A Criterion for Topological Close-Packed Phase Formation in High Entropy Alloys

The stability of topological close-packed (TCP) phases were found to be well related to the average value of the d-orbital energy level ( Md ) for most reported high entropy alloys (HEAs). Excluding some HEAs that contain high levels of the elements aluminum and vanadium, the results of this study indicated that the TCP phases form at Md > 1.09. This criterion, as a semi-empirical method, can play a key role in designing and preparing HEAs with high amounts of transitional elements.


Introduction
New types of alloy, which include high entropy alloys (HEAs) and multi-component alloys with equiatomic and near-equiatomic compositions, are garnering much research attention in the metallic materials community [1][2][3].Compared to conventional metallic alloys based on one or two major elements, these new alloys generally contains 5-13 principal elements [2].These alloys were named multi-component alloys by Cantor et al. [1] and defined as HEAs by Yeh et al. [2].To date, many promising properties have been reported, including highly wear-resistant HEAs like Co1.5CrFeNi1.5Tiand Al0.2Co1.5CrFeNi1.5Ti[4,5]; high-strength body-centered-cubic (BCC) structured AlCoCrFeNiTi0.5 and AlCrFeNiMo0.2HEAs (at room temperature) [6,7], and NbMoTaV HEA (at elevated temperatures) [8,9], and high corrosion resistant Cu0.5NiAlCoCrFeSi HEA [10].Researchers are drawn to HEAs due to their unique compositions, microstructures, and adjustable properties.To date, many studies have been performed on HEAs, and they have become a hot topic for research in the materials community .
HEAs generally consist of simple solid solutions of face centered cubic (FCC) structures, body centered cubic (BCC) structurs or a combination of the two.Research findings show that a small quantity of intermetallic compound phase/metastable particles often form in HEAs after solidification or the completion of an aging treatment at intermediate temperatures [3,25,33].These intermetallic compound phases/metastable particles, especially the topological close-packed (TCP) phases, are closely related to the mechanical properties of HEAs.Therefore, it is necessary to predict the occurrence of TCP phases which include the Laves phase, σ phase, R phase, μ phase, P phase, and so on.
In the past, most HEAs contained the elements Co, Fe, and Ni as their primary elements.Usually, some alloying elements, such as Al, B, Cr, Mo, Nb, Si, Ti, V, are added to HEAs that contain Co, Fe and Ni, forming alloy systems such as Al0.5CoCrCuFeNiTix [34], Al0.5CoCrCuFeNiBx [35], Al0.5CoCrCuFeNiVx [36], AlCoCrFeNiMox [37], AlCoCrCuFeNiMox [38], AlCoCrFeNiTix [6], AlCoCrFeNiNbx [39], AlCoCrFeNiSix [40], and AlCoCrFeNiTi0.5 [41].It is worth noting that when the concentrations of Al, B, Cr, Mo, Nb, Si, Ti, and V exceed a critical value, some ordered phases might form, which could deteriorate the plasticity of the HEAs.For example, the plastic strain of as-cast AlCoCrFeNiNb0.25 alloy is remarkably lower than that of as-cast AlCoCrFeNi alloy due to the formation of the Laves phase [39].Taking AlCoCrFeNiNbx alloys [39] as an example again, with the increase of x, the crystal structure of the alloys can transform from BCC to BCC + Laves phase.Although both the yield strength and hardness increase, the plasticity decreases dramatically.Another case to illustrate this point includes the structural transformation sequence for the CoCrFeNiMox system, where with the increase of x, it has been observed that FCC → FCC + σ → FCC + σ + μ [42].The formation of the σ phase is considered to be due to two factors, one of which is the identical crystal structure and the similar electron configuration between Cr and Mo, and the other is the substitution capability among Co, Fe, and Ni elements.Similarly, in AlCoCrFeNiNbx and CoCrFeNiTix HEAs, the TCP phase forms with an increased content of Nb and Ti elements, which is explained by the large atomic size of Nb and Ti elements as well as its large negative enthalpies of mixing with other alloying elements [6,43].However, the experimental results are far from proving this satisfactorily.
Recently, the valence electron concentration (VEC) was used to predict the existence of the σ phase, a TCP phase, in Cr-and/or V-containing HEAs [33] where the criterion worked well.Electronegativity was also successfully used to predict TCP phases in HEAs [44].Yang et al. successfully used a simple thermodynamic rule, i.e., the entropic and the average heat of mixing, to predict phase selection in multicomponent alloys [45].Liu et al. proposed an effective atomic size parameter to predict the phase stability in multicomponent systems [46].These semi-empirical modes have achieved great success in predicting the stability of solid solution phases of HEAs.The d-orbital energy level (Md) physically correlates with electronegativity and metallic radius of elements, both of which are parameters used in the classical approach by Hume-Rothery and Darken and Gurry [47].For example, the Md value will increase with the increase of metallic radius, and it will reduce with the increase of electronegativity, so the parameter Md has great potential for dealing with the stability of TCP, especially in those HEAs containing high mounts of certain transitional elements.Therefore, in this paper, another parameter was used: the average value of the energy level of d-orbital ( Md ) parameters to predict all types of TCP phases of HEAs.

