often been driven by either pioneering research methodologies or the creation of materials endowed with distinctive structural features and functionalities. Building upon these concepts, a revolutionary strategy, known as “high entropy,”has gained prominence in recent times. The high-entropy approach presents a novel avenue for exploring and investigating properties of materials that were previously unexplored, spanning the entire periodic table. Designing materials based on the highentropy concept entails the combination of five or more principal elements to create a single system. This approach differs from a doped system, where the concentration of the dopant is restricted, and structural destabilization can occur if the concentration surpasses a specific threshold. High-entropy approach is currently a line of development of new alloys, known as high-entropy alloys, HEAs. In these systems, ideally crystallizing in simple phases with monoatomic base, the configurational entropy due to disordered positions of the atoms compensates the increase in energy of the disordered structure. HEAs, proposed almost 20 years ago by Cantor et al. [1] and Yeh et al. [2] are multielement systems with at least five principal elements, which can be defined as a function of its configurational entropy. In an ideal solid solution, the calculated ideal configuration entropy of mixing per mole, ΔS mix , can be calculated as [2] ΔSmix ¼RX N i1 xiln xi(1) where Rdenotes the gas constant, x i signifies the molar fraction of the i-th element, and Ncorresponds to the total number of the constituent elements. While the value for an equiatomic quinary alloy yields ΔS mix =1.61R, a compound is generally considered as HEA when ΔS mix >1.5R≈12.5 J K 1 mol 1 . However, this estimation is valid only for single-phase solid solution (SPSS). When a mixture of phases appears, the configurational entropy of each phase must be weighted to evaluate ΔS mix in the complete system. Miracle and Senkov [3] proposed different terms, such as complex concentrated alloys and multiprincipal component alloys (MCA), to define those systems that do not develop a HEA but a more complex microstructure. J. S. Blázquez, A. Vidal-Crespo, R. Caballero-Flores, J. J. Ipus, C. F. Conde Dpto. Física de la Materia Condensada ICMSE-CSIC Universidad de Sevilla P.O. Box 1065, 41080 Sevilla, Spain E-mail: [email protected] A. F. Manchón-Gordón Instituto de Ciencia de Materiales de Sevilla ICMSE CSIC-Universidad de Sevilla C. Américo Vespucio 49, 41092 Sevilla, Spain The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adem.202401148. © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. DOI: 10.1002/adem.202401148 Stability descriptors for the formation of solid solutions can be divided into two categories: inspired by Hume–Rothery rules (HRR) and derived from thermodynamic approaches. Herein, HRRs are extended from binary to high-entropy alloys (HEAs) focusing on compositions prepared by ball milling. Parameters describing stability criteria are interrelated and implicitly account for the microstrains’storage energy, more determinant than entropy increase in stabilization of HEAs and more effective in bcc structures than close-packed ones (fcc and hcp). An effective temperature, T eff ,isdefined as the ratio between increase in metallic bonding energy of solid solutions with respect to segregated pure constituents and configurational entropy. This versatile parameter is used as a threshold for stabilization of HEAs at equilibrium and out of equilibrium. When T eff is below room temperature, HEA would be stable at equilibrium. When T eff is below melting temperature, HEA would be obtained by rapid quenching. Limitations related to electronegativity differences remain valid in mechanically alloyed solid solutions. However, ball milling broadens the allowed differences in atomic size to form HEA. Moreover, thermodynamic criteria can be surpassed in these systems, allowing the formation of single-phase solid solutions beyond the compositional range predicted by those criteria. REVIEW www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (1 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH Many reviews on HEAs are available in the literature, including those that analyze key aspects of HEAs, such as the high-entropy-based nomenclature and the role of entropy and related thermodynamic parameters in phase stabilization. Moreover, structural and functional features have already been highlighted in review articles, some of which can be found in the reference list. The strategical interest on applicability has predominantly driven them, particularly concerning mechanical properties. On the other hand, the very name of these materials (very successfully broadcasted) should be specifically assigned to SPSSs, as already claimed by Miracle and Senkov, [3] and even the high-entropy reasoning for stabilization is far from being determinant. Moreover, several conditions, well established for binary solutions, are taken for granted when extended to HEAs. The aim of this work is to reasoning on the success of high-entropy strategy to develop single-phase systems in MCA, with main focus on the viability of production by milling and particularly by mechanical alloying (MA). Generally, powder production by MA or milling is followed by consolidation techniques, which may alter the microstructure of the as-milled powder. This will imply some characteristic features related to the stability of as-milled powder. However, this aspect is out of the scope of this work. Therefore, although thermal behavior and kinetic of transformation of ball-milled powders is of interest, our main interest is devoted to those as-milled (generally metastable) systems. Strategies such as low-temperature consolidation using binding material or flash annealing to prevent phase transformation would be necessary when a final bulk sample is needed. 