2024 Volume 65 Issue 9 Pages 988-994
Some high-entropy alloys (HEAs) with the face-centered cubic structure have an excellent strength-ductility balance. While unique deformation modes such as fine twinning patterns other than dislocation glide contribute to the mechanical properties, it is still unclear what properties and features of HEAs cause such unique deformation process. In the present study, fundamental properties responsible for the excellent mechanical properties of CoCrFeNiMn and its subsystems were explored comprehensively by the first-principles calculations. The local lattice distortion reaches almost 2% of the Burgers vector, which contributes to improving strength in HEAs. Furthermore, the stacking fault energy (SFE) was significantly low in random solid solution, while it increases around some domains where the short-range order (SRO) is formed. The increase in the SFE is caused by the disturbance of the chemical SRO and the spin order due to the SF formation. Our calculations suggest that low and high SFE domains distributed in a solid solution region unique to HEAs lead to successive activation of various deformation modes (Plaston), which achieves excellent strength-ductility balance.
Fundamental properties responsible for the mechanical properties in equiatomic CoCrFeNiMn, CrFeNiMn and CoCrNi. Large lattice distortion and low and high SF domains associated with the degree of SRO formation contribute to the excellent strength-ductility balance in FCC-HEAs.
While great efforts have been made throughout the long history of metallurgy to achieve a desirable balance between high strength, ductility and toughness, the trade-off relationship has made it difficult to manage these conflicting mechanical properties [1–3]. High-entropy alloys (HEAs) that appeared in the 2000s have been extensively studied as new materials with great potential to strike a good balance between strength and ductility [4–12]. In HEAs, high mixing entropy derived from multi-component alloys is utilized to stabilize the solid solution states, contributing to free energy. In practice, however, using only entropy effects cannot achieve the solid solution alloys at will, so various alloy systems have been explored within the enormous search space of alloy systems [13–15]. Non-equiatomic, multi-phase HEAs and even medium-entropy alloys (MEAs) have become new research targets in the search for better mechanical properties.
The well-studied HEA with a face-centered cubic (FCC) structure is a single-phase solid solution of equiatomic CoCrFeNiMn, commonly known as a Cantor alloy [16]. CoCrFeNiMn has excellent mechanical properties at room and cryogenic temperatures [7], overcoming the trade-off relationship to exhibit both high strength and good ductility [17, 18]. Chemical short-range order (SRO), one of the specific characteristics of HEA has been considered as an efficient guideline for the fundamental properties. More recently, the excellent mechanical properties in FCC-HEA are derived from twinning-induced plasticity (TWIP) [7, 19–22] and transformation-induced plasticity (TRIP) [22–25]. In general, the fundamental features in FCC metals, such as the stability and activity of dislocations, twinning deformation, and FCC to hexagonal close-packed (HCP) martensitic transformation, are determined by the stacking fault energy (SFE) [26]. As the SFE consequently dominates the deformation mode in most FCC alloys, including HEAs, the SFE is one of the essential properties for understanding the macroscopic mechanical properties associated with the microstructure under deformation in FCC alloys. It was expected that CoCrFeNiMn and its derivatives have low SFE and that the excellent mechanical properties of some FCC-HEAs are effectively associated with the transition of the deformation mechanism from the conventional dislocation glide to other modes, such as deformation twinning [7, 27–30]. More recently, some derivatives of CoCrFeNiMn have achieved higher strength, keeping better ductility by utilizing the strategy for controlling SFE with additional solutes [31, 32].
