Intermetallic Phases in High-Entropy Alloys: Statistical Analysis of their Prevalence and Structural Inheritance

Strengthening high entropy alloys (HEAs) via second phases is a very effective approach. However, the design of intermetallic (IM) phases in HEAs is challenging, mainly because our understanding of IM phases in HEAs is still very limited. Here, a statistical approach is used to enhance our understanding towards IM phases in HEAs. A database consisting of 142 IM-containing HEAs was constructed. Our aim is twofold. The first is to reveal the most common IM phase types in published HEAs. The second is to understand whether HEAs inherit their IM structures from their binary/ternary subsystems, or whether they tend to form new structures irrelevant to their subsystems. The results show that the five most prevalent IM structures in the HEAs surveyed here are Laves, σ, B2, L12, and L21. This trend is evidently different from the overall trend among known binary/ternary IMs. As for structural inheritance, all the IM phases contained in the alloys are existing structures in the binary/ternary subsystems of the respective alloys. This suggests that the compositional complexity in HEAs does trigger additional complexity in IM structure formation. These findings have important implications in the future design and development of HEAs.

A very effective approach for strengthening alloys is the incorporation of second phases.Therefore, recently, some researchers proposed that instead of focusing on SSS alloys, more attention should be paid to HEAs that contain intermetallic (IM) phases [2, 19,20].In fact, incorporating second phases was even considered one of the three key strategies for the future development of HEAs [21].Indeed, experimental results have shown that IM phases can improve the mechanical properties of HEAs.For example, the outstanding wear resistances in the Al 0.2 Co 1.5 CrFeNi 1.5 Ti and Al 0.3 CrFe 1.5 MnNi 0.5 alloys are the results of the formation of η and σ phases, respectively [22,23].Very recently, it has been demonstrated that through carefully designed second phases, high strength and good ductility can even be achieved simultaneously [24][25][26][27].For instance, the (FeCoNiCr) 94 Ti 2 Al 4 alloy shows a yield strength of over 1 GPa, while keeping a high tensile ductility of 17%.This is due to the engineering of γ' precipitates through various thermomechanical processes [24].
However, the tailoring of IM phases in HEAs for property enhancement is still very challenging.This is because compared with SSS phases; our fundamental understanding of IM phases in HEAs is still very limited.For example, there have been more than 20 publications regarding the formation criterion and stability of SSS phases in HEAs [6,[28][29][30][31], although some of the earlier works have limited accuracy due to imprecise experimental data.In contrast, papers discussing IM phases in HEAs are very rare.Currently there are only one criterion on the general selection of IM phase type [32], and three stability criterions on two specific IM phase types (σ and Laves phases) [33][34][35].
To predict the stability of IM phases in HEAs, the reasonable first step is to identify the types of phases that could form in the alloy, and then compare the relative stabilities of these candidates.However, even the first step is difficult.Unlike the case in SSS phases, where only three types of structures are concerned (FCC, BCC, and HCP), there are more than two thousand structure types for IM phases [36].A plausible assumption is that the candidate IM phases in HEAs are the IM phases in their subsystems (for quinary HEAs this means all quaternary, ternary, and binary subsystems).This is similar to the case in many ternary systems, where the IM phases are all inherited from the three binary subsystems.Nevertheless, even in ternary alloys this assumption is not necessarily true (i.e., there could be new structures that do not exist in the three binary subsystems).In HEAs, the composition is apparently more complex, which renders the above assumption even more questionable.In addition, in HEAs a new class of IM phases called high entropy intermetallics (HEI) have been reported [21].These new IM phases show completely new compositions that have not reported before, which further complicates the determination of the candidate IM phases.
This work is the first part of an attempt to understand the IM phases in HEAs from a macroscopic viewpoint.A database of 142 IM-containing HEAs is constructed.Care is taken to reduce alloys with similar composition so that the diversity in the database is enhanced.The alloys and their respective IM structures are statistically analyzed.Our goal is twofold.The first is to reveal the most common IM phase types in HEAs.The results in HEAs are then compared with the general trends in binary and ternary IM phases and discussed.The second is to understand the inheritance of IM structures in HEAs.The IM structure types in the HEAs are compared with those in their subsystems to see if the structures are inherited from the subsystems of the alloys.The frequency which irrelevant (i.e., not existing in corresponding subsystems) structure types occur is also probed and discussed.

