CPA + Exact Muffin-Tin Orbitals (EMTO)

The CPA method, which was developed to simulate random media, relies on the mean field approach to consider the chemical disorder in HEAs. In most cases, DFT calculations based on CPA are implemented with exact muffin-tin orbitals (EMTO) [Reference Tian, Varga, Chen, Delczeg and Vitos38] (although the Korringa–Kohn–Rostoker technique [Reference Kohn and Rostoker39] has also been applied occasionally [Reference Tian, Wang, Irving and Vitos40]). Within the muffin-tin approach, the potential for each atom is assumed to be spherically symmetric in the muffin-tin region (around lattice sites) and constant in-between [see Fig. 2(a)]. Wave functions are obtained by matching solutions of the Schrödinger equation within each muffin-tin sphere with plane-wave solutions in the intermediary region.

Figure 2:

(a) Schematic diagram of muffin-tin potential technique. The muffin-tin regions around every atom are outlined by circles. (b) CPA for the equiatomic HEAs. The symbol P represents the real alloy potential, $\tilde{P}$ represents the coherent potential, and P _A, P _B, P _C, P _D, and P _E are the potentials of the constituent alloying elements. Adapted from [Reference Tian41] under Creative Commons Attribution License. (c) SQS cell for quinary Mo₂₀Nb₂₀Ta₂₀V₂₀W₂₀ BCC HEA. Reprinted from Wang et al. Computation of entropies and phase equilibria in refractory V–Nb–Mo–Ta–W high-entropy alloys, Acta Materialia, 143, 88–101, Copyright (2018), with permission from Elsevier [Reference Wang, Yan, Zhu, Wang, Wu, Hui, Otis, Shang, Liu and Chen164]. (d) SLAE cell for equiatomic BCC (top) and FCC (bottom) HEAs. Reprinted from [Reference Tian41] under Creative Commons Attribution License.

In the CPA approach, each alloy atom is considered to be embedded in an ordered effective lattice, the properties of which are determined self-consistently using the mean field approach. An average occupation of lattice sites by each type of alloying atoms is calculated, and the local potentials for each type of constituent atom are presumed to be the same throughout the sample lattice; thus, all the local effects and variations in the atomic environment are lost [see Fig. 2(b)].

The CPA-based DFT calculations effectively mimic the chemical and magnetic disorder. They have a relatively low computational cost in comparison with the supercell method as described below [Reference Vitos, Abrikosov and Johansson34]. The list of advantages of CPA also includes its efficiency in handling of point and linear defects. The ground-state properties (lattice parameters, bulk moduli, and mixing enthalpies) of many HEAs can be calculated accurately using the CPA method. Since extraction of the energetics for HEA samples is relatively easy, this method is suitable for multiscale simulations. In particular, it can be efficiently coupled with atomistic simulations based on the Monte-Carlo sampling algorithm [Reference Vitos, Abrikosov and Johansson34]. The main disadvantage of the CPA method is its inability to include the local lattice distortions, since all the local atomic variations are ignored within CPA, and the difficulty in computing of interatomic forces, which hinders accurate modeling of alloy dynamics with DFT-based MD simulations [Reference Ikeda, Grabowski and Körmann33].

Supercell method

Similar local atomic environment (SLAE)

The most recent approach to construct HEA supercells is based on the idea of a SLAE structure [Reference Song, Tian, Hu, Vitos, Wang, Shen and Chen37]. A SLAE supercell [see Fig. 2(d)] can be constructed by setting the local atomic environments of all the lattice sites similar to each other. The local atomic environments are defined according to the radial distribution functions for each type of alloying atoms in the HEA. Using an iterative scheme, in which atomic pairs within the supercell are rearranged, the SLAE conditions can be progressively approximated to the desired accuracy. Song et al. [Reference Song, Tian, Hu, Vitos, Wang, Shen and Chen37] used SLAE supercells to study local lattice distortion induced by the large atomic size mismatch in FCC and BCC HEAs. They found that the local lattice distortion of the BCC HEAs is much more significant than that of FCC HEAs.

Comparison between first-principles techniques

In conclusion, the main advantage of the SQS, SSOS, and SLAE supercell techniques is that they explicitly include local lattice distortion in HEAs. However, the computational cost is relatively higher in comparison with the CPA approach, which is orders of magnitude less expensive [Reference Vitos, Abrikosov and Johansson34]. Furthermore, as the number of alloying elements in HEAs increases, it becomes increasingly problematic to find a small-sized SQS or SLAE cell adequately representing their chemical randomness. Consequently, size effects become evident in the calculated results; for example, Tian et al. [Reference Tian, Hu, Yang, Zhao, Johansson and Vitos48] found that the elastic constants of the studied BCC HEAs computed using the SQS method depended on the supercell size. Lastly, an introduction of defects into a HEA supercell can break the perfectly matched correlation functions since the atomic arrangements around the hosted defects are strongly affected, and these effects are not captured by the correlated functions for the defect-free bulk.

Existing interatomic potentials

The most important aspect of MD simulations is the interatomic potential. Without construction of novel interatomic potentials, HEA MD simulations must solely rely upon existing potentials. This can be challenging because of the large number of alloying elements in HEAs. Only a few ternary potentials are available for MD, much less quaternary or greater. Early MD studies of HEAs used the COMPASS force field to simulate scratch tests of equiatomic AlCoCrFeNi [Reference Huang49]. A mixture of EAM and Lennard-Jones potentials has also been used to investigate chemical heterogeneity in an Al_0.1CrCoFeNi HEA [Reference Sharma, Singh, Johnson, Liaw and Balasubramanian50].

A strategy to study HEAs via MD has thus been to use a combination of existing binary and ternary potentials. Elements that compose common HEAs include Co, Cr, Fe, Mn, Ni, Al, Cu, Ti, Ta, Mo, Nb, V, W, and Zr. The most readily available potentials are for the CoCrFeMnNi system as several of its suballoys have been developed to study Co- [Reference Dong, Kim, Ko, Lee and Lee51] and Ni-based superalloys [Reference Kim, Jung and Lee52], as well as high-Mn steels [Reference Kim, Shin and Lee53]. Second nearest-neighbor modified embedded-atom method (2NN MEAM) interatomic potentials have been developed for unary Co [Reference Dong, Kim, Ko, Lee and Lee51], Fe [Reference Lee, Baskes, Kim and Koo Cho54], Cr [Reference Lee, Baskes, Kim and Koo Cho54], Mo [Reference Lee, Baskes, Kim and Koo Cho54], W [Reference Lee, Baskes, Kim and Koo Cho54], V [Reference Lee, Baskes, Kim and Koo Cho54], Nb [Reference Lee, Baskes, Kim and Koo Cho54], Ta [Reference Lee, Baskes, Kim and Koo Cho54], Mn [Reference Kim, Shin and Lee53], Cu [Reference Lee, Shim and Baskes55], Ag [Reference Lee, Shim and Baskes55], Au [Reference Lee, Shim and Baskes55], Ni [Reference Lee, Shim and Baskes55], Pd [Reference Lee, Shim and Baskes55], Pt [Reference Lee, Shim and Baskes55], Al [Reference Lee, Shim and Baskes55], and Pb [Reference Lee, Shim and Baskes55]; for binary Co–Cr [Reference Choi, Kim, Seol and Lee56], Co–Fe [Reference Choi, Kim, Seol and Lee56], Co–Mn [Reference Choi, Kim, Seol and Lee56], Cr–Mn [Reference Choi, Kim, Seol and Lee56], Mn–Ni [Reference Choi, Kim, Seol and Lee56], Co–Al [Reference Dong, Kim, Ko, Lee and Lee51], Fe–Mn [Reference Kim, Shin and Lee53], Ni–Co [Reference Kim, Jung and Lee52], Fe–Cr [Reference Lee, Shim and Park57], Fe–Ni [Reference Wu, Lee and Su58], and Cr–Ni [Reference Wu, Lee and Su58]; and for ternary Ni–Al–Co [Reference Kim, Jung and Lee52] and Fe–Cr–Ni [Reference Wu, Lee and Su58]. These potentials cover all interactions in equiatomic CoCrFeMnNi (also referred to as the Cantor alloy) and have been used to study diffusion and lattice stability in it [Reference Choi, Jo, Sohn, Lee and Lee59]. A FeNiCr interatomic potential has previously been developed to study plasticity [Reference Bonny, Terentyev, Pasianot, Poncé and Bakaev60] and aging under irradiation [Reference Bonny, Castin and Terentyev61] in stainless steels. This potential has been applied to study the critical stress for dislocation motion in HEAs [Reference Varvenne, Luque and Curtin25] as well as to study the separation distance of dissociated, mixed dislocations and stacking fault energy [Reference Smith, Hooshmand, Esser, Otto, McComb, George, Ghazisaeidi and Mills19].

