Simulation and Calculation for Predicting Structures and Properties of High-Entropy Alloys

Abstract

High-entropy alloys (HEAs) have attracted the attention of scholars due to their outstanding properties such as excellent fracture, and irradiation resistance for various applications. However, the complex composition space hinders the exploration of new HEAs. The traditional experimental trial-and-error method has a long periodicity and is difficult to understand the complexity of the structural characteristics of HEAs. With the rise of the “Materials Genome Initiative”, simulation methods play an important role in accelerating the development of new materials and speeding up the design process of new HEAs. In this chapter, some of the multi-scale simulation methods, such as density functional theory (DFT) calculations and molecular dynamics (MD) methods, used in designing HEAs and predicting their properties are reviewed. The advantages and limitations of these methods are discussed, and the role of computational simulation methods in guiding experiments is illustrated. This study aims to promote the rapid development of computational simulation methods in HEAs.

1. Introduction

Metal materials play an essential role in aerospace, transportation, national defense equipment, and other important areas of the national economy, and the development of science and technology has put forward higher requirements for new metal materials. Traditional alloys such as aluminum alloys [1, 2] and magnesium alloys [3] are mainly based on 1 or 2 elements, and the properties are changed or optimized by adding small amounts of other elements. The traditional alloy preparation technique and its performance have become mature and stable after years of research and development, and new alloys are urgently required to alleviate the bottleneck. In 2004, high-entropy alloys (HEAs) were first proposed [4, 5], breaking away from the traditional alloy single-element-based design concept. Because of their excellent properties and wide potential for application, HEAs have gained considerable attention in recent years and have become a hot field of research in materials science. HEAs are new multi-principal metallic materials with a predominantly configurational entropy. In HEAs, there is a wide variety of primary elements, and no element dominates, so the mixing entropy value is high. According to Boltzmann hypothesis, the mixing entropy ΔS_mix of n-component alloys is

$$∆S_{\mathit{mix}}=-R\sum _i{C}_i{\mathit{Lnc}}_i$$E1

where R is gas constant, C_i is the i^th element molar fraction.

The thermophysical parameter calculation is based on the “Hume-Rothery criterion.” This rule is extended to the field of HEAs, and a variety of related parameters are proposed for predicted phase formation, which may not be applicable to all HEAs. Zhang et al. [6] summarized the factors of the atomic-size difference, δ, and the enthalpy of mixing, ΔH_mix, of the multi-component alloys:

$$\delta =\sqrt{\sum _{i=1}^N{x}_i{\left(1-x_i/\sum _{j=1}^N{x}_j{r}_j\right)}^2}$$E2

$$∆H_{\mathit{mix}}=\sum _{i\ne j}4x_i{x}_j∆H_{\mathit{ij}}$$E3

where N is the number of the elements in HEAs, x_i or x_j is the atomic percentage of the i^th or j^th component, r_j is the atomic radius of the j^th component, and ΔH_mix is the mixing enthalpy for i and j element.

Subsequently, to further understand the connection between ΔH_mix and ΔS_mix, Zhang and Yang [7] proposed a new parameter, Ω, defined by:

$$\Omega =T_m∆S_{\mathit{mix}}/\mid ∆H_{\mathit{mix}}\mid$$E4

where T_m is the melting temperature of the N-component alloy. Zhang et al. [8] summarized the published HEAs and suggested a phase-formation rule using the δ and Ω with Ω ≥ 1.1 and δ ≤ 6.6% as shown in Figure 1. The Ω criterion enables simple and convenient phase structure prediction by combining the parameters that affect HEAs: size difference, mixing enthalpy, and mixing entropy. However, the FCC-type phase-forming δ shows a significant overlap with that of the BCC-type phase, which means new rules or parameters need to be considered for the phase formation.

Figure 1.

The relationship between parameters δ and Ω for multi-component alloys [7]. Copyright © 2011 Elsevier.

Since FCC and BCC phases overlap, from the perspective of alloy design, Guo et al. [9] proposed the valence electron concentration (VEC) to determine the formation of FCC or BCC solid solution in HEAs.

$$\mathit{VEC}=\sum _{i=1}^n{c}_i{\left(\mathit{VEC}\right)}_i$$E5

where c_i and VEC_i are the atomic percentage and VEC of the i^th component.

