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

*Yong Zhang and Yuanying Yue*

## **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.

**Keywords:** high-entropy alloys, simulation and calculation, density functional theory calculations, molecular dynamics, phases, properties

## **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 Δ*Smix* of n-component alloys is

$$
\Delta \mathbf{S}\_{\rm mix} = -R \sum\_{i} \mathbf{C}\_{i} L n c\_{i} \tag{1}
$$

where *R* is gas constant, *Ci* 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, Δ*Hmix*, of the multi-component alloys:

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

$$
\Delta H\_{\text{mix}} = \sum\_{i \neq j} 4 \varkappa\_i \varkappa\_j \Delta H\_{ij} \tag{3}
$$

where *N* is the number of the elements in HEAs, *xi* or *xj* is the atomic percentage of the *i* th or *j* th component, *rj* is the atomic radius of the *j* th component, and Δ*Hmix* is the mixing enthalpy for *i* and *j* element.

Subsequently, to further understand the connection between Δ*Hmix* and Δ*Smix*, Zhang and Yang [7] proposed a new parameter, *Ω*, defined by:

$$
\Omega = T\_m \Delta \text{S}\_{mix} / |\Delta H\_{mix}| \tag{4}
$$

where *Tm* 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

*Simulation and Calculation for Predicting Structures and Properties of High-Entropy Alloys DOI: http://dx.doi.org/10.5772/intechopen.105963*

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.

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.

$$\text{VEC} = \sum\_{i=1}^{n} c\_i (\text{VEC})\_i \tag{5}$$

where *ci* and *VECi* 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 solidsolution 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 firstprinciples density functional theory (DFT), molecular dynamics (MD), the calculation of phase diagram (CALPHAD), and high-throughput methods [13–20]. 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 selfconsistent. 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 multielectron 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 quasirandom 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 TixVNbMo refractory high entropy alloy (RHEA) by the VCA in conjunction with the equation of state (EOS) equilibrium equation of state and the quasi-harmonic

### *Simulation and Calculation for Predicting Structures and Properties of High-Entropy Alloys DOI: http://dx.doi.org/10.5772/intechopen.105963*

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 VMoNbTaWMx (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, VMoNbTaWCr1.75 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 (PA, PB, PC, PD, PE) 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 *P*~, as shown in **Figure 2** [27].

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

#### **Figure 2.**

*Two-dimensional illustration of the CPA for the equimolar ABCDE HEAs (HEAs). The symbol P represents the real alloy potential, P for coherent potential, P* ~ *A, PB, PC, PD, and PE are the potentials of the alloying elements [27]. © 2017 Tian.*

composed of transition metals [35]. Niu et al. [35] calculated Δ*Hmix*, 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 CrxAlFeCoNi alloys (x = 0, 0.5, 1.0, 1.5). Singh et al. [38] examined the total energy of Ti0.25CrFeNiAlx 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

## *Simulation and Calculation for Predicting Structures and Properties of High-Entropy Alloys DOI: http://dx.doi.org/10.5772/intechopen.105963*

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 [45–47]. 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,

#### **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.*

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 antiferromagnetism 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 FeCoNiSi0.2M0.2 (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.

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 [50–52]. 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

*Simulation and Calculation for Predicting Structures and Properties of High-Entropy Alloys DOI: http://dx.doi.org/10.5772/intechopen.105963*

**Figure 4.** *Nanoindentation models of (a) the single crystal HEA, and (b) Cu [57].*

simulations combined with EAM potential to study microstructural evolution and mechanical properties of AlxCoCrFeNi HEAs under uniaxial tension. Higher aluminum content was found to deteriorate Young's modulus, yield stress, and yield strain of AlxCoCrFeNi 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.

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

**Figure 5.** *HEA films deposited on Si(100) substrate [17]. Copyright © 2015 Elsevier.*

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.

## **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.

Because of their high hardness and excellent wear resistance, light-weight HEAs are suitable as protective coatings for machine components and tools [61]. Sanchez

*Simulation and Calculation for Predicting Structures and Properties of High-Entropy Alloys DOI: http://dx.doi.org/10.5772/intechopen.105963*

#### **Figure 6.**

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

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 (*Tm*), Δ*Smix*, and Δ*Hmix* 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 singlephase 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 Δ*χ*, *δ*, Δ*Smix*, and Δ*Hmix*. 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 crossvalidation 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, Tbreak, 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 Tbreak for optimizing the high temperature properties of the alloys.

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

*Simulation and Calculation for Predicting Structures and Properties of High-Entropy Alloys DOI: http://dx.doi.org/10.5772/intechopen.105963*

#### **Figure 7.**

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



#### **Table 1.**

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

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.

## **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 multicomponent 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 timeconsuming 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.

