A Simple Phenomenological Model to Describe Stability of Homogeneous Solid Solutions in High Entropy Alloys from Metallic Bonding Potential

MTLA 28 (2023) 101744 Available online 14 March 2023 2589-1529/© 2023 The Author(s). Published by Elsevier B.V. on behalf of Acta Materialia Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Full Length Article A simple phenomenological model to describe stability of homogeneous solid solutions in high entropy alloys from metallic bonding potential J.S. Bl´ azquez * , N. García-Pinto, C.F. Conde Dpto. Física de la Materia Condensada, ICMSE-CSIC Universidad de Sevilla, P.O. Box 1065, Sevilla 41080, Spain ARTICLE INFO Keywords: High entropy alloys Metallic bonding Solid solutions ABSTRACT A simple model based on the potential parameters used to describe metallic bonding is extended to solid solutions. A figure of merit (an effective temperature, Teff ) is proposed using a simple average over the potential coefficients to discern whether a homogeneous solid solution is expected to be stable or not in high entropy alloy compositions. Teff is calculated as the ratio between the solid solution excess in bonding energy over the average mixture divided by the configurational entropy. Application to the sexinary AlCrCuFeNiCo system stablishes a stability threshold for Teff <500 K. The model can successfully describe both the deviations from Vegard’s law observed in binary alloys and the differences in this parameter between B2 ordered and bcc disordered phases considering average potential coefficients over the different possible atomic couples. Compositional tuning needed to optimize the feasibility, stability, microstructure, phase transitions and functional properties of new alloys generally yields multicomponent alloys (MCAs) with four or more elements [1]. This line of research led to the discovery and development of the so-called high entropy alloys (HEA) proposed by Cantor [2] and Yeh et al. [3] in 2004. These HEA systems are MCAs with close to equiatomic compositions with enhanced configurational entropy. This configurational entropy, ΔSconf , is expected to facilitate the formation of simple metallic phases (monoatomic bcc, fcc and hcp structures, or even amorphous systems) [4–6]. The interest in HEA opened a vast compositional range for alloy design, which phase diagram extension is huge in comparison with those corresponding to the conventional explored regions (close to corners and edges of the multicomponent phase diagrams). In order to clarify this vast compositional region, simulations and theoretical studies [7–11] are very useful to optimize the experimental research time, and helped to point for the most interesting regions of the vast phase diagram depending on the property or functionality of interest. Citing the words of D. B. Miracle (2019) “…exploring the enormous number of HEA and CCA [complex, concentrated alloys] compositions and their microstructures is currently the biggest challenge.” [12]. This author poses the development of HEA as a new reinforcement in the alloys design activity but shows the tremendous effort needed to explore the compositional range. Despite the accepted main role of ΔSconf , this is not the only parameter to consider for single phase formation in MCAs. The present situation is still far from being clear [4,13], although several phenomenological rules have been proposed. These rules are based on different parameters such as mixing enthalpy [8,9,14], valence electron concentration [15], size factor [16,17] and structural misfit [18]. These parameters are generally based on reasoning followed to understand binary alloys [14,15,19] and summarized in the empirical Hume-R- othery’s rules [20] and semiempirical Miedema’s model [21]. More recently, King et al. [19,22] have developed a software (Alloy Search and Predict, ASAP [23]) for a rapid estimation of the single phase HEA systems based on the Miedema’s values of enthalpy of mixing. In these papers the authors define a parameter ϕ= − ΔGss/|ΔGmax|, where ΔGss is the Gibbs free energy for the solid solution and ΔGmax is the extreme value of any binary system among the constituents (negative and minimum for intermetallic, positive and maximum for segregation). The use of the absolute value considers both destabilization ways of the solid solution: the affinity to form an intermetallic phase as well as the positive enthalpy of mixing between the elements leading to segregation. King et al. [19] proposed a value of ϕ≥1 as indicative of stable solid solution. On the other hand, Gorban et al. [15] proposed the electron concentration per atom, Csd, as the main factor to classify the microstructures of HEA compositions: bcc solid solution (4.2<Csd <7.2), fcc solid solution (Csd ≥7.5), two-phase solid solutions (7.2<Csd <8.0), Laves phase (4.4<Csd <8.1), σ phase (6.3<Csd <7.2), μ phase (Csd ∼7.3). * Corresponding author. E-mail address: [email protected] (J.S. Bl´ azquez). Contents lists available at ScienceDirect Materialia journal homepage: www.elsevier.com/locate/mtla https://doi.org/10.1016/j.mtla.2023.101744 Received 30 November 2022; Accepted 11 March 2023 Materialia 28 (2023) 101744 2 In this work, we develop a simple model from the potential energy for metallic bonding and extend it to alloys by assuming a near neighbors approximation. The proposed approach is able to describe the deviation of lattice parameter from linearity of Vegard’s law as well as the differences between ordered and disordered phases. Moreover, a figure of merit is proposed as an effective temperature that can distinguish between HEA compositions leading to multiphase systems or single phase ones. Atomic bonding in solids is described taking into account two contributions to the cohesion energy: an attractive term and a repulsive one. In the case of metallic bonding, the attractive term is due to the interaction between the electron cloud dispersed in the solid and the lattice ions whereas the repulsive term is due to the increase in the energy of the fermionic electron cloud when its volume is reduced (see Supplemental content). This leads to the following general expression for the cohesion energy per atom [24]: U N= − A r+B r2(1) where A and B are parameters of the solid and r is the distance between first neighbors. In order to obtain those two parameters for each metal, we can use the experimental values of the lattice parameter, a, and the compressibility, κ. The former is related to the equilibrium distance between first neighbors (dU/dr|r0=0): r0=2B A= α a(2) Where α = 3 √/2 for monoatomic body centered cubic (bcc), 1/ 2 √for monoatomic face centered cubic (fcc), and 1 for hexagonal close packed (hcp) structures (for simplification we have assumed perfect hcp structures). Therefore, the cohesion energy per atom in a pure metal at equilibrium distance is: U0 N= − A2 4B(3) The inverse of the compressibility (bulk modulus) is related to the second derivative of the cohesion energy: 1 κ= − VdP dVeq =N2βr3 0(−2A r3 0+6B r4 0)( 1 3Nβr2 0)2 =1 144β A5 B4(4) Where the volume of the solid has been used: V=Nβr3, with β=4/ 27 √ for bcc and β=1/ 2 √for fcc and hcp. Taking into account Eqs. (2) and Fig. 1. Estimated bonding energy from Eq. (3) as a function of melting temperature for different pure metals as a function of their crystalline structure. Fig. 2. (left panels) Lattice parameter as a function of the composition for different binary solid solutions. Data taken from [26] are plotted as squares; broken lines correspond to Vegard’s law; blue lines correspond to average model using 〈A〉 = xA M1+(1−x)AM2 and 〈B〉 = xBM1+ (1−x)BM2; red lines correspond to using Eq. (8) taking AMi and BMi from Table 1S (supplemental content) and fitting AM12 and BM12. (Right panels) Fitted AM12 and BM12 parameters vs. corresponding average values 〈A〉 = xAM1+(1−x)AM2 and 〈B〉 = xBM1+ (1−x)BM2. Red lines correspond to the identity AM12 = 〈A〉and BM12 = 〈B〉. J.S. Bl´ azquez et al. Materialia 28 (2023) 101744 3 (4), it is possible to obtain the values of A and B for each metal from the experimental values of a and 1/κ as: A=9β α 4a4 κ(5) and B=9 2β α 5a5 κ(6) Table 1S (Supplemental content) collects the lattice parameter (taken from [25,26]), bulk modulus [27] and A and B parameters for different pure metals crystallizing in bcc, fcc, hcp or double hcp (Dhcp) structures. In the Supplemental content, simple models for A and B are described to associate them with the valence electrons, as estimated by Pauling [28] (See Fig. 1S in Supplemental content). From the data of Table 1S and using Eq. (3) it is possible to estimate the equilibrium energy of the bonding, U0. Fig. 1 shows the absolute value of this energy per mol as a function of the melting temperature, Tm, for the different metallic elements grouped by their crystalline structures. A general linear trend is observed validating the potential used to describe the metallic bonding. Eq. (2) relates the lattice parameter with the bonding coefficients A and B in pure metals. This can be extended to binary systems assuming average values of these interaction coefficients. Considering that interactions are finally ascribed to couples of atoms, we can approach the average value of bonding coefficients from averaging the possible pairs of atoms, for a solid solution M1xM21−x, with x the atomic fraction: 〈A〉 = x2AM1+2x(1−x)AM12 +(1−x)2AM2 〈B〉 = x2BM1+2x(1−x)BM12 +(1−x)2BM2 (7) Where AMi and BMi are the corresponding coefficients of the pure Mi metal. Expressions reduce to 〈A〉 = xAM1+(1−x)AM2 and 〈B〉 = xBM1 +(1−x)BM2 when AM12 =1 2(AM1+AM2)and BM12 =1 2(BM1+BM2), respectively. Therefore, under this approach, the lattice parameter