The analysis of experimental data leads to the following conclusions: BCC structure of HEA is easier to predict than FCC structure; VEC < 6.8 will form BCC structure solid solution; if 6.8 < VEC < 7.8, FCC + BCC structure solid solution will be formed; VEC > 7.8, FCC structure solid solution will be formed. Therefore, it is the potential to separate FCC and BCC phases by the VEC criterion, but it is not probable to determine whether there is intermetallic compound formation. The above parameters that emerged during the development of HEAs are important conclusions for researchers in their quest to accelerate alloy development. The prediction of thermodynamic parameters improves the research efficiency and gives strong theoretical headings for the experiment, thus reducing waste. Due to the huge composition space of HEAs, these parameter judgments cannot satisfy every possible composition, and therefore, researchers are eager to have dependable databases and high-performance calculations in order to increase the efficiency of alloy design.

It has been particularly notable that the Materials Genome Initiative (MGI) was announced in 2011 to accelerate the pace of materials discovery, design, and implementation through the integration of experimentation, theory, and computation in a highly integrated, high-throughput manner [10]. By integrating both computational and experimental data, as well as high-throughput computations and multi-scale simulations, this project aims to change the research and design culture of materials and advance material development methods and approaches [11]. MGI project has contributed to the development of HEAs. Even though predictive computational modeling of HEAs is challenging primarily due to the complex multi-component system and disordered solid-solution structure, HEAs are challenging systems to model. Although computational modeling of HEAs is becoming increasingly popular as a tool for studying the structure (including defects, dislocation), thermodynamics, kinetics, and mechanical properties [12]. In the material simulation, we can simulate the material from various scales, and qualitatively as well as quantitatively describe the characteristics of the material and promote our understanding of it from multiple perspectives. For materials with different scale-space, there are corresponding material calculation methods, including the first-principles density functional theory (DFT), molecular dynamics (MD), the calculation of phase diagram (CALPHAD), and high-throughput methods [13, 14, 15, 16, 17, 18, 19, 20]. Hence, from the microscopic to the macroscopic scale, this chapter reviews the limitations and potentials of different simulation methods by summarizing in a targeted manner the characteristics and application areas of different simulation methods. It also looks at database-driven machine learning, as well as the use of multi-scale simulation methods in the future to aid in the design, development, and performance tuning of new HEAs.

2. DFT calculations

In comparison with other computational techniques, first-principles calculations are an effective method for predicting the physical and structural properties of materials. It is calculated only by parameters such as the number of atoms inherent to the material. The essence is to obtain the various properties of the material by solving the Schrӧdinger equation. However, the state of motion of an electron corresponds to a Schrӧdinger equation, which can be solved for simple single-electron systems and is hard to solve for complex multi-electron systems. Kohn and Sham [21] considered that the particle density function of a multi-particle system can be achieved by a simple single-particle wave equation, and the Kohn-Sham equation [22] is self-consistent. Scientists usually use scientific approximations to simplify the Schrödinger equation to reach an exact solution. One of the most widely used first-principles calculations based on DFT [23]. The DFT calculation process converts the multi-electron problem into a single-electron problem by describing the physical properties of the electron density of states. DFT calculations usually include only fundamental physical constants such as speed of light, Planck’s constant, electron, and charge mass as input parameters [24]. Solving the Schrӧdinger equation is an iterative process, given an initial electron number density iteration to determine if it converges, to obtain the total energy. Then calculate fundamental material properties such as lattice constants, elastic constants, stacking fault energies, vacancy formation energies and migration barriers, and cohesive energies as a function of composition and crystal structure [25, 26]. The first-principles approach referred here deals with the DFT. Although the DFT simplifies the Schrödinger equation, the computation process is still challenging because HEAs have multiple principal components. Thus, special quasi-random structure (SQS) modeling, coherent potential approximation (CPA), and virtual crystal approximation (VCA) calculations are used for the DFT calculation of HEA [16, 27]. Common software used for DFT calculations is VASP (Vienna Ab Initio Simulation Package, Vienna, Austria) [28], CASTEP (Cambridge Sequential Total Energy Package) [29], and SIESTA [30].