A study [Reference Zhou, Johnson and Wadley62] in 2004 provided a new framework, which forms the basis for construction of an arbitrarily complex interatomic potential. This framework was originally created to model misfit dislocations in multilayered structures. It is an EAM-type potential containing modified pair potential energy and embedding energy functions that vary smoothly between atomic neighbors of different species. The original elemental set contained Cu, Ag, Au, Ni, Pd, Pt, Al, Pb, Fe, Mo, Ta, W, Mg, Co, Ti, and Zr, but has been extended to include Cr [Reference Granberg, Nordlund, Ullah, Jin, Lu, Bei, Wang, Djurabekova, Weber and Zhang63, Reference Lin, Johnson and Zhigilei64]. This potential has been used to simulate dislocation behavior in a FCC Co₃₀Fe_16.67Ni_36.67Ti_16.67 alloy [Reference Rao, Woodward, Parthasarathy and Senkov65] and a BCC Co_16.67Fe_36.67Ni_16.67Ti₃₀ alloy [Reference Rao, Varvenne, Woodward, Parthasarathy, Miracle, Senkov and Curtin66]. It has also been used to simulate HEA thin film growth [Reference Xie, Brault, Thomann, Yang, Zhang and Shang67] and HEA cluster annealing on a Si substrate [Reference Xie, Brault, Thomann and Bauchire68]. The quantitative accuracy of these potentials has not been verified and the original authors cautioned against applying the potential to metal compound formation, but the ability to create interatomic potentials of arbitrary complexity offers an opportunity to explore multiple principal component alloys, where traditional alloying models do not apply.

Machine learning for potential development

To expand the set of available elements and ensure accurate potentials to simulate HEAs in MD, new interatomic potentials need to be developed. Among the proposed frameworks to develop new interatomic potentials, the most prominent ones feature incorporation of machine learning (ML) techniques to guide potential development. All ML frameworks share two fundamental features: representation of atomic configurations that obeys required symmetries (descriptor) and an independent covariate or controlled variable to model the potential (regressor). Along with this framework, a robust set of high-fidelity reference data to train the regressor and validate the model is required. Below we review several approaches, focusing mainly on bulk metallic systems that are most relevant to HEAs (interested reader may refer to [Reference Sosso, Deringer, Elliott and Csányi69, Reference Behler70, Reference Handley and Behler71] for a more detailed review of ML-based research in interatomic potential development).

There are commonalities in the choice of the descriptor among the different frameworks that must address required symmetries and other conditions. These include: invariance with respect to chemical permutation, translations, rotations, and smoothness of atomic density near neighborhood cutoff boundary (smooth variation in atomic density is able to avoid large numerical errors when training the regressor). The descriptor also must be a continuous one-to-one mapping between the atomic configurations and points on the potential energy surface. In classical potential models, these descriptors include quantities such as bond length, dihedral angle, coordination number, bond type, atomic species, etc.

It must also be noted that any ML model of the potential energy surface cannot be accurately extrapolated to regions of configuration space that have not been included in the training dataset. This requires atomic configuration data to train the ML-based potential that covers all expected atomic configurations, including bulk, surface, liquid configurations, and all possible chemical environments for heterogeneous systems.

Artificial neural network potentials

Artificial neural networks (ANNs) are among the most developed approaches to modeling the potential energy surface of atomic systems with several reviews available [Reference Behler72, Reference Behler73, Reference Behler74, Reference Handley and Popelier75]. ANNs are models consisting of layers of connected nodes. Each layer is connected to adjacent layers via weights (i.e., the fitting parameters) by training on the reference dataset. It has been suggested that the flexibility of ANNs can be used to model the potential energy surface of atomic configurations [Reference Behler72, Reference Behler and Parrinello76]. In the case of ANN-based potentials, the input includes an atomic configuration and the output yields the energy of such configuration. The layers of nodes between the input and output represent the complex, nonlinear relationship between the atomic configuration of the system and the system energy. Because the input nodes include every atom in the system, this potential is limited to small systems and is not easily transferable to other systems. These potentials have been successfully used for molecular systems and to a smaller extent for bulk metallic systems [Reference Behler73].

To address these difficulties of system size, Behler and Parrinello [Reference Behler and Parrinello76] proposed to use an atomic-level ANN (in contrast to a system-level ANN discussed above), to predict the local atomic energy (and thus is independent of system size). The atomic-level ANN takes a set of descriptors that account for all atomic interactions within a given cutoff radius and include radial and angular functions. The atomic potential energy is then predicted by evaluation of this trained atomic-level ANN for each atom in the system. The total energy of the system is obtained by summing up the evaluated atomic potential energies. Figure 3 provides a diagram for such an ANN. To cite an example, 8200 DFT simulations were used to train an ANN potential for bulk Si [Reference Behler and Parrinello76]. With regard to large systems, ANN potentials have also been used to simulate TiO₂ [Reference Artrith and Urban77], bulk Si [Reference Behler and Parrinello76, Reference Behler, Martoňák, Donadio and Parrinello78, Reference Behler, Martoňák, Donadio and Parrinello79], Cu [Reference Artrith and Behler80], C [Reference Khaliullin, Eshet, Kühne, Behler and Parrinello81, Reference Khaliullin, Eshet, Kühne, Behler and Parrinello82], Na [Reference Eshet, Khaliullin, Kühne, Behler and Parrinello83, Reference Eshet, Khaliullin, Kühne, Behler and Parrinello84], zinc oxide [Reference Artrith, Morawietz and Behler85], Cu on zinc oxide [Reference Artrith, Hiller and Behler86], and GeTe [Reference Sosso, Miceli, Caravati, Behler and Bernasconi87, Reference Sosso, Donadio, Caravati, Behler and Bernasconi88, Reference Sosso, Behler and Bernasconi89, Reference Sosso, Miceli, Caravati, Giberti, Behler and Bernasconi90]. Evaluation of ANN potentials may be slower than classical force fields, but they are orders of magnitude faster than DFT.

Figure 3:

Diagram describing the implementation of an atomic-level ANN potential to calculate the system energy in an atomic simulation. Reprinted figure with permission from Artrith and Behler, Physical Review B, 85, 045439, 2012, Copyright (2012) by the American Physical Society, https://doi.org/10.1103/PhysRevB.85.045439 [Reference Artrith and Behler80].

Vacancies

According to the second law of thermodynamics, the existence of vacancies is inevitable in an equilibrium state at finite temperatures. High configurational entropy in HEAs leads to greatly enhanced concentration of vacancies at equilibrium. Vacancy concentration in HEAs is substantially larger than those in pure metals and simple binary alloys since it increases exponentially $\left( {\sim{{\exp \left( {n - 1} \right)} \over n}} \right)$ with the number, n, of alloying elements [Reference Wang, Liu and Dou113]. This unusually high vacancy concentration changes both the thermodynamics and kinetics of HEAs, and, in particular, explains their extraordinary high radiation tolerance [Reference Wang, Liu and Dou113].

Vacancy formation energy

The presence of chemical disorder in HEAs, manifested in significant variation of the local atomic environment, makes theoretical description of vacancies and interstitials in HEAs very challenging. Definitions of defect structure and energetics are less clear for HEAs than for simple metals and their binary alloys. Since the defect structure and energetics as well as defect behavior depend on the local environment, which varies from atom to atom, a statistical approach based on sampling and averaging is required.