2. Supersaturated Solutions of Binary Alloys Prepared by Mechanical Alloying It is increasingly accepted that SPSS in HEAs are generally metastable systems, [4–7] where production is promoted by both using rapid quenching techniques and the slow diffusion in such complex compositions. [8] However, this core effect, so-called sluggish diffusion effect and considered beneficial for HEA formation, is not free from debate and even increase in diffusivity in HEAs is reported for some elements. [9–12] HEAs, as SPSS, can be understood as supersaturated solid solutions, due to their very probable metastable character. These types of metastable systems can be obtained by MA in a wide compositional range. The main difference between samples produced from molten alloy and those prepared by ball milling is the departure state of the system. Whereas melt solidification starts from the molten alloy, in MA the raw materials generally consist of a mixture of pure powders in equilibrium at room temperature. This is mainly affecting the estimation of the contribution of configurational entropy, which in the case of melt-solidified samples is approximated to complete disorder at melting temperature. However, in supersaturated solutions obtained by milling, entropy has never contributed to the energy balance at such high temperature. In their highly referenced work, [13] Suryanarayana collected the different solubilities reported in binary systems by MA at room temperature, RT, and in comparison with the corresponding maximum solubility in thermodynamic equilibrium. Figure 1 shows the representation of such values for mechanically alloyed samples as a function of the equilibrium solubility at RT as well as versus its maximum value in solid state in thermodynamic equilibrium. Two features are remarkable in Figure 1: first, the ability of MA to develop a complete solid solution for immiscible systems at RT, as reported for AgCu, CoCu, and MgTi; [13] second, the lack of this achievement at intermediateand even for some highequilibrium solubilities. In fact, when representing the improvement in solubility by MA (estimated as the difference between solubility measured in MA and maximum equilibrium), with respect to maximum equilibrium solubility, a rough linear decay is observed yielding even large negative values for highequilibrium solubility values, as shown in right panel of Figure 1. This figure identifies the maximum solubility values obtained by MA with a larger hollow black square. Figure 1. Reported solubility for mechanically alloyed binary alloys versus equilibrium solubility at A) RT and B) maximum equilibrium solubility. Small solid squares correspond to literature data and maximum values reported for MA are enclosed inside open squares. Difference between solubility reported for MA binary systems (small red solid squares correspond to literature data and maximum values reported for MA are enclosed in hollow black squares) against C) maximum equilibrium solubility. Data taken from. [13] www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (2 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH 15272648, 2025, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adem.202401148 by Readcube (Labtiva Inc.), Wiley Online Library on [10/04/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 3. Application of Hume–Rothery Rules to Mechanically Alloyed Binary Systems Traditionally, Hume–Rothery rules (HRR) are considered for understanding the formation of a solid solution, independently of the production process employed to obtain it (i.e., solidifying from the melt or MA). The four HRRs can be summarized as: 1) coincidence of crystalline structure of the pure elements forming the solid solution; 2) similar atomic size, r, with viable solid solution for jΔrj r hi <15%; 3) maximum solubility for equal valence electron number; and 4) similar electronegativities, χ P (Pauling scale is considered in this work). Concerning first rule, Figure 2 shows a scheme with the different elements of interest grouped attending to their crystalline structure at RT and ambient pressure. Intersections are allowed among the sets to represent the possible allotropes of the elements. [14] For example, fcc allotrope of Fe matches the structure when alloyed with Al or Ni and bcc allotrope when alloyed with Cr or Nb. Figure 3 shows a schematic diagram of the binary alloys with equilibrium solid solutions in full compositional range, indicating their structures (AsSb, BiSb, with rhombohedral structure; and GeSi, with diamond structure are not included). This figure supplies a general view of those pairs of elements with the highest affinity to develop solid solutions. Figure 4 shows MA solubility with respect to (A) difference in Pauling electronegativity between the elements ΔχP¼jχPð1ÞχPð2Þj; (B) relative change in the radius calculated as Δr=r hi¼j2ðr1r2Þ=ðr1þr2Þj (i.e., 4 th and 2 nd HRR, respectively); (C) valence electron concentration (VEC, values taken from HEAPS software [15] ); and (D) VEC differences (ΔVEC =VEC(1)VEC(2), i.e., 3 rd HRR) between first and second component of the binary for those alloys with full range of equilibrium miscibility in solid state. Figure 5 shows equivalent Figure 2. Scheme to identify those elements that fulfill the first HRR. Monoatomic body-center-cubic (bcc: Im3m space group), face-center-cubic (fcc: Fm3m), close-packed-hexagonal (hcp: P63=mmc), double closepacked-hexagonal (dhcp: P63=mmc), diamond (Fd3m), white tin (I41=amd), and arsenic-like rhombohedral (R3m) structures are considered. In the intersections among the sets, the color of the elements is chosen in agreement with the structure observed at RT and ambient pressure. Figure 3. Schematic diagram of the different binary systems with