of a solid solution is predicted to be: a(x) = 2〈B〉 α 〈A〉=2(x2BM1+2x(1−x)BM12 +(1−x)2BM2) α (x2AM1+2x(1−x)AM12 +(1−x)2AM2)(8) Leading to a nonlinear relationship between the lattice parameter and the atomic fraction unlike the linear one predicted by Vegard’s law which is based on the hard spheres approximation. The approximate character of Vegard’s law [29] is long time known [30] and independent of the metallic bonding character. In fact, deviations from Vegard’s law are also reported for covalent [31,32] and ionic [33,34] bonding. Friedel [30] ascribed these deviations in binary solid solutions to the difference in the compressibility of the pure elements. Magadenov [35] recently proposed a method for obtaining lattice parameter of binary alloys at different pressures taking into account the differences in compressibility and specific volume between pure systems. However, in general, continuum elasticity models [36] do not correctly predict the deviations found due to the atomistic character of the problem [37]. King [38] proposed that deviations from Vegard’s law are indicative of electron transfer between the constituent atoms leading to changes in their effective radii. Lattice parameter of several binary solid solutions (data taken from [26]) were fitted to Eq. (8) and shown in Fig. 2. This figure also shows the Vegard’s law (dashed lines) and the non-free parameter curves obtained using AM12 =1 2(AM1+AM2)and BM12 =1 2(BM1+BM2)(blue curves); in the following named as average model. In general, it is worth mentioning that non-free parameter curves correctly predict the sense of deviation of the data from Vegard’s law. Concerning the magnitude of the deviation, this can be overestimated (e.g. Pd-Au) or underestimated (e.g. Ag-Au). The particular case of Ag-Au system is indicative of the limitations of the average model as it cannot predict intermediate compositions beyond the lattice parameters of both pure metals. However, deviations from the average model in most cases are not as high as left panels of Fig. 2 suggest. In fact, it must be taken into account that the parameters used derive from experimental data of lattice parameter and bulk modulus of pure metals. Particularly, the latter presents large errors [27]. Fig. 2 also shows the comparison between fitted AM12 and BM12 parameters and the corresponding average values 〈A〉 = xAM1+(1−x)AM2 and 〈B〉 = xBM1+ (1−x)BM2. Values are generally close to the identity line. However, Ag-Au and Cu-Ni systems clearly deviate. In fact, average model would not be able to explain differences in the lattice parameter between ordered and disordered phases. However, although limited to first neighbors, the proposed model can explain why lattice parameter differs between ordered and disordered phases, considering that AM12 and BM12 are not necessarily equal to 〈A〉and 〈B〉, respectively. In the case of Fe(Al), ordered B2 phase appears in the compositional range from 22 to 54 at.% Al for temperatures above 660 C and below 550 C a DO 3 structure appears [39] in a compositional range from 23 to 31 at.% [40]. Fig. 3 shows the value for the lattice parameter of Fe 1-x Al x alloys for B2 ordered and bcc disordered phases (data taken from [40, 41]). When average values for AM12 and BM12 parameters are considered, no change between the lattice parameter of ordered and disordered phases is predicted (blue line in Fig. 3). In order to obtain the potential coefficients, BM12/AM12 =rBA ratio is taken from the experimental value of the lattice parameter of Fe 0.5 Al 0.5 alloy, using Eq. (2) and considering a perfect order for this stoichiometric composition. To obtain individual values for AM12 and BM12, we took two choices: as a first choice we use the average value for AM12 = (AFe +AAl)/2 and then calculate BM12 = rBAAM12. Analogously, a second choice was using BM12 = (BFe +BAl)/2 and calculate AM12 =BM12/rBA. Curves can be now calculated assuming perfect disorder (continuous lines) and maximum order (dash-dotted lines), i.e. Al is always in correct positions and Fe excess is placed in Al sites. No significant differences are found whether the first or second choice is taken. Despite the expected errors in lattice parameter, the curves agree with the experimental data in the range of stability of disordered bcc and ordered B2, whereas between 0.22 <x<0.35, in the range of DO 3 ordered structure (where half the lattice parameter of the fcc phase is shown [40]), the lattice parameter is almost constant. However, the model, using just first neighbors approach, is not expected to represent this more complex system. In the frame of the proposed model, it is possible to discuss on the Fig. 3. Lattice parameter of ordered and disordered Fe 1-x Al x alloys. Experimental data taken from [41] (black squares) and [40] (hollow triangles: red corresponds