*Simulation and Calculation for Predicting Structures and Properties of High-Entropy Alloys DOI: http://dx.doi.org/10.5772/intechopen.105963*

## **Acknowledgements**

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.

## **Author details**

Yong Zhang\* and Yuanying Yue State Key Laboratory of Advanced Metals and Materials, University of Science and Technology Beijing (USTB), Beijing, China

\*Address all correspondence to: drzhangy@ustb.edu.cn

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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## CALPHAD as a Toolbox to Facilitate the Development of HEAs

*Wei Wang and Zhou Li*

## **Abstract**

This chapter would be a comprehensive review of the successful application of the CALPHAD method in the research and development of high-entropy alloys (HEAs) in recent years. The structure of this chapter is obedient to the conventional framework in materials science, and several topics would be covered, including composition and structure design, heat treatment and solidification processing, and some properties and performances, such as mechanical and electrochemical corrosion behaviors and high-temperature oxidation characteristics. A conclusion and outlook of the CALPHAD method are also provided at the end of this chapter. Hope this chapter could offer a thermodynamic and kinetic perspective for facilitating the field of HEA research and inspire other researchers to contribute more valuable ideas and works.

**Keywords:** high-entropy alloys, CALPHAD, materials design, processing optimization, mechanical properties, electrochemical corrosion, high-temperature oxidation

## **1. Introduction**

The concept of high-entropy alloys (HEAs), where the configurational entropy plays a significant role, is closely related to the content of thermodynamics. As a phenomenological methodology of computational thermodynamics, CALPHAD could be the most direct tool for the design of HEAs [1]. CALPHAD stands for the abbreviation of CALculation of PHAse Diagram, which is a computational approach proposed in the early 1970s to bridge the gap between the theoretical calculations and the experimental phase equilibrium information [2]. The Gibbs energy function occupies the core of the CALPHAD method. Thermodynamic properties of the materials could be described by the Gibbs energy of each phase. The phase diagrams are constructed by minimizing the total Gibbs energy of the system. The model parameters of Gibbs energy, which is the main constitute of the CALPHAD databases, are assessed by fitting the input data via CALPHAD-based software, such as Thermo-Calc [3], Pandat [4], and FactSage [5]. The experimental results would be the first choice for the input, and the first-principle calculations would be a good complement wherever there is a lack of experimental input [6]. The Gibbs energy of an element can be written as follows [7]:

$$G\_i^{\theta} - H\_i^{\text{SER}} = a + bT + cT \ln\left(T\right) + dT^2 + dT^{-1} + fT^3 + \dots \tag{1}$$

where *HSER <sup>i</sup>* represents the enthalpy of the pure element *i* in its reference state standard element reference (SER). This reference state is defined as the most stable state of the pure element at 298.15 K and 1 bar. Coefficients *a*, *b*, *c*, *d*, … are the parameters to be optimized.

To deal with the case containing more elements, several models were proposed to describe the atomic interactions, such as compound energy formalism (CEF) [8], cluster variation method [9], quasi-chemical model [10], and so on. In this chapter, the compound energy formalism (CEF) as the most widely used model was introduced. In the CEF, the structure of a phase is considered to be constituted by a set of sublattices represented by the formula (A, C)*x*(B, D)*y*, with A, C mixing on the first sublattice and B, D occupying the second one [6].

$$\mathcal{G}\_m^a = \mathrm{^{Srf}}\mathcal{G}\_m + T^{\mathrm{efg}}\mathcal{S}\_m + {^{\mathrm{phy}}}\mathcal{G}\_m + {^{\mathrm{Ex}}}\mathcal{G}\_m$$

$$\mathcal{G} = \sum P\_{I\_0}(Y)\mathcal{G}\_{md}\stackrel{\circ}{+} + RT\sum\sum n'y'\_j\ln\left(y'\_j\right) + {^{\mathrm{phy}}}\mathcal{G}\_m + \sum P\_{I\_1}(Y)L\_{I\_1} + \sum P\_{I\_2}(Y)L\_{I\_2} \dots \tag{2}$$

*SrfGm* is named "surface of reference," which is a linear combination of the Gibbs energy of end-members. *cfgSm* denotes the configurational entropy of the phase, considering the ideal mixing only. The sum of *Srf Gm* and *TcfgSm* is the so-called ideal solution model. *PhyGm* describes the physical effect (usually the magnetic effect), which was treated separately. *ExGm* represents the excess Gibbs energy describing the interactions between the constituents in different sublattices. *PI*<sup>0</sup> ð Þ *Y* refers to the product of site fraction in *I*0, which means an array of zeroth order including only one constituent in each sublattice. *PIn* ð Þ *Y* corresponds to the *n*th-order array, which contains *n* extra constituents in sublattices 1, 2, … , *n* – 1. *n<sup>s</sup>* is the number of sites in sublattice *s* and *y<sup>s</sup> <sup>j</sup>* is the site fraction of element *j* in sublattices. *L* denotes the interaction parameter, which is a Redlich-Kister polynomial.