solid solubility in the full compositional range. The RT structures of pure elements, along with their possible allotropic phases, are indicated. Lines are drawn between the two components of a full range solid solution. The color of the lines indicates whether crystalline structure of the solid solution is bcc (blue), fcc (red), or hcp (green). Dotted green lines correspond to double-hcp structures (dhcp) or other variants (see the cases of DyNd and GdNd). www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (3 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH 15272648, 2025, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adem.202401148 by Readcube (Labtiva Inc.), Wiley Online Library on [10/04/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License to Figure 4 but shows the data corresponding to binary alloys with <1 at% equilibrium miscibility at RT. The data points for fully miscible systems in equilibrium are randomly distributed in Figure 4 (maximum ΔχP¼0.35 for CuMn; maximum Δr r hi¼9.7% for BiSb) but the reported MA solubility is higher when RT crystalline phase is common to both elements (green symbols) and lower when full range of solubility requires high temperature/pressure allotropic phases (orange symbols). The only exception is BiSb, with average electronegativity 2.06, indicating a stronger covalent character. Concerning immiscible systems, solubilities are below 40% for ΔχP>0.4 and for Δr=r hi >20%, independently of whether crystalline structures matches at RT or allotropic phases allow for this match. Comparing atomic size difference and electronegativity, Suryanarayana [16] did not find Durken–Gurry plots (representing Δχ P vs Δr=r hi ) satisfactory to apply HRR to MA extension of solubility. The parameter to consider for the 3 rd HRR is ΔVEC. Full solubility range, in both equilibrium and by MA, can be observed only for some systems with |ΔVEC| ≤2. Meanwhile, alloys with full solubility range are spread over 2 <VEC hi <12. This parameter must be distinguished from itinerant electrons per atom, which in the frame of HEA is named e/a(see HEAPS [15] ). However, reader must be aware that in other contexts, for example, martensitic alloys, the acronym used for VEC is e/a, from electrons per atom, instead of VEC. Van Arkel-Ketelaar triangle [17] can be used in order to further appreciate the 4 th HRR for equilibrium and binary MA systems and combined with the type of bonding. Figure 6 shows the distribution of systems with full range of miscibility in equilibrium (panel A) and those systems that achieve at least the equiatomic solid solution by MA (panel B). It can be observed that data corresponding to equilibrium are more dispersed than MA data. The former values extend from fully metallic character to Figure 4. Maximum MA solubility as a function of A) difference in Pauling electronegativity Δχ; B) relative difference of atomic radii Δr=r hi ; C) average VEC; and D) difference in VEC between the first and the second component for those binary alloys which are fully miscible in solid state. Green symbols correspond to those binary alloys whose crystalline structure of pure elements at RT fulfills the 1 st HRR. Orange symbols correspond to those binary systems for which 1 st HRR is fulfilled by the existence of allotropic phases. Figure 5. Maximum MA solubility as a function of A) difference in Pauling electronegativity Δχ; B) relative difference of atomic radii Δr=rhi; C) average VEC; and D) difference in VEC between the first and the second component for those binary alloys whose solubility at RT is below 1 at%. Green symbols correspond to those binary alloys whose crystalline structure of pure elements at RT fulfills the 1 st HRR. Orange symbols correspond to those binary systems for which 1 st HRR would be fulfilled by the existence of allotropic phases. Red symbols correspond to systems that do not fulfill 1 st HRR. www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (4 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH 15272648, 2025, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adem.202401148 by Readcube (Labtiva Inc.), Wiley Online Library on [10/04/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License midpoint of the triangle. It is worth noticing that systems with immiscibility gap are placed at average electronegativities above 1.8. Figure 6C shows in a color scale the equilibrium solubility for those binary systems with Δr=rhi<8%. Figure 6D represents, also in a color scale, the maximum solubility reported for MA binary alloys. The general ideas derived from binary systems would be extrapolated to MCAs to discern the differences between SPSS produced by solidifying from the melt and by MA. 4. Families of High-Entropy Alloys Several authors classify the different compositions suitable for HEAs in different families: [3] 3D transition metals HEAs; refractory metals HEAs; low-density HEAs; HEA brasses and bronzes; precious metals HEAs; and rare-earth HEAs. Attending to Figure 2 and 3, we can identify the core of such families with those sets of elements for which each element has full solubility with, at least, two other elements of the set. Cantor alloys 3d transition metals HEAs are represented by CrMnFeCoNi. [18] In this alloy, with fcc structure, the only element that departs from the 1 st HRR is chromium (as shown in Figure 3) but even CrFeCoNi and CrCoNi yield single fcc phase. [19] However, Zhang et al. [20] found irreversible transition from fcc to hcp under a pressure of 22.1 GPa in the very prototype CrMnFeCoNi Cantor alloy, the hcp phase the stable one at low temperatures and fcc the polymorph at high temperatures. This is particularly interesting as first-principle calculations conclude hcp phase exhibits enhanced properties with respect to fcc one. [21] In contradiction with Zhang et al. [20] Ahmad