to disordered bcc, orange to half lattice parameter of ordered DO 3 phase, and purple to ordered B2 phase). Blue line corresponds to the calculated curve using AM12 = (AAl +AFe)/2 and BM12 = (BAl +BFe)/2 and predicted lattice parameters for ordered and disordered phases are indistinguishable. Green curves correspond to calculated curves taken rBA =BM12/AM12 = ( 3 √ 4)a(x=0.5). Maximum ordered (dash-dotted lines) and fully disordered (solid lines) phases are now distinguishable. J.S. Bl´ azquez et al. Materialia 28 (2023) 101744 4 stability of solid solutions. In fact, using Eq. (3), we can calculate the difference between the energy of an equiatomic solid solution and that of the corresponding mixture of pure components as: ΔU0 N= − 〈A〉2 4〈B〉−1 2(−A2 M1 4BM1−A2 M2 4BM2) = − (AM1+AM2)2 8(BM1+BM2)−1 2(−A2 M1 4BM1−A2 M2 4BM2)(9) Fig. 4 shows the calculated ΔU0/N values using data from Table 1S as a function of the maximum solubility (taken from [39]) of different elements in Ag and vice versa. In the average model, calculated values are always positive as Eq. (9) can be written as: ΔU0 N=1 8(AM1BM2−AM2BM1)2 BM1BM2(BM1+BM2)>0(10) However, configurational entropy should lead to stable solid solutions when available. As a general rule, solubility increases as ΔU0 decreases and ΔU0>20 kJ/mol can be used as a rule of thumb to predict insolubility between different elements. All these features pose a very simple way to predict stability for solid solutions, which can be straightforwardly extended to more complex systems such as HEAs. Once composition is known, ΔSconf (per mol) can be calculated as: ΔSconf = − R∑ i xilog(xi)(11) Where R is the gas constant and xi is the atomic concentration of element i with ∑ i xi=1. Thus ΔSconf is enhanced in equiatomic systems and furthermore by addition of new elements. However, this does not necessarily lead to single phase systems. In fact, whereas CrCuFeNi 2 shows a single fcc phase, addition of Al in Al x CrCuFeNi 2 (0.8<x<1.5) leads to the formation of fcc and bcc phases [42], despite ΔSconf is enhanced almost 20%. In the frame of the model described here, we can calculate the energy difference per mol between the average over those of pure element phases and the corresponding complete solid solution by extending Eq. (9) to p number of elements: ΔU0 NA= − 〈A〉2 4〈B〉+∑ p i xi A2 Mi 4BMi (12) Applying this to the previous case, ΔU0 increases from 0.6 kJ/mol for CrCuFeNi 2 system to 13 kJ/mol for Al 1.5 CrCuFeNi 2 . The relative change in ΔU0 clearly exceeds that of ΔSconf and can explain why a single solid solution is not formed for the Al containing composition but the elements partition among three different solid solutions (a bcc and two fcc phases [42]). Energy dispersive X-ray analysis can be found in the literature showing Cu-rich composition for the fcc phases and Fe-rich for the bcc one [42–44]. However, the semiquantitative character of this technique prevents the use of the reported composition data to appreciate a decrease in the relative effect of ΔU0 and ΔSconf . This can be parametrized in the figure of merit Teff =ΔU0/ΔSconf , which is an effective temperature at which excess energy and thermal contribution of configurational entropy compensate each other. Singh et al. [45] applied 3D atom probe analysis (compositional probe within atomic scale) to AlCrCuFeCoNiCo equiatomic alloy and reported the presence of 40 nm sized regions with Al 3 Cr 40 Cu 1 Fe 31 Ni 6 Co 19 composition. The average equiatomic composition should lead to ΔU0=10 kJ/mol and ΔSconf =14.9 J/molK, whereas in the nanometric regions the corresponding values are ΔU0=2.3 kJ/mol and ΔSconf =11.4 J/molK. Therefore, Teff reduces from 670 K in the average composition to 200 K in the nanocrystalline phase. In order to supply a more complete view, Fig. 5 shows Teff values for the 53,130 compositions Al x1 Cr x2 Cu x3 Fe x4 Ni x5 Co 100-Σxi (stepped by Δxi=5 at.%). Al-free and Al-containing binary alloys as well as sexinary alloys and Al-free quinary alloys are distinguished in the figure. The plot exemplifies the formidable problem to be solved when exploring the field for HEA compositions. It can be observed that the presence of Al in sexinary compositions increases Teff with respect to Alfree quinary alloys, explaining why destabilization of single phase solid solutions occurs as it is experimentally observed. Experimental data on Al x1 Cr x2 Cu x3 Fe x4 Ni x5 Co 100-Σxi compositions (70 alloys with around 50 different compositions taken from [42–58]) are also shown in Fig. 5. In this figure, it is indicated whether the alloy appears as a single fcc solid solution (solid orange circles), as bcc plus B2 phases (yellow filled circles with orange border) or as a mixture of fcc and bcc phases (hollow orange circles). It is worth mentioning that in some papers a single bcc