# Thermodynamic Properties as a Function of Temperature of AlMoNbV, NbTaTiV, NbTaTiZr, AlNbTaTiV, HfNbTaTiZr, and MoNbTaVW Refractory High-Entropy Alloys from First-Principles Calculations

^{*}

## Abstract

**:**

## 1. Introduction

#### Literature Review

_{25}Nb

_{50}Ta

_{25}and TiZrNbTa exist in a single-phase BCC solution with a coarse-grain microstructure, showing that laser metal deposition is a suitable processing tool [48]. Recently, Bhandari et al. published a paper on elastic properties of RHEAs predicted through machine learning, wherein their method was validated by comparing results of RHEAs to datasets. The RHEAs included MoNbTiVZr, MoNbTiZr, and NbTiVZr [49].

## 2. Materials and Methods

#### 2.1. Equation of State: Energy vs. Volume and Debye Model

#### 2.2. Thermodynamic Properties

#### 2.3. Computational Details

## 3. Results and Discussion

#### 3.1. Methodology Validation on BCC Ta and FCC Al

#### 3.2. AlMoNbV—HEA Validation and Atomic Configuration Permutations

#### 3.3. Thermodynamic Properties of RHEAs

#### 3.3.1. Quaternary RHEAs

#### 3.3.2. Quinary RHEAs

#### 3.4. Comparing NbTaTiZr, NbTaTiV, and AlNbTaTiV

## 4. Conclusions

- Using DFT + SQS + Debye is a reliable method to use when calculating the finite temperature thermodynamic properties of an RHEA. This is because the difference between the highest and lowest calculated properties of entropy and heat capacity from all of the atomic configuration permutations of the quaternary system AlMoNbV vary by at most 1.7%. Only a few atomic permutation calculations are needed, depending on the number of chemical species in the alloy, which saves computational time and resources.
- The presence of Al and Zr elements with lower VEC in the RHEAs contributes to higher thermal expansion, while the presence of Mo, V, and W, elements with higher VEC contribute to lower thermal expansion, further validating the theory that lower VEC can lead to higher ductility in BCC refractory alloy design.
- In the RHEAs, the presence of Hf, an element with a VEC of four electrons, in the same way as other ductile metals such as Zr contributes to higher entropy and a higher ${C}_{V}$ compared to systems without Hf, especially at lower temperatures.
- At higher temperatures, Al contributes to the highest ${C}_{V}$, which could be attributed to its capability for thermal expansion and low VEC of 3.
- V, Mo, and W elements with high VEC are the most structurally stable with the lowest thermal expansion, lowest ${C}_{V}$, and entropy.
- The comparison of the compositionally similar systems as well as the comparison of quaternary and quinary systems offers valuable guidance for subsequent theoretical and experimental endeavors.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Correction Statement

## Appendix A

**Table A1.**The 24 permutations are shown with their ground-state energy (GSE). GSE values are at 0 K and 0 GPa in terms of eV/atom and the ground-state volume is in terms of (Å

^{3}/atom).

Type | GSE (eV/Atom) | ${\mathit{V}}_{0}$ (Å^{3}/Atom) |
---|---|---|

AlMoNbV | −8.547 | 15.60 |

AlMoVNb | −8.556 | 15.64 |

AlNbMoV | −8.552 | 15.63 |

AlNbVMo | −8.558 | 15.61 |

AlVMoNb | −8.567 | 15.62 |

AlVNbMo | −8.560 | 15.57 |

MoAlNbV | −8.565 | 15.61 |

MoAlVNb | −8.556 | 15.66 |

MoNbAlV | −8.558 | 15.60 |

MoNbVAl | −8.551 | 15.59 |

MoVAlNb | −8.563 | 15.62 |

MoVNbAl | −8.565 | 15.57 |

NbAlMoV | −8.559 | 15.62 |

NbAlVMo | −8.574 | 15.61 |

NbMoAlV | −8.550 | 15.61 |

NbMoVAl | −8.561 | 15.61 |

NbVAlMo | −8.555 | 15.60 |

NbVMoAl | −8.554 | 15.58 |

VAlMoNb | −8.554 | 15.64 |

VAlNbMo | −8.575 | 15.57 |

VMoAlNb | −8.553 | 15.64 |

VMoNbAl | −8.567 | 15.61 |

VNbAlMo | −8.552 | 15.62 |

VNbMoAl | −8.546 | 15.61 |

AVG | −8.559 | 15.61 |

STD | 0.007 | 0.023 |

Max | −8.546 | 15.66 |

Min | −8.575 | 15.57 |

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**Figure 1.**Calculated (

**a**) entropy and (

**b**) enthalpy values as a function of temperature for pure Ta and calculated (

**c**) entropy and (

**d**) enthalpy as a function of temperature for pure Al. Calculated values are solid lines, and NIST-JANAF [66] data are points. Enthalpy is considered with reference to 298.15 K, where $H=H\left(T\right)$ − $H\left(298.15\right)$.

**Figure 2.**(

**a**) Entropy and (

**b**) ${C}_{V}$ as a function of temperature from 0 to 1600 K comparing this work to the values Ge et al. [27] reported for the AlMoNbV RHEA. The red solid line is the average of the 24 permutations of the AlMoNbV system, while the purple dashed-dotted line is the highest dataset from the 24 permutations and the blue dotted line is the lowest dataset.

**Figure 3.**(

**a**) Entropy, (

**b**) enthalpy, (

**c**) ${C}_{V}$, and (

**d**) $\alpha $, linear thermal expansion as a function of temperature from 0 K to 1800 K of quaternary systems NbTaTiZr, AlMoNbV, NbTaTiV calculated in the present work [27].

**Figure 4.**(

**a**) Entropy, (

**b**) enthalpy, (

**c**) ${C}_{V}$, and (

**d**) $\alpha $, linear thermal expansion, as a function of temperature from 0 K to 1800 K for quinary systems AlNbTaTiV, HfNbTaTiZr, and MoNbTaVW.

**Figure 5.**(

**a**) Entropy, (

**b**) enthalpy, (

**c**) ${C}_{V}$, and (

**d**) $\alpha $, linear coefficient of thermal expansion as a function of temperature from 0 K to 1800 K for systems NbTaTiZr, NbTaTiV, and AlNbTaTiV.

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**MDPI and ACS Style**

Moreno, D.E.; Hargather, C.Z.
Thermodynamic Properties as a Function of Temperature of AlMoNbV, NbTaTiV, NbTaTiZr, AlNbTaTiV, HfNbTaTiZr, and MoNbTaVW Refractory High-Entropy Alloys from First-Principles Calculations. *Solids* **2023**, *4*, 327-343.
https://doi.org/10.3390/solids4040021

**AMA Style**

Moreno DE, Hargather CZ.
Thermodynamic Properties as a Function of Temperature of AlMoNbV, NbTaTiV, NbTaTiZr, AlNbTaTiV, HfNbTaTiZr, and MoNbTaVW Refractory High-Entropy Alloys from First-Principles Calculations. *Solids*. 2023; 4(4):327-343.
https://doi.org/10.3390/solids4040021

**Chicago/Turabian Style**

Moreno, Danielsen E., and Chelsey Z. Hargather.
2023. "Thermodynamic Properties as a Function of Temperature of AlMoNbV, NbTaTiV, NbTaTiZr, AlNbTaTiV, HfNbTaTiZr, and MoNbTaVW Refractory High-Entropy Alloys from First-Principles Calculations" *Solids* 4, no. 4: 327-343.
https://doi.org/10.3390/solids4040021