et al. [22] only found, under a pressure of about 20 GPa, the appearance of tiny extra diffraction peaks, that they identify as bcc, with the perseverance of fcc maxima up to 48.9 GPa and after heating up to 1100 K. In this HEA family, Ti [23,24] leads to single-phase fcc, as well as Mo substituting for Co in Cu-added alloys. [25] Other elements, such as V [26] or Nb, [1] have also been added to original Cantor alloy CrMnFeCoNi in equiatomic fractions, and single fcc phase is reported. However, other studies identify the presence of precipitates of sigma [27] and C14 Laves [28] phases, respectively, in Mn-free compounds. On the other hand, elements such Al can only be added to CrMnFeCoNi up to below 9 at% to avoid multiphase formation. [29] Interestingly, it was recently shown that MA can help to overcome these limitations in both CrFeCoNiAl 0.75 and MnFeCoNiAl 0.75 (above 15 at%) forming metastable fcc single phase. [30] Figure 6. Van Arkel-Ketelaar triangles using K, Cl, and KCl as extremes for metallic, covalent, and ionic bondings. A) Binary systems with complete solubility in solid state, green symbols correspond to RT structures matching 1 st HRR (full solid symbols, full solubility at RT; half-filled symbols, miscibility gap at RT; crossed symbols, ordered or intermetallic phases at RT). B) Binary alloys obtained by MA as solid solution for equiatomic composition. Color scale as C) maximum equilibrium solubility for binary alloys with Δr r hi≤8% and as D) maximum solubility for binary alloys obtained by MA. www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (5 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH 15272648, 2025, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adem.202401148 by Readcube (Labtiva Inc.), Wiley Online Library on [10/04/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Refractory metal HEAs with bcc single phase were proposed by Senkov et al. [31] with the development of W 21.1 Nb 20.6 Mo 21.7 Ta 15.6 V 21.0 composition in 2010. Figure 3 shows that all couples within these elements present full miscibility range. Their high melting temperatures make these HEAs particularly interesting for high-temperature applications. [32] Group 4 elements Ti, Zr, and Hf (hcp at RT but bcc allotropes have full solubility in binaries with W, Nb, Mo, Ta, and V, for Ti; and with Nb and Ta, for Zr and Hf) can be used to preserve bcc single phase in as-cast equiatomic TiZrHfNbV. However, Laves phase precipitates after 10 min annealing at 1173 K in this alloy; and Laves phases appear in a large amount even for as-cast samples in the case of TiZrHfNbCr. [33] These Laves phases are favored by their topologically close-packed structure. [34] However, recent results for VNbMoTaW compositions identify the presence of two different bcc phases (partitioning V and Nb to one of the phases), which for equiatomic composition present very similar lattice parameters. [35] Aranda et al. [36] also reported two bcc structures in TiNbTaZrMo as well as Gao et al. [37] in spark-plasma-sintered Ti x AlV 0.5 CrMo (x=0.5–2) (bcc plus ordered B2) although single bcc was reported for asmilled powder. Lilensten et al. [38] reported orthorhombic–hexagonal martensitic transformation in Ti 35 Zr 27.5 Hf 27.5 Nb 5 Ta 5 .Alis used to develop lightweight HEAs, preserving bcc single phase in AlCrTiNbMo. [39] Some refractory metal HEAs are placed as lowdensity HEAs, generally containing Al and the lighter refractory elements Sc, Ti, V. Al x TiNbVZr (0.5 ≤x≤1.5) alloys crystallize in bcc structure [40] but hcp is found in (AlSc) 25 Ti 25 Hf 25 Zr 25[12] and Ti 30 Zr 30 Hf 30 Re 5 Al 5[41] compositions, for which, with the exception of Al, all the pure elements exhibit hcp at RT (see Figure 2). Close packed hexagonal structure was also found for Re 56 Nb 11 Ti 11 Zr 11 Hf 11 , [42] with superconducting properties but which could be considered a medium-entropy alloy (ΔS mix =1.3R) clearly rich in rhenium. Low-density HEAs, apart from multiphasic Al 20 Be 20 Fe 10 Si 15 Ti 35 ρ=3.91 g cm 3 , [43] can be divided in two groups: refractory metals, described above (ρ<6gcm 3 ), and those containing lighter elements such as Mg, [40,44,45] which also generally crystallize in multiphase systems. In 2015, Al 20 Li 20 Mg 10 Sc 20 Ti 30 was reported as the first single-phase HEA with low density, 2.67 g cm 3 . [46] It was produced by MA in single fcc phase but annealing leads to the polymorphic transformation to hcp, although O and N contamination preserves fcc phase. In Al 11 Li 10 Mg 23 Ti 21 Zr 35 , hcp structure is formed but precipitating intermetallic AlZr 2 . [47] Intermetallics are also produced after Si addition, being MCA with ρ<2gcm 3 . [48,49] HEA bronzes and brasses can be identified as an extension of Cu–Mn–Ni ternary system with addition of Al, Sn or Zn. [50] From these elements, in quaternary alloys (CuMnNi) 100x X x , only X=Zn preserves single-phase fcc in the equiatomic composition. Below x=25 at% and for X =Sn, L2 1 ordered structure is also formed and becomes the single phase in equiatomic quaternary composition. For X =Al, it progressively yields the formation of bcc for x=10 at%, and L2 1 phase for x=20 at% along with fcc. Higher Al content leads to the formation of ordered B2 along with L2 1 and the disappearance of fcc phase. Quinary alloys also show multiphase microstructure. Precious metals HEAs were predicted by Kube and Schroers [6] based on the analysis of atomic size misfit dependence at r hi ≈140 pm. However, results on this family of HEAs are scarce. Thiel et al. [51] observed single fcc phase in AuCuNiPdPt and analyzed the strength and plasticity of this system. The same research group further studied (AuCuNiPt) 100x Pd x and (AuCuNiPd) 100x Pt x (0 ≤x≤20) finding single fcc phase in all cases. [52] Concerning the family of rare-earth HEAs, in 2014, Takeuchi et al. [53] reported a single hcp phase in YGdTbDyLu and GdTbDyTmLu compositions, contemporary to the work of Feuerbacher et al. [54] on HoDyYGdTb. Both were claimed to be the first obtained hcp HEA. Pressure-induced transformations were reported for HoDyYGdTb alloy to Sm-type (4.4 GPa), to dhcp (26.7 GPa), and to distorted fcc (40.2 GPa). [55] Although Zhao et al. [56] and Soler et al. [57] reported the mechanical properties of single-hcp-phase rare-earth HEA, and Wang et al. [58] described the deformation mechanism of these hcp HEAs, these systems have deserved interest mainly due to their magnetic properties, [59–62] including magnetocaloric effect. [63,64] Combining Gd, Y, and Sc with Co, Ti, and Zr, which lays between refractory and rare-earth HEA, segregates in hcp refractory (ScTiZr) and rare-earth (GdScY) ternaries with the presence of cubic phases and intermetallics. [65] Figure 3 also suggests the possibility of identifying other HEA families based on the full miscibility between couples of hcp elements: RuReOsCo. In fact, several articles appear in the literature that can be related to this family. For example, Yusenko et al. [66] produced Ir 19 Os 22 Re 21 Rh 20 Ru 19 compound, with single-hcp phase stable after 1500 K annealing and an applied pressure of 45 GPa, while Ter-Isahakyan et al. [67] produced a combinatorial thin film to sweep the OsRuWCo system with microstructures varying from amorphous to single hcp phase, in addition to single-phase hcp HEAs in bulk form for (Co 36.6 Os 20.5 Ru 24.8 W 18.2 ) 75.6 X 24.4 with X =Fe and Ir. Due to highly reactive and the poising character of OsO 4 , Os-free compositions have been also studied (Fe 12 Ir 20 Re 20 Rh 20 Ru 28 and Ir 25 Re 25 Rh 25 Ru 25 ), characterized by a hcp structure. [68] Metallic products from the fission of UO 2 , aka five-metal particles, with composition Mo x Pd 9 Rh 9 Ru 28 Tc 14 (0 ≤x≤87 at%), have been considered as HEA [69] and could be placed into this family. In this compound, technetium stabilizes the single-phase hcp structure. 5. Stability Criteria for High-Entropy Alloys Stability descriptors for the formation of a random solid solution can be divided in two different categories: 1) Those inspired on HRRs: That is, the atomic size misfit, δ r , the electronegativity difference, Δχ P , and the difference in the ΔVEC; 2) Those derived from a thermodynamic approach: The enthalpy of mixing, ΔH mix , the minimum enthalpy of formation for suitable intermetallics, ΔH int , and entropy contribution, ΔS mix . In fact, it was ΔS mix the property that gave birth to HEA name. The importance of this parameter, already identified by Yeh et al. [2] was generally maximized, considering its contribution at melting temperature. Nowadays it is more cautiously taken to determine the formation of SPSS. The latter criteria, developed for systems obtained from the melt, are not directly applicable to HEAs produced by milling www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (6 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH 15272648, 2025, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adem.202401148 by Readcube (Labtiva Inc.), Wiley Online Library on [10/04/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License but to those compositions produced by quenching. [30] However, HRR criteria are generally assumed to be valid for HEAs formation by milling. [70] In fact, HRRs do not explicitly consider thermodynamic assumptions, which makes them suitable criteria to identify possibility of SPSS development independently of the production technique. 5.1. HRR-Based Criteria Although HRRs were developed to understand the formation of binary solid solutions, they were extended to more complex microstructures. [71] Particularly, Poletti and Battezzati [72] derived HRR representations for MCA to discern the SPSS formation in HEAs. In this section we will revisit these rules and the restrictions they imply. Finally, their applicability to MA systems will be discussed. 5.1.1. Structure Match. 1 st HRR To discuss on the 1 st HRR, we can refer to Figure 2 and 3. If we consider those elements that have full solubility with at least three other elements with the same structure, we can distinguish different groups of possible compositions and different structures. In particular, these possible combinations contemplated two bcc groups (I-bcc) CrVNbMoTaWTiZrHf (corresponding to refractory-HEA family, which can develop hcp structure when Ti–Zr–Hf is combined with other hcp elements) and (II-bcc) EuBaCaSrYb (bulk metallic high-entropy glass for CaSrYb-based Sr 20 Ca 20 Yb 20 (Li 0.55 Mg 0.45 ) 20 Zn 20 ,Sr 20 Ca 20 Yb 20 Mg 20 Zn 10 Cu 10 , and Sr 20 Ca 20 Yb 20 Mg 20 Zn 20[45,73] ). Moreover, one fcc group can be identified, (I-fcc) NiCuRhPdIrPtAuFeCoMn corresponding to both Cantor alloy family and precious metals HEA family. Finally, two hcp groups can be expected, (I-hcp) RuReOsCo and (II-hcp) rare-earth (including Sc and Y) systems. The I-fcc group justifies that HRR can be coherent with fcc structure for Cantor alloy once allotropic forms are considered, unlike interpretation of Guo & Liu. [74] 5.1.2. Atomic Size Match. 