Computational approaches based on electronic structure calculations have the advantage of capturing the fundamental properties directly related to local composition and configuration. The first-principles calculations were applied to the SFE calculations in some FCC-HEAs, showing that the SFEs of FCC-MEA and FCC-HEA are significantly lower than conventional FCC metals [33–39]. The type of element and its concentration were, thus, expected to have a significant effect on the SFE. However, most SFE calculations have been performed assuming an ideal random solid solution, in which the effect of chemical SRO on the SFE has not been considered. Moreover, most FCC-HEAs and FCC-MEAs derived from CoCrFeNiMn have complicated magnetic states, which makes it difficult to implement DFT calculations for these systems. In the present study, CoCrFeNiMn and its subsystems were targeted, and the effects of magnetic state and SRO formation on the local lattice distortion and the SFE related to the mechanical properties were comprehensively discussed.
In this study, equiatomic CoCrFeNiMn, CrFeNiMn, and CoCrNi corresponding to FCC-HEA and FCC-MEA were taken as typical examples of Cantor alloy and its subsystems. The atomic models were constructed to reproduce equiatomic quinary, quaternary, and ternary alloys, where a statistically random solid solution for each composition was achieved by special quasi-random structures (SQS) generated using the “mcsqs” function in alloy theoretic automated toolkit (ATAT) [40]. In the present study, pair correlation within the third nearest neighbor was considered to make random configurations. 180-atom supercell with dimensions a a = 5e′1, b = 3e′2 and c = 2e′3 was constructed. Here, the basis vectors in the system are defined as $\mathbf{e}'{}_{1} = a_{0}[\bar{1}10]$, $\mathbf{e}'{}_{2} = a_{0}[11\bar{2}]$, e′3 = a0[111] with the lattice constant a0.
First-principles calculations were performed using the Vienna ab initio simulation package (VASP) [41, 42]. Projector augmented wave potentials [43] were employed with the Perdew–Burke–Ernzerhof generalized gradient approximation exchange-correlation density functional [44]. The Brillouin-zone gamma-centered k-point samplings were chosen based on the Monkhorst–Pack algorithm [45], where a 3 × 3 × 3 grid was used. A cut-off in plane-wave energy of 400 eV was applied using a first-order Methfessel–Paxton scheme that employed a smearing parameter of 0.1 eV. The magnetic effect was considered on the assumption of collinear spin-polarized conditions. The initial spin states were determined to make the system magnetically neutral (paramagnetic states), where the up and down spin conditions were randomly distributed. The total energy was converged within 10−5 to 10−4 eV for all calculations. The conjugate gradient method was employed to optimize the atomic configuration and to search for the relaxed configurations that terminated the search when the force on all atoms was reduced to 0.02 to 0.025 eV/Å. The rougher threshold value was used for unstable configurations. The atomic configurations shown in this study were visualized by VESTA [46] and Atomeye [47] software.
The atomic model to evaluate the SFE was constructed according to the following procedure. The uniform displacement of bP was applied to the upper half of the perfect crystal along the $[11\bar{2}]$ direction, where $\mathbf{b}_{P} = a_{0}[11\bar{2}]/6$; a0 is the lattice constant. If a rectangular periodic cell is used, two SF surfaces are introduced at the bottom and center along the (111) plane. To eliminate the SF interface at the bottom layer, the shape of the supercell was modified by giving a shear component equal to the magnitude of bP, i.e. h32 = −|bP|. Accordingly, only one SF plane is inserted into the model of the orthorhombic periodic cell. The SFE and generalized SFE correspond to the excess energy when the uniform displacement is applied to the atoms on the upper half of the model. All atoms were relaxed only along the direction perpendicular to the slip plane. Note that the VASP code was modified to make the constraint optimization constrain along the direction based on the Cartesian coordinate system. Thus, the SFE is evaluated by the energy difference between the SF model and the perfect crystal as γSF = (ESF − E0)/ASF, where ASF is the area of the SF interface.