Construction of the Alloy Database
The HEAs in this research were collected using the following principles: (1) Only alloys that contain IM phases were collected because our target is the IM phases.
(2) Alloys that have unidentified or uncertain IM phase types were excluded.In other words, either the formula (e.g., Fe 2 Ti) or the common/trivial name (e.g., σ) of all the compounds in the alloy has to be reported.This is because including these alloys leads to difficulty in our analysis.
(3) Alloys with similar composition and identical IM phase constituents were listed only once (i.e., only one of the alloys was included).Two alloys are considered to have similar compositions if they have the same composing elements, and only the relative concentrations of the elements are different.This is because many studies fabricate series of alloys by systematically changing the content of a particular element.For example, He et al. [37] conducted detailed study on the effect of Al on the structure and properties of Al x CoCrFeMnNi alloys.They fabricated a total of 14 alloys (x = 0.02, 0.03, 0.04, 0.08, 0.09, 0.1, 0.11, 0.12, 0.14, 0.15, 0.16, 0.18, 0.19, and 0.25).Ten of the 14 alloys have the same IM phase constituent: B2.This similarity is not surprising; the alloys are composed of the same elements and the relative concentrations of these elements do not change significantly.If these alloys are all included, they and the IM phase they contain will have an unreasonably high weighting in the database, which will lead to biased results in subsequent analyses.Therefore, only one of the ten alloys was included in the database.
(4) If the same alloy contains different IM phases under different processing conditions, each condition with different IM phase constituents will be recorded separately.
However, in subsequent analyses these phases will be combined into one entry and counted only once.Assume the phase constituent of an alloy in the as-cast, 1100 • C annealed, and 900 • C annealed states are (α, β), (α, β, γ), and (α, δ), respectively.These three states will all be recorded in the database.However, when calculating, for example, the number of occurrences for each IM phase, the three records will be combined.The alloy will be regarded as containing four phases (α, β, γ, and δ) and counted only once.
Based on the above principles, a total of 142 IM-containing HEAs with sufficient compositional distinction are collected.These principles enhance the diversity of our database, and can markedly reduce the bias from compositional repetition.As an example, Miracle and Senkov published a comprehensive review on HEAs recently [3].In their review, an HEA database with a total of 408 unique alloys was constructed.247 of the 408 alloys contain IM phases.However, using the above principles, only 74 of the 247 alloys (~30%) will be included our database-70% of the alloys are excluded because of compositional similarity.This clearly shows the efficacy of the above principles.