2.1 Modeling methods

2.1.1 VCA

The VCA is based on the mean-field theory. Commonly, atomic potentials representing atoms of two or more elements are averaged. This is an oversimplified approach to substitutional solid solutions [31]. As there is no need to construct a supercell, the calculation time can be reduced considerably. In most cases, the VCA can be applied and are effective when the alloying elements are neighbors on the periodic table [27, 32] (i.e. TiVNbMo [33]). Nonetheless, it remains to be seen whether the VCA can be applied to other HEAs. The VCA was used to investigate the effect of alloying elements on phase stability, elastic and thermodynamic properties of random Nb-Ti-V-Zr HEAs. Liao et al. [32] found that the lattice constant, elastic constant, and thermal expansion coefficient of NbTiVZr were in agreement with other calculations and experiments, confirming that the VCA scheme was suitable for random Nb-Ti-V-Zr systems. A similar study was conducted by Chen et al. [14] which focused on the phase structure, elastic constants, and thermodynamic properties of Ti_xVNbMo refractory high entropy alloy (RHEA) by the VCA in conjunction with the equation of state (EOS) equilibrium equation of state and the quasi-harmonic Deby-Grüneisen model. Researchers are involved in the exploration of new HEA systems suitable for use in the VCA. Gao et al. [14] explored the elastic constants and elastic properties of VMoNbTaWM_x (M = Cr, Ti) RHEA by using the first principle and VCA method. In addition, they found that, when Cr content was raised, the bulk modulus B, Young’s modulus E, and the shear modulus G increased, while the Pugh ratio B/G and Poisson’s ratio ν fluctuated to some extent. Among them, VMoNbTaWCr_1.75 had the highest plasticity and VMoNbTaWCr₂ had the highest strength, respectively. It is important to note that VCA is computationally compact, highly efficient, and easy to model, yet the influence of the environment on the system is ignored, thereby resulting in a somewhat limited application.

2.1.2 CPA

The CPA rests on the assumption that the alloy may be replaced by an ordered effective medium, which is self-consistent in its parameters. The single-site approximation is applied to the impurity problem, which is a description of a single impurity embedded in an effective medium and no extra information is given about the individual potential and charge density beyond the sphere or polyhedron around the impurity. The CPA relies on two main approximations. One is to presume that the local potentials (P_A, P_B, P_C, P_D, P_E) around an atom from the alloy are the same, resulting in the disregard of local environment effects. Accordingly, a similar approximation can be made by replacing the system with a monoatomic medium described by the site-independent coherent potential $\stackrel{\sim }{P}$, as shown in Figure 2 [27].

Figure 2.

Two-dimensional illustration of the CPA for the equimolar ABCDE HEAs (HEAs). The symbol P represents the real alloy potential, $\stackrel{\sim }{P}$ for coherent potential, P_A, P_B, P_C, P_D, and P_E are the potentials of the alloying elements [27]. © 2017 Tian.