In this study, the formation of the SRO was explored with considering magnetic states. Monte Carlo (MC) simulations were carried out to find more stable configurations than random solid solutions. Swapping atoms was done according to the probability of acceptance associated with the difference in total energy of two configurations based on the Metropolis algorithm [48], where the total energy at each MC step was evaluated by the first-principles calculations. The procedure was shown in section 2, but 10−4 eV convergence condition was used, and only Γ point calculations were done for MC simulations to save computation time. Atomic relaxation did not apply to the present MC simulations. The SRO formation at two temperatures (300 and 800 K) was investigated. The SRO formation was analyzed using the Warren-Cowley SRO parameter [49]. Based on the modification [50], the following definition was used for the SRO parameter of each pair of elements. Note that the SRO parameter Δδ used in the present analysis has the opposite sign to the definition in Ref. [50], i.e., the SRO parameter is positive when the probability of the pair is higher than the ideally random configuration.
| \begin{equation} \Delta \delta_{ij}^{n} = N_{ij}^{n}(p_{ij} - p_{ij}^{\text{rand}}), \end{equation} | (1) |
where $N_{ij}^{n}$ is the coordination number within the n-th nearest neighbor shell. For example, $N_{ij}^{1}$ corresponds to 12. pij is the probability of the i-j atom pair after MC simulations, and $p_{ij}^{\text{rand}}$ is the probability of the same atom pair in an ideally random solution at a given composition. The SRO parameter of the 1st nearest neighbor pair was discussed in this paper. The SRO parameters of each constituent element for CoCrFeNiMn, CrFeNiMn, and CoCrNi were summarized in Fig. 1. Each radar chart includes a random solid solution and SRO-formed models after the MC simulations at 800 and 300 K. If there is no SRO formation corresponding to the ideally random solid solution, the value should be zero, as shown by the red circle. The initial configuration prepared by SQS (blue line) is confirmed to be close to this red circle. Focusing on the strong trend of SRO formation after MC simulations in CoCrFeNiMn, the Fe-Cr, Cr-Ni, Ni-Mn, Co-Mn, and Co-Cr pairs increase while Fe-Mn, Cr-Cr and Co-Ni decrease. These trends also apply to other subsystems. A particularly noticeable tendency from several MC calculations with the same compositions is that the Fe-Cr, Cr-Ni, and Ni-Mn pairs increase while the Cr-Cr pair decreases.
SRO parameter of atom pair within 1st nearest neighbor shell for a random solid solution and SRO-formed models at 800 and 300 K in (a) CoCrFeNiMn, (b) CrFeNiMn and (c) CoCrNi. The red circle indicates zero, corresponding to the probability of an ideally random solid solution.
The magnetism should significantly influence the SRO formation and mechanical properties. The distributions of the magnetic moment of all constituent elements in CoCrFeNiMn, CrFeNiMn, and CoCrNi alloys are summarized in Fig. 2, where random solid solution and SRO-formed models at 300 K were provided. Five random and three SRO-formed models were used for the statistically sufficient analysis. The magnetic moment distribution for each element shows similar trends for all alloys; Fe and Mn have relatively large values, while Co and Ni have small ones. It was also found that the magnetic moment distributes widely for a random solid solution, which agrees well with other first-principles calculations [34]. On the other hand, as shown in Fig. 2(d), (e) and (f), the magnetic moment becomes slightly biased when the SRO is formed. As if dragged by Fe, which has a strong magnetic moment, Co, Ni, and Mn are distributed with the same sign as that of Fe, while Cr has the opposite sign as that of Fe. This magnetic moment arrangement significantly affects the SRO formation of the Fe-Cr pairs within the 1st nearest neighbor, as discussed above, which also strongly influences the relationship between SFE and SRO, which will be discussed later in detail.
Distribution of magnetic moment of each element for the random models in (a) CoCrFeNiMn, (b) CrFeNiMn and (c) CoCrNi. (d)–(e) Same as (a)–(c) but for SRO-formed models.