One of the fundamental characteristics of point defects is their formation energy. For example, vacancy formation energy (VFE) provides the essential information on the phase stability and tendency of HEA atoms to segregate to form nanosized inclusions. Since each atom in a HEA has its own specific local atomic environment, VFE exhibits a distribution, rather than a single value as in a simple metal. The distribution of VFE and its average value can be calculated either by using DFT or MD. The vacancy formation energy, E _v, is calculated by summing the energy, E _df, of a defective supercell containing a vacancy (where one of the constituent atoms is removed) with the value of the chemical potential for the removed atom, μ_a, and subtracting the energy of a perfect supercell, E _pf : E _v = E _df + μ_a − E _pf. The chemical potential μ_a is a measure of variation in the total energy of the supercell when the atom is removed. The value of the chemical potential can be obtained by using Widom-type substitution techniques [Reference Zhao, Stocks and Zhang114]. In particular, an atom A in a perfect supercell is substituted by an atom B, and the energy, E _A→B of the modified supercell is calculated. The difference between the chemical potentials of the selected atoms is calculated as: μ_A − μ_B = E _A→B − E _pf [Reference Zhao, Stocks and Zhang114], where E _pf is the energy of the perfect supercell. The difference is calculated for the all possible pairs of different atoms in the supercell. The energy of the perfect supercell can also be written as ${E_{{\rm{pf}}}} = \sum\limits_a^M {{N_a}{\mu _a}}$, where a indicates a specific type of constituent HEA atom, N _a is the number of atoms of the selected type in the supercell, and μ_a is the corresponding chemical potential. To obtain the chemical potentials of the constituent atoms in HEAs, one has to solve the combined set of the equations [Reference Zhao, Stocks and Zhang114], and average over all constituent atoms of the same type located within the supercell. In contrast, in pure metals, one can take energy of an isolated atom as its chemical potential in VFE calculation.

The vacancy formation energies in equiatomic Cr₂₅Co₂₅Fe₂₅Ni₂₅ were studied by using first-principles calculations [Reference Middleburgh, King, Lumpkin, Cortie and Edwards115, Reference Zhao, Egami, Stocks and Zhang116, Reference Chen, Ding, Feng, Liu, Liu, Lu, Li, Li, Liu and Chen117]. The obtained distributions of VFE for the specific atomic vacancies are shown in Fig. 4(a). The VFE for atoms of Cr₂₅Co₂₅Fe₂₅Ni₂₅ varies over a wide range from 0.72 to 2.89 eV for Fe, from 0.88 to 2.9 eV for Co, from 0.78 to 3.09 eV for Cr, and from 0.91 to 2.95 eV for Ni due to large lattice distortions and evident chemical disorder. The average value of VFE from least to greatest is: Fe (1.58 eV) < Cr (1.61 eV) < Co (1.7 eV) < Ni (1.89 eV). The VFEs for specific atomic vacancies are all higher than those in their pure metallic counterparts (in contrast, the formation energies of self-interstitials are lower than in pure metals [Reference Middleburgh, King, Lumpkin, Cortie and Edwards115, Reference Zhao, Egami, Stocks and Zhang116, Reference Chen, Ding, Feng, Liu, Liu, Lu, Li, Li, Liu and Chen117]). For example, the formation energy for a Ni vacancy in pure Ni (1.45 eV) is lower than in Cr₂₅Co₂₅Fe₂₅Ni₂₅ (1.89 eV). The relatively high values of VFE in HEAs are in agreement with the experiments in which Cr₂₅Co₂₅Fe₂₅Ni₂₅ exhibited an outstanding radiation tolerance in comparison with pure nickel [Reference Chen, Ding, Feng, Liu, Liu, Lu, Li, Li, Liu and Chen117]. Due to the high VFE values, void formation in the irradiated samples was significantly suppressed even at elevated temperatures [Reference Chen, Ding, Feng, Liu, Liu, Lu, Li, Li, Liu and Chen117].

Figure 4:

(a) Distribution of vacancy formation energies for Ni, Co, Cr, and Fe vacancies in Cr₂₅Co₂₅Fe₂₅Ni₂₅ HEA. Reprinted figure with permission from Zhao et al., Physical Review Materials, 2, 013602, 2018, Copyright (2018) by the American Physical Society, https://doi.org/10.1103/PhysRevMaterials.2.013602 [Reference Zhao, Egami, Stocks and Zhang116]. (b, c) A schematic depiction of a random (b) and an energetically favorable (c) packing around a vacancy in HEAs. Reprinted figure with permission from Wang et al., Physical Review Materials, 1, 043601, 2017 Copyright (2017), by the American Physical Society [Reference Wang, Liu and Dou113]. (d) Distribution of vacancy and self-interstitial migration barriers for Ni, Co, Cr, and Fe in Cr₂₅Co₂₅Fe₂₅Ni₂₅ HEA. Reprinted figure with permission from Zhao et al., Physical Review Materials, 2, 013602, 2018, Copyright (2018) by the American Physical Society, https://doi.org/10.1103/PhysRevMaterials.2.013602 [Reference Zhao, Egami, Stocks and Zhang116]. (e) Self-interstitial in Ni_33.3Co_33.3Cr_33.3 HEA: a [010] Ni–Cr dumbbell viewed from (100) plane. Reprinted figure with permission from Zhao et al., Physical Review Materials, 2, 013602, 2018, Copyright (2018) by the American Physical Society, https://doi.org/10.1103/PhysRevMaterials.2.013602 [Reference Zhao, Egami, Stocks and Zhang116]. (f) Formation energies of interstitial dumbbells in Cr₂₅Co₂₅Fe₂₅Ni₂₅ HEA. Reprinted figure with permission from Zhao et al., Physical Review Materials, 2, 013602, 2018, Copyright (2018) by the American Physical Society, https://doi.org/10.1103/PhysRevMaterials.2.013602 [Reference Zhao, Egami, Stocks and Zhang116].

Structure and energetics

Besides vacancies, another important type of point defect in HEAs, namely self-interstitials, has been recently studied by first-principles technique. In comparison with vacancies, the formation energy of self-interstitials is relatively high; hence, self-interstitials are likely to emerge only under specific nonequilibrium conditions, for example under intensive radiation.

The fundamental characteristic of self-interstitials is their formation energy, E _i, which is calculated in a similar way to vacancy formation energy, by summing the energy, E _df, of a defective supercell containing a self-interstitial (where the alloying atom is inserted in a perfect HEA supercell) and subtracting from this the value of chemical potential for the inserted atom, μ_a, and the energy of the perfect supercell, E _pf,: E _i = E _df − μ_a − E _pf. The value of μ_a can be obtained using the same Widom-type substitution techniques as described previously for vacancies [Reference Zhao, Stocks and Zhang114, Reference Zhao, Egami, Stocks and Zhang116].

Using DFT, Middleburgh et al. [Reference Middleburgh, King, Lumpkin, Cortie and Edwards115] and Zhao et al. [Reference Zhao, Egami, Stocks and Zhang116] calculated the distribution of formation energy values for various self-interstitials in equiatomic four-element Co₂₅Cr₂₅Fe₂₅Ni₂₅. They found that Co self-interstitials possess the lowest average formation energy. The resulting order in the calculated average energies for self-interstitial formation was found to be: Co (−2.63 eV) < Fe (−2.38 eV) < Ni (−2.18 eV) < Cr (−0.91 eV) [Reference Middleburgh, King, Lumpkin, Cortie and Edwards115]. As in pure metals with a FCC lattice, a split interstitial dumbbell is formed in the Co₂₅Cr₂₅Fe₂₅Ni₂₅ HEA when an extra atom is embedded in its FCC lattice. For example, a [010] Ni–Cr split interstitial dumbbell obtained by Zhao et al. [Reference Zhao, Egami, Stocks and Zhang116] is shown in Fig. 4(e). In contrast to the vacancy formation energies, which are higher in HEAs than in corresponding pure metals, the formation energies of self-interstitials are lower. As can be seen in Fig. 4(f), the formation energies of the split interstitial dumbbells containing Ni atom in the Co₂₅Cr₂₅Fe₂₅Ni₂₅ HEA are smaller than in pure Ni (3.96 eV), while the formation energies of Ni vacancy are higher in HEAs than in pure Ni (1.45 eV) [see Fig. 4(a)].

Diffusion

In addition to the formation energy of self-interstitials, first-principles calculations can be used to obtain the distribution of migration energy barriers for self-interstitials hosted in HEAs. According to DFT calculations [Reference Zhao, Egami, Stocks and Zhang116], the average values for the interstitial migration energy barriers in HEAs are higher than in pure metals of the alloy’s constituent elements. In particular, the migration barriers in Co₂₅Cr₂₅Fe₂₅Ni₂₅ are higher than in pure Ni (0.15 eV) [Reference Zhao, Egami, Stocks and Zhang116]; therefore, diffusion of self-interstitials is notably slowed down relative to pure Ni. In addition, the distributions of values for migration energy barriers of different self-interstitials in Co₂₅Cr₂₅Fe₂₅Ni₂₅ considerably overlap [see Fig. 4(e)]; hence, the diffusion rates for different self-interstitials are rather similar. Besides split interstitial dumbbells, a larger cluster of self-interstitials can be formed in HEAs under irradiation. Lu et al. [Reference Lu, Niu, Chen, Jin, Yang, Xiu, Zhang, Gao, Bei, Shi, He, Robertson, Weber and Wang125] used MD simulations to study the magnitude of migration energy barriers for clusters of self-interstitials in Co₂₅Cr₂₅Fe₂₅Ni₂₅ and Ni₂₀Co₂₀Fe₂₀Cr₂₀Mn₂₀ alloys. They found that the average magnitude of migration energy barriers increases directly with the interstitial cluster size in HEAs, while it is almost independent of the cluster size in pure Ni.