2 nd HRR The 2 nd HRR is quantified through the atomic size mismatch, δ r , defined as δr¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n i¼1 xi1ri r  2 s(2) where x i corresponds to the molar fraction of the element i,δ r parameter is ubiquitously identified as a limit to distinguish between SPSS and multiphase systems or amorphous formation for δ r >8.5%. However, Guo et al. [75] reduced this value to ≈6.6% and some bulk metallic glasses can be found at smaller values. [3,26,74–77] Data dispersion for amorphous structures can occur as glass formation is strongly dependent on the quenching rate. Thus the limits between compositions forming crystalline or amorphous microstructures are not well defined when quenching rate is not specified. Kube and Schroers [6] focused on δ r and identified only two compositional HEA regions suitable to be stable (or mildly stable) SPSS: at rhi≈125 pm, which corresponds to the transition metal HEA family; and at r hi ≈140 pm, which corresponds to the two HEA families: precious and refractory metals. For the first system derived from Figure 3, (I-bcc) CrVNbMoTaWTiZrHf, δ r =7.64%, but this can be reduced to δ r =6.6%, excluding Cr of the set (the smallest atom), and even δ r =3.57%, when excluding the larger too (Zr and Hf ). For the second group (II-bcc) identified in Figure 3, EuBaCaSrYb, δ r =4.6%. However, bulk amorphous are obtained when CaSrYb ( r hi ≈202 pm) are combined with much smaller atoms: Li (r=151.9 pm), Mg (r=160.1 pm), and/or Zn (r=139.5 pm). The I-fcc group, NiCuRhPdIrPtAuFeCoMn, presents δ r =5.57% if all the elements are considered. However, we should distinguish between MnFeCoNiCu (Cantor alloys), with δ r =3.18%, and RhPdIrPtAu (precious metal HEAs), with δ r =2.54%. For hcp HEAs, I-hcp with RuReOsCo elements yields quite small δ r =3.56%, whereas II-hcp (rare earth-based HEAs excluding Eu and Yb) LaCePrNdSmGdTbDyHoErTmLu yields an even smaller value of δ r ≈2%. These values are smaller than the limit observed in Figure 4, δ r <10%, for full-range solid solutions in binary alloys. Therefore, 2 nd HRR just exclude few elements from the groups derived from Figure 3. 5.1.3. Electron Valence Match. 3 rd HRR As commented above, concerning the 3 rd HRR, two parameters are used to classify MCA compositions, VEC and number of free electrons per atom (e/a, reader must be aware of misleading by the use of this acronym instead of VEC in Heusler alloys [78] ). Whereas VEC is an atomic parameter, e/amay strongly depend on the band structure and different values can be found in the literature. [79] However, Poletti & Battezzati [72] showed that boundaries between fcc and bcc phases can be found based on a plot of e/aversus VEC with nice extrapolations to pure elements. It is useful to observe how Mituzani et al. [79] used (e/a)/ VEC ratio to evaluate the fraction of near free electrons in a valence band, to understand the difference between these parameters. The closer this ratio to 1, the less the difference between e/aand VEC, and the better the interpretation of the band in the nearly free electron model. Therefore, for simplicity, we will consider ΔVEC to test the 3 rd HRR. Concerning the I-bcc group identified from Figure 3, CrVNbMoTaWTiZrHf, VEC values are between 4 and 6, thus not excluding any element as ΔVEC ≤2. Moreover, e/achanges from 0.9 for V to 1.76 for Zr, using the data supplied by Mizutani and Sato. [80] For the II-bcc group EuBaCaSrYb, ΔVEC ≤1, considering VEC =3 for lanthanides, respecting 3 rd HRR. For I-fcc HEAs, MnFeCoNiCu Cr-free Cantor alloy 7≤VEC ≤11 (Mn and Cu being the extreme values), which is a range much larger than solid solutions observed for binary alloys. However, e/arange is much more restricted than for bcc group (between e/a=1 for Cu and e/a=1.16 for Ni). Values for the fcc precious metals subgroup, RhPdIrPtAu, are 9≤VEC ≤11, with a broader distribution of e/afrom 1 for Au and Rh to 1.63 for Pt. Finally, I-hcp HEAs RuReOsCo yields 7 ≤VEC ≤9, thus without excluding any element for suitable solid solution. In II-hcp, even smaller differences are expected. www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (7 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH 15272648, 2025, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adem.202401148 by Readcube (Labtiva Inc.), Wiley Online Library on [10/04/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 5.1.4. Electronegativity Match. 4 th HRR The last HRR concerns electronegativity difference, Δχ P . For binary alloys, Δχ P <0.4 when complete miscibility is achieved in equilibrium (see Figure 4A). This requirement persists for MA binary samples and reported maximum solubility is below ≈30% when Δχ P ≥0.4 (see the cases of Cu in Ta and W in Fe in Figure 5A). For selected I-bcc, VNbMoTaWTi, with low δ r , we can distinguish between V, Nb, Ta, and Ti, with χ P ≈1.6 and χ P <0.1 for any couple between the set; and for Mo and W with χ P =2.16 and 2.36, respectively, thus Δχ P >0.4 among them and any other element of the previous set. In II-bcc, EuBaCaSrYb (with χ P =1.20, 0.89, 1.00, 0.95, and 1.10, respectively), none of the element pairs departs from Δχ P <0.4 rule. For fcc HEAs, CrMnFeCoNiCu Cantor alloy, extreme values for Mn and Ni yield a difference inside but close to the limit: Δχ P =0.36. However, it reduces to 0.08 when excluding Mn and Cr. For RhPdIrPtAu, χ P =2.20 for Pd, Ir and Pt with slight difference for Rh, although for Au, χ P increases to 2.54, but within the limit: Δχ P <0.4. Finally, for hcp HEA RuReOsCo, Δχ P ≈0.3 between Ru or Os with Re or Co, and rare-earth metals are within Δχ P ≤0.17, including Y. Taking into account the performed analysis, it can be concluded that the extension of HRR to the different set of elements derived from Figure 3 (from 1 st HRR) would not exclude any of those sets from suitable SPSS as HEAs. Close compact structures can be found such as fcc in Cantor alloys and precious metal HEAs, and hcp for rare-earth HEAs and some lightweight HEAs. However, truly bcc solid solutions are only found for refractory metal HEAs, and when this less compact phase is observed in other families, such as Cantor alloys, it is suspicious of being a multiphase system with elemental partitioning in ordered and disordered phases. 