# A Review of Multi-Scale Computational Modeling Tools for Predicting Structures and Properties of Multi-Principal Element Alloys

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## Abstract

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## 1. Introduction

## 2. Phase Equilibria and Crystal Structures of MPE Alloys

_{x}NbTiV, using TCHEA1, TCHEA2, TCNI7, TCNI8, and TTTI3 databases of Thermo-Calc, respectively. Moreover, B. Gwalani et al. [38] have comprehensively studied the effects of thermo-mechanical processes on microstructures of Al

_{0.3}CoCrFeNi MPE alloy and their discrepancies from thermodynamic predictions by TCHEA database of Thermo-Calc. According to this study, nanoscale ordered L1

_{2}phase was formed experimentally when the alloy was solutionized at 1150 °C before annealing. This meta-stable L1

_{2}phase was not captured in the equilibrium thermodynamic calculations due to different driving forces for nucleation, as well as different nucleation barriers in different phases [38]. According to the recent comprehensive study of Abu-Odeh et al. [39], the calculated phase equilibria of 71% of alloys (153 alloys out of 216 total studied alloys) using TCHEA1 (the database deigned for MPE alloys) matched the experiments. In another study, Tancret et al. [40] have reported the predictive capability of TCNI7, TTNI8, TCFE8, and SSOL5 databases of Thermo-Calc. According to this study, TCFE8 was the most successful database for MPE alloys, while SSOL5 was the most accurate for non-MPE alloys [40]. Figure 5 shows the accuracy of different CALPHAD databases in predicting the phase formations in MPE and non-MPE alloys [41]. It should be noted that current databases of Thermo-Calc are not able to capture the phase formations in most of the experimentally studied MPE alloys.

_{2}alloys using the PanHEA database of Pandat. Nevertheless, some researchers prefer to develop their own thermodynamic databases [43,44] or use the available CALPHAD-based databases (e.g., SGTE) directly without using commercial software [45,46]. This will allow these frameworks to study MPE alloys for which there are not enough experimental data available in the databases of commercial software. A statistical study was performed by Miracle and Senkov [1] illustrating the predictive capability of the CALPHAD calculations with respect to the experimental results. They compared two CALPHAD datasets (f

_{AB}= 1 and f

_{AB}= All) at two different temperatures (600 °C and melting temperature (T

_{m})). According to their findings, CALPHAD predicts more phases in the microstructures of the MPE alloys than the experimental observations. Also, some phases are over predicted while some phases are neglected. In addition to predicting structures and phases of MPE alloys, thermodynamic methods and approaches were also used to calculate the mixing enthalpy, configurational entropy, mismatch entropy, and other thermodynamic aspects of high entropy bulk metallic glasses (HE-BMGs) [30,31].

_{20}Cr

_{20}Fe

_{40-x}Mn

_{20}Ni

_{x}MPE alloys, Heidelmann et al. [58] determined the stable structures for matrix and inter-grain phases of ZrNbTiTaHf, and Mu et al. [59] computed the lattice constants of five different refractory MPE alloys, all using first-principles calculations.

_{1−y}Al

_{y}MPE alloys by calculating Gibbs free energies and structural energies of different phases as a function of Al content, Middleburgh et al. [62] studied the segregation and migration of species in CrCoFeNi alloys by calculating the vacancy energy and defect energy versus lattice binding energies, Yu et al. [63] investigated the nano-scale phase separation in some fcc MPE alloys using ground state formation energies, and Leong et al. [64] applied the rigid band approximation (RBA), a simplification of density functional theory approach, to investigate the phase formation behaviors in MPE alloys and several phases were successfully predicted.

_{X}CrCoFeNi alloy, using MD simulations with considering second nearest-neighbor modified embedded atom method (2NN-MEAM) and Lennard-Jones (LJ) potentials, respectively. The main problem in using the MD simulations is generally the unavailability of the interatomic potentials. If the interatomic potentials are not specifically developed for the target materials or the temperature of interest, the results may not be useful or accurate. Developing interatomic potentials can be a time-consuming and computationally expensive process.

_{1.33}CoCrFeNi MPE alloy (Figure 6) [79]. The large amounts of the required data in Monte Carlo simulations usually increase the complexity of such simulations.