Atomistic simulations of self-interstitial diffusion in Co₂₅Cr₂₅Fe₂₅Ni₂₅ and Ni₂₀Co₂₀Fe₂₀Cr₂₀Mn₂₀ revealed that self-interstitials and their small clusters migrate randomly in all possible directions in HEAs, while in pure Ni (and NiCo binary alloy) they exhibit relatively fast quasi-one-dimensional diffusion along the close-packed rows of atoms [Reference Lu, Niu, Chen, Jin, Yang, Xiu, Zhang, Gao, Bei, Shi, He, Robertson, Weber and Wang125]. The directional migration in HEAs is prevented by the lattice distortion. The lattice distortion is related to the differences in the atomic size of the alloying elements: the atomic radius of Co deviates from Ni only by +1.8%, while Cr by +10.3%, Fe by +10.6%, and Mn by sizable +23.2% [Reference Lu, Niu, Chen, Jin, Yang, Xiu, Zhang, Gao, Bei, Shi, He, Robertson, Weber and Wang125]. The lattice distortion enhances atomic scattering, which randomizes the focused directional motion of self-interstitials along the preferable directions in HEAs, and reduces their mobility. In contrast to vacancies, most of the self-interstitials were immobile and remained in defective regions created during irradiation [Reference Lu, Niu, Chen, Jin, Yang, Xiu, Zhang, Gao, Bei, Shi, He, Robertson, Weber and Wang125]. The suppressed migration of self-interstitials in HEAs prevents them from leaving defective regions; hence, they recombine at a high rate with relatively mobile vacancies. The combined effect of the lattice distortion, sluggish vacancy diffusion, and relative immobility of self-interstitials is mainly responsible for outstanding radiation tolerance of HEAs.

Recently, a novel HEA containing carbon interstitials has been experimentally developed by Li et al. [Reference Li, Tasan, Springer, Gault and Raabe126] relying on first-principles calculations. The addition of carbon interstitials into the Mn₃₀Fe_49.5Co₁₀Cr₁₀C₅ alloy increased its stacking fault energy and therefore its phase stability. The newly designed HEA benefits profoundly from additional interstitial-induced strengthening, which is linked to much higher lattice distortion (caused by the carbon interstitials) and the interaction between extrinsic defects and dislocations. The new Mn₃₀Fe_49.5Co₁₀Cr₁₀C₅ unified all known metallurgical strengthening effects in single bulk material [Reference Li, Tasan, Springer, Gault and Raabe126].

Dislocations

Post-deformation experimental investigations have indicated that plasticity in HEAs is chiefly dislocation dominated, but there are currently no reported in situ observations of dislocation behavior in HEAs. Transmission electron microscopy (TEM) studies also have not yet provided atomistic-scale details of dislocation structures in HEAs. Theories of strengthening in HEAs [Reference Wu, Bei, Pharr and George10, Reference Varvenne, Luque and Curtin25, Reference Toda-Caraballo and Rivera-Díaz-del-Castillo127] are built upon the heterogeneous environment in HEAs. Atomistic simulations include techniques by which we can provide atomistic-level details of dislocation structures and their dynamic behavior and also allow us to investigate the physical mechanisms underlying these proposed strengthening theories. Here, we provide an overview of those published atomistic studies on dislocations in HEAs.

Dislocation structure

Several experimental reports indicate the presence of planar, dissociated dislocations in deformed FCC HEAs [Reference Laplanche, Kostka, Reinhart, Hunfeld, Eggeler and George11, Reference Otto, Dlouhý, Somsen, Bei, Eggeler and George18, Reference Smith, Hooshmand, Esser, Otto, McComb, George, Ghazisaeidi and Mills19, Reference Okamoto, Fujimoto, Kambara, Kawamura, Chen, Matsunoshita, Tanaka, Inui and George20, Reference Kireeva, Chumlyakov, Pobedennaya and Vyrodova21]. In these reports, a commonly noted feature is the high variability of the dislocation dissociation distance. To study the dislocation dissociation distance and core structure in a HEA, a relatively larger number of atoms have to be included in a computational supercell. Therefore, MD method is the most suitable. Atomistic simulations of dislocations in HEAs show similar results of stable equilibrium configuration and provide insight into the atomistic nature of the variability in separation distance and core characteristics.

The core structure of edge and screw dislocations in the four-element Co₃₀Fe_16.67Ni_36.67Ti_16.67 FCC alloy was examined by Rao et al. [Reference Rao, Woodward, Parthasarathy and Senkov65] using atomistic simulations. The obtained equilibrium configuration of an edge dislocation core is illustrated in Fig. 5(a). The most striking feature of the core structure is its irregular shape. As can be seen in Fig. 5(a), the distance between the two Shockley partials varies significantly along the dislocation line (in pure metals, the distance between Shockley partials is constant). The large variation in the separation distance (with an amplitude of ∼0.3 nm) is due to chemical disorder and lattice distortion in the HEA [Reference Rao, Woodward, Parthasarathy and Senkov65]. The average separation distance between Shockley partials in the Co₃₀Fe_16.67Ni_36.67Ti_16.67 is ∼8 nm for the edge dislocations, and ∼4.5 nm for the screw dislocations.

Figure 5:

(a) The relaxed core of 1/2 [Reference Shapeev110] edge dislocation in FCC HEA Co₃₀Fe_16.67Ni_36.67Ti_16.67. The substantial fluctuations in Shockley partial splitting distance along the dislocation are shown. Reprinted from Rao et al. Atomistic simulations of dislocation behavior in a model FCC multicomponent concentrated solid solution alloy, Acta Materialia, 134, 188–194, Copyright (2017), with permission from Elsevier [Reference Rao, Woodward, Parthasarathy and Senkov65]. (b) Mixed, edge, and screw dislocations in a Cantor equiatomic FCC HEA (Cr₂₀Mn₂₀Fe₂₀Co₂₀Ni₂₀) observed at cryogenic temperatures (T = 77 K). Reprinted from [Reference Okamoto, Fujimoto, Kambara, Kawamura, Chen, Matsunoshita, Tanaka, Inui and George20] under Creative Commons Attribution 4.0 International License. (c) A variety of dislocation types in Cr₂₀Mn₂₀Fe₂₀Co₂₀Ni₂₀ FCC HEA. Reprinted from Wang et al. Investigation into nanoscratching mechanical response of AlCrCuFeNi high-entropy alloys using atomic simulations, Applied Surface Science, 416, 470–481, Copyright (2017), with permission from Elsevier [Reference Wang, Li, Fang, Liu and Zhang129]. (d) The core of 1/2 [Reference Podryabinkin and Shapeev111] screw dislocation in BCC HEA Co_16.67Fe_36.67 Ni_16.67Ti₃₀. The screw dislocation core spreads on several (110) planes. Reprinted from Rao et al. Atomistic simulations of dislocations in a model BCC multicomponent concentrated solid solution alloy, Acta Materialia, 125, 311–320, Copyright (2017), with permission from Elsevier [Reference Rao, Varvenne, Woodward, Parthasarathy, Miracle, Senkov and Curtin66].