CPA has proven to be a very successful and popular technique that has been used extensively in the calculation of total energy, density of states, conductivity, and other electronic structure properties of random alloys [20]. The Exact Muffin-Tin Orbital (EMTO) in conjunction with the CPA method is demonstrated to be effective for a series of HEA systems including refractory HEAs [34] and HEA systems composed of transition metals [35]. Niu et al. [35] calculated ΔH_mix, lattice parameter (a₀), bulk modulus (B), and shear modulus (G) by the exact EMTO-CPA for over 2700 compositions of the NiFeCrCo alloy as a single-phase solid solution in paramagnetic state. An application of the CPA method is a mean-field approximation that is computationally small and, therefore, has an advantageous efficiency. The application can describe phenomena such as magnetic disorder and lattice vibrations. Rao et al. [36] studied the abnormal magnetic behavior of FeNiCoMnCu HEAs using DFT implemented in the EMTO-CPA formalism. Cu played a significant role in stabilizing the ferromagnetic order of Fe, they found. The calculated magnetization and Curie temperatures of alloys closely match the experimental results. Furthermore, comparing SQS with CPA, where there were no convergence problems, the results were very similar. It is important to emphasize that the difficulties associated with treating different magnetic states in the supercell approach further emphasize the advantages of the EMTO-CPA method for the present study. Through the Korringa-Kohn-Rostoker (KKR-CPA) method, Cieslak et al. [37] calculated total energy electronic accounting for chemical disorder effects of high entropy Cr_xAlFeCoNi alloys (x = 0, 0.5, 1.0, 1.5). Singh et al. [38] examined the total energy of Ti_0.25CrFeNiAl_x and found increasing Al stabilized the BCC phase and the FCC phase became stable above %65-Al. However, there are also certain disadvantages associated with the CPA method. The first problem is that the effective atom is fiction, and the resulting uniformity of the environment is not correct. Second, since all surrounding lattice sites are identically occupied, each atom is in a position of high symmetry, i.e. there is no force that would normally cause it to move from its own lattice site, thus, there is no lattice distortion [39].

2.1.3 SQS

SQS is a special periodic structure that is constructed using a small number of atoms per unit cell. The correlation functions within the first few nearest-neighbor shells are designed to approach the periodic functions of a random alloy to ensure that periodicity errors occur only among more distant neighbors. An SQS can be considered to be the best unit cell possible representing random alloys since interactions between distant neighbors generally contribute less to the system energy than interactions between near neighbors [40]. There are two approaches to generate SQSs. One is to generate exhaustively all possible combinations of supercells for a given cell size, and then select the one that best mimics the correlation functions of the random alloy. The second method involves performing Monte Carlo simulations to locate the optimal SQS. The two methods have been, respectively, realized in the gensqs and mcsqs codes within the Alloy Theoretic Automated Toolkit (ATAT) developed by Axel van de Walle and coworkers [41, 42]. While the gensqs code can only be used to generate smaller SQS, the mcsqs code [43] is more powerful and can be used to produce large SQS containing hundreds of atoms per unit cell. The SQS method combined with VASP software can be used to calculate properties such as lattice constants, layer misalignment energy, phase stability, elastic constants, magnetic properties, and electronic density of states of HEAs. According to Zhang et al. [44], stacking fault energies (SFEs) were computed in FCC and HCP HEAs utilizing the first-principles method combined with the SQS technique, revealing the mechanism for the formation of stacking faults and nanodiamonds. According to their findings, the negative SFEs are related to the energetic preference for HCP stacking and the metastability of FCC structures at low temperatures. During the past decade, a series of ferromagnetic HEAs systems such as Fe-Co-Ni-Al-Si, Fe-Co-Ni-Mn-Al, Fe-Co-Ni-Mn-Ga, Fe-Co-Ni-Cr-Si, Fe-Co-Ni-Mn-Si, and Fe-Co-Ni-Cu-Si. have been reported [45, 46, 47]. As well, non-ferromagnetic elements can significantly influence the magnetic properties of HEAs [36]. The importance of understanding the influences of non-ferromagnetic elements on the magnetic behavior of HEAs cannot be overstated. This can be achieved by using a first-principle method in conjunction with the SQS technique. Zuo et al. [19] used the SQS approach to create the structures of CoFeMnNi, CoFeMnNiCr, and CoFeMnNiAl, with the DFT calculations conducted at 0 K through the VASP. The DFT calculations on the electronic and magnetic structures reveal that the anti-ferromagnetism of Mn atoms in CoFeMnNi is suppressed especially in the CoFeMnNiAl HEAs, because Al changes the Fermi level and itinerant electron-spin coupling that leads to ferromagnetism as illustrated in Figure 3. Furthermore, Wei et al. [19] investigated the mechanism of the magnetic behavior of FeCoNiSi_0.2M_0.2 (M = Cr, Mn) HEAs using first-principles calculations combined with the SQS method. It was found that doping the Mn resulted in a reduction in the number of spin-down electrons, which ultimately led to the transition of the Mn from an antiferromagnetic to ferrimagnetic state. SQS method may also be used for investigating short-range order effects in the chemical environment.