## 3. Properties of MPE Alloys

#### 3.1. Mechanical Properties

#### 3.1.1. Elastic Properties

_{x}MPE alloys [90]. Moreover, they indicated that the VEC value of about 4.72 is critical for the elastic isotropy in these refractory MPE alloys. In another study, Li et al. [91] utilized first-principles calculations to determine the effects of crystallographic directions on the Young’s modulus of some fcc and hcp MPE alloys (see Figure 8).

_{x}MoNbTiV alloys by employing EMTO-CPA, in which the fraction of Al was controlled between x = 0 and x = 1.5. They have shown that the υ, B/G ratio and ab-initio Cauchy pressure of these alloys decreased with the increase of Al content. Moreover, these alloys were predicted to become isotropic when VEC ≈ 4.82 or x ≈ 0.4. Employing the EMTO-CPA method, Li et al. [93] studied the equilibrium volume, the ideal tensile strength, and elastic properties of some bcc-phase MPE alloys, ZrVTiNb, ZrNbHf, ZrVTiNbHf and ZrTiNbHf. The obtained results were expected to provide a guideline to design refractory MPE alloys with controlled strength level [93]. Furthermore, Li et al. [94] investigated the variation behaviors of the ideal tensile strength (ITS) and the ideal shear strength (ISS) in terms of the composition of elements. The ab-initio EMTO-CPA calculations were combined with the quasi-harmonic Debye-Grüneisen model by Ge et al. [95] to investigate the equilibrium bulk properties, thermo-elastic properties, and the Curie temperature of ferromagnetic and paramagnetic CoCrFeMnNi alloys. In their study, the elastic moduli were found to linearly decrease and the ductility increase with the temperature increase [95]. All of these studies demonstrated that integrated the EMTO-CPA method is capable of determining elastic properties of MPE alloys with five or fewer elements.

_{24}Cr

_{19}Fe

_{24}Ni

_{19}Al

_{8}(Ti,Si,C)

_{6}Compositionally Complex Alloys (CCA), which is derived from single phase CoCrFeNi-based MPE alloys. They qualitatively studied the local microstructure evolutions, such as the fragmentation of brittle phase and plastic deformation of the ductile phases, with respect to 3D numerical modelling of local strains and stresses [105]. Their approach was proposed to be helpful in the initial screening of the composition of alloy during the design of new MPE alloys [105].

#### 3.1.2. Plastic Deformation

_{0.3}CoCrFeN MPE alloy, using a combination of experiments and MD simulations. According to this study, σ and B2 intermetallic compounds endorsed deformation twining and strain hardening of the fcc matrix. They used ~900 K atoms with an EAM interatomic potential in their simulations [113]. In a three dimensional MD simulation study, Wang et al. [114] investigated the strengthening mechanism and plastic deformation behaviors of AlCrCuFeNi MPE alloys. They used 1.5 M atom simulations utilizing a combination of EAM and Morse potential, and determined the surface topography, friction coefficient, dislocation density and subsurface damaged structure during the process of nanoscale scratching, and the dynamic evolution of scratching forces, and compared them with those in pure metals [114]. In another study, the strain-induced transformation plasticity of single-crystal and nanocrystalline Co

_{25}Ni

_{25}Fe

_{25}Al

_{7.5}Cu

_{17.5}MPE alloy during fcc to bcc phase transition was investigated using MD simulations by Li et al. [115]. With 2.3M atom simulations using a normalized EAM potential, they concluded that the fcc to bcc phase transition provides an alternative approach in designing novel MPE alloys with improved strength and ductility [115]. Sharma and Balasubramanian [99] have investigated the deformation mechanisms of a single phase Al

_{0.1}CoCrFeNi, under tension by employing MD simulations. They considered an EAM-LJ hybrid potential with 62,500 atoms. Their simulation results attempted to offer insights on the nucleation and dynamic evolution of defects, which cannot be achieved by experiments and not unfeasible by first-principles calculations [99]. Finally, by employing MD simulations focusing on dislocation motion behavior, and utilizing an EAM potential in 1.4M atom simulations, Smith et al. [116] demonstrated that the SFE in equiatomic CrMnFeCoNi MPE alloys should be considered as a spatially local property instead of global variable. In Table 2 the calculated SFE values of several MPE alloys by DFT, MD, and experiment are compared, demonstrating that calculations of SFE is mostly ignored by the most of previous MD studies, therefore one can’t confidently utilize such MD simulations to study plastic deformation in MPE alloys.