To compare against experimental TEM analysis of mixed dissociated dislocations in the Cr₂₀Mn₂₀Fe₂₀Co₂₀Ni₂₀ alloy, atomistic simulations on mixed dislocations in Ni, NiFe2% and NiFe20%Cr26% were performed [Reference Smith, Hooshmand, Esser, Otto, McComb, George, Ghazisaeidi and Mills19]. It was reported that fluctuations in the dislocation line and dissociation distance increased with the number of alloying elements, while the local stacking fault energy decreased. Atomistic calculations indicated that the variable separation distance of dissociated dislocations was correlated with the local stacking fault energy. The variable separation distance agreed well with TEM observations [Reference Laplanche, Kostka, Horst, Eggeler and George14, Reference Otto, Dlouhý, Somsen, Bei, Eggeler and George18, Reference Smith, Hooshmand, Esser, Otto, McComb, George, Ghazisaeidi and Mills19, Reference Okamoto, Fujimoto, Kambara, Kawamura, Chen, Matsunoshita, Tanaka, Inui and George20, Reference Kireeva, Chumlyakov, Pobedennaya, Kuksgauzen, Karaman and Sehitoglu128] although the atomistic level fluctuations would be difficult to detect using diffraction contrast imaging. A high variability in the distance between partials was reported for mixed dislocations in the Cantor FCC HEA, which dissociated into Shockley partials [see Fig. 5(b)], whose separation distance varies within 5–8 nm range [Reference Okamoto, Fujimoto, Kambara, Kawamura, Chen, Matsunoshita, Tanaka, Inui and George20]. The observed dislocations are smoothly curved without any specific line orientation, which indicates an isotropy in the mobility of their edge and screw segments [see Fig. 5(b)].

The core structure of dislocations in FCC HEAs is planar since all the core atoms are located within a single (111) plane [Reference Rao, Woodward, Parthasarathy and Senkov65]. The fraction of perfect dislocations in FCC HEAs is relatively low in comparison with simple metals with a FCC lattice. Recent MD simulations have revealed a large variety of dislocation types [Shockley, Hirth, Frank, stair rods, and others as shown in Fig. 5(c)] in FCC HEAs [Reference Wang, Li, Fang, Liu and Zhang129].

Simulations on 1/2 [Reference Podryabinkin and Shapeev111] screw dislocations in the Co_16.67Fe_36.67 Ni_16.67Ti₃₀ BCC HEA showed that the dislocation core is nonplanar and spreads on several (110) planes [see Fig. 5(d)], similar to pure BCC metals [Reference Rao, Varvenne, Woodward, Parthasarathy, Miracle, Senkov and Curtin66]. Although the shape of the BCC screw dislocation is visibly irregular, it lacks the significant waviness along the length of the dislocation as observed in the FCC screw dislocations. The edge dislocation in the BCC HEA displays a planar core for the majority of the dislocation line although it has been reported that spreading on successive (110) planes does occur along some segments of the BCC HEA edge dislocation [Reference Rao, Varvenne, Woodward, Parthasarathy, Miracle, Senkov and Curtin66]. Further, the edge dislocation in the BCC HEA shows waviness similar to the observed dislocations in FCC HEAs.

Dislocation shearing simulations

There are still open questions as to the role that chemical and structural heterogeneity plays in the observed strengthening of HEAs. MD simulations of dynamic shearing of isolated dislocations can be used to examine strengthening in HEAs, reveal how dislocations interact with the HEA lattice and local heterogeneity, and inform proposed theories of strengthening in HEAs. MD and molecular statics (MS) simulations have been used to study the critical shear stress of isolated dislocations in FCC and BCC HEAs [Reference Varvenne, Leyson, Ghazisaeidi and Curtin24, Reference Varvenne, Luque and Curtin25, Reference Choi, Jo, Sohn, Lee and Lee59, Reference Rao, Woodward, Parthasarathy and Senkov65, Reference Rao, Varvenne, Woodward, Parthasarathy, Miracle, Senkov and Curtin66]. Common results among these studies are that dislocations in HEAs have critical stresses that are significantly greater than their pure element counterparts, and this strengthening correlates with the atomic-scale fluctuations in the dislocation line.

MD has been used to perform shear simulations on isolated edge and screw dislocations in a FCC Co₃₀Fe_16.67Ni_36.67Ti_16.67 HEA [Reference Rao, Woodward, Parthasarathy and Senkov65]. Based on the beginning of dislocation motion at 0 K, the reported Peierls stress for screw dislocations is 0.0025–0.00375μ, which is at least an order of magnitude greater than that in pure FCC metals. Similarly for edge dislocations, the reported Peierls stress is 0.00375–0.005μ, which is at least one to two orders of magnitude higher than that in pure FCC metals. The dislocations are observed to maintain their wavy morphology, and glide occurred by continuous expansion and contraction of local stacking fault splitting widths. The authors note that some portions of the dislocation can move a significant amount before attaining the critical stress. By increasing the temperature from 5 to 300 K, the average critical stress for motion of a screw dislocation decreased from 0.0028μ to 0.0015μ and the average critical stress for motion of an edge dislocation decreased from 0.0046μ to 0.0028μ.

Similar MD shearing simulations have been performed on isolated edge and screw dislocations in BCC Co_16.67Fe_36.67Ni_16.67Ti₃₀ [Reference Rao, Varvenne, Woodward, Parthasarathy, Miracle, Senkov and Curtin66]. The reported Peierls stress for a screw dislocation in Co_16.67Fe_36.67Ni_16.67Ti₃₀ was between 0.035μ to 0.04μ at 0 K. As for screw dislocations in pure BCC metals, the high Peierls stress is attributed to the nonplanar core. At 0 K, the Peierls stress of the edge dislocation in the BCC HEA is 0.025μ, several orders of magnitude higher than edge dislocations in a typical BCC metal. With increasing temperature from 0 to 300 K, the critical resolved shear stress of a screw dislocation decreased from 0.035–0.04μ to 0.02–0.027μ while the average critical resolved shear stress of an edge dislocation decreased from 0.015μ to 0.01μ.

It was reported that at a subcritical resolved shear stress, there are several kink segments in the HEA BCC screw dislocation that lie on (110) and (112) planes [Reference Rao, Varvenne, Woodward, Parthasarathy, Miracle, Senkov and Curtin66]. Collision of these kinks resulted in immobile jogs, which serve as barriers to slip and also leave behind debris as the dislocation glides. This is in contrast to typical BCC screw dislocation motion that proceeds via double-kink nucleation and kink propagation. Using a standard BCC kink-pair model, the authors estimated that the BCC HEA has a kink-pair activation energy between 7.25 and 9.89 eV. It is interesting to note that the high critical resolved shear stresses (CRSS) of the screw and edge dislocations in this BCC HEA gives a ratio of CRSS significantly lower than that in pure BCC metals. This is important because the anisotropy of dislocation strength and mobility plays the dominant role in dislocation evolution in BCC metals. Experimental observations of BCC HEAs subjected to uniaxial strain indicate that screw dislocations mainly control plastic deformation [Reference Couzinié, Lilensten, Champion, Dirras, Perrière and Guillot130, Reference Lilensten, Couzinié, Perrière, Hocini, Keller, Dirras and Guillot131, Reference Feuerbacher, Heidelmann and Thomas132]. Screw dislocations in various BCC HEAs appear as straight lines, and their mobility is relatively low compared to pure BCC metals and their binary alloys. They form localized bands of high dislocation density separated by dislocation-free zones. Furthermore, the presence of dislocation debris (loops, dipoles) suggests strong pinning effects in BCC HEAs [Reference Couzinié, Lilensten, Champion, Dirras, Perrière and Guillot130, Reference Lilensten, Couzinié, Perrière, Hocini, Keller, Dirras and Guillot131, Reference Feuerbacher, Heidelmann and Thomas132]. Therefore, a high resolved shear stress is required to move them [Reference Feuerbacher, Heidelmann and Thomas132].

To identify the role that chemical and structural heterogeneity plays in these reported simulations, a methodology to create an analogous pure, “average” interatomic potential from a more complex potential has been proposed [Reference Varvenne, Luque, Nöhring and Curtin133]. Such a potential reproduces bulk properties such as elastic constants, stacking fault energies, and lattice constant of the HEA, but contains a single atomic species. Comparison between this “average” alloy and its HEA counterpart has shown that waviness observed in the dislocation line in the true random alloy arises due to the atomic heterogeneity within the HEA [Reference Varvenne, Luque, Nöhring and Curtin133]. Dislocations in the “average” alloy remain atomically straight. Furthermore, an “average” alloy that shows a stable, dissociated dislocation with constant dissociation distance of 15 Å, while the dissociated dislocation in the true random alloy shows a variable dissociation distance along the dislocation line [Reference Rao, Woodward, Parthasarathy and Senkov65]. An average potential created to compare against the FCC Co₃₀Fe_16.67Ni_36.67Ti_16.67 HEA showed that the critical stress for both screw and edge dislocations in the average alloy is less than <10 MPa, which is similar to simple FCC metals [Reference Rao, Woodward, Parthasarathy and Senkov65]. Correspondingly, the critical stress for a screw dislocation in the average BCC analogue of the BCC Co_16.67Fe_36.67Ni_16.67Ti₃₀ HEA displays shear stresses between 0.02μ to 0.025μ, nearly a factor of 2 less than in the random alloy. The results from atomistic simulations suggest that in both FCC and BCC HEAs, compositional fluctuations play an important role in the strengthening of HEAs.