Lattice distortion is especially important characteristic of HEAs. The mechanical properties were found to be associated well with the lattice distortion using the δ parameter [4, 7]. Atomistic simulations have the advantage of measuring local lattice distortion, i.e., atomic displacement from the original lattice point of each atom can be measured directly by the first-principles calculations. In the present study, the mean square atomic displacement (MSAD) [51] defined by the following equation was used to evaluate the local lattice distortion of each element and the whole system:
| \begin{equation} \mathrm{MSAD} = \frac{1}{N}\sum_{i = 1}^{N}|r_{i} - r_{i}^{0}|^{2}, \end{equation} | (2) |
where N is the number of atoms, ri is the coordinate of atom i after relaxation, and $r_{i}^{0}$ is the initial lattice point. To evaluate the MSAD, fully atomic relaxation was performed for both random and SRO-formed models after MC simulations at 300 K. The square root MSAD scaled by the magnitude of Burgers vector b is shown in Fig. 3. At first, the effect of magnetism was investigated using nonmagnetic and paramagnetic states for CoCrFeNiMn as shown in Fig. 3(a). There is no significant difference between nonmagnetic and paramagnetic calculations, which means the magnetic states do not influence the local lattice distortion. The MSADs in random solid solution and SRO-formed models in CoCrFeNiMn, CrFeNiMn, and CoCrNi alloys are shown in Fig. 3(b). Each element deviates from its ideal lattice points to a greater or lesser degree for all alloys. It is commonly seen in all alloys that there are apparent differences in the MSAD between elements, with relatively large lattice distortion for Cr and Mn and small for Ni. It was also found that the MSAD decreases after the SRO is formed indicating that such ordering occurs not only by chemical interaction but also by reducing elastic strain energy. More interestingly, the MSAD has little correlation with the diversity of constituent elements, i.e., configurational entropy, and tends to be larger when Cr and Mn, which are highly displaced from the lattice point, are included with higher concentration. As for an overall feature in Cantor and its subsystems, the average value of MSAD reaches almost 2% of the Burgers vector, which should influence the friction stress for dislocation motion.
Local lattice distortion using MSAD values for each constitutive element and a whole system, where the square root of the MSAD normalized by the Burgers vector was given. (a) Comparison between nonmagnetic and paramagnetic states in CoCrFeNiMn. (b) MSAD for the random solid solution and SRO-formed models in CoCrFeNiMn, CrFeNiMn and CoCrNi.
In addition to the local lattice distortion, the SFE is recognized as another crucial factor dominating dislocation motion, twin formation, and FCC-HCP martensitic transformation in FCC-HEAs. As the atomic model used in the present study has six layers along [111] direction, six different types of configurations for the SF model were created by inserting the SF layer into each layer as shown in Fig. 4. As was done for the lattice distribution, the effect of magnetism on the SFE was investigated using nonmagnetic and paramagnetic states for CoCrFeNiMn as shown in Fig. 5. Unlike the MSAD or local lattice distortion, the SFE value is significantly influenced by the magnetism. The value was quite low when calculated for nonmagnetic states. Moreover, the value remained low even if the SRO was formed. On the other hand, as it is known that the magnetism increased the SFE [34], the SFE became higher when paramagnetic states were considered. Surprisingly, the SFE was found to be abnormally large after the SRO was formed.
Atomic models of periodic supercell for SFE calculations.
SFEs calculated using random solid solution and SRO-formed models with and without considering magnetism.
Subsequently, the difference in the SFE between alloy types is shown in Fig. 6, where three random solid solution and SRO-formed models considering magnetism were used to calculate the stable SFE, respectively. As mentioned above, the SF layer can be inserted into every [111] layer, and thus, 18 SFE data can be collected for each case. Obviously, the SFE depends on the type and concentration of the constituent elements. For example, focusing on the differences in alloy types, the SFE of CrFeNiMn is the highest due to the lack of Co, stabilizing the HCP phase. Furthermore, it is commonly found that the SFE tends to increase significantly when the SRO is formed, and the trend becomes more significant as the degree of ordering increases. On the other hand, the trend depending on the SRO formation is not significant in CoCrNi, as shown in Fig. 6(c). What causes the increase in the SFE and the difference between alloy types associated with the SRO formation remains unexplained.