5.2. Thermodynamic Criteria 5.2.1. Conventionally used Thermodynamic Criteria This criteria combine the original parameter from which HEAs were named, ΔS mix , with the mixing enthalpy, ΔH mix , and the enthalpy of formation of the different intermetallics and compounds to establish an energy balance for determination of which is the stable microstructure. In fact, George et al. [81] identified the enthalpic contribution as more important than the entropic one and discussed on the more appropriateness of “low enthalpy”rather than “high entropy”naming for HEAs. Martin et al. [15] developed a free software (High-Entropy Alloys Predicting Software) which collects different criteria and useful parameters to describe the stability of a single-phase HEA in a given composition. We have used this software to obtain the different parameters here discussed, except for e/a, which were taken from Mizutani & Sato. [80] The most widely used among those parameters are described below. The Ωparameter, proposed by Yang et al. [77] is defined as Ω¼TmΔSmix jΔHmixj(3) where T m is the melting temperature (generally estimated as the average melting temperature among those of the pure elements). Single-phase HEAs are predicted for Ω≥1.1 in combination with δ r ≤6.6%. Small absolute values of ΔH mix are convenient as ΔH mix ≫0 yields to segregation and ΔH mix ≪0 yields intermetallic formation.The Λparameter, proposed by Singh et al. [82] can be expressed as Λ¼ΔSmix δr2(4) where entropic advantage is compared with the stored elastic energy that should destabilize the solid solution. Single-phase HEAs are predicted for Λ≥0.95 J mol 1 K 1 in combination with 5 kJ mol 1 <ΔH mix <0. The ϕ(f) parameter, proposed by Ye et al. [83] depends on the packing fraction f, and is defined as ϕðfÞ¼ ΔSmix ΔHmix Tm jSxsðfÞj (5) that represents the entropy ratio between the mixing and the excess entropy due to atomic size misfit, S xs (f). This excess entropy results from the preference of large atoms to be surrounded by small ones, deviating the real behavior from the statistical random distribution assumed for ΔS mix . As a consequence, this parameter depends on the packing fraction f, being enhanced in close compact structures. Thus ϕ>20 would lead to the formation of single-phase HEAs. Despite the calculation of S xs (f)iscomplex, [83] a good linearity is observed with the square of atomic size mismatch S xs (f)=mδ r2 ,asshowninFigure 7A. Slight changes of the linear parameters are observed, between different HEA families. Moreover, both entropy excesses are linearly correlated as shown in Figure 7B with S xs (fcc) =1.6163(3) S xs (bcc). 5.2.2. The Effective Temperature Criterion: Comparison with Other Parameters Recently, some of the authors of this work analyzed the stability of HEAs after comparison of ΔS mix with the energy increase of average bonding metallic potential with respect to segregated systems. This relationship can be represented as an effective temperature, T eff , [84] defined as Teff ¼ΔU0 ΔSmix (6) where the energy increase is calculated as ΔU0¼U0X i xiUi o ¼PixiAi r0 þPixiBi r02X i xiAi r0i þBi r0i2 (7) where A i and B i are the phenomenologically obtained potential coefficients and r 0i is the equilibrium distance between first neighbors for pure metals, which can be found in supplemental content of ref. [84]; r 0 is the equilibrium distance between first neighbors in solid solution and can be obtained as www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (8 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH 15272648, 2025, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adem.202401148 by Readcube (Labtiva Inc.), Wiley Online Library on [10/04/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License r0¼2PixiBi PixiAi (8) A value of T eff <500 K was proposed as suitable threshold to describe the formation of SPSS. However, a more suitable interpretation, considering metastability in HEAs, should be to compare T eff with T m for the different families of alloys. Figure 8 shows T eff as a function of ΔS mix for different series of alloys: Fe 1 Co x Ni y Cu z Mn u Cr v ;Nb 1 Mo x Ta y W z Cr u V v ; Ti 1 Sc x V y Cr z Zr u Hf v ;Al 1 Li x Mg y Sc z Ti u ;Au 1 Cu x Ni y Cu z Mn u Cr v ; Ru 1 Re x Os y Co z Ir u Rh v (0.2 ≤x,y,z,u,v≤1). The horizontal solid lines indicate the corresponding average melting temperature T m and the horizontal dot-dashed lines, the maximum and minimum melting temperatures in each series (data calculated using HEAPS software). This figure shows that increasing entropy facilitates SPSS formation from cooling in Cantor alloys and high-density-refractory HEAs, as T eff values of suitable compositions (ΔS mix >12.5 J K 1 mol 1 ) lie below their corresponding melting temperatures. This also occurs for the less studied RuReOsCoIrRh system and lightweight alloys. On the other hand, T eff in precious metals HEAs are close to their melting points and for lightweight refractory metals, difference between T eff and T m can be more than 1000 K. Figure 8 also evidences that increasing ΔS mix does not always yield an enhancement of SPSS stability as the tip of the data does not coincide with the minimum T eff in any of the families considered here. In fact, we can distinguish two extremes in composition tailoring. The upper side of the data cloud describes compositions that become entropy-stabilized SPSS as increasing ΔS mix decreases T eff . On the other hand, those compositions placed at the lower side of the data cloud correspond to entropy destabilized SPSS as increasing ΔS mix increases T eff . T eff , derived from a phenomenological analysis of the metallic bonding, contains information of both thermodynamic and atomic size misfits. Relationship between T eff and δ r is shown in Figure 9 for the different considered families of HEAs. The log–log plot indicates a power relationship with an exponent 2 between T eff and δ r . The considered values of T eff (<500 K) and δ r (<6.6%) to identify the formation of SPSS microstructures are indicated also in the figure by horizontal and vertical lines, respectively. It