An extension of the Labusch model [Reference Okamoto, Fujimoto, Kambara, Kawamura, Chen, Matsunoshita, Tanaka, Inui and George20, Reference Toda-Caraballo and Rivera-Díaz-del-Castillo127] for solid solution strengthening in a random alloy has been proposed to explain strengthening in FCC HEAs [Reference Varvenne, Luque and Curtin25]. To compare against the proposed theory, molecular statics simulations were performed on dislocations in a FeNiCr alloy [Reference Varvenne, Leyson, Ghazisaeidi and Curtin24, Reference Varvenne, Luque and Curtin25]. The MS simulations show a good agreement with theoretically predicted shear strength as a function of dislocation glide distance, and match well with the theoretically predicted shear strength at 0 K as a function of Cr content. By comparing to the average alloy [Reference Varvenne, Luque, Nöhring and Curtin133], which displays Peierls stresses similar to single-element FCC metals, it was concluded that strengthening in the HEAs arises from the structural heterogeneity and chemical disordering.

The effect of alloying element has also been demonstrated through atomistic simulations of edge dislocations in equiatomic FCC Co₂₅Cr₂₅Fe₂₅Ni₂₅, Co₂₅Cr₂₅Mn₂₅Ni₂₅, Co₂₀Cr₂₀Fe₂₀Mn₂₀Ni₂₀, Fe₅₀Ni₅₀, Co₂₅Fe₂₅Mn₂₅Ni₂₅, and Co₅₀Ni₅₀ [Reference Choi, Jo, Sohn, Lee and Lee59]. By interpolating between the strengthening effects of different alloying elements, they derived an empirical equation to predict the CRSS and used this empirical equation to design a nonequiatomic HEAs with yield strengths greater than the equiatomic Co₂₀Cr₂₀Fe₂₀Mn₂₀Ni₂₀.

MD simulations of defect growth in irradiated alloys observed that HEAs display a reduction in defect sizes as compared to their pure elemental and binary alloy analogues [Reference Granberg, Nordlund, Ullah, Jin, Lu, Bei, Wang, Djurabekova, Weber and Zhang63]. This reduction in defect sizes was attributed to the mobility of dislocations and their ability to combine. Dynamic dislocation shearing simulations revealed that the mobility of dislocations in the HEA decreased with increasing the number of constituent elements in a Ni-based FCC alloy. Mobility of edge dislocations in Ni-based ternary alloy (Ni_33.3Co_33.3Cr_33.3) is less than that in binary (Ni₅₀Fe₅₀), which is lower than in pure Ni, but the lowest dislocation mobility was found in equiatomic four-component Mn₂₅Ni₂₅Co₂₅Cr₂₅ FCC HEA [Reference Granberg, Nordlund, Ullah, Jin, Lu, Bei, Wang, Djurabekova, Weber and Zhang63].

Dislocation mechanisms

There have not been any systematic studies on dislocation mechanisms such as cross-slip, dislocation–planar defect interaction, dislocation reactions, etc., but in context of the reported low SFE in HEAs, it is expected that these mechanisms may depend on or be influenced by SFE.

It has been noted that cross-slip can occur spontaneously in dislocation shearing simulations [Reference Rao, Woodward, Parthasarathy and Senkov65]. Based on the simulation at a temperature of 300 K, it was estimated that the cross-slip activation energy is on the order of 0.3 eV. Considering that the predicted cross-slip energy for the “average” alloy was 3.8–4.3 eV, this suggested that local chemical fluctuations in the HEA can result in an effectively low cross-slip energy for certain segments of the dislocation. Cross-slip in FCC solid solutions has been studied through atomistic simulations and has shown that the cross-slip energy can be significantly decreased in a random alloy [Reference Nöhring and Curtin134].

Nanopillar simulations

The studies discussed above focus on existing dislocations in an isolated environment. Atomistic simulations can also be used to explore such phenomena as dislocation nucleation and dislocation interaction. Nanopillar experiments provide a technique to study the mechanical properties of small volumes and isolated microstructural features at nanoscale [Reference Greer and De Hosson135]. They also provide a system that comes closest to bridging the length-scale gap between MD simulations and experiments.

Atomistic simulations of quasistatic tension of a single-crystalline FCC CoCrFeMnNi nanopillar have been performed to investigate dominant deformation mechanisms in a representative FCC HEA [Reference Choi, Jo, Sohn, Lee and Lee59]. Tension was applied along the high-symmetry 〈110〉 direction of a nanopillar with cross-section dimensions of ∼5000 Ų and a length of ∼300 Å. It was noted that dislocations formed most easily in the FCC HEA rather than in the binary or unary alloy, and this was attributed to the “severe lattice distortion” of the HEA.

Small-scale compression experiments allow characterization of the mechanical properties of individual phases of a multiphase HEA. Micro-pillar compression experiments have been performed on the FCC and BCC phases of the Al_0.7CoCrFeNi HEA at room temperature [Reference Giwa, Liaw, Dahmen and Greer136].

Stacking faults

A stacking fault (SF) is a type of planar defect characterizing an irregularity in the periodic sequence of crystalline planes (typically the close-packed planes) affecting a finite region of crystal. This is seen as insertion of a plane into the stacking sequence (creating an extrinsic stacking fault) or removal of a plane from the stacking sequence (creating an intrinsic stacking fault). Stacking faults are also closely related to twinning as successively faulted planes can result in a twinned microstructure. The main characteristic of SF is the stacking fault energy (SFE), γ_sf, calculated according to:

(1)$${\gamma _{{\rm{sf}}}} = {{{E_{{\rm{SF}}}} - {E_{{\rm{PF}}}}} \over A}\quad ,$$

where E _SF is the energy of the lattice structure with a stacking fault, E _PF is the energy of the perfect lattice, and A is the cross-sectional area of the stacking fault, i.e., the interface area over which the stacking fault extends in the specific plane [(111) for FCC lattice, (112) for BCC]. The main challenges related to the calculation of SFE in HEAs by ab initio or atomistic simulations are the following: first, a relatively large supercell is required due to the inherent multielement character and accompanying chemical randomness in HEAs; second, a relatively high accuracy in SFE calculation is required since even small variation of the order a few ∼1 mJ/m² in SFE may affect the dominant mechanism of plastic deformation, especially at low temperatures [Reference Li, Körmann, Grabowski, Neugebauer and Raabe137]; third, the value of SFE can strongly depend on temperature [Reference Li, Körmann, Grabowski, Neugebauer and Raabe137]; and finally, contributions associated with lattice vibrations [Reference Zhao, Stocks and Zhang138] and magnetic moments of constituent atoms [Reference Huang, Li, Lu, Tian, Shen, Holmström and Vitos139] must be properly taken into account.

A stacking fault structure in four-component equiatomic FCC HEA (Ni₂₅Co₂₅Cr₂₅Fe₂₅) is illustrated in Fig. 6(a). One of the most intriguing characteristics of stacking faults in HEAs is its relatively low value, especially at cryogenic temperatures. DFT calculations revealed that the values of SFE drop with an increase in the number of alloying atoms in equiatomic FCC HEAs [Reference Beyramali Kivy and Asle Zaeem140]. Moreover, in nonequiatomic HEAs, SFE can be further reduced: for example, Co₂₀Cr₂₆Fe₂₀Mn₂₀Ni₁₄ shows a lower SFE than its equiatomic counterpart Co₂₀Cr₂₀Fe₂₀Mn₂₀Ni₂₀ [Reference Niu, Zaddach, Koch and Irving46]. Introduction of interstitial atoms (especially carbon and nitrogen atoms) further reduces the value of SFE [Reference Li, Tasan, Springer, Gault and Raabe126]. The SFE is also sensitive to temperature [Reference Huang, Li, Lu, Tian, Shen, Holmström and Vitos139] and element-specific segregation around SF [Reference Abbasi, Dick, Hickel and Neugebauer141, Reference Medvedeva, Park, Van Aken and Medvedeva142].