solid solution is reported (e.g. Al x CrFeNiCo (x≥1.25) in [48,54]) but in those cases dendritic microstructure evidences the presence of partitioning in two different phases [53] and the presence of a mixture of B2 ordered phase and bcc disordered phase is frequently reported [51,53]. Therefore, it can be clearly appreciated a boundary between single phase systems and multiphase systems around Teff ∼500 K. This value is reasonably comparable with room temperature. Moreover, experimental data show that research on HEAs is focused mainly on those alloys with the highest ΔSconf (see the Fig. 4. Differences between the equilibrium energy of an equiatomic supersaturated solid solution and that corresponding to the average of the pure metals for different binary alloys with silver as a function of the maximum solubility (taken from [39]) of the element in Ag fcc phase and vice versa. Fig. 5. T eff for 53,130 Al x1 Cr x2 Cu x3 Fe x4 Ni x5 Co 100-Σxi compositions with increments of Δxi=5 at.%. Binary alloys (red symbols) as well as sexinary alloys (blue circles) and Al-free quinary alloys (green circles) are distinguished in the figure. Teff for 70 alloys from literature [42–58] are shown as large orange circles for single phase systems (solid symbols) and multiphase systems: fcc+bcc (hollow symbols) and bcc+B2 (yellow filled symbols). J.S. Bl´ azquez et al. Materialia 28 (2023) 101744 5 experimental data covering the right extreme values in Fig. 5), which is not always the correct route when single phase solid solutions are the desired goal. In order to test the model in a broader range of elements, the value of Teff for the quinary and sexinary alloys collected by Gorban et al. [15] are shown in Fig. 6, distinguishing between those compositions leading to fcc solid solutions, bcc solid solutions, mixture of them or including intermetallic compounds (Laves, sigma, or mu phases). 16 elements (Al, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zr, Nb, Mo, Hf, Ta, W, and Re) are combined in the 46 alloys shown. We found that fcc solid solutions appear below Teff ∼600 K (a limit roughly in agreement but above that found for AlCrCuFeCoNiCo system). Moreover, only three exceptions are found: two of them correspond to about 50% fcc plus 50% sigma phase (equiatomic quinary VCrFeCoNi and sexinary VCrFeCoNiCu), and ReMoWNbTa alloys with bcc structure. Concerning the former exceptions, intermetallic phases are not considered in the proposed model. Concerning the latter exception, it is interesting that B2 ordered structure can frequently appear in these systems to accommodate differences in the atomic radii and ReMoWNbTa shows a small 2% value of the parameter δ=2.1%, well below the limit predicted by Yang et al. (δ <6.6%) [14] to develop an ordered phase. Moreover, this limit was revised to be δ<4% [4,16]. The other alloys analyzed in our study and exhibiting bcc structures showed a higher value of δ >4% and an almost linear correlation can be found between this parameter and Teff (see Supplementary content, figure 3S). Supplemental content also shows a comparison between the predicted and experimental lattice parameter for compositions with bcc and fcc structures (Fig. 4S). Conclusions A model to describe the stability of homogeneous solid solutions in high entropy alloys is developed from the potential parameters used to describe metallic bonding. The model assumes a first neighbor approach for solid solutions by averaging the potential coefficients over the contributions of the different atomic couples in the alloy. In the simple average model, the average potential coefficients can be obtained by simply averaging over the values of the corresponding elements. The model is able to describe the deviations of the lattice parameter from the linear law due to Vegard. When potential coefficients are obtained averaging over the different couples of atoms, a good agreement with the experimental data can be obtained. Moreover, differences in the lattice parameter between ordered and disordered phases can be well reproduced. Using the simple average model, a figure of merit is proposed as an effective temperature defined by the increment in the solid solution bonding energy over the mixture of pure phases divided by the configurational entropy (Teff =ΔU0/ΔSconf ). In the particular case of the Al- Cr-Cu-Fe-Ni-Co system, a value of Teff <500 K would lead to the formation of single phase solid solutions, whereas Teff >500 K would lead to the partitioning of atoms in two or more different solid solutions. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by PAI of the Regional Government of Andalucía and the VII PPIT of University of Sevilla. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mtla.2023.101744. References [1] A. Inoue, F.L. 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