Characterization of the Database
Table 1 shows the alloy database constructed in this research.It lists the composition, processing condition, and the solid solution and IM phases of each alloy.The abbreviations used in Table 1 are explained in the footnote of the table.Because the target of this research is the IM phase, each IM phase in the alloy is listed in a separate row.Three types of structure information are recorded (if possible).The first is the phase name.This is the common name for each structure (if it exists).Common name are either based on Strukturbericht notation or common trivial names [38] for certain structure types (most notably the topologically closed packed phases).For example, B2 is the Strukturbericht notation for cP2-CsCl and σ is the trivial name for tP30-CrFe.The second is the compound name reported in the original work (if reported).The third is the structure prototype of the phase.In many cases, the original publication only reports the common name of the phase without indicating the exact compound.In such cases the "compound reported" column will be left empty.Contrarily, if only the compound is reported, the common name of the phase will be identified based on the structure prototype.Moreover, to probe the structural inheritance in the alloys, the binary and ternary subsystems of each alloy are examined using various handbooks [39][40][41] to see whether the observed IM phases already exist in the subsystems.Binary subsystems are examined first.If the target IM phase can be found in any of the binary subsystems, the search is stopped and the subsystem will be recorded in the "IM Structure Found in" column.If the IM phase does not exist in any binary subsystem, the search will be continued in ternary ones and then recorded (if found).It should be noted that one IM phase can often be found in multiple subsystems.However, due to space limit only one of the subsystems is recorded.L1 0 = tP2-AuCu; L1 2 = cP24-AuCu 3 ; µ = hR13-Fe 7 W 6 ; R = hR21-TiNi 2 ; σ = tP30-CrFe.Some reported compounds have two different structures, both structures are listed and separated with a slash.For such compounds, "phase name" will be left empty.In addition, structure prototypes of certain compounds are yet unknown.This is abbreviated as Unkn.
The 142 IM-containing alloys use a total of 25 elements.This includes 1 alkali metal (Li); 1 alkaline earth metal (Mg); 18 transition metals (Ag, Co, Cr, Cu, Fe, Hf, Mn, Mo, Nb, Ni, Pt, Ta, Ti, V, W, Y, Zn, Zr); 2 basic metals (Al, Sn); 2 metalloids (B, Si) and 1 non-metal (C).To understand the prevalence of these elements in our alloy database, the relative frequency of usage of each element is calculated and plotted in Figure 1.Co, Cr, Fe, and Ni are the most common elements in this study and have relative frequencies higher than 66%.Al (56%) and Ti (49%) also have very high relative frequencies.Three other elements, including Cu (28%), V (25%), and Zr (21%), have relative frequencies higher than 20%.All other elements have relative frequencies below 20%.Miracle and Senkov published a comprehensive review on HEAs very recently [3].They constructed an HEA database with a total of 408 unique alloys, regardless of the phase type present in the alloy.Thus, their alloys can serve as a good comparison group to ours.The frequency with which each element is used in their database has been calculated [3].This data is converted to relative frequency and plotted in Figure 1.As seen in Figure 1, Co, Cr, Fe, and Ni still have the highest relative frequencies (74% or higher).Other elements having relative frequencies higher than 20% include Al (67%), Cu (46%), Ti (30%), and Mn (25%).Therefore, the seven most common elements in both databases are the same, but the relative sequence is slightly different.The popularity of these elements in both databases is largely because most of them belong to the two classic HEA systems [2]: Al-Co-Cu-Cr-Fe-Ni and Al-Co-Cr-Fe-Ni.These systems and their derivative systems (systems having two or fewer different principal elements) constitute the largest HEA family [2,3].Thus, the popularity of these elements is not unexpected.
Comparing the relative frequencies of the same element in the two databases also helps to reveal the differences between the two databases.Based on Figure 1, only six elements have lower relative frequencies in our database than in Miracle and Senkov's, namely Al, Fe, Co, Ni, Cu, and Mn.Other elements all have higher relative frequencies of usage in our database than in Miracle's.The reason for the former is two-fold.Firstly, in our database, alloys with similar compositions and identical phase constituents are only listed once (see Section 2).Many alloys containing Al, Fe, Co, Ni, Cu, and Mn are thus excluded, leading to lower relative frequencies.For example, Miracle and Senkov's database lists 29 IM-containing alloys in the Al-Co-Cr-Cu-Fe-Ni system, but ours lists only two.The second is that many simple solid solution alloys are also composed of these elements.They are also excluded in the present study.The reason for the latter is evident-the majority of elements typically lead to the formation of IM phases.Thus, in a database of IM-containing alloys, they exhibit higher relative frequencies.
The alloys in our database can be roughly categorized into three groups.The first is the CoCrFeNi-based alloys.These are defined here as alloys containing more than 60 at.% of Co, Cr, Fe, and Ni.The second is the refractory alloys, defined here as alloys containing more that 60 at.% of refractory elements (Cr, Hf, Mo, Nb, Ta, Ti, V, W, and Zr).The third is other alloys, which includes all alloys that do not belong to the previous groups.In our database, there are 86 CoCrFeNi-based alloys (60.6%), 25 refractory alloys (17.6%), and 31 other alloys (21.8%).
It is also worth noting that processing route can have an evident effect on the type of IM phase present in the alloys.For example, the IM phases in arc-melted [42] and cladded AlCoCrFeNiTi 1.5 [43] alloys are different.Another example is the Al 0.3 CoCrFeNi alloy [44], in which different annealing temperature can lead to completely different IM phase type, and thus different mechanical properties.