Figure 6:

(a) Staking fault configuration in four-element equiatomic Co₂₅Cr₂₅Fe₂₅Ni₂₅ FCC HEA. Reprinted from Huang et al. Temperature-dependent stacking fault energy of FeCrCoNiMn HEA, Scripta Materialia, 108, 44–47, Copyright (2015), with permission from Elsevier [Reference Zhang, Zhuang, Hu, Kai and Liu143]. (b) Stacking fault energy in quinary equiatomic Co₂₀Cr₂₀Fe₂₀Mn₂₀Ni₂₀ FCC HEA as a function of temperature (left panel). The chemical, magnetic, and strain energy contributions to the stacking fault energy of Co₂₀Cr₂₀Fe₂₀Mn₂₀Ni₂₀ as a function of temperature (right panel). Reprinted with permission from [Reference Huang, Li, Lu, Tian, Shen, Holmström and Vitos139].

According to first-principles calculations, the SFE value of equiatomic Cantor HEA is γ_sf = 3.4 mJ/m² at low temperatures [Reference Huang, Li, Lu, Tian, Shen, Holmström and Vitos139], and it is even negative at zero temperature [Reference Zhang, Zhuang, Hu, Kai and Liu143]. In FCC HEAs, the stacking fault can be considered as a local HCP stacking sequence embedded in an FCC lattice. The SFE value is related to a free energy difference between FCC and HCP lattice structures. The negative SFE value at T = 0 K indicates the instability of the FCC lattice, which is less energetically favorable than the HCP one. The low SFE value of the Cantor HEA is due to the significant contribution of lattice distortion at low temperatures [Reference Bhattacharjee, Sathiaraj, Zaid, Gatti, Lee, Tsai and Yeh144], and the presence of specific atoms near the stacking fault (e.g., Co atoms located near a SF reduces its SFE value by ∼55% compared to when they are located away from it [Reference Patriarca, Ojha, Sehitoglu and Chumlyakov145]). The value of the SFE gradually rises with an increase in temperature and reaches γ_sf = 21 mJ/m², (which is comparable to SFE in stainless steels [Reference Whelan, Hirsch, Horne and Bollmann146]), at room temperature. The slope of SFE as a function of temperature slightly decreases with temperature showing a tendency to even out at high temperatures [see the left panel of Fig. 6(b)].

The contribution of chemical, magnetic, and strain effects to the SFE value of the Cantor HEA was studied by Huang et al. [Reference Huang, Li, Lu, Tian, Shen, Holmström and Vitos139] using DFT calculations. They found that the chemical, magnetic, and strain contributions to the value of SFE are comparable. The chemical contribution is calculated as the free energy difference between the faulted and perfect lattice divided by the area of the stacking fault, and it is routinely obtained by first-principles calculations and atomistic simulations. The chemical contribution to SFE increases almost linearly with temperature [see the right panel of Fig. 6(b)]. The magnetic contribution can only be obtained by first-principles calculations since it includes the magnetic entropy of a FCC HEA containing a stacking fault. The magnetic contribution is at the same order of magnitude as the chemical one, and it rapidly increases at low temperatures, while slightly decreasing at high temperatures [see the right panel of Fig. 6(b)]. The strain effect contribution is related to the structural relaxation around a stacking fault. The strain effect contribution is relatively small in comparison with the chemical and magnetic ones [see the right panel of Fig. 6(b)], and it slightly decreases with increasing temperature [Reference Huang, Li, Lu, Tian, Shen, Holmström and Vitos139].

Another intriguing characteristic of SF in HEAs is that the SFE value varies across the HEA samples due to chemical disorder and local lattice distortion associated with it. It has been proposed that the value of SFE varies throughout the HEA [Reference Smith, Hooshmand, Esser, Otto, McComb, George, Ghazisaeidi and Mills19]. To show the local variation in SFE in atomistic simulations, the cross-sectional area of SF was partitioned into columns [Reference Smith, Hooshmand, Esser, Otto, McComb, George, Ghazisaeidi and Mills19]. For each of the columns, the excess energy was calculated and divided by its area. Based on this measure, it was demonstrated that the stacking fault energy in these alloys can have a standard deviation that is almost ∼158% of the mean value. This clearly indicates heterogeneity in the local planar SFE within the HEA and explains why dissociated dislocations show a wavy structure along their length. DFT calculations also revealed that the SFE can display a standard deviation of 75.5% of the mean value [Reference Liu and Wei147].

Twinning

Deformation twinning has been proposed as the deformation mechanism underlying the increasing strain hardening and fracture toughness of the Cantor HEA at cryogenic temperatures [Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie13, Reference Laplanche, Kostka, Horst, Eggeler and George14]. Deformation twins are assumed to act as barriers to dislocation slip leading to dynamic strain hardening. TEM of deformed samples suggests that the primary deformation mechanisms in BCC HEAs are both dislocations and mechanical twinning [Reference Senkov, Scott, Senkova, Miracle and Woodward148]. Twinning has also been suggested to contribute to increased ductility in BCC refractory equiatomic TiNbMoTaW HEA [Reference Han, Chen, Zhao, Fan, Yang, Shao and Yao149] and an equiatomic TaNbHfZrTi HEA [Reference Senkov, Scott, Senkova, Meisenkothen, Miracle and Woodward150]. Twinning in FCC HEAs is assisted by the formation of successive neighboring faulted planes and hence dependent on the value of SFE. For the FCC HEA, the stacking fault plane is (111) and the displacement direction is $\left\langle {11\bar{2}} \right\rangle$ [Reference Choi, Jo, Sohn, Lee and Lee59]. It has been observed experimentally that the number of formed twin boundaries decreases with increasing the number of the constituent alloying elements in HEAs [Reference Wu, Song, Ni, Wang, Liu, Liu and Liao151] since the high-entropy effect reduced the twin boundary energy.

Atomistic simulations of quasistatic tension of a single-crystalline equiatomic FCC CoCrFeMnNi nanopillar reported the formation of twins along the planes of extended partial dislocations [Reference Choi, Jo, Sohn, Lee and Lee59]. Formation and growth of deformation twins in these simulations is shown in Fig. 7(a). Twinning is experimentally observed in a CoCrFeMnNi FCC HEA during tensile deformation [Reference Otto, Dlouhý, Somsen, Bei, Eggeler and George18, Reference Joo, Kato, Jang, Moon, Tsai, Yeh and Kim152] typically at cryogenic temperatures [see Fig. 7(b)]. The formation of twins in this FCC HEA was attributed to the negative energy difference between HCP and FCC stacking at 0 K. This implies that the HCP crystal structure is more stable than the FCC crystal at 0 K, thus giving the HEA a low stacking fault energy. The reported low stacking fault energy in FCC HEAs and the experimental presence of deformation twins and stacking faults suggest that twinning in FCC HEAs follows typical FCC twin formation mechanisms.

Figure 7:

(a) MD simulations of tension in a FCC CoCrFeMnNi HEA showing the formation of deformation twins, reprinted from [Reference Choi, Jo, Sohn, Lee and Lee59] under Creative Commons Attribution 4.0 International License. (b) High resolution TEM images showing the existence of deformation twins in a deformed HEA, reprinted from Otto et al. The influences of temperature and microstructure on the tensile properties of a CoCrFeMnNi high-entropy alloy, Acta Materialia, 61, 5743–5755, Copyright (2013), with permission from Elsevier [Reference Otto, Dlouhý, Somsen, Bei, Eggeler and George18].

Deformation twinning in BCC HEAs has been observed experimentally [Reference Senkov, Scott, Senkova, Miracle and Woodward148, Reference Han, Chen, Zhao, Fan, Yang, Shao and Yao149, Reference Senkov, Scott, Senkova, Meisenkothen, Miracle and Woodward150], but no crystallographic information of active twinning systems has been reported.

Grain boundaries

A grain boundary (GB) is an interface between two adjacent grains with the same lattice structure in a polycrystalline material, but with two different lattice orientations. GBs in HEAs are commonly formed in the interior of a material during crystallization or processing of HEAs when there is a large misfit between crystal lattices in adjacent grains. However, in some HEAs (e.g., in equiatomic Co₂₅Cr₂₅Fe₂₅Ni₂₅ and Co₂₀Cr₂₀Fe₂₀Mn₂₀Ni₂₀ alloys [Reference Wu, Song, Ni, Wang, Liu, Liu and Liao151]), twin boundaries, formed during twinning deformation, can gradually transform into high-angle GBs [Reference Wu, Song, Ni, Wang, Liu, Liu and Liao151]. It may be one of the reasons why, unlike in simple metals and their binaries, the fraction of high-angle GBs dominates in HEAs [Reference Wu, Song, Ni, Wang, Liu, Liu and Liao151].