Results and Discussion
It is generally accepted that d-electron bonding is of great significance to the metallurgical properties of transition metals.The average value of the d-orbital energy level was proposed as: 1 ( ) where Ci is the atomic fraction of component i in the alloy, and (Md)i is the d-orbital energy level of element i in the M-element centered cluster in the i-M binary solid solution alloy, in which i is a solvent and M is a solute.This was proposed to design superalloys by Morinaga, et al. [47][48][49][50][51], who found that all of the Ni-based, Co-based, and Fe-based superalloys with the σ phase formed displayed higher Md values (about 0.915) than those of alloys without forming the σ phase.To data, the Md parameter has been successfully used to design superalloys.In HEAs, most principle elements are transition elements similar to superalloys.Therefore, in this paper, the parameters Md can be adopted to study the stability of the TCP phase in HEAs that contain more transitional elements.In HEAs, the matrix and commonly added alloying elements, including Ni, Fe, Co, Cr, and so on, have FCC, BCC, or a combination of the two as a structure.Previous research shows that the (Md)i value of transition metals in the BCC-structured Fe is slightly higher than that of transition metals in the FCC-structured Ni3Al, although they display the same variation trend [50,52].It seems possible to establish a semi-empirical model to predict the presence of TCP phases in HEAs.When a new HEA is designed, the accurate structure was not known before testing, so in order to calculate its Md value, it was necessary to assume a single structure for HEAs.According to previous research results, whether adopting the FCC structure or BCC structure, the (Md)i value was almost the same.Therefore, the assumed single structure may be a FCC or BCC structure, regardless of the real structure the HEA.For calculation convenience, all HEAs in this study were assumed to have a single FCC Ni3Al structure when the Md values were calculated.Thereunto, the (Md)i value was the individual element which was calculated based on the FCC Ni3Al structure, which is shown in Figure 1a.An Al atom is surrounded by twelve Ni atoms as its nearest neighbours, and surrounded by six Al atoms in its second nearest neighbours.The cluster used in the calculation was [MNi12Al6], which is shown in Figure 1b.An Al atom in the center was substituted for the various alloying elements of M (M = Ti, Cr, Mn, Fe, Co, Ni, Cu, V, Mo, and so on).More details can be found in Refs.[47,48].Although the (Md)i values calculated in BCC/(BCC + FCC)-structured HEAs were slightly lower than the actual data, this approach could successfully predict the stability of the TCP phase by providing a reasonable guideline.
According to the methods reported by Morinaga [47][48][49][50], the value of the parameter Md was calculated for the reported HEA system that contained TCP phases.The values are shown in Table 1 [5,6,34,[36][37][38]39,42,43,[53][54][55][56][57][58][59][60][61][62][63][64][65].It was found that all the HEAs in Table 1 were synthesized by arc melting and then solidified in a water-cooled copper hearth.As can be seen, a rule emerges in that TCP phases occurred when the value of Md reached 1.09.To further illustrate this point, the Md values for the HEA systems containing TCP phases were plotted in Figures 2 and 3, which clearly show the capability of Md to quantitatively predict the formation of TCP phases in HEAs.It was found that when Md > 1.09, stable TCP phases appeared except in a few exceptions where no TCP phases formed at Md < ~0.95.In the range of Md = ~0.95 to 1.09, the existence of TCP phases was unknown.This is a novel method to predict the formation of TCP phases of HEAs.It must be pointed out that the theory may work well only in HEAs with high amounts of transition elements, especially, when these transition elements are Co, Cr, Cu, Fe, Ni, Mo, Ti, Mn, and Nb elements.It was found that the Laves phase vanished in AlxCoCrFeNiTi0.5 alloys with high Al contents [66] and that the σ phase vanished in Al0.5CoCrCuFeNiVx alloys with high V contents [36].It should be noted that all statistic HEAs are nominal compositions in the paper.Therefore, in part of designed HEAs' composition there may be a deviation in the actual active content after melting.The target of this paper is to use some parameters to predict the phase formation in high-entropy alloys, and from the prediction point of view only nominal compositions can be used, as actual compositions are only known after the materials are prepared.In this sense, the difference between the nominal composition and the actual composition is not a concern for the prediction we intend to make.

Conclusions
In this study, the average value of the energy level of the d-orbital ( Md ) was used to predict the stability of TCP phases in HEAs.A TCP phase formation rule was proposed by calculating the Md value for HEAs containing Al, Co, Cr, Cu, Fe, Ni, Mo, Si, Ti, Nb and other transition elements.TCP phases were found to be stable at 1.09 Md > except for some HEAs containing high levels of the elements Al and V. Specifically, it was discovered that the applicable alloying elements were limited to some transition elements (such as Co, Cr, Cu, Fe, Ni, Mn, Mo, Ti, and Nb), particularly as these elements easily formed the TCP phase.A novel method was used to predict the formation of TCP phases in HEAs.However, the significant impact of large amounts of Al and V elements on the stability of the TCP phase remains unclear, and therefore, more research is needed in order to fully understand it.

Figure 1 .
Figure 1.The calculation of (Md)i value: (a) Crystal structure of Ni3Al; and (b) Cluster model used in the calculation.

Figure 2 .
Figure 2. Relationship between the Md and the TCP phase stability for the HEA systems containing Al, Co, Cr, Fe, Ni, and Mo elements.Note on the legend: fully closed symbols and top-half closed symbols represent solid solution phase plus TCP phase; fully open symbols represent sole solid solution phases.

Figure 3 .
Figure 3. Relationship between the Md and the TCP phase stability for the HEA systems containing Al, Co, Cr, Cu, Fe, Ni, Mo, Si, Ti, V, Nb, and etc. elements.Note on the legend: fully closed symbols and top-half closed symbols represent solid solution phase plus TCP phase; fully open symbols represent sole solid solution phases.

Table 1 .
The crystal structure and the parameters, Md , for typical reported multi-component HEAs.