Figure 3.

(a) Spin-polarized total DOS; (b) Co d partial DOS; (c) Fe d partial DOS; (d) Mn d partial DOS; (e) Ni d partial DOS; and (f) Al s, p and Cr d partial DOS for the FCC CoFeMnNi, FCC CoFeMnNiCr, and BCC CoFeMnNiAl from DFT calculations at zero temperature; and (g) Mn d-orbital decomposed partial DOS. The vertical dotted lines indicate the Fermi level [48]. © 2017 Acta Materialia Inc.

The SQS method can obtain a more realistic disordered distribution of HEAs and to consider the influence of the local atomic environment within the alloy matrix on the physical and chemical properties of the alloy. However, the complexity of the constituent elements of HEAs leads to the construction of SQS supercells considering more correlation functions among the principal elements, which has some influence on the efficiency and accuracy of the calculation. Therefore, how to reduce the difficulty of supercell construction is also a pressing issue for researchers to address.

3. MD calculations

Molecular dynamics (MD) methods rely heavily on Newtonian mechanics to calculate the properties and structure of molecules at the molecular level by simulating molecular motion. It is derived from samples that originate from a whole system made up of different states of the molecular system. The configuration of the system is then derived using calculation. Since computer computing power has increased rapidly, the research system of the MD method has also progressed to a larger spatial and temporal scale. A gap exists between the mechanical property values derived from simulation and the actual macroscopic mechanical properties of materials. However, this does not hinder the systematic study of the microstructural evolution of materials using MD methods. The main software groups currently used for molecular dynamics simulations are Lammps [49], Gromacs, Amber, Material studio, etc. Constructing models is the basis of molecular dynamics studies, and models are generally constructed by the random occupation of lattice sites by constituent atoms. Potential functions describe interatomic interactions, and the accuracy of the MD simulation results is dependent on the potential function describing the interatomic interactions. The main potential functions commonly used are Lennard-Jones (L-J) potential, Embedded atom method (EAM) potential, and Average-atom potential [50, 51, 52]. Currently, the most extensive description of interatomic interactions in HEAs is the EAM potential, which compensates for the shortcomings of the pair potential by forming a many-body potential function. Nevertheless, the EAM potential does not consider the covalent bond directionality. Based on this, the researchers also proposed the modified embedded-atom method (MEAM) potential and the EAM-Morse potential [53, 54], etc.

Qi et al. [55] used the MD method with MEAM potential to simulate the microstructure evolution and mechanical properties of CoCrFeMnNi HEAs under nano scratching. Several new behaviors were found in HEAs, such as twin boundary migration and dislocation locks. In HEAs, MD methods have been applied to mechanical properties, irradiation damage, thermal stability, and film growth by studying the microstructure evolution of the alloy and its mechanism. Jiang et al. [56] used MD simulations combined with EAM potential to study microstructural evolution and mechanical properties of Al_xCoCrFeNi HEAs under uniaxial tension. Higher aluminum content was found to deteriorate Young’s modulus, yield stress, and yield strain of Al_xCoCrFeNi HEAs. However, the dislocation density declined with increasing temperature. The high Al concentration suppressed the decrease of tensile properties with increasing temperature. Nanoindentation experiments utilize an indenter of a specific shape to apply a load to the surface of a material in a vertical direction and through computer control of the variation in load, determine the depth of the indentation in real-time. To gain a better understanding of the surface properties of HEAs, it is imperative to learn more about the deformation mechanism of indentation; however, due to the limitations of instruments and means of observation, experiments generally yield loaddisplacement curves, elastic moduli, hardness, and other macroscopic properties, and the microstructure and deformation mechanism cannot be examined at the nanoscale. MD simulation has proved to be a useful tool for analyzing and predicting the evolution of tissue and mechanical properties of HEAs under indentation. Therefore, more MD research on nanoindentation has been conducted. Based on MD simulations, LUO et al. [57] constructed the nanoindentation models of single-crystal FeCoCrNiCu HEA and Cu as shown in Figure 4. MD models included (i) FeCoCrNiCu HEA workpiece + virtual indenter and (ii) Cu workpiece + virtual indenter. Compared to Cu, the FeCoCrNiCu HEA exhibited a high dislocation density and high loading force during indentation. These findings indicated that the HEA had high strength.