GBs in polycrystalline HEAs strongly affect their mechanical properties. Overall, GBs are weak spots, where GB sliding occurs or intergranular fracture initiates. However, GBs can also be the sources and sinks of dislocations, and barriers disrupting their motion. Thus, a systematic reduction in the average grain size is a proper way to strengthen HEAs according to the Hall–Petch relation [Reference Pande and Cooper153]. In experiments, HEAs with nanosized grains exhibit high strength and superior hardness [Reference Zhou, Hu, Huang and Luo154].

The most interesting special feature of GBs in HEAs is their low grain boundary energy (GBE) compared to simple metals and their binary alloys. The value of GBE in HEAs is significantly reduced by the inherent high-entropy effects [Reference Zhou, Hu, Huang and Luo154, Reference Zhou, Hu and Luo155, Reference Liu, Wu, He, Nieh and Lu156]. The remarkable thermal stability of polycrystalline HEAs, exhibiting strong resistance to grain coarsening during thermal annealing, is an apparent manifestation of the low value of GBE. This sluggish grain coarsening was experimentally observed in Cantor and other Ni-based HEAs [Reference Zhou, Hu, Huang and Luo154, Reference Zhou, Hu and Luo155, Reference Liu, Wu, He, Nieh and Lu156]. The sluggish migration of GBs in HEAs is a result of the slow cooperative diffusion of many atoms limited by the diffusion rate of the slowest ones [Reference Liu, Wu, He, Nieh and Lu156]. Ideally, grain coarsening in HEAs at elevated temperatures can be further reduced or even prevented. GB migration can be hindered by Zener pinning [Reference Novikov157] or by adsorption of specific atoms (Zr, Mo, Nb) at GBs, which further reduces GBE, thus decreasing the corresponding driving force for grain coarsening in HEAs (Ni₂₉Fe₂₃Co₂₃Cr₂₃Zr₂ and Ni₂₅Fe₂₃Co₂₃Cr₂₃Mo₂Nb₂Zr₂) [Reference Zhou, Hu, Huang and Luo154,Reference Zhou, Hu and Luo155].

In addition, GBs in HEAs also provide new pathways for fast atomistic diffusion [Reference Paul, Belova and Murch158, Reference Skouras, Burganos and Payatakes159] since they are usually characterized by an increased free volume and lower density. The density of GBs declines with an increase in the number of alloying elements in multicomponent HEAs since the high-entropy effects reduce GBE [Reference Wu, Song, Ni, Wang, Liu, Liu and Liao151]. Thus, HEAs exhibit a low GB density, low GBE, and slow GB migration [Reference Wu, Song, Ni, Wang, Liu, Liu and Liao151].

An incomplete list of the interesting questions related to GBs in HEAs, which can be studied by first-principles and atomistic simulations, includes: the structure, energetics and migration of grain boundaries, and diffusion along the grain boundaries. Triple and multiple junctions can be investigated. A great deal can be learned about the interaction of GBs with dislocations and their role in mechanical deformation, creep, thermal resistance to coarsening, strengthening, and hardening of HEAs. Evidently, the research in this promising field is still in its initial phase.

Phase boundaries

Most HEAs contain only a single phase in their as-cast state due to high cooling rates; however, at elevated temperatures during prolonged thermal annealing, new phases (BCC, σ, η, LI₁, etc.) may arise [Reference Miracle and Senkov118]. Since long-range diffusion, required for phase separation in HEAs, is sluggish, whereas the interaction between the inter-diffusing atoms is strong, only a few ultrafine crystallites of secondary phase nucleate [Reference Miracle and Senkov118]. A typical microstructure of a dual-phase HEA is a host matrix with uniformly dispersed embedded inclusions (nanocrystalline grains of secondary phase).

Early research was almost exclusively focused on single-phase HEAs since high configurational entropy of alloys makes a single phase preferable. It was also subtly assumed that single-phase HEAs possess superior mechanical properties, while the availability of multiple phases may lead to the formation of brittle intermetallic compounds and reduce the solid solution hardening effect in HEAs [Reference Li, Tasan, Pradeep and Raabe160]. However, recent experiments have revealed that the mechanical properties of single-phase HEAs are not necessarily superior [Reference Li, Tasan, Springer, Gault and Raabe126, Reference Li, Tasan, Pradeep and Raabe160, Reference Li, Pradeep, Deng, Raabe and Tasan161].

It is well known that FCC HEAs exhibit low strength and high plasticity, while BCC HEAs show high strength and low plasticity. Therefore, a dual-phase HEA containing both FCC and BCC phases may display an unprecedented combination of high tensile ductility and fracture strength. An interesting example is Al_xCoCrFeNi HEA, which transforms from a FCC phase to a mixture of FCC and BCC phases with increasing Al content (0.5 < x < 0.8) [Reference Miracle and Senkov118, Reference Basu, Ocelík and De Hosson162]. This dual-phase HEA possesses the benefits of both constituent phases: the ductility of the FCC phase along with the strength of the BCC phase. Another example includes dual-phase nonequiatomic HEAs (Co₂₀Cr₂₀Fe₃₄Mn₂₀Ni₆ [Reference Li, Körmann, Grabowski, Neugebauer and Raabe137], Fe₅₀Mn₃₀Co₁₀Cr₁₀ [Reference Li, Pradeep, Deng, Raabe and Tasan161], Fe_49.5Mn₃₀Co₁₀Cr₁₀C_0.5 [Reference Li, Tasan, Springer, Gault and Raabe126]), which are also capable of overcoming the strength–ductility trade-off. The extraordinary properties and phase stability of these HEAs had been initially predicted by DFT calculations, and then they were subsequently synthesized experimentally. These HEAs contain two constituent phases (FCC matrix and laminate HCP inclusions) and exhibit multiple strengthening mechanisms (strengthening due to multiple principal elements, strain-induced transformation from FCC to HCP phase, formation of stacking faults and dislocation slip in FCC phase, increase in grain and interphase boundary density, mechanical twinning, dislocation hardening, and formation of stacking faults in HCP phase), which are sequentially activated during mechanical loading [Reference Li, Tasan, Pradeep and Raabe160].

The presence of interphase boundaries between different compositional and crystallographic phases may serve as an additional strengthening mechanism in multiphase HEAs. As a general rule, the strain transfer between two adjacent BCC and FCC phases depends on the geometric structure of the interphase between them. Some BCC–FCC interfaces in Al_0.7CoCrFeNi HEA permit easy strain transfer, which results in abundant dislocation slip, while others act as barriers to dislocation motion, which completely blocks the strain transfer and generates sharp stress gradients between adjacent phases [Reference Basu, Ocelík and De Hosson162]. The reduction in ductility and hence the strengthening effect at the BCC–FCC phase boundary are predominantly due to misfit stresses, lattice parameter, and chemical mismatch between nearby phases. The interfacial strengthening effect in the Al_0.7CoCrFeNi HEA is at least four times larger than that of conventional BCC–FCC interfaces [Reference Basu, Ocelík and De Hosson162].

On the other hand, phase boundaries can reduce hardness of multiphase HEAs. For example, BCC–FCC interfaces in Al_0.6CoCrFeNi are incoherent interfaces with relatively low interfacial shear strengths [Reference Basu, Ocelík and De Hosson162]. The incoming gliding dislocations passing from BCC to FCC grains in HEAs frequently dissociate into in- and out-of-plane components: the in-plane ones glide freely along the interface and subsequently shear it, while the out-of-plane components contribute to vacancy climb mechanisms and gradually lead to normal displacement of the interphase boundary into the BCC grain. The in-plane segments of dislocations absorbed at the interphase boundary subsequently lead to a local drop in hardness near the phase interface. The presence of the regions around phase interfaces with lower values of hardness was experimentally confirmed [Reference Basu, Ocelík and De Hosson162]. Thus, the mechanism of interfacial shear by lattice dislocations is the key mechanism behind the weakening effect due to phase interfaces in HEAs.

Both first-principles and atomistic simulations can contribute significantly to our understanding of phase boundaries: their structure and energetics, as well as their role in deformation and failure of multiphase HEAs. It is noted that the research of interphase boundaries based on computer simulations is in its early stages.