The Most Prevalent IM Structures
Figure 2a shows the relative frequencies of occurrence for the most common IM structures in the present study.The sequence, in order of decreasing prevalence, is (values in parentheses indicate number of occurrence): Laves (44), σ (38), B2 (37), L1 2 (15), L2 1 (8), µ (5), Ti 2 Ni (5), D8 4 (4), χ (4), η (4), and D8 8 (4).The Laves phase includes three closely related structure types: C14, C15 and C36.Their respective values are indicated in the corresponding bars.From Figure 2a, Laves, σ, B2, L1 2 and L2 1 are the five most prevalent structures, with Laves, σ, and B2 appearing in more than 25% of the alloys.L1 2 and L2 1 , on the other hand, appear in approximately 5-10% of the alloys.All other structures have relative frequencies lower than 5%.In other words, the popularity of the structures is quite non-uniform.This is not unexpected because among all identified IM structure types, the distribution of popularity is also quite non-uniform.For example, about 50% of all known IM compounds belong to 44 most popular structure types [38]-that is less than 2% of the known structure types!Additionally, as shown in Figure 1, the usage of the elements in our HEAs is also quite non-uniform.This will further concentrate the structure of the IM phases to certain structure types because of compositional similarity.As mentioned previously, it is meaningful to compare the results of the present database and Miracle and Senkov's.Therefore, the relative frequencies of common IM structures in their alloy database are also calculated and plotted in Figure 3a.The most common IM structures in order of decreasing prevalence are (values in parentheses indicate number of occurrence): B2 (177), σ (60), Laves (58) [C14 (50), C15 (8)], L1 2 (15), E9 3 (6), Ti 2 Ni (5), D0 2 (4), and L2 1 (4).Thus, the top four most common structures in the two databases are the same, but the relative sequence is different.This difference largely originates from the exclusion of compositionally similar alloys in the present study.For examples, there are a total of 334 IM-containing microstructure reports in Miracle and Senkov's study.However, the Al-Co-Cr-Fe-Ni alloy system alone accounts for 86 (or 25.7%) of them!This is because Al-Co-Cr-Fe-Ni is one of the two "classic" systems that have been studied most thoroughly.Thus, the effect of composition has been probed in considerable detail.For instance, Al x CoCrFeNi alloys with x = 0.3, 0.44, 0.7, 0.75, 0.8, 0.876, 0.9, 1, 1.2, 1.25, 1.5, 1.8, 2, 2.08, 2.5 and 3 have all been reported.Besides composition, the effect of processing route is also probed, leading to even more microstructure reports.Unfortunately, the IM phases in the Al-Co-Cr-Fe-Ni system are very similar-46 of them contain B2.Due to these reasons, the frequency of the B2 phase is evidently overestimated.This probably explains why B2 tops the list in Miracle and Senkov's database.Although their work is comprehensive and of great value, we believe our analysis better represents the overall reality by reducing the bias from similar alloys.Ferro and Saccone [38] calculated the most common structure types based on Villars and Calver's collection of about 26,000 known IM phases [36].It is meaningful to compare our results with this over trend in all IM phases.Figure 3b shows the most common structure types among all binary and ternary IM phases [38].Please note that the NaCl structure is omitted in Figure 3b because it is typically associated with non-metallic elements such as O and N (i.e., mostly ceramic phases).The five most common structures in order of decreasing prevalence are: Laves (including both C14 and C15 structures), BaAl 4 , C23, B2, and L1 2 .Compared with the top-five list in our database, it is seen that the Laves, B2, and L1 2 phases remain popular in HEAs, but the BaAl 4 and Co 2 Si structures do not appear in our HEA database at all.This distinction is not difficult to understand if we take a closer look at compounds with these structures.BaAl 4 -type compounds almost always contain elements in the lanthanide and actinide series, which are still very rarely used in HEAs [36].The case for the C23 structure is similar-C23-type compounds are typically formed in compounds containing lanthanide, IIA, and IVA elements [36].As will be shown later, IM structures formed in HEAs are always structures that existing in their subsystems.Consequently, if the necessary component of a particular structure is absent in an alloy, it is unlikely that such structure will form.By and large, the differences between the common IM structures in the present study and that in all binary/ternary IM compounds is probably because the explored compositional space in HEAs is still extremely limited.
For the (Co, Cr, Fe, Ni)-based alloys, the five most common phases (Figure 2b) listed in order of decreasing prevalence are: σ (32), B2 (23), Laves (20), L1 2 (12), and L2 1 (6).The common phases are same as that in all alloys, but the relative popularity is different.This again suggests that the IM phases present are still dependent on the composition.As will be shown later, the IM phases in HEAs are basically the IM phases that already exist in its binary and/or ternary subsystems.Thus, the choice of element is directly related to the type of IM phase.