Figure 4.

Nanoindentation models of (a) the single crystal HEA, and (b) Cu [57].

HEA coatings have attracted more and more attention from researchers, especially HEA hard coatings, which can be used for cutting tools used in harsh environments, etc., to significantly improve their service life. This suggests that the exploration of HEA high wear-resistant coatings has a high prospect of application. The growth mode of thin films influences their structure and properties, so molecular dynamics simulations of thin film growth can be used to study the mechanism of film growth. Xie et al. [17] studied AlCoCrCuFeNi HEA coatings. Figure 5 showed the deposition of HEA coatings on Si(100) substrate. The atoms in Al₂Co₉Cr₃₂Cu₃₉Fe₁₂Ni₆ and Al₃Co₂₆Cr₁₅Cu₁₈Fe₂₀Ni₁₈ were arranged in a crystalline structure, while Al₃₉Co₁₀Cr₁₄Cu₁₈Fe₁₃Ni₆ formed an amorphous structure over the entire thickness. The simulation results show that the differences in the number of elements and atomic sizes have a significant effect on the atomic configuration, and a tendency to develop from solid solution to bulk amorphous is predicted by calculating the parameters of the HEAs.

Figure 5.

HEA films deposited on Si(100) substrate [17]. Copyright © 2015 Elsevier.

4. CALPHAD methods

In terms of phase diagrams, they are a geometric representation of a system in equilibrium and thus serve as the basis for the study of solidification, phase transformation, crystal growth, and solid-phase transformation. Traditional phase diagrams such as binary or ternary phase diagrams can rely on experimental determination but for multivariate systems, the experimental approach is not desirable. To overcome the obstacles, CALPHAD methods based on thermodynamic theory and thermodynamic databases are created. CALPHAD methods estimate the Gibbs free energy of each phase, such as solid solution (SS) phase, intermetallic compound (IM), etc., by calculating the mixing enthalpy and the conformational entropy. It is now possible to develop new materials on a more reliable basis. Figure 6 shows the steps of the HEAs design using CALPHAD methods [15]. HEAs can be developed from these thermodynamic models directly or HEA databases can be created from the models specifically for HEAs. Once the parameters have been optimized, the relevant thermodynamic information can then be derived, such as the composition of each phase, phase ratio, activity, and mixing enthalpy. Several companies offer databases and associated software tools, notably PANDAT, FactSage, and Thermo-Calc [58], each of which has developed databases geared toward the study of HEAs. The main thermodynamic databases that have been developed for HEAs are PanHEA [59, 60] and TCHEA [13, 58] etc.

Figure 6.

The steps of HEAs compositional design using CALPHAD [15]. © 2020 Li, Xie, Wang, Liaw, and Zhang.

Because of their high hardness and excellent wear resistance, light-weight HEAs are suitable as protective coatings for machine components and tools [61]. Sanchez et al. [62] designed low-density and inexpensive Al₄₀Cu₁₅Cr₁₅Fe₁₅Si₁₅, Al₆₅Cu₅Cr₅Si₁₅Mn₅Ti₅, and Al₆₀Cu₁₀Fe₁₀Cr₅Mn₅Ni₅Mg₅ alloys by the CALPHAD method in conjunction with thermodynamic database TCAL5. CALPHAD thermodynamic modeling was successful in predicting the constituent phases, which were in close agreement with experimental results. But there remains a gap between Thermo-Calc calculations and the experimental results. Overall, this database has been proven to be an appropriate technique for designing Al-based HEAs. The growth of microelectronics has highlighted the importance of silicide materials - high entropy silicides (HES) [63, 64] that are especially promising due to their potential for use in microelectronics. ThermoCalc Software equipped with the TCHEA3 HEA thermodynamic database was used for complex HES compositions with targeted phase stability by Vyatskikh et al. [65]. Two single-phase HES materials were identified, the ternary (CrMoTa)Si₂ and quinary (CrMoTaVNb)Si₂. Both materials were identified using the CALPHAD method. It could be decided that both the ternary and quinary alloys were predicted to exhibit a single phase with a C40 hexagonal crystal structure.