One of the aims of the CALPHAD method is to develop the reliable databases for the users of interest. The descriptions of stable binaries and ternaries are the building blocks for the multicomponent CALPHAD databases. The CALPHAD method uses extrapolation to extend the range from these stable constituent lower systems to the unstable region and higher order systems. The semiempirical nature and the unique structure of the database (**Figure 1**) make CALPHAD an accurate and self-consistent technique and gain the popularity in the research of structural materials [11]. Therefore, the CALPHAD has been considered as an important part of the Integrated Computational Materials Engineering (ICME) and Materials Genome Initiative (MGI) [12].

With the combination of thermodynamic and kinetic databases, the CALPHAD method could not only depict the characteristics of thermodynamics (phase diagrams and thermophysical properties) but also simulate the kinetic process (precipitation and diffusion) by solving the diffusion equations numerically. The following sections will show how the CALPHAD method serves as a powerful toolbox to facilitate the development of the HEAs.

## **2. Thermodynamic calculations in the composition and phase design of HEAs**

The emergence of the HEA provides a new alloying strategy for the materials scientists. Different from the conventional way of adding a handful of alloying

**Figure 1.**

*The structure of the CALPHAD thermodynamic databases.*

#### **Figure 2.**

*The contour map of entropy of mixing on an illustrated ternary alloy. The blue regions represent the conventional alloys with only one principal element and the bright area indicates the high-entropy region [13].*

elements to a primary one, the HEA creates a mixture of the multiple principle elements with (near) equiatomic concentration [13, 14]. The most intuitive representation of this strategy would be the composition map of the phase diagram (**Figure 2**). The conventional alloys locate at the corner regions (blue region), and the HEAs

occupy the central region of the phase diagram. Therefore, the name of "central alloys" is perhaps a good choice to eliminate the ambiguity caused by the confusing use of "high-entropy alloys," "medium-entropy alloys," "complex concentrated alloys (CCAs)," "multiprincipal element alloys (MPEAs)," etc. Its vastness of the composition space and the complex combination of elements open the possibilities of discovering novel alloys with exceptional properties [15]. Meanwhile, it is a more challenging task of the HEA design to find the desired point in this composition sea. As the sole technique of visualizing the phase diagram directly, the CALPHAD method that is expert in the composition and phase design for the multicomponent systems could serve as an equipped ship for the HEA researchers. Hundreds of published works have applied the CALPHAD method in the phase prediction of HEAs [16, 17]. In this section, the advantage and limitations of the CALPHAD method will be discussed with several examples.

#### **2.1 Application of CALPHAD databases in the HEA design**

Phase diagrams have been considered as the road map for the research of materials science, which act as a guiding role to design the experiments and understand the behavior of the materials. As mentioned above, the accuracy of the calculated phase diagram depends on the reliability of the CALPHAD databases. Owing to the lack of specific HEA databases, the early works usually uses the Ni-base and, sometimes, the Fe-base databases to explore the alloy composition of the desired phase. Choi et al. [18] used the TCFE2000 database [19] to design a new face-centered cubic (FCC) phase HEA with nonequiatomic composition. **Figure 3** shows that the prepared samples fall well on the FCC single-phase region, which verifies the validity of the

**Figure 3.** *Calculated FCC phase region of 10Co-15Cr-10 V-65 (Fe, Mn, Ni) [18].*

**Figure 4.**

*Pseudobinary phase diagram of CoCrFeNiCux [20].*

CALPHAD calculation. Zhang et al. [20] employed the Thermo-Calc software [3] with database TCNI8 [19] to calculate the phase diagram of the CoCrFeNi-based HEAs. In **Figure 4**, the pseudobinary phase diagram of CrCoFeNi-Cux was calculated to show the alloying effect of Cu on the phase stability of the HEA. Miscibility gaps of FCC and liquid phase begin to form with the increasing Cu content. This calculated result has been experimentally confirmed by the work of Wu et al. [21]. Butler et al. [22] studied the phase stability of AlNiCoCrFe alloy using the CALPHAD approach. **Figure 5d** shows their prediction on the phase fractions of the Al15 HEA calculated by using the TCNi8 database. They found that the predicted phase stability was generally in agreement with the experimental measurements except the FCC phase. These findings indicate that the CALPHAD could be a powerful tool for the design of HEAs, but the quality of the databases needs to be improved. Therefore, the specific thermodynamic databases, TCHEA [23] and PanHEA [24], have been tailored in recent years to accelerate the development of HEAs. Feng et al. [25] extensively investigated the phase stability of the lightweight HEAs using both the experimental and computational methods. They calculated the phase fraction of each in Al1.5CrFeMnTi alloy (**Figure 6a**) and the isopleth of Al1.5CrFeMnTix (**Figure 6b**) using Pandat software with the PanHEA database. A reasonable agreement has been reported between the experimentally determined and predicted compositions. One of the disagreements is that the nanosized L21 phase was experimentally observed at higher annealing temperature than the predicted one. MacDonald et al. [26] thoroughly discussed the FCC phase decomposition of the equiatomic CoCuFeMnNi alloy with the aid of the CALPHAD method. They performed the calculation on the equilibrium step diagram of CoCuFeMnNi alloy at the temperature range from 400 to 1600°C employing the Thermo-Calc software with database TCHEA3 and compared the experimentally observed compositions of three phases with the CALPHAD-predicted ones at 500°C. It can be seen in **Figure 7** that the composition predicted by CALPHAD reasonably agrees with the measured ones except the case in the Cu-rich FCC phase. Although the