Structure Inheritance in HEAs
Before going further, it is useful here to explain the term "structural inheritance" using ternary systems as examples.IM phases in ternary phase diagrams are very often extended from the boundary of the diagrams.In other words, they are often solid solutions of binary IM phases.Thus, the IM structure already exists in at least one of the three binary subsystems.In this work, this is described as the inheritance of the IM structure in the ternary alloy from its binary subsystems.In some cases, the IM phase region can even connect from one edge of the triangle to another, as illustrated in the imaginary phase diagram in Figure 4a.Similarly, in a quaternary system, the IM structures can be inherited from binary and ternary subsystems.However, there are cases where the IM phase cannot be found in any of the binary subsystems.Therefore, the IM phase is not inherited and is original to the ternary alloy.In such cases, the phase region of the IM phase is in the interior of the phase diagram, as illustrated in Figure 4b.Similar situations can be seen in quaternary systems, too.For example, in the Al-Co-Fe-Ti system, the TiFeCoAl phase (cF16-LiMgPdSn) is original and does not exist in any of its binary or ternary subsystems.What about the case in HEAs?Are the IM structures in HEAs inherited from their subsystems?Do original structures exist?If so, what are they and how often do they appear?
Table 1 shows the IM structures in each HEA collected in this study.We performed a detailed analysis on these structures to identify whether or not the structure is inherited.In some trivial cases, this can be confirmed directly from the chemical formula.For example, a Fe 2 Ti phase in the CoCrCuFeNiTi alloy is clearly inherited from the Fe-Ti binary system.In other cases, this is done by comparing the structure prototypes of the IM phases in an HEA and those in its subsystems.The result is indicated in the "IM Structure Found in" column in Table 1.As shown in the table, all the IM structures in the 142 alloys can be found in the subsystems of the respective alloy.In fact, in more than 90% of the alloys (129 out of 142 alloys), the IM structures are found in binary subsystems.In the remaining 13 alloys, the structures cannot be found in binary subsystems and are seen only in ternary ones.A typical reason for this is that the structure is intrinsically ternary (i.e., three different elements are required to form that structure).This is typically reflected by a ternary prototype.The best example is the L2 1 or Heusler phase (cF16-AlCu 2 Mn). 8 of the 13 remaining alloys contain the L2 1 phase.There are also structures that are not intrinsically ternary, but are still found in ternary subsystems only.The most notable case is the χ phase (found in the Cr-Fe-Ti system).
Based on the above, for all the HEAs in this study, the IM structures are inherited from their respective binary/ternary subsystems.This information has important implications.To predict the IM phases in HEAs, whether via empirical or CALPHAD approaches, a list of candidate phases is necessary so that the relative stabilities of the candidate phases can be compared to determine the stable phase.If the IM structures in an HEA are frequently irrelevant to its binary and ternary subsystems, given that phase diagrams of quaternary and higher-order systems are very rare, the prediction of IM phases in HEAs would be extremely difficult.Our results; however, show that the candidate IM phases basically resides in the binary and ternary subsystems, so phase prediction is feasible.Another way of looking at this is that if the composing elements in an HEA do not form a particular IM structure, it is unlikely that that phase will appear in the alloy.
Our results also suggest that when considering the candidate IM phases in an HEA, it is reasonable to consider just the phases in binary subsystems, plus certain important structures from known ternary combinations.This is because in more than 90% of the alloys, the IM phases exist in their binary subsystems.Moreover, those existing only in ternary subsystems are highly focused on certain element combinations.For example, the L2 1 phase is related to the Al-Ni-Ti combination, and the χ phase is related to the Cr-Fe-Ti combination.Further analysis of the dependence of IM phase type on composition is underway and will be published later.