Therefore, we conclude that the CALPHAD methodology is capable of formulating compositionally complex, HEA systems, and overcoming obstacles associated with certain experiments (such as high-temperature and high-pressure environments). To realize the rapid scientific design of materials, it is possible to use the component that is easy to calibrate by experiment to predict the component that is difficult to calibrate. Still, the non-equilibrium solidification structures observed in experiments and the CALPHAD calculations based on equilibrium have some differences.

5. Machine learning

Considering the complex elemental composition of HEAs, the use of an empirical “trial and error” material design paradigm may result in significant time and cost overruns, and thus a new material development paradigm is urgently required to guide the design of HEAs. Computing power and the development of computing platforms have led to an increase in computational materials science that has promoted the development of materials research and development from a trial-and-error mode to a computation-driven process. As of late, the importance of MGI has resulted in the development of big data on materials and the full application of artificial intelligence to the development of HEAs. Among them, machine learning models such as support vector machine (SVM), principal component analysis (PCA), and cluster analysis play an important role in the construction and screening of HEA features and the prediction and classification of phase structures, and the prediction of HEA properties can be performed with the help of artificial neural networks, linear regression, and logistic regression. At the same time, active learning strategies based on Bayesian optimization and genetic algorithms are applied to the inverse optimization design of HEAs, which makes them have better comprehensive performance. Zhang et al. [66] used atomic radius, melting temperature, mixing entropy, and empirical parameters of HEA phase formation as features, and established a high-precision HEA phase classification model using a combination of genetic algorithm screening material features and machine learning models. Based on 322 data samples of cast HEAs, Li and Guo [67] built a support vector machine classification model by screening five material factors: VEC, δ, melting temperature (T_m), ΔS_mix, and ΔH_mix as features by sequential selection method, and the model achieved more than 90% accuracy in classifying HEAs as BCC single-phase, FCC single-phase and non-forming single-phase solid solution. Huang et al. [68] developed K-nearest neighbor (KNN), SVM, and artificial neural network (ANN) classification models based on 401 HEA data samples featuring VEC, electronegativity difference Δχ, δ, ΔS_mix, and ΔH_mix. And then the prediction of whether the HEA formed SS, IM, and mixed SS and IM (SS + IM). The prediction accuracies of the three models obtained from cross-validation for the three classifications were 68.6%, 64.3%, and 74.3%, respectively. The reason for the low classification accuracy was found to be the unclear interphase boundary between SS and SS + IM by the self-organizing mapping (SOM) neural network. Then, the binary classification models of SS and IM, SS + IM and IM, SS and SS + IM were developed using multi-layer feed-forward neural network (MLFFNN) with classification accuracies of 86.7%, 94.3%, and 78.9%, respectively. Zhao et al. [69] applied machine learning to combine elemental characteristics with long-term ordering and established 87% of prediction accuracy. A deep neural network classification model for HEAs was developed by Lee et al. [70]. To compensate for the lack of experimental data, additional HEA data were generated using a conditional generative adversarial network (GAN), which improved the classification accuracy from 84.75% to 93.17%, exceeding the prediction accuracy of previous literature. Machine learning is generally based on big data, and data mining and cleaning are difficult. The quality of data will also directly determine the accuracy of prediction. Therefore, is it possible to make accurate predictions based on high-quality and relatively small data sets? A bilinear log model based on 21 HEA compositions was proposed by Steingrimsson et al. [71], and the break temperature, T_break, was introduced to predict the ultimate strength of temperature-dependent body-centered-cubic HEAs. They derived the ultimate strength as a function of composition and temperature by using Figure 7 at high temperatures and defined the key T_break for optimizing the high temperature properties of the alloys.

Figure 7.

Identification of compositions with the ability to retain strengths at high temperatures [71].