#### 3.1.3. Solute Strengthening

_{x}MPE alloys. They have predicted that this theoretical model could be used in studying the yield stress, and also in the design of the other bcc MPE alloys.

#### 3.2. Thermo-Chemical Properties

_{2}(Co,Ni)

_{3}(Al,Mo,Nb) phases by Yao et al. [71], providing data for the design of novel tungsten-free high-temperature Co based MPE alloys. In another study and by employing the EMTO-CPA method, Cao et al. [92] predicted that the addition of Al slightly decreased the thermodynamic stability of bcc Al

_{x}MoNbTiV MPE alloys [92]. Furthermore, Löffler et al. [130] assessed the heat capacity of the quaternary AlCuMgSi Q phase precipitation by the combination of experiment and first-principles calculations.

#### 3.3. Magnetic Properties

_{C}) of some equiatomic MPE alloys [136]. An integrated computational study of the mean field, and DFT calculations was also conducted to compute Curie temperatures of MPE alloys [137]. Some candidate MPE alloys with good magnetic properties were revealed, including CoFeNiCrAg

_{0.37}, CoFeNiCr

_{0.8}Cu

_{0.64}or CoFeNiCrAu

_{0.29}. It was also concluded that the hypothetical T

_{C}maps can be directly used in creating ferromagnetic MPE alloys with well-defined target magnetizations and T

_{C}’s. Figure 10 shows the calculated local magnetic moments in Al

_{x}CrMnFeCoNi (0 ≤ x ≤ 5) MPE alloys using a first principles approach [138].

## 4. Summary

## Author Contributions

## Conflicts of Interest

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**Figure 2.**The most common elements used in MPE alloys [1] and their volatility of prices over a period of 10 years. Black elements have a low price (~$0.5/mole of average 50 years), blue elements have an intermediate price (~$5/mole of average 50 years), and the red element has a high price (~$100/mole of average 50 years) [11].

**Figure 3.**Relationship between property/price ratio and the configurational entropy of common alloys (this figure is from Reference [14]; IMs: Intermetallics or metallic compounds).

**Figure 4.**Microstructures of MPE alloys and Bulk Metallic Glasses as a function of enthalpy of mixing and atomic size difference (this figure is from Reference [17]).

**Figure 5.**Capability of four different CALPHAD databases in predicting the formation of MPE alloys (HEA) and others (non-HEA). Dark sectors show success and light sectors represent the failure of the data bases (this figure is from Reference [41]).

**Figure 6.**(

**a**) Simulated structure of Al

_{1.33}CoCrFeNi at T = 300 K viewed along the [001] direction. (

**b**) and (

**c**) Partial radial distribution functions g

_{αβ}(r) showing the atomic distributions around Al and Cr atoms, respectively (this figure is from Reference [79]).

**Figure 7.**Fracture toughness as a function of yield strength for a wide variety of materials, including MPE alloys (high entropy alloys) (this figure is from Reference [86]).

**Figure 8.**Directional dependence of Young’s modulus E (in GPa) of some MPE alloys in hcp and fcc structures (this figure is from Reference [91]).

**Figure 9.**(

**a**) fcc supercell structure used for calculating generalized stacking fault energy curves and surface energies of CoCrFeNi-based single phase MPE alloys. (

**b**) Calculated GSFE curves for CoCrFeNi and CoCrFeNiAl0.3Ti0.1 by considering two different fault planes shown in (

**a**); subset pictures show twin boundary formation for these two cases (this figure is from Reference [109]).

**Figure 10.**Theoretical local magnetic moments at 0 K for paramagnetic bcc (solid symbols) and fcc (hollowed symbols) Al

_{x}CrMnFeCoNi (0 ≤ x ≤ 5) MPE alloys as a function of Al content (this figure is from Reference [138]).