Conclusions
A database of IM-containing HEAs has been constructed by examining related publications.The database consists of 142 unique alloys.Care is taken to avoid compositional repetition so that the diversity in the database is enhanced and statistical bias is reduced.The usage of elements in these alloys is very non-uniform and highly concentrated to Co, Cr, Fe, Ni, Al, and Ti, all appearing in 50% or more of the alloys.
Based on statistical analysis of the database, it is found that the five most prevalent IM structures in all HEAs are Laves, σ, B2, L1 2 , and L2 1 .Those in (Co, Cr, Fe, Ni)-based HEAs are the same, but with a different sequence.These results are evidently different from the overall trend in known binary/ternary IM structures mainly because the non-uniform usage of the elements in the alloys, which prevents the formation of many structures.
All the IM structures in the HEAs surveyed here are existing structures in the binary/ternary subsystems of the respective alloys.This means that HEAs tend to inherit IM structures from their subsystems, instead of forming new structures irrelevant to their subsystems.It seems that the compositional complexity in HEAs does not trigger additional structural complexity.Therefore, when predicting IM phases in HEAs, it is probably reasonable to consider only the IM phase types existing in the binary and ternary subsystems of the alloys, and ignore irrelevant phases.These findings have important implications in the future design and development of HEAs.

Figure 1 .
Figure 1.Relative frequencies of usage of various elements in the present study and in Miracle and Senkov's work, data from [3].

Figure 2 .
Figure 2. Relative frequencies of occurrence for the most common IM structures in this study: (a) Based on all alloys in the database; (b) Based on (Co, Cr, Fe, Ni)-based alloys only.

Figure 3 .
Figure 3. (a) Relative frequencies of occurrence for the most common IM structures in Miracle and Senkov's work, data from [3]; (b) Frequency of occurrence for the most common IM structures in all binary/ternary IM phases, data from [38].

Figure 4 .
Figure 4. Imaginary phase diagrams illustrating the inheritance of IM phases in ternary systems.Colored areas represent single phase regions.(a) A phase inherited from two binary subsystems (the ν phase); (b) A phase that is original to the ternary systems (the σ phase).

Table 1 .
Composition, processing condition, phase constituents, and structural inheritance of the 142 IM-containing HEAs considered in this study.

Composition Processing Condition 1 Solution Phases 2 IM Phases 3 IM Structure Found in Ref PhaseName Compound Reported Structure Prototype
Co 17 Cr 14 Cu 8 Fe 17 Mo 0.1 Ni 34.8 TiW 0.1