Distribution of SFEs using the random solid solution and SRO models formed at 800 and 300 K in (a) CoCrFeNiMn, (b) CrFeNiMn and (c) CoCrNi.
The detailed investigation of the relationship between magnetic moment distribution associated with SRO formation and SFE is shown in Fig. 7. In Fig. 7(a), the magnetic moments of atoms in different (111) planes for the stable FCC phase with SRO formation at 300 K were given. In common with the three (111) planes, the atoms with large magnetic moments with the same sign are not located within the 1st nearest neighbor position. As seen in the dotted triangle, these atoms tend to be aligned one atom apart. Such a configuration mainly corresponds to the formation of the Fe-Cr pair, as shown in Fig. 1, in which the Fe and Cr atoms located within the 1st nearest neighbor position have opposite signs of magnetic moments, as shown in Fig. 2(b). However, the atoms with large magnetic moments come to the first nearest neighbor position when the SF layer is formed, as seen in the area surrounded by the solid circle. This pair formed in this way, such as Fe-Fe and Cr-Cr, is unstable because it disrupts not only the stable chemical SRO but also the ordering of the magnetic moment, resulting in a higher SFE. Finally, the distribution of the change in magnetic moments for all atoms of SRO-formed models is shown in Fig. 7(c), where the horizontal axis corresponds to the coordinates along the ⟨111⟩ direction, aligned so that the SF position is at the center. It is confirmed in the figure that the magnetic moment changes significantly near the SF layer or is accompanied by spin-flip in some cases. Note that there is room for further spin-flip to decrease the SFE. As described above, it can be concluded that the formation of the SRO contributes significantly to the higher SFE due to the change in the chemical stability of SRO and the magnetic moment of some pairs within the 1st nearest neighbor position. The slight increase in the SFE due to SRO formation in CoCrNi, which is composed of elements with small magnetic moments, is also explained by this mechanism.
Magnetic moment of each atom in the (111) plane for SRO-formed model at 300 K in CoCrFeNiMn. (a) Typical images of magnetic moment in each (111) plane in a perfect crystal. (b) Magnetic moment of each atom belonging to two (111) planes next to each other across an SF layer, where the images before and after SF formation were shown. (c) Distribution of magnetic moment of all atoms in terms of the distance from the SF layer.
In the real materials, the near-random solid solution state and the SRO-formed areas are distributed heterogeneously in various variations, indicating that the SFE value varies from domain to domain while the alloys seem to be in a single-phase solid solution. Even if the TWIP mode is activated in some areas with low SFE, the twin growth is prevented by the existence of the high SFE domain. At the same time, new TWIP or TRIP deformation occurs in other low SFE domains. The situation indicates that the localization of plastic deformation is less likely to occur, which is the key factor in the excellent strength-ductility balance. Such a deformation process is not seen in other materials with uniformly low SFE and is unique to the HEAs. Thus, the “Plaston” strategy [52], in which the plastic deformation occurs successively at different stress levels, is realized effectively in HEAs.
In summary, fundamental properties responsible for the excellent mechanical properties of FCC-HEA and FCC-MEA were investigated by the first-principles calculations. Equiatomic CoCrFeNiMn, CrFeNiMn and CoCrNi were targeted as typical examples of Cantor alloy and its subsystems, and the effects of SRO formation and magnetic state on the local lattice distortion and the SFE were explored comprehensively.
This work was supported by Grant-in-Aids for Scientific Research on Innovative Areas on High Entropy Alloys through the grant numbers JP18H05453, in part by JSPS KAKENHI JP23K23030 and JST FOREST JPMJFR213P. Simulations were performed on the large-scale parallel computer system of HPE SGI 8600 at JAEA.