HEAs experimental data has increased dramatically over the past two decades, and ML provides a means to utilize this information. In particular, ML in HEAs is currently focused on the prediction of phases, and there are 13 commonly used criteria as shown in Table 1 [72]. In the future, a focus of machine learning will be to identify new unified criteria for phase formation in HEAs. Combined with simulation methods and machine learning already can accelerate the compositions and procedures of HEAs compositions and procedures. ML will play an important role in addressing challenges that are too difficult for relationships among phases/structures, the processing structure-property, the microstructure, and the performance of materials.

Parameters	Formula
Mean atom radius	$a={\sum }_{i=1}^n{c}_i{r}_i$
Atomic size difference	$\delta =\sqrt{{\sum }_{i=1}^n{c}_i{\left(1-\frac{r_i}{a}\right)}^2}$
Average of the melting points of constituent elements	$T_m={\sum }_{i=1}^n{c}_i{T}_{\mathit{mi}}$
Standard deviation of melting temperature	${\sigma }_T=\sqrt{{\sum }_{i=1}^n{c}_i{\left(1-\frac{T_i}{T_m}\right)}^2}$
Average mixing enthalpy	$∆H_{\mathit{mix}}=4{\sum }_{i\ne j}{c}_i{c}_j{H}_{\mathit{ij}}$
Standard deviation of mixing enthalpy	${\sigma }_{∆H}=\sqrt{{\sum }_{i\ne j}{c}_i{c}_j\left(H_{\mathit{ij}}-∆H_{\mathit{mix}}\right)}$
Ideal mixing entropy	$S_{\mathit{id}}=-k_B{\sum }_{i=1}^n{c}_{i}ln{c}_i$
Electronegativity	$\chi ={\sum }_{i=1}^n{c}_i{\chi }_i$
Standard deviation of electronegativity	$∆\chi =\sqrt{{\sum }_{i=1}^n{c}_i{\left({\chi }_i-\chi \right)}^2}$
Average VEC	$\mathit{VEC}={\sum }_{i=1}^n{c}_i\mathit{VE}{C}_i$
Standard deviation of VEC	${\sigma }_{\mathit{VEC}}=\sqrt{{\sum }_{i=1}^n{c}_i{\left({\mathit{VEC}}_i-\mathit{VEC}\right)}^2}$
Mean bulk modulus	$K={\sum }_{i=1}^n{c}_i{K}_i$
Standard deviation of bulk modulus	${\sigma }_K=\sqrt{{\sum }_{i=1}^n{c}_i{\left(K_i-K\right)}^2}$

Table 1.

The 13 design parameters and the corresponding formula [72].

6. Conclusions

As computers have developed rapidly, materials have faced new challenges and opportunities. Developing new alloys is no longer a time-consuming and laborious trial-and-error process, but rather a method for efficiently exploring alloys using computations. Contrary to conventional alloys, HEAs are positioned in the center of the phase diagram. Many primary elements indicate a large composition space, which presents both impediments and challenges for the development. A number of HEAs are studied including RHEAs, light-weight HEAs, and others, all of which have great industrial applications. Based on the proposed simulations and calculations, researchers can target the exploration of alloy compositions based on properties in order to develop new HEAs. The focus of this chapter is on reviewing the simulation tools at different scales and summarizing cutting-edge research. The use of alloy design calculations based on DFT or MD calculations of alloy properties and multi-component phase diagrams calculated by CALPHAD is an effective method for saving time and reducing costs. However, the multi-element (more than 5) and microstructure (solid solution) of HEAs make the calculation process more complex and time-consuming than that of conventional alloys. In addition, there is a gap between the phase composition of HEAs determined by the experimental method and that predicted by the CALPHAD method. Therefore, combining more than two computational methods is a focus for future simulations. An example is the combination of DFT and CALPHAD methods. CALPHAD is often limited in scope due to the lack of reliable data. DFT method can calculate various thermodynamic properties, such as formation energy, heat of formation, etc. to supplement the data to provide support for phase diagrams. Last, it is also vital that a robust and comprehensive database be established for HEAs through the MGI project.

Acknowledgments

Yong Zhang acknowledges support from (1) Guangdong Basic and Applied Basic Research Foundation (2019B1515120020); and (2) Creative Research Groups of China (No.51921001).

Conflict of interest

The authors declare no conflict of interest.