Revisit the VEC criterion in high entropy alloys (HEAs) with high-throughput ab initio calculations: A case study with Al-Co-Cr-Fe-Ni system
Introduction
Since the last few years, high entropy alloys (HEAs) have attracted significant attention due to their remarkable properties such as excellent strength, high hardness, high wear resistance, and superior corrosion resistance [1], [2], [3], [4]. Contrary to the traditional alloys, which rely on one dominant base metal and small additions of other elements, HEAs typically comprise 5 or more principal elements forming single solution phases [5], [6]. Most of these single solution alloys are composed of face-centered cubic (FCC), body-centered cubic (BCC), or hexagonal-closed packed (HCP) crystallographic structures [7]. Unlike the single-component FCC, BCC, and HCP pure metal, which exhibits no lattice distortion, a five components solution phase, especially for HEAs, typically shows severe lattice distortion because each atom in the matrix is surrounded by different kinds of atoms and suffers lattice strain and stress [8]. In general, the lattice distortion effect on microstructure and mechanical properties are positive and encouraging [9].
The exploration of phase stability is essential for the development of novel HEAs. Efforts utilizing scientific criteria and approaches for the phase prediction were applied to HEA-related topics, such as entropy of mixing (△Smix) [10], atomic size differences (δ), melting points (Tm) [11], the enthalpy of mixing (△Hmix) [12], the average number of itinerant electrons per atom ratio (e/a), parameter Ω (the scale ratio of △Smix to △Hmix) [13], and valence electron concentration (VEC) [14], [15]. Out of all these parameters, the VEC rule proposed by Guo et al. [15] has been adopted as one of the most reliable [16] and frequently used rules for the determination of phase stabilities in HEAs [17], [18], [19], [20], [21], [22], [23]. It was claimed that the FCC phase is stable at VEC≥ 8, BCC is stable at VEC≤ 6.87, and a mixture of FCC and BCC phases exists at 6.87 ≤VEC≤ 8 [15], based on the experimental observation of around 20 HEAs.
However, it has been reported that the empirical VEC rule has its own limitations, such as the neglection of the temperature effect [23], the argument of the VEC threshold value [8], [24], [25], [26], and the effectiveness on non-transition metal (TM) elements [13]. Recently, Yang et al. [23] systematically revisited the empirical VEC rule in Al-Co-Cr-Fe-Ni high entropy alloy with high-throughput CALPHAD (HT-CALPHAD) approach. It was suggested that more than 90% of compositions are observed to have BCC structures when 5.7 ≤VEC≤ 7.2% and 100% FCC are found at VEC≥ 8.4. Importantly, with the adoption of CALPHAD modeling, the effect of temperature could be considered as well.
However, there are still several additional issues with the empirical VEC rule. While the HT-CALPHAD approach is able to provide the temperature range of BCC and FCC structures, most of these single-phase structures are stable only at high temperatures or specific temperature ranges. It is hard to draw a conclusion if the VEC criterion works at low temperature when both BCC and FCC are thermodynamically not stable. Second, although the reliability of the VEC rule in controlling the FCC- or BCC-type solid solution phases has been experimentally [15], [27] and theoretically [23], [28], [29] verified in multi-components systems (>5 elements), its validity in low-order systems, especially for binaries and ternaries, has not been investigated.
In the present work, the HT-ab initio calculations were employed to investigate the relative stability of BCC and FCC structures with the change of the VEC value. The Special Quasi-Random Structures (SQS) model [30] is used to deal with the modeling of random BCC and FCC alloys. The classical Al-Co-Cr-Fe-Ni quinary systems will be applied as a case study to revisit the validity of the empirical VEC rule at the ground state. The prediction will start with pure elements and will be continued with binaries, ternaries, and even quaternary compositions. Meanwhile, the impact of each element on the stability of BCC and FCC structures will be explored as well.
Section snippets
The Selection of SQSs in the Al-Co-Cr-Fe-Ni HEAs
The SQS method attempts to find the small-unit-cell periodic supercell that for as many figures as possible [31], [32], [33], where is the correlation function of a random alloy. The structures that exhibit the least correlation function mismatch will be selected for the ab initio calculations as these structures are closest to random alloys. The SQSs supercells were generated by the alloy-theoretic automated toolkit (ATAT) package [34] with the mcsqs code [35]. The
Difference of formation enthalpy () between BCC and FCC phase
The formation enthalpy, , is a thermochemical parameter that determines how favorable to form a particular phase compared to the pure element stable state at 0 K. The of FCC and BCC phase can be defined by the Eq. (2) [44], [45]:where is the ground-state total energy of FCC/BCC alloys and represents the energy of each element in its stable phase at 0 K.
The difference of between the FCC and BCC phase can be interpreted as an
Conclusions
The present work systematically examined the reliability of the empirical VEC rule through the HT-ab initio calculations. The classical Al-Co-Cr-Fe-Ni quinary system was applied as a case study to investigate the ground state formation enthalpy of FCC and BCC structure, considering pure elements, binary, ternary, and quaternary compositions. Particularly, we established a new VEC criterion based on (FCC)-(BCC), which can be applied to define the relative stability of FCC and BCC
CRediT authorship contribution statement
Songge Yang: Constructed the idea of the HT calculations and had primary writing responsibilities. Guangchen Liu: Visualization. Yu Zhong: Conceptualized, guided all the aspects, and led the project as well as extensively revised the manuscript.
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.
Acknowledgment and disclaimer
This material is based upon work supported by the Department of Energy under Award Number DE-FE0030585 and Extreme Science and Engineering Discovery Environment (XSEDE) Award Numbers TG-DMR190004. The authors would like to thank the support and guidance from the DOE National Energy Technology Laboratory program manager, Maria M. Reidpath. This paper was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency
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2022, Journal of Alloys and CompoundsCitation Excerpt :The probable phase to be formed can be predicted using the VEC rule. However, an argument to the contrary was made in a recent study by Yang et al. [26], who proposed a new approach known as the effectiveness of the VEC rule (EVEC). However, this approach cannot predict the stability of dual-phase regions such as the FCC + BCC phases.
Strengthening mechanisms in high entropy alloys: A review
2022, Materials Today CommunicationsCitation Excerpt :Yang et al. concluded that the most compositions of AlCoCrFeNi system at VEC≥ 8 have negative value of Hf(FCC)-Hf(BCC) and are FCC stabilizer, while the majority of compositions at VEC≤ 6.87 show the positive value of Hf(FCC)-Hf(BCC) and are BCC stabilizer, which is consistent with Guo’s criteria. However, Guo’s criteria [39] did not work effectively for the ground state compositions at VEC < 5, as the FCC is stable as compared to the BCC structure for the most light-weight and transition elements at VEC < 5 as shown in Fig. 4 [49]. Therefore, VEC and ∆Hf between FCC and BCC phase could be considered as an effective approach for the phase prediction in HEAs.