is observed that T eff parameter is more restrictive than δ r . However, only Cantor alloys are well below these limits. Except for light refractory alloys, all other families lie below δ r limit. However, T eff limit excludes Co-containing RuReOsCo(IrRh), NbTa-based ternary refractory HEAs with low content of W or Mo, and Scand Mg-free low-density alloys. The general trend helps us to discern that entropy-stabilized alloys (upper regions of the data clouds of Figure 8) correspond to those elements whose addition leads to a decrease in δ r . Analogously, those element additions leading to increase δ r will imply entropy destabilized alloys and the entropy criteria would be only beneficial when δ r decreases. With respect to electronegativity (Figure 10), as discussed above, all systems lie below Δχ p <0.4 but the most unfavorable families are precious metals HEAs and some heavy refractory compositions. A general increase in T eff is observed as Δχ p increases for Cantor alloys, low-density refractory alloys, and precious metal HEAs. However, heavy refractory HEAs, low-density HEAs, and RuReOsCoIrRh HEAs are almost insensitive to this parameter. No correlation is observed between T eff and average VEC, VEC hi ,(Figure 11A). However, this parameter is useful to distinguish between fcc ( VEC hi >7.5) and bcc ( VEC hi <6.87), as already proposed in the literature. [72] Ye et al. [83] constrained these limits for fcc to 7.84 <VEC hi <9.5 and bcc to 4.3 <VEC hi <5.7. Moreover, hcp is found for 2.6 <VEC hi <3. The HEA families described here follow this classification except for RuReOsCoIrRh with high VEC hi ≈9, typical of fcc, but with hcp structure when SPSS is reported. [66] Lightweight refractory metals with bcc and hcp structures would imply the high VEC hi limit for hcp to shift to higher values than 3. As it occurs with ΔχP, all simulated systems respect the threshold jΔVECj≤2 derived from HRR (Figure 11B), being precious metals and refractory metals HEA the most favorable families and Cantor alloys, the most unfavorable, which extend close to the limit when Mn and/or Cr are present. Comparison of T eff with respect to ΔH mix is shown in Figure 12. The corresponding limits identified for SPSS (5<ΔHmix <0) are also drawn. Low-density HEAs show higher deviation to large negative ΔH mix , indicating a strong Figure 7. A) Excess entropy for fcc and bcc systems as a function of square of the atomic size mismatch for Fe 1 Co x Ni y Cu z Mn u Cr v ; Nb 1 Mo x Ta y W z Cr u V v ;Ti 1 Sc x V y Cr z Zr u Hf v ;Al 1 Li x Mg y Sc z Ti u ;Au 1 Cu x Ni y Cu z Mn u Cr v ;Ru 1 Re x Os y Co z Ir u Rh v (x,y,z,u,vbetween 0.2 and 1). Linear fittings to the different alloy series are shown. Values obtained from HEAPS. [15] B) Slopes of fcc versus bcc structures from linear fitting of left panel. www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (9 of 19) © 2024 The Author(s). 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[185] M. Huang, J. Jiang, Y. Wang, Y. Liu, Y. Zhang, Mater. Des. 2022,217, 110637. Javier S. Blázquez has been a full professor since 2018, Department Physics of Condensed Matter, Faculty of Physics, University of Sevilla. He belongs to the research group of noncrystalline solids (FQM121 Research Group at the University of Sevilla). His fields of interest are metastable soft magnetic materials; crystallization kinetics; magnetocaloric effect. Alejandro F. Manchón-Gordón has been a postdoctoral reseacher since 2021, Institute of Materials Science of Sevilla (CSIC). His research group is reactivity of solids. His fields of interest include highentropy materials; thermomagnetic transitions; mecanochemistry; and reactive flash sintering. Antonio Vidal-Crespo is a Ph.D. student, Department Physics of Condensed Matter, Faculty of Physics, University of Sevilla. His research group is noncrystalline solids (FQM-121 Research Group at the University of Sevilla). His fields of interest include metastable soft magnetic materials; crystallization kinetics; magnetocaloric effect. Rafael Caballero-Flores has been an associate professor (tenured) since 2022. He works at Department Physics of Condensed Matter, Faculty of Physics, University of Sevilla. He is part of the research group noncrystalline Solids (FQM-121 Research Group at the University of Sevilla). His fields of interest are soft magnetic materials and magnetocaloric effect. Jhon J. Ipus has been an associate professor (tenured) since 2021, Department Physics of Condensed Matter, Faculty of Physics, University of Sevilla. He is part of the research group noncrystalline solids (FQM-121 Research Group at the University of Sevilla). His fields of interest include metastable soft magnetic materials; mechanical alloying; and magnetocaloric effect. www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (18 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH 15272648, 2025, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adem.202401148 by Readcube (Labtiva Inc.), Wiley Online Library on [10/04/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Clara F. Conde is a full professor and has been an emeritus professor since 2022, Department Physics of Condensed Matter, Faculty of Physics, University of Sevilla. She is the head of the research group noncrystalline solids (FQM-121 Research Group at the University of Sevilla). Her fields of interest are metastable soft magnetic materials; crystallization kinetics; and magnetocaloric effect. www.advancedsciencenews.com www.aem-journal.com Adv. Eng. Mater. 2025,27, 2401148 2401148 (19 of 19) © 2024 The Author(s). Advanced Engineering Materials published by Wiley-VCH GmbH 15272648, 2025, 6, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adem.202401148 by Readcube (Labtiva Inc.), Wiley Online Library on [10/04/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License