Alloys | E_{DFT} | E_{MD} | E_{exp.} |
---|---|---|---|

Al_{0.1}CoCrFeNi | - | 199 [99] | 203 [99] |

CoCrFeNi | 275.7 [94], 274.1 [89], 225 [100], 196 [100] | - | 226 [101] |

CoCrMnNi | 265.6 [89] | - | 171 [100] |

CoFeMnNi | 267.2 [95] | - | - |

CoCrFeMnNi | 262.4 [94], 279.7 [95], 207 [100] | - | 215 [102], 137 [100] |

Cr_{10}Mn_{40}Fe_{40}Co_{10} | 328.1 [94] | - | - |

TiZrVMoTaNb | 71.9 [59] | - | - |

TiZrVMoTaNbCr | 130.9 [59] | - | - |

TiZrVMoTaNbCrW | 166.7 [59] | - | - |

NbVTiZrAl | 118.0 [98] | - | - |

ZrNbHf | 95.4 [93] | - | - |

ZrVTiNb | 95.1 [98], 117.5 [93] | - | 80 [103], 101 [104] |

ZrTiNbHf | 88.9 [93] | - | - |

ZrVTiNbHf | 97.1 [93] | - | - |

TiZrNbMoV_{x} | 141.1 (x = 1) [90], 127.8 [96] | - | - |

Al_{x}MoNbTiV | 174.4 (x = 1) [92], 185.4 [96] | - | - |

TaNbHfZrTi | 185.4 [96] | - | 78.5 [103], 87 [104] |

NbTaTiWV | 257.3 [96] | - | - |

WNbMoTaV | 218.0 [96] | - | - |

MoNbTaTiV | 130.5 [96] | - | - |

MoTiZrNbHfTa | 136.6 [96] | - | - |

Alloys | γ_{SFE-DFT} | γ_{SFE-MD} | γ_{SFE-exp} |
---|---|---|---|

FeCrCoNiMn | 21 [111], 27.3 [100], 29.7 [109] | - | 25 [117], 26.5 [118], 19 [100] |

CoCrFeNi | 31.6 [109], 31.7 [100] | - | 27 [118] |

CoCrFeNiCu_{0.5} | 29.0 [109] | - | - |

CoCrFeNiCu | 27.5 [109] | - | - |

CoCrFeNiCuAl_{0.5} | 32.0 [109] | - | - |

CoCrFeNiCuTi_{0.5} | 37.4 [109] | - | - |

CoCrFeNiAl_{0.3} | 35.2 [109] | - | - |

CoCrFeNi | 31.6 [109] | - | 26.8 [117] |

CoCrFeNiCu_{0.5} | 29.0 [109] | - | - |

CoCrFeNiCu | 27.5 [109] | - | 49.0 [115] |

CoCrFeNiCuAl_{0.5} | 32.0 [109] | - | - |

Co_{20}Cr_{26}Fe_{20}Ni_{14}Mn_{20} | - | - | 3.5 [117] |

Co_{15}Cr_{20}Fe_{20}Ni_{25}Mn_{20} | - | - | 38 [118] |

Co_{26}Cr_{18.5}Fe_{18.5}Ni_{18.5}Mn_{18.5} | - | - | 9.7 [100] |

(CoCrFeNi)_{86}Mn_{14} | - | - | 29 [118] |

(CoCrFeNi)_{94}Mn_{6} | - | - | 28 [118] |

FeCrNi | - | 20 [114] | - |

**Table 3.**The reviewed computational modeling methods used to study the structures and properties of MPE alloys.

Method | First-Principles | Monte-Carlo | MD | Microscale (e.g., PFM simulations) | FEM | Thermodynamics | |
---|---|---|---|---|---|---|---|

Structures/Phases | a | b | c | × | × | d | |

Properties | Mechanical | e | × | f | × | g | × |

Thermo-Chemical | h | × | × | × | i | × | |

Magnetic | j | × | × | × | × | × |

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Beyramali Kivy, M.; Hong, Y.; Asle Zaeem, M.
A Review of Multi-Scale Computational Modeling Tools for Predicting Structures and Properties of Multi-Principal Element Alloys. *Metals* **2019**, *9*, 254.
https://doi.org/10.3390/met9020254

**AMA Style**

Beyramali Kivy M, Hong Y, Asle Zaeem M.
A Review of Multi-Scale Computational Modeling Tools for Predicting Structures and Properties of Multi-Principal Element Alloys. *Metals*. 2019; 9(2):254.
https://doi.org/10.3390/met9020254

**Chicago/Turabian Style**

Beyramali Kivy, Mohsen, Yu Hong, and Mohsen Asle Zaeem.
2019. "A Review of Multi-Scale Computational Modeling Tools for Predicting Structures and Properties of Multi-Principal Element Alloys" *Metals* 9, no. 2: 254.
https://doi.org/10.3390/met9020254