**Figure 5.** *BSE image of Al15 HEA heated treating at (a) 700°C and (b) 1050°C, (c) XRD spectra of Al 15 HEA, and (d) calculated phase fractions for the Al 15 HEA [22].*

CALPHAD has been successfully applied to the phase and composition design of HEAs, its power has not been fully explored due to the imperfection of the databases. The difficulties of developing a high-quality HEA database could be attributed to several reasons. As we noted earlier, the traditional alloy databases were developed based on the binaries and ternaries with only one principal component, while the HEAs focus on the central regions of the alloy systems. Therefore, one of the difficulties is that there is lack of experimental data on this region as the input for the CALPHAD assessment [27]. Another thing that should be noted is that the ternary interaction could be significant at the highly concentrated region, so all the ternaries should be carefully evaluated [28]. Those barriers will burden the workload of assessment exponentially and make it almost a "mission impossible" to develop a perfect HEA database in a short term [27].

## **2.2 High-throughput CALPHAD (HT-CALPHAD) calculations for the HEA design**

Alternatively, the high-throughput CALPHAD method and the machine learning (ML) models provide us a promising way of accelerating the design of advanced HEAs. Feng et al. [29] utilized the CALPHAD-based high-throughput calculation to screen the optimal composition of the lightweight HEAs. The vast composition space has been narrowed to a small range by meeting the criteria on phase fraction and temperature, with eight candidate alloys surviving out finally (**Figure 8**). In **Figure 8b–e**, region 1 meets only one criterion, region 2 meets two of criteria, and

*CALPHAD as a Toolbox to Facilitate the Development of HEAs DOI: http://dx.doi.org/10.5772/intechopen.105191*

**Figure 6.** *Calculated phase diagram of the (a) alloy Al1.5CrFeMnTi and (b) isopleth of Al1.5CrFeMnTix [25].*

region 3 meets three. This approach significantly reduces the cost of experimental work. Zeng et al. [30] proposed several phase selection rules by combining the machine learning method and the CALPHAD calculations. The flowchart of this chapter is summarized in **Figure 9**. More than 300,000 of entries of phase equilibrium information were generated by the CALPHAD method. Then, the XGBoost method was employed to explore five most important features to depict the composition space spanned by the generated data. Based on the trained ML model, five-phase selection rules were established, which provides an efficient approach of designing a singlephase HEA. The CALPHAD method has accumulated tons of high-quality phase equilibria and thermochemical data in decades, which would be a valuable resource for the machine learning models. At the same time, ML could help CALPHAD extend its application and establish the structure–property connection quantitatively. A recent review of the HT-CALPHAD method could be found in [31].

Besides, thermodynamic analysis on the phase stability of single solid solution (SSS) phases in HEA has also been extensively studied by using the CALPHAD method [14, 32]. Unexpectedly, the configurational entropy does not play the

#### **Figure 7.**

*(a) Calculated equilibrium step diagram of the CoCuFeMnNi alloy from 400–1600°C compared with experimentally measured phase composition from STEM EDS, (b) Fe-Co rich B2 phase, (c) Cu-rich FCC phase, and (d) the matrix FCC phase [26].*

#### **Figure 8.**

*(a) Flowchart of the HT-CALPHAD, (b) Al-Cr projection, (c) Al-Fe projection, (d) Al-Mn projection, and (e) Al-Ti projection [29].*

**Figure 9.** *The road map for the discovery of phase selection rules [30].*

dominant role in many cases [1]. Other entropic contribution and enthalpy should also be valued in the analysis because the phase stability of the SSS depends on the Gibbs energy (G = H – TS) of its competing intermetallic phases [14, 15]. A widely accepted framework of this analysis is needed for further research.

## **3. CALPHAD-based prediction in processing optimization**

#### **3.1 Heat treatment processing optimization**

Known as the idiom "the refined steel, softly winds fingers," the condensed words were refined as the most significant information about heat treatment in the present perspective. For metallic materials, heat treatment represents some thermal as well as thermochemical processes to bring a series of complicated transformations in physical metallurgy, mechanical, and corrosion resistance performances. Therefore, to understand heat treatment, it is necessary to study the thermal phenomena, microstructure, phase stability/transformation, properties, and so on. As a kind of promising material, HEAs like other ferrous and nonferrous alloys, heat treatment is also an important process for improving formability, machinability, strength, and ductility.

As the name implies, the CALPHAD method is a technique of calculation for phase diagram initially. Therefore, in the field of the phase diagram, the CALPHAD method is inherently associated with a unique advantage in comparison with other techniques. One of the most important advantages of the CALPHAD method is to predict highorder thermodynamic information via extrapolation from the corresponding loworder subsystems. To realize the CALPHAD method, it is required to combine both computational platforms and some thermodynamic and mobility databases, and so on. Some commercial software is available to perform various calculation and simulation, including Thermo-Calc software (also add-on diffusion and precipitation module,

Dictra & TC-Prisma) [3, 19], Pandat [33], and FactSage [34]. Some commercial databases also need to be matched in these software to provide various thermodynamic and mobility information, including TCHEA5 and MOBHEA2 [35], PanHEA\_TH, PanHEA\_MB, and PanHEA\_MV [36]. The continuous update databases, which contained dozens of elements and hundreds of subsystems, provide a set of self-consistence thermodynamic and kinetic parameters to assist a high-quality prediction. With the development of thermodynamic and kinetic modeling, the CALPHAD method has played a more and more important role in materials design, processing optimization, phase transformation, mechanical and corrosion behavior investigations, and so on.

In this section, some CALPHAD-based research on the heat treatment of HEAs was collected to aim at highlighting the significant role of the CALPHAD method in the thermodynamic, kinetic, and precipitation predictions, which are the three main parts of this section.

#### *3.1.1 Thermodynamic analysis of heat treatment*

The phase diagram is one of the most important components in the field of materials science and technology, obviously in the field of heat treatment. Thus, the CALPHAD method has an inherent advantage due to the technique that was initially originated to couple the phase diagrams and thermochemistry, especially due to the extrapolation characteristic. For "multiprincipal element alloy (MPEA)" [37, 38] or "complex concentrated alloy (CCA)" [39] systems, the extrapolation characteristic from the known low-order subsystems to the unknown high-order HEAs system becomes an outstanding superiority of the CALPHAD method, particularly in comparison with the conventional one or two principal metallic materials. To complete a high-quality thermodynamic prediction needs an important prerequisite, i.e., a set of critical, reliable, and self-consistence thermodynamic modeling, especially in the whole composition and at a wide temperature range, due to the equiatomic or nearequiatomic composition region of the HEA system.

The application of the CALPHAD-based thermodynamic analysis in heat treatment will be introduced as follows.

The equilibrium mapping and stepping calculations of the phase diagram, viz., calculated isothermal/isoplethal sections and calculated equilibrium phase fraction at various temperatures, are two major applications in materials design and also in heat treatment optimization. The basic investigation approach is to study from the subsystems (i.e., binary and ternary) to the multicomponent HEA systems; some examples for determining the temperature of annealing heat treatment for a proposed alloy are shown in (**Figures 10** and **11**) [27, 40, 41].

For casting, a low liquidus temperature and a narrow window between liquidus temperature (*T*liquidus) and solidus temperature (*T*solidus) are desirable [42]. However, for high-temperature alloys (i.e., superalloys) and high-temperature HEAs (also known as high-entropy superalloys (HESA) [1]), a higher *T*liquidus and *T*solidus and also a narrow window between the two temperatures are beneficial. For homogenization heat treatment, a wide temperature range of the one-phase region at high temperature (i.e., high-temperature solid solution phase) is preferable (**Figure 10**). Meanwhile, for precipitation heat treatment, a wider window in-between *T*solidus and the solvus temperature of the precipitation phase (*T*solvus) is favorable. Therefore, in certain or designed compositions for an HEA system, these typical temperatures of heat treatment can be predicted by using the

*CALPHAD as a Toolbox to Facilitate the Development of HEAs DOI: http://dx.doi.org/10.5772/intechopen.105191*

**Figure 10.**

*The calculated Co-Cr phase diagram (a1) and the phase fractions at various temperatures for Co-15Cr (a1–1) and Co-15Cr (a1–2) [40].*

CALPHAD-based thermodynamic calculations, which are very important for facilitating various heat treatment processing optimization.

#### *3.1.2 Kinetic analysis of heat treatment*

The CALPHAD-based thermodynamic calculation is usually performed as a significant starting point for materials design, processing optimization, and so on. Sequentially, kinetic simulation tends to become an important link.

To perform kinetic simulation, it is necessary to combine a simulation module, which has been implanted in some specific kinetic models (e.g., 1D homogenization models [43–45]), and various thermodynamic and kinetic databases (Section 3.1). The simulation of diffusion is usually both time- and space-dependent and is particularly suitable for solving one-phase problems and moving boundary problems. Kinetic simulation for assisting to optimize processing parameters of homogenization heat treatment is an important application.

For homogenization heat treatment, the kinetic simulation is a valuable method to predict and optimize the specific processing parameter, especially for the relationship

#### **Figure 11.**

*Calculated phase diagram of the Co-Cr-Ni system. The isoplethal section along with the Cr-Cr0.5Ni0.5 pseudobinary section (a); and the isothermal section at 1200°C (b) [27].*

**Figure 12.**

*Solidification curves and homogenization heat treatment of M5 alloy (thermodynamic and kinetic database: TCHEA2 and MOBNI2) [40] (This figure was not displayed in this reference).*

between temperature, time, and distance in some given phases. Meanwhile, various heat treatments can be optimized by CALPHAD-based simulations, for instance, to facilitate appropriate heat treatment procedures for manipulating solid solution strengthening and precipitation hardening of HEAs [46], and so on.

#### **3.2 Solidification and precipitation simulations**

The Scheil-Gulliver solidification simulation [47, 48], also called Scheil solidification simulation, is usually performed on the basis of the CALPHAD method. For

analyzing solidification processes, some equations were used by assuming that the diffusivity in the solid phases is extremely slow to be treated as zero and the diffusion in the liquid phase is very fast. With the assumption, a type of nonequilibrium transformation can be treated as a local equilibrium state. An example of the Scheil solidification simulation can be found in [40] (**Figure 12**).

Various modules and packages mentioned in Section 3.1, such as TC-Prisma and Pandat, have been developed especially to predict the precipitation problems. Taking TC-Prisma as an example; for predicting the concurrent nucleation, growth, and coarsening of dispersed precipitate phases, the module built by the Langer-Schwartz theory [49] also adopted the Kampmann-Wagner numerical method [50] and many models [51]. To simulate multimodal particle size distribution of precipitate phase is a typical application.

As shown in **Figure 13**, to investigate the precipitation kinetics and to predict the temperature–time-transformation (TTT) diagram for high-entropy superalloys with a typical structure of γ matrix phase (i.e., disordered structure A1) and γ' coherent precipitate phase (i.e., ordered structure L12), various simulations can be performed by TC-Prisma with an available software development kits (SDKs) of TC-Python language combined with many thermodynamic and kinetic databases [52].

#### **Figure 13.**

*Experimental temperature profile for the alloys conducted by homogenization and solid solution heat treatment at 1500 K for 20 h and then air cooling to room temperature (a); based on the experimental temperature profile, simulated particle size distribution and cubic factor of the γ' precipitate phase for the alloys by continuous cooling from the solvus temperature (b); and the simulated TTT diagram for the γ' precipitate phase from the γ matrix phase (c and d) (The details can be found in [52]).*

## **4. Theoretical understanding of materials properties and performances**

#### **4.1 Mechanical properties**

Until now in this chapter, the application of the CALPHAD method in composition and structure design and the processing optimization has been briefly described in the previous sections. On the basis of the classical framework and the conventional research route in materials science, only left the last section here, i.e., the CALPHADbased calculation and simulations for facilitating the theoretical understanding of materials properties and performances. The materials science computational modeling is a key crosslink point to connect the other three points, including composition and structure, processing, and properties and performances. In this section, the CALPHAD-based prediction in understanding mechanical properties, electrochemical corrosion, and high-temperature oxidation performances is described as follows.

As described previously, CALPHAD calculations and simulations are a coupling technique for experimental and theoretical phase diagrams, thermodynamic and kinetic information, and so on. Therefore, the thermodynamic and kinetic databases are compiled by a set of expressions of Gibbs energy and atomic mobility on the basis of a number of composition/temperature- and time/distance-dependent functions (also contain other influence factors, e.g., pressure, volume, magnetic, etc.). In this framework, predicting mechanical properties is probably not the main direction of the CALPHAD method. However, the characteristic of the thermodynamic and kinetic database also provides some fundamental data and information to establish various mechanical databases to predict the corresponding mechanical performances. Take CALPHAD-based prediction for the Young's moduli of the Ti-Nb-Zr-Ta/Mo system as an example [53–59]. Based on Young's moduli data through experimental investigation and/or theoretical calculations, the composition-dependent Young's moduli database can be built also from low-order systems to high-order systems like other thermodynamic and kinetic databases. In **Figure 14**, the predicted data are in good agreement with the experimental results for Young's moduli of the body-centered cubic (BCC) Ti-Nb-Zr-Mo system. It shows that the CALPHAD-based prediction for Young's moduli on the basis of the accurate database is a reliable method.

#### **4.2 Electrochemical corrosion performance**

From the corrosion mechanism perspective, electrochemical corrosion and hightemperature oxidation are the most important two types of corrosion. Therefore, both of the main kinds of corrosion predicted by the CALPHAD-based calculations and simulations will be introduced in the last two sections.

The electrochemical corrosion is usually treated as wet corrosion, i.e., the metal undergoes some reactions in various specific aqueous solutions. Therefore, understanding the electrochemical corrosion mechanism in thermodynamics and kinetics are two key points. The CALPHAD-based prediction is a powerful technique to study the thermodynamic theory of electrochemical corrosion; a typical representative application is the calculation of the Pourbaix diagram.

The Pourbaix diagram, i.e., potential-pH diagrams, derived by M. Pourbaix, collected thermodynamic information for the relevant electrochemical and chemical reactions. With the development of the experimental and theoretical thermodynamic data for the Pourbaix diagram, the calculation of the Pourbaix diagram can be applied *CALPHAD as a Toolbox to Facilitate the Development of HEAs DOI: http://dx.doi.org/10.5772/intechopen.105191*

#### **Figure 14.**

*The diagram of Young's moduli of the BCC Ti-Nb-Zr-Mo system. The data (red line) [53] were predicted from Young's modulus database in comparison to the experimental results [53, 54] (blue and green points).*

not only in corrosion science but also in hydrometallurgy and electrodeposition processing. In the field of corrosion science, the potential-pH domain represents two significant physical and chemical processes. That is, potential is an important characteristic parameter to reflect the metal either kept in an immunity state or reacted to various specific oxidation or ionic states; pH is a significant typical parameter to mirror the activities of H+ or OH in corrosive media or environment. The two parameters usually represent the anodic and cathodic reactions, respectively. Therefore, the calculation of the Pourbaix diagram is a unique method to predict electrochemical corrosion behaviors.

Like other thermodynamic calculations, the calculation of the Pourbaix diagram also needs many thermodynamic databases especially to combine aqueous database (e.g., TCAQ3 database [60]). A calculated Pourbaix diagram and a diagram of the potential and the phase amount of compounds for CoCrFeNi alloy at pH = 7 are shown in **Figure 15** [20]. From the calculation results, some important information can be obtained, such as the main stable oxides and spinels in the passive layers, the specific potential and pH series for different oxidation and reduction reactions, and so on. It needs to notice that the potential in the Pourbaix diagram and in the real corrosion system represents equilibrium and nonequilibrium potential, respectively. Therefore, the equilibrium and non-equilibrium states should be distinguished when analyzing the corrosion behaviors.

#### **4.3 High-temperature oxidation performance**

Compared with the metallic materials suffering various types of destruction at room temperature, the high-temperature alloys (i.e., superalloys) undergo much more heavily deterioration at elevated temperatures. To take the turbine as an example, as the most important core component, like the skeleton for a human, in fossil-energy/nuclear power plants and vehicles, especially in aircraft, the

#### **Figure 15.**

*The calculated Pourbaix diagram (a) and the diagram of potential and the phase amount of oxides and spinels for CoCrFeNi alloy at pH = 7 (b) [20].*

high-temperature metallic materials should conquer a series of complex chemical, physical, and metallurgical process, for instance, high corrosion, wear, fatigue, stress, and creep resistance performances, particularly for the interaction between these mechanics and corrosion factors. Therefore, the superalloys with high performances at elevated temperatures are the most significant materials in high-temperature components.

For the corrosion behaviors at room temperature and elevated temperature, the basic theories have some common points, such as thermodynamic and kinetic theories. However, the corrosion mechanisms of the high-temperature corrosion, usually called high-temperature oxidation, still have some specific theories including thermodynamics and kinetics in comparison with the theories for electrochemical corrosion at room temperature. To draw inferences from the above sections, perform the CALPHAD-based calculations, and simulations for high-temperature oxidation also focus on the thermodynamic and kinetic computation. That is, some predictions can be obtained from the above sections in this chapter, and some calculations and simulations can usually be performed especially to predict the behaviors of hightemperature oxidation. Some common examples are described in the following.

Some thermodynamic predictions for different usages based on the materials in specific environments or working conditions can be applied here. For instance, isothermal and isoplethal sections can be calculated for various HEA systems, not only for the alloying elements systems but also especially for the alloying element–oxygen systems. **Figure 16** shows the calculated stable phases of the Fe-Cr-O system at 650°C at various oxygen activities [61]. A spinel miscibility gap appears in the S1 + S2 region (S1: Fe3-*x*Cr*x*O4, S2: FeCr2-*x*Fe*x*O4), S1 and S2 represent the spinel phase between Fe3O4 and FeCr2O4 phase, and the miscibility gap will be disappeared above 665°C.

In the calculated phase diagram of the alloying element-oxygen systems, the oxygen can be exchanged for carbon, nitrogen, or sulfur, because the processes of oxidization, carbonization, nitridation, and vulcanization are treated as generalized oxidization.

In Sections 3.1.2 and 3.2, the application of the kinetic and precipitation simulations on heat treatment has been introduced. As described above, the thermodynamic calculations can be applied to study materials design, processing optimization, and

*CALPHAD as a Toolbox to Facilitate the Development of HEAs DOI: http://dx.doi.org/10.5772/intechopen.105191*

#### **Figure 16.**

*The calculated stable phases of the Fe-Cr-O system at 650°C at various oxygen activity. C: Corundum-type M2O3, S1: Fe3-*x*Cr*x*O4, S2: FeCr2-*x*Fe*x*O4, W: Fe1-*x*O, BCC: Ferrite structure [61].*

#### **Figure 17.**

*A Dictra simulation of high-temperature oxidation. The oxide thickness time was dependent on the Fe-Cr-Al system for 24 h at 600°C. the simulation sets an assumption that the bulk composition in alloys only is affected by the grain size and grain boundary diffusion in the oxide.*

various properties and performance; however, each prediction has some different specific emphases for different studies. The kinetic and precipitation simulation is the same situation.

Kinetic simulation has the capability to predict the formation of oxide layers during the high-temperature oxidation. The thickness of the oxide layers formed with time for the Fe-Cr-Al system for 24 h at 600°C was simulated by Dictra, as shown in **Figure 17** [62].

The simulation sets an assumption that the bulk composition in alloys only is affected by the grain size and grain boundary diffusion in the oxide layers. Meanwhile, precipitation simulation is capable of predicting particle size distribution of precipitate phases. Until now, the studies of the precipitation simulation for the HEAs are not very common like steels. The reason probably is that the typical precipitation and inclusion in different HEA systems are not like M7C3 or M23C6 in steels. However, much stronger support for the precipitation simulation of the HEAs will be provided in the future resulting in more and more experimental investigation of HEAs [63–66].

## **5. Conclusions**

In this chapter, the applications of the CALPHAD-based calculations and simulations in the recent advances and the new perspectives of HEAs were introduced briefly. The framework of the chapter is followed by the classical components, i.e., modeling, composition and structure, processing, and properties and performances. Here, the CALPHAD-based thermodynamic and kinetic modeling play the key role to connect the other three components. As can be seen, some examples were not only limited in the CALPHAD prediction in HEAs, but also contained some examples in different alloy systems, probably because not so much corresponding research was found. To draw inferences from the similar research, it is no problem to apply the similar thermodynamic and kinetic calculations and simulations in the specific HEAs systems if the specific alloy systems have the matched databases. Besides the content described in this chapter, there still exist a number of other research studies on the basis of the CALPHAD computations, and many techniques can also couple with the CALPHAD data or combined with CALPHAD software programs. These new trends extremely extend the boundaries of the CALPHAD method. For instance, the CALPHAD modeling can couple with first principle, phase field, machine learning, etc.; meanwhile, many conventional CALPHAD software can interact with Python or MATLAB through specific application programming interfaces (APIs).

HEA is a fast-growing field attractive to the scientists of different backgrounds. Many emerging applications of CALPHAD have not been included in this chapter, such as the refractory HEAs, high-entropy ceramics (HECs), additive manufacturing of HEAs, etc. These areas would offer new chances of exploring the unexpected and exciting features of HEAs. Hope CALPHAD could play a more important role in the new journey of discovering advanced HEAs.

## **Acknowledgements**

W. Wang would like to acknowledge the funding from a Key R & D item in Science and Technology Development Project / International Collaborative in Science and Technology Development Project, Science and Technology Department of Jilin Province in China (Grant No. 20200401106GX and 20210402061GH). W. Wang would like to acknowledge the funding from the Opening Project of Shanxi Key Laboratory of Controlled Metal Solidification and Precision Manufacturing, North University of China (Grant No. MSPM202001). Z. Li would like to acknowledge the National Natural Science Foundation for Youth of China (Grant No. 52001098).

## **Conflict of interest**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this chapter.

## **Author details**

Wei Wang<sup>1</sup> \* and Zhou Li<sup>2</sup> \*

1 Northeast Electric Power University, Jilin, China

2 Shandong First Medical University, Jinan, China

\*Address all correspondence to: wei6@kth.se and lizhou\_alfred2011@hotmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Section 4 